Finite-Time Tracking Control of Rigid Spacecraft Under Actuator Saturations and Faults

Abstract

In this chapter, an adaptive fast terminal sliding mode control law (AFTSMCL) is presented to resolve attitude tracking control problem for rigid spacecraft, which can provide finite-time convergence, strong robustness, and fault-tolerant control. A rigorous proof is achieved first. Simulation results are presented to illustrate the effectiveness of presented control law.

Appendix

Proof of Theorem 8.1 Consider the Lyapunov candidate function:

$$\begin{aligned} V_{ss}=\frac{1}{2}[S^T(t)J S(t)+\sum _{i=1}^3\frac{1}{p_i\varrho }\bar{c}_i^2]\end{aligned}$$

(8.34)

where $\bar{c}_i=c_i-\varrho \hat{c}_i$. $\varrho :=\gamma \cdot \zeta$ in which $\gamma :=\min \{\delta _i\}, i=1,2,3$. $\zeta$ is designed in (8.52). Its time derivative is’

$$\begin{aligned}&\dot{V}_{ss}=S^T(t)J\dot{S}(t)-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i\nonumber \\&\quad =S^T(t)[-(\varOmega _e+C\varOmega _d)^\times J\varOmega +J(\varOmega _e^\times C\varOmega _d-C\dot{\varOmega }_d)\nonumber \\&\qquad \, +\frac{1}{2}J(K_2+K_1F_e)(e_4I_3+e^\times _v)\varOmega _e]+S^T(t)DF(u_{ss})\nonumber \\&\qquad \,+S^T(t)d-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i\nonumber \\&\quad =S^T(t)[-(S-K_2e_v-K_1S_{au}+C\varOmega _d)^\times J\varOmega \nonumber \\&\qquad \, +J((\varOmega -C\varOmega _d)^\times C\varOmega _d-C\dot{\varOmega }_d)\nonumber \\&\qquad \, +\frac{1}{2}J(K_2+K_1F_e)(e_4I_3+e^\times _v)(\varOmega -C\varOmega _d)]\nonumber \\&\qquad \, +S^T(t)DF(u_{ss})+S^T(t)d-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i\nonumber \end{aligned}$$

Noting the property of operator $^\times$ which implies $S^T S^{\times }={0}^T$ and $(C\varOmega _d)^\times C\varOmega _d={0}$ obtains

$$\begin{aligned}&\dot{V}_{ss}=S^T(t)[(K_2e_v+K_1S_{au}-C\varOmega _d)^\times J\varOmega +J(\varOmega ^\times C\varOmega _d\nonumber \\&\qquad \, -C\dot{\varOmega }_d)+\frac{1}{2}J(K_2+K_1F_e)(e_4I_3+e^\times _v)(\varOmega -C\varOmega _d)] \nonumber \\&\qquad \,+S^T(t)DF(u_{ss})+S^T(t)d -\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i\nonumber \\&\quad \le \Vert S\Vert \Vert G(t)\Vert +S^T(t)DF(u_{ss})-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i\nonumber \end{aligned}$$

Using Assumption 8.3 yields

$$\begin{aligned} \dot{V}_{ss}\le&\Vert S(t)\Vert [c_{1}+c_{2}\Vert \varOmega (t)\Vert +c_{3}\Vert \varOmega (t)\Vert ^2]\nonumber \\&\,\, +S^T(t)DF(u_{ss})-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i \end{aligned}$$

(8.35)

Case 1: If $S_i(t)>0$, then $F(u_{ssi})=-u_M (1-e^{u_{ssi}})$, according to Lagrange mean value theorem, it yields that

$$\begin{aligned} F(u_{ssi})=u_M\cdot e^{\nu _{1i}}\cdot u_{ssi},~~~~~~ \nu _{1}\in (u_{ssi}, 0) \end{aligned}$$

(8.36)

Taking (8.36) into (8.35) gives

$$\begin{aligned} \dot{V}_{ss}\le&\Vert S(t)\Vert [c_{1}+c_{2}\Vert \varOmega (t)\Vert +c_{3}\Vert \varOmega (t)\Vert ^2]\nonumber \\&+S^T(t)\cdot D \cdot u_M\cdot e^{\nu _{1}}\cdot u_{ss}-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i \end{aligned}$$

(8.37)

where $e^{\nu _{1}}={\text {diag}}\{e^{\nu _{11}}, e^{\nu _{12}}, e^{\nu _{13}}\}$.

Case 2: If $S_i(t)<0$, then $F(u_{ssi})=u_M (1-e^{-u_{ssi}})$, based on Lagrange mean value theorem again, it gets that

$$\begin{aligned} F(u_{ssi})=u_M\cdot e^{-\nu _{2i}}\cdot u_{ssi}, ~~~~~~\nu _{2}\in (0, u_{ssi}) \end{aligned}$$

(8.38)

Synthesizing inequalities (8.35) and (8.38) yields

$$\begin{aligned} \dot{V}_{ss}\le&\Vert S(t)\Vert [c_{1}+c_{2}\Vert \varOmega (t)\Vert +c_{3}\Vert \varOmega (t)\Vert ^2]\nonumber \\&+S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{2}}\cdot u_{ss}-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i \end{aligned}$$

(8.39)

where $e^{\nu _{2}}={\text {diag}}\{e^{-\nu _{21}}, e^{-\nu _{22}}, e^{-\nu _{23}}\}$.

Case 3: Consider other 6 mixed cases, that is

$$\begin{aligned} (1)S_1>0, S_2>0, S_3<0; \end{aligned}$$

(8.40)

$$\begin{aligned} (2) S_1>0, S_2<0, S_3>0; \end{aligned}$$

(8.41)

$$\begin{aligned} (3) S_1>0, S_2<0, S_3<0; \end{aligned}$$

(8.42)

$$\begin{aligned} (4) S_1<0, S_2>0, S_3>0; \end{aligned}$$

(8.43)

$$\begin{aligned} (5) S_1<0, S_2>0, S_3<0; \end{aligned}$$

(8.44)

$$\begin{aligned} (6) S_1<0, S_2<0, S_3>0. \end{aligned}$$

(8.45)

Denote $SU(t):=S^T(t)DF(u_{ss})$. Based on (8.36) and (8.38), it follows from every mixed case, (8.40)–(8.45), respectively, that

$$\begin{aligned} (1)SU(t)=S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{3}}\cdot u_{ss} \end{aligned}$$

(8.46)

where $e^{\nu _{3}}={\text {diag}}\{e^{\nu _{11}}, e^{\nu _{12}}, e^{-\nu _{23}}\}$.

$$\begin{aligned} (2)SU(t)=S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{4}}\cdot u_{ss} \end{aligned}$$

(8.47)

where $e^{\nu _{4}}={\text {diag}}\{e^{\nu _{11}}, e^{-\nu _{22}}, e^{\nu _{13}}\}$.

$$\begin{aligned} (3)SU(t)=S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{5}}\cdot u_{ss} \end{aligned}$$

(8.48)

where $e^{\nu _{5}}={\text {diag}}\{e^{\nu _{11}}, e^{-\nu _{22}}, e^{-\nu _{23}}\}$.

$$\begin{aligned} (4)SU(t)=S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{6}}\cdot u_{ss} \end{aligned}$$

(8.49)

where $e^{\nu _{6}}={\text {diag}}\{e^{-\nu _{21}}, e^{\nu _{12}}, e^{\nu _{13}}\}$.

$$\begin{aligned} (5)SU(t)=S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{7}}\cdot u_{ss} \end{aligned}$$

(8.50)

where $e^{\nu _{7}}={\text {diag}}\{e^{-\nu _{21}}, e^{\nu _{12}}, e^{-\nu _{23}}\}$.

$$\begin{aligned} (5)SU(t)=S^T(t)\cdot D\cdot u_M \cdot e^{\nu _{8}}\cdot u_{ss} \end{aligned}$$

(8.51)

where $e^{\nu _{8}}={\text {diag}}\{e^{-\nu _{21}}, e^{-\nu _{22}}, e^{\nu _{13}}\}$.

Based on Case 1–Case 3, it follows from inequalities (8.37), (8.39) and equalities (8.46)–(8.51) that

$$\begin{aligned} \dot{V}_{ss} \le&\Vert S(t)\Vert [c_{1}+c_{2}\Vert \varOmega (t)\Vert +c_{3}\Vert \varOmega (t)\Vert ^2]\nonumber \\&+S^T(t)\cdot \varrho \cdot u_{ss}-\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i \end{aligned}$$

(8.52)

where $\varrho :=\gamma \cdot \zeta$, $\zeta :=u_M\cdot c\in (0,u_M)$ with $c:=\min (e^{\nu _{1i}}, e^{-\nu _{2i}}), i=1,2,3$, $\nu _{1i}\in (u_{ssi}, 0) (u_{ssi}<0), \nu _{2i}\in (0, u_{ssi}) (u_{ssi}>0), i=1,2,3$. Based on Assumption 8.1, it is shown that $c\in (0,1)$, then $\zeta \in (0,u_M)$. The proposed control law in equations (8.21) and (8.22) are readily shown to satisfy (8.9), and therefore

$$\begin{aligned} S^T(t)\cdot u_{ss}\le 0 \end{aligned}$$

(8.53)

In addition, $u_M$, $e^{\nu _{1i}}(i=1,2,3)$ and $e^{-\nu _{2i}}(i=1,2,3)$ are positive constants. Then, $S^T(t)\cdot u_M \cdot e^{\nu _{l}}\cdot u_{ss}\le 0, l=1,2,3\ldots 8$. It follows from the definition of $\varrho$ that it is the minimum value of $\delta _iu_Me^{\nu _{1i}}(i=1,2,3)$ and $\delta _iu_Me^{-\nu _{2i}}(i=1,2,3)$. Hence, $S^T(t)\cdot \varrho \cdot u_{ss}$ is the maximum value of SU(t). What has been discussed earlier guarantees that inequality (8.52) is satisfied and reasonable. There are two cases for the following analyses.

Case A. For the case of $\Vert S(t)\Vert \hat{\sigma }(t)>\epsilon$. Substituting the AFTSMCL (8.21) and the ACL (8.23)–(8.25) into inequality (8.52) obtains

$$\begin{aligned} \dot{V}_{ss} \le&-\varrho S^T(t)\tau S(t)-\varrho S^T(t)\rho {\text {sgn}}^{\frac{1}{2}}(S(t))\nonumber \\&+\Vert S(t)\Vert [c_{1}+c_{2}\Vert \varOmega (t)\Vert +c_{3}\Vert \varOmega (t)\Vert ^2]\nonumber \\&-\varrho \hat{c}_1(t)\Vert S(t)\Vert -\varrho \hat{c}_2(t)\Vert S(t)\Vert \Vert \varOmega (t)\Vert \nonumber \\&-\varrho \hat{c}_3(t)\Vert S(t)\Vert \Vert \varOmega (t)\Vert ^2- c_1\Vert S(t)\Vert \nonumber \\&- c_2\Vert S(t)\Vert \Vert \varOmega (t)\Vert - c_3\Vert S(t)\Vert \Vert \varOmega (t)\Vert ^2\nonumber \\&+\varrho \Vert S(t)\Vert [ \hat{c}_1(t) +\hat{c}_2(t)\Vert \varOmega (t)\Vert +\hat{c}_3(t)\Vert \varOmega (t)\Vert ^2]\nonumber \\&+\sum _{i=1}^3\frac{o_i}{p_i}(c_i-\varrho \hat{c}_i)\hat{c}_i-\varrho \sum _{i=1}^3\frac{o_i}{4p_i} \hat{c}_i\nonumber \\ \le&-\varrho \sum _{i=1}^{3}(\tau _{i}S^2_i+\rho _i|S_i|^{\frac{3}{2}}+\frac{o_i}{4p_i}\hat{c}_i)\nonumber \\&+\sum _{i=1}^3\frac{o_i}{p_i}(c_i-\varrho \hat{c}_i)\hat{c}_i \end{aligned}$$

(8.54)

Consider the fact that

$$\begin{aligned} \sum \limits _{i=1}^3&\frac{o_i}{p_i}(c_i-\varrho \hat{c}_i)\hat{c}_i\nonumber \\&=-\sum _{i=1}^3\frac{o_i}{p_i\varrho }(c_i-\varrho \hat{c}_i)(c_i-\varrho \hat{c}_i)+\sum _{i=1}^3\frac{o_i}{p_i\varrho }(c_i-\varrho \hat{c}_i)c_i\nonumber \\&=-\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\bar{c}_i^2 +\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\bar{c}_i(c_i+\varrho \hat{c}_i)\nonumber \\&\le -\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\bar{c}_i^2 +\sum _{i=1}^3\frac{o_i}{2p_i\varrho }c^2_i \end{aligned}$$

(8.55)

Substituting (8.55) into (8.54) that

$$\begin{aligned} \dot{V}_{ss}\le&-\varrho \sum _{i=1}^{3}(\tau _{i}S^2_i+\rho _i|S_i|^{\frac{3}{2}}+\frac{o_i}{4p_i}\hat{c}_i)\nonumber \\&-\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\bar{c}_i^2 +\sum _{i=1}^3\frac{o_i}{2p_i\varrho }c^2_i \end{aligned}$$

(8.56)

Adding $\sum \limits _{i=1}^3\frac{o_i}{2p_i\varrho }\sqrt{\varrho }|\bar{c}_i|^{\frac{3}{2}}-\sum \limits _{i=1}^3\frac{o_i}{2p_i\varrho }\sqrt{\varrho }|\bar{c}_i|^{\frac{3}{2}}$ to the inequality (8.56) implies

$$\begin{aligned} \dot{V}_{ss}\le&-\varrho \sum _{i=1}^{3}(\tau _{i}S^2_i+\rho _i|S_i|^{\frac{3}{2}}+\frac{o_i}{4p_i} \hat{c}_i)-\sum _{i=1}^3\frac{o_i}{4p_i\varrho }\bar{c}_i^2\nonumber \\&-\sum _{i=1}^3\frac{o_i}{4p_i\varrho }(|\bar{c}_i|-\sqrt{\varrho }\sqrt{|\bar{c}_i|})^2 -\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\sqrt{\varrho }|\bar{c}_i|^{\frac{3}{2}}\nonumber \\&+\sum _{i=1}^3\frac{ o_i}{4\varrho p_i}(\varrho |\bar{c}_i|+2c^{2}_i)\nonumber \\ \le&-\varrho \sum _{i=1}^{3}(\tau _{i}S^2_i+\rho _i|S_i|^{\frac{3}{2}}+\frac{o_i}{4p_i} \hat{c}_i)-\sum _{i=1}^3\frac{o_i}{4p_i\varrho }\bar{c}_i^2\nonumber \\&+\sum _{i=1}^3\frac{o_i}{4p_i}(c_i+\varrho \hat{c}_i)+\sum _{i=1}^3\frac{o_i}{2\varrho p_i}c^{2}_i-\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\sqrt{\varrho }|\bar{c}_i|^{\frac{3}{2}}\nonumber \end{aligned}$$

Indeed, based on Property 8.3, it is shown that

$$\begin{aligned} \dot{V}_{ss}\le&-\varrho (2\tau _{\min }/J_M)\frac{1}{2}S^T(t)JS(t)-\frac{1}{2}\sum _{i=1}^3\frac{o_i}{2p_i\varrho }\bar{c}^{2}_i\nonumber \\&-\varrho \rho _{\min }(2/J_M)^{\frac{3}{4}}(\frac{1}{2}S^T(t)JS(t))^{\frac{3}{4}}\nonumber \\&-\sum _{i=1}^3 o_i (2p_i)^{\frac{-1}{4}}\varrho ^\frac{1}{4}(\frac{1}{2p_i\varrho }\bar{c}^2_i)^{\frac{3}{4}}+\eta _\delta \end{aligned}$$

(8.57)

where $\tau _{\min }:=\min \{\tau _i\}$, $\rho _{\min }:=\min \{\rho _i\}$ and $\eta _\delta :=\sum \limits _{i=1}^3\frac{o_i}{2p_i\varrho }c^2_i+\sum \limits _{i=1}^3\frac{o_i}{4p_i}c_i$. Based on (8.57), it is shown that when $|S_i|\ge \varDelta _1 (\varDelta _1:=\min \{\sqrt{\frac{\eta _\delta \cdot J_M}{\varrho \tau _{\min }\cdot J_m}}, \sqrt{(\frac{\eta _\delta }{\varrho \rho _{\min }} (\frac{J_M}{J_m})^\frac{3}{4})^3}\})$, the following inequality (8.58) holds:

$$\begin{aligned}&\dot{V}_{ss}+\iota _3V_{ss}+\iota _4V_{ss}^{\frac{3}{4}}\le 0 \end{aligned}$$

(8.58)

where $\iota _3:=\min \{\varrho (2\tau _{\min }/J_M), \frac{o_i}{2}\}$ and $\iota _4:=\min \{ \varrho \rho _{\min }(2/J_M)^{\frac{3}{4}}, o_i (2p_i)^{\frac{-1}{4}}\varrho ^\frac{1}{4}\}$. It follows from (8.58) and Lemma 1.4 that $\lim \limits _{t\rightarrow t_{f}}|S_i|\le \varDelta _1$ and the finite time is given by

$$\begin{aligned} t_f\le \frac{4}{\iota _{3}}\ln \frac{\iota _{3}\root 4 \of {V_{ss}(0)}+\iota _{4}}{\iota _{4}} \end{aligned}$$

(8.59)

Case B. For the case of $\Vert S(t)\Vert \hat{\sigma }(t)\le \epsilon$. It follows from inequality (8.52), AFTSMCL (8.21) and ACL (8.23)–(8.25) that

$$\begin{aligned}&\,\, \dot{V}_{ss} \le -\varrho (S^T(t)\tau S(t)+ S^T(t)\rho {\text {sgn}}^{\frac{1}{2}}(S(t)))\nonumber \\&\qquad \quad -\varrho \frac{\Vert S(t)\Vert ^2}{\epsilon }\hat{\sigma }^2 -\sum _{i=1}^3\frac{1}{p_i}(c_i-\varrho \hat{c}_i)\dot{\hat{c}}_i\nonumber \\&\qquad \quad +\Vert S(t)\Vert [c_{1}+c_{2}\Vert \varOmega (t)\Vert +c_{3}\Vert \varOmega (t)\Vert ^2]\nonumber \\&\le -\varrho S^T(t)\tau S(t)-\varrho S^T(t)\rho {\text {sgn}}^{\frac{1}{2}}(S(t))\nonumber \\&\,\, -(\sqrt{\varrho }\frac{\Vert S(t)\Vert }{\sqrt{\epsilon }}\hat{\sigma }-\sqrt{\varrho }\sqrt{\epsilon }/2)^2+\frac{\varrho \epsilon }{4}+ \sum _{i=1}^3\frac{o_i}{p_i}(c_i-\varrho \hat{c}_i)\hat{c}_i\nonumber \end{aligned}$$

Using (8.55) again yields

$$\begin{aligned} \dot{V}_{ss} \le&-\varrho _\epsilon (2\tau _{\min }/J_M)\frac{1}{2}S^T(t)JS(t)-\sum _{i=1}^3\frac{o_i}{2p_i\varrho _\epsilon }\bar{c}_i^2\nonumber \\&-\varrho _\epsilon \rho _{\min }(2/J_M)^{\frac{3}{4}}(\frac{1}{2}S^T(t)JS(t))^{\frac{3}{4}} + \eta _{\epsilon } \end{aligned}$$

(8.60)

where $\eta _{\epsilon }=\sum \limits _{i=1}^3\frac{o_i}{2p_i\varrho }c^2_i +\frac{\varrho \epsilon }{4}$. Based on (8.60), it is shown that when $|S_i|\ge \varDelta _2 (\varDelta _2:=\min \{\sqrt{\frac{\eta _\epsilon \cdot J_M}{\varrho \tau _{\min }\cdot J_m}}, \sqrt{(\frac{\eta _\epsilon }{\varrho \rho _{\min }} (\frac{J_M}{J_m})^\frac{3}{4})^3}\})$, it obtains the following (8.61):

$$\begin{aligned} \dot{V}_{ss}\le - \iota _5 V_{ss}\le 0 \end{aligned}$$

(8.61)

where $\iota _5:=\min \{ \varrho (2\tau _{\min }/J_M), o_i\}$. Based on (8.60), it also yields directly

$$\begin{aligned}&\dot{V}_{ss} \le - \iota _5 V_{ss}+\eta _{\epsilon } \end{aligned}$$

(8.62)

Based on (8.61) or (8.62), it is shown that the attitude signals of the spacecraft system are uniformly ultimately bounded (UUB) [276].

If $\Vert S(t)\Vert \hat{\sigma }(t)>\epsilon$, it follows from (8.58) that the tracking errors of the spacecraft system can reach into a small region of FTSMS $\varDelta _1$ in finite time. If $\Vert S(t)\Vert \hat{\sigma }(t)\le \epsilon$, based on (8.61) and (8.62), it is shown that the attitude tracking errors of the spacecraft system are UUB. In other words, the states cannot escape from $\varDelta _2$ after entering it. Based on the analyses of Case A and Case B, it is concluded that $\varDelta :=\max \{\varDelta _1,\varDelta _2\}$ is an attractive region for any initial state $(e_i(0), \varOmega _{ei}(0))$ and can be reached in finite time, which means there exists a finite time $t_{f}$, $|S_i|\le \varDelta$ for $\forall t\ge t_{f}$. Also, $t_{f}$ is defined in (8.59). There are two cases for the following analyses.

Case I: Consider the case of $|e_i|\ge \varepsilon$. Because of $|S_i | \le \varDelta (i = 1,2,3)$, it yields

$$\begin{aligned} \varOmega _{ei}+k_{2i}e_i+k_{1i}e_i^r=\varsigma _i, |\varsigma _i|\le \varDelta (i=1,2,3) \end{aligned}$$

(8.63)

the Eq. (8.63) can be rewritten as:

$$\begin{aligned} \varOmega _{ei}+(k_{2i}-\frac{\varsigma _i}{2e_i})e_i+(k_{1i}-\frac{\varsigma _i}{2e^r_i})e_i^r=0 (i=1,2,3) \end{aligned}$$

(8.64)

It follows from Lemma 8.2 that $\varOmega _{ei}+k_{2i}e_i+k_{1i}e_i^r=0 (k_{1i}>0, k_{2i}>0)$ is FTSM. As the similar analysis of that in [17, 124, 217], as long as $k_{2i}-\frac{\varsigma _i}{2e_i}>0$ and $k_{1i}-\frac{\varsigma _i}{2e^r_i}>0$, the Eq. (8.64) is still kept in the form of FTSMS. It follows from $k_{2i}-\frac{\varsigma _i}{2e_i}>0$ and $k_{1i}-\frac{\varsigma _i}{2e^r_i}>0$ that $|e_i|>\frac{\varDelta }{2 k_{2i}}$ and $|e_i|>\root r \of {\frac{\varDelta }{2 k_{1i}}}$ can guarantee that Eq. (8.64) is in the form of FTSM. According to $|e_{i}| \ge \varepsilon$, it shows that the attitude tracking error $e_{i}$ can converge to the region

$$\begin{aligned} \varDelta _e:=\max \left( \varepsilon , \frac{\varDelta }{2k_{2i}}, \root r \of {\frac{\varDelta }{2k_{1i}}}\right) \end{aligned}$$

(8.65)

in finite time. Furthermore, with the FTSM dynamics (8.63), the velocity tracking error $\varOmega _{ei}$ converges to the region

$$\begin{aligned} |\varOmega _{ei}|\le & {} |\varsigma _i|+k_{2i}|e_i|+k_{1i}|e_i|^r (i=1,2,3)\nonumber \\\le & {} \varDelta +\tilde{k}_2\varDelta _e+\tilde{k}_1\varDelta _e^r=\varDelta _\omega \end{aligned}$$

(8.66)

in finite time, where $\tilde{k}_2=\max \{k_{2i}\}, \tilde{k}_1=\max \{k_{1i}\}(i=1,2,3)$.

Case II: Consider the case of $|e_i|<\varepsilon$. The attitude tracking error $e_i(i=1,2,3)$ has been in the region $\varDelta _e$, then based on (8.66), it is shown that the velocity tracking error $\varOmega _{ei}$ converges to the region $\varDelta _\omega$ in finite time.