6轴机器人运动学正解,逆解2

其他 2019-10-25 19:52:31 阅读次数: 0

逆解 
逆解计算方法可以参考以下书籍 
机器人学导论——分析、系统及应用 电子工业出版社 
机器人学导论第3版 机械工业出版社 
机器人学建模、规划与控制 西安交通大学出版社

对于关节1,2,3可以从运动方程手工推导出各个关节旋转角度的计算公式

逆解求解的结果并不是唯一的 可能有多组解
 

/*计算逆解 根据机器人坐标计算机器人关节角度 
 *关节参数在文件 param_table中
 *机器人坐标在文件 xyzrpy中
 *计算结果在屏幕输出 */



#include <stdio.h>
#include <math.h>
#include <string.h>

#define XYZ_F_3D "./xyzrpy"
#define DESIGN_DT "./param_table"
#define XYZ_F_TOOL "./tool_xyz"

#define PI (3.1415926535898)
#define ANG2RAD_EQU(N) (N *= (180.0/3.1415926535898) )
#define ANG2RAD(N) ( (N) * (180.0/3.1415926535898) )
#define RAD2ANG (3.1415926535898/180.0)
#define IS_ZERO(var) if(var < 0.0000000001 && var > -0.0000000001){var = 0;} 
// #define IS_ZERO(var) ( (var) < 0.0000000001 && (var) > -0.0000000001 )?0 :1 
#define JUDGE_ZERO(var) ( (var) < 0.0000000001 && (var) > -0.0000000001 )

#define MATRIX_1 1
#define MATRIX_M 4
#define MATRIX_N 4

#define ANGLE_OFFSET_J2 90
#define ANGLE_OFFSET_J3 90


typedef struct {
    double joint_v;  //joint variable
    double length;
    double d;
    double angle;
}param_t;

param_t param_table[6] ={0};
double worldx =0, worldy =0, worldz =0, 
       worldrr =0, worldrp =0, worldry =0;
double z_offset=0;

void printmatrix(double matrix[MATRIX_N][MATRIX_N], int m, int n);

int matrix_mul(double matrix_a[MATRIX_N][MATRIX_N],
            double matrix_b[MATRIX_N][MATRIX_N],
            double matrix_result[MATRIX_N][MATRIX_N], int m, int n);
void matrix_copy(double matrix_a[MATRIX_N][MATRIX_N],
            double matrix_b[MATRIX_N][MATRIX_N], int m, int n);

void calculate_matrix_R(double worldrr, double worldrp, double worldry,
            double (*matrix_R)[MATRIX_N]);
void calculate_matrix_A(double matrix[MATRIX_N][MATRIX_N], 
            param_t *p_param);
int judge(double j1, double j2, double j3);
void matrix_translate(double matrix[MATRIX_M][MATRIX_N], int m, int n);
void fun_zyz(double matrix_R[MATRIX_N][MATRIX_N], 
            double *p_r,  double *p_p, double *p_y);
int fun_j456(double  j1, double j2, double j3,
            param_t *p_table,double p_matrix_R[MATRIX_N][MATRIX_N],
            double *p_j4, double *p_j5, double *p_j6);

int fun_j2(double j1, double *p_j2, 
            double a1, double a2, double a3, double d4,   
            double px, double py, double pz )
{//计算关节2的角度 
    double v1_c, v1_s, v2_c, v2_s;
    double var_M, var_K, tmp;
    double var_sqrt[2] = {0};

    v1_c =cos(j1);
    IS_ZERO(v1_c);
    v1_s =sin(j1);
    IS_ZERO(v1_s);
    var_M = v1_c*px + v1_s*py - a1;
    var_K = (d4*d4 + a3*a3 - a2*a2 - pz*pz - var_M*var_M) / (-2 * a2);
    tmp = var_M*var_M + pz*pz - var_K*var_K;
    IS_ZERO(tmp);
    if( tmp >=0 ){
    //if( (var_M*var_M + pz*pz - var_K*var_K) >=0){
        //var_sqrt[0] = sqrt(var_M*var_M + pz*pz - var_K*var_K);
        var_sqrt[0] = sqrt(tmp);
        var_sqrt[1] = -var_sqrt[0];
    }else{
        printf("m^2 + z^2 - k^2 <0 : %lf\n", tmp);
        p_j2[0] =0, p_j2[1] =0;
        return 0;
    }

    p_j2[0] = -atan2(var_M, pz) + atan2(var_K, var_sqrt[0]);
    p_j2[1] = -atan2(var_M, pz) + atan2(var_K, var_sqrt[1]);
    return 1;
}

int fun_j3(double j1, double j2, double *p_j3,
            double a1, double a3, double d4,
            double px, double py, double pz)
{//计算关节3的角度 
    double var_K, tmp;
    double var_sqrt[2];
    double v1_c, v1_s, v2_c, v2_s;
    v1_c = cos(j1);
    IS_ZERO(v1_c);
    v1_s = sin(j1);
    IS_ZERO(v1_s);
    v2_c = cos(j2);
    IS_ZERO(v2_c);
    v2_s = sin(j2);
    IS_ZERO(v2_s);

    var_K = -v2_s*v1_c*px - v1_s*v2_s*py + v2_c*pz + v2_s*a1;
    IS_ZERO(var_K);

    tmp = d4*d4 + a3*a3 - var_K*var_K;
    IS_ZERO(tmp);

    if( tmp >=0 ){
        var_sqrt[0] = sqrt(tmp);
        var_sqrt[1] = -var_sqrt[0];
        p_j3[0] = atan2(d4, a3) + atan2(var_K, var_sqrt[0]);
        p_j3[1] = atan2(d4, a3) + atan2(var_K, var_sqrt[1]);
    }else{
        printf("m^2 + z^2 - k^2 <0 : %lf\n", d4*d4 + a3*a3 - var_K*var_K);
        p_j3[0] =0; p_j3[1] = 0;
        return 0;
    }
    return 1;
}

/* 计算过程 根据运动方程 计算矩阵 列出等式 计算 j1 j2 j3
 * 计算旋转矩阵 根据 j1 j2 j3 计算T3 并转置 与旋转矩阵相乘 3*3
 * 计算zyz 就是 j4 j5 j6 */
int main()
{
    double matrix_R[MATRIX_N][MATRIX_N];

    double j1[2] = {0};  //元素值 >=360 度或 < -360 度 表示角度无效
    double j2[4] = {0};
    double j3[8] = {0};
    double j4[8] = {0};
    double j5[8] = {0};
    double j6[8] = {0};

    int i, j;
//  double z_offset=0;
//  memset(param_table, 0, sizeof(param_table) );

    FILE * fp=NULL;
    fp=fopen(XYZ_F_3D, "r");
    if(fp== NULL){
        perror("open xyzrpy file error\n");
        return 0;
    }
    fscanf(fp, "%lf%lf%lf%lf%lf%lf",
                &worldx, &worldy, &worldz, &worldry, &worldrp, &worldrr);
    fclose(fp);

    printf("worldx: %lf worldy: %lf worldz: %lf\nworldry: %lf worldrp: %lf worldrr: %lf\n",
          worldx, worldy, worldz, worldry, worldrp, worldrr);

    fp=fopen(DESIGN_DT, "r");
    if( fp== NULL){
        perror("open param_table file error\n");
        return 0;
    }

    for(i=0; i<6; i++){
        fscanf(fp, "%lf%lf%lf",
                    &param_table[i].length,
                    &param_table[i].d,
                    &param_table[i].angle );
    }
    fscanf(fp, "%lf", &z_offset );
    fclose(fp);

    param_table[0].angle *= RAD2ANG;
    param_table[1].angle *= RAD2ANG;
    param_table[2].angle *= RAD2ANG;
    param_table[3].angle *= RAD2ANG;
    param_table[4].angle *= RAD2ANG;
    param_table[5].angle *= RAD2ANG;

    calculate_matrix_R(worldrr, worldrp, worldry, matrix_R);
    matrix_R[0][3] = worldx;
    matrix_R[1][3] = worldy;
    matrix_R[2][3] = worldz-z_offset;
    matrix_R[3][0] = 0;
    matrix_R[3][1] = 0;
    matrix_R[3][2] = 0;
    matrix_R[3][3] = 1;
    printmatrix(matrix_R, MATRIX_N, MATRIX_N);

    //double var_M, var_K;
    //double var_sqrt[2];
    double a1 = param_table[0].length;
    double a2 = param_table[1].length;
    double a3 = param_table[2].length;
    double d4 = param_table[3].d;
    double px = matrix_R[0][3];
    double py = matrix_R[1][3];
    double pz = matrix_R[2][3];

    double v1_c, v1_s, v2_c, v2_s;

    //计算 j1
    j1[0] = atan2(worldy, worldx); 
    IS_ZERO( j1[0] );
    //ANG2RAD_EQU(j1[0]);
    j1[1] = j1[0] +PI;
    JUDGE_ZERO(j1[1] -2*PI)? (j1[1] = 0) : 1;
    //j1[1] = JUDGE_ZERO(j1[1] -2*PI)? j1[1] = 0: 1;
    printf("j1: \n  %lf , %lf\n", ANG2RAD(j1[0]), ANG2RAD(j1[1]) );

    //计算 j2
    int v_bool;
    v_bool = fun_j2(j1[0], j2, a1, a2, a3, d4, px, py, pz);
    if(v_bool)
        printf("j2: %lf, %lf\n", ANG2RAD(j2[0])-90, ANG2RAD(j2[1])-90 );
    else{
        printf("this j2 invalid\n");
        j2[0] =2*PI; j2[1] =2*PI;
//      j2[0]>0 ? (j2[0] += 2*PI): (j2[0] -= 2*PI) ;    
//      j2[1]>0 ? (j2[1] += 2*PI): (j2[1] -= 2*PI) ;    
    }

    v_bool = fun_j2(j1[1], j2+2, a1, a2, a3, d4, px, py, pz);
    if(v_bool)
      printf("j2: %lf, %lf\n", ANG2RAD(j2[2])-90, ANG2RAD(j2[3])-90 );
    else{
        printf("this j2 invalid\n");
        j2[2] =2*PI; j2[3] =2*PI;
    }

    //计算 j3
    for(i=0; i<8; i+=2){
        v_bool = fun_j3(j1[i/4], j2[i/2], j3+i, a1, a3, d4, px, py, pz);
        if(v_bool)
          printf("j3: %lf, %lf\n", 
                      ANG2RAD(j3[i])-90, ANG2RAD(j3[i+1])-90 );
        else {
            printf("this j3 invalid\n");
            j3[i] =2*PI; j3[i+1] =2*PI;
            //j3[k]>0 ? (j3[k] += 2*PI): (j3[k] -= 2*PI) ;
            //j3[k+1]>0 ? (j3[k+1] += 2*PI): (j3[k+1] -= 2*PI) ;
        }
    }

printf("judge\n");
    for(i=0; i<8; i++){
        printf("j1[%d]: %lf, j2[%d]: %lf, j3[%d]: %lf\n",
                    i/4, j1[i/4], i/2, j2[i/2], i, j3[i]);

        //if(j1[i/4]==2*PI || j2[i/2]==2*PI || j3[i]==2*PI) continue;

        if( !judge(j1[i/4], j2[i/2], j3[i]) ) { 
            j3[i]>=0 ? (j3[i] += 2*PI): (j3[i] -= 2*PI) ; }
    }

    printf("\nj1: %lf, %lf\nj2: %lf, %lf, %lf, %lf\n", 
            ANG2RAD(j1[0]), ANG2RAD(j1[1]),
            ANG2RAD(j2[0])-90, ANG2RAD(j2[1])-90, 
            ANG2RAD(j2[2])-90, ANG2RAD(j2[3])-90 );
    printf("j3:\n");
    for(i=0; i<8; i++){
        printf(" %lf ", ANG2RAD(j3[i])-90);
        if( (i+1)%4 ==0 )printf("\n");
    }

//计算 j4 j5 j6   
    for(i=0, j=0; i<8; i++){
        if(j3[i] >= 2.0*PI || j3[i] < -2.0*PI) continue;
        printf("\n----j1[%d]: %lf j2[%d]: %lf j3[%d]: %lf\n", 
                    i/4, ANG2RAD(j1[i/4]), 
                    i/2, ANG2RAD(j2[i/2])-90, 
                    i, ANG2RAD(j3[i])-90 );

        fun_j456(j1[i/4], j2[i/2], j3[i], param_table, matrix_R, 
                    &j4[j], &j5[j], &j6[j]);
        printf("j4: %lf, %lf\nj5: %lf, %lf\nj6: %lf, %lf\n",
                    ANG2RAD(j4[j]), ANG2RAD(j4[j+1]),
                    ANG2RAD(j5[j]), ANG2RAD(j5[j+1]), 
                    ANG2RAD(j6[j]), ANG2RAD(j6[j+1]) );
        j +=2;
    }

}

void calculate_matrix_R(double angle_r, double angle_p, double angle_y,
            double (*matrix_R)[MATRIX_N])
{
/*计算旋转矩阵 */
    int i,j;
    double mtmp;
    double r_c, r_s, p_c, p_s, y_c, y_s;

    angle_r *= RAD2ANG;
    angle_p *= RAD2ANG;
    angle_y *= RAD2ANG;

    r_c = cos( angle_r );
    IS_ZERO(r_c);
    r_s = sin( angle_r );
    IS_ZERO(r_s);
    p_c = cos( angle_p );
    IS_ZERO(p_c);
    p_s = sin( angle_p );
    IS_ZERO(p_s);
    y_c = cos( angle_y );
    IS_ZERO(p_c);
    y_s = sin( angle_y );
    IS_ZERO(y_s);

    matrix_R[0][0] = r_c * p_c;
    matrix_R[0][1] = r_c * p_s * y_s - r_s * y_c;
    matrix_R[0][2] = r_c * p_s * y_c + r_s * y_s;

    matrix_R[1][0] = r_s * p_c;
    matrix_R[1][1] = r_s * p_s * y_s + r_c * y_c;
    matrix_R[1][2] = r_s * p_s * y_c - r_c * y_s;

    matrix_R[2][0] = -p_s;
    matrix_R[2][1] = p_c * y_s;
    matrix_R[2][2] = p_c * y_c;

}

int judge(double j1, double j2, double j3)
{
    /* j1 j2 j3 是弧度 j2 j3 已加90度 */
    double x, y, z, tmp;
    j2 -= 0.5*PI;
    j3 -= 0.5*PI;

    //计算x
    tmp = -sin(j2);
    IS_ZERO(tmp);
    x = tmp * param_table[1].length;

    tmp = cos(j2+j3);
    IS_ZERO(tmp);
    x -= param_table[2].length * tmp;

    tmp = -sin(j2+j3);
    IS_ZERO(tmp);
    x +=tmp* param_table[3].d;
    x += param_table[0].length; 
    y = x;

    tmp =cos(j1);
    IS_ZERO(tmp);
    x *=tmp;
    //计算y
    tmp =sin(j1);
    IS_ZERO(tmp);
    y *=tmp;
    //计算z
    tmp = cos(j2);
    IS_ZERO(tmp);
    z = param_table[1].length*tmp;

    tmp = sin(j2+j3);
    IS_ZERO(tmp);
    z -=param_table[2].length*tmp;

    tmp = cos(j2+j3);
    IS_ZERO(tmp);
    z += param_table[3].d *tmp +z_offset;

    //printf("%lf %lf %lf\n", x, y, z);
    tmp = x - worldx;
    if( tmp > 0.0000000001 || tmp < -0.0000000001 ) return 0;
//  if( !(tmp < 0.0000000001 && tmp > -0.0000000001) ) return 0;
    tmp = y - worldy;
    if( tmp > 0.0000000001 || tmp < -0.0000000001 ) return 0;
    tmp = z - worldz;
    if( tmp > 0.0000000001 || tmp < -0.0000000001 ) return 0;
    return 1;

}

int fun_j456(double  j1, double j2, double j3, 
            param_t *p_table, double p_matrix_R[MATRIX_N][MATRIX_N], 
            double *p_j4, double *p_j5, double *p_j6)
{
    double matrix_a[MATRIX_N][MATRIX_N], matrix_b[MATRIX_N][MATRIX_N];
    double matrix_tmp[MATRIX_N][MATRIX_N];

    //printf("j1: %lf j2: %lf j3:  %lf\n", j1, j2, j3);
    p_table[0].joint_v = j1;
    p_table[1].joint_v = j2;
    p_table[2].joint_v = j3;

    calculate_matrix_A(matrix_a, p_table+0);
    calculate_matrix_A(matrix_b, p_table+1);
    matrix_mul(matrix_a, matrix_b, matrix_tmp, MATRIX_N, MATRIX_N);

    calculate_matrix_A(matrix_b, p_table+2);
    matrix_mul(matrix_tmp, matrix_b, matrix_a, MATRIX_N, MATRIX_N);
    matrix_translate(matrix_a, MATRIX_N-1, MATRIX_N-1);

    matrix_mul(matrix_a, p_matrix_R, matrix_b, MATRIX_N-1, MATRIX_N-1);

    fun_zyz(matrix_b, p_j4, p_j5, p_j6);    

}

void fun_zyz(double matrix_R[MATRIX_N][MATRIX_N], 
            double *p_r,  double *p_p, double *p_y)
{
    double mtmp =sqrt(matrix_R[0][2]*matrix_R[0][2] + 
                matrix_R[1][2]*matrix_R[1][2]);

//  printf("ZYZ  \n--- > -pi  and < 0\n");
    p_r[0] = atan2( matrix_R[1][2], matrix_R[0][2]);
    p_p[0] = atan2( mtmp, matrix_R[2][2]);
    p_y[0] = atan2( matrix_R[2][1], -matrix_R[2][0] );

//  printf("ZYZ  \n--- > -pi  and < 0\n");
    p_r[1] = atan2( -matrix_R[1][2], -matrix_R[0][2]);
    p_p[1] = atan2( -mtmp, matrix_R[2][2]);
    p_y[1] = atan2( -matrix_R[2][1],  matrix_R[2][0] );

}

void calculate_matrix_A(double matrix[MATRIX_N][MATRIX_N], param_t *p_param)
{//根据关节参数计算矩阵 
    double *pmatrix=(double *)matrix;
    double value, var_c, var_s, angle_c, angle_s;

    var_c = cos(p_param->joint_v);
    IS_ZERO(var_c);
    var_s = sin(p_param->joint_v);
    IS_ZERO(var_s);
    angle_c = cos(p_param->angle);
    IS_ZERO(angle_c);
    angle_s = sin(p_param->angle);
    IS_ZERO(angle_s);

    *pmatrix++ = var_c;
    *pmatrix++ = -var_s * angle_c;
    *pmatrix++ = var_s * angle_s;
    *pmatrix++ = p_param->length * var_c;

    *pmatrix++ = var_s;
    *pmatrix++ = var_c * angle_c;
    *pmatrix++ = -var_c *angle_s;
    *pmatrix++ = p_param->length * var_s;

    *pmatrix++ =0;
    *pmatrix++ = angle_s;
    *pmatrix++ = angle_c;
    *pmatrix++ = p_param->d;

    *pmatrix++ =0;
    *pmatrix++ =0;
    *pmatrix++ =0;
    *pmatrix =1;

}

void matrix_copy(double matrix_a[MATRIX_N][MATRIX_N],
            double matrix_b[MATRIX_N][MATRIX_N], int m, int n)
{
    int i,j;
    for(i=0; i<m; i++){
        for(j=0; j<n; j++){
            matrix_b[i][j] = matrix_a[i][j];
        }
    }
}

int matrix_mul(double matrix_a[MATRIX_N][MATRIX_N],
            double matrix_b[MATRIX_N][MATRIX_N],
            double matrix_result[MATRIX_N][MATRIX_N], int m, int n)
{
    int i,j,k;
    double sum;
    double matrix_tmp[MATRIX_N][MATRIX_N]={0};

    /*嵌套循环计算结果矩阵（m*p）的每个元素*/
    for(i=0; i<m; i++)
      for(j=0; j<n; j++)
      {
          /*按照矩阵乘法的规则计算结果矩阵的i*j元素*/
          sum=0;
          for(k=0; k<n; k++)
            sum += matrix_a[i][k] * matrix_b[k][j];
          matrix_tmp[i][j] = sum;
      }
    matrix_copy(matrix_tmp, matrix_result, MATRIX_N, MATRIX_N);
    return 0;
}

void matrix_translate(double matrix[MATRIX_M][MATRIX_N], int m, int n)
{//矩阵转置
    double m_tmp;
    int i, j, k;

    for(i=0, j=0; i<m; i++, j++){
        for(k=j; k<n; k++){
            if(i == k) continue;
            m_tmp = matrix[i][k];
            matrix[i][k] = matrix[k][i];
            matrix[k][i] = m_tmp;
        }
    }
}

void printmatrix(double matrix[MATRIX_N][MATRIX_N], int m, int n)
{
    int i, j;

    for(i=0; i<m; i++){
        for(j=0; j<n; j++){
            printf(" %lf ", matrix[i][j]);
        }
        printf("\n");
    }
    printf("\n");
}

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