sentis2005control

Control of Free-Floating Humanoid Robots Through Task Prioritization

Authors: Sentis, Khatib · Year: 2005 · Venue: IEEE International Conference on Robotics and Automation (ICRA), Barcelona
Raw: md

Summary

Sentis and Khatib extend the Operational Space Formulation and its prioritized multi-task (whole-body) control hierarchy to robots whose base link is free-floating (uncontrolled, momentum-conserving), motivated by humanoids in free-space. The key construct is an Extended Generalized Jacobian that folds the dynamic coupling between joint motion and base motion into each prioritized task Jacobian, so that every task objective is controlled in the null space of all higher-priority objectives using only joint torques. The paper also gives a singularity-robust controller via eigen-decomposition of the task inertia for runtime task-feasibility handling, demonstrated in the SAI simulator on a 24-DOF humanoid.

Key Claims

• For a free-floating system with no actuation on the 6-DOF base, conservation of base (linear + angular) momentum lets the operational-space velocity be written from joint velocities alone via the Generalized Jacobian $J=J_r-J_b{A}_{bb}^{-1}{A}_{br}$, equivalently $J=J_{sys}{\overline{S}}_q$ (eqs. 13–14).
• The joint-space inertia of the free-floating robot is the Schur complement of the base block: $A=A_{rr}-A_{rb}{A}_{bb}^{-1}{A}_{br}$ (eq. 6), also called the Generalized Inertia Tensor; Coriolis/gravity transform analogously (eqs. 7–8).
• Prioritized tasks are realized by an Extended Generalized Jacobian $J_{k|prev(k)}=J_{sys(k)}{\overline{S}}_q{N}_{prev(k)}$ (eq. 30) projected into the accumulated null space, with the unique dynamically consistent null space $N_{prev(k)}=I-{\sum }_{i=1}^{k-1}{\overline{J}}_{i|P(i)}{J}_{i|P(i)}$ (eq. 33).
• The paper characterizes for the first time the virtual mass properties (${\Lambda }_{k|prev(k)}$) of prioritized tasks for free-floating robots, enabling precise task control and hybrid position/force control of free-floating behaviors.
• When a task Jacobian $J_{k|prev(k)}$ drops rank (infeasible task), an SVD/eigen-decomposition of ${\Lambda }_{k|prev(k)}^{-1}$ yields a singularity-robust controller (eq. 38) that achieves decoupling only in the controllable directions $U_{r(k)}$; feasibility can thus be monitored and the strategy switched at runtime.

Method

Model assumptions: a humanoid with $n$ actuated joints plus an unactuated 6-DOF base ($x_{base}$). The system dynamics (eq. 1) have a zero torque entry on the base block,
$A_{sys}\left(\begin{array}{c}{\ddot{x}}_{base}\\ \ddot{q}\end{array}\right)+b_{sys}+g_{sys}=\left(\begin{array}{c}0\\ \Gamma \end{array}\right),$
which is the defining signature of the free-floating (free-floating, not free-flying) regime: the base cannot be commanded directly, only through reaction dynamics.

A joint selection matrix $S_q=[0I]$ (eq. 2) and its dynamically consistent generalized inverse ${\overline{S}}_q=A_{sys}^{-1}{S}_q^{T}A$ (eq. 3) project the full dynamics onto the joint-space dynamics $A\ddot{q}+b+g=\Gamma$ (eq. 4). Block partitioning (eq. 5) into base ($b$) and robot ($r$) components yields the Schur-complement inertia (eq. 6).

The operational-space task uses $\dot{x}=J_{sys}({\dot{x}}_{base}^T{\dot{q}}^T{)}^T$ with $J_{sys}=(J_b{J}_r)$ (eqs. 9–10). Momentum conservation (eq. 12), $A_{bb}{\dot{x}}_b+A_{br}\dot{q}=\text{const}$ (taken zero), collapses this to $\dot{x}=J\dot{q}$ with the Generalized Jacobian (eqs. 13–14). Operational-space dynamics $\Lambda \ddot{x}+\mu +p=F$ follow with $\Lambda =(J{A}^{-1}{J}^T{)}^{-1}$, $\overline{J}=A^{-1}{J}^T\Lambda$, null space $N=I-\overline{J}J$ (eqs. 15–17), and decoupling torque ${\Gamma }_{task}=J^T(\Lambda {\ddot{x}}_{ref}+\mu +p)$ (eq. 16). Posture/secondary tasks are stacked in $N^T$ via an extended posture Jacobian $J_{p|t}=J_{sys(p)}{\overline{S}}_{q}N$ (eq. 21). The full prioritized hierarchy nests null-space projections (eqs. 26–36), with the singularity-robust variant (eqs. 37–38).

Note the notation clash flagged by OCR: eq. (1) text labels $\ddot{x}$ the ”$n\times 1$ vector of joint accelerations” though the symbol should be $\ddot{q}$; the matrix entries themselves are consistent.

Regime: strictly free-floating (momentum conservation is invoked to eliminate base velocities). A free-FLYING base would carry actuation on the top block of eq. (1) and would not rely on $A_{bb}{\dot{x}}_b+A_{br}\dot{q}=\text{const}$.

Relevance to thesis

This is a foundational template for our work but in the opposite actuation regime: it derives task-prioritized operational-space control assuming an unactuated base, so the Generalized Jacobian and Schur-complement inertia absorb reaction coupling that, in our free-flying manipulator, we instead actively control. The Extended Generalized Jacobian and the unique dynamically consistent null space (eq. 33) give a clean recipe for stacking constraints (e.g., joint limits, friction) above the end-effector task — directly transferable to our planning/control stack if we restore the base actuation block. The eigen-decomposition feasibility/singularity controller (eq. 38) is a candidate runtime safeguard for the risk layer.

Connections

Topics: generalized_jacobian · dynamic_coupling · task_prioritization · operational_space_control

Key Equations / Quotes

“Umetami and Yoshida introduced an extended end-effector Jacobian named Generalized Jacobian … which combines the dynamic interactions between the joint velocities and the velocities of the free-floating base.” (Sec. I)

Generalized Jacobian (eq. 13–14):
$J=J_r-J_b{A}_{bb}^{-1}{A}_{br}=J_{sys}{\overline{S}}_q,{\overline{S}}_q=\left(\begin{array}{c}-A_{bb}^{-1}{A}_{br}\\ I\end{array}\right).$

Schur-complement joint inertia (eq. 6):
$A=A_{rr}-A_{rb}{A}_{bb}^{-1}{A}_{br}.$

Unique dynamically consistent prioritized null space (eq. 33):
$N_{prev(k)}=I-{\sum }_{i=1}^{k-1}{\overline{J}}_{i|P(i)}{J}_{i|P(i)}.$

Singularity-robust task torque (eq. 38) using the rank-truncated inverse $U_{r(k)}{\Sigma }_{r(k)}^{-1}{U}_{r(k)}^T$.

“Notice that the right leg stretches out to preserve angular momentum and the robot’s body moves downward to preserve the center of mass position. Both of these behaviors are due to free-floating dynamics.” (Sec. V)

Open Questions

• The derivation sets the initial spatial momentum (constant in eq. 12) to zero; how do the Generalized Jacobian and task feasibility change under nonzero stored momentum, which is the operationally relevant case?
• The feasibility test thresholds eigenvalues of ${\Lambda }_{k|prev(k)}^{-1}$ at exactly zero; no condition-number / near-singular margin is given for robust runtime switching.
• Stability of the priority switching (full-rank controller eq. 36 ↔ rank-truncated eq. 38) is asserted via decoupling but not formally proven.