# Multi-Mode Compound Grasping Robot Finger Driven by Linkage
## 3 Mechanical Analysis and Simulation  
In this section, we will discuss the parameters that affect MCG fingers in detail. The kinematic and dynamic model of the MCG finger is shown in Figure 11. The grasping force of the MCG finger when grasping an object is analyzed to evaluate the stability of grasping. The simulation results are given by MATLAB, which shows the grasping function of the MCG hand. \(\theta_{1}\)--Angle of the first linkage concerning the base, which is the angle between AG and the negative direction of the x-axis, rad;  
\(\theta_{2}\)--Angle of the second linkage concerning base, which is the angle between AE and the negative direction of the x-axis, rad;  
\(\alpha_{1}\)--Rotational angle of the transition shaft, which is the angle between AB and the negative direction of the x-axis, rad;  
\(\alpha_{2}\)--Angle of the first phalanx concerning the second phalanx, which is the angle between AB and BC, rad;  
Figure 11: Kinematic and dynamic model of the MCG finger. \(m_{1-7}\)--Mass of the proximal, middle, and distal phalanx, second, fourth, first, and third linkage, respectively, kg;  
\(l_{1-7}\)--Length of AB, BC, CD, AE, EF, AG, and GF, respectively, m.  
\(\beta\)--The angle between the line from the distal shaft to the centroid of the distal phalanx and the line CF in the distal phalanx, rad. \(k_{1-3}\)--Torsion coefficient of the first, second, and third spring, respectively, N/rad. \(\xi_{1-3}\)--The pre-rotation angle of the first, second, and third spring, respectively, rad. \(\tau_{1-2}\)--The torque applied by the first motor, and the second motor, respectively, N/rad. The MCG finger controls three degrees of freedom through two actuators, which is an underactuated system. In the simulation analysis of the MCG finger, it is found that when the first and second links are driven to control the motion of the MCG finger, the simulation results are extremely complex. This is because the torque exerted by the first driver on the first linkage acts on the first phalanx and the second phalanx comprehensively, so the solution is more complex. To simplify the simulation while obtaining the kinematics and dynamics analysis of the MCG finger, this paper chooses to place the first and second virtual drivers at the proximal and middle joint axes, respectively, to equivalently replace the effect of the first motor on the proximal and middle phalanx. In real motion, the kinematics curve of the first driver can be inversely solved by using the motion angle of two virtual motors during simulation. If the parameters of the first driver solved are used as input, the results equivalent to the simulation can be obtained. Take point A (Proximal shaft) as the origin; the horizontal direction is the x-axis, the vertical direction is the y-axis, and the vertical point A outward is the z-axis to establish the coordinate system as shown in Figure 11. The coordinate of point A can be obtained directly:  
\[\mathbf{r_{A}}=[0\quad 0\quad 0] \tag{1}\]  
The length of AB (proximal phalanx) is \(l_{1}\), the angle between AB and the negative direction of the x-axis is \(\alpha_{1}\), so the coordinate of B is:  
\[\mathbf{r_{B}}=[-l_{1}\cos(\alpha_{1})\quad l_{1}\sin(\alpha_{1})\quad 0] \tag{2}\]  
The length of BC (proximal phalanx) is \(l_{2}\), the angle between AB and BC is \(\alpha_{2}\), the calculated coordinate of B can be obtained:  
\[\mathbf{r_{C}}=[-l_{2}\cos(\alpha_{1}+\alpha_{2})-l_{1}\cos(\alpha_{1})\quad l _{2}\sin(\alpha_{1}+\alpha_{2})+l_{1}\sin(\alpha_{1})\quad 0] \tag{3}\]  
The length of AE (second linkage) is \(l_{2}\), the angle between AB and BC is \(\theta_{2}\), the coordinate of E can be obtained:  
\[\mathbf{r_{E}}=[-l_{4}\cos(\theta_{1})\quad l_{4}\sin(\theta_{1})\quad 0] \tag{4}\]  
The parameters of the MCG finger are mainly determined by the second driver and the first and second virtual drives. Therefore, after the coordinates of points A, B, C, and E are determined, the parametric coordinates of all linkage positions of the MCG finger can be solved by the geometric method.