digital

Line and Circular Interpolation on XY Plane

Tips: This is my homework of “Computer Control System” course, so mistakes do happen.

This article will discuss an interpolation algorithms for line and circular. And it provides an implement of C language.

Interpolation Algorithms

Many numerical control machines are powered by stepping motors. When a pulse is sent to a stepping motor, the stepping motor alters its position by a unit step. Two motors can be used to control the XY movements of an arm or tool over a working plane.

If the pulse are generated by a device which can remember or generate a specified train of pulses, repetitive operations such as grinding, painting, or cutting can be performed hunderds of times with virtually no variation. A microcomputer is an obvious choice to gernerate and remember the pulse.

Since stepper motors can move only in discrete steps, we must approximate the actual curve by a series of small XY motions. Many algorithms rely upon parametric functions such as sine and cosine to perform the necessary calculations. Parametric functions, however, typically require a high degree of numeric precision. Calculating sine and cosine values with a microcomputer can be too time-consuming to be useful in a realtime application.

The following two algorithms require no parametric functions. This makes them ideally suited to the
computation and memory capacities of microcomputers. Since these algorithms do not require a large amount of complex mathematical calculation, they are fast enough to be used in real-time applications.

C Program

main.c
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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>

char type[10];
char data1[5],data2[5],data3[5],data4[5],data5[5];
/* consider that the point may be decimal, so use "int" */
int x_0,y_0;  // starting point
int x_1,y_1;  // endpoint
int xe,ye;    // relative endpoint
int xi,yi;    // relative position
int x,y;      // real position
int dx,dy;    // the value of impulse
int fxy;      // value of function
int step;     // consider that the step may be decimal
int rad;      // the radius of the circle
int x_dir,y_dir; // direction of impulse
int p2x;         // Circle: when fxy is positive, decide step towards x or not
int cw;          // present clockwise
int check,requirement = 1; // make sure endpoint locates at the desired one

void doline();
void getdir();
void docircle(); // according to requirement, the arc is 1/4 circle
void clockwise();
void counterclockwise();

int main(void)
{
    printf("Please input command line (ie: 'G01 (2,3,7,8) 1') :\n");
    scanf("%s (%[^,],%[^,],%[^,],%[^)]) %s",type,data1,data2,data3,data4,data5);

    x_0 = atof(data1);
    y_0 = atof(data2);
    x_1 = atof(data3);
    y_1 = atof(data4);
    x = x_0;
    y = y_0;
    step = atof(data5);
    getdir();

    printf("\n\nFor code '%s' with step = %d",type,step);
    printf("\nGoing from (%d,%d) to (%d,%d):\n",x_0,y_0,x_1,y_1);
    printf("\nXi\tYi\tFxy\tDx\tDy\tX\tY");

    if(!strcmp(type,"G01")) doline();
    else
    {
        if(!strcmp(type,"G02")) cw = 1;
        else cw = 0;
        docircle();
    }


    printf("\n%d\t%d\t\t\t\t%d\t%d",xi,yi,x,y);

    //system("pause");

    return 0;
}

void getdir()
{
    if(x_1>=x_0) x_dir = 1;
    else x_dir = -1;
    if(y_1>=y_0) y_dir = 1;
    else y_dir = -1;
}

void doline()
{
    xe = x_1 - x_0;
    ye = y_1 - y_0;
    xi = yi = 0.0;
    p2x = 1;

    while((xi<xe)||(yi<ye))
    {
        fxy = abs(xe)*abs(yi) - abs(ye)*abs(xi);
        impulse();
    }
}

void docircle()
{
    rad = abs(x_1 - x_0);
    if(cw) clockwise();
    else counterclockwise();

    while(requirement)
    {
        fxy = (xi*xi) + (yi*yi) - (rad*rad);
        impulse();
        switch(check)
        {
            case 1: if((xi<=xe)&&(yi>=ye)) requirement = 0;break;
            case 2: if((xi>=xe)&&(yi>=ye)) requirement = 0;break;
            case 3: if((xi>=xe)&&(yi<=ye)) requirement = 0;break;
            case 4: if((xi<=xe)&&(yi<=ye)) requirement = 0;break;
        }
    }
}

void impulse()
{
    if(fxy>=0)
    {
        dx = x_dir*p2x*step;
        dy = y_dir*(!p2x)*step;
    }
    else
    {
        dx = x_dir*(!p2x)*step;
        dy = y_dir*(p2x)*step;
    }

    printf("\n%d\t%d\t%d\t%d\t%d\t%d\t%d",xi,yi,fxy,dx,dy,x,y);

    xi = xi + dx;
    yi = yi + dy;
    x = x + dx;
    y = y + dy;
}

void clockwise()
{
    if((x_1>x_0)&&(y_1<y_0))
    {
        p2x = 0;
        xe = rad;
        ye = 0.0;
        xi = 0.0;
        yi = rad;
        check = 3;
    }
    if((x_1>x_0)&&(y_1>y_0))
    {
        p2x = 1;
        xe = 0.0;
        ye = rad;
        xi = -rad;
        yi = 0.0;
        check = 2;
    }
    if((x_1<x_0)&&(y_1>y_0))
    {
        p2x = 0;
        xe = -rad;
        ye = 0.0;
        xi = 0.0;
        yi = -rad;
        check = 1;
    }
    if((x_1<x_0)&&(y_1<y_0))
    {
        p2x = 1;
        xe = 0.0;
        ye = -rad;
        xi = rad;
        yi = 0.0;
        check = 4;
    }
}
void counterclockwise()
{
    if((x_1<x_0)&&(y_1>y_0))
    {
        p2x = 1;
        xe = 0.0;
        ye = rad;
        xi = rad;
        yi = 0.0;
        check = 1;
    }
    if((x_1<x_0)&&(y_1<y_0))
    {
        p2x = 0;
        xe = -rad;
        ye = 0.0;
        xi = 0.0;
        yi = rad;
        check = 4;
    }
    if((x_1>x_0)&&(y_1<y_0))
    {
        p2x = 1;
        xe = -rad;
        ye = 0.0;
        xi = 0.0;
        yi = -rad;
        check = 3;
    }
    if((x_1>x_0)&&(y_1>y_0))
    {
        p2x = 0;
        xe = rad;
        ye = 0.0;
        xi = 0.0;
        yi = -rad;
        check = 2;
    }
}

Result

G01

G02

Reference

“XY INTERPOLATION ALGORITHMS” By Kenneth and Melvin Goldberg

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