[Robotic Arm]

Project mission

Our mission is to revolutionize precision and efficiency in automation through our advanced robotic arm technology. By integrating cutting-edge trajectory planning and manipulation capabilities, we aim to empower industries with the ability to perform complex tasks with unprecedented accuracy and reliability. Our robotic arm is designed to be versatile, adapting seamlessly to various applications ranging from manufacturing and assembly to research and healthcare. Through innovative engineering and a commitment to quality, we strive to push the boundaries of what is possible in robotic automation, enhancing productivity and fostering technological advancement.

A robotic arm's trajectory and manipulation involve the planned path and movements it executes to reach a target position and perform specific tasks, like picking up objects or assembling parts. This process involves several key elements:

Trajectory Planning: Determining the optimal path the robotic arm should take from its starting point to the end position. This involves calculating joint angles, velocities, and accelerations that the arm will follow over time.
Manipulation: Refers to the actual interaction of the robotic arm with objects in its environment, which could include grasping, lifting, moving, or assembling objects. Manipulation is highly dependent on the arm's end effector (like a gripper or tool) and is controlled by precise movements and force adjustments.
Control Systems: These are essential for both trajectory planning and manipulation, ensuring the arm follows the planned path accurately and interacts with objects effectively. Control systems use feedback from sensors to adjust movements in real time.
Kinematics and Dynamics: These are the mathematical foundations that describe the motion of the robotic arm without regard to the forces that cause the motion (kinematics) and the forces involved (dynamics), respectively. These principles help in designing the trajectory and controlling the arm.

Overall, robotic arm trajectory and manipulation involve complex, coordinated movements designed to perform tasks efficiently and accurately in automated and industrial applications.

Effort 1

Import Libraries

The script starts by importing necessary libraries:

numpy for numerical operations, especially handling arrays and mathematical functions.
matplotlib.pyplot for plotting graphs.
Axes3D from mpl_toolkits.mplot3d to enable 3D plotting capabilities.

Effort 2

Arm Specifications

link1_length and link2_length set to 1 unit each, representing the lengths of the first and second segments of the robotic arm.
joint_angles defined as [np.pi/4, np.pi/4], setting the initial angles for the two joints in radians.

Forward Kinematics Function

forward_kinematics(joint_angles) computes the (x, y) coordinates of the tip of the robotic arm using the joint angles.
The positions are calculated based on trigonometric functions, which account for the cumulative effect of each joint on the overall orientation of the arm.

Plotting the Arm

plot_arm(joint_angles) visualizes the arm on a 3D plot:
- It computes the end position of the arm using the forward_kinematics function.
- It plots the segments of the arm from the origin to the calculated end point, marking joints and end effector (tip of the arm).
- The plot settings ensure that all parts of the arm are visible within the axes limits.

Animation of Arm Movement

interpolate_joint_angles(start_angles, end_angles, steps=10) animates the transition of the arm's joint angles from a starting position to an end position.
It generates a sequence of linearly spaced angles between the start and end angles using numpy.linspace.
For each set of joint angles in this trajectory, the arm is plotted and briefly paused to visualize the motion.

Example Usage

new_joint_angles = [np.pi/2, np.pi/3] sets a new target position for the joints.
interpolate_joint_angles(joint_angles, new_joint_angles) animates the arm from the initial position to this new position.

Report Conclusion

The script effectively demonstrates the use of mathematical concepts (like trigonometry and linear interpolation) and programming (with Python and Matplotlib) to simulate and visualize the motion of a robotic arm. It's a straightforward example of how robots can be controlled and visualized in software, which is crucial for applications in robotics, automation, and educational purposes. The use of 3D plotting provides a clear and intuitive way to observe the mechanical movements and transitions between different joint configurations.

Simulation and Visualization of Kinova® Gen3 Robot Trajectories

Objective

The objective of this MATLAB script is to simulate and visualize the control trajectories for a Kinova® Gen3 robot in both task-space and joint-space. This involves setting initial conditions, defining motion models, calculating trajectories, and graphically displaying these trajectories.

Methods

Robot Model and Configuration
- The Kinova® Gen3 robot model is loaded with specific settings including gravity directed in the negative z-direction to simulate a typical Earth environment.
- The robot is initialized in its home configuration.
End-Effector and Trajectory Parameters
- The script identifies the end-effector and specifies parameters such as the time step and tool speed, which influence the resolution and pace of the simulation.
Pose Initialization
- Initial and final poses of the end-effector are set up using transformation matrices, defining the start and target positions and orientations.
Task-Space Trajectory Generation
- A linear trajectory is computed between the initial and final poses, interpolated at specified times to create waypoints. This defines the path that the end-effector should follow in task-space.
Simulation of Task-Space Control
- A task-space motion model is used to implement a PD (Proportional-Derivative) controller that drives the robot's end-effector along the planned trajectory. The ODE solver ode15s is used due to the stiff nature of the robotic system dynamics.
Joint-Space Trajectory Calculation
- The script employs inverse kinematics to determine the necessary joint configurations to achieve the desired end-effector poses. It then interpolates these configurations over time to form a smooth trajectory in joint-space.
Simulation of Joint-Space Control
- Similar to task-space control, a joint-space motion model with PD control is simulated using the ode15s solver to follow the trajectory defined by the joint configurations.
Visualization
- The robot's motion is visualized by updating its configuration at each simulation timestep and plotting the end-effector's position. This is done separately for task-space and joint-space to visually compare their differences.

Results

The script successfully generates and simulates two types of trajectories for the Kinova® Gen3 robot:

Task-Space Trajectory: Directly controls the end-effector's position and orientation.
Joint-Space Trajectory: Controls the positions of the robot's joints to achieve the desired end-effector position.

Both trajectories are visualized in a MATLAB figure, showing the paths that the end-effector takes in each control scheme. The robot's behavior under task-space and joint-space control can be compared visually to assess differences in path efficiency and smoothness.

Conclusion

This MATLAB script effectively demonstrates the capabilities of the Kinova® Gen3 robot in following both task-space and joint-space trajectories. It provides a comprehensive example of how different control strategies can be implemented and visualized in MATLAB using the Robotics System Toolbox™. Future enhancements could include integrating sensor feedback for adaptive control, optimizing trajectory planning algorithms, and improving computational efficiency.

Recommendations

Optimize Control Parameters: Tuning the PD controllers to minimize overshoot and settle time.
Incorporate Real-time Feedback: Utilizing sensors to adjust trajectories dynamically based on environmental interaction.
Error and Exception Handling: Enhance robustness by managing potential errors in joint limits and trajectory execution.

Matlab Code

% Load the Kinova® Gen3 robot model

robot = loadrobot('kinovaGen3','DataFormat','row','Gravity',[0 0 -9.81]);

% Set initial robot configuration to the home position

currentRobotJConfig = homeConfiguration(robot);

% Get number of joints and the end-effector frame

numJoints = numel(currentRobotJConfig);

endEffector = "EndEffector_Link";

% Define the trajectory time step and desired tool speed

timeStep = 0.1; % seconds

toolSpeed = 0.1; % m/s

% Set the initial and final end-effector poses

jointInit = currentRobotJConfig;

taskInit = getTransform(robot, jointInit, endEffector);

taskFinal = trvec2tform([0.4, 0, 0.6]) * axang2tform([0 1 0 pi]);

% Compute task-space trajectory waypoints

distance = norm(tform2trvec(taskInit) - tform2trvec(taskFinal));

initTime = 0;

finalTime = distance / toolSpeed;

trajTimes = initTime:timeStep:finalTime;

timeInterval = [trajTimes(1); trajTimes(end)];

% Interpolate between taskInit and taskFinal

[taskWaypoints, taskVelocities] = transformtraj(taskInit, taskFinal, timeInterval, trajTimes);

% Create a task space motion model

tsMotionModel = taskSpaceMotionModel('RigidBodyTree', robot, 'EndEffectorName', endEffector);

tsMotionModel.Kp(1:3, 1:3) = 0;

tsMotionModel.Kd(1:3, 1:3) = 0;

% Define initial states for task-space control

q0 = currentRobotJConfig;

qd0 = zeros(size(q0));

% Simulate the robot motion in task space

[tTask, stateTask] = ode15s(@(t, state) exampleHelperTimeBasedTaskInputs(tsMotionModel, timeInterval, taskInit, taskFinal, t, state), timeInterval, [q0; qd0]);

% Create an inverse kinematics solver

ik = inverseKinematics('RigidBodyTree', robot);

ik.SolverParameters.AllowRandomRestart = false;

weights = [1 1 1 1 1 1];

% Calculate the initial and final joint configurations using IK

initialGuess = jointInit;

jointFinal = ik(endEffector, taskFinal, weights, initialGuess);

wrappedJointFinal = wrapToPi(jointFinal);

% Interpolate between the initial and final joint configurations

ctrlpoints = [jointInit', wrappedJointFinal'];

jointConfigArray = cubicpolytraj(ctrlpoints, timeInterval, trajTimes);

% Create a joint space motion model

jsMotionModel = jointSpaceMotionModel('RigidBodyTree', robot, 'MotionType', 'PDControl');

% Simulate the robot motion in joint space

[tJoint, stateJoint] = ode15s(@(t, state) exampleHelperTimeBasedJointInputs(jsMotionModel, timeInterval, jointConfigArray, t, state), timeInterval, [q0; qd0]);

% Visualize the task-space and joint-space trajectories

show(robot, currentRobotJConfig, 'PreservePlot', false, 'Frames', 'off');

hold on

axis([-1 1 -1 1 -0.1 1.5]);

for i = 1:length(trajTimes)

% Task-space visualization

configNow = interp1(tTask, stateTask(:, 1:numJoints), trajTimes(i));

poseNow = getTransform(robot, configNow, endEffector);

show(robot, configNow, 'PreservePlot', false, 'Frames', 'off');

taskSpaceMarker = plot3(poseNow(1,4), poseNow(2,4), poseNow(3,4), 'b.', 'MarkerSize', 20);

% Joint-space visualization

configNow = interp1(tJoint, stateJoint(:, 1:numJoints), trajTimes(i));

poseNow = getTransform(robot, configNow, endEffector);

show(robot, configNow, 'PreservePlot', false, 'Frames', 'off');

jointSpaceMarker = plot3(poseNow(1,4), poseNow(2,4), poseNow(3,4), 'r.', 'MarkerSize', 20);

drawnow;

end

% Add a legend and title

legend([taskSpaceMarker, jointSpaceMarker], {'Defined in Task-Space', 'Defined in Joint-Space'});

title('Manipulator Trajectories');

% Load robot model from URDF filelbr = importrobot('iiwa14.urdf'); % 14 kg payload versionlbr.DataFormat = 'row';gripper = 'iiwa_link_ee_kuka';
% Define the cup dimensions and positioncupHeight = 0.2;cupRadius = 0.05;cupPosition = [-0.5, 0.5, cupHeight/2];
% Add a fixed body to represent the cupbody = rigidBody('cupFrame');setFixedTransform(body.Joint, trvec2tform(cupPosition));addBody(lbr, body, lbr.BaseName);
% Define the number of waypoints and the home configurationnumWaypoints = 5;q0 = homeConfiguration(lbr);qWaypoints = repmat(q0, numWaypoints, 1);
% Create the inverse kinematics solver with necessary constraintsgik = generalizedInverseKinematics('RigidBodyTree', lbr, ...    'ConstraintInputs', {'cartesian','position','aiming','orientation','joint'});
% Create constraintsheightAboveTable = constraintCartesianBounds(gripper);heightAboveTable.Bounds = [-inf, inf; ...                           -inf, inf; ...                           0.05, inf];
distanceFromCup = constraintPositionTarget('cupFrame');distanceFromCup.ReferenceBody = gripper;distanceFromCup.PositionTolerance = 0.005;
alignWithCup = constraintAiming('iiwa_link_ee');alignWithCup.TargetPoint = [0, 0, 100];
limitJointChange = constraintJointBounds(lbr);
fixOrientation = constraintOrientationTarget(gripper);fixOrientation.OrientationTolerance = deg2rad(1);
% Intermediate position before the final approachintermediateDistance = 0.3;limitJointChange.Weights = zeros(size(limitJointChange.Weights));fixOrientation.Weights = 0;distanceFromCup.TargetPosition = [0,0,intermediateDistance];
% Solve for second waypoint[qWaypoints(2,:),solutionInfo] = gik(q0, heightAboveTable, ...                                     distanceFromCup, alignWithCup, ...                                     fixOrientation, limitJointChange);
% Prepare for remaining waypointslimitJointChange.Weights = ones(size(limitJointChange.Weights));fixOrientation.Weights = 1;alignWithCup.Weights = 0;
fixOrientation.TargetOrientation = tform2quat(getTransform(lbr,qWaypoints(2,:),gripper));finalDistanceFromCup = 0.05;distanceFromCupValues = linspace(intermediateDistance, finalDistanceFromCup, numWaypoints-1);maxJointChange = deg2rad(10);
% Solve for remaining waypointsfor k = 3:numWaypoints    distanceFromCup.TargetPosition(3) = distanceFromCupValues(k-1);    limitJointChange.Bounds = [qWaypoints(k-1,:)' - maxJointChange, ...                               qWaypoints(k-1,:)' + maxJointChange];    [qWaypoints(k,:),solutionInfo] = gik(qWaypoints(k-1,:), ...                                         heightAboveTable, ...                                         distanceFromCup, alignWithCup, ...                                         fixOrientation, limitJointChange);end
% Visualize the trajectoryframerate = 15;r = rateControl(framerate);tFinal = 10;tWaypoints = [0,linspace(tFinal/2,tFinal,numWaypoints-1)];numFrames = tFinal * framerate;qInterp = pchip(tWaypoints,qWaypoints',linspace(0,tFinal,numFrames))';
% Compute the gripper positiongripperPosition = zeros(numFrames, 3);for k = 1:numFrames    gripperPosition(k,:) = tform2trvec(getTransform(lbr,qInterp(k,:), gripper));end
% Show the initial configuration with the table and cupfigure;show(lbr, qWaypoints(1,:), 'PreservePlot', false);hold on;exampleHelperPlotCupAndTable(cupHeight, cupRadius, cupPosition);plot3(gripperPosition(:,1), gripperPosition(:,2), gripperPosition(:,3), 'k.');

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