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Deep Kinematic Models for Kinematically Feasible Vehicle Trajectory Predictions
Henggang Cui, Thi Nguyen, Fang-Chieh Chou, Tsung-Han Lin, Jeff Schneider, David Bradley, Nemanja Djuric
TL;DR
Predicting surrounding vehicles matters for autonomous driving, but unconstrained trajectory models may produce inaccurate headings and kinematically infeasible motion. The paper embeds two-axle vehicle kinematics in a deep prediction model. On real-world data, the method outperformed the state of the art, and it was tested onboard SDVs.
Problem
Unconstrained vehicle-trajectory methods do not explicitly encode kinematics, although accurate future motion and headings are important for autonomous-vehicle planning.
Source: §I · p. 1Method
The deep kinematic model embeds two-axle vehicle kinematics in a learning model’s output layer to jointly predict positions, headings, and velocities.
Source: §I · p. 1Results
The proposed method outperformed the state of the art in extensive real-world experiments, and the framework was subsequently tested onboard SDVs.
Source: §V · p. 6Takeaways & Limitations
Abstract
from arXiv · showhide
Self-driving vehicles (SDVs) hold great potential for improving traffic safety and are poised to positively affect the quality of life of millions of people. To unlock this potential one of the critical aspects of the autonomous technology is understanding and predicting future movement of vehicles surrounding the SDV. This work presents a deep-learning-based method for kinematically feasible motion prediction of such traffic actors. Previous work did not explicitly encode vehicle kinematics and instead relied on the models to learn the constraints directly from the data, potentially resulting in kinematically infeasible, suboptimal trajectory predictions. To address this issue we propose a method that seamlessly combines ideas from the AI with physically grounded vehicle motion models. In this way we employ best of the both worlds, coupling powerful learning models with strong feasibility guarantees for their outputs. The proposed approach is general, being applicable to any type of learning method. Extensive experiments using deep convnets on real-world data strongly indicate its benefits, outperforming the existing state-of-the-art.
I. INTRODUCTION
Predicting surrounding vehicles’ future movement is important for autonomous driving, but unconstrained predictions can yield inaccurate headings and kinematically infeasible trajectories. The paper proposes a deep kinematic model that embeds vehicle kinematics in trajectory prediction.
I. INTRODUCTION
- SDVs may bring traffic-safety, transportation-access, and environmental benefits.Source: §I · p. 1
- Surrounding-vehicle motion prediction is a key component of safe autonomous operation, supporting estimates of future traffic states for planning.Source: §I · p. 1
- Unconstrained models can produce poor vehicle headings and kinematically infeasible motion, undermining occupancy reasoning and potentially affecting the SDV’s future motion plans.Interpolating headings from positions is sensitive to position errors, and a vehicle’s center-motion direction may differ from its true heading.Source: §I · p. 1
- The proposed deep kinematic model embeds two-axle vehicle kinematics in a learning model’s output layer and jointly predicts positions, headings, and velocities.The paper presents a CNN using rasterized vehicle context as an example; the approach is described as applicable to any deep architecture.Source: §I · pp. 1–2
- The method outperformed the state of the art on large-scale real-world SDV-fleet data, and was subsequently tested onboard SDVs.The reported predictions were accurate and kinematically feasible.Source: §I · p. 2
II. RELATED WORK
Earlier autonomous-driving prediction approaches include physical propagation and map-based lane following. Their assumptions and coverage limit their usefulness across prediction horizons and unusual traffic scenarios.
A. Motion prediction in autonomous driving
- For short-term prediction, a common approach propagates the current estimated vehicle state with a physical model under an assumption of unchanged driver controls.Source: A. Motion prediction in autonomous… · p. 2
- The constant-control assumption becomes increasingly unrealistic over longer horizons, degrading this approach’s performance.Source: A. Motion prediction in autonomous… · p. 2
- Map-based lane-following methods generate paths from vehicle state and lane topology, but unusual traffic scenarios are difficult to model.They first associate vehicles with lanes and then generate paths for each vehicle-lane pair.Source: A. Motion prediction in autonomous… · p. 2
B. Learned prediction models
Learned motion-prediction methods range from classical models to deep networks, but many predict only vehicle centers without enforcing kinematics. The paper proposes an architecture-agnostic kinematic layer to address this limitation.
B. Learned prediction models
- Classical machine-learning methods, including hidden Markov models, Bayesian networks, and Gaussian processes, require manually designed features and lack state-of-the-art performance.Source: B. Learned prediction models · p. 2
- Deep approaches use recurrent networks, rasterized scene context, social layers, or generative models; some generative methods require many trajectory samples at inference.The cited Social-GAN example may require as many as 20 samples.Source: B. Learned prediction models · p. 2
- Many learned methods predict center positions without kinematic constraints, while heading interpolation or independently predicted headings can produce infeasible states.TrafficPredict and TraPHic include actor type and size but still do not enforce kinematics.Source: B. Learned prediction models · p. 2
- The proposed vehicle kinematic layer is architecture- and learning-method-agnostic, replacing a trajectory decoder or policy model to improve accuracy and guarantee kinematic feasibility.Source: B. Learned prediction models · p. 2
C. Vehicle motion models
Vehicle motion models include kinematic models that assume no wheel slip and dynamic models that represent more complex tire-force interactions. The paper contrasts its learned approach with optimization-based model predictive control.
C. Vehicle motion models
- Kinematic vehicle models assume no wheel slip, whereas dynamic models attempt to represent interactions when tire forces cannot overcome inertia.Source: C. Vehicle motion models · p. 3
- Prior model predictive control work indicates that a simple kinematic bicycle model can represent non-articulated vehicles performing normal driving maneuvers.Source: C. Vehicle motion models · p. 3
- Model predictive control uses known costs and dynamics to optimize control inputs, while this approach leaves costs and optimization implicit in a deep neural network.Source: C. Vehicle motion models · p. 3
III. PROPOSED APPROACH
The approach assumes tracked actor states and map data are available, then predicts future actor states over a fixed-step horizon in actor-centric coordinates.
III. PROPOSED APPROACH
- An onboard tracking system supplies actor states, including bounding boxes, position, velocity, acceleration, heading, and turning rate, alongside map data.The assumed map includes road locations, lane boundaries, and other relevant information.Source: §III · p. 3
- Given map data and actor state estimates through time tj, the task is to predict that actor’s next H states.The time steps are equally spaced; the example uses a 0.1-second gap for a 10 Hz tracker.Source: §III · p. 3
- Past and future states use actor-centric coordinates, with forward and left defining the x- and y-axes and the bounding-box center as origin.Source: §III · p. 3
A. Unconstrained motion prediction
The unconstrained baseline predicts future positions directly, derives other state quantities from them, and can extend the prediction to multiple trajectory modes.
A. Unconstrained motion prediction
- The Unconstrained Model predicts future x- and y-positions directly, then derives headings and velocities from displacement vectors without vehicle-kinematic constraints.This can produce inaccurate or infeasible positions and higher derivatives.Source: A. Unconstrained motion prediction · p. 3
- The model takes map and state information encoded as an overhead raster and trains against Euclidean distance between predicted and observed positions.It predicts H trajectory points at Δt-second intervals.Source: A. Unconstrained motion prediction · p. 3
- For multimodal prediction, the model outputs M trajectory modes with softmax probabilities and selects the mode closest to ground truth during training.The selected mode is indexed by m*.Source: A. Unconstrained motion prediction · p. 3
- The multimodal objective trades off winning-mode trajectory error against mode-selection cross-entropy, updating positions only for the winning mode and probabilities for all modes.The objective is optimized across actors and times to learn the model parameters; this predictor is called the Unconstrained Model (UM).Source: A. Unconstrained motion prediction · pp. 3–4
B. Deep kinematic model
The Deep Kinematic Model adds a differentiable vehicle-kinematics layer that predicts controls and rolls them forward into feasible actor states.
B. Deep kinematic model
- DKM inserts a sequence of H kinematic nodes between UM’s final hidden layers and output, with each node applying the same vehicle-model update equations.Source: B. Deep kinematic model · p. 4
- At each horizon, a node predicts longitudinal acceleration and steering angle, then uses them to roll out the actor’s next state.The modeled two-axle vehicle follows the kinematics introduced in.Source: B. Deep kinematic model · p. 4
- The actor state comprises position, heading, and velocity; fixed kinematic parameters include wheel distances and maximum allowed acceleration and steering angle.Source: B. Deep kinematic model · p. 4
- The first node starts from the current state, later nodes use the preceding node’s output, and predicted controls are clipped to allowed ranges.Source: B. Deep kinematic model · p. 4
- DKM remains differentiable and uses UM’s losses without control labels; its kinematics can be adapted to other vehicle types and its layer to other learning methods or architectures.The paper notes these extensions are beyond the scope of its work.Source: B. Deep kinematic model · p. 4
IV. EXPERIMENTS
The experiments compare learned and constrained motion-prediction methods on real-world vehicle data, including the proposed DKM and several baselines and variants.
IV. EXPERIMENTS
- The evaluation uses 240 hours of 10Hz driving data, with 7.8 million moving-vehicle points and prediction horizons up to 6s.The data were collected across varying conditions and several US cities, then split 3:1:1 into train, validation, and test sets.Source: §IV · p. 4
- The comparison includes a UKF that propagates tracked states and Social-LSTM, which models interactions between actors.The UKF is non-learned; Social-LSTM considers social interactions.Source: §IV · p. 4
- Polynomial baselines fit n-degree curves, while UM-velo predicts unconstrained velocities and UM-LSTM recursively predicts positions.Poly-n infers scalar polynomial parameters in the last hidden layer; UM-LSTM adds a 128-unit LSTM layer.Source: §IV · pp. 4–5
- The study reports prediction errors in Table I, with lower errors indicating better performance.Maximum acceleration and steering angle are set to 8m/s^2 and 45°, respectively, roughly following midsize-sedan characteristics.Source: §IV · p. 4, Table I · p. 5
- UM-heading additionally predicts headings, and CTRA applies the Constant Turning Rate and Acceleration model at the output.The DKM is also evaluated as the proposed deep vehicle kinematic model.Source: §IV · p. 5
- The models share a backbone and training setup, and most use a multimodal loss to output three trajectories and their probabilities.UKF and Social-LSTM are exceptions; poly-n, UM, UM-velo, and UM-LSTM predict positions and derive headings through interpolation.Source: §IV · p. 5
A. Results
The results compare prediction accuracy and kinematic feasibility across unconstrained and constrained methods. DKM performs best overall, while other methods show trade-offs between position accuracy, heading quality, and feasibility.
A. Results
- Higher-degree polynomial models improve position errors over poly-1, but degrade heading and feasibility metrics.Poly-1 is a constant-velocity model that guarantees feasible trajectories but does not capture real-world behavior well; all metrics improve over the non-learned model.Source: A. Results · p. 5
- Unconstrained models perform strongly on center-position prediction, but UM and UM-velo have about 30% kinematically infeasible trajectories.Their infeasibility is mostly associated with heading interpolation for slow-moving or stopping actors.Source: A. Results · p. 5
- UM-heading improves heading and infeasibility metrics over UM, yet around 20% of its predictions remain kinematically infeasible.The passage attributes the remaining infeasibility to independently predicted positions and headings without kinematic constraints.Source: A. Results · pp. 5–6
- DKM gives the best performance, improves heading errors by a large margin, and guarantees kinematically feasible trajectories.Table I compares prediction errors; the evaluation also reports the percentage of kinematically infeasible trajectories.Source: A. Results · pp. 5–6
- At the 6s horizon, DKM’s acceleration and turning-rate distributions are closer to ground truth than UM’s, especially for turning rate.The turning-rate Wasserstein distance is more than two times smaller; the histograms use log-scale axes and ground-truth tails represent rare events and outliers.Source: Figure 5 · p. 5, A. Results · p. 6
B. Case studies
The case studies compare UM, poly-2, and DKM in a right turn and a stopping scenario, examining trajectories and predicted state profiles.
B. Case studies
- In the right-turn case, all three methods predict the turn, but DKM has smooth acceleration and heading profiles while UM’s acceleration and turning rate are noisy.Poly-2 also produces smooth profiles but cuts the corner and overshoots the turn; the authors note constrained approaches may struggle with complex maneuvers.Source: B. Case studies · p. 6
- In the stopping case, all methods predict a stop, but poly-2 briefly predicts reverse motion and DKM’s smooth deceleration best matches ground truth.UM again produces noisy acceleration and heading estimates.Source: B. Case studies · p. 6
V. CONCLUSION
The method combines deep learning with robotics vehicle models to guarantee kinematically feasible motion predictions. It is general across machine-learning algorithms; real-world fleet experiments outperformed the state of the art, and the framework was tested onboard SDVs.
V. CONCLUSION
- The approach combines deep-learning algorithms with robotics vehicle models to guarantee kinematically feasible predicted motion.The method is described as usable with any machine-learning algorithm.Source: §V · p. 6
- The approach is general across machine-learning algorithms, while the evaluation used convolutional networks as an example.The paper identifies both convolutional and recurrent deep networks as possible algorithms.Source: §V · p. 6
- Real-world, large-scale SDV-fleet experiments indicated benefits over the existing state of the art, and the framework was subsequently tested onboard SDVs.Onboard testing followed extensive offline testing.Source: §V · p. 6