Lyapunov functions and regions of attraction for spherically constrained relative orbital motion

Abstract

This paper considers the problem of a deputy spacecraft constrained to remain a fixed distance from another spacecraft. The relative dynamics of the deputy spacecraft are derived from a velocity-dependent potential function. A constant of motion is presented that aids in the development of multiple Lyapunov functions and stabilizing control laws. Through both numerical and analytical methods, estimates of the region of attraction are presented for each of the control laws. These regions of attraction serve as domains on which it is conclusively known that the equilibrium points are stabilizable. Lastly, a control law with a spatially maximal region of attraction is presented that can be used to track a time-varying trajectory.

Appendix A

To develop a coordinate system where the singularities are moved to a different set of points, an alternate set of spherical coordinates is introduced. Again, we use the variables $\rho \in {\mathbb {R}}$, $\theta \in {\mathbb {R}}$, and $\phi \in {\mathbb {R}}$ to define the transformation as

$$\begin{aligned} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \rho \begin{bmatrix} c_\phi s_\theta \\ s_\phi \\ c_\phi c_\theta \end{bmatrix} \end{aligned}$$

(112)

The singular points are again at $\phi = \pm \pi /2$, but these now correspond to the points $(0,\pm R,0)$ in Cartesian coordinates. In accordance with the change of variables, the potential energy function U becomes

$$\begin{aligned} U = -2\rho ^2\omega ^2c_{\phi }^2\left( \frac{\dot{\phi }s_{\theta }}{\omega } - c_{\theta }^2 + \frac{3}{4}\right) -\rho \dot{\rho }\omega s_{2\phi } s_{\theta }\nonumber \\ \end{aligned}$$

(113)

The kinetic energy remains

$$\begin{aligned} T = \frac{1}{2}\left( \rho ^2 \left( \dot{\phi }^2 + \dot{\theta }^2c^2_{\phi }\right) + \dot{\rho }^2 \right) \end{aligned}$$

(114)

The transformation between control accelerations is given by the equation

$$\begin{aligned} \begin{bmatrix} u_\rho \\ u_\theta \\ u_\phi \end{bmatrix} = \begin{bmatrix} c_\theta &{} 0 &{} -s_\theta \\ -s_\theta s_\phi &{} c_\phi &{} -c_\theta s_\phi \\ s_\theta c_\phi &{} s_\phi &{} c_\theta c_\phi \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \end{aligned}$$

(115)

Following the procedure outlined in Sect. 3.2, the equations of motion become

$$\begin{aligned} \ddot{\theta }&= 2 \dot{\theta }\dot{\phi } \tau _\phi + 2 \omega ^2 s_{2\theta } + 2 \dot{\phi } \omega c_\theta + \frac{u_\theta }{R c_\phi } \end{aligned}$$

(116a)

$$\begin{aligned} \ddot{\phi }&= \frac{1}{2}(4 \omega ^2 c^2_\theta - \dot{\theta }^2 - 3 \omega ^2) s_{2\phi } - 2 \omega \dot{\theta } c^2_\phi c_\theta + \frac{u_\phi }{R} \end{aligned}$$

(116b)

We now analyze the equilibrium points $(0,0,\pm R)$ corresponding to $(\theta ,\phi ) = (0,0)$ and $(\pi ,0)$. For a generic equilibrium point $(\theta _e,\phi _e)$, translated spherical coordinates

$$\begin{aligned} \vartheta = \theta - \theta _e, \quad \varphi = \phi - \phi _e \end{aligned}$$

(117)

are introduced so that, in any case, the origin is the equilibrium point in $(\vartheta ,\varphi )$ coordinates. The nonlinear differential equations with zero input are then

$$\begin{aligned} \ddot{\vartheta } =&\ 2 \dot{\vartheta } \dot{\varphi } \tau _{(\varphi +\phi _e)} + 2 \omega ^2 s_{(2\vartheta +2\theta _e)} + 2 \dot{\varphi } \omega c_{(\vartheta +\theta _e)} \end{aligned}$$

(118a)

$$\begin{aligned} \ddot{\varphi } =&\ \frac{1}{2} \left( 4 \omega ^2 c^2_{(\vartheta +\theta _e)} - \dot{\vartheta }^2 - 3 \omega ^2 \right) s_{(2\varphi +2\phi _e)} \nonumber \\&\ -2 \omega \dot{\vartheta } c^2_{(\varphi +\phi _e)} c_{(\vartheta +\theta _e)} \end{aligned}$$

(118b)

1.1 Linear control and numerical RoA

Paper [15] suggests asymptotically stabilizing control law for the $(\theta ,\phi ) = (0,0)$ and $(\pi ,0)$ equilibrium points are

$$\begin{aligned} u_\theta&= -(4+k_p) R \omega ^2 \vartheta - 2 k_d R \omega \dot{\vartheta }, \quad k_p,k_d > 0 \end{aligned}$$

(119a)

$$\begin{aligned} u_\phi&= -(1+k_p) R \omega ^2 \varphi - 2 k_d R \omega \dot{\varphi }, \quad k_p,k_d > 0 \end{aligned}$$

(119b)

Following the procedure outlined in Sect. 4.2, the results of the numerical RoA estimation are outlined in Table 2.

Table 2 Numerical RoA parameters

The results outlined in Table 2 are for values of $k_d = k_p = 1$. Varying the magnitudes of $k_d$ and $k_p$ will change the estimated RoA. The points (0, 0) and $(\pi ,0)$ have the same RoA. Figure 11 depicts the RoA on the zero velocity slice of the domain D, i.e., $D|_{\dot{\vartheta } = \dot{\varphi } = 0}$.

1.2 Equilibrium points (0, 0) and $(\pi ,0)$

When $\theta _e \in \{0,\pi \}$ and $\phi _e = 0$, the system dynamics are defined as

$$\begin{aligned} \ddot{\vartheta }&= 2 \dot{\vartheta } \dot{\varphi } \tau _{\varphi } + 2 \omega ^2 s_{2\vartheta } + 2 \dot{\varphi } \omega c_{(\vartheta +\theta _e)} + \frac{u_{\vartheta }}{R c_{\varphi }} \end{aligned}$$

(120a)

$$\begin{aligned} \ddot{\varphi }&= \frac{1}{2} \left( 4 \omega ^2 c^2_{\vartheta } - \dot{\vartheta }^2 - 3 \omega ^2 \right) s_{2\varphi } - 2 \omega \dot{\vartheta } c^2_{\varphi } c_{(\vartheta +\theta _e)} + \frac{u_{\varphi }}{R} \end{aligned}$$

(120b)

Given the nonlinear control laws

$$\begin{aligned} u_\vartheta&= -\left( 3\omega ^2s_{2\vartheta } + k_d\dot{\vartheta }\right) R c_{\varphi },\quad k_d > 0 \end{aligned}$$

(121a)

$$\begin{aligned} u_\varphi&= -\left( \omega ^2 s_{2\varphi }\left( 3 c_{\vartheta }^2 - \frac{3}{4}\right) + k_d \dot{\varphi }\right) R\quad k_d > 0 \end{aligned}$$

(121b)

the closed-loop system becomes

$$\begin{aligned} \ddot{\vartheta }&= 2\dot{\varphi }\dot{\vartheta } \tau _{\varphi } + 2\dot{\varphi }\omega c_{(\vartheta + \theta _e)} -\omega ^2 s_{2\vartheta } - k_d \dot{\vartheta } \end{aligned}$$

(122a)

$$\begin{aligned} \ddot{\varphi }&= -s_{2\varphi }\left( \frac{\dot{\vartheta }^2}{2} + \omega ^2\left( c_{\vartheta }^2 + \frac{3}{4}\right) \right) - 2\dot{\vartheta }\omega c_{\varphi }^2 c_{(\vartheta + \theta _e)} \nonumber \\&\quad \,- k_d \dot{\varphi } \end{aligned}$$

(122b)

Fig. 11

The estimated RoA for the PD control law for the equilibrium points (0, 0) and $(\pi ,0)$

where $k_d > 0$. For this closed-loop system on the domain D, as defined in Sect. 5, we propose the Lyapunov candidate function

$$\begin{aligned} V = \dot{\varphi }^2 + (\dot{\vartheta }^2 - \frac{3}{2}\omega ^2 - 2\omega ^2 c^2_{\vartheta })c^2_{\varphi } + 3.5\omega ^2 \end{aligned}$$

(123)

Proposition 11

Given the dynamics described in Eqs. (122a)–(122b) on the domain D, the function V is a Lyapunov function.

Proof

The proof is similar to that of Proposition 5. $\square$

Note that V is also continuous on the domain ${\bar{D}}$. The domain ${\bar{D}}$ is defined in Sect. 5.

Proposition 12

On the domain $\partial {\bar{D}} \subset {\mathbb {R}}^4$ where $\alpha = \sqrt{3.5}\omega$, the function

$$\begin{aligned} V = \dot{\varphi }^2 + (\dot{\vartheta }^2 - \frac{3}{2}\omega ^2 - 2\omega ^2 c^2_{\vartheta })c^2_{\varphi } + 3.5\omega ^2 \end{aligned}$$

has a minimum value of $\beta = 2\omega ^2$.

Proof

The proof is similar to that of Proposition 6. $\square$

Theorem 8

Given the closed loop system defined in (122a)–(122b), a RoA is the set

$$\begin{aligned} \varOmega = \left\{ (\vartheta ,\varphi ,\dot{\vartheta },\dot{\varphi }) \in D: V \le \gamma \right\} \end{aligned}$$

(124)

where $\gamma \in (0,2\omega ^2)$.

Proof

The proof is similar to that of Theorem 4. As seen in Fig. 12, the ROA of the nonlinear control law encompasses the estimated ROA of the PD control law. $\square$

Fig. 12

A RoA for the PD control law and nonlinear (NL) control law of the equilibrium points (0, 0) and $(\pi ,0)$

1.3 Analytical RoA for equilibrium points (0, 0) and $(\pi ,0)$

When $\theta _e \in \{0,\pi \}$ and $\phi _e = 0$, the system dynamics are defined as

$$\begin{aligned} \ddot{\vartheta }&= 2 \dot{\vartheta } \dot{\varphi } \tau _{\varphi } + 2 \omega ^2 s_{2\vartheta } + 2 \dot{\varphi } \omega c_{(\vartheta +\theta _e)} + \frac{u_{\vartheta }}{R c_{\varphi }} \end{aligned}$$

(125a)

$$\begin{aligned} \ddot{\varphi }&= \frac{1}{2} \left( 4 \omega ^2 c^2_{\vartheta } - \dot{\vartheta }^2 - 3 \omega ^2 \right) s_{2\varphi } - 2 \omega \dot{\vartheta } c^2_{\varphi } c_{(\vartheta +\theta _e)} \nonumber \\&\quad + \frac{u_{\varphi }}{R} \end{aligned}$$

(125b)

Given the nonlinear control laws

$$\begin{aligned} u_\vartheta&= -\frac{R}{4}\left( \omega ^2\left( 8 s_{2\vartheta } + s_{\vartheta }\right) + 2 k_d \dot{\vartheta }\right) c_{\varphi }, \quad k_d > 0 \end{aligned}$$

(126a)

$$\begin{aligned} u_\varphi&= -\frac{R}{4}\left( \omega ^2 \left( 8 c^2_{\vartheta } + c_{\vartheta } - 5 \right) s_{2\varphi } + 2 k_d \dot{\varphi }\right) ,\quad k_d > 0 \end{aligned}$$

(126b)

the closed-loop system becomes

$$\begin{aligned} \ddot{\vartheta }&= -\frac{\omega ^2}{4}s_{\vartheta } + 2 \omega \dot{\varphi }c_{(\vartheta + \theta _e)} + 2\dot{\varphi }\dot{\vartheta }\tau _{\varphi } - \frac{k_d}{2}\dot{\vartheta } \end{aligned}$$

(127a)

$$\begin{aligned} \ddot{\varphi }&= -\frac{1}{2}s_{2\varphi }\left( \dot{\vartheta }^2 + \frac{\omega ^2}{2}\left( c_{\vartheta } + 1\right) \right) - 2\omega \dot{\vartheta }c_{\varphi }^2c_{(\vartheta + \theta _e)}\nonumber \\&\quad - \frac{k_d}{2}\dot{\varphi } \end{aligned}$$

(127b)

For this closed-loop system on the domain $D_\pi$, as defined in Sect. 6, we propose the Lyapunov candidate function

$$\begin{aligned} V = \dot{\varphi }^2 + (\dot{\vartheta }^2 - \omega ^2 c^2_{\vartheta /2}) c^2_{\varphi } + \omega ^2 \end{aligned}$$

(128)

Proposition 13

Given the dynamics described in Eqs. (127a)–(127b) on the domain $D_{\pi }$, the function V is a Lyapunov function.

Proof

The proof is similar to that of Proposition 5. $\square$

Note that V is also continuous on the domain ${\bar{D}}_\pi$. The domain ${\bar{D}}_\pi$ is defined in Sect. 6.

Theorem 9

Given the closed loop system defined in (127a)–(127b), a RoA is the set

$$\begin{aligned} \varOmega = \left\{ (\vartheta ,\varphi ,\dot{\vartheta },\dot{\varphi }) \in D_\pi : V \le \gamma \right\} \end{aligned}$$

(129)

where $\gamma \in (0,\omega ^2)$.

Proof

The proof is similar to that of Theorem 4. $\square$

Corollary 3

As $\gamma \rightarrow \omega ^2$, $\varOmega |_{\dot{\vartheta } = \dot{\varphi } = 0} \rightarrow D_{\pi }|_{\dot{\vartheta } = \dot{\varphi } = 0}$

Proof

See Corollary 2. $\square$

Figure 13 compares the RoA of the three control laws investigated for the equilibrium points (0, 0) and $(\pi ,0)$. As seen in the figure, the RoA for the spatially maximal (SpatMax) control law’s RoA covers the entire sphere at zero velocity except for the equilibrium point $(0,\pi )$ and singular points $(0,\pm \frac{\pi }{2})$.

Fig. 13

Comparison of control law’s RoAs for equilibrium points (0, 0) and $(\pi ,0)$