## 3 Control Algorithm Design
### Attitude Controller Design
#### 3.3.4 Spin-Stabilized Phase
\begin{array}{rl}\dot{V}(t)=&-K_{5}\mathbf{e}_{p}^{b}{}^{\mathrm{T}}\Big{(}\mathbf{ S}^{b}\times\mathbf{\omega}_{bi}^{b}\Big{)}-K_{8}\Big{\|}\mathbf{\omega}_{bi}^{b}- \omega_{spin}\mathbf{S}^{b}\Big{\|}^{2}+\Big{(}\mathbf{\omega}_{bi}^{b}-\omega_{spin} \mathbf{S}^{b}\Big{)}^{\mathrm{T}}J^{-1}\mathbf{d}\\ &+\Big{(}\mathbf{\omega}_{bi}^{b}-\omega_{spin}\mathbf{S}^{b}\Big{)}^{\mathrm{T}}\Big{(} K_{5}\mathbf{e}_{p}^{b}\times\mathbf{S}^{b}+K_{6}\mathbf{S}_{prev}^{b}\times\mathbf{S}^{b}-K_{7}\mathbf{S}^{b} \times\mathbf{\omega}_{bi}^{b}\times\mathbf{S}^{b}\Big{)}\\ =&-K_{5}\mathbf{e}_{p}^{\mathrm{T}}\Big{(}\mathbf{S}^{b}\times\mathbf{\omega}_{bi}^{b} \Big{)}+K_{5}\mathbf{\omega}_{bi}^{b}{}^{\mathrm{T}}\Big{(}\mathbf{e}_{p}^{b}\times \mathbf{S}^{b}\Big{)}+\mathbf{\omega}_{bi}^{b}{}^{\mathrm{T}}\Big{(}K_{6}\mathbf{S}_{prev}^{b }\times\mathbf{S}^{b}-K_{7}\mathbf{S}^{b}\times\mathbf{\omega}_{bi}^{b}\times\mathbf{S}^{b} \Big{)}\\ &-K_{8}\|\mathbf{e}_{2}\|^{2}+\mathbf{e}_{2}^{\mathrm{T}}J^{-1}\mathbf{d}\end{array} \, \tag{23}\]  
where \(\mathbf{S}_{prev}^{b}\times\mathbf{S}^{b}\) can be approximately written as:  
\[ \begin{array}{rl}\mathbf{S}_{prev}^{b}\times\mathbf{S}^{b}&=\Big{(}\mathbf{S}_{prev}^{b }-\mathbf{S}^{b}\Big{)}\times\mathbf{S}^{b}\\ &=-\frac{\dot{\mathbf{S}}^{b}}{\Delta T}\mathbf{S}^{b}\times\mathbf{S}^{b}\\ &=-\frac{1}{\Delta T}\mathbf{S}^{b}\times\mathbf{\omega}_{bi}^{b}\times\mathbf{S}^{b}\end{array} . \tag{24}\]  
Obviously, it follows that \(\mathbf{\omega}_{bi}^{b}{}^{\mathrm{T}}\Big{(}K_{6}\mathbf{S}_{prev}^{b}\times\mathbf{S} ^{b}-K_{7}\mathbf{S}^{b}\times\mathbf{\omega}_{bi}^{b}\times\mathbf{S}^{b}\Big{)}\leq 0\). At the same time, according to the cross product rule, we know that \(\mathbf{e}_{p}^{b}{}^{\mathrm{T}}\Big{(}\mathbf{S}^{b}\times\mathbf{\omega}_{bi}^{b} \Big{)}=\mathbf{\omega}_{bi}^{b}{}^{\mathrm{T}}\Big{(}\mathbf{e}_{p}^{b}\times\mathbf{S}^{ b}\Big{)}\). Then,  
\[ \begin{array}{rl}\dot{V}(t)&\leq-K_{8}\|\mathbf{e}_{2}\|^{2}+\mathbf{e}_{2}{}^{ \mathrm{T}}J^{-1}\mathbf{d}\\ &\leq\|\mathbf{e}_{2}\|\Big{(}-K_{8}\|\mathbf{e}_{2}\|+\frac{\varepsilon}{\min(J_{s},J_{s},J_{s})}\Big{)}\end{array} . \tag{25}\]  
It follows that \(\dot{V}(t)<0\) if \(\|\mathbf{e}_{2}\|>\frac{\varepsilon}{K_{8}\min(I_{s},J_{s},J_{s})}\). Therefore, \(V(t)\) is strictly monotonous decreasing when \(\mathbf{e}_{2}\) is outside the region \(\mathfrak{D}\triangleq\Big{\{}\mathbf{e}_{2}\big{\|}\|\mathbf{e}_{2}\|\leq\frac{ \varepsilon}{K_{8}\min(I_{s},J_{s},J_{s})}\Big{\}}\), and ultimately ensures that \(\mathbf{e}_{2}\) enters the region \(\mathfrak{D}\). Once \(\mathbf{e}_{2}\) enters the region \(\mathfrak{D}\), we know that \(\mathbf{e}_{2}\) will remain in the region \(\mathfrak{D}\) afterward according to the fact that \(\dot{V}(t)<0\). Therefore, it can be concluded that the control error \(\mathbf{e}_{2}\) is UUB and will eventually converge to the region \(\mathfrak{D}\)[33]. It can be then proven that \(\mathbf{e}_{1}\) is also bounded. In practical applications, the upper bound of the control errors can be controlled in an acceptable range by selecting appropriate control coefficients. To this end, the proof is completed.