A Discrete Geometric Optimal Control Framework for Systems with Symmetries

Kobilarov, Marin; Desbrun, Mathieu; Marsden, Jerrold E.; Sukhatme, Gaurav S.

doi:10.15607/rss.2007.iii.021

Cited by 19 publications

(23 citation statements)

References 32 publications

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“…with given boundary conditions g 0 and g N , where h ∈ R >0 is the time step, τ : g → G is the retraction map (see, e.g., [10,11,22,23,24]), and the functions C d and V d satisfy assumptions (i)-(v). We can solve (P2) using a discrete analogue of the Lagrange-Pontryagin variational principle.…”

Section: Discrete Lagrange-pontryagin Principlementioning

confidence: 99%

Optimal Control Problems with Symmetry Breaking Cost Functions

Bloch¹,

Colombo²,

Gupta³

et al. 2017

SIAM J. Appl. Algebra Geometry

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Abstract. We investigate symmetry reduction of optimal control problems for left-invariant control affine systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuoustime formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler-Poincaré equations from a variational principle. By using a Legendre transformation, we recover the Lie-Poisson equations obtained by Borum and Bretl [IEEE Trans. Automat. Control, 62 (2017), pp. 3209-3224] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie-Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.

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Section: Discrete Lagrange-pontryagin Principlementioning

confidence: 99%

Optimal Control Problems with Symmetry Breaking Cost Functions

Bloch¹,

Colombo²,

Gupta³

et al. 2017

SIAM J. Appl. Algebra Geometry

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“…So far, DMOC has been successfully applied to problems in space mission design (see e.g. Dellnitz et al 2006a;Junge et al 2006) and robotics (see for instance Kanso and Marsden 2005;Leyendecker et al 2007;Kobilarov et al 2007). …”

Section: Discretizationmentioning

confidence: 99%

A multi-objective approach to the design of low thrust space trajectories using optimal control

Dellnitz

Ober‐Blöbaum

Post

et al. 2009

Celest Mech Dyn Astr

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In this article, we introduce a novel three-step approach for solving optimal control problems in space mission design. We demonstrate its potential by the example task of sending a group of spacecraft to a specific Earth L 2 halo orbit. In each of the three steps we make use of recently developed optimization methods and the result of one step serves as input data for the subsequent one. Firstly, we perform a global and multi-objective optimization on a restricted class of control functions. The solutions of this problem are (Pareto-)optimal with respect to V and flight time. Based on the solution set, a compromise trajectory can be chosen suited to the mission goals. In the second step, this selected trajectory serves as initial guess for a direct local optimization. We construct a trajectory using a more flexible control law and, hence, the obtained solutions are improved with respect to control effort. Finally, we consider the improved result as a reference trajectory for a formation flight task and compute trajectories for several spacecraft such that these arrive at the halo orbit in a prescribed relative configuration. The strong points of our three-step approach are that the 123 34 M. Dellnitz et al.challenging design of good initial guesses is handled numerically by the global optimization tool and afterwards, the last two steps only have to be performed for one reference trajectory.

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“…It is quite surprising to note that there has been almost no investigation into a maximum principle for discrete-time optimal control problems defined on manifolds despite the fact that smooth manifolds arise quite naturally in many practical control problems such as robotics (see, for example, [21,22] and references therein) and spacecraft attitude control (see, for example, [15,18,31] and references therein).…”

Section: Introductionmentioning

confidence: 99%

The Discrete-Time Geometric Maximum Principle

Kipka¹,

Gupta²

2019

SIAM J. Control Optim.

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We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. These results are organized around a new theorem on critical and approximate critical points for discrete-time geometric control systems. We show that this theorem can be used to derive Lie group variational integrators in Hamiltonian form; to establish a maximum principle for control problems in the absence of state constraints; and to provide sufficient conditions for exact penalization techniques in the presence of state or mixed constraints. Exact penalization techniques are used to study sensitivity of the optimal value function to constraint perturbations and to prove necessary optimality conditions, including in the form of a maximum principle, for discrete-time geometric control problems with state or mixed constraints.

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A Discrete Geometric Optimal Control Framework for Systems with Symmetries

Cited by 19 publications

References 32 publications

Optimal Control Problems with Symmetry Breaking Cost Functions

Optimal Control Problems with Symmetry Breaking Cost Functions

A multi-objective approach to the design of low thrust space trajectories using optimal control

The Discrete-Time Geometric Maximum Principle

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