## 2 Constraint Analysis at the Velocity Level
### Analytic Constraint Equation  
Utilizing a reciprocity relation is a well-known approach to get the structural constraint information, and there are geometric, numeric, and analytic techniques to obtain the reciprocal screw's basis elements. Geometric and numeric approaches suffer from two contradicting limitations, as discussed in the previous sections. On the one hand, the geometric method is well related to the physical meaning, but it cannot be consistently applied to complicated systems. On the other hand, the numerical method cannot be easily interpreted, but it can solve any type of screw system. This study uses an analytic approach introduced by [PERSON] and [PERSON] [25; 26; 27], since it can give a consistent derivation process with physical and geometrical meanings. Each manipulator limb is isolated and is considered a serial mechanism to obtain the manipulator's structural information. Then, limb motion generators, that span the limb IMS, are used to get the limb restriction screw. For the manipulators shown in Figures 1-3, the limb IMS is spanned by \(4\xi_{0}-1\xi_{\infty}\) screws, where \(\xi_{0}\) and \(\xi_{\infty}\) are zero and infinite screws, respectively. Hence, the Jacobian matrix of the \(i\)th leg is:  
\[\mathbf{J}_{i}= \begin{bmatrix}\mathbf{0}&\mathbf{s}_{2i_{\parallel}}&\mathbf{s}_{3i_{ \parallel}}&\mathbf{s}_{4i_{\parallel}}&\mathbf{s}_{5i_{\parallel}}\\ \mathbf{s}_{1i_{\parallel}}&\mathbf{s}_{2i_{\parallel}}\times\mathbf{I}_{i}&\mathbf{0}&\mathbf{0 }&\mathbf{0}\end{bmatrix} . \tag{13}\]  
describe the parasitic motion. The parasitic motion can also be incorporated at the velocity-level without deriving any coupling relation as suggested in [29] by projecting arbitrary motion onto the null space of the constraint matrix. In this section, the coupling relation between parasitic and independent terms is formulated analytically. After detecting and identifying the parasitic terms it can be written as a function of the independent terms. This is called the coupling relationship of the task space variables. For the example manipulators used in this paper, the variables \(v_{z}\), \(\omega_{x}\), and \(\omega_{y}\) were identified as the independent variables, whereas \(v_{x}\), \(v_{y}\), and \(\omega_{z}\) were the parasitic terms [12; 18; 22], and the following general relation can describe it. \[ \begin{bmatrix}v_{x}\\ v_{y}\\ \omega_{z}\end{bmatrix} =f\left( \begin{bmatrix}\omega_{x}\\ \omega_{y}\\ v_{z}\end{bmatrix} \right). \tag{20}\]  
To rearrange the manipulator's task motion variables in the format shown in Equation (20), the general constraint given in Equation (19) is used. Then, we can obtain the velocity-level analytic coupling relation between the parasitic and independent velocity terms by substituting the manipulator's structural constraint information in Equation (19). Obtaining the coupling relation between dependent and independent variables enables us to understand the parasitic motion in-depth. It also can be used to understand the geometric parameters that cause the parasitic motion for eventual optimization of the structure. The method of obtaining the coupling relation from the relation in Equation (19) is advantageous over the position-level approach when the manipulator is designed at the velocity-level. Moreover, obtaining \(\mathbf{G}_{c}\) is a well-established procedure at the velocity-level compared to the geometric-based point-plane constraint approach. This section briefly shows the detailed derivation of parasitic terms as a function of the independent motion by manipulating the velocity constraint element-wise. From the Equation (19), we have the \(i\)th limb constraint's relation with the task velocity and can be rewritten as:  
\[\mathbf{s}_{2i_{1}}^{T}\mathbf{v}+[\mathbf{s}_{2i_{1}}\times]\mathbf{a}_{i}\mathbf{\omega}=0, \tag{21}\]  
where  
\[[\mathbf{s}_{2i_{1}}\times]= \begin{bmatrix}0&0&c\xi_{i}\\ 0&0&s\xi_{i}\\ -c\xi_{i}&-s\xi_{i}&0\end{bmatrix} . \tag{22}\]  
Then, by distributing the elements of \(\mathbf{s}_{2i\parallel}\) we can get:  
\[