R. W. Brockett. Control theory and singular riemannian geometry. In New Directions in Applied Mathematics, pages 11--27. Springer, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....We will later demonstrate a cost functional derived from the L # norm of the actuator forces that results in continuous velocity profile and hence reduce errors due to dead reckoning. There are many approaches to the generation of open loop trajectories for nonholonomic systems. Brockett [6] derived the optimal control inputs for canonical systems in which the tangent space to the configuration space manifold is spanned by the input vector fields and their Lie brackets. Murray and Sastry [7] studied a more general class of nonholonomic systems that can be converted to the so called ....

R. W. Brockett, "Control theory and singular Riemannian geometry," in New Directions in Applied Mathematics, pp. 11--27, New York: Springer-Verlag, 1981.

....controllers. 1 Introduction Stabilization of nonholonomic mobile robots has been a subject of intense research in the past years [1, 2, 3] The implications of nonholonomic constraints on the kind of admissible control inputs for this class of systems has made the problem particularly challenging [4]. Many approaches have been proposed to address the issue of nonholonomic stabilization and can be broadly characterized as open loop strategies [5, 6] time varying feedback designs [7, 8, 9] and discontinuous static feedback methods [10, 11, 12] Although nonholonomic systems arise mainly from ....

....lower part of (2) q = Gu (6) u = v (7) where v is now the new control input. If (3) was smooth, then by backstepping it through the integrators (7) one could stabilize the complete dynamic system (2) However, the kinematic control law (3) is discontinuous to conform with Brockett s condition [4]. This motivated the extension of the technique of integrator backstepping to handle the case where the original control law that stabilizes the subsystem is non smooth. The result obtained can be used in general for systems that are described by discontinuous ordinary di#erential equations: ....

R. Brockett, Control Theory and Singular Riemannian Geometry, New Directions in Applied Mathematics. Springer, 1981.

....path for such a holonomic system as this series of integrators that describe the manipulator, may not be so difficult. The nonholonomic nature of the mobile base, however, poses severe restrictions on the kind of potential motion planning schemes: no smooth, time invariant feedback law would do [8]. Therefore, the adoption of feedback leaves only two options: either nonsmooth [9] or time varying strategies [10] Both of these have been explored in the literature and significant results have been presented. The authors feel that a nonsmooth, time invariant scheme would suit better, since ....

R. Brockett, Control Theory and Singular Riemannian Geometry, New Directions in Applied Mathematics. Springer, 1981.

....means that it is a redundant mechanism, with all the inherent capabilities and problems of such systems. However, as it is always the case, there is a price to pay for advantages: more difficulty in control. This difficulty is due mainly to a class of motion constraints called nonholonomic [20] [1], 14] 6] 23] These constraints are equations involving the generalized coordinates and their derivatives in a way that makes them non integrable. Thus, the dimension of the configuration space can not be reduced. Under the effect of such constraints the system maintains its controllability ....

R. Brockett, Control theory and singular riemannian geometry, New Directions in Applied Mathematics, Springer, 1981.

....its two main components, namely, the mobile base and the manipulator mounted on top. Due to the kinematic constraints imposed on the wheels of the base, the mobile manipulator is a nonholonomic system. As a concequence, no continuous static feedback motion planning scheme can be applied (Brockett, 1981); it has to be either open loop, time varying (Murray and Sastry, 1990) or discontinuous (de Wit and Sordalen, 1991) Motion planning for nonholonomic systems has traditionally been divided into two stages: path planing and trajectory planning. This decomposition into two separate concequent ....

Brockett, R. (1981). Control Theory and Singular Riemannian Geometry. New Directions in Applied Mathematics. Springer.

....reference, approximating homogeneous expansions are defined and their control theoretical application studied. Regarding the stabilzation problem for nonlinear driftless control systems, the use of time varying (or discontinuous) feedbacks is proven necessary by the famous Brockett s theorem [6]. On the other hand smooth time periodic stabilizers with garanteed asymptotic stability show rather slow convergence rates, so that the application of homogeneous feedbacks becomes appropriate. A comprehensive description of this succesfull approach is described in [25] We here concer ourselves ....

....2 ) and the approximating vector fields are d dt 2 4 1 2 3 3 5 = 2 4 1 2 1 2 ( 1 2 Gamma 2 1 ) 3 5 = 2 4 1 0 1 2 2 3 5 1 2 4 0 1 Gamma 1 2 1 3 5 2 : 6. 14) Once again, we are therefore dealing with the standard Brockett s nonholonomic system [6]. 40 BULLO AND MURRAY Regarding the standard SE(3) system, by applying the same transformation of the previous subsection (that is going from, p to z = R T p) we (once again) have R = Rb z = v Gamma Theta z As instructive example, we assume to be dealing with an underactuated ....

R. W. Brockett. Control theory and singular Riemannian geometry. In Peter Hilton and Gail Young, editors, New Directions in Applied Mathematics, 1981.

....nilpotent Lie group (i.e. its Lie algebra g satisfies [g, g,g] 0) and D is a left invariant distribution such that D Phi[g,g] g (Theorem 4.1) Note that the latter condition is satisfied for graded nilpotent Lie algebras. This extends known results on the so called Gaveau Brockett problem [1], 3] Our methods rely on deriving the equations for the normal and abnormal curves purely in terms of the structure constants. These equations appear in Section 2. We also found the methods in [7] useful in our investigations. We are very grateful to Richard Montgomery for many useful ....

....the above assumption, any minimizer through 0 is normal in some subgroup of G (for the induced Carnot metric) and hence any minimizer is smooth. Remark In the case where G is a free nilpotent Lie group of 2 step, one can prove that any minimizer is in fact normal, see Gaveau [3] and Brockett [1]. Proof: We need to show that any abnormal minimizer through 0 is normal in some subgroup of G. We will proceed by induction on the dimension of G. The main step of the induction is given by the following lemma, whose proof we postpone. 4.2 Lemma: Any abnormal curve through 0 (if it exists) is ....

R.W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics (P.J. Hilton and G.S. Young, eds.), Springer-Verlag, 1981.

....many different perspectives [Latombe, 1991] the focus of this paper is the combination of geometric methods and optimal control methods to solve the motion planning problem. Early work by Bailleuil [Baillieul, 1978] studied optimal controls for nonlinear systems in a geometric setting. Brockett [Brockett, 1982, Brockett and Dai, 1993] derived the optimal controls for a special class of nonholonomic systems and showed that the optimal input patterns can be described by sinusoidal and elliptical functions. Murray and Sastry [Murray and Sastry, 1993] 1 later proposed the use of sinusoids as sub optimal ....

Brockett, R. W. (1982). Control theory and singular Riemannian geometry. In Hilton, P. J. and Young, G. S., editors, New Directions in Applied Mathematics, pages 11--27. Springer-Verlag, New York.

....of open loop control laws is convenient because their design is relatively simple and the obtained trajectories are predictable. Among the proposed techniques, we cite the use of: Sinusoidal inputs (Murray and Sastry, 1993) inspired by Brockett s work on optimal control for a class of systems (Brockett, 1981). Piecewise constant inputs, as in multirate digital control (La#erriere and Sussmann, 1991; Monaco and Normand Cyrot, 1992) Mixed piecewise constant and polynomial inputs (Tilbury, Murray, and Sastry, 1993) in order to obtain a smoother behavior. Below, we recall the latter method, that ....

Brockett, R. W. 1981. Control theory and singular Riemannian geometry. New directions in applied mathematics, eds. P. Hinton and G. Young. Springer-Verlag, pp. 11--27.

....different viewpoints: probabilistic road maps by Kavraki et al. 67] differential geometric methods by Sussmann [118] and behavior based planning by Manikonda [88] are some of these examples. There are many approaches to the generation of open loop trajectories for nonholonomic systems. Brockett [17] derived the optimal control inputs for canonical systems in which the tangent space to the configuration space manifold is spanned by the input vector fields and their Lie brackets. Murray and Sastry [93] studied a more general class of nonholonomic systems that can be converted to the so called ....

R. W. Brockett. Control theory and singular Riemannian geometry. In New Directions in Applied Mathematics, pp. 11--27, Springer-Verlag, New York, 1981.

....schemes can be found in [84] and [138] At the end of this review we mention a class of continuous motion planning methods of a rather different character that are an alternative to variational methods. These techniques, usually known as steering, grew from research in nonlinear control [18, 139] and can also be viewed as constructive proofs of controllability. They have been very successfully applied to nonholonomic systems [77] Murray and Sastry [93] showed that a large class of nonholonomic systems can be steered to a desired configuration using sinusoids. A more general theory was ....

R. W. Brockett. Control theory and singular Riemannian geometry. In P. Hilton and G. Young, editors, New directions in Applied Mathematics, pages 11--27. SpringerVerlag, New York, 1981.

.... Caratheodory metrics have attracted much attention. It appears to be that both geometric phases in physics [16] and nonholonomic motion planning in robotics [3] 10] can be treated from the unified point of view of sub Riemannian geometry. On the other hand, the classical isoperimetric problems [1] and their generalizations [2] 6] not only occupy a very important place in the subRiemannian geometry but also, under certain conditions, admit complete mathematical characterization of all their extrema. At the same time, calculation of sub Riemannian geodesics arisen in the generalized ....

....of all their extrema. At the same time, calculation of sub Riemannian geodesics arisen in the generalized isoperimetric problems is a challengeable problem for the modern geometrical control theory, since even for simple examples the calculation of length minimizers is not a trivial exercise [1], 2] 6] The state of the art in the field of sub Riemannian geometry until 1985 is outlined in the paper [17] More recent information about this subject can be found in [9] 22] Many interesting and important results on the sub Riemannian geodesics are already proved [12] 13] 18] 22] ....

R.Brockett, Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics, P.Hilton and G.Young, ed., Springer-Verlag, New Holland, 1990, pp.11-27.

....planning, i.e. motion planning subject to nonintegrable constraints on the velocities. Li and Canny [1] proposed a path finding algorithm for an embedded manifold rolling on a flat plane, using the case where the embedded manifold is a unit ball for illustration. Based on the results of Brockett [2], Murray and Sastry [3, 4, 5] developed a nonholonomic motion planning algorithm using sinusoids at integrally related frequencies. Another approach to nonholonomic motion planning has been the use of a highly oscillatory control, 6] These are among several important contributions to ....

R. W. Brockett, "Control theory and singular riemannian geometry", in New Directions in Applied Mathematics, P. J. Hilton and G. S. Young, Eds., pp. 11--27. Springer-Verlag, 1982.

....motion planning, i.e. motion planning subject to nonintegrable constraints on the velocities. Li Canny (1990) proposed a path finding algorithm for an embedded manifold rolling on a flat plane, using the case where the embedded manifold is a unit ball for illustration. Motivated by results of Brockett (1982), Murray Sastry (1990, 1991, 1993) developed a nonholonomic motion planning algorithm using sinusoids at integrally related frequencies. Another approach to nonholonomic motion planning has been the use of a highly oscillatory control, Gurvits Li 1993) These are among several important ....

Brockett, R. W. (1982), Control theory and singular riemannian geometry, in P. J. Hilton & G. S.

....1. Introduction. The standard Dido s problem can be formulated as follows: find the optimal form of a lot of land of the maximum area for a given perimeter. Clearly, its solution is a circle [1] The Dido s problem can be interpreted as distance measuring in some sub Riemannian geometry [2]. This interpretation leads to fascinating generalizations of Dido s problem. Pioneer results in this direction are presented in [2] where the solution of certain multidimensional extension of Dido s problem was established. The works [3] 5] have the first formulation of Dido s problem with a ....

....maximum area for a given perimeter. Clearly, its solution is a circle [1] The Dido s problem can be interpreted as distance measuring in some sub Riemannian geometry [2] This interpretation leads to fascinating generalizations of Dido s problem. Pioneer results in this direction are presented in [2], where the solution of certain multidimensional extension of Dido s problem was established. The works [3] 5] have the first formulation of Dido s problem with a fixed center of mass: find the minimal length curve through the origin which envelopes the given area with a fixed center of mass. ....

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R. Brockett, Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics, ed. R. Brockett, New York, 1981, pp. 11-27.

.... in the field of sub Riemannian geometry find their numerous applications in the theory of geometric phases [17] 22] and in nonholonomic motion planning [8] 13] On the other hand, analysis of sub Riemannian minimizers is a challengeable problem for the modern geometrical control theory [1], 2] 10] The state of the art in the field of sub Riemannian geometry until 1985 is outlined in the paper [23] More recent information about this subject can be found in [7] 12] 17] 25] Although many interesting and important results on sub Riemannian geodesics are already obtained ....

....for the following hamiltonian system. d d x( p H(x ( p ( d d p( Gamma 1 2 x H(x ( p ( where H(x; p) 1 2 fi fi B T (x)p fi fi 2 : Recall the following standard notations: ffl C k [0; 1] the set of k Gamma times continuously differentiable on [0,1] functions. ffl k Delta k k denotes the uniform norm on C k [0; 1] i.e. kuk k = max t2[0;1] 0ik j ( d dt ) i u(t) j for u(t) 2 C k [0; 1] d dt ) 0 u(t) is another notation for u(t) ffl C 1 [0; 1] denotes k C k [0; 1] where the intersection is taken over all ....

R.Brockett, Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics, P.Hilton and G.Young, ed., Springer-Verlag, New Holland, 1990, pp.11-27.

....theory may be extremely useful. These tools can aid us in answering basic questions about the optimality of biological gait patterns that have developed through evolution. More practically, they have already seen use in choosing appropriate control inputs to guide a desired motion (for example, Brockett [1981]) As we proceed, we will describe some of the ways in which an understanding of the geometric nature of the problem can provide insights into why cyclic control motions are so tightly coupled with locomotion generation. 2 Connections and Bundles As we have indicated, one of the fruitful ideas ....

Brockett, R.W. [1981] Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics , P.J. Hilton and G.S. Young (eds.), Springer.

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R. W. Brockett. Control theory and singular riemannian geometry. In New Directions in Applied Mathematics, pages 11--27. Springer, 1981.

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R. W. Brockett. Control theory and singular riemannian geometry. In P. J. Hilton and G. S Young, editors, New Directions in Applied Mathematics, pages 11--27. Springer-Verlag, 1982.

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R. Brockett, Control Theory and Singular Riemannian Geometry,New Directions in Applied Mathematics. Springer, 1981.

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R. W. Brockett. Control theory and singular riemannian geometry. In New Directions in Appl. Math., pages 11--27. Springer, 1981.

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R. W. Brockett, Control theory and singular Riemannian geometry, In New Directions in Applied Mathematics, pp. 13--27. Springer-Verlag, 1982. 33

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R.W. Brockett, Control theory and singular Riemannian geometry , in: New directions in applied mathematics, P.J. Hilton and G.S. Young (eds.), Springer-Verlag, New York, 1982.

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R.W. Brockett. Control theory and singular Riemannian geometry. In New Directions in Applied Mathematics, pages 11--27. Springer-Verlag, 1981.

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R. W. Brockett [1981], Control theory and singular Riemannian geometry, in New Directions in Applied Mathematics, P. Hilton and G. Young, eds., Springer-Verlag, New York, pp. 1127.

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