Title:

Convergence analysis of ILC algorithms with application to compass gait bipedal walking robot

At an early age, i.e., up to about 12 years, humans learn to walk and subsequently develop a robust and flexible gait. This is learned by repetitively taking similar steps and the experience is stored in the muscle/reflexive memory. Over the last 30 years, a variety of humanoid bipedal robots have been developed to copy the human gait. However, walking/locomotion is still a relatively difficult control problem due to its complex hybrid nature because of nonsmooth dynamics. Although, simple walking comprises of single support in which one leg swings forward, then it impacts with ground for a brief double support phase and further transition of the other support leg to start a new swing. The steps are repeated again and again in a similar manner for walking over an even surface. As the swinging leg strikes the ground, it is a nonlinear impact which poses a challenge since it causes nonzero initial state errors for each step which depend on the error in the gait at last moment for previous step. The usual bipedal control relies on complex techniques based on inverse kinematics, ZMP (ZeroMoment Pole) and COP (Centre Of Pressure) to generate the required control inputs for the joints. However, a basic cognitive assumption is that walking is a relatively simple task which can be learned and the biological systems have achieved it by simple repetitions. This has been overlooked in these control techniques. In the past, ILC has been proposed to solve the repetitive learning problems. The Iterative Learning Controller learns to generate the desired set of input signals to compensate for the output tracking errors in a sequential manner such that in the initial iterations, the signals values at earlier time indices have faster rate than the later ones. So, at the last time index the convergence is achieved after all the earlier ones. ILC learns/adapts the joint control for repetitive gaits. In this thesis it has been proposed to be used as a muscle memory where control signals are learnt for a repetitive batch. Thus, ILC equates to “learning a sequence of action by muscles”. Due to the transfer of state error in a cyclic manner from the end of a previous step/repetition to the recent step/repetition, the convergence has to be established in joint control and state space. Similar is the case of continuous walking where the ground impacts transfer part of the error in the gait to the start of a new step representing an impacting Cyclic ILC scenario. Hence, the ILC problem is changed from finite to an infinite horizon. The second problem occurs with the nonconstant length of the iteration due to change in step size. The two scenarios have been considered: Firstly, when the control input is updated using ILC with identical initial conditions at the start of each repetition. Secondly, control input update under varying initial conditions leading to Cyclic ILC. The batch to batch evolution of control inputs at each sample time within a batch is formulated. The sequential convergence of control input generated by ILC algorithms has been investigated. The exact relationship for the rate of convergence of the control input has been formulated down to the sampletime level. This provides deeper insight about the ILC algorithms and hence exact factors affecting the convergence could be established. Limits of the learning process have been clearly demonstrated as well. Although, simpler DILC converges for zero initial error but for cyclic nonzero initial errors, it has offset error which corresponds to the initial state error. With proportional part, the PDILC algorithm has eliminated the offset error which has been illustrated for a damped pendulum and further implemented to bipedal locomotion. For reasons of energy efficiency, passive dynamics has been chosen for the compass gait model of the biped. The walking problem for the compass gait robot has been solved using the modified PDILC which utilizes the acceleration error term as well. The steady gait has been achieved for the compass gait robot on flat surface which has been verified by the phase portraits.
