3-D Dynamic Modeling and Validation of Human Arm for Torque Determination During Eating Activity Using Kane’s Method

Abstract

Upper limb disability is one of the major adversities faced by post-stroke patients. Eating is one of the fundamental activities of survival for all living beings. The robotic rehabilitation systems for people with upper limb disabilities must have the capability of assisting the patients, providing appropriate forces/torques, during various eating activities. In this study, a 3-D, four-DOF dynamic, mathematical model of human arm, including wrist and elbow joints, focusing on elbow flexion/extension motion, forearm pronation/supination, wrist flexion/extension and wrist adduction/abduction is formulated, for predicting the torques during different eating activities. A simulation study and experimental validation has been conducted involving five different food types and using two types of cutlery, which are, a fork and a spoon, to study their effect on the corresponding torques produced. It was observed that the maximum torque is obtained in both wrist and elbow joint when the subject digs into the food and eats (event B) in the majority of the eating tasks. The accuracy of the model, in terms of torque prediction, was compared to that of the load cell, for all eating activities, using RMSE as a statistical measure, to the test the performance of the model. The results indicate that 3-D dynamic model formulated fits all the torques for all eating activities very well, with the average RMSE of 0.05 Nm and the performance of the model is good. These results verify that the proposed Kane’s model, successfully models the HUL, during different eating tasks and using different types of cutlery.

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Acknowledgements

The authors would like to thank the Ministry of Education, Malaysia for supporting this research under the Fundamental Research Grant Scheme (FRGS14-107-0348).

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Correspondence to Norsinnira Zainul Azlan.

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Appendices

Appendix A

Elements of matrix \(M{\vec{\ddot{Q}}}\) can be written as:

$$\begin{aligned} k_{11} & = - m_{B} \rho_{{B_{3}^{2} }} - m_{C} \left( {r^{2} + \rho_{{C_{3} }} c_{\varphi } c_{2}^{2} + rs_{2}^{2} \rho_{{C_{3} }} c_{\varphi } s_{2} } \right) \\ & \quad - m_{D} \left[ {r^{2} + \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } c_{2}^{2} - \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } s_{2}^{2} + rc_{2}^{2} \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } + \frac{1}{2}l_{{C_{2} }} s_{\varphi } } \right] \\ & \quad - I_{B}^{ * } - I_{C}^{ * } - I_{D}^{ * } \left[ {c_{\varphi }^{2} + 2c_{\varphi } s_{2} c_{4} s_{\varphi } c_{3 - \varphi } s_{4} + c_{\varphi } s_{\varphi } c_{2} s_{3 - \varphi } c_{3 - \varphi } + s_{\varphi } c_{3 - \varphi } s_{4}^{2} c_{\varphi } c_{2} s_{3 - \varphi } } \right. \\ & \quad + s_{\varphi }^{2} c_{3 - \varphi }^{2} s_{4}^{2} + s_{\varphi } c_{3 - \varphi } c_{\varphi } c_{2} s_{3 - \varphi } + s_{\varphi }^{2} c_{3 - \varphi }^{2} - 2c_{\varphi } c_{2} c_{3 - \varphi } s_{\varphi } s_{3 - \varphi } \\ & \quad \left. +{ s_{\varphi }^{2} s_{3 - \varphi }^{2} - s_{\varphi } c_{3 - \varphi } c_{4} c_{\varphi } s_{2} s_{4} + l_{{c_{2} }}^{2} c_{\varphi }^{2} s_{2}^{2} } \right]. \\ \end{aligned}$$
(46)
$$\begin{aligned} k_{12} & = \frac{1}{2}m_{D} l_{{C_{2} }} + I_{C}^{ * } s_{\varphi } \\ & \quad + I_{D}^{ * } \left[ {c_{\varphi } c_{2} s_{3 - \varphi } c_{3 - \varphi } + s_{\varphi } c_{3 - \varphi }^{2} + s_{\varphi } c_{3 - \varphi }^{2} s_{4}^{2} - c_{\varphi }^{2} c_{2} c_{3 - \varphi } s_{3 - \varphi } + s_{\varphi } s_{3 - \varphi }^{2} c_{\varphi } } \right] \\ \end{aligned}$$
(47)
$$k_{13} = - I_{D}^{ * } \left[ { - c_{\varphi } s_{2} - s_{\varphi } c_{3 - \varphi } s_{4} c_{4} } \right]$$
(48)
$$k_{14} = - I_{D}^{ * } \left[ {c_{\varphi } c_{2} c_{3 - \varphi } - s_{\varphi } s_{3 - \varphi } } \right]$$
(49)
$$\begin{aligned} k_{21} & = - m_{D} \left[ {\frac{1}{2}rl_{{C_{2} }} c_{2} + l_{{C_{2} }} \left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } c_{2} + l_{{C_{2} }}^{2} s_{\varphi } + s_{3 - \varphi } \rho_{{D_{3} }} rc_{2} c_{4} - \rho_{{D_{3} }} rs_{2} s_{4} } \right. \\ & \quad + \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } c_{2} c_{4} + \frac{1}{2}l_{{C_{2} }} s_{\varphi } c_{4} + \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } s_{2} s_{3 - \varphi } s_{4} \\ & \left. {\quad - \frac{1}{2}c_{3 - \varphi } s_{4} l_{{C_{2} }} c_{\varphi } s_{2} + \frac{1}{2}s_{3 - \varphi } l_{{C_{2} }} c_{\varphi } s_{2} } \right] \\ & \quad + I_{C}^{ * } - I_{D}^{ * } \left[ { - c_{3 - \varphi } c_{\varphi } c_{2} s_{3 - \varphi } - c_{3 - \varphi }^{2} s_{4}^{2} s_{\varphi } + s_{3 - \varphi } c_{\varphi } c_{2} c_{3 - \varphi } - s_{3 - \varphi }^{2} s_{\varphi } - c_{3 - \varphi }^{2} c_{4}^{2} s_{\varphi } } \right] \\ \end{aligned}$$
(50)
$$k_{22} = - m_{D} \left[ { - l_{{C_{2} }}^{2} + \frac{1}{2}l_{{C_{2} }} s_{\varphi } c_{4} } \right] - I_{C}^{ * } - I_{D}^{ * } \left[ {c_{3 - \varphi }^{2} + s_{3 - \varphi }^{2} c_{\varphi } } \right]$$
(51)
$$k_{ 2 3} = k_{ 3 2} = k_{ 3 4} = 0$$
(52)
$$k_{24} = - I_{D}^{ * } s_{3 - \varphi }$$
(53)
$$k_{31} = m_{D} c_{4} \rho_{{D_{3} }} \left[ { - rs_{2} c_{3 - \varphi } + \frac{1}{2}s_{3 - \varphi } l_{{C_{2} }} c_{\varphi } s_{2} } \right] - I_{D}^{ * } c_{\varphi } s_{2}$$
(54)
$$k_{33} = - I_{D}^{ * }$$
(55)
$$\begin{aligned} k_{41} & = - m_{D} \left[ {\rho_{{D_{3} }} rc_{2} c_{4} - r\rho_{{D_{3} }} s_{2} s_{3 - \varphi } s_{4} + \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } c_{2} c_{4} \rho_{{D_{3} }} } \right. \\ & \left. {\quad + \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)\rho_{{D_{3} }} c_{\varphi } s_{2} s_{3 - \varphi } s_{4} - \frac{1}{2}\rho_{{D_{3} }} c_{3 - \varphi } s_{4} l_{{C_{2} }} c_{\varphi } s_{2} } \right] \\ & \quad - I_{D}^{ * } \left[ {c_{\varphi } c_{2} c_{3 - \varphi } - s_{\varphi } s_{3 - \varphi } } \right] \\ \end{aligned}$$
(56)
$$k_{ 4 4} = 1$$
(57)

Elements of \(\vec{G}\) matrix can be written as follows:

$$j_{11} = - \rho_{{B_{3} }} s_{1}$$
(58)
$$j_{12} = - c_{\varphi } s_{2} \rho_{{C_{3} }} s_{1} s_{2} - c_{\varphi } s_{2} \rho_{{C_{3} }} c_{1} c_{2} s_{\varphi }$$
(59)
$$\begin{aligned} j_{13} & = - rs_{1} + \frac{1}{2}l_{{C_{2} }} s_{\varphi }^{2} c_{1} s_{2} + \frac{1}{2}l_{{C_{2} }} c_{1} s_{\varphi } s_{2} \\ & \quad - \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)s_{1} c_{2} - \frac{1}{2}l_{{C_{2} }} s_{\varphi } s_{1} c_{2} - \frac{1}{2}l_{{C_{2} }} s_{1} c_{2} \\ & \quad - \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } s_{2}^{2} s_{1} + \frac{1}{2}l_{{C_{2} }} c_{\varphi }^{2} s_{2} c_{3} c_{1} \\ \end{aligned}$$
(60)
$$j_{ 2 1} = j_{ 2 2} = j_{ 3 1} = j_{ 3 2} = j_{ 4 1} = j_{ 4 2} = 0$$
(61)
$$\begin{aligned} j_{23} & = \frac{1}{2}l_{{C_{2} }} c_{1} s_{\varphi } s_{2} - \frac{1}{2}l_{{C_{2} }} s_{1} c_{2} + s_{3 - \varphi } \rho_{{D_{3} }} c_{4} c_{1} c_{2} \\ & \quad + s_{3 - \varphi } \rho_{{D_{3} }} c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi }^{2} \rho_{{D_{3} }} s_{4} c_{1} s_{2} - s_{3 - \varphi }^{2} \rho_{{D_{3} }} s_{4} s_{1} s_{\varphi } c_{2} \\ & \quad - s_{3 - \varphi } \rho_{{D_{3} }} s_{1} c_{\varphi } c_{3 - \varphi } s_{4} + c_{3 - \varphi }^{2} s_{4} \rho_{{D_{3} }} s_{1} s_{2} \\ & \quad + c_{3 - \varphi }^{2} s_{4} \rho_{{D_{3} }} c_{1} c_{2} s_{\varphi } + c_{3 - \varphi } s_{4} \rho_{{D_{3} }} s_{3 - \varphi } c_{1} c_{\varphi } \\ \end{aligned}$$
(62)
$$j_{33} = - c_{4} \rho_{{D_{3} }} c_{3 - \varphi } s_{1} s_{2} - c_{4} \rho_{{D_{3} }} c_{3 - \varphi } c_{1} c_{2} s_{\varphi } - c_{4} \rho_{{D_{3} }} c_{1} c_{\varphi }$$
(63)
$$j_{43} = \rho_{{D_{3} }} \left( { - c_{4} s_{1} c_{2} + c_{4} c_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} s_{1} s_{2} + s_{3 - \varphi } s_{4} c_{1} c_{2} s_{\varphi } - c_{\varphi } c_{1} c_{3 - \varphi } s_{4} } \right)$$
(64)

Elements of \(\vec{V}\) matrix can be written as follows:

$$z_{11}=-m_{C}\left[rc_{\varphi}s_{2}\rho_{C_{3}}s_{\varphi}c_{2}+rs_{2}c_{\varphi}c_{2}\rho_{C_{3}}s_{\varphi}\right]-m_{D}\left[\frac{1}{2}c_{2}^{2}c_{\varphi}^{2}l_{C_{2}}s_{2}-\frac{1}{2}\left(l_{CB}+l_{C_{3}}\right)c_{2}c_{\varphi}s_{2}s_{\varphi}+\frac{1}{2}l_{C_{2}}s_{\varphi}+\frac{1}{2}l_{C_{2}}s_{2}^{3}c_{\varphi}^{2}+\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)s_{\varphi}c_{\varphi}c_{2}s_{2}+\frac{1}{2}l_{C_{2}}s_{\varphi}^{2}s_{2}+c_{\varphi}^{2}s_{2}^{2}s_{4}c_{2}c_{4}^{2}\rho_{D_{3}+}c_{\varphi}^{2}s_{4}^{2}s_{2}c_{2}^{2}c_{4}\rho_{D_{3}}s_{3-\varphi}+c_{\varphi}s_{2}s_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}c_{2}c_{4}-c_{\varphi}^{2}s_{2}^{3}s_{4}^{2}s_{3-\varphi}\rho_{D_{3}}c_{4}-c_{\varphi}^{2}s_{2}^{2}s_{4}^{3}s_{3-\varphi}^{2}\rho_{D_{3}}c_{2}-c_{\varphi}^{2}s_{2}^{2}s_{4}^{3}s_{3-\varphi}^{2}\rho_{D_{3}}c_{2}-c_{\varphi}^{2}s_{2}s_{4}^{3}\rho_{D_{3}}s_{3-\varphi}s_{\varphi}c_{3-\varphi}-c_{\varphi}^{2}c_{2}^{2}c_{4}^{3}s_{3-\varphi}\rho_{D_{3}}s_{2}-c_{\varphi}^{2}c_{2}^{3}s_{3-\varphi}^{3}c_{4}\rho_{D_{3}}s_{4}-c_{2}^{2}c_{\varphi}s_{3-\varphi}c_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}s_{4}+c_{\varphi}^{2}c_{2}s_{3-\varphi}^{2}c_{4}^{2}s_{2}^{2}s_{4}\rho_{D_{3}}+c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}^{2}c_{4}s_{2}s_{4}^{2}\rho_{D_{3}}+c_{\varphi}c_{2}s_{3-\varphi}^{2}c_{4}s_{2}s_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}-s_{\varphi}c_{3-\varphi}c_{4}^{3}c_{2}\rho_{D_{3}}c_{\varphi}s_{2}-s_{\varphi}c_{3-\varphi}c_{4}^{2}c_{2}^{2}\rho_{D_{3}}c_{\varphi}s_{3-\varphi}s_{4}-s_{\varphi}^{2}c_{3-\varphi}^{2}c_{4}^{2}\rho_{D_{3}}s_{4}c_{2}+s_{\varphi}c_{3-\varphi}c_{4}^{2}s_{2}^{2}s_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}+s_{\varphi}c_{3-\varphi}c_{4}s_{2}s_{3-\varphi}^{2}s_{4}^{2}\rho_{D_{3}}c_{\varphi}c_{2}+s_{\varphi}^{2}c_{3-\varphi}^{2}c_{4}s_{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}-c_{\varphi}^{2}c_{2}c_{3-\varphi}^{2}\rho_{D_{3}}s_{2}^{2}s_{4}+c_{\varphi}^{2}c_{2}^{2}c_{3-\varphi}^{2}s_{2}\rho_{D_{3}}s_{3-\varphi}c_{4}+c_{\varphi}c_{2}c_{3-\varphi}^{3}s_{2}s_{\varphi}c_{4}\rho_{D_{3}}+s_{\varphi}s_{3-\varphi}\rho_{D_{3}}s_{2}^{2}c_{3-\varphi}c_{\varphi}s_{4}-s_{\varphi}s_{3-\varphi}^{2}s_{2}c_{3-\varphi}\rho_{D_{3}}c_{\varphi}c_{2}c_{4}-s_{\varphi}^{2}s_{3-\varphi}s_{2}c_{3-\varphi}^{2}c_{4}\rho_{D_{3}}+c_{\varphi}^{2}s_{2}c_{4}^{2}c_{2}s_{4}^{2}\rho_{D_{3}}+c_{\varphi}^{2}s_{2}c_{4}c_{2}^{2}s_{4}^{2}\rho_{D_{3}}s_{3-\varphi}+c_{\varphi}s_{2}c_{4}c_{2}s_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}+c_{\varphi}^{2}s_{2}^{2}c_{4}^{3}c_{3-\varphi}\rho_{D_{3}}s_{4}+c_{\varphi}^{2}s_{2}^{2}c_{4}^{2}c_{3-\varphi}\rho_{D_{3}}c_{2}s_{3-\varphi}s_{4}+c_{\varphi}s_{2}^{2}c_{4}^{2}c_{3-\varphi}^{2}\rho_{D_{3}}s_{\varphi}s_{4}+c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}s_{4}^{3}\rho_{D_{3}}c_{4}+c_{\varphi}^{2}c_{2}^{3}s_{3-\varphi}^{2}s_{4}^{3}\rho_{D_{3}}+c_{\varphi}c_{2}^{2}s_{3-\varphi}s_{4}^{3}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}+c_{\varphi}^{2}c_{2}s_{3-\varphi}s_{4}^{2}s_{2}c_{3-\varphi}c_{4}^{2}\rho_{D_{3}}+c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}^{2}s_{4}^{2}s_{2}c_{3-\varphi}c_{4}\rho_{D_{3}}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{2}s_{2}c_{3-\varphi}^{2}c_{4}\rho_{D_{3}}s_{\varphi}+s_{\varphi}c_{3-\varphi}c_{2}s_{4}^{2}\rho_{D_{3}}c_{\varphi}c_{4}+s_{\varphi}c_{3-\varphi}s_{4}^{3}c_{2}^{2}\rho_{D_{3}}c_{\varphi}s_{3-\varphi}+s_{\varphi}^{2}c_{3-\varphi}^{2}c_{2}s_{4}^{2}\rho_{D_{3}}+s_{\varphi}c_{3-\varphi}^{2}s_{2}c_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{4}^{2}+s_{\varphi}c_{3-\varphi}^{2}s_{4}^{2}s_{2}c_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}+s_{\varphi}^{2}c_{3-\varphi}^{3}s_{4}^{2}s_{2}c_{4}\rho_{D_{3}}-c_{\varphi}^{2}c_{2}^{2}c_{3-\varphi}^{2}\rho_{D_{3}}+c_{\varphi}c_{2}c_{3-\varphi}s_{3-\varphi}s_{\varphi}\rho_{D_{3}}+s_{3-\varphi}s_{4}c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}-s_{3-\varphi}^{2}s_{\varphi}^{2}\rho_{D_{3}}+c_{4}^{2}c_{\varphi}^{2}s_{2}^{2}\rho_{D_{3}}s_{4}-\rho_{D_{3}}c_{4}^{4}c_{\varphi}^{2}s_{2}c_{2}s_{3-\varphi}-\rho_{D_{3}}c_{4}^{4}c_{\varphi}s_{2}s_{\varphi}c_{3-\varphi}-\rho_{D_{3}}c_{4}^{4}c_{\varphi}s_{2}s_{\varphi}c_{3-\varphi}+\rho_{D_{3}}c_{\varphi}^{2}c_{4}c_{2}s_{3-\varphi}s_{4}^{2}s_{2}-\rho_{D_{3}}c_{\varphi}^{2}c_{4}^{3}c_{2}^{2}s_{3-\varphi}^{2}s_{4}-\rho_{D_{3}}c_{\varphi}c_{4}^{3}c_{2}s_{3-\varphi}s_{4}s_{\varphi}c_{3-\varphi}+\rho_{D_{3}}s_{\varphi}c_{3-\varphi}s_{4}^{2}c_{4}-\rho_{D_{3}}s_{\varphi}c_{3-\varphi}s_{4}c_{4}^{3}c_{\varphi}c_{2}s_{3-\varphi}-\rho_{D_{3}}s_{\varphi}^{2}c_{3-\varphi}^{2}s_{4}c_{4}^{3}+\rho_{D_{3}}c_{\varphi}^{2}s_{4}^{3}c_{4}^{2}s_{2}+\rho_{D_{3}}c_{\varphi}^{2}s_{4}c_{4}c_{2}s_{3-\varphi}+\rho_{D_{3}}c_{\varphi}s_{4}^{4}c_{4}s_{\varphi}c_{3-\varphi}+\rho_{D_{3}}c_{\varphi}^{2}c_{2}s_{3-\varphi}s_{2}s_{4}^{3}c_{4}+\rho_{D_{3}}c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}^{2}s_{4}^{4}+\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{4}s_{\varphi}c_{3-\varphi}+\rho_{D_{3}}s_{4}^{3}s_{\varphi}c_{3-\varphi}c_{\varphi}s_{2}c_{4}+\rho_{D_{3}}s_{4}^{4}s_{\varphi}c_{3-\varphi}c_{\varphi}c_{2}s_{3-\varphi}+\rho_{D_{3}}s_{4}^{4}s_{\varphi}^{2}c_{3-\varphi}^{2}-c_{\varphi}^{2}c_{2}^{2}c_{3-\varphi}^{2}\rho_{D_{3}}{+c}_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}s_{\varphi}s_{3-\varphi}+s_{3-\varphi}s_{\varphi}\rho_{D_{3}}c_{\varphi}c_{2}c_{3-\varphi}-s_{3-\varphi}^{2}s_{\varphi}^{2}\rho_{D_{3}} \right]$$
(65)
$$z_{12}=-m_{C}\left[-rc_{\varphi}s_{2}\rho_{C_{3}}c_{2}-\rho_{C_{3}}c_{\varphi}s_{2}c_{2}-rs_{2}c_{\varphi}c_{2}\rho_{C_{3}}+rs_{2}\rho_{C_{3}}c_{\varphi}c_{2} \right]-m_{D}\left[-\left(l_{CB}+l_{C_{3}} \right)c_{2}s_{2}c_{\varphi}+l_{C_{2}}s_{\varphi}s_{2}-c_{\varphi}s_{2}s_{4}^{2}\rho_{D_{3}}c_{3-\varphi}c_{2}c_{4}+c_{\varphi}s_{2}^{2}s_{4}^{3}s_{3-\varphi}\rho_{D_{3}}c_{3-\varphi}+c_{\varphi}^{2}c_{2}s_{3-\varphi}c_{4}^{2}\rho_{D_{3}}c_{3-\varphi}s_{4}-c_{\varphi}c_{2}s_{3-\varphi}^{2}c_{4}s_{2}s_{4}^{2}\rho_{D_{3}}c_{3-\varphi}+c_{3-\varphi}c_{4}^{3}c_{2}\rho_{D_{3}}c_{\varphi}s_{2}+c_{3-\varphi}c_{4}^{2}c_{2}^{2}\rho_{D_{3}}c_{\varphi}s_{3-\varphi}s_{4}+c_{3-\varphi}^{2}c_{4}^{2}\rho_{D_{3}}s_{\varphi}s_{4}c_{2}-c_{3-\varphi}c_{4}^{2}s_{2}^{2}s_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}-c_{3-\varphi}c_{4}s_{2}s_{3-\varphi}^{2}s_{4}^{2}\rho_{D_{3}}c_{\varphi}c_{2}-c_{3-\varphi}^{2}c_{4}s_{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}s_{\varphi}+s_{\varphi}c_{3-\varphi}^{2}c_{4}^{2}\rho_{D_{3}}s_{4}c_{2}-s_{\varphi}c_{3-\varphi}^{2}c_{4}s_{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}-c_{\varphi}c_{2}c_{3-\varphi}^{3}s_{2}\rho_{D_{3}}c_{4}-s_{3-\varphi}\rho_{D_{3}}s_{2}^{2}c_{3-\varphi}c_{\varphi}s_{4}+s_{3-\varphi}^{2}s_{2}c_{3-\varphi}\rho_{D_{3}}c_{\varphi}c_{2}c_{4}+s_{3-\varphi}s_{2}c_{3-\varphi}^{2}s_{\varphi}c_{4}\rho_{D_{3}}+s_{\varphi}s_{3-\varphi}s_{2}c_{3-\varphi}^{2}\rho_{D_{3}}c_{4}-c_{\varphi}s_{2}c_{4}c_{2}s_{4}^{2}\rho_{D_{3}}c_{3-\varphi}-c_{\varphi}s_{2}^{2}c_{4}^{2}c_{3-\varphi}^{2}\rho_{D_{3}}s_{4}+c_{\varphi}c_{2}^{2}s_{3-\varphi}s_{4}^{3}\rho_{D_{3}}c_{3-\varphi}-c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{2}s_{2}c_{3-\varphi}^{2}c_{4}\rho_{D_{3}}-c_{3-\varphi}s_{4}^{3}c_{2}\rho_{D_{3}}c_{\varphi}c_{4}-c_{3-\varphi}s_{4}^{3}c_{2}^{2}\rho_{D_{3}}c_{\varphi}s_{3-\varphi}-c_{3-\varphi}^{2}s_{4}^{3}c_{2}\rho_{D_{3}}s_{\varphi}-c_{3-\varphi}^{2}s_{4}^{2}s_{2}c_{4}^{2}\rho_{D_{3}}c_{\varphi}-c_{3-\varphi}^{2}s_{4}^{2}s_{2}c_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}-c_{3-\varphi}^{3}s_{4}^{2}s_{2}c_{4}\rho_{D_{3}}s_{\varphi}-s_{\varphi}c_{3-\varphi}^{2}s_{4}^{3}c_{2}\rho_{D_{3}}-s_{\varphi}c_{3-\varphi}^{3}s_{4}^{2}s_{2}c_{4}\rho_{D_{3}}-2c_{\varphi}c_{2}^{2}c_{3-\varphi}s_{3-\varphi}\rho_{D_{3}}s_{4}-2c_{\varphi}c_{2}c_{3-\varphi}^{2}s_{3-\varphi}\rho_{D_{3}}s_{2}c_{4}+s_{3-\varphi}^{2}s_{\varphi}\rho_{D_{3}}c_{2}s_{4}+s_{3-\varphi}^{2}s_{\varphi}\rho_{D_{3}}s_{2}c_{3-\varphi}c_{4}+s_{3-\varphi}^{2}s_{\varphi}c_{2}s_{4}\rho_{D_{3}}+s_{2}c_{3-\varphi}c_{4}s_{3-\varphi}^{2}s_{\varphi}\rho_{D_{3}}+\rho_{D_{3}}c_{4}^{4}c_{\varphi}s_{2}c_{3-\varphi}+\rho_{D_{3}}c_{\varphi}c_{4}^{3}c_{2}s_{3-\varphi}s_{4}c_{3-\varphi}-\rho_{D_{3}}c_{3-\varphi}s_{4}^{2}c_{4}c_{\varphi}c_{4}s_{2}+\rho_{D_{3}}c_{3-\varphi}s_{4}c_{\varphi}c_{2}s_{3-\varphi}c_{4}^{3}+\rho_{D_{3}}c_{3-\varphi}^{2}s_{4}s_{\varphi}c_{4}^{3}+\rho_{D_{3}}s_{\varphi}c_{3-\varphi}^{2}s_{4}c_{4}^{3}-\rho_{D_{3}}c_{\varphi}s_{4}^{4}c_{4}c_{3-\varphi}-\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{4}c_{3-\varphi}-\rho_{D_{3}}c_{3-\varphi}s_{4}^{3}c_{\varphi}s_{2}c_{4}-\rho_{D_{3}}c_{3-\varphi}s_{4}^{4}c_{\varphi}c_{2}s_{3-\varphi}-\rho_{D_{3}}s_{4}^{4}s_{\varphi}c_{3-\varphi}^{2}-c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}s_{3-\varphi}s_{4}-s_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}c_{2}c_{3-\varphi}+s_{3-\varphi}^{2}\rho_{D_{3}}s_{\varphi}s_{4}+s_{4}s_{3-\varphi}^{2}s_{\varphi}\rho_{D_{3}}+\frac{1}{2}l_{C_{2}}\left(l_{CB}+l_{C_{3}} \right)s_{\varphi}s_{2}c_{\varphi}-\rho_{D_{3}}c_{3-\varphi}^{2}s_{4}^{4}s_{\varphi}+\left(s_{3-\varphi}s_{4} \right)\left(c_{\varphi}s_{2}c_{4}^{2}\rho_{D_{3}}c_{3-\varphi}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}\rho_{D_{3}}c_{3-\varphi}c_{4}-c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{2}+c_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}c_{4}+c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}s_{\varphi}c_{4}+s_{\varphi}c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}c_{4} \right)+c_{3-\varphi}\left(c_{\varphi}s_{2}s_{4}s_{3-\varphi}\rho_{D_{3}}-c_{\varphi}c_{2}s_{3-\varphi}^{2}c_{4}\rho_{D_{3}}+c_{3-\varphi}^{2}c_{4}c_{\varphi}c_{2}\rho_{D_{3}}-c_{3-\varphi}c_{4}s_{\varphi}s_{3-\varphi}\rho_{D_{3}}-s_{\varphi}c_{3-\varphi}c_{4}s_{3-\varphi}\rho_{D_{3}}\right)-s_{3-\varphi}c_{4}\rho_{D_{3}}\left(\rho_{D_{3}}c_{\varphi}s_{4}^{2}c_{4}c_{3-\varphi}-\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{2}c_{3-\varphi}-\rho_{D_{3}}c_{3-\varphi}s_{4}c_{\varphi}s_{2}c_{4}-\rho_{D_{3}}c_{3-\varphi}s_{4}^{2}c_{\varphi}c_{2}s_{3-\varphi}-\rho_{D_{3}}c_{3-\varphi}^{2}s_{4}^{2}s_{\varphi}-\rho_{D_{3}}s_{\varphi}c_{3-\varphi}^{2}s_{4}^{2}-c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}s_{3-\varphi}-s_{3-\varphi}\rho_{D_{3}}c_{\varphi}c_{2}c_{3-\varphi}+s_{3-\varphi}^{2}\rho_{D_{3}}s_{\varphi}+s_{3-\varphi}s_{\varphi}\rho_{D_{3}} \right)+\frac{1}{2}l_{C_{2}}c_{\varphi}s_{2}\left[\left(-c_{3-\varphi}s_{4} \right)\left(c_{\varphi}s_{2}c_{4}^{2}\rho_{D_{3}}c_{3-\varphi}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}\rho_{D_{3}}c_{3-\varphi}c_{4}-c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{2}+c_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}c_{4}+c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}s_{\varphi}c_{4}+s_{\varphi}c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}c_{4} \right)+s_{3-\varphi}\left(c_{\varphi}s_{2}s_{4}s_{3-\varphi}\rho_{D_{3}}-c_{\varphi}c_{2}s_{3-\varphi}^{2}c_{4}\rho_{D_{3}}+c_{3-\varphi}^{2}c_{4}c_{\varphi}c_{2}\rho_{D_{3}}-c_{3-\varphi}c_{4}s_{\varphi}s_{3-\varphi}\rho_{D_{3}}-s_{\varphi}c_{3-\varphi}c_{4}s_{3-\varphi}\rho_{D_{3}}\right)+c_{3-\varphi}c_{4}\left(\rho_{D_{3}}c_{\varphi}s_{4}^{2}c_{4}c_{3-\varphi}-\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{2}c_{3-\varphi}-\rho_{D_{3}}c_{3-\varphi}s_{4}c_{\varphi}s_{2}c_{4}-\rho_{D_{3}}c_{3-\varphi}s_{4}^{2}c_{\varphi}c_{2}s_{3-\varphi}-\rho_{D_{3}}c_{3-\varphi}^{2}s_{4}^{2}s_{\varphi}-\rho_{D_{3}}s_{\varphi}c_{3-\varphi}^{2}s_{4}^{2}-c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}s_{3-\varphi}-s_{3-\varphi}\rho_{D_{3}}c_{\varphi}c_{2}c_{3-\varphi}+s_{3-\varphi}^{2}\rho_{D_{3}}s_{\varphi}+s_{3-\varphi}s_{\varphi}\rho_{D_{3}} \right) \right]-I_{D}^{\ast}\left[c_{\varphi}^{2}c_{4}^{2}c_{2}s_{2}+c_{\varphi}^{2}c_{2}^{2}c_{4}s_{3-\varphi}s_{4}+s_{\varphi}c_{3-\varphi}s_{4}c_{\varphi}c_{2}c_{4}+s_{\varphi}c_{3-\varphi}s_{4}c_{\varphi}c_{2}c_{4}-c_{\varphi}^{2}s_{2}^{2}c_{4}s_{3-\varphi}s_{3}-c_{\varphi}^{2}c_{2}s_{{3-\varphi}^{2}}s_{2}s_{3}-s_{\varphi}c_{3-\varphi}s_{4}c_{\varphi}s_{2}s_{3-\varphi\varphi}s_{3}-c_{\varphi}s_{2}s_{3-\varphi}s_{3}s_{\varphi}c_{3-\varphi}s_{4}-c_{\varphi}^{2}c_{2}s_{2}c_{3-\varphi}^{2}+s_{\varphi}s_{3-\varphi}c_{\varphi}s_{2}c_{3-\varphi}+c_{\varphi}^{2}c_{2}s_{2}s_{4}^{2}-c_{\varphi}^{2}c_{2}^{2}s_{4}c_{2}s_{3-\varphi}c_{4}-s_{\varphi}c_{3-\varphi}c_{4}c_{\varphi}c_{2}s_{4}+c_{\varphi}^{2}s_{2}^{2}s_{4}c_{4}s_{3-\varphi}-c_{\varphi}^{2}c_{2}s_{2}s_{3-\varphi}c_{4}^{2}-s_{\varphi}c_{\varphi}c_{3-\varphi}s_{2}s_{3-\varphi}c_{4}^{2} \right] \right]$$
(66)
$$z_{13}=-m_{D21}\left[c_{\varphi}s_{2}s_{4}c_{2}\rho_{D_{3}}c_{4}^{2}-c_{\varphi}s_{2}^{2}s_{4}^{2}s_{3-\varphi}\rho_{D_{3}}c_{4}-c_{\varphi}c_{2}^{2}s_{3-\varphi}c_{4}^{3}\rho_{D_{3}}+c_{\varphi}c_{2}s_{3-\varphi}^{2}c_{4}^{2}s_{2}s_{4}\rho_{D_{3}}-c_{3-\varphi}c_{4}s_{2}s_{3-\varphi}s_{4}\rho_{D_{3}}c_{4}-s_{\varphi}c_{3-\varphi}c_{4}^{3}c_{2}\rho_{D_{3}}+s_{\varphi}c_{3-\varphi}c_{4}^{2}s_{2}s_{3-\varphi}s_{4}\rho_{D_{3}}+s_{4}c_{2}c_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{2}+s_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}c_{2}c_{4}-s_{4}^{2}s_{2}^{2}s_{3-\varphi}\rho_{D_{3}}c_{\varphi}c_{4}-s_{4}^{3}s_{2}s_{3-\varphi}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}-c_{\varphi}c_{2}c_{3-\varphi}^{2}s_{2}s_{4}\rho_{D_{3}}+s_{\varphi}s_{3-\varphi}s_{2}c_{3-\varphi}s_{4}\rho_{D_{3}}+c_{\varphi}s_{2}c_{4}c_{2}s_{4}\rho_{D_{3}}c_{4}+c_{\varphi}s_{2}^{2}c_{4}^{3}c_{3-\varphi}\rho_{D_{3}}+c_{\varphi}c_{2}^{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}c_{4}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}s_{2}c_{3-\varphi}c_{4}^{2}\rho_{D_{3}}+s_{\varphi}c_{3-\varphi}c_{2}s_{4}\rho_{D_{3}}c_{4}+s_{\varphi}c_{3-\varphi}^{2}s_{4}s_{2}c_{4}^{2}\rho_{D_{3}}+c_{4}^{2}c_{2}s_{4}^{2}\rho_{D_{3}}c_{\varphi}+c_{4}c_{2}^{2}\rho_{D_{3}}c_{\varphi}s_{3-\varphi}+c_{4}c_{2}s_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}+c_{4}^{3}s_{2}c_{3-\varphi}\rho_{D_{3}}c_{\varphi}s_{4}+c_{4}^{2}s_{2}c_{3-\varphi}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}+c_{4}s_{2}c_{3-\varphi}^{2}c_{4}\rho_{D_{3}}s_{\varphi}s_{4}+\rho_{D_{3}}c_{4}^{2}c_{\varphi}s_{2}s_{4}+\rho_{D_{3}}c_{\varphi}c_{4}^{2}c_{2}s_{3-\varphi}s_{4}^{2}+\rho_{D_{3}}s_{\varphi}c_{3-\varphi}s_{4}^{2}c_{4}^{2}+\rho_{D_{3}}c_{4}^{3}c_{\varphi}s_{2}s_{4}-\rho_{D_{3}}c_{4}^{4}c_{\varphi}c_{2}s_{3-\varphi}-\rho_{D_{3}}c_{4}^{4}s_{\varphi}c_{3-\varphi}+\rho_{D_{3}}c_{\varphi}s_{4}^{3}c_{4}^{2}+\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{3}c_{4}+\rho_{D_{3}}s_{4}^{3}s_{\varphi}c_{3-\varphi}c_{4}+\rho_{D_{3}}c_{4}^{2}s_{4}^{2}c_{\varphi}s_{2}+\rho_{D_{3}}c_{4}s_{4}^{3}c_{\varphi}c_{2}s_{3-\varphi}+\rho_{D_{3}}s_{4}^{3}c_{4}s_{\varphi}c_{3-\varphi}+\rho_{D_{3}}c_{4}s_{4}^{2}c_{2}^{2}+s_{3-\varphi}s_{4}\left(c_{\varphi}s_{2}c_{4}\rho_{D_{3}}s_{4}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}+s_{\varphi}c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}+c_{4}\rho_{D_{3}}c_{\varphi}s_{2}s_{4}-c_{4}^{2}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}-c_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi} \right)+c_{3-\varphi}\left(s_{4}c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}-s_{\varphi}s_{3-\varphi}\rho_{D_{3}}s_{4} \right)-s_{3-\varphi}c_{4}\rho_{D_{3}}\left(c_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{4}+c_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}+c_{4}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}s_{4}+\rho_{D_{3}}c_{4}^{2}c_{\varphi}s_{2}+\rho_{D_{3}}c_{4}s_{\varphi}c_{3-\varphi}s_{4} \right)+\frac{1}{2}l_{C_{2}}c_{\varphi}s_{2}\left[\left(-c_{3-\varphi}s_{4} \right)\left(c_{\varphi}s_{2}c_{4}\rho_{D_{3}}s_{4}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}+s_{\varphi}c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}+c_{4}\rho_{D_{3}}c_{\varphi}s_{2}s_{4}-c_{4}^{2}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}-c_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi} \right)+s_{3-\varphi}\left(s_{4}c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}-s_{\varphi}s_{3-\varphi}\rho_{D_{3}}s_{4} \right)+c_{3-\varphi}c_{4}\left(c_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{4}+c_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{4}+c_{4}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}s_{4}+\rho_{D_{3}}c_{4}^{2}c_{\varphi}s_{2}+\rho_{D_{3}}c_{4}s_{\varphi}c_{3-\varphi}s_{4} \right) \right] \right]-I_{D}^{\ast}\left[\left(c_{\varphi}s_{2}c_{4}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}+s_{\varphi}c_{3-\varphi}s_{4}+s_{\varphi}c_{3-\varphi}s_{4} \right)\left(c_{\varphi}c_{2}c_{3-\varphi}s_{4}-s_{\varphi}s_{3-\varphi}s_{4} \right)+\left(c_{\varphi}c_{2}c_{3-\varphi}-s_{\varphi}s_{3-\varphi} \right)\left(-c_{\varphi}c_{2}s_{3-\varphi}-s_{\varphi}c_{3-\varphi} \right)+\left(c_{\varphi}s_{2}s_{4}-c_{\varphi}c_{2}s_{3-\varphi}c_{4}-s_{\varphi}c_{3-\varphi}c_{4} \right)\left(-c_{\varphi}c_{2}c_{3-\varphi}c_{4}+s_{\varphi}s_{3-\varphi}c_{4} \right) \right]$$
(67)
$$\begin{aligned} z_{14} & = - m_{D} \left[ { - \rho_{{D_{3} }} s_{2} c_{3 - \varphi } c_{\varphi } s_{2} s_{4} + s_{2} c_{3 - \varphi } \rho_{{D_{3} }} c_{\varphi } c_{2} s_{3 - \varphi } c_{4} + s_{2} c_{3 - \varphi }^{2} s_{\varphi } c_{4} \rho_{{D_{3} }} } \right. \\ & \quad + 2s_{3 - \varphi } s_{\varphi } \rho_{{D_{3} }} \left( {c_{2} s_{4} + s_{2} c_{3 - \varphi } c_{4} } \right) - s_{4} c_{\varphi } c_{2} c_{3 - \varphi } \rho_{{D_{3} }} - s_{4} \rho_{{D_{3} }} c_{\varphi } c_{2} + s_{4} \rho_{{D_{3} }} s_{\varphi } s_{3 - \varphi } \\ & \quad + c_{3 - \varphi } \left( {c_{\varphi } s_{2} s_{4} \rho_{{D_{3} }} - \rho_{{D_{3} }} c_{\varphi } c_{2} s_{3 - \varphi } c_{4} - s_{\varphi } c_{3 - \varphi } c_{4} \rho_{{D_{3} }} 22} \right) \\ & \quad - s_{3 - \varphi } c_{4} \rho_{{D_{3} }} \left( {\rho_{{D_{3} }} s_{3 - \varphi } s_{\varphi } - \rho_{{D_{3} }} c_{\varphi } c_{2} c_{3 - \varphi } - \rho_{{D_{3} }} s_{\varphi } s_{3 - \varphi } } \right) \\ & \quad + \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} \left[ {s_{3 - \varphi } \left( {c_{\varphi } s_{2} s_{4} \rho_{{D_{3} }} - \rho_{{D_{3} }} c_{\varphi } c_{2} s_{3 - \varphi } c_{4} - s_{\varphi } c_{3 - \varphi } c_{4} \rho_{{D_{3} }} } \right)} \right. \\ & \quad \left. {\left. { + c_{3 - \varphi } c_{4} \left( {\rho_{{D_{3} }} s_{3 - \varphi } s_{\varphi } - \rho_{{D_{3} }} c_{\varphi } c_{2} c_{3 - \varphi } - \rho_{{D_{3} }} s_{\varphi } s_{3 - \varphi } } \right)} \right]} \right] \\ & \quad - I_{D}^{ * } \left[ {\left( {c_{\varphi } s_{2} c_{4} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} } \right)\left( { - c_{\varphi } s_{2} s_{4} + c_{\varphi } c_{2} s_{3 - \varphi } c_{4} + s_{\varphi } c_{3 - \varphi } c_{4} } \right)} \right. \\ & \quad \left. { + \left( {c_{\varphi } s_{2} s_{4} - c_{\varphi } c_{2} s_{3 - \varphi } c_{4} - s_{\varphi } c_{3 - \varphi } c_{4} } \right)\left( {c_{\varphi } s_{2} c_{4} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} } \right)} \right] \\ \end{aligned}$$
(68)
$$\begin{aligned} z_{15} & = - m_{D} \left[ {\frac{1}{2}l_{{C_{2} }} s_{2} - c_{3 - \varphi }^{2} c_{4}^{2} \rho_{{D_{3} }} s_{4} c_{2} + c_{3 - \varphi }^{2} c_{4} s_{2} s_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }} } \right. \\ & \quad - s_{3 - \varphi } s_{2} c_{3 - \varphi }^{2} \rho_{{D_{3} }} c_{4} + c_{3 - \varphi }^{3} s_{4}^{2} s_{2} c_{4} \rho_{{D_{3} }} - s_{3 - \varphi }^{2} \rho_{{D_{3} }} \left( {c_{2} s_{4} + s_{2} c_{3 - \varphi } c_{4} } \right) \\ & \quad - \rho_{{D_{3} }} c_{3 - \varphi }^{2} s_{4} c_{4}^{3} + \rho_{{D_{3} }} c_{3 - \varphi }^{2} s_{4}^{4} - s_{3 - \varphi }^{2} \rho_{{D_{3} }} s_{4} + l_{{C_{2} }} \left( {l_{CB} + l_{{C_{3} }} } \right)s_{2} c_{\varphi } \\ & \quad - s_{3 - \varphi } s_{{4^{2} }} c_{3 - \varphi }^{2} \rho_{{D_{3} }} c_{4} + c_{3 - \varphi }^{2} s_{3 - \varphi } \rho_{{D_{3} }} c_{4} - s_{3 - \varphi } c_{4} \rho_{{D_{3} }} \left( {\rho_{{D_{3} }} c_{3 - \varphi }^{2} s_{4}^{2} - s_{3 - \varphi }^{2} \rho_{{D_{3} }} } \right) \\ & \quad \left. { + \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} \left[ {c_{3 - \varphi }^{3} s_{4}^{2} \rho_{{D_{3} }} c_{4} + c_{3 - \varphi } s_{3 - \varphi }^{2} \rho_{{D_{3} }} c_{4} + c_{3 - \varphi } c_{4} \left( {\rho_{{D_{3} }} c_{3 - \varphi }^{2} s_{4}^{2} - s_{3 - \varphi }^{2} \rho_{{D_{3} }} } \right)} \right]} \right] \\ \end{aligned}$$
(69)
$$\begin{aligned} z_{16} & = - m_{D} \left[ {c_{3 - \varphi } c_{4}^{4} c_{2} \rho_{{D_{3} }} - s_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } c_{2} c_{4} + s_{4}^{3} s_{2} s_{3 - \varphi } \rho_{{D_{3} }} c_{3 - \varphi } } \right. \\ & \quad - s_{3 - \varphi } s_{2} c_{3 - \varphi } s_{4} \rho_{{D_{3} }} - \rho_{{D_{3} }} c_{3 - \varphi } s_{4}^{3} c_{2} c_{4} - \rho_{{D_{3} }} c_{3 - \varphi }^{2} s_{4} s_{2} c_{4}^{2} \\ & \quad - c_{4} c_{2} s_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } - c_{4}^{2} s_{2} c_{3 - \varphi }^{2} \rho_{{D_{3} }} s_{4} - \rho_{{D_{3} }} c_{3 - \varphi } s_{4}^{2} c_{4}^{2} + \rho_{{D_{3} }} c_{4}^{4} c_{3 - \varphi } \\ & \quad - \rho_{{D_{3} }} c_{3 - \varphi } s_{4}^{3} c_{4} - \rho_{{D_{3} }} c_{4} s_{4}^{3} c_{3 - \varphi } + s_{3 - \varphi } s_{4} \left( {c_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } - s_{4}^{2} c_{3 - \varphi } \rho_{{D_{3} }} } \right) \\ & \quad + s_{3 - \varphi } c_{3 - \varphi } s_{4} \rho_{{D_{3} }} - \rho_{{D_{3} }} s_{3 - \varphi } c_{4} \left( {\rho_{{D_{3} }} c_{3 - \varphi } s_{4} c_{4} + \rho_{{D_{3} }} c_{4} c_{3 - \varphi } s_{4} } \right) \\ & \quad + \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} \left[ { - c_{3 - \varphi } s_{4} \left( {c_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } - s_{4}^{2} c_{3 - \varphi } \rho_{{D_{3} }} } \right) + s_{3 - \varphi }^{2} s_{4} \rho_{{D_{3} }} + c_{3 - \varphi } c_{4} \left( {\rho_{{D_{3} }} c_{3 - \varphi } s_{4} c_{4} + \rho_{{D_{3} }} c_{4} c_{3 - \varphi } s_{4} } \right)} \right] \\ & \quad - I_{D}^{ * } \left[ {s_{3 - \varphi } s_{4} \left( {c_{\varphi } s_{2} c_{4} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} } \right) + c_{\varphi } c_{3 - \varphi } \left( {c_{\varphi } c_{2} c_{3 - \varphi } - s_{\varphi } s_{3 - \varphi } } \right)} \right. \\ & \quad \left. {\left. { - s_{3 - \varphi } c_{4} \left( {c_{\varphi } s_{2} s_{4} - c_{\varphi } c_{2} s_{3 - \varphi } c_{4} - s_{\varphi } c_{3 - \varphi } c_{4} } \right)} \right]} \right] \\ \end{aligned}$$
(70)
$$\begin{aligned} z_{17} & = - m_{D} \left[ { - s_{2} c_{3 - \varphi }^{2} \rho_{{D_{3} }} c_{4} - 2s_{3 - \varphi } \rho_{{D_{3} }} \left( {c_{2} s_{4} + s_{2} c_{3 - \varphi } c_{4} } \right)} \right. \\ & \quad - 2s_{3 - \varphi } \rho_{{D_{3} }} s_{4} + \rho_{{D_{3} }} c_{3 - \varphi }^{2} c_{4} + \rho_{{D_{3} }}^{3} s_{3 - \varphi }^{2} c_{4} \\ & \quad + \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} \left[ {s_{3 - \varphi } c_{3 - \varphi } c_{4} \rho_{{D_{3} }} + 2c_{3 - \varphi } c_{4} s_{3 - \varphi } \rho_{{D_{3} }} } \right] \\ & \quad - I_{D}^{ * } \left[ { - c_{3 - \varphi } c_{4} \left( {c_{\varphi } s_{2} c_{4} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} } \right)} \right. \\ & \quad \left. {\left. { - c_{3 - \varphi } s_{4} \left( {c_{\varphi } s_{2} s_{4} - c_{\varphi } c_{2} s_{3 - \varphi } c_{4} - s_{\varphi } c_{3 - \varphi } c_{4} } \right)} \right]} \right] \\ \end{aligned}$$
(71)
$$\begin{aligned} z_{18} = - m_{D} \left[ {\rho_{{D_{3} }} s_{4} c_{4}^{2} c_{2} + s_{4}^{3} s_{2} s_{3 - \varphi } \rho_{{D_{3} }} c_{4} + c_{4}^{2} c_{2} s_{4} \rho_{{D_{3} }} + c_{4}^{3} s_{2} c_{3 - \varphi } \rho_{{D_{3} }} } \right. \hfill \\ + \rho_{{D_{3} }} c_{4}^{3} s_{4} + \rho_{{D_{3} }} c_{4}^{2} s_{4}^{2} + s_{3 - \varphi } c_{4} s_{4}^{2} \rho_{{D_{3} }} - s_{3 - \varphi } c_{4}^{3} \rho_{{D_{3} }} \hfill \\ \left. { + \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} \left[ { - \rho_{{D_{3} }} c_{3 - \varphi } c_{4} s_{4}^{2} + \rho_{{D_{3} }} c_{3 - \varphi } c_{4}^{3} } \right]} \right] \hfill \\ \end{aligned}$$
(72)
$$\begin{aligned} z_{19} & = - m_{D} \left[ { - s_{2} c_{3 - \varphi } s_{4} \rho_{{D_{3} }} + c_{3 - \varphi } s_{4} \rho_{{D_{3} }} + \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} \rho_{{D_{3} }} s_{3 - \varphi } s_{4} } \right] \\ & \quad - I_{D}^{ * } \left[ { - s_{4} \left( {c_{\varphi } s_{2} c_{4} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} } \right) + c_{4} \left( {c_{\varphi } s_{2} s_{4} - c_{\varphi } c_{2} s_{3 - \varphi } c_{4} - s_{\varphi } c_{3 - \varphi } c_{4} } \right)} \right] \\ \end{aligned}$$
(73)
$$z_{20} = - m_{D} \left[ { - \rho_{{D_{3} }} \left( {c_{2} s_{4} + s_{2} c_{3 - \varphi } c_{4} + s_{4} + s_{3 - \varphi } c_{4} - \frac{1}{2}l_{{C_{2} }} c_{\varphi } s_{2} c_{3 - \varphi } c_{4} } \right)} \right]$$
(74)
$$z_{21}=-m_{D}\left[- \frac{1}{2}l_{{C_{2}}} rs_{\varphi} s_{2} + l_{{C_{2}}}^{2} c_{\varphi}^{2} c_{2} s_{2} - l_{{C_{2}}} \left({l_{CB} + l_{{C_{3}}}} \right)c_{\varphi} s_{2} s_{\varphi} + \frac{1}{2}l_{{C_{2}}}\left[c_{4}^{2}s_{2}^{2}c_{\varphi}^{2}\rho_{D_{3}}s_{4}-c_{4}^{3}c_{\varphi}^{2}s_{2}c_{2}s_{3-\varphi}\rho_{D_{3}}-c_{4}^{3}c_{\varphi}s_{2}s_{\varphi}c_{3-\varphi}\rho_{D_{3}}+c_{4}c_{\varphi}^{2}c_{2}s_{3-\varphi}s_{2}s_{4}^{2}\rho_{D_{3}}-c_{4}^{2}c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}^{2}s_{4}\rho_{D_{3}}-c_{4}^{2}c_{\varphi}c_{2}s_{3-\varphi}s_{4}s_{\varphi}\rho_{D_{3}}+c_{4}s_{\varphi}c_{3-\varphi}s_{4}^{2}c_{\varphi}s_{2}\rho_{D_{3}}-c_{4}^{2}s_{\varphi}c_{3-\varphi}s_{4}c_{\varphi}c_{2}s_{3-\varphi}\rho_{D_{3}}-c_{4}^{2}s_{\varphi}^{2}c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}+2s_{4}^{2}c_{\varphi}^{2}c_{4}^{2}s_{2}\rho_{D_{3}}+2s_{4}^{2}\rho_{D_{3}}c_{\varphi}c_{4}s_{\varphi}c_{3-\varphi}+s_{4}^{2}c_{\varphi}^{2}c_{2}s_{3-\varphi}s_{2}c_{4}\rho_{D_{3}}+s_{4}^{3}\rho_{D_{3}}c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}^{2}+2s_{4}^{3}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}s_{\varphi}c_{3-\varphi}+s_{4}^{3}\rho_{D_{3}}s_{\varphi}^{2}c_{3-\varphi}^{2}-c_{\varphi}^{2}c_{2}^{2}c_{3-\varphi}^{2}\rho_{D_{3}}s_{4}+c_{\varphi}c_{2}c_{3-\varphi}s_{4}s_{\varphi}s_{3-\varphi}\rho_{D_{3}}+s_{3-\varphi}s_{\varphi}s_{4}c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}-s_{3-\varphi}^{2}s_{\varphi}^{2}s_{4}\rho_{D_{3}} \right]+s_{3-\varphi}\rho_{D_{3}}\left[rc_{\varphi}c_{3-\varphi}s_{4}+c_{4}c_{\varphi}^{2}c_{2}\frac{1}{2}l_{C_{2}}s_{2}-\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)c_{4}c_{\varphi}s_{2}s_{\varphi}+s_{3-\varphi}s_{4}\left(-\frac{1}{2}l_{C_{2}}c_{\varphi}^{2}s_{2}^{2}-\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)s_{\varphi}c_{\varphi}c_{2}-\frac{1}{2}l_{C_{2}}s_{\varphi}^{2} \right)-c_{3-\varphi}s_{4}\left(-\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)c_{\varphi}^{2}-\frac{1}{2}l_{C_{2}}s_{\varphi}c_{\varphi}c_{2}+c_{\varphi}^{2}s_{2}^{2}c_{4}s_{4}\rho_{D_{3}}-c_{\varphi}^{2}s_{2}c_{4}^{2}\rho_{D_{3}}c_{2}s_{3-\varphi}-c_{\varphi}s_{2}c_{4}^{2}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}+c_{\varphi}^{2}c_{2}s_{3-\varphi}s_{4}^{2}\rho_{D_{3}}s_{2}-c_{\varphi}^{2}c_{2}^{2}s_{3-\varphi}^{2}s_{4}\rho_{D_{3}}c_{4}-c_{\varphi}c_{2}s_{3-\varphi}s_{4}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}c_{4}+s_{\varphi}c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{2}-s_{\varphi}c_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}c_{4}-s_{\varphi}^{2}c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}c_{4} \right) \right]+c_{3-\varphi}s_{4}\rho_{D_{3}}\left[-r\left(s_{\varphi}c_{2}c_{3-\varphi}+c_{\varphi}s_{3-\varphi} \right)-\frac{1}{2}l_{C_{2}}c_{\varphi}^{2}s_{2}^{2}c_{3-\varphi}+\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)s_{\varphi}c_{\varphi}c_{2}c_{3-\varphi}+\frac{1}{2}l_{C_{2}}s_{\varphi}^{2}c_{3-\varphi}-\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)c_{\varphi}^{2}s_{3-\varphi}-\frac{1}{2}l_{C_{2}}s_{\varphi}c_{\varphi}c_{2}s_{3-\varphi}+\rho_{D_{3}}c_{\varphi}^{2}c_{2}c_{3-\varphi}s_{2}s_{4}-\rho_{D_{3}}c_{\varphi}^{2}c_{2}^{2}c_{3-\varphi}s_{3-\varphi}c_{4}-\rho_{D_{3}}c_{\varphi}c_{2}c_{3-\varphi}^{2}s_{\varphi}c_{4}-\rho_{D_{3}}s_{\varphi}s_{3-\varphi}c_{\varphi}s_{2}s_{4}+\rho_{D_{3}}s_{\varphi}s_{3-\varphi}^{2}c_{\varphi}c_{2}c_{4}+\rho_{D_{3}}s_{\varphi}^{2}s_{3-\varphi}c_{3-\varphi}c_{4} \right] \right]$$
(75)
$$z_{22}=-m_{D}\left[\frac{1}{2}l_{C_{2}}\left[-\left(l_{CB}+l_{C_{3}} \right)c_{\varphi}s_{2}+c_{4}^{3}c_{\varphi}s_{2}\rho_{D_{3}}c_{3-\varphi}+c_{4}^{2}c_{\varphi}c_{2}s_{3-\varphi}s_{4}c_{3-\varphi}\rho_{D_{3}}-c_{4}c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{2}+c_{4}^{2}c_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}+c_{4}^{2}c_{3-\varphi}^{2}s_{4}s_{\varphi}\rho_{D_{3}}+c_{4}^{2}s_{\varphi}c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}-s_{4}^{3}\rho_{D_{3}}c_{\varphi}c_{4}c_{3-\varphi}-s_{4}^{3}c_{\varphi}c_{2}s_{3-\varphi}c_{3-\varphi}\rho_{D_{3}}-s_{4}^{2}\rho_{D_{3}}c_{3-\varphi}c_{\varphi}s_{2}c_{4}-s_{4}^{2}\rho_{D_{3}}c_{3-\varphi}c_{\varphi}c_{2}s_{3-\varphi}s_{4}-s_{4}^{3}\rho_{D_{3}}c_{3-\varphi}^{2}s_{\varphi}-s_{4}^{3}\rho_{D_{3}}s_{\varphi}c_{3-\varphi}^{2}-c_{\varphi}c_{2}c_{3-\varphi}\rho_{D_{3}}s_{4}s_{3-\varphi}-s_{3-\varphi}\rho_{D_{3}}s_{4}c_{\varphi}c_{2}c_{3-\varphi}+s_{3-\varphi}^{2}s_{\varphi}\rho_{D_{3}}s_{4} \right]+\rho_{D_{3}}s_{3-\varphi}\left[-\left(l_{CB}+l_{C_{3}} \right)c_{\varphi}s_{2}+l_{C_{2}}s_{\varphi}+c_{\varphi}s_{2}c_{4}^{2}\rho_{D_{3}}c_{3-\varphi}+c_{\varphi}c_{2}s_{3-\varphi}s_{4}\rho_{D_{3}}c_{3-\varphi}c_{4}-c_{3-\varphi}s_{4}^{2}\rho_{D_{3}}c_{\varphi}s_{2}+c_{3-\varphi}s_{4}\rho_{D_{3}}c_{\varphi}c_{2}s_{3-\varphi}c_{4}+c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}s_{\varphi}c_{4}+s_{\varphi}c_{3-\varphi}^{2}s_{4}\rho_{D_{3}}c_{4} \right]+c_{3-\varphi}s_{4}\rho_{D_{3}}\left[c_{3-\varphi}l_{C_{2}}c_{\varphi}+\rho_{D_{3}}c_{\varphi}c_{2}c_{3-\varphi}^{2}c_{4}+\rho_{D_{3}}s_{3-\varphi}c_{\varphi}s_{2}s_{4}-\rho_{D_{3}}s_{3-\varphi}^{2}c_{\varphi}c_{2}c_{4}-{2\rho}_{D_{3}}s_{3-\varphi}s_{\varphi}c_{3-\varphi}c_{4} \right] \right]-I_{D}^{\ast}\left[{-c}_{3-\varphi}s_{4}\left(c_{\varphi}c_{2}c_{4}-c_{\varphi}s_{2}s_{3-\varphi}s_{3} \right)-s_{3-\varphi}c_{\varphi}s_{2}c_{3-\varphi}+c_{3-\varphi}c_{4}\left(c_{\varphi}c_{2}s_{4}+c_{\varphi}s_{2}s_{3-\varphi}c_{4} \right) \right]$$
(76)
$$\begin{aligned} z_{23} & = - m_{D} \left[ {\frac{1}{2}l_{{C_{2} }} \left[ {\rho_{{D_{3} }} s_{4} c_{4}^{2} c_{\varphi } s_{2} + \rho_{{D_{3} }} c_{4} c_{\varphi } c_{2} s_{3 - \varphi } s_{4}^{2} } \right.} \right. \\ & \quad + \rho_{{D_{3} }} c_{4} s_{\varphi } c_{3 - \varphi } s_{4}^{2} + \rho_{{D_{3} }} c_{4}^{2} c_{\varphi } s_{2} s_{4} - \rho_{{D_{3} }} c_{4}^{3} c_{\varphi } c_{2} s_{3 - \varphi } \\ & \quad - c_{4}^{3} \rho_{{D_{3} }} s_{\varphi } c_{3 - \varphi } + s_{4}^{2} c_{\varphi } c_{4}^{2} \rho_{{D_{3} }} + \rho_{{D_{3} }} c_{4} s_{4}^{2} c_{\varphi } c_{2} s_{3 - \varphi } \\ & \quad \left. { + s_{4}^{2} \rho_{{D_{3} }} s_{\varphi } c_{3 - \varphi } c_{4} + s_{4} \rho_{{D_{3} }} c_{4}^{2} c_{\varphi } s_{2} + s_{4}^{2} \rho_{{D_{3} }} c_{4} c_{\varphi } c_{2} s_{3 - \varphi } + s_{4}^{2} \rho_{{D_{3} }} c_{4} s_{\varphi } c_{3 - \varphi } } \right] \\ & \quad + s_{3 - \varphi } \rho_{{D_{3} }} \left[ {c_{\varphi } s_{2} c_{4} \rho_{{D_{3} }} s_{4} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }} + s_{\varphi } c_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }} } \right. \\ & \quad \left. { + \rho_{{D_{3} }} c_{\varphi } s_{2} s_{4} c_{4} - c_{4}^{2} \rho_{{D_{3} }} c_{\varphi } c_{2} s_{3 - \varphi } - c_{4}^{2} \rho_{{D_{3} }} s_{\varphi } c_{3 - \varphi } } \right] \\ & \quad \left. { + c_{3 - \varphi } s_{4} \rho_{{D_{3} }} \left[ {\rho_{{D_{3} }} c_{\varphi } c_{2} c_{3 - \varphi } s_{4} - \rho_{{D_{3} }} s_{4} s_{\varphi } s_{3 - \varphi } } \right]} \right] \\ & \quad - I_{D}^{ * } \left[ { - c_{3 - \varphi } s_{4} \left( {c_{\varphi } c_{2} c_{3 - \varphi } s_{4} - s_{\varphi } s_{3 - \varphi } s_{4} } \right) + s_{3 - \varphi } \left( { - c_{\varphi } c_{2} s_{3 - \varphi } - s_{\varphi } c_{3 - \varphi } } \right)} \right. \\ & \quad \left. { + c_{3 - \varphi } c_{4} \left( { - c_{\varphi } c_{2} c_{3 - \varphi } c_{4} + s_{\varphi } s_{3 - \varphi } c_{4} } \right)} \right] \\ \end{aligned}$$
(77)
$$\begin{aligned} z_{24} & = - m_{D} \left[ {\frac{1}{2}l_{{C_{2} }} \left[ { - c_{\varphi } c_{2} c_{3 - \varphi } \rho_{{D_{3} }} s_{4} + s_{3 - \varphi } s_{\varphi } \rho_{{D_{3} }} s_{4} - \rho_{{D_{3} }} s_{4} c_{\varphi } c_{2} c_{3 - \varphi } + \rho_{{D_{3} }} s_{4} s_{\varphi } s_{3 - \varphi } } \right]} \right. \\ & \left. { + \rho_{{D_{3} }} c_{3 - \varphi } s_{4} \left[ {\rho_{{D_{3} }} c_{\varphi } s_{2} s_{4} - \rho_{{D_{3} }} s_{\varphi } c_{3 - \varphi } c_{4} } \right]} \right] \\ & - I_{D}^{ * } \left[ { - c_{3 - \varphi } s_{4} \left( { - c_{\varphi } s_{2} s_{4} + c_{\varphi } c_{2} s_{3 - \varphi } c_{4} + s_{\varphi } c_{3 - \varphi } c_{4} } \right)} \right. \\ & \left. { + c_{4} c_{3 - \varphi } \left( {c_{\varphi } s_{2} c_{1} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} + s_{\varphi } c_{3 - \varphi } s_{4} } \right)} \right] \\ \end{aligned}$$
(78)
$$\begin{aligned} z_{25} & = - m_{D} \left[ {\frac{1}{2}l_{{C_{2} }} \left[ { - c_{4}^{2} c_{3 - \varphi }^{2} s_{4} \rho_{{D_{3} }} + s_{4}^{3} c_{3 - \varphi }^{2} \rho_{{D_{3} }} - s_{3 - \varphi }^{2} \rho_{{D_{3} }} s_{4} } \right]} \right. \\ & \quad + \rho_{{D_{3} }} s_{3 - \varphi } \left[ {\frac{1}{2}l_{{C_{2} }} s_{3 - \varphi } s_{4} - c_{3 - \varphi }^{2} s_{4} c_{4} \rho_{{D_{3} }} } \right] \\ & \quad \left. { + \rho_{{D_{3} }} c_{3 - \varphi } s_{4} \left[ {\frac{1}{2}l_{{C_{2} }} c_{3 - \varphi } + \rho_{{D_{3} }} s_{3 - \varphi } c_{3 - \varphi } c_{4} } \right]} \right] \\ \end{aligned}$$
(79)
$$\begin{aligned} z_{26} & = - m_{D} \left[ {\frac{1}{2}l_{{C_{2} }} \left[ { - c_{4} c_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }} + \rho_{{D_{3} }} c_{4}^{3} c_{3 - \varphi } - s_{4}^{3} \rho_{{D_{3} }} c_{3 - \varphi } c_{4} - s_{4}^{2} \rho_{{D_{3} }} c_{4} c_{3 - \varphi } } \right]} \right. \\ & \left. {\quad + s_{3 - \varphi } \rho_{{D_{3} }} \left[ {c_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } } \right] + c_{3 - \varphi } \rho_{{D_{3} }}^{2} s_{4}^{2} s_{3 - \varphi } } \right] \\ & \quad - I_{D}^{ * } \left[ { - c_{3 - \varphi } s_{3 - \varphi } s_{4}^{2} + s_{3 - \varphi } c_{\varphi } c_{3 - \varphi } + s_{3 - \varphi } c_{4}^{2} c_{3 - \varphi } } \right] \\ \end{aligned}$$
(80)
$$z_{27} = - m_{D} \left[ { - l_{{C_{2} }} \rho_{{D_{3} }} s_{3 - \varphi } s_{4} + \rho_{{D_{3} }}^{2} c_{3 - \varphi }^{2} c_{4} s_{4} } \right]$$
(81)
$$z_{28} = - m_{D} \left[ {l_{{C_{2} }} c_{4}^{2} s_{4} \rho_{{D_{3} }} + s_{3 - \varphi } \rho_{{D_{3} }}^{2} c_{4} s_{4} } \right]$$
(82)
$$z_{29} = - m_{D} \left[ {c_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }}^{2} } \right] - I_{D}^{ * } c_{3 - \varphi }$$
(83)
$$z_{30} = - \frac{1}{2}l_{{C_{2} }} m_{D} \rho_{{D_{3} }} s_{4}$$
(84)
$$z_{31}=m_{D}c_{4}\rho_{D_{3}}\left[- r\left({s_{\varphi} c_{2} c_{3 - \varphi} + c_{\varphi} s_{3 - \varphi}} \right) + c_{3 - \varphi} \left({- \frac{1}{2}l_{{c_{2}}} c_{\varphi}^{2} s_{2}^{2} + \frac{1}{2}\left({l_{CB} + l_{{C_{3}}}} \right)s_{\varphi} c_{\varphi} c_{2} + \frac{1}{2}l_{{C_{2}}} s_{\varphi}^{2}} \right) +s_{3-\varphi}\left(-\frac{1}{2}\left(l_{CB}+l_{C_{3}} \right)c_{\varphi}^{2}-\frac{1}{2}l_{C_{2}}s_{\varphi}c_{\varphi}c_{2}+\rho_{D_{3}}c_{\varphi}^{2}c_{2}c_{3-\varphi}s_{2}s_{4}-\rho_{D_{3}}c_{\varphi}^{2}c_{2}^{2}c_{3-\varphi}s_{3-\varphi}c_{4}-\rho_{D_{3}}s_{\varphi}c_{\varphi}c_{2}c_{4}+\rho_{D_{3}}s_{\varphi}c_{\varphi}s_{2}s_{4}s_{3}-\rho_{D_{3}}s_{\varphi}^{2}s_{3-\varphi}c_{3-\varphi}c_{4} \right) \right]$$
(85)
$$\begin{aligned} z_{32} & = m_{D} c_{4} \rho_{{D_{3} }} \left[ {c_{3 - \varphi } l_{{C_{2} }} s_{\varphi } + \rho_{{D_{3} }} c_{\varphi } c_{2} c_{4} c_{3 - \varphi }^{2} + \rho_{{D_{3} }} s_{3 - \varphi } c_{\varphi } s_{2} s_{4} - c_{\varphi } c_{2} c_{4} \rho_{{D_{3} }} s_{3 - \varphi }^{2} } \right. \\ & \left. {\quad - \rho_{{D_{3} }} s_{\varphi } c_{3 - \varphi } c_{4} s_{3 - \varphi } + \rho_{{D_{3} }} s_{\varphi } c_{3 - \varphi } c_{4} s_{3 - \varphi } } \right] \\ & \quad - I_{D}^{ * } \left[ {c_{4}^{2} c_{\varphi } c_{2} - c_{\varphi } s_{2} s_{3 - \varphi } s_{3} + c_{\varphi } c_{2} s_{4}^{2} + c_{\varphi } s_{2} s_{3 - \varphi } c_{4} s_{4} } \right] \\ \end{aligned}$$
(86)
$$z_{33} = m_{D} c_{4} \rho_{{D_{3} }} \left[ {\rho_{{D_{3} }} s_{4} c_{\varphi } c_{2} c_{3 - \varphi } + \rho_{{D_{3} }} s_{4} s_{3 - \varphi } s_{\varphi } } \right] - I_{D}^{ * } 2s_{4} c_{\varphi } c_{2} c_{3 - \varphi } c_{4}$$
(87)
$$\begin{aligned} z_{34} & = m_{D} c_{4} \rho_{{D_{3} }} \left[ {\rho_{{D_{3} }} s_{4} c_{\varphi } s_{2} - \rho_{{D_{3} }} c_{4} c_{\varphi } c_{2} s_{3 - \varphi } - \rho_{{D_{3} }} c_{4} s_{\varphi } c_{3 - \varphi } } \right] \\ & \quad - I_{D}^{ * } \left[ {c_{\varphi } s_{2} s_{4} c_{4} + c_{\varphi } c_{2} c_{4}^{2} s_{3 - \varphi } + s_{\varphi } c_{3 - \varphi } c_{4}^{2} + s_{4} c_{\varphi } s_{2} c_{1} + s_{4}^{2} c_{\varphi } c_{2} s_{3 - \varphi } + s_{4}^{2} s_{\varphi } c_{3 - \varphi } } \right] \\ \end{aligned}$$
(88)
$$z_{35} = m_{D} c_{4} \rho_{{D_{3} }} \left[ {\frac{1}{2}l_{{C_{2} }} + \rho_{{D_{3} }} s_{3 - \varphi } c_{3 - \varphi } c_{4} } \right]$$
(89)
$$z_{36} = m_{D} c_{4} \rho_{{D_{3} }} \left[ {\rho_{{D_{3} }} s_{3 - \varphi } s_{4} } \right]$$
(90)
$$z_{37} = m_{D} c_{4} \rho_{{D_{3} }} \left[ {\rho_{{D_{3} }} c_{3 - \varphi } c_{4} } \right] + I_{D}^{ * } c_{3 - \varphi }$$
(91)
$$z_{ 3 8} = 0$$
(92)
$$z_{39} = m_{D} c_{4} \rho_{{D_{3} }}^{2} s_{4} - 2I_{D}^{ * } s_{4} c_{4}$$
(93)
$$z_{ 4 0} = 0$$
(94)
$$\begin{aligned} z_{41} & = - m_{D} \rho_{{D_{3} }} \left[ {rc_{\varphi } c_{3 - \varphi } s_{4} + \frac{1}{2}l_{{C_{2} }} c_{\varphi }^{2} s_{2} c_{2} c_{4} - \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)s_{\varphi } c_{\varphi } s_{2} c_{4} } \right. \\ & \quad - \frac{1}{2}l_{{C_{2} }} c_{\varphi }^{2} s_{2}^{2} s_{4} s_{3 - \varphi } + \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)s_{\varphi } c_{\varphi } c_{2} c_{4} s_{4} s_{3 - \varphi } \\ & \quad + \frac{1}{2}l_{{C_{2} }} s_{\varphi }^{2} s_{4} s_{3 - \varphi } - c_{3 - \varphi } s_{4} \left( { - \frac{1}{2}\left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi }^{2} - \frac{1}{2}l_{{C_{2} }} s_{\varphi } c_{\varphi } c_{2} } \right) \\ & \quad + c_{\varphi }^{2} s_{2}^{2} c_{4} s_{4} \rho_{{D_{3} }} - c_{\varphi }^{2} c_{4}^{2} s_{2} c_{2} s_{3 - \varphi } \rho_{{D_{3} }} - c_{4}^{2} s_{2} s_{\varphi } c_{\varphi } \rho_{{D_{3} }} c_{3 - \varphi } \\ & \quad + c_{\varphi }^{2} s_{2} c_{2} s_{4}^{2} s_{3 - \varphi } \rho_{{D_{3} }} - c_{\varphi }^{2} c_{2}^{2} s_{3 - \varphi }^{2} c_{4} s_{4} \rho_{{D_{3} }} - s_{4} s_{3 - \varphi } c_{\varphi } c_{2} \rho_{{D_{3} }} \\ & \quad \left. { + s_{4}^{2} s_{2} s_{\varphi } c_{\varphi } \rho_{{D_{3} }} c_{3 - \varphi } - c_{2} s_{\varphi } c_{\varphi } \rho_{{D_{3} }} c_{3 - \varphi } s_{3 - \varphi } c_{4} s_{4} } \right] \\ \end{aligned}$$
(95)
$$\begin{aligned} z_{42} & = - m_{D} \rho_{{D_{3} }} \left[ { - \left( {l_{CB} + l_{{C_{3} }} } \right)c_{\varphi } s_{2} c_{4} + l_{{C_{2} }} s_{\varphi } s_{4} s_{3 - \varphi } } \right. \\ & \quad + c_{\varphi } s_{2} c_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } + 2s_{4} s_{3 - \varphi } c_{\varphi } c_{2} \rho_{{D_{3} }} c_{3 - \varphi } c_{4} \\ & \quad \left. { - c_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }} c_{\varphi } s_{2} + 2c_{3 - \varphi }^{2} s_{4} \rho_{{D_{3} }} s_{\varphi } c_{4} } \right] \\ & \quad + I_{D}^{ * } c_{\varphi } s_{2} c_{3 - \varphi } \\ \end{aligned}$$
(96)
$$\begin{aligned} z_{43} & = - m_{D} \rho_{{D_{3} }} \left[ {c_{\varphi } s_{2} c_{4} s_{4} \rho_{{D_{3} }} + c_{\varphi } c_{2} s_{4}^{2} \rho_{{D_{3} }} s_{3 - \varphi } + s_{\varphi } s_{2} s_{4}^{2} \rho_{{D_{3} }} c_{3 - \varphi } } \right. \\ & \left. {\quad + c_{\varphi } s_{2} c_{4} \rho_{{D_{3} }} s_{4} - c_{4}^{2} c_{2} c_{\varphi } \rho_{{D_{3} }} s_{3 - \varphi } - c_{4}^{2} s_{\varphi } \rho_{{D_{3} }} c_{3 - \varphi } } \right] \\ & \quad + I_{D}^{ * } \left[ {c_{\varphi } c_{2} s_{3 - \varphi } + s_{\varphi } c_{3 - \varphi } } \right] \\ \end{aligned}$$
(97)
$$i_{ 4 4} = 0$$
(98)
$$i_{45} = m_{D} \rho_{{D_{3} }}^{2} c_{4} s_{4} c_{3 - \varphi }^{2}$$
(99)
$$z_{46} = - m_{D} \rho_{{D_{3} }} \left[ {c_{3 - \varphi } s_{4}^{2} \rho_{{D_{3} }} + c_{4} c_{\varphi } s_{2} s_{4} \rho_{{D_{3} }} } \right] - I_{D}^{ * } c_{3 - \varphi } c_{\varphi }$$
(100)
$$z_{ 4 7} = 0$$
(101)
$$z_{48} = - m_{D} \rho_{{D_{3} }} c_{4} s_{4} \rho_{{D_{3} }}$$
(102)
$$z_{ 4 9} { = 0}$$
(103)
$$z_{ 5 0} = 0$$
(104)

The elements of matrix \(\vec{E}\) can be written as:

$$\begin{aligned} f_{{I_{1} }} & = rc_{1} + \left( {\frac{1}{2}\left( {l_{CB} + l_{{c_{3} }} } \right)c_{\varphi } c_{2} + \frac{1}{2}l_{{c_{2} }} s_{\varphi } } \right)\left( {c_{1} c_{2} + s_{1} s_{\varphi } s_{2} } \right) \\ & \quad - \left( {\frac{1}{2}\left( {l_{CB} + l_{{c_{3} }} } \right)c_{\varphi } c_{2} } \right)\left( {s_{1} s_{\varphi } c_{2} - c_{1} s_{2} } \right) \\ & \quad + \frac{1}{2}l_{{c_{2} }} c_{\varphi }^{2} s_{2} s_{1} + \left( {l_{{D_{3} }} c_{\varphi } c_{2} c_{3 - \varphi } - l_{{D_{3} }} s_{\varphi } s_{3 - \varphi } - c_{\varphi } s_{2} s_{4} l_{{D_{2} }} + c_{\varphi } c_{2} s_{3 - \varphi } c_{4} l_{{D_{2} }} + s_{\varphi } c_{3 - \varphi } c_{4} l_{{D_{2} }} } \right) \\ & \quad \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right) \\ & \quad + \left( { - c_{\varphi } s_{2} c_{4} l_{{D_{3} }} - c_{\varphi } c_{2} s_{3 - \varphi } s_{4} l_{{D_{3} }} - s_{\varphi } c_{3 - \varphi } s_{4} l_{{D_{3} }} } \right)\left( { - c_{3 - \varphi } c_{1} s_{2} + c_{3 - \varphi } s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } s_{3 - \varphi } } \right) \\ & \quad + \left( {c_{\varphi } s_{2} c_{4} l_{{D_{2} }} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} l_{{D_{2} }} + s_{\varphi } c_{3 - \varphi } s_{4} l_{{D_{2} }} } \right)\left( {s_{4} c_{1} c_{2} + s_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } c_{4} c_{1} s_{2} - s_{3 - \varphi } c_{4} s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } c_{3 - \varphi } c_{4} } \right) \\ \end{aligned}$$
(105)
$$\begin{aligned} f_{{T_{1} }} & = rc_{1} + \left( {\frac{1}{2}\left( {l_{CB} + l_{{c_{3} }} } \right)c_{\varphi } c_{2} + \frac{1}{2}l_{{c_{2} }} s_{\varphi } } \right)\left( {c_{1} c_{2} + s_{1} s_{\varphi } s_{2} } \right) \\ & \quad - \left( {\frac{1}{2}\left( {l_{CB} + l_{{c_{3} }} } \right)c_{\varphi } c_{2} } \right)\left( {s_{1} s_{\varphi } c_{2} - c_{1} s_{2} } \right) + \frac{1}{2}l_{{c_{2} }} c_{\varphi }^{2} s_{2} s_{1} \\ & \quad + \left( {l_{{M_{3} }} c_{\varphi } c_{2} c_{3 - \varphi } - l_{{M_{3} }} s_{\varphi } s_{3 - \varphi } - c_{\varphi } s_{2} s_{4} l_{{M_{2} }} + c_{\varphi } c_{2} s_{3 - \varphi } c_{4} l_{{M_{2} }} + s_{\varphi } c_{3 - \varphi } c_{4} l_{{M_{2} }} } \right) \\ & \quad \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right) \\ & \quad + \left( { - c_{\varphi } s_{2} c_{4} l_{{M_{3} }} - c_{\varphi } c_{2} s_{3 - \varphi } s_{4} l_{{M_{3} }} - s_{\varphi } c_{3 - \varphi } s_{4} l_{{M_{3} }} } \right)\left( { - c_{3 - \varphi } c_{1} s_{2} + c_{3 - \varphi } s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } s_{3 - \varphi } } \right) \\ & \quad + \left( {c_{\varphi } s_{2} c_{4} l_{{M_{2} }} + c_{\varphi } c_{2} s_{3 - \varphi } s_{4} l_{{M_{2} }} + s_{\varphi } c_{3 - \varphi } s_{4} l_{{M_{2} }} } \right)\left( {s_{4} c_{1} c_{2} + s_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } c_{4} c_{1} s_{2} - s_{3 - \varphi } c_{4} s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } c_{3 - \varphi } c_{4} } \right) \\ \end{aligned}$$
(106)
$$\begin{aligned} f_{{I_{2} }} & = \frac{1}{2}l_{{c_{2} }} \left( {c_{1} c_{2} + s_{1} s_{\varphi } s_{2} } \right) + \left( {s_{3 - \varphi } l_{{D_{3} }} - c_{3 - \varphi } c_{4} l_{{D_{2} }} } \right) \\ & \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right) \\ & + \left( {c_{3 - \varphi } s_{4} l_{{D_{3} }} + c_{3 - \varphi } c_{4} l_{{D_{1} }} } \right)\left( { - c_{3 - \varphi } c_{1} s_{2} + c_{3 - \varphi } s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } s_{3 - \varphi } } \right) \\ & + \left( { - c_{3 - \varphi } s_{4} l_{{D_{3} }} } \right)\left( {s_{4} c_{1} c_{2} + s_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } c_{4} c_{1} s_{2} - s_{3 - \varphi } c_{4} s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } c_{3 - \varphi } c_{4} } \right) \\ \end{aligned}$$
(107)
$$\begin{aligned} f_{{T_{2} }} & = \frac{1}{2}l_{{c_{2} }} \left( {c_{1} c_{2} + s_{1} s_{\varphi } s_{2} } \right) \\ & \quad + \left( {s_{3 - \varphi } l_{{M_{3} }} - c_{3 - \varphi } c_{4} l_{{M_{2} }} } \right)\left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right) \\ & \quad + \left( {c_{3 - \varphi } s_{4} l_{{M_{3} }} + c_{3 - \varphi } c_{4} l_{{M_{1} }} } \right)\left( { - c_{3 - \varphi } c_{1} s_{2} + c_{3 - \varphi } s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } s_{3 - \varphi } } \right) \\ & \quad + \left( { - c_{3 - \varphi } s_{4} l_{{M_{3} }} } \right)\left( {s_{4} c_{1} c_{2} + s_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } c_{4} c_{1} s_{2} - s_{3 - \varphi } c_{4} s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } c_{3 - \varphi } c_{4} } \right) \\ \end{aligned}$$
(108)
$$\begin{aligned} f_{{I_{3} }} & = - s_{4} l_{{D_{2} }} \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right) \\ & - c_{4} l_{{D_{3} }} \left( { - c_{3 - \varphi } c_{1} s_{2} + c_{3 - \varphi } s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } s_{3 - \varphi } } \right) \\ & + c_{4} l_{{D_{2} }} \left( {s_{4} c_{1} c_{2} + s_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } c_{4} c_{1} s_{2} - s_{3 - \varphi } c_{4} s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } c_{3 - \varphi } c_{4} } \right) \\ \end{aligned}$$
(109)
$$\begin{aligned} f_{{T_{3} }} & = - s_{4} l_{{M_{2} }} \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right) \\ & \quad - c_{4} l_{{M_{3} }} \left( { - c_{3 - \varphi } c_{1} s_{2} + c_{3 - \varphi } s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } s_{3 - \varphi } } \right) \\ & \quad + c_{4} l_{{M_{2} }} \left( {s_{4} c_{1} c_{2} + s_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } c_{4} c_{1} s_{2} - s_{3 - \varphi } c_{4} s_{1} s_{\varphi } c_{2} + s_{1} c_{\varphi } c_{3 - \varphi } c_{4} } \right) \\ \end{aligned}$$
(110)
$$f_{{I_{4} }} = l_{{D_{3} }} \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right)$$
(111)
$$f_{{T_{4} }} = l_{{M_{3} }} \left( {c_{1} c_{2} c_{4} + c_{4} s_{1} s_{\varphi } s_{2} + s_{3 - \varphi } s_{4} c_{1} s_{2} - s_{3 - \varphi } s_{4} s_{1} s_{\varphi } c_{2} - s_{1} c_{\varphi } c_{3 - \varphi } s_{4} } \right)$$
(112)

Appendix B

Subject 2

Cereal activity (Fig. 14)

Fig. 14
figure14

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during cereal eating activity of Subject 2

Noodle activity (Fig. 15)

Fig. 15
figure15

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during noodle eating activity of Subject 2

Rice activity (Fig. 16)

Fig. 16
figure16

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during rice eating activity of Subject 2

Soup activity (Fig. 17)

Fig. 17
figure17

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\), comparison plots, during soup eating activity of Subject 2

Vegetable activity (Fig. 18)

Fig. 18
figure18

a Motion trajectories, b Elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\), comparison plots, during vegetable eating activity of Subject 2

Subject 3

Cereal activity (Fig. 19)

Fig. 19
figure19

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during cereal eating activity of Subject 3

Noodle activity (Fig. 20)

Fig. 20
figure20

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during noodle eating activity of Subject 3

Rice activity (Fig. 21)

Fig. 21
figure21

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\), comparison plots, during rice eating activity of Subject 3

Soup activity (Fig. 22)

Fig. 22
figure22

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during soup eating activity of Subject 3

Vegetable activity (Fig. 23)

Fig. 23
figure23

a Motion trajectories, b elbow flexion/extension \((T_{1} )\), c forearm pronation/supination \((T_{2} )\), d wrist adduction/abduction \((T_{3} )\), and e wrist flexion/extension \((T_{4} )\) comparison plots, during vegetable eating activity of Subject 3

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Hussain, Z., Azlan, N.Z. 3-D Dynamic Modeling and Validation of Human Arm for Torque Determination During Eating Activity Using Kane’s Method. Iran J Sci Technol Trans Mech Eng 44, 661–694 (2020). https://doi.org/10.1007/s40997-019-00299-8

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Keywords

  • Dynamic modeling
  • Wrist
  • Elbow
  • Eating
  • Kane’s method
  • RMSE