Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles

Brennan McCann; Morad Nazari

doi:10.3934/jgm.2022002

Article Contents

2022, Volume 14, Issue 1: 29-55. Doi: 10.3934/jgm.2022002

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Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles

Brennan McCann^, and
Morad Nazari

Aerospace Engineering, Embry-Riddle Aeronautical University, 1 Aerospace Boulevard, Daytona Beach, FL 32114, USA

^* Corresponding author: Brennan McCann
^* Corresponding author: Brennan McCann

Received: April 30, 2021

Revised: August 31, 2021

Published: March 2022

The first author is supported by FIRST and GAANN

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Abstract

Presented herein are a class of methodologies for conducting constrained motion analysis of rigid bodies within the Udwadia-Kalaba (U-K) formulation. The U-K formulation, primarily devised for systems of particles, is advanced to rigid body dynamics in the geometric mechanics framework and a novel development of U-K formulation for use on nonlinear manifolds, namely the special Euclidean group $ {\mathsf{SE}(3)}$ and its second order tangent bundle ${\mathsf{T}^2\mathsf{SE}(3)} $, is proposed in addition to the formulation development on Euclidean spaces. Then, a Morse-Lyapunov based tracking controller using backstepping is applied to capture disturbed initial conditions that the U-K formulation cannot account for. This theoretical development is then applied to fully-constrained and underconstrained scenarios of rigid-body spacecraft motion in a lunar orbit, and the translational and rotational motions of the spacecraft and the control inputs obtained using the proposed methodologies to achieve and maintain those constrained motions are studied.

Keywords:

Constrained motion,
geometric mechanic Udwadia-Kalaba formalism,
Morse-Lyapunov function,
spacecraft pose control,
rigid body tracking problem.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. Inertial $ \mathcal N $, perifocal $ \mathcal P $, and body $ \mathcal B $ reference frames

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Figure 2. Fully-constrained translational motion comparison between formulation on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$

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Figure 3. Fully-constrained rotational motion comparison between formulation on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$

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Figure 4. Fully-constrained control input comparison between formulations on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$

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Figure 5. Translational motion in underconstrained (UC) case versus that in the fully-constrained (FC) case

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Figure 6. Rotational motion in underconstrained (UC) case versus that in the fully-constrained (FC) case

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Figure 7. Control inputs in underconstrained (UC) case versus those in the fully-constrained (FC) case

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Figure 8. Underconstrained translational motion comparison between formulation on $ \mathbb{R}^{6} $ and $ {\mathsf{T}^2\mathsf{SE}(3)}$

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Figure 9. Underconstrained rotational motion comparison between formulation on $ \mathbb{R}^{6} $ and ${\mathsf{T}^2\mathsf{SE}(3)} $

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Figure 10. Underconstrained control inputs comparison between formulation on $ \mathbb{R}^{6} $ and ${\mathsf{T}^2\mathsf{SE}(3)} $

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Figure 11. Position response using U-K and M-L control with disturbed ICs

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Figure 12. Velocity response using U-K and M-L control with disturbed ICs

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Figure 13. Attitude response using U-K and M-L control with disturbed ICs

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Figure 14. Angular velocity response using U-K and M-L control with disturbed ICs

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Figure 15. Total control input using U-K and M-L control with disturbed ICs

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