Operational Space Formulation

Definition

The operational space formulation (Khatib 1987) writes a manipulator’s equations of motion directly in task (end-effector / Cartesian) coordinates rather than in joint coordinates, so that the controller reasons about the quantity it actually wants to regulate. Its central object is the task-space (operational-space) inertia $\Lambda$ — the apparent mass/inertia felt at the end-effector — obtained as a congruence transform of the joint-space inertia, which when combined with exact nonlinear feedback makes the end-effector behave as a free-floating unit mass. For kinematically redundant arms it supplies the dynamically-consistent generalized inverse and an associated null-space projector that inject internal/posture motion without disturbing the task, and it reframes a kinematic singularity as local redundancy about the lost direction. It is the terrestrial, fixed-base antecedent of the circumcentroidal coordinated formulation used in this thesis; the regime caveat below is load-bearing.

Key Equations

Throughout, symbols are rendered in canonical notation (notation.md): $\mathbf{M}$ is the inertia (Khatib’s $A$), ${\mathbf{J}}_{{\nu }_e}$ the task/end-effector Jacobian (Khatib’s $J$). The task-space inertia $\Lambda$, the dynamically-consistent inverse $\bar{\mathbf{J}}$, and the projector $\mathbf{N}$ have no notation.md row yet (see warnings / inconsistency flag).

Task-space inertia (nonredundant, $n=6$). The end-effector apparent inertia is the inverse-Jacobian congruence of the joint-space inertia:

$$\Lambda ={\mathbf{J}}_{{\nu }_e}^{-T}\mathbf{M}{\mathbf{J}}_{{\nu }_e}^{-1}\text{(Khatib eq 18, terrestrial fixed base).}$$

Task-space inertia (general / redundant, $n>6$). When ${\mathbf{J}}_{{\nu }_e}^{-1}$ does not exist, the same quantity is the inverse of the inertia-weighted Gram matrix:

$$\Lambda =({\mathbf{J}}_{{\nu }_e}{\mathbf{M}}^{-1}{\mathbf{J}}_{{\nu }_e}^T{)}^{-1}\text{(Khatib 1995 eq 17; Sentis–Khatib eq 15).}$$

Fundamental force map. Operational (task) forces $\mathbf{F}$ and joint torques relate by the Jacobian transpose:

$$\mathbf{\tau }={\mathbf{J}}_{{\nu }_e}^T\mathbf{F}\text{(Khatib eq 28).}$$

Decoupled (unit-mass) closed loop. With exact nonlinear compensation the task command $\mathbf{F}=\Lambda {\mathbf{F}}^{\ast }$ makes the end-effector a configuration-independent unit mass driven by the linear servo ${\mathbf{F}}^{\ast }$:

$$\ddot{\mathbf{x}}={\mathbf{F}}^{\ast }\text{(Khatib eqs 30–31).}$$

Dynamically-consistent generalized inverse. The unique inverse of ${\mathbf{J}}_{{\nu }_e}$ whose null-space torques produce zero task acceleration is the inertia-weighted inverse:

$$\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}_{{\nu }_e}^T\Lambda ={\mathbf{M}}^{-1}{\mathbf{J}}_{{\nu }_e}^T({\mathbf{J}}_{{\nu }_e}{\mathbf{M}}^{-1}{\mathbf{J}}_{{\nu }_e}^T{)}^{-1}\text{(Khatib 1987 eqs 51–52; Khatib 1995 Thm 1).}$$

It minimizes the instantaneous kinetic energy among all generalized inverses — see dynamically_consistent_inverse.

Null-space torque decomposition. Joint torque splits into a task part and an internal-motion part annihilated by the task:

$$\mathbf{\tau }={\mathbf{J}}_{{\nu }_e}^T\mathbf{F}+\underset{{\mathbf{N}}^T}{\underbrace{(\mathbf{E}-{\mathbf{J}}_{{\nu }_e}^T{\bar{\mathbf{J}}}^T)}}{\mathbf{\tau }}_0\text{(Khatib 1987 eq 55; Khatib 1995 eq 27),}$$

where ${\mathbf{N}}^T=\mathbf{E}-{\mathbf{J}}_{{\nu }_e}^T{\bar{\mathbf{J}}}^T$ is the dynamically-consistent null-space projector; ${\mathbf{\tau }}_0$ stabilizes posture without disturbing the end-effector — see null_space_control.

Singularity as local redundancy. Inside a neighborhood of a singular configuration ${\mathcal{D}}_s=\{\mathbf{q}:|s(\mathbf{q})|\le s_0\}$ (with $s$ the determinant factor of the lost direction), the arm is controlled as redundant with respect to the $(m-1)$-dimensional subspace orthogonal to the singular direction; motion along the lost direction is driven by null-space torques (Khatib 1987 eq 70 and surrounding text). The apparent inertia $\Lambda$ diverges along the singular direction while staying finite in the orthogonal complement.

Regime: terrestrial fixed base (and free-floating extension), not free-flying

Khatib’s $\mathbf{M}$ ($=A$) is a fixed-base joint-space inertia: a grounded, inertial base with no reaction coupling and no momentum-conservation constraint. Khatib (1995) maps the free-flying robot (actuated 6-DOF base, our regime) onto the macro-/mini paradigm by citation only (Russakow & Khatib 1992); Sentis–Khatib (2005) instead works strictly free-floating — the base block of the dynamics carries zero torque and the Generalized Jacobian/Schur-complement inertia absorb the reaction coupling we actively control. The $\Lambda$ / $\bar{\mathbf{J}}$ machinery transfers to our system only after the coupled base+arm inertia replaces $\mathbf{M}$ and the base actuation block is restored.

Source Support

• khatib1987unified — origin paper: task-space EOM, $\Lambda ={\mathbf{J}}^{-T}\mathbf{M}{\mathbf{J}}^{-1}$, fundamental force map, unified motion/force law, redundant $\bar{\mathbf{J}}$, null-space stabilization, singularity-as-local-redundancy. Regime: fixed-base PUMA 560.
• khatib1995inertial — object-level synthesis: uniqueness of the dynamically-consistent inverse (Thm 1), reduced-effective-inertia (Thm 2), augmented object model (Thm 3); explicitly frames the free-flying base as a macro structure (cited, not derived).
• sentis2005control — extends the formulation to a free-floating base via the Generalized Jacobian $\mathbf{J}={\mathbf{J}}_r-{\mathbf{J}}_b{\mathbf{M}}_{bb}^{-1}{\mathbf{M}}_{br}$ and Schur-complement inertia, plus prioritized (task-priority) null-space stacking and an SVD singularity-robust task controller.

Related Topics

• dynamically_consistent_inverse — the inertia-weighted inverse $\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^T\Lambda$ defined and motivated here.
• null_space_control — the projector ${\mathbf{N}}^T=\mathbf{E}-{\mathbf{J}}^T{\bar{\mathbf{J}}}^T$ that injects posture/internal motion without task disturbance.
• operational_space_control — the closed-loop controller built on this formulation (unit-mass decoupling, prioritized whole-body control).
• generalized_jacobian — the free-floating analogue (Umetani 1989; Sentis–Khatib) that folds momentum conservation into the task Jacobian; distinct object/regime from our circumcentroidal ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.
• task_priority_control — the prioritized multi-task hierarchy (Sentis–Khatib) that stacks objectives in successive dynamically-consistent null spaces.

Open Questions

• Does the kinetic-energy-minimizing interpretation of $\bar{\mathbf{J}}$ survive when the fixed-base $\mathbf{M}$ is replaced by the coupled base+arm inertia of a free-flying system? (Left open by Khatib 1987.)
• How does the singularity-as-local-redundancy gate ${\mathcal{D}}_s$ relate to the project’s threshold cascade on ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ (singularity_threshold_cascade)? Both localize behavior near rank loss but via different gating variables.
• Formal stability of the transition controller inside ${\mathcal{D}}_s$ is shown by simulation, not proven (Khatib 1987); likewise the priority-switching stability in Sentis–Khatib (2005).