## 4 Optimization of Parasitic Motion  
The parasitic motion has been recognized as one of the major drawbacks of the lower mobility PMs. Consecutively, several researchers have tried to eliminate it from the workspace. However, most attempts are entirely geometry dependent and applied at the position-level while most parallel manipulators control scheme is designed at the velocity-level. In this paper, the parasitic terms in the constraint-compatible velocity are optimized using the velocity-level relation derived in Section 3. From Equation (28), we can see that parasitic motions are dependent on two generic parameters, i.e, \(\omega_{x}\) and \(\omega_{y}\). The angles \(\xi_{i}\) and \(\alpha_{i}\) are simultaneously altered during the optimization procedure; the algorithm simultaneously changes the distal and proximal joint of the limbs, to determine the optimal configuration. Thus, all of them are in the vector \(\delta_{i}\). i.e, \(\delta_{i}= \begin{bmatrix}\xi_{i}&\alpha_{i}\end{bmatrix} ^{T}\). By denoting \(\mathbf{C}_{1}= \begin{bmatrix}\mathbf{v}_{0}&\mathbf{v}_{1}&\mathbf{v}_{2}\end{bmatrix} \), the inverse of \(\mathbf{C}_{1}\) can be rewritten in the form shown in Appendix A. Then, Equation (28) can be described in analytic form as:  
\[ \begin{bmatrix}v_{x}\\ v_{y}\\ \omega_{z}\end{bmatrix} =\frac{1}{\mathbf{v}_{0}\cdot(\mathbf{v}_{1}\times\mathbf{v}_{2})} \begin{bmatrix}(\mathbf{v}_{1}\times\mathbf{v}_{2})^{T}\\ (\mathbf{v}_{2}\times\mathbf{v}_{0})^{T}\\ (\mathbf{v}_{0}\times\mathbf{v}_{1})^{T}\end{bmatrix}\begin{bmatrix}-a_{12}\xi\xi_{1}& -a_{12}\xi_{1}\\ -a_{22}\xi_{2}&-a_{22}\xi_{2}\\ -a_{32}\xi_{3}&-a_{32}\xi_{3}\end{bmatrix}\begin{bmatrix}\omega_{x}\\ \omega_{y}\end{bmatrix} . \tag{30}\]  
Figure 4: Numerical simulation. where  
\[\mathbf{v}_{0} = \left[c\xi_{1}\quad c\xi_{2}\quad c\xi_{3}\right]^{T}\] \[\mathbf{v}_{1} = \left[-s\xi_{1}\quad-s\xi_{2}\quad-s\xi_{3}\right]^{T}\] \[\mathbf{v}_{2} = \left[-(a_{1x}c\xi_{1}+a_{1y}s\xi_{1})\quad-(a_{2x}c\xi_{2}+a_{2y }s\xi_{2})\quad-(a_{3x}c\xi_{3}+a_{3y}s\xi_{3})\right]^{T}.\]  
The expression for variable \(\xi_{i}\) is given in Equation (4). The detail elements of the vector \(\mathbf{a}_{i}\) given in Equation (2) can be expanded as:  
\[\mathbf{a}_{i} = \left[a_{ix}\quad a_{iy}\quad a_{iz}\right]^{T}, \tag{31}\]  
\[a_{ix} = R_{11}r_{p}c\alpha_{i}+R_{12}r_{p}s\alpha_{i}=r_{p}((c\theta c \phi+s\psi s\theta s\phi)ca_{i}+(s\psi s\theta c\phi-c\theta s\phi)sa_{i})\] \[a_{iy} = R_{21}r_{p}c\alpha_{i}+R_{22}r_{p}s\alpha_{i}=r_{p}(c\psi s\phi c \alpha_{i}+c\psi c\phi s\alpha_{i})\] \[a_{iz} = R_{31}r_{p}c\alpha_{i}+R_{32}r_{p}s\alpha_{i}=r_{p}((s\psi c \theta s\phi-s\theta c\phi)ca_{i}+(s\phi s\theta+s\psi c\theta c\phi)sa_{i})\]  
where \(\alpha_{i}\) is the angle to locate the spherical joints from the x-axis of frame \(\{O^{\prime}\}\) as in Equation (3). Our objective in this section is to remove the parasitic motion from the workspace. Through careful observation of the parasitic motion equation, only \(\xi_{i}\) and \(\alpha_{i}\) are the relevant geometric parameters that can be meaningfully altered to get the configuration of the manipulator without parasitic motion. The radius \(r_{p}\) of the moving platform does linearly relate to the parasitic motion, and from the equation, it is observed that the optimal solution is trivial, which is \(r_{p}=0\). Thus, only angle \(\xi_{i}\) and \(\alpha_{i}\) will be considered as feasible design parameters and are denoted by vector \(\delta_{i}\). By obtaining optimum \(\delta_{i}\) for limbs 2 and 3 from the x-axis, the minimum parasitic motion can be achieved for all choices of \(r_{p}\). Thus, the choice of \(r_{p}\) can be left to the designers. Subsequently, it is required to transform Equation (30) in the form that is suitable to the selected optimization algorithm. The algorithms can be single-objective such as Quasi-Newton or multi-objective such as pattern-search or Pareto-front.