## 3 Comparative Analysis of Inverse Kinematics Methods
The link \(r_{a,b}\) connects the reference frames \(o_{a}\) and \(o_{b}\), while the vector \(r_{a,c}\) connects the system \(o_{a}\) with the points along the given trajectory. The following models for comparison, which assess the effectiveness of the Directed Angular Restitution (DAR) method, are all structured according to this generalized setup. The primary objective is to determine whether the vectors \(r_{a,b}\) and \(r_{a,c}\) are collinear after applying the rotational transformation as a function of \(\Delta q\). Collinearity is not about making \(r_{a,b}\) and \(r_{a,c}\) identical but ensuring they point in the same direction. This collinearity is visually represented by the orange dashed lines relative to the link \(r_{a,b}\). Two variants are analyzed for each trajectory type: cumulative displacement and non-cumulative displacement. The cumulative variant involves updating the vector \(r_{a,b}\), with each transformation, changing its initial condition with each angular displacement \(\Delta q\). In the non-cumulative variant, \(r_{a,b}\) does not accumulate transformations, retaining its initial condition throughout, thus requiring larger angular displacements to achieve collinearity. For each trajectory and its respective variants, the error between the desired joint trajectory and the calculated joint trajectory is presented. This error analysis is crucial for evaluating the accuracy and effectiveness of the different methods in ensuring that the vectors remain collinear after the transformations are applied. By comparing the errors across the different methods, the significance of the DAR method in maintaining precise and stable motion is underscored. Figures 2-4 present the results of non-cumulative trajectory tracking for circular, oval, and logarithmic trajectories, respectively. In these simulations, it is observed that without the DAR functions, the trajectories exhibit inconsistencies and abrupt changes in the direction of rotation. In contrast, when the DAR functions are applied, the trajectories become smoother and demonstrate a continuous and consistent direction of rotation. This improvement is particularly notable with the directed inverse cosine function (DAR), which provides correct results due to its range of \([0,\pi]\), compared to the inverse sine and tangent functions, which have ranges of \([-\pi/2,\pi/2]\). The discrepancy in directional changes is clearly visible in the orange dashed lines that represent the connection between the systems \(o_{b}\) and \(o_{c}\). These lines should be collinear with the link represented by the black lines, which remain of constant magnitude. The collinearity of the orange dashed lines indicates a correct calculation of the joint variable. Figure 3: Non\(-\)cumulative semi-oval trajectory tracking. Figure 2: Non\(-\)cumulative circular trajectory tracking. The importance of employing DAR functions in ensuring smooth and precise movement in robotic applications is underscored by these comparisons. The associated errors between the desired and calculated joint trajectories are shown in Figures 5-7. These error analyses further highlight the effectiveness of the DAR method in maintaining accurate trajectory tracking, with significantly lower errors when the DAR functions are applied. Figure 4: Non\(-\)cumulative logarithmic trajectory tracking. Figure 5: Error analysis for non-cumulative circular trajectory tracking (Figure 2). Figure 8-10 present the results of cumulative trajectory tracking for circular, oval, and logarithmic trajectories, respectively. In practical trajectory planning, large displacements are uncommon as they tend to induce high acceleration values or uncertainties. Instead, small deltas are preferred to ensure smooth transitions and avoid over-accelerations. In this new simulation, the conditions from the previous experiment were maintained. However, in the cumulative variant, the vector \(r_{a,b}\) undergoes continuous transformations, with its initial condition changing at each iteration.