A nonlinear H-infinity control method for multi-DOF robotic manipulators

Abstract

The paper proposes a new nonlinear H-infinity control method for multi-DOF robotic manipulators. At first stage local linearization of the robotic model is performed round its present operating point. The approximation error that is introduced to the linearized model due to truncation of higher-order terms in the performed Taylor series expansion is represented as a disturbance. The control problem is now formulated as a mini–max differential game in which the control input tries to minimize the state vector tracking error while the disturbances affecting the model try to maximize it. Using the linearized description of the robot’s dynamics, an H-infinity feedback controller is designed through the solution of a Riccati equation at each step of the control algorithm. The inherent robustness properties of H-infinity control assure that the disturbance effects will be eliminated and the robot’s state variables will converge to the desirable setpoints. The proposed method stands for a reliable solution to the problem of nonlinear control and stabilization for multi-DOF robotic manipulators. It is also a novel approach, comparing to control of a robotic manipulator based on global linearization of its dynamics. Its efficiency is further confirmed through simulation experiments.

Appendices

Appendix 1: Dynamic model of the multi-DOF robotic system

The dynamic model of the 2-DOF rigid-link robot which is depicted again in Fig. 15 and under the assumption that the masses of the links are concentrated at the links’ end is given by

$$\begin{aligned} D(\theta )\ddot{\theta }+h(\theta ,\dot{\theta })+g(\theta )=\tau \end{aligned}$$

(53)

where $D(\theta )$ is the inertia matrix

$$\begin{aligned} D(\theta )=\begin{pmatrix} (m_1+m_2){l_1^2}+{m_1}{l_2^2}+2{m_2}{l_1}{l_2}\hbox {cos}(\theta _2) &{}\quad {m_2}{l_2^2}+{m_2}{l_1}{l_2}\hbox {cos}(\theta _2) \\ {m_2}{l_2^2}+{m_2}{l_1}{l_2}\hbox {cos}(\theta _2) &{}\quad {m_2}{l_2}(l_1+l_2)\hbox {cos}(\theta _2) \end{pmatrix} \end{aligned}$$

(54)

$h(\theta ,\dot{\theta })$ is the Coriolis and centrifugal forces vector

$$\begin{aligned} h(\theta ,\dot{\theta })= \begin{pmatrix} -{m_2}{l_1^2}\hbox {sin}(\theta _2)\dot{\theta }_2^2-2{m_2}{l_1^2}\hbox {sin}(\theta _2){\dot{\theta }_1}{\dot{\theta }_2} \\ {m_2}{l_1^2}\hbox {sin}(\theta _2)\dot{\theta }_1^2 \end{pmatrix} \end{aligned}$$

(55)

$g(\theta )$ is the gravitational forces vector

$$\begin{aligned} g(\theta )= \begin{pmatrix} (m_1+m_2)g{l_1}\hbox {cos}(\theta _1)+{m_2}g{l_2}\hbox {cos}(\theta _1+\theta _2) \\ {m_2}g{l_2}\hbox {cos}(\theta _1+\theta _2) \end{pmatrix} \end{aligned}$$

(56)

and $\tau (t)$ is the control inputs vector consisting of the torques that are generated by the motors mounted on the robot’s joints.

Fig. 15

A 2-DOF rigid-link robotic manipulator

It holds that

$$\begin{aligned} D^{-1}(\theta )={1 \over {detD}} \begin{pmatrix} {m_2}{l_2}(l_1+l_2)\hbox {cos}(\theta _2) &{} -{m_2}{l_2^2}+{m_2}{l_1}{l_2}\hbox {cos}(\theta _2) \\ -{m_2}{l_2^2}+{m_2}{l_1}{l_2}\hbox {cos}(\theta _2) &{} \quad (m_1+m_2){l_1^2}+{m_2}{l_2^2}+2{m_2}{l_1}{l_2}\hbox {cos}(\theta _2) \end{pmatrix} \end{aligned}$$

(57)

where the determinant of D is

$$\begin{aligned} detD= & {} [(m_1+m_2){l_1^2}+{m_2}{l_2^2}\nonumber \\&+\,2{m_2}{l_1}{l_2} \hbox {cos}(\theta _2)][{m_2}{l_2}(l_1+l_2)\hbox {cos}(\theta _2)]\nonumber \\&-\,[{m_2}{l_2^2}+{m_2}{l_1}{l_2}\hbox {cos}(\theta _2)]^2\nonumber \\ detD= & {} (m_1+m_2){m_2}{l_1^2}{l_2}(l_1+l_2)\hbox {cos}(\theta _2)\nonumber \\&+\,{m_2^2}{l_2^3}(l_1+l_2)\hbox {cos}(\theta _2)\nonumber \\&+\,2{m_2^2}{l_1}{l_2^2}(l_1+l_2)\hbox {cos}^2(\theta _2)-\nonumber \\&-\,{m_2^2}{l_2^4}-{m_2^2}{l_1^2}{l_2^2}\hbox {cos}^2(\theta _2)\nonumber \\&-\,2{m_2^2}{l_1}{l_2^3}\hbox {cos}(\theta _2) \end{aligned}$$

(58)

Without loss of generality, the following parameters’ values are assumed: $m_1=1\,{\hbox {kg}}$, $m_2=1\,{\hbox {kg}}$, $l_1=1{\hbox {m}}$, $l_2=1{\hbox {m}}$, and $g=10m/sec^2$. Thus, the inverse of the inertia matrix $D(\theta )$ becomes

$$\begin{aligned} D^{-1}(\theta )={1 \over {detD}} \begin{pmatrix} 2\hbox {cos}(\theta _2) &{} -1+\hbox {cos}(\theta _2) \\ -1+\hbox {cos}(\theta _2) &{} 2+2\hbox {cos}(\theta _2) \end{pmatrix} \end{aligned}$$

(59)

with

$$\begin{aligned} detD=4\hbox {cos}(\theta _2)+3\hbox {cos}^2(\theta _2)-1 \end{aligned}$$

(60)

For the previously given values of the parameters in the robot’s model, one has

$$\begin{aligned} h(\theta ,\dot{\theta })= & {} \begin{pmatrix} -\hbox {sin}(\theta _2)\dot{\theta }_2^2-2\hbox {sin}(\theta _2)\dot{\theta _1}\dot{\theta _2} \\ \hbox {sin}(\theta _2)\dot{\theta }_1^2 \end{pmatrix} \end{aligned}$$

(61)

$$\begin{aligned} g(\theta )= & {} \begin{pmatrix} 10\hbox {cos}(\theta _1)+10\hbox {cos}(\theta _1+\theta _2) \\ 10\hbox {cos}(\theta _1+\theta _2) \end{pmatrix} \end{aligned}$$

(62)

Using the above one gets

$$\begin{aligned}&-\,D^{-1}(\theta )h(\theta ,\dot{\theta })\nonumber \\&\quad ={-{1} \over {detD}} \begin{pmatrix} 2\hbox {cos}(\theta _2) &{} -1+\hbox {cos}(\theta _2) \\ &{} \\ -1+\hbox {cos}(\theta _2) &{} 2+2\hbox {cos}(\theta _2) \end{pmatrix}\nonumber \\&\qquad \times \,\begin{pmatrix} -\hbox {sin}(\theta _2)\dot{\theta }_2^2-2\hbox {sin}(\theta _2){\dot{\theta }_1}{\dot{\theta }_2} \\ {\hbox {sin}(\theta )_2}{\dot{\theta }_1^2} \end{pmatrix} \end{aligned}$$

(63)

or equivalently

$$\begin{aligned}&-\,D^{-1}(\theta )h(\theta ,\dot{\theta }) ={-{1} \over {detD}}\nonumber \\&\begin{pmatrix} 2\hbox {cos}(\theta _2)[-\hbox {sin}(\theta _2)\dot{\theta }_2^2-2\hbox {sin}(\theta _2){\dot{\theta }_1}{\dot{\theta }_2}]\\ +\,[-1+\hbox {cos}(\theta _2)]\hbox {sin}(\theta )_2{\dot{\theta }_1^2} \\ [-1+\hbox {cos}(\theta _2)][-\hbox {sin}(\theta _2)\dot{\theta }_2^2-2\hbox {sin}(\theta _2){\dot{\theta }_1}{\dot{\theta }_2}]\\ +\,[2+2\hbox {cos}(\theta _2)]{\hbox {sin}(\theta )_2}{\dot{\theta }_1^2} \end{pmatrix}\nonumber \\ \end{aligned}$$

(64)

and also

$$\begin{aligned}&-\,D^{-1}(\theta )g(\theta )\nonumber \\&\quad =-{{1} \over {detD}} \begin{pmatrix} 2\hbox {cos}(\theta _2) &{} -1+\hbox {cos}(\theta _2) \\ -1+\hbox {cos}(\theta _2) &{} 2+2\hbox {cos}(\theta _2) \end{pmatrix}\nonumber \\&\quad \begin{pmatrix} 10\hbox {cos}(\theta _1)+10\hbox {cos}(\theta _1+theta_2) \\ 10\hbox {cos}(\theta _1+\theta _2) \end{pmatrix} \end{aligned}$$

(65)

or equivalently

$$\begin{aligned}&-\,D^{-1}(\theta )g(\theta ) ={-{1} \over {detD}}\nonumber \\&\begin{pmatrix} 2\hbox {cos}(\theta _2)[10\hbox {cos}(\theta _1)+10\hbox {cos}(\theta _1+\theta _2)]\\ +\,[-1+\hbox {cos}(\theta _2)]10\hbox {cos}(\theta _1+\theta _2) \\ [-1+\hbox {cos}(\theta _2)][10\hbox {cos}(\theta _1)+10\hbox {cos}(\theta _1+\theta _2)]\\ +\,[2+2\hbox {cos}(\theta _2)]10\hbox {cos}(\theta _1+\theta _2) \end{pmatrix}\nonumber \\ \end{aligned}$$

(66)

Next, by defining the state variables $x_1=\theta _1$, $x_2=\dot{\theta }_1$, $x_3=\theta _2$ and $x_4=\dot{\theta }_2$ one gets

$$\begin{aligned}&D^{-1}(\theta )\nonumber \\&\quad = \begin{pmatrix} {{2\hbox {cos}(x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} &{} {{-1+\hbox {cos}(x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \\ {{-1+\hbox {cos}(x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} &{} {{2+2\hbox {cos}(x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{pmatrix}\nonumber \\ \end{aligned}$$

(67)

and also

$$\begin{aligned} -D^{-1}(\theta )h(\theta ,\dot{\theta }) = -\begin{pmatrix} {{2\hbox {cos}(x_3)[-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[-1+\hbox {cos}(x_3)]\hbox {sin}(x_3){x_2^2}} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \\ \\ {{[1+\hbox {cos}(x_3)][-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[2+2\hbox {cos}(x_3)]\hbox {sin}(x_3){x_2^2}} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{pmatrix} \end{aligned}$$

(68)

and similarly

$$\begin{aligned} -D^{-1}(\theta )g(\theta )=-\begin{pmatrix} {{2\hbox {cos}(x_3)[10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]+[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1+x_3)]} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \\ \\ {{[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]+[2+2\hbox {cos}(x_3)]10\hbox {cos}(x_1+x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{pmatrix} \end{aligned}$$

(69)

The state-space equations of the robotic model are

$$\begin{aligned} \begin{pmatrix} \ddot{x}_1 \\ \ddot{x}_3 \end{pmatrix}=-{D^{-1}(\theta )}h(\theta ,\dot{\theta })-{D^{-1}(\theta )}g(\theta )-{D^{-1}(\theta )}{\tau } \end{aligned}$$

(70)

and using that the state vector is $x=[x_1,x_2,x_3,x_4]^T=[\theta _1,\dot{\theta }_1,\theta _2,\dot{\theta }_2]^T$ and the control inputs vector is $\tau =[\tau _1,\tau _2]^T=[u_1,u_2]^T$ this is also written in the form

$$\begin{aligned} \dot{x}=f(x)+g_a(x){u_1}+g_b(x)u_2 \end{aligned}$$

(71)

or equivalently

$$\begin{aligned} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \end{pmatrix}= \begin{pmatrix} f_1 \\ f_2 \\ f_3 \\ f_4 \end{pmatrix}+ \begin{pmatrix} g_{a_1} \\ g_{a_2} \\ g_{a_3} \\ g_{a_4} \end{pmatrix}{u_1}+ \begin{pmatrix} g_{b_1} \\ g_{b_2} \\ g_{b_3} \\ g_{b_4} \end{pmatrix}{u_2} \end{aligned}$$

(72)

with

$$\begin{aligned}&f_1=x_2 \end{aligned}$$

(73)

$$\begin{aligned}&f_2=-{{2\hbox {cos}(x_3)[-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[-1+\hbox {cos}(x_3)]\hbox {sin}(x_3){x_2^2}} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}}-\nonumber \\&\qquad \quad -\,{{2\hbox {cos}(x_3)[10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]+[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1+x_3)]} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(74)

$$\begin{aligned}&f_3=x_4 \end{aligned}$$

(75)

$$\begin{aligned}&f_4=-{{[1+\hbox {cos}(x_3)][-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[2+2\hbox {cos}(x_3)]\hbox {sin}(x_3){x_2^2}} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}}\nonumber \\&\qquad \quad -\,{{[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]+[2+2\hbox {cos}(x_3)]10\hbox {cos}(x_1+x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(76)

Moreover, one has

Appendix 2: Linearization of the robot’s dynamics

The Jacobian matrices of the model of the 2-DOF robotic manipulator are defined as

$$\begin{aligned} {\nabla _x}f\mid _{(x^{*},u^{*})}= & {} \begin{pmatrix} {{{\partial }{f_1}} \over {{\partial }{x_1}}} &{} {{{\partial }{f_1}} \over {{\partial }{x_2}}} &{} {{{\partial }{f_1}} \over {{\partial }{x_3}}} &{} {{{\partial }{f_1}} \over {{\partial }{x_4}}} \\ {{{\partial }{f_2}} \over {{\partial }{x_1}}} &{} {{{\partial }{f_2}} \over {{\partial }{x_2}}} &{} {{{\partial }{f_2}} \over {{\partial }{x_3}}} &{} {{{\partial }{f_2}} \over {{\partial }{x_4}}} \\ {{{\partial }{f_3}} \over {{\partial }{x_1}}} &{} {{{\partial }{f_3}} \over {{\partial }{x_2}}} &{} {{{\partial }{f_3}} \over {{\partial }{x_3}}} &{} {{{\partial }{f_3}} \over {{\partial }{x_4}}} \\ {{{\partial }{f_4}} \over {{\partial }{x_1}}} &{} {{{\partial }{f_4}} \over {{\partial }{x_2}}} &{} {{{\partial }{f_4}} \over {{\partial }{x_3}}} &{} {{{\partial }{f_4}} \over {{\partial }{x_4}}} \end{pmatrix}_{(x^{*},u^{*})} \end{aligned}$$

(78)

$$\begin{aligned} {\nabla _x}{g_a}\mid _{(x^{*},u^{*})}= & {} \begin{pmatrix} {{{\partial }{{g_a}_1}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_a}_1}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_a}_1}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_a}_1}} \over {{\partial }{x_4}}} \\ {{{\partial }{{g_a}_2}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_a}_2}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_a}_2}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_a}_2}} \over {{\partial }{x_4}}} \\ {{{\partial }{{g_a}_3}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_a}_3}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_a}_3}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_a}_3}} \over {{\partial }{x_4}}} \\ {{{\partial }{{g_a}_4}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_a}_4}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_a}_4}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_a}_4}} \over {{\partial }{x_4}}} \end{pmatrix}_{(x^{*},u^{*})} \end{aligned}$$

(79)

$$\begin{aligned} {\nabla _x}{g_b}\mid _{(x^{*},u^{*})}= & {} \begin{pmatrix} {{{\partial }{{g_b}_1}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_b}_1}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_b}_1}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_b}_1}} \over {{\partial }{x_4}}} \\ {{{\partial }{{g_b}_2}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_b}_2}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_b}_2}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_b}_2}} \over {{\partial }{x_4}}} \\ {{{\partial }{{g_b}_3}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_b}_3}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_b}_3}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_b}_3}} \over {{\partial }{x_4}}} \\ {{{\partial }{{g_b}_4}} \over {{\partial }{x_1}}} &{} {{{\partial }{{g_b}_4}} \over {{\partial }{x_2}}} &{} {{{\partial }{{g_b}_4}} \over {{\partial }{x_3}}} &{} {{{\partial }{{g_b}_4}} \over {{\partial }{x_4}}} \end{pmatrix}_{(x^{*},u^{*})} \end{aligned}$$

(80)

where

$$\begin{aligned}&\begin{array}{cccc} {{{\partial }{f_1}} \over {{\partial }{x_1}}} =0&{{{\partial }{f_1}} \over {{\partial }{x_2}}}=1&{{{\partial }{f_1}} \over {{\partial }{x_3}}} =0&{{{\partial }{f_1}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(81)

$$\begin{aligned}&\begin{array}{c} {{{\partial }{f_2}} \over {{\partial }{x_1}}} =-{{{2\hbox {cos}(x_3)}[10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)] +[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1+x_3)]} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{array} \end{aligned}$$

(82)

$$\begin{aligned}&{{{\partial }{f_2}} \over {{\partial }{x_2}}}=-{{{-2\hbox {cos}(x_3)}{2\hbox {sin}(x_3)}{x_4}+[-1+\hbox {cos}(x_3)]\hbox {sin}(x_3){2{x_2}}} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \nonumber \\&- {{2\hbox {cos}(x_3)10\hbox {sin}(x_1+x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(83)

$$\begin{aligned}&{{{\partial }{f_2}} \over {{\partial }{x_3}}}=-\left\{ {{-2\hbox {sin}(x_3)[-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+2\hbox {cos}(x_3)[\hbox {cos}(x_3){x_4^2}-2\hbox {cos}(x_3){x_2}{x_4}]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]^2}}\right. \nonumber \\&\quad \left. + {{[-\hbox {sin}(x_3)]\hbox {sin}(x_3){x_2^2}+[-1+\hbox {cos}(x_3)]\hbox {cos}(x_3){x_2^2}} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]^2}} \right\} [4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1]\nonumber \\&\quad -\left\{ {{{2\hbox {sin}(x_3)[10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]-2\hbox {cos}(x_3)10\hbox {sin}(x_1+x_3)}} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]^2}}\right. \nonumber \\&\quad \left. + {{[-\hbox {sin}(x_3)][10\hbox {cos}(x_1+x_3)]-[-1+\hbox {cos}(x_3)]\hbox {sin}(x_1+x_3)} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]^2}} \right\} [4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1] \end{aligned}$$

(84)

$$\begin{aligned}&\left\{ +{{2\hbox {cos}(x_3)[-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[-1+\hbox {cos}(x_3)]\hbox {sin}(x_3){x_2^2}} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]^2}}\right. \nonumber \\&\quad \left. + {{2\hbox {cos}(x_3)[10\hbox {cos}(x_1){+}10\hbox {cos}(x_1{+}x_3)]{+}[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1{+}x_3)]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]^2}} \right\} [-4\hbox {sin}(x_3)-6\hbox {cos}(x_3)\hbox {sin}(x_3)]\nonumber \end{aligned}$$

$$\begin{aligned}&{{{\partial }{f_2}} \over {{\partial }{x_4}}}={{2\hbox {cos}(x_3)[-\hbox {sin}(x_3)2{x_4}-2\hbox {sin}(x_3){x_2}]} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(85)

$$\begin{aligned}&\begin{array}{cccc} {{{\partial }{f_3}} \over {{\partial }{x_1}}}=0&{{{\partial }{f_3}} \over {{\partial }{x_2}}}=0&{{{\partial }{f_3}} \over {{\partial }{x_3}}}=0&{{{\partial }{f_3}} \over {{\partial }{x_4}}}=1 \end{array} \end{aligned}$$

(86)

$$\begin{aligned}&{{{\partial }{f_4}} \over {{\partial }{x_1}}}=-{{[-1+\hbox {cos}(x_3)][-10\hbox {sin}(x_1)-10\hbox {sin}(x_1+x_3)]-[2+2\hbox {cos}(x_3)]10\hbox {sin}(x_1+x_3)} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(87)

$$\begin{aligned}&{{{\partial }{f_4}} \over {{\partial }{x_2}}}=-{{[1+\hbox {cos}(x_3)][-2\hbox {sin}(x_3){x_4}]+[2+2\hbox {cos}(x_3)]\hbox {sin}(x_3)2{x_2}} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(88)

$$\begin{aligned}&{{{\partial }{f_4}} \over {{\partial }{x_3}}}=-\{ {{-\hbox {sin}(x_3)[-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[1+\hbox {cos}(x_3)][\hbox {cos}(x_3){x_4^2}-2\hbox {cos}(x_3){x_2}{x_4}]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2}\nonumber \\&\qquad \qquad + {{-2\hbox {sin}(x_3)\hbox {sin}(x_3){x_2^2}+[2+2\hbox {cos}(x_3)]\hbox {cos}(x_3){x_2^2}} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2} \} [4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1]\nonumber \\&\qquad \qquad -\left\{ {{-\hbox {sin}(x_3)[10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]+[-1+\hbox {cos}(x_3)][-10\hbox {sin}(x_3)-10\hbox {sin}(x_1+x_3)]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2}\right. \nonumber \\&\qquad \qquad \left. + {{-2\hbox {sin}(x_3)10\hbox {cos}(x_1+x_3)-[2+2\hbox {cos}(x_3)]10\hbox {sin}(x_1+x_3)} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2}\right\} [4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1]\end{aligned}$$

(89)

$$\begin{aligned}&\left\{ {{[1+\hbox {cos}(x_3)][-\hbox {sin}(x_3){x_4^2}-2\hbox {sin}(x_3){x_2}{x_4}]+[2+2\hbox {cos}(x_3)]\hbox {sin}(x_3){x_2^2}} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2}\right. \nonumber \\&\quad \left. + {{[-1+\hbox {cos}(x_3)][10\hbox {cos}(x_1)+10\hbox {cos}(x_1+x_3)]+[2+2\hbox {cos}(x_3)]10\hbox {cos}(x_1+x_3)} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2}\right\} [-4\hbox {sin}(x_3)-6\hbox {cos}(x_3)\hbox {sin}(x_3)]\nonumber \\&{{{\partial }{f_4}} \over {{\partial }{x_4}}}={{[1+\hbox {cos}(cx_3)][-\hbox {sin}(x_3)2{x_4}-2\hbox {sin}(x_3){x_2}]} \over {4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}} \end{aligned}$$

(90)

In a similar manner, one computes

$$\begin{aligned}&\begin{array}{cccc} {{{\partial }{{g_a}_1}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_a}_1}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_a}_1}} \over {{\partial }{x_3}}}=0&\quad {{{\partial }{{g_a}_1}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(91)

$$\begin{aligned}&\begin{array}{ccc} {{{\partial }{{g_a}_2}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_a}_2}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_a}_2}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(92)

$$\begin{aligned}&{{{\partial }{{g_a}_2}} \over {{\partial }{x_3}}}= {{-2\hbox {sin}(x_3)[4\hbox {cos}(x_3)+4\hbox {cos}^2(x_3)-1]+2\hbox {cos}(x_3) [-4\hbox {sin}(x_3)-6\hbox {sin}(x_3)\hbox {cos}(x_3)]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2} \end{aligned}$$

(93)

$$\begin{aligned}&\begin{array}{cccc} {{{\partial }{{g_a}_3}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_a}_3}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_a}_3}} \over {{\partial }{x_3}}}=0&\quad {{{\partial }{{g_a}_3}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(94)

$$\begin{aligned}&\begin{array}{ccc} {{{\partial }{{g_a}_2}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_a}_2}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_a}_2}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(95)

$$\begin{aligned}&{{{\partial }{{g_a}_2}} \over {{\partial }{x_3}}}= {{-\hbox {sin}(x_3)[4\hbox {cos}(x_3)+4\hbox {cos}^2(x_3)-1]+[-1+\hbox {cos}(x_3)] [4\hbox {sin}(x_3)+6\hbox {sin}(x_3)\hbox {cos}(x_3)]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2} \end{aligned}$$

(96)

while one also obtains

$$\begin{aligned}&\begin{array}{cccc} {{{\partial }{{g_b}_1}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_b}_1}} \over {{\partial }{x_2}}}=0&{{{\partial }{{g_b}_1}} \over {{\partial }{x_3}}}=0&\quad {{{\partial }{{g_b}_1}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(97)

$$\begin{aligned}&\begin{array}{ccc} {{{\partial }{{g_b}_2}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_b}_2}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_b}_2}} \over {{\partial }{x_3}}}=0 \end{array} \end{aligned}$$

(98)

$$\begin{aligned}&\begin{array}{c} {{{\partial }{{g_b}_2}} \over {{\partial }{x_3}}}= {{-\hbox {sin}(x_3)[4\hbox {cos}(x_3)+4\hbox {cos}^2(x_3)-1]+[-1+\hbox {cos}(x_3)][4\hbox {sin}(x_3) +6\hbox {sin}(x_3)\hbox {cos}(x_3)]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2} \end{array}\end{aligned}$$

(99)

$$\begin{aligned}&\begin{array}{cccc} {{{\partial }{{g_b}_3}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_b}_3}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_b}_3}} \over {{\partial }{x_3}}}=0&\quad {{{\partial }{{g_b}_3}} \over {{\partial }{x_4}}}=0 \end{array} \end{aligned}$$

(100)

$$\begin{aligned}&\begin{array}{ccc} {{{\partial }{{g_b}_4}} \over {{\partial }{x_1}}}=0&\quad {{{\partial }{{g_b}_4}} \over {{\partial }{x_2}}}=0&\quad {{{\partial }{{g_b}_4}} \over {{\partial }{x_3}}}=0 \end{array} \end{aligned}$$

(101)

$$\begin{aligned}&\begin{array}{c} {{{\partial }{{g_b}_4}} \over {{\partial }{x_3}}}= {{-2\hbox {sin}(x_3)[4\hbox {cos}(x_3)+4\hbox {cos}^2(x_3)-1]+[2+2\hbox {cos}(x_3)] [4\hbox {sin}(x_3)+6\hbox {sin}(x_3)\hbox {cos}(x_3)]} \over {[{4\hbox {cos}(x_3)+3\hbox {cos}^2(x_3)-1}]}^2} \end{array} \end{aligned}$$

(102)