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franka_ik_pybind.cpp
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// Analytical Franka inverse kinematics using q7 as redundant parameter
// - Yanhao He, February 2020
// c++ -O3 -Wall -shared -std=c++11 -fPIC $(python3 -m pybind11 --includes) franka_ik_pybind.cpp -o franka_ik_pybind$(python3-config --extension-suffix)
// https://www.linyuanshi.me/post/pybind11-array/
#ifndef FRANKA_IK_HE_HPP
#define FRANKA_IK_HE_HPP
#include <array>
#include <cmath>
#include "Eigen/Dense"
#include <pybind11/pybind11.h>
#include <pybind11/eigen.h>
#include <pybind11/stl.h>
#include <pybind11/functional.h>
#include <pybind11/stl_bind.h>
namespace py = pybind11;
// inverse kinematics w.r.t. End Effector Frame (using Franka Hand data)
std::array< std::array<double, 7>, 4 > franka_IK_EE ( Eigen::Matrix3d R_EE,
Eigen::Vector3d z_EE,
Eigen::Vector3d p_EE,
double q7,
std::array<double, 7> q_actual_array )
{
const std::array< std::array<double, 7>, 4 > q_all_NAN = {{ {{NAN, NAN, NAN, NAN, NAN, NAN, NAN}},
{{NAN, NAN, NAN, NAN, NAN, NAN, NAN}},
{{NAN, NAN, NAN, NAN, NAN, NAN, NAN}},
{{NAN, NAN, NAN, NAN, NAN, NAN, NAN}} }};
const std::array<double, 7> q_NAN = {{NAN, NAN, NAN, NAN, NAN, NAN, NAN}};
std::array< std::array<double, 7>, 4 > q_all = q_all_NAN;
const double d1 = 0.3330;
const double d3 = 0.3160;
const double d5 = 0.3840;
const double d7e = 0.2104;
const double a4 = 0.0825;
const double a7 = 0.0880;
const double LL24 = 0.10666225; // a4^2 + d3^2
const double LL46 = 0.15426225; // a4^2 + d5^2
const double L24 = 0.326591870689; // sqrt(LL24)
const double L46 = 0.392762332715; // sqrt(LL46)
const double thetaH46 = 1.35916951803; // atan(d5/a4);
const double theta342 = 1.31542071191; // atan(d3/a4);
const double theta46H = 0.211626808766; // acot(d5/a4);
const std::array<double, 7> q_min = {{-2.8973, -1.7628, -2.8973, -3.0718, -2.8973, -0.0175, -2.8973}};
const std::array<double, 7> q_max = {{2.8973, 1.7628, 2.8973, -0.0698, 2.8973, 3.7525, 2.8973}};
if (q7 <= q_min[6] || q7 >= q_max[6])
return q_all_NAN;
else
for (int i = 0; i < 4; i++)
q_all[i][6] = q7;
// compute p_6
Eigen::Vector3d p_7 = p_EE - d7e*z_EE;
Eigen::Vector3d x_EE_6;
x_EE_6 << std::cos(q7 - M_PI_4), -std::sin(q7 - M_PI_4), 0.0;
Eigen::Vector3d x_6 = R_EE*x_EE_6;
x_6 /= x_6.norm(); // visibly increases accuracy
Eigen::Vector3d p_6 = p_7 - a7*x_6;
// compute q4
Eigen::Vector3d p_2;
p_2 << 0.0, 0.0, d1;
Eigen::Vector3d V26 = p_6 - p_2;
double LL26 = V26[0]*V26[0] + V26[1]*V26[1] + V26[2]*V26[2];
double L26 = std::sqrt(LL26);
if (L24 + L46 < L26 || L24 + L26 < L46 || L26 + L46 < L24)
return q_all_NAN;
double theta246 = std::acos((LL24 + LL46 - LL26)/2.0/L24/L46);
double q4 = theta246 + thetaH46 + theta342 - 2.0*M_PI;
if (q4 <= q_min[3] || q4 >= q_max[3])
return q_all_NAN;
else
for (int i = 0; i < 4; i++)
q_all[i][3] = q4;
// compute q6
double theta462 = std::acos((LL26 + LL46 - LL24)/2.0/L26/L46);
double theta26H = theta46H + theta462;
double D26 = -L26*std::cos(theta26H);
Eigen::Vector3d Z_6 = z_EE.cross(x_6);
Eigen::Vector3d Y_6 = Z_6.cross(x_6);
Eigen::Matrix3d R_6;
R_6.col(0) = x_6;
R_6.col(1) = Y_6/Y_6.norm();
R_6.col(2) = Z_6/Z_6.norm();
Eigen::Vector3d V_6_62 = R_6.transpose()*(-V26);
double Phi6 = std::atan2(V_6_62[1], V_6_62[0]);
double Theta6 = std::asin(D26/std::sqrt(V_6_62[0]*V_6_62[0] + V_6_62[1]*V_6_62[1]));
std::array<double, 2> q6;
q6[0] = M_PI - Theta6 - Phi6;
q6[1] = Theta6 - Phi6;
for (int i = 0; i < 2; i++)
{
if (q6[i] <= q_min[5])
q6[i] += 2.0*M_PI;
else if (q6[i] >= q_max[5])
q6[i] -= 2.0*M_PI;
if (q6[i] <= q_min[5] || q6[i] >= q_max[5])
{
q_all[2*i] = q_NAN;
q_all[2*i + 1] = q_NAN;
}
else
{
q_all[2*i][5] = q6[i];
q_all[2*i + 1][5] = q6[i];
}
}
if (std::isnan(q_all[0][5]) && std::isnan(q_all[2][5]))
return q_all_NAN;
// compute q1 & q2
double thetaP26 = 3.0*M_PI_2 - theta462 - theta246 - theta342;
double thetaP = M_PI - thetaP26 - theta26H;
double LP6 = L26*sin(thetaP26)/std::sin(thetaP);
std::array< Eigen::Vector3d, 4 > z_5_all;
std::array< Eigen::Vector3d, 4 > V2P_all;
for (int i = 0; i < 2; i++)
{
Eigen::Vector3d z_6_5;
z_6_5 << std::sin(q6[i]), std::cos(q6[i]), 0.0;
Eigen::Vector3d z_5 = R_6*z_6_5;
Eigen::Vector3d V2P = p_6 - LP6*z_5 - p_2;
z_5_all[2*i] = z_5;
z_5_all[2*i + 1] = z_5;
V2P_all[2*i] = V2P;
V2P_all[2*i + 1] = V2P;
double L2P = V2P.norm();
if (std::fabs(V2P[2]/L2P) > 0.999)
{
q_all[2*i][0] = q_actual_array[0];
q_all[2*i][1] = 0.0;
q_all[2*i + 1][0] = q_actual_array[0];
q_all[2*i + 1][1] = 0.0;
}
else
{
q_all[2*i][0] = std::atan2(V2P[1], V2P[0]);
q_all[2*i][1] = std::acos(V2P[2]/L2P);
if (q_all[2*i][0] < 0)
q_all[2*i + 1][0] = q_all[2*i][0] + M_PI;
else
q_all[2*i + 1][0] = q_all[2*i][0] - M_PI;
q_all[2*i + 1][1] = -q_all[2*i][1];
}
}
for (int i = 0; i < 4; i++)
{
if ( q_all[i][0] <= q_min[0] || q_all[i][0] >= q_max[0]
|| q_all[i][1] <= q_min[1] || q_all[i][1] >= q_max[1] )
{
q_all[i] = q_NAN;
continue;
}
// compute q3
Eigen::Vector3d z_3 = V2P_all[i]/V2P_all[i].norm();
Eigen::Vector3d Y_3 = -V26.cross(V2P_all[i]);
Eigen::Vector3d y_3 = Y_3/Y_3.norm();
Eigen::Vector3d x_3 = y_3.cross(z_3);
Eigen::Matrix3d R_1;
double c1 = std::cos(q_all[i][0]);
double s1 = std::sin(q_all[i][0]);
R_1 << c1, -s1, 0.0,
s1, c1, 0.0,
0.0, 0.0, 1.0;
Eigen::Matrix3d R_1_2;
double c2 = std::cos(q_all[i][1]);
double s2 = std::sin(q_all[i][1]);
R_1_2 << c2, -s2, 0.0,
0.0, 0.0, 1.0,
-s2, -c2, 0.0;
Eigen::Matrix3d R_2 = R_1*R_1_2;
Eigen::Vector3d x_2_3 = R_2.transpose()*x_3;
q_all[i][2] = std::atan2(x_2_3[2], x_2_3[0]);
if (q_all[i][2] <= q_min[2] || q_all[i][2] >= q_max[2])
{
q_all[i] = q_NAN;
continue;
}
// compute q5
Eigen::Vector3d VH4 = p_2 + d3*z_3 + a4*x_3 - p_6 + d5*z_5_all[i];
Eigen::Matrix3d R_5_6;
double c6 = std::cos(q_all[i][5]);
double s6 = std::sin(q_all[i][5]);
R_5_6 << c6, -s6, 0.0,
0.0, 0.0, -1.0,
s6, c6, 0.0;
Eigen::Matrix3d R_5 = R_6*R_5_6.transpose();
Eigen::Vector3d V_5_H4 = R_5.transpose()*VH4;
q_all[i][4] = -std::atan2(V_5_H4[1], V_5_H4[0]);
if (q_all[i][4] <= q_min[4] || q_all[i][4] >= q_max[4])
{
q_all[i] = q_NAN;
continue;
}
}
return q_all;
}
// "Case-Consistent" inverse kinematics w.r.t. End Effector Frame (using Franka Hand data)
std::array<double, 7> franka_IK_EE_CC ( Eigen::Matrix3d R_EE,
Eigen::Vector3d z_EE,
Eigen::Vector3d p_EE,
double q7,
std::array<double, 7> q_actual_array )
{
const std::array<double, 7> q_NAN = {{NAN, NAN, NAN, NAN, NAN, NAN, NAN}};
std::array<double, 7> q;
// Eigen::Map< Eigen::Matrix<double, 4, 4> > O_T_EE(O_T_EE_array.data());
// constants
const double d1 = 0.3330;
const double d3 = 0.3160;
const double d5 = 0.3840;
const double d7e = 0.2104;
const double a4 = 0.0825;
const double a7 = 0.0880;
const double LL24 = 0.10666225; // a4^2 + d3^2
const double LL46 = 0.15426225; // a4^2 + d5^2
const double L24 = 0.326591870689; // sqrt(LL24)
const double L46 = 0.392762332715; // sqrt(LL46)
const double thetaH46 = 1.35916951803; // atan(d5/a4);
const double theta342 = 1.31542071191; // atan(d3/a4);
const double theta46H = 0.211626808766; // acot(d5/a4);
const std::array<double, 7> q_min = {{-2.8973, -1.7628, -2.8973, -3.0718, -2.8973, -0.0175, -2.8973}};
const std::array<double, 7> q_max = {{ 2.8973, 1.7628, 2.8973, -0.0698, 2.8973, 3.7525, 2.8973}};
// return NAN if input q7 is out of range
if (q7 <= q_min[6] || q7 >= q_max[6])
return q_NAN;
else
q[6] = q7;
// FK for getting current case id
double c1_a = std::cos(q_actual_array[0]); double s1_a = std::sin(q_actual_array[0]);
double c2_a = std::cos(q_actual_array[1]); double s2_a = std::sin(q_actual_array[1]);
double c3_a = std::cos(q_actual_array[2]); double s3_a = std::sin(q_actual_array[2]);
double c4_a = std::cos(q_actual_array[3]); double s4_a = std::sin(q_actual_array[3]);
double c5_a = std::cos(q_actual_array[4]); double s5_a = std::sin(q_actual_array[4]);
double c6_a = std::cos(q_actual_array[5]); double s6_a = std::sin(q_actual_array[5]);
std::array< Eigen::Matrix<double, 4, 4>, 7> As_a;
As_a[0] << c1_a, -s1_a, 0.0, 0.0, // O1
s1_a, c1_a, 0.0, 0.0,
0.0, 0.0, 1.0, d1,
0.0, 0.0, 0.0, 1.0;
As_a[1] << c2_a, -s2_a, 0.0, 0.0, // O2
0.0, 0.0, 1.0, 0.0,
-s2_a, -c2_a, 0.0, 0.0,
0.0, 0.0, 0.0, 1.0;
As_a[2] << c3_a, -s3_a, 0.0, 0.0, // O3
0.0, 0.0, -1.0, -d3,
s3_a, c3_a, 0.0, 0.0,
0.0, 0.0, 0.0, 1.0;
As_a[3] << c4_a, -s4_a, 0.0, a4, // O4
0.0, 0.0, -1.0, 0.0,
s4_a, c4_a, 0.0, 0.0,
0.0, 0.0, 0.0, 1.0;
As_a[4] << 1.0, 0.0, 0.0, -a4, // H
0.0, 1.0, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0,
0.0, 0.0, 0.0, 1.0;
As_a[5] << c5_a, -s5_a, 0.0, 0.0, // O5
0.0, 0.0, 1.0, d5,
-s5_a, -c5_a, 0.0, 0.0,
0.0, 0.0, 0.0, 1.0;
As_a[6] << c6_a, -s6_a, 0.0, 0.0, // O6
0.0, 0.0, -1.0, 0.0,
s6_a, c6_a, 0.0, 0.0,
0.0, 0.0, 0.0, 1.0;
std::array< Eigen::Matrix<double, 4, 4>, 7> Ts_a;
Ts_a[0] = As_a[0];
for (unsigned int j = 1; j < 7; j++)
Ts_a[j] = Ts_a[j - 1]*As_a[j];
// identify q6 case
Eigen::Vector3d V62_a = Ts_a[1].block<3, 1>(0, 3) - Ts_a[6].block<3, 1>(0, 3);
Eigen::Vector3d V6H_a = Ts_a[4].block<3, 1>(0, 3) - Ts_a[6].block<3, 1>(0, 3);
Eigen::Vector3d Z6_a = Ts_a[6].block<3, 1>(0, 2);
bool is_case6_0 = ((V6H_a.cross(V62_a)).transpose()*Z6_a <= 0);
// identify q1 case
bool is_case1_1 = (q_actual_array[1] < 0);
// IK: compute p_6
// Eigen::Matrix3d R_EE = O_T_EE.topLeftCorner<3, 3>();
// Eigen::Vector3d z_EE = O_T_EE.block<3, 1>(0, 2);
// Eigen::Vector3d p_EE = O_T_EE.block<3, 1>(0, 3);
Eigen::Vector3d p_7 = p_EE - d7e*z_EE;
Eigen::Vector3d x_EE_6;
x_EE_6 << std::cos(q7 - M_PI_4), -std::sin(q7 - M_PI_4), 0.0;
Eigen::Vector3d x_6 = R_EE*x_EE_6;
x_6 /= x_6.norm(); // visibly increases accuracy
Eigen::Vector3d p_6 = p_7 - a7*x_6;
// IK: compute q4
Eigen::Vector3d p_2;
p_2 << 0.0, 0.0, d1;
Eigen::Vector3d V26 = p_6 - p_2;
double LL26 = V26[0]*V26[0] + V26[1]*V26[1] + V26[2]*V26[2];
double L26 = std::sqrt(LL26);
if (L24 + L46 < L26 || L24 + L26 < L46 || L26 + L46 < L24)
return q_NAN;
double theta246 = std::acos((LL24 + LL46 - LL26)/2.0/L24/L46);
q[3] = theta246 + thetaH46 + theta342 - 2.0*M_PI;
if (q[3] <= q_min[3] || q[3] >= q_max[3])
return q_NAN;
// IK: compute q6
double theta462 = std::acos((LL26 + LL46 - LL24)/2.0/L26/L46);
double theta26H = theta46H + theta462;
double D26 = -L26*std::cos(theta26H);
Eigen::Vector3d Z_6 = z_EE.cross(x_6);
Eigen::Vector3d Y_6 = Z_6.cross(x_6);
Eigen::Matrix3d R_6;
R_6.col(0) = x_6;
R_6.col(1) = Y_6/Y_6.norm();
R_6.col(2) = Z_6/Z_6.norm();
Eigen::Vector3d V_6_62 = R_6.transpose()*(-V26);
double Phi6 = std::atan2(V_6_62[1], V_6_62[0]);
double Theta6 = std::asin(D26/std::sqrt(V_6_62[0]*V_6_62[0] + V_6_62[1]*V_6_62[1]));
if (is_case6_0)
q[5] = M_PI - Theta6 - Phi6;
else
q[5] = Theta6 - Phi6;
if (q[5] <= q_min[5])
q[5] += 2.0*M_PI;
else if (q[5] >= q_max[5])
q[5] -= 2.0*M_PI;
if (q[5] <= q_min[5] || q[5] >= q_max[5])
return q_NAN;
// IK: compute q1 & q2
double thetaP26 = 3.0*M_PI_2 - theta462 - theta246 - theta342;
double thetaP = M_PI - thetaP26 - theta26H;
double LP6 = L26*sin(thetaP26)/std::sin(thetaP);
Eigen::Vector3d z_6_5;
z_6_5 << std::sin(q[5]), std::cos(q[5]), 0.0;
Eigen::Vector3d z_5 = R_6*z_6_5;
Eigen::Vector3d V2P = p_6 - LP6*z_5 - p_2;
double L2P = V2P.norm();
if (std::fabs(V2P[2]/L2P) > 0.999)
{
q[0] = q_actual_array[0];
q[1] = 0.0;
}
else
{
q[0] = std::atan2(V2P[1], V2P[0]);
q[1] = std::acos(V2P[2]/L2P);
if (is_case1_1)
{
if (q[0] < 0.0)
q[0] += M_PI;
else
q[0] -= M_PI;
q[1] = -q[1];
}
}
if ( q[0] <= q_min[0] || q[0] >= q_max[0]
|| q[1] <= q_min[1] || q[1] >= q_max[1] )
return q_NAN;
// IK: compute q3
Eigen::Vector3d z_3 = V2P/V2P.norm();
Eigen::Vector3d Y_3 = -V26.cross(V2P);
Eigen::Vector3d y_3 = Y_3/Y_3.norm();
Eigen::Vector3d x_3 = y_3.cross(z_3);
Eigen::Matrix3d R_1;
double c1 = std::cos(q[0]);
double s1 = std::sin(q[0]);
R_1 << c1, -s1, 0.0,
s1, c1, 0.0,
0.0, 0.0, 1.0;
Eigen::Matrix3d R_1_2;
double c2 = std::cos(q[1]);
double s2 = std::sin(q[1]);
R_1_2 << c2, -s2, 0.0,
0.0, 0.0, 1.0,
-s2, -c2, 0.0;
Eigen::Matrix3d R_2 = R_1*R_1_2;
Eigen::Vector3d x_2_3 = R_2.transpose()*x_3;
q[2] = std::atan2(x_2_3[2], x_2_3[0]);
if (q[2] <= q_min[2] || q[2] >= q_max[2])
return q_NAN;
// IK: compute q5
Eigen::Vector3d VH4 = p_2 + d3*z_3 + a4*x_3 - p_6 + d5*z_5;
Eigen::Matrix3d R_5_6;
double c6 = std::cos(q[5]);
double s6 = std::sin(q[5]);
R_5_6 << c6, -s6, 0.0,
0.0, 0.0, -1.0,
s6, c6, 0.0;
Eigen::Matrix3d R_5 = R_6*R_5_6.transpose();
Eigen::Vector3d V_5_H4 = R_5.transpose()*VH4;
q[4] = -std::atan2(V_5_H4[1], V_5_H4[0]);
if (q[4] <= q_min[4] || q[4] >= q_max[4])
return q_NAN;
return q;
}
// pybind11 wrapper for franka_IK
py::array_t<double> franka_IK(Eigen::Ref<Eigen::MatrixXd> targetHandPosition,
py::array_t<double>& tHO,
double q7,
py::array_t<double>& cP)
{
py::buffer_info buf2 = tHO.request();
py::buffer_info buf3 = cP.request();
auto result = py::array_t<double>(28);
result.resize({4, 7});
py::buffer_info buf4 = result.request();
double* targetHandOrientation = (double*)buf2.ptr;
double* currentPosition = (double*)buf3.ptr;
double* ptr4 = (double*)buf4.ptr;
Eigen::Quaterniond q(targetHandOrientation);
Eigen::Matrix3d targetRotation = q.normalized().toRotationMatrix();
Eigen::Vector3d targetPosition = targetHandPosition;
Eigen::Vector3d offset(0.0, 0.0, 0.1034);
targetPosition = targetRotation * offset + targetPosition;
std::array<double, 7> q_actual_array;
for (int i = 0; i < 7; i++) q_actual_array[i] = currentPosition[i];
std::array< std::array<double, 7>, 4 > ans = franka_IK_EE(targetRotation, targetRotation.block<3, 1>(0, 2), targetPosition, q7, q_actual_array);
for (int i = 0; i < 4; i++)
{
for(int j = 0; j < 7; j++)
{
ptr4[i*7+j] = ans[i][j];
}
}
return result;
}
// pybind11 wrapper for franka_IK_EE_CC
py::array_t<double> franka_IKCC(Eigen::Ref<Eigen::MatrixXd> targetHandPosition,
py::array_t<double>& tHO,
double q7,
py::array_t<double>& cP)
{
py::buffer_info buf2 = tHO.request();
py::buffer_info buf3 = cP.request();
auto result = py::array_t<double>(7);
py::buffer_info buf4 = result.request();
double* targetHandOrientation = (double*)buf2.ptr;
double* currentPosition = (double*)buf3.ptr;
double* ptr4 = (double*)buf4.ptr;
Eigen::Quaterniond q(targetHandOrientation);
Eigen::Matrix3d targetRotation = q.normalized().toRotationMatrix();
Eigen::Vector3d targetPosition = targetHandPosition;
Eigen::Vector3d offset(0.0, 0.0, 0.1034);
targetPosition = targetRotation * offset + targetPosition;
std::array<double, 7> q_actual_array;
for (int i = 0; i < 7; i++) q_actual_array[i] = currentPosition[i];
std::array<double, 7> ans = franka_IK_EE_CC(targetRotation, targetRotation.block<3, 1>(0, 2), targetPosition, q7, q_actual_array);
for (int i = 0; i < 7; i++)
{
ptr4[i] = ans[i];
}
return result;
}
PYBIND11_MODULE(franka_ik_pybind, m)
{
m.doc() = "Analytical IK";
m.def("franka_IK", &franka_IK);
m.def("franka_IKCC", &franka_IKCC);
}
#endif // FRANKA_IK_HE_HPP