H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Di#erential Geometric Control Theory, pages 1--116. Birkhauser, Boston, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that all the planners described in the previous section are probabilistically complete. First we describe the concept of local controllability (in the literature also referred to as smalltime local controllability or local local controllability) adopting the terminology introduced by Sussman [Sus83] Given a robot A, let Sigma A be its control system. That is, Sigma A describes the velocities that A can attain in configuration space. For a configuration c of a robot A, the set of configurations that A can reach within time T is denoted by A Sigma A ( T; c) A is defined to be locally ....

H.J. Sussmann. Lie brackets, real analyticity and geometric control. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, 1983.

....Y M (x) the set of points which can be reached from x in time less than with a control u 2 UM . We say that the control system is small time locally controllable (STLC) at x if Y M (x) contains a neighborhood of x for every 0. The following result is standard in control theory, see e.g. [23] or [19] for a proof. Proposition 2.6 Consider the control system Eq. 76) with u 2 UM . Let x 0 be an equilibrium point of Y (x) i.e. Y (x 0 ) 0. If the linear span of the brackets ad k (Y ) X i ) x) i = 1; m ; k = 0; 1; 2; 78) has rank n at x 0 then Eq. 76) is STLC at ....

....(47) is. We note OE u ( OE u ) the solution of Eq. 47) Eq. 83) and Y M T (x) Y M T (x) the set of reachable points for the control system (47) 83) Using the convexity of the set of values the control can assume and the continuous dependence of OE u on u, it is easy to see ([23], Prop. 2.3.1) that Y M T (x) is a compact set. We choose now M and T such that M 2 T h. Since Y M T (x) and Y M T (x) are compact, there is ae 0 0 such that Y M T (x) Y M T (x) ae B(2ae 0 =3) 84) Furthermore we may choose ae 0 arbitrarily small by choosing M and or T ....

[Article contains additional citation context not shown here]

H. J. Sussmann, Lie brackets, real analyticity and geometric control. In Differential Geometric Control Theory, Proc. conf. Michigan, 1--116 Birkauser, (1983).

....bySchrodinger equation with the control multiplying the state variable. This is a right invariant bilinear system whose state varies on a compact Lie group. Controllability of systems on Lie groups was dealt with in the classical paper [12] and in a number of following papers. The survey article [18] and the book [11]give an up to date account of the main results and we refer to them for further references on this topic. It is a problem of great fundamental and practical importance to characterize the set of states that can be obtained in arbitrary small time. This set is in general not the ....

....in these cases, if one proves that not every state can be reached in arbitrary time then it immediately follows A = e B . There havebeenmany studies concerning the property of Small Time Local Controllability for agiven point in the state space of a nonlinear system. Many results (see e.g. [18]) deal with the case where the point is an equilibrium point. The following simple criterion, based on the Maximum Principle [1] 18] will be applied for the systems of interest here. The proof is a generalization of the oneusedin[4] for the case of two level quantum systems. Theorem 4.5. Assume ....

[Article contains additional citation context not shown here]

H. J. Sussmann. Lie brackets, real analyticity and geometric control, in Differential Geometric Control Theory, R.W. Brockett, R. S. Millman and H. J. Sussmann eds., pp. 1-116, Birkhauser, Boston, 1983.

....) the set of points which can be reached from x in time less than # with a control u # UM . We say that the control system is small time locally controllable (STLC) at x if Y M ##( contains a neighborhood of x for every # 0. The following result is standard in control theory, see e.g. [21,17] for a proof. Proposition 2. Consider the control system Eq. 27) with u # UM . Let x 0 be a critical point ofY( x) i.e. Y( x 0 ) 0. If the linear span of the brackets ad k( i) i = 1, m, k= 0, 1, 2, has rank n at x 0 then Eq. 27) is STLC at x 0 . 12 L. Rey Bellet, ....

Sussmann, H.J.: Lie brackets, real analyticity and geometric control. In: Differential Geometric Control Theory, Proc. Conf. Michigan, Basel--Boston: Birkuser, 1983, pp. 1--116

....controllability: for practical applications, the set of trim trajectories will be much richer. Notice that the uncontrollable car in [40] which can only turn left with different turning radii, is in fact controllable according to our definition, even though it is not small time controllable [41]. 3 Motion planning The hybrid control architecture lends itself to computationally efficient solutions of many problems of interest for practical applications. Most often the results that are achievable using nonlinear control theory involve stability or tracking performance. Ensuring the ....

H. Sussmann. Lie brackets, real analyticity and geometric control. In R. Brockett, R. Millman, and H.Sussmann, editors, Differential Geometric Control Theory, volume 27 of Progress in Mathematics, pages 1--116. Birkhauser, 1982.

....at x if the set of points reachable from x by an admissible curve (i.e. tangent at each point to the distribution Sigma) contains a neighborhood of x. This is equivalent in our case (i.e. analytic or algebraic vector fields) to the condition L r(x) x) R n (theorem of Chow, see [Ch] or [Sus]) 2.1 Remark Let L k be the set of combinations of iterated brackets of length k of combinations of V 1 ; V s with coefficients in the ring of polynomials R[X 1 ; Xn ] Let L k (x) be the subspace of T x M which consists of the values V (x) taken, at the point x, by the ....

....= x; y) we have dimL 1 (q) 1 if x = 0, dimL 1 (q) 2 if x 6= 0, so all points on the line x = 0 are singular and the others are regular. For other examples, arising in the context of mobile robots with trailers, and for more details, see [La] 3 We state without proofs (see for instance [Sus]) 2.4 Proposition Fix x 2 M . i) For each k, there exists a neighbourhood U of x such that dimL k (y) dimL k (x) for all y 2 U . ii) Suppose the system is controllable at x. There exists a neighbourhood V of x such that the system is controllable at y, and r(y) r(x) for all y 2 V . ....

Sussmann H.J., Lie brackets, Real Analyticity and Geometric control, in "Differential Geometric Control Theory", Brockett, Millman, Sussmann Eds, Birkauser (1982), 1-116. 14

....of a multibody manipulator [63] 1.2. Terminology. The central issue of this paper is showing whether the 1JOC an underactuated system is rich enough to serve as a parts feeder. To characterize the system precisely, we borrow definitions of reachable sets from nonlinear control theory [61]. A part s configuration q = x, y,#) T is controllable from q if, starting from q, the part can reach every configuration in the configuration space. The part is controllable to q if q is reachable from every configuration. The part is accessible from q if the set of configurations reachable ....

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, Boston, MA, 1983.

....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 ....

....called normal [119] Without further assumptions, we can not say whether the optimal solution exists and whether it is unique. However, Pontryagin et al. 113] showed that if the optimal solution exists it must satisfy the necessary conditions known as the Pontryagin s minimum principle: 1 See [139] for a more general setting of the problem. 2 The choice is usually between so called strong and weak topologies [119] Theorem 2.2 (Pontryagin s minimum principle) Define the Hamiltonian: H(x; u; t) L(x; u; t) T f(x; u; t) 2:4) If a control u (t) is optimal and it generates a ....

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. Millman, R. W. Brockett, and H. J. Sussmann, editors, Differential Geometric Control Theory, pages 1--116. Birkhauser, Boston, 1983.

....that all the planners described in the previous section are probabilistically complete. First we describe the concept of local controllability (in the literature also referred to as smalltime local controllability or local local controllability) adopting the terminology introduced by Sussman [Sus83] Given a robot A, let Sigma A be its control system. That is, Sigma A describes the velocities that A can attain in configuration space. For a configuration c of a robot A, the set of configurations that A can reach within time T is denoted by A Sigma A ( T; c) A is defined to be locally ....

H.J. Sussmann. Lie brackets, real analyticity and geometric control. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, 1983.

....is whether the 1JOC is rich enough to manipulate parts, or whether the two constraints on the motion of the fence (pivot is fixed) overly constrain the set of reachable configurations of the part. To be precise, we borrow definitions of reachable sets from nonlinear control theory (Sussmann [173]) A part s configuration q = x; y; OE) T is controllable from q if, starting from q, the part can reach every configuration in the configuration space. The part is controllable to q if q is reachable from every configuration. The part is accessible from q if the set of configurations ....

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, 1983.

....of the completeness claim. 9. 1 Local controllability and the local topological property First we describe the concept local controllability (in the literature also referred to as smalltime local controllability or local local controllability) adopting the terminology introduced by Sussman [Sus83] Given a robot A, let Sigma A be its control system. That is, Sigma A describes the velocities that A can attain in C space. For a configuration c of a robot A, the set of configurations that A can reach within time T is denoted by A Sigma A ( T ; c) A is defined to be locally controllable ....

H.J. Sussmann. Lie brackets, real analyticity and geometric control. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, 1983.

....(Trinkle et al. 63] 1.2 Terminology The central issue of this paper is showing whether the 1JOC an underactuated system is rich enough to serve as a parts feeder. To characterize the system precisely, we borrow definitions of reachable sets from nonlinear control theory (Sussmann [61]) A part s configuration q = x;y;f) T is controllable from q if, starting from q, the part can reach every configuration in the configuration space. The part is controllable to q if q is reachable from every configuration. The part is accessible from q if the set of configurations reachable ....

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, 1983.

....g i s inherit these properties. The outline of this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43] In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5] we discuss the use of sinusoidal inputs for steering systems of first order, i.e. systems where controllability is achieved after just one ....

H. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Differential Geometric Control Theory, pages 1--116. Birkhauser, 1983.

....is whether the 1JOC is rich enough to manipulate parts, or whether the two constraints on the motion of the fence (pivot is fixed) overly constrain the set of reachable configurations of the part. To be precise, we borrow definitions of reachable sets from nonlinear control theory (Sussmann [32]) A part s configuration q = x; y; f) T is controllable from q if, starting from q, the part can reach every configuration in the configuration space. The part is controllable to q if q is reachable from every configuration. The part is accessible from q if the set of configurations reachable ....

H.J. Sussmann. Lie brackets, real analyticity and geometric control. In Differential Geometric Control Theory, Brockett, Millman, and Sussman, eds. Birkhauser, 1983.

....THE CONTINUATION METHOD 7 For X 2 V (M ) we use HX to denote the Hamiltonian function, or momentum function corresponding to X, i.e. the function HX : T M IR given by HX (x; z) hz; X(x)i; for x 2 M; z 2 T x M: 3. 2) For X 2 V (M ) we let TX, the variational vector field of X (cf. [21]) be the element of V (TM ) which is the infinitesimal generator of the 1 parameter pseudogroup of maps f Phi (1) t g t2IR , where f Phi t g t2IR is the 1 parameter pseudogroup of maps Phi t : M M generated by the vector field X, and Phi (1) t (x; y) Phi t (x) D Phi t (x) y) for (x; ....

.... fi 1 rF 1 fi 2 rF 2 fi 3 rF 3 (4.26) with fi 1 ; fi 2 ; fi 3 2 C k 2 b (T Sigmad Omega Gamma and ffi 1 ; ffi 2 ; ffi 3 2 C k 2 b (T Sigmad Omega Gamma 2 . Then belongs to A k 2 (T Sigmad Omega ; i) 5. The critical estimate for the strongly bracket generating case. In [21], the MPP was solved by using the following theorem (see also [3] for another proof) Theorem 5.1. Let Sigma = M; f ) be a smooth driftless control affine system with dynamical equation x = m X i=1 u i f i (x) whose state space M is a smooth connected n dimensional manifold. Assume that ....

H.J. Sussmann, Lie brackets, real analyticity and geometric control, in Differential Geometric Control Theory, R.W. Brockett, R.S. Millman and H.J. Sussmann Eds., Birkhauser, Boston, 1983, pp. 1-115.

No context found.

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Di#erential Geometric Control Theory, pages 1--116. Birkhauser, Boston, 1983.

No context found.

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Differential Geometric Control Theory. Birkhauser, Boston, MA, 1983.

No context found.

H. J. Sussmann. Lie brackets, real analyticity and geometric control. In R. W. Brockett, R. S. Millman, and H. J. Sussmann, editors, Di#erential Geometric Control Theory, pages 1--116. Birkhauser, Boston, 1983.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC