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115
Nonholonomic motion planning: Steering using sinusoids
 IEEE fins. Auto. Control
, 1993
"... AbstractIn this paper, we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vec ..."
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Cited by 285 (14 self)
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AbstractIn this paper, we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vector fields and their first order Lie brackets. Using Brockett’s result as motivation, we derive suboptimal trajectories for systems which are not in canonical form and consider systems in which it takes more than one level of bracketing to achieve controllability. These trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. We define a class of systems which can be steered using sinusoids (chained systems) and give conditions under which a class of twoinput systems can be converted into this form. I.
A Differential Geometric Approach to Motion Planning
 NONHOLONOMIC MOTION PLANNING
, 1993
"... We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the sy ..."
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Cited by 80 (0 self)
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We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the righthand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of nonnilpotent systems for which the algorithm, in cas...
Motion Planning For Controllable Systems Without Drift
 In Proceedings of the IEEE International Conference on Robotics and Automation
, 1991
"... We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the sy ..."
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Cited by 73 (8 self)
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We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the righthand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of nonnilpotent systems for which the algorithm, in cascade with a precompensator, produces an exact solution in a finite number of steps. We also ...
Limits Of Highly Oscillatory Controls And The Approximation Of General Paths By Admissible Trajectories
, 1991
"... We describe sufficient conditions, extending earlier work by Kurzweil and Jarnik, for a sequence of inputs u j = (u j 1 ; : : : ; u j m ) 2 L 1 ([0; T ]; IR m ) to be such that, for every mtuple (f 1 ; : : : ; f m ) of smooth vector fields, the trajectories of x(t) = P m k=1 u j k (t)f ..."
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Cited by 69 (8 self)
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We describe sufficient conditions, extending earlier work by Kurzweil and Jarnik, for a sequence of inputs u j = (u j 1 ; : : : ; u j m ) 2 L 1 ([0; T ]; IR m ) to be such that, for every mtuple (f 1 ; : : : ; f m ) of smooth vector fields, the trajectories of x(t) = P m k=1 u j k (t)f k (x(t)) converge to those of an "extended system" x(t) = P r k=1 v k (t)f k (x(t)), where the new vector fields fm+1 ; : : : ; f r are Lie brackets of the original f k 's. Using these conditions, we can solve the inverse problem: given a trajectory fl of the extended system, find trajectories of the original system that converge to fl. This is done by means of a universal construction that only involves a knowledge of the v's. These results can be applied to solve the problem of approximate tracking for a controllable system without drift. 1. Introduction In this paper we report a number of results on the relation between the solutions of an equation x(t) = m X k=1 u k (t)f k (x(...
Guidelines in nonholonomic motion planning for mobile robots
 ROBOT MOTION PLANNNING AND CONTROL
, 1998
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Trajectory Generation for the NTrailer Problem Using Goursat Normal Form
, 1995
"... In this paper, we develop the machinery of exterior differenllai forms, more particularly the Gourset normal form for a Ffaffian system, tor solving nonsoloMwic motion phdng probkms, &.e., motion planning for systems with lloniatcgrable velocity constraints. We use tbis technique to solve the p ..."
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Cited by 64 (9 self)
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In this paper, we develop the machinery of exterior differenllai forms, more particularly the Gourset normal form for a Ffaffian system, tor solving nonsoloMwic motion phdng probkms, &.e., motion planning for systems with lloniatcgrable velocity constraints. We use tbis technique to solve the problem of rbxing a mobile robot WMI R trailers. We present an algorithm for finding a family of ~WIS~~~OM whicb will convert the system of rolling constraints on the wheels of the robot with n traiten into the GoaFapt canonical form..nRo of these transformations are studied in detail. The Gomt normal form for exterior diffemtial systems is dual to the socalled chainedform for vector fields that bas been studied previously. Consequently, we are able to give the state feedback law aad change o € e00rdinaW tovert the Ntrai4r system id0 chained form. Tllree metbods for for chainedform systems using shrosoidg and polynomiPls aa inputs are presented. The motion prpnnhag strategy Is therefore to the Ntrailer system into Gonrsat form, use this to lind the cboinedform coordinates, plan a path for the corresponding cimkdform system, and then transform the resalting traje.ctory back into the original coordinates. Simulations and h.ames of mode animations of the Ntnder system for parallel parking and backing into a loading dock using this strategy are included.
Motion Control of DriftFree, LeftInvariant Systems on Lie Groups
 IEEE Transactions on Automatic Control
, 1995
"... In this paper we address the constructive controllability problem for driftfree, leftinvariant systems on finitedimensional Lie groups with fewer controls than state dimension. We consider small (ffl) amplitude, lowfrequency, periodically timevarying controls and derive average solutions for sys ..."
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Cited by 62 (7 self)
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In this paper we address the constructive controllability problem for driftfree, leftinvariant systems on finitedimensional Lie groups with fewer controls than state dimension. We consider small (ffl) amplitude, lowfrequency, periodically timevarying controls and derive average solutions for system behavior. We show how the pthorder average formula can be used to construct openloop controls for pointtopoint maneuvering of systems that require up to (p \Gamma 1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p = 2; 3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(ffl ) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs.
Shortest Paths For The ReedsShepp Car: A Worked Out Example Of The Use Of Geometric Techniques In Nonlinear Optimal Control.
, 1991
"... We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion coul ..."
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Cited by 55 (5 self)
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We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion could always be achieved by means of trajectories of a special kind, namely, concatenations of at most five pieces, each of which is either a straight line or a circle, and (b) that these concatenations can be classified into 48 threeparameter families. We show how these results fit in a much more general framework, and can be discovered and proved by applying in a systematic way the techniques of Optimal Control Theory. It turns out that the "classical" optimal control tools developed in the 1960's, such as the Pontryagin Maximum Principle and theorems on the existence of optimal trajectories, are helpful to go part of the way and get some information on the shortest paths, but do not suffice ...
Shortest paths for subRiemannian metrics on ranktwo distributions
, 1995
"... We study lengthminimizing arcs in subRiemannian manifolds (M;E;G) whose metric G is defined on a ranktwo bracketgenerating distribution E. It is well known that all lengthminimizing arcs are extremals, and that these extremals are either "normal" or "abnormal." Normal extrem ..."
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Cited by 48 (1 self)
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We study lengthminimizing arcs in subRiemannian manifolds (M;E;G) whose metric G is defined on a ranktwo bracketgenerating distribution E. It is well known that all lengthminimizing arcs are extremals, and that these extremals are either "normal" or "abnormal." Normal extremals are locally optimal, in the sense that every sufficiently short piece of such an extremal is a minimizer. The question whether every lengthminimizer is a normal extremal remained open for several years, and was recently settled by R. Montgomery, who exhibited a counterexample. But Montgomery's geometric optimality proof depends heavily on special properties of his example and still leaves open the question whether abnormal minimizers are an exceptional phenomenon or a common occurrence. We present an analytic technique for proving local optimality of a large class of abnormal extremals that we call "regular." Our technique is based on (a) a "normal form theorem," stating that, locally, a regular abnormal e...