# A Method for Precise Tracking Control of Pneumatic Artificial-Muscle-Driven Exoskeletal Robot
## 2 Pneumatic-Muscle-Driven Upper Limb Exoskeletal Robot System
### Kinematic and Dynamic Analysis of Upper Limb Exoskeleton during Lifting Process  
In this paper, the Lagrangian formulation is adopted for dynamic modeling, providing a description of the coupling between the torque and structural motion of the elbow joint and pneumatic muscles in the Upper Limb Exoskeleton. The main principle involves establishing a center-of-mass coordinate system C at the position of the load's center of mass and a body-fixed coordinate system J at the axis of the left and right elbow joints. This enables the coordinate transformation of the center-of-mass coordinate system relative to the world coordinate system W, as shown in Figure 8. After the elbow joint rotates by an angle \(\theta\), the new coordinate axis is as shown by the red coordinate axis. The left and right arms are represented by 1 and 2, respectively. The position of the origin of J1 relative to W is denoted as \( \begin{pmatrix}0&0&-L\end{pmatrix} ^{T}\), where \(q_{1},\hat{q}_{1},\hat{q}_{1}\) represent the rotational angle, angular velocity, and angular acceleration of J1 relative to W, respectively. The position of the origin of C1 relative to J1 is denoted as \( \begin{pmatrix}^{1}C_{x}&^{1}C_{y}&^{1}C_{z}\end{pmatrix} ^{T}\). Taking the left arm of the exoskeleton robot with the upper limb as an example, when the joint rotation angle is q1, the orientation matrix \({}_{J1}^{W}\mathrm{T}\) of the conjoined body coordinate system J1 is relative to the world coordinate system W, the orientation matrix \({}_{C1}^{I1}T\) of the centroid coordinate system C1 is relative to the conjoined body coordinate system J1, and the orientation matrix \({}_{C1}^{W}T\) and rotation matrix \({}_{C1}^{W}R\) of the centroid coordinate system C1 are relative to the world coordinate system W.  
\[ \begin{smallmatrix}W\\ l_{1}\end{smallmatrix} T= \begin{pmatrix}\cos(q_{1})&-\sin(q_{1})&0&0\\ \sin(q_{1})&\cos(q_{1})&0&0\\ 0&0&1&-L\\ 0&0&0&1\end{pmatrix}  \tag{32}\]Based on the pose matrix \(\overset{W}{C1}T\), the Roll-Pitch-Yaw (RPY) Euler angles \(\alpha_{1},\beta_{1},\gamma_{1}\) of the centroid coordinate system C1 with respect to W can be obtained. \[\alpha_{1}=\arctan\left( \begin{matrix}\frac{W}{C1}T(3,2)\\ \frac{W}{C1}T(3,3)\end{matrix} \right),\beta_{1}=\arcsin\left(-\overset{W}{C1 }T(3,1)\right),\gamma_{1}=\arctan\left( \begin{matrix}\frac{W}{C1}T(2,1)\\ \frac{W}{C1}T(1,1)\end{matrix} \right) \tag{36}\]  
The Jacobian matrix \(\overset{W}{C1}J^{O}\) of the centroid coordinate system C1 with respect to the angular velocity of W is as follows:  
\[\overset{W}{C1}J^{O}=\left( \begin{matrix}\frac{\partial\pi_{1}}{\partial\eta_ {1}}&\frac{\partial\pi_{1}}{\partial\eta_{2}}\\ \frac{\partial\beta_{1}}{\partial\eta_{1}}&\frac{\partial\pi_{1}}{\partial \eta_{2}}\\ \frac{\partial\pi_{1}}{\partial\eta_{1}}&\frac{\partial\pi_{1}}{\partial\eta_{ 2}}\end{matrix} \right) \tag{37}\]  
Therefore, the angular velocity \(\overset{W}{C1}\omega\) of the centroid coordinate system C1 with respect to W is:  
\[\overset{W}{C1}\omega=\left( \begin{matrix}W\\ C1\end{matrix} J^{O}\right)\,\tilde{q} \tag{38}\]  
Based on the pose matrix \(\overset{W}{C1}T\), the position \((x_{1},y_{1},z_{1})\) of the centroid coordinate system C1 with respect to W is:  
\[x_{1}=\overset{W}{C1}T(1,4),y_{1}=\overset{W}{C1}T(2,4),z_{1}=\overset{W}{C1 }T(3,4) \tag{39}\]  
The Jacobian matrix \(\overset{W}{C1}J^{P}\) of the centroid coordinate system C1 with respect to the linear velocity of W is:  
\[\overset{W}{C1}J^{P}=\left( \begin{matrix}\frac{\partial x_{1}}{\partial\eta_ {1}}&\frac{\partial x_{1}}{\partial\eta_{2}}\\ \frac{\partial y_{1}}{\partial\eta_{1}}&\frac{\partial y_{1}}{\partial\eta_{2} }\\ \frac{\partial z_{1}}{\partial\eta_{1}}&\frac{\partial z_{1}}{\partial\eta_{2} }\end{matrix} \right) \tag{40}\]  
Figure 8: Establishment of Upper Limb Exoskeleton coordinate axis. (**a**) Physical map. (**b**) Structure diagram.