Spectral and Conformal: Neural Network Methods for Calibrated Trajectory Prediction and Control-Point Intervals

Jasmine Ratchford; Raffaele Romagnoli

eess.SYOct 30, 2026

Spectral and Conformal: Neural Network Methods for Calibrated Trajectory Prediction and Control-Point Intervals

Authors: Jasmine Ratchford, Raffaele Romagnoli

Organizations: CMU Software Engineering Institute (former affiliation) · Duquesne University

Abstract

Autonomous systems that navigate their environment need methods to quickly assess the feasibility and safety of desired trajectories. Traditional solutions that rely on numerical integration can be computationally heavy, while fast data-driven surrogates lack the formal guarantees that numerical approaches provide. This work demonstrates a framework that uses (1) a neural network with a spectral backbone to learn trajectories and quantile bands with built-in constraints (initial conditions and stability); (2) a modified conformal quantile regression to provide calibrated, joint trajectory-level coverage; and (3) interval arithmetic to extend noise-free coverage guarantees to control points. After refactoring the network for inference, the trajectory inference time for a quadrotor under a closed-loop linear quadratic regulator is 3.45~~ms on a Raspberry Pi Zero 2 W. The total pipeline time, including trajectory inference time, is 9.81~~ms, or approximately 100~Hz, consistent with typical control rates.

Figures & tables

Figure 1: A schematic of the trajectory network showing the lift, evolve, and project and decode portions of the network in sequence. The initial condition $\vec{x}_{0}^{\,d}$ is concatenated with the output of a SiLU encoder $L_{\theta}$ to form the lifted state $\vec{\chi}_{0}^{\,\ell}$ , which the spectral layer $Q_{\theta}e^{\Lambda_{\theta}\tau}Q_{\theta}^{H}$ evolves linearly to any query time $\tau$ . The projection $P\vec{\chi}^{\,\ell\tau}$ plus a mean-correction head yields $\vec{x}^{\,d}(t)$ ; two ReLU-gated quantile heads, added around the mean, yield $\vec{x}_{\text{lo}}^{\,d}(t)$ and $\vec{x}_{\text{hi}}^{\,d}(t)$ . All heads act on the stop-gradient (sg) residual $\bar{P}(\vec{\chi}^{\,\ell\tau}-\vec{\chi}_{0}^{\,\ell})$ , with gate $\alpha$ enabled after warmup; at $\tau{=}0$ the output equals $x_{0}$ by construction.

Target	Raw coverage		Post-CQR coverage		Raw width	Post-CQR width	$Q_{\eta}$
	cell	joint	cell	joint	(mean)	(mean)
95%	94.94%	1.10%	99.95%	95.31%	0.00327	0.00775	0.00093
99%	99.05%	22.64%	99.99%	99.00%	0.00433	0.00855	0.00087

Table 1: Conformal quantile regression on the baseline trajectory model (

\ell{=}128

m{=}251

, seed 42), trained at two coverage targets. Raw cell-level coverage is close to the trained nominal in both cases, but raw trajectory-level (joint) coverage is far below the target. After conformal correction, the trajectory-level coverage reaches the target, with mean band width roughly

2{\times}

the raw band width. Per-dimension weights

\omega[d]

reflect difficulty in

R

-normalized units: body rates (

p,q

) and angles (

\phi,\theta

) are hardest, while translational positions (

x,y

) are easiest.

Figure 2: Conformal-calibrated trajectory bands for the worst-case hold-out trajectory prior to conformal calibration under the 95% quantile model (baseline: $\ell{=}128$ , $m{=}251$ , seed 42). Solid gray is the true trajectory; red dashed is the model’s mean prediction; gold is the raw quantile band before calibration; blue is the per-dim conformal-calibrated band at 95% joint coverage. The "worst case" is the test trajectory whose raw band has the largest miss before calibration; the true trajectory is contained at every $(t,d)$ .

	$u/v$	$x/y$	$w$	$z$	$p/q$	$\phi/\theta$	$r$	$\psi$
	[m/s]	[m]	[m/s]	[m]	[rad/s]	[rad]	[rad/s]	[rad]
$\epsilon_{\text{knot}}$ ( $\times 10^{-4}$ )	5.23	1.36	7.23	1.39	14.8	2.36	1.80	0.19
$\epsilon_{\text{knot}}^{p95}$ ( $\times 10^{-3}$ )	1.07	0.28	1.41	0.28	3.75	0.46	0.70	0.07
Traj. hw ( $\times 10^{-2}$ )	1.28	1.01	2.06	1.65	1.11	0.28	0.16	0.15
$\epsilon_{\text{knot}}^{p95}/\text{Traj}_{hw}$ (%)	8	3	7	2	34	16	43	5

Table 2: The sup-norm (per-trajectory maximum) residual between the true trajectory and its own least-squares spline fit, summarized as a mean (

\epsilon_{\text{knot}}

) and 95th percentile (

\epsilon_{\text{knot}}^{p95}

) across trajectories, compared against the calibrated trajectory half-width (

\text{Traj}_{hw}

). The ratio is

\epsilon_{\text{knot}}^{p95}/\text{Traj}_{hw}

. Results from a 5000-trajectory test run. Some channels (e.g.,

u

and

v

) are combined due to the similarity of their values.

Stage	Description	Mean (ms)	Median (ms)	95th pct (ms)
1	Activity Model (optional to get knot locations)	2.07	2.04	2.12
2	Spline Basis Construction	1.48	1.46	1.55
3	Trajectory Prediction (ML - This Paper)	3.45	3.41	3.49
4	Control-Point Calculation	2.81	2.77	2.97
Total with ML (1+2+3+4)		9.81	9.69	10.00

Table 3: Per-stage inference latency on the Raspberry Pi Zero 2 W. The focus of this paper is the trajectory prediction component, part 3. The activity model is a method for estimating an optimal knot density that neither affects nor is affected by the method described in this paper ( Ratchford and Romagnoli, 2026 ) .

Figure 3: Calibrated trajectory bands and control-point intervals for one test trajectory, across six of twelve state dimensions $(u,x,v,y,w,z)$ ( $\ell{=}128$ , $m{=}251$ , 95% joint coverage). Gray solid is the true trajectory; red dashed is the mean prediction; the blue band is the per-dim conformal-calibrated band; black dots with vertical bars show the derived control points ( $c_{i,\text{mid}}^{d}$ ) and their intervals $[c_{i,\text{lo}}^{d},c_{i,\text{hi}}^{d}]$ at the Greville abscissae of the knot vector. The control-point intervals contain the spline coefficients of any band-respecting trajectory.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Quadrotor Dynamics
Parameter	Value
Total system mass	0.600 kg
Quadrotor body mass	0.410 kg
Propeller radius	0.0504 meters
Arm length	0.108 meters
Propeller mass	0.00311 kg

Appendix

Table 4: Parameter values used in data generation and processing. Table repeated from Ratchford and Romagnoli (2026) .

Layer Type	Input	Output	Param.
Linear / SiLU	12	128	1664
Linear / SiLU	128	64	8256
Linear / SiLU	64	128	8320
Linear (lift)	128	116	14964
Spectral decomposition	128	(2, 128)	33024
Decoder: mean (Linear)	244	12	2928

Appendix

Table 5: The baseline architecture used to predict the trajectory and quantile bands from initial conditions. The spectral decomposition evolves an

\ell{=}128

lifted state; decoders read the

2\ell-d=244

non-state lifted channels and are bias-free.

Figure 4: Validation loss curves for the baseline model ( $\ell{=}128$ , $m{=}251$ ): trajectory loss (mean MSE, black, left axis) and total loss including pinball band terms (red dotted, right axis). Vertical lines mark the $\alpha$ warmup window (epochs 400-500), during which band decoders activate and the loss briefly shows some instability before stabilizing and converging slowly. There are 62,500 samples in the validation dataset. These samples are generated once (unlike training data that is regenerated every 5 epochs) such that model performance is evaluated across the same data each epoch.

Baseline (lifted = 128, rollout = 251, 3 seeds)
Architecture		Validation Loss
Lifted	Rollout	Trajectory Only	Total
Dim	Steps	(mean MSE, $\times 10^{-6}$ )	(total, $\times 10^{-4}$ )
128	251	$1.12(13)$	$1.050(94)$
Lifted dimension sweep (rollout = 251, 3 seeds)
64 ∗	251	$1.69(19)$	$1.144(64)$

Appendix

Table 6: Effect of lifted dimension, encoder shape, rollout length, physics consistency, and trained quantile interval on fit. Architecture rows report mean

\pm

std over three seeds; ablation rows are single-seed. Trajectory Only is the MSE on the mean prediction (interval-invariant); Total is the total validation loss including pinball band terms. ∗ Trained for 1200 epochs rather than 900. † The validation loss at the 99% interval is not directly comparable with the 95% rows: the pinball weights

q

1{-}q

are smaller at more extreme quantile levels.

Figure 5: Full twelve-dimension version of Figure 2 : conformal-calibrated trajectory bands for the worst-case hold-out trajectory prior to conformal calibration under the 95% quantile model (baseline: $\ell{=}128$ , $m{=}251$ , seed 42). Solid gray is the true trajectory; red dashed is the model’s mean prediction; gold is the raw quantile band before calibration; blue is the per-dim conformal-calibrated band at 95% joint coverage.

Figure 6: Same as Figure 5 , but for the 99% quantile model. The raw quantile bands (gold) are wider than at the 95% target, and the conformal correction is correspondingly smaller; the calibrated band (blue) achieves 99% joint coverage at a band width comparable with that of the 95% case.

Figure 7: Full twelve-dimension version of Figure 3 : calibrated trajectory bands and control-point intervals for one test trajectory ( $\ell{=}128$ , $m{=}251$ , 95% joint coverage). Gray solid is the true trajectory; red dashed is the mean prediction; the blue band is the per-dim conformal-calibrated band; black dots with vertical bars show the computed control points and their intervals at the Greville abscissae of the knot vector. There is no simulated noise on the initial condition. The generation process is identical to Figure 5 with the added computation of spline control points and their intervals, per Section 5.7 .

Figure 8: Same as Figure 7 with the same number of knots, but now with a uniform distribution. Notably, the control intervals are wider and farther from the trajectory.

Target	Raw coverage		Post-CQR coverage		Raw width	Post-CQR width	$Q_{\eta}$
	cell	joint	cell	joint	(mean)	(mean)
95%	87.61%	0.00%	99.95%	94.77%	0.01076	0.03518	0.00537

Appendix

Table 7: Conformal quantile regression on the RNN model (

\ell{=}128

m{=}251

, seed 42) trained for the 95% coverage target.

Figure 9: Using an RNN instead of the spectral backbone. Compare to Figure 5 and Figure 6 . Similar to those figures, this represents the “worst” hold-out sample based on its accuracy prior to CQR.

Figure 10: Input-noise propagation for one initial condition with a small amount of noise in each dimension. Black is truth; red dashed is the mean prediction; gold is the raw quantile band; blue is the per-dim conformal-calibrated band; green dashed is the marginal band from perturbing the IC with representative sensor-noise magnitudes and taking mixture quantiles across the perturbation set. This does not provide coverage guarantees in the sense of Tibshirani et al. (2019) , and we intend a full study with a justified model for the distribution shift.

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This paper proposes a constant time-delay trajectory tracking method for vehicle convoys operating without inter-vehicle communication, a common coordinate system, or global positioning. The method integrates a probabilistic sequence-to-sequence (Seq2Seq) neural network with an invariant extended Kalman filter (IEKF) to warm-start the prediction process, allowing accurate estimation of a leader vehicle's relative trajectory on the SE(2) manifold. A geometric model predictive controller is further incorporated to fully exploit the manifold-based trajectory predictions for improved control performance. The system can handle arbitrary nonlinear trajectories with varying speeds and motion profiles while reducing the need for expert-based domain knowledge for the design of trajectory following systems, even under long trajectory delays. The effectiveness of the method is validated through comparisons with a pure IEKF baseline, learning-based methods, and the ground-truth trajectory in kinematic simulations, as well as in experiments using real robotic vehicles.

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