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This is to certify that the

dissertation entitled

DEVELOPMENT OF MINIATURE
CLIMBING ROBOTS - MODELING,
CONTROL AND MOTION PLANNING

presented by

JIZHONG XIAO

has been accepted towards fulfillment
of the requirements for

PHD degreein ELECTRICAL ENGINEERING

 

Major professor

Date 7/30/02

 

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6/01 c:/CIRC/DateDuap65-p. 15

 

DEVELOPMENT OF MINIATURE
CLIMBING ROBOTS — MODELING,
CONTROL AND MOTION PLANNING

By

J IZHONG XIAO

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

2002

ABSTRACT

Development of Miniature Climbing Robots
—Mode1ing, Control and Motion Planning
By

J izhong Xiao

An increasing interest in the development of special climbing robots has been
witnessed in the last decade. Cleaning high-rise buildings, spray painting and sand
blasting of gas tanks, inspecting and maintaining nuclear facilities, assisting fire fight-
ing and rescue operations, remote monitoring of hazardous environment, etc. — all
of these practical problems have an immediate need of automation. Climbing robots,
with their ability to adhere to wall surfaces and move around carrying appropriate
sensors or tools, are the best candidates for these kinds of jobs. The special character-
istics and capabilities of climbing robots would not only allow them to replace human
workers in these dangerous duties but also eliminate costly erection of scaffolding.

This dissertation describes the development of miniature bipedal climbing robots,
with an emphasis on the CRAWLER robot. This robot is the smallest such robot
to date, able to climb walls, walk on ceilings, travel through pipes, and transit be-
tween two inclined surfaces. Two active suction feet, where pump motors vacuum
air out of suction cups, are used to support the robot on surfaces. The robot adopts
the bipedal structure and an under-actuated mechanism to provide the robot with
. versatile mobility and multiple locomotion modes. The under-actuated mechanism
reduces the number of motors required and thereby the robot size, weight, and power
consumption, but it imposes challenges on robot control and motion planning.

The robot kinematic model and dynamic model are derived. The analysis of the
dynamic model reveals that the robot link gravity is a dominant term in robot dy-

namic effects. A joint level PD (proportional + derivative) control plus feedforward

gravity compensation method is proposed. This method outperforms the conven-
tional PD control method because it not only achieves the joint level control but also
compensates for the gravity effects which depend on the configuration of all the robot
links. A DSP-based embedded control system is designed and installed on-board to
control the robot. By using the DSP (digital signal processor) chip, the number of
electrical components is minimized and the control system is self-contained.

In motion planning analysis, a hybrid configuration space concept is proposed
which incorporates the continuous configuration space with the discrete motion sta-
tus space (i.e., standing foot, motion mode). The hybrid configuration space is an
effective tool to analyze the motion planning of the climbing robot since the robot
motion is uniquely determined in the hybrid configuration space framework. Based
on motion pattern analysis, a motion planning method is developed that consists of a
global planner and a local planner. The global planner generates a possible path by
simplifying the robot as a rectangular rigid object with no kinematic constraints. By
using an approach called trapezoidal decomposition and a searching algorithm known
as A‘, it is easy for the global planner to smooth the possible path and allow the
robot to move effectively in translation mode. The possible path describes the global
motion of the robot and minimizes the turns. The purpose of the local planner is to
generate a feasible motion sequence around the turning point by considering the robot
constraints. A cost function is defined based on the motion status information and a
number of heuristics to help the search of an optimal motion sequence. This approach
using global and local levels of refinement reduces the overall complexity and simpli-
fies implementation. Experiments and simulations demonstrate the effectiveness of

our robot system.

Copyright by@

Jizhong Xiao
2002

To my parents, my wife—Jing Ye, and my son—Bowen Xiao.

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Dr. N ing Xi, for his
insightful guidance, enthusiastic encouragement and kind support during the research
and the development of this dissertation. Without his assistance and advice, this
research would not be possible.

I would like to thank Dr. R. Lal Thmmala, who is the principal investigator of
the micro-robot project, for his supervision, and helpful feedback and suggestions
after reading many of my papers. I am very grateful to other guidance committee
members: Dr. Fathi Salam, and Dr. Gang Bao. Their insightful comments and
suggestions will enhance the technical soundness of this dissertation. I also would
like to thank Dr. Zengyu Yu from Texas Instruments Inc. for his valuable help on
the TI DSP implementation.

Special thanks are dedicated to Dr. Hans Dulimarta for his help on the software
coding and the collaborative contributions to the work on motion planning. I am
grateful to other members of the micro-robot project and the colleagues from Robotics
and Automation Lab at Michigan State University. The discussions with Mark Minor,
Meng Yue, Jindong Tan, Weihua Sheng, Yu Sun, Heping Chen, Amit Goradia, and
Imad Elhajj, substantially contributed to my work and broadened my knowledge.

I also would like to thank the facilitators of the NSC84O writing course, Dr. Renat
Snider, Dr. Ruth Barton, Dr. James Rash, and Dr. Timothy Grotjohn, for their great
help in polishing my papers and dissertation.

I am greatly indebted to my parents for their continuous encouragement and
support throughout my academic life. Thanks also go to my wife, Jing Ye, for her

patience, understanding and support during my time in graduate school.

vi

 

TABLE OF CONTENTS

1 INTRODUCTION 1
1.1 Background ................................ 1
1.2 Literature Review ............................. 2

1.2.1 Climbing Robots in Literatures ................. 2
1.2.2 Overview of Robot Motion Planning .............. 5
1.3 Our Climbing Robots ........................... 8
1.4 Challengas and Difficulties ........................ 11
1.5 Outline of the Dissertation ........................ 12

2 MODELING OF THE CRAWLER ROBOT 14

2.1 Mechanical Structure ........................... 14
2.1.1 Description of Mechanism .................... 14
2.1.2 Locomotion Modes ........................ 16
2.1.3 Robot Foot ............................ 18

2.2 Kinematic Model ............................. 21
2.2.1 Coordinate Assignment ...................... 22
2.2.2 Forward Kinematics ....................... 25
2.2.3 Inverse Kinematics ........................ 31
2.2.4 Eulerian Jacobian ......................... 34
2.2.5 Simulation ............................. 36

2.3 Locomotion Analysis ........................... 37

2.4 Dynamic Model .............................. 42
2.4.1 Lagrange-Euler Method ..................... 43
2.4.2 Derivation of the Dynamic Model ................ 49

2.5 Summary ................................. 60

vii

3 EMBEDDED CONTROL SYSTEM DESIGN 61

3.1 Control System Overview ........................ 61
3.1.1 Actuators and Sensors ...................... 61

3.1.2 Control System Structure .................... 62

3.2 Hardware Design ............................. 65
3.2.1 TMS32OLF2407 DSP Overview ................. 65
3.2.2 DSP Implementation of the Controller ............. 68

3.3 Control Strategy ............................. 70
3.3.1 Motor Dynamics ......................... 70
3.3.2 Control Algorithm ........................ 73

3.4 Software Development .......................... 79
3.4.1 Software Modules ......................... 79
3.4.2 Encoder Feedback ......................... 81
3.4.3 Servo Control Module ................. - ..... 82

3.5 Experiments ................................ 84
3.6 Summary ................................. 85
4 MOTION PLANNING OF THE CRAWLER ROBOT 88
4.1 Preliminaries ............................... 89
4.1.1 Configuration Space ....................... 90
4.1.2 Exact Cell Decomposition .................... 95
4.1.3 A‘ Algorithm ........................... 97

4.2 Motion Planning Analysis ........................ 98
4.2.1 Motion Status Space ....................... 99
4.2.2 Motion Pattern Analysis ..................... 100
4.2.3 Hybrid Configuration Space ................... 106

4.3 Motion Planning Algorithm ....................... 107
4.3.1 Problem formulation ....................... 107

viii

4.3.2 Global Planner .......................... 108

4.3.3 Local Planner ........................... 109
4.3.4 Cost Function ........................... 111
4.4 Performance Evaluation ......................... 113
4.4.1 Simulation ............................. 113
4.4.2 Experiment ............................ 115
4.5 Summary ................................. 116
CONCLUDING REMARKS 118
5.1 Conclusion ................................. 118
5.2 Future Work ................................ 119

ix

2.1
2.2
2.3

3.1
3.2

4.1

LIST OF TABLES

Suction cup test results .......................... 20
Link coordinate parameters: LFS.translation mode .......... 24
Link coordinate parameters: LFSspin mode .............. 24
Actuators and sensors used in the CRAWLER robot ......... 63
Motor parameters: escap 17S78-208P from API-Portescap ...... 73
Motion status fields ............................ 99

1.1
1.2
1.3

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19

3.1
3.2
3.3

LIST OF FIGURES

Picture of the FLIPPER robot ......................
Flipping mode of locomotion .......................

Picture of the CRAWLER robot .....................

Exploded view of the CRAWLER robot .................
Exploded view of the CRAWLER rack/ leg pair ............
Locomotion modes of the CRAWLER robot ..............
Schematics of the suction foot ......................
Suction foot testing configuration ....................
Whereabouts of the CRAWLER robot .................
Coordinate frames: LFS.translation mode ...............
Coordinate frames: LFS_spin mode ...................
Block diagram of kinematic model simulation .............
Simulation results of robot kinematic model in LFS_spin .......
Simulation results of robot kinematic model in LFS.translation . .
Locomotion sequence to accomplish “Move forward” ..........
Locomotion sequence to accomplish “Make a left turn” ........
Locomotion sequence to transit from ground to a wall surface
CRAWLER robot assembly .......... - .............
Gravity analysis when the CRAWLER robot sticks on a ceiling . . . .
Gravity analysis when the CRAWLER robot walks on a floor .....
Gravity analysis when the CRAWLER robot climbs up a wall
Gravity analysis when the CRAWLER robot climbs down a wall . . .

Sensors at robot foot ...........................
Picture of the remote controller .....................

Control system block diagram of the climbing robot ..........

xi

9
9
10

15
16
17
19
20
21

24
36
37
37
39
4O
42
50
56
56
57
57

62
63
64

3.4 Functional block diagram of the LF2407 DSP controller ........ 66

3.5 Block diagram of the DSP-based controller ............... 68
3.6 Equivalent circuit of DC servo motor .................. 71
3.7 PD control block diagram ........................ 74
3.8 PD+gravity compensation control block diagram ........... 76
3.9 Encoder pulse and motor direction ................... 81
3.10 Servo control interrupt service routine flowchart ............ 82
3.11 Velocity trajectory of motion profile ................... 83
3.12 Motion profile flowchart ......................... 84
3.13 The CRAWLER robot moving forward on a wall ............ 85
3.14 The CRAWLER robot making turns around a corner ......... 86
3.15 The CRAWLER robot moving between surfaces ............ 87

4.1 C—obstacle in a translational case; B.- is an obstacle, by superimpose
the shape of a robot (R) all through the peripheral of the obstacle, the
C-obstacle CE,- is constructed. ...................... 94
4.2 C-obstacle in translational and rotational case. B,- is an obstacle, CB,-
is a slice corresponding to the Minkowski sum of the obstacle and the

robot R whose shape rotates continuously from 9 = 0 to 0 = 360°.

The C-obstacle is a 3-D stack of the slices. ............... 94
4.3 Tfapezoidal decomposition method ................... 96
4.4 Motion pattern of the CRAWLER robot in LFS phase ........ 101
4.5 Motion pattern of the CRAWLER robot in RFS phase ........ 102

4.6 Motion pattern of the CRAWLER robot making right turn in LFS phase 104
4.7 Motion pattern of the CRAWLER robot making right turn in RFS phase105

4.8 The CRAWLER robot collides with obstacles ............. 107
4.9 Block diagram of the motion planner .................. 108
4.10 Generated path by the A" algorithm .................. 114

xii

4.11 Smoothed path after removal of sharp turns .............. 115

4.12 Robot navigating in a maze ....................... 116

5.1 Aerodynamic attraction and a prototype wheeled climbing robot . . . 121

xiii

CHAPTER 1
INTRODUCTION

1.1 Background

Interest in the development of special climbing and walking robots is growing rapidly.
To increase the operation efficiency and to protect human health and safety in haz-
ardous tasks are the motivations behind the development of climbing robot. Climbing
robots with the ability to adhere to wall surfaces and move around carrying appro— '
priate tools are currently being strongly requested by various industries in order to
perform dangerous operations such as cleaning of high-rise buildings, spray painting
and sand blasting of gas tanks, maintenance of nuclear facilities, aircraft inspection,
surveillance in hostile environments, assistance for fire fighting and rescue operations,
etc. Such capabilities of climbing robots would not only allow them to replace human
workers in those dangerous duties but also eliminate costly scaffolding.

By now, applications of climbing robots have been found in many industries. The
chemical and nuclear industries are the two primary fields where climbing robots are
expected to perform remote inspection and maintenance tasks in highly contaminated
areas or radioactive environments. Application examples include: the retrieval of
irradiated material samples [20], the inspection of storage tanks [76], the handling
and manipulation of nuclear fuel [12] [65], and the dismantling of radioactive facilties
by underwater climbing robots [5].

In construction and shipbuilding industries, applications of climbing robots are
also apparent. The most relevant examples are the cleaning of windows or walls [76],
the painting or sanding of ship hulls, the inspection of the metallic skeletons of bridges
[1]. In military applications, climbing robots have been used for the inspection of the
exterior of large aircraft [6].

In addition to those areas, other fields have benefited from the development of

climbing robots. It has been shown that tiny pipe and duct climbing robots are very
useful in weld inspection, leak detection, checking and cleaning of unwanted deposits
in gas ducts, and sewage pipes [36], [50], [68, 69], [72].

Low-cost miniature climbing robots, with versatile maneuverability and the ca-
pability to operate in areas where access is tight, have drawn much attention from
national research agencies. This dissertation is based on the research work of a
micro-robot project at Michigan State University funded by DARPA (Defense Ad-
vanced Research Project Agency) Distributed Robotics Program. The purpose of
the project is to design, build and test prototype micro-robots that incorporate con-
trol, sensing and planning to perform different missions. The mission scope includes
inspection in constrained environments, such as pipes and ventilation ducts, and gath-
ering information about a hostile situation within a building (’Ihmmala et a1) [71].
Thus, the robots must be able to crawl through pipes, travel on surfaces with varying
inclinations, such as floors, walls, and ceilings, and be able to traverse between such

surfaces.

1.2 Literature Review

1.2.1 Climbing Robots in Literatures

A number of climbing robots have been developed in recent years. Motivations are
typically inspection or maintenance in dangerous environments like the exteriors of
tall buildings, airplanes or ships; or in nuclear facilities or pipelines.

One of the most important issues for climbing robots is to design a proper adhesive
mechanism to ensure that the robot sticks to wall surfaces reliably. So far, three kinds
of adhesive devices have been used: magnetic devices, vacuum suckers and attraction
force generators based on aerodynamic principles. Magnetic adhesive devices are the

most promising for robots moving around on steel structures. Robots using permanent

magnets or electromagnets can be found in [23], [24], [27], [76] for climbing large steel
structures and in [36], [68, 69] for the internal inspection of iron pipes. However, their
applications are limited to steel walls due to the nature of magnets.

In applications for non-ferromagnetic wall surfaces, robots most generally use
vacuum suckers to produce the adhesive force. Example of such robots include the
Robug [19, 20, 43, 44] at University of Portsmouth, UK, NINJA-1 robot [26, 49]
at Tokyo Institute of Technology, Japan, and ROBIN [57] at Vanderbilt University,
USA. Besides those robots built in academic institutes, some robots have been put into
practical use. For example, MACS robots [6] at the Jet Propulsion Laboratory (J PL)
use suction cups for surface adherence when inspecting the exterior of large military
aircraft; Robicen robots [12, 65] use pneumatic actuators and suction pads for remote
inspection in nuclear power plants; SADIE robots [72] use a sliding frame mechanism
and vacuum gripper feet for weld inspection of gas duct internals at nuclear power
stations. A wall climbing robot with scanning type suction cups is reported in [77].
An underwater climbing robot with vacuum grippers for contact arc metal drilling
and cutting is reported in [5]. Other climbing robots based on pneumatic actuators
and vacuum suction cups are reported in [64] and [76]. The great defect of the
suction foot is limited locomotion speed, but advantages are apparent and outweigh
this drawback. A suction foot is capable of producing a powerful adhesive force and
their pneumatic components have an excellent weightzpower ratio. A climbing robot
with articulate structure and equipped with suction feet is capable of moving between
surfaces by using the degrees of motion freedom of the legs.

Apart from the aforementioned adhesive mechanisms, the third choice is to create
attraction force based on aerodynamics including the use of propeller [52, 53, 54] and
reversedhovercraft technique. In this case, the robot is normally driven by wheels and
is suitable to move quickly on the wall surface. But it has two major disadvantages.

The first one is that it is difficult to produce a sufficient adhesive force due to the

aperture between the robot and the wall. The second is that the transition of robot
between surfaces is very difficult if not impossible.

Besides the suction mechanism, mechanical structure and mobility are other design
factors determined by specific applications. Climbing up large, nearly flat surfaces
like ship hulls or reactor tanks or exterior of aircraft will demand simple machines
with fewer degrees of freedom in their motion. Wheeled vehicles with vacuum grippers
[12] [76] or robots using sliding frame systems [5] [6] may satisfy this requirement.
On the other hand, in more complicated tasks where robots are required to transit
from a floor to a wall or from a wall to a ceiling, articulated limbed structures with
two to eight legs are predominant. Bipedal robots with the ability to move between
surfaces include ROSTAM III [7], ROBIN [57], and robots designed by Nishi [51].
As alternatives for increased safety and load capacity, quadruped and more legged
robots are adopted. ROBUG II [43] and NINJA-1 [26, 49] are quadruped robots
featuring multiple degree-of-freedom (DOF) legs protruding from a central body and
carrying suction feet. With even greater complexity and larger size, the robot in
[23] has hexapod configuration and the ROBUG III [19, 20] use eight legs to provide
increased stability. A spider-like pipe robot is developed in [50]. More limbs typically
provide redundant support and often increase load capacity and safety. However,
these benefits are achieved at the cost of increased complexity, size and weight.

Thus, when compactness and efficiency are critical, as in the case of miniature
robots, a structure with minimal weight and complexity is best applied. For these
reasons, the biped format is an excellent candidate. Among the three adhesive mech-
anisms, each of them has its advantages and disadvantages. The vacuum sucker is
the most versatile device and is commonly used in the climbing robot. Our miniature
climbing robots adopt suction foot and bipedal articulated structure to achieve good

balance between compactness and desired mobility.

1.2.2 Overview of Robot Motion Planning

The analysis techniques and algorithms for motion planning have become quite valu-
able in many applications such as robotics, virtual reality systems, graphical ani-
mation, and computer-aided design. A robot with motion planning ability can au-
tonomously determine the detailed motion sequences to accomplish a particular task
without collision with obstacles. This relieves the human operators from tediously
issuing the detailed commands for the robot.

The configuration space concept deveIOped by LozanO-Perez [59], [60], and J .C.
Latombe [39], etc, served as a powerful representational tool for both the development
and the analysis of motion planning algorithms. In the configuration space fi'amework,
it becomes possible to develop algorithms that are generalizable and adaptable to a
wide variety of applications. There exists a large number of techniques for solving the
basic motion planning problem and they typically follow one of the three approaches:
roadmap, cell decomposition and artificial potential field (see Latombe [39] for a
complete survey).

Cell decomposition methods partition the configuration space into regions within
which a collision-free path is easy to determine, and this path can be easily connected
to collision-free paths of adjacent regions. A solution to the motion planning problem
is determined by searching for a sequence of collision-free cells that initially include
the initial configuration and finally includes goal configuration. Approaches to con-
figuration space partitioning can be categorized either as exact cell decomposition,
in which cells are typically constructed by determining critical sets in an algebraic
description of the configuration space [39, 66], or approximate cell decomposition, in
which each cell is assumed to be of the same basic shape, and the configuration space
is usually decomposed in a hierarchical manner [8, 58].

In a roadmap approach, a one-dimensional network of paths is constructed that

preserves the connectivity of the free configuration space. A solution path is con-

structed by connecting initial and goal configurations to the roadmap and performing
a graph search on the extended roadmap. The visibility graph approach generates
a roadmap by connecting certain vertices of the boundary of the free configuration
space, and is primarily suitable for two-dimensional, polygonal configuration-space
planning problems [17] [21] [61] [63]. The topological retraction Operation has been
used in a roadmap generation approach that continuously retracts free configura-
tion space onto its generalized Voronoi diagram [56]. Other roadmap methods are
described in [13, 14, 15].

In the prior two approaches, a path is constructed by analyzing the global structure
of free configuration space; however, in most artificial potential field methods, the
robot moves locally according to some forces that are defined as the negative gradient
of an artificial potential function. This approach was taken in [37] for real-time
collision avoidance. The potential function is typically defined on free configuration
space as the sum of an attractive potential that pulls the robot toward the goal, and
a repulsive potential that pushes the robot away from obstacles. One of the primary
concerns in this approach is the existence of local minima in the potential function.
In [30] heuristic search is performed over nominal paths at a global level, and in
[9, 10] Brownian motion is combined with motion planning to escape local extrema.
In [62] it has been shown that for some problems, a global navigation function can
be constructed that ensures the existence of a single minimum that lies at the goal
configuration.

Approaches to the basic motion planning problem are often evaluated with regard
to completeness and computational complexity. An algorithm is considered to be com-
plete if it is guaranteed to find a solution path whenever one exists. The exact cell
decomposition method is complete, but extremely expensive with doubly-exponential
time complexity in the dimension of the configuration space. The best known com-

plete method constructs a roadmap through geometric stratifications and has time

complexity that is exponential in dimension [14]. Approximate cell decomposition
methods usually achieve resolution completeness, which means that the algorithm
can dynamically adjust the resolution until a solution is found; however, due to the
hierarchical partitioning of n—dimensional cells, the time and space complexity grows
quickly with the dimension of the configuration space [39]. Potential field methods
are typically far more computationally efficient than the other approaches; however,
completeness is usually sacrificed.

In recent years, random sampling has emerged as a promising new approach for
motion planning. Randomized motion planners are capable of solving complex mo—
tion planning problems in high-dimensional spaces. Planners based on random sam-
pling are not complete. Some of them satisfy a weaker, but still interesting property
called probabilistic completeness: if a path exists, the probability that the planner
finds it converges to 1 quickly, as running time increases. Two of the most popu-
lar approaches include the probabilistic roadmap (PRM) algorithm [3, 35], and the
Rapidly-Exploring Random Tree (RRT) [38, 41] algorithm.

Probabilistic roadmap method proceeds in two stages. In the preprocessing stage,
it samples collision-free configurations at random and connects them by simple canon-
ical paths, thus creating a probabilistic roadmap. In the query stage, it connects the
two query configurations to the roadmap and searches the roadmap for a path. The
first PRM planners [28, 34, 70] use a straightforward uniform distribution for sampling
new configuration, possibly followed by an enhancement step to increase the sampling
density in critical regions [34]. Recently a number of other sampling strategies have
been developed, including shrinking the obstacles [29], sampling near the free space
boundaries [2] or medial axis of the configuration space [73], using “guards” to reject
unwanted samples [55]. The “lazy PRM” [11] is the variation of PRM method built
with all collision checking postponed until a query is presented. The RRT method

starts from a single collision free node and grows out from it in fixed-size randomly

directed steps until the unexplored space is filled. RRT is particularly suited for
problems that involve differential constraints.

Recently, there is a trend to expand configuration space framework to handle more
complicated motion planning problems, such as planning under uncertainties, plan-
ning with kinematic and dynamic constraints, and multiple robot motion planning,
etc. LaValle [40, 42] conducted a systematic research and extended the basic motion
planning problem with game theory. He advocated the expansion from configuration
space to state space and to broader space — information space to cover general motion
strategy problems, involving uncertainty in sensing and control, environment uncer-
tainties, and the coordination of multiple robots. Sharma and Sutanto [67] proposed
a vision-based motion planning framework in which they unified the configuration
space and an image feature space and considered sensors as an integral part of the
motion goal. In [39], the dynamic motion planning problem, where the obstacles in

the workspace are moving, was studied in configuration-time space.

1.3 Our Climbing Robots

We have built two prototype climbing robots. The first generation robot is called
“FLIPPER” and its picture is shown in Figure 1.1. The mechanical design of the
FLIPPER robot is reported in [45, 46] by Minor, Dulimarta, et a1. Dangi, Starn
and Aslam reported the design and testing of the Smart Robotic Foot in [16]. The
kinematic model of the FLIPPER is reported by Yue, Minor et al. in [79]. The DSP—
based controller design of the FLIPPER robot is reported in [74] by Xiao et a1. When
the robot stands vertically, it measures approximately 45mm x 45mm x 248mm and
weights 335 grams (11.8 Oz).

The FLIPPER is a biped robot with two suction feet located at the robot extrem-
ities. The middle joint is of revolute type. The motion of the FLIPPER is achieved

by sticking one foot on the surface and “flipping” its body as shown in Figure 1.2.

m

I! ." U
l
I.

 

Figure 1.1: Picture of the FLIPPER robot

The flipping mode of locomotion requires space twice of the height of the robot leg,
and thus it is not suitable when the robot has to travel through confined spaces, such

as ventilation ducts.

 

 

Figure 1.2: Flipping mode of locomotion

In response to the requirement of performing missions in a constrained environ-

ment, the second generation of the microrobot, “CRAWLER”, has been constructed.

The picture of the CRAWLER is shown in Figure 1.3.

_1..

 

Figure 1.3: Picture of the CRAWLER robot

The CRAWLER robot achieves linear motion through extension and contraction
of the robot legs. It is used to travel through inside pipes. The size and weight of the
robot are minimized through under-actuated mechanism, wherein 5 joints are driven
by only 3 motors. This special mechanical structure reduces robot size and weight
but poses challenges in control and motion planning. The dimension of the robot is
approximately 80 mm in height, and 50 mm in width. It weights about 450 grams
(15.8 Oz). Our robots are the smallest of the bipedal climbing robots found in the
literatures. The next larger climber is Yano’s robot [77], which measures 380mmx
250mmx 170mm and weighs 10 kg.

This dissertation will only present the modeling, control and motion planning of

the CRAWLER robot.

10

1.4 Challenges and Difficulties

To design a miniature robot that can climb walls, walk on ceilings, crawl through
pipes, and transit between different inclined surfaces is a challenging task. The re-
quirements of small size, light weight and low power consumption must be satisfied.

Unlike normal wheeled vehicle robots, climbing robots must have a proper ad-
hesive mechanism to ensure that the robots grip wall surfaces firmly and without
sacrificing mobility. The desired capabilities of the robot, such as the transition be-
tween different surfaces and traveling through pipes, imposes further difficulties on
the design and control. To achieve its mission, a robot with multiple forms of loco-
motion is imperative. By adopting suction feet, bipedal articulated structure, and
an under-actuated mechanism, our climbing robots achieve a good balance between
compactness and maneuverability.

As a self-contained system, the climbing robot needs to carry its own power source,
sensors, control system and associated hardware. Thus minimization of power con-
sumption and weight is critical to prolonged operation. The challenges in designing
the robot controller are to use as few components as possible and take any measures
to reduce power usage. Thus the selection of proper sensors, controller chips, compo-
nents and the design of small and efficient electrical circuit becomes very important.
By efficiently utilizing the resources of a DSP chip, our embedded control system
minimizes the component count, reduces the power consumption, and becomes self-
contained and tetherless.

From a mechanical design point of view, an under-actuated mechanism reduces
both weight and power consumption of a robot, since it requires fewer motors than a
conventional robot. However, the under-actuated mechanism increases control com-
plexity and imposes challenges in motion planning.

In order to generate smooth locomotion, the robot requires a motion controller

capable of computing the robot kinematic model in real-time and generating syn-

11

chronized motion control of multiple joints. The effects of gravity on the robot joints
change with varying inclinations as the robot moves across different surfaces. This
makes gravity compensation much more difficult than for fixed-base robots or mobile
robots moving on the ground. We derive the robot kinematic and dynamic models
and utilize them to plan the robot motion and propose a PD+gravity compensation
method to achieve a good performance.

The motion planning of climbing robots with under-actuated mechanism is more
complicated than that of wheeled robots. Due to the constraints imposed by the
mechanical structure and the under-actuation, the climbing robot can perform only
restricted motions even in the absence of obstacles. The coupling between motion
joints and the switching between different motion modes further complicate motion
planning. We introduce a hybrid configuration space concept and propose an effective
motion planning approach to generate a path and a motion sequence allowing the

robot to travel through an environment without collision with obstacles.

1.5 Outline of the Dissertation

The scope of this dissertation is to investigate the modeling, control and motion
planning of the CRAWLER climbing robot. .

Chapter 2 describes the modeling of the CRAWLER robot. The mechanical struc-
ture and locomotion modes of the robot are introduced first. Then, the robot kine-
matic and dynamic models are derived. Locomotion analysis is also presented in this
chapter.

Chapter 3 discusses the embedded control system design based on a DSP chip.
Overview of the control system architecture, the hardware design and software devel-
opment are presented. A joint level PD+gravity compensation method is proposed.
This control method outperforms the conventional PD controller because it not only

achieves joint level control but also compensates for the gravity effects which is de-

12

termined by the configuration of all the robot links. The experimental results are
provided with snapshots showing the robot capabilities.

Chapter 4 studies the motion planning of the CRAWLER robot. The hybrid
configuration space concept is proposed to incorporate the continuous configuration
space with discrete motion status (i.e., standing foot, motion mode) imposed by the
under-actuated mechanism of the robot. The motion pattern of the CRAWLER robot
is analyzed under the hybrid configuration space framework. A motion planning
algorithm is develOped which consists of a global planner and a local planner to
generate a collision-free path and a feasible motion sequence to travel along the path.
A cost function is defined based on the motion status information to guide the search
for an optimal path. Simulations and experiments have been conducted and the
results verified the efficency of the method.

Chapter 5 summarizes the contributions of the research and concludes the disser-
tation with recommendations for the improvement of the CRAWLER robot and for

future research directions of climbing robots.

13

CHAPTER 2
MODELING OF THE CRAWLER ROBOT

In this chapter, the mechanical structure of the CRAWLER robot is introduced, the
robot kinematic model and dynamic model are derived, and the locomotion analysis

is presented.

2.1 Mechanical Structure

The purpose of the climbing robot is semi-autonomous reconnaissance in confined en-
vironments. The robot will be deployed on the exterior or interior surfaces of buildings
and structures. It will traverse horizontal and vertically inclined surfaces and climb
between them in order to provide video surveillance or deliver reconnaissance sensors
to specified locations. The robot must be sufficiently small to travel through pipes,
ventilation ducts and to avert visual detection. For effective and prolonged opera-
tion, the robot needs to be small in size, light in weight, and consume as less power
as possible.

The CRAWLER robot is designed as a bipedal robot with an under-actuated
mechanism to achieve good balance between compactness and maneuverability. The
robot is supported by two suction feet which can hold on anticipated smooth and
non-porous surfaces. The dimension of the prototype robot is approximately 80 mm
in height and 50 mm in width. It weights 450 grams. It has the capability to climb
on a wall, walk on a horizontal surface, crawl through pipes, and transit between two

inclined surfaces.

2.1.1 Description of Mechanism

The mechanical structure of the CRAWLER robot is shown in Figure 1.3. The

under-actuated mechanism enables the robot to drive 5 joints using only 3 motors

l4

thus reducing both the weight and the power consumption of the robot. Motors
1 and 3 independently drive joints 1 and 5, respectively; thereby adjusting the tilt
angle of the suction foot 1 and foot 2 so that the robot can grip the surface firmly.
Motor 2 is responsible for controlling joints 2, 3, and 4. Joint 2 and 4 are revolute
joints providing steering capability of the feet relative to the legs. Joint 3 represents
the prismatic motion of the legs that allows the robot expanding and contracting its
legs. The clock-wise (CW) rotation of motor 2 causes contraction, i.e., both legs slide
into the robot body; while the counter-clock-wise (CCW) rotation of motor 2 causes

expansion, i.e., both legs translate out of the robot body.

 

Figure 2.1: Exploded view of the CRAWLER robot

A partially exploded view of the CRAWLER robot is illustrated in Figure 2.1. The
robot consists of identical pairs of legs, racks, ankles and suction feet. Motor 2 drives
the helical gear, through the drive pinion, making the two racks sliding in opposite
directions. The innovation of the design lies in the structure of the rack/ leg pair and

the lock-pin cams. Each rack has a log-pin notch (see Figure 2.2). In the normal case,

15

the lock pin passes through a slot on the leg and engages the notch by an elastic plate,
resulting in the rack/ leg pair to slide together. When a rack/ leg combination slides in
and pulls the lock-pin bearing into the cam slot, the lock-pin cams contained within
the upper and lower halves of the body force the bearing to move along the special
curve and push the lock pin outside the notch. This disengaging effect separates the
leg / rack pair and prevents the leg from moving but allows the rack to continue sliding.

The rack then drives the corresponding robot foot to rotate along joint 2 or joint 4.

 

lock-pin and beating—r” lock-pin slot
upper plate
lock-pin notch
\— gearedrack
,... .. “ _/' I
' _ " -1 @ lower plate
I ' - " '9 ’/
“I “V" \ Iock-pinbearing
' ' {If/II space! s
\\‘ij)/' . .
F6 ankle pinion
I $9 -—- ankle bracket
-—— bevel gear

Figure 2.2: Exploded view of the CRAWLER rack/ leg pair

2.1.2 Locomotion Modes

The CRAWLER robot has three motion modes and the capability to switch between
them. Motor 2 drives a set of joints (joint 2,3,4) but not all of them simultaneously.
In each of the three modes, a particular subset of joints is driven and the remaining
joints are locked to prevent rotation. Figure 2.3 shows the top-down view of the

CRAWLER robot and their locomotion modes. Notice that the legs are essentially

16

identical, with the exception of their lock-pin locations. In the case of leg 1, the pin
is adjacent to the ankle and enters its cam slot when both legs are contracted. In the
case of leg 2, the lock-pin is mounted at the end opposite from the ankle and it enters
its cam slot when both legs are extended. The switching between motion modes is

achieved by the engaging/ disengaging of the lock-pin.

 

(a) Translation Mode
lock-pins couple both legs with racks

 

Joint 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lock-pin unlock the Rack 1 from Leg 1

 

 

 

Figure 2.3: Locomotion modes of the CRAWLER robot

0 'h‘anslation Mode: When both the lock-pin bearings are outside the cam,
i.e., the legs and their corresponding racks are locked together, joint 2 and 4
are prevented from rotating; thereby the rotation of motor 2 causes translation

motion of the legs. A CCW rotation of motor 2 causes the legs to extend while

17

a CW rotation causes them to contract. If the translation motion continues
beyond a certain range, both in extension and contraction, one of the lock-pins
will enter its cam slot on the body, causing the mode switch from the translation

mode to spin-l mode or spin-2 mode.

0 Spin-1 Mode: When lock-pin on leg 1 enters its cam-slot during contraction,
it disengages the rack 1 from leg 1 and allows the CW rotation of foot 1 relative
to leg 1 about joint 2. Meanwhile, since the lock-pin on leg 2 still couples the

leg and rack motion, leg 2 will continue to contract and joint 4 is held fixed.

0 Spin-2 Mode: If the legs of the robot keep extending in translation mode, the
lock-pin bearing on leg 2 will enter its cam slot and unlock the rack 2 from leg
2 causing the CCW rotation of foot 2 about joint 4. At mean time, legl/rackl

pair continues to extend along joint 3 while joint 2 is held fixed.

2.1.3 Robot Foot

The two legs of the robot are supported by the suction feet that provide the robot
with the ability to walk on a horizontal surface as well as climb a vertical wall.
Unlike the large suction feet reported in [12], [65], where the power and air are
supplied externally through a tether, our miniature suction feet are self-contained
and tetherless without limitations in motion range [16] (Dangi, Starn and Aslam).
The suction foot is shown schematically in Figure 2.4. Its main components are a
diaphragm-type vacuum pump, a suction cup, a pressure sensor and a micro machined
shape memory alloy valve. The pump is connected to the suction cup through a
custom designed aluminum connector. The connector integrates the foot components
and serves as a mounting platform for the robot body. The 40 mm diameter suction
cup with cleats is used for adherence. The cleats increase the rigidity of the grip and

provide a larger contact area to reduce slippage. The pressure sensor monitors the

18

pressure level inside the suction cup to ensure that the foot is firmly attached to the
object surface. The foot is released through the actuation of the valve by a signal
from the robot controller. Several touch sensors are also attached to the suction cup
in different radial directions. This gives information on which part of the suction cup
has touched the contact surface and facilitates the robot in adjusting the suction foot

orientation. The fully assembled foot measures 40mm x 40mm x 25mm and weighs

 

 

 

35 grams.
Motor
0 9 a
Pum " w
p Connector § 5
O 7 m
Micro-valve

Figure 2.4: Schematics of the suction foot

Comprehensive tests were conducted on different types of surfaces with different
cup diameters and the results were reported in [16] by Dangi, Stam and Aslam. The
foot was tested in both parallel and perpendicular configurations. In the parallel
configuration, as shown in Figure 2.5 (a), the load is applied via a rod at a distance
D from the vertical testing surface. In the perpendicular configuration, the load

direction was kept perpendicular to the horizontal surface as shown in Figure 2.5 (b).

Table 2.1 shows the experimental results of the suction foot with 40 mm diameter
cup. Three types of sample surfaces viz. glass, painted steel cabinet, and spray
painted plastic, with increasing order of surface roughness, were used for testing. The
maximum load value is obtained by gradually increasing the load until the suction

foot fails to support it. The surface conditions were alternated between dry and wet.

19

 

a) Parallel configuration b) Perpendicular configuration

Figure 2.5: Suction foot testing configuration

The wet condition was simulated by spraying of deionized water from a distance
of 150 mm. It can be seen that for the same surface condition (e. g. dry) the load
supported decreased with increasing surface roughness. This can be attributed to loss
of vacuum with rougher surfaces. It can also be seen that a wet surface supports more
weight than a dry surface. A possible explanation for this phenomenon can be that
water occupies the gap between the cup and test surface resulting in lower vacuum

loss than dry condition.

Table 2.1: Suction cup test results

Glass Metal Cabinet Painted Plastic

 

Considering the 450 grams self weight of the robot, the extra holding capacity
of one foot is in the range of 650-3200 grams, more than enough to carry controller
hardware, a 9V battery (40 grams), miniature reconnaissance sensors such as wireless
pin-hole camera, micro-phone and transmitters, etc. Figure 2.6 are some snapshots

showing where the climbing robot can go in the engineering building at Michigan

20

State University.

 

I-
60an up stalrs Walklng on a celllng

Figure 2.6: Whereabouts of the CRAWLER robot

2.2 Kinematic Model

In robot systems, the control is realized in the joint space, whereas the task level
commands are normally expressed in world coordinate space. For the climbing ro-
bot, the reconnaissance camera is mounted at one of its feet to permit the robot to

either look through a glass window or to use the camera like a periscope when the

21

alternative foot supports the robot. The tilt angle of the camera is determined by the
position/ orientation of the robot’s free foot relative to its standing foot. Thus it is
imperative to derive the robot kinematic model which describes the relation between
the robot joint variables and the position/ orientation of the robot’s free foot with re-
spect to a fixed reference coordinate frame located at the center of standing foot. The
kinematic model is also very important in solving the robot motion planning problem.

This section derives the kinematic model of the CRAWLER climbing robot.

2.2.1 Coordinate Assignment

Because the structure of the CRAWLER robot requires that at least one foot remain
in contact with the surface at all times, the setup of the coordinate frames is conducted
in the three-dimensional space with respect to right-foot supporting (RFS) phase and
left-foot supporting (LFS) phase. In each phase, the robot has two motion modes.
One is the translation mode, which means both rack and leg pairs are locked together
and thus the middle joint motor 2 drives the two legs sliding in Opposite direction.
In this mode, only two rotational joints (J 1 and J5) and the prismatic joint J3 move.
The other is spin mode, which occurs when one of the racks is separated from its leg
pair, resulting in the corresponding foot spinning while the other leg and rack pair
sliding. In spin mode, two rotational joints (J 1, J 5), one spin joint (J2 or J4) and the
sliding joint (J3) are involved in the motion. Since the robot is symmetric, we only
analyze the kinematics in LFS phase. The kinematic model is the same for both RFS
and LFS phases. The assignment of coordinate frames based on Denavit-Hartenberg
(D—H) method [18] for LFS_translation mode is shown in Figure 2.7.
LFS_translation mode: In LFS.translation mode, the reference frame is at-
tached to the left foot which is fixed on the ground surface. The base coordinate
frame is at joint 1 with the Z0 axis aligned with .11 rotation axis. The Z1 and Z2

axes are aligned with the sliding motion axis of J3 and the rotary motion axis of J5

22

 

 

 

 

 

 

 

Luv.
................ X"‘._\
................ 44
J3 Zl Jslfifl
x2 I! 22
M YOII Y1 {
x0 .. Y3
.1
I .1 a <
20 ”It 23
Yrefll
Xref
// Zref

Figure 2.7: Coordinate frames: LFS.translation mode

respectively. The right foot can move freely with the “end-effector frame” attached
at the center of suction cup. The L is the robot height and M is the distance between
robot ankle to the leg. The prismatic distance between the centers of the robot feet
is denoted as d.

The link coordinate parameters in LFS.translation mode is shown in Table 2.2,

where a and ,3 and d are joint variables. The four geometric parameters associated

with each link in Table 2.2 are defined as follows:

0 I9,- is the joint angle from the X -_1 axis to the X,- axis about the Z -_1 axis (using

the right-hand rule).

0 d,- is the distance from the origin of the (i — 1)th coordinate frame to the

intersection of the Z.-_1 axis with the X.- axis along the Z -_1 axis.

0 a,- is the offset distance from the intersection of the Z,-1 axis with the X ,- axis to
the origin of the ith frame along the X, axis (or the shortest distance between

the Zi_1 and Z; 8X68).

e a.- is the offset angle from the Z,-_1 axis to the Z, axis about the X,- axis (using

the right-hand rule).

23

Table 2.2: Link coordinate parameters: LFS_translation mode

 

 

 

 

 

 

 

 

Motion Joint 6,- oz, a, d,-
J 1 rotate a 90" M 0
J3 translate 180° 90° M d
J5 rotate fl 0 L—M O

 

 

 

LFSspin mode: The assignment of coordinate frames for LFSspin mode is

shown in Figure 2.8.

 

 

 

 

 

 

 

 

wan-h
............. INK.
................. ,1“.
........................... / d ”fl";
~-...--------'N2 L w *3
J Z2 .19” 3
J1
M Y0“Z1 "L x3" 23
1 71 X1 Y4
LI, x0 "-21
L x1 J1 a
X4 v 24
Yrefll 20
Net
// 4..

Figure 2.8: Coordinate frames: LFSspin mode

In LFSspin mode, four joints involved in the motion. Those are the rotational
joint J1 and J5, prismatic slide motion of J3 and spin motion of J2. Note that Z1 is
aligned with the spin motion axis of J 2. The link coordinate parameters in LFS_spin

mode is shown in Table 2.3, where a, [3, 71 and d are joint variables.

Table 2.3: Link coordinate parameters: LFS_spin mode

 

 

 

 

 

 

 

 

Motion Joint 9; l a,- [ a,- ] diJ
J 1 rotate 0: —90° 0 0
J2 spin 71 90° M
J3 translate —90° 90° M d
J5 rotate ,6 0 L-M O

 

 

 

 

 

 

24

In LFS_spin mode, the slide motion of joint 3 and the spin motion of joint 2 are
coupled, both are driven by the motor 2. The relation is expressed as d = k'yl, where

k is a constant.

2.2.2 Forward Kinematics

Once the D-H coordinate system has been established for each robot link, a homoge—
neous transformation matrix “ 54,- can easily be developed relating the ith coordinate

frame to the (i — 1)th coordinate frame as follows:

, -
cos 6, — cos a,- sin 6,- sin a,- sin 0,- a,~ cos 0,-
i_ 1 sin 0,- cos (1,- cos 9, — sin (1,- cos 0,- a,- sin 6,-
i (2. 1)
0 sin 0:,- cos a,- d,
0 o 0 1 ]

 

 

The homogeneous matrix "’7".- which specifies the location of the ith coordinate
frame with respect to the reference coordinate system is the chain product of succes-

sive coordinate transformation matrices of “54,-, and is expressed as
ref]: : "(4094122 . . . “$4, (2.2)

The robot kinematic model is expressed by the 4 x 4 homogeneous transformation
matrix T='°’T,,, n is the number of robot moving joints. The T matrix has the

combined effect of rotation, translation, perspective, and global scaling, and can be

25

expressed as:

 

 

 

 

[- 1
n;- 33: ax p1:
T: 77 §' 51: I; = ny 3y 03/ py : R3x3 P3x1 , (2.3)
0 0 O 1 n, 32 a, p, f1x3 1x1
0 0 0 1

 

 

 

where the upper right submatrix P3 x1 represents the end-effector position with re-
spect to the reference frame; the upper left submatrix R3x3 is the rotation matrix,
representing the orientation of the end-effector frame; the lower left submatrix f1x3
represents perspective transformation which is useful for computer vision. The vector
ii is the normal vector which is aligned with the direction of X axis of the end-effector
frame and is orthogonal to the suction cup surface. The sliding vector 5' is pointing
in the direction of the Y axis of the end-effector frame which aligns with the robot
slide motion direction. The approach vector 6 is aligned with the direction of Z of
the end-effector.

LFS_translation mode: By plugging in the parameter value in Table 2.2
to Equation 2.1, the transformation matrices for adjacent coordinate frames can be

derived as follows:

10 D O
010 L—M
"£40: (2.4)
D 01 D
0 0 0 1

 

 

26

 

cosa 0 sina Mcosa
0 sina 0 —cosa M sina
A1:
0 1 O O
0 0 O 1
-1 0 O —M
1 0 O 1 0
A2:
0 1 0 d
O 0 O 1
Fcosfl -sinfi O (L—M)cosfi
$43: srnfl cosfl O (L—M)smfl
0 0 1 O
0 O O 1

 

_

 

 

 

 

.l

According to Equation 2.2, the robot kinematics matrix is derived as:

nranalation

="f T3 = "540941912343

p

 

I.

—cos(a+fl) sin(a+fl) O —(L-M)cos(a+fl)+dsina
-sin(a+fl) .—cos(a+5) O —(L—M)[sin(a+fi)—l]—dcosa
0 0 l 0
0 0 0 1

(2.5)

(2.6)

(2.7)

(2.8)

 

d

The rotation submatrix R3x3 needs nine elements to completely describe the end-

effector orientation of robot. It is necessary to find a convenient expression. A set

of Euler angles (11)), (6), (o), are used to describe the orientation with respect to a

27

fixed reference frame. There are many different types of Euler angle representation
[18]. Here the roll(Rz,¢), pitch(R,,,9), yaw (Raw) were used to represent the robot

orientation. They correspond to the following rotations in sequence:

1. A rotation of 1,!) about the Xref axis (RM/Q.
2. A rotation of 0 about the Yref axis (Ry/,0)-

3. A rotation of o about the Z"; axis (Rm).

The resultant composite rotation matrix is:

Rae» = R2.¢Rv.9R¢.¢ (29)
005450039 cos¢sin08in¢ — sinrtcosz/J cos¢sin9cosz/J + sinqbsint/J

- sin¢cos€ sindsinOsintl) + cos¢costb sin¢sindcosib — cosrtsintb

 

 

—-sin0 cosdsinzp cosdcoszp

b -

Comparing this matrix with the rotation submatrix R3,“; in Equation 2.8, we have

the following relationship between the joint angle and Euler angle.

¢=0
6:0
¢=a+fl—n

LFSspin mode: By plugging in the parameter value in Table 2.3 to Equation
2.1, the transformation matrices for adjacent coordinate frames in LFS_spin mode

can be derived as follows:

[- '1
1 0 O 0
0 1 0 L — M
”640 = (2.10)
0 0 1 0
O 0 0 1

 

 

28

‘144

r -
cos a O —— sin a 0
0 sin a 0 cos a 0
A1 =
0 —1 0 0
0 0 0 1
cos 71 0 sin 71 0
£42 = sin 71 0 — cos 71 0
O 1 0 M
0 0 0 1
li 0 O —1 O
2 —1 O O -M
A3 =
O 1 0 d
O O O 1 d

 

 

 

 

 

 

 

cosfi —sinfi O (L—M)cosfiq

sinfl cosfl O (L—M)sinfl
0 0 l 0
O 0 0 1

 

(2.11)

(2.12)

(2.13)

(2.14)

According to Equation 2.2, the robot kinematics matrix in LFS_spin mode is derived

as:

Tspin :ref T4 = reMqul l[122}13I344

29

(2.15)

sin a cos [3 + cos a sin ,8 sin 71 — sin a sin 6 + cos 0 cos 5 sin '71 — cos 0 cos 71
— cos (1 cos 6 + sin a sin fl sin 71 cos a sin 5 + sin oz cos 6 sin 71 -— sin 01 cos '71
sin fl cos '71 cos ,8 cos '71 sin 71
_ O 0 0

 

where

p, = (L — M)(sinacosfl + cosasinflsin 71) + dcosasinyl
1),, = (L—M)(1—cosacos,3+sinozsinflsin'yl)+dsinasin71
pz = (L—M)sinflcos'71+dcosyl

d=k71

It is easy to use Euler angles system I [18] to represent the robot orientation. They

correspond to the following sequence of rotations:
1. A rotation of (,6 angle about the Zn, axis (Rm).
2. A rotation of 9 about the rotated Xfief axis (R159).
3. Finally, a rotation of 1]) about the rotated Zia, axis (Egg).

The resultant composite rotation matrix is:

R¢,0.¢ = R,,¢R,,,9R,,,,, (2-15)

cos¢coszp — sinocosdsinI/J —cos¢sinz,b - sin¢cos€cos¢ sin¢sin9

= sin¢cos¢+cos¢cosl9sintb -sin¢sinz/J+cos¢cosl9cosw,b —cos¢sin6

 

 

L sin 9 sin a sin 0 cos 1/) cos I9

Comparing this matrix with the rotation submatrix 83x3 in Equation 2.15, we

30

Pa:

Py

Pz
1

 

have the following relationship between the joint angle and Euler angle.

¢=a—7r/2
0=7T/2—’71
¢=fl

Thus far, the robot transformation matrix has been deduced for both motion modes.
The solution of forward kinematic problem is obtained by evaluating each element
in T matrix for a given set of joint variables. The robot orientation can also be '

represented by the Euler angles.

2.2. 3 Inverse Kinematics

In this section, the inverse kinematics are studied, i. e., for a given robot position and
orientation, derive the corresponding joint variables which can make the robot reach
the desired configuration.

The solution of inverse kinematics can be obtained in many forms. In the motion
planning and control of the robot, the orientation vectors play a critical role. In order
to make the suction foot grip the surface firmly, we must ensure that the normal vector
it be perpendicular to the contact surface. By specifying the position vector p‘ and the
foot normal vector ii, the relation between the suction foot and the contact surface is
determined. Therefore, it will bring convenience in the implementation if the solution
of inverse kinematics only contains the two vectors it and 15'.

Among the possible solutions of robot inverse kinematics, some representation
of the solution may be inconsistent and ill-conditioned. For example, the arc cosine
function does not behave well since its accuracy in determining the angle is dependent
on the angle, i.e., cos0 = cos (—6). In order to evaluate 0 for —7r 3 0 S 1r, an

arc tangent function, atan2(y,x), which returns tan-1(y/x) adjusted to the proper

31

quadrant is defined and will be used in the solution of inverse kinematics.

0" S 6 S 90° for + a: and +31

90° S 6 S 180° for — :c and +3]
6 = atan2(y, :17) =

—180° S 6 S -90° for - :r and —y

—90° S 6 S 0° for + a: and —y

(2.17)

Inverse kinematics analysis is considered in LFS_translation and LFSspin mode re-

spectively.

LFS-translation mode: The transformation matrix is rewritten as in the fol-

 

lowing.
] n: 3:: ax P:
n s a
T = v y y Pu
n, s, a, pz
0 0 0 1
—cos(a+6) sin(a+fi) 0 —(L—M)cos(a+)6)+dsina

0
O

O
0

1
0

—sin(a+fi) —cos(a+fl) O —(L—M)[sin(a+fl)—l]—dcosa

0
1

 

b

By comparing both sides of this Equation, we have:

2,: __ sin(a + 6)

nz - cos(a+fi) =

a + B = arctan (ny, nx)

 

. _ __ 3—(L-M)nz
d sum — p: - (L - M )m 2) tan“ — -P:+(L—M)(ny+1)
dcosa=—py+(L—M)(ny+1) d=‘”‘—“.%E£-%

32

(2.18)

 

(2.19)

(2.20)

From Equation 2.19 and 2.20, the joint variables are solved using position vector p‘

and the foot normal vector 71':

a = atan2 [p1, - (L — M)nx, — py + (L — M)(ny +1)]
2 atan2 [ny, nx] — a (2-21)
(1 _ p1—]L-—M)n,

sin a

LFS.spin mode: By analyzing the Equation 2.15, we have:

 

 

x: L—Mn,+dcosasin _ __
Pu:(L-M)(1+ny)+dsinasinryl Pz-( — )nz
1:: L-an+dcosasin _ _
P ( ) 71 => tan ’71 = p; L(LMM)n3 (2.23)
P: = (L - M)nz + dcos 71 (Pz -( - )nz) cosa
n, = sin ficos '71 nz
=> tanfi = — (2.24)

sz = cos ,6 cos 7, 3‘
Fiorn Equations 2.22, 2.23 and 2.24, the joint variables are solved as follows:

a = atan2[py -— (L — M)(ny +1), p; — (L —- M)nz]
71 = atan2[px — (L — M)nx, (p, — (L — M)n,,) cos 01] (2.25)
= 1‘71

fl = atan2(nz, 3,)

So far, the robot inverse kinematics has been derived for both motion modes. The
solutions of the inverse kinematics will be used in the implementation of the robot

motion controller.

33

2. 2.4 Eulerian Jacobian

This section derives the Eulerian Jacobian, which relates the linear velocity and the
Euler angular velocities to the joint velocities. By differentiating the Eulerian angle
and the position vector p‘ with respect to time t,' the Jacobian matrix is derived as

follows.

LFS_translation mode:

£5? = (L ‘ M)sin(a+6)a+ 2(100806! + (L — M)sin(oz+6)6+ 23inad
if} = —(L - M) cos(a + 6)d + 2dsin ad — (L - M) cos(o + )6),6 — Zoos ad(2.26)

wz=d+6

In matrix form, we have

- r .
T 1,: .5.

fly = J3x3 d (2°27)
L”: . . 6 .

 

 

 

 

p

(L-M)sin(a+6)+2dcosa 2sina (L—M)sin(oz+6)
J3x3 = —(L - M) cos(a + 6) + 2dsina —2 cosa —(L — M) cos(a + 6)
1 0 1

 

 

b

andpz=0,wz=0,wy=0.

34

LFS_spin mode: The Jacobian matrix is derived as

 

 

 

 

,
a]
15., [a
I? = J6,“ 7.1 (2.28)
()5 d
. -3 .

b 1L d

where

J[1,1] = (L — M)(cosozc036 — sinasin6sin 71) — dsinasin 'yl
J[1,2] = (L-M)cosasinflcosyl+dcosacosyl

J [1,3] = cosasin 71

J[1,4] = (L—M)(—sinasin,6+cosacosfisin'yl)

J[2, 1] = (L - M)(sinac056 + cosasinfisin '71) + dcosasin '71
J[2,2] = (L—M)sinasin6cosyl+dsinacos'71

J [2, 3] = sinasin 71

J[2,4] = (L— M)(cosasin)6+sinacosfisinyl)

J[3, 1] = O

J[3,2] = —(L—M)sin,63in'yl—dsin'yl

J[3, 3] = cos y,

J[3, 4] = (L — M) cos6cos'71)

J[4]=[1000] J[5]=[0 —100] J[6]=[0001]

35

2. 2. 5 Simulation

Forward kinematics deals with the following question: given the joint variables (rotary
or translation) and the geometric link parameters, what is the position and orientation
of the robot end-effector. On the other hand, the inverse kinematics addresses the
problem: given a desired position and orientation of the end-effector and the robot
link parameter, what is the corresponding joint variables which make the robot reach
the desired prescribed configuration.

Computer simulation is conducted to verify the validity of the robot kinematic
model as shown in Figure 2.9. The software initially generates locations in the joint
space within the joint variable limits. These joint variables are inputed into the
forward kinematic model to obtain the transformation matrix T. The T matrix is
then fed into the inverse kinematic model to obtain the joint angle solution which

should agree to the desired joints angles previously fed into the direct kinematics

 

 

 

 

 

 

 

routine.
, _ Position and orientation
Joint V3" ables of end-effector
r Forward kinematics =
'l' Tmatn'x
Error
Inverse kinematics _

 

 

 

 

 

 

Figure 2.9: Block diagram of kinematic model simulation

The Tables in Figure 2.10 and 2.11 show some list of the simulation results. Based
on the desired joint variables, the forward kinematics calculate the robot position
and orientation of its free foot related to the reference frame located at the center of
its standing foot. Then for each position and orientation, the inverse kinematics is
employed to obtain the corresponding joint variables which lead to that configuration.

Due to the mechanical stOp, each joint variable has its motion limits. The simulation

36

is conducted within the following joint variable ranges: 0 S a S 60°; 0 S 71 S 60°,
-30° S 6 S 90°. The coupling between spin angle and robot length in LFS_spin
mode is expressed as (94 — 0.2 x 71) S d S 94 mm. The limit of translation distance

in LFS_translation mode is 94 S d S 140 mm.

 

] Simulation results of robot kinematic model in LFS_spin mode

 

 

 

 

 

 

 

 

 

 

 

[13:11:66,061 variable Position Qiqmm’m Inverse kinematics
,(ggree) (mm) (mm) X Y Z (degree) (mm)
01 Y, [3 x y 2 [X052] [X082] [3952] 01’ 7', [3’ d'
o 0 o 94.0 94.0 0 o [0,-1,0] [1,0,0] [0,0,1] 0 o o 94.0
0 0 V3 94.0 131.24 21.5 o [0.866,-o.5,o] [0.5,0.866,0] [0,0,1] 0 0 11/3 94.0
o 0 n/Z 94.0 137.0 43.9 0 [1.0.0] [0.1.0] [0.0.1] o o 7r/2 94.0
0 m4 0 93.84 66.36 0 '66-36 [0,4,0] [.707,o,-.707] [.707,0,.707] o n/4 0 93.84
0 «I4 n/6 93.84 81.56 5.76 -81.56 [.3536,-.866,-.3536][.612,.5,-.612][.707,0,.707] 0 71/4 11/6 93.84
W6 0 0 94.0 102.91 52.76 0 [o.5,-0.866,0] [0.866,0.5,0] — [0,0,1] x/6 o o 94.0
hi6 o -1r/6 94.0 31.41 47.0 0 [0,4,0] [1,0,0] [0,0,1] n/6 o —n/6 94.0
PUG 0 m 94.0 124.41 90.0 0 [1,0,0] [0,1,0] [0,0,1] «I6 0 4/3 94.0

 

 

h/G «IS an 93.79 6149 57.14 _113.5[.625,-.217,-.75][-.217,.875,-.433][.75,.433,.5] n/6 11/3 m 93.79
[m zI3 110 93.79 34.2 10223 -118. [.65,.125,-.75] [-.625,.65,-.433] [.433,.75,.5]1r/3 71/3 71/2 93.79

 

 

 

 

 

Figure 2.10: Simulation results of robot kinematic model in LFS_spin

 

Simulation results of robot kinematic model in LFS_translation mode

 

 

 

 

 

 

llDesined joint variable; Position Qicmatlon Inverse kinematics
(degree) (mm) (mm) x Y 2 (degree) (mm)
a. B d X y 2 [x933 Z] [XQY’ Z] LXDY’ Z] (1’ B ’ d ,
0 o 94.0 94.0 o 0 [0.4.0] [1.0.0] [0.0.1] 0 o 94.0
0 0 100.0 100.0 0 0 {0,-1.0} [1,0,0] [0,0,1] 0 0 100.0
It/6 0 100.0 108.1 55.76 0 [03,-0.866,0] [0.866,0.5,0] [0,0,1] n/6 0 100.0
x/6 -n/6 100.0 86.60 50.0 0 [0,4,0] [1,0,0] [0,0,1] n/6 —7t/6 100.0

 

n/4 n/6 120,0 126.39116.7 o [0.966,-0.259,0][0.259,o.966,o][0,0,1] n/4 x/6 120,0
n/3 n/2 140.0 91.5 201.48 0 [0.5,0.866,0] [-0.866,0.5,0] [0,01] n73 7;;2 140,0

 

 

 

 

 

 

 

Figure 2.11: Simulation results of robot kinematic model in LFS_translation

The simulation results show that the desired joint variables and the calculated
joint variables matches well. This verifies the correctness of the robot kinematic

model.

37

2.3 Locomotion Analysis

In order to generate smooth locomotion, the robot is required to solve the robot
kinematics in real-time and generate synchronized motion control of multiple joints.
The key point in the motion control is to determine which motion modes should
be applied to and combined together to accomplish the task. In order to distinguish
between different motion modes, a contact switch is installed on each leg to determine
whether the leg and rack are locked together. The contact switches give us the
information on motion mode switching so that different kinematic models are used
to plan the robot trajectory.

For example, in order to accomplish “move forward” walking task, LFS.translation
and RFS_translation motion modes get involved. Assuming foot 1 is initially the
standing foot, and foot 2 is the front foot, the sequence of the joint motions to

accomplish one step of moving forward is illustrated in Figure 2.12.

1. Rotate J1 to lift up the front foot off the ground, and slide J3 in LFS.translation

mode to extend the robot by a stride of d.
2. Rotate .11 again in opposite direction to make the front foot touch the ground.

3. After switching the standing foot, rotate J5 to lift up the rear foot off the
ground, and slide J3 in RFS.translation mode to contract the robot and bring

the rear foot forward by a distance of d.

4. Rotate J5 again to make the rear foot touch the surface.

With the completion of this sequence, the standing foot 1 of the robot has moved
forward d distance in straight line. Stages 1 is governed by LFS.translation motion
mode and stage 3 by RFS.translation motion mode, thus, Equations 2.8 and 2.21

should be used in the trajectory planning and motion control.

38

 

 

7 // /////
foot 1 StageO foot 2
(rear foot) (front foot)
Intel‘d
66:50

 

 

 

 

// //
Stage 1
————4
// // //////
Stage 2
contract

 

 

 

//////
Stage 3
a: :
///// /7/7/

 

 

Stage 4

Figure 2.12: Locomotion sequence to accomplish “Move forward”

In order to change the direction of its walk, the robot needs to steer one of its

feet. Steering is achieved by performing the following sequence and is illustrated in

[III/l l/7777
/ a. 2 51/
Amarl‘obom (fronttoot

Figure 2.13.

 

 

 

cotflac‘

 

/////7

e‘
as
w“? .. I'.-...n‘ '
u f' ..V...-......’- . .4
/////1
/ , . / 1 .

 

 

    

 

 

 

 

 

 

Figure 2.13: Locomotion sequence to accomplish “Make a left turn”

1. Rotate J 1 to lift up the front foot off the ground, and slide J3 in LFS.translation

mode to contract the robot body until the leg1 / rackl pair is separated.

2. Continue to contract in LFS_spin mode causing the robot body to spin around

J2 axis. In this case, foot 1 is wound up.

3. After switching the standing foot, rotate J5 to lift up foot 1 off the surface, and

slide J3 in opposite direction in RFSspin mode to unwind foot 1 until the leg

40

1 and rack 1 are locked.

4. Rotate J5 again in opposing direction to bring the rear foot in contact with the

surface.

Step 1 is controlled in LFS-translation motion mode, which is governed by Equa-
tion 2.8. Step 2 is controlled in LFS_spin motion mode, which is governed by Equation
2.15. Step 3 is controlled in RF S_spin motion mode. After this motion sequence, the
robot has made a left turn and is ready to move along the new direction.

In one step, the robot cannot reach arbitrarily desired position with the desired
orientation. When the robot cannot post the foot to the desired position, it needs to
take several steps and adjusts its standing foot properly. The increased complexity
of motion planning is the penalty paid for the under-actuation structure.

The robot also has the ability to transit between differently inclined surfaces.
Figure 2.14 shows the locomotion sequence when the robot move from a floor to a
wall surface. The most important thing the robot needs to do is to locate itself to a

proper position in front of the wall and then start to transit.

Al

 

 

 

 

 

 

 

 

///// \ ///// \ ///// \
toot1 Stage 1 Stage 2 Stage 3 Stage 4
t t t
t z A
\ \ \ I
//////// /77777777777777 /77777777777777
Stage 5 Stage 6 Stage 7 Stage 8

Figure 2.14: Locomotion sequence to transit from ground to a wall surface

41

. The robot walks to the proper location close to a wall with the robot main axis

perpendicular to the wall surface.
. Rotate J1 to lift up the front foot to its maximum allowed value.

. Slide J3 in LFS-translation mode to extend the robot to its maximum length

and rotate .15 to align the front foot with the wall surface.

. Rotate J1 in opposite direction to put the robot down to the wall surface and

adjust J5 until the front foot touch the wall seamlessly.
. Release the suction on the rear foot, and make the front foot the standing foot.

. Slide J3 in RFS.translation mode until the robot contract to its minimum

length, and rotate J1 to adjust the rear foot.
. Rotate J5 to bring the rear foot in contact with the wall surface.

. Execute the suction of rear foot and then the robot is ready to move around on

the wall.

2.4 Dynamic Model

The robot dynamic model is a set of mathematical equations describing the dynamic

behavior of the robot, i.e., the relationship between the robot joint forces/ torques

and the robot joint positions, velocities and accelerations. Such equations of motion

are useful for the design of suitable control laws or strategies to achieve a desired

system performance. The dynamic model of a robot can be obtained from known

physical laws such as the laws of Newtonian mechanics and Lagrangian mechanics.

The derivation of the robot dynamic model based on the Lagrange-Euler (L-E) for-

mulation is straight forward and systematic. This section derives the dynamic model

of the CRAWLER robot based on the L—E method.

42

2.4.1 Lagrange-Euler Method

In the L—E formulation, the system’s dynamic behavior is described in terms of work
and energy using generalized coordinates. All the workless forces and constraint
forces are automatically eliminated in this method. By directly using the L—E formu-
lation and the D-H link coordinate representation of a robot arm, the resultant robot
dynamic model is a compact matrix description of the robot equations of motion.
Assuming a robot has n links, the L-E formulation is expressed in the following

form

d 6L 6L
— — — — = . . = ... .2
(341') aqz- F, i 1, 2, ,n (2 9)

where

L: Lagrangian function: kinetic energy K — potential energy P

K = total kinetic energy of the robot

P= total potential energy of the robot

0 q,-= generalized coordinates of the robot

qi=first time derivative of the generalized coordinate, q,-

F,- = generalized force(or torque) applied to the robot at joint i to drive link i.

The L—E formulation requires knowledge of the kinetic energy of the physical robot
system, which in turn requires knowledge of the velocity of each joint. Let ir, be a
point fixed in a link i and expressed in homogeneous coordinates with respect to ith

link coordinate frame,

43

17', = yi (2.30)

Zr

 

 

Let 0r, be the same point ‘7‘, with respect to the base coordinate frame, “34, the
homogeneous coordinate transformation matrix which relates the spatial displacement
of the ith link coordinate frame to the (i — 1)th link coordinate frame, and 94, the
coordinate frame matrix which relates the ith coordinate frame to the base coordinate

frame; then 0r, is related to the point 1r, by

= 9437‘; = 941242 ' ° ' “Mfr, (2.31)

The velocity of 1"r, expressed in the base coordinate frame can be written as

ND

’0; vi dt07'( i) =%:94 i7" 1') (2-32)

%i}42 ' ' ' i_A"7" + dtoAiMQ ' ' ’i—Miiri + ' ° ° + 941’ ' ' t-Mi iTi + 9437.},

= (z.—:. 0"“

The partial derivative of 21, with respect to q- can be easily calculated with the
J

help of a matrix Q, which, for a revolute joint, is defined as

44

 

 

 

 

 

0 —1 O O
1 0 0 0
Q.- =
0 0 0 0
_ O 0 0 0 J
and, for a prismatic joint, as
0 0 0 0
O 0 O 0
Q.- =
0 0 O 1
O O 0 0
It then follows that
ai-IA, _ Q,_,A
aqi — 3 I

Let us define

60A, (Angi—lAi for j S i

[1,]. a =
6g,- 0 for j > i

 

(2.33)

(2.34)

(2.35)

(2.33)

Equation 2.36 can be interpreted as the effect of the motion of joint j on all the

points on link i. Using this notation, v, can be expressed as

45

i 6942' . i

The interaction effects of the motion of joint 3' and joint I: on all the points on

link i can be expressed as

 

94j—1ij—lAk—1Qkk-Mi for i Z k 2 J
6U,- .
Uijk é 6g): = (Mk—lek—Mj—leJ-Mi f0?" iJZ Z k (2'38)
0 . for j < j or i < k

After obtaining the joint velocity of each link, we need to find the kinetic energy
of link i. Let dK, be the kinetic energy of a particle with differential mass dm in link

i, then

dK, = éTr(v,v,-T)dm (2.39)

P i r T
= éTr 102:; Uipdpiri (Z: Uirqriri) dm
= éTTi Z 2‘; U,p(1‘, dm TiT )Uir TqPqT]

_p=l r=1

 

where the trace operator is the sum of diagonal elements of a square matrix, i.e.,
Tr(A)-— Z": a,,, where A is any square matrix with elements of a,,-.

Then the kinetic energy of link i is

46

1 p i i ' t' T . .
K,- =/dK, = -2-Tr ZZU,p(/'r,r, dm)U,,quq,.] (2.40)

_p=1 r=l

= éTr :ZU,pJ,U,,chpcj,]

Lp=l r=1

 

where J, is the inertia of all the points on link i. J, is dependent on the mass
distribution of link i and is expressed with respect to the ith coordinate frame. Hence,
J, is a coefficient term needs be computed only once and can be expressed in inertia

tensor

f zfdm f x,y,dm f :c,z,dm f 2:,dm

= (2.41)
fx,z,dm fy,z,dm fzfdm fz,dm '

_ f x,dm f y,dm f z,dm f dm J

- - T fxiyidm fyfdm fyizidm fy,dm
J, = / ‘r,’r, dm

 

 

-1314: +1" Izy Ixz mifi

131; W 11,: migi

- 1:2 1112 MP miii
"lift miy—i 777451 mi ..

 

where I", 11m and I“ are known as the moments of inertia of the link about the axes

2:, y and 2, respectively, which are defined as

In = f(y2 + 22)dm
IW = f(:r2 + 22)dm (2-42)
I,z = f(:l:2 -l- y2)dm

47

Ixy, In, and Iyz, are known as the products of inertia which are defined as

I“, = In = fxydm
I“ = In = fxzdm (2-43)
1,,z = 12,, = fyzdm

and where 1'7", = [27,, 37,, 2,, 1]T is the center of mass vector of link i expressed in the
ith link coordinate frame.

Hence, the total kinetic energy K of the robot is

n i i

(2.44)

K
II
M
3
ll
NI 1—4
M
M
M
E
5
S4
S
29
5°.
:9.

The total potential energy of the robot can be obtained by summing all the po-

tential energies in each link, P,

n n

P = 2P; = 2 -mig(9\ITI) (2-45)

i=1 i=1

where g = [gm g,, g,, 0] is a gravity row vector expressed in the base coordinate
system, and [g]: g = 9.8062m/sec2 is the gravitational constant.

The lagrangian function L = K — P is given by

L = $22: [Tr(U,,-J.-U.-.T)q,-q.] + ZWECAITI) (2.43)

i=1 j=l k=l i=1

Applying the L-E formulation to this Lagrangian function yields the generalized

48

force required to drive the ith link of the robot, for i = 1, 2, - - - , n

 

d 6L 6L
F.- = — — — .

dt 3,.) aq. (2 47)
n .l n .l .l n

= Z 2 measure + :3 z 2 measures. - : mngjt’fj
[=1 k=1 [=1 k=1m=1 5::

= Z 0.th + Z Z htkmqkqm + c.-
k=l k=1 m=l

In a matrix form, the robot dynamics can be expressed as
F(t) = D(a(t))<i(t) + h(QO). 4(0) + C(00)) (2-48)
where q(t), q(t), q(t) are n x 1 vectors of generalized joint variable, velocity, and ac-
celeration, respectively, and
o F(t) is an n x 1 generalized torque vector applied at joints i = 1, 2, - - - ,n.
o D(q) is an n x n inertial acceleration-related symmetric matrix.

0 h(q, (j) is an n x 1 nonlinear Coriolis and centrifugal force vector.

0 G(q) is an n x l gravitational force vector.

2.4.2 Derivation of the Dynamic Model

This section derives the dynamic model of the CRAWLER robot. Since the CRAWLER
robot is symmetric, with identical leg/ rack pairs and foot assemblies, the robot dy-
namics is analyzed in LFS phase only. The dynamic model is the same for both RFS
and LFS phases.

Figure 2.15 shows the assembly of the CRAWLER robot. Assuming foot 1 sup—

ports the robot, the coordinate frame is set in the same way as in Figure 2.7, with

49

 

 

 

 

 

 

Figure 2.15: CRAWLER robot assembly

XoYoZo as the base coordinate frame. The robot consists of three links driven by
three motors. Link 1 consists of legl/rackl pair and the body plate, this combination
rotates around axis Z0; link 2 is leg2/rack2 assembly, which slides along Z, axis; link
3 is foot 2 assembly, which rotates around axis Z2. The location of the center of mass
of each link, and the setup of coordinate frames are illustrated in Figure 2.15. When
the robot moves in LFS.translation mode, the generalized joint variables are a, d and
)6.

As discussed in section 2.2.2, the homogeneous coordinate transformation matri-
ces, "54,, i = 1,2,3, are shown in Equations 2.5 to 2.7. By plugging in these matrices

to Equation 2.36, we have

50

U31

6941 60A,

 

 

U11 = i=3;
p—sina O cosa —Msina
_ cosa 0 sina M ccsa
— 0 0 0 O
b 0 0 0 0
‘11 30
r sinoz cosa O dcosoz-
_ —cosa sina 0 dsina
_ O 0 0 0
_ 0 0 0 O _
ages
6g, 60:
- sin(a+6) cos(a+6) O (L-M)sin(a-.l-,6)+dcosa

0 O

 

O 0

51

 

 

—cos(a+6) sin(a+6) 0 —(L—M)cos(a+6)+dsina
0
0

q

(2.49)

(2.50)

(2.51)

 

 

 

 

 

 

 

U22 = an (id (2.52)
0 0 0 sinoz
0 0 0 —cosa
0 O O 0
0 O 0 0
60A, 6%
U32 _ 562-79?— (2.53)
O 0 0 sina
0 0 0 —cosa
O O O O
O 0 O 0
6%, 6943
U33 — Eg—EF (2.54)
sin(a+6) cos(a+6) 0 (L—M)sin(a+6)
_ -—cos(a+,6) sin(a+6) 0 —(L-M)OOS(0+5)
0 0 O 0
0 0 0 0
By using
Di]: : Z UijjUjiT (255)
j=m(ivk)

the elements of the 3 x 3 inertia matrix D(q) are

52

D11 = Tr(U11J1U11T)+ Tr(U21J2U21T) + Tr(U31J3U31T) (2.56)
D12 = D21 = TT(U22~12U21T) + Tr(U32J3U31T)

D13 = D31 = TT(U33J3U31T)

1),, = Tr(U22J2U22T) + Tr(U32J3U32T)

D23 = Dsz = TT(U33']3U32T)

D33 = TT(U3313U33T)

To derive the 3 x 1 Coriolis and centrifugal force vector, h(q, (j) = [h], ’12, h3]T, we

us the following equation

3 3 3 3 3
h.- = E Z hamqtqm = Z Z I Z Tr(U,-ka,U,-,T)Iqtqm (2.57)

k=l m==l k=l m=l j=ma.1:(i,k,m)

It is worth noting that the inertial coefficient in J, can be obtained by using the
AutoCAD design parameters and Equation 2.41. Some of the coefficient may be zero,
therefore simplifies the expression of D(q) and h(q, rj) greatly.

Since the gravity force plays a dominant role in robot dynamies when the robot
moves at slow speeds, we derive the gravity-related terms in detail as follows.

The elements of the gravity term, G(q) = [c1, c2, c3]T, are

—(mngj,’i‘j (2.58)

TI'Mw

Noticing the coordinate system in Figure 2.15, the center of mass of each link j

53

expressed in the jth link coordinate frame can be simplified as

 

b

q r- - p- -

 

 

 

 

 

0 —M 4:33]
0 - T 0
272 = lyzl 31:3 =
'51] 0 0
1 l l
.. L . . .I

The gravity representations in base coordinate frame are different when the CRAWLER

robot walks in different places, which are illustrated in Figures 2.16 to 2.19. When

the robot walks on a floor, we have g= [0, -g, 0, 0]T. Similarly, g= [0, g, 0, 0]T when

the robot sticks on a ceiling, g= {-g, 0, O, O]T when the robot climbs up along a wall,

and g: [9, 0,0, 0]T, when the robot climbs down along a wall.

By plugging the necessary parameters in Equation 2.58, the corresponding gravity

terms in those cases is derived.

When the CRAWLER robot moves on a ceiling, we have

C1:

—(mlgU111f1 + ngUziaiz + maSUsrafsl (2-59)
[mlglifl] sin a + mlgMcosa]
+[mggMcosa + mgg(d — [372]) sin a] _

+[mgg(|i3| — (L — M)) cos(a + 6) + mggdsin a]

C2 = —(m2gU222E2+m38U323f3) (2-60)

= —(mg + m3)gcosa

54

Cs

—m3gU33373

magllial - (L - M )l 008(a + 6)

When the robot moves on a floor, we have

CI

“(mlgulllfl + magUzrafz + mngsiafs)

—[mlg|21|sina + mlgM cos a]

—[mggM cosa + ng(d - [372]) sin a]

—[m3g(|:53| — (L -— M)) cos(a + 6) + m3gdsin a]

Ca

62

-(ngU222F2 + mng323f3)

(mg + m3)g cos a

-m38U38373

-maglli‘3l - (L - M )1 008(6): + 6)

55

(2.61)

(2.62)

(2.33)

(2.64)

 

 

 

 

\\\\\\\

- . 417.7

 

 

\\\\\\\\\\\\\

Figure 2.17: Gravity analysis when the CRAWLER robot walks on a floor

56

 

\\N\\\\\\\\\\\\\\

 

Figure 2.18: Gravity analysis when the CRAWLER robot climbs up a wall

4.4... A.
A A '\

 

\\\\\\\\\\\\\\\\\ _

 

Figure 2.19: Gravity analysis when the CRAWLER robot climbs down a wall

57

When the robot climbs up on a wall, we have

Cl = —(mlgU11‘f1 + ngU312F3 + mggU313f3) (2.65)
= lmlglzll cosa — mlgM sin a]
+[—m2gM sina + ng(d — [372D cos a]

+[—mgg(|i:3[ + (L - M)) sin(a + 6) — m3gdcos al

62‘ = -(TTI2gU32272 + mng323f3) (2.66)
= (mg + m3)g sina
03 = ’m3gU333f3 (2-67)

= -m,g[|23| - (L - M )l Sin(or + 5)

When the robot climbs down on a wall, we have

01 = 4711181111171 + nguaiaia + mngsrafs) (2-68)
= —[mlg|21|cosa — mlgM sin a]
—[—mggM sin a + mgg(d — [52]) cosa]

—[—mgg(|i:3| + (L — M)) sin(a + 6) — mggdcos a]

58

C2 = —(m2gU323f3+m3gU33373) (2.69)

= —(m2 + m3)g sin a

C3 = —m3gU333’fs (2.70)

= mag[|§:3| — (L — M)] sin(a + 6)

The generalized force can be identified by considering the virtual work done by
non-conservative forces acting on the robot. For the CRAWLER robot case, F =
Tor — f, where Tor = [tor1,tore,,tor3]T is a 3 x 1 vector, and tor, is the torque
executed on joint i by motor i. f = [f1, f2, f3]T is a vector representing friction

terms, with
f.- = K;.sgn(ci.-) (2.71)
where K I, is a constant friction coefficient. sgn(a:) is a signum function defined as

+1 if 2:20
sgn(a:) =
-1 if z<0

Hence, the dynamic equation of the CRAWLER robot is given in matrix form by

TOTO) = D(q)<'1'(t) + h(q, (i) + 0(0) + f (ti) (2-72)

where the generalized joint variables are q, E a, q; E d, and q3 E 6.

59

This equation and its corresponding terms shown in Equations 2.55 to 2.71 de-
scribe the dynamic behavior of the CRAWLER robot in terms of the relationship
between the input joint torques and the output motion. In practical situation, the
CRAWLER robot is unable to move at fast speed. Therefore, the acceleration—related
inertia matrix D(q) and the velocity-related Coriolis and centrifugal force vector
h(q, q') are neglectable. The gravitational effects of the links play a dominant role
in robot dynamics. A control algorithm which compensate the gravitational effects

will be discussed in chapter 3.

2.5 Summary

This chapter presents the modeling of the CRAWLER robot. The mechanical struc-
ture and locomotion modes of the robot are introduced. The kinematic model that
describes the relation between the joint variables and the position/ orientation of the
robot’s free foot with respect to its standing foot is derived. The robot kinematic
model is governed by different equations in different motion modes. Therefore, it is
important to use the corresponding kinematic equations when plan the robot motion.
In order to study the dynamic behavior of the CRAWLER robot, the dynamic model
is derived based on the L-E formulation. The dynamic model describes the relation
between the input joint torques and the output motion. The gravitational effects play
a dominant role in robot dynamics.

In next chapter, an embedded control system for the CRAWLER robot will be
introduced. The control system will use the derived kinematic model and compensate

the gravitational effects to achieve a good performance.

60

CHAPTER 3
EMBEDDED CONTROL SYSTEM DESIGN

As a self-contained embedded control system, the climbing robot needs to carry its
own power source, sensors, control system, and associated hardware. Thus mini-
mization of weight and power consumption is critical to prolonged operation. The
TMS320LF2407 digital signal processor (DSP) from Texas Instruments (TI) Inc. is
an ideal candidate for an embedded controller because of its high-speed performance,
its support for multi—motor control, and its low power consumption. This chapter

describes the control system design and implementation based on a TI DSP.

3.1 Control System Overview

3.1.1 Actuators and Sensors

To minimize the weight and complexity, the climbing robot has a limited number of
actuators and primary sensors. The actuators include three DC servo motors with en-
coder feedback, two suction pump motors, and two micro valves. The primary sensor
components include two pressure sensors and six touch sensors located at the suction
feet. In order to distinguish between different motion modes, two contact switches are
installed on robot legs to determine whether the leg and rack are locked. For remote
control operation the robot has a wireless receiver module, which communicates with
the transmitter module in a remote controller. All the signals from those components
and sensors need to be processed and integrated into an on—board control system.
Apart from the primary sensors which are critical for Operation, additional appli-
cation sensors can be installed on the robot as payloads when requested by specific
tasks. The application sensors include, but are not limited to, wireless cameras,

sonar sensors, inclinometers, microphones and infrared sensors. For reconnaissance

61

purpose, a wireless pin-hole camera is installed and the video images are transmitted
to and processed at a host computer.

Figure 3.1 is the close-up of the robot foot where the sensors are shown. The touch
sensors are attached to the suction cup in different radial directions. They consist of
an outer tube and an inner super-elastic wire isolated by a silicon tube. When the
inner wire touches the outer tube the switch closed. In such a way, the touch sensors
provide a set of digital inputs to DSP to facilitate the robot in adjusting the suction

foot orientation.

 

Figure 3.1: Sensors at robot foot

The major components of the robot and their functions are listed in Table 3.1.

To enable teleoperation, a remote controller is built around an RF (radio fre-
quency) transmitter module which is operated at 418 Mhz. The picture of the 16-key
remote controller is shown in Figure 3.2. The key sequences encode a set of operator

commands.

3.1.2 Control System Structure

The control system structure is illustrated in the block diagram as shown in Figure

3.3. The physical actuators and sensors are represented in the right block. Other

62

Table 3.1: Actuators and sensors used in the CRAWLER robot

sensors

toa

to cup
facilitate the of the suction foot orientation

between different motion modes

remote controller

 

Figure 3.2: Picture of the remote controller

63

blocks represent the on-board software modules including command interpreter, task

level scheduler, trajectory planner, joint level controller and motion planner.

Crawler Olln'blng Robot
lnltiel OM noel Feasible motion E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OOO'iOUMiOM Mdion sequence Encoder
_——__.m
Planner
a
Teek -
Commend Mohon Stetus
r Level w |§:::=:=
l
"mm“ esk level Scheduler
commends
H Motor 3
Trajectory
Motion steps Planner M
Operator ‘__’ Contact
Comnd Joht M
Level
Deslred lolnt Controller
Remote 009'“ — Touch
Controller ‘ Sensors Co
Closed loop line St “1:“
motion control ‘ Pns'm
Sensors
m

 

 

 

 

Figure 3.3: Control system block diagram of the climbing robot

The operator commands, such as “move forward” , “make left turn”, are trans-
mitted from the remote controller held by a human Operator and decoded by the
on—board command interpreter. The generated task level commands are then fed into
the task level scheduler. The task level scheduler uses a finite state machine to keep
track of robot motion status and decompose the command into several motion steps.
The trajectory planner solves the inverse kinematic model and interpolates the path
to generate a set of desired joint angles. The digital motor controller then drives each
joint to the desired set points so that the foot is placed to the desired location.

Placing the foot on a surface consists of gross and fine motion controls. The gross
motion control is based on the desired solution of the inverse kinematics and it brings
the free foot to the neighborhood of the desired position and orientation. However,

the results of the actual foot placement may not be sufficiently accurate due to the

64

existence of uncertainties and disturbances caused by backlash, gear friction, sensor
error, and varying gravitational effects. Thus, after the gross motion control, the joint
level controller conducts the fine motion control. The fine motion control utilizes the
signal from the touch sensors and pressure sensors as a feedback to adjust the robot
foot to ensure that it grips the contact surface reliably. Adjustment of the robot
foot is a closed loop control process and only requires varying the tilt angle in small
increments to align the foot with the contact surface.

When the robot has motion planning ability, which is described in the next chap-
ter, the motion planner generates a feasible motion sequence and transmits it to the
task level scheduler. After the motion sequence has been executed, the robot is able
to travel from its initial configuration to its goal configuration, while avoiding the

obstacles in the environment.

3.2 Hardware Design

This section describes the hardware implementation of a self-contained embedded
robot control system based on a TI DSP chip — TMS320LF2407.

- 3. 2.1 TMS320LF2407 DSP Overview

The TMS320LF2407 device [31, 32, 78] is a new member of the C2000 family of TI
DSP controllers. It is targeted to meet the needs of control system applications. By
integrating the high performance of a DSP core and the on-chip peripherals into a
single-chip solution, the LF2407 device is a low—cost alternative to traditional micro-
controller units (MCUs) and expensive multichip designs. This chip provides all the
resources needed to build a self—contained embedded control system. The functional
block diagram of the LF2407 DSP controller is shown in Figure 3.4.

The LF2407 DSP controller is based on the 16—bit, fixed-point, low-power C2xx

65

 

     
   
 

411:1

IIIIIIIII

.‘

 

l 1

till lllllllll ELL Ll

 

 

 

 

 

 

 

 

 

Figure 3.4: Functional block diagram of the LF2407 DSP controller

DSP core. The parallel architecture of the central processing unit (CPU) enables the
DSP to perform in high speed at 30 million instructions per second (MIPS). Thus
it allows the DSP to often process algorithm in real time rather than approximate
results with look-up tables.

By integrating memory and peripherals onto a single chip, the DSP devices reduce
system cost and save circuit board space. LF2407 DSP provides 32K words on—chip
flash memory, which offers a reprogrammable solution useful for the initial prototyping

of applications. The 2.5K words on-chip random access memory (RAM) includes

66

544 words of Dual-Access RAM (DARAM) and 2K words of Single-Access RAM
(SARAM), which are configurable as data or program memory. With the help of an
external memory interface the LF2407 chip can offer up to 64K program memory and
64K data memory space.

The LF2407 DSP has two event manager modules (EVA and EVB) that have been
optimized for digital motor control. Capabilities of each module include two 16-bit
general-purpose timers, three capture units, and eight 16-bit pulse-width modulation
(PWM) channels. Two of the capture units have built-in quadrature encoder pulse
(QEP) circuits, which are used to obtain the motor encoder readings easily.

The high performance analog-to-digital converter (ADC) has 10-bits resolution,
a minimum conversion time of 500 ns, and up to 16 channels of analog input. The
auto-sequencing capability of the ADC allows a maximum of 16 conversions to take
place in a single conversion session without any CPU overhead.

A serial communications interface (SCI) is integrated on the device to provide
asynchronous communication capability. For systems requiring additional communi-
cation interfaces, the LF2407 offers a synchronous serial peripheral interface (SP1)
and a controller area network (CAN) communications module. To maximize device
flexibility, functional pins are configurable as general purpose inputs / outputs (GPIO).
Thus, up to 40 multiplexed digital I/O pins are available. JTAG (Joint Test Action
Group) module provides non-intrusive real-time debugging capability, which is helpful
to reduce the system development time.

The LF2407 DSP is based on low-power 3.3V CMOS technology and it integrates
many power management features. It has three power down modes and the abil-
ity to power-down each peripheral module independently. These features make the
LF2407 DSP a desirable device to be used in an embedded control system with power

constraints.

67

3.2.2 DSP Implementation of the Controller

Due to the limitations of size, weight, and power consumption, one of the challenging
tasks in designing the controller for the miniature robot is to use as few compo-
nents as possible. Figure 3.5 illustrates the controller block diagram based on a
TMSBZOLF2407 DSP chip.

 

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TMS320LF2407 DSP
CAPS

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960m"

 

 

 

 

 

 

ll
so as

 

 

 

 

 

 

 

 

 

Aocmo lOPFZ I2 en
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van: l IOPOI emsmo
M? r at

Iowa: -- 3v

Comet IOPL- A Eu
W H OOPCI M at ev

XINTl IOPB 6.1 IOPF 4.5
Decoder K4 Received

 

Figure 3.5: Block diagram of the DSP-based controller

Two quadruple half-H driver SN754410 from TI are used to drive the three servo
motors and the two pump motors. The servo motors are driven by the PWM outputs
PWM 1, 2, PWM 3, 4, and PWM 5, 6 of EVA via the H-bridge driver. Timer 1 is used
as the time base to generate the PWM signal with a frequency of 20 KHz. Timer 3 is
used to generate the servo—control sampling rate of 1 KHz. During each servo sampling
period (1 ms), the compare registers, CMPR 1, CMPR 2, and CMPR 3 are updated
for Motor 1, Motor 2, and Motor 3, respectively according to the calculated joint
control value. The LF2407 DSP has two built-in quadrature encoder pulse (QEP)

circuits. The encoder readings of servo Motor 1 and Motor 2 are easily obtained

68

using the QEPl/QEP2 and QEP3/QEP4 of the event manager modules with Timer
2 and Timer 4 as the time base, respectively. However, for Motor 3, an alternative
method needs to be employed to get the encoder reading. The encoder channel A of
servo Motor 3 is connected to capture unit pin 3 (CAP3), and the capture unit 3 is
enabled to detect the rising edge of the encoder pulse. Channel B is connected to the
CAP6/IOPF1 pin and this shared function pin is configured as digital input IOPFl.
Whenever the capture interrupt is triggered in CAP 3, an interrupt service routine
detects the digital level of the encoder channel B to determine the rotation direction
and adjusts the encoder pulse count value. Apart from serving as the time base for
PWM waveform, Timer 1 is also used as the time base for capture 3 operation. These
two functions do not affect each other. A software solution for getting the encoder
reading of Motor 3 using the capture unit is described in section 3.4.2.

The two pump motors require only binary switches to drive them. So the shared
function pins PWM7/IOPE1 and PWM8/IOPE2 are configured as digital output
pins. By using the H-bridge driver the pump can be turned on by setting the output
high. The micro valves are also controlled by digital outputs from the DSP by using
two transistors as drivers.

The touch sensors and contact switches produce high-level or low-level digital
signals that are connected directly to the I/O pins of the DSP. The analog inputs
from the pressure sensors are converted by the ADC to determine whether the suction
foot is securely attached to a flat surface. A receiver module and a decoder chip are
used for remote control operation. The impulse signal from the decoder triggers the
external interrupt pin of the DSP. The interrupt service routine processes the four
digital I/O outputs from the decoder and translates the remote control signal into
proper commands. _

It is clear that the resources of the DSP chip are efliciently utilized. As a result,

the components count of the embedded control system is very small and the circuit

69

is very compact.

3.3 Control Strategy

It is convenient to control the CRAWLER robot by using joint level PD (proportional
plus derivative) controllers, where each joint of the robot is treated as a servomech-
anism. However, this method is ineffective because the joint level controller doesn’t
take into consideration the dynamic effects of the whole robot links. Our control
strategy is to combine a PD controller with a feedforward nonlinear compensation
term to minimize the gravity loading effects on each joint. This control strateg
can achieve better performance than the conventional PD control method because
it not only achieves the joint level control but also compensates for the nonlinear
gravitational effects which are determined by the configuration of the whole robot
links.

3. 3. 1 Motor Dynamics

Each joint of the CRAWLER robot is driven by a DC servo motor. This section
introduces the motor transfer function from which control algorithms will be obtained.
The equivalent circuit of a permanent magnet DC servo motor [18]is shown in

Figure 3.6 based on the following variables:
e V, Applied voltage
e L Armature inductance

e R Armature resistance

Vb Back EMF (electromotive force)

e Jm Moment of inertia of the motor

70

 

 

Annature leCUlt

 

 

Gear traln
8. Load

 

Figure 3.6: Equivalent circuit of DC servo motor
e fm Viscous friction coefficient of the motor
e qm Angular displacement of the motor shaft
e Torm Torque de lived by the motor

Kb Back EMF constant

K 0 Torque constant of the motor

The following equations describe the motor characteristics.

Torm(t) = Kaia(t) (3-1)
Vb“) : Kb‘lm“)

_ . as)
v.(t> — Rz.(t)+L d,

TOTm(t) = JmilmU) + fmde)

 

+ Vb(t)

Taking the Laplace transform of these equations we have:

71

Torm(s) = KaIa(s) (3.2)

(3.3)

 

Terms) = sszqmo) + sfmqmls) (3.4)

By manipulating terms, we obtain the transfer function from the applied voltage

to the angular displacement of the motor shaft

qm(3) _ Kc

VJ?) ’ s[s2JmL + (L f... + 12.1...)3 + R f... + Kara] (3'5)

 

Since the electrical time constant of the motor is much smaller than the mechanical
time constant, we can neglect the armature inductance effect, L. Also assume that

the friction is zero,i.e., fm = 0, the above equation is simplified as

qm(s) _ Kc K

Va(s) — s(sRJm + KaKb) = s(Tms + 1) (3'6)

 

 

where

K é l/K,

is called the motor gain constant, and

 

the motor time constant.

Table 3.2 lists the actual parameters of the motors [47] used in the CRAWLER

robot which are selected from API Motion company.

Table 3.2: Motor parameters: escap 17S78-208P from API-Portescap

 

 

 

 

 

 

 

Parameters variable unit value
Back EMF constant Kb V/1000rpm (Vs/rad) 0.56 (0.0053)
Torque constant K, mN m / A 5.3
Terminal resistance R ohm 6.9
Rotor inductance L mH 0.15
Rotor inertia Jm lrcgm2.10"7 0.50
Mechanical time constant Tm ms 12

 

 

 

 

 

 

3. 3. 2 Control Algorithm

When the motors are used to drive the robot joints, gear trains are installed to increase
the motor load capability and to couple the motor shaft with the joint shaft. If the
gear ratio n = g: < 1, where qL is the angular displacement of the joint shaft, then

we have

qL(s) _ nKa
V3?) ‘ s(sRJ,,. + Kara) (3'7)

'

The purpose of the joint level controller is to servo the motor so that the actual
angular displacement of the joint will track a desired angular displacement specified
by a preplanned trajectory. The PD feedback controller is designed as follows

73

W) = Kplqmt)—qL(t)l:KvlddL(t)—qr(t)l (3.3)

 

er(t) + Kve(t)

 

where KP, K” are the positional feedback gain and error derivative feedback gain,
respectively. The gear ratio n is included to compute the applied voltage referred
to the motor shaft. qu(t), qu(t) are the desired position and velocity, respectively,
which are precomputed from a trajectory planner.

Taking the Laplace transform of Equation 3.8 and substituting into Equation 3.7
yield the closed-loop transfer function relating the actual angular displacement to the

desired angular displacement as:

qL(s) _ KaKvs + KaKp

_ 7 3.9
qu(s) 32RJm + s(KaKb + KaKv) + KaKp ( )

 

 

Figure 3.7 shows the block diagram of this closed-loop PD control system.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tor,(.r)
932/ r x 1 :13)“ 1 . 1 1 .(3 9,0)
+3 ' w — r — w K. r — - n i
Wail ' I WK R 705(3)” SJ. 3
Kb

 

 

 

 

 

 

Figure 3.7: PD control block diagram

Notice that the torque generated at the motor shaft has to compensate for the

load torque, thus from Equation 3.4 we have

74

Torm(s) = sszqm(s) + sfmqm(s) + TorL(s) (3.10)

where TorL(s) is the Laplace transform of the load torque referred to the motor shaft.
The transfer function relating the load torque to the actual joint displacement is

given by

qL(S_____)__ ________| —nR
TorL(s) 4%“): 0: 32521... + s KaKb + K,K.,) + KaK
P

 

(3.11)

Using the superposition principle and Equations 3.9 and 3.11, we can obtain the

actual displacement of the joint from these two inputs as follows:

(s)- (K K s+K K)p)q“L(s ~nRTorL(s)
q” 3212.]... + s(K K, + K K.) + K K,,

 

(3.12)

The robot dynamic model has been derived in section 2.4 and we restate it as

below in Equation 3.13.

Tor(t) = D(Q)q'(t) + h(q,r1)+G(Q) +f(<I) (3-13)

It reveals that four dynamic terms contribute to the torques applied at robot
joints. In practical implementation, the CRAWLER robot is only able to move at
slow speeds. Therefore, the dynamic effects related to the joint acceleration and
velocity are very small and neglectable. The dynamic effects other than the gravity
torques can be assumed to be a very small constant value. Thus, Equation 3.13 is

simplified as

75

Tor(t) = G(t) + Te (3.14)

where G(t) = G(q(t)) and Te is a small constant.
Considering the gear train, the load torque applied at the shaft of the driving

motor is
TorL(t) = nTor(t) = nG(t) + nTe (3.15)

where n < 1 is the gear ratio.

The corresponding Laplace transform of Equation 3.15 is

TorL(s) = nG(s) + 1:3 (3.16)

The gravity effects are determined by the configuration of all the robot links and
plays a dominant role in robot dynamic model. In order to compensate for gravity
effects, we can precompute these torque values based on the desired trajectory and
feed the computed torques into the controller to minimize the effects as shown in

Figure 3.8. This is called the PD+gravity compensation method.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

forum)
Torque Computation . Tor (s)
l.
y (s) ‘5 L _(: 41(3)
1
. 25,- 19+“. '- ""}v '- K. ; T 1 - n H»
91(3) ll_ ' \l- R Tor_ (.r) S ' '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Kb

 

 

 

 

 

 

Figure 3.8: PD+gravity compensation control block diagram

76

The computation of Torm, depends on the dynamic model. Since the gravity is

the dominant term in robot dynamics, Tormp only computes the gravity term as
Torcomp(t) = non) (3.17)

The element of C (t) is c,-. It represents the gravity effects exerted at joint i and
can be calculated using Equations 2.59 to 2.70 in section 2.4.

Now we have the following proposition

Proposition 3.3.1 When a motor rotates in a direction to overcome the gravity
effects of its load, the motor control system using PD+gravity compensation method
achieves a smaller steady-state error in response to a step input than the system using

the conventional PD control method.

Proof. We prove this proposition by comparing the steady-state errors of the
conventional PD controller and our proposed PD+gravity compensation controller.
Using Equation 3.12, the system error e(t) = qu(t) — qL(t) in the Laplace trans-

form domain can be expressed as

3(3) = qu(3) - qL(3) (3-18)
(sszR + sKaKb)qfi(s) + nRTorL(s)
3212],” + s(KaKb + KaKu) + KaKp

 

For a step input of magnitude A, qu(t) = A, its Laplace transform is qu(s) =
A/s. Using the final value theorem, the steady-state error of the PD control system

due to the step input can be expressed as

77

tlim e(t) = lirn) sE(s) (3.19)

8 lim (sszR + SKaKb)A/3 + nRTorL(s)
._.o 3212.]... + s(K,K., + K.K.,) + KaKp
slim nRTOTL(S)
.—.0 3212.1". + 3(KaKb + KaKv) + KGKP
lim n2R[sG(s) + Te]
a—+0 32R]m + 3(KaKb + KaKv) + KaKp
sn2RG(s) anTe

1.
.1333 mm, + s(K,,Kb + K,K.,) + KaKp + K.Kp

 

 

 

 

 

By investigating the gravity terms in Equations 2.59 to 2.70, and their correspond-

ing Laplace transfer functions, we know that the term

lim sn2RG(s)
a—oO szRJm + 3(KaKb + KaKu) + KaKp

 

is non-zero. Therefore, the steady-state error of the PD control system is a function

of the gravity effects. In the case when the motor rotates to overcome the gravity

force, this term is positive resulting in large steady-state errors.

On the other hand, the error equation of the control system using PD+gravity

compensation method is

E(s) =

qu(3) — (11.03) (3-20)
(s2JmR + sKaKb)qfi(s) + nR[TorL(s) — Tormp(s)]
s2RJm + 3(KaKb + KaKv) + KaK,

 

78

The steady-state position error corresponding to the step input becomes

e = slim nR[TorL(s) — Tormp(s)]
5’ s-+0 32R]m + s(KaKb + KaKv) + KaKp
= lim snR[nG(s) + nTe/s — nC(s)]
...o sZRJm + s(KaKb + KaKv) + KaK,

(3.21)

 

 

If the computed torque Tormp is equivalent to the gravity loading of the links,
i.e., C(s) = 0(5), then the steady-state position error reduces to

anTe

K.Kp (3.22)

 

eu =

which is small because TC is assumed to be very small and other parameters have
bounded positive values. [:1

In real digital control implementation, the computed torque Tormp is converted
to the applied voltage through a voltage-torque characteristic coefficient K m. A
typical voltage-torque curve can be found in [18]. The coefficient K a is obtained

from experiments. Then the PD+gravity compensation method is expressed as

er(t) + Kvé(t) Tormp
+ —_
12 Km

 

V..(t) = (3.23)

3.4 Software Development

3.4.1 Software Modules

Software modules have been developed in C and TI DSP assembly language to control
the climbing robots. They work well for both the FLIPPER and the CRAWLER

robots. The main software modules running on the embedded controller are:

79

e Task level scheduler

e Trajectory planner

Joint level servo controller

Remote command interpreter

e Communication module

Task level scheduler detects the motion status and decomposes the task level
commands into several motion steps. A finite state machine is developed to keep
track of the robot motion status, such as the switching between different motion
modes, the changing of standing foot, and the states of the suction pumps.

Tl'ajectory planner activates different kinematic models according to different mo—
tion modes and solves the inverse kinematics to plan the joint level trajectory. The
interpolation is conducted to generate desired joint angles.

Joint level servo controller implements the PD+gravity compensation control al-
gorithm to drive the motors to desired angles.

Command interpreter is used to decode the commands sent by a human operator
through a remote transmitter. The receiver chip on the embedded controller board
triggers an external interrupt whenever a button in the remote controller panel is
pushed. The interrupt service routine processes four digital I / O outputs from the
receiver and translates the remote control signal into proper commands.

Communication module handles the RS—232 serial communication between em-
bedded controller board and a host computer. A command interface is implemented

to facilitate the testing of the robot control system.

80

3.4. 2 Encoder Feedback

Accurate digital motor control is the basic requirement for the robot to accomplish
certain tasks. To achieve this, the servo controller requires position information from
the encoder as feedback. As mentioned before, it is easy to obtain encoder reading for
motor 1 and motor 2 using the two built-in QEP circuit of DSP. However, for motor
3, software solution is adopted to get the encoder reading because adding additional

circuits is not desirable for a miniature robot limited in size and weight.

ChB

DIR j |

Figure 3.9: Encoder pulse and motor direction

 

 

The analysis of the magnetic encoder waveform of the servomotor shown in Figure

3.9 indicates that:

1. The phase shift of the encoder pulse sequences between two channels (A and
B) is 90 degrees.

2. The logic levels of channel B corresponding to the rising edges of channel A

alternate when the motor changes direction.

These properties are utilized to obtain the encoder reading. The encoder channel
A of servo Motor 3 is connected to CAP3 pin and the capture unit 3 is enabled to
detect the rising edge of the encoder pulse. Channel B is connected to input digital
I / 0 pin IOPFl. Whenever a rising edge is detected, a capture interrupt is triggered.
In the interrupt service routine, the logical level of channel B is detected through
IOPFl. When channel B is logic low, it means that the motor rotates in a positive

direction, then the encoder pulse count variable CAP3.cnt increments by one. When

81

channel B is logic high, it means the motor has changed the rotation direction, then
CAP3.cnt decrements by one. In such a way, CAP3.cnt updates its value by +1 or -1
according to the rotation direction every time the encoder pulse triggers. Thus, the
value of CAP3.cnt indicates the motor angle expressed in encoder counts. CAP3.cnt
is set as a global variable and its value can be accessed by servo control routine to

accomplish feedback control.

3.4.3 Servo Control Module

Servo control module is the fundamental and most important part for a robot control
system. A specific interrupt service routine (ISR) processes the servo control and
motion profile calculation. Timer 3 is selected to generate servo control sampling
period of 1 ms by setting the associated period register, T3PR, with proper value.
Figure 3.10 shows the flowchart of the servo ISR.

 

U pdate motor
posllon

l

Calculate
motlon promo

1

Update
posltlon error

i

PD + grevlty
cornpensatlon

i

Update PWM
duty cycle

Figure 3.10: Servo control interrupt service routine flowchart

 

 

 

 

 

 

 

 

 

 

 

For smooth motion control, a motion profile is necessary to control the motor

82

acceleration and deceleration. When a motor is commanded to rotate a certain angle,
the desired position for each servo step must be calculated by the motion profile rou-
tine. Without the motion profile, motion will be abrupt, causing excessive wear on the
mechanical components and degrading the performance of the control algorithm. For
our application, a linear piecewise trapezoidal/ triangular velocity trajectory (Figure
3.11) is adopted to implement the motion profile. It consists of acceleration, constant
motion, and deceleration sections for the motor rotating a large angle. When the

motor rotates a small angle, it only consists of acceleration and deceleration motion.

 

 

“Veloclty
vue— phasel -e-ue— phase2 —e-
V_llmlt- Long
move
Slope=- -,
Ame, W, \a Short
. move
'1 : : Tlme

 

 

 

Figure 3.11: Velocity trajectory of motion profile

The flowchart for implementing the motion profile is shown in Figure 3.12. The
value of the commanded move distance is divided by two and placed in variable
ph1.dist. The motion destination (or final desired position) is the initial position of
the motor plus the commanded move distance. The desired distance, desired velocity
and desired position for each servo step is placed in variable, d..dist, d..vel and d.pos
respectively. The flat.cnt variable is used to determine when the constant velocity
section starts or stops. The first.half is the flag indicating whether the motion is
in phase 1 or phase 2. When the desired position reaches the motion destination,
the move_in.progress flag is cleared ignoring further motion profile calculation until
the move is completed and this flag is set. The output of the motion profile routine
is the desired position for each servo step. Those desired positions guide the motor

rotating from acceleration stage through constant motion stage to deceleration stage

83

until settling down at the destination position.

   
 

d_pos=rlnel
desired postuon

  

 

 

 

 

o_pos-
d_pos-d_dlst

 

 

u_pos= d_pos:
d_pos-d_dlst d_pos+d_ulst

d_pos-
d_poe4-d_dlst

 

 

 

 

 

 

 

Dalila poellon
' for each servo up

Figure 3.12: Motion profile flowchart

3.5 Experiments

The CRAWLER robot and its embedded control system based on TMS320LF2407
DSP chip has been successfully designed and implemented. Experiments were con-
ducted to evaluate the performance of the robot.

Figure 3.13 shows a sequence of snapshots of the CRAWLER moving one step
forward on a wall surface.

Figure 3.14 shows the CRAWLER robot walking around a corner. Figure 3.14
(1-4) show that the CRAWLER makes a 45° left turn. After the CRAWLER moves

84

 

Figure 3.13: The CRAWLER robot moving forward on a wall

forward by two steps (5—8), it makes a second 45° left turn (9—10), and then continues
moving forward (ll-12) along the new direction. The 4th snapshot shows that the
robot unwinds the rear foot after a turn to make the robot ready to move in translation
mode.

Figure 3.15 shows that the robot is capable of moving between surfaces.

3.6 Summary

This chapter describes the embedded control system design of the climbing robot.
By using TI DSP, the number of electrical components is minimized. A efficient
circuit board is designed, and installed on the robot body, which makes the control

system self-contained and tetherless. A joint level PD+gravity compensation method

85

 

Figure 3.14: The CRAWLER robot making turns around a corner

is proposed. This control method outperforms the conventional PD controller because
it not only achieves the joint level control but also compensates for the gravity effects
which is determined by the configuration of all the robot links. Software modules
of the control system are also presented in this chapter. Experimental results are

presented with snapshots showing the capabilities of the robot.

86

 

Figure 3.15: The CRAWLER robot moving between surfaces

87

CHAPTER 4
MOTION PLANNING OF THE CRAWLER ROBOT

With the installation of the robot controller, the climbing robot now can be tele-
operated by human operators. However, in order to make the robot smart, we need to
provide the robot with motion planning ability. A robot with motion planning ability
is capable of determining autonomously a motion path to accomplish a particular
task without collision with obstacles. This automatic determination of a path relieves
the user from the burden of specifying tediously detailed instructions for the robot.
Therefore, the user can specify tasks more declaratively, by stating what he wants
done rather than how to do it.

There are two different situations when the robot planes its motion. In the first
situation, the robot has exact knowledge of the environmental settings and the loca—
tion and shape of obstacles. The purpose of the motion planing in this situation is
to find a collision-free path connecting the initial and goal position. It is called the
basic motion planning. In the second situation, the robot has no knowledge about
the environment. The robot uses its own sensors to detect and navigate through
the environment. It is called sensor-based motion planning or robot navigation. In
this chapter, we only study the basic motion planning problem by assuming that the
environment is known.

The goal of basic motion planning is to compute a motion path, which brings a
robot from an initial configuration to a goal configuration while avoiding obstacles
in the robot’s environment. However, finding a motion path is not enough for the
CRAWLER robot. Motion planner of the CRAWLER robot must be able to gener-
ate detailed motion sequences to guide the robot travel through the path. Due to
the under-actuated mechanism and mechanical constraints, the motion planning of
the CRAWLER robot is extremely difficult. The robot can perform only restricted

motions even in the absence of obstacles. Since the standing foot alternates when

88

the robot moves, the motion planner must be able to schedule correct foot place-
ment for each motion step. For the CRAWLER robot, motor 2 drives the robot
forward/ backward and makes turn. Since only one control input is available, the
translation motion and rotation motion are highly coupled. This makes the motion
planning of the CRAWLER robot more complicated than that of the car-like robot,
where two separate control inputs are available to control the translation motion and
rotation motion, respectively. The switching between different motion modes further
complicated the motion planning of the CRAWLER robot.

In this chapter, motion planning problems and their solutions of our bipedal
CRAWLER robot are presented. To deal with the constraints imposed by the kine-
matic structure, the configuration space is broadened by introducing the robot motion
status to form a hybrid configuration space. The discrete motion status — stand-
ing foot and motion mode, is used to decompose the continuous configuration space
into several submanifolds. Based on the study of the robot motion pattern, a motion
planning method which consists of a global planner and a local planner is devel-
Oped. The motion status information enables us to define a cost function which is
used in the motion planning algorithm to generate a feasible path and an Optimal
motion sequence. Simulation and experimental results have verified the theoretical

development.

4.1 Preliminaries

The primary goal Of this section is to provide the necessary background for investi-
gating the robot motion planning problem. Some preliminary mathematical concepts
are introduced first, and then the exact cell decomposition motion planning method
and the A" searching algorithm are described which will be used in our own motion

planning approach in following sections.

89

4 . 1.1 Configuration Space

The motion planning problem has received a lot of attention over the years. The
configuration space concept proposed by Lozano-Perez [59, 60, 58] and J .C. Latombe
[39], etc, served as a powerful representational tool for both the development and the
analysis of motion planning algorithms. Robot position and orientation are character-
ized by just one point in configuration space. With the configuration space framework,
it becomes possible to develop algorithms that are generalizable and adaptable to a

wide variety Of applications.

Definition 4.1.1 (Topological Space) [48]: A topology on a set X is a collection
T of subsets of X having the following properties:

1. Both X and the empty set (0 are in T.
2. The union of elements of any subcollection of T is in T.
3. The intersection of the elements of any finite subcollection of T is in T.

A set X for which a topology T has been specified is called a topological space.

If X is a topological space with topologr T, we say that a subset U Of X is an
open set of X if U belongs to the collection T.

Definition 4.1.2 (Homeomorphism) [39]: Let f : X —» Y be a bijective function
between topological spaces X and Y. Since f is bijective, the inverse f"1 exists. If

both f and f‘1 are continuous, then f is a homeomorphism.

Definition 4.1.3 (Manifolds) [48]: A topological space M is a manifold if for

every :1: E M, an open set 0 C M exists such that:
1. a: E 0.

2. O is (locally) homeomorphic to R".

90

 

3. nisfixedforalleM.

The dimension n is referred to as the dimension of the manifold, M.

If X and Y are topological spaces, the Cartesian product X x Y = {(23, y) : a: E
X and y 6 Y} defines a new topological space.

As defined by Latombe [39], the configuration space concept is presented as follows.
Let R be a rigid object — the robot — moving in a Euclidean space W, called
workspace, represented as R", with N=2 or 3. W is equipped with a fixed Cartesian

coordinate frame, denoted by fw. A moving reference frame, .733, is attached to R.

Definition 4.1.4 (Configuration Space) [39]: A configuration q of R is a specifi-
cation of the position and orientation of .773 with respect to fw. The configuration

space of R is the space C of all the possible configurations of R.

The configuration space C is intrinsically independent of the choice of the frames
f'w and .73. Only the representation of C depends on these frames. A configuration
can be described by a list of real parameters. For 2-D rigid objects, where N=2, a
configuration q of R can be bijectively represented by three parameters, 2:, y and
0, where (x, y) determines the position of the origin Of f}; in fw and 0 E [0, Zn)
is the rotation parameter. For 3—D rigid objects, a configuration can be represented
by position parameters (2:, y, z) and 3 Euler angles d), 6, and 1b, which determine the
orientation of fg’s axes with respect to fw.

The subset of W occupied by R at configuration q is denoted by R(q). A point
a Of R is denoted by a(q) in W when R is at configuration q. Thus, for any two
configurations q and q’, a(q) and a(q’) are the same point in R, but in general do
not coincide in W.

By defining the topology in terms of a metric, the configuration space is a topo—

logical space generated by the set of all possible configurations. The robot displace—

91

ment metric is defined as:

d(q, (1’) = 1513 || a(<1) - a((1') II

where [I a: — :c’ [I denotes the Euclidean distance between any two points a: and a" in
B”. This metric represents the distance between two configurations q and q’ and it

tends toward zero when the regions R(q) and R(q’) tend to coincide, and conversely.

Definition 4.1.5 (Path in configuration space C ) [3.9]: A path of the robot R
from the initial configuration Qinit to the goal configuration (1900; is a continuous map

7' : [0,1] ——+ C with 7(0) = qinu and r(l) = (10001.

The continuity of r entails that for all so 6 [0,1], lim,,_.,,o mantra; || r(r(s)) —
r(r(so)) II: 0, with 8 taking its values in [0,1].
If no kinematic or dynamic constraints limit the motion of R, we say that R is a

free-flying Object.

Definition 4.1.6 (Minkowski Operator)[39]: Let X be an afl‘ine space whose ori-
gin is 0, and A and B be any two subsets of X . The Minkowski operator GB (addition)
and e (subtraction) are defined by:

A®B = {x|r=a+b,a€A,b€B},

eB {—blb e B},

.498 = Ae(eB).

Let obstacles Bl, - -- [3,, be fixed rigid Objects distributed in W. Every obstacle

3,, i = 1 to p, in the workspace W maps in configuration space C to a region:

681' = {q 6 CWQ) 0 8.- 7e 0}

92

which is called a C-obstacle. The union of all the C-Obstacles:

Up

i=1

CB,

is called the C-obstacle region, and the set:

Circe = C\Uf=1CB, = {q E C|R(Cl) fl (Uf=1Bi) = 0}

is called the free space. Any configuration in C free is called a free configuration.

Assume that both the geometry of R and 81 - - - B, and the location of the [3,-’3
in W are accurately known. The basic motion planning problem is: Given an
initial and goal configuration q,-,,,-t, (19m), generate a free path 1' between the two
configurations, specifying a continuous sequence of position and orientations Of R
avoiding contact with the [3,-’8 if they belong to the same connected component of
C fm, and to report failure otherwise.

For a point robot, the workspace and configuration space are the same. The C-
Obstacles are identical to the Obstacles in the workspace. When robot R is a planar
object allowed to translate freely without rotation, the configuration space is R“, but
the C-obstacles are Obtained by “growing” the Obstacles by the shape Of the robot
as illustrated in Figure 4.1, which corresponding to the Minkowski sum of robot and
Obstacle.

When the robot can translate and rotate in a planar work space, the configuration
space is 3-dimensional, with (2:, y, 0) 6 R2 x [0, 360°). The C-obstacle looks like
a twisted pillar as shown in Figure 4.2. Now imagine sweeping a horizontal plane
upwards through the 0 direction of configuration space. Each slice is a Minkowski
sum whose shape changes continuously from 0 = 0 to 6 = 360°. The stack of slices

constitutes the C—Obstacle.

93

  

 

 

 

 

/////

 

 

 

>6

Figure 4.1: C-obstacle in a translational case; B,- is an obstacle, by superimpose the
shape of a robot (R) all through the peripheral of the Obstacle, the C-Obstacle C8,- is

constructed.

 

//x
7%

 

 

 

 

 
  

 

s

    

    

\\\\\\\\\\

/////
/////,’//
7//////// /////’/7/2

   

Figure 4.2: C—obstacle in translational and rotational case. B; is an obstacle, CE.- is
a slice corresponding to the Minkowski sum of the obstacle and the robot R whose
shape rotates continuously from 9 = 0 to 6 = 360°. The C-Obstacle is a 3-D stack of

the slices.

94

 

Now consider a more complex articulated manipulator robot. This robot R is
made of 11: moving rigid Objects R1, - - -, R,c connected by mechanical joints (revolute
or prismatic joints) which constrain their relative motion. Let f3, be the frame
attached to R,~,i E [1, k]. The configuration Of R is a specification of the position
and orientation of every frame $3,, i = 1 to k, with respect to fw. Therefore, the
configuration space of a fixed—base articulated robot with k revolute and prismatic

joints and no kinematic loop is a k-dimensional smooth manifold.

4.1.2 Exact Cell Decomposition

There are three general approaches to the basic motion planning problem [39]: cell
decomposition methods, roadmap methods, and artificial potential field methods. In
this section, an exact cell decomposition method is presented in the simple case where
C = R2 and the C-Obstacle region forms a polygonal region in C.

The decomposition of C ,m and the associated connectivity graph are defined as

follows:

Definition 4.1.7 convex polygonal decomposition/3.9]: A convex polygonal de-
composition IC of C 1m is a finite collection of convex polygons, called cells, such that
the interiors of any two cells do not intersect and the union of all the cells is equal to
the closure of Cine. Two cells k and k’ in [C are adjacent if and only if k n k’ is a

line segment of non-zero length.

Definition 4.1.8 connectivity graph (39/: The connectivity graph g associated
with a convex polygonal decomposition [C of Cfm is the non-directed graph specified
as follows: 1) g ’3 nodes are the cells in IC. 2) Two nodes is g are connected by a link

if and only if the corresponding cells are adjacent.

The exact cell decomposition algorithm for planning a free path connecting q...“

and (1,001 is the following:

95

 

1. Generate a convex polygonal decomposition [C of C free.
2. Construct the connectivity graph 9’ associated with IC.
3. Search 9 for a sequence of adjacent cells between q,-,,,-¢ and qgoa).

4. If the search terminates successfully, return the generated sequence of cells;

otherwise, return failure.

The output Of the algorithm is a sequence k1, m, kp of cells called a channel,
such that qi'nit 6 k1, qgmz E kID and for every j E [1,p — 1], k,- and k,“ are adjacent.
A continuous free path can be computed from the channel, for example by connect-
ing the initial configuration to the goal configuration through the midpoints of the

intersections Of every two successive cells.

gm 1 0

 

11 19

 

  

   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3: TIapezoidal decomposition method

An efficient exact cell decomposition method called trapezoidal decomposition
is illustrated in Figure 4.3. The free space is externally bounded by a polygon and

96

internally bounded by three polygonal obstacles. The first step Of the method consists
of exactly decomposing the free space into trapezoidal and triangular cells. These
cells are built by drawing vertical rays from the C-Obstacle vertices. Two cells are
adjacent if they share a portion Of an edge of non-zero length. The second step is
to construct the connectivity graph representing the adjacency relation between the
cells and search this graph for a path (thick lines in Figure 4.3 B). A path in the
connectivity graph determines a channel in free space. The channel is the sequence of
cells labeled as 3, 4, 7, 8, 9, 12, 14, 16, 17 and 18. The third step of the method consists
Of transforming this channel into a free path, by connecting the initial configuration
to the goal configuration through the midpoints of the intersections of every two

successive cells.

4.1.3 A“ Algorithm

In many cases, motion planning involves searching a graph to find a minimum-cost
path connecting the initial node and the goal node. A“ Algorithm [25, 39, 33]is the
most popular method for graph searching, guaranteed to return a path of minimum
cost whenever the path exists.

A‘ explores a weighted graph G iteratively by following paths originating at Ninit-
The cost of a path is defined as the sum of the costs Of its arcs. A" employs an additive
evaluation function f (N) = g(N) + h(N), where g(N) is the cost of the currently
evaluated path from Ninit to N, and h(N) is a heuristic estimate Of the cost of the
path remaining between N and goal node N900). A‘ constructs a tree T of selected
paths of C using the elementary operation Of node-expansion, i.e., generating all
successors Of a given node. T is represented by associating to each visited node N
a pointer to its parent node in the current T. Starting with Nina, A‘ selects for
expansion the leaf node of T that has the lowest f value, and only maintains the

lowest f path to any given node. It is known that if f (N) is a lower bound to the

97

cost of any continuation path from N to N900), then A“ is admissible, that is, it is

guaranteed to find the optimal path. A‘ algorithm is described as follows:

1.

Put the start node, Nina, on a list called OPEN of unexpanded nodes.

. If OPEN is empty, exit with failure; no solution exists.

Remove from OPEN a node, N, at which f(N)=g(N)+h(N) is minimum (break
ties arbitrarily, but in favor of a goal node) and place it on a list called CLOSED

to be used for expanded nodes.

. If N is a goal node, exit successfully with the solution Obtained by tracing back

the path along the pointers from N to NW}, ( pointers are assigned in steps 5
and 6).

. Expand node N, generating all its successors with pointers back to N.

. For every successor N’ of N:

e Calculate f(N’)

e If N’ was neither in OPEN nor in CLOSED, then add it to OPEN. Assign
the newly computed f (N ’ ) to node N’.

e If N’ already resided in OPEN or CLOSED, compare the newly computed
f (N ’ ) with that previously assigned to N’. If the new value is lower, substi-
tute it for the old (N ’ now points back to N instead of to its predecessor).
If the matching node N’ resided in CLOSED, move it back to OPEN.

7. GO to (2).

4.2

Motion Planning Analysis

For a given robot and set of obstacles, the basic motion planning techniques under

configuration space framework only consider geometric issues to determine a collision-

98

free path. For more complicated problems, it is beneficial to broaden configuration
space framework to handle sensory information, kinematic and dynamic constraints,
etc. Sharma and Sutanto [67] proposed a vision-based motion planning framework
in which they unified the configuration space and an image feature space and con-
sidered sensors as an integral part of the motion goal. In [39], the dynamic motion
planning problem — where the Obstacles in the workspace are moving, was studied
in configuration-time space. The approach is to add a dimension —- time — to
the robot’s configuration space and take the specificity of the time dimension into
account. LaValle [42, 40] introduced a game-theory—based mathematical foundation
upon which analyzes and algorithms can be developed for a broad class of motion
planning problems. He advocated the expansion from configuration space to state
space and to broader space — information space to cover general motion strategy
problems, involving uncertainty in sensing and control, environment uncertainties,
and the coordination of multiple robots. The framework is general enough to encom-
pass every motion planning issue; however, more work needs to be done to tailor the
idea to specific applications.

In this section, “hybrid configuration space” concept is proposed to incorporate
the continuous configuration space with discrete motion status space. The broadened

space is a more effective tool to analyze the motion planning of the climbing robot.

4.2.1 Motion Status Space

The robot motion status has two fields as shown in Table 4.1. It represents precisely

which foot is supporting the robot and in what motion mode the robot is moving.

Table 4.1: Motion status fields

 

[ Standing Foot ] Motion Mode ]

 

99

Notice that there exist only two choices for the first field, either LFS or RF S.
Neither foot grips the floor is an invalid motion status, because in this case the robot
cannot climb a wall and the motion direction is undetermined. Both feet grip the
floor is also a invalid status because the robot cannot move except to release one of
the foot. We assign three values (-—1,0, +1) to represent each of the motion modes,
i. e. 0 represents translation mode, —1 represents spin mode 1, and +1 represents
spin mode 2. The combination Of the fields makes six motion statuses, namely LFS-
1, LFSO, LFS+1, RFS-1, RFSO and RFS+1. For example, LFS-1 means the left foot
supports the robot while the right root is free to move, and the robot is under spin-1

mode. The motion status space is defined as follows:

Definition 4.2.1 (Motion Status Space) Let’s define a motion status set S whose
elements are the six motion statuses, i.e. S ={LFS-1, LFSO, LFS+1, RFS-1, RFSO,
RFS +1}. The collection of all subsets of S is a topology on S and is called the discrete

topology. We say the topological space S the motion status space.

4.2.2 Motion Pattern Analysis

When the CRAWLER robot walks on a 2-D plane, motor 2 controls the robot to
move forward/backward or make turns. Motors 1 and 3, which control the robot
tilt angles a and ,6, only help the robot to lift the free foot up from or down to the
contact surface to reduce the friction. For the ease Of explanation, we assume that
the friction between the free foot and the contact surface is zero when the foot slide
along the surface. Thus motors 1 and 3 can be ignored when studying the motion
planning on a 2-D plane, and motor 2 is the only driving motor. Throughout the

paper, foot 1 is assumed to be the left foot while foot 2 is the right foot.

100

Fixed-base Case

We first study the motion when the robot is standing on one of its feet. Since the
base is fixed at the center Of the standing foot, the configuration space Of the robot is
one-dimensional, represented by the driving angle of motor 2, 1b. The motion pattern
Of the robot and the mapping between the configuration space and robot workspace

in LFS phase and RFS phase is shown in Figure 4.4 and Figure 4.5, respectively.

robot 9 Motor angle V
main axis: Spin 1 Translate Spin 2

a@s@5@a

Configuration Space
P(d)

P(O) ©
1

ii ®

I
Archimedes“ Spiral

 

    
      
   
 
  
 

 
 

 
  
 
  
    

  
    
     
   
  
   

  

Archimedes' Spiral ' 5

cu
e$y T’- ..... ..- .-
‘oo‘° ..

  

P(b)

     
  

   

G)

  
    

LFS

VVorkspaceor'robotlnLFS anaemia

Figure 4.4: Motion pattern of the CRAWLER robot in LFS phase

In Figure 4.4, the shaded circle represents the left foot which supports the robot,
while the dotted circles represent the right foot which moves freely. In the one-
dimensional configuration space, 1/2 = a represents the motor angle corresponding to
the mechanical extreme when the robot contracts; ’l/) = (1 represents the motor angle
corresponding to the mechanical extreme when the robot expands; it = b and 1b = c
are the switching points from the translation mode to the spin-1 mode and spin-2

mode respectively. P(a), P(b), P(c), and P(d) represent the positions of the center

101

of foot 2 in the robot workspace corresponding to the different values of motor angle
1b. In range [b, c], the robot moves in the translation mode, i. e., both lock-pins are
outside the cam and the legs contract or extend. When motor 2 rotates CW and it
falls in [a, b), the robot moves in spin-1 mode. In this range, leg 2 contracts and foot
1 rotates, making the free foot move along an Archimede’s spiral curve. When motor
2 rotates in CCW direction and it falls in (c, d], the robot moves in spin-2 mode. In
this range, leg 1 extends and foot 2 winds up.

Similarly, Figure 4.5 shows the motion pattern of the robot in RFS phase. When
the motor angle v is in the range [b, c], it moves in the translation mode. The CCW
rotation of motor 2 puts if) within (c, d], and the robot moves in spin-2 mode. In this
mode, since foot 2 supports the robot, the rotation of foot 2 and the extending of leg
1 causes the robot to move along a spiral curve. The CW rotation of motor 2 makes

the robot move in spin-1 mode where leg 2 contracts and foot 1 winds up.

Motor angle V
_, Spin 1 Translde Spin 2

 

   

r ,o .

      

 

1' .' pm ......... ' =
I ' ' E).u___————'
. tgpend ®c°mfl _ b)""‘“"
E‘W‘ll . ® k =
Arohimedes' curve " Aron edes'Splrel —
pm, .......... ‘ p“) 33%.: —
I
Worltapace of robot in RFS Sweep Area

Figure 4.5: Motion pattern of the CRAWLER robot in RFS phase

Let 2 be the distance between the centers of the two feet, then the robot motion

102

in LFS and RF S can be modeled as follows:

"71 = ("A

é —_.— k1], ifasw<b (4.1)
’72 = 0

’71 = 0

2 = 21ml ifbswsc (4-2)
72 = 0

’71 = 0

z = k.) ifc<1pgd (4.3)
”'72 '-" ("A

where a < b < c < d, k, and h are constants. 71 and 72 are the rotation angles of
foot 1 and foot 2, respectively. When foot 1 (or foot 2) is the standing foot, '71 (or

'72) represents the robot swing angle relative to the robot main axis.

Definition 4.2.2 (Robot Main axis) The main axis of the CRAWLER robot is
the straight line connecting the center of the two feet when both '71 = 0 and '72 = 0.

It is the motion direction when the robot moves in translation mode.

The following property is observed which will be used in the robot motion plan-

ning:

Property 4.2.1 The sweep area of the robot moving in RFS phase is much larger
than that in LFS phase.

Due to the under-actuated mechanism, the CRAWLER robot needs to switch the

standing foot to make a right turn. Figure 4.6 shows the motion pattern of the robot

103

to accomplish a right turn when the left foot is the standing foot. In the first step,
motor 2 rotates CW causing the robot to contract until it reaches P(b). At this point
the robot motion switches from the translation mode to spin-l mode. The difference
between making a left turn or right turn is in step 2. TO make a right turn, the
robot needs to switch the standing foot to the right foot (foot 2). The continuous
CW rotation of motor 2 in RFS phase will wind up foot 1 to the desired angle '71,
and make the robot contract. In step 3, the standing foot switches back to the left
foot (foot 1), and motor 2 rotates in CCW direction. The robot still moves in spin-1
mode, but the CCW rotation of motor 2 makes the robot extend and swing to the
right by 71 until motor 2 angle reaches b.

 

 

 

 

 

r0 130![ l: 8'0: M otor angle V
”7+ . - Spin1 ATrenslete‘ SW12
$.51 " 5 Q) 5 Q) 3 ® "
53$- [I Y ‘. Configuration Space
pc).. 1 Contract
r, iL .‘
O'.‘ 0.
® 1 fl ’
y—I'h /'
a. .. /
P(bi /t:

 

 

 

 

 

 

 

.

\ O
----.-.-.-.-.-.-n-.

\ v

 

 

 

 

 

Figure 4.6: Motion pattern of the CRAWLER robot making right turn in LFS phase

Similarly, Figure 4.7 shows the motion pattern of the robot to accomplish a right
turn when the right foot is the standing foot. The CCW rotation of motor 2 makes
the robot extend until it reaches P(c). In step 2, the robot switches standing foot to
left foot and moves in spin-2 mode. The further CCW rotation of motor 2 makes the

robot extend, and foot 2 wind up to a desired angle 72. After switching the standing

104

foot to the right foot, motor 2 rotates in CW direction, making the robot contract
and swing to the right by 72.

Motor enple V

RFS
Spin 1 Translate Spin 2

, 5 (Des Q) o (s) a
_.. ® .. onfiguratlonSpece

 

 

71

I
r=t

 
 

W T3,]
- - - -.-.-.-...-.-.-.-.::.-._,,. “ll-(33.-
i

 

. Contract

 

 

 

 

1:.
23...}.

 

Aronunedes' spiral curve

 

 

 

 

 

 

 

A
to 1.... p.-'r.— -
I

 

Step 3

Figure 4.7: Motion pattern of the CRAWLER robot making right turn in RFS phase

In practical implementation, the switching of the standing foot is a procedure that
cost a lot Of battery power. The robot needs to turn on the pump motor of its free
foot; adjust tilt angle to ensure that the suction cup grips the flat surface firmly;
and then turn Off the pump motor of the original standing foot. The robot needs to

minimize this switching to decrease the power consumption.

Ree Motion Case

At the second stage, we study the robot motion when it walks on a plane. Let W be a
2-D Euclidean space R“, representing the robot physical workspace, equipped with a
fixed frame XoOYo as reference. The robot reference coordinate frame is attached at
the center of foot 1 and is moving when the robot walks in W. At each specific location
of W, the robot may swing and its length may change when the robot contracts or
extends. The term robot posture is used to describe the robot length, and the swing

angle related to the robot main axis. Now we have the following proposition:

105

Proposition 4.2.1 The configuration space of the CRAWLER robot walking on a
2-D plane is 4-dimensional and can be parameterized by (x, y, 0, 11)). (x, y) and 0 rep-
resent the position and orientation of the moving robot reference frame with respect to
X0 0Y0. w is the motor 2 angle which determines the robot posture. The configuration

of the CRAWLER robot is a specification of position, orientation and posture.

4.2.3 Hybrid Configuration Space

The configuration of the robot cannot uniquely determine the robot motion. For
example, in Figure 4.8 (A) and (B), although the robot has the same position and
orientation in the workspace XOOYO, the current standing foot will determine which
obstacle the robot will hit when it extends. In LFS phase, the extension of the robot
will cause it to collide with Obstacle 1. On the other hand, if the right foot is the
standing foot, the robot may hit Obstacle 2. Moreover, even when the robot has the
same position, orientation, and standing foot, the motion mode will determine which
obstacle it will hit. In Figure 4.8 (C), the contraction of the robot makes it move
in spin-l mode, and therefore it will collide with Obstacle 2. On the other hand, the
extension of the robot will make it hit obstacle 1.

Therefore we define the hybrid configuration space concept to integrate the
robot configuration with the motion status, and analyze the motion planning under

this framework.

Definition 4.2.3 (Hybrid Configuration Space) Let two topological spaces C
and S be the robot configuration space and the motion status space, respectively. The
product of these two topology spaces (C x S) is defined as the hybrid configuration

space.

By adding the robot motion status dimension to the configuration space descrip-

tion, we can explicitly control the robot standing foot and motion mode; thereby

106

 

 

 

 

 

 

 

Figure 4.8: The CRAWLER robot collides with Obstacles

uniquely determine the robot motion. The configuration space is decomposed into
two submanifolds, one for each standing foot. Each submanifold is further partitioned
into three subspaces associated with the motion modes. Each of the three subspaces
has different characteristics as described by Equation 4.1 to Equation 4.3. Unlike
the stratified configuration space proposed in [22], where the configuration space is
decomposed into submanifolds upon which the system has different sets Of equations
of motion, the hybrid configuration space enables us to more selectively choose among
the six submanifolds associate with the motion status. The introduction of the motion
status variable indicating the standing foot and motion modes allows us to describe

the motion planning approach for the climbing robot more efficiently.

4.3 Motion Planning Algorithm .

4 . 3. 1 Problem formulation

The motion planning of the CRAWLER robot can be described as follows: given

the initial and final configurations (position (x, y), orientation 9, posture it) and the

107

 

status (standing foot, motion mode) of the robot, find a collision- free path P and a
feasible motion sequence to allow the robot to navigate through P.
A motion planning approach consisting of a global planner and a local planner is

developed and shown in Figure 4.9.

 

Simulate robot Motion Planner
workspace

[ Pofiible path i

Feasibility test
I around
intersection point

 

 

r

 

 

 

 

 

Trapezoidal
decomposition

 

 

  
  
 

 

 

 

   

 

 
  

 

 

 

Global * .Rfemqgf Local
planner m OBSI a planner
V A' algorithm to node
I generate path L_ very-turn IS
' ' - No teaSIble?
I Collusron detection
* 13.5....

 

 

 

Feasible path§
Generate motion

 

 

Path sm oothing

 

 

 

 

 

 

 

 

 

 

mollision detection sequence
...—.d
Posslb a path Motion sequence

Figure 4.9: Block diagram of the motion planner

4. 3.2 Global Planner

The motion planning approach adopts a two-level refinement. The global planner
simplifies the robot as a rectangular body and searches the path in a 3-D sub-space
(x, y, 0) of the hybrid configuration space without taking the 1,1) and motion status
dimensions into account. The robot is regarded as a free-flying rigid object, i. e. the
motion Of the robot is not limited by any kinematic constraints except the Obstacles.
The shape of the robot is assumed to be enclosed in a rectangular boundary and the
representation of the robot is reduced to just the location (x, y) and orientation 6.

The width of the rectangle has the same physical dimension as the robot’s width,

108

while the length is P(b), i. e., the smallest length value in translation mode. The
local frame Of the robot is fixed at the axis of foot 1. The purpose of the global
planner is to find a collision-free path by initially ignoring the details of the robot
constraints. The global planner uses the trapezoidal decomposition method and the
A“ algorithm described in section 4.1.3 to generate a collision-free minimal length
path. The collision detection is achieved by “growing” the obstacle boundary with
the robot width. After that, the global planner smooths out the path by eliminating
sharp turns and combining the line segments to form a straight line whenever possible.
The constructed collision-free path after smoothing is named the possible path.

The trapezoidal decomposition method generates paths that can be smoothed
relatively easily, but this is not necessarily the case for many current randomized
algorithms (e.g., the probabilistic roadmap algorithm or Rapidly-Exploring Random
'Itee algorithm). For randomized algorithms, the time required to smooth paths to
the degree required by the local planner can be large. This is the reason we have

selected the trapezoidal decomposition method rather than randomized methods.

4. 3. 3 Local Planner

The possible path describes the global motion of the simplified robot but may not
be feasible for the real robot to travel. Any straight line in a possible path is always
feasible. However, intersections of the line segments where the robot makes turns
may be non-traversable. The purpose of the local planner is to make a feasibility
test in the region around each intersection point by searching for a motion sequence
which can accomplish the turn. If a motion sequence exists, the turn is feasible but
the path may contain deviations from the possible path. The new path which passed
the test is named the feasible path. If a motion sequence cannot be found due to the
Obstacles crowding around the intersection point, the path passing though this point

is infeasible and ruled out. In this case, after removing the infeasible points, the global

109

planner is called again to generate another smooth possible path. The feasibility test
of the new possible path by the local planner is repeated. This interactive process
between the global planner and the local planner continues until a feasible path is
constructed or the motion planner returns a failure.

The feasibility test is conducted locally and considers the robot constraints. The
process is basically a search of a decision graph guided by a cost function. The initial
and goal nodes Of the graph are the starting and final configurations of the turn.
At each node, a collision detection is performed and a decision is made among four
possible actions the robot can take from the current configuration. Those are ( 1)
switch foot, (2) translate, (3) turn left, and (4) turn right.

Since the standing foot and robot configuration at each motion step is predetermined
by the prior node, the collision detection of the current node is conducted in the one
dimensional configuration space 1]). The projection Of the local obstacles into the 1-D
configuration space is made by taking advantage of the sweep area and the motion
status. For example, in Figure 4.4, if an obstacle maps outside the range Of 1,0 6 [a, d]
of the configuration space, then it will not collide with the robot. If an obstacle is
located in the sweep area, it will be mapped to somewhere within range if) E [a, b].
The collision will possibly occur when the robot moves in spin-1 mode. Thus, the
decision will be make to either avoid turn left or to turn left just with a limited angle
avoiding the collision. When the required turning angle is larger than the maximum
angle the robot can spin, it becomes necessary for the robot to make several small
intermediate turns.

The time taken by the local planner to examine the feasibility of the possible path
in the vicinity of the turning intersections is minimized because the global planner has
smoothed the path and reduced the number of turns. Another reason for the motion
planner to smooth the path is that the CRAWLER robot is designed to prefer straight

line motion.

110

The last task of the motion planner is to generate a motion sequence to travel
along the feasible path. The prefered motion sequence to travel in the straight line is
moving in translation mode with full stride. Since the motion sequence around every
intersection has already been produced when the local planner made its feasibility
test, the combination of the straight line and intersection motion sequence composes
the whole motion sequence. The connection between the straight line motion sequence
and the intersection motion sequence is a noticeable issue. The motion at full stride
may reach the intersection on a wrong foot or at a wrong step size. There are two
solutions to this problem. The first one is to adjust the last foot step only and the
second one is to share the step size difference among all the translation strides. Either

solution is easy to implement.

4. 3.4 Cost Function

The complete feasibility test of the local planner requires an exhaustive search and is
practically infeasible. So some restrictions and heuristics are introduced to guide the

search for a feasible path and motion sequence:

1. The robot is designed to prefer translation motion. Hence, the motion planner
should generate straight line paths and minimize the number of turns whenever

possible.

2. The sweep area of the robot in RFS is much larger than that in LFS. This leads
to the selection of left foot as the standing foot because the robot is prone to

collide with obstacles in RFS.

3. The right turn Of the robot involves the switching of the standing foot which
consumes a large amount Of power. Therefore, the motion planner should avoid

right turns whenever possible.

111

4. The robot should use maximum stride length in translation mode, i.e., travel the

maximum slide distance from P(b) to P(c) to avoid unnecessary foot switching.

However, the construction of a decision graph still leaves a large number Of possible
motion sequences even if the graph is not fully expanded. A criteria to determine
the “optimal motion sequence” among those possibilities is required. [Taking into
consideration the above heuristic rules, the optimal motion sequence is the one that
minimizes the foot switching, mode switching, and renders the shortest feasible path.

In order to mathematically describe the optimal motion sequence and the heuristic
rule, we first quantify the motion status and then define a cost function to guide the
search. As mentioned before, the motion status has been quantified by assigning
three values (—1, 0, +1) to represent each of the motion modes, i.e., 0 represents the
translation mode, -1 represents the spin-1 mode, and +1 represents the spin-2 mode.
Phrther, we define LFS é —1, and RFS 2 l.

The cost function of each motion step is defined as follows:

Cost(k) = We - dist(k) + W1 - foot(k) + W2 - |foot(k) — foot(k — 1)| (4.4)
+W3 - |mode(k) — mode(k — 1)|

 

where, dist(k) = \/(x(k) — x(k — 1))2 + (y(k) — y(k — 1))2, and where W0, W1, W2,
and W3 are weights.

At each step 1:, the motion planner keeps track of the robot location, standing foot
and motion mode, and uses the corresponding value to calculate the cost function.
The motion sequence which minimizes the total cost is selected as the Optimal one.
By changing the weights of the cost function, we can change the definition Of an

optimal path.

112

The first term of the right hand side of the Equation 4.4 shows that the cost
of a short path is smaller than that of a long path. It is noted that the actual path
traveled by the robot using the motion sequence has some variations from the original
possible path. The second term reflects that the motion planner prefers LFS because
it reduces the cost function value when foot(k) = LFS = —1. The third term reveals
that the robot favors minimal foot switching. The cost of one foot switching is 2W2.
If the robot travels the same distance with more foot switchings, then the cost will
be higher. The fourth term discourages mode switching. For example, the cost of the
robot remaining in the same motion mode is 0, while the cost of changing from the
translation mode to the spin-l mode is l, and switching from spin-l to spin-2 is 2.
The minimization Of the third and fourth terms helps to reduce the power usage.

The smoothing of the possible path by the global planner also implicitly penalizes

the sharp turns and reflects the preference to linear motion over rotation motion.

4.4 Performance Evaluation

TO Observe the behavior of our motion planning approach, we performed simulations
of the robot to navigate among polygonal Obstacles in 2—D space as well as an actual

experiment in which the robot moves in a maze.

4.4.1 Simulation

A motion planning software is developed under Linux on a host computer. The sim-
ulated robot workspace is easily produced using xfig drawing utility and saved in a
.fig file. The global motion planner then reads the file and uses the polygon trape-
zoidalization method to generate a robot path. An O(n log n) sweeping-line based
algorithm is used to horizontally decompose the free space in the robot workspace.

A set of edges is produced by connecting the midpoints of adjacent parallel edges

113

of trapezoid free-space. The A‘ algorithm is used to generate a distance-optimal
path. During the process, a simple collision detection is performed by considering
the width and length of the robot boundary. For a better performance of the local
planner, a smoothing algorithm is employed to eliminate unnecessary turns and check
if the removal Of a turn will cause the new resulting path to collide with any of the
Obstacles. Figures 4.10 and 4.11 show the path generated by the A‘ algorithm and

the smoothed possible path, respectively.

 

 

 

 

 

 

 

Figure 4.10: Generated path by the A" algorithm

The local planner then makes a feasibility test and refines the robot motion within
the region surrounded by a circle centered at each intersection point. The radius of
the circle is determined by the robot turning angle. If the two line segments form an
acute angle where the robot will take a sharp turn, then the corresponding radius is
set to a large value. If the two line segments form an obtuse angle, the local planner
selects a small radius. The two intersection points between the circle and the two line

segments are the starting and final points of a turn. The local planner determines

114

 

 

.1

9.;7. I I
’ l

a:

‘0’:

S

..

 

 

 

 

 

 

Figure 4.11: Smoothed path after removal of sharp turns

the turning direction (left or right) based on the relative position Of the incoming
and outgoing line segments of the turning point. By following the heuristics and
complying with some restrictions, the local planner constructs and searches a graph
which describes the motion sequences used to accomplish the turn. The output of the
motion planner is a feasible path and a motion sequence which minimizes the cost

function.

4 .4 .2 Experiment

In a real experiment, a CRAWLER robot is required to move from the initial config-
uration at the right lower part of a maze to the goal configuration at the upper left
part of the maze as shown in Figure 4.12. The environment of the maze is simulated
using the drawing utility. Once a user selects a destination via the user interface,
the motion planner running on the host PC reads the description of the environment

from a .fig file and generates a motion sequence. The sequence Of commands are

115

imported to the physical robot to navigate in the maze. A snapshots showing the

robot traveling through the maze is presented in Figure 4.12.

 

Figure 4.12: Robot navigating in a maze

4.5 Summary

This chapter investigates the motion planning of the CRAWLER robot. The under-
actuated mechanism makes the motion planning extremely difficult because the robot
can only perform restricted motions even in the absence of Obstacles. Based on a de-
tailed analysis of the robot motion pattern, a hybrid configuration space concept is
proposed to deal with the constraints imposed by the kinematic structure. Under
hybrid configuration space framework, the robot motion can be uniquely determined
by incorporating the continuous configuration space with discrete motion status. A
motion planning algorithm is developed which consists of a global planner and a local

planner. A cost function is defined based on the motion status information to guide

116

the search for an Optimal path. With the motion planning capability, the CRAWLER
robot is able to autonomously determine a path and a set of motion sequence to travel
through an environment without collision with Obstacles. Simulation and experimen-

tal results verified the efficency of the motion planning method.

117

CHAPTER 5
CONCLUDING REMARKS

5.1 Conclusion

This dissertation describes the development of the CRAWLER climbing robot. The
climbing robot uses two suction feet to support itself on surfaces. It adOpts a bipedal
structure and an under-actuated mechanism to achieve a good balance between com-
pactness and mobility. This bipedal structure provides the robot with the capability
to transit between two inclined surfaces. The under-actuated mechanism contributes
to the small size and light weight Of the robot because it uses less motors to drive
more joints. A small penalty paid for under actuation is increased complexity in
modeling, control and motion planning.

The mechanical structure and locomotion modes of the robotare analyzed. The
kinematic model that describes the relation between the joint variables and the po-
sition/ orientation of the robot’s free foot with respect to its standing foot is derived.
The robot kinematic model is governed by different equations in different motion
modes. Therefore, it is important to use the corresponding kinematic equations when
planning the robot motion. In order to study the dynamic behavior of the climbing
robot, the dynamic model is derived based on the Lagrange-Euler formulation. The
dynamic model describes the relation between the input joint torques and the out-
put motion. It reveals that the gravitational effects play a dominant role in robot
dynamics when the robot moves at slow speeds.

An embedded control system is designed and implemented based on a TI DSP
chip. The self-contained controller hardware is built, which efficiently utilizes the
DSP resources and minimizes the component count. A joint level PD+gravity com-
pensation method is proposed. This control method outperforms the conventional

PD controller because it not only achieves joint level control but also compensates for

118

the gravity loading, which is determined by the configuration of all the robot links.
Software modules are developed to effectively control the robot. Experimental results
demonstrate that the robot has the capability to climb walls, walk on ceilings, crawl
through pipes, and transit between two inclined surfaces.

In motion planning analysis, a hybrid configuration space concept is proposed.
The hybrid configuration space incorporates the continuous configuration space with
discrete motion status (i.e., standing foot, motion mode), and it is a more effective
tool to analyze the motion planning of the climbing robot. Based on the motion
pattern analysis, a motion planning method is developed to generate a collision-free
path and feasible motion sequence to allow the robot to travel along the path. The
motion planning approach consists of a global planner and a local planner. This
approach of two levels Of refinement reduces the overall complexity and simplifies
implementation. A cost function is defined based on the motion status information
and a number of heuristics to help the search for an Optimal motion sequence. Ex-
periments and simulations demonstrate the effective performance of the CRAWLER

robot system.

5.2 Future Work

In this section, some recommendations for the improvement of the CRAWLER robot
and for future research directions of climbing robots are made.

As a prototype robot, the CRAWLER robot has some mechanical malfunctions
and hence needs improvement. It is Observed that the shafts of joint 1 and joint 5 slip
(see Figure 1.3), resulting in degradation of performance. Sometimes the joints are
not able to lift up the robot body due to the slipping of the joint shafts. We suggest
that the joints 1 and 5 be fixed tO their respective ankles with screws passing through
their shafts (Figure 2.1). Another problem we experienced is the difficulty in getting

accurate information on when the leg/ rack pairs are disengaged. This information

119

determines the motion mode switching. Currently, we have installed two contact
switches on the robot legs to determine when the lock-pins slide inside the lock-pin
cams. But this indirect scheme is not sufficiently accurate to determine the motion
mode switching. The suggested solution is to install a contact switch inside each cam
to get more accurate information on motion mode switching.

When a CRAWLER robot climbs in different places, the gravity effects change
with the varying inclinations. It would be beneficial to install a miniature inclinometer
to facilitate the compensation of gravity effects.

Basic motion planning assumes that the robot knows the environment a prior.
This assumption is reasonable but more work needs to be done to obtain the envi-
ronment settings by processing video images or using a global laser scanner. One
application scenario is Of a FLIPPER robot that climbs on a Office ceiling to detect
and transmit video images of the targeted Office environment to a host computer.
The host computer processes the video images to obtain the environment settings
and then executes the motion planning algorithm. The motion planner on the host
computer generates a feasible motion sequence and commands the actual CRAWLER
robot to travel through the environment.

Climbing robots using suction feet have some constraints. The first constraint
is that they are unable to move at fast speeds. This limitation is due to the fact
that it takes time to vacuum the air out of the suction cup until a proper pressure
is established or to release the foot by opening the valve until the suction cup can
be peeled off from the surface. The second constraint is that the robots with suction
feet can only walk on flat and non-porous surfaces. In order to overcome those
problems, great efforts should be made to exploit novel adhesive mechanisms which
are adaptable to different kinds of surfaces without sacrificing the robot motion speed.
Inspired by the exceptional ability of gecko lizards who are able to climb rapidly on

nearly any vertical surfaces, some scientists are now investigating the structure of the

120

gecko foot [4] in hopes of providing the key to robotic climbing.

Another promising adhesive mechanism is based on aerodynamic attraction. This
technique has been demonstrated at the 2000 DARPA Distributed Robotics Program
PI (Principal Investigator) Meeting and is illustrated in Figure 5.1. A rotating column
of air is created by a rotating disk containing miniature turbo plates located on the
inside rim of the disk. This cylindrical column of air has properties similar to a
vortex in that the interior air pressure is much lower than the ambient air pressure,
and as a consequence, a resultant force is generated which attracts the disk towards
the ceiling. The climbing robot based on this adhesive technique is driven by wheels,
leading to fast speed when moving on a wall. However, it is difficult for the robot to
transit between different inclined surfaces, such as from a floor to a wall, or from a
wall to a ceiling. A future research focus is to combine the two adhesive mechanisms
(suction foot and aerodynamic attraction) and exploit the advantages of both the
wheeled locomotion and articulate structure to achieve the quick motion and smooth

transition.

 

Figure 5.1: Aerodynamic attraction and a prototype wheeled climbing robot

121

[1]

[2]

[3]

[4]

[5]

[6]

[7]

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