## 1 Introduction  
In recent decades, robot applications have been extensively studied [1] and numerous improvements have been developed [2]. These advances are becoming more robust today since they mainly focus on work with repetitive tasks to increase productivity, such as industrial applications [3, 4] and applications in the medical area [5]. Therefore, robot arms (manipulators) must be precise to be used by these applications and many others [6]. Usually, these robots can work in dangerous environments, in places where humans cannot access to perform dangerous tasks [7]; at the same time, these robot arms must navigate obstacles because the environment of a robotic arm is often very complicated. For these reasons, the robotic arm's motion must be precise and fulfill some specific characteristics that are defined depending on the environment and the application. In addition, it is necessary to thoroughly study trajectories and kinematics (direct and inverse) to verify that the robotic arm does not show any complications while performing a motion [8]. For example, the most famous methods to calculate the trajectory of a robotic arm [9, 10] are cubic polynomials [11], trapezoidal trajectory [12], and the Euler angles [10], amongst others. In addition, many studies, methods, algorithms, and designs (electronic and mechanical) help to obtain a better performance in a robotic arm motion [13, 14, 15, 16, 17, 18]. Nevertheless, some of these methods are complicated to implement. Others are feasible in exhibiting an unnatural motion, infinite jerk (third time derivative of position), or require other resources, such as an optimal timing solution or specific PID (proportion-al-integral-derivative) control to generate the optimal trajectory. For example, the LSPB method (Linear Segment with Parabolic Blends) has improved the trajectory's performance [19]. However, it requires more calculations. This is because the LSPB divides the trajectory into segments and may only sometimes produce the most optimal solution in terms of the executed time [20]. Additionally, using this method can result in undesirable acceleration profiles, limiting the flexibility of the resulting motion [21]. Concerning n-order polynomials, they also have some disadvantages. According to some authors [11; 22; 23; 24], higher-order polynomials are more complex and may require more computation to evaluate, which can be a drawback in real-time control systems. In addition, it presents a local minima or maxima, which can lead to unexpected or undesired behavior if the trajectory is not carefully designed. Finally, this method may not be as flexible as other types of trajectories, such as splines, in accurately following the desired path. In summary, high-order polynomials must be carefully designed using other techniques to optimize this method. In another example, the authors in [25] present trajectories with a total time of 11 s using a radial basis function (RBF) neural network. This method is reliable and an excellent option for performing motion planning. However, the trajectories present many big-long oscillations through time, which means a greater energy consumption and unsafe motion planning to avoid collisions [26]. The 3-5-3 interpolation polynomial method presented in [27] has the same problem mentioned previously in [25]. Other works [28; 29; 30; 31] use numerical methods with homotopy continuation to generate optimal trajectories in manipulators. The authors mentioned that this approach is a favorable option for generating trajectories because these are versatile and fulfill the characteristics of the mechanical system. Other related works that use homotopic functions for optimal trajectory planning focus mainly on mobile robots [32], humanoid robots [33], dynamics, and control problems [34]. In previous work, a novel algorithm introduced in [35; 36] was presented as a trajectory planning approach with more characteristics which had not been mentioned before and others which had been overlooked.