Robust adaptive terminal sliding mode control on SE(3) for autonomous spacecraft rendezvous and docking

Abstract

This paper addresses the relative position and attitude tracking control in the framework of geometric mechanics for autonomous rendezvous and docking of two spacecraft where the relative motion of the leader and follower spacecraft tracks a desired time-varying trajectory. Using exponential coordinates on the Lie group $\mathrm {SE(3)}$, which is the set of positions and orientations in three-dimensional Euclidean space, and the adjoint operator on the Lie algebra $\mathfrak {se}(3)$, the relative coupled translational and rotational dynamics is modeled. Based on the terminal sliding mode, a robust adaptive terminal sliding mode control scheme on $\mathrm {SE(3)}$ is proposed to ensure the finite-time convergence of the relative motion tracking errors using limited control inputs despite the presence of unknown disturbances and moment of inertia uncertainty. The control scheme is then applied to a situation where the follower spacecraft synchronizes its attitude motion with the leader, while maintaining a constant relative position with respect to the leader. The robustness of the controller is established using Lyapunov stability theory. Simulation results of close range rendezvous and docking verify that the proposed control scheme can achieve faster and more accurate tracking performance while consuming less control energy than the conventional terminal sliding mode control method.

Appendix

The adjoint operator on the Lie algebra $\mathfrak {se}(3)$ can be obtained as a derivative of the adjoint action of $\mathrm {SE(3)}$ on $\mathfrak {se}(3)$ via the exponential map below (Eq. 75). Let ${\varvec{g}} = \exp (\epsilon {\varvec{Y}})$ where ${\varvec{Y}} \in \mathfrak {se}(3)$ and $\epsilon \in \mathbb {R}$ and let ${\varvec{X}} \in \mathbb {R}^6$. From (12);

$$\begin{aligned} \mathrm {Ad}_{{\varvec{g}}} {\varvec{X}}= \left( \mathrm {expm}(\epsilon {\varvec{Y}}) {\varvec{X}} \exp (-\epsilon {\varvec{Y}})\right) = {\varvec{g}} {\varvec{X}} {\varvec{g}}^{-1}. \end{aligned}$$

(75)

The derivative of this adjoint evaluated at $\epsilon =0$ is defined to be the adjoint operation on the Lie algebra, and is given by

$$\begin{aligned}&\mathrm {ad}_{{\varvec{Y}}} {\varvec{X}}: = \frac{d}{d\epsilon } (\mathrm {expm}(\epsilon {\varvec{Y}}) {\varvec{X}} \mathrm {expm}(-\epsilon {\varvec{Y}}))|_{\epsilon =0}\nonumber \\&\quad = {\varvec{Y}}{\varvec{X}} - {\varvec{X}}{\varvec{Y}}. \end{aligned}$$

(76)

If ${\varvec{X}}={\varvec{\eta }}^\vee \in \mathfrak {se}(3)$ for some ${\varvec{\eta }}\in \mathbb {R}^6$, and if

$$\begin{aligned} {\varvec{Y}} = {\varvec{\zeta }}^\vee = \begin{bmatrix}{\varvec{{\varTheta }}}^\times&{\varvec{\beta }}\\ 0_{1\times 3}&0 \end{bmatrix}\in \mathfrak {se}(3), \end{aligned}$$

(77)

then this operator can be expressed by the matrix

$$\begin{aligned} \mathrm {ad}_\zeta = \begin{bmatrix}{\varvec{{\varTheta }}}^\times&0_{3\times 3}\\ {\varvec{\beta }}^\times&{\varvec{{\varTheta }}}^\times \end{bmatrix}, \end{aligned}$$

(78)

such that

$$\begin{aligned} \mathrm {ad}_{{\varvec{\zeta }}}{\varvec{\eta }} = \Big ({\varvec{Y}} {\varvec{X}} - {\varvec{X}} {\varvec{Y}}\Big )^| . \end{aligned}$$

(79)

The time derivative of the adjoint action $\mathrm {Ad}_{h^{-1}}{\varvec{\xi }}^0$ in (32) is derived from the adjoint operation on the Lie algebra. Taking the time derivative of $\mathrm {Ad}_{h^{-1}}{\varvec{\xi }}^0$

(80)

Since $\frac{\text{ d }}{{\text{ d }}t} {\varvec{h}}^{-1} = -{\varvec{h}}^{-1}\dot{{\varvec{h}}} {\varvec{h}}^{-1} = -{\tilde{{\varvec{\xi }}}}^\vee {\varvec{h}}^{-1}$,

(81)

Thus, Eq. (81) is arranged by using (79) and (12) as

$$\begin{aligned} \frac{\text{ d }}{{\text{ d }}t}{\mathrm {Ad}_{{\varvec{h}}^{-1}}{\varvec{\xi }}^0} =&-\mathrm {ad}_{{\tilde{{\varvec{\xi }}}}} \mathrm {Ad}_{ {\varvec{h}}^{-1}}{\varvec{\xi }}^0 + \mathrm {Ad}_{{\varvec{h}}^{-1}}\dot{{\varvec{\xi }}}^0 \nonumber \\ =&-\mathrm {ad}_{{\varvec{\xi }}} \mathrm {Ad}_{{\varvec{h}}^{-1}}{\varvec{\xi }}^0 + \mathrm {Ad}_{{\varvec{h}}^{-1}}\dot{{\varvec{\xi }}}^0. \end{aligned}$$

(82)