We study the tracking of a trajectory for a nonholonomic system by recasting the problem as a constrained optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a nonholonomic system and the desired reference trajectory, both evolving on the distribution which defines the nonholonomic constraints. The problem is studied from a geometric framework. Optimality conditions are determined by the Pontryagin Maximum Principle and also from a variational point of view, which allows the construction of geometric integrators. Examples and numerical simulations are shown to validate the results.
This letter presents two bearing-constrained formation tracking control protocols for multiple nonholonomic agents based respectively on the bearing vectors and displacements between the agents. The desired formation pattern of the system is specified by the desired inter-agent bearing vectors. In the proposed control schemes, there are two or more leaders moving with the same constant velocity; the other follower agents do not have the information of the leaders' velocity nor communicate variables with their neighbors. Under both the proposed control laws, the system achieves the moving targe…
The paper considers a motion control problem for kinematic models of nonholonomic wheeled systems. The class of maneuverable wheeled systems is defined consisting of systems that can follow any sufficiently smooth non-stop trajectory on the plane. A sufficient condition for maneuverability is obtained. The design of control law that stabilizes motion along the desired trajectory on the plane is performed in two steps. On the first step the trajectory on the configuration manifold of the system and the input function are constructed that ensure the exact reproduction of the desired trajectory o…
We consider some generalizations of the classical nonholonomic integrator and give a geometric approach to characterize controllability for these systems. We use Stokes' theorem and results from complex analysis to obtain necessary and sufficient conditions for controllability. Furthermore, we show that optimal trajectories of certain minimum energy optimal control problems defined on these systems can be identified with the trajectory of a charged particle in an electromagnetic field.
The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems whose degree of nonholonomy is equal to 1. It is shown that the trajectories of the closed-loop system converge exponentially to any given neighborhood of the reference curve provided that the solutions are defined in the sense of sampling. This conver…
In this work, we investigate the state stabilization and trajectory tracking problems of underactuated surface ships with full state model of having non-diagonal inertia and damping matrices. By combining the novel state transformations, the direct Lyapunov approach, and the nonlinear time-varying tools, the stabilization and the trajectory tracking controllers are developed respectively guaranteeing global uniform asymptotic convergence of the state to the desired set point and global exponential convergence to the desired reference trajectory via mild persistent exciting conditions. Simulati…
We consider tracking control for multibody systems which are modelled using holonomic and nonholonomic constraints. Furthermore, the systems may be underactuated and contain kinematic loops and are thus described by a set of differential-algebraic equations that cannot be reformulated as ordinary differential equations in general. We propose a control strategy which combines a feedforward controller based on the servo-constraints approach with a feedback controller based on a recent funnel control design. As an important tool for both approaches we present a new procedure to derive the interna…
This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered jumps in the state evolution, potentially accompanied by mode switching in the control vector field as well. For a given trajectory with state jumps, we show how to construct an approximation of a nearby perturbed trajectory corresponding to a small variation of the initial condition and input. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal…
We develop an optimization-based framework for joint real-time trajectory planning and feedback control of feedback-linearizable systems. To achieve this goal, we define a target trajectory as the optimal solution of a time-varying optimization problem. In general, however, such trajectory may not be feasible due to , e.g., nonholonomic constraints. To solve this problem, we design a control law that generates feasible trajectories that asymptotically converge to the target trajectory. More precisely, for systems that are (dynamic) full-state linearizable, the proposed control law implicitly t…
In this paper we consider the problem of stabilization and tracking of desired state trajectory for a wide range of nonlinear control problems with disturbances. We present the sufficient conditions for the existence of $C^k$ state feedback controllers and the process of their mathematical designing is described.
A novel dynamic model-based trajectory tracking control law is proposed for a four-wheel differentially driven mobile robot using a backstepping technique that guarantees the Lyapunov stability. The present work improves the work of Caracciolo et al. (ICRA 1999), a dynamic feedback linearization approach, by reducing the number of required assumptions and the number of state terms. We also thoroughly investigate on a gain tuning procedure which is often overlooked for nonlinear controllers. Finally, the performance of the proposed controller is compared with the dynamic feedback linearization …
We treat the periodic trajectory tracking problem: given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group $G$ and an initial condition in $G$, find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood of some point of the periodic trajectory) when $G$ is semisimple and the system is Lie-determined (i.e. controllable), and only for a class of periodic trajectories (which we call regular).…
This work presents new methods and algorithms for tracking the shape and trajectory of moving reflecting obstacles with broken rays, or rays reflecting at an obstacle. While in tomography the focus of the reconstruction method is to recover the velocity structure of the domain, the shape and trajectory reconstruction procedure directly finds the shape and trajectory of the obstacle. The physical signal carrier for this innovative method are ultrasonic beams. When the speed of sound is constant, the rays are straight line segments and the shape and trajectory of moving objects will be reconstru…
Motion planning for autonomous vehicles requires spatio-temporal motion plans (i.e. state trajectories) to account for dynamic obstacles. This requires a trajectory tracking control process which faithfully tracks planned trajectories. In this paper, a control scheme is presented which first optimizes a planned trajectory and then tracks the optimized trajectory using a feedback-feedforward controller. The feedforward element is calculated in a model predictive manner with a cost function focusing on driving performance. Stability of the error dynamic is then guaranteed by the design of the fe…
