Robustness of Adaptive Control under Time Delays for Three-Dimensional Curve Tracking

Abstract

We analyze the robustness of a class of controllers that enable three-dimensional curve tracking by a free moving particle. The free particle tracks the closest point on the curve. By building a strict Lyapunov function and robustly forward invariant sets, we show input-to-state stability under predictable tolerance and safety bounds that guarantee robustness under control uncertainty, input delays, and a class of polygonal state constraints, including adaptive tracking and parameter identification under unknown control gains. Such an understanding may provide certified performance when the control laws are applied to real-life systems. Introduction.Curve tracking is a central problem in control and path planning for mobile robots [1,8,27,36,37]. The work presented in [39] used a nonstrict Lyapunov function to find feedback controllers that ensure that a robot moves parallel to a two-dimensional (2D) curve. See also [32] for chained form systems that capture the nonholonomic dynamics of a large variety of mobile robots, such as a kinematic car with trailers, which led to extended curve tracking control laws, and [12] for a reformulation of 2D curve tracking in the three-dimensional (3D) case. Curve tracking is also important for cooperative controllers that track motion for multiple mobile robots; see [38] for applications to ocean sensing. The controls in the works cited above have been found to give reliable performance, even under severe perturbations, in farming [18], obstacle avoidance in corridors [41], and ocean sampling [38].Given a curve tracking control, we are interested in deriving predictable safety and tolerance bounds under several types of uncertainty and input delays, and in identifying unknown control gains. To this end, this paper presents novel 3D analogs of the robustness results that we derived for 2D in [20,21]. For 2D tracking, [20] proved robustness using input-to-state stability (ISS) with respect to actuator errors and robust forward invariance, which gave predictable tolerance and safety bounds under time delays and a class of polygonal state constraints. Briefly described, robust forward invariance means forward invariance in the usual sense of differential equations, but that is also preserved under perturbations; see section 3 for the precise definition. Actuator errors can be modeled as additive perturbations on controls, and communication delays are also common [6, 16], e.g., in marine robotics where there may only be intermittent communications under unfriendly sea conditions. In

Tracking and parameter identification for model reference adaptive control

“…In its basic form, the PE condition is the requirement that the reference trajectory is such that the regressor satisfies a PE inequality when evaluated along the given reference trajectory; see Section 4.1 for our relaxed PE condition. For two‐dimensional curve tracking for gyroscopic models, our works proved globally asymptotically stable tracking and parameter identification results using a novel barrier Lyapunov function approach that ensured robustness with respect to actuator uncertainties under polygonal state constraints; see also the work of Malisoff and Zhang for three‐dimensional (3D) analogs. The adaptive control work provides time‐varying gains to ensure exponential convergence for some nonlinear systems and to ensure convergence of the parameter estimates to a constant vector (which might not be the true value of the unknown parameter vector).…”

Stability and robustness analysis for human pointing motions with acceleration under feedback delays

“…However, using sublevel sets of ISS Lyapunov functions to find bounds on the allowable perturbations can lead to bounds that are conservative. Therefore, we also use a robust forward invariance approach from [16,17], to generate state performance bounds that are less conservative.An advantage of the robust forward invariance approach is that it chooses the state constraints to facilitate finding maximal allowable perturbation sets under which the state constraint sets are strongly forwardly invariant; see our definitions in Section 3. This contrasts with the usual approaches to state constrained problems, where the state space is generally fixed and where the goal was to prevent the state from exiting the given fixed state constraint set; for example, see the interesting robust positive invariance approach in [18], where the goal is to construct sets of initial states for discrete time dynamics such that the given state constraint set is never violated under perturbations, but where no maximal allowable perturbation sets are found under delays.…”

“…This contrasts with the usual approaches to state constrained problems, where the state space is generally fixed and where the goal was to prevent the state from exiting the given fixed state constraint set; for example, see the interesting robust positive invariance approach in [18], where the goal is to construct sets of initial states for discrete time dynamics such that the given state constraint set is never violated under perturbations, but where no maximal allowable perturbation sets are found under delays. However, the methods from [16,17] are limited to two-dimensional or three-dimensional curve tracking dynamics that arise in marine robotics, so the robust forward invariant sets we provide in this work are completely different from those in [16,17]. Hence, other advantages of the method of this paper compared with the ones of [16,17] are that: (i) we extend the approaches of [16,17] to a very different dynamical system; and (ii) the new dynamical system that we consider here leads to a completely different trapezoidal shaped class of robust forward invariant sets, which could not be handled using the analysis in [17], which was limited to hexagonally shaped robustly forwardly invariant sets for curve tracking dynamics.…”

“…However, the methods from [16,17] are limited to two-dimensional or three-dimensional curve tracking dynamics that arise in marine robotics, so the robust forward invariant sets we provide in this work are completely different from those in [16,17]. Hence, other advantages of the method of this paper compared with the ones of [16,17] are that: (i) we extend the approaches of [16,17] to a very different dynamical system; and (ii) the new dynamical system that we consider here leads to a completely different trapezoidal shaped class of robust forward invariant sets, which could not be handled using the analysis in [17], which was limited to hexagonally shaped robustly forwardly invariant sets for curve tracking dynamics.…”

“…However, using sublevel sets of ISS Lyapunov functions to find bounds on the allowable perturbations can lead to bounds that are conservative. Therefore, we also use a robust forward invariance approach from [16,17], to generate state performance bounds that are less conservative.…”