Geodesics and an Optimal Control Algorithm C. Yal cn Kaya
J. Lyle Noakes
School of Mathematics University of South Australia The Levels, S.A. 5095 Australia
[email protected]
Abstract Recently, a global algorithm for nding a geodesic joining given two points on a Riemannian manifold has been developed. It is simple to implement, and works well in practice, with no need for an initial guess! An optimal control problem is much more general than nding a geodesic. Nevertheless, a generalization seems now possible, based on recent numerical experiments.
1 Introduction 1.1 The optimal control problem Consider the general control system _ ( ) = ( ( ) ( ))
x t
f x t u t
where the state vector x 2 Rn , the control vector u 2 Rm, and time t 2 R+. The vector eld f in Rn is smooth. We are concerned with the following class of optimal control problems.
8> Z 1 < min ( u 0 P : > subject : such thatto
) _= ( ) (0) = 0 (1) = f Suppose that an optimal control can explicitly be found from the necessary conditions of optimality 10] in other words, one can nd a feedback law o = ( ), where : Rn Rn ;! Rm is continuous in and 2 Rn . This results in the following equations, which all together are called here the control TPBVP. L x u
dt
x
f x u
x
x
x
x
u
k
k x x
8 _= ( ( >< Pc : > _ = ; ( : with (0) = x
)) (
f x k x @f
@x
x
)) (1) = f
x k x
0
x
x
x
where (0) and (1) represent the point boundary conditions. Note that solving problem (Pc ) is in general very dicult. In the numerical solution of problem (Pc ), two main approaches are used. These are namely the sequential quadratic programming (SQP), and the direct solution of TPBVPs incorporating some x
x
Department of Mathematics University of Western Australia Nedlands, W.A. 6907 Australia
[email protected]
combination of multiple shooting, collocation methods, and continuation (or homotopy) methods. More details about these methods can be found in 4, 11, 2, 3]. Our major focus is TPBVPs.
1.2 Approximations and Geodesics
Lane and Riesenfeld 6] presented two fast subdivision algorithms (which we call here the L&R algorithms) for the evaluation of B-spline and Bernstein curves and surfaces. Noakes gave in 7] a Riemannian generalization of a special case of the L&R algorithms, called the nonlinear corner-cutting algorithm, where the line segments in the L&R algorithm are replaced by geodesic segments, and Euclidean space by a Riemannian manifold. Recall that A geodesic is a parameterized curve on a Riemannian manifold whose velocity vector is constant (or parallel) with respect to the Riemannian structure 1]. A geodesic between points (0) and (1) solves the following TPBVP. 8< _ i = i P Pg : : _ i = ; njk=1 ;ijk ( ) j k with (0) = (0) (1) = (1) for = 1 , where the nonlinear functions ;ijk of are the Christoel symbols, which are not so easy to nd in general (unless (0) and (1) are nearby). y
x
x
x
i
y
y
x
y
:::n
x
y
y
2 The Leap-Frog Algorithm The works by Noakes 7, 8] motivated the leap-frog algorithm for nding geodesics (as a solution to Problem (Pg )). If the initial and nal points (0) and (1) are not nearby, then simple shooting method does not work well, in which case usually multiple shooting is used. The leap-frog algorithm resembles multiple shooting in that the interval 0 1] is subdivided and geodesics are found separately over each subinterval. The most important dierences between the leap-frog algorithm and multiple shooting are given as: (i) each step of the leapfrog algorithm updates only real variables at a time, (ii) the curves of the leap-frog algorithm satisfy both boundary conditions at every step of the iteration, and (iii) The leap-frog algorithm always converges, without the need for an initial guess. y
y
n
p. 1
z2(2)
x
z1(2) z2(1)
z(1) 1
ω1 z0
z(0) 1
z3(1)
z(0) 2
z4
z(0) 3
x
n
z(2) 3
ω2
The initial partition points were taken to be on a straight line between (0) and (1). First, the number of partitions p was taken to be 2 (which in this case is equivalent to simple shooting), but it was seen to fail. This is not unexpected, given the nonlinearity of the system. Application of the algorithm succeeded when p was set to 4. In the computer program written, the values of p was taken to be powers of 2, for the ease of bookkeeping. The cost at the rst iteration was 8211, which converged to the solution cost, 1475, at the 15th iteration. At each iteration, p = 2 (the simple shooting) was tried rst, but this eventually only worked in the last iteration. n
n
Figure 1: The leap-frog algorithm
n
An illustration of the algorithm for four partitions and two iterations is given in Figure 1. For full mathematical details, the reader should refer to 9]. In Figure 1, 0 = (0), 4 = (1), i(0) are the initial partition points, and i(k) are the new partition points at the -th iteration. The ( +1)st iteration is described as follows. Given a set of partition points, i(k) , rst a geodesic between 0 and 2(k) is found, and 1(k) is moved to the mid-point of the geodesic and labelled as 1(k+1) . Now a geodesic between 1(k+1) and 3(k) is found and 2(k) is moved to the mid-point of the geodesic and labelled as (k+1) 2 . This procedure is repeated once more between (k+1) and so as to get (k+1) . 4 2 3 z
y
z
y
z
z
k
k
z
z
z
z
z
z
z
z
z z
z
z
3 An Optimal Control Application At present, there is no available theory that suggests the leap-frog algorithm would work for the control TPBVP given by (Pg ), let alone for a general TPBVP. Nevertheless numerical experiments give optimism that the leap-frog algorithm would work for the problem (Pg ) as well. The following simulation exemplies this.
8 1Z 1 >> min 2 >> u 2 0 < subject to P:> >> such that >: u
dt
_1 = 2 _ 2 = 3 + (1 ; 21 ) 1 _ 3 = 2 sin 2 + (0) = ;1 0 0]T (1) = 1 0 0]T
x
x
x
x
x
x
x
x
x
x
x
u
:
For this problem an optimal control input can explicitly be solved as o = ; 3 . Then, problem (Pc ) for this example becomes u
8 _1 = 2 >> >> _ 2 = 3 + (1 ; 21) 1 < _ 3 = 2 sin 2 ; 3 Pc : > _ 1 = ;(1 ; 3 21 ) 2 >> _ 2 = ;( 1 + 3 (sin 2 + 2 cos 2)) >: _ 3 = ; 2 with (0) = ;1 0 0]T (1) = 1 x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
0 0]T
References
1] W.M. Boothby, An Introduction to Dierentiable Manifolds and Riemannian Geometry. Academic Press, London,1986. 2] Y. Chen, & A.A. Desrochers, \Minimum-time control laws for robotic manipulators," International Journal of Control, 57(1), pp. 1-27, 1993. 3] Y. Chen, J. Huang, & J.T.Y. Wen, \A continuation method for time-optimal control synthesis for robotic point-to-point motion," Proceedings of the 32nd Conference on Decision and Control, San Antonio, Texas, U.S.A., December 1993, pp. 1628-1633. 4] P.E. Gill, W. Murray, & M.A. Saunders, \Largescale SQP methods and their application in trajectory optimization," In: Computational Optimal Control, Eds: Bulirsch, R., & Kraft, D., Birkh auser Verlag, Basel, pp. 29-42, 1994. 5] H.B. Keller, Numerical Methods for Two-Point Boundary-Value Problems. Blaisdell Publishing Co, 1968. 6] J.M. Lane, & R.F. Riesenfeld (1980). A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-2(1), pp. 35-46, 1980. 7] J.L. Noakes, \Nonlinear Corner-Cutting," Advances in Computational Mathematics, (to appear in). 8] J.L. Noakes, \Riemannian quadratics," Proceedings of Chamonix, 1996. (to appear in). 9] J.L. Noakes, \A global algorithm for geodesics," preprint. 10] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, & E.F. Mishenko, the Mathematical Theory of Optimal Processes. Interscience, New York, 1962. 11] O. von Stryk, & M. Schlemmer, \Optimal control of the industrial robot Manutec r3," In: Computational Optimal Control, Eds: Bulirsch, R., & Kraft, D., Birkh auser Verlag, Basel, pp. 367-382, 1994.
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