# Design, Simulation, and Comparison of Advanced Control Strategies for a 3-Degree-of-Freedom Robot
## 4 Design of Control Strategies
### Controllers
#### 4.4.1 Computed Torque Family Controller  
The following Computed Torque Control family is presented:  
Proportional-Derivative Computed Torque Control  
Computed Torque Control, a nonlinear control method based on the dynamic model, is particularly suited for robot control. This approach, which represents a special application of feedback linearization in nonlinear systems, has become increasingly relevant in modern system theory. Its effectiveness is optimal when robot arm parameters are accurately known. A key challenge in robot control involves ensuring the manipulator accurately follows a previously planned trajectory. Precise positioning is prerequisite to any useful task. High-performance operation in uncertain environments requires modern design  
Figure 6: 3-DoF Scorbot robot. Figure 7: Scorbot robot considering the link lengths. techniques. Additionally, consideration must be given to robot movement in free space, where contact with the environment generates external forces that may affect joint or gear flexibility, potentially risking both human safety and robot integrity. For a desired trajectory \(q_{d}(t)\), tracking error \(e\) is defined to ensure accurate path following. Rewriting the 3-DoF robot dynamic equation in (19), we obtain (39) and (40) [3]. \[N\big{(}q,\dot{q}\big{)}=C\big{(}q,\dot{q}\big{)}+G(q)+F\big{(}\dot{q}\big{)} \tag{39}\]  
\[\tau=M(q)\ddot{q}+N\big{(}q,\dot{q}\big{)}+\tau_{d} \tag{40}\]  
In (41), the tracking error is defined as follows:  
\[e=q_{d}-q \tag{41}\]  
Based on the equation in (13), where \(q_{d}\) is the desired trajectory and q is the actual trajectory, the controller must optimize the robot's movement from an initial to a final point, ensuring the end-effector moves quickly and safely. Figure 8 illustrates the 3-DoF Scrobot model robot implementing the dynamic equation. To demonstrate the influence of the input t on the tracking error, it is derived twice to obtain (42) and (43). \[\dot{e}=\dot{q}_{d}-\dot{q} \tag{42}\]  
\[\ddot{e}=\ddot{q}_{d}-\ddot{q} \tag{43}\]  
Solving \(\ddot{q}\) in (40) and substituting in Equation (43), we obtain (44). \[\ddot{e}=\ddot{q}_{d}+M^{-1}(N+\tau_{d}-\tau) \tag{44}\]  
Defining the control input Function (45)  
\[u=\ddot{q}_{d}+M^{-1}(N-\tau) \tag{45}\]  
and the disturbance Function (46)  
\[\mathrm{w}=M^{-1}\tau_{d} \tag{46}\]  
we obtain (47)  
\[x=\left[ \begin{array}{c}e\\ \dot{e}\end{array} \right] \tag{47}\]  
Therefore, the PD Computed Torque Control is defined in Equation (48)  
\[\tau=M(q)\big{(}\ddot{q}_{d}+K_{d}\dot{e}+K_{p}e\big{)}+N\big{(}q,\dot{q}\big{)} \tag{48}\]  
where  
\(K_{p}\): proportional gain. \(K_{d}\): derivative gain. Figure 9 shows the control scheme that represents Equation (48). Figure 8: 3-DoF robot. Initially, the parameter values used were selected according to experiments cited in the study articles in order to establish a set of configurations that allowed the controller to function through modification and improvement. One limitation was the complexity of linking the model of the robot designed in Simscape to Simulink for performing 3D animations. The parameter values used with the Computed Torque PD controller were selected from work [1]. For the simulation, the following considerations were established, which are indicated in Table 5.  
Proportional-Integral-Derivative Computed Torque Control  
Similarly, the PID Computed Torque Control is defined in (49). \[\tau=M(q)\big{(}\tilde{q}_{d}+K_{d}\dot{e}+K_{p}e+K_{i}\dot{e}\big{)}+N\big{(}q, \dot{q}\big{)} \tag{49}\]  
where  
\(K_{i}\): integral gain. Analysis of the PID Computed Torque scheme reveals its similarity to the PD Computed Torque, with the addition of an integral component. However, its connection configuration at the summation point deviates from the classic PID control form due to its structural characteristics. Similarly, the parameter values used with the Computed Torque PID controller were selected from work [1].