Distributed Feedback Control Algorithms for Cooperative Locomotion: From Bipedal to Quadrupedal Robots

This thesis synthesizes general and scalable distributed nonlinear control algorithms with application to legged robots. It explores both naturally decentralized problems in legged locomotion, such as the collaborative control of human-lower extremity prosthesis and the decomposition of high-dimensional controllers of a naturally centralized problem into a net- work of low-dimensional controllers while preserving equivalent performance. In doing so, strong nonlinear interaction forces arise, which this thesis considers and sufficiently addresses. It generalizes to both symmetric and asymmetric combinations of subsystems. Specifically, this thesis results in two distinct distributed control algorithms based on the decomposition approach.

Towards synthesizing the first algorithm, this thesis presents a formal foundation based on de- composition, Hybrid Zero Dynamics (HZD), and scalable optimization to develop distributed controllers for hybrid models of collaborative human-robot locomotion. This approach con- siders a centralized controller and then decomposes the dynamics and parameterizes the feedback laws to synthesize local controllers. The Jacobian matrix of the Poincaré map with local controllers is studied and compared with the centralized ones. An optimization problem is then set up to tune the parameters of the local controllers for asymptotic stability. It is shown that the proposed approach can significantly reduce the number of controller parameters to be optimized for the synthesis of distributed controllers, deeming the method computationally tractable. To evaluate the analytical results, we consider a human amputee with the point of separation just above the knee and assume the average physical parameters of a human male. For the lower-extremity prosthesis, we consider the PRleg, a powered knee-ankle prosthetic leg, and together, they form a 19 Degrees of Freedom (DoF) model. A multi-domain hybrid locomotion model is then employed to rigorously assess the performance of the afore-stated control algorithm via numerical simulations. Various simulations involving the application of unknown external forces and altering the physical parameters of the human model unbeknownst to the local controllers still result in stable amputee loco- motion, demonstrating the inherent robustness of the proposed control algorithm.

In the later part of this thesis, we are interested in developing distributed algorithms for the real-time control of legged robots. Inspired by the increasing popularity of Quadratic programming (QP)-based nonlinear controllers in the legged locomotion community due to their ability to encode control objectives subject to physical constraints, this thesis exploits the idea of distributed QPs. In particular, this thesis presents a formal foundation to systematically decompose QP-based centralized nonlinear controllers into a network of lower-dimensional local QPs. The proposed approach formulates a feedback structure be- tween the local QPs and leverages a one-step communication delay protocol. The properties of local QPs are analyzed, wherein it is established that their steady-state solutions on periodic orbits (representing gaits) coincide with that of the centralized QP. The asymptotic convergence of local QPs' solutions to the steady-state solution is studied via Floquet theory. Subsequently, to evaluate the effectiveness of the analytical results, we consider an 18 DoF quadrupedal robot, A1, as a representative example. The network of distributed QPs mentioned earlier is condensed to two local QPs by considering a front-hind decomposition scheme. The robustness of the distributed QP-based controller is then established through rigorous numerical simulations that involve exerting unmodelled external forces and intro- ducing unknown ground height variations. It is further shown that the proposed distributed QPs have reduced sensitivity to noise propagation when compared with the centralized QP.

Finally, to demonstrate that the resultant distributed QP-based nonlinear control algorithm translates equivalently well to hardware, an extensive set of blind locomotion experiments on the A1 robot are undertaken. Similar to numerical simulations, unknown external forces in the form of aggressive pulls and pushes were applied, and terrain uncertainties were introduced with the help of arbitrarily displaced wooden blocks and compliant surfaces. Additionally, outdoor experiments involving a wide range of terrains such as gravel, mulch, and grass at various speeds up to 1.0 (m/s) reiterate the robust locomotion observed in numerical simulations. These experiments also show that the computation time is significantly dropped when the distributed QPs are considered over the centralized QP.