Added plots to delta robot and jacobian math
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Sharwin24
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@ -15,6 +15,9 @@ repo: https://github.com/Sharwin24/DeltaRobot
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---
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An open source ROS package for controlling delta robots with forward and inverse kinematics, trajectory generation, and visualization. Designed for public use and easy integration with new delta robot designs and applications.
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<div align="center">
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<img src="workspace.png" alt="Delta Robot Workspace" style="border-radius: 15px; height: 200px; margin-left: 5px; display: inline-block;">
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</div>
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## Robot Kinematics
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The robot's forward and inverse kinematics were first implemented in a [jupyter notebook](https://github.com/Sharwin24/DeltaRobot/blob/main/delta_kinematics.ipynb) to visualize the robot's configuration space and workspace.
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@ -140,6 +143,155 @@ The forward and inverse kinematics were then implemented in C++ following the ap
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</details>
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</div>
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### The Jacobian
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The Modern Robotics [2] textbook details the process of deriving the Jacobian for a delta robot. The Jacobian is useful for converting end effector velocities to joint velocities.
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<div style="overflow-x: auto; width: 100%;">
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\[
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\dot{\theta} = J_{\Theta} \cdot \underbrace{\dot{p}}_{EE Velocity}
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\]
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</div>
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<div>
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<details>
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<summary>Jacobian ROS C++ Implementation</summary>
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```cpp
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std::pair<std::vector<double>, std::vector<double>> DeltaKinematics::calcAuxAngles(double theta1, double theta2, double theta3) {
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// First determine the end effector position using FK
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Point position = this->deltaFK(theta1, theta2, theta3);
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const double UP = (sqrt3 / 3) * this->SP;
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const Eigen::Vector3d P = {position.x, position.y, position.z};
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const Eigen::Vector3d D = {UP - this->AL, 0, 0};
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Eigen::Matrix3d C;
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for (int i = 0; i < 3; ++i) {
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double phi_i = this->phi[i];
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Eigen::Matrix3d R;
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R << std::cos(phi_i), std::sin(phi_i), 0,
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-std::sin(phi_i), std::cos(phi_i), 0,
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0, 0, 1;
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Eigen::Vector3d c_i = R * P + D;
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// Set the i-th column of C to c_i.
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C.col(i) = c_i;
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}
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double C_x2 = C(0, 1);
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double C_y2 = C(1, 1);
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double C_z2 = C(2, 1);
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double C_x3 = C(0, 2);
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double C_y3 = C(1, 2);
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double C_z3 = C(2, 2);
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// C_squared = c_xi^2 + c_yi^2 + c_zi^2
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double C_sqrd_2 = C_x2 * C_x2 + C_y2 * C_y2 + C_z2 * C_z2;
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double C_sqrd_3 = C_x3 * C_x3 + C_y3 * C_y3 + C_z3 * C_z3;
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// theta_3i = arccos(C_yi / PL)
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double t32 = acos(C_y2 / this->PL);
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double t33 = acos(C_y3 / this->PL);
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// k_numerator = c_xi ^ 2 + c_yi ^ 2 + c_zi ^ 2 - L ^ 2 - ELL ^ 2
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// k_denominator = 2 * L * ELL * sin(theta_3i)
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// theta_2i = arccos(k_numerator / k_denominator)
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double t22_numerator = C_sqrd_2 - this->AL * this->AL - this->PL * this->PL;
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double t22_denominator = 2 * this->AL * this->PL * sin(t32);
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double t23_numerator = C_sqrd_3 - this->AL * this->AL - this->PL * this->PL;
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double t23_denominator = 2 * this->AL * this->PL * sin(t33);
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double t22 = acos(t22_numerator / t22_denominator);
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double t23 = acos(t23_numerator / t23_denominator);
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// theta_1i is the actuated angles which were passed into the function
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// We only need to return the auxiliary angles
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return std::make_pair(std::vector<double>{theta2, t22, t23}, std::vector<double>{theta3, t32, t33});
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}
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Eigen::Matrix3d DeltaKinematics::calcJacobian(double theta1, double theta2, double theta3) {
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// The Jacobian matrix has 2 components: JTheta and Jp
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// Since this Jacobian will be used to compute the joint velocities, we need the inverse of JTheta
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// Jp * p_dot = JTheta * theta_dot -> theta_dot = JTheta_inv * Jp * p_dot
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// Obtain auxiliary angles
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auto aux_angles = this->calcAuxAngles(theta1, theta2, theta3);
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const std::vector<double> t1 = {theta1, theta2, theta3};
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const std::vector<double> t2 = aux_angles.first;
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const std::vector<double> t3 = aux_angles.second;
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double t22 = t2[1];
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double t23 = t2[2];
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double t32 = t3[1];
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double t33 = t3[2];
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// Jp Calculation
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auto J_ix = [this, &t1, &t2, &t3](int i) -> double {
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return sin(t3[i]) * cos(t2[i] + t1[i]) * cos(this->phi[i]) + cos(t3[i]) * sin(this->phi[i]);
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};
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auto J_iy = [this, &t1, &t2, &t3](int i) -> double {
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return -sin(t3[i]) * cos(t2[i] + t1[i]) * sin(this->phi[i]) + cos(t3[i]) * cos(this->phi[i]);
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};
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auto J_iz = [this, &t1, &t2, &t3](int i) -> double {
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return sin(t3[i]) * sin(t2[i] + t1[i]);
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};
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Eigen::Matrix3d Jp;
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for (int i = 0; i < 3; ++i) {
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Jp(i, 0) = J_ix(i);
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Jp(i, 1) = J_iy(i);
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Jp(i, 2) = J_iz(i);
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}
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// JTheta Calculation
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Eigen::Matrix3d JTheta;
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// Populate the diagonals with AL*sin(t2[i])*sin(t3[i])
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JTheta(0, 0) = this->AL * sin(theta2) * sin(theta3);
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JTheta(1, 1) = this->AL * sin(t22) * sin(t23);
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JTheta(2, 2) = this->AL * sin(t32) * sin(t33);
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// Invert JTheta
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Eigen::Matrix3d JTheta_inv = JTheta.inverse();
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return JTheta_inv * Jp;
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}
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DeltaJointVels DeltaKinematics::calcThetaDot(double theta1, double theta2, double theta3, double x_dot, double y_dot, double z_dot) {
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Eigen::Matrix3d J = this->calcJacobian(theta1, theta2, theta3);
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Eigen::Vector3d p_dot(x_dot, y_dot, z_dot);
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Eigen::Vector3d theta_dot = J * p_dot;
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DeltaJointVels joint_vels;
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joint_vels.theta1_vel = theta_dot(0);
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joint_vels.theta2_vel = theta_dot(1);
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joint_vels.theta3_vel = theta_dot(2);
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return joint_vels;
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}
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std::vector<EEVelocity> DeltaKinematics::computeGradient(const std::vector<Point>& position_data, double dt) {
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size_t n = position_data.size();
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std::vector<EEVelocity> velocities(n, {0.0, 0.0, 0.0});
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if (n == 0 || n == 1) return velocities;
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// Use forward difference for the first point
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velocities[0].x_vel = (position_data[1].x - position_data[0].x) / dt;
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velocities[0].y_vel = (position_data[1].y - position_data[0].y) / dt;
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velocities[0].z_vel = (position_data[1].z - position_data[0].z) / dt;
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// Use central difference for interior points
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for (size_t i = 1; i < n - 1; ++i) {
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velocities[i].x_vel = (position_data[i + 1].x - position_data[i - 1].x) / (2 * dt);
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velocities[i].y_vel = (position_data[i + 1].y - position_data[i - 1].y) / (2 * dt);
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velocities[i].z_vel = (position_data[i + 1].z - position_data[i - 1].z) / (2 * dt);
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}
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// Use backward difference for the last point
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velocities[n - 1].x_vel = (position_data[n - 1].x - position_data[n - 2].x) / dt;
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velocities[n - 1].y_vel = (position_data[n - 1].y - position_data[n - 2].y) / dt;
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velocities[n - 1].z_vel = (position_data[n - 1].z - position_data[n - 2].z) / dt;
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return velocities;
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}
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```
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</details>
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</div>
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Using the Jacobian we can convert end-effector position trajectories into Joint velocity trajectories.
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<div align="center" style="overflow-x: auto; width: 100%;">
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<img src="circle_pos_vel.png" alt="Circle Trajectory " style="border-radius: 15px; height: 200px; margin-left: 5px; display: inline-block;">
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</div>
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## End-Effector Sensors
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| Sensor | Image | Description |
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@ -149,4 +301,5 @@ The forward and inverse kinematics were then implemented in C++ following the ap
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## References
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- [Delta Robot Kinematics](https://hypertriangle.com/~alex/delta-robot-tutorial/)
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1. [Delta Robot Kinematics](https://hypertriangle.com/~alex/delta-robot-tutorial/)
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2. [Modern Robotics](http://hades.mech.northwestern.edu/index.php/Modern_Robotics)
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