## 4 Kinematics, Workspace, Singularity, and Simulation
### Singularity Analysis Using Geometric Algebra  
In this section, the singularity analysis of the 3 UPS/S spherical wrist (Figure 2) is presented. The singularity analysis is performed using the apparatus of geometric algebra and it is based on the approach introduced in [42; 43]. Firstly, a brief introduction to geometric algebra is presented. Geometric algebra could be considered a further development of Clifford algebra. Clifford algebra was created in the 19 th century, and in the second half of the 20 th century, this algebra has been further developed into a unified language named geometric algebra. A single basic kind of multiplication called the geometric product between two vectors is defined in geometric algebra. The geometric product (ab) is composed of the inner product (a\(\cdot\)b) and outer product (a \(\wedge\)b) of two vectors, i.e.,  
\[\text{ab}=\text{a}\cdot\text{b}+\text{a}\wedge\text{b}\]  
Figure 11: Slices of rotational workspace considering ROM of spherical joints. The inner product a-b is scalar-valued (grade 0). The geometric algebra \(G_{n}\) is a \(2^{n}\)-dimensional algebra, i.e., it generates exactly \(2^{n}\) linearly independent elements. The elements of geometric algebra are as follows: scalar (grade 0); vector (grade 1); bivector (grade 2) is the result of the other product of two vectors; and \(k\) blade (\(a_{1}\wedge a_{2}\wedge\ldots\wedge a_{k}\)) is the result of the outer product of \(k\) vectors. The integer \(k\) is named grade. The geometric algebra contains nonzero blades of maximum grade which are called pseudoscalars. For example, the unit pseudoscalar of \(G_{3}\) of a 3D Euclidean metric space with the standard orthonormal basis \(\{e_{1},e_{2},e_{3}\}\) could be written as  
\[I_{3}=e_{1}\wedge e_{2}\wedge e_{3}=e_{1}e_{2}e_{3},\]  
and the inverse of this pseudoscalar is  
\[I_{3}^{-1}=e_{3}\wedge e_{2}\wedge e_{1}\]  
In this paper, the basis vectors obey the following rules:  
\[e_{i}\cdot e_{j}=\delta_{\bar{i}j};\ \delta_{\bar{i}j}=1\ (i=j);\ \delta_{\bar{i}j}=0 \ (i\
eq j);\ e_{i}\wedge e_{i}=0\]  
A generic element of geometric algebra is called a multivector, which can be written as  
\[M=\sum_{k=0}^{n}\langle M\rangle_{k},\]  
where \(\langle M\rangle_{k}\) denotes the \(k\)-vector part of the multivector \(M\). For the purpose of the singularity analysis, the screws are represented in terms of geometric algebra. In geometric algebra \(G_{3}\), a general screw can be written as a multivector [42], i.e.,  
\[s=u+r\wedge u+hI_{3}u=v_{1}e_{1}+v_{2}e_{2}+v_{3}e_{3}+b_{1}e_{2}\wedge e_{3}+ b_{2}e_{3}\wedge e_{1}+b_{3}e_{1}\wedge e_{2}, \tag{8}\]  
where '\(u\)' is the direction of the screw axis; '\(r\)' is the position vector of a point of the screw axis; '\(h\)' is the pitch of the screw; \(v_{i}\) and \(b_{i}\) are scalar coefficients; and \(I_{3}\) is the unit pseudoscalar \(I_{3}=e_{1}e_{2}e_{3}\) of the geometric algebra \(G_{3}\). The same screw can be written as a six-dimensional vector in \(G_{6}\)[42]:  
\[S=v_{1}e_{1}+v_{2}e_{2}+v_{3}e_{3}+b_{1}e_{4}+b_{2}e_{5}+b_{3}e_{6} \tag{9}\]  
The 3 UPS/S parallel mechanism has four legs, and each of the UPS legs has full mobility while the fourth leg (1S) does not possess full mobility. To characterize this leg as a full mobility one, it was supposed that the remaining degrees of freedom are represented by dummy joints, which are considered driven but locked joints. In other words, three dummy-driven joints were introduced. The notations of screws in this paper are as follows: small letter '\(s\)' indicates a screw written in the geometric algebra \(G_{3}\), while a screw given in \(G_{6}\) is denoted with capital letter '\(S\)'. The screws associated with the joints of each UPS leg can be noted as \({}^{i}S_{1}\) (revolute joint), \({}^{i}S_{2}\) (revolute joint), \({}^{i}S_{3}\) (prismatic joint--driven), \({}^{i}S_{4}\)(revolute joint), \({}^{i}S_{5}\)(revolute joint), \({}^{i}S_{6}\)(revolute joint), and \(j=1,2,3\)--leg number (Figure 12).