# Design of an Active and Passive Control System of Hand Exoskeleton for Rehabilitation
## 4 Active and Passive Control
### Active Control  
The active rehabilitation mode is that the patients do flexion and extension of the finger actively and the exoskeleton tracks and adapts to the motion status of the patients automatically. The controller does not directly control the exoskeleton to reach the desired configuration, but control it to follow the patient's motion according to the motor ability. According to the impedance model and the input of the force signals on the tips, the trajectory of the exoskeleton finger is planned in real time. The actual motion status parameters are obtained by detecting the output \(\theta\) of the exoskeleton, thus we have the control variable \(e(t)\). The position controller is used to realize the real-time track of the patient's fingers. The control block diagram is shown in Figure 7.  
The relationship between the joint space of the exoskeleton and the Cartesian space is  
\[\left\{\begin{array}{l}\mathbf{P}=\mathbf{f}(\mathbf{\theta})\\ \dot{\mathbf{P}}=\mathbf{f}(\mathbf{\theta})\dot{\mathbf{\theta}}\end{array}\right., \tag{10}\]  
where \(\mathbf{f}(\mathbf{\theta})\) is the forward kinematic relationship of the exoskeleton. Taking the derivative of Equation (10) and we have the relationship between the accelerations in joint space and Cartesian space:  
\[\ddot{\mathbf{P}}=\mathbf{J}(\mathbf{\theta})\ddot{\mathbf{\theta}}+\dot{\mathbf{J}}(\mathbf{\theta}) \dot{\mathbf{\theta}}. \tag{11}\]  
During the contact of finger and exoskeleton, the relationship between finger and exoskeleton is a second order dynamic relation, and we simplify it as a linear spring model. In Cartesian space, the impedance model is as follows:  
\[\mathbf{M}_{t}(\ddot{\mathbf{P}}-\ddot{\mathbf{P}}_{d})+\mathbf{B}_{t}(\dot{\mathbf{P}}-\dot{\mathbf{P }}_{d})+\mathbf{K}_{t}(\mathbf{P}-\mathbf{P}_{d})=\mathbf{F}_{ext}, \tag{12}\]  
where \(\mathbf{M}_{t}\), \(\mathbf{B}_{t}\), \(\mathbf{K}_{t}\) are objective inertia matrix, objective damping matrix, and objective stiffness matrix, respectively. \(\mathbf{P}\) and \(\mathbf{P}_{d}\) are respectively the actual position and desired position of the exoskeleton fingertip. \(\mathbf{F}_{ext}\) is the contact force between the finger and the exoskeleton.  
Rewriting the impedance model Equation (12) and substituting it into Equations (10) and (11), we have:  
\[\ddot{\mathbf{\theta}}_{d}=\ddot{\mathbf{\theta}}-\mathbf{J}^{-1}\mathbf{M}_{t}{}^{-1}[\mathbf{F}_ {ext}-\mathbf{B}_{t}(\dot{\mathbf{P}}-\dot{\mathbf{P}}_{d})-\mathbf{K}_{t}(\mathbf{P}-\mathbf{P}_{d})] +\mathbf{J}^{-1}\dot{\mathbf{j}}\dot{\mathbf{\theta}}. \tag{13}\]  
Due to the integral of Equation (13) and \(\mathbf{\theta}_{d}\) has been obtained, the PID control method is adopted in the position control loop. In the active rehabilitation mode, the position error \(e(t)\) is used to make the motor output torque \(\mathbf{\tau}\) actuate the exoskeleton to follow the desired trajectory, and the relationship between \(e(t)\) and \(\mathbf{\tau}\) is as follows:  
\[\mathbf{\tau}=\mathbf{K}_{P}\cdot\mathbf{e}(t)+\mathbf{K}_{t}\cdot\int_{0}^{t}e(t)dt+\mathbf{K}_{D }\cdot\frac{d\mathbf{e}(t)}{dt}, \tag{14}\]  
Figure 7: The active control block diagram.  
where \(\mathbf{K}_{P}\), \(\mathbf{K}_{I}\), \(\mathbf{K}_{D}\) are the proportional, integral, and differential gains of the controller, respectively.