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18
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 251 (15 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.
Nilpotent Bases for a Class of NonIntegrable Distributions with Applications to Trajectory Generation for Nonholonomic Systems
 Math. Control Signals Systems
, 1994
"... . This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of findin ..."
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Cited by 46 (3 self)
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. This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using "chained form" and cast that work into a coordinatefree setting. 1. Introduction This work is motivated by the recent interest in trajectory generation for mechanical systems with nonholonomic constraints. Consider the problem of steering a mechanical system with ndimensional configuration space M from an initial configuration x 0 to a final configuration x 1 , subject to a set of indep...
Local Motion Planning for Nonholonomic Control Systems Evolving on Principal Bundles
 In Proc. Mathematical Theory of Networks and Systems
, 1998
"... this paper are as follows. First, we construct an expansion for the system's group displacement that arises from small periodic motions in the base space. This expansion is a generalization of the work of Leonard and Krishnaprasad [6], who developed an analogous formula for case when the local conne ..."
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Cited by 12 (3 self)
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this paper are as follows. First, we construct an expansion for the system's group displacement that arises from small periodic motions in the base space. This expansion is a generalization of the work of Leonard and Krishnaprasad [6], who developed an analogous formula for case when the local connection form A is constant. Kolmanovsky et al., [10] have developed a less structured version of this formula that expands directly in the group rather than in its Lie algebra. In order to develop an intrinsic geometric understanding of these systems we next relate the terms in the expansion to the infinitesimal holonomy algebra of the bundle and to the controllability distribution. In doing so we introduce the covariant derivative on the associated adjoint bundle as a simple means to calculate the terms in the expansion and controllability formulas. In doing so we develop formulas to test the smalltimelocalcontrollability of these systems. These results represent a sharpening and intrinsic restatement of the controllability results of Kelly and Murray [4]. We sum up by showing that we can write our expansion to any order as a reduction of a series for general a#ne control systems given by Sussmann [11].
Motion Control Algorithms for Simple Mechanical Systems with Symmetry
"... We treat underactuated mechanical control systems with symmetry taking the viewpoint of the aftinc connection formalism. We first review the appropriate notions and tests of controllability associated with these systems, including that of fiber controllability. Secondly, we present a series expansio ..."
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Cited by 12 (3 self)
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We treat underactuated mechanical control systems with symmetry taking the viewpoint of the aftinc connection formalism. We first review the appropriate notions and tests of controllability associated with these systems, including that of fiber controllability. Secondly, we present a series expansion describing the evolution of the trajectories of general mechanical control systems starting from nonzero velocity. This series is then used to investigate the behavior of the system under smallamplitude periodic forcing. On this basis, motion control algorithms are designed for systems with symmetry to solve the tasks of pointtopoint rcconfiguration, static interpolation and stabilization problems. Several examples are given and the performance of the algorithms is illustrated in the blimp system.
A characterization of the Lie Algebra Rank Condition by transverse periodic functions
 Journal of Control and Optimization
, 2000
"... The Lie Algebra Rank Condition plays a central role in nonlinear systems control theory. We show that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof here outlined details can be found in [4 ..."
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Cited by 8 (3 self)
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The Lie Algebra Rank Condition plays a central role in nonlinear systems control theory. We show that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof here outlined details can be found in [4] is constructive and provides a method for the determination of such functions. This is illustrated by an example.
Lie Bracket Extensions And Averaging: The SingleBracket Case
, 1997
"... We explain a general approximation technique for nonholonomic systems by discussing in detail a special example, chosen so as to illustrate some of the technical aspects of the general construction. The example considered is that of an extension of a twoinput system obtained by adding a single brac ..."
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Cited by 6 (2 self)
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We explain a general approximation technique for nonholonomic systems by discussing in detail a special example, chosen so as to illustrate some of the technical aspects of the general construction. The example considered is that of an extension of a twoinput system obtained by adding a single bracket of degree five. This bracket is sufficiently complicated to exhibit some phenomena, such as multiplicity, that do not occur for brackets of lower degree.
The combinatorics of nonlinear controllability and noncommuting flows, Lectures given at
 the Summer School on Mathematical Control Theory
, 2001
"... These notes accompany four lectures, giving an introduction to new developments in, and tools for problems in nonlinear control. Roughly speaking, after the successful development, starting in the 1960s, of methods from linear algebra, complex analysis and functional analysis for solving linear cont ..."
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Cited by 5 (2 self)
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These notes accompany four lectures, giving an introduction to new developments in, and tools for problems in nonlinear control. Roughly speaking, after the successful development, starting in the 1960s, of methods from linear algebra, complex analysis and functional analysis for solving linear control problems, the 1970s and 1980s saw the emergence of differential geometric tools that were to mimic that success for nonlinear systems. In the past 30 years this theory has matured, and now connects with many other branches of mathematics. The focus of these notes is the role of algebraic combinatorics for both illuminating structures and providing computational tools for nonlinear systems. On the control side, we focus on problems connected with controllability, although the combinatorial tools obviously have just as much use for other control problems, including e.g. pathplanning, realization theory, and observability.
Combinatorics of Realizations of Nilpotent Control Systems
 In: Nonlinear Control Systems Design, Selected papers of IFAC symposium (M. Fliess, Ed
, 1993
"... This article gives a very simple algorithm which allows to immediately write down a canonical nonlinear representation of a free nilpotent Lie algebra. Specically, it denes a set of local coordinates and gives a formula for the components of a set of system vector elds in terms of these coordinates. ..."
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Cited by 3 (0 self)
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This article gives a very simple algorithm which allows to immediately write down a canonical nonlinear representation of a free nilpotent Lie algebra. Specically, it denes a set of local coordinates and gives a formula for the components of a set of system vector elds in terms of these coordinates. The components of iterated Lie brackets of the system vector elds can also be read o easily without any further dierentiation. The formulae given here are very close to Sussmann's product expansion of the ChenFliess series and to the chronological calculus introduced by Agrachev and Gamkrelidze.
Optimal Controls for Nilpotent Systems
 Theory of Networks and Systems, Beghi, Finesso, Picci, eds
, 1999
"... This note develops explicit formulas for normal forms of the Hamiltonian lifts of trajectories of free nilpotent control systems. These normal forms utilize notions from chronological algebras and rely on combinatorial results about unique factorizations into nonincreasing sequences of generalized H ..."
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Cited by 2 (1 self)
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This note develops explicit formulas for normal forms of the Hamiltonian lifts of trajectories of free nilpotent control systems. These normal forms utilize notions from chronological algebras and rely on combinatorial results about unique factorizations into nonincreasing sequences of generalized Hall words. The normal forms of the Hamilton Jacobi equations exhibit a natural bigraded structure. They open the door for a systemic study from a perturbations pointofview which may yield apriori estimates of growth rates of the costates which govern the oscillatory behaviour of extremals. 1 Introduction This article brings a different point of view to the classical research program of analyzing the optimal controls for finite dimensional nonlinear control systems, and whose eventual goal is a regular optimal synthesis. Research of the last two decades have made it clear how difficult this problem is for general systems, and we now have much more modest aims, e.g. looking at specific...
A General Method for Motion Planning for QuasiStatic Legged Robotic Locomotion
, 1997
"... We present a general motion planning scheme for a class of "kinematic" legged robots. The method does not depend upon the number of legs, nor is it based on foot placement concepts. Instead, our method is based on an extension of a nonlinear motion planning algorithm for smooth systems to the legged ..."
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Cited by 2 (2 self)
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We present a general motion planning scheme for a class of "kinematic" legged robots. The method does not depend upon the number of legs, nor is it based on foot placement concepts. Instead, our method is based on an extension of a nonlinear motion planning algorithm for smooth systems to the legged case, where the relevant dynamics are not smooth. Our extension is based on the realization that legged robot configuration spaces are stratified. The algorithm is illustrated with a simple example. 1 Introduction The motion planning problem for a legged robot is the problem of determining control inputs (e.g., mechanism joint variable trajectories) which will steer the robot from a starting configuration to a desired final configuration. This paper presents a general motion planning scheme for a class of kinematic legged robots. The method is independent of the number of legs and other aspects of a robot's morphology. One important feature of this method is that it is distict from "tradit...