Appendix
Symbol | Item | Symbol | Item |
|---|
\(\widetilde{m}_0\)
| Mass of the main body |
R
| Radius of the main body |
\(m_0\)
| Mass of solar panel BE, DF
|
L
| Length of BE, DF
|
\(m_1 \)
| Mass of robotic arm AP
|
\(L_1\)
| Length of robotic arm AP
|
\(m_2\)
| Mass of robotic arm PQ
|
\(L_2\)
| Length of robotic arm PQ
|
\(m_3\)
| Mass of robotic arm QR
|
\(L_3\)
| Length of robotic arm QR
|
\(\rho \)
| Density of the solar panels |
\(\varvec{\gamma }\)
|
\(\gamma _k=\int _0^L\rho \xi \Lambda _k(\xi ) d\xi \)
|
K
| Stiffness matrix of BE, DF
|
\(\varvec{\beta }\)
|
\(\beta _k=\int _0^L\rho \Lambda _k(\xi ) d\xi \)
|
\(\varvec{\Lambda }\)
| First n flexural modes of BE, DF
|
\(\varvec{\alpha }\)
|
\(\alpha _k=\int _0^L\rho \Lambda _k(\xi )^2 d\xi \)
|
$$\begin{aligned} M_0=\widetilde{m}&_0+2m_0~~~~~~~~~J_0=\frac{\widetilde{m}_0R^2}{2}+2m_0(R^2+RL+\frac{L^2}{3})~~~~~\kappa _1^2=\frac{1}{3}L_1^2\\ \kappa _2^2=\frac{1}{3}&L_2^2~~~~\kappa _3^2=L_3^2(\frac{1}{3}m_3 + m)/(m_3+m)~~~~~~~~~\widetilde{\theta }=\theta +\theta _1+\theta _2+\theta _3\\ M\left( 1,1\right)&= M_0 + m_1 + m_2 + m_3 + m~~~~~~~~~~~~M\left( 1,2\right) = 0 \\ M\left( 1,3\right)&= - \left( \frac{m_2}{2} + m_3 + m\right) L_2\cos \left( \theta +\theta _1+\theta _2\right) - \left( \frac{m_3}{2} + m\right) L_3 \cos \widetilde{\theta } \\ {}&\quad - \,\left( m_1 + m_2 + m_3 + m\right) R\cos \theta - \left( \frac{m_1}{2}+ m_2 + m_3 + m\right) L_1\cos \left( \theta +\theta _1\right) \\ M\left( 1,4\right)&= - \left( \frac{m_1}{2} + m_2 + m_3 + m\right) L_1 \cos \left( \theta +\theta _1\right) - \left( \frac{m_2}{2} + m_3 + m\right) \\ {}&~~~\Big [ L_2 \cos \left( \theta +\theta _1+\theta _2\right) \Big ] - \left( \frac{m_3}{2} + m\right) L_3 \cos \widetilde{\theta } \\ M\left( 1,5\right)&= - \left( \frac{m_2}{2} + m_3 + m\right) L_2 \cos \left( \theta +\theta _1+\theta _2\right) \\ {}&\quad -\, \left( \frac{m_3}{2} + m\right) L_3 \cos \widetilde{\theta }\\ M\left( 1,6\right)&= - \left( \frac{m_3}{2} + m\right) L_3 \cos \widetilde{\theta }\\ M\left( 2,2\right)&= M_0 + m_1 + m_2 + m_3 + m \\ \end{aligned}$$
$$\begin{aligned} M\left( 2,3\right)&= - \left( \frac{m_2}{2} + m_3 + m\right) L_2 \sin \left( \theta +\theta _1+\theta _2\right) - \left( \frac{m_3}{2} + m\right) L_3 \sin \widetilde{\theta } \\&\quad -\, \left( m_1 + m_2 + m_3 + m\right) R \sin \theta - \left( \frac{m_1}{2}+ m_2 + m_3 + m\right) L_1 \sin \left( \theta +\theta _1\right) \\ M\left( 2,4\right)&= - \left( \frac{m_1}{2}+ m_2 + m_3 + m\right) L_1 \sin \left( \theta +\theta _1\right) \\ {}&\quad -\, \left( \frac{m_2}{2} + m_3 + m\right) L_2 \sin \left( \theta +\theta _1+\theta _2\right) - \left( \frac{m_3}{2} + m\right) L_3 \sin \widetilde{\theta } \\ M\left( 2,5\right)&= - \left( \frac{m_2}{2} + m_3 + m\right) L_2 \sin \left( \theta +\theta _1+\theta _2\right) - \left( \frac{m_3}{2} + m\right) L_3 \sin \widetilde{\theta }\\ M\left( 2,6\right)&= - \left( \frac{m_3}{2} + m\right) L_3 \sin \widetilde{\theta } \\ M\left( 3,3\right)&= J_0 + m_1 \left( R^2+\kappa _1^2\right) + m_2 \left( R^2+L_1^2+\kappa _2^2\right) \\ {}&\quad +\, 2 \left( \frac{m_1}{2}+ m_2 + m_3 + m\right) R L_1 \cos \theta _1+ \left( m_3+m\right) \left( R^2+L_1^2+L_2^2+\kappa _3^2\right) \\ {}&\quad + 2 \left( \frac{m_2}{2} + m_3 + m\right) L_2 \left[ R \cos \left( \theta _1+\theta _2\right) +L_1 \cos \theta _2\right] \\ {}&\quad +\, 2 \left( \frac{m_3}{2} + m\right) L_3 \left[ R \cos \left( \theta _1+\theta _2+\theta _3\right) + L_1 \cos \left( \theta _2+\theta _3\right) +L_2 \cos \theta _3\right] \\ M\left( 3,4\right)&= m_1 \kappa _1^2 + \left( \frac{m_1}{2}+ m_2 + m_3 + m\right) R L_1 \cos \theta _1 + \left( m_3+m\right) \left( L_1^2+L_2^2+\kappa _3^2\right) \\ {}&\quad +\,m_2 \left( L_1^2+\kappa _2^2\right) + \left( \frac{m_2}{2} + m_3 + m\right) L_2 \left[ R \cos \left( \theta _1+\theta _2\right) +2 L_1 \cos \theta _2\right] \\ {}&\quad +\, \left( \frac{m_3}{2} + m\right) L_3 \left[ R \cos \left( \theta _1+\theta _2+\theta _3\right) + 2 L_1 \cos \left( \theta _2+\theta _3\right) +2 L_2 \cos \theta _3\right] \\ M\left( 3,5\right)&= m_2 \kappa _2^2 + \left( m_3+m\right) \left( L_2^2+\kappa _3^2\right) \\ {}&\quad +\, \left( \frac{m_2}{2} + m_3 + m\right) L_2 \left[ R \cos \left( \theta _1+\theta _2\right) +L_1 \cos \theta _2\right] \\ {}&\quad +\, \left( \frac{m_3}{2} + m\right) L_3 \left[ R \cos \left( \theta _1+\theta _2+\theta _3\right) + L_1 \cos \left( \theta _2+\theta _3\right) +2 L_2 \cos \theta _3\right] \\ M\left( 3,6\right)&= \left( m_3+m\right) \kappa _3^2 \\ {}&\quad +\, \left( \frac{m_3}{2} + m\right) L_3 \left[ R \cos \left( \theta _1+\theta _2+\theta _3\right) + L_1 \cos \left( \theta _2+\theta _3\right) +L_2 \cos \theta _3\right] \\ M\left( 4,4\right)&= m_2 \left( L_1^2+\kappa _2^2\right) + \left( m_3+m\right) \left( L_1^2+L_2^2+\kappa _3^2\right) + 2 \left( \frac{m_2}{2} + m_3 + m\right) \\ {}&~~~~L_1 L_2 \cos \theta _2 + 2 \left( \frac{m_3}{2} + m\right) L_3 \left[ L_1\cos \left( \theta _2+\theta _3\right) +L_2 \cos \theta _3\right] m_1 \kappa _1^2 \\ M\left( 4,5\right)&= m_2 \kappa _2^2 + \left( m_3+m\right) \left( L_2^2+\kappa _3^2\right) + \left( \frac{m_2}{2} + m_3 + m\right) L_1 L_2 \cos \theta _2 \\ {}&\quad +\, \left( \frac{m_3}{2} + m\right) L_3 \left[ L_1 \cos \left( \theta _2+\theta _3\right) +2 L_2 \cos \theta _3\right] \\ M\left( 4,6\right)&= \left( m_3+m\right) \kappa _3^2 + \left( \frac{m_3}{2} + m\right) L_3 \left[ L_1 \cos \left( \theta _2+\theta _3\right) +L_2 \cos \theta _3\right] \\ M\left( 5,5\right)&= m_2 \kappa _2^2 + \left( m_3+m\right) \left( L_2^2+\kappa _3^2\right) + 2 \left( \frac{m_3}{2} + m\right) L_2 L_3 \cos \theta _3 \\ M\left( 5,6\right)&= \left( m_3+m\right) \kappa _3^2 + \left( \frac{m_3}{2} + m\right) L_2 L_3 \cos \theta _3 \\ M\left( 6,6\right)&= \left( m_3+m\right) \kappa _3^2 ~~~~~~~~~~~~M(7:(6+n),(7+n):(6+2n))= \varvec{0}_{n\times n}\\ M(1,7:&(6+n)) = -\varvec{\beta }^T \sin \theta ~~~~~M(1,(7+n):(6+2n)) = \varvec{\beta }^T \sin \theta \\ M(2,7:&(6+n))= \varvec{\beta }^T \cos \theta ~~~~~~~M(2,(7+n):(6+2n))= -\varvec{\beta }^T\cos \theta \\ M(3,7:&(6+2n)) = \varvec{\sigma }^T~~~~~~~~~~~~M(4:6,7:(6+2n))= \varvec{0}_{3\times 2n}\\ M(7:(6&+n),7:(6+n))=M((7+n):(6+2n),(7+n):(6+2n)) =diag\{\varvec{\alpha }\} \\ \end{aligned}$$
$$\begin{aligned} f\left( 1\right)&= \left( m_1+m_2+m_3+m\right) R \dot{\theta }^2 \sin \theta +\varvec{\beta }^T(\dot{\varvec{b}}-\dot{\varvec{a}}) \dot{\theta } \cos \theta \\&\quad +\, \left( \frac{m_1}{2}+m_2+m_3+m\right) L_1 \left( \dot{\theta }+\dot{\theta }_1\right) ^2 \sin \left( \theta +\theta _1\right) \\&\quad +\, \left( \frac{m_2}{2}+m_3+m\right) L_2\left( \dot{\theta }+\dot{\theta }_1+\dot{\theta }_2\right) ^2 \sin \left( \theta +\theta _1+\theta _2\right) \\&\quad +\, \left( \frac{m_3}{2}+m\right) L_3 \left( \dot{\theta }+\dot{\theta }_1+\dot{\theta }_2+\dot{\theta }_3\right) ^2 \sin \widetilde{\theta }\\ f\left( 2\right)&= -\left( m_1+m_2+m_3+m\right) R \dot{\theta }^2 \cos \theta +\varvec{\beta }^T(\dot{\varvec{b}}-\dot{\varvec{a}}) \dot{\theta } \sin \theta \\&\quad -\, \left( \frac{m_1}{2}+m_2+m_3+m\right) L_1 \left( \dot{\theta }+\dot{\theta }_1\right) ^2 \cos \left( \theta +\theta _1\right) \\&\quad -\, \left( \frac{m_2}{2}+m_3+m\right) L_2\left( \dot{\theta }+\dot{\theta }_1+\dot{\theta }_2\right) ^2 \cos \left( \theta +\theta _1+\theta _2\right) \\&\quad -\, \left( \frac{m_3}{2}+m\right) L_3 \left( \dot{\theta }+\dot{\theta }_1+\dot{\theta }_2+\dot{\theta }_3\right) ^2 \cos \widetilde{\theta }\\ f\left( 3\right)&= -\left( \frac{m_1}{2}+m_2+m_3+m\right) R L_1 \dot{\theta }_1 \left( 2 \dot{\theta }+\dot{\theta }_1\right) \sin \theta _1\\&\quad -\, \left( \frac{m_2}{2}+m_3+m\right) R L_2\left( 2 \dot{\theta }+2 \dot{\theta }_1+\dot{\theta }_2\right) \left( \dot{\theta }_1+\dot{\theta }_2\right) \sin \left( \theta _1+\theta _2\right) \\&\quad -\, \left( \frac{m_2}{2}+m_3+m\right) L_1 L_2\left( 2 \dot{\theta }+2 \dot{\theta }_1+\dot{\theta }_2\right) \dot{\theta }_2 \sin \theta _2\\&\quad -\, \left( \frac{m_3}{2}+m\right) R L_3 \left( 2 \dot{\theta }+\dot{\theta }_1+\dot{\theta }_2+\dot{\theta }_3\right) \left( \dot{\theta }_1+\dot{\theta }_2+\dot{\theta }_3\right) \sin \left( \theta _1+\theta _2+\theta _3\right) \\&\quad -\, \left( \frac{m_3}{2}+m\right) L_1 L_3 \left( 2 \dot{\theta }+2 \dot{\theta }_1+\dot{\theta }_2+\dot{\theta }_3\right) \left( \dot{\theta }_2+\dot{\theta }_3\right) \sin \left( \theta _2+\theta _3\right) \\&\quad -\, \left( \frac{m_3}{2}+m\right) L_2L_3 \left( 2 \dot{\theta }+2 \dot{\theta }_1+2 \dot{\theta }_2+\dot{\theta }_3\right) \dot{\theta }_3 \sin \theta _3\\&\quad +\,\varvec{\beta }^T(\dot{\varvec{a}}-\dot{\varvec{b}})(\dot{x}\cos \theta +\dot{y}\sin \theta )\\ f\left( 4\right)&= \Big [\left( \frac{m_1}{2}+m_2+m_3+m\right) L_1 \sin \theta _1+ \left( \frac{m_2}{2}+m_3+m\right) L_2\sin \left( \theta _1+\theta _2\right) \\&\quad +\, \left( \frac{m_3}{2}+m\right) L_3 \sin \left( \theta _1+\theta _2+\theta _3\right) \Big ] R \dot{\theta }^2\\&\quad -\, \left( \frac{m_2}{2}+m_3+m\right) L_1 L_2 \left( \dot{\theta }+2 \dot{\theta }_1+\dot{\theta }_2\right) \dot{\theta }_2 \sin \theta _2\\&\quad -\, \left( \frac{m_3}{2}+m\right) L_2 L_3\left( 2 \dot{\theta }+2 \dot{\theta }_1+2 \dot{\theta }_2+\dot{\theta }_3\right) \dot{\theta }_3 \sin \theta _3\\&\quad -\, \left( \frac{m_3}{2}+m\right) L_1 L_3\left( 2 \dot{\theta }+2 \dot{\theta }_1+\dot{\theta }_2+\dot{\theta }_3\right) \left( \dot{\theta }_2+\dot{\theta }_3\right) \sin \left( \theta _2+\theta _3\right) \\ f\left( 5\right)&= \left[ \left( \frac{m_2}{2}+m_3+m\right) L_2 \sin \left( \theta _1+\theta _2\right) +\left( \frac{m_3}{2}+m\right) L_3\sin \left( \theta _1+\theta _2+\theta _3\right) \right] R \dot{\theta }^2\\&\quad +\, \left[ \left( \frac{m_2}{2}+m_3+m\right) L_2 \sin \theta _2+\left( \frac{m_3}{2}+m\right) L_3\sin \left( \theta _2+\theta _3\right) \right] L_1 \left( \dot{\theta }+\dot{\theta }_1\right) ^2\\&\quad -\, \left( \frac{m_3}{2}+m\right) L_2 L_3\left( 2 \dot{\theta }+2 \dot{\theta }_1+2 \dot{\theta }_2+\dot{\theta }_3\right) \dot{\theta }_3 \sin \theta _3\\ f\left( 6\right)&= \left( \frac{m_3}{2}+m\right) L_3\Big [R \dot{\theta }^2 \sin \left( \theta _1+\theta _2+\theta _3\right) +L_1 \left( 2 \dot{\theta }+\dot{\theta }_1\right) \dot{\theta }_1 \sin \left( \theta _2+\theta _3\right) \\&\quad +\, L_2 \left( \dot{\theta }+\dot{\theta }_1+\dot{\theta }_2\right) ^2 \sin \theta _3\Big ]\\ f(7:&(6+n)) = -\varvec{\beta }\left( \dot{x} \cos \theta +\dot{y} \sin \theta \right) \dot{\theta } + K\varvec{a}\\ f((7&+n):(6+2n)) = \varvec{\beta }\left( \dot{x} \cos \theta +\dot{y} \sin \theta \right) \dot{\theta } + K\varvec{b}\\ \end{aligned}$$
