Where Can a Robot Hand Move Fast?

A robot arm's Jacobian turns joint speeds into hand speed. Feed in every joint-speed direction of size 1 and the hand speeds trace an ellipse, the manipulability ellipse. Its axes are the singular vectors and their lengths the singular values: the directions the hand moves fast and slow.

How it works

The singular value decomposition

In one sentence

Every $m \times n$ matrix factors as $A = U\Sigma V^T$ with $U$ , $V$ orthogonal and $\Sigma$ diagonal with $\sigma_1 \ge \sigma_2 \ge \cdots \ge 0$ ; $A$ maps the unit sphere onto an ellipsoid with semi-axes $\sigma_i\mathbf{u}_i$ .

General formula

{\htmlData{sym=0}{\dot{\mathbf{p}}}} = {\htmlData{sym=1}{J}}({\htmlData{sym=2}{\mathbf{q}}})\,{\htmlData{sym=3}{\dot{\mathbf{q}}}}, \qquad {\htmlData{sym=1}{J}} = {\htmlData{sym=4}{U}}{\htmlData{sym=5}{\Sigma}} {\htmlData{sym=6}{V}}^T, \quad {\htmlData{sym=5}{\Sigma}} = \operatorname{diag}({\htmlData{sym=7}{\sigma_1}}, {\htmlData{sym=8}{\sigma_2}})

Any linear map is a rotation, then a stretch along perpendicular axes, then another rotation. The SVD finds those axes and stretches.

What the symbols mean

$\dot{\mathbf{p}}$
Hand velocity
How fast the gripper moves, and which way.
$J$
Jacobian
Matrix turning joint speeds into hand velocity.
$\mathbf{q}$
Joint angles
Shoulder and elbow angles; J depends on them.
$\dot{\mathbf{q}}$
Joint speeds
How fast each joint turns.
$U$
Output directions
Columns: the ellipse's axes, in hand space.
$\Sigma$
Singular values
Diagonal of σ₁, σ₂: the ellipse's half-axes.
$V$
Input directions
Columns: joint-speed mixes that map onto the axes.
$\sigma_1$
Largest stretch
Fastest hand speed per unit joint speed.
$\sigma_2$
Smallest stretch
Slowest direction; 0 at a singularity.

For a two-link arm, the Jacobian $J$ is the $2\times2$ matrix of how the hand position changes with each joint angle.
Joint speeds on the unit circle, $\|\dot{\mathbf{q}}\| = 1$ , map to an ellipse of hand speeds.
The SVD $J = U\Sigma V^T$ names it: the columns of $U$ are the ellipse's axes and the singular values $\sigma_1 \ge \sigma_2$ their half-lengths.
When the arm stretches straight, $\sigma_2 \to 0$ and the ellipse flattens to a line: a singularity, where the hand can't move in one direction at all.

Learn the math

Diagonalization of Symmetric MatricesLin Alg · Lesson 37, Symmetric Matrices and Quadratic Forms The Singular Value DecompositionLin Alg · Lesson 39, Symmetric Matrices and Quadratic Forms Principal Component AnalysisLin Alg · Lesson 40, Symmetric Matrices and Quadratic Forms