## New Coordinate Measuring Machine Featuring a Parallel Mechanism*

### Takaaki OIWA**

* Received May 15, 1997

** Faculty of Engineering, Shizuoka University: 3-5-1 Johoku, Hamamatsu 432

**Key words: **coordinate measuring machine, parallel mechanism, 3 DOF mechanism, kinematic analysis, dimensional metrology

### 1. Introduction

In recent years, coordinate measuring machines (CMMs) have been widely used for precision measurement in various fields. Such the conventional CMM uses an *XYZ* mechanism consisting of three mutually orthogonal slide mechanisms. It appears, however, the machineÕs accuracy and efficiency are already at their limit due to several of its characteristics: (1) violation of the Abbe's principle, which is the basis of precision measurement; (2) a weak (cantilever) beam structure in which deflection is often generated by minimal bending force; (3) accumulation of measurement errors led from each axis; and (4) low traverse speed due to a mass accumulation. In short, these problems, which limit the precision of the CMM, are inherent in its stacked architecture or serial mechanism.

This paper proposes a new coordinate measuring machine based on a parallel mechanism that avoids the problems above. The use of this new mechanism will potentially improve the stiffness, accuracy, and efficiency of the CMM. This paper describes the fundamentals and synthesis of the new machine, on which it also performs a kinematic analysis.

### 2. Fundamentals

**Figure 1** depicts the proposed CMM. The touch trigger probe attached to the stage is connected to three prismatic joints (struts) through the revolutionary joints. Each prismatic joint is connected to the overhead base through three spherical joint and contains within it the length measuring instruments (scales) and actuators to expand and contract itself. Variations in the length of the three struts move the stage in three-dimensional space. When the probe touches the measuring object, the probeÕs position (coordinate) and attitude can be derived absolutely from the strut lengths.

The proposed CMM has a number of advantages over the conventional CMM: (1) because the probe is in sensitive directions of the scales, the motion error of the stage has little effect on the measured value; (2) the truss structure has a high stiffness because its members are subject to very few bending forces; (3) the systematic error produced by each of the scales is averaged with the other two; and (4) the small inertial mass enables high moving speed.

### Number synthesis

In general, the degree of freedom for a closed-loop spatial mechanism can be obtained by using Grubler's formula^{1)} as follows:

where *F *is the degree of freedom, *l* is the number of bodies including the stage and the base, *n *is the number of joints, and *f*_{i} is the freedom of the *i*th joint. Using equation 1, possible geometrical arrangements with 3 DOF were searched so that each loop had the same arrangement of joints, and so that the number of closed loop or struts was 3. The obtained arrangements and degrees of freedom for each joint are shown in **Figure 2** . It can be seen that the sum of the degrees of freedom of the joints is always 15, regardless of the number of joints (the arrangement of the mechanism depicted in Figure 1 is 3-1-1).
### Forward kinematic analysis

In general, kinematic analysis on the parallel mechanism is much more complicated than the inverse kinematic analysis usually easy. Moreover it is almost impossible to derive the solutions analytically. In the proposed mechanism, not only the forward kinematics but also the inverse kinematics are complex.
First, in the local coordinate system * °*_{S } shown in **Figure 3** , the position vectors of the joints* C*_{i }are as follows:

Second, in the other base coordinate system * °*_{B} the position vectors are as follows:

Using the translation vector ^{S}**P**_{B} and the rotation matrix ^{S}R_{B}, the above equation can be rewritten in the base coordinates as

This simultaneous equation has 9 unknowns because *i* ranges from 1 to 3. However, it is difficult to solve the nonlinear equation analytically. Thus, only the rotation angles of the struts *b*_{i} were calculated numerically using equations 2 and 3. When the rotation matrix ^{S}R_{B} is expressed by the direction cosine, equation 4 becomes a linear simultaneous equation with 9 unknowns. This equation can be solved easily. Consequently, the coordinate ^{B}**P**_{S} and the orientation of the probe ^{B}R_{S} can be obtained from the following equations:

### Measuring space

The measuring space of the new CMM was estimated by using the forward kinematic analysis described above. When the strut lengths ranged from 800mm to 1250mm, the coordinates of the probe were plotted without considering the angle limitations of the spherical joints. **Figure 4** provides an overhead view of the measuring space. **Figure 5** gives a vertical section in the X-Z plane. Generally, it is difficult to estimate the working space of the parallel mechanism because it is not rectangular. However, we can see that the working space of the proposed CMM is not less than 500*500*250.

### Prototype of 3 DOF manipulator

An experimental manipulator with 3 DOF was constructed in order to examine the validity of the new mechanism, as shown in **Figure 6** . The stage moves in three-dimensional space in correspondence with the actuation of the prismatic joints by three individual servo motors. Because the manipulator is not equipped with length measurement instruments, aside from the rotary encoders built into the motors, the coordinates of the probe canÕt be measured with high accuracy.

### Conclusion

A new 3-D CMM based on parallel architecture and therefore distinct from the conventional orthogonal type was proposed. The fundamentals of the new CMM were described. A kinematic analysis was performed to demonstrate the derivation of probe coordinates from the strut length. The results were then used to estimate the measurement space. Finally, an experimental 3 DOF manipulator was made and actuated in three dimensional space.

### Acknowledgments

This research was supported in part by a grant from the Japan Foundation for Promotion of Advanced Automation Technology.

### References

1) K.H.Hunt: Structural Kinematics of In-Parallel-Actuated Robot-Arms, Trans. ASME, J. Mechanisms, Transmissions and Automation in Design, 105, (1983) 706.