# Motion Control and External Force Estimation of a Pneumatically Driven Multi-DOF Robotic Forceps
## 3 Control Scheme  
### Kinematics  
Figure 4 presents the schematic of the driving system, including the coordinate system. Forceps motion angles \(\mathbf{q}=\left(q_{1},q_{2},q_{3}\right)^{T}\) are intuitively used to represent the motion of the whole forceps, where \(q_{1}\) and \(q_{2}\) respectively denote the bending angles of the first and the second joints, and \(q_{3}\) denotes the open angle of the gripper.  
As shown in Figure 4, the rotational angles of the two gripper fingers (\(\varphi_{\text{gL}}\) and \(\varphi_{\text{gR}}\)) determine both \(q_{2}\) and \(q_{3}\) by independently controlling the two gripper fingers. Hence, in this work, we used the mechanical joint angles \(\mathbf{\varphi}=\left(\varphi_{1},\varphi_{\text{gL}},\ \varphi_{\text{gR}} \right)^{T}\) instead of the forceps motion angles \(\mathbf{q}\) for motion control. The relationship between the forceps motion angles \(\mathbf{q}\) and the mechanical joint angles \(\mathbf{\varphi}\) can be geometrically expressed as follows:  
\[\mathbf{q}=\left[\begin{array}{c}q_{1}\\ q_{2}\\ q_{3}\end{array}\right]=J_{\text{link}}\mathbf{\varphi}=\left[\begin{array}{ccc}1 &0&0\\ 0&1/2&1/2\\ 0&1&-1\end{array}\right]\left[\begin{array}{c}\varphi_{1}\\ \varphi_{\text{gL}}\\ \varphi_{\text{gR}}\end{array}\right] \tag{1}\]  
Figure 3: Mechanism of the wire route: (**a**) variation of the path of the wires with the static pulley base and (**b**) variation of the path of the wires with the rotatable pulley base.  
\(J_{\rm link}\) is the Jacobian matrix that translates the mechanical joint angles \(\mathbf{\varphi}\) to the forceps motion angles \(\mathbf{q}\). As mentioned in Section 2, we used three pneumatic cylinders as actuators with linear actuation motions. The actuator displacements \(\mathbf{x}=(x_{1},x_{2},x_{3})^{T}\) denote the linear positions of each cylinder rod, which are measured by the potentiometers (RDC1010A12, Travel: 10 mm, Alps Alpine Co., Ltd., Tokyo, Japan) connected to the cylinder rods shown in Figure 4. Additionally, the relationship between \(\mathbf{x}\) and \(\mathbf{\varphi}\) is expressed by the following equation:  
\[\mathbf{\varphi}=\left[\begin{array}{c}\varphi_{1}\\ \varphi_{\rm gL}\\ \varphi_{\rm gR}\end{array}\right]=\mathbf{J}_{\mathbf{x}}\mathbf{x}=\left[\begin{array}[ ]{ccc}\frac{1}{r}_{1}&0&0\\ 0&\frac{1}{r_{\rm gL}}&0\\ 0&0&\frac{1}{r_{\rm gR}}\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array}\right] \tag{2}\]  
where the Jacobian matrix \(\mathbf{J}_{x}\) in (2) translates the displacement of the cylinder rods \(\mathbf{x}\) to the mechanical joint angles \(\mathbf{\varphi}\). Parameters \(r_{1},r_{\rm gL},~{}r_{\rm gR}\) in \(\mathbf{J}_{x}\) represent the radius of the wire path about the joint rotation center, as presented in Figure 4.  
Moreover, the relationship between the driving force output by each cylinder \(f_{\rm cyl}=(f_{1},f_{2},f_{3})\) and the torque of each mechanical joint \(\mathbf{\tau}_{\varphi}=\left(\tau_{1},\tau_{\rm gL},\tau_{\rm gR}\right)\) can be expressed as (3) using the principle of virtual work.  
\[\mathbf{\tau}_{\varphi}=\left(\mathbf{J}_{x}^{\rm T}\right)^{-1}\!\!f_{\rm cyl} \tag{3}\]