Differential flatness based design, planning and control for classes of under-actuated systems

Under-actuated systems arise in numerous situations. In certain applications, such as walking robots it is unavoidable as there are phases in walking cycle where the leg tips along the heel or the toes. Under-actuation can be a better design choice for robots in space and industrial applications due to cost and dead-weight considerations. Another instance where under-actuation finds application is in restoring operation in spite of actuator failure.Control of nonlinear under-actuated systems is an area of ongoing research. In general, for an under-actuated system, not all state trajectories are dynamically feasible and it is hard to characterize feasible trajectories analytically. Even if a feasible trajectory is found, designing a controller for an under-actuated system is also a difficult task. Differential flatness, if applicable, provides a systematic unified approach to (i) plan dynamically feasible trajectories and (ii) design a controller that can track those trajectories. However, a nonlinear under-actuated system may not be differentially flat. This work presents an approach to design under-actuated systems to be differentially flat enabling a systematic trajectory planning and control. The design methodology has two parts: (i) a recursive inertia distribution scheme that places the center-of-mass (COM) of links at specific locations and (ii) an actuator and torque spring placement scheme.This approach for Design, Planning and Control is applied to two classes of under-actuated systems: (i) Planar Open-Chain Manipulators and (ii) Bipedal Walking Robots. Feasible trajectories are constructed using SQP based numerical optimization. The optimization algorithm allows to find trajectories that satisfy motion constraints such as limit on torques for serial chain manipulator, ground clearance of the swinging leg for walking robot, etc. A linear full state feedback controller is designed in the flat output domain to track desired trajectories. Results from trajectory planning and dynamic simulations of flatness based tracking are presented for both systems. Based on the design methodology experimental prototypes of (i) a three degree-of-freedom (DOF) under-actuated manipulator and (ii) a four-link bipedal robot have been fabricated. The flatness based control methodology is experimentally demonstrated using the 3-DOF robotic arm. Effect of two kinds of non-idealities on the flatness based controller is studied (i) parametric uncertainties and (ii) unmodeled viscous friction at unactuated joints. For parametric uncertainties, it is shown that under certain conditions a robust controller can be designed. For viscous friction, it is shown that (i) for the original set of flat outputs a stable internal dynamics is induced and (ii) the system remains differentially flat with an alternate set of outputs. Results from tracking simulations for both conditions are presented.This work essentially integrates the Planning and Control of Under-Actuated Mechanical Systems with their Design. It has been demonstrated by simulations and experiments that certain classes of under-actuated systems can be designed to be differentially flat enabling a systematic trajectory planning and control. It is also shown that certain types of non-idealities can be compensated with a robust control strategy or a modification in the flat outputs. With additional design features, such as locks at unactuated joints, these designs can potentially provide a cheaper alternative for fully actuated robots in applications where point to point motion is desired. This work suggests that it can be beneficial to design a system not just from the perspective of the actual task at hand but also from the perspective of Planning and Control.