# Design and Experiment of Dry-Ice Cleaning Mechanical Arm for Insulators in Substation
## 4 Kinematics and Simulation Analysis of Dry-Ice Cleaning Mechanical Arm  
### Kinematics Analysis of Dry-Ice Cleaning Mechanical Arm  
The principle of dry-ice cleaning mechanical arm is the same as that of the industrial stacking mechanical arm. Considering that the working track of the robot arm is mainly concentrated in the vertical lifting or stretching in a single plane, the schematic diagram of dry-ice cleaning mechanical arm is shown in Figure 7[43; 44]. Establish the coordinate system. The origin of the coordinate is O. The length of the main boom OQ, the connecting rod OM, the auxiliary boom MN, the small arm NP, and the end tool PT are \(L1\), \(L2\), \(L1\), \(L3\), and \(L0\), respectively. The definition of the angle is negative clockwise around the \(Z\)-axis and positive counterclockwise. At the beginning, the tool at the end of the mechanical arm is at the lowest point (\(Y=0\)). After a little time, the corners of the main arm and the connecting rod reach \(A\) and \(B\), respectively. The relationship between nozzle T (\(X\), \(Y\)) and each rod length (\(L1\), \(L2\), \(L3\), and \(L0\)) and the angle of main and auxiliary motors (\(A\) and \(B\)) is analyzed. The coordinates of each joint are: O (0, 0), A (\(X1\), \(Y1\)), B (\(X2\), \(Y2\)), C (\(X3\), \(Y3\)). Among them:  
1. Forward kinematics analysis: \[\left\{ \begin{array}{c}X1=L1\cos A\\ Y1=L1\sin A\end{array} \right.\] (1) \[\left\{ \begin{array}{c}X2=L1\cos A+(L3-L2)\cos(B-\pi)\\ Y2=L1\sin A+(L3-L2)\sin(B-\pi)\end{array} \right.\] (2) \[\left\{ \begin{array}{c}X=L1\cos A+(L3-L2)\cos(B-\pi)+L0\\ Y=L1\sin A+(L3-L2)\sin(B-\pi)\end{array} \right.\] (3)
2. Inverse kinematics analysis As shown in the mechanism diagram 3-1, in \(\Delta\) OQP: \[\left\{ \begin{array}{c}X2=X-L0\\ Y2=Y\end{array} \right.\] (4) \[L1^{2}+(L3-L2)^{2}-2L1(L3-L2)\cos(B-A)=(X-L0)^{2}+Y^{2}\] (5)  
Figure 7: Schematic diagram of dry-ice cleaning robot. \[\cos(B-A)=\frac{\left(X-L0\right)^{2}+Y^{2}-L1^{2}-\left(L3-L2\right)^{2}}{-2L1 1\left(L3-L2\right)} \tag{6}\]  
\[B-A=\arccos\left(\frac{\left(X-L0\right)^{2}+Y^{2}-L1^{2}-\left(L3-L2\right)^{2} }{-2L1\left(L3-L2\right)}\right) \tag{7}\]  
It can also be calculated:  
\[L1^{2}+\left(X-L0\right)^{2}+Y^{2}-2L1\sqrt{\left(X-L0\right)^{2}+Y^{2}}\cos D =\left(L3-L2\right)^{2} \tag{8}\]  
\[\cos D=\frac{L1^{2}+\left(X-L0\right)^{2}+Y^{2}-\left(L3-L2\right)^{2}}{2L1 \sqrt{\left(X-L0\right)^{2}+Y^{2}}} \tag{9}\]  
\[D=\arccos\left(\frac{L1^{2}+\left(X-L0\right)^{2}+Y^{2}-\left(L3-L2\right)^{2 }}{2L1\sqrt{\left(X-L0\right)^{2}+Y^{2}}}\right) \tag{10}\]  
\[\tan C=\frac{Y}{X-L0} \tag{11}\]  
\[C=\arctan\left(\frac{Y}{X-L0}\right) \tag{12}\]  
According to formula (5) to (12), we can get:  
\[\left\{ \begin{array}{l}A=\arccos\left(\frac{L1^{2}+\left(X-L0\right)^{2}+Y^ {2}-\left(L3-L2\right)^{2}}{2L1\sqrt{\left(X-L0\right)^{2}+Y^{2}}}\right)+ \arctan(\frac{Y}{X-L0})\\ B=\arccos\left(\frac{L1^{2}+\left(X-L0\right)^{2}+Y^{2}-\left(L3-L2\right)^{2 }}{2L1\sqrt{\left(X-L0\right)^{2}+Y^{2}}}\right)+\arctan(\frac{Y}{X-L0})+\\ \arccos\left(\frac{L1^{2}-\left(X-L0\right)^{2}-Y^{2}+\left(L3-L2\right)^{2} }{2L1\left(L3-L2\right)}\right)\end{array} \right. \tag{13}\]  
The parameters in formula (13) are: \(L0=375\) mm, \(L1=1285\) mm, \(L2=230\) mm, \(L3=1310\) mm. By establishing the kinematic equation of the dry-ice cleaning mechanical arm, the relationship between the position and pose of mechanical arm end nozzle tool and the joint (O, M, N, P, and Q) variables and the connecting rod parameters (\(L1\), \(L2\), and \(L3\)) is obtained. With the help of MATLAB mupad tool, the inverse kinematics of the mechanical arm is solved, and the motion law of each joint is calculated. Finally, the end motion track of the mechanical arm is a straight line that moves vertically up and down. The motion data of each joint is imported into ADAMS, and spline curve functions are generated, respectively.