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