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Solve Inverse Kinematics for a Four-Bar Linkage

This example shows how to solve inverse kinematics for a four-bar closed-chain linkage. Robotics System Toolbox™ does not directly support closed-loop mechanisms. However, the loop-closing joints can be approximated using kinematic constraints. This example shows how to setup a rigid body tree for a four-bar linkage, specify the kinematic constraints, and solve for a desired end-effector position.

Initialize the four-bar linkage rigid body tree model.

robot = rigidBodyTree('Dataformat','column','MaxNumBodies',7);

Define body names, parent names, joint names, joint types, and fixed transforms in cell arrays. The fixed transforms define the geometry of the four-bar linkage. The linkage rotates in the xz-plane. An offset of -0.1 is used in the y-axis on the 'b4' body to isolate the motion of the overlapping joints for 'b3' and 'b4'.

bodyNames = {'b1','b2','b3','b4','b5','b6'};
parentNames = {'base','b1','b2','base','b4','b5'};
jointNames = {'j1','j2','j3','j4','j5','j6'};
jointTypes = {'revolute','revolute','fixed','revolute','revolute','fixed'};
fixedTforms = {eye(4), ...
                trvec2tform([0 0 0.5]), ...
                trvec2tform([0.8 0 0]), ...
                trvec2tform([0.0 -0.1 0]), ...
                trvec2tform([0.8 0 0]), ...
                trvec2tform([0 0 0.5])};

Use a for loop to assemble the four-bar linkage:

  • Create a rigid body and specify the joint type.

  • Specify the JointAxis property for any non-fixed joints.

  • Specify the fixed transformation.

  • Add the body to the rigid body tree.

for k = 1:6

    b = rigidBody(bodyNames{k});
    b.Joint = rigidBodyJoint(jointNames{k},jointTypes{k});
    if ~strcmp(jointTypes{k},'fixed')
        b.Joint.JointAxis = [0 1 0];

Add a final body to function as the end-effector (handle) for the four-bar linkage.

bn = 'handle';
b = rigidBody(bn);
setFixedTransform(b.Joint,trvec2tform([0 -0.15 0]));

Specify kinematic constraints for the GeneralizedInverseKinematics object:

  • Position constraint 1 : The origins of 'b3' body frame and 'b6' body frame should always overlap. This keeps the handle in line with the approximated closed-loop mechanism. Use the -0.1 offset for the y-coordinate.

  • Position constraint 2 : End-effector should target the desired position.

  • Joint limit bounds : Satisfy the joint limits in the rigid body tree model.

gik = generalizedInverseKinematics('RigidBodyTree',robot);
gik.ConstraintInputs = {'position',...  % Position constraint for closed-loop mechanism
                        'position',...  % Position constraint for end-effector 
                        'joint'};       % Joint limits
gik.SolverParameters.AllowRandomRestart = false;

% Position constraint 1
positionTarget1 = constraintPositionTarget('b6','ReferenceBody','b3');
positionTarget1.TargetPosition = [0 -0.1 0];
positionTarget1.Weights = 50;
positionTarget1.PositionTolerance = 1e-6;

% Joint limit bounds
jointLimBounds = constraintJointBounds(gik.RigidBodyTree);
jointLimBounds.Weights = ones(1,size(gik.RigidBodyTree.homeConfiguration,1))*10;

% Position constraint 2
desiredEEPosition = [0.9 -0.1 0.9]'; % Position is relative to base.
positionTarget2 = constraintPositionTarget('handle');
positionTarget2.TargetPosition = desiredEEPosition; 
positionTarget2.PositionTolerance = 1e-6;
positionTarget2.Weights = 1;

Compute the kinematic solution using the gik object. Specify the initial guess and the different kinematic constraints in the proper order.

iniGuess = homeConfiguration(robot);
[q, solutionInfo] = gik(iniGuess,positionTarget1,positionTarget2,jointLimBounds);

Examine the results in solutionInfo. Show the kinematic solution compared to the home configuration. Plots are shown in the xz-plane.

loopClosingViolation = solutionInfo.ConstraintViolations(1).Violation;
jointBndViolation = solutionInfo.ConstraintViolations(2).Violation;
eePositionViolation = solutionInfo.ConstraintViolations(3).Violation;

title('Home Configuration')
view([0 -1 0]);
title('GIK Solution')
view([0 -1 0]);

Figure contains 2 axes objects. Axes object 1 with title Home Configuration contains 15 objects of type patch, line. These objects represent base, b1, b2, b3, b4, b5, b6, handle. Axes object 2 with title GIK Solution contains 15 objects of type patch, line. These objects represent base, b1, b2, b3, b4, b5, b6, handle.

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