Realtime Safety Control for Bipedal Robots to Avoid Multiple Obstacles via CLF-CBF Constraints

An artificial potential field considers a robot as a charged particle and a goal as the opposite charged particle so that the robot can be attracted to the goal [borenstein1989real, hwang1992potential, valavanis2000mobile]. The farther the robot is from the goal, the greater the attraction force. On the other hand, obstacles presented within the environment are the same charge as the robot, which will generate the repelling force to push the robot away from them. In the end, the potential field should be defined everywhere with a force to keep the robot moving toward the goal without colliding with any obstacles.

However, there are local minima issues where attracting fields and repelling field gets superimposed and trap the robot in an unknown state other than the goal. Moreover, oscillating issues happen inside narrow passages or nearby the obstacles. Adaptive potential fields with heuristic functions have been proposed to avoid local minima [IntroLR:PotentialField3, IntroLR:PotentialField4, IntroLR:PF5localminima]. Combining potential fields with gradient functions to reduce oscillating problem is also proposed in [IntroLR:PF6oscillation, IntroLR:PF7oscillation2]. Recently, reference [IntroLR:PFvsCBF] demonstrates that a CBF can be considered an advanced potential field method with improved stability for obstacle avoidance tasks.

Control barrier functions (CBF) have been introduced as a crucial means to provide safety guarantees for nonlinear systems [LR:CBFvsRA, LR:CBFSafety-Critical]. Model Predictive Control (MPC) repeatedly solves constrained finite-horizon nonlinear optimization problems. However, solving such problems in real-time still remains challenging because it is not trivial to predict whether the system will remain within the safe region after applying the control input from the last step [LR:MPClongHorizon1, LR:MPClongHorizon2]. Combining a CBF together with Model Predictive Control (MPC) [LR:CBFMPC1, LR:CBFMPC2, LR:CBFMPC3] enables an MPC-CBF system to take maximum advantage of both approaches.

To further improve the robustness of the safety system in practice, managing unknown input disturbance and unmodeled dynamics are proposed in [LR:CBFdisturbance1, LR:CBFdisturbance2, LR:CBFdisturbance3, LR:CBFdisturbance4]. CBFs are also widely used in multi-agent systems for collision avoidance and intersection crossing scenarios [LR:CBF-Multi-Interaction1, LR:CBF-Multi-Interaction2, LR:CBF-Multi-Interaction3, LR:CBF-Multi-Interaction4]. Control Lyapunov functions guarantee the stability and convergence for the control system, and have been widely combined with CBFs to address safety-critical problems [LR:CLFsontag1989universal, LR:CBFCLFInsp, LR:CLFCBF1, LR:CLFCBF2, LR:CLFCBF3, LR:CLFCBF4, LR:CLFCBF5, LR:CLFCBF6, LR:CLFCBF7discrete]. Reference [LR:CLFCBF1] focuses on using CLF-CBF-QP to achieve stable walking for bipedal robots. Applications to solve trajectory planning problems under spatiotemporal and control input constraints are discussed in [LR:CLFCBF4]. Both [LR:CLFCBF2, LR:CLFCBF3] extend their algorithm with point-wise feasibility and asymptotic stability; they, however, still are not able to provide solutions to maintain stability while avoiding obstacles. Moreover, the robot dynamics in [LR:CLFCBF5, LR:CLFCBF6, LR:CLFCBF7discrete] are not applicable to our bipedal robot.

Usually a control barrier function in a control system is designed for a single obstacle. When there are multiple obstacles in the control system, the barrier functions for each obstacle are combined to provide safety guarantees. Reference [rauscher2016constrained] uses multiple CBF constraints for the designed control system. The multiple CBF functions can also be combined to obtain a single CBF function [romdlony2016stabilization, chen2021backup] which can keep robot avoiding obstacles.

An equilibrium point of a system is where the robot state remains unchanged. Reference [LR:Equilibria] shows that although CBF-CLF-QP allows the combination of stability and afwty \jzl? in navigation problems, the QP formulation can also lead to undesired equilibrium points in the closed-loop system. With this awareness, [LR:EquilibriaTan] analyzes and proves the existence of equilibrium points in closed-loop systems and demonstrated a modified quadratic program formulation. In addition, [LR:EquilibriaCollisionCone, LR:EquilibriaBall] shows that the unwanted equilibrium point can be removed by transforming the system into a convex manifold, or by increasing the complexity of system states such as including velocities into the system state.

An equilibrium point is also considered as a “deadlock” situation in the multi-agent control system. Reference [LR:EquilibriaMultiAgent] discusses a deadlock scenario with two agents in an one dimensional environment, which is not applicable in our multidimensional system and multi-obstacle environment. In addition, [LR:grover2019deadlock] analyze deadlocks for CBF-QP systems where the robot dynamics is assumed as a double-integrator. However, the analysis is still not applicable to our system due to the different robot dynamics.

The present of multiple obstacles is common in practice. Most work applies more CBF constraints to align with the shrink of the safety space [xu2016control, xu2018constrained, LR:multi-CBF]. However, the increasing number of CBF constraints results in high complexity of the QP and quickly become intractable. In this work, we propose a theorem to show that the product of valid CBFs is still a valid CBF for our system, in which we avoid adding CBF constraints so that the complexity of the QP remains constant when the number of obstacles increases.