Robots have become increasingly popular due to their ability to perform complex tasks and operate in unknown and hazardous environments. Many robotic systems are underactuated i.e., they have fewer control inputs than their degrees-of-freedom (DOF). Common examples of underactuated robotic systems are legged robots such as bipeds, flying robots such as quadrotors, and swimming robots. Due to limited control authority, underactuated systems are prone to instability. This work includes impulsive inputs in the set of admissible controls to address several challenging control problems. It has already been shown that continuous-time approximation of impulsive inputs can be realized in physical hardware using high-gain feedback.Stabilization of an equilibrium point is an important control problem for underactuated systems. The ability of the system to remain stable in the presence of disturbances depends on the size of the region of attraction of the stabilized equilibrium. The sum of squares and trajectory reversing methods are combined to generate a large estimate of the region of attraction. This estimate is then effectively enlarged by applying the impulse manifold method to stabilize equilibria from points lying outside the estimated region of attraction. Simulation results are provided for a three-DOF tiptoebot and experimental validation is carried out on a two-DOF pendubot. Impulsive inputs are also utilized to control the underactuated inertia-wheel pendulum (IWP). When subjected to impulsive inputs, the dynamics of the IWP can be described by algebraic equations. Optimal sequences of inputs are designed to achieve rest-to-rest maneuvers and the results are applied to the swing-up control problem. The novel problem of juggling a devil-stick using impulsive inputs is also investigated. Impulsive forces are applied to the stick intermittently and the impulse of the force and its point of application are modeled as inputs to the system. A dead-beat design for one of the inputs simplifies the control problem and results in a discrete linear time invariant system. To achieve symmetric juggling, linear quadratic regulator (LQR) and model predictive control (MPC) based designs are proposed and validated through simulations.A repetitive motion is described by closed orbits and therefore, stabilization of closed orbits is important for many applications such as bipedal walking and steady swimming. We first investigate the problem of energy-based orbital stabilization using continuous inputs and intermittent impulsive braking. The orbit is a manifold where the active generalized coordinates are fixed and the total mechanical energy of the system is equal to some desired value. Simulation and experimental results are provided for the tiptoebot and the rotary pendulum, respectively. The problem of orbital stabilization using virtual holonomic constraints (VHC) is also investigated. VHCs are enforced using a continuous controller which guarantees existence of closed orbits. A Poincare section is constructed on the desired orbit and the orbit is stabilized using impulsive inputs which are applied intermittently when the system trajectory crosses the Poincare section. This approach to orbital stabilization is general, and has lower complexity and computational cost than control designs proposed earlier.
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