Pseudoinverse Jacobian

Definition

The pseudoinverse (Moore–Penrose generalized inverse) Jacobian is the standard velocity-level inverse-kinematics map used when the manipulator Jacobian $\mathbf{J}$ is non-square or rank-deficient, so an ordinary inverse does not exist. For a redundant (wide, $m\le n$) Jacobian it returns the minimum-norm joint-rate solution; for an over-determined (tall, $m\ge n$) Jacobian it returns the least-squares solution (holt1994inertial). Holt’s setup is a 6-DOF PUMA arm on a disturbed 3-DOF platform tracking an inertial-space target — a fixed-base / mobile-platform manipulator in a terrestrial space-station-assembly experiment, not a free-floating or free-flying space base; the kinematic pseudoinverse itself is regime-agnostic, but the disturbance-rejection framing assumes the base motion is an exogenous, externally measured signal rather than a reaction to arm motion.

Key Equations

Symbols per notation.md.

The Jacobian maps differential joint motion to a differential Cartesian (linear + angular) displacement $\mathrm{d}\mathbf{u}\in {\mathbb{R}}^6$:

$\mathrm{d}\mathbf{u}=\mathbf{J}(\mathbf{q})\mathrm{d}\mathbf{q}.$

The Moore–Penrose pseudoinverse ${\mathbf{J}}^{+}$ (Holt writes $J^{†}$; the ${}^{+}$ superscript matches the convention in damped_least_squares, where $({\mathbf{J}}^{\oplus }{)}^{+}$ is used) selects the minimum-norm / least-squares branch by the shape of $\mathbf{J}\in {\mathbb{R}}^{m\times n}$:

${\mathbf{J}}^{+}\triangleq \{\begin{array}{ll}{\mathbf{J}}^{\top }(\mathbf{J}{\mathbf{J}}^{\top }{)}^{-1},& m\le n(\text{redundant, min-norm})\\ {\mathbf{J}}^{-1},& m=n\\ ({\mathbf{J}}^{\top }\mathbf{J}{)}^{-1}{\mathbf{J}}^{\top },& m\ge n(\text{least-squares})\end{array}$

This closed form still inverts a matrix, so it does not by itself cure singularities — $\Vert {\mathbf{J}}^{+}{\Vert }_2\to \infty$ as the smallest singular value ${\sigma }_{\min }(\mathbf{J})\to 0$, which motivates damping or an SVD-based treatment (holt1994inertial §2–3, Fig. 1). Holt’s own remedy is an approximate pseudoinverse ${\mathbf{J}}^{‡}$ that pseudoinverts the $3\times 3$ block submatrices $\mathbf{B},\mathbf{E}$ of the partitioned PUMA Jacobian separately — four-times cheaper than a full SVD, exact when $\mathbf{J}$ is non-singular, but it does not satisfy the Moore–Penrose conditions at a singularity.

Source Support

• holt1994inertial — defines the pseudoinverse Jacobian by Jacobian shape (Eq. 5), shows $\Vert {\mathbf{J}}^{+}\Vert$ blows up near a singularity, and develops the block-wise approximate pseudoinverse ${\mathbf{J}}^{‡}$ as a real-time-cheap, singularity-defined alternative (PUMA-on-platform experiment).

Related Topics

• damped_least_squares — the Tikhonov-regularized fix for the pseudoinverse’s blow-up: weights each SVD direction by $s_i/(s_i^2+{\lambda }^2)$ so the gain stays finite as ${\sigma }_{\min }\to 0$.
• singularity_robust_inverse — the singularity-robust family the damped pseudoinverse belongs to; a robust replacement for ${\mathbf{J}}^{+}$ inside a singular region.
• resolved_motion_rate_control — RMRC is the velocity-level control loop in which ${\mathbf{J}}^{+}$ is the workhorse map from desired task rate to joint rate.
• kinematic_singularity — the rank-loss configuration where ${\mathbf{J}}^{+}$ becomes ill-conditioned, the exact failure mode this page guards against.
• generalized_jacobian — the free-floating analogue (GJM) that folds momentum conservation into the Jacobian; its pseudoinverse plays the same role but for a reaction-coupled base, a regime distinct from Holt’s externally-driven platform and from our free-flying system.

Open Questions

• Holt’s pseudoinverse is the fixed-base manipulator Jacobian with the platform disturbance treated as an external signal; in our free-flying system the base is actuated and coordinated with the arm — does the relevant inverse become the circumcentroidal Jacobian inverse $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{+}$ rather than the bare manipulator ${\mathbf{J}}^{+}$?
• The approximate pseudoinverse ${\mathbf{J}}^{‡}$ trades Moore–Penrose optimality for a 4× speed-up via $3\times 3$ block SVDs; does any analogous block structure exist in our coupled $\mathbf{\Gamma }$ / ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ to make the singularity-robust inverse cheaper online?
• For a redundant ($m<n$) arm the minimum-norm branch discards the null space; how should a null-space term (posture / reaction-null-space objective) be superposed on ${\mathbf{J}}^{+}\mathrm{d}\mathbf{u}$ for the 7-DOF case?