chiaverini1994review

Review of the Damped Least-Squares Inverse Kinematics with Experiments on an Industrial Robot Manipulator

Authors: Chiaverini, Siciliano, Egeland · Year: 1994 · Venue: IEEE Transactions on Control Systems Technology 2(2), pp. 123–134
Raw: md

Summary

The canonical experimental review of damped least-squares (DLS) inverse kinematics for driving a
manipulator through kinematic singularities. The core scheme replaces the ill-conditioned Jacobian
(pseudo-)inverse with a DLS inverse whose damping factor is raised only in the neighbourhood of a
singularity, trading exact tracking for bounded joint velocities. The paper adds: a weighted DLS
variant that allocates the accuracy loss to user-chosen end-effector directions, an on-line estimate
of closeness to a singularity (driven by the smallest singular value of the Jacobian), and a
feedback correction term that restores closed-loop tracking convergence. Validated on the six-joint
ABB IRb2000 through shoulder and wrist singularities — i.e. theory backed by hardware, not just simulation.

Key Claims

• A constant damping factor cannot serve both accuracy (far from singularities) and feasibility (at
  singularities); the damping must be varied as a function of singularity proximity.
• Singularity proximity is measured on-line by the smallest singular value ${\sigma }_{\min }$ of the
  Jacobian (an estimate avoiding a full SVD each step), which schedules the damping.
• A weighted DLS confines the inevitable accuracy degradation to selected operational-space
  directions, preserving accuracy along the others.
• A closed-loop feedback error term is required: the open-loop DLS solution drifts; adding a task
  error feedback makes the scheme track-convergent.

Method

Regime — fixed-base industrial manipulator (ABB IRb2000, 6 joints; terrestrial). No floating/flying
base and no momentum coupling. The contribution is purely the singularity-robust velocity-level
inverse kinematics layer.

The damped least-squares solution damps the Jacobian inverse near rank loss:
$\dot{\mathbf{q}}={\mathbf{J}}^{\top }{\left(\mathbf{J}{\mathbf{J}}^{\top }+{\lambda }^2\mathbf{I}\right)}^{-1}\dot{\mathbf{x}},$
with damping ${\lambda }^2$ ramped up as ${\sigma }_{\min }(\mathbf{J})$ falls below a threshold (zero far
from singularities ⇒ exact pseudoinverse; positive near them ⇒ bounded $\dot{\mathbf{q}}$). The
weighted variant inserts an output weighting so the accuracy loss is steered to chosen directions, and
a feedback term ${\dot{\mathbf{x}}}_d+\mathbf{K}\mathbf{e}$ (task error $\mathbf{e}$)
replaces the bare reference rate to guarantee tracking convergence.

Relevance to thesis

Supporting — the foundational reference for our singularity-handling layer. The thesis’s
conditioning cascade applies damped least-squares to the circumcentroidal transform $\mathbf{\Gamma }$
and the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, scheduling the Tikhonov/DLS damping on
${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ — exactly the proximity-driven variable damping this
paper introduced (for a fixed base). The ${\sigma }_{\min }$-as-proximity-scalar idea and the variable-damping
schedule transfer directly; the free-flying difference is which matrix is damped (the coupled
$\mathbf{\Gamma }$ / ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, not a fixed-base geometric Jacobian).

Connections

Topics: damped_least_squares · singularity_robust_inverse · pseudoinverse_jacobian · kinematic_singularity · manipulability_measure

Key Equations / Quotes

“The basic scheme adopts a damped least-squares inverse of the manipulator Jacobian with a varying
damping factor acting in the neighborhood of singularities.” (Abstract)

$\dot{\mathbf{q}}={\mathbf{J}}^{\top }{\left(\mathbf{J}{\mathbf{J}}^{\top }+{\lambda }^2\mathbf{I}\right)}^{-1}\dot{\mathbf{x}}$

“An on-line estimation algorithm is employed to measure closeness to singular configurations.”
(Abstract) — the smallest singular value ${\sigma }_{\min }(\mathbf{J})$ schedules ${\lambda }^2$.

Open Questions

• The damping schedule ${\lambda }^2({\sigma }_{\min })$ is tuned empirically here; what objective fixes
  the accuracy-vs-feasibility tradeoff on the coupled free-flying $\mathbf{\Gamma }$ rather than a
  fixed-base Jacobian?
• The closeness estimate avoids a full SVD; on the time-varying circumcentroidal block, is a cheap
  ${\sigma }_{\min }$ estimator stable enough to schedule the Tikhonov damping in real time?