Auxiliary Control to Avoid Undesirable Equilibria in Constrained Quadratic Programs for Trajectory Tracking Applications

Control Lyapunov function (CLF) and control barrier function (CBF) based quadratic programs (QPs) may create undesirable local equilibria in control systems. One recent solution utilizes a rotating nonradial CLF to avoid such equilibria in regulation applications. For trajectory tracking applications, a nominal feedback tracking control can be incorporated into the QP cost function to improve the tracking performance of the CLF–CBF–QP. However, the direction of the steepest descent curve of the CLF can differ from that of the nominal feedback control, which may compromise the tracking performance. Moreover, the design of a CLF is system-specific and generally not easy to realize. This article proposes a tracking–CBF–QP formulation, where a nominal tracking control is incorporated in the cost function of a CBF-based QP. If the nominal control conflicts with the CBF condition, undesirable local equilibria may be induced in the closed-loop system. This work theoretically investigates the occurrence of such undesirable local equilibria and provides an auxiliary control approach to prevent the system from falling into such equilibria. The auxiliary control is activated only when a conflict is projected between the nominal control and the CBF constraint, allowing the nominal control to guide the system otherwise. The proposed tracking–CBF–QP is utilized to design a novel leader–follower algorithm, and its effectiveness is verified experimentally.