Multi-Layered Safety for Legged Robots via Control Barrier Functions and Model Predictive Control

Abstract

The problem of dynamic locomotion over rough terrain requires both accurate foot placement together with an emphasis on dynamic stability. Existing approaches to this problem prioritize immediate safe foot placement over longer term dynamic stability considerations, or relegate the coordination of foot placement and dynamic stability to heuristic methods. We propose a multi-layered locomotion framework that unifies Control Barrier Functions (CBFs) with Model Predictive Control (MPC) to simultaneously achieve safe foot placement and dynamic stability. Our approach incorporates CBF based safety constraints both in a low frequency kino-dynamic MPC formulation and a high frequency inverse dynamics tracking controller. This ensures that safety-critical execution is considered when optimizing locomotion over a longer horizon. We validate the proposed method in a 3D stepping-stone scenario in simulation and experimentally on the ANYmal quadruped platform.

Predictive Control Barrier Functions for Online Safety Critical Control

“…Thus, in every case, a function α ∈ K satisfying α(m(λ)) ≥ m ′ (λ), ∀λ ∈ [0, T ] and α(m(T )) ≥ γ will satisfy condition (3), so H * is indeed a CBF. Note that these are sufficient, not necessary conditions on α.…”

“…However, one drawback of CBFs is that the CBF condition is reactive (frequently called myopic [2]); i.e., CBFs only consider the safety of the current state and state derivative, rather than considering future objectives of the system. For this reason, the CBF condition may allow trajectories to reach states where large control inputs are required to maintain safety [2], [3], or when control inputs are constrained, CBF safe sets are often overly conservative, requiring large evasive actions (e.g., [4]). For instance, CBFs may cause two cars meeting at an intersection to both decelerate to a complete stop to maintain a required safety distance, whereas it would be more efficient for only one car to decelerate slightly, allowing both cars to traverse the intersection without stopping.…”

“…PCBFs are closely related to Model Predictive Control (MPC) [3], [5]- [9] and to the notion of Backup CBFs [10]- [14], which both also consider safety on a receding horizon, but PCBFs provide distinct advantages. Unlike MPC, the CBF condition is always control-affine, even when the dynamics and constraints are nonlinear.…”

“…Thus, the control input under a PCBF can be computed using a Quadratic Program (QP) with small dimension and affine constraints, which can be solved more easily than nonlinear MPC (e.g. [3], [7]). Moreover, as the PCBF approach does not introduce a fixed discretization, safety is guaranteed at all times in the prediction horizon rather than just at the MPC sampled times.…”

Safe Stochastic Model Predictive Control

“…Further approaches have recently been proposed to address safety within MPC. A combination of MPC and control barrier functions allows considering safety similarly to how Lyapunov functions are used for stability [19], [20]. However, guaranteeing recursive feasibility in the presence of uncertainty remains a challenge.…”