Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

Lee, T.; Leok, Melvin; McClamroch, N. Harris

doi:10.1007/s10883-008-9047-7

Cited by 58 publications

(64 citation statements)

References 16 publications

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“…The contribution of the external control moment and the Carnot energy loss term are denoted by u d k and Q d k . They are defined to approximate the additional term in the variational principle (20) that arises due to a discontinuity:…”

Section: Lie Group Variational Integratormentioning

confidence: 99%

“…These provide a realistic mathematical model for tethered systems and a reliable numerical simulation tool that characterizes the nonlinear coupling between the string dynamics, the rigid body dynamics, and the reel mechanism accurately. The proposed high-fidelity computational framework can be naturally extended to formulating and solving control problems associated with string deployment, retrieval, and vibration suppression as in [20]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

2010

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A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid body motion, elastic wave dynamics in the string, and the disturbances introduced by the reeling mechanism. Such interactions are dynamically important in many engineering problems, but tend be obscured in lower fidelity models.

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Section: Lie Group Variational Integratormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

2010

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“…Then geometric PD controllers on SO(3) and SE(3) were designed and applied on a quadrotor [2,[11][12][13]. Moreover, Lee et al provided the computational optimal geometric method on SO(3) [14,15] and employed the Pontryagin maximum principle to attain solutions of an open loop time-optimal problem. Spindler provided the differential equations which the optimal controls must satisfy via Pontryagin maximum principle.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic Optimal Control for Mechanical Systems on Lie Group

Liu

Tang

Guo

2017

Advances in Mathematical Physics

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The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the HamiltonJacobi-Bellman equation. For a special case on SO(3), the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.

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“…However, not so many numerical methods are available for solving optimal control problems for dynamical systems whose configuration space is a differentiable manifold. Noticeable and interesting exceptions are the discrete-time methods presented in [12], [13], [14], [15] for mechanical systems on Lie groups. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Constrained motion planning for multiple vehicles on SE(3)

Saccon

Aguiar

Häusler

et al. 2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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Abstract-This paper proposes a computational method to solve constrained cooperative motion planning problems for multiple vehicles undergoing translational and rotational motions. The problem is solved by means of the Lie group projection operator approach, a recently developed optimization strategy for solving continuous-time optimal control problems on Lie groups. State constraints (for collision avoidance) are handled by means of a set of barrier functions, turning the optimization approach into an interior point method. A sample computation is shown to demonstrate the effectiveness of the method.

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Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

Cited by 58 publications

References 16 publications

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

Intrinsic Optimal Control for Mechanical Systems on Lie Group

Constrained motion planning for multiple vehicles on SE(3)

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