## 2 The Model of the Planar 3-RRR Parallel Robot
### The Kinematic Model of the 3-RRR Planar Parallel Robot
The physical model of the three-phase asynchronous motor is shown in Figure 3.  
The model of the three-phase asynchronous motor can be established by coordinate transformation. The transformation matrix from the three-phase coordinate system to the two-phase stationary coordinate system through the geometric relationship can be expressed as follows [27],  
\[\text{C}_{36/25}=\sqrt{\frac{2}{3}}\left[ \begin{array}{ccc}1&-\frac{1}{2}&- \frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{array} \right] \tag{13}\]  
Figure 3: Physical model of the three-phase asynchronous motor. The transformation matrix from the two-phase stationary coordinate system to the two-phase rotation coordinate system can be expressed as follows [27],  
\[C_{2s/2r}=\left[ \begin{array}{cc}\cos\theta_{e}&\sin\theta_{e}\\ -\sin\theta_{e}&\cos\theta_{e}\end{array} \right] \tag{14}\]  
**3. Virtual Bench Setup and Genetic Algorithm Controller**  
To verify the control effect of the PID controller optimized by the genetic algorithm, a virtual bench is constructed in SimMechanics. At the same time, to simulate the gap error between the two revolute joints in the actual system in the establishment of a three-dimensional model, the axis distance between the two revolute joints is set to 0.5 mm. As Figure 4 shows, the virtual bench of the planar 3-RRR parallel industrial robot is translated from its Pro/E form into Simulink in the SimMechanics toolbox. According to the industrial robot structure in Figure 1, a virtual bench can be established as shown in Figure 4. The parallel industrial robot is composed of three elements, six links, one moving platform, and three driving joints. The moving platform is connected by three branches which consist of two individual links. Each of the two links are connected by revolute joints. The three driving joints on the static platform can constitute an equilateral triangle. Similarly, the three connection joints on the moving platform can also constitute an equilateral triangle. The control scheme is significant for achieving the good stability and precise positioning needed by the 3-RRR planar parallel industrial robot. As we already know, the traditional PID controllers have good control of linear objects and their great advantages are simplicity and convenience. However, when the control reference is complicated in the trajectory tracking control, or when the dynamical parameters are changeable and complex, the classic PID control may not rapidly present a good result. In order to make the controller perform better, the genetic algorithm is introduced in this paper. It is a random search algorithm of natural selection and genetic mechanism which possesses the advantage of simplicity, versatility, robustness, and speed compared to the conventional method. The design flow chart of the genetic algorithm controller is shown in Figure 5.  
The design process of the genetic algorithm is as follows:  
1. Parameter encoding: Due to the need-tuning parameters in the real domain, the simple binary encoding and encoder, six sub-strings can be used as the six parameters influencing each other. They then formed a chromosome. The precision of the PID solution parameters is set to 4 digits after the decimal point, and the solution space of the PID parameters can be divided into \((1000-0)\times(10^{4})\) = 10,000,000 points. Because \(2^{23}<\) 10,000,000 \(<2^{24}\), 24-bit binary numbers are needed to represent these solutions. In other words, a solution code is a 24-bit binary string. Figure 4: The virtual bench of the planar 3-RRR parallel industrial robot. Initially, these binary strings are randomly generated. One such binary string represents a chromosome string where the length of the chromosome string is 24. 2. The fitness function is the basis for evaluating the selection, where the error integral can be used as the performance of the system, such as Integrator Error (IE), Integrator Absolute Error (IAE), Integral Square Error (ISE), Integration Time and Absolute Error (ITAE), etc. Each formula has its own focus.