Marginal stability in ℒ₁ -adaptive control of manipulators

This paper presents a preliminary analysis of the robustness of a recently-proposed adaptive controller for robot manipulators based on the ℒ₁ control paradigm. Here, the use of a lowpass filter in the control input decouples the estimation loop from the control loop, thereby facilitating an arbitrary increase of estimation rates (limited only by hardware) without sacrificing robustness. Tuning of the filter also allows for shaping the nominal response and enhancing the system's robustness. The paper further demonstrates improvements in the critical time delay associated with a static reference input achieved through the introduction of time delay in the state-predictor formulation. Guided by results from the theory of single-input-single-output ℒ₁ control systems, a linear, time-invariant system is proposed in order to derive a conservative lower bound on the system's actuator critical time delay for static reference input in the limit of large estimation gains, as an indicator of robustness. Finally, a numerical method is proposed for quantifying the robustness against time delay of the system's response to a given static reference input based on techniques of parameter continuation. This method computes the critical time delay at which local stability is lost in a Hopf bifurcation. The dependence of this critical time delay on control parameters, such as adaptive gains and filter bandwidth, is here obtained using advanced algorithms for computing approximate covers of implicitly defined manifolds.