## 5 Simulations and Comparison
### Vehicle Models and Test Conditions
#### 5.1.2. Kambara Test Using Proposed Algorithm  
The vehicle data come from [60,63]. Conditions of the investigation were as follows: the forces and torques values were limited here, based on references, i.e., \(|\tau_{u}|\leq 140\) N, \(|\tau_{\rm r}|\leq 8\) Nm, and the velocities \(u_{max}=1\) m/s, \(v_{max}=0.2\) m/s. For the controller used, the inaccuracy of the model parameters W = 0.2 (20%) and the following set of control parameters were assumed:  
\[k_{x}=25,\ \ k_{\psi}=1.2,\ \ \lambda_{1}=\lambda_{2}=0.1,\ \ \Gamma_{1}=\Gamma_{2}=0.1,\ \ \beta_{1}=0.2,\ \ \beta_{2}=0.01. \tag{70}\]  
The set of coefficients was chosen to ensure that the tracking task is accomplished under the constraints introduced. Figure 6 shows the results obtained for linear trajectory (62) tracking and constant disturbances (64). From Figure 6a-e it can be observed that the position tracking task is realized correctly and the velocities \(u,r\) tend towards the desired ones. The velocities \(u,v,r\) are below the limits (Figure 6f) as well as the force \(\tau_{u}\) and torque \(\tau_{r}\) are limited with exception the first phase of motion (Figure 6g). Using time-varying disturbances (65)-(67), the controller also accomplished the trajectory tracking task as shown in Figure 7a-g, but the effect of these disturbances is clearly visible for position, velocity, and force and moment errors (fluctuating variables). Next, in Figure 8a-g, simulation results for the cycloid trajectory (63) and constant disturbances (64) are given. These are similar to those obtained for the linear trajectory and perturbation constants shown in Figure 6a-g. However, such movement causes some signal changes during trajectory tracking. For the variable disturbances (65)-(67), the performance shown in Figure 9a-g was very close to that obtained for the time-varying disturbances as can be noted from Figure 7a-g.  
Figure 6. Simulation results for Kambara—IQV controller and linear trajectory (constant disturbances): (**a**) desired and realized trajectory; (**b**) position errors (E-frame); (**c**) position errors (B-frame) and angular error; (**d**) desired and realized velocity \((u_{d},u)\); (**e**) desired and realized velocity \((r_{d},r)\); (**f**) realized velocities; (**g**) applied force and torque. Figure 8: Simulation results for Kambara—IQV controller and cycloid trajectory (constant disturbances): (**a**) desired and realized trajectory; (**b**) position errors (E-frame); (**c**) position errors (B-frame) and angular error; (**d**) desired and realized velocity \((u_{d},u)\); (**e**) desired and realized velocity \((r_{d},r)\); (**f**) realized velocities; (**g**) applied force and torque. Figure 7: Simulation results for Kambara—IQV controller and linear trajectory (variable disturbances): (**a**) desired and realized trajectory; (**b**) position errors (E-frame); (**c**) position errors (B-frame) and angular error; (**d**) desired and realized velocity \((u_{d},u)\); (**e**) desired and realized velocity \((r_{d},r)\); (**f**) realized velocities; (**g**) applied force and torque. For a more objective evaluation, criteria (68) were used and summarized in Table 3.  
It can be observed that for the Kambara vehicle, the mean position and velocity errors are smaller when a cycloid trajectory is used instead of the linear trajectory. The mean force \(\tau_{u}\) and the mean kinetic energy have however greater values. It may be observed that the differences in system response in the presence of fixed and time-varying disturbances are small. However, as can be seen from the figures shown there are differences in the variables obtained (oscillation of some signals).