Dynamically Consistent Inverse

Definition

For a kinematically redundant manipulator ($n$ joints, $m<n$ task DOF), the dynamically
consistent (generalized) inverse $\bar{\mathbf{J}}$ of the task Jacobian is the inertia-weighted
right inverse $\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^{\top }\mathbf{\Lambda }$, where
$\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$ is the operational-space
(task) inertia. Khatib’s defining property is force-domain consistency, not minimum joint-velocity:
joint forces drawn from its null-space projector $\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }$
produce zero operational (end-effector) force, so secondary/self-motion torques never disturb the
decoupled task dynamics (khatib1987unified, §VI–VII). Equivalently,
on the velocity side it resolves redundancy by minimizing the manipulator’s instantaneous kinetic
energy rather than the unweighted joint-velocity norm of the Moore–Penrose
pseudoinverse. The source is a fixed-base, terrestrial rigid
manipulator; $\mathbf{M}$ is the joint-space inertia of that arm with no base-reaction coupling.

Key Equations

Symbols per notation.md.

Operational-space (task) inertia and the dynamically consistent inverse (Khatib eqs. 51–52):

$$\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1},\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^{\top }\mathbf{\Lambda }.$$

Every joint-force command decomposes into a task term plus a null-space term that the end-effector
cannot feel (Khatib eq. 55; ${\mathbf{\tau }}_0$ arbitrary):

$$\mathbf{\tau }={\mathbf{J}}^{\top }\mathbf{F}+\underset{\mathbf{N}}{\underbrace{(\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top })}}{\mathbf{\tau }}_0.$$

Notation flags. (i) Khatib writes the joint-space inertia as $A$ and the task inertia as
${\Lambda }_r$; here these are the canonical $\mathbf{M}$ and $\mathbf{\Lambda }$ of
notation.md. (ii) Khatib uses $\Gamma$ for the joint-force vector; this is
renamed $\mathbf{\tau }$ here to avoid colliding with the load-bearing coordinate transform
$\mathbf{\Gamma }$ in notation.md. $\mathbf{F}$ is the operational (task) force.

Source Support

• khatib1987unified — primary and sole source: defines $\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^{\top }\mathbf{\Lambda }$ (eq. 52), proves its uniqueness as the inverse consistent with the end-effector/manipulator dynamics (eqs. 55–57), and uses its null-space projector to add stabilizing self-motion torques without disturbing the decoupled task (eqs. 63–67). Fixed-base terrestrial arm; the redundant-mechanism results are illustrated on a 3-DOF planar manipulator (the PUMA 560 is the non-redundant experimental platform for the operational-space implementation, not the redundancy demonstration).

Related Topics

• operational_space_formulation — the end-effector equations of motion in which $\bar{\mathbf{J}}$ arises; $\mathbf{\Lambda }$ is its task inertia.
• operational_space_control — the controller that commands $\mathbf{F}$ and folds the consistent null-space torques in for asymptotic stabilization.
• null_space_projection — $\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }$ is the dynamically consistent instance of a null-space projector (idempotent, but inertia-weighted rather than orthogonal).
• generalized_inertia_matrix — $\mathbf{M}$, the joint-space inertia that weights the inverse; the weighting is what makes the inverse dynamically (not just kinematically) consistent.
• pseudoinverse_jacobian — the Moore–Penrose inverse $\bar{\mathbf{J}}$ reduces to when $\mathbf{M}=\mathbf{E}$; contrast: it minimizes joint-velocity norm and its null space is not force-consistent.

Open Questions

• The source assumes a fixed-base arm. For our free-flying system the relevant inertia is the coupled base–arm $\mathbf{M}\in {\mathbb{R}}^{(6+n)\times (6+n)}$ (or its circumcentroidal reduction $\breve{\mathbf{M}}$); does the consistent-inverse construction carry over to that Jacobian, and is the resulting null-space then the reaction_null_space rather than the kinematic one?
• Computing $\bar{\mathbf{J}}$ requires inverting $\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$, which is ill-conditioned near a kinematic singularity; how does this interact with the damped/Tikhonov regularization the project applies to $\mathbf{\Gamma }$ and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$?