# Design of a Predictive RBF Compensation Fuzzy PID Controller for 3D Laser Scanning System
## 2 Lidar Pitching Motion Model  
As shown in Figure 1, the pitching device drives the 2D Lidar to swing up and down and scan the target object or environment. The planar 2D data from the 2D Lidar and the pitching angle of the encoder can be collected. Through an improved simplex calibration algorithm, the 2D data and the pitching angle information are converted to construct 3D data of the point cloud image of the real environment.  
The 3D laser scanning system can be simplified to a schematic diagram to better analyze the Lidar motion process, which is shown in Figure 2. The input signal \(u(t)\) of the driver controls the output torque of the DC servo motor. Through the reducer, the output shaft of the motor is connected to Pole 1, which drives the 2D Lidar to pitch. The 2D Lidar is connected to the encoder through Pole 2, and the output pitching angle information can be gained. In the pitching motion, the 2D Lidar will be subjected to the torque of gravity and friction torque. Thus, the relationship between the input signal \(u(t)\) and the output pitching angle \(\theta\) is  
\[\ddot{\theta}=\frac{\mathrm{K_{u}K_{m}}u(t)}{\mathrm{JR}}+\frac{\mathrm{Gros \theta}}{\mathrm{J}}-\frac{F_{\mathrm{f}}(t)}{\mathrm{J}} \tag{1}\]  
where \(\theta\) is the pitching angle, \(\mathrm{K_{u}}\) is the amplification coefficient of the pulse width modulation (PWM) power amplifier, \(\mathrm{K_{m}}\) is the torque coefficient of the motor, \(\mathrm{R}\) is the armature resistance, \(\mathrm{G}\) is the gravity of the 2D Lidar, \(\mathrm{r}\) is the distance from the center of Lidar to the center of rotation, \(\mathrm{J}\) is the moment of inertia of the Lidar around the system rotation center, and \(F_{\mathrm{f}}(t)\) is a friction torque based on the Stribeck friction model.  
Figure 1: 3D Laser scanning system.  
Figure 2: (**a**) Schematic diagram of the 3D laser scanning system; (**b**) schematic diagram of the Lidar’s gravity center and rotation center.  
As shown in Figure 3, the Stribeck curve indicates the relationship between friction torque and velocity at different friction stages. It can be expressed as  
\[\begin{split} F_{\mathrm{f}}(t)=\left\{\begin{array}{ll}\mathrm{F }_{\mathrm{m}}&F(t)>\mathrm{F}_{\mathrm{m}}\\ F(t)&-\mathrm{F}_{\mathrm{m}}<F(t)<\mathrm{F}_{\mathrm{m}}\\ -\mathrm{F}_{\mathrm{m}}&F(t)<-\mathrm{F}_{\mathrm{m}}\\ \end{array}\right.\quad\left(\left|\dot{\theta}(t)\right|<\alpha\right)\\ F_{\mathrm{f}}(t)=\left(\mathrm{F}_{\mathrm{c}}+(\mathrm{F}_{\mathrm{m}}- \mathrm{F}_{\mathrm{c}})\mathrm{e}^{-\alpha_{1}\left|\dot{\theta}(t)\right|} \right)\mathrm{sgn}\left(\dot{\theta}(t)\right)+\mathrm{k}_{\mathrm{v}}\dot{ \theta}(t)&\left(\left|\dot{\theta}(t)\right|>\alpha\right)\end{split} \tag{2}\]  
where \(F(t)\) is the driving torque, \(\mathrm{F}_{\mathrm{m}}\) is the maximum static friction torque, \(\mathrm{F}_{\mathrm{c}}\) is the Coulomb friction torque, \(\mathrm{k}_{\mathrm{v}}\) is the proportional coefficient of the viscous friction torque, \(\mathrm{\theta}\) is the pitching angle, \(\alpha\) is the critical angular velocity, and \(\alpha_{1}\) is 0.1.