A Physics-Informed CNN-LSTM for Speed and Heading-Rate Prediction from LiDAR Road Geometry Toward Safe Curve Negotiation

1. Introduction

Curved road segments concentrate a large share of the difficulty of automated driving: the vehicle must anticipate how sharply to turn and how fast it can travel before it enters the curve, using only what its sensors perceive of the road ahead. Light detection and ranging (LiDAR) sensors provide dense three-dimensional measurements of the road surface and its surroundings [1], and public datasets such as PandaSet [2] make it possible to study these problems with real sensor data [3].

Autonomous Vehicles (AVs) are expected to improve road safety by mitigating human factors such as distraction and cognitive errors [4], and automated systems can anticipate hazardous situations through trajectory generation and prediction [5]. To do so, an AV must plan and execute maneuvers while accounting for road infrastructure, vehicle dynamics, and actuator constraints [6]. Road geometry is one of the most significant constraints in motion planning [7,8], and it becomes critical on curves, where the admissible speed depends on the curve radius, the banking of the road, and tire–road friction.

Two broad families of methods address this problem. Model-based approaches, such as Model Predictive Control (MPC), predict the vehicle response over a horizon and optimize steering and speed commands subject to explicit constraints [6,9]; they are interpretable but depend on accurate vehicle models. Learning-based approaches, from convolutional perception pipelines [10] to deep reinforcement learning [11,12], learn the mapping from sensor data to driving actions, but they offer no guarantee that their outputs respect physical limits. Physics-informed machine learning [13,14,15] offers a middle ground: physical knowledge is introduced into the training objective so that the learned model is discouraged from producing physically implausible outputs.

A useful predictor of future heading rate and speed must also be compared with an appropriate reference. Over short horizons, both signals change slowly, so simply repeating the last observed value (persistence) is already a strong predictor, and a learned model is only useful if it improves on it. The advantage that LiDAR can offer over persistence is anticipation: it perceives the geometry of the road ahead before the vehicle reaches it.

This work follows these ideas. A multi-task CNN-LSTM receives a 2 s window of features that include a preview of the road geometry extracted from LiDAR at several look-ahead distances, and it predicts the change of the heading rate, used as a steering proxy, and of the speed over horizons of 1, 2, and 3 s. A physics-informed term in the loss penalizes predicted speeds above the maximum cornering speed of a banked curve with friction. The main contributions are the following:

• A feature-extraction pipeline that derives the same road-geometry descriptors, including a look-ahead preview of the road, from the LiDAR point clouds of two datasets (PandaSet and KITTI), after transforming the PandaSet points from world coordinates to a vehicle-aligned frame.
• A multi-task CNN-LSTM that predicts changes with respect to the last observed values and is trained with a one-sided physics-informed loss based on the banked-curve speed limit.
• An evaluation over five sequence-wise splits and three prediction horizons that compares the model with a persistence baseline and with an ablation of the physics term, both on unseen PandaSet sequences and, without retraining, on KITTI [16], reporting where the model improves on the baseline and where it does not.
• An open implementation of the full pipeline to support reproducibility.

The remainder of the paper is organized as follows. Section 2 reviews related work. Section 3 describes the datasets, the physical background, the feature extraction, the model, and the evaluation protocol. Section 4 presents the results, Section 5 discusses them together with the limitations of the study, and Section 6 concludes the paper.

2. Related Work

Recent work on autonomous driving over curved paths has focused on curvature-aware planning and predictive tracking under dynamic and geometric constraints. Liu et al. [17] formulate a two-layer MPC architecture in which an outer layer generates a curvature-adaptive reference and an inner layer solves a constrained optimal control problem for path tracking, reducing lateral error at high speeds by incorporating curvature-dependent bounds on steering rate and lateral acceleration. Li et al. [18] perform trajectory planning directly in Cartesian coordinates instead of the Frenet frame, embedding vehicle kinematics and collision constraints in a nonlinear optimization that remains well posed on highly curved references. For large-curvature roads, Wang et al. [19] model the driver–vehicle–road interaction with a kinematic vehicle model coupled with an artificial potential field, producing smooth, collision-free trajectories when curvature varies abruptly. In unstructured environments, Min et al. [20] improve the A* algorithm by refining its heuristic, smoothing the resulting path, and incorporating vehicle contour and motion constraints. Yang et al. [21] augment a path-tracking controller with a prediction model of future lateral deviation, reducing steady-state error on curved segments.

Probabilistic models are also widely used to anticipate driver behavior: hidden Markov models have been applied to driver-intention prediction [22] and lane-change prediction [23], and Kalman filtering is a standard tool for estimating vehicle states from noisy measurements [24]. These methods model temporal dependencies explicitly but rely on hand-designed state representations.

On the learning side, physics-informed neural networks introduce governing equations into the loss function [14], and broader surveys describe how scientific knowledge can be integrated with machine learning through the architecture, the loss, or the data [13,15]. The present work combines these lines: it learns the mapping from LiDAR-derived geometry and vehicle dynamics to future heading rate and speed with a CNN-LSTM, and it constrains the predicted speed through a loss term derived from the banked-curve speed limit, without solving an optimization problem online.

3. Materials and Methods

Figure 1 summarizes the complete pipeline, from the raw sensor data to the evaluation. Each stage is described in the following subsections. All experiments were run on Google Colab [25] with an NVIDIA T4 GPU, using TensorFlow/Keras.

3.1. Datasets

PandaSet [2] contains more than 100 driving sequences of about 8 s each, recorded at 10 Hz with a 360° mechanical spinning LiDAR (Pandar64), a forward-facing LiDAR (PandarGT), six cameras, and GPS/IMU localization. In this work, the Pandar64 point clouds and the corresponding LiDAR sensor poses were used. A total of [XX] sequences with complete data and at least 51 frames were processed. Figure 2 shows a LiDAR frame of sequence 001.

KITTI raw [16] provides synchronized Velodyne HDL-64E point clouds and high-precision GPS/IMU (OXTS) measurements at 10 Hz. Three sequences of the Road category (2011_09_26, drives 0015, 0027, and 0028) were used exclusively for zero-shot evaluation; no KITTI data were used for training, validation, or fitting any preprocessing step.

3.2. Physical Background: Maximum Speed on a Banked Curve

For a vehicle of mass m moving along a curve of radius $\rho$, the normal component of Newton’s second law gives

$$\mathrm{\Sigma }{F}_n=m{a}_n=m{\displaystyle \frac{v^2}{\rho }}.$$

(1)

On a road banked by an angle $\theta$ and without friction, the vertical and normal force balances on the vehicle, with normal reaction N and weight $W=mg$, are

$$N\cos \theta -W=0,N\sin \theta ={\displaystyle \frac{W}{g}}{\displaystyle \frac{v^2}{\rho }},$$

(2)

which yields the rated speed of the curve,

$$v^2=g\rho \tan \theta .$$

(3)

When tire–road friction with coefficient $\mu$ is included and the vehicle is at the limit of sliding outward, the maximum admissible speed becomes

$$v_{max}(\rho ,\theta )=\sqrt{{\displaystyle \frac{g\rho (\tan \theta +\mu )}{1-\mu \tan \theta }}}.$$

(4)

Equation (4) is the physical constraint used in the loss function (Section 3.6). Note that the mass cancels out: the speed limit depends only on the curve radius, the banking angle, and friction. In this study, $\mu$ is fixed to a conservative value of 0.7, representative of dry pavement.

3.3. Feature Extraction

Eight features are computed for every frame: four road-preview offsets, the banking angle, the road grade, the vehicle speed v, and the heading rate $\dot{\psi }$. The turning radius R and the physical speed limit $v_{max}$ are also computed for the physics-informed loss but are not network inputs.

Coordinate frame. PandaSet stores its point clouds in world coordinates. This was verified by comparing, along sequence 001, the center of the point cloud with the sensor position: both moved by about 84 m between the first and last frames. Each frame is therefore transformed to a vehicle-aligned frame with its origin at the sensor, the x axis along the direction of travel, the y axis to the left, and the z axis kept as the world vertical, so that banking and grade are measured with respect to gravity. The yaw ${\psi }_k$ is estimated from the sensor trajectory as the direction of the displacement between frames $k-1$ and k; it is held constant when the displacement is below 0.05 m (vehicle practically stopped) and unwrapped to avoid $\pm {180}^{\circ }$ jumps. KITTI Velodyne points are already expressed in the sensor frame with the same axis convention.

Road geometry from LiDAR. The same procedure is applied to both datasets. Points within 0–50 m ahead and $\pm 15$ m laterally are kept, and ground points are selected in a band from $-0.15$ to $+0.30$ m around the 10th percentile of their height. From these points:

• The road preview at distance $d\in \{10,20,30,40\}$ m is the lateral position of the center of the ground extent in a 5 m slice centered at d, computed as the midpoint of the 10th and 90th percentiles of y. As the road bends, these offsets shift toward the inside of the curve, increasingly with distance.
• The grade is $G=100({\overline{z}}_{far}-{\overline{z}}_{near})/30$ (%), using ground points beyond 40 m and within 10 m.
• The banking angle is

  $$\theta =atan2\left({\overline{z}}_r-{\overline{z}}_l,\max \left(|{\overline{y}}_r-{\overline{y}}_l|,0.5\right)\right),$$

  where the subscripts r and l denote the ground points within 20 m to the right and left of their median lateral position.

Missing values (for example, a slice with too few ground points) are filled with the previous valid value. Figure 3 illustrates the ground points and the road boundaries in the vehicle frame.

Vehicle dynamics. In PandaSet, the speed is computed from the three-dimensional displacement between consecutive sensor poses, $v_k=3.6\parallel {\mathbf{p}}_k-{\mathbf{p}}_{k-1}\parallel /\Delta t$ (km/h) with $\Delta t=0.1$ s, and the heading rate ${\dot{\psi }}_k$ is the frame-to-frame difference of the yaw smoothed with a trailing three-frame moving average (only past frames are used). Figure 4 shows the speed profile of sequence 001. In KITTI, the speed is the OXTS forward velocity and the heading rate is computed in the same way from the unwrapped OXTS yaw. The turning radius follows from circular-motion kinematics, $R=v/\omega$ with $\omega =|\dot{\psi }|f$ and $f=10$ Hz, clipped to 5–1000 m, and $v_{max}$ is obtained from Eq. (4) with R and $\theta$.

3.4. Sequence Construction

The per-frame features are converted into supervised samples with a sliding window. Each input ${\mathbf{X}}_i\in {\mathbb{R}}^{20\times 8}$ contains the eight features of 20 consecutive frames (2.0 s of history). For a horizon of $H\in \{10,20,30\}$ frames (1, 2, and 3 s), the targets are the mean heading rate (deg/frame) and the mean speed (km/h) over the following H frames. The network does not predict these values directly but their change with respect to the last observed frame of the window, $\Delta \dot{\psi }$ and $\Delta v$; the final prediction is the last observed value plus the predicted change. A predicted change of zero corresponds to the persistence baseline. For the physics term, the most restrictive limit within the horizon, $\min v_{max}$, is stored with each sample.

3.5. CNN-LSTM Architecture

The network (Table 1) combines one-dimensional convolutions along the time axis, which extract local temporal patterns from the input window, with an LSTM layer, which models longer temporal dependencies. Causal padding ensures that each convolutional output depends only on current and past frames. Dropout is used to limit overfitting, and two linear output heads predict the heading-rate and speed changes.

3.6. Physics-Informed Loss

The physics term is a one-sided (hinge) penalty that is zero whenever the predicted speed ${\widehat{v}}_i=v_{last,i}+\Delta {\widehat{v}}_i$ is below the physical limit:

$${\mathcal{L}}_{phys}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\left[ReLU\left(v_{last,i}+\Delta {\widehat{v}}_i-v_{max,i}\right)\right]}^2,$$

(5)

and the total loss is

$$\mathcal{L}={\mathrm{MSE}}_{\Delta \dot{\psi }}+{\mathrm{MSE}}_{\Delta v}+\lambda {\mathcal{L}}_{phys},\lambda =0.1,$$

(6)

with $g=9.81$ m/s² and $\mu =0.7$ in Eq. (4) (the denominator is bounded below by 0.05 to avoid numerical blow-up). The penalty is computed in the standardized space of the speed change: the limit $v_{max,i}-v_{last,i}$ is standardized with the same scaler as $\Delta v$ and supplied through an auxiliary input that bypasses the CNN-LSTM backbone. The constraint therefore acts on the gradients during training and is not a network input. Because it involves only the speed output, it is not expected to affect the heading-rate prediction directly.

3.7. Training and Evaluation Protocol

Figure 5 summarizes the evaluation protocol. The PandaSet sequences were split at random, by whole sequences, into training (about 68%), validation (about 12%), and test (20%) sets, so that windows of the same sequence never appear in more than one set. Inputs and targets were standardized with scalers fitted on the training sequences only, and the same scalers were applied, without refitting, to the validation and test sequences and to KITTI. Early stopping on the validation loss selected the weights of each model. To assess the sensitivity of the results to the choice of test sequences, the entire procedure was repeated for five different random splits, and results are reported as the mean and standard deviation over the splits.

Three predictors were compared:

• Persistence baseline: predicts that the mean heading rate and speed over the horizon equal their last observed values. It requires no training.
• CNN-LSTM without PIDL: the architecture of Table 1 trained with $\lambda =0$ (ablation).
• CNN-LSTM with PIDL: the proposed model ($\lambda =0.1$).

Performance is reported with the mean squared error (MSE) and the coefficient of determination ($R^2$) of each output in its original units. Because all predictors are evaluated on the same samples within each split, the main comparison is the relative change of MSE with respect to persistence, computed per split and then averaged, together with the number of splits in which the network obtains a lower MSE. The percentage of predicted speeds above $v_{max}$ is also reported.

4. Results

Each test split contained 8 PandaSet sequences (408, 328, and 248 samples at horizons of 1, 2, and 3 s), and the three KITTI sequences provided 828, 798, and 768 samples. Unless otherwise stated, values are the mean ± standard deviation over the five splits.

4.1. Training Behavior

Figure 6 shows the training and validation loss of the proposed model in the first split at the 1 s horizon. The validation loss reached its minimum after four epochs and increased afterwards, while the training loss continued to decrease; early stopping retained the weights of the best validation epoch, which prevents the overfitting observed when training for a fixed number of epochs.

4.2. Comparison with the Persistence Baseline

Table 2 reports the MSE of the persistence baseline and of the proposed model, together with the relative change of MSE and the number of splits in which the model was better; Figure 7 shows the relative changes graphically.

On unseen PandaSet sequences, the model reduced the speed MSE with respect to persistence by 15.7% at 1 s (lower error in 4 of 5 splits), 23.0% at 2 s, and 27.0% at 3 s (5 of 5 splits in both cases). The heading-rate MSE decreased by 3.7% at 1 s (5 of 5 splits), 16.4% at 2 s (4 of 5), and 13.7% at 3 s (3 of 5), with a large variability across splits at the longer horizons.

On KITTI, without retraining, the heading-rate MSE was on par with persistence at all horizons (relative changes between $-1.3\%$ and $+5.4\%$). The speed MSE, in contrast, was higher than that of persistence by 24.4%, 40.8%, and 42.9% at 1, 2, and 3 s, and the model was better in at most one of the five splits.

4.3. Coefficient of Determination

Table 3 reports $R^2$ for completeness. The speed $R^2$ is high for all predictors because the speed varies little within a few seconds. The heading-rate $R^2$ on PandaSet varies strongly between splits (standard deviation of about 0.6–0.85) and is negative at the longer horizons for both predictors: the test sequences of some splits contain mostly straight driving, where the variance of the heading rate is very small and $R^2$ becomes unstable. For this reason, the relative MSE comparison of Table 2 is used as the main criterion.

Figure 8 compares predicted and actual values in the first split at the 1 s horizon, and Figure 9 shows the speed along the first KITTI sequence. In KITTI, the predicted speed closely follows the persistence baseline, with a lag of about one horizon with respect to the actual mean speed.

4.4. Ablation of the Physics Term

In all 30 runs (five splits and three horizons), the CNN-LSTM trained with and without the physics term produced identical metrics to four decimal places, and no predicted speed exceeded $v_{max}$ in either configuration. The physics penalty was therefore never active during training: all target and predicted speeds were well below the cornering limit given by Eq. (4) with $\mu =0.7$.

5. Discussion

Within PandaSet, the CNN-LSTM consistently predicted speed better than persistence, and the advantage grew with the horizon, from 15.7% at 1 s to 27.0% at 3 s. This pattern is consistent with the role of the road-geometry preview: over one second, the future speed is close to the current one and there is little to gain, whereas over two or three seconds the geometry ahead provides information about upcoming decelerations and accelerations that the last observed value cannot contain. The heading-rate error also decreased with respect to persistence, but less consistently across splits; the heading rate is a noisier target, and the preview features are a coarse description of the road, based on the extent of the ground points rather than on detected lane markings.

The zero-shot results on KITTI show the limits of this generalization. Even though the road-geometry features are now computed with the same procedure from the LiDAR of both datasets, the model did not improve on persistence: the heading-rate error was on par with it, and the speed error was clearly higher. A plausible explanation is a shift in the speed distribution: the KITTI sequences reach speeds close to 80 km/h, whereas the PandaSet training data are mostly below 50 km/h, so the speed changes learned on PandaSet do not extrapolate well. The two datasets also differ in LiDAR sensor, mounting height, and driving environment, which affect the ground filtering and the preview offsets. Training on more diverse data, or adapting the model to the target domain, would be required to obtain a benefit over persistence in a new dataset.

The physics-informed term did not influence the results. In the tested data, the predicted speeds never approached the physical cornering limit, so the one-sided penalty remained zero and did not contribute to the gradients. In this setting, the term acts only as a safeguard that would become active if the network predicted physically inadmissible speeds; its effect on accuracy can only be evaluated on data that include speeds close to the limit, such as sharper curves, higher speeds, or lower friction.

Table 4 places the proposed approach qualitatively next to representative methods from the literature. Each method was evaluated with its own data and conditions, so the comparison describes the characteristics of each approach rather than a benchmark on common data.

5.1. Limitations

Several limitations should be considered when interpreting these results:

• Size of the evaluation. Each test split contains only eight PandaSet sequences, and the zero-shot evaluation uses three KITTI sequences. The variability across splits is large for the heading rate, and more data would be needed to establish the size of the improvements more precisely.
• Physics term. The physics-informed penalty was never active in the tested data, so its effect on accuracy could not be assessed.
• Cross-dataset generalization. The model did not improve on persistence when applied to KITTI without retraining, and its speed error was higher than that of persistence.
• Road preview. The preview offsets are derived from the extent of the ground points, without lane or road-boundary detection, and can be affected by sidewalks, parked vehicles, or open areas next to the road.
• Steering proxy. The heading rate of the vehicle is used as a proxy for steering; the steering-wheel angle itself is not available in the datasets used.
• Physical model. The friction coefficient is fixed at 0.7 rather than estimated, and only the lateral cornering limit is modeled, not the combined longitudinal and lateral friction ellipse.
• Offline validation. All experiments are offline. Closed-loop control, hardware-in-the-loop testing, actuator delays, and deployment on a vehicle were not evaluated.

6. Conclusions

This study presented a multi-task CNN-LSTM that predicts the heading rate and speed of a vehicle over horizons of 1 to 3 s from a LiDAR-based preview of the road geometry and the recent vehicle dynamics, trained with a physics-informed loss based on the maximum cornering speed of a banked curve with friction, and evaluated against a persistence baseline over five sequence-wise splits. On unseen PandaSet sequences, the model reduced the speed error with respect to persistence by 15.7% to 27.0%, with an advantage that grew with the horizon, and the heading-rate error by 3.7% to 16.4%. Without retraining, it did not improve on persistence on KITTI, where its speed error was higher. The physics-informed term was never active because all speeds remained well below the physical limit.

These results indicate that a LiDAR preview of the road geometry contains useful information to anticipate speed changes beyond a simple persistence model within the same domain, but that this benefit does not transfer directly to a different dataset. They also show the importance of comparing learned predictors with simple baselines and across several data splits.

Future work will focus on (i) training on larger and more diverse datasets and evaluating domain adaptation, (ii) improving the road preview with lane and road-boundary detection, (iii) evaluating the physics-informed term on data closer to the physical limits and with a dynamically estimated friction coefficient, (iv) incorporating the steering-wheel angle where available, and (v) validating the system in closed loop and on an instrumented vehicle, including deployment on embedded hardware such as the NVIDIA Jetson platform. In the long term, this line of work aims to contribute to advanced driver assistance systems for curve negotiation, in line with the goals of the Latin New Car Assessment Programme (Latin NCAP) of promoting active safety technologies in the region.