## 4 Optimization of Parasitic Motion
This is demonstrated in the simulation and is shown in Figure 4(a)-c. Figure 4(a) shows the configuration of zero x-axis parasitic motion for the PhRS manipulator. Figure 4(b),c shows the configuration (limb arrangement) that eliminates the x-axis parasitic motion for the PvRS and RPS manipulators, respectively. Note that the limb number convention is consistently applied for all figures and counted counterclockwise starting from the +ve x-axis throughout the paper. The results shown in Figure 5 is the leg (limb) arrangement of the manipulator for zero parasitic motion along the x-axis, and these parameters are \(\xi_{1}=\alpha_{1}=0^{\circ}\), \(\xi_{2}=\alpha_{2}=90^{\circ}\), \(\xi_{3}=\alpha_{3}=-90^{\circ}\). Substituting the angular values and all the corresponding elements in Equation (32) proves the result is true. Based on the result obtained, the following optimality condition can be drawn:  
Conditions for \(v_{x}=0\): To eliminate the parasitic motion along the x-axis, the center of the second and third limb's spherical joints must be collinear, and both should be orthogonal to the first limb. Similarly, the appropriate limb arrangement or configuration for the zero parasitic motion along the y-axis is obtained by optimizing Equation (35). Accordingly, the zero parasitic motion along this particular axis is obtained at \(\xi_{1}=\alpha_{1}=0^{\circ}\), \(\xi_{2}=\alpha_{2}=90^{\circ}\) and \(\xi_{3}=\alpha_{3}=\pm 180^{\circ}\) for all three manipulators. Likewise, the following optimality condition can be drawn for \(v_{y}=0\). Conditions for \(v_{y}=0\): To eliminate the parasitic motion along the y-axis, the center of the first and the third limb's spherical joints must be collinear. i.e., \(\xi_{1}=0^{\circ}\), \(\xi_{2}=90^{\circ}\) and \(\xi_{3}=\pm 180^{\circ}\). Pattern-search algorithm: Considering the Quasi-Newton algorithm is suitable only for a single-digit output objective function, the non-gradient vector optimization (multi-objective algorithm) called pattern-search is used to simultaneously optimize \(v_{x}\) and \(v_{y}\). The pattern-search method is a family of derivative free nonlinear numerical optimization. To reach to the optimal point, it looks a long certain specified direction, and evaluate the cost function at a given step length each of these directions. These points form a frame around the current iteration. Depending on whether any point within the pattern has a lower cost function value than the current point, the frame shrinks or expands in the next iteration. The search stops after the minimum pattern size is reached. The key condition to choose a search direction is that at least one direction in this set should give a descent direction for the cost function whenever the current point is not a stationary point. From the sequential optimization of \(v_{x}\) and \(v_{y}\), we have noticed that the parasitic motion along the x- and y-axes are antagonistic functions. Therefore, the pattern-search algorithm achieved a simultaneous optimization of parasitic motions along the x- and y-axes by coinciding the center of the second limb's spherical joint with the center of the third limb's spherical joint, as shown in Figure 5(a). Although the resulting configuration seems meaningless, it accurately tells the condition for eliminating both \(v_{x}\) and \(v_{y}\) parasitic motions from the workspace. Figure 5: Optimal configuration for \(v_{x}=0\). As can be seen from Figure 5(a), the limb arrangement that eliminates both parasitic motions have two coincident spherical joints. This is because of the constant length radius of the moving platform. In general, the result tells, all spherical joints need to be collinear in order for the manipulator to perform without parasitic motion. Thus, the following condition can be drawn. Condition for \(v_{x}=v_{y}=0\): To eliminate the parasitic motion in all directions, the three joints on the moving platform must be collinear.