89 research outputs found
A Global Steering Method for Nonholonomic Systems
In this paper, we present an iterative steering algorithm for nonholonomic
systems (also called driftless control-affine systems) and we prove its global
convergence under the sole assumption that the Lie Algebraic Rank Condition
(LARC) holds true everywhere. That algorithm is an extension of the one
introduced in [21] for regular systems. The first novelty here consists in the
explicit algebraic construction, starting from the original control system, of
a lifted control system which is regular. The second contribution of the paper
is an exact motion planning method for nilpotent systems, which makes use of
sinusoidal control laws and which is a generalization of the algorithm
described in [29] for chained-form systems
Exponential stabilization of driftless nonlinear control systems using homogeneous feedback
This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers
Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus
We propose in this paper a constructive procedure that transforms locally,
even at singular configurations, the kinematics of a car towing trailers into
Kumpera-Ruiz normal form. This construction converts the nonholonomic motion
planning problem into an algebraic problem (the resolution of a system of
polynomial equations), which we illustrate by steering the two-trailer system
in a neighborhood of singular configurations. We show also that the n-trailer
system is a universal local model for all Goursat structures and that all
Goursat structures are locally nilpotentizable.Comment: LaTeX2e, 23 pages, 4 figures, submitted to International journal of
contro
Flat systems, equivalence and trajectory generation
Flat systems, an important subclass of nonlinear control systems introduced
via differential-algebraic methods, are defined in a differential
geometric framework. We utilize the infinite dimensional geometry developed
by Vinogradov and coworkers: a control system is a diffiety, or more
precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold
equipped with a privileged vector field. After recalling the definition of
a Lie-Backlund mapping, we say that two systems are equivalent if they
are related by a Lie-Backlund isomorphism. Flat systems are those systems
which are equivalent to a controllable linear one. The interest of
such an abstract setting relies mainly on the fact that the above system
equivalence is interpreted in terms of endogenous dynamic feedback. The
presentation is as elementary as possible and illustrated by the VTOL
aircraft
Motion planning and stabilization of nonholonomic systems using gradient flow approximations
Nonlinear control-affine systems with time-varying vector fields are
considered in the paper. We propose a unified control design scheme with
oscillating inputs for solving the trajectory tracking and stabilization
problems. This methodology is based on the approximation of a gradient like
dynamics by trajectories of the designed closed-loop system. As an intermediate
outcome, we characterize the asymptotic behavior of solutions of the considered
class of nonlinear control systems with oscillating inputs under rather general
assumptions on the generating potential function. These results are applied to
examples of nonholonomic trajectory tracking and obstacle avoidance.Comment: submitte
Symbolic computation for mobile robot path planning
AbstractMotion planning for mobile robots is an arduous task. Among the various methods that have been proposed for the solution of this problem in its open loop version is the LafferriereāSussmann method, which is based on differential geometry and employs piecewise constant inputs. This paper gives a succinct description of the method and of a freely available software tool, called the Lie Algebraic Motion PlannerāLAMP and written in Mathematicaā¢, which automates motion planning based on this technique
Optimal path planning for nonholonomic robotics systems via parametric optimisation
Abstract. Motivated by the path planning problem for robotic systems this paper considers nonholonomic path planning on the Euclidean group of motions SE(n) which describes a rigid bodies path in n-dimensional Euclidean space. The problem is formulated as a constrained optimal kinematic control problem where the cost function to be minimised is a quadratic function of translational and angular velocity inputs. An application of the Maximum Principle of optimal control leads to a set of Hamiltonian vector field that define the necessary conditions for optimality and consequently the optimal velocity history of the trajectory. It is illustrated that the systems are always integrable when n = 2 and in some cases when n = 3. However, if they are not integrable in the most general form of the cost function they can be rendered integrable by considering special cases. This implies that it is possible to reduce the kinematic system to a class of curves defined analytically. If the optimal motions can be expressed analytically in closed form then the path planning problem is reduced to one of parameter optimisation where the parameters are optimised to match prescribed boundary conditions.This reduction procedure is illustrated for a simple wheeled robot with a sliding constraint and a conventional slender underwater vehicle whose velocity in the lateral directions are constrained due to viscous damping
Flat systems, equivalence and trajectory generation
3rd cycleIntroduction : Control systems are ubiquitous in modern technology. The use of feedback control can be found in systems ranging from simple thermostats that regulate the temperature of a room, to digital engine controllers that govern the operation of engines in cars, ships, and planes, to flight control systems for high performance aircraft. The rapid advances in sensing, computation, and actuation technologies is continuing to drive this trend and the role of control theory in advanced (and even not so advanced) systems is increasing..
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