Augmented Object Model

Definition

The augmented object model (Khatib 1995) extends the operational space formulation to a system of $N$ manipulators rigidly grasping a common object. A single point on the object is chosen as the operational point; under the rigid-grasp assumption it is fixed with respect to every end-effector, so its operational coordinates $\mathbf{x}$ describe the whole multi-effector/object system. The system’s kinetic energy is then a quadratic form in the operational velocities whose inertia matrix — the augmented-object kinetic-energy matrix ${\mathbf{\Lambda }}^{\oplus }$ — equals the sum of the individual effector matrices plus the object’s own load matrix, all reflected to the operational point (the additive property). The whole assembly thus behaves dynamically as one virtual rigid object that can be feedback-decoupled to a unit mass/inertia, exactly as a single manipulator is in the operational-space approach. Khatib develops it for fixed-base terrestrial arms but explicitly cites its use to coordinate a free-flying base (macro) with a lightweight manipulator (mini) via Russakow & Khatib 1992.

Key Equations

Symbols per notation.md.

Operational-point velocity relative to the object’s center-of-mass velocity, with $\mathbf{r}$ the vector from operational point to CoM (here $\mathbf{v},\mathbf{\omega }$ are operational-point linear/angular velocity; ${\mathbf{v}}_c,{\mathbf{\omega }}_c$ the CoM velocities). The rigid-body relation $\mathbf{v}={\mathbf{v}}_c+{\mathbf{\omega }}_c\times ({\mathbf{p}}_O-{\mathbf{p}}_c)={\mathbf{v}}_c+[\mathbf{r}{]}^{\wedge }{\mathbf{\omega }}_c$ fixes the off-diagonal sign for $\mathbf{r}$ defined operational-point$\to$CoM (Khatib eq. 60):

$$\left[\begin{array}{c}\mathbf{v}\\ \mathbf{\omega }\end{array}\right]=\left[\begin{array}{cc}\mathbf{E}& [\mathbf{r}{]}^{\wedge }\\ 0& \mathbf{E}\end{array}\right]\left[\begin{array}{c}{\mathbf{v}}_c\\ {\mathbf{\omega }}_c\end{array}\right]$$

Additive property of the augmented-object inertia: the augmented matrix is the sum of the $N$ effector operational-space inertias ${\mathbf{\Lambda }}_i$ (each reflected to the operational point) plus the object/load matrix ${\mathbf{\Lambda }}_L$ (Theorem 3):

$${\mathbf{\Lambda }}^{\oplus }(\mathbf{x})={\mathbf{\Lambda }}_L(\mathbf{x})+\sum _{i=1}^N{\mathbf{\Lambda }}_i(\mathbf{x}).$$

Notation flag. Khatib writes this matrix $A^{\oplus }$ (joint-space inertia $A$). Per the convention already adopted on operational_space_formulation, inertia is $\mathbf{M}$ and the operational-space inertia is $\mathbf{\Lambda }$ (notation.md “Additions” row, $\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$). The superscript $\oplus$ here marks “augmented object”, not the circumcentroidal $\oplus$ of the thesis frames — and this $\mathbf{\Lambda }$ is unrelated to the helix waypoint set $\mathbf{\Lambda }=\{{\mathbf{\lambda }}_i\}$ in notation.md. Both clashes are local; do not conflate.

The Coriolis/centrifugal vector ${\mathbf{\mu }}^{\oplus }$ and gravity vector ${\mathbf{p}}^{\oplus }$ share the same additive property, so the closed-form decoupled model is ${\mathbf{\Lambda }}^{\oplus }\ddot{\mathbf{x}}+{\mathbf{\mu }}^{\oplus }+{\mathbf{p}}^{\oplus }={\mathbf{F}}^{\oplus }$, with ${\mathbf{F}}^{\oplus }$ the net operational force partitioned among the redundant actuators.

Source Support

• khatib1995inertial — primary; states the augmented object construct (Theorem 3, additive property of the kinetic-energy matrix), its redundant-manipulator extension via the dynamically-consistent inverse, and the macro-/mini- (free-flying base) application.

Related Topics

• operational_space_formulation — the single-manipulator framework that the augmented object model generalizes to $N$ grasping arms.
• dynamically_consistent_inverse — required to project each redundant arm’s dynamics into the $m$-dimensional operational space; the augmented inertia then depends on the full multi-arm configuration.
• macro_mini_manipulation — the inertial-properties analysis (reduced effective inertia) that motivates treating a free-flying base as a macro structure carrying a lightweight mini-manipulator.
• coordinated_control — the augmented object is the model on which decoupled, coordinated motion/force control of the grasping system is built.
• generalized_inertia_matrix — the augmented-object kinetic-energy matrix is the multi-effector generalization of the operational-space (task) inertia at the grasp point.

Open Questions

• Khatib’s augmented object assumes a rigid grasp with the operational point fixed to all end-effectors; our inspection task has a single non-contact end-effector imaging a target — does the construct offer anything beyond the single-arm operational-space inertia in that regime, or only when a future capture/grasp phase is added?
• The model is derived for fixed-base arms and (via Russakow & Khatib 1992) a free-flying base treated as a macro structure. Does the macro/mini inertial decomposition coincide with, or differ from, the circumcentroidal CoM/attitude+EE split used in this thesis for a comparably-massed base and arm?
• The additive property reflects each effector inertia to the operational point. For a free-flying base whose actuated motion is itself part of the kinematic chain, is the reflected base inertia still configuration-additive, or does base actuation break the clean sum?