Guidelines in nonholonomic motion planning for mobile robots

Laumond, Jean–Paul; Sekhavat, S.; Lamiraux, Florent

doi:10.1007/bfb0036070

Cited by 272 publications

(197 citation statements)

References 68 publications

Supporting

Mentioning

189

Contrasting

Unclassified

Order By: Relevance

“…This problem has important applications in robotics [8] and vision [13]. On the other hand, this is the simplest sub-Riemannian problem where the conjugate and cut loci differ one from another in the neighborhood of the initial point.…”

Section: Introductionmentioning

confidence: 99%

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Моисеев¹,

Sachkov²

2009

ESAIM: COCV

114

149

View full text Add to dashboard Cite

Abstract.The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained. Mathematics Subject IntroductionProblems of sub-Riemannian geometry have been actively studied by geometric control methods, see books [2,3,7,10]. One of the central and hard questions in this domain is a description of cut and conjugate loci. Detailed results on the local structure of conjugate and cut loci were obtained in the 3-dimensional contact case [1,6]. Global results are restricted to symmetric low-dimensional cases, primarily for left-invariant problems on Lie groups (the Heisenberg group [5,22], the growth vector (n, n(n + 1)/2) [9,11,12], the groups SO(3), SU(2), SL(2) and the Lens Spaces [4]).The paper continues this direction of research: we start to study the left-invariant sub-Riemannian problem on the group of motions of a plane SE(2). This problem has important applications in robotics [8] and vision [13]. On the other hand, this is the simplest sub-Riemannian problem where the conjugate and cut loci differ one from another in the neighborhood of the initial point.The main result of the work is an upper bound on the cut time t cut given in Theorem 5.4: we show that for any sub-Riemannian geodesic on SE(2) there holds the estimate t cut ≤ t, where t is a certain function defined on the cotangent space at the identity. In a forthcoming paper [21] we prove that in fact t cut = t. The bound on the cut time is obtained via the study of discrete symmetries of the problem and the corresponding Maxwell points -points where two distinct sub-Riemannian geodesics of the same length intersect one another.This work has the following structure. In Section 1 we state the problem and discuss existence of solutions. In Section 2 we apply Pontryagin Maximum Principle to the problem. The Hamiltonian system for normal extremals is triangular, and the vertical subsystem is the equation of mathematical pendulum. In Section 3Keywords and phrases. Optimal control, sub-Riemannian geometry, differential-geometric methods, left-invariant problem, Lie group, Pontryagin Maximum Principle, symmetries, exponential mapping, Maxwell stratum. * The second author is partially supported by Russian Foundation for Basic Research, and we endow the cotangent space at the identity with special elliptic coordinates induced by the flow of the pendulum, and integrate the normal Hamiltonian system in these coordinates. Sub-Riemannian geodesics are parameterized by Jacobi's functions. In Section 4 we construct a discrete group of symmetries of the exponential mapping by continuation of reflections in the phase cylinder of the pendulum. In the main Section 5 we obtain an explicit description of Maxwell strata corresponding to the group of discrete symmetries, and prove the upper ...

show abstract

Section: Introductionmentioning

confidence: 99%

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Моисеев¹,

Sachkov²

2009

ESAIM: COCV

114

149

View full text Add to dashboard Cite

show abstract

“…Since the studied non-holonomic robot has two inputs (its leg lengths) and three generalized coordinates (its orientation parameters), its kinematics can be formulated in chained form and, as a consequence, its path planning problem can be solved using well-established procedures (see [12] and the references therein). Using one of these analytical methods, the motion planning problem for the studied robot is solved in [7] using control functions given by truncated trigonometric series.…”

Section: Introductionmentioning

confidence: 99%

Geometric Path Planning Without Maneuvers for Nonholonomic Parallel Orienting Robots

Grosch

Thomas

2016

IEEE Robot. Autom. Lett.

View full text Add to dashboard Cite

Abstract-Current geometric path planners for non-holonomic parallel orienting robots generate maneuvers consisting of a sequence of moves connected by zero-velocity points. The need for these maneuvers restrains the use of this kind of parallel robots to few applications. Based on a rather old result on linear time-varying systems, this paper shows that there are infinitely differentiable paths connecting two arbitrary points in SO(3) such that the instantaneous axis of rotation along the path rest on a fixed plane. This theoretical result leads to a practical path planner for non-holonomic parallel orienting robots that generates single-move maneuvers. To present this result, we start with a path planner based on three-move maneuvers, and then we proceed by progressively reducing the number of moves to one, thus providing a unified treatment with respect to previous geometric path planners.

show abstract

“…This problem is closely connected to models of vision [5,10,11] and robotics [8]. On the other hand, this problem is important for understanding properties of the heat kernel of the diffusion equation on the group SE(2), see [4].…”

Section: Introductionmentioning

confidence: 99%

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Sachkov¹

2009

ESAIM: COCV

View full text Add to dashboard Cite

Abstract.The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.Mathematics Subject Classification. 49J15, 93B29, 93C10, 53C17, 22E30.

show abstract

Guidelines in nonholonomic motion planning for mobile robots

Cited by 272 publications

References 68 publications

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Geometric Path Planning Without Maneuvers for Nonholonomic Parallel Orienting Robots

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Contact Info

Product

Resources

About