Parametric FIR filtering for G-code interpolation with corner smoothing and zero circular contour error for NC systems

Abstract

Motion planning is an important topic in numerical control systems. It has two main schemes: acceleration/deceleration (acc/dec) before interpolation (ADBI) and acc/dec after interpolation (ADAI). An ADBI can interpolate command without contour error, but it causes a velocity discontinuity problem when passing through a corner. Meanwhile, an ADAI can generate a smooth cornering profile without discontinuous velocity, but it has contour error along circular trajectories. To have the benefits of both schemes and avoid their disadvantages, this study proposes a new scheme for G-code motion planning: parametric acc/dec interpolation (PADI). The PADI is a new scheme different from the ADAI and ADBI schemes. The PADI first plans a motion profile in a parametric space like an ADAI and then interpolates the blended parametric command in the working space like an ADBI. Through this approach, it can generate a smooth cornering profile like an ADAI and interpolate a circular command with zero contour error like an ADBI. These advantages can be observed in simulations and experiments. The proposed PADI scheme provides a brand-new motion planning strategy with multiple advantages that traditional methods cannot simultaneously achieve.

Appendix: Acceleration-based FIR filtering [1]

1.1 A.1 Filter types

An FIR filter can be expressed as

$$Y[k]=\sum_{i=0}^{{N}_{f}-1}{h}_{i}X[k-i]$$

(A.1)

where X is the input, Y is the output, k is the index of time, ${h}_{i}$ is the coefficient of the FIR filter, and ${N}_{f}$ is the size of the filter (or the number of the filter’s taps). There are two types of FIR filters used in this paper: 1st-type and 2nd-type FIR filters. A 1st-type filter is a moving average filter, while a 2nd-type FIR filter is composed of two 1st-type FIR filters linked up in series. The impulse responses of a 1st-type FIR filter and a 2nd-type FIR filter are shown in Fig. 

Fig. 18

(a) First-type and (b) second-type FIR filter

18(a) and (b), respectively. Equation (A.2) defines the coefficients of a 1st-type FIR filter, equation (A.3) defines the coefficients of an odd-size 2nd-type FIR filter, and equation (A.4) defines the coefficients of an even-size 2nd-type FIR filter:

$${h}_{i}=1/{N}_{f}$$

(A.2)

$${h}_{i}=\left\{\begin{array}{cc}\frac{4i+4}{{N}_{f}^{2}+2{N}_{f}},& i=0,\dots ,\frac{{N}_{f}-1}{2}\\ \frac{4{N}_{f}-4i}{{N}_{f}^{2}+2{N}_{f}},& i=\frac{{N}_{f}+1}{2},\dots ,{N}_{f}-1\end{array}\right.$$

(A.3)

$${h}_{i}=\left\{\begin{array}{cc}\frac{4i+4}{{N}_{f}^{2}+2{N}_{f}+1},& i=0,\dots ,\frac{{N}_{f}}{2}-1\\ \frac{4{N}_{f}-4i}{{N}_{f}^{2}+2{N}_{f}+1},& i=\frac{{N}_{f}}{2},\dots ,{N}_{f}-1\end{array}\right.$$

(A.4)

1.2 A.2 A-FIR scheme

An A-FIR ADAI scheme is illustrated in Fig. 

Fig. 19

A-FIR ADAI scheme

19. The tangent acceleration profile ${A}_{F}$ separates to ${A}_{F,X}$ and ${A}_{F,Y}$ along x- and y-axes, respectively. Then, they convolute with FIR filters to have ${A}_{X}$ and ${A}_{Y}$. Finally, the integrators generate position commands x and y from the filtered acceleration profiles. By designing the tangent acceleration profiles and FIR filters, one can generate command trajectories for G01 trajectories.

For a convolution process of an acceleration pulse in an A-FIR, the final velocity value can be any desired value. This allows the A-FIR to connect several convolution processes in a series without stopping the planning process. Because these convolution processes are individual processes, the constraints and the target velocities in these processes can be different. This is the key feature of A-FIRs that V-FIRs cannot achieve.

1.3 A.3 Acceleration profile for G01 trajectories

Figure 

Fig. 20

n-segment trajectory

20 is an n-segment trajectory with n + 1 points ${P}_{0}$, ${P}_{1}$,…, ${P}_{n}$, where the black line is the reference trajectory and the yellow line is the convoluted trajectory. An index $i=0,\dots ,n-1$ is used for simplifying the notation. For this trajectory, the segment length is ${S}_{i}$, the unit vector along each segment is ${\stackrel{\rightharpoonup}{v}}_{i}$, and the corner angle is ${\theta }_{i}$:

$${S}_{i}=\left|\overrightarrow{{P}_{i}{P}_{i+1}}\right|$$

(A.5)

$${\stackrel{\rightharpoonup}{v}}_{i}=\overrightarrow{{P}_{i}{P}_{i+1}}/{S}_{i}$$

(A.6)

$${\theta }_{i}={\mathrm{cos}}^{-1}\left({-\stackrel{\rightharpoonup}{v}}_{i-1}\bullet {\stackrel{\rightharpoonup}{v}}_{i}\right)$$

(A.7)

A tangent acceleration profile ${\stackrel{\rightharpoonup}{A}}_{F}$ can be written as follows:

$${A}_{F}\left[k\right]=\sum {A}_{i}\left[k\right]{\stackrel{\rightharpoonup}{v}}_{i}$$

where ${A}_{i}$ is the acc/dec component along ${\stackrel{\rightharpoonup}{v}}_{i}$. An acc/dec component is displayed in Fig. 

Fig. 21

Acc/dec component

21, which is composed of one acceleration pulse, one gap width, one deceleration pulse, and one delay width in sequence. For the i-th acc/dec component, the acceleration pulse width is ${N}_{i,0}$, the gap width is ${N}_{i,1}$, the deceleration pulse width is ${N}_{i,2}$, and the delay width is ${N}_{i,3}$, while the magnitude of the acceleration and deceleration pulses are ${A}_{i,0}$ and ${A}_{i,1}$, respectively, or

$${A}_{i}\left[k\right]=\left\{\begin{array}{cc}{A}_{i,0}& 0\le k-{N}_{i-1}\le {N}_{i,0}\text{-1}\\ 0& {N}_{i,0}\le k-{N}_{i-1}\le {N}_{i,1}\text{-1}\\ -{A}_{i,1}& {N}_{i,1}\le k-{N}_{i-1}\le {N}_{i,2}\text{-1}\\ 0& {N}_{i,2}\le k-{N}_{i-1}\le {N}_{i,3}\text{-1}\end{array}\right.$$

where

$$\left\{\begin{array}{l}N_i=N_{i-1}+\sum_{j=0}^3N_{i,j}\\N_{-1}=0\end{array}\right.$$

The magnitudes of ${A}_{\mathrm{i},1}$ and ${A}_{i+\mathrm{1,0}}$ around the same corner point ${P}_{i+1}$ are the same, which are denoted as ${A}_{i+1,c}$:

$${A}_{i,1}={A}_{i+\mathrm{1,0}}={A}_{i+1,c}$$

(A.8)

There are n + 1 FIR filters used in motion planning, in which sizes are ${N}_{0,f}$, ${N}_{1,f}$,…, ${N}_{n,f}$, individually. The filter with size ${N}_{i,f}$ convolutes the pulses with value ${-A}_{i-\mathrm{1,1}}$ and ${A}_{i,0}$. In Fig. 20, there are smooth cornering trajectories around corners because two adjacent segments’ trajectories are blended together after filtering. Set the blending time as

$${N}_{i,b}={N}_{i+1,f}-{N}_{i,3}$$

(A.9)

Then, the corner error ${\varepsilon }_{i}$ around point ${P}_{i}$ is the shortest distance from ${P}_{i}$ to the trajectory. The corner error equation is

$$\varepsilon_i=A_{i,c}\cos\left(\frac{\theta_i}2\right)T^2f_b\left(N_{i-1,b}\right)/f_f\left(N_{i,f}\right)$$

(A.10)

where ${f}_{b}\left({N}_{b}\right)$ is a function only depending on blending width and filter type, and ${f}_{f}\left({N}_{f}\right)$ is a function only depending on filter size and type:

$${f}_{b}\left({N}_{b}\right)=\left\{\begin{array}{cc}({N}_{b}^{3}-{N}_{b})/24& {1}^{\mathrm{st}}\text{-type, }{N}_{b}\in {Z}_{o}\\ ({N}_{b}^{3}-4{N}_{b})/24& {1}^{\mathrm{st}}\text{-type, }{N}_{b}\in {Z}_{e}\\ \frac{\left(\begin{array}{c}{N}_{b}^{4}+4{N}_{b}^{3}\\ +2{N}_{b}^{2}-4{N}_{b}-3\end{array}\right)}{192}& {2}^{\mathrm{nd}}\text{-type, }{N}_{b}\in {Z}_{o}\\ \frac{\left(\begin{array}{c}{N}_{b}^{4}+4{N}_{b}^{3}\\ -4{N}_{b}^{2}-16{N}_{b}\end{array}\right)}{192}& {2}^{\mathrm{nd}}\text{-type, }{N}_{b}\in {Z}_{e}\end{array}\right.$$

$${f}_{f}\left({N}_{f}\right)=\left\{\begin{array}{cc}{N}_{f},& {1}^{\mathrm{st}}\text{-type}\\ \frac{{N}_{f}^{2}+2{N}_{f}+1}{4},& {2}^{\mathrm{nd}}\text{-type, }{N}_{f}\in {Z}_{o}\\ \frac{{N}_{f}^{2}+2{N}_{f}}{4},& {2}^{\mathrm{nd}}\text{-type, }{N}_{f}\in {Z}_{e}\end{array}\right.$$

where ${Z}_{o}$ stands for odd numbers and ${Z}_{e}$ stands for even numbers. Besides, the jerk around point ${P}_{i}$ is bounded as

$${J}_{i,\mathrm{sup}}=\underset{0\le j<{N}_{i,f}}{\mathrm{max}}\left({h}_{i,j}\right)\mathrm{max}\left(\mathrm{1,2sin}\left(\frac{{\theta }_{i}}{2}\right)\right)\frac{{A}_{i,c}}{T}$$

(A.11)

In corner smoothing, two types of filters can generate a smooth profile. The 2nd-type FIR filter, however, can generate a smoother profile than a 1st-type FIR filter and is more suitable for corners with sharper angles. Therefore, one can set ${\theta }_{F}$ as a threshold to determine which type of filter is suitable for the corner smoothing around a corner point.