Robust and Adaptive Dynamic Walking of Bipedal Robots
Legged locomotion has several interesting challenges that need to be addressed, such as the ability of dynamically walk over rough terrain like stairs or stepping stones, as well as the ability to adapt to unexpected changes in the environment and the dynamic model of the robot. This thesis is driven towards solving these challenges and makes contributions on theoretical and experimental aspects to address: dynamic walking, model uncertainty, and rough terrain. On the theoretical front, we introduce and develop a unified robust and adaptive control framework that enables the ability to enforce stability and safety-critical constraints arising from robotic motion tasks under a high level of model uncertainty. We also present a novel method of walking gait optimization and gait library to address the challenge of dynamic robotic walking over stochastically generated stepping stones with significant variations in step length and step height, and where the robot has knowledge about the location of the next discrete foothold only one step ahead. On the experimental front, our proposed methods are successfully validated on ATRIAS, an underactuated, human-scale bipedal robot. In particular, experimental demonstrations illustrate our controller being able to dynamically walk at 0.6 m/s over terrain with step length variation of 23 to 78 cm, as well as simultaneous variation in step length and step height of 35 to 60cm and -22 to 22cm respectively. In addition to that, we also successfully implemented our proposed adaptive controller on the robot, which enables the ability to carry an unknown load up to 68 lb (31 kg) while maintaining very small tracking errors of about 0.01 deg (0.0017 rad) at all joints. To be more specific, we firstly develop robust control Lyapunov function based quadratic program (CLFQP) controller and L1 adaptive control to handle model uncertainty for bipedal robots. An application is dynamic walking while carrying an unknown load. The robust CLF-QP controller can guarantee robustness via a quadratic program that can be extended further to achieve robust safety-critical control. The L1 adaptive control can estimate and adapt to the presence of model uncertainty in the system dynamics. We then present a novel methodology to achieve dynamic walking for underactuated and hybrid dynamcal bipedal robots subject to safety-critical constraints. The proposed controller is based on the combination of control Barrier functions (CBFs) and control Lyapunov functions (CLFs) implemented as a state-based online quadratic program to achieve stability under input and state constraints. The main contribution of this work is the control design to enable stable dynamical bipedal walking subject to strict safety constraints that arise due to walking over a terrain with randomly generated discrete footholds. We next introduce Exponential Control Barrier Functions (ECBFs) as means to enforce high relativedegree safety constraints for nonlinear systems. We also develop a systematic design method that enables creating the Exponential CBFs for nonlinear systems making use of tools from linear control theory. Our method creates a smooth boundary for the safety set via an exponential function, therefore is called Exponential CBFs. Similar to exponential stability and linear control, the exponential boundary of our proposed method helps to have smoother control inputs and guarantee the robustness under model uncertainty. The proposed control design is numerically validated on a relative degree 4 nonlinear system (the two-link pendulum with elastic actuators and experimentally validated on a relative degree 6 linear system (the serial cart-spring system). Thanks to these advantages of Exponential CBFs, we then can apply the method to the problem of 3D dynamic walking with varied step length and step width as well as dynamic walking on time-varying stepping stones. For the work of using CBF for stepping stones, we use only one nominal walking gait. Therefore the range of step length variation is limited ([25 : 60](cm)). In order to improve the performance, we incorporate CBF with gait library and increase the step length range significantly ([10 : 100](cm)). While handling physical constraints and step transition via CBFs appears to work well, these constraints often become active at step switching. In order to resolve this issue, we introduce the approach of 2-step periodic walking. This method not only gives better step transitions but also offers a solution for the problem of changing both step length and step height. Experimental validation on the real robot was also successful for the problem of dynamic walking on stepping stones with step lengths varied within [23 : 78](cm) and average walking speed of 0:6(m=s). In order to address the problems of robust control and safety-critical control in a unified control framework, we present a novel method of optimal robust control through a quadratic program that offers tracking stability while subject to input and state-based constraints as well as safety-critical constraints for nonlinear dynamical robotic systems under significant model uncertainty. The proposed method formulates robust control Lyapunov and barrier functions to provide guarantees of stability and safety in the presence of model uncertainty. We evaluate our proposed control design on different applications ranging from a single-link pendulum to dynamic walking of bipedal robot subject to contact force constraints as well as safety-critical precise foot placements on stepping stones, all while subject to significant model uncertainty. We conduct preliminary experimental validation of the proposed controller on a rectilinear spring-cart system under different types of model uncertainty and perturbations. To solve this problem, we also present another solution of adaptive CBF-CLF controller, that enables the ability to adapt to the effect of model uncertainty to maintain both stability and safety. In comparison with the robust CBF-CLF controller, this method not only can handle a higher level of model uncertainty but is also less aggressive if there is no model uncertainty presented in the system.