## 6 Experimental Studies
### Encoder-Based Tracking  
In the first scenario, the performance of the LTV MPC scheme is evaluated using feedback from the encoders. The considered trajectory of the end-effector is designed as described in Section 3.2 and the path is illustrated in Figure 10, whereas the complete motion of the manipulator can be seen in the deposited video ([[https://www.itm.uni-stuttgart.de/en/research/vision-based-control-of-a-powercube-robot/](https://www.itm.uni-stuttgart.de/en/research/vision-based-control-of-a-powercube-robot/)]([https://www.itm.uni-stuttgart.de/en/research/vision-based-control-of-a-powercube-robot/](https://www.itm.uni-stuttgart.de/en/research/vision-based-control-of-a-powercube-robot/))). Note that the trajectory is motivated by an obstacle avoidance and includes motion reversals in both joints to investigate the control performance on friction phenomena caused by the Harmonic Drive gears that underly highly nonlinear friction effects. Figure 11 illustrates experimental results of the considered motion. Figure 10: Desired end-effector trajectory for the motion in the first scenario. Therefore, the measured values of generalized coordinates and velocities are compared with the desired trajectories. In addition, the applied motor currents determined by the MPC, that is, \(I_{1,\text{appl}}\) and \(I_{2,\text{appl}}\), are compared with the preliminarily calculated feed-forward controller based on Equation (12). For validation purpose, the control error of the end-effector is defined as  
\[e_{\text{ee,enc}}=\left\|\mathbf{r}_{\text{ee,enc}}-\mathbf{r}_{\text{ee,des}}\right\| \tag{33}\]and shown in Figure 12. Thereby, \(\mathbf{r_{\rm ee,enc}}=\mathbf{r_{\rm ee}}(\alpha_{\rm enc},\beta_{\rm enc})\) and \(\mathbf{r_{\rm ee,des}}=\mathbf{r_{\rm ee}}(\alpha_{\rm des},\beta_{\rm des})\) are the end-effector positions computed by the forward kinematics with the generalized coordinates measured by the encoders or with the desired generalized coordinates, respectively. Obviously, the proposed LTV MPC scheme works well and can track the predefined trajectory with high accuracy in terms of the angle and angular velocities. Moreover, the proposed control structure can easily handle the predefined input constraints, which can be seen during the period 6.0 s \(\leq t\leq\) 7.5 s as the second joint reaches its input constraints whereby the motor in fact has to provide higher torques to track the desired trajectory. As a result, the tracking error of the end-effector increases, see Figure 12. In contrast to a PID controller, no wind-up effects occur, no additional anti-wind-up control structure is necessary to handle input constraints, and due to the centralized control structure all states are taken into account to compute the inputs to reduce the control error in an optimal manner. The controller also deals with reversing motor directions and treats the occurring friction phenomena, which occur at sign changes in the velocities. Nevertheless, by comparing the implemented and the feed-forward motor currents, it becomes clear that there is a notable deviation. This deviation originates from uncertainties and disturbances since in the case of a perfect model and no disturbances the feed-forward controller (12) would be sufficient. Despite the distinct model uncertainties, the proposed LTV MPC scheme compensates these uncertainties well and thus reveals its robustness. Consequently, the modeling and parameter identification efforts can be extremely reduced with the proposed LTV MPC scheme compared to a pure feed-forward controller. Furthermore, the command response, as well as changes in the input, can easily be adjusted and weighted by a modification of the weighting parameters for the states or inputs in the cost functional, which is much more intuitive and evident than tuning the parameters of a classical PD or PID controller.