## 4 Controller Design  
### Desired Path  
First, the reference signal needs to be determined. In this system, the input signal should be given according to the driving characteristics of the CLRV. The performance test of the LVR should include the ability to accelerate, decelerate, drive at a uniform speed, move forward, reverse, turn, and climb over slopes of about \(20^{\circ}\). Suppose the maximum speed of the pre-researched CLRV is \(v_{rmax}=4\) m/s with \(a_{rmax}=1\) m/s\({}^{2}\).  
Additionally, the signal-to-noise ratio of \(a_{rx}\) and \(a_{ry}\) is 20 dB. After integrating the accelerations \(a_{rx}\) and \(a_{ry}\), the speeds \(v_{rx}\) and \(v_{ry}\) can be obtained. Then, integrating the speeds \(v_{rx}\) and \(v_{ry}\), the position information \(X_{r}\) and \(Y_{r}\) of the CLRV can be obtained, respectively. Figure 4 illustrates the position \(X_{r}\) as well as the speed \(v_{rx}\) of path one of the CLRV in the \(X\) direction. When the CLRV is going uphill or downhill, its speed in the \(X\) direction quickly decreases before stabilizing as it ascends, and quickly increases before stabilizing as it descends.  
Path two is the trajectory of the CLRV moving forward and backward in the \(Y\)-direction. The speed \(v_{ry}\) and position \(Y_{r}\) changes in the \(Y\) direction of path two are shown in Figure 5.  
Figure 4: Desired path one for track-following servo control is plotted in this figure. The initial position of the CLRV is \((-40,0)\). Path one is the trajectory of the CLRV moving in the \(X\)-direction. The trajectory consists of uniform acceleration, uniform deceleration, uniform motion, and changes in \(X\)-directional speed due to uphill and downhill slopes.  
Path three represents the trajectory of the CLRV as it makes a turn with a radius of 10 m and a velocity of 3 m/s. Path four is the trajectory of the CLRV navigating a turn while avoiding obstacles. The speeds of the CLRV in the \(X\) and \(Y\) directions for paths three and four are also displayed in Figures 6 and 7, respectively.  
Figure 5: Desired path two for track-following servo control is plotted in this figure. The initial position of the CLRV is \((-40,0)\).  
Figure 6: CLRV’s motion trajectory and speeds in the \(XY\) directions for desired path three. The initial position of the CLRV is \((0,-9)\). This path mainly consists of three parts: a section of straight motion with uniform acceleration, a turning curve with a radius of 10 m, and a section of straight motion with uniform deceleration.  
In order to accurately control the crane, the controller must handle digital values obtained from sensors rather than continuous values of the states. Therefore, it is necessary to convert the desired path signal from continuous time to discrete time for more accurate representation. This system uses a monocular vision system to measure the position and orientation of lifting points, with a range of \(-40\leq X\leq 40\) m (or \(-10\leq Y\leq 10\) m) and positioning accuracy of \(\pm 10\) mm. The processing frequency of the visual system is 20 Hz. The present positions of the crane and the trolley in the \(X\) and \(Y\) directions are measured by the high-precision magnetic scale, the response time is 1 ms, and the measurement error is better than \(\pm 10\)\(\mu\)m.  
The position measurement noise is added to the output of the desired path with Gaussian white noise. The Gaussian noise amplitude is 0.01 m and the sampling time is 0.05 s. The sampling period of the desired path is 0.05 s, the discrete signal is delayed by one sampling period, and then the signal is passed to the position servo loop controller.