Abstract
A ridable ballbot (RB) is a car robot with novel inverse pendulum structures, wherein a body consisting of a cabin and a person dynamically balances on top of a ball. One of the interesting but challenging features of the RB is unstable, underactuated, and second-order constraints. The underactuated and unstable dynamics of the RB result in challenging planning and control tasks. To address these concerns, a full three-dimensional (3D) model of the RB and a nonlinear control approach for the complicated operation duties including balancing, heading, and transferring tasks have been proposed in this study. An Euler–Lagrange method is applied to achieve the full 3D dynamics of the RB. By applying an approach to transform the full 3D dynamics into two subsystems. Thus, a nonlinear feedback controller can be applied to control the RB system based on stabilizing the transformed two subsystems. The proposed control quality and the stability analysis of the system are discussed. Simulation results are provided to demonstrate the superior performance of the proposed controller. Meanwhile, the experimental performance of the proposed controller is validated on an actual RB. Comparative simulation is conducted to demonstrate the merit features of the proposed control scheme.
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Appendices
Appendix 1: Element of mass matrix \(\mathbf{M}\left(\mathbf{q}\right)\)
The components of \(\mathbf{M}\left(\mathbf{q}\right)\) matrix are given by
$${m}_{11}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}\left({m}_{a}+{m}_{k}+{I}_{k}{r}_{k}^{-2}\right){r}_{w}^{2}+1.5{I}_{w}{\mathrm{cos}}^{2}\alpha \\ +3{I}_{w}\left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)-3{I}_{w}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ -4.5{I}_{w}{\mathrm{cos}}^{2}\alpha \left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)-6{I}_{w}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +4.5{I}_{w}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+9{I}_{w}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{I}_{w}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\\ -1.5{I}_{w}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\end{array}\right),\quad {m}_{12}={m}_{12}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}1.5{I}_{w}\mathrm{sin}2{\theta }_{z}-1.5{I}_{w}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-2.25{I}_{w}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}\\ +1.5{I}_{w}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+2.25{I}_{w}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +3{I}_{w}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{I}_{w}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ -3{I}_{w}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}-4.5{I}_{w}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -2.25{I}_{w}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${m}_{13}={m}_{31}=\left(\begin{array}{l}{m}_{a}l\left(\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{z}-\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\right)\\ -1.5{I}_{w}{r}_{k}{r}_{w}^{-2}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),\quad {m}_{14}={m}_{41}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}{m}_{a}l{r}_{w}^{2}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{I}_{w}{r}_{k}{\mathrm{sin}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +3{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{z}-4.5{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +1.5{I}_{w}{r}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\end{array}\right),$$
$${m}_{15}={m}_{51}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}{m}_{a}l{r}_{w}^{2}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{z}-{m}_{a}l{r}_{w}^{2}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +1.5{I}_{w}{r}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -1.5{I}_{w}{r}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -2.25{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +2.25{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),\quad {m}_{22}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}\left({m}_{a}+{m}_{k}+{I}_{k}{r}_{k}^{-2}\right){r}_{w}^{2}+3{I}_{w}{\mathrm{sin}}^{2}\alpha \\ -3{I}_{w}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)\\ +6{I}_{w}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right){\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{I}_{w}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right){\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +1.5{I}_{w}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${m}_{23}={m}_{32}=\left(\begin{array}{l}1.5{I}_{w}{r}_{k}{r}_{w}^{-2}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -{m}_{a}l\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-{m}_{a}l\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{z}\end{array}\right),\quad {m}_{24}={m}_{42}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}{m}_{a}l{r}_{w}^{2}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +3{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}-3{I}_{w}{r}_{k}{\mathrm{sin}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -4.5{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -1.5{I}_{w}{r}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\end{array}\right),$$
$${m}_{25}={m}_{52}=\frac{1}{{r}_{w}^{2}}\left(\begin{array}{l}{m}_{a}l{r}_{w}^{2}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{z}+{m}_{a}l{r}_{w}^{2}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +1.5{I}_{w}{r}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +1.5{I}_{w}{r}_{k}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right){\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -2.25{I}_{w}{r}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),\quad {m}_{33}={I}_{x}+{m}_{a}l+3{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}{\mathrm{cos}}^{2}\alpha ,$$
$${m}_{44}=\left(\begin{array}{l}+\left({I}_{y}+{m}_{a}l\right){\mathrm{cos}}^{2}{\theta }_{x}+3{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}{\mathrm{sin}}^{2}\alpha \\ +{I}_{z}{\mathrm{sin}}^{2}{\theta }_{x}-3{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}{\mathrm{cos}}^{2}{\theta }_{x}\left(1-1.5{\mathrm{cos}}^{2}\alpha \right)\end{array}\right),\quad {m}_{45}={m}_{54}=\frac{1}{2}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\left(\begin{array}{l}{I}_{y}-{I}_{z}+{m}_{a}l-3{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}\\ +4.5{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}{\mathrm{cos}}^{2}\alpha \end{array}\right),$$
$${m}_{55}=\left(\begin{array}{l}m{l}^{2}+1.5{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}{\mathrm{cos}}^{2}\alpha +{I}_{y}{\mathrm{sin}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+{I}_{x}{\mathrm{sin}}^{2}{\theta }_{y}\\ +\left({I}_{z}-{m}_{a}{l}^{2}+3{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}-4.5{I}_{w}{r}_{k}^{2}{r}_{w}^{-2}{\mathrm{cos}}^{2}\alpha \right){\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\end{array}\right).$$
Appendix 2: Element of centrifugal and damping matrix \(\mathbf{C}\left(\mathbf{q},\dot{\mathbf{q}}\right)\)
Elements of centrifugal and damping matrix of dynamic Eq. (3) are as follows:
$${c}_{11}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-3{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{z}-3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}+3{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}+4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}+3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +6{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ -4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ -4.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}-9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ -3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ -4.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-6{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -6{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +4.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -1.5{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}(1-1.5{\mathrm{cos}}^{2}\alpha )+{b}_{x}\end{array}\right),$$
$${c}_{12}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha -3{\dot{\theta }}_{z}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\\ -4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}-6{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}-6{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}-9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ -12{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}+9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}+9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{z}\mathrm{sin}{\theta }_{y}\\ +12{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +18{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-1.5{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -18{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ +2.25{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}-9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}-1.5{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -1.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}-4.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{c}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+2.25{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +2.25{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${c}_{13}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}-3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}-6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ -6{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{z}-{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -3{r}_{k}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +12{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +3{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+6{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-18{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +4.5{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}-1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +1.5{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}+2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{14}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}+3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}-3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}{x}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}\\ +2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}-6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-1.5{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}+1.5{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-2.25{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${c}_{15}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha -3{\dot{y}}_{k}+6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}+6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{z}\\ -9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}-9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -3{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-12{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -3{r}_{k}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}\\ +4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+18{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}\\ +6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}-9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ -3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ -6{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}-1.5{r}_{k}{\dot{\theta }}_{z}\mathrm{cos}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+1.5{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -2.25{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{21}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha -3{\dot{\theta }}_{z}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\\ -4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}-6{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}-6{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}-9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ -12{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}+9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}+9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +12{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +18{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-1.5{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}+6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -18{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ +2.25{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}-9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -4.5{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}-1.5{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -1.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}-4.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+2.25{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +2.25{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${c}_{22}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}+3{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{z}-3{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-3{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}-4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}-6{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ -6{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}+9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}+3{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}+4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +6{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+6{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -9{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -4.5{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-4.5{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -4.5{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+1.5{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -2.25{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+{b}_{y}\end{array}\right),$$
$${c}_{23}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}+3{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}-6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ -6{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ +3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+12{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -6{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\\ +9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -18{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ +4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+4.5{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ -1.5{\dot{x}}_{k}\mathrm{sin}2{x}_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-1.5{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\end{array}\right),$$
$${c}_{24}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}-{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +3{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{z}-1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ +3{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}+3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +4.5{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -1.5{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -1.5{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +2.25{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${c}_{25}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha -3{\dot{x}}_{k}+6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}+6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{z}+3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{z}\\ -9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}-9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ -12{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{z}+{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -3{r}_{k}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{z}\\ +4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+18{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}\\ -6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}+9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}-2{m}_{a}l{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +3{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +6{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -1.5{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}+1.5{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-1.5{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +2.25{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\end{array}\right),$$
$${c}_{31}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}-3{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}+4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}+6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +6{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}-9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-12{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -1.5{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}-1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ -3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+18{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -4.5{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}+1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -1.5{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}-2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{32}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}-3{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}-1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}+6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +6{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}-9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-12{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +1.5{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}+3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ -1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+18{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}-2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -3{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -4.5{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}+2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{z}+1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +1.5{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -2.25{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\end{array}\right),$$
$${c}_{34}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}0.5\left({I}_{y}-{I}_{z}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}+\left({I}_{y}-{I}_{x}-{I}_{z}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\\ +2\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}+0.5{m}_{a}{l}^{2}{I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\\ -1.5{r}_{k}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}-3{r}_{k}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}-3{r}_{k}{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +3{r}_{k}{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+3{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}+6{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +3{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}+3{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -3{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+4.5{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\\ -9{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}-4.5{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -4.5{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}-9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{35}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}{r}_{w}^{2}\left({I}_{y}-{I}_{x}-{I}_{z}\right){I}_{w}^{-1}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{y}+2{I}_{w}^{-1}{r}_{w}^{2}\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +0.5{I}_{w}^{-1}{r}_{w}^{2}\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-3{r}_{k}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{y}+3{r}_{k}{\dot{x}}_{k}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +3{r}_{k}{\dot{y}}_{k}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+3{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}+6{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ -6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +1.5{r}_{k}^{2}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-9{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -2.25{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+1.5{r}_{k}{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -1.5{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}+2.25{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -2.25{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\end{array}\right),$$
$${c}_{41}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}-6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ +1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}-1.5{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}+1.5{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +3{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}-2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +4.5{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +0.75{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +1.5{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +3{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -4.5{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-1.125{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${c}_{42}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{z}{\mathrm{sin}}^{2}\alpha \mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -1.5{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ -3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +4.5{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}+2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -0.75{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +1.5{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-4.5{r}_{k}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +1.125{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}2{\theta }_{z}\end{array}\right),$$
$${c}_{43}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}+\left({I}_{x}-{I}_{y}+{I}_{z}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\\ +2\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +3{r}_{k}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}+3{r}_{k}^{2}{\dot{\theta }}_{z}\mathrm{cos}{\theta }_{y}\\ +3{r}_{k}{\dot{y}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{x}}_{k}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\\ -6{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}-3{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -3{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +9{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}+4.5{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +4.5{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{z}+9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-4.5{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\end{array}\right),$$
$${c}_{44}=\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}+3{I}_{w}{r}_{w}^{-2}{r}_{k}^{2}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}-4.5{I}_{w}{r}_{w}^{-2}{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}+{b}_{ry},$$
$${c}_{45}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}0.5\left({I}_{y}-{I}_{x}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{y}+\left({I}_{x}-{I}_{y}+{I}_{z}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\\ +2\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}+0.5\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\\ -3{r}_{k}{\dot{y}}_{k}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{x}}_{k}\mathrm{sin}{\theta }_{z}+3{r}_{k}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\\ +3{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}\\ +6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}-6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}-3{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\\ -6{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}-9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}+9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +1.5{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}+9{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +3{r}_{k}{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}-2.25{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\\ +3{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-4.5{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -4.5{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{51}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{\dot{y}}_{k}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha -6{\dot{y}}_{k}\left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)+1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}{\theta }_{z}\\ +9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)+3{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +12{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-18{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ -6{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-1.5{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}\\ -3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}+4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ +1.5{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}+9{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}+3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}-2.25{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +4.5{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ +1.5{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}-2.25{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{52}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}3{\dot{x}}_{k}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha -6{\dot{x}}_{k}\left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)-1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{z}+3{r}_{k}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{z}\\ +9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \left({\mathrm{cos}}^{2}{\theta }_{x}+{\mathrm{cos}}^{2}{\theta }_{z}\right)+3{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +12{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-18{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{z}\\ -6{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{x}_{y}{\mathrm{cos}}^{2}{\theta }_{z}+1.5{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}+2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{z}\\ +3{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}-4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{z}\\ -1.5{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}+9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ +6{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}+9{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-3{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ -3{r}_{k}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}+4.5{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}{\mathrm{cos}}^{2}{\theta }_{z}\\ +2.25{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}2{\theta }_{z}+3{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -9{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}-9{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +4.5{r}_{k}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}-4.5{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}2{\theta }_{z}\\ -1.5{r}_{k}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}+2.25{r}_{k}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\end{array}\right),$$
$${c}_{53}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}\left({I}_{z}-{I}_{y}-{I}_{z}-2{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{y}\\ +2\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +3{r}_{k}^{2}{\dot{\theta }}_{y}\mathrm{cos}{\theta }_{y}-3{r}_{k}{\dot{x}}_{k}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -3{r}_{k}{\dot{y}}_{k}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}-6{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}-6{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -3{r}_{k}^{2}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}+9{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ -9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{cos}{\theta }_{z}-9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +4.5{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}-1.5{r}_{k}{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +1.5{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}-2.25{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ +2.25{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}2{\theta }_{y}\mathrm{cos}{\theta }_{z}\end{array}\right),$$
$${c}_{54}=\frac{{I}_{w}}{{r}_{w}^{2}}\left(\begin{array}{l}\left({I}_{x}-{I}_{y}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}\mathrm{sin}2{\theta }_{y}+\left({I}_{z}-{I}_{y}-{I}_{x}-2{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\\ +2\left({I}_{y}-{I}_{z}-{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\\ +0.5\left({I}_{z}-{I}_{y}-{m}_{a}{l}^{2}\right){I}_{w}^{-1}{r}_{w}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\\ +3{r}_{k}{\dot{y}}_{k}\mathrm{cos}{\theta }_{z}-3{r}_{k}{\dot{x}}_{k}\mathrm{sin}{\theta }_{z}+3{r}_{k}^{2}{\dot{\theta }}_{x}\mathrm{cos}{\theta }_{y}\\ -3{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{z}+3{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}{\theta }_{z}\\ -6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}+6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}\\ -6{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{cos}{\theta }_{y}-6{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}\\ +1.5{r}_{k}^{2}{\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}+9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{z}\\ -9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}{\theta }_{z}+6{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ -6{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}-3{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\\ +9{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{cos}{\theta }_{y}-3{r}_{k}{\dot{y}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\\ -2.25{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}-9{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +9{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\mathrm{sin}{\theta }_{z}+4.5{r}_{k}^{2}{\dot{\theta }}_{z}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}\\ -3{r}_{k}{\dot{x}}_{k}\mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}+4.5{r}_{k}{\dot{x}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{cos}{\theta }_{z}\\ +4.5{r}_{k}{\dot{y}}_{k}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}\mathrm{sin}{\theta }_{y}\mathrm{sin}{\theta }_{z}\end{array}\right),$$
$${c}_{55}=\left(\begin{array}{l}\left({I}_{x}-{I}_{y}\right){\dot{\theta }}_{y}\mathrm{sin}2{\theta }_{y}+\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}\\ +\left({I}_{y}-{I}_{z}+{m}_{a}{l}^{2}\right){\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}+{b}_{rz}\\ -\frac{3{I}_{w}{r}_{k}^{2}{\dot{\theta }}_{x}\mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}}{{r}_{w}^{2}}-\frac{3{I}_{w}{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}}{{r}_{w}^{2}}\\ +\frac{4.5{I}_{w}{r}_{k}^{2}{\dot{\theta }}_{x}{\mathrm{cos}}^{2}\alpha \mathrm{sin}2{\theta }_{x}{\mathrm{cos}}^{2}{\theta }_{y}}{{r}_{w}^{2}}+\frac{4.5{I}_{w}{r}_{k}^{2}{\dot{\theta }}_{y}{\mathrm{cos}}^{2}\alpha {\mathrm{cos}}^{2}{\theta }_{x}\mathrm{sin}2{\theta }_{y}}{{r}_{w}^{2}}\end{array}\right).$$
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Tham, B.C., Tran, T.N., Xuan-Mung, N. et al. Dynamic modeling and nonlinear feedback control of a full 3D ridable ballbot. J Braz. Soc. Mech. Sci. Eng. 44, 486 (2022). https://doi.org/10.1007/s40430-022-03768-5
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DOI: https://doi.org/10.1007/s40430-022-03768-5