Dexterity Analysis of a 4-UPS-RPS Parallel Mechanism

A novel 4-UPS-RPS spatial 5 degrees of freedom parallel mechanism is introduced and its dexterity measures are analysed. The stationary platform of the parallel mechanism is connected with the moving platform by the same four UPS (universal joints-prismatic pairs-spherical joints) driving links and a RPS (revolute joints-prismatic pairs-spherical joints) link. The velocity Jacobian matrix and the force Jacobian matrix of the parallel mechanism are derived, and the dimensionless velocity Jacobian matrix is established. On this basis, the three dexterity measures, including average conditional number, the smallest singular value and operation, are obtained. According to these, the square mean dexterity coefficient, which is used to comprehensively evaluate the dexterity of various configurations, is proposed. In addition, the distribution of different dexterity measures above in the workspace is analysed respectively. This research can provide an important theoretical base for the dimensional synthesis and trajectory planning for a parallel mechanism.


Introduction
The parallel mechanism, as a new type of mechanism, has recently been paid more attention and is being utilized for many practical applications due to its high stiffness, good dynamic performance, no accumulated precision errors and tight construction 1 .
The dexterity measures of a parallel mechanism are an important index for measuring the transmitting accuracy between the input and output motion (force), the evaluation of kinematics performance, dimensional synthesis and trajectory planning of a parallel mechanism. At present several researchers are studying the dexterity measures of parallel mechanisms2-7 and many achievements have been made. The relevant research include analytic study of local dexterity, closed form solution to the local dexterity and isotropy of parallel manipulators, dexterity index and its distribution, etc. However, the previous studies are oriented to one or a couple of dexterity measures 8 , which has limitations in addition to the fact that the comprehensive performance cannot be evaluated. Therefore, the question of how to expound effective dexterity measures and establish new comprehensive evaluation standards for performance is the key problem for this paper. This paper takes the 4-UPS-RPS parallel mechanism (see Figure 1), which consists of a stationary platform, a moving platform, four UPS (universal joint-prismatic pair-spherical joint) actuating limbs, a RPS (revolution joints-prismatic pairs-spherical joints) actuating limb and a working head, as an example, and the Jacobian matrix is established. The distribution of dexterity measures, including average conditional number, the smallest singular value, operation and square mean dexterity coefficient are analysed.

The Jacobian matrix of THE 4-UPS-RPS parallel Mechanism
The velocity Jacobian matrix and force Jacobian matrix are derived based on the following definition: As shown in Fig. 2 Then the equation is obtained as follows For the spherical joint 1 S , we can get From the RPS branch , we can get Where uA J is constraint mapping matrix, Put the equation (6) into equation (7), we can get For all the five actuated limbs, the relationship between the velocity of the moving platform and the velocity of the actuated limbs is given by So the drive mapping matrix 1A J can be expressed by Combine equation (4) with equation (8), we can get where 6 6 J R   called the velocity Jacobian matrix is the transfer matrix between driving velocity and six-dimension velocity of the moving platform.
From equation (12), we can get where the Jacobian matrix is the transfer matrix between six-dimension velocity of the moving platform and driving velocity.
The angular velocity of the moving platform  is expressed by the differential coefficient with respect to time of the orientation parameter denoted with In the mechanism 202 cos cos , R is the distribution radius of the spherical joint on the moving platform.
As is well-known, there is a dual relationship between the velocity Jacobian matrix and the force Jacobian matrix. So we can get the force transfer matrix by transposing the velocity transfer matrix J and the force transfer matrix ' J     is given by

The dexterity measures of THE 4-UPS-RPS parallel Mechanism
In the 4-UPS-RPS parallel mechanism, the distribution position angle of the universal joint on the stationary platform is 90°, the distribution position angle between the front revolution joint and the neighbouring universal joint on the stationary platform is 45°, the distribution position angle of the spherical joint on the moving platform is 72°. The distance between the front revolution joint and the centre point on the stationary platform is 717mm, the distribution radius of the spherical joints on the stationary platform is 645mm, the distribution radius of the spherical joint on the moving platform is 202mm.

The average conditional number
The average conditional number is given by

The operation
The operation is given by