First steps toward formal controller synthesis for bipedal robots with experimental implementation

Abstract

Bipedal robots are prime examples of complex cyber-physical systems (CPS). They exhibit many of the features that make the design and verification of CPS so difficult: hybrid dynamics, large continuous dynamics in each mode (e.g., 10 or more state variables), and nontrivial specifications involving nonlinear constraints on the state variables. In this paper, we propose a two-step approach to formally synthesize controllers for bipedal robots so as to enforce specifications by design and thereby generate physically realizable stable walking. In the first step, we design outputs and classical controllers driving these outputs to zero. The resulting controlled system evolves on a lower dimensional manifold and is described by the hybrid zero dynamics governing the remaining degrees of freedom. In the second step, we construct an abstraction of the hybrid zero dynamics that is used to synthesize a controller enforcing the desired specifications to be satisfied on the full order model. Our two step approach is a systematic way to mitigate the curse of dimensionality that hampers the applicability of formal synthesis techniques to complex CPS. Our results are illustrated with simulations showing how the synthesized controller enforces all the desired specifications and offers improved performance with respect to a classical controller. The practical relevance of the results is illustrated experimentally on the bipedal robot AMBER 3. High-level motion primitives were used on MABEL to allow walking over rough ground without tripping. While tools for automatic low-level control algorithm synthesis are well developed, at the state machine level, all tuning was done by hand for lack of appropriate tools. "Probable correctness" was established through extensive simulation and experiments.

An essential model for generating walking motions for humanoid robots

“…the frame Σ 0 . The constraint of the contact wrench to avoid takeoff, sliding and rotation of the support foot can be expressed based on equations (10) and (11) for any humanoid robot.…”

“…On the other hand, a walking gait can be an "automatic" task if the floor is flat and empty, or a task precisely defined by the environment. In the first case, the assumption that the motion of the robot depends only on its internal states is made [7,8,9,10,11]. In the second case, the modeling must be based on a reference linked to the environment to impose a precise pose of the landing foot even in presence of perturbations.…”

Walking With Confidence: Safety Regulation for Full Order Biped Models

“…By varying α continuously over time, the user can change the hybrid zero dynamics manifold to modify the robot behaviour in real-time. The idea of modifying the HZD manifold in real-time is similar to [23] in which the desired hip velocity v acts as an input to the manifold.…”

“…This reliance is restrictive, as it precludes behaviors that would leave the neighborhood of the limit cycle. Recent work has been done to extend the range of safe HZD behaviours beyond a single limit cycle neighborhood [21], [22], [23]. In [21] and [22], the controller is allowed to discretely switch between a family of periodic gaits.…”

Feedback Regularization and Geometric PID Control for Robust Stabilization of a Planar Three-link Hybrid Bipedal Walking Model

“…Different methods incorporating feedback linearization have been used extensively to control biped robots on a flat perfectly horizontal ground [3], [4], [5], [6], [7], [8], [9] as well as on rough terrain [10], [11], [12]. Decentralized control methods have been used in [13] for stabilization of periodic orbits in walking gaits.…”