posture_oracle_gap

The Posture Oracle Gap: From a Kinematic Existence Proof to a Buildable Posture Lever

Statement

A kinematics-only “oracle” rollout of the demanded camera schedule on the 7-DOF
redundant arm exhibits a singularity-free route: it tracks the demand with median error
$\approx 0$ and keeps its worst ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ above the
singular floor at every flown speed (worst ${\sigma }_6\ge 0.05$, never within a factor of
two of the pothole threshold $0.025$). The live free-flying system, flying the same
demand, instead spends up to $27\%$ of its steps below ${\sigma }_6=0.025$. So a wall-clear
route through joint space exists; the live policy simply does not take it.

This result names and bounds the oracle gap — the amount of the oracle’s clean route
that survives once the oracle’s three idealizations are removed: (i) it moves the arm
kinematically (no forces, no momentum bookkeeping); (ii) it holds the base exactly on
its guidance trajectory; and (iii) it picks postures with a policy (the damped
pseudo-inverse) the live controller does not run — the live system reconstructs velocities
on the self-motion (${\mathbf{v}}_n=0$) section, parking the spare degree of freedom
rather than steering it (dynamic_singularity,
singularity_threshold_cascade).

Claim (ours). The oracle gap decomposes into three offline-computable utilization
bounds, each gating the next, each computable from data already logged (no new
simulations), and each cheap to fail. Writing ${\mathbf{q}}_O(s)$ for the oracle’s
witness route (the wall-clear thread it exhibited) and ${\mathbf{q}}_L(s)$ for the live
route under the ${\mathbf{v}}_n=0$ section, the lever “bias the live route toward
${\mathbf{q}}_O$” is admissible iff all three pass:

$$U_1=\underset{s}{\max }\max \left(\frac{{\tau }_{\mathrm{req}}(s)}{{\tau }_{\mathrm{sat}}},\frac{{\dot{q}}_{\mathrm{req}}(s)}{{\dot{q}}_{\lim }}\right)\le 1\text{(dynamical admissibility),}$$ $$U_2=\underset{s}{p99}\frac{L_J\Vert \delta \mathbf{q}(\text{base error})\Vert }{0.025}<1\text{(margin robustness),}$$ $$U_3=\underset{s}{p99}\frac{|\mathrm{d}{\phi }_O/\mathrm{d}t-d(s,{\mathbf{q}}_O)|}{v_{n,\max }}<1\text{(reachability under one spare DOF).}$$

Only if $U_1,U_2,U_3$ all pass does an implementation get written; the full-helix A/B
remains the adoption gate, as for every lever. This converts a metaphysical question —
“could the 7-DOF ever avoid these walls?” — into three engineering inequalities, and turns
the kinematic existence proof into a buildable posture lever.

Assumptions

• Free-flying regime, 7-DOF redundant arm (one spare degree of freedom). The 6-DOF
  arm has no spare DOF, so the construction does not apply to it (per the per-arm split
  of the 7-DOF derivation). Ours.
• A route is a continuous map $\mathbf{q}(s)\in F(s)$ where $s\in [0,s_{\mathrm{end}}]$
  is arclength along the demanded schedule and the fiber
  $F(s)=\{\mathbf{q}:\mathbf{q}\text{ realizes the demanded camera pose }g(s)\}$
  is generically a one-dimensional self-motion curve.
• The witness route ${\mathbf{q}}_O(s)$ is logged from the schedule-$\kappa$ oracle
  rollout with ${\sigma }_6({\mathbf{q}}_O)\ge 0.05$ throughout and joint rates inside the
  live $50rad/s$ clip; it is the route to be tracked.
• Each bound consumes only already-available data: $U_1$ from inverse dynamics along the
  logged ${\mathbf{q}}_O$ using the controller’s own reduced objects
  $\breve{\mathbf{M}},\breve{\mathbf{C}}$
  (circumcentroidal_decoupling); $U_2$ from logged base
  errors plus a numerically estimated local Lipschitz constant $L_J$ of
  $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{+}$ along ${\mathbf{q}}_O$ plus Weyl’s inequality; $U_3$
  from the kernel direction and task velocities the robot code already produces.
• The witness-route headroom is ${\sigma }_6({\mathbf{q}}_O)-0.025\ge 0.025$ (it never dips
  below $0.05$, and the pothole threshold is $0.025$).
• One spare DOF, three claimants (posture lever, envelope, pan-centering): $v_{n,\max }$
  is the self-motion rate granted to the lever, not the full null-space authority.

Proof sketch

Full derivation / methodology: posture_proof.md.
This is a methodology and a set of bounds, not yet a closed theorem — hence
status: developing; the proof file notes elaboration is pending for the constants
($L_J$, drift $d$, granted $v_{n,\max }$).

Geometric framing — each demanded camera pose admits a one-parameter family of arm
configurations (the fiber / self-motion circle); as the schedule sweeps the pose, the
fibers sweep a tube in joint space. A route is a thread through the tube on the moving
fiber; a section is a rule picking one point per fiber. The live ${\mathbf{v}}_n=0$ rule
is a valid ghost-motion suppressor but makes the route an accident of history — nothing
steers it from walls. The lever replaces/biases that section to track the wall-clear
witness route. The gap is everything obstructing that convergence, split three ways:

1. Gap 1 — dynamical admissibility ($U_1$, direct/constructive). Differentiate the
   logged ${\mathbf{q}}_O(s)$ along $s$ at mission speed ($\dot{s}=v$) to get the demanded
   joint rates/accelerations; the reduced dynamics $\breve{\mathbf{M}},\breve{\mathbf{C}}$
   convert these to required torques and base reactions. Pre-register utilization $U_1$;
   if $U_1>1$ the lever is dead on arrival at zero simulation cost. Subtlety: differentiate
   the sampled route without amplifying noise (use the analytic schedule rate, smooth
   ${\mathbf{q}}_O^{\prime }$ as surface paths are smoothed).
2. Gap 2 — margin robustness ($U_2$, perturbation bound). Chain three links.
   (a) Weyl’s inequality for singular values: ${\sigma }_{\min }(\mathbf{A}+\mathbf{E})\ge {\sigma }_{\min }(\mathbf{A})-\Vert \mathbf{E}{\Vert }_2$ — a perturbation of
   spectral norm $\Vert \mathbf{E}{\Vert }_2$ cannot drop the smallest singular value by
   more than $\Vert \mathbf{E}{\Vert }_2$. (b) Jacobian Lipschitz smoothness:
   $\Vert ({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{+}(\mathbf{q}+\delta \mathbf{q})-({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{+}(\mathbf{q}){\Vert }_2\le L_J\Vert \delta \mathbf{q}\Vert$,
   with $L_J$ estimated by finite differences along ${\mathbf{q}}_O$. (c) Base-error →
   configuration perturbation: to first order $\delta \mathbf{q}$ is the arm Jacobian
   inverse applied to the logged base-pose error. Against headroom $0.025$ this gives $U_2$;
   the lever survives only if measured base error cannot eat the entire witness margin.
3. Gap 3 — reachability ($U_3$, 1-D reduction). Parameterize each fiber by a scalar
   coordinate $\phi$ (angle along the self-motion circle); every route is a scalar
   $\phi (s)$ and the witness route a known reference ${\phi }_O(s)$. To first order
   $\dot{\phi }=v_n+d(s,\mathbf{q})$, where $v_n$ is the chosen self-motion velocity
   (currently pinned to $0$) and $d$ is the task-induced self-motion drift (the projection
   of the task-consistent joint velocity onto the kernel direction). Pre-register $U_3$
   against the granted $v_{n,\max }$; if $U_3<1$ a simple proportional law on $\phi$
   tracks ${\phi }_O$ with available authority. By construction $v_n$ moves the arm only
   along the fiber (the M-orthogonal $z_a,\hat{k}$ machinery), answering “does the null-space
   term fight the task?” structurally — it cannot.

Ordering is load-bearing: each bound is cheaper than what it gates, and any failure kills
the lever for free. Honest caveat for registration: the witness route came from one
policy (the oracle’s damped inverse). A failure of $U_1$–$U_3$ kills this witness, not
the idea — a different wall-clear route might still pass.

Source / provenance

• Ours — the gap decomposition, the three utilization bounds, the fiber/section
  formalism, and the “buildable posture lever” framing are this thesis’s own (7-DOF
  free-flying work). See posture_proof.md.
• Standard tool reused in Gap 2: Weyl’s inequality for singular values (singular-value
  perturbation; e.g. Horn & Johnson, Matrix Analysis). One inequality, three-line
  derivation from the variational characterization of ${\sigma }_{\min }$.
• Diagnostics behind the motivating numbers ($27\%$ of live steps below ${\sigma }_6=0.025$;
  oracle worst ${\sigma }_6\ge 0.05$): the schedule-$\kappa$ oracle rollout and the live
  conditioning audit feeding singularity_threshold_cascade.
• Notation: notation.md — ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$,
  self-motion / null-space reconstruction (${\mathbf{v}}_n$), reduced objects
  $\breve{\mathbf{M}},\breve{\mathbf{C}}$, schedule progress $s$.

Caveats

• Methodology, not yet a theorem. The three inequalities are pre-registrable bounds;
  the constants ($L_J$, drift $d$, granted $v_{n,\max }$) and the closed reachability proof
  are pending elaboration in posture_proof.md
  (hence status: developing).
• 7-DOF only. No spare DOF on the 6-DOF arm ⇒ no fiber to steer; the construction is
  empty there.
• Single witness. All three bounds are evaluated on the oracle’s one damped-inverse
  route. A bounded “no” (the witness fails) is a defensible thesis lemma but does not
  prove no wall-clear route is buildable.
• Empirical thresholds. ${\sigma }_6=0.025$ (pothole) and ${\sigma }_6\ge 0.05$ (witness
  headroom) are configuration-specific; the gap must be re-bounded for any other
  arm/speed/schedule.
• First-order links. The base-error → $\delta \mathbf{q}$ map and the fiber-coordinate
  ODE are first-order; large base excursions or near-freeze-floor regions (where $\phi$
  is ill-defined, ${\sigma }_6$ below the kernel-freeze floor) need higher-order or guarded
  treatment.
• Adoption gate unchanged. Passing $U_1$–$U_3$ only licenses a build; the full-helix
  A/B remains the adoption gate.

Related

• Sibling results: singularity_threshold_cascade —
  the live conditioning stack and the $\sigma$ floors the witness route must clear ·
  circumcentroidal_decoupling — supplies
  $\mathbf{\Gamma }$, the hat/breve distinction, and the reduced
  $\breve{\mathbf{M}},\breve{\mathbf{C}}$ used in Gap 1 ·
  singularity_geometry — the joint-space
  geometry of the singular walls the fibers must thread.
• Topics: dynamic_singularity ·
  motion_planning.
• Full derivation: posture_proof.md.
• Notation: notation.md.