# Virtual and Physical Prototyping of Reconfigurable Parallel Mechanisms with Single Actuation
## 3 Mobility Analysis
### Analysis for a Single Kinematic Chain  
First, we consider one kinematic chain and the unit twists that correspond to it. We will study only the spatial part: the planar part of each chain is a 1-DOF coulisse mechanism, and the rotational angle of the driving wheel uniquely determines the carriage position. To begin with, we define coordinate system \(OXYZ\), of which center \(O\) is in the center of the carriage spherical joint, as shown in Figure 1c. Let axis \(X\) be tangent to the circular guide, axis \(Y\) be directed along the swinging arm to its rotation axis and axis \(Z\) be orthogonal to the circular guide plane. Let \(\mathbf{\hat{s}}_{i}=\left[ \begin{array}{cccc}s_{i}^{x}&s_{i}^{y}&s_{i}^{z} \end{array} \right]^{\mathrm{T}}\) be a unit vector directed from one spherical joint to the other, where \(i=1\,\dots\,n_{i}\) is an index of a kinematic chain, and \(n_{i}=3\,\dots\,6\) is a number of chains depending on the mechanism structure shown in Figure 2a-d. Then, the platform twist system for one kinematic chain can be written as follows in this coordinate system:  
\[ \begin{array}{ccccc}\mathbf{\underline{\xi}}_{i1}=\left[\begin{array}{cccc }0&0&1&R&0&0\end{array}\right]^{\mathrm{T}},\\ \mathbf{\underline{\xi}}_{i2}=\left[\begin{array}{cccccccc}1&0&0&0&0&0\end{array} \right]^{\mathrm{T}},\\ \mathbf{\underline{\xi}}_{i3}=\left[\begin{array}{cccccccc}0&1&0&0&0&0\end{array} \right]^{\mathrm{T}},\\ \mathbf{\underline{\xi}}_{i4}=\left[\begin{array}{cccccccc}0&0&1&0&0&0\end{array} \right]^{\mathrm{T}},\\ \mathbf{\underline{\xi}}_{i5}=\left[\begin{array}{cccccccc}1&0&0&0&L_{i}s_{i }^{z}&-L_{i}s_{i}^{y}\end{array}\right]^{\mathrm{T}},\\ \mathbf{\underline{\xi}}_{i6}=\left[\begin{array}{cccccccc}0&1&0&0&-L_{i}s_{i }^{z}&0&L_{i}s_{i}^{x}\end{array}\right]^{\mathrm{T}},\\ \mathbf{\underline{\xi}}_{i7}=\left[\begin{array}{cccccccc}0&0&1&L_{i}s_{i }^{y}&-L_{i}s_{i}^{x}&0\end{array}\right]^{\mathrm{T}},\end{array}  \tag{1}\]  
where \(\mathbf{\underline{\xi}}_{i1}\) corresponds to the carriage motion along the circular guide; \(\mathbf{\underline{\xi}}_{i2}\), \(\mathbf{\underline{\xi}}_{i3}\) and \(\mathbf{\underline{\xi}}_{i4}\) relate to the spherical motion in the carriage joint; \(\mathbf{\underline{\xi}}_{i5}\), \(\mathbf{\underline{\xi}}_{i6}\) and \(\mathbf{\underline{\xi}}_{i7}\) relate to the spherical motion  
Figure 2: CAD models of the synthesized mechanisms with a reconfigurable design: (**a**) tripod; (**b**) quadropod; (**c**) pentapod; (**d**) hexapod. in the platform joint; \(R\) is a radius of the circular guide; and \(L_{i}\) is a distance between the spherical joints (leg length). We can compose matrix \(\mathbf{T}_{i}\), whose columns are twists (1), and examine its rank to determine if the kinematic chain constrains the platform motion or not. To obtain the rank, we can perform a Gaussian elimination procedure [33] (p. 97) and transform initial matrix \(\mathbf{T}_{i}\) to the following upper triangle \(\mathbf{U}_{i}\):  
\[\mathbf{U}_{i}=\left[ \begin{array}{cccccccc}1&1&0&0&1&0&0\\ 0&-R&0&0&-R&-L_{i}\tilde{s}_{i}^{x}&L_{i}\tilde{s}_{i}^{y}\\ 0&0&1&0&0&1&0\\ 0&0&0&1&0&0&1\\ 0&0&0&0&L_{i}\tilde{s}_{i}^{z}&0&-L_{i}\tilde{s}_{i}^{x}\\ 0&0&0&0&0&L_{i}\tilde{s}_{i}^{x}&-L_{i}\tilde{s}_{i}^{x}s_{i}^{y}/\tilde{s}_{ i}^{z}\\ \end{array} \right],\ \text{if}\ \tilde{s}_{i}^{z}\ \
eq\ 0, \tag{2}\] \[\mathbf{U}_{i}=\left[ \begin{array}{cccccccc}1&1&0&0&1&0&0\\ 0&-R&0&0&-R&0&L_{i}\tilde{s}_{i}^{y}\\ 0&0&1&0&0&1&0\\ 0&0&0&1&0&0&1\\ 0&0&0&0&-L_{i}\tilde{s}_{i}^{y}&L_{i}\tilde{s}_{i}^{x}&0\\ 0&0&0&0&0&0&-L_{i}\tilde{s}_{i}^{x}\\ \end{array} \right],\ \text{if}\ \tilde{s}_{i}^{z}=0.\]  
For a general chain configuration, there are no zero rows in matrix \(\mathbf{U}_{i}\); hence, the rank equals six. This means the chain does not impose any constraints on the platform motion (see Equation (A3) in Appendix A: only a zero wrench will satisfy it). However, in some configurations, the matrix can be rank deficient.