# A Method for Precise Tracking Control of Pneumatic Artificial-Muscle-Driven Exoskeletal Robot
## 2 Pneumatic-Muscle-Driven Upper Limb Exoskeletal Robot System
### Establishment of Control Law for Upper Limb Exoskeleton  
In order to complete the power-assisted handling task of the Upper Limb Exoskeleton, the elbow joint angle, angular velocity and angular acceleration are subject to target trajectory planning as the reference input of the control system. In this paper, the target angle process is constructed by the fifth-order polynomial, the total time is 5 s, the control period is 0.01 s, and the lifting angle changes from 0 to 0.3 rad. Taking the left arm as an example, the target angles of the left and right arms are shown in Figure 9. The fifth polynomial is used as the target movement trajectory for the following reasons: (1) The time and angle of conformation to human-body handling process habits. (2) The fifth polynomial can ensure a continuous change in the trajectory angle process, and the corresponding angular velocity and angular acceleration are also continuous changes, which can ensure safety and comfort. (3) It can be set to specify the initial and end angle, angular speed, and angular acceleration. Therefore, the fifth polynomial is a good target trajectory function for complex human dynamics.  
In the process of the Upper Limb Exoskeleton carrying objects, the pressure value of the pneumatic muscle is controlled according to the change in the load and the angle error value to maintain the tracking of the elbow angle. Therefore, it is necessary to establish a control rate according to the difference between the actual angle, angular velocity, and angular acceleration and the target value during the rotation process. In order to realize the tracking process of the angle of the handling process, as long as \(\ddot{q}\to\tilde{s}\ddot{q}\), there must be \(q\to\tilde{s}q,\dot{q}\to\tilde{s}\dot{q}\); according to the previous dynamic model, it can be determined that:  
\[{}^{a}\tau=M(q)\times\ddot{q}+C\big{(}q,\dot{q}\big{)}+C(q)-p_{\tau},\ddot{q} \to\tilde{s}\ddot{q} \tag{52}\]  
The error function is constructed as follows:  
\[e=\tilde{s}q-q,\dot{e}=\dot{s}\dot{q}-\dot{q},\tilde{e}=\tilde{s}\ddot{q}-\ddot {q} \tag{53}\]  
When \(K_{d},K_{p}\) is a positive diagonal matrix, the equation \(\ddot{e}+K_{d}\dot{e}+K_{p}e=0\) is asymptotically stable in a large range. Therefore, when \(K_{d},K_{p}\) is a positive diagonal matrix, \(\ddot{e}\to 0,\dot{e}\to 0,e\to 0\). The following formula can be obtained by substituting \(\ddot{e}=\tilde{s}\ddot{q}-\ddot{q}\) into the dynamic model. It is also proven that the control method can realize tracking control and the tracking process is convergent.  
\[{}^{a}\tau=M(q)\big{(}\tilde{s}\ddot{q}+K_{d}\dot{e}+K_{p}e\big{)}\ddot{q}+C \big{(}q,\dot{q}\big{)}+G(q)-{}^{p}\tau \tag{54}\]  
The control torque of pneumatic muscle obtained via discretization is:  
\[{}^{a}\tau_{k}=M(q_{k-1})\big{(}\tilde{s}\ddot{q}_{k}+K_{d}\dot{e}_{k-1}+K_{p }e_{k-1}\big{)}+C\big{(}q_{k-1},\dot{q}_{k-1}\big{)}+G(q_{k-1})-{}^{p}\tau_{k-1} \tag{55}\]  
According to the principle of torque equivalence, the torque required by the pneumatic muscle can be converted into the output force of the pneumatic muscle. Combined with  
Figure 9: Target trajectory planning.  
the length of the pneumatic muscle, the corresponding air pressure value of PAM can be obtained according to the inverse model of the output force of the pneumatic muscle.