# Design and Accuracy Analysis of a Micromachine Tool with a Co-Planar Driving Mechanism
## 2 Principle of the Co-Plane Platform
### Kinematic Analysis  
Kinematic equations were derived, and thereafter used to analyze the following motion characteristics: (a) the relationship between feed resolution of platform and tilt-drive angle; (b) the relationship between the X and Z movement of the platform and the tilt-drive angle; and (c) the relationship between the rotation resolution of the platform and the range of the rotation angle. 1. Relationship between feed resolution of platform and tilt-drive angle While the platform carries out the axial feed, the resolution is decided by the tilt-drive angle. The feed resolution can be calculated as follows \[R_{P}=R_{B}\cdot\ \cos\theta\] (1) where \(R_{P}\) is the feed resolution of the platform, \(R_{B}\) is the movement resolution of the ballscrew, and \(\theta\) is the angle between the sliding rod and the ballscrew 2. Relationship between the X and Z movement of platform and the tilt-drive angle The co-planar axial linear movement of the platform depends on the length of the ballscrew and the tilt angle. The relationship can be expressed as follows \[D_{P}=\ D_{B}\cdot\ \cos\theta\] (2) where \(D_{P}\) is the platform movement, \(D_{B}\) is the ballscrew movement, and \(\theta\) is the angle between the sliding rod and the ballscrew 3. The relationship between the rotation resolution of the platform and the range of the rotation angle To understand the relationship between the rotation resolution of the platform and the rotation angle, it is necessary to first derive the relationship between the travel distance of the ballscrew nut and the rotational angle of the platform. When the platform is rotated, the travel distance of the tilt-drive servo sliding block (shown in Figure 3) can be calculated as follows \[l_{1}=\ \big{|}f\big{(}\sin\theta^{\prime\prime}-\sin\theta^{\prime}\big{)} \big{|}\] (3) where \(l_{1}\) is the travel distance of the servo-drive sliding block when the platform rotates about the B-axis,\(f\) is the distance from the connection point to the center of the platform, \(\theta^{\prime}\) is the angle before rotation, and \(\theta^{\prime\prime}\) is the angle after rotation. The difference between \(\theta^{\prime\prime}\)and \(\theta^{\prime}\) is the rotation degree of the platform. After substituting Equation (3) into Equation (4), the travel distance of the ballscrew nut can be calculated. \[D_{B1}=\ l_{1}/\cos\theta \tag{4}\]  
where \(D_{B1}\) is the travel distance of the ballscrew nut, and \(\theta\) is the tilt-drive angle. The shortest ballscrew travel gives the smallest rotation of the platform. Thus, the ballscrew feed resolution can be used to calculate the movement resolution of the platform by using Equation (1). By substituting Equation (1) into Equation (4), the relationship between rotation resolution and rotation range can be obtained using Equations (5) and (6). \[R_{P}=l=l_{2}-l_{1}=\left|f\big{(}\sin\theta^{\prime\prime}-\sin\theta^{\prime} \big{)}\right| \tag{5}\]  
\[\sin^{-1}(l_{2}/f)-\sin^{-1}(l_{1}/f)=\theta^{\prime\prime}-\theta^{\prime}= \theta_{\rm r} \tag{6}\]  
where \(\theta_{r}\) represents the rotation resolution of platform. According to the equations explained above, a kinematic analysis was performed. Figure 4 shows the relationship between platform feed resolution and tilt-drive angle \(\theta\). It is noted that the larger the tilt-drive angle, the higher the feed resolution. By contrast, the smaller the tilt-drive angle, the lower the feed resolution. Figure 5 shows the relationship between the linear travel length of the platform and the tilt-drive angle. The results show that larger tilt-drive angles provide finer movement resolution, but the platform travel length will be smaller as a consequence. Conversely, a smaller tilt-drive angle provides coarser movement resolution and larger platform travel length. Figure 6 shows the relationship between the rotation resolution of the platform and the rotation angle. It is noted that rotation resolution will be decreased when the platform rotation angle is increased.