# Kinematic Analysis and Verification of a New 5-DOF Parallel Mechanism
## 2 Design and Verification of the Scheme
### Singularity Analysis  
Parallel mechanism has the characteristics of strong bearing capacity, large stiffness, and high motion accuracy. However, in practical application, it shows stiffness degradation, motion instability, bearing capacity reduction, and other phenomena when it is at some special positions, which is revealed in the experimental verification. In the study of this paper, the uncertainty of degree of freedom appears when the moving platform is directly  
Figure 3: Sketch of limb 6.  
above the zero point. The reduction of mechanism performance is not only related to the design, manufacture, and assembly of the mechanism but also closely related to the unique singularity and the dynamic stability of the singularity. The singularity of the mechanism generally refers to the situation that the inverse kinematics solution of the mechanism does not exist, the mechanism motion is unstable, and the driving force in the limb suddenly increases at some positions [35]. The analysis of singularity is the basis to verify the design and avoid the motion in singular space. The essence of singularity of parallel mechanism is force screw failure of partial limbs acting on the moving platform. Based on this, the force of limbs acting on the moving platform is defined as line vector. The linear correlation between these line vectors is analyzed by using Grassmann line geometry method and the maximum linear independent number of these line vectors is defined as rank of line cluster. The rank of the line cluster can be divided into 1 to 5 [36]. When the rank of the line cluster is 1, there is a line in the three-dimensional space. When the rank of a line cluster is 2, it includes two types of spatial distribution: (a) A pair of skew lines; (b) A flat pencil of lines. When the rank of the line cluster is 3, there are 4 kinds of spatial distribution: (a) A regulus; (b) All lines through a point; (c) The union of two flat pencils having a line in common but lying in distinct planes and with distinct centers; (d) All lines in a plane. When the rank of the line cluster is 4, the spatial distribution is four kinds: (a) Elliptic congruence; (b) Parabolic congruence; (c) Hyperbolic congruence; (d) Degenerate congruence. When the rank of the line cluster is 5, there are two spatial distributions: (a) Non singular complex: generated by five skew lines ; (b) Singular complex: all the lines meeting one given line. The singularity of this scheme occurs when the moving platform is parallel to the fixed platform and the moving platform is directly above the zero point; the force line vector extension lines of the five limbs intersect at one point, as shown in Figure 4. In this case, the singular form of the rank of the line cluster belongs to the (b) spatial distribution with rank 3 of the line cluster. It is the case that all lines go through a point. At this time, the rank of the five force line vectors is 3. In theory, for a compound screw system composed of force line vectors and couple-vectors, if the force line vectors are linearly independent of each other, and the couple-vectors are independent of each other, after the force line vectors are translated, the new force couple system composed of the resulting additional force couple and the original force couple system is also linearly independent. However, the specific research object of this paper is five force line vectors and a couple-vector. From the specific form of plucker coordinate, the rank of the composite screw system is equal to the rank of a directional vector matrix composed of five force line vectors plus one couple-vector rank [37]. Therefore, the rank of the five force line vectors is 3, plus the rank 1 of the couple-vector, so that the rank of the final composite screw system is 4. The corresponding physical meaning is that the movement of the moving platform along the \(X\), \(Y\), and \(Z\) axes and the rotation around the \(Z\)-axis are effectively constrained while the rotation around the \(X\) and \(Y\) axes is not limited, resulting in a singularity.