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.
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Inertial
Fully-constrained translational motion comparison between formulation on
Fully-constrained rotational motion comparison between formulation on
Fully-constrained control input comparison between formulations on
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
Underconstrained rotational motion comparison between formulation on
Underconstrained control inputs comparison between formulation on
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