Compute Joint Torques To Balance An Endpoint Force and Moment

Initialize Robot

The twoJointRigidBodyTree robot is a 2-D planar robot. Joint configurations are output as column vectors.

twoJointRobot = twoJointRigidBodyTree("column");

Problem Setup

The endpoint force eeForce is a column vector with a combination of the linear force and moment acting on the end-effector body ("tool"). Note that this vector is expressed in the base coordinate frame and is shown below.

fx = 2; 
fy = 2;
fz = 0;
nx = 0;
ny = 0;
nz = 3;
eeForce = [nx;ny;nz;fx;fy;fz];
eeName = "tool";

Specify the joint configuration of the robot for the balancing torques.

q = [pi/3;pi/4];
Tee = getTransform(twoJointRobot,q,eeName);

Geometric Jacobian Method

Using the principle of virtual work [1], find the balancing torque using the geometricJacobian object function and multiplying the transpose of the Jacobian by the endpoint force vector.

J = geometricJacobian(twoJointRobot,q,eeName);
jointTorques = J' * eeForce;
fprintf("Joint torques using geometric Jacobian (Nm): [%.3g, %.3g]",jointTorques);

Joint torques using geometric Jacobian (Nm): [1.41, 1.78]

Inverse Dynamics for Spatially-Transformed Force

Using another method, calculate the balancing torque by computing the inverse dynamics with the endpoint force spatially transformed to the base frame.

Spatially transforming a wrench from the end-effector frame to the base frame means to exert a new wrench in a frame that happens to collocate with the base frame in space, but is still fixed to the end-effector body; this new wrench has the same effect as the original wrench exerted at the ee origin. In the figure below, ${\mathbf{f}}_{\mathbf{ext}}$ and ${\mathbf{n}}_{\mathbf{ext}}$ are the endpoint linear force and moment respectively, and the ${\mathbf{f}}_{\mathbf{ee}}^{\mathbf{base}}$and ${\mathbf{n}}_{\mathbf{ee}}^{\mathbf{base}}$are the spatially transformed forces and moments, respectively. In the snippet below, fbase_ee is the spatially transformed wrench.

r = tform2trvec(Tee);
fbase_ee = [cross(r,[fx fy fz])' + [nx;ny;nz]; fx;fy;fz];
fext = -externalForce(twoJointRobot, eeName, fbase_ee);
jointTorques2 = inverseDynamics(twoJointRobot, q, [], [], fext);
fprintf("Joint torques using inverse dynamics (Nm): [%.3g, %.3g]",jointTorques2)

Joint torques using inverse dynamics (Nm): [1.41, 1.78]

Inverse Dynamics for End-Effector Force

Instead of spatially transforming the endpoint force to the base frame, use a third method by expressing the end-effector force in its own coordinate frame (fee_ee). Transform the moment and the linear force vectors into the end-effector coordinate frame. Then, specify that force and the current configuration to the externalForce function. Calculate the inverse dynamics from this force vector.

eeLinearForce = Tee \ [fx;fy;fz;0];
eeMoment = Tee \ [nx;ny;nz;0];
fee_ee = [eeMoment(1:3); eeLinearForce(1:3)];
fext = -externalForce(twoJointRobot,eeName,fee_ee,q);
jointTorques3 = inverseDynamics(twoJointRobot, q, [], [], fext);
fprintf("Joint torques using inverse dynamics (Nm): [%.3g, %.3g]",jointTorques3);

Joint torques using inverse dynamics (Nm): [1.41, 1.78]

References

[1]Siciliano, B., Sciavicco, L., Villani, L., & Oriolo, G. (2009). Differential kinematics and statics. Robotics: Modelling, Planning and Control, 105-160.

[2]Harry Asada, and John Leonard. 2.12 Introduction to Robotics. Fall 2005. Chapter 6 Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.