Abstract
We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Agrachev and A. Sarychev, On abnormal extremals for Lagrange variational problems.J. Math. Syst., Estimation and Control (to appear).
J. Baillieul, Geometric methods for nonlinear optimal control problems.J. Optimiz. Theory and Appl. 25 (1978), No 4, 519–548.
A. Bliss, Lectures on calculus of variations.Univ. Chicago Press, 1946.
A. Bloch and P. Crouch, Nonholonomic and Vakonomic control systems on Riemannian manifolds.Fields Inst. Commun. 1 (1993).
R. Brockett, Control theory and singular Riemannian geometry. In: New Directions in Applied Mathematics.Springer Verlag, New York, 1981, 11–27.
G. Brunnett and P. Crouch, Elastic curves on the sphere.Adv. in Comput. Math. 2 (1994), 23–40.
R. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions.Duke Univ. Preprint. (1993), No 5.
M. Camarinha, F. Silva Leite, and P. Crouch, On the relation between higher-order variational problems on Riemannian manifolds (to appear)
P. Chang, C. Lin, and J. Luh, Formulation and optimization of cubic polynomial joint trajectories for industrial robots.IEEE Trans. Autom. Control AC28 (1983), 1066–1074.
P. Crouch, Y. Yan, F. Silva Leite, and G. Brunnett, On the construction of spline elements on spheres.Proc. Second Europ. Control Conf., Groningen, June 28–July1,4 (1993), 1930–1934.
P. Crouch and J. Jackson, A non-holonomic dynamic interpolation problem.Proc. Conf. on Analysis of Control Dynam. Syst., Lyons, France, 1990,Birkhäuser, ser.Progress in Syst. and Control (1991), 156–166.
—, Dynamic interpolation and application to flight control.J. Guidance, Control and Dynam. 14 (1991), 814–822.
P. Crouch, Dynamic interpolation for linear systems.Proc. 29th IEEE Conf. Decis. and Control, Hawaii, 1990, 2312–2314.
P. Crouch and F. Silva Leite, Geometry and the dynamic interpolation problem.Proc. Am. Control Conf., Boston, 26–29 July, 1991. 1131–1136.
H. Erzberger, J. McLean, and J. Barmon, Fixed-range optimal trajectories for shot haul aircraft.NASA TN. D-8115. Dec., 1975.
H. Erzberger and H. Lee, Constrained optimum trajectories with specified range.J. Guidance, Control and Dynam. 3 (1980), 78–85.
H. Erzberger and J. McLean, Fuel-conservative guidance system for powered lift aircraft.J. Guidance, Control and Dynam. 4 (1981), 253–261.
L. Faibusovich, Explicitly solvable nonlinear optimal control problems.Int. J. Control 48, (1988), No. 6, 2507–2526.
S. Helgason, Differential geometry, Lie groups and symmetric spaces.Academic Press, 1978.
N. Hicks, Notes on differential geometry.Van Nostrand Reinhold Math. Stud. 3 1971, 1978.
J. Jackson, Dynamic interpolation and application to flight control. PhD Thesis.Arizona State Univ., 1990.
V. Judjevic, Non Euclidean elastica. (1993).Am. J. of Math. (to appear).
C. Liu and J. Luh, Approximate joint trajectories for control of industrial robots along cartesian paths.IEEE Trans. Syst., Man and Cybernet. 14 (1984), 444–450.
J. Milnor, Morse theory.Ann. Math. Stud., Princeton Univ. Press 51 1969.
R. Montgomery, On the singularities in the space of curves tangent to a nonintegrable distribution and their relation to characteristics (to appear).
L. Noakes, G. Heinzinger, and B. Paden, Cubic splines on curved spaces.IMA J. Math. Control and Inf. 6, (1989), 465–473.
K. Nomizu, Invariant affine connections on homogeneous spaces.Am. J. Math. 76 (1954), 33–65.
T. Okubo, Differential geometry. In: Monographs and textbooks in pure and applied mathematics.112 Marcel Dekker, Inc., New York, 1987.
W. Poor, Differential geometric structures.McGraw Hill, 1981.
S. Sasaki, On the differential geometry of the tangent bundle on Riemannian manifolds. I.Tohôko Math. J. 10 (1958), 338–354.
W. Shadwick, On the geometry of the Lagrange problem.Preprint, 1989.
H. Sussmann, A cornucopia of abnormal sub-Riemannian minimizers. (to appear).
R. Strichartz, Sub-Riemannian geometry.J. Differ. Geom. 24, (1986), 221–263.
Author information
Authors and Affiliations
Additional information
Work supported in part by NSF grant No. 89-14643 and NATO project CRG 910926.
Work supported by ISR, SCIENCE project ERB-SC1*CT90-0433 and NATO project CRG 910926.
Rights and permissions
About this article
Cite this article
Crouch, P., Leite, F.S. The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces. Journal of Dynamical and Control Systems 1, 177–202 (1995). https://doi.org/10.1007/BF02254638
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02254638