singularity_geometry

The Neutral/Elbow Singularity is a Posture Singularity — Determinant Factorization, Mass- and Length-Independent

Statement

For the 6R arm of our free-flying space manipulator, the arm Jacobian determinant
factorizes into a single constant times three variable-separated trigonometric factors,
one each for the reach/shoulder, elbow, and wrist families:

$$\det \mathbf{J}=c\underset{\text{reach / shoulder}}{\underbrace{[\cos {\theta }_2+\frac{A_3}{A_2}\cos ({\theta }_2+{\theta }_3)+\frac{D_5}{A_2}\sin ({\theta }_2+{\theta }_3+{\theta }_4)]}}\cdot \underset{\text{elbow}}{\underbrace{\sin {\theta }_3}}\cdot \underset{\text{wrist}}{\underbrace{\sin {\theta }_5}},$$

with $c$ a pure constant and the bracketed link-length ratios $(1,A_3/A_2,D_5/A_2)$
entering the reach factor only as coefficients, never as roots. Because a product is
zero iff one factor is zero, the determinant separates the rank loss into three
independent singularity families.

The key consequence: the neutral (all-joints-zero) configuration is singular because the
elbow factor $\sin {\theta }_3$ vanishes at ${\theta }_3=0$ (the arm is dead straight). This is a
configuration (posture) singularity — its zero set is fixed by joint angle alone. No
link length and no link mass appears in the $\sin {\theta }_3$ factor, so neither lengthening
the links nor counterweighting the base can move that zero. Empirically the bare-arm
conditioning is linear in the vanishing factor,

$${\sigma }_{\min }(\mathbf{J})\approx 0.107|\sin {\theta }_3|,$$

so clearing the derate floor ${\sigma }_{\min }\ge 0.025$ requires $|\sin {\theta }_3|\gtrsim 0.23$,
i.e. an elbow margin $|{\theta }_3|\gtrsim 13$–$14^{\circ }$ off straight. This is the
kinematic root cause of why the system spends time near singularity and is the geometric
companion to the singularity_threshold_cascade, which
schedules conditioning off ${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ (notation.md).

Assumptions

• Bare 6R arm Jacobian $\mathbf{J}$ (alias the fixed-base manipulator Jacobian
  ${\mathbf{J}}_{{\nu }_e}\equiv {\mathbf{J}}_m$, notation.md), built from
  the symbolic DH parameters of the MJCF model (link lengths a scaled UR3, $\times 1.18$).
• The factorization is symbolic in the joint angles ${\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5$
  with link lengths numeric; it is a kinematic determinant identity, independent of dynamics.
• This statement concerns the kinematic configuration singularity of the arm. It is
  distinct from the dynamic_singularity of a
  free-floating base (Papadopoulos $\det {\mathbf{J}}^{\ast }=0$); see Caveats.
• Free-flying regime (fully-actuated 6-DOF base). At the neutral configuration the
  coupled circumcentroidal map $\mathbf{\Gamma }$ does not rescue rank: its lower-right
  block is ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, and ${\sigma }_G<10^{-4}$ at neutral.
• Mass-/length-independence of the elbow zero is a property of the $\sin {\theta }_3$ factor; the
  separate away-from-neutral CoM-cancellation question is addressed in the full proof.

Proof sketch

1. Assemble the 6R arm Jacobian from symbolic DH, angles symbolic and link lengths numeric;
   verify against Pinocchio (cpin) to $2\times 10^{-16}$.
2. Take $\det \mathbf{J}$ and reduce symbolically (SymPy as_independent + bracket
   extraction, identity-checked to $0$). The determinant collapses to one constant times
   $[\text{reach bracket}]\cdot \sin {\theta }_3\cdot \sin {\theta }_5$.
3. Read off the three variable-separated factors. Confirm the reach bracket’s normalized
   coefficients equal the link-length ratios $(1,A_3/A_2,D_5/A_2)$, so lengths live only in
   that factor and only as coefficients (never roots).
4. Evaluate at neutral $\mathbf{q}=0$: ${\theta }_3=0\Rightarrow \sin {\theta }_3=0\Rightarrow \det \mathbf{J}=0$. Establish the empirical linear law ${\sigma }_{\min }\approx 0.107|\sin {\theta }_3|$
   over an elbow sweep; invert for the $|{\theta }_3|\gtrsim 14^{\circ }$ reach margin.
5. Conclude length-/mass-independence at neutral: scaling all links by $\alpha$ scales
   $\det \mathbf{J}$ by ${\alpha }^3$ (homogeneous) but leaves every singular surface fixed;
   counterweighting changes mass/CoM but the $\sin {\theta }_3$ factor is mass-blind. Both
   classical “fixes” act on factors that are not the one vanishing.

Full derivation (factorization, the $0.107|\sin {\theta }_3|$ elbow law, the broken
CoM-cancellation study, and the licensed levers):
singularity_geometry.md.

Subtlety carried in the full proof

At neutral, length-independence is exact (above). The legacy “lengthening helps the bare
arm but the circumcentroidal CoM shift cancels it” claim — about a held standoff away from
neutral — was tested and found BROKEN: ${\sigma }_{\min }(\mathbf{\Gamma })$ rises nearly in
lockstep with the bare arm; the CoM coupling is a bounded $\sim 26\%$ haircut, not a
cancellation. The historical failure of lengthening was a reach-margin/posture effect
(fixed-standoff elbow over-fold + operating near ${\theta }_3\approx 0$), not a CoM cancellation.

Source / provenance

• Ours. Symbolic DH/Jacobian determinant factorization of the 6R arm; the elbow law
  ${\sigma }_{\min }\approx 0.107|\sin {\theta }_3|$; the neutral-singularity diagnosis; and
  the refutation of the link-length / counterweight escape.
• Full derivation + regeneration: singularity_geometry.md
  (regen validation/singularity_geometry.py, pinned by validation/test_singularity_geometry.py,
  4 PASS; CoM-cancellation verdict from a 7-agent proof workflow, logs/logs_Jun18_26/CLAIMS.md).
• Classical configuration-singularity framing (fixed-base serial arms; the determinant /
  posture-singularity picture): siciliano1990tutorial.
• Contrast (free-floating dynamic singularity, a different object):
  papadopoulos1993dynamic.
• Notation: notation.md.

Caveats

• This is the kinematic configuration singularity of the bare arm. The
  dynamic_singularity of a free-floating base (where
  $\det {\mathbf{J}}^{\ast }=0$ depends on mass distribution) is a distinct phenomenon; our
  free-flying base does not lose actuation there, and the elbow result above is mass-blind by
  construction.
• The factorization constants $(c,A_3/A_2,D_5/A_2)$ and the slope $0.107$ are
  configuration-specific to the scaled-UR3 arm; the structure (variable separation, the
  $\sin {\theta }_3$ elbow factor) is general to this 6R geometry but the numbers must be
  re-derived for any other arm.
• The ${\sigma }_{\min }\approx 0.107|\sin {\theta }_3|$ law is empirical (fit over an
  elbow sweep), not an analytic identity; only the determinant factorization itself is exact.
• Operating-posture caveat (log-confirmed): in closed loop the elbow parks near the $-\pi$
  fold at $|\sin {\theta }_3|\approx 0.31$ and the operating slope is
  ${\sigma }_{\min }\approx 0.048|\sin {\theta }_3|$ — about half the bare-arm $0.107$, the
  same Γ/arm coupling haircut. A deterministic off-neutral ${\mathbf{q}}_0$ seed sets a good
  start, but the operating floor is set by posture, not the seed.
• Not a lever: link length / base counterweight as a neutral-config fix (cannot move the
  ${\theta }_3=0$ zero). The structurally correct escape is redundancy: a 7th joint makes
  $\mathbf{J}$ $6\times 7$ and lets the null space steer ${\theta }_3$ off $0$ without moving
  the end-effector.

Related

• Sibling results: singularity_threshold_cascade — why the
  near-singular regime matters and how conditioning is scheduled off ${\sigma }_G$;
  circumcentroidal_decoupling — supplies $\mathbf{\Gamma }$
  and the ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ block that inherits this rank loss in the free-flying
  coupled map.
• Topics: dynamic_singularity ·
  manipulability_measure ·
  damped_least_squares.
• Notation: notation.md.