Coupled Control Systems: Periodic Orbit Generation with Application to Quadrupedal Locomotion

A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this letter presents a novel formulation for nonlinear control systems that are subject to coupling constraints via virtual “coupling” inputs that abstractly play the role of Lagrange multipliers. The main contribution of this letter is a process—mirroring solving for Lagrange multipliers in robotic systems—wherein we isolate subsystems free of coupling constraints that provably encode the full-order dynamics of the coupled control system from which it was derived. This dimension reduction is leveraged in the formulation of a nonlinear optimization problem for the isolated subsystem that yields periodic orbits for the full-order coupled system. We consider the application of these ideas to robotic systems, which can be decomposed into subsystems. Specifically, we view a quadruped as a coupled control system consisting of two bipedal robots, wherein applying the framework developed allows for gaits (periodic orbits) to be generated for the individual biped yielding a gait for the full-order quadrupedal dynamics. This is demonstrated on a quadrupedal robot through simulation and walking experiments on rough terrains

Coupled Control Systems: Periodic Orbit Generation with Application to Quadrupedal Locomotion

A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this letter presents a novel formulation for nonlinear control systems that are subject to coupling constraints via virtual “coupling” inputs that abstractly play the role of Lagrange multipliers. The main contribution of this letter is a process—mirroring solving for Lagrange multipliers in robotic systems—wherein we isolate subsystems free of coupling constraints that provably encode the full-order dynamics of the coupled control system from which it was derived. This dimension reduction is leveraged in the formulation of a nonlinear optimization problem for the isolated subsystem that yields periodic orbits for the full-order coupled system. We consider the application of these ideas to robotic systems, which can be decomposed into subsystems. Specifically, we view a quadruped as a coupled control system consisting of two bipedal robots, wherein applying the framework developed allows for gaits (periodic orbits) to be generated for the individual biped yielding a gait for the full-order quadrupedal dynamics. This is demonstrated on a quadrupedal robot through simulation and walking experiments on rough terrains

Verifying Safe Transitions between Dynamic Motion Primitives on Legged Robots

Functional autonomous systems often realize complex tasks by utilizing state machines comprised of discrete primitive behaviors and transitions between these behaviors. This architecture has been widely studied in the context of quasi-static and dynamics-independent systems. However, applications of this concept to dynamical systems are relatively sparse, despite extensive research on individual dynamic primitive behaviors, which we refer to as "motion primitives." This paper formalizes a process to determine dynamic-state aware conditions for transitions between motion primitives in the context of safety. The result is framed as a "motion primitive graph" that can be traversed by standard graph search and planning algorithms to realize functional autonomy. To demonstrate this framework, dynamic motion primitives -- including standing up, walking, and jumping -- and the transitions between these behaviors are experimentally realized on a quadrupedal robot

Preference-Based Learning for User-Guided HZD Gait Generation on Bipedal Walking Robots

This paper presents a framework that unifies control theory and machine learning in the setting of bipedal locomotion. Traditionally, gaits are generated through trajectory optimization methods and then realized experimentally -- a process that often requires extensive tuning due to differences between the models and hardware. In this work, the process of gait realization via hybrid zero dynamics (HZD) based optimization problems is formally combined with preference-based learning to systematically realize dynamically stable walking. Importantly, this learning approach does not require a carefully constructed reward function, but instead utilizes human pairwise preferences. The power of the proposed approach is demonstrated through two experiments on a planar biped AMBER-3M: the first with rigid point feet, and the second with induced model uncertainty through the addition of springs where the added compliance was not accounted for in the gait generation or in the controller. In both experiments, the framework achieves stable, robust, efficient, and natural walking in fewer than 50 iterations with no reliance on a simulation environment. These results demonstrate a promising step in the unification of control theory and learning

Automated gap-filling for marker-based biomechanical motion capture data

Marker-based motion capture presents the problem of gaps, which are traditionally processed using motion capture software, requiring intensive manual input. We propose and study an automated method of gap-filling that uses inverse kinematics (IK) to close the loop of an iterative process to minimize error, while nearly eliminating user input. Comparing our method to manual gap-filling, we observe a 21% reduction in the worst-case gap-filling error (p < 0.05), and an 80% reduction in completion time (p < 0.01). Our contribution encompasses the release of an open-source repository of the method and interaction with OpenSim