Singularities and Manipulability Ellipsoids

At some poses an arm cannot move its hand in certain directions, no matter how fast the joints spin. These poses are singularities. The manipulability ellipse shows, at each pose, how easily the hand can move in every direction.

θ₁ θ₂ (elbow) Show the force ellipse instead Sweep the elbow

The manipulability ellipse

Joint speeds on the unit circle, $\lVert\dot{\boldsymbol\theta}\rVert = 1$, map through the Jacobian to an ellipse of hand speeds. Its axes come from $J J^\top$, and its area gives Yoshikawa's manipulability measure:

$$w = \sqrt{\det\left(J J^\top\right)} = L_1 L_2\,\lvert\sin\theta_2\rvert$$

What to notice

As the elbow straightens, $\theta_2 \to 0$, the ellipse collapses into a line. The hand can still move sideways, but it cannot move outward at all, since the arm is already fully stretched. The same happens when the arm is folded at $\theta_2 = 180^\circ$. Near a singularity, inverse kinematics asks for huge joint speeds to make small hand motions, which is why controllers avoid them.

The force ellipse is the opposite shape. Where the arm moves easily, it pushes weakly, and where it barely moves, it can push hard. A straight arm holds a large load along its length, like a locked elbow when you push a door.