## 2 Materials and Methods
### Register Control Algorithm Based on Pulse Width Subdivision
#### 2.2.3 Analysis of Detection Reliability and Influencing Factors  
The printing quality and geometric shape of the register color mark are two important factors that affect detection stability. Figure 8 shows the color mark shape positioning model, in which \(p_{rise}\) and \(p_{down}\) indicate the rising and falling edge of the color mark pulse, respectively, and \(w_{m}\) is the detected color mark width. Generally, these shapes are considered rectangular and triangular and are taken to detect the machine direction and cross-direction register errors. However, the rectangular color mark can only be used for machine direction register control. Because the triangular mark has a bevel, its detection and positioning results are affected by both directions. Therefore, the right-angle side of the triangle mark can be used to achieve machine direction register control, and the triangle mark width can be used as the basis for detecting cross-direction register errors.  
According to the proposed encoder pulse-width subdivision positioning model in Figure 5, the color mark electrical detection error comes from the pulse edge jitter. When positioning the rectangular color mark, its edge jitter can present a symmetrical relationship between left and right, and its positioning accuracy is better than the single-edge-detection method after taking the average of the rising and falling-edge positions. However, due to the existence of the bevel, the jitter of the color mark bevel can be significantly higher than the positioning accuracy of the right-angle edges, and there is also crosstalk between the transverse and longitudinal registrations (machine direction and cross-direction).  
The qualified and unqualified register color mark patterns and detection-edge jitter oscilloscope display waveforms, as shown in Figure 9. It can be seen that color marks (a), (c) and (e) are unqualified printing patterns, the colors of the patterns are not full enough and the edge is irregular. Color marks (b), (d) and (f) are qualified printing patterns with full color and clear and regular edges. On the right, (g) and (h) are the pulse waveforms measured using an oscilloscope, including four pulse signals; from top to bottom, these are color mark pulse (blue), encoder pulse A (green), B (purple) and Z (yellow). According to the operation principle of the incremental rotary encoder, A and B pulses are two orthogonal pulse signals with a phase difference of 90 degrees [40]. Based on the phase relationship, the frequency doubling and direction discrimination function can be realized, and Z is the starting signal of each printing cycle. The real-time monitoring waveforms are presented in (g) and (h). When the encoder pulse signal remains stable without fluctuation, the width-time of an encoder pulse is 25 microseconds according to the oscilloscope display, and its register error fluctuation range is plus or minus one encoder pulse accuracy. Therefore, low-quality printed patterns obviously cannot meet the requirements of control system detection accuracy.  
Figure 8: Color mark shape positioning model.