steady_state_error_floor

Steady-State (Cruise-Lag) Pose-Error Floor

Statement

For the coordinated control law tracking a moving reference at constant cruise, the stacked
outer-loop pose error $\tilde{\mathbf{x}}=[{\tilde{\mathbf{x}}}_b;{\tilde{\mathbf{x}}}_e]\in {\mathbb{R}}^9$
(notation.md) does not decay to zero: it settles to a non-zero standing offset
${\tilde{\mathbf{x}}}_{ss}$, the cruise-lag floor. When the error-rate Jacobian
${\mathbf{J}}_{\tilde{x}}$ is invertible,

\tilde{\boldsymbol{x}}_{ss}=-\big(\boldsymbol{J}_{\tilde{x}}^\top\breve{\boldsymbol{K}}\big)^{-1}\Big[\breve{\boldsymbol{C}}\,\breve{\boldsymbol{v}}_d+\big(\boldsymbol{C}_c+\breve{\boldsymbol{D}}\,\breve{\boldsymbol{G}}_{v_c}\big)\dot{\tilde{\boldsymbol{x}}}_c\Big]. \tag{final.tex eq x_ss / current_sota eq 4.14}

This is the equilibrium of the working closed-loop attitude+EE block (eq 4.12). The floor is set by
the proportional restoring term ${\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}$ balancing two
forcing terms: the cruising arm’s Coriolis force $\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d$,
and the CoM coupling $({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c$.

Coriolis-only idealization. If the CoM is tracked perfectly (${\dot{\tilde{\mathbf{x}}}}_c=0$),
the coupling term drops and the floor reduces to

\tilde{\boldsymbol{x}}_{ss}=-\big(\boldsymbol{J}_{\tilde{x}}^\top\breve{\boldsymbol{K}}\big)^{-1}\breve{\boldsymbol{C}}\,\breve{\boldsymbol{v}}_d. \tag{current_sota eq 4.14, Coriolis-only limit}

The operational metric is the magnitude of the EE-position part,
$p_e=\Vert {\tilde{\mathbf{x}}}_{e,\text{position}}\Vert$ (current_sota §7), which under finite-difference
feedforward settles near $\sim 0.15\mathrm{m}$ and is predicted to drop toward this analytic floor once
the analytic acceleration feedforward (current_sota §5.6) replaces the finite-differenced demand. Ours:
this floor formula and its cruise-lag interpretation are the project’s reading of the Giordano closed loop,
not a result stated in the paper (which analyzes the regulator, eq 4.15-4.16).

Assumptions

• Steady state along the reference: $\dot{\tilde{\mathbf{x}}}=0$, so when
  ${\mathbf{J}}_{\tilde{x}}^{-1}$ exists the velocity matches the demand, $\breve{\mathbf{v}}={\breve{\mathbf{v}}}_d$,
  and (by differentiating $\dot{\tilde{\mathbf{x}}}=0$) the inertial acceleration
  terms $\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}$, $\breve{\mathbf{M}}{\dot{\breve{\mathbf{v}}}}_d$
  and the velocity-error damping $\breve{\mathbf{D}}(\breve{\mathbf{v}}-{\breve{\mathbf{v}}}_d)$
  all cancel from eq 4.12. What survives is the algebraic balance
  $0=-\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d-{\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}{\tilde{\mathbf{x}}}_{ss}-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c$.
• ${\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}$ invertible: requires the error-rate
  Jacobian ${\mathbf{J}}_{\tilde{x}}=\mathrm{blkdiag}({\mathbf{J}}_{{\tilde{x}}_b},{\mathbf{J}}_{{\tilde{x}}_e})$
  nonsingular (attitude errors away from the quaternion singular set) and $\breve{\mathbf{K}}$ SPD.
• Non-zero forcing at the operating point: ${\breve{\mathbf{v}}}_d\ne 0$ (the arm is
  cruising), so $\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d\ne 0$. A floor exists only
  because the controller is asked to track motion, not hold a setpoint — at rest the regulator
  (eqs 4.15-4.16) drives $\tilde{\mathbf{x}}\to 0$.
• Circumcentroidal reduction in force: the $9\times 9$ attitude+EE block ($\breve{\mathbf{M}},\breve{\mathbf{C}}$,
  reduced) is the relevant subsystem, the CoM $3\times 3$ having decoupled — see
  circumcentroidal_decoupling. The hat/breve distinction is
  load-bearing here: $\breve{\mathbf{C}}$ in eq 4.14 is the reduced $9\times 9$ Coriolis, not
  the full $12\times 12$ $\hat{\mathbf{C}}$.

Why the CoM-coupling term is kept

The Coriolis-only limit assumes perfect CoM tracking, ${\dot{\tilde{\mathbf{x}}}}_c=0$.
In practice the inner CoM loop is held off equilibrium by residual forcing: the measured CoM error
settles near $10^{-3}\mathrm{m}$, not zero. The CoM error obeys the homogeneous damped second-order
system $m{\ddot{\tilde{\mathbf{x}}}}_c+{\mathbf{D}}_c{\dot{\tilde{\mathbf{x}}}}_c+{\mathbf{K}}_c{\tilde{\mathbf{x}}}_c=0$
(eq 4.9), but with a small standing ${\dot{\tilde{\mathbf{x}}}}_c\ne 0$ the coupling
term $({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c$
remains a genuine (if small) contributor to the EE floor, so it is retained in eq 4.14.

Proof sketch

No dedicated proof yet (Phase-D deepening home). The floor formula is used in
derate_impedance_vs_speed.md (the
${\tilde{\mathbf{x}}}_{ss}\to \frac{1}{\gamma }{\tilde{\mathbf{x}}}_{ss}$ relocation under impedance
softening) and referenced in threshold_derivation.md;
a dedicated generated/math/steady_state_error_floor.md write-up is planned for Phase D.

1. Start from the project’s working control equation (current_sota eq 4.12), the expanded closed loop
   for the reduced attitude+EE block with damping-on-error and acceleration feedforward.
2. Impose $\dot{\tilde{\mathbf{x}}}=0$. Through the rate map (eq 4.3),
   $\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}+{\mathbf{J}}_{\tilde{x}}{\breve{\mathbf{G}}}_{v_c}{\dot{\tilde{\mathbf{x}}}}_c$,
   conclude $\breve{\mathbf{v}}={\breve{\mathbf{v}}}_d$ at steady state (when ${\mathbf{J}}_{\tilde{x}}^{-1}$
   exists), cancelling the inertial and velocity-error damping terms.
3. Solve the residual algebraic balance for ${\tilde{\mathbf{x}}}_{ss}$, giving eq 4.14.
4. Specialize ${\dot{\tilde{\mathbf{x}}}}_c=0$ for the Coriolis-only idealization; bound
   the kept coupling term from the measured $\sim 10^{-3}\mathrm{m}$ CoM error.

The deepening (sensitivity of ${\tilde{\mathbf{x}}}_{ss}$ to $\breve{\mathbf{K}}$, the
$\gamma$-derate relocation ${\tilde{\mathbf{x}}}_{ss}\to \frac{1}{\gamma }{\tilde{\mathbf{x}}}_{ss}$
under impedance softening, and the analytic-FF floor prediction) is Phase D work.

Source / provenance

• Literature: giordano2019coordinated supplies the closed-loop
  structure (eqs 22, 26-31, 34) and the regulator stability result (eqs 36-38). The paper analyzes the
  regulator (setpoint), where $\tilde{\mathbf{x}}\to 0$; it does not state a cruise-lag
  floor. Regime: Giordano 2019 is a free-flying (thrusters + arm, actuated base) formulation —
  the same regime as ours; this is not a free-floating result.
• Ours: the steady-state floor ${\tilde{\mathbf{x}}}_{ss}$ (eq 4.14), its cruise-lag reading, the
  retention of the CoM-coupling term against the $\sim 10^{-3}\mathrm{m}$ measured CoM error, and the
  analytic-FF floor prediction are the project’s own contributions, built on the project’s working
  control equation (current_sota eq 4.12, final.tex eq 34b).
• Builds on circumcentroidal_decoupling (the $\breve{\cdot }$ reduced
  block in which eq 4.14 lives).

Caveats

• Breaks where ${\mathbf{J}}_{\tilde{x}}$ is singular (large attitude errors, quaternion
  singular set): the inversion $({\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}{)}^{-1}$ fails and
  $\breve{\mathbf{v}}={\breve{\mathbf{v}}}_d$ no longer follows. The formula is a small-error,
  on-reference statement.
• Distinct from singularity collapse. The cruise-lag floor is a controller-stiffness phenomenon, not
  a kinematic-singularity one. Near a circumcentroidal singularity the relevant object is
  ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ and the floor can relocate under the impedance derate
  (${\tilde{\mathbf{x}}}_{ss}\to \frac{1}{\gamma }{\tilde{\mathbf{x}}}_{ss}$, current_sota §6.3) — the
  arm goes slack rather than driving toward the singular direction. That relocation is a separate
  analysis (singularity-handling layer).
• Coriolis-only is optimistic. It assumes perfect CoM tracking; the kept coupling term is the honest
  correction for the off-equilibrium inner loop. Quantifying its contribution is Phase D.
• Operational floor vs analytic floor. The measured $p_e\sim 0.15\mathrm{m}$ is under
  finite-difference feedforward; the analytic floor of eq 4.14 is the predicted limit under the
  analytic acceleration feedforward (current_sota §5.6). The gap closure is unverified pending the
  Phase D deepening.

Implementation (sims wiki)

External — into the code wiki via the sims_wiki/ symlink (resolves in Obsidian, not GitHub).

• breve_controller — the closed loop whose equilibrium is this floor ${\tilde{\mathbf{x}}}_{ss}$.
• base_guidance — the outer cruise demand that produces the lag.