Propagating Elevation-Map Uncertainty Through the Contact Maximum in Closed Form

Aleš Kučera; Karel Zimmermann

cs.ROOct 5, 2026

Propagating Elevation-Map Uncertainty Through the Contact Maximum in Closed Form

Authors: Aleš Kučera, Karel Zimmermann

Organizations: Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic

Abstract

Risk-aware planners score paths on uncertain elevation maps using the path cost's mean and standard deviation. Modeling rigid contact, however, requires computing a maximum over several uncertain cells. First-order propagation loses accuracy here by differentiating at only a single cell, while Monte Carlo sampling requires a full path evaluation per draw. We compute the moments of that contact maximum in closed form using Clark's pairwise recursion. By tracking each contact's covariance against the shared map cells, we propagate the smooth remainder using exact Gaussian quadratic-form identities. A contest-depth calibration, fitted once on two design traverses, closes the aggregate standard-deviation shortfall that remains. On 317 held-out rover path segments, scored against a Monte Carlo reference from the same belief, every pre-registered criterion was met. The corrected Clark fold cuts the median error of the mean from linearization's 2.3% to 0.19% and attains the lowest error in the conditional value at risk (CVaR) at the 90% level of every method tested. A benchmark plan costs just 5.5 microseconds on a GPU. These accuracy gains concentrate at contested contacts. While they seldom change which path is chosen on this terrain, the fold still selects the reference-best path in 98% of decisions against linearization's 93 to 96%. On a second dataset the mean transfers, though the risk number does not.

Figures & tables

Fig. 1: (a) A wheel over Gaussian terrain heights (per-cell mean and

\pm 2\sigma

band) settles on the cell that maximizes its elevation, adjusted for wheel geometry. The contact height

e

is thus a maximum over underlying cells, five of the seven here contesting it; the five are ringed, and in pink the one highest in the mean map, the only cell linearization considers. (b) The distribution of

e

(a constructed example, its two standard-deviation ratios matched to the medians at this contest depth, Section III-H ). The bars are

200{,}000

draws; the curve is the Gaussian carrying their mean and standard deviation (

19.6\pm 3.1

cm). Linearization misses both, underestimating the mean by

3.8

cm and overestimating the standard deviation by

29\%

(

15.8\pm 4.0

); the Clark fold reproduces both. The deficit the corrected fold repairs is not here but in the aggregation over a plan (Section III-G ).

Fig. 2: (a) A synthetic 1-D environment (

35

cm step,

25

cm drop) sampled with

\sigma=2

cm range noise (

25

samples per

10

cm cell) and rasterized into a Gaussian belief. At the step, samples straddle the discontinuity; the mean lands mid-air, and the standard deviation grows eightfold. (b) Error in

\mathbb{E}[\max]

against a

20{,}000

-draw Monte Carlo reference for a flat

3

-cell footprint swept along the same strip, so the step in (a) stands over the cells it spoils. The mean map is systematically optimistic, and log-sum-exp at its best-fit temperature (

\tau=1.6

cm) returns the mean map’s own value to within

0.002

cm, its weight reading the gap between the means rather than the spread of the margin (Section III-C ). RMS error:

0.14

cm for the fold,

0.59

for LSE,

1.00

for the mean map.

Fig. 3: The standard deviation deficit and correction on held-out windows. These plots show the ratio of each estimator’s standard deviation to the Monte Carlo reference, plotted against the window’s contest depth

\alpha_{w}

. The data is visualized as medians and interquartile ribbons across six

\alpha_{w}

-quantile bins, with the fitted correction law

d(\alpha_{w})

overlaid. (a) Foresight: the estimators and the deficit correction under the causal map available at decision time. (b) Hindsight: the same under the fully fused map. In (b) the uncorrected fold’s medians sit above the frozen law by

0.004

0.025

; in (a) they cross it.

Fig. 4: The corrected fold on a BASEPROD elevation map. (a) The belief mean across a $4$ m foresight window of a design traverse, rendered orthographically at true vertical scale. The path is the window’s own track (the $35$ of $39$ steps that pass the quality flags), colored by the corrected fold’s $\mathrm{CVaR}_{0.9}$ at each step. A $3$ D model of the robot is placed where this risk peaks, its wheels seated on the support heights the fold returns there: the rear wheel’s cylinder element evaluates $K=7$ cells, where the top two candidates are nearly tied, $\alpha=0.01$ standard deviations of their difference apart. The vehicle’s $20^{\circ}$ roll, derived from these three supports, is computed but not drawn. (b) A $1.96$ m wide rear view of the contested rear contact. It reveals the ground beneath that wheel, which the chassis and the front wheels obscure in panel (a). The ring in (a) marks this contact. (c) The per-step mean $\mathbb{E}[\ell_{t}]$ alongside its $\pm 1\sigma$ band, the ribbon colored by the $\mathrm{CVaR}_{0.9}$ risk. The risk spikes to $0.24$ rad 2 at the contested step, against a median of $0.0094$ along the plan.

	med. rel. err. $\mathbb{E}[\ell]$	med. rel. err. $\mathrm{sd}[\ell]$
clark-corr (ours)	$\mathbf{1.9\!\cdot\!10^{-3}}$ / $\mathbf{8.5\!\cdot\!10^{-4}}$	$\mathbf{0.028}$ / $\mathbf{0.018}$
clark	$\mathbf{1.9\!\cdot\!10^{-3}}$ / $\mathbf{8.5\!\cdot\!10^{-4}}$	$0.061$ / $0.035$
fosm	$2.3\!\cdot\!10^{-2}$ / $1.5\!\cdot\!10^{-2}$	$0.038$ / $0.023$
mc-32	$1.2\!\cdot\!10^{-2}$ / $9.4\!\cdot\!10^{-3}$	$0.093$ / $0.097$

TABLE I: Claim 1 : median relative error against the MC reference, foresight / hindsight (

317

323

windows). Bold marks the best in each column; the two Clark arms share one mean.

method	F med.	F p95	H med.	H p95
clark-corr (ours)	$\mathbf{0.0045}$	$\mathbf{0.124}$	$\mathbf{0.0022}$	$\mathbf{0.152}$
clark	$0.0088$	$0.234$	$0.0054$	$0.257$
mc-32	$0.0085$	$0.322$	$0.0068$	$0.484$
fosm	$0.012$	$0.823$	$0.0074$	$0.778$
lse	$0.015$	$0.179$	$0.0090$	$0.196$
ut	$0.027$	$1.216$	$0.016$	$1.028$

TABLE II: Claim 2 :

|\mathrm{CVaR}_{0.9}(\mathrm{method})-\mathrm{CVaR}_{0.9}(\mathrm{MC})|

(rad 2 ), median and 95th percentile, foresight (F) and hindsight (H), on the

317

323

held-out windows (reference median

0.50

0.63

); the lse and ut rows use all

363

368

windows and a re-drawn reference, where clark-corr is

0.0045

0.0019

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