Balancing and Step Recovery Capturability via Sums-of-Squares Optimization

Abstract

Abstract-A fundamental requirement for legged robots is to maintain balance and prevent potentially damaging falls whenever possible. As a response to outside disturbances, fall prevention can be achieved by a combination of active balancing actions, e.g. through ankle torques and upper-body motion, and through reactive step placement. While it is widely accepted that stepping is required to respond to large disturbances, the limits of active motions on balancing and step recovery are only well understood for the simplest of walking models. Recent advances in convex optimization-based verification and control techniques enable a more complete understanding of the limits and capabilities of more complex models. In this work, we present an algorithmic approach for formal analysis of the viable-capture basins of walking robots, calculating both inner and outer approximations and corresponding push recovery control strategies. Extending beyond the classic Linear Inverted Pendulum Model (LIPM), we analyze a series of centroidal momentum based planar walking models, examining the effects of center of mass height, angular momentum, and impact dynamics during stepping on capturability. This formal analysis enables an explicit calculation of the differences between these models, and assessment of whether the simplest models ultimately sacrifice capability, and thus stability, when designing push recovery control policies.

Walking With Confidence: Safety Regulation for Full Order Biped Models

“…From Section III-B, the dimension of the reduced order manifold (our state space) is twice the sum of the degree of underaction and the number of shaping parameters. In [12], the authors show that a 6 dimensional state space is tractable for similar sums-of-squares programs. Our approach can thus currently handle a maximum of three degrees of underactuation and/or shaping parameters.…”

“…However, the state-space dimension of realistic robot models far exceeds the limits of this tool. For instance, the state-space of the benchmark model Rabbit [2] has dimension 10, while many sums-of-squares problems become computationally challenging above dimension 6 [12]. In this section, we show how the the state-space dimension can be reduced to a feasible size using the idea of hybrid zero dynamics [15].…”

“…This technique for ensuring hybrid invariance is similar to the procedure given in [26]. The guard of our HZD manifold Z is given asŜ = {x | θ = θ max ,θ > 0} and the reset is defined as in (12).…”

“…We use an optimization similar to [12] to find the largest possible viability domainV ⊂ Z \ Z F for our hybrid zero dynamics system. We representV as the zero super-levelset of a polynomial functionv : 0}), and represent the autonomous viable controller using a polynomial functionû s : Z → R nα .…”

“…This approach uses semi-definite programming to identify the limits of safety in the state space of a system as well as associated controllers for a broad class of nonlinear [5], [6], [7] and hybrid systems [8], [9]. These safe sets can take the form of reachable sets (sets that can reach a known safe state) [10], [9], [5] or invariant sets (sets whose members can be controlled to remain in the set indefinitely) in state space [11], [8], [12]. However, the representation of each of these sets in state space severely restricts the size of the problem that can be tackled by these approaches.…”