Arm Singularity Lab

What is modelled

A planar arm with two equal-mass links moves its hand out and back along a radial line. A closed-loop inverse-kinematics law turns the wanted hand velocity into joint velocities. Near full extension the arm loses the ability to move its hand along the arm direction: the Jacobian becomes singular.

Equations

• Forward kinematics x = L1 cos q1 + L2 cos(q1+q2), y = L1 sin q1 + L2 sin(q1+q2).
• Jacobian determinant det J = L1 L2 sin q2.
• Naive: q̇ = J−1 v with v = vd + Kp (xd − x).
• Damped: q̇ = JT (J JT + λ2 I)−1 v, which is bounded by |v| / (2λ).
• On a radial line dq2/dt = − r (dr/dt) / (L1 L2 sin q2), which blows up as sin q₂ goes to 0.

Checks run in the page

• Analytic Jacobian against finite differences.
• Determinant against L1 L2 sin q2.
• The damped solver tends to the exact inverse when lambda goes to 0.
• The damping bound |v| / (2 lambda).
• The radial-line elbow speed law.

Honest simplifications

Kinematics only: no link mass, no motor torque, no sensor noise, explicit Euler steps of 1 ms and simple per-joint speed clipping. The damped solver gives up a little tracking accuracy near the singularity in exchange for bounded speeds; that trade is the point of the lab.

• Press Near the singularity and compare the red and green speed bars.
• Press Exactly full reach: the naive solver saturates the joint speed limit and loses track of the target.
• Raise lambda and see the tracking error of the damped solver grow while its peak speed falls.