"Control of dynamic motion is an important subject of study in robotics as it is desirable for robots to have a specific motion pattern rather then moving to a set point. The motions of robots also involve changing dynamic behaviors due to interaction with the environment, such as during contact, and this leads to hybrid system dynamics. A popular example of a hybrid dynamical system is a legged robot; the hybrid dynamics is due to the periodic switching of swing and stance legs and impulsive dynamics due to ground contacts. Legged robots require control of a dynamic trajectory defined by the walking gait or running motion. For legged robots, the spring loaded inverted pendulum (SLIP) model is commonly used to describe the dynamic motion in a simplified manner. The SLIP model has also been used for control of hopping robots and a fundamental limitation of the model is that it fails to account for impact with the ground; this is due to its single degree-of-freedom in the vertical direction. We investigate the control of a hopping robot starting from a more general two-mass model and then expand the theory to planar multi-link robot systems. The investigation involves two ground contact models, rigid and elastic, for the objective of apex height control. In the rigid case, the ground is assumed to provide an impulsive force to the hopping robot resulting in an inelastic collision. A hybrid control strategy is designed to deal with the hybrid dynamical system: a continuous controller based on partial feedback linearization is used in conjunction with a discrete controller that updates a control parameter at each hop to achieve the control objective. In the elastic case, the ground acts as a massless spring, which deflects as the robot exerts a force upon contact. In this case, we show that a continuous controller based on the backstepping algorithm can ensure asymptotic convergence to the desired apex height. Several robot configurations are considered, and for each configuration the complete hybrid dynamics is taken into account while designing the controller. The controllers compensate for the impulsive dynamics as well as higher order dynamics that are ignored in simplified models such as the SLIP model. Experimental validation of apex height control of a two-mass hopping robot on a rigid foundation is provided"--Pages ii-iii.
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