# American Institute of Mathematical Sciences

March  2022, 14(1): 29-55. doi: 10.3934/jgm.2022002

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

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

* Corresponding author: Brennan McCann

Received  May 2021 Revised  September 2021 Published  March 2022 Early access  January 2022

Fund Project: The first author is supported by FIRST and GAANN

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.

Citation: Brennan McCann, Morad Nazari. Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles. Journal of Geometric Mechanics, 2022, 14 (1) : 29-55. doi: 10.3934/jgm.2022002
##### References:

show all references

##### References:
Inertial $\mathcal N$, perifocal $\mathcal P$, and body $\mathcal B$ reference frames
Fully-constrained translational motion comparison between formulation on $\mathbb{R}^{6}$ and ${\mathsf{T}^2\mathsf{SE}(3)}$
Fully-constrained rotational motion comparison between formulation on $\mathbb{R}^{6}$ and ${\mathsf{T}^2\mathsf{SE}(3)}$
Fully-constrained control input comparison between formulations on $\mathbb{R}^{6}$ and ${\mathsf{T}^2\mathsf{SE}(3)}$
Translational motion in underconstrained (UC) case versus that in the fully-constrained (FC) case
Rotational motion in underconstrained (UC) case versus that in the fully-constrained (FC) case
Control inputs in underconstrained (UC) case versus those in the fully-constrained (FC) case
Underconstrained translational motion comparison between formulation on $\mathbb{R}^{6}$ and ${\mathsf{T}^2\mathsf{SE}(3)}$
Underconstrained rotational motion comparison between formulation on $\mathbb{R}^{6}$ and ${\mathsf{T}^2\mathsf{SE}(3)}$
Underconstrained control inputs comparison between formulation on $\mathbb{R}^{6}$ and ${\mathsf{T}^2\mathsf{SE}(3)}$
Position response using U-K and M-L control with disturbed ICs
Velocity response using U-K and M-L control with disturbed ICs
Attitude response using U-K and M-L control with disturbed ICs
Angular velocity response using U-K and M-L control with disturbed ICs
Total control input using U-K and M-L control with disturbed ICs
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