Short answer: no. The floor formula is untouched.
The two results live on different layers — the floor is the
equilibrium of the continuous closed loop, the \(\omega_b\) ripple is a property of the
explicit integrator — and a continuous-time quantity cannot be
moved by the choice of time-stepping. There is exactly one genuine seam
between them (the steady-state assumption \(\breve{\bm v}\to\breve{\bm v}_d\)), and the
\(\omega_b\) note’s own near-zero-mean
finding is what keeps that seam from biasing the floor. Separately, one
prose sentence in error_floor.tex now sits in
tension with the \(\omega_b\) verdict
and should be reconciled (it is an attribution issue, not a formula
error).
Sources compared: - Floor: error_floor.tex eq
x_ss; steady_state_error_floor
(current_sota eq 4.14 / final.tex eq 34b), built on giordano2019coordinated.
- Ripple: omega_b_forward_euler_instability.md;
discretization_stability_omega_b.
It pays to write each result as what it actually is, because the confusion only arises when both are described loosely as “the near-singularity oscillation.”
The floor is a fixed point. Impose the steady-state condition \(\dot{\tilde{\bm x}}=\bm 0\) on the reduced attitude+EE closed loop (current_sota eq 4.12). When the error-rate Jacobian \(\bm J_{\tilde x}\) is invertible this forces \(\breve{\bm v}=\breve{\bm v}_d\), which cancels the inertial terms and the velocity-error damping \(\breve{\bm D}(\breve{\bm v}-\breve{\bm v}_d)\). What survives is an algebraic balance whose solution is
\[ \tilde{\bm x}_{ss}=-\big(\bm J_{\tilde x}^\top\breve{\bm K}\big)^{-1}\Big[\breve{\bm C}\,\breve{\bm v}_d+\big(\bm C_c+\breve{\bm D}\,\breve{\bm G}_{v_c}\big)\dot{\tilde{\bm x}}_c\Big]. \]
Read the right-hand side: it is built from \(\bm J_{\tilde x}\), \(\breve{\bm K}\), \(\breve{\bm C}\), \(\breve{\bm v}_d\), and the CoM-coupling block. There is no \(dt\) anywhere in it. The floor is a property of the continuous closed-loop ODE — specifically of its equilibrium manifold — and is defined whether you integrate the loop with forward Euler, backward Euler, or solve it in closed form.
The ripple is a multiplier of the integrator. Linearize the velocity error \(\bm e=\breve{\bm v}-\breve{\bm v}_d\) about the demand and keep the dominant damping mode, \(\dot{\bm e}=-\breve{\bm M}^{-1}\breve{\bm D}\,\bm e\). In continuous time every mode decays monotonically, \(e_i(t)=e_i(0)\,e^{-\mu_i t}\) with \(\mu_i>0\) — unconditionally stable. Forward Euler replaces that decay with the recurrence \(e_i^{k+1}=(1-dt\,\mu_i)\,e_i^k\), whose multiplier \(\rho_i=1-dt\,\mu_i\) crosses \(-1\) — a growing period-2 cycle — exactly when
\[ dt\,\mu_{\max}>2,\qquad \mu_{\max}=\max\operatorname{eig}\!\big(\breve{\bm M}^{-1}\breve{\bm D}\big). \]
Read this right-hand side: it is all \(dt\) and spectrum of \(\breve{\bm M}^{-1}\breve{\bm D}\). It says nothing about where the loop settles, only about whether the explicit step tracks the continuous decay or overshoots and reverses sign each step.
So the floor answers “where does \(\tilde{\bm x}\) settle?” and the ripple answers “does the integrator converge to the continuous trajectory or limit-cycle around it?” Different questions, different variables (\(\tilde{\bm x}\) vs \(\bm e\)), different parameters (\(\breve{\bm K}\) vs \(dt\)).
The clean argument is the one already made: \(\tilde{\bm x}_{ss}\) is a function of
continuous-time quantities with no \(dt\) dependence, and the \(\omega_b\) instability is a statement
purely about \(dt\). Changing \(dt\) — even across the stability boundary
\(dt^\star=2/\mu_{\max}\) — changes
whether the transient rings, not the equilibrium it
rings about. Setting \(dt=0.01\) (the
causal re-run, spec omega_ripple_dt010) removed the ripple
while leaving the closed-loop ODE, and hence its floor, identical. That
is the cleanest possible evidence the two are independent: a
manipulation that annihilates one leaves the other’s defining equation
unchanged.
A small quantitative sanity check makes the separation concrete. The ripple is a near-zero-mean period-2 oscillation in \(\breve{\bm v}\) of amplitude \(\sim0.3\); integrating a \(\sim16.7\,\mathrm{Hz}\) (\(=1/2dt\) at \(dt=0.03\)) velocity wobble of that size gives a pose wobble on the order of \(0.3/(2\pi\cdot16.7)\approx 3\times10^{-3}\,\mathrm{rad}\approx0.16^\circ\) — a fraction of the \(\sim0.8^\circ\) pointing budget. So in the simulator the measured pose error is
\[ \tilde{\bm x}(t)\;\approx\;\underbrace{\tilde{\bm x}_{ss}}_{\text{floor: continuous equilibrium}}\;+\;\underbrace{\tilde{\bm x}_{\text{ripple}}(t)}_{\text{zero-mean, } \mathcal O(0.16^\circ),\ dt\text{-artifact}} . \]
The two contributions add; they do not multiply or shift one another. The floor is the mean; the ripple is fluctuation about it.
There is a place the two results actually touch, and it is worth stating precisely rather than waving away. The floor derivation’s load-bearing step is \(\breve{\bm v}\to\breve{\bm v}_d\) at steady state — this is what cancels the damping term and collapses the dynamics to the algebraic balance. But the \(\omega_b\) result says that under forward Euler at \(dt>dt^\star\), the velocity error \(\bm e=\breve{\bm v}-\breve{\bm v}_d\) does not decay to zero — it limit-cycles. So does the floor’s premise fail in the discrete sim?
No, and the reason is the property the \(\omega_b\) note already establishes for an unrelated purpose: the limit cycle is near-zero-mean. The floor is an equilibrium statement, which in the rippling discrete system corresponds to the time-averaged balance. Taking the time average of the velocity error over a period-2 cycle gives \(\langle\bm e\rangle\approx\bm 0\), so \(\langle\breve{\bm v}\rangle\approx\breve{\bm v}_d\) and the averaged damping term \(\breve{\bm D}\langle\bm e\rangle\) still vanishes. The algebraic balance that defines \(\tilde{\bm x}_{ss}\) is recovered in the mean. The ripple injects a zero-mean fluctuation but no DC offset, so the floor — a DC quantity — survives intact.
This is the precise condition under which the two results stay decoupled: the floor is robust to the discretization artifact exactly because the artifact is zero-mean. Were the period-2 cycle ever to acquire a non-zero mean (the note flags that the bounding mechanism and the zero-mean property would need re-checking if the ripple approached the velocity clip \(v_{\max}=50\), which it does not here), that DC component would bias the measured offset away from the analytic floor. It does not, so the floor stands — but now we know why it stands, not merely that it does.
error_floor.tex (not a
formula problem)One sentence in error_floor.tex (the paragraph after eq
x_ss) now sits awkwardly against the \(\omega_b\) verdict:
“when the arm and base are enabled … Tikhonov damping is applied homogeneously to all coordinates. This causes spikes in \(\tilde{\bm x}_c\) and \(\dot{\tilde{\bm x}}_c\) near singularity.”
Two things to check against the new result, both of which need
your eyes on the sim data (I have not edited the
.tex — per the LaTeX rule I only flag, I do not rewrite
your derivations):
Attribution. The sentence pins the near-singular spikes on the Tikhonov / \(\gamma\)-regularized damping. The \(\omega_b\) note explicitly exonerates the \(\gamma\)-regularized reconstruction (“a passive, sign-stable SPD map … cannot self-oscillate”) and reassigns near-singular oscillation to the forward-Euler integration of \(\breve{\bm M}^{-1}\breve{\bm D}\), noting that raising damping makes it worse (it increases \(\breve{\bm D}\), pushing \(dt\,\mu\) past 2). If the CoM spikes share that mechanism, “Tikhonov damping causes the spikes” is at best incomplete and at worst the wrong suspect.
Which loop. The spikes in the .tex
are in \(\tilde{\bm x}_c,\dot{\tilde{\bm
x}}_c\) — the decoupled \(3\times3\) CoM loop (\(m\ddot{\tilde{\bm x}}_c+\bm D_c\dot{\tilde{\bm
x}}_c+\bm K_c\tilde{\bm x}_c=\bm 0\)). The \(\omega_b\) instability is derived for the
reduced \(9\times9\)
\([\bm\omega_b;\bm\nu_e^\oplus]\)
block, a different subsystem with its own well-conditioned
inertia. So the two may genuinely be separate phenomena — in which case
the .tex is fine as written and simply describes a CoM-loop
effect the \(\omega_b\) note does not
cover. But the same large-\(dt\) / large-\(\mu\) mechanism would afflict the CoM loop
too if \(\bm M_c^{-1}\bm D_c\) ever
developed a fast mode near singularity. The decisive check is the one
the \(\omega_b\) note already
pioneered: the \(dt\)-independent lag-1
autocorrelation of the logged \(\dot{\tilde{\bm x}}_c\) — if it reads \(\approx-1\) at \(dt=0.03\) and flips to \(\approx+1\) at \(dt=0.01\), the CoM spikes are the same
forward-Euler artifact and the “Tikhonov damping” sentence should be
rewritten to match.
Either way, the floor formula eq x_ss is
unaffected — this is about the explanatory sentence
beneath it, not the equation.
error_floor.tex with the
forward-Euler attribution, using the lag-1 autocorrelation test on the
CoM channel to decide whether the CoM spikes are the same artifact.The moral. Continuous and discrete stability are different theorems about the same loop: continuous stability is necessary but not sufficient for stability under explicit integration (discretization_stability_omega_b makes this its closing caveat). The cruise-lag floor lives entirely on the continuous side, so the discrete-side artifact leaves it formally untouched — but the only reason the simulator measures the right floor is the empirical zero-mean of the discrete artifact. Keep the layers separate in the writing, and the one quantitative thing linking them (zero-mean) explicit.