A unified energy-based control framework for tethered spacecraft deployment

Abstract

This paper proposed a unified energy-based control framework for fast, stable, and precision deployment of underactuated TSS. The tension controller with partial state feedback is derived from an artificial potential energy function and a dissipative function, where the control objectives and requirements are transformed into the necessary and sufficient conditions for these functions. The feedback law can be either linear or nonlinear, depending on the construction of the artificial potential energy function and the dissipative function. The controllability of the underactuated TSS is proved which is the original contribution of this work. The energy-based tension control is proved asymptotically stable by the Lyapunov technique and LaSalle’s invariance principle. Furthermore, the constraints on positive tension and nonnegative tether deploy velocity are incorporated into the energy-based tension controller by control gain optimization using optimal control. Four controllers are developed based on the proposed control framework to demonstrate the effectiveness and robustness of the proposed energy-based framework using numerical simulation.

Appendix: Stability analysis of two equilibrium states

A1. The horizontal equilibrium states $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 ) =(c_1^h ,0,0,1,0)$ and $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 )=(c_1^h ,0,0, -1,0)$ are unstable.

Proof

Assume there is any small perturbation of $\varepsilon \ne 0$ on $y_3$. From Eq. (26), there exist $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 )=(c_1^h ,0,\varepsilon ,\sqrt{1-\varepsilon ^{2}},0)$ and $V_\varepsilon =-\frac{3}{2}(c_1^h )^{2}\varepsilon ^{2}+U_a (c_1^h )<U_a (c_1^h )$. Since V is non-increasing as shown in Eq. (27), its trajectory will move away from the equilibrium state $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 )=(c_1^h ,0,0,1,0)$. Thus, this equilibrium state is unstable. Similar perturbation analysis can be preceded with the other equilibrium $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 )=(c_1^h ,0,0,-1,0)$ with the same conclusion. $\square$

A2: The vertical equilibrium states $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 ) =(c_1^v ,0,1,0,0)$ and $(y_1 ,y_2 ,y_3 ,y_4 ,y_5 )=(c_1^v ,0,-1,0,0)$ are stable.

Proof

Note that ${{\mathcal {U}}}$ represents the union of two vertical equilibriums set ${{\mathcal {U}}}_+ =\{y | y=(c_1^v ,0,1,0,0)\}$ and $\mathcal{U}_- =\{y | y=(c_1^v ,0,-1,0,0)\}$, respectively. It follows that from Eq. (27) the Lyapunov function $V=-\frac{3}{2}(c_1^v )^{2}y_3^2 +U_a (c_1^v )$ obviously has the minimums only and only if the $y_3^2 =1$. Thus, the states in the neighborhood of ${{\mathcal {U}}}_+$ will converge to ${{\mathcal {U}}}_+$ . This implies the states in ${{\mathcal {U}}}$ are locally asymptotically stable. Similarly, the states in the neighborhood of ${{\mathcal {U}}}_-$ will converge to ${{\mathcal {U}}}_-$. $\square$