A parametric piecewise-linear approach to laser projection

Abstract

A parametric approach to sketching two-dimensional piecewise linear curves by a scanning galvo mirror system is presented. The proposed method uses a decomposed piecewise linear model that is based on the one-dimensional description of Chua-Kang. The curves to be drawn are treated as a parametric system composed of two positional equations, $X$ and $Y$, which are related to each other by an artificial parameter $\mu$. To verify the effectiveness of this approach, the proposed formulation is applied to a dual-axis motor/mirror assembly Scanner Galvo Mirror System model LP20 showing its capability to sketch accurately any piecewise linear curve.

Appendix A

Let the two-dimensional piecewise linear curve of Fig. 15 be described by $L$ linear segments, and $(L+1)$-coordinates denoted as ($X_{i},Y_{i}$), for $i=1,2,\ldots ,L+1$.

Fig. 15

A $L$-segments piecewise linear curve

Each segment has a slope as follows:

• Region $(1)$:    $J^{(1)}$ for $x\le \beta _{1}$

• Region $(2)$:    $J^{(2)}$ for $\beta _{1}< x\le \beta _{2}$, and consecutively

• Region $(L-1)$:    $J^{(L-1)}$ for $\beta _{L-2}< x\le \beta _{L-1}$

• Region $(L)$:    $J^{(L)}$ for $\beta _{L-1}< x$

$$\begin{aligned} J^{(i)}=\frac{Y_{i+1}-Y_{i}}{X_{i+1}-X_{i}},\forall i=\left\{ 1,2,\ldots ,L\right\} \end{aligned}$$

(18)

Beginning from the left to the right, the piecewise linear characteristic can be decomposed into the sum of $L$ components as shown in Fig. 16: a straight line passing through the origin with slope $m_{0}$, a concave characteristic which starts at $\beta _{1}$ with a negative slope $m_{1}$, and consecutively, the concave characteristics that start at $\beta _{L-2}$ with slope $m_{L-2}$, and $\beta _{L-1}$ with slope $m_{L-1}$, respectively.

Fig. 16

Concave segments that compose the piecewise linear curve of Fig. 15

Adding the $L$ characteristics, it can be obtained an explicit expression $y(x)=y_{0}+y_{1}$ $+\cdots +y_{L-1}$ to describe the curve of Fig. 15.

The slopes $m_{0}, m_{1}, \ldots , m_{L-1}$ must satisfy the following:

• Region $(1)$:    $m_{0}=J^{(1)}$

• Region $(2)$:    $m_{0}+m_{1}=J^{(2)}$ , and consecutively

• Region $(L)$:    $m_{0}+\cdots +m_{L-1}=J^{(L)}$ or $\sum ^{L-1}_{i=1}m_{i}=J^{(L)}$

Thus, $m_{0}=J^{(1)}, m_{1}=J^{(2)}-J^{(1)}, \ldots , m_{L-1}=J^{(L)}-J^{(L-1)}$.

After using the normalized equivalence of $y_{i}=\frac{1}{2}m_{i}\left[ \left| x-\beta _{i}\right| +\left( x-\beta _{i}\right) \right]$, (In this case, de normalized case $m_{i}=1$) which is graphically depicted in Fig. 17, it can be obtained.

$$\begin{aligned} y_{0}&= m_{0}x \\ y_{1}&= \frac{1}{2}m_{1}\left[ \left| x-\beta _{1}\right| +\left( x-\beta _{1}\right) \right] \\ \vdots \\ y_{L-1}&= \frac{1}{2}m_{L-1}\left[ \left| x-\beta _{L-1}\right| +\left( x-\beta _{L-1}\right) \right] \end{aligned}$$

Fig. 17

Graphical representation for $y_{i}=\frac{1}{2}m_{i}\left[ \left| x-\beta _{i}\right| +\left( x-\beta _{i}\right) \right]$, normalized $m_{i}=1$

It must be noted that each $y_{i}$ equivalence has been scaled by the corresponding slope.

After that, with $y(x)=y_{0}+y_{1}+\cdots +y_{L-1}$, it can also be obtained

$$\begin{aligned} y(x)=\left( -\frac{1}{2}\sum ^{L-1}_{i=1}m_{i}\beta _{i}\right) +\left( m_{0}+\frac{1}{2}\sum ^{L-1}_{i=1}m_{i}\right) x+\left( \frac{1}{2}\sum ^{L-1}_{i=1}m_{i}\left| x-\beta _{i}\right| \right) \end{aligned}$$

(19)

This may be written in the following general form:

$$\begin{aligned} y(x)=a+bx+\sum _{i=1}^{L-1}c_{i}\left| x-\beta _{i}\right| \end{aligned}$$

(20)

where

$$\begin{aligned} b&= m_{0}+\frac{1}{2}\sum ^{L-1}_{i=1}m_{i}=J^{(1)}+\frac{1}{2}\sum ^{L-1}_{i=1}\left( J^{(i+1)}-J^{(i)}\right) =\frac{J^{(1)}+J^{(L)}}{2},\end{aligned}$$

(21)

$$\begin{aligned} c_{i}&= \frac{1}{2}\sum ^{L-1}_{i=1}m_{i}=\frac{J^{(i+1)}-J^{(i)}}{2},\end{aligned}$$

(22)

$$\begin{aligned} a&= -\frac{1}{2}\sum ^{L-1}_{i=1}m_{i}\beta _{i}=-\sum ^{L-1}_{i=1}\frac{J^{(i+1)}-J^{(i)}}{2}\beta _{i}=-\sum ^{L-1}_{i=1}c_{i}\beta _{i} \end{aligned}$$

(23)

Two extensions of Eq. (23) can be provided in reference (Chua et al. 1987). The first one considers the case when the initial segment of Fig. 15 does not pass through the origin, in this case Eq. (23) needs simply to be completed by adding a constant as

$$\begin{aligned} a=y(0)-\sum _{i=1}^{\sigma }c_{i}\beta _{i} \end{aligned}$$

(24)

The second one considers a more generalized situation by extending the piecewise linear curve domain as running from the second to the first quadrant. This allows recast Eq.(23) as

$$\begin{aligned} a=y(0)-\sum _{i=1}^{\sigma }c_{i}\left| \beta _{i}\right| \end{aligned}$$

(25)