Optimal control and integrability on Lie groups

Conference on Decision and Control, 2006

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E 3 , the spheres S 3 and the hyperboloids H 3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

2008

This paper considers left-invariant control systems defined on the Lie groups SU (2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is twofold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

2007

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a threedimensional space. A Lie group formulation arises naturally and the vehicles are modelled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically the three-dimensional space forms Euclidean space E 3 , the sphere S 3 and the hyperboloid H 3. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). The Maximum Principle of optimal control, shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It is then shown that the projections of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Proceedings of the 44th IEEE Conference on Decision and Control, 2005

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E 3 , the spheres S 3 and the hyperboloids H 3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Most of the configuration space of mechanical and non-mechanical problems in physical world involve matrix Lie groups. Especially these Lie groups provide a mathematically rich formulation for studying a variety of motion control problems involving rotating and translating bodies. While control systems have been specialized to different matrix Lie groups, study of optimal control of these systems by minimizing the input cost function has been a tempting research area for physicists and mathematicians. The smallest exceptional Lie group G_2 and its Lie algebra g_2 act locally as the symmetries when a ball performs rolling motion on a larger ball restricted by the radius of the smaller ball being one third of the bigger one. Here we take a left-invariant control system specified on G_2 and study its optimal control as well as stability of the resulting dynamics.

IEEE Transactions on Automatic Control, 2000

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a threedimensional space. A Lie group formulation arises naturally and the vehicles are modelled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically the three-dimensional space forms Euclidean space E 3 , the sphere S 3 and the hyperboloid H 3 . The corresponding frame bundles are equal to the Euclidean group of motions SE , the rotation group SO(4) and the Lorentz . The Maximum Principle of optimal control, shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It is then shown that the projections of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

waset.org

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.

This paper analyzes the Riemannian cubic polynomials's problem from a Hamiltonian point of view. The description of the problem on compact Lie groups is particulary explored. The state space of the second order optimal control problem considered is the tangent bundle of the Lie group which also has a group structure. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the tangent bundle. Using these control geometrical tools, the equivalence between the Hamiltonian approach developed here and the known variational one is verified. Moreover, the equivalence allows us to deduce two invariants along the cubic polynomials which are in involution.

This thesis tackles the Motion Planning Problem (MPP) for nonholonomic systems defined on matrix Lie groups. The methodology used to tackle this problem is based on optimal control theory, where the cost function to be minimized is the control energy of the system. An application of the Maximum Principle to this optimal control problem leads naturally to the Hamiltonian formalism and to the language of symplectic geometry. The solutions to the corresponding Hamiltonian vector fields, called extremals, yield necessary conditions for optimality.

We consider the equivalence of cost-extended control systems corresponding to certain invariant optimal control problems. We prove that two equivalent cost-extended systems have the same optimal controlled trajectories and the same extremal curves (up to a diffeomorphism). We also prove that equivalent cost-extended systems are lifted (via the Pontryagin Maximum Principle) to linearly equivalent Hamilton-Poisson systems. A few illustrative examples are discussed.