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

  • * Corresponding author: Brennan McCann

    * Corresponding author: Brennan McCann 

The first author is supported by FIRST and GAANN

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  • 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.

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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