314 research outputs found
Path planning for simple wheeled robots : sub-Riemannian and elastic curves on SE(2)
This paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group SE(2). An application of Pontryaginâs maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals and the expression for the position reduced to quadratures. Reachable sets are defined in the arbitrary speed case and a numerical plot of the time-limited reachable sets presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment
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
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
Stabilization of non-admissible curves for a class of nonholonomic systems
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
convergence property is also illustrated numerically by several examples of
nonholonomic systems of degrees 1 and 2.Comment: This is the author's version of the manuscript accepted for
publication in the Proceedings of the 2019 European Control Conference
(ECC'19
Dynamics and control of a class of underactuated mechanical systems
This paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems, defined as systems with fewer inputs than degrees of freedom. Control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified. Controllability and stabilizability results are derived for this class of underactuated systems. Examples are included to illustrate the results; these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable
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