Although there has been a significant amount of research in designing and analyzing impulsive control systems, there are very few applications of impulsive control in mechanical systems. In the first part of this dissertation, we investigate new control strategies for underactuated mechanical systems based on impulsive inputs. The control problem of underactuated systems is more challenging since such systems have fewer actuators than the number of their degrees of freedom. We first address the important concern related to application of impulsive control in mechanical systems, namely, implementation of impulse-like control inputs using standard hardware. This is done through experimental verification of an impulsive control algorithm for swing-up control of the Pendubot; the control algorithm was developed earlier in our research group. Showing the effectiveness of the impulsive control algorithm in experiments, we develop impulsive control algorithms for swing-up control of the Acrobot; the impulsive control algorithms have distinct advantages over existing algorithms in terms of the time required for swing-up and maximum control torque used by the continuous controller. Impulsive inputs cause jumps in velocity states of mechanical systems, and consequently, produce jumps in Lyapunov function candidates used in control design. This attribute is used to enlarge the region of attraction of equilibria using impulsive inputs at discrete instants of time. Several case studies of underactuated mechanical systems have been presented to demonstrate this benefit of using impulsive control. Another advantage of using impulsive inputs is that such inputs can significantly alter the dynamics of the system in a very short period of time. This property is used to design a safe fall algorithm for humanoid robots undergoing a fall, i.e., after the continuous controller has failed to keep the system trajectories confined to a fixed region around the equilibrium. The algorithm uses impulsive inputs to change the fall direction of the robot to minimize the damage to people and objects in the vicinity, as well as to its own self. In many instances, external disturbances or impact from interaction with the environment can have an adverse effect on system performance. In the second part of this dissertation, we develop control algorithms to mitigate these effects in underactuated biped robots. We first develop a disturbance rejection algorithm for the synthetic-wheel biped to mitigate the effects of external disturbances. A continuous controller is then designed for the synthetic-wheel biped to generate an impact-free walking gait. This gait consumes zero energy in the ideal case and the necessary conditions to achieve this gait are shown to be general and applicable to a range of bipedal robotic systems. An underactuated mechanical system can be non-minimum-phase if its zero dynamics is unstable. We finally investigate the output-tracking problem for linear non-minimum-phase systems. Intermittent output tracking is achieved under the condition of finite preview of the reference trajectory; the control algorithm uses switched inputs, which can be approximated by impulsive inputs in the limit.
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