## Appendix A  
The terms of the Lagrange equation are as follows [12]:  
\[m_{1} = m_{bs}+1.5(3S_{\alpha}^{2}S_{\varphi}^{2}-S_{\alpha}^{2}+C_{ \varphi}^{2})I_{wr},\] \[m_{2} = 0.75c_{1}S_{2\varphi}S_{\theta},\] \[m_{3} = 1.5(c_{1}C_{\varphi}^{2}-2S_{\varphi}^{2})r_{s}+lm_{b}C_{\varphi }C_{\theta},\] \[m_{4} = -lm_{b}S_{\varphi}S_{\theta},\] \[m_{5} = m_{bs}+1.5(C_{\alpha}^{2}C_{\theta}^{2}+C_{\alpha}^{2}S_{\varphi }^{2}S_{\theta}^{2}+2C_{\varphi}^{2}S_{\alpha}^{2}S_{\theta}^{2})I_{wr},\] \[m_{6} = -1.5c_{1}C_{\varphi}S_{\varphi}S_{\theta}I_{wr}r_{s},\] \[m_{7} = 1.5r_{s}C_{\alpha}^{2}C_{\theta}I_{wr}-lm_{b}C_{\varphi},\] \[m_{8} = I_{by}C_{\varphi}^{2}+m_{b}I^{2}C_{\varphi}^{2}+1.5(3S_{\alpha} ^{2}S_{\varphi}^{2}-S_{\alpha}^{2}+C_{\varphi}^{2})I_{wr}r_{s},\] \[m_{9} = I_{bx}+m_{b}I^{2}+1.5r_{s}^{2}C_{\alpha}^{2}I_{wr},\] \[c_{11} = -3S_{2\varphi}(3C_{2\alpha}-1)I_{wr}\phi/8,\] \[c_{12} = 0.75c_{1}I_{wr}(S_{\theta}C_{2\varphi}\phi+c_{2}\theta),\] \[c_{13} = 0.75c_{1}S_{2\varphi}I_{wr}\dot{x}+0.75c_{1}C_{2\varphi}S_{ \theta}I_{wr}\dot{y}-lm_{b}C_{\varphi}S_{\theta}\phi-(1.5r_{s}C_{\varphi}c_{ 1}-lm_{b}C_{\theta})S_{\varphi}I_{wr}\dot{\theta},\] \[c_{14} = 0.75c_{1}C_{2\varphi}S_{\theta}I_{wr}\dot{y}-(1.5c_{1}I_{wr}r_{s }C_{\varphi}+lm_{b}C_{\theta})S_{\varphi}\dot{\varphi}-lm_{b}C_{\varphi}S_{ \theta}\dot{\theta},\] \[c_{21} = 0.75c_{1}C_{2\varphi}S_{\theta}I_{wr}\dot{\varphi}+0.75c_{1}C_{ \varphi}C_{\theta}S_{\theta}I_{wr}\dot{\theta},\] \[c_{22} = 1.5c_{1}C_{\varphi}^{2}C_{\theta}S_{\theta}I_{wr}\dot{\theta}-1.5 c_{1}C_{\varphi}S_{\theta}^{2}S_{\varphi}I_{wr}\dot{\varphi},\] \[c_{23} = 0.75c_{1}C_{2\varphi}S_{\theta}I_{wr}\dot{x}-0.75c_{1}S_{2\varphi }S_{\theta}^{2}I_{wr}\dot{y}+lm_{b}S_{\varphi}\dot{\varphi}-0.5(c_{1}C_{ \varphi}^{2}+2C_{2\varphi})r_{s}S_{\theta}I_{wr}\dot{\theta},\] \[c_{24} = 0.75c_{1}C_{\varphi}C_{\theta}S_{\varphi}I_{wr}\dot{x}+1.5c_{1}C _{\varphi}^{2}C_{\theta}S_{\theta}I_{wr}\dot{y}+(3-3c_{1}C_{\varphi}^{2}-2C_{ \alpha}^{2})S_{\theta}I_{wr}r_{s}\dot{\varphi}/2\] \[-1.5c_{1}c_{2}I_{wr}r_{s}\dot{\theta},\] (A1)\[c_{31} = 3(3C_{2\alpha}-1)S_{2\varphi}C_{\theta}I_{\text{ur}}\dot{y}/16+3(3C_{ 2\alpha}-1)S_{2\varphi}I_{\text{ur}}r_{s}\dot{\varphi}/8,\] \[c_{32} = 3I_{\text{ur}}(c_{3}S_{\theta}r_{s}\dot{\varphi}/2-c_{1}c_{2}\dot{ x}/4-c_{1}C_{\theta}^{2}C_{\theta}S_{\theta}\dot{y}/2),\] \[c_{33} = -0.75c_{1}S_{2\varphi}I_{\text{ur}}r_{s}\dot{x}+1.5c_{3}S_{\theta} I_{\text{ur}}r_{s}\dot{y}+(3r_{s}^{2}c_{1}-2m_{b}l^{2}-2I_{by})S_{2\varphi}I_{ \text{ur}}\dot{\theta}/4,\] \[c_{34} = -(4m_{b}l^{2}+4I_{by}+3(3C_{2\alpha}-1)S_{2\varphi}I_{\text{ur}}r_ {s}^{2})\dot{\varphi}/8,\] \[c_{41} = -3c_{1}S_{2\varphi}I_{\text{ur}}\dot{x}-3c_{1}S_{\theta}C_{2 \varphi}I_{\text{ur}}\dot{y}+0.75c_{1}S_{2\varphi}I_{\text{ur}}r_{s}\dot{ \theta},\] \[c_{42} = 0.75I_{\text{ur}}(c_{1}S_{2\varphi}^{2}S_{2\varphi}\dot{y}-c_{1} C_{2\varphi}S_{\theta}\dot{x}-2c_{3}S_{\theta}r_{s}\dot{\theta}),\] \[c_{44} = 0.75c_{1}S_{2\varphi}r_{s}I_{\text{ur}}\dot{x}-1.5c_{3}S_{\theta} r_{s}I_{\text{ur}}\dot{y}+(2m_{b}l^{2}-3c_{1}I_{\text{ur}}r_{s}^{2}+4I_{by})S_{2 \varphi}\dot{\theta}/4,\] \[g_{3} = -glm_{b}C_{\varphi}S_{\theta},\] \[g_{4} = -glm_{b}C_{\theta}S_{\varphi},\]  
where \(C_{(\cdot)}\) and \(S_{(\cdot)}\) denote \(\cos{(\cdot)}\) and \(\sin{(\cdot)}\), respectively, and \(c_{1}=(2-3C_{\alpha}^{2}),c_{2}=C_{\varphi}C_{\theta}S_{\varphi},\)\(c_{3}=(S_{\alpha}^{2}-c_{1}C_{\varphi}^{2}),I_{\text{ur}}=3I_{\text{ur}}C_{ \alpha}^{2}/(2r_{\text{ur}}^{2}),\) and \(m_{bs}=m_{b}+m_{s}+I_{s}/r_{s}^{2}.\)