## 3 Comparative Analysis of Inverse Kinematics Methods  
In the field of robotics, the accuracy and robustness of inverse kinematics solutions are considered critical for ensuring precise motion and control. Traditional inverse kinematics methods are typically based on trigonometric functions such as inverse sine (sin\({}^{-1}\)), inverse cosine (cos\({}^{-1}\)), and inverse tangent (tan\({}^{-1}\)) to compute the angles between vectors. These angles are fundamental for determining the joint configurations required to achieve a desired end-effector position. However, inherent limitations in these standard trigonometric functions often lead to significant issues in practical applications, particularly when dealing with complex trajectories or when the system is sensitive to initial conditions. In this section, a detailed comparative analysis is presented between traditional trigonometric approaches and the Directed Angular Restitution (DAR) method enhanced with spin functions. The objective of this comparison is to highlight the challenges associated with the standard methods and to demonstrate the superior performance of the DAR method in addressing these issues. The traditional methods of inverse kinematics, rooted in geometric analysis, rely heavily on the use of trigonometric functions. These methods are prevalent because they offer straightforward solutions based on the geometric relationships between joint vectors and desired positions. However, despite their simplicity, these approaches are prone to several significant drawbacks, including:  
1. Limited Range: Trigonometric functions like cos\({}^{-1}\), sin\({}^{-1}\), and tan\({}^{-1}\) are restricted by their mathematical domains, which can result in incorrect angle calculations when the actual angles fall outside these ranges. 2. Sign Ambiguity: Determining the correct sign during the computation of these angles is critical for ensuring the correct direction of motion. A misinterpretation can lead to incorrect rotational directions and potential instability in the robot's movement. 3. Sensitivity to Initial Conditions: The performance of these methods is highly sensitive to initial conditions. Even slight variations in the starting configuration can lead to significant discrepancies in the resulting joint trajectories. The analysis in this section focuses on two primary scenarios:  
1. Initial Trajectory Calculations: In these simulations, the inverse kinematics is performed without updating the initial position of the vector \(r_{a,b}\). Each point in the desired trajectory is analyzed independently, simulating a non-cumulative approach where the current state does not influence subsequent calculations. 2. Cumulative Trajectory Calculations: In contrast, these simulations involve updating the position of the vector \(r_{a,b}\) as the system evolves over time. This cumulative approach reflects a more realistic scenario where the position of the vector is continuously adjusted based on prior movements. The comparison with standard trigonometric functions is crucial because these functions are widely used in basic geometric analyses for solving inverse kinematics problems. However, they are prone to the issues previously mentioned, which can undermine the accuracy and stability of robotic motion. By comparing these traditional methods with the DAR method, the analysis aims to underscore the importance of addressing these limitations to achieve more accurate and stable motion in robotic applications. The following comparative analysis is based on the generalized system structure illustrated in Figure 1. This figure, presented in Section 2, outlines the arrangement of the four reference frames: \(o_{0}\), \(o_{a}\), \(o_{b}\), and \(o_{c}\). The system \(o_{c}\) represents the points of three specific trajectories--circular, oval, and logarithmic--each of which is chosen to highlight particular characteristics that are essential for demonstrating the advantages and disadvantages of each function being analyzed.