Optimal Control on the Rotation Group SO (3)

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 minimizes the integral of the Lorentz inner product 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.

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.

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.

SIAM Journal on Control and Optimization, 2005

Consider the control system (Σ) given byẋ = x(f + ug), where x ∈ SO(3), |u| ≤ 1 and f, g ∈ so(3) define two perpendicular left-invariant vector fields normalized so that f = cos(α) and g = sin(α), α ∈]0, π/4[. In this paper, we provide an upper bound and a lower bound for N (α), the maximum number of switchings for time-optimal trajectories of (Σ). More precisely, we show that N S (α) ≤ N (α) ≤ N S (α) + 4, where N S (α) is a suitable integer function of α such that N S (α) ∼ α → 0 π/(4α). The result is obtained by studying the time optimal synthesis of a projected control problem on RP 2 , where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere S 2. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

2021

The problem of attitude tracking using rotation matrices is addressed using an approach which combines inverse optimality and ℒ_2 disturbance attenuation. Conditions are provided which solve the inverse optimal nonlinear H_∞ control problem by minimizing a meaningful cost function. The approach guarantees that the energy gain from an exogenous disturbance to a specified error signal respects a given upper bound. For numerical simulations, a simple problem setup from literature is considered and results demonstrate competitive performance.

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.

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.

2011

This paper discusses key implementation details required for computing the solution of a continuous-time optimal control problem on a Lie group using the projection operator approach. In particular, we provide the explicit formulas to compute the time-varying linear quadratic problem which defines the search direction step of the algorithm. We also show that the projection operator approach on Lie groups generates a sequence of adjoint state trajectories that converges, as a local minimum is approached, to the adjoint state trajectory of the first order necessary conditions of the Pontryagin's Maximum Principle, placing it between direct and indirect optimization methods. As illustrative example, an optimization problem on SO(3) is introduced and numerical results of the projection operator approach are presented, highlighting second order converge rate of the method.

IEEE Transactions on Automatic Control, 2013

Many nonlinear systems of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. Examples range from aircraft and underwater vehicles to quantum mechanical systems. In this paper, we develop an algorithm for solving continuous time optimal control problems for systems evolving on (noncompact) Lie groups. This algorithm generalizes the projection operator approach for trajectory optimization originally developed for systems on vector spaces. Notions for generalizing system theoretic tools such as Riccati equations and linear and quadratic system approximations are developed. In this development, the covariant derivative of a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. An example optimal control problem on SO(3) is provided to highlight implementation details and to demonstrate the effectiveness of the method.