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完成正运动学解算和逆运动学中两个关节的解算
2024-10-03 21:43:35 +08:00
/**
* @brief Description: Algorithm module of robotics, according to the
* book[modern robotics : mechanics, planning, and control].
* @file: RobotAlgorithmModule.c
* @author: Brian
* @date: 2019/03/01 12:23
* Copyright(c) 2019 Brian. All rights reserved.
*
* Contact https://blog.csdn.net/Galaxy_Robot
* @note:
*@warning:
*/
#include "RobotAlgorithmModule.h"
#include <math.h>
#include <string.h>
#include <stdlib.h>
#include <stdio.h>
/**
*@brief Description:use GrublersFormula calculate The number of degrees of
* freedom of a mechanism with links and joints.
*@param[in] N the number of links( including the ground ).
*@param[in] m the number of degrees of freedom of a rigid body.
* (m = 3 for planar mechanisms and m = 6 for spatial mechanisms).
*@param[in] J the number of joints.
*@param[in] f the number of freedoms provided by each joint..
*@return The number of degrees of freedom of a mechanism
*@note: This formula holds only if all joint constraints are independent.
* If they are not independent then the formula provides a lower
* bound on the number of degrees of freedom.
*@warning:
*/
int GrublersFormula(int m, int N, int J, int *f)
{
int i;
int dof;
int c = 0;
for (i=1;i<=J;i++)
{
c = c + f[i-1];
}
dof = m*(N - 1 - J) + c;
return dof;
}
/**
*@brief Description: Make matrix b equal to matrix a.
*@param[in] a a 3 x 3 matrix.
*@param[out] b result,b=a.
*@return No return value.
*@note:
*@warning:
*/
void Matrix3Equal(double a[][3], double b[][3])
{
int i;
int j;
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
b[i][j] = a[i][j];
}
}
return;
}
/**
*@brief Description: Make matrix b equal to matrix a.
*@param[in] a a 4 x 4 matrix.
*@param[out] b result,b=a.
*@return No return value.
*@note:
*@warning:
*/
void Matrix4Equal(double a[][4], double b[][4])
{
int i;
int j;
for (i = 0; i < 4; i++)
{
for (j = 0; j < 4; j++)
{
b[i][j] = a[i][j];
}
}
return;
}
/**
*@brief Description:Calculate a 3 x 3 matrix add a 3 x 3 matrix.
*@param[in] a a 3 x 3 matrix.
*@param[in] b a 3 x 3 matrix.
*@param[out] c result of a+b.
*@return No return value.
*@note:
*@warning:
*/
void Matrix3Add(double a[][3], double b[][3], double c[][3])
{
int i;
int j;
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
c[i][j] = a[i][j] + b[i][j];
}
}
return;
}
/**
*@brief Description:Calculate a 4 x 4 matrix add a 4 x 4 matrix.
*@param[in] a a 4 x 4 matrix.
*@param[in] b a 4 x 4 matrix.
*@param[out] c result of a+b.
*@return No return value.
*@note:
*@warning:
*/
void Matrix4Add(double a[][4], double b[][4], double c[][4])
{
int i;
int j;
for (i = 0; i < 4; i++)
{
for (j = 0; j < 4; j++)
{
c[i][j] = a[i][j] + b[i][j];
}
}
return;
}
/**
*@brief Description:Calculate a 3 x 3 matrix Subtract a 3 x 3 matrix.
*@param[in] a a 3 x 3 matrix.
*@param[in] b a 3 x 3 matrix.
*@param[out] c result of a-b.
*@return No return value.
*@note:
*@warning:
*/
void Matrix3Sub(double a[][3], double b[][3], double c[][3])
{
int i;
int j;
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
c[i][j] = a[i][j] - b[i][j];
}
}
return;
}
/**
*@brief Description:Calculate a 4 x 4 matrix Subtract a 4 x 4 matrix.
*@param[in] a a 4 x 4 matrix.
*@param[in] b a 4 x 4 matrix.
*@param[out] c result of a-b.
*@return No return value.
*@note:
*@warning:
*/
void Matrix4Sub(double a[][4], double b[][4], double c[][4])
{
int i;
int j;
for (i = 0; i < 4; i++)
{
for (j = 0; j < 4; j++)
{
c[i][j] = a[i][j] - b[i][j];
}
}
return;
}
/**
*@brief Description: Calculate two 3 x 3 matrix multiplication.
*@param[in] a a 3 x 3 matrix.
*@param[in] b a 3 x 3 matrix.
*@param[out] c result of a*b.
*@return No return value.
*@note:
*@warning:
*/
void Matrix3Mult(double a[][3],double b[][3],double c[][3])
{
int i;
int j;
int k;
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
c[i][j] = 0.0;
for (k=0;k<3;k++)
{
c[i][j] = c[i][j] + a[i][k] * b[k][j];
}
}
}
return;
}
/**
*@brief Description: Calculate two 4 x 4 matrix multiplication.
*@param[in] a a 4 x 4 matrix.
*@param[in] b a 4 x 4 matrix.
*@param[out] c result of a*b.
*@return No return value.
*@note:
*@warning:
*/
将外设的初始化放到单独的文件中,增加bsp_can
2024-10-06 19:49:15 +08:00
void Matrix4Mult(double a[][4], double b[][4], double c[][4])
完成正运动学解算和逆运动学中两个关节的解算
2024-10-03 21:43:35 +08:00
{
int i;
int j;
int k;
for (i = 0; i < 4; i++)
{
for (j = 0; j < 4; j++)
{
c[i][j] = 0.0;
for (k = 0; k < 4; k++)
{
c[i][j] = c[i][j] + a[i][k] * b[k][j];
}
}
}
return;
}
/**
*@brief Description: Calculate 3 x 3 matrix multiply a value.
*@param[in] a a 3 x 3 matrix.
*@param[in] Value a scalar value.
*@param[out] c result of a*Value.
*@return No return value.
*@note:
*@warning:
*/
void Matrix3MultValue(double a[][3], double Value, double c[][3])
{
int i;
int j;
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
c[i][j] = a[i][j] * Value;
}
}
return;
}
/**
*@brief Description: Calculate 4 x 4 matrix multiply a value.
*@param[in] a a 4 x 4 matrix.
*@param[in] Value a scalar value.
*@param[out] c result of a*Value.
*@return No return value.
*@note:
*@warning:
*/
void Matrix4MultValue(double a[][4], double Value, double c[][4])
{
int i;
int j;
for (i = 0; i < 4; i++)
{
for (j = 0; j < 4; j++)
{
c[i][j] = a[i][j] * Value;
}
}
return;
}
/**
*@brief Description:Computes the result of a 3 x 3 Matrix multiply a 3-vector.
*@param[in] R a 3 x 3 Matrix.
*@param[in] vec1 an input of 3-vector.
*@param[out] vec2 the output result of 3-vector.
*@return No return value.
*@note:
*@warning:
*/
void Matrix3MultVec(double R[3][3], double vec1[3], double vec2[3])
{
int i;
int j;
for (i=0;i<3;i++)
{
vec2[i] = 0;
for (j=0;j<3;j++)
{
vec2[i] = vec2[i] + R[i][j] * vec1[j];
}
}
return;
}
/**
*@brief Description:Computes a 3-vector add a 3-vector.
*@param[in] vec1 a 3-vector.
*@param[in] vec2 a 3-vector.
*@param[out] vec3 result of vec1 + vec2;
*@return No return value.
*@note:
*@warning:
*/
void Vec3Add(double vec1[3], double vec2[3], double vec3[3])
{
vec3[0] = vec1[0] + vec2[0];
vec3[1] = vec1[1] + vec2[1];
vec3[2] = vec1[2] + vec2[2];
return;
}
/**
*@brief Description:Computes result of two 3-vector cross product.
*@param[in] vec1 a first 3-vector.
*@param[in] vec2 a first 3-vector.
*@param[out] vec3 result of vec1 cross product vec2.
*@return No return value.
*@note:
*@warning:
*/
void Vec3Cross(double vec1[3],double vec2[3],double vec3[3])
{
vec3[0] = vec1[1] * vec2[2] - vec2[1] * vec1[2];
vec3[1] = -(vec1[0] * vec2[2] - vec2[0] * vec1[2]);
vec3[2] = vec1[0] * vec2[1] - vec2[0] * vec1[1];
return;
}
/**
*@brief Description:Computes a 3-vector multiply a scalar value.
*@param[in] vec1 a 3-vector.
*@param[in] value a scalar value.
*@param[out] vec2 result of vec1 * value.
*@return No return value.
*@note:
*@warning:
*/
void Vec3MultValue(double vec1[3], double value, double vec2[3])
{
vec2[0] = vec1[0] * value;
vec2[1] = vec1[1] * value;
vec2[2] = vec1[2] * value;
return;
}
/**
*@brief Description: Computes the norm of a 3-vector.
*@param[in] vec an input 3-vector.
*@return the norm of a an input 3-vector.
*@note:
*@warning:
*/
double Vec3Norm2(double vec[3])
{
return sqrt(vec[0] * vec[0] + vec[1] * vec[1] + vec[2] * vec[2]);
}
/**
*@brief Description: Computes the 2-norm of a n-vector.
*@param[in] n dimension of vector.
*@param[in] vec an input n-vector.
*@return the norm of a an input 3-vector.
*@note:
*@warning:
*/
double VecNorm2(int n,double *vec)
{
int i;
double tmp=0.0;
for (i=0;i<n;i++)
{
tmp = tmp + vec[i] * vec[i];
}
if (tmp<0.0)
{
tmp = 0;
}
return sqrt(tmp);
}
/**
*@brief Copy matrix from a to b .
*@param[in] a First matrix.
*@param[in] m rows of matrix a.
*@param[in] n columns of matrix a.
*@param[in] b Second matrix.
*@return No return value.
*@note:
*@warning:
*/
void MatrixCopy(const double *a, int m, int n, double *b)
{
memcpy(b, a, m*n * sizeof(double));
return;
}
/**
*@brief Computes Matrix transpose.
*@param[in] a a matrix.
*@param[in] m rows of matrix a.
*@param[in] n columns of matrix a.
*@param[out] b transpose of matrix a.
*@return No return value.
*@note:
*@waring:
*/
void MatrixT(double *a, int m, int n, double *b)
{
int i, j;
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
{
b[j*m + i] = a[i*n + j];
}
}
return;
}
/**
*@brief Computes matrix multiply.
*@param[in] a First matrix.
*@param[in] m rows of matrix a.
*@param[in] n columns of matrix b.
*@param[in] b Second matrix.
*@param[in] l columns of matrix b.
*@param[out] c result of a*b.
*@return No return value.
*@note:
*@waring:
*/
void MatrixMult(const double *a, int m, int n, const double *b, int l, double *c)
{
int i, j, k;
for (i = 0; i < m; i++)
{
for (j = 0; j < l; j++)
{
c[i*l + j] = 0.0;
for (k = 0; k < n; k++)
{
c[i*l + j] = c[i*l + j] + a[i*n + k] * b[k*l + j];
}
}
}
return;
}
/**
* @brief Description: Algorithm for Computing the roll-pitch-yaw angles(rotate around fix reference X,Y,Z axis).
* @param[in] R Rotation matrix.
* @param[out] roll Angles for rotate around fix reference X axis.
* @param[out] pitch Angles for rotate around fix reference Y axis.
* @param[out] yaw Angles for rotate around fix reference Z axis.
* @return No return value.
* @note:
*@warning:
*/
void RotToRPY(double R[3][3],double *roll,double *pitch,double *yaw)
{
if (fabs(1.0 + R[2][0]) < ZERO_ELEMENT)
{
*yaw = 0;
*pitch = PI / 2.0;
*roll = atan2(R[0][1], R[1][1]);
}
else if (fabs(1.0 - R[2][0]) < ZERO_ELEMENT)
{
*yaw = 0;
*pitch = -PI / 2.0;
*roll = atan2(R[0][1], R[1][1]);
}
else
{
*yaw = atan2(R[1][0], R[0][0]);
*pitch = atan2(-R[2][0], sqrt(R[0][0] * R[0][0] + R[1][0] * R[1][0]));
*roll = atan2(R[2][1], R[2][2]);
}
return;
}
/**
* @brief Description: Algorithm for Computing the rotation matrix of the roll-pitch-yaw angles.
* @param[in] roll Angles for rotate around fix reference X axis.
* @param[in] pitch Angles for rotate around fix reference Y axis.
* @param[in] yaw Angles for rotate around fix reference Z axis.
* @param[out] R Rotation matrix.
* @return No return value.
* @note:
*@warning:
*/
void RPYToRot(double roll, double pitch, double yaw, double R[3][3])
{
double alpha = yaw;
double beta = pitch;
double gamma = roll;
R[0][0] = cos(alpha)*cos(beta);
R[0][1] = cos(alpha)*sin(beta)*sin(gamma) - sin(alpha)*cos(gamma);
R[0][2] = cos(alpha)*sin(beta)*cos(gamma) + sin(alpha)*sin(gamma);
R[1][0] = sin(alpha)*cos(beta);
R[1][1] = sin(alpha)*sin(beta)*sin(gamma) + cos(alpha)*cos(gamma);
R[1][2] = sin(alpha)*sin(beta)*cos(gamma) - cos(alpha)*sin(gamma);
R[2][0] = -sin(beta);
R[2][1] = cos(beta)*sin(gamma);
R[2][2] = cos(beta)*cos(gamma);
return;
}
/**
*@brief Computes the inverse of the rotation matrix R.
*@param[in] R rotation matrix .
*@param[out] InvR inverse matrix of R.
*@return No return value.
*@note Input R must be a 3 x 3 rotation matrix.
*/
void RotInv(double R[3][3], double InvR[3][3])
{
int i;
int j;
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
InvR[i][j] = R[j][i];
}
}
return;
}
/**
*@brief Description:Returns the 3 x 3 skew-symmetric matrix corresponding to omg.
*@param[in] omg a 3-vector.
*@param[out] so3Mat the 3 x 3 skew-symmetric matrix corresponding to omg.
*@return No return value.
*@note:
*@warning:
*/
void VecToso3(double omg[3],double so3Mat[3][3])
{
so3Mat[0][0] = 0.0; so3Mat[0][1] = -omg[2]; so3Mat[0][2] = omg[1];
so3Mat[1][0] = omg[2]; so3Mat[1][1] = 0; so3Mat[1][2] = -omg[0];
so3Mat[2][0] = -omg[1]; so3Mat[2][1] = omg[0]; so3Mat[2][2] = 0.0;
return;
}
/**
*@brief Description:Returns the 3-vector corresponding to the 3 x 3 skew-symmetric matrix so3mat.
*@param[in] so3Mat a 3 x 3 skew-symmetric matrix so3mat.
*@param[out] omg the 3-vector.
*@return No return value.
*@note:
*@warning:
*/
void so3ToVec(double so3Mat[3][3], double omg[3])
{
omg[0] = so3Mat[2][1]; omg[1] = so3Mat[0][2]; omg[2] = so3Mat[1][0];
return;
}
/**
*@brief Description:Extracts the unit rotation axis omghat and the corresponding
* rotation angle theta from exponential coordinates omghat*theta for rotation, expc3.
*@param[in] expc3
*@param[out] omghat the unit vector of rotation axis .
*@param[out] theta rotation angle.
*@return No return value.
*@note:
*@warning:
*/
void AxisAng3(double expc3[3],double omghat[3],double *theta)
{
int i;
int ret = 0;
*theta = sqrt(expc3[0] * expc3[0] + expc3[1] * expc3[1] + expc3[2] * expc3[2]);
if (*theta<ZERO_ANGLE)
{
omghat[0] = 0.0;
omghat[1] = 0.0;
omghat[2] = 0.0;
*theta = 0.0;
return;
}
for (i=0;i<3;i++)
{
omghat[i]=expc3[i]/(*theta);
}
return ;
}
/**
*@brief Description:Computes the rotation matrix R in SO(3) corresponding to
* the matrix exponential of so3mat in so(3).
*@param[in] so3Mat [omghat]*theta,matrix exponential of so3mat in so(3).
*@param[out] R rotation matrix R in SO(3).
*@return No return value.
*@note:
*@warning:
*/
void MatrixExp3(double so3Mat[3][3],double R[3][3])
{
double omgtheta[3];
double omghat[3] = { 0 };
double theta = 0;
int ret = 0;
int i;
int j;
double MatI3[3][3] =
{
1,0,0,
0,1,0,
0,0,1
};
so3ToVec(so3Mat, omgtheta);
AxisAng3(omgtheta, omghat, &theta);
if (theta<ZERO_ANGLE)
{
Matrix3Equal(MatI3, R);
return ;
}
else
{
//calculate formula(3.51) in book [modern robotics : mechanics,planning,and control]
double omgmat[3][3];
double temp[3][3];
Matrix3MultValue(so3Mat, 1.0 / theta, omgmat);
Matrix3Mult(omgmat, omgmat, temp);
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
R[i][j] = MatI3[i][j] + sin(theta)*so3Mat[i][j] / theta + (1 - cos(theta))*temp[i][j];
}
}
}
return;
}
/**
*@brief Description:Computes the matrix logarithm so3mat in so(3) of the rotation matrix R in SO(3).
*@param[in] R the rotation matrix.
*@param[out] so3Mat matrix logarithm.
*@return No return value.
*@note:
*@warning:
*/
void MatrixLog3(double R[3][3],double so3Mat[3][3])
{
double omg[3] = { 0 };
double acosinput = (R[0][0] + R[1][1] + R[2][2] - 1.0) / 2.0;
if (fabs(acosinput - 1.0)<ZERO_VALUE)
{
memset(so3Mat, 0, 9 * sizeof(double));
}
else if (acosinput<=-1.0)
{
if ((1.0+R[2][2])>= ZERO_VALUE)
{
omg[0] = 1.0 / sqrt(2 * (1.0 + R[2][2]))*R[0][2];
omg[1] = 1.0 / sqrt(2 * (1.0 + R[2][2]))*R[1][2];
omg[2] = 1.0 / sqrt(2 * (1.0 + R[2][2]))*(1.0 + R[2][2]);
}
else if ((1.0+R[1][1]>= ZERO_VALUE))
{
omg[0] = 1.0 / sqrt(2 * (1.0 + R[1][1]))*R[0][1];
omg[1] = 1.0 / sqrt(2 * (1.0 + R[1][1]))*(1.0 + R[1][1]);
omg[2] = 1.0 / sqrt(2 * (1.0 + R[1][1]))*R[2][1];
}
else
{
omg[0] = 1.0 / sqrt(2 * (1.0 + R[0][0]))*(1.0 + R[0][0]);
omg[1] = 1.0 / sqrt(2 * (1.0 + R[0][0]))*R[1][0];
omg[2] = 1.0 / sqrt(2 * (1.0 + R[0][0]))*R[2][0];
}
omg[0] = PI*omg[0];
omg[1] = PI*omg[1];
omg[2] = PI*omg[2];
VecToso3(omg, so3Mat);
}
else
{
int i;
int j;
double theta = acos(acosinput);
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
so3Mat[i][j] = 1.0 / (2.0*sin(theta))*(R[i][j] - R[j][i])*theta;
}
}
}
return;
}
/**
*@brief Description:Builds the homogeneous transformation matrix T corresponding to a rotation
* matrix R in SO(3) and a position vector p in R3.
*@param[in] R a rotation matrix R in SO(3).
*@param[in] p a position vector p in R3.
*@param[out] T the homogeneous transformation matrix T.
*@return No return value.
*@note:
*@warning:
*/
void RpToTrans(double R[3][3], double p[3], double T[4][4])
{
int i;
int j;
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
T[i][j] = R[i][j];
}
}
T[0][3] = p[0];
T[1][3] = p[1];
T[2][3] = p[2];
T[3][0] = 0.0;
T[3][1] = 0.0;
T[3][2] = 0.0;
T[3][3] = 1.0;
return;
}
/**
*@brief Description: Extracts the rotation matrix R and position vector p
from a homogeneous transformation matrix T.
*@param[in] T a homogeneous transformation matrix.
*@param[out] R the rotation matrix.
*@param[out] p position vector.
*@return No return value.
*@note:
*@waring:
*/
void TransToRp(double T[4][4], double R[3][3], double p[3])
{
int i;
int j;
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
R[i][j] = T[i][j];
}
}
p[0] = T[0][3];
p[1] = T[1][3];
p[2] = T[2][3];
return;
}
/**
*@brief Description:Computes the inverse of a homogeneous transformation matrix T.
*@param[in] T a homogeneous transformation matrix.
*@param[out] InvT the inverse of T.
*@return No return value.
*@note:
*@warning:
*/
void TransInv(double T[4][4],double InvT[4][4])
{
double R[3][3];
double InvR[3][3];
double p[3];
double p2[3];
TransToRp(T, R, p);
RotInv(R, InvR);
Matrix3MultValue(InvR, -1.0, R);
Matrix3MultVec(R, p, p2);
RpToTrans(InvR, p2, InvT);
return;
}
/**
*@brief Description:Computes the se(3) matrix corresponding to a 6-vector twist V.
*@param[in] V a 6-vector twist V.
*@param[out] se3Mat the se(3) matrix.
*@return No return value.
*@note:
*@warning:
*/
void VecTose3(double V[6], double se3Mat[4][4])
{
double so3Mat[3][3];
double omg[3];
int i;
int j;
omg[0] = V[0];
omg[1] = V[1];
omg[2] = V[2];
VecToso3(omg, so3Mat);
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
se3Mat[i][j] = so3Mat[i][j];
}
}
se3Mat[0][3] = V[3];
se3Mat[1][3] = V[4];
se3Mat[2][3] = V[5];
se3Mat[3][0] = 0.0;
se3Mat[3][1] = 0.0;
se3Mat[3][2] = 0.0;
se3Mat[3][3] = 0.0;
return;
}
/**
*@brief Description:Computes the 6-vector twist corresponding to an se(3) matrix se3mat.
*@param[in] se3Mat an se(3) matrix.
*@param[out] V he 6-vector twist.
*@return No return value.
*@note:
*@warning:
*/
void se3ToVec(double se3Mat[4][4],double V[6])
{
V[0] = se3Mat[2][1];
V[1] = se3Mat[0][2];
V[2] = se3Mat[1][0];
V[3] = se3Mat[0][3];
V[4] = se3Mat[1][3];
V[5] = se3Mat[2][3];
return;
}
/**
*@brief Description:Computes the 6 x 6 adjoint representation [AdT ]
of the homogeneous transformation matrix T.
*@param[in] T a homogeneous transformation matrix.
*@param[out] AdT the 6 x 6 adjoint representation [AdT ].
*@return No return value.
*@note:
*@warning:
*/
void Adjoint(double T[4][4], double AdT[6][6])
{
double R[3][3];
double p[3];
double so3Mat[3][3];
int i;
int j;
TransToRp(T, R, p);
VecToso3(p, so3Mat);
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
AdT[i][j] = R[i][j];
}
}
for (i = 0; i < 3; i++)
{
for (j = 3; j < 6; j++)
{
AdT[i][j] = 0.0;
}
}
for (i=3;i<6;i++)
{
for (j = 3; j < 6; j++)
{
AdT[i][j] = R[i-3][j-3];
}
}
double mat[3][3];
Matrix3Mult(so3Mat, R, mat);
for (i = 3; i < 6; i++)
{
for (j = 0; j < 3; j++)
{
AdT[i][j] = mat[i-3][j];
}
}
return;
}
/**
*@brief Description: Computes a normalized screw axis representation S of a screw described by
* a unit vector s in the direction of the screw axis, located at the point q, with pitch h.
*@param[in] q a point lying on the screw axis.
*@param[in] s a unit vector in the direction of the screw axis.
*@param[in] h the pitch of the screw axis.
*@param[in] S a normalized screw axis representation.
*@return No return value.
*@note:
*@warning:
*/
void ScrewToAxis(double q[3], double s[3], double h,double S[6])
{
double v[3];
double temp[3];
Vec3Cross(s, q, temp);
Vec3MultValue(temp, -1.0, temp);
Vec3MultValue(s, h, v);
Vec3Add(v, temp, v);
S[0] = s[0];
S[1] = s[1];
S[2] = s[2];
S[3] = v[0];
S[4] = v[1];
S[5] = v[2];
return;
}
/**
*@brief Description: Extracts the normalized screw axis S and the distance traveled along the screw
theta from the 6-vector of exponential coordinates S*theta.
*@param[in] expc6 the 6-vector of exponential coordinates.
*@param[out] S the normalized screw axis.
*@param[out] theta the distance traveled along the screw.
*@return No return value.
*@note:
*@warning:
*/
void AxisAng6(double expc6[6],double S[6],double *theta)
{
*theta = Vec3Norm2(expc6);
if (*theta<ZERO_ANGLE)
{
*theta = Vec3Norm2(&expc6[3]);
if (*theta<ZERO_DISTANCE)
{
*theta=0.0;
//S is undefine,no motion at all.
S[0] = 0.0; S[1] = 0.0; S[2] = 0.0; S[3] = 0.0; S[4] = 0.0; S[5] = 0.0;
return;
}
}
Vec3MultValue(expc6, 1.0 / (*theta), S);
Vec3MultValue(&expc6[3], 1.0 / (*theta), &S[3]);
return ;
}
/**
*@brief Description: Computes the homogeneous transformation matrix T in SE(3) corresponding to
* the matrix exponential of se3mat in se(3).
*@param[in] se3mat the matrix exponential of se3mat in se(3).
*@param[out] T the homogeneous transformation matrix T in SE(3).
*@return No return value.
*@note:
*@warning:
*/
void MatrixExp6(double se3Mat[4][4], double T[4][4])
{
int i;
int j;
double so3mat[3][3];
double omgmat[3][3];
double temp[3][3];
double Gtheta[3][3];
double p[3];
double v[3];
double omgtheta[3];
double omghat[3];
double theta;
double MatI3[3][3] =
{
1.0,0.0,0.0,
0.0,1.0,0.0,
0.0,0.0,1.0
};
TransToRp(se3Mat, so3mat, p);//extracts so3mat from se3mat
so3ToVec(so3mat, omgtheta);
if (Vec3Norm2(omgtheta)<ZERO_ANGLE)
{
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
T[i][j] = MatI3[i][j];
}
}
T[0][3] = se3Mat[0][3];
T[1][3] = se3Mat[1][3];
T[2][3] = se3Mat[2][3];
T[3][0] = 0.0;
T[3][1] = 0.0;
T[3][2] = 0.0;
T[3][3] = 1.0;
}
else
{
AxisAng3(omgtheta, omghat, &theta);
MatrixExp3(so3mat, temp);
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
T[i][j] = temp[i][j];
}
}
Matrix3MultValue(so3mat, 1.0/theta,omgmat);
Matrix3Mult(omgmat, omgmat, temp);
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
Gtheta[i][j] = MatI3[i][j] * theta + (1.0 - cos(theta))*omgmat[i][j] + (theta - sin(theta))*temp[i][j];
}
}
v[0] = p[0] / theta;
v[1] = p[1] / theta;
v[2] = p[2] / theta;
Matrix3MultVec(Gtheta, v, p);
T[0][3] = p[0]; T[1][3] = p[1]; T[2][3] = p[2];
T[3][0] = 0.0;
T[3][1] = 0.0;
T[3][2] = 0.0;
T[3][3] = 1.0;
}
return;
}
/**
*@brief Description: Computes the matrix logarithm se3mat in se(3) of
the homogeneous transformation matrix T in SE(3)
*@param[in] T the homogeneous transformation matrix.
*@param[out] se3Mat the matrix logarithm of T.
*@return No return value.
*@note:
*@warning:
*/
void MatrixLog6(double T[4][4], double se3Mat[4][4])
{
int i;
int j;
double R[3][3];
double so3mat[3][3];
double p[3];
TransToRp(T, R, p);
MatrixLog3(R, so3mat);
int flag = 0;
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
if (fabs(so3mat[i][j])<ZERO_ELEMENT)
{
continue;
}
else
{
flag = 1;
break;
}
}
}
if (!flag)
{
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
se3Mat[i][j] = 0.0;
}
}
se3Mat[0][3] = T[0][3];
se3Mat[1][3] = T[1][3];
se3Mat[2][3] = T[2][3];
se3Mat[3][0] = 0.0;
se3Mat[3][1] = 0.0;
se3Mat[3][2] = 0.0;
se3Mat[3][3] = 0.0;
}
else
{
double MatI3[3][3] =
{
1.0,0.0,0.0,
0.0,1.0,0.0,
0.0,0.0,1.0
};
double v[3];
double temp[3][3];
double InvGtheta[3][3];
double theta = acos((R[0][0] + R[1][1] + R[2][2] - 1.0) / 2.0);
for (i=0;i<3;i++)
{
for (j=0;j<3;j++)
{
se3Mat[i][j] = so3mat[i][j];
}
}
Matrix3Mult(so3mat, so3mat, temp);
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
InvGtheta[i][j] = MatI3[i][j] - 0.5*so3mat[i][j] + (1.0 / theta - 0.5*1.0 / tan(0.5*theta))*temp[i][j] / theta;
}
}
Matrix3MultVec(InvGtheta, p, v);
se3Mat[0][3] = v[0];
se3Mat[1][3] = v[1];
se3Mat[2][3] = v[2];
se3Mat[3][0] = 0.0;
se3Mat[3][1] = 0.0;
se3Mat[3][2] = 0.0;
se3Mat[3][3] = 0.0;
}
return ;
}
/**
*@brief Description: Computes the end-effector frame given the zero position of the end-effector M,
* the list of joint screws Slist expressed in the fixed-space frame, and the list of joint values thetalist.
*@param[in] M the zero position of the end-effector expressed in the fixed-space frame.
*@param[in] Joint Num the number of joints.
*@param[in] Slist the list of joint screws Slist expressed in the fixed-space frame.
* in the format of a matrix with the screw axes as the column.
*@param[in] thetalist the list of joint values.
*@param[out] T the end-effector frame expressed in the fixed-space frame.
*@return No return value.
*@note: when Slist is a matrix ,make sure that columns number of Slist is equal to JointNum,
* rows number of Slist 6 .The function call should be written as
* FKinBody(...,(double *)Slist,...).
*@warning:
*/
void FKinSpace(double M[4][4],int JointNum,double *Slist, double *thetalist,double T[4][4])
{
int i, j, k, l;
double se3mat[4][4];
double T2[4][4];
double exp6[4][4];
double V[6];
Matrix4Equal(M, T);
for (i= JointNum-1;i>=0;i--)
{
for (l=0;l<6;l++)
{
V[l] = Slist[l*JointNum+i];//get each column of Slist.
}
VecTose3(V, se3mat);
for (j=0;j<4;j++)
{
for (k=0;k<4;k++)
{
se3mat[j][k] = se3mat[j][k] * thetalist[i];
}
}
MatrixExp6(se3mat, exp6);
Matrix4Mult(exp6, T, T2);
Matrix4Equal(T2, T);
}
return ;
}
/**
*@brief Description:Computes the end-effector frame given the zero position of the end-effector M,
* the list of joint screws Blist expressed in the end-effector frame, and the list of joint values thetalist.
*@param[in] M the zero position of the end-effector expressed in the end-effector frame.
*@param[in] JointNum the number of joints.
*@param[in] Blist the list of joint screws Slist expressed in the end-effector frame.
* in the format of a matrix with the screw axes as the column.
*@param[in] thetalist the list of joint values.
*@param[out] T the end-effector frame expressed in the end-effector frame.
*@return No return value.
*@note: when Blist is a matrix ,make sure that columns number of Slist is equal to JointNum,
* rows number of Slist 6 .The function call should be written as
* FKinBody(...,(double *)Blist,...).
*@warning:
*/
void FKinBody(double M[4][4], int JointNum, double *Blist, double thetalist[], double T[4][4])
{
int i, j, k, l;
double se3mat[4][4];
double T2[4][4];
double exp6[4][4];
double V[6];
Matrix4Equal(M, T);
for (i = 0; i < JointNum ; i++)
{
for (l = 0; l < 6; l++)
{
V[l] = Blist[l*JointNum + i];//get each column of Slist.
}
VecTose3(V, se3mat);
for (j = 0; j < 4; j++)
{
for (k = 0; k < 4; k++)
{
se3mat[j][k] = se3mat[j][k] * thetalist[i];
}
}
MatrixExp6(se3mat, exp6);
Matrix4Mult(T, exp6, T2);
Matrix4Equal(T2, T);
}
return ;
}
/**
*@brief Description: Computes the body Jacobian Jb(theta) in 6×n given a list of joint screws Bi
expressed in the body frame and a list of joint angles.
*@param[in] Blist The joint screw axes in the end - effector frame when the manipulator is
* at the home position, in the format of a matrix with the screw axes as the row.
*@param[in] thetalist A list of joint coordinates.
*@param[out] Jb Body Jacobian matrix.
*@return No return value.
*@note: when Blist and Jb are matrixes ,make sure that columns number of Slist or Jb is equal to JointNum,
* rows number of Slist or Jb is 6 .The function call should be written as
* JacobianSpace(JointNum,(double *)Slist,thetalist,(double *)Jb).
*@warning:
*/
void JacobianBody(int JointNum,double *Blist, double *thetalist,double *Jb)
{
int i;
int j;
int k;
double T1[4][4];
double T2[4][4];
double se3mat[4][4];
double V[6];
double AdT[6][6];
double T[4][4] = {
1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1
};
//Fist column of Jbn.
for (i=0;i<6;i++)
{
Jb[i*JointNum+JointNum -1] = Blist[i*JointNum + JointNum-1];
}
//Jbi for i=n-1,n-2,...,1.
for (i= JointNum -2;i>=0;i--)
{
for (j=0;j<6;j++)
{
V[j] = -1.0*Blist[j*JointNum+i+1];
}
VecTose3(V, se3mat);
Matrix4MultValue(se3mat, thetalist[i + 1], se3mat);
MatrixExp6(se3mat, T1);
Matrix4Mult(T, T1,T2);
Matrix4Equal(T2, T);
Adjoint(T, AdT);
for (j=0;j<6; j++)
{
Jb[j*JointNum + i] = 0;
for (k=0;k<6;k++)
{
Jb[j*JointNum + i] = Jb[j*JointNum + i] + AdT[j][k] * Blist[k*JointNum + i];
}
}
}
return ;
}
/**
*@brief Description:Computes the space Jacobian Js(theta) in R6 x n given a list of joint screws Si
* expressed in the fixed space frame and a list of joint angles.
*@param[in] JointNum joints number.
*@param[in] Slist The joint screw axes expressed in the fixed space frame when the manipulator is
* at the home position, in the format of a matrix with the screw axes as the column.
*@param[in] thetalist A list of joint coordinates.
*@param[out] Js Space Jacobian matrix.
*@return No return value.
*@note: when Slist and Js are matrices ,make sure that columns number of Slist or Js is equal to JointNum,
* rows number of Slist or Js is 6 .The function call should be written as
* JacobianSpace(JointNum,(double *)Slist,thetalist,(double *)Js).
*@warning:
*/
void JacobianSpace(int JointNum, double *Slist, double *thetalist, double *Js)
{
int i;
int j;
int k;
double T1[4][4];
double T2[4][4];
double se3mat[4][4];
double V[6];
double AdT[6][6];
double T[4][4] = {
1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1
};
//Fist column of Js.
for (i = 0; i < 6; i++)
{
Js[i*JointNum + 0] = Slist[i*JointNum + 0];
}
//Jsi for i=2,3,...,n.
for (i = 1; i <JointNum; i++)
{
for (j = 0; j < 6; j++)
{
V[j] = Slist[j*JointNum + i - 1];
}
VecTose3(V, se3mat);
Matrix4MultValue(se3mat, thetalist[i - 1], se3mat);
MatrixExp6(se3mat, T1);
Matrix4Mult(T, T1, T2);
Matrix4Equal(T2, T);
Adjoint(T, AdT);
for (j = 0; j < 6; j++)
{
Js[j*JointNum + i] = 0;
for (k = 0; k < 6; k++)
{
Js[j*JointNum + i] = Js[j*JointNum + i] + AdT[j][k] * Slist[k*JointNum + i];
}
}
}
return;
}
#define SIGN(a,b) ((b) >= 0.0 ? fabs(a) : -fabs(a))
#define MAX(a,b) (a>b?a:b)
#define MIN(a,b) (a<b?a:b)
/**
*@brief Computes (a^2+b^2)^(1/2),
*@param[in] a
*@param[in] b
*@note:
*@warning:
*/
double pythag(const double a, const double b)
{
double absa, absb;
absa = fabs(a);
absb = fabs(b);
if (absa > absb) return absa*sqrt(1.0 + (absb / absa)*(absb / absa));
else return (absb == 0.0 ? 0.0 : absb*sqrt(1.0 + (absa / absb)*(absa / absb)));
}
void reorder(double *u,int m,int n,double *w,double *v)
{
/*Given the output of decompose, this routine sorts the singular values, and corresponding columns
of u and v, by decreasing magnitude.Also, signs of corresponding columns are flipped so as to
maximize the number of positive elements.*/
int i, j, k, s, inc = 1;
double sw;
double *su = (double *)malloc(m * sizeof(double));
if (su==NULL)
{
return;
}
double *sv = (double *)malloc(n * sizeof(double));
if (sv==NULL)
{
free(su);
return;
}
do { inc *= 3; inc++; } while (inc <= n);// Sort.The method is Shell’s sort.(The work is negligible as compared to that already done in decompose.)
do {
inc /= 3;
for (i = inc; i < n; i++) {
sw = w[i];
for (k = 0; k < m; k++) su[k] = u[k*n + i];
for (k = 0; k < n; k++) sv[k] = v[k*n + i];
j = i;
while (w[j - inc] < sw) {
w[j] = w[j - inc];
for (k = 0; k < m; k++) u[k*n + j] = u[k*n + j - inc];
for (k = 0; k < n; k++) v[k*n + j] = v[k*n + j - inc];
j -= inc;
if (j < inc) break;
}
w[j] = sw;
for (k = 0; k < m; k++) u[k*n + j] = su[k];
for (k = 0; k < n; k++) v[k*n + j] = sv[k];
}
} while (inc > 1);
for (k = 0; k < n; k++) {
//Flip signs.
s = 0;
for (i = 0; i < m; i++) if (u[i*n + k] < 0.0) s++;
for (j = 0; j < n; j++) if (v[j*n + k] < 0.0) s++;
if (s > (m + n) / 2) {
for (i = 0; i < m; i++) u[i*n + k] = -u[i*n + k];
for (j = 0; j < n; j++) v[j*n + k] = -v[j*n + k];
}
}
free(su);
free(sv);
return;
}
/**
*@brief Given the matrix a,this routine computes its singular value decomposition.
*@param[in/out] a input matrix a or output matrix u.
*@param[in] m number of rows of matrix a.
*@param[in] n number of columns of matrix a.
*@param[in] tol any singular values less than a tolerance are treated as zero.
*@param[out] w output vector w.
*@param[out] v output matrix v.
*@return No return value.
*@note: input a must be a m rows and n columns matrix,w is n-vector,v is a n rows and n columns matrix.
*@warning:
*/
int svdcmp(double *a, int m, int n, double tol, double *w, double *v)
{
int flag, i, its, j, jj, k, l, nm;
double anorm, c, f, g, h, s, scale, x, y, z;
double *rv1 = (double *)malloc(n * sizeof(double));
if (rv1==NULL)
{
return 1;
}
g = scale = anorm = 0.0;
for (i=0;i<n;i++)//Householder reduction to bidiagonal form.
{
l = i + 2;
rv1[i] = scale*g;
g = s = scale = 0.0;
if (i<m)
{
for (k = i; k < m; k++)scale += fabs(a[k*n + i]);
if (scale!=0.0)
{
for (k = i; k < m;k++)
{
a[k*n + i] /= scale;
s += a[k*n + i] * a[k*n + i];
}
f = a[i*n + i];
g = -SIGN(sqrt(s), f);
h = f*g - s;
a[i*n + i] = f - g;
for (j=l-1;j<n;j++)
{
for (s = 0.0, k = i; k < m;k++)
{
s += a[k*n + i] * a[k*n + j];
}
f = s / h;
for (k = i; k < m;k++)
{
a[k*n + j] += f*a[k*n + i];
}
}
for (k = i; k < m;k++)
{
a[k*n + i] *= scale;
}
}
}
w[i] = scale*g;
g = s = scale = 0.0;
if (i+1 <= m && (i+1)!=n)
{
for (k=l-1;k<n;k++)
{
scale += fabs(a[i*n + k]);
}
if (scale!=0.0)
{
for (k=l-1;k<n;k++)
{
a[i*n + k] /= scale;
s += a[i*n + k] * a[i*n + k];
}
f = a[i*n + l - 1];
g = -SIGN(sqrt(s), f);
h = f*g - s;
a[i*n + l - 1] = f - g;
for (k=l-1;k<n;k++)
{
rv1[k] = a[i*n + k] / h;
}
for (j = l - 1; j < m; j++)
{
for (s = 0.0, k = l - 1; k < n; k++)
{
s += a[j*n + k] * a[i*n + k];
}
for (k = l - 1; k < n; k++)
{
a[j*n + k] += s*rv1[k];
}
}
for (k=l-1;k<n;k++)
{
a[i*n + k] *= scale;
}
}
}
anorm = MAX(anorm, (fabs(w[i]) + fabs(rv1[i])));
}
for (i=n-1;i>=0;i--)//Accumulation of right-hand transformations.
{
if (i<n-1)
{
if (g!=0.0)
{
for (j = l;j<n;j++)//Double division to avoid possible underflow.
{
v[j*n + i] = (a[i*n + j] / a[i*n + l]) / g;
}
for (j = l;j<n;j++)
{
for (s = 0.0, k = l;k<n;k++)
{
s += a[i*n + k] * v[k*n + j];
}
for (k = l;k<n;k++)
{
v[k*n + j] += s*v[k*n + i];
}
}
}
for (j = l;j<n;j++)
{
v[i*n + j] = v[j*n + i] = 0.0;
}
}
v[i*n + i] = 1.0;
g = rv1[i];
l = i;
}
for (i=MIN(m,n)-1;i>=0;i--)//Accumulation of left-hand transformations.
{
l = i + 1;
g = w[i];
for (j = l;j<n;j++)
{
a[i*n + j] = 0.0;
}
if (g!=0.0)
{
g = 1.0 / g;
for (j = l;j<n;j++)
{
for (s = 0.0, k = l; k < m;k++)
{
s += a[k*n + i] * a[k*n + j];
}
f = (s / a[i*n + i])*g;
for (k = i; k < m;k++)
{
a[k*n + j] += f*a[k*n + i];
}
}
for (j = i; j < m;j++)
{
a[j*n + i] *= g;
}
}
else
{
for (j = i; j < m;j++)
{
a[j*n + i] = 0.0;
}
}
++a[i*n + i];
}
for (k=n-1;k>=0;k--)
{
/* Diagonalization of the bidiagonal form: Loop over
singular values, and over allowed iterations. */
for (its=0;its<30;its++)
{
flag = 1;
for (l = k;l>=0;l--)//Test for splitting.
{
nm = l - 1;
if (l==0 || fabs(rv1[l])<= tol*anorm)
{
flag = 0;
break;
}
if (fabs(w[nm]) <=tol*anorm)break;
}
if (flag)//Cancellation of rv1[l], if l > 0.
{
c = 0.0;
s = 1.0;
for (i=l;i<k+1;i++)
{
f = s*rv1[i];
rv1[i] = c*rv1[i];
if (fabs(f) <= tol*anorm)break;
g = w[i];
h = pythag(f, g);
w[i] = h;
h = 1.0 / h;
c = g*h;
s = -f*h;
for (j = 0; j < m;j++)
{
y = a[j*n + nm];
z = a[j*n + i];
a[j*n + nm] = y*c + z*s;
a[j*n + i] = z*c - y*s;
}
}
}
z = w[k];
if (l==k)//Convergence
{
if (z<0.0)//Singular value is made nonnegative..
{
w[k] = -z;
for (j=0;j<n;j++)
{
v[j*n + k] = -v[j*n + k];
}
}
break;
}
//if (its==29)
//{
// printf("no convergence in 30 svdcmp iterations\n");
//}
x = w[l]; //Shift from bottom 2-by-2 minor.
nm = k - 1;
y = w[nm];
g = rv1[nm];
h = rv1[k];
f = ((y - z)*(y + z) + (g - h)*(g + h)) / (2.0*h*y);
g = pythag(f, 1.0);
f = ((x - z)*(x + z) + h*((y / (f + SIGN(g, f))) - h)) / x;
c = s = 1.0;
for (j = l;j<=nm;j++)//Next QR transformation:
{
i = j + 1;
g = rv1[i];
y = w[i];
h = s*g;
g = c*g;
z = pythag(f, h);
rv1[j] = z;
c = f / z;
s = h / z;
f = x*c + g*s;
g = g*c - x*s;
h = y*s;
y *= c;
for (jj=0;jj<n;jj++)
{
x = v[jj*n + j];
z = v[jj*n + i];
v[jj*n + j] = x*c + z*s;
v[jj*n + i] = z*c - x*s;
}
z = pythag(f, h);
w[j] = z;
if (z)//Rotation can be arbitrary if z = 0.
{
z = 1.0 / z;
c = f*z;
s = h*z;
}
f = c*g + s*y;
x = c*y - s*g;
for (jj = 0; jj < m;jj++)
{
y = a[jj*n + j];
z = a[jj*n + i];
a[jj*n + j] = y*c + z*s;
a[jj*n + i] = z*c - y*s;
}
}
rv1[l] = 0.0;
rv1[k] = f;
w[k] = x;
}
}
free(rv1);
reorder(a, m, n, w, v);//对特征值按降序排列，且调整对应矩阵
return 0;
}
double u[PINV_MAX * PINV_MAX] = { 0 };
double w[PINV_MAX] = { 0 };
double vw[PINV_MAX * PINV_MAX] = { 0 };
double v[PINV_MAX * PINV_MAX] = { 0 };
double uT[PINV_MAX * PINV_MAX] = { 0 };
/**
*@brief Computes Moore-Penrose pseudoinverse by Singular Value Decomposition (SVD) algorithm.
*@param[in] a an arbitrary order matrix.
*@param[in] m rows.
*@param[in] n columns.
*@param[in] tol any singular values less than a tolerance are treated as zero.
*@param[out] b Moore-Penrose pseudoinverse of matrix a.
*@return 0:success,Nonzero:failure.
*@retval 0 success.
*@retval 1 failure when use malloc to allocate memory.
*@note:
*@warning:
*/
int MatrixPinv(const double *a, int m, int n, double tol, double *b)
{
int i, j;
int ret;
memcpy(u, a, m*n * sizeof(double));
ret=svdcmp(u, m, n, tol,w, v);
for (i = 0; i < n; i++)
{
if (w[i] <= tol)
{
w[i] = 0;
}
else
{
w[i] = 1.0 / w[i];
}
}
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
{
vw[i*n + j] = v[i*n + j] * w[j];
}
}
MatrixT(u, m, n, uT);
MatrixMult(vw, n, n, uT, m, b);
return 0;
}
/**
*@brief Description:use an iterative Newton-Raphson root-finding method.
*@param[in] JointNum Number of joints.
*@param[in] Blist The joint screw axes in the Body frame when the manipulator
* is at the home position, in the format of a matrix with the
* screw axes as the columns.
*@param[in] M The home configuration of the end-effector.
*@param[in] T The desired end-effector configuration Tsd.
*@param[in] thetalist0 An initial guess of joint angles that are close to satisfying Tsd.
*@param[in] eomg A small positive tolerance on the end-effector orientation error.
* The returned joint angles must give an end-effector orientation error less than eomg.
*@param[in] ev A small positive tolerance on the end-effector linear position error. The returned
* joint angles must give an end-effector position error less than ev.
*@param[in] maxiter The maximum number of iterations before the algorithm is terminated.
*@param[out] thetalist Joint angles that achieve T within the specified tolerances.
*@return 0:success,Nonzero:failure
*@retval 0 success.
*@retval 1 The maximum number of iterations reach before the algorithm is terminated.
*@retval 2 failure to allocate memory for Jacobian matrix.
*@note:
*@warning:
*/
int IKinBodyNR(int JointNum, double *Blist, double M[4][4], double T[4][4], double *thetalist0, double eomg, double ev, int maxiter, double *thetalist)
{
int i, j;
double Tsb[4][4];
double iTsb[4][4];
double Tbd[4][4];
double se3Mat[4][4];
double Vb[6];
int ErrFlag;
double *Jb = (double *)malloc(JointNum * 6 * sizeof(double));
if (Jb == NULL)
{
return 2;
}
double *pJb = (double *)malloc(JointNum * 6 * sizeof(double));
if (pJb == NULL)
{
free(Jb);
return 2;
}
double *dtheta = (double *)malloc(JointNum * sizeof(double));
if (dtheta == NULL)
{
free(Jb);
free(pJb);
return 2;
}
MatrixCopy(thetalist0, JointNum, 1, thetalist);
FKinBody(M, JointNum, Blist, thetalist, Tsb);
TransInv(Tsb, iTsb);
Matrix4Mult(iTsb,T,Tbd);
MatrixLog6(Tbd, se3Mat);
se3ToVec(se3Mat, Vb);
ErrFlag = VecNorm2(3, &Vb[0]) > eomg || VecNorm2(3, &Vb[3]) > ev;
i = 0;
#if(DEBUGMODE)
///////////////////Debug///////////
printf("iteration:%4d thetalist: ", i);
int k;
for (k = 0; k < JointNum; k++)
{
printf("%4.6lf\t", thetalist[k]);
}
printf("\n");
////////////////////////////////////
#endif
while (ErrFlag && i<maxiter)
{
JacobianBody(JointNum, Blist, thetalist, Jb);
MatrixPinv(Jb, 6, JointNum, 2.2E-15, pJb);
MatrixMult(pJb, JointNum,6 , Vb, 1, dtheta);
for (j=0;j<JointNum;j++)
{
thetalist[j] = thetalist[j] + dtheta[j];
}
i++;
FKinBody(M, JointNum, Blist, thetalist, Tsb);
TransInv(Tsb, iTsb);
Matrix4Mult(iTsb, T, Tbd);
MatrixLog6(Tbd, se3Mat);
se3ToVec(se3Mat, Vb);
ErrFlag = VecNorm2(3, &Vb[0]) > eomg || VecNorm2(3, &Vb[3]) > ev;
#if (DEBUGMODE)
///////////////////////DEBUG///////////
printf("iteration:%4d thetalist: ", i);
for (k = 0; k < JointNum; k++)
{
printf("%4.6lf\t", thetalist[k]);
}
printf("\n");
//////////////////////////////////////
#endif
}
free(Jb);
free(pJb);
free(dtheta);
return ErrFlag;
}
/**
*@brief Description:use an iterative Newton-Raphson root-finding method.
*@param[in] JointNum Number of joints.
*@param[in] Slist The joint screw axes in the space frame when the manipulator
* is at the home position, in the format of a matrix with the
* screw axes as the columns.
*@param[in] M The home configuration of the end-effector.
*@param[in] T The desired end-effector configuration Tsd.
*@param[in] thetalist0 An initial guess of joint angles that are close to satisfying Tsd.
*@param[in] eomg A small positive tolerance on the end-effector orientation error.
* The returned joint angles must give an end-effector orientation error less than eomg.
*@param[in] ev A small positive tolerance on the end-effector linear position error. The returned
* joint angles must give an end-effector position error less than ev.
*@param[in] maxiter The maximum number of iterations before the algorithm is terminated.
*@param[out] thetalist Joint angles that achieve T within the specified tolerances.
*@return 0:success,Nonzero:failure
*@retval 0
*@note :
*@warning:
*/
int IKinSpaceNR(int JointNum, double *Slist, double M[4][4], double T[4][4], double *thetalist0, double eomg, double ev, int maxiter,double *thetalist)
{
int i, j;
double Tsb[4][4];
double iTsb[4][4];
double Tbd[4][4];
double AdT[6][6];
double se3Mat[4][4];
double Vb[6];
double Vs[6];
int ErrFlag;
double *Js = (double *)malloc(JointNum * 6 * sizeof(double));
if (Js == NULL)
{
return 2;
}
double *pJs = (double *)malloc(JointNum * 6 * sizeof(double));
if (pJs == NULL)
{
free(Js);
return 2;
}
double *dtheta = (double *)malloc(JointNum * sizeof(double));
if (dtheta == NULL)
{
free(Js);
free(pJs);
return 2;
}
MatrixCopy(thetalist0, JointNum, 1, thetalist);
FKinSpace(M, JointNum, Slist, thetalist, Tsb);
TransInv(Tsb, iTsb);
Matrix4Mult(iTsb, T, Tbd);
MatrixLog6(Tbd, se3Mat);
se3ToVec(se3Mat, Vb);
Adjoint(Tsb, AdT);
MatrixMult((double *)AdT, 6, 6, Vb, 1, Vs);
ErrFlag = VecNorm2(3, &Vs[0]) > eomg || VecNorm2(3, &Vs[3]) > ev;
i = 0;
#if(DEBUGMODE)
///////////////////Debug///////////
printf("iteration:%4d thetalist: ", i);
int k;
for (k = 0; k < JointNum; k++)
{
printf("%4.4lf, ", thetalist[k]);
}
printf("\n");
////////////////////////////////////
#endif
while (ErrFlag && i < maxiter)
{
JacobianSpace(JointNum, Slist, thetalist, Js);
MatrixPinv(Js, 6, JointNum, 2.2E-15, pJs);
MatrixMult(pJs, JointNum, 6, Vs, 1, dtheta);
for (j = 0; j < JointNum; j++)
{
thetalist[j] = thetalist[j] + dtheta[j];
}
i++;
FKinSpace(M, JointNum, Slist, thetalist, Tsb);
TransInv(Tsb, iTsb);
Matrix4Mult(iTsb, T, Tbd);
MatrixLog6(Tbd, se3Mat);
se3ToVec(se3Mat, Vb);
Adjoint(Tsb, AdT);
MatrixMult((double *)AdT, 6, 6, Vb, 1, Vs);
ErrFlag = VecNorm2(3, &Vs[0]) > eomg || VecNorm2(3, &Vs[3]) > ev;
#if (DEBUGMODE)
///////////////////////DEBUG///////////
printf("iteration:%4d thetalist: ", i);
for (k = 0; k < JointNum; k++)
{
printf("%4.4lf, ", thetalist[k]);
}
printf("\n");
//////////////////////////////////////
#endif
}
free(Js);
free(pJs);
free(dtheta);
return ErrFlag;
}
/**
* @brief Description: Algorithm for Computing the ZYX Euler Angles according to rotation matrix.
* @param[in] R Rotation matrix.
* @param[out] alpha Angles for rotation around Z axis.
* @param[out] beta Angles for rotation around Y axis.
* @param[out] gamma Angles for rotation around X axis.
* @return No return value.
* @note:
*@warning:
*/
void RotToZYXEulerAngle(double R[3][3], double *alpha, double *beta, double *gamma)
{
if (fabs(1.0 + R[2][0])<ZERO_ELEMENT)
{
*alpha = 0;
*beta = PI / 2.0;
*gamma = atan2(R[0][1], R[1][1]);
}
else if (fabs(1.0 - R[2][0])<ZERO_ELEMENT)
{
*alpha = 0;
*beta = -PI / 2.0;
*gamma = atan2(R[0][1], R[1][1]);
}
else
{
*alpha = atan2(R[1][0], R[0][0]);
*beta = atan2(-R[2][0], sqrt(R[0][0] * R[0][0] + R[1][0] * R[1][0]));
*gamma = atan2(R[2][1], R[2][2]);
}
return;
}
/**
* @brief Description: Algorithm for Computing the rotation matrix of ZYX Euler Angles.
* @param[in] alpha Angles for rotation around Z axis.
* @param[in] beta Angles for rotation around Y axis.
* @param[in] gamma Angles for rotation around X axis.
* @param[out] R Rotation matrix.
* @return No return value.
* @note:
* @waring:
*/
void ZYXEulerAngleToRot(double alpha, double beta, double gamma, double R[3][3])
{
R[0][0] = cos(alpha)*cos(beta);
R[0][1] = cos(alpha)*sin(beta)*sin(gamma) - sin(alpha)*cos(gamma);
R[0][2] = cos(alpha)*sin(beta)*cos(gamma) + sin(alpha)*sin(gamma);
R[1][0] = sin(alpha)*cos(beta);
R[1][1] = sin(alpha)*sin(beta)*sin(gamma) + cos(alpha)*cos(gamma);
R[1][2] = sin(alpha)*sin(beta)*cos(gamma) - cos(alpha)*sin(gamma);
R[2][0] = -sin(beta);
R[2][1] = cos(beta)*sin(gamma);
R[2][2] = cos(beta)*cos(gamma);
return;
}
/**
* @brief Description: Computes the unit vector of Euler axis and rotation angle corresponding to rotation matrix.
* @param[in] R A rotation matrix.
* @param[out] omghat the unit vector of Euler axis .
* @param[out] theta the rotation angle.
* @return No return value.
* @retval 0
* @note: if theta is zero ,the unit axis is undefined and set it as a zero vector [0;0;0].
*@warning:
*/
void RotToAxisAng(double R[3][3],double omghat[3],double *theta)
{
double tmp;
double omg[3] = { 0 };
double acosinput = (R[0][0] + R[1][1] + R[2][2] - 1.0) / 2.0;
if (fabs(acosinput-1.0)<ZERO_VALUE)
{
memset(omghat, 0, 3 * sizeof(double));
*theta = 0.0;
}
else if (acosinput <= -1.0)
{
if ((1.0 + R[2][2]) >= ZERO_VALUE)
{
omg[0] = 1.0 / sqrt(2 * (1.0 + R[2][2]))*R[0][2];
omg[1] = 1.0 / sqrt(2 * (1.0 + R[2][2]))*R[1][2];
omg[2] = 1.0 / sqrt(2 * (1.0 + R[2][2]))*(1.0 + R[2][2]);
}
else if ((1.0 + R[1][1] >= ZERO_VALUE))
{
omg[0] = 1.0 / sqrt(2 * (1.0 + R[1][1]))*R[0][1];
omg[1] = 1.0 / sqrt(2 * (1.0 + R[1][1]))*(1.0 + R[1][1]);
omg[2] = 1.0 / sqrt(2 * (1.0 + R[1][1]))*R[2][1];
}
else
{
omg[0] = 1.0 / sqrt(2 * (1.0 + R[0][0]))*(1.0 + R[0][0]);
omg[1] = 1.0 / sqrt(2 * (1.0 + R[0][0]))*R[1][0];
omg[2] = 1.0 / sqrt(2 * (1.0 + R[0][0]))*R[2][0];
}
omghat[0] = omg[0];
omghat[1] = omg[1];
omghat[2] = omg[2];
*theta=PI;
}
else
{
*theta = acos(acosinput);
tmp = 2.0*sin(*theta);
omghat[0] = (R[2][1] - R[1][2]) / tmp;
omghat[1] = (R[0][2] - R[2][0]) / tmp;
omghat[2] = (R[1][0] - R[0][1]) / tmp;
}
return;
}
/**
* @brief Description: Computes the unit quaternion corresponding to the Euler axis and rotation angle.
* @param[in] omg Unit vector of Euler axis.
* @param[in] theta Rotation angle.
* @param[in] q The unit quaternion
* @return No return value.
* @note:
*@warning:
*/
void AxisAngToQuaternion(double omg[3],double theta, double q[4])
{
q[0] = cos(theta / 2.0);
q[1] = omg[0] * sin(theta / 2.0);
q[2] = omg[1] * sin(theta / 2.0);
q[3] = omg[2] * sin(theta / 2.0);
return;
}
/**
* @brief Description:Computes the unit quaternion corresponding to a rotation matrix.
* @param[in] q Unit quaternion.
* @param[out] R Rotation matrix.
* @return No return value.
* @note:
* @warning:
*/
void QuaternionToRot(double q[4], double R[3][3])
{
R[0][0] = q[0] * q[0] + q[1] * q[1] - q[2] * q[2] - q[3] * q[3];
R[0][1] = 2.0*(q[1] * q[2] - q[0] * q[3]);
R[0][2] = 2.0*(q[0] * q[2] + q[1] * q[3]);
R[1][0] = 2.0*(q[0] * q[3] + q[1] * q[2]);
R[1][1] = q[0] * q[0] - q[1] * q[1] + q[2] * q[2] - q[3] * q[3];
R[1][2] = 2.0*(q[2] * q[3] - q[0] * q[1]);
R[2][0] = 2.0*(q[1] * q[3] - q[0] * q[2]);
R[2][1] = 2.0*(q[0] * q[1] + q[2] * q[3]);
R[2][2] = q[0] * q[0] - q[1] * q[1] - q[2] * q[2] + q[3] * q[3];
return;
}
/**
* @brief Description: Computes the unit quaternion corresponding to the rotation matrix.
* @param[in] R The rotation matrix.
* @param[out] q The unit quaternion.
* @return No return value.
* @note:
* @warning:
*/
void RotToQuaternion(double R[3][3], double q[4])
{
double omghat[3];
double theta;
RotToAxisAng(R, omghat, &theta);
AxisAngToQuaternion(omghat, theta, q);
return;
}
/**
* @brief 单位四元数转换为旋转矩阵。
* @param[in] q 单位四元数[ox,oy,oz,ow].
* @param[out] R 旋转矩阵.
* @return 无返回值.
* @note:
* @warning: (1)此函数不对输入参数做检查，需保证输入为单位四元数。
(2)四元数顺序为[ox,oy,oz,w]。
*/
void q2rot(double q[4], double R[3][3])
{
double q1[4] = { 0 };
q1[0] = q[3];
q1[1] = q[0];
q1[2] = q[1];
q1[3] = q[2];
QuaternionToRot(q1, R);
return;
}
/**
* @brief 旋转矩阵转单位四元数。
* @param[in] R 旋转矩阵
* @param[out] q 单位四元数.[ox,oy,oz,ow].
* @return 无返回值.
* @note:
* @warning: (1)此函数不对输入参数做检查，需保证输入为旋转矩阵。
(2)四元数顺序为[ox,oy,oz,w]。
*/
void rot2q(double R[3][3], double q[4])
{
double q1[4] = { 0 };
RotToQuaternion(R, q1);
q[0] = q1[1];
q[1] = q1[2];
q[2] = q1[3];
q[3] = q1[0];
return;
}
/**
* @brief Description: Computes the parameters of orientation interpolation between two orientations.
* @param[in] Rs Start orientation.
* @param[in] Re End orientation.
* @param[in] Param structure of orientation interpolation parameters..
* @return No return value.
* @note:
* @warning:
*/
void InitialOrientInpParam(double Rs[3][3],double Re[3][3], OrientInpParam *Param)
{
double InvR[3][3];
MatrixCopy((double *)Rs, 3, 3, (double *)Param->Rs);
MatrixCopy((double *)Re, 3, 3, (double *)Param->Re);
RotInv(Rs, InvR);
MatrixMult((double *)InvR, 3, 3, (double *)Re, 3, (double *)Param->R);
RotToAxisAng(Param->R, Param->omg, &Param->theta);
MatrixCopy((double *)Param->R, 3, 3, (double *)Param->Ri);
Param->thetai = 0.0;
Param->InpFlag = 1;
return;
}
/**
* @brief Description: Computes orientations in each interpolation cycle.
* @param[in] Param Interpolation parameter structure.
* @param[out] dtheta angle need to rotate from previous orientation to next orientation in next time step.
* @return Ri1 orientations in next interpolation cycle.
* @retval 0
* @note:
* @warning:
*/
void QuaternionOrientInp(OrientInpParam *Param, double dtheta, double Ri1[3][3])
{
double q[4];
double R[3][3];
Param->InpFlag = 2;
Param->thetai = Param->thetai + dtheta;
if (Param->thetai >= Param->theta)
{
Param->thetai = Param->theta;
Param->InpFlag = 3;
}
AxisAngToQuaternion(Param->omg, Param->thetai, q);
QuaternionToRot(q, R);
MatrixMult((double *)Param->Rs, 3, 3, (double *)R, 3, (double *)Ri1);
MatrixCopy((double *)Ri1, 3, 3, (double *)Param->Ri);
return;
}
/**
* @brief Description:Computes the parameters of line path for interpolation.
* @param[in] p1 Coordinates of start point.
* @param[in] p2 Coordinates of end point.
* @param[out] p Line path parameters structure.
* @return No return value.
* @note:
* @warning:
*/
void InitialLinePathParam(double p1[3],double p2[3], LineInpParam *p)
{
int i;
for (i=0;i<3;i++)
{
p->p1[i] = p1[i];
p->p2[i] = p2[i];
p->pi[i] = p1[i];
}
p->L = sqrt((p2[0] - p1[0])*(p2[0] - p1[0]) + (p2[1] - p1[1])*(p2[1] - p1[1]) + (p2[2] - p1[2])*(p2[2] - p1[2]));
if (p->L<ZERO_DISTANCE)
{
p->t[0] = 0.0;
p->t[1] = 0.0;
p->t[2] = 0.0;
}
else
{
p->t[0] = (p2[0] - p1[0]) / p->L;
p->t[1] = (p2[1] - p1[1]) / p->L;
p->t[2] = (p2[2] - p1[2]) / p->L;
}
p->InpFlag = 1;
p->Li = 0;
return;
}
/**
* @brief Description:Computes the line path interpolation coordinates in each interpolation cycle.
* @param[in] p Line path parameters structure.
* @param[in] dL step length in next interpolation cycle.
* @param[in] pi1 coordinates in next interpolation cycle.
* @return No return value.
* @note:
* @warning:
*/
void LinePathInp(LineInpParam *p, double dL, double pi1[3])
{
p->InpFlag = 2;
if (p->Li + dL>=p->L)
{
pi1[0] = p->p2[0];
pi1[1] = p->p2[1];
pi1[2] = p->p2[2];
p->Li = p->L;
p->InpFlag = 3;
}
else if ( p->L- p->Li < 2.0*dL)
{
//avoid distance of final step is too small.
dL = 0.5*dL;
pi1[0] = p->pi[0] + p->t[0] * dL;
pi1[1] = p->pi[1] + p->t[1] * dL;
pi1[2] = p->pi[2] + p->t[2] * dL;
p->Li = p->Li+dL;
}
else
{
pi1[0] = p->pi[0] + p->t[0] * dL;
pi1[1] = p->pi[1] + p->t[1] * dL;
pi1[2] = p->pi[2] + p->t[2] * dL;
p->Li = p->Li + dL;
}
p->pi[0] = pi1[0];
p->pi[1] = pi1[1];
p->pi[2] = pi1[2];
return;
}
/**
* @brief Description: Computes the parameters of both line path and orientation for interpolation.
* @param[in] p1 Start coordinates,including x,y,z coordinates and orientation angles roll-pitch-yaw angles.
* @param[in] p2 End coordinates,including x,y,z coordinates and orientation angles roll-pitch-yaw angles.
* @param[out] LPO Parameters of both line path and orientation for interpolation.
* @return No return value.
* @note:
* @warning:
*/
void InitialLinePOInpParam( double p1[6], double p2[6], LinePOParam *LPO)
{
double Rs[3][3];
double Re[3][3];
RPYToRot(p1[3], p1[4], p1[5], Rs);
RPYToRot(p2[3], p2[4], p2[5], Re);
InitialLinePathParam( p1, p2, &(LPO->Line));
InitialOrientInpParam(Rs, Re, &(LPO->Orient));
RpToTrans(Rs, p1, LPO->Ts);
RpToTrans(Rs, p1, LPO->Ti);
RpToTrans(Re, p2, LPO->Te);
LPO->InpFlag = 1;
return;
}
/**
* @brief Description:Computes the line path interpolation coordinates and orientation in each interpolation cycle.
* @param[out] LPO Line path and orientation parameters structure.
* @param[in] dL Line path interpolation step length.
* @param[out] dtheta angle interpolation step length for Orientation interpolation.
* @return No return value.
* @note:
* @warning:
*/
void LinePOInp(LinePOParam *LPO,double dL,double dtheta,double Ti[4][4])
{
double pi[3];
double Ri[3][3];
LPO->InpFlag = 2;
LinePathInp(&LPO->Line, dL, pi);
QuaternionOrientInp(&LPO->Orient, dtheta, Ri);
if (LPO->Line.InpFlag==3 && LPO->Orient.InpFlag==3)
{
LPO->InpFlag = 3;
}
RpToTrans(Ri, pi, Ti);
MatrixCopy((double *)Ti, 4, 4, (double *)LPO->Ti);
return;
}