Robot Arm Joint Torques

What is modelled

A robot arm holds its payload up by torque at every joint. The torque at a joint is the weight of everything beyond it times the horizontal distance from the joint, so an outstretched arm needs far more torque than a folded one, and the shoulder carries the most. When the arm accelerates, each joint must also supply the torque to turn the inertia of everything beyond it. A motor cannot supply more than its limit, which sets the heaviest payload the pose allows.

Equations

• Gravitational potential energy U = g Σ mi yi and holding torque τj = ∂U/∂qj = g Σi≥j mi (xi − xj).
• Mass matrix M = Σ (mi JiT Ji + Ii wi wiT) with link inertia I = m l²/12.
• Torque for a start from rest τ = G(q) + M(q) α.
• Payload limit from |τj(mp)| = Tj,max, which is linear in the payload.

Checks run in the page

• The analytic holding torques against numerical differentiation of the potential energy.
• The kinetic energy from link velocities against the mass matrix, and its symmetry and positive definiteness.
• The payload limit put back in so the most loaded joint sits exactly at its limit.
• The work of the joint torques along a path against the potential energy gained.

Honest simplifications

The arm is a planar three-link chain in a vertical plane (a real six-axis arm adds rotations out of this plane), links are uniform rods with the mass proportional to the square of the size scale, motors are given torque limits in a fixed ratio, the acceleration torque is for a start from rest so Coriolis and centrifugal terms are left out, and gearbox friction and motor inertia are not included.

• Press Reaching out and then Folded, easy to see how the shoulder torque falls when the arm is folded in.
• Press Heavy payload and read which joint overloads first.
• Press Fast move and Big arm to compare the acceleration torque with the holding torque, and watch the payload limit shrink.