Core 11 min 2 of 3

The Jacobian

A linear map from joint velocities to tip velocities, whose singular values are the arm’s instantaneous capability.

What this repairs

Knowing that $\dot{x} = J\dot{q}$ without a picture of what the columns, the rank, or the singular values mean physically.

Assumes

Forward kinematics as composition

Differentiate forward kinematics with respect to the joint variables:

$\dot{\mathbf{x}} = J(\mathbf{q})\,\dot{\mathbf{q}}, \qquad J_{ij} = \frac{\partial x_i}{\partial q_j}.$

Read it column by column. Column $j$ is the tip velocity produced by moving joint $j$ at unit speed with all others locked. For a revolute joint that is $\mathbf{z}_j \times (\mathbf{p}_{\text{tip}} - \mathbf{p}_j)$ — the familiar " $\boldsymbol{\omega} \times \mathbf{r}$ ", so the column's length grows with the distance from the joint to the tip.

Which explains something that is otherwise a surprise: the same joint speed moves the tip faster when the arm is extended. Capability is not a property of the robot alone; it is a property of the robot in a configuration.

The image of the unit ball in joint-velocity space under $J$ is an ellipsoid in task space — the manipulability ellipsoid. Its axes are the singular values of $J$ and their directions the left singular vectors. Long axis: a direction the tip moves easily. Short axis: a direction that costs a lot of joint speed.

Yoshikawa's scalar measure is $\mu = \sqrt{\det(JJ^\top)}$ , the ellipsoid's volume, which for a square $J$ is $\lvert\det J\rvert$ . It is a useful single number and it hides the shape: an arm with $\mu$ well above zero can still be nearly unable to move in one particular direction, and "one particular direction" is often the one the task needs.

Insight A singularity is a lost direction, not a bad number

At a singularity $J$ loses rank: the ellipsoid collapses to a lower-dimensional disc, and there exists a task-space direction the tip cannot move in at any joint velocity. The determinant going to zero is the symptom; the lost direction is the thing.

Nearby, the inverse $J^{-1}$ has enormous entries, so a resolved-rate controller commands joint velocities that exceed the actuators' limits and the arm lurches. The standard remedy is damped least squares:

$\dot{\mathbf{q}} = J^\top\left(JJ^\top + \lambda^2 I\right)^{-1}\dot{\mathbf{x}}_{\text{desired}},$

which bounds the joint velocity at the cost of tracking accuracy in the collapsed direction. That trade is not avoidable — the direction really is unavailable — and $\lambda$ is the choice of how gracefully to fail.

Check yourself

A two-link arm has $q_2 = 0$ (fully extended). Which tip motions are still possible?

Show answer

Only motion perpendicular to the arm — the tangential direction. Both joints now produce tip velocity along the same line, so the two Jacobian columns are parallel, the rank drops to one, and the radial direction (in or out along the arm) is unreachable at any joint speed. Try to command it and $J^{-1}$ blows up. The manipulability ellipse in the figure flattens to a segment at exactly this configuration.