Singularities during trajectory tracking — a research study

Date: Jun 12, 2026. Question: what causes the sharp condition-number (κ) peaks during orbit tracking, how do we alleviate them, reduce error, smooth the trajectory, and harden the robot — and are we asking the right questions?

Companions: this study complements (does not repeat) the empirical forensics in generated_reports/GNC/Jun12_26/chain10_kappa_forensics_verdict.md (+ offline tables, codepath audit) and the chain docs tasks/daily_log/Jun12_26/chain10_kappa_forensics_{design,plan}.md. Those documents establish what happens in our data; this one asks what the literature says it is and what to do about it.

Sources: the 14 markdown-converted papers in docs/Most_Relevant/papers/ (citekeys below match folder names and docs/Most_Relevant/references.bib), the CHAIN_10 evidence set, the s_min_G/s_min_J study, the Phase 00 round-1 verdict, generated_reports/math/derivation_7dof.md, and a code audit of the live singularity-handling stack (file:line refs throughout). Method: per-source structured extraction by parallel readers, synthesis by the advisor, then an adversarial verification pass checking every load-bearing citation against its source file.

Legend (decode before reading)

Full vocabulary lives in wiki/terminology.md. The minimum to read this report:

1. What we already know (internal evidence, two paragraphs)

CHAIN_10 (pre-registered, offline, bit-exact replays of 8 runs) killed the numerics hypothesis H1 (the finite-difference path tangent is discarded at its only call site — ee_guidance.py:574 in the current snapshot; the CHAIN_10 docs cite line 620, which predates the CHAIN_11/12 mesh refactor) and the surface-curvature hypothesis H2 (curvature at peaks indistinguishable from calm windows; Cliff’s δ never near the +0.5 bar). It promoted H3 — the schedule walks the arm near its extension limit in schedule-dependent phase bands — on the striking evidence that the 7-DOF’s hot band moves wholesale under ±2% speed (phases 0.29–0.35 at −2%; startup-only at +2%, with zero post-startup peaks), while the registered driver-level test using the crude reach_b proxy stayed honest-inconclusive (δ = +0.82 on the 7-DOF +2% run and +0.60 on the 6-DOF nominal, but −0.42 on the 7-DOF nominal). H4 got the SEED verdict: entries 23/14/2 — the schedule changes how often we fall in, not how long an unchanged set of falls lasts. Pointing error is EVENT-dominant on the 7-DOF (δ up to +0.94 inside peak windows; the 6-DOF anti-correlates because its peaks occur while crawling).

Context from earlier work: the near-singular regime is the arm at full extension — the reach/shoulder factor R(θ) of det(J_arm) = R·sinθ₃·sinθ₅ — not base mass (factorization and the 0.64–0.80 derate residual: mission_1_success.md; the +1.0 s_min_G/extension correlation: the CHAIN_10 design and verdict docs; base mass ruled out by the Jun-8 counterweight sweep recorded in high_level_plans/decisions.md). The 6-DOF lives saturated on that floor (dwell 614–655 s at every speed, derate fraction 0.64–0.80); the 7-DOF lives marginally near it, which is exactly why ±2% swings it by orders of magnitude (phase00_round1_verdict.md). The s-metric question is settled: s_min_G and s_min_J are redundant witnesses here (agreement 1.0000 on derate decisions).

2. What the κ in our logs actually is (code ground truth)

Worth stating precisely once, because the literature’s claims attach to specific matrices:

• kJ, s_min_J ≡ σ₆: condition number / smallest singular value of **J⁺ = J_νe − R_eb0·J̄_v** (utils/robot.py:238-243) — the arm’s joint columns with the CoM-translation contribution removed. This is the partial-base generalized Jacobian of our controller’s source paper [giordano2019coordinated], eq. (14): the matrix whose σ_min defines that paper’s stability region Ω.
• kG, s_min_G: same quantities for the full 12×(6+n) coordinated map Γ (utils/robot.py:250-255). A code comment (recording finding F4) at breve_controller.py:305-306 states Γ is singular iff the arm map is — a claim our redundancy study confirmed numerically.
• The live mitigation stack has four layers with four different thresholds:
  1. derate ramp on s_min_G — scale 1.0 above 0.025, floor 0.25 below 0.005, linear between (utils/sampling.py:70-78, parameters.yaml:82-87); throttles CoM velocity (×s), feedforward accel (×s²), ω_b, base wrench, and arm gain blocks;
  2. Tikhonov regularization of the Γ velocity solve — λ = max(10⁻⁴, 0.05² − s_min_G²), i.e. engaging already below σ = 0.05, plus a ±50 velocity clip (breve_controller.py:498-516);
  3. damped pseudo-inverse of J⁺ — Tikhonov below σ = 0.02, hold-last-healthy-inverse below 0.005 (a hard switch; utils/robot.py:173-210);
  4. kernel freeze (7-DOF) — binary at σ₆ = 0.025 (utils/robot.py:312-323).

Layer 2 deserves a flag: between σ = 0.05 and 0.025 the solve is already being damped while the derate has not yet engaged. Damping of this isotropic kind has a known, quantified cost in the literature — tracking error injected into all directions, not just the dying one (see §4.3). That makes layer 2 a prime suspect for the event-dominant pointing error inside peak windows, alongside freeze thaw jumps (measured p99 = 82.9°).

3. What the spikes are — the literature’s answer

3.1 The singular set is fixed in configuration space; the schedule chooses the path through it

[papadopoulos1993dynamic] proves that for a floating-base manipulator, all singularities of the generalized Jacobian are functions of the joint angles q alone — independent of base attitude — and that in the workspace the same end-effector pose can be singular or not depending on the path taken to reach it (the nonholonomy of momentum). The workspace splits into a Path-Independent Workspace (PIW: no path can be singular) and a Path-Dependent Workspace (PDW: singularity is path-conditional). Each singular hypersurface maps to a radial annulus of reach around the system CoM; in their planar worked example the outer annulus extends exactly to the full-extension boundary.

[calzolari2020singularity] sharpens this: the singular surfaces are fixed walls in configuration space — of complex topology (Calzolari’s summary of Rybus et al. 2012, their ref [33]; their own Figs. 2–3 show the planar singular curve “is not a line”) — and sometimes thin enough that a locally-constrained optimizer “can fail to detect an unfeasible step” — i.e. a local manipulability or σ threshold can step straight across a wall undetected. Singularity avoidance done properly is therefore a global, geometric problem (their S-Maps turn it into collision avoidance against precomputed configuration-space boxes), not a local-trigger problem.

The synthesis with CHAIN_10 is one sentence: the walls don’t move; the schedule’s joint-space path does. A ±2% retiming of the aim clock re-routes q(t) relative to immovable singular surfaces, changing the number of close encounters — which is precisely the SEED verdict (entries 23/14/2) and the hot band relocating. The 6-DOF doesn’t show it because its path is pinned against the wall at every speed (saturated floor).

3.2 Which wall: full reach, and why our heavy base makes it “kinematic”

The lineage matters here. [umetani1989resolved] derived the Generalized Jacobian J* = J_M − J_S·I_S⁻¹·I_M (the matrix family our kG instruments) and noted the heavy-base limit: as base inertia grows, the coupling correction vanishes and J* degenerates to the fixed-base arm Jacobian. ([wilde2018equations], the canonical modern construction, never states the limit — but it reads off directly from that paper’s explicit build, eqs. 52 and 110: cite Umetani–Yoshida for the point, Wilde for the machinery.) Consequence for us: with a spacecraft base much heavier than a UR3, the generalized map’s conditioning inherits the arm’s ordinary kinematic singular structure — full extension first. That is the structural reason s_min_G tracks the arm-extension factor at +1.0 and why the s_min_G/s_min_J study found the two interchangeable. The exotic “dynamic singularities” of the free-floating literature (mass-distribution-shifted singular sets, [papadopoulos1993dynamic]; coupling-matrix degeneracies, [nenchev2013reaction], [sone2016reactionless]) are largely suppressed for us — additionally so because our base is actuated: [papadopoulos1993dynamic] §VI shows constant base attitude reduces the singular set to the kinematic one. One sign-caution worth recording: in [sone2016reactionless]’s planar specimen model the coupling-inertia determinant is maximal — farthest from their dynamic singularities — at full arm extension (tied to manipulator-CoM-to-base-CoM distance, with their explicit caveat that spatial joint-axis geometry also matters): the opposite tail from the arm Jacobian’s extension singularity. Citing that paper carelessly could appear to contradict H3; in fact it corroborates that our spikes are arm-kinematic, not coupling-driven. And it closes the PIW/PDW question for our system: in this heavy, attitude-controlled limit the path-dependence subtlety collapses, so “which workspace is the helix in?” reduces to “how close does the demanded standoff reach approach the outer (full-extension) annulus” — precisely the quantity the schedule oracle of §4.1 computes.

3.3 Independent reproduction of H3’s mechanism

Two papers reproduce our promoted mechanism almost verbatim:

• [bruschi2025singularity], second simulation: when the demanded EE trajectory is not compatible with the base trajectory (and is feasible only for a finite window), then under their equal-priority combined design — the class our controller belongs to — “the robotic arm tends to reach a fully extended position, thereby entering a singular configuration.” (Their own task-priority law tracks straight through the same demand; that contrast is §4.2’s subject.) Their Definitions 1–2 make feasibility/compatibility a property of the demanded trajectory pair — not the controller. Our aim clock is that demand generator.
• [giordano2018workspace], momentum-dumping experiment: the EE goal drifts toward the boundary of the CoM-centered reachable workspace, the arm stretches to keep holding the pose, and generalized-Jacobian manipulability decays to “a practically singular configuration.” Their driver is contact-induced CoM drift; ours is the schedule sweeping the standoff geometry — the stretch→conditioning-collapse mechanism is identical.

And the controller’s own papers concede the gap our derate fills: [giordano2019coordinated] proves stability (for the nonredundant n = 6 case — the 7-DOF extension is our own, derivation_7dof.md) only inside Ω = {σ_min(J⁺) > 0} with zero margin and no enforcement mechanism; [giordano2020coordination] states the practical remedy is “suitably pre-planned trajectories [18,43] … to limit the occurrence of singularities” — its reference [43] being exactly [calzolari2020singularity]. The burden of staying away from singularities is, by the designers’ own admission, on trajectory/schedule design. Our κ bands are what that unfunded mandate looks like in flight.

3.4 Why the error concentrates inside the peaks (event dominance)

Four mutually consistent explanations, all from the fixed-base robustness literature:

• 1/σ amplification: joint velocity along the dying direction scales as 1/σ_m; near σ = 0.01 a 0.1 m/s command implies a 10 rad/s joint-rate demand ([chiaverini1997singularity], who sizes the singular-region threshold ε from exactly this ratio — our 0.025 caps amplification at ~40×).
• Loss of authority, not just noise: at singularity the wrench/velocity component along the singular direction maps to nothing — “loss of actuation in a singular direction” ([giordano2019coordinated] §VI-A); transpose/pseudo-inverse schemes move along the projection of the demand, “the result may be large end-effector errors” ([papadopoulos1993dynamic]).
• Region, not point: [nenchev2013reaction] observes — of the Operational Space Formulation’s inertia-weighted Jacobian inverse, his contrast case to RNS, but the argument is the inverse-of-a-quadratic-form nonlinearity that any Jacobian-inverting scheme shares — that “excessive fluctuation in the joint velocity can be expected” in the “(not necessarily small) vicinity” of kinematic singularities. Peaks are windows, not points.
• Feedback build-up: ill-conditioned inverse errors “build up when fed back” — closed-loop versions of non-robust schemes could not even run with significant gains ([chiaverini1997singularity] §VI). Our controller is exactly such a closed kinematic loop.

There is also a self-inflicted event class the literature warns about: hard thresholds make their own transients. Truncated-SVD inversion “generates discontinuous joint velocities due to the threshold”, and in the same case study a sudden primary-task error jump appears at the exit of an algorithmic singularity — driven by ill-conditioned null-space velocities integrated across the crossing ([chiaverini1997singularity] Fig. 2); a too-small threshold lets the schedule drive through the singularity and chatter ([torres1992minimizing]*, see §7 caveat); [sone2016reactionless] add a quartic term to their damping schedule specifically to keep the entry/exit smooth. Our stack has two hard switches (freeze at 0.025; hold-last-inverse at 0.005) and one measured exit transient (thaw jump p99 = 82.9°). The CHAIN_10 recommendation — audit freeze/derate handling before reconstruction math — is exactly what this literature would order.

4. What to do about it — a layered mitigation taxonomy

Ordered from most upstream (cheapest, most preventive) to most downstream (last-resort robustness). The right posture is layered: plan so entries are rare, spend redundancy to make them rarer, and make the inversion graceful for the residue.

4.1 Schedule layer — make avoidance a planning property (strongest leverage)

The controller-design literature itself sends the problem here: [giordano2020coordination]’s stated practical remedy is pre-planned singularity-aware trajectories; [calzolari2020singularity]’s whole program is planning-side; and flight practice agrees — ETS-VII’s arm trajectories were generated offline and executed feedforward ([nenchev2013reaction]; that was for momentum management rather than singularity avoidance, but it is the operational precedent for offline trajectory certification). The deliberate counterpoint is [bruschi2025singularity], which moves the handling into the control law instead (§4.2’s subject).

• Offline κ-along-schedule oracle. We can already predict the hot bands before flying: evaluate J⁺(q_des(s)) along the aim-clock schedule for each candidate path speed and locate the near-singular phase bands offline. [wilde2018equations] supplies closed-form joint-angle derivatives of every matrix involved if an analytic route is ever wanted; pragmatically, our existing bit-exact strided replay (r1_kernel_continuity.kappa_series) over a desired-path rollout is the same oracle. This is the conditioning twin of the T1 coverage oracle, and it converts ±2% schedule sensitivity from a hazard into a design degree of freedom: if +2% produces zero post-startup peaks (it does — CHAIN_10), then schedule selection/re-timing is a zero-hardware mitigation, gated only by full-helix validation of coverage/tracking (the round-3 lesson: validate on the full helix, short gates mislead). Adjacent precedent: [das2025understanding] re-plan joint trajectories in an SVD basis — of the coupling matrix, targeting base disturbance rather than conditioning — the same shape of fix applied to a different matrix.
• Reach corridor / standoff design. The compatible-trajectory construction following [bruschi2025singularity]’s Def. 2 gives the principled core of our “reach margin ≤ 70%” idea: verify offline that a chosen constant base–EE offset admits an arm IK solution “as if the base were fixed on the ground.” (Their check is a single IK-existence test on one constant offset; extending it per-pose along our varying standoff path is our generalization.) Our lever is literally one knob: the standoff radius camera_radius = radius_scale(0.55) × r_reach (parameter_loader.py:183) and the reach clamp at ee_guidance.py:766-784. [giordano2018workspace] gives the same insight from the workspace side: keep the aim target inside a favorable region of the CoM-centered reachable workspace — in their experiments manipulability then stayed favorable, with their own caveat that unfavorable base attitudes could still reach zero manipulability (moot for our attitude-controlled base, but their guarantee is conditional). A full Calzolari S-Map is overkill for one arm with one dominant singular factor — but their point-cloud pruning stage (approximate — it locates near-singular regions, certifies nothing), or simply our oracle above, captures the working benefit without the 170-hour formal computation; only the full S-Map buys the guarantee.

4.2 Architecture layer — who carries the task near a wall

• Task-priority restructure ([bruschi2025singularity], extending [chiaverini1997singularity] to space robots): pseudo-invert the full base+arm Jacobian [J_b J_m] — which has rank 6 for every q because the base alone provides 6-DOF EE mobility — and demote base pose to a transpose-based null-space task. The control law is then non-singular on the whole configuration manifold (their Property 1): the base carries the EE through the arm’s singular band, paying in base-pose excursion (and unmodeled fuel) instead of in dwell. This is the structural alternative to derating: our derate trades time (dwell cliff 431/231/3.27 s); their priority swap trades base tracking inside the band. For a sustained 30-rev inspection (vs their seconds-long capture), the fuel/pointing cost of repeated base excursions is the open question — a candidate for a costed A/B, not an adoption.
• Spend the 7th joint, don’t just freeze it. [papadopoulos1993dynamic] §VI suggests redundancy may provide the runtime escape (explicitly flagged there as requiring further research); [giordano2020coordination] proposes null-space control as the mitigation for their own 7-DOF arm’s unmanaged-null-space disturbance (deferred to future work); [nenchev2013reaction] frames the spare-DOF null channel as something to use for a secondary task — by analogy only, since his channel is the Reaction Null Space (kernel of the coupling-inertia matrix, a dynamic object), not the kinematic task-null direction L2 climbs. This is exactly the pre-registered round-2 lever L2 (σ-gradient null-space pull): drive the spare DOF uphill in σ₆, away from the extension wall, at zero primary-task cost. The kernel freeze remains the right numerical guard (a frozen basis beats a 69°/step garbage basis), but it is conditioning-neutral; L2 is the conditioning-active complement. Two cautions from the literature: (i) stacked tasks create algorithmic singularities — rank loss of the projected map while each matrix is individually healthy ([chiaverini1997singularity] §IV; [sone2016reactionless] observed them as counter-rotating wrist joints in narrow windows). The diagnostic is cheap: log the determinant/σ_min of the restricted (null-space-projected) map alongside kJ/kG and check which one collapses (Sone’s three-determinant fingerprint). (ii) If L2 is built on the classic (J_C(I−J⁺J))⁺ inversion it adds singularities; Chiaverini’s reformulated law (his eq. 34: project the secondary task’s own pseudo-inverse) avoids inverting the projected matrix and degrades by shrinking the conflicting null-space motion instead of amplifying it.
• One DOF, three claimants. derivation_7dof.md already flags that σ-climbing, envelope clearance, and pan-centering compete for the single null DOF. The literature does not resolve allocation; it does say the budget is real ([nenchev2013reaction]: under the full momentum constraint a 7-DOF arm’s reactionless manifold is 1-D, relaxing to n−3 = 4 if only base orientation is protected; the kinematic spare DOF available to L2 is exactly one either way — you cannot buy everything with one joint).

4.3 Inversion layer — make the residue graceful (audit before adding)

The published menu maps one-to-one onto layers we already run, which turns this section into an audit rather than a shopping trip:

• Speed scaling is the singularity-consistent choice — and only part of our derate is speed scaling. [sone2016reactionless] compare damped least squares (DLS) against the Singularity-Consistent method (slow down along the path, keep the velocity direction exact): DLS corrupts speed AND direction inside the singular region; SC’s “effect … is only upon the speed.” The derate’s velocity/accel-scaling component (base_guidance.py:118-141) is SC-shaped — that component’s cost surfaces as schedule lag (dwell, traversal debt), not pointing error, and this is the strongest published justification for keeping it. But the same ramp also scales the arm gain blocks and the base wrench (breve_controller.py:313-316, 326-327) — a feedback-stiffness reduction in all task directions, which is itself a DLS-type cost in Sone’s taxonomy. So the sharpened audit question: in-window pointing error must be attributed among (a) the Tikhonov layer (σ < 0.05), (b) the damped J⁺ inverse, (c) switching transients (freeze/thaw, hold-last-inverse), and (d) the derate’s own gain-scaling component — only the speed-scaling component is exonerated by the SC argument. [chiaverini1997singularity] eq. 14 quantifies the DLS cost: damping injects error into all task directions, scaling as λ²/(σᵢ²+λ²).
• If damping must stay, make it directional. Numerical filtering (originated by Maciejewski & Klein 1988; analyzed as eq. 16 of [chiaverini1997singularity]) damps only along the near-singular output direction u_m, preserving pseudo-inverse accuracy in every healthy direction — the right shape for protecting our tight pe_p99 gate. Cost: an on-line estimate of u_m (~100 flops, recursive); with two simultaneous singular factors (our documented case) the β-augmented variant (eq. 20 there, same origin) applies.
• Smooth every engage/exit. Variable damping with a smooth ramp (his eq. 15 — our derate ramp already has this shape, good), Sone’s quartic-smoothed λ(σ) schedule, and the Holt chattering warning all say the same thing: hard switches at σ boundaries manufacture events exactly where the system is most fragile. Our two binary switches (freeze at 0.025, hold-last-inverse at 0.005) and the measured 83° thaw jump are the audit targets. The round-2 lever L4 (thaw policy) is the pre-registered home for this.
• Threshold sizing has a principled recipe. [chiaverini1997singularity] sizes the singular-region threshold from the joint-rate amplification ratio 1/ε (his 0.01 ⇒ 10 rad/s per 0.1 m/s commanded, offered “for the sake of argument”); [torres1992minimizing]* states the same principle qualitatively — bound the commanded joint step against joint-level saturation limits. Together they support a defensible re-derivation of 0.025 for the thesis, replacing “it worked.”

4.4 Metric layer — are we measuring the right thing? (mostly yes)

• σ_min is the literature’s standard proximity measure ([chiaverini1997singularity]; [giordano2019coordinated]’s Ω; [giordano2020coordination]’s Ω) — our s-metric-first, derate-fraction-headline convention is well aligned. κ is nowhere used as a trigger in these papers; treating kJ as a logged witness (noisy, scale-free) rather than a control signal is correct.
• One candidate addition, with homework: adapting the SVD-spectrum entropy H_norm of [das2025understanding] (their eq. 29 — defined on the base–arm coupling matrix, where low entropy flags one dominant mode) to our Jacobian spectrum as a one-collapsing-direction vs broadband discriminator is our extrapolation, not their claim — and a single near-zero σ among healthy ones is a weak entropy signature (H_norm ≈ 0.92 of max for n = 7). Free to compute since the spectrum is already in hand, but verify its discriminating power offline before trusting it as a metric.
• One structural caution: a local σ threshold cannot see a thin singular wall coming ([calzolari2020singularity] Fig. 4 — a manipulability-constrained step crossing a singularity undetected). Locally we are fine (the derate ramp reacts), but predictively only the schedule-level oracle (§4.1) closes that gap.

5. Direct answers to the prompt

What is causing the spikes? The aim-clock schedule, in specific orbit-phase bands, demands standoff poses that pull the arm toward its full-extension singular surface — a fixed wall in joint space ([papadopoulos1993dynamic], [calzolari2020singularity]). Because our base is heavy and actuated, the generalized map’s conditioning inherits the arm’s kinematic reach factor ([umetani1989resolved] heavy-base limit) — hence s_min_G ≡ arm extension (corr +1.0). A ±2% retiming re-routes the joint-space path relative to the walls, moving the bands and the entry count (SEED). The κ spike is the symptom; the cause is a schedule-feasibility deficit in the sense of [bruschi2025singularity] Defs. 1–2. Numerics (H1) and surface curvature (H2) are exonerated both empirically (CHAIN_10) and structurally (no paper offers a mechanism for either).

How do we alleviate them? In order of leverage: (1) re-time/re-shape the schedule offline against a κ-along-schedule oracle so the helix never demands near-wall reach (§4.1); (2) tighten the standoff/reach corridor — one existing knob (§4.1); (3) spend the 7th joint’s null motion climbing σ₆ (lever L2), with the algorithmic-singularity diagnostic logged (§4.2); (4) keep the derate’s speed-scaling component (the published singularity-consistent design), attribute the in-window pointing cost among the four inversion-layer suspects — including the derate’s own gain-scaling component — smooth the two hard switches (L4), and if damping is implicated, switch it to directional filtering (§4.3).

How do we reduce error and smooth the trajectory? The event-dominant verdict says the error budget is spent at entries/exits, not in cruise — so L1 (velocity-lag feedforward) must be designed and evaluated against the startup/hot-band events (CHAIN_10 follow-up 4), and the smoothing wins are at the σ-boundary transitions (thaw policy, hold-last-inverse), not in the EE path itself. Note the reframe: it is not the end-effector trajectory that needs smoothing — it is the arm’s route through configuration space, which posture/null-space action shapes.

How do we harden the robot? Make schedule sensitivity a designed quantity — with a margin band, not a point. The clean +2% arm sits beside a non-monotone dwell cliff (431/231/3.27 s), so choosing it because it was clean at one setting is exactly what a committee pokes. The oracle should sweep a 2-D neighborhood (speed × standoff) and certify that a whole band around the chosen operating point stays wall-clear; the residual sensitivity then feeds the Phase-A CVaR axis (already pre-registered as a first-class risk source), and the fragility campaign measures noise on top of a system whose deterministic singular encounters are planned out, not accidental.

6. Are we asking the right questions?

Mostly yes — three confirmations and three reframes.

Confirmed right: (1) pre-registered falsification with progress-indexed (not time-indexed) peaks was the correct instrument: the literature’s central fact (fixed walls, moving path) is only visible in phase/configuration coordinates. (2) Preferring σ_min over κ as the control signal matches the entire robustness literature. (3) The event-vs-broadband pointing discriminator was exactly the right question, and its EVENT answer is what [chiaverini1997singularity] predicts for a closed-loop pseudo-inverse kinematic loop — with [giordano2019coordinated] supplying the loss-of-actuation-at-the-event mechanism (stated there for its transpose-type realization).

Reframe 1 — from “why does κ spike” to “where does the demanded path graze the wall.” κ is a consequence. The upstream object is the demanded joint-space path q_des(s) and its clearance from the fixed singular set. That question is answerable offline, deterministically, per schedule candidate — which is a stronger position than diagnosing peaks after flight. H3’s closure (swap reach_b for the true extension margin / strided s_min replay) is this reframe executed on logged data.

Reframe 2 — schedule sensitivity is a design DOF, not just a risk. CHAIN_10 treated ±2% as a perturbation axis; the +2% arm (zero post-startup peaks, dwell 3.27 s) is also an existence proof that a clean schedule exists nearby. The question “how do we harden against ±2%?” should be paired with “which retiming do we deliberately choose?” — gated, as always, on full-helix coverage/tracking validation.

Reframe 3 — the 6-DOF and 7-DOF are different problems wearing the same metric. The 6-DOF sits saturated on the reach floor (64–80% derated; bands immobile): its fix is geometric (standoff/reach corridor — §4.1), and no inversion-layer tuning will move it. The 7-DOF is marginal (entries swing 10× under ±2%): its fixes are schedule re-timing and null-space σ-climbing. One number (derate fraction) headlines both, but the treatments do not transfer across arms — worth keeping separate in round-2 scoring.

One question we were not asking: does our own robustness stack create events? The literature’s threshold-transient warnings (§3.4) plus our measured 82.9° thaw jumps and the σ<0.05 Tikhonov layer suggest an attribution audit within the mitigation stack — which layer owns the in-window pointing error — before any new machinery is added. That is a sharper version of CHAIN_10’s “review freeze/derate handling before the reconstruction math.”

7. Source-hygiene alerts (action needed before thesis citation)

1. torres1992minimizing is the wrong paper under that citekey. The markdown (and per the sweep, the staged source) is Holt & Desrochers, “Inertial-Space Disturbance Rejection for Space-Based Manipulators” (NASA N94-26281) — the Torres–Dubowsky enhanced-disturbance-map paper appears only as its reference [4]. The Holt content is genuinely useful (the σ_min threshold trade-off and chattering warning cited in §3.4/§4.3 are theirs), but nothing from this folder may be cited as Torres–Dubowsky. Re-fetch the real JGCD 1992 paper; the bib note already says the NTRS report was “staged as source PDF” — that staging grabbed the wrong report.
2. Bib ↔︎ folder mismatches (the bib header promises citekey == folder name): folders giordano2020wholebody and giordano2021compliant have no bib entries (and no markdown conversions — not swept); bib entries nenchev1999reaction, nakamura1986inverse, yoshikawa1985manipulability, yoshikawa1985dynamic, chiaverini1994review, wampler1986manipulator, vafa1990kinematics, whitney1969resolved have no folders. (Loose PDFs for at least the Nenchev 1999 paper sit in docs/Most_Relevant/ unfoldered.) Additionally, Maciejewski & Klein 1988 — the origin of directional/filtered damping, per [chiaverini1997singularity]’s own attribution — is in neither the bib nor the folders; add it if §4.3’s filtering route is pursued.
3. Conversion fidelity notes from the sweep: chiaverini1997singularity.md drops scattered inline math symbols (display equations intact); umetani1989resolved.md is OCR of the Japanese JRSJ version (abstract and the load-bearing GJM equations 14, 16–23 readable; eq. 8’s subscripts and the eq. 13 block garbled — re-derive from neither); giordano2019coordinated.md reproduces the two known published typos (eq. 31 sign, eq. 34d block order — already in project memory) and garbles eq. 4 / 34b — do not re-derive from those transcriptions. The two textbooks (siciliano2009robotics, spong2006robot) were deliberately not page-read in this study; cite them only as background.

8. Named follow-ups (smallest convincing test, mapped to existing harness)

Deliberately not recommended now: a full Calzolari S-Map (formal machinery disproportionate to one dominant singular factor — YAGNI), adopting Bruschi’s priority swap outright (fuel/base-pose cost unmodeled for a 30-rev inspection; would need a costed A/B), and any new runtime metric — even H_norm enters as an offline pilot first.

References (citekey → source; all in docs/Most_Relevant/papers/<key>/<key>.md unless noted)

Internal evidence: chain10_kappa_forensics_verdict.md, chain10_offline_tables.md, chain10_codepath_audit.md, sming_vs_sminj_study.md, phase00_round1_verdict.md, round_2_questions.md, chain10_kappa_forensics_design.md, generated_reports/math/derivation_7dof.md, generated_reports/GNC/Jun10_26/mission_1_success.md, plus code refs as cited (snapshot of utils/robot.py, GNC/breve_controller.py, GNC/base_guidance.py, GNC/com_guidance.py, GNC/guidance/ee_guidance.py, YAMLs_by_domain/parameters.yaml at Jun 12, 2026).