**7.3 The simulation of cascaded SISO closed loop for high-accuracy tool manipulation control system**

In this section, we present the simulation results of the feedforward and the feedback control system designed for the robot tool manipulator. As discussed, the plant linearization method is used in the feedback controller design. The controlled system is simulated with the original nonlinear tool robot arm and the camera sensor model of Eqs. (77) and (78). Furthermore, we illustrate how varying the damping ratio ζ affects the responses. We chose the bandwidth of the inner-loop control as 100*rad=s* and the bandwidth of the outer-loop control as 10*rad=s*. The intrinsic camera parameters are chosen as following: *f <sup>u</sup>* ¼ 2*:*8 *mm* (Focal length) and *α* ¼ 120° (Angle of view). Other parameters are chosen as: *L* ¼ 1*m*, and *Lt* ¼ 0*:*135*m*.

We compare the performance of the feedforward-feedback control system and the feedback-only control system in two different scenarios. One scenario is simulated when the tool is kept inside the camera angle of view during the entire run time. It should be noted that both the feedback and the feedforward controllers are active during the entire simulation. The second scenario is simulated when the tool is outside the camera angle of view during the entire simulation time. When the tool pose is outside the camera angle of view, the feedback signal is replaced by the estimation signal from the model, as shown in **Figure 22**. Furthermore, for each scenario, we show how varying the damping ratio **ζ** affects the responses. Assuming the initial angle of the tool, *qm***<sup>0</sup>** ¼ **0** in the inertial base frame, we vary the pose or the rotational angle of the camera, *qv*, in the inertial base frame, for each simulation scenario.

**Figures 23**–**26** show step responses of the joint angle *qmf ^* and the output tool pose *PTX* (only X coordinate of the six dofs pose of the tool *PT*) for the two scenarios. Each response is simulated with four different damping ratios: ζ¼2 (Blue), **ζ** ¼ 1.5 (Green), **ζ** ¼ 1 (Purple), and **ζ** ¼ 0.7 (Black). With the same damping ratio (same color), the response of the feedback-only control system is shown in the dashed line and the response of the feedforward-feedback control system is illustrated in the solid line.

In the first scenario, the camera rotates 5° counterclockwise with respect to the inertial frame. Then, the tool stays in the camera angle of view with any joint angle *qmf ^ <sup>ϵ</sup>* ½ � �**180**°, **<sup>180</sup>**° . In addition, a disturbance *dqm* ¼ �**10**°, is added to this joint angle. The responses in **Figures 23** and **24** illustrate that both the feedback-only and the feedforward-feedback control systems can reach stability and are robust to the disturbances. With varying the damping ratios, the feedforward-feedback system

*Role of Uncertainty in Model Development and Control Design for a Manufacturing Process DOI: http://dx.doi.org/10.5772/intechopen.104780*

**Figure 23.** *Step responses of qm <sup>f</sup> ^ . Scenario 1: qv* ¼ �5°, *dqm* ¼ �10°*.*

**Figure 24.** *Step responses of PTX. Scenario 1: qv* ¼ �5°, *dqm* ¼ �10°*.*

responses are faster in transient when compared to the response of the feedback-only system. The feedforward-feedback system response, when the damping ratio is small (**ζ** ¼ 0.7), results in a less overshoot when compared to the response of the feedbackonly system. The optimal damping ratio, **ζ***opt* = 1, (as discussed in Section 6.4), results in the fastest response and no overshoots, both for the feedforward-feedback and the feedback-only control systems.

**Figure 25.** *Step responses of qm <sup>f</sup> ^ . Scenario 2: qv* ¼ �65°, *dqm* <sup>¼</sup> <sup>15</sup>°*.*

**Figure 26.** *Step responses of PTX. Scenario 2: qv* ¼ �65°, *dqm* ¼ 15°*.*

In the second scenario, the camera rotates 65° counterclockwise with respect to the inertial frame. Using geometry, we can calculate that the tool is out of the camera range of view when *qmf ^* <sup>∉</sup> ½ � **<sup>35</sup>***:***21**°, **<sup>134</sup>***:***79**° . Staring from the initial angle *qm***<sup>0</sup>** <sup>¼</sup> **<sup>0</sup>**°

*Role of Uncertainty in Model Development and Control Design for a Manufacturing Process DOI: http://dx.doi.org/10.5772/intechopen.104780*

and move to the target joint angle *qmf* ¼ **50**°, there is a range *qmf ^ <sup>ϵ</sup>* <sup>½</sup>**0**°, **<sup>35</sup>***:***21**°] that camera cannot detect the tool but estimation of the pose is required to drive the tool to the target. Even perturbed with the disturbance *dqm* ¼ **15**°, all the simulation responses shown in **Figures 25** and **26** reach the target within a second. In this scenario, the feedforward-feedback control system still converges faster in transient but generates bigger overshoots compared to the feedback-only control system. The large overshoots may come from accumulated disturbances that cannot be eliminated by the feedforward control without the intervention of the feedback control. A feedforward controller may drive the tool away from its target even faster when the disturbance appears in the loop. Perhaps, a possible solution, which will be investigated in the future, would be the use of a switching algorithm, switching from a feedforward to a feedback controller, rather than the use of a continuous feedforwardfeedback controller.

In summary, the responses from a continuous feedforward-feedback system are more vulnerable to the disturbances especially when the starting position of the tool is far from its target. Although the disturbances will be eliminated as soon as the tool moves inside the camera range of view, the overshoots are more severe if more disturbances are accumulated in the process. In the real-world applications, the feedback-only control solutions are slower than the simulation results as the camera requires extra time, which is not considered in these simulations, to take pictures. Therefore, a feedforward controller, which compensates for the speed limitation of the feedback-only control, becomes indispensable in real manufacturing environments. The issue of the overshoots can be dealt with either by upgrading the camera with a wider range of view or as mentioned previously, the use of a switching algorithm, such as switching from a feedforward to a feedback controller, rather than the use of a continuous feedforward-feedback controller.
