Hover-Height Stability of Civilian Repulsorlift Landspeeders over Uneven Ground under Payload Changes: A Second-Order Model Tested against Corellian Test-Track Logs, 30–26 BBY

1. Introduction

Landspeeders are a familiar civilian repulsorlift vehicle, in use across the Core and the Outer Rim, including on desert worlds where the surface offers no made road. They ride a short distance above the ground instead of on it. This paper, written in 25 BBY, concerns one property of that arrangement which operators feel and designers have seldom quantified: how steadily a craft holds its hover height when the ground beneath it steps or ripples, and when the load it carries changes.

Hover height is the only margin between hull and ground, and a landspeeder that dips too far strikes its skid plate. Bench work has measured how lift varies with clearance for a planar coil (Bexmar, 31 BBY), and trials over broken ground have reported the damping of civilian skimmers (Threnn & Vollarp, 29 BBY). An operator survey has recorded where ride and clearance complaints cluster (Halcrow, 28 BBY), but no account ties payload, ground profile and speed to clearance in one predictive relation, and payload matters because a hire craft may carry nothing in the morning and half its own weight by noon.

Two features of the record shape our approach. Published bench data fix the lift of a coil at a few clearances and say little about a whole hull in motion, and the field of a repulsor coil does not respond instantly; a lag of the order of a hundredth of a standard second has been reported for step demand (Ostrahl, 30 BBY). Any model of hover dynamics is therefore an idealisation. We make no claim that the lift of a real coil follows a power law, and every parameter below is a finding about the four test hulls examined here, not a constant of repulsorlift engineering.

We aimed to calibrate a second-order model of hover height on logged runs, to test it on a hull type and a corrugation course absent from calibration, and to use it, with its errors stated, to map how a landspeeder fails.

2. System Description

Hulls and clearance. The test fleet of the Corellian Engineering Corporation's Research Division supplied four civilian landspeeder hull types, labelled A to D. Each carries a paired repulsor array, treated for heave motion as a single lift unit. Mean unloaded static clearance was 0.150 m (hull A), 0.155 m (B), 0.148 m (C) and 0.152 m (D). All lengths and times are standard metres and standard seconds. The skid plate of each hull sits at a clearance of 0.03 m, and we call clearance below that value ground contact.

Loads. Payload was fixed ballast, expressed as a fraction p of the hull's curb mass and set to 0, 0.25 or 0.5, with weighed loads departing from nominal by about 0.01 in standard deviation. At p = 0.5 static clearance was between 0.121 and 0.127 m.

Courses and logging. The trials ran on Corellian test tracks between 30 and 26 BBY, on courses later codified by the Test Range Office (Corellian Engineering Corporation Test Range Office, 26 BBY). Clearance was logged by an underside range sensor at 200 samples per standard second, with logger noise of 1.5 mm (standard deviation) measured on the bench. A surface step was a smooth 1.5 m ramp rising 0.06 m or 0.16 m, crossed at 6, 12 or 18 m/s. The corrugation course was a sinusoidal ridge pattern of 8 m wavelength and 0.02 m amplitude, entered through a taper and run for 20 standard seconds at 10, 13, 16 or 19 m/s with a payload fraction of 0.25. Ridges of a few metres' wavelength are familiar on unmade routes on arid worlds (Perrelan & Desmarr, 32 BBY).

Design and sample flow. All 88 issued run sheets yielded complete logs; none was excluded. Step runs formed a full factorial of 4 hulls, 3 payload fractions, 2 step heights and 3 speeds, one run per cell (72 runs), and corrugation runs crossed 4 hulls with 4 speeds (16 runs). Hull D was designated as the held-out hull in the test plan before any calibration, because bench checks had shown its field to respond more slowly than the others. Calibration used only the 54 step runs on hulls A to C (Table 1).

Response measures. For each step run we extracted four quantities from the clearance trace after a five-sample moving average. The dip fraction is the greatest fall of clearance below its static value divided by step height, and the overshoot fraction is the highest clearance above the static value after the dip, divided likewise. Settling time is the last instant, from the start of the ramp, at which clearance departs from its static value by more than 10% of step height. Minimum clearance is the lowest filtered value. For corrugation runs the response is the root-mean-square clearance deviation over the final 10 standard seconds.

Non-independence. Runs share hulls. Intervals below treat runs as independent and come from a percentile bootstrap over runs with 10,000 resamples (Ilvarin, 31 BBY), so they understate the uncertainty from having only four hulls.

3. Analysis / Model

The idealisation. Let h be the clearance, z(t) the height of the ground under the craft and p the payload fraction. We assume that the field supplies a lift per unit curb mass of K/hⁿ, so that lift falls as clearance grows, and that a damping force proportional to the rate of change of clearance acts with rate constant γ per unit curb mass. Motion is heave only, with the ground level across the craft's width. With g the gravity constant of the track, calibrated at 9.7 m/s², and H the static clearance, the equation of motion is d²h/dt² = g[(H/h)ⁿ − 1] − γ/(1 + p) · dh/dt − d²z/dt². The damping force is assumed not to grow with payload, so the acceleration it produces falls as mass is added. That assumption is ours. Predictions integrate the equation by fourth-order Runge–Kutta with a step of 0.5 ms; halving the step moved minimum clearance by under 1 mm. The ground rises as a cubic smoothstep in time over the ramp length divided by speed, and a raised-cosine ramp changes the prediction for a 0.16 m step at 18 m/s unloaded by under 2 mm. Predictions are unfiltered and use each run's measured pre-step clearance for H, whereas the observed traces were filtered as described in Section 2.

Static consequence. Setting the right-hand side to zero gives H = (K/[g(1 + p)])^1/n. Static clearance should therefore fall with payload such that the slope of ln H against ln(1 + p) is −1/n. This gives an estimate of the exponent from clearance alone. In the dynamic predictions H is taken from the measured clearance, which follows this static relation with exponent 2.00, not from the calibrated n.

Linearisation. For small excursions about H the craft is a second-order system with natural angular frequency ω, where ω² = ng/H, and damping ratio ζ = γ/(2(1 + p)ω). Because H falls with payload, ω rises slightly, while ζ falls because the same damping acts on a larger mass. For ground displacement a sin(Ωt), where Ω = 2πv/λ for speed v over ridges of wavelength λ, the amplitude of clearance deviation is a times Ω²/|ω² − Ω² + 2iζωΩ|, a gain that peaks above the natural frequency.

Calibration. The two free parameters, n and γ, were chosen by grid search to minimise the summed squared errors in dip fraction and overshoot fraction over the 54 calibration runs, each scaled by its sample standard deviation. Settling time and minimum clearance were not fitted. The grid had steps of 0.05 in n and 0.25 per standard second in γ. The best fit was n = 2.15 and γ = 7.0 per standard second. Percentile intervals from 1,000 bootstrap refits on the same grid were 2.05–2.25 for n and 6.5–7.5 for γ. Refitting on all 72 step runs gave n = 2.20 and γ = 6.75, inside both intervals.

A discrepancy. The static relation gives a separate estimate. Regressing log pre-step clearance on ln(1 + p), with an intercept per hull, on the 12 hull-and-payload cell means (7 residual degrees of freedom) gave a slope of −0.499 and a static exponent of 2.00 (95% CI 1.94–2.07). The dynamic exponent is 7% larger, and the intervals overlap only at 2.05–2.07. We cannot separate the causes. One candidate is the field lag noted by Ostrahl (30 BBY), which the model omits and which could shift both effective stiffness and effective damping. That mixing of a static H with a dynamic n is a further reason to read n and γ as effective values that absorb such effects.

Implied dynamics. At the calibrated values the natural frequency is 1.87 cycles per standard second unloaded, 1.98 at p = 0.25 and 2.07 at p = 0.5, while the damping ratio falls from 0.30 to 0.23 and then 0.18. On the corrugation course the peak gain is 1.76 unloaded, 2.28 at p = 0.25 and 2.83 at p = 0.5, so half a curb mass of cargo raises the resonant amplification by 61%. The peak lies at 16.5, 16.7 and 17.1 m/s on ridges of 8 m wavelength. Across payloads it sits near 2.1 cycles per standard second of ground forcing, so the resonant speed is roughly 2.1 per standard second times the ridge wavelength.

4. Validation Against Field Data

Steps on the held-out hull. Table 2 compares predictions with the 18 runs on hull D, which played no part in calibration. Errors are root-mean-square differences with bootstrap intervals. The baseline predicts every run by the calibration-run mean. The dip fraction was predicted with an error of 0.018 (95% CI 0.012–0.023) against a baseline of 0.139 (0.113–0.161), and the bias of −0.006 (−0.014 to 0.002) is indistinguishable from zero. Settling time, which was not fitted, was predicted to 0.12 standard seconds (0.07–0.16) against 0.27 (0.18–0.37), again with no detectable bias. Minimum clearance was predicted to 2.3 mm (1.3–3.1), with a small positive bias of +1.2 mm (0.3–2.1): the model puts the lowest point slightly too high. Overshoot fraction was the weakest quantity: its error of 0.29 (0.04–0.49) is wide and its interval overlaps that of the baseline, 0.79 (0.33–1.22), so we cannot claim a reliable improvement there.

The calibration hulls. On hulls A to C the model gave errors of 0.018 (0.013–0.024) in dip fraction, 0.18 (0.10–0.26) in overshoot fraction and 0.121 standard seconds (0.094–0.148) in settling time. Dip-fraction error by hull was 0.018, 0.023, 0.012 and 0.018 for A to D, so the held-out hull was no worse than the others on the fitted measure.

What the runs show. Across all 72 step runs, dip fraction rose with speed, averaging 0.30, 0.51 and 0.62 at 6, 12 and 18 m/s, and hardly changed with payload, averaging 0.47, 0.48 and 0.48 at fractions 0, 0.25 and 0.5. Payload acted instead on recovery: mean settling time was 0.69, 0.92 and 1.15 standard seconds across the three fractions, a rise of about two thirds from empty to half load. Overshoot depended strongly on step height, averaging 0.59 for 0.06 m steps and 1.28 for 0.16 m steps. Overshoot beyond the step height is what a lift that stiffens steeply as clearance shrinks would produce, and a linear model cannot show it.

Linearisation. The linearised model was markedly worse (Table 2). On hull D it over-predicted the dip fraction by 0.056 (0.039–0.074) and under-predicted minimum clearance by 7.4 mm (4.3–10.7 mm). Over all 72 runs, in-sample for the calibration hulls, its dip-fraction error grew from 0.041 at 0.06 m steps to 0.095 at 0.16 m steps, while the full model's error was 0.021 and 0.015.

Corrugation. Observed root-mean-square deviations ranged from 8.7 to 30.6 mm, mean 20.4 mm. The model's error was 3.3 mm (2.1–4.2), a mean absolute percentage error of 11%, with a correlation of 0.97. The bias was +1.8 mm (0.5–3.2), so the model over-predicts, and the slope of observed on predicted was 0.76 (0.66–0.87). The over-prediction arises at resonance: mean observed and predicted values were 9.2 and 8.5 mm at 10 m/s, 20.0 and 19.4 mm at 13 m/s, 27.7 and 31.7 mm at 16 m/s, and 24.8 and 29.1 mm at 19 m/s. Every hull responded most at 16 m/s, as predicted, though speeds 3 m/s apart cannot locate the peak more finely.

5. Failure Modes

Ground contact. Minimum filtered clearance over the 72 step runs ranged from 29.6 to 137 mm; twelve runs fell below 50 mm and six below 40 mm. Two runs, one on hull A and one on hull D, touched the skid line, at 29.6 and 29.9 mm, both at payload fraction 0.5, step height 0.16 m and 18 m/s. The model predicted 32.7 and 34.9 mm and flagged neither. The shortfalls, 3.1 and 5.0 mm, exceeded the held-out error of 2.3 mm. A 5 mm margin on the threshold, chosen after seeing the data, would have flagged three runs, both contacts and one other. That margin is not a validated rule.

The contact map. Table 3 gives, at each payload's mean static clearance, the step height that the model predicts would carry clearance to the skid line. Payload lowers it by 28% at 12 m/s and 27% at 18 m/s, from 0.24 to 0.17 m in the latter case. At 6 m/s it exceeds 0.6 m at every payload and is not tabulated. All tabulated critical heights lie beyond the largest step actually run, 0.16 m, so they are extrapolations. The nearest cell, at half load and 18 m/s, has a predicted critical height of 0.17 m against a tested 0.16 m, and contact was observed there.

Resonance over periodic roughness. The linear gain implies that ridges of 8 m wavelength taken at about 17 m/s amplify their own height by 1.76 to 2.83 times, depending on payload. Combined with static clearance, it puts the ridge amplitude that would bring clearance to the skid line at 0.069 m unloaded, 0.046 m at p = 0.25 and 0.033 m at p = 0.5. The tested amplitude was 0.02 m and the lowest clearance in any corrugation run was 93 mm. These linear extrapolations are more likely too low than too high, since the model over-predicted amplitude near resonance, but the trials cannot check that.

Payload transients. Applied without logged runs to test it, the model has a sudden load of half a curb mass carry clearance from 151.2 mm to a new static value of 123.5 mm, undershooting to 110.6 mm and settling in 0.82 standard seconds. Unloading gives a peak of 163.9 mm. The heave transient is benign, far above the skid line.

Shift of load. The hazard from load shifting lies in pitch and roll, which the model excludes. If the array is taken as two pads obeying the same law, a pad carrying share s of the total load behaves like a single unit at an effective payload fraction 2s(1 + p) − 1. At p = 0.5 and s = 0.6 the effective fraction is 0.8, static pad clearance falls to 112.8 mm, and the model puts minimum clearance for a 0.16 m step at 18 m/s at 27.8 mm, below the skid line. At s = 0.8 the effective fraction of 1.4 is nearly three times the largest tested. The pad decomposition is an assumption that no logged run tests.

6. Conclusion

A one-dimensional model with a power-law lift and a linear damping term reproduces the clearance loss of landspeeder hulls over surface steps with a root-mean-square error of about 0.02 of step height on a hull it was not fitted to, and predicts settling time to about 0.12 standard seconds without having been fitted to it. It does less well on overshoot and on resonant corrugation amplitude, and its linearised form fails once a step is comparable to the clearance. The two calibrated constants, an exponent of 2.15 and a damping rate of 7.0 per standard second, are effective values for these four hulls; the exponent departs modestly from the static value of 2.00, which we attribute provisionally to the field lag the model omits.

In this model, the practical effect of payload is on recovery and resonance: added load leaves the initial dip unchanged, lengthens recovery by about two thirds from empty to half load, lowers the damping ratio from 0.30 to 0.18 and raises the resonant gain over periodic roughness by 61%.

Shifted load remains unsettled, and the results rest on four hulls, one run per cell and a single held-out hull. Later work should log the field's lag, test ridges above 0.02 m and add pitch before any critical height in Table 3 is used as an operating limit.