# Design, Simulation, and Comparison of Advanced Control Strategies for a 3-Degree-of-Freedom Robot
###### Abstract  
This study presents the design, simulation, and comparative analysis of three advanced control strategies applied to a 3-Degree-of-Freedom (DoF) robot manipulator. The controllers investigated are a variant from the Computed Torque Control family, a Proportional-Derivative-Integral with fuzzy logic (PD-PI + fuzzy) controller, and a Model Predictive Control (MPC) scheme. The controller performance is evaluated through the tracking of predefined trajectories in the three-dimensional space. The results are analyzed through XYZ coordinate motion graphs and 3D trajectories. To quantify performance, three error indicators are employed: Residual Mean Square (RMS) with a value of 0.0720 for the Computed Torque Controller, Residual Standard Deviation (RSD), and Index of Agreement (IA). The results demonstrate that the proposed controllers achieve accurate trajectory tracking, with IA values close to unity, demonstrating a high degree of concordance between the desired and executed trajectories. a 2024 14 11010 11  
planned and generated for the robot to execute, considering various scenarios and disturbances. Performance evaluation is conducted through graphical analysis and calculation of quantitative performance indices. Fuzzy logic represents a mathematical approach for modeling imprecise or approximate reasoning and ideas. In fuzzy set theory, an element can belong to a set to varying degrees, allowing partial membership. A fuzzy rule expresses a cause-and-effect relationship through linguistic terms. Its fuzzy nature stems from the use of imprecise and relative adjectives: if X belongs to A, then Y belongs to B, where X and Y are elements and A and B are fuzzy sets [4]. While fuzzy inference systems are universal nonlinear mappers, they still require linear intermediate operations. The following section describes the most common types of fuzzy inference systems [4]:  
* Mamdani: Calculates the truth value for each system input. The system employs T-norms, typically using the minimum between inputs or their product. During defuzzification, the membership function is truncated at the T-norm value level. * Takagi-Sugeno: This system enhances [PERSON]'s approach by reducing computational processing. Instead of output membership functions, it uses concrete linear combinations of input values. The final output is computed as a weighted average between the linear output combinations and T-norm values. * Tsukamoto: Similarly to [PERSON], this system uses injective output membership functions to obtain the pre-image of the T-norm value. The authors in [5] present novel fuzzy metrics following [PERSON] and [PERSON]'s approach. They classify these examples according to their construction and established fuzzy metrics, demonstrating their application to color image processing with potential extensions to other engineering problems. Equations (1)-(3) present well-known generic fuzzy metrics [5]. \[M(x,y,t)=\left(\frac{\left(min\{f(x),f(y)\}\right)^{\alpha}+g(t)}{\left(max\{ f(x),f(y)\}\right)^{\alpha}+g(t)}\right)^{\beta} \tag{1}\]  
\[M(x,y,t)=\frac{g(t)}{g(t)+m*d(x,y)} \tag{2}\]  
\[M(x,y,t)=e^{(-d(x,y)/g(t))} \tag{3}\]  
Subsequently, the authors in [6] implement a linear MPC scheme for a Shunk Power-Cube robot model, achieving reduced computational load and semi-automatic trajectory execution. The system integrates camera-based feedback to demonstrate vision-based control effects and improve control performance. Their experimental validation includes two experiments, highlighting the advantages and limitations of their vision-based time-varying linear MPC scheme implemented on standard computing hardware. In [7], the authors developed an MPC with vision-based servo control to achieve high-precision control for hand-like object manipulation, addressing the complexities of modeling grippers, objects, and contacts. Their control scheme demonstrates superior performance compared to conventional vision-based servo control.