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AUTONOMOUS VEH IC LES

Department of Mechanical Engineering, Stanford University, Building 530, 440Escondido Mall, Stanford, CA 94305, USA.*Corresponding author. Email: [email protected]

Spielberg et al., Sci. Robot. 4, eaaw1975 (2019) 27 March 2019

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Neural network vehicle models for high-performanceautomated drivingNathan A. Spielberg*, Matthew Brown, Nitin R. Kapania, John C. Kegelman, J. Christian Gerdes

Automated vehicles navigate through their environment by first planning and subsequently following a safetrajectory. To prove safer than human beings, they must ultimately perform these tasks as well or better thanhuman drivers across a broad range of conditions and in critical situations. We show that a feedforward-feedback control structure incorporating a simple physics-based model can be used to track a path up tothe friction limits of the vehicle with performance comparable with a champion amateur race car driver. Thekey is having the appropriate model. Although physics-based models are useful in their transparency and in-tuition, they require explicit characterization around a single operating point and fail to make use of the wealthof vehicle data generated by autonomous vehicles. To circumvent these limitations, we propose a neuralnetwork structure using a sequence of past states and inputs motivated by the physical model. The neuralnetwork achieved better performance than the physical model when implemented in the same feedforward-feedback control architecture on an experimental vehicle. More notably, when trained on a combination of datafrom dry roads and snow, the model was able to make appropriate predictions for the road surface on whichthe vehicle was traveling without the need for explicit road friction estimation. These findings suggest that thenetwork structure merits further investigation as the basis for model-based control of automated vehicles overtheir full operating range.

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INTRODUCTIONAutonomous vehicles promise to revolutionize humanmobility and ve-hicle safety. This promise stems from eliminating the need for a humandriver, markedly reducing the cost of mobility and, ideally, eliminatingthe 94% of crashes attributable to human recognition, decision, orperformance error (1). This, in turn, requires automated vehicles capa-ble of navigating safely through the world more skillfully than a humandriver. Many automated vehicle designs incorporate a commonarchitecture of generating a safe, collision-free trajectory through theenvironment with a high-level planning layer and subsequent trackingof this trajectory using lower-level controllers (2–5). Navigating safely,therefore, requires not only the planning of collision-free trajectories butalso the ability to tightly track the desired path, ideally within tens ofcentimeters. Furthermore, as deployment of automated vehiclesexpands, these path-following controllers must handle a wide varietyof conditions, including safely navigating icy roads when friction is re-duced or emergency collision-avoidance maneuvers when the needarises. Each of these cases is an example of everyday driving in whichoperation at the vehicle’s limits becomes critical.

Although there has been substantial work developing control tech-niques for automated vehicles, much of this effort has focused oncontrolling the vehicle under normal driving conditions with gentlemaneuvers on high-friction, dry road surfaces (6, 7). Research oncontrolling vehicles in more critical maneuvers near the friction limitshas illuminated many associated challenges (8–11). Fundamentally, asthe vehicle approaches the limits of tire-road friction, it becomes eitherunstable (if the rear tire reaches the limits) or uncontrollable (if the fronttire reaches the limits). To track a path at the limits of the vehicle’s cap-abilities, some estimate of the vehicle’s road-tire friction coefficient isnecessary for both trajectory design and determining the appropriatesteering commands. Obtaining such estimates is challenging in general

(12) and complicated further by the facts that tire-road friction oftenquickly changes and portions of the road may consist of differentsurfaces. Aside from difficulty in estimating this critical parameter, de-veloping an accurate dynamicmodel that can be used for trajectory gen-eration and following at the limits is a difficult task because theequations of motion become highly nonlinear. The designer must fur-ther choose the appropriate level of fidelity, deciding whether or not toinclude effects such as weight transfer across tires due to acceleration orthe lag in tire force generation with rapid steering movement.

As challenging as the limits of handling are for the control systemdesigner, they are even more challenging for the average driver and asubstantial factor in many crashes. Yet, more experienced drivers, par-ticularly those with racing experience even at an amateur level, have theability to safely control the vehicle at the full limits of its capabilities (13).In racing, this is demonstrated by producing low and consistent laptimes, but, in a critical maneuvering situation, this capability meansthe ability to harness all of the tire-road friction to avoid a crash. Ifwe want automated vehicles that can maneuver better than anexperienced driver in critical situations, the bar on the performanceof a tracking controller must be set rather high.

Here, we show that a simple path-tracking architecture can enablean automated vehicle to track a path accurately while using the tire-roadfriction as fully as a champion amateur race car driver. The key is an ap-propriatemodel.Using a physics-based dynamicsmodel for feedforwardcontrol, a simple linear feedback controller, and a trajectory designedfrom the vehicle’s modeled friction limits, the car could drive withmeanpath-tracking errors below 40 cm at themodeled friction limits. Becausethe model represents only an estimate of what the true limits are, webenchmarked our automated vehicle performance against a championamateur race driver and compared lap times over segments on the track.This novel comparison demonstrates that the controller’s operation atthe modeled friction limits is comparable in friction utilization withthe real-world ability of an experienced race car driver.

To achieve this level of performance, of course, the simple physics-based model must be well characterized for the particular condition of

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driving on this dry high-friction race course. The question of how com-parablemodels could be developed for on-road automated vehicles thennaturally arises. Although obtaining parameters for different vehicleswould be feasible as part of a typical development process, severalparameters vary strongly with the road condition. Although research-ers have demonstrated online parameter estimation that can adapt tochanging road conditions (14–16), such techniques have not maturedto the point of commercial deployment in automobiles or theperformance required of a safety-critical system. In addition, real-timeestimation does not leverage the vast quantity of data that current ve-hicles generate, and future automated vehicles will conceivably be ableto share. Nor does it address the question of model fidelity given thatparameter estimation grows more challenging as the model com-plexity increases. Ideally, the model generation process should be ca-pable of leveraging data about surfaces with varying levels of frictionand reduce the number of a priori modeling decisions while stillcapturing the accuracy and performance of physics-based modelstuned for specific conditions.

These requirements motivated investigating neural network modelsfor vehicle control. Because of their universal function approximationproperties, neural network models have shown numerous recentachievements, such as benchmarks in image recognition andmasteringthe game of Go (17–19). Early research in neural network modelsillustrated that these models are capable of vehicle control and dynamicmodel identification (20, 21). Neural network vehicle models haveshown success in numerous robotics applications fromquadcopter con-trol to control of scale rally racing vehicles (22, 23). These models havebeen successfully used for vehicle dynamic model identification buthave yet to be used to capture changing vehicle dynamics from drivingat the limits on multiple friction surfaces (24, 25). Additionally, neuralnetworkmodels can use history information to capture time-varying or

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higher-order effects, as shown in bothmodel helicopter and robotics applica-tions (26–28).
We present a feasibility study inwhichwe used the states and inputs of thephysics-based model as guidance to de-velop a two-layer feedforward neuralnetwork capable of learning vehicle dy-namic behavior on a range of differentsurfaces. The network involves a combi-nation of current measurements andhistory information from three previoustime steps. The history informationenabled the network to provide predic-tions of behavior at different friction levelswithout the need for an explicit frictionestimation scheme. When trained on acombination of high- and low-frictiondata, the model made predictions appro-priate to the surface described by thehistory information. By foregoing the stepof friction estimation, the neural networkwith history information fused estimationand prediction capabilities, simplifyingthe task of vehicle control. This additionalfunctionality did not come at the expenseof performance. We show increased pathtracking performance at the limits when


compared with the tuned physics-based model. Furthermore, asimulation study demonstrates that the neural network model couldcapture a range of dynamic behaviors that were not included in thesimple physics-based model.

RESULTSTo investigate the performance of a path tracking architecture at thelimits of a vehicle’s handling capabilities, we designed an experimentalcomparison with an experienced human race driver. An appropriatebenchmark for our automated vehicle in this case is a proficient humandriver who has substantial driving and amateur racing experience andextensive familiarity with the test course. In this experiment, we used aphysics-based feedforward-feedback controller (as shown in Fig. 1) im-plemented on an automated 2009 Audi TTS (Shelley) (Fig. 2A) (29).The controller tracked a desired path while a separate controllermatched the desired vehicle speed through brake, throttle, and gear shiftcommands. The path and speed profile were designed through optimi-zation techniques tominimize the time required to drive the track basedon the vehicle model (30).

The model used for feedback-feedforward control generates an ap-propriate steer angle to apply for a given path curvature and vehicle lon-gitudinal speed. The accuracy of this input has a substantial effect onboth the resulting path tracking error and the required feedback effort.The feedforward steering command here was derived from the equa-tions of motion for a planar single-track or “bicycle” model, acommonly used model in the vehicle dynamics community derivedfromNewtonian physics. For this paper, mention of the “physics-basedmodel” explicitly refers to the planar bicycle model. To calculate afeedforward steering input from these equations of motion, we usedsteady-state operating conditions to determine feedforward tire forces.

Control

SteeringFeedforward

SteeringFeedback

Automated Vehicle

Κ(s)

Ux(s)

++

δfb

δffwδ

e, ΔΨ

Modeling

Neural Network Model

Bicycle Model

Contains Model

Uy Uxr

Fxfδ

Fig. 1. Simple feedforward-feedback control structure used for path tracking on an automated vehicle. Pos-sible models for use in generating feedforward steering commands consist of the physics-based model or a neuralnetwork model.

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Subsequently, these steady-state tire forces were converted to a desiredsteering input through the use of a physics-based tire model, whichexplicitly accounted for the effect of tire force generation and saturation.To compensate for inaccuracies in generated feedforward commandsand disturbances, we used a simple path-based steering feedback con-troller to track a desired trajectory. This controller is based on e, thevehicle’s lateral deviation from the desired trajectory, and DY, the ve-hicle’s heading deviation from the desired trajectory, as shown in Fig. 1.The tire parameters of the physics-basedmodel were fit using nonlinearleast squares with experimental vehicle data.

To compare the automated approach with the experienced driver,we created a closed course study of racing performance consisting ofthe first five turns of Thunderhill Raceway Park inWillows, California.Both the automated vehicle and human participant attempted tocomplete the course in the minimum amount of time. This consistedof driving at accelerations nearing 0.95gwhile tracking aminimum time


racing trajectory at the physical limits of tire adhesion. At this combinedlevel of longitudinal and lateral acceleration, the vehicle was able to ap-proach speeds of 95miles per hour (mph) on portions of the track. Bothautomated vehicle and human participant participated in 10 trials ofdriving around the closed course. Tests were conducted under the sameconditions, including ballasting the car to equate themass of the vehicleduring automated andhuman-driven testing. Even under these extremedriving conditions, the controller was able to consistently track the ra-cing line with themean path tracking error below 40 cm everywhere onthe track (Fig. 2D).

To investigate the consistency of path following, we examined themean absolute deviation from the median (MADmedian) path disper-sion, which is a robust measure of each driven trajectory’s deviationfrom the track centerline. The experienced driver displayed a muchgreater path dispersion on average between laps than the automated ve-hicle (Fig. 2D). These data are also represented as a projection onto the

A

D

E

B C

Fig. 2. Automated and human driving. (A) “Shelley,” Stanford’s autonomous Audi TTS designed to race at the limits of handling. (B) Human driver’s MAD median pathprojected onto the first five turns of Thunderhill Raceway Park in Willows, California. (C) Shelley’s MAD median path scaled by a factor of 4 to highlight relativedifferences. (D) MAD median path for both the human driver and Shelley (in red) along with Shelley’s mean absolute deviation from the intended path (in blue).(E) Segment times from the champion amateur race driver benchmarked to Shelley.

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trackmap in Fig. 2 (B and C), where N represents the north direction ofthe test course. The controller’s consistency of path deviation demon-strates that the control approach was not only accurate but also precise.The lowpath dispersion of the automated vehicle can be explained by itsobjective to follow a precomputed trajectory using a highly accurateGPS-based localization system. As discussed later, the higher dispersionof the human-driven paths suggests that the human driver adopted adifferent strategy than the automated vehicle. Thus, the human andautomated vehicle cannot be compared in terms of tracking accuracyor variability. They can, however, be compared in terms of time.

We used the metric of section time to compare both automated ve-hicle and human driver because this is the metric that both the racingdriver and the automated vehicle’s desired trajectory attempted tomin-imize. To compare section times on our closed racing course,we dividedthe course into three sections. Figure 2E shows section times recorded


during the combined trials of both humanparticipant and automated vehicle atThunderhill RacewayPark.As thenotchedbox and whiskers plots indicate, Shelley’stimes through each section of the trackare well within the range of section timesfrom the proficient human driver, whichshows that the performance of themodel-based controller is comparable with thatof an experienced race driver at the limitsof the vehicle’s capabilities. On each box,the central line is the median, the pointmarked with a black diamond is themean, the edges of the box are the 25thand 75th percentiles, the whiskers extendto the adjacent sample no further than 1.5times the interquartile range (IQR) be-yond the edges, and possible “outliers”are plotted individually. The notchedsections provide visual comparison inter-vals and were calculated as median±1:57� IQR=

ffiffiffin

p. The comparable lap

times indicate comparable friction utiliza-tion by the simple feedforward-feedbackcontroller with its physics-based model.The low path dispersion and comparablesection times relative to a human driverresulted from a model tuned to a partic-ular road surface. Having established aperformance baseline relative to an ex-perienced human driver, we could thenuse this controller performance as abaseline for the neural network model.

Motivated by the states and controlsconsidered in the physics-based model,we chose to use a feedforward neuralnetwork with the inputs shown in Fig. 3.The neural network model consists oftwo hidden layers with 128 units ineach layer and makes use of three de-layed input states for each model stateor control. Similar to the physics-basedmodel, the network predicts the vehi-cle’s yaw rate and lateral velocity deri-

vatives. The network was initially trained in a supervised manner toreplicate the physics-based model. After training on 200,000 trajec-tories over the full range of the physics-based model’s input space,we updated the neural network with experimental vehicle datacollected in high- and low-friction testing. High-friction testing wasconducted at Thunderhill Raceway Park, and low-friction testingwas conducted on a mixture of ice and snow at a test track nearthe Arctic Circle.

Although the neural network model can be used in a wide variety ofcontrol schemes, we wanted to compare it with the benchmarkprovided by the physics-based feedforward-feedback controller. There-fore, we used the learned neural networkmodel to generate feedforwardcommands, making the same steady-state assumptions as the physics-basedmodel. To generate feedforward steering commands, we foundequilibrium points of the neural network dynamics model using a

tt-10ms

t-20mst-30ms

δ

Steering Angle

tt-10ms

t-20mst-30ms

Lateral VelocityUy

tt-10ms

t-20mst-30ms

Longitudinal VelocityUx

tt-10ms

t-20mst-30ms

Front Longitudinal Force

Fxf

tt-10ms

t-20mst-30ms

Yaw Rater

Input Layer

FC1 FC2

Output Layer

Uy

r

˙

Fig. 3. Neural network dynamics model with input design based on the physics-based model. FC1 and FC2denote fully connected layers in our two-layer feedforward neural network dynamics model.

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second-order nonlinear optimization method. Measured speed andpath curvature were used as inputs to the optimization to specify thecorrect feedforward command. This optimization was implementedonline on the vehicle, and feedforward steering commands werecalculated from the network at a rate of 20 Hz. To compensate fordisturbances and model mismatch, we used the same simple path-based feedback controller structure in both cases for the comparisonbetween controllers.

We compared both controllers by implementing them on an auto-nomous Volkswagen GTI (Fig. 4B), which we had the opportunity touse to obtain data on snow as it was prepared for automated driving.Figure 4A demonstrates the oval track on the skid pad at ThunderhillRaceway Park used to evaluate the two controllers. Both controllerschemes used the same longitudinal speed profile and longitudinalcontroller and were tested at the limit of the vehicle’s capability. Whencompared, we found that on turn entry, labeled “1” in Fig. 4C, theneural network controller learned to apply more steering than the


physics-based model, resulting in lower tracking error in the middleof the turn. In the middle of the turn, the tracking error was influencedby the amount of road-tire friction available, with negative errorshowing the vehicle exceeding the grip limit. Furthermore, the neuralnetwork controller commanded less steering on turn exit (“3”) becauseof its closer proximity to the desired path. The peaks shown on the exitand the straight sections were influenced by the steering feedbackparameters, such as the controller gain and lookahead distance. Wefound that the neural network controller was able to visibly achieve adifferent distribution of lateral errors over the course of a lap at the lim-its (Fig. 4D). The count shown in this distribution is dependent on thebin size of 2.4 cm, where the number of bins chosen was 25. Theseresults suggest that the neural network model is, in this case, able toachieve a higher degree of model fidelity than the physics-basedmodel when implemented under the same set of steady-state assump-tions and control architecture. That is, the controller satisfied thedesired performance benchmark on this course.



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A B

C D

Fig. 4. Experimental model comparisons. (A) Experiment track map, showing corresponding sections 1, 2, and 3 in experimental data plots. (B) Picture of VolkswagenGTI experimental autonomous race vehicle. (C) Experimental comparison between physics-based controller and neural network controller showing lower tracking errorat the limits on oval test track. (D) Histogram showing difference in lateral error distributions of neural network and physics-based controllers on oval test track.

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The true power of the data-drivenmodel, however, is not just to pro-vide comparable performance to the physics-based approach. The neu-ral network model also has the potential to incorporate higher-orderdynamic effects and learn vehicle behavior on different road surfaces.To determine whether our learned model (Fig. 3) displays these char-acteristics, we examined predictions incorporating higher-fidelity vehi-cle dynamics modeling and multiple surface friction values in a pair ofadditional studies.

To demonstrate the modeling capability of the network relative tothe simplified physics-based model, we used different fidelity dynamic


models to generate training data based on a uniform random controlpolicy. These data were used to both train the network and identify thebest-fit parameters for the physics-based model, enabling their predic-tive capability to be compared. In the first comparison, the physics-based model itself generated the data, so there was no model mismatchbetween the physics-based model that generated the simulation dataand the physics-based model that was learned. In this case, “No Mis-match” (Fig. 5A), the physics-based model substantially outperformedthe neural network model and recovered the parameter set used forsimulation. This is understandable, because the physics-based model



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A B

C D

Fig. 5. Training and testing. (A) Training process for simulated data including multiple effects of model mismatch between data generation and optimization models.(B) Testing process for simulated data, showing generalization capability of learned models. (C) Training process for real collected vehicle data under various frictionconditions. (D) Testing process for real vehicle data showing generalization capabilities of learned models.

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represents the true model form behind the data, whereas the neuralnetwork was attempting to learn an approximate model.

The picture changed, however, when models of different fidelitygenerated the training data. We generated simulated data usingphysics-based models that were augmented to include the effectsof longitudinal weight transfer, tire relaxation length, and multipleroad surface friction values (Fig. 5A). When the data were fit to thesimple physics-based model, these additional effects of model mis-match resulted in biased parameter values. We found that in all ofthese cases of model mismatch, the neural network model outper-formed the physics-based model in prediction (Fig. 5A). Furthermore,we found that these results extended to held-out simulation data (Fig.5B). These results are consistentwith the insights fromphysics thatwereused to design the neural network predictive model. For example, inlearning the effects of tire relaxation, the network was able to capturechanging slip angle dynamics by including multiple delayed stages ofstates and inputs, whereas the physics-based model only used the cur-rent input and states to predict the vehicle’s dynamics.

Motivated by the neural network’s ability to capture a rich array ofdynamics in simulation, we designed an additional study to evaluatethe model’s ability to make predictions on different road surfaces inreal-world conditions. To do this, we collected both manually drivenand automated data using the Volkswagen GTI platform (Fig. 4B).Additionally, we collected data from both high-friction driving ondry asphalt and low-friction driving on snow and ice. To illustratethe capability of the neural network to learn dynamicsmodels for bothlow- and high-friction conditions, we separately trained and verifiedmodels for each condition individually (Fig. 5C). The results indicatean advantage for the neural network structure over the physics-basedmodel in both the high- and low-friction cases. The data from bothconditions could further be combined and used to train a single neuralnetwork and physics-based model. We found that this results in thehighest training and testing errors for the physics-based model be-cause of its inability to capture two different friction conditions asshown in Fig. 5C. The identified physics-based model characteristicsrepresented approximately the average road surface condition, whereasthe hidden nodes of the neural network model were able to implicitlyrepresent and apply different conditions. As a result, the neural networkoutperformed the physics-based model by over an order of magnitudein both training and testing. These results indicate that the neuralnetwork model had greater predictive performance on both mixedand isolated road surface data, a characteristic that also generalizes toheld-out test data as shown in Fig. 5D.

DISCUSSIONThe results demonstrate that, with an appropriate model, a simplefeedforward-feedback controller can provide path-tracking perform-ance at the limits of a vehicle’s friction capabilities, with friction utilizationcomparable with an experienced human race car driver. In addition, ourfeasibility study demonstrates that a neural network can provide the nec-essary model for such an approach, achieving performance that is betterthan a simple, but carefully tuned, static physics-based model. Most no-tably, such a model could predict performance on different frictionsurfaces without having to explicitly identify friction and demonstratedrobustness when higher-fidelity vehicle dynamics characteristics wereconsidered. The testing presented here shows that such neural networkstructures are viable candidates for dynamic models of automated vehi-cles and merit further investigation.


Benchmarking a tracking controller such as the one presented hereagainst human performance is challenging. As demonstrated by the lev-el of path dispersion, the human driver did not formulate the problemin terms of exact path tracking. Rather, the human driver tended toanchor the desired path at particular points, such as the apex of a turn,and focused on pushing the car to the friction limits (13). Becausethe approach of the humandriver was fundamentally different from thetypical automated vehicle architecture, section time seemed to be thefairest comparison of the two approaches. Both the human and the de-sired trajectory were operating with the intent to minimize time. Giventhe extreme sensitivity that section time displays to friction utilization,we can infer comparable friction utilization from comparable time.

Additionally, although our champion amateur driver is fast, profes-sional drivers are faster still, suggesting an even greater capability to usefriction. Thus, we have demonstrated capability comparable with an ex-pert human, but we have not demonstrated the capability to exceed thehigh end of human performance. Accomplishing this will, in all likeli-hood, involve adopting some of the human driver’s willingness to devi-ate from the path to more fully use friction and reduce time.

The results comparing the control performance of the neuralnetworkmodel and the physics-basedmodel indicate that the controllerusing the neural network model had better path tracking performanceon the path chosen for testing. The controller using the physics-basedmodel attained a largermagnitude of lateral error, operating near 50 cmduring cornering. However, on the basis of typical lane widths between2.7 and 3.6m and a typical vehicle width of 2m, both controllers wouldkeep a vehicle within a lane boundary even at the friction limits (31).The cornering speeds on this path did not exceed 26 mph, so this ex-periment reflects a reasonable model of an emergency maneuver for ur-ban or suburban driving. Although further verification with othermaneuvers is necessary before deployment, these results demonstratethe feasibility of a neural network approach to vehicle control at the limits.

When using the neural network model, the controller’s feedfor-ward computation only uses a small portion of the model’s statespace (where the vehicle is at steady state), whereas the neuralnetwork model has the capability to learn transient dynamicseffects, as shown by its one-step prediction error. Therefore, thetrue potential of the neural network for control has not been rea-lized in this particular control architecture. In addition, by makingsteady-state assumptions to generate commands from the neuralnetwork model, the state history of the network must be con-strained. The feedforward controller did not make full use of theneural network’s capabilities to simultaneously estimate and pre-dict for variable friction surfaces. Other control structures, suchas model predictive control, could make use of the estimation cap-abilities of the network, providing a simple way to tie together si-multaneous estimation and control. Similarly, it would be possibleto use a more complex physics model or estimate parameters on-line. However, this sequence of comparisons seemed to be theclearest way to establish a benchmark for the quality of the neuralnetwork model relative to more conventional approaches and hu-man performance.

When learning the neural network model of the vehicle’s dynam-ics, the learning process was efficient and only involved 35-min worthof data from the physical vehicle. Gathering additional data for otherroad surfaces, conditions, and tires is therefore possible without muchadditional cost. Additional investigation is required to establish theextent to which different conditions can be properly encoded in thisneural network structure.

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MATERIALS AND METHODSPhysics-based modelThe proposed structure of the physics-based control design begins withthe assumption that the vehicle dynamics are given by the planar single-track model, also referred to as the bicycle model. A schematic of thevehicle states is shown in Fig. 6A, and relevant parameters are describedin Table 1. The planar bicyclemodelmakes the key assumption that theleft and right tires act to produce a single combined lateral force, result-ing in just two lateral forces,Fyf andFyr, acting at the front and rear axles.Actuation of steer angle d at the front tire results in generation of thelateral tire forces through the slip angles af and ar. The two resultingstates that evolve are vehicle yaw rate r, which describes the vehicle an-gular rotation, and sideslip b, which is the ratio of the lateral velocityUy

to longitudinal velocity Ux. Because we are primarily interested in thelateral control of the vehicle, the longitudinal velocityUx is considered atime-varying parameter and not a vehicle state.

In addition to the two vehicle states b and r, two additional states arerequired to show the vehicle’s position relative to the desired path. Theseare also shown in Fig. 6A. The lateral path deviation, or lateral error e, isthe distance from the vehicle center of gravity to the closest point on thedesired path. The vehicle heading error DY is defined as the angle be-tween the vehicle’s centerline and the tangent line drawn on the desiredpath at the closest point.

The equations of motion for the states shown in Fig. 6A are given by

_Uy ¼Fyr þ Fyf cosdþ Fxf sind

m� rUx ð1AÞ

_r ¼ aFyf cosdþ aFxf sind� bFyr

Izð1BÞ

b ¼ arctanUy

Ux

� �ð1CÞ

_e ¼ UxsinDYþ UycosDY ð1DÞ

D _Y ¼ r � _sk ð1eÞ


To obtain equations of motion for feedforward controller design interms of derivatives of the error states, we can take time derivatives of _eand D _Y, set Fxf = 0, and substitute from Eqs. 1A and 1B, yielding

€e ¼ Fyf þ Fyr

m� Uxk_s ð2AÞ

D€Y ¼ aFyf � bFyr

Iz� k€s� _k _s ð2BÞ

The bicycle model dynamics are abbreviated as shown in Eq. 3B asfBike for describing the vector-valued learned bicycle model. The tireparameters (Cf, Cr, m) in this model are learned to predict the bicycle

Table 1. Physics model definitions.

Parameter
Symbol Units
Front axle to CG
a m
Rear axle to CG
b m
Front lateral force
Fyf N
Front longitudinal force
Fxf N
Front tire slip
af rad
Rear lateral force
Fyr N
Rear tire slip
ar rad
Steer angle input
d rad
Yaw rate
r rad/s
Sideslip
b rad
Lateral path deviation
e m
Heading deviation
DY rad
Longitudinal velocity
Ux m/s
Lateral velocity
Uy m/s
Fig. 6. Physics-basedmodel and tiremodel. (A) Schematic of planar bicycle model including error states, referred to in this paper as the physics-based model. (B) Frontand rear tire curves, with brush Fiala model fit to empirical tire data.

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model dynamics from measured vehicle data. The inputs to the lateralbicycle model at a time t are denoted by xt as shown in Eq. 3A.

xt ¼ ðr;Uy;Ux; d; Fxf Þt ð3AÞ

_r_Uy

� �¼ f Bikeðxt ;Cf ;Cr; mÞ ð3BÞ

Lookahead controllerThe physics-based controller used as a benchmark for control isbased on a feedforward-feedback architecture, a block diagram ofwhich is shown in Fig. 1. Inputs to the feedforward steering angledFFW are current path curvature k and the forward velocity Ux. In-puts to the feedback steering angle dFB are the error states e andDY. The total steering command d is the sum of the feedback andfeedforward inputs.

The objective of the steering feedforward is to provide an estimate ofthe steer angle required to traverse a path with a known path curvatureand velocity profile. This minimizes the level of compensation requiredby the steering feedback, reducing tracking errors and allowing for lessoverall control effort. The feedforward steering angle should dependonly on the desired trajectory and should be independent of the ac-tual vehicle states.

To design the feedforward steering controller from the physics-based model, we made the simplifying assumption that the vehicle op-erates under steady-state cornering conditions. This assumption hasbeen shown (32) to reduce oscillation of the controller’s yaw rate re-sponse. Setting _s ¼ Ux and €s ¼ _k ¼ 0 in Eq. 2 yields the followingsteady-state front and rear tire forces:

FFFWyf ¼ mb

LU2

xk ð4AÞ

FFFWyr ¼ ma

LU2

xk ð4BÞ

Under steady-state conditions and assuming small angles, thefeedforward steering angle of the vehicle relates to the front and rearlateral tire slip af and ar and path curvature k by vehicle kinematics

dFFW ¼ Lk� aFFWf þ aFFWr ð5Þ

where aFFWf and aFFWr are the lumped front and rear feedforward tireslip angles.

The choice of feedforward tire slip angles is related to the tire forcesin Eq. 4 via a tire model Fy = f(a). To account for saturation of tire forcewith increasing tire slip magnitude, a single friction coefficient brushFiala model (33) maps lateral tire slip angles into tire force as follows

Fy⋄ ¼�C⋄tana⋄ þ C2

⋄

3mFz⋄jtana⋄jtana⋄

� C3⋄

27m2F2z⋄tan3a⋄; ja⋄j < arctan

3mFz⋄

C⋄

� �

�mFz⋄sgn a⋄; otherwise

8>>>><>>>>:

ð6Þ


where the symbol ⋄ ∈ [ f, r] denotes the lumped front or rear tire, m is thesurface coefficient of friction, and C⋄ and Fz⋄ are the correspondingcornering stiffness and normal load parameters. The cornering stiffnessesand friction coefficient were determined by using nonlinear least squaresto fit experimental data, as shown in Fig. 6B.

With the feedforward design complete, the remaining step is todesign the feedback controller. The goal of the feedback controlleris to minimize a lookahead error eLA, which is the vehicle trackingerror projected a distance xLA in front of the vehicle (Fig. 6A).

The lookahead error and resulting feedback control law are given by

dFB ¼ �kPðeþ xLAsinðDYþ bssÞÞ ð7Þ

bss ¼ aFFWr þ bk ð8Þ

with proportional gain kp. Note that a key feature of this feedback con-troller is incorporation of steady-state sideslip information bss. In (32),accounting for sideslip in the feedback control law eliminated steady-state path tracking errors, assumingnoplant-modelmismatch. Further-more, a linear systems analysis shows that using steady-state sideslipinstead of measured vehicle sideslip allows the controller to maintainrobust stability margins.

Comparison with human driverTo provide a comparison for benchmarking the physics-based control-ler, we compared Shelley’s performance with that of a proficient driver.The proficient driver has years of amateur racing experience, includingworking with the research team. In addition, this driver has extensiveexperiencewith the course. The physics-based lookahead controller wasimplemented on Shelley, a 2009 Audi TTS. Shelley is equipped with anactive brake booster, throttle-by-wire, and electronic power steering forfull automated control. In addition, Shelley used an integrated naviga-tion system aided by differential global navigation satellite systemsignals to obtain multicentimeter accuracy of position measurements.The system included a dSPACE MicroAutoBox II, which was usedto record data from the vehicle as well as execute control commandsat 200 Hz.

The objective of comparing differences in racing performance be-tween the physics-based lookahead controller and the proficient driveris shown through comparisons of section times and path dispersion.Experimental tests were performed at Thunderhill Raceway Park, lo-cated in Willows, California. Both the human participant and Shelleydrove 10 consecutive trials of turns 2 to 6 on the East track. In each trial,the vehicle’s total mass was consistent. GPSmarkers for each section wereused to segment each trial into three portions for analysis. For both parti-cipants, the driven path recorded from GPS was used to calculate the lat-eral deviation and distance along the track centerline. To characterize thedispersion of each participant’s trajectories, we chose the MAD median.

Learning a global neural network modelAs opposed to learning parameters in our physics-based model, welearned a neural network model capable of foregoing the step ofmodeling and identifying latent states, such as vehicle road friction in-teraction. Inmodeling the unknown or changing dynamics, the design-er must condense all unknown or unmodeled effects into a givenparameter set of predefined dimension. By using a neural networkmodel using both control and state history denoted with a vector ht,

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we imposed less structure on the system identification task, allowingthe network model to identify its own internal representations of time-varying dynamics. In the expanded network input space, a given pointPt = ht can be used to fully construct the system’s latent state, given thatthe number of delayed stages T is long enough. This is demonstrated byTakens’s theorem and further demonstrated in learning complex heli-copter dynamics models (27, 34).

We learn a neural network dynamics model of the form shown inEq. 9, where q represents the learned network weight parameters, xt re-presents each stage of delayed state and control inputs, ht denotes thehistory of state and control inputs, and fNN is the abbreviation for thevector-valued two–hidden layer neural network dynamics model usingsoftplus activation functions. In the network equations, a representslayer activations and z represents the weighted input to a given layer.

xt ¼ ðr;Uy;Ux; d; Fxf Þt ð9AÞ

ht ¼ ½xt ;…; xt�T � ð9BÞ

z1 ¼ WT1 ht þ b1 ð9CÞ

a1 ¼ logð1þ ez1Þ ð9DÞ

z2 ¼ WT2 a1 þ b2 ð9EÞ

a2 ¼ logð1þ ez2Þ ð9FÞ

_r_Uy

� �¼ WT

3 a2 þ b3 ð9GÞ

q ¼ ½W1; b1;W2; b2;W3; b3� ð9HÞ

_r_Uy

� �¼ f NNðht ; qÞ ð9IÞ

The measured targets for the network in training consist of the nextmeasured yaw rate (r) and lateral velocity (Uy) states. Measured inputsto the network were delayed 10 ms per stage xt. To predict target states,we learned the state derivatives with the network and then used Eulerintegration with a 10-ms time step (Dt) as shown below. The predictedtargets for the physics-based model are also similarly calculated usingEuler integration with a 10-ms time step, where t+1 denotes the nextsampled time step in Eq. 10.

rU y

� �tþ1

¼ rUy

� �t

þ Dt⋅ fNNðht ; qÞ ð10Þ

Simulated dataTo study the neural network’s capability to learn representations forthese effects, we designed a simulation study using the physics-basedmodel with varying degrees of additional model complexity. To dem-onstrate the capability of a neural network model for modeling vehicledynamics, we used the single-track vehicle model and the Fiala tire


model as previously described. To generate a dataset using the physics-based model as shown in Eq. 1, we first sampled the initial startingconditions (r, Uy, Ux) uniformly at random in the space of stable initialstates.We also sampled the initial controls (d,Fxf) uniformly at random inthe space of possible inputs. To create a single trajectory of length T usedas an input to the network for training, we used a uniform random con-trol policy for both d and Fxf with physics-based model dynamics to de-termine next states for the rest of the control trajectory. In addition tousing the high-friction physics-based model to generate training data,we also modeled the following additional dynamics effects.

Weight transferDuring high-performance driving, one common effect that influencesthe vehicle’s dynamics is longitudinal weight transfer. This effect im-pacts the vehicle’s dynamics by increasing or decreasing the amountof normal force experienced on each tire as a result of acceleration orbraking. This is shown in Eq. 11, where h is the height to the center ofgravity of the vehicle,m is the vehicle mass, and g is acceleration due togravity. When coupled with the Fiala tire model, weight transfer acts toincrease or decrease the force capability of a given tire. We generated asimulated dataset with a dynamics model consisting of the physics-based model plus longitudinal weight transfer.

Fzf ¼ bLmg � h

Lmax ð11AÞ

Fzr ¼ aLmg þ h

Lmax ð11BÞ

Tire relaxationDuring low-speed driving, another dominant effect to model is tire re-laxation length. Tire relaxation length can be modeled as a delay inthe amount of lateral force experienced by each tire as shown belowin Eq. 12 (35). The amount of delay is characterized by the tire relaxa-tion length s, a property of the tire, as well as the magnitude of the ve-locity V of the vehicle. One simulated dataset was generated with aphysics-based model including the effects of tire relaxation length.

_af ¼ Vsf

arctanUy þ ar

Ux

� �� d� af

� �ð12AÞ

_ar ¼ Vsr

arctanUy � br

Ux

� �� ar

� �ð12BÞ

Varying tire frictionTo verify that the network model is capable of high predictive per-formance inmultiple environmental conditions, we generated one data-set that consists of simulated high- and low-friction data.We did this bymodeling both low- and high-friction surfaces explicitly as the frictionparameter in the Fiala tire curve, which have the following values asshown in Eq. 13. We generated a simulated dataset for training consist-ing of 200,000 sampled trajectories, where half of the dataset wascollected at high friction and half was collected at low simulated friction.

mlow ¼ 0:3 ð13AÞ

mhigh ¼ 1:0 ð13BÞ

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Last, one simulated dataset was generated where all of these effectswere added to the physics-basedmodel, and 200,000 samples were gen-erated for training.

Experimental dataTo demonstrate the capability of network models in real predictiontests, we collected vehicle data under both high- and low-frictiondriving conditions. Collected real vehicle data encompassed eachof the modeled effects shown in simulation, including both highand low friction, plus additional difficult-to-model effects such assuspension geometry or internal control loops. Collected data con-sist of 214,629 trajectory samples or about 35 min of driving datasplit roughly equally between high- and low-friction driving. Datawere collected using an automated Volkswagen GTI with a similararchitecture to Shelley, as previously described. Low-friction datawere collected in a testing track near the Arctic Circle, consistingof a variety of speeds up to the limits of the vehicle’s capabilities on alow-friction test track. High-friction data were collected from a varietyof maneuvers at the handling limits, with data collected at ThunderhillRaceway Park.

An integrated navigation system was used to measureUx, Uy, r, andax, where ax denotes the longitudinal acceleration of the vehicle. Mea-surements of the vehicle’s steering angle were acquired from the vehicle’scontroller area network (CAN).Measurementswere recorded at 100Hzon a dSPACEMicroAutoBox. For the learning task, recorded data werefiltered using a second-order Butterworth low-pass filter with a 6-Hzcutoff to only learn relevant vehicle dynamics and not higher-frequencyeffects such as suspension vibration.

Optimization and trainingIn preparing data for learning, each experimental dataset was divided70% train, 15% held out for development, and 15% held out for modeltesting. To break temporal correlations within the dataset, we random-ized the data so that each sample consisted of a time-correlated trajec-tory but any given two samples were not correlated. To comparecapabilities of both the physics-based model and the neural networkmodel, we optimized bothmodels to fit the observed dynamics trainingdata. We used the mean squared error objective for training as shownbelow, where U y and r denote a predicted quantity from a model;otherwise, Uy and r are measured target quantities, and N denotesthe number of training examples. By training the network, we aresolving the optimization problem shown in Eq. 14, where q denotesthe weights of the network.

minimizeq

1N∑N

i¼1jj r

Uy

� �tþ1

� rU y

� �tþ1

jj2

subject torU y

� �tþ1

¼ rUy

� �t

þ Dt⋅ fNNðht ; qÞ ð14Þ

To optimize the physics-based model to observed data, we similarlyformed an optimization model to train the model parameters. Theparameters learned for the physics-basedmodel consist of the tire friction(m) and the tire front and rear cornering stiffnesses (Cf and Cr). Becauseof model mismatch in reality, any observed model mismatch will resultin learned parameter variation of the model. Similarly, in the data gen-erated from physics-based models including additional unmodeled


effects, model mismatch also resulted in parameter variation. This ledto solving the optimization problem (Eq. 15) for training the physics-based model of recorded data.

minimizeCf ;Cr ;m

1N∑N

i¼1jj r

Uy

� �tþ1

� rU y

� �tþ1

jj2

subject torU y

� �tþ1

¼ rUy

� �t

þ Dt⋅ f Bikeðxt ;Cf ;Cr; mÞ ð15Þ

Parameters for the physics-based model were initialized from arandom Gaussian distribution for training, and parameters for theneural networkmodel were initialized using Xavier uniform initializa-tion. We used the Adam optimization method with default parameterinitializations to perform a first-order optimization of the learnedparameters in both the physics-based model and the neural networkmodel (36). Training was performed using mini batches of 1000samples for each update. The learning framework was implementedin TensorFlow using graphics processing unit parallelization in Pythonand trained with a computing cluster with an Intel i7 processor andNvidia 1080 graphics card (37). For a single training dataset, the learningprocess took about 25 min.

Control using learned modelsTo control the vehicle using a learned neural network model, we de-veloped a control method that leverages the lookahead feedforward-feedback control architecture. In this case, we used the learned neuralnetwork model to develop the approximate feedforward steering andsideslip commands. The neural network model was first trained onhigh-friction simulated data from a bike model simulation sampledwith 200,000 trajectories over the state space of the model. Once theinitial learning process converged, we used real experimental data inthe next step of the training process inspired by the technique of sub-sequently increasingmodel fidelity simulators (38).We supplementedour datasets with both high- and low-friction experimental test data.Additional experimental data were obtained from tracking our oval testtrack at a range of lateral accelerations with the baseline feedback-feedforward controller to ensure that there was similarity in the testdistribution of our controller. Once these data were combined, theneural network vehicle model was retrained to fit the newly collectedreal data as shown in Fig. 7. Once the model was optimized on realcollected data, the model was used to generate feedforward steeringand sideslip commands.

From the kinematics, which are not learned in the neural networkdynamics model, we found that in steady state, we arrived at

rss ¼ k_s ð16AÞ

Fx;ss ¼ �mrssUy ð16BÞ

These steady-state values were used as inputs to the network in anoptimization problem to find a stationary point in the model forfeedforward control. Conditions for finding a stationary point areillustrated in the network model by finding a point where the statederivatives are zero for a given set of control and state inputs. For the

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neural networkwith history, this consists of each of the delayed state andcontrol inputs being constrained to be the same as shown in Eq. 17B.

_r_Uy

� �¼ f NNðr;Uy;Ux; d; Fx;…; r;Uy;Ux; d; FxÞ ð17AÞ

00

� �¼ f NNðrss;Uy;ss;Uxmeasured ; dss; Fx;ss;…; rss;Uy;ss;Uxmeasured ; dss; Fx;ssÞ

ð17BÞ

To compute the feedforward values as in the physics-based control-ler, both a speed and path curvature must be provided. Speed wasmeasured from the vehicle as an input to the network, and curvaturewas provided from the precomputed trajectory to follow. During thecontrol process, curvature was calculated using a map-matchingalgorithm online on the vehicle.

Last, the controller computes the feedforward commands dffw andUy,ffw that best achieve equilibrium within the actuation limits (du,dl).The solution was optimized with a constrained second-order interiorpoint optimization method. This process used CasADi and IPOPT tosolve the following nonlinear optimization problem shown in Eq. 18(39, 40).

minimizedffw ;Uy;ffw

_r2 þ _U2y

subject to d≤dud≥dl

_r_Uy

� �¼ f NNðdffw;Uy;ffwÞ

ð18Þ

This optimization problem aims to find the closest conditions andinputs for steady state of the learned model in a least-squares sense. Inpractice, the optimization finds solutions that are numerically close tosteady state, attaining amaximum value of the cost function of 1.0072 ×10−18 during testing on the vehicle. When used for control, this optimi-


zation problem was solved every 50 ms on a computer with an Intel i7processor. The kinematic steering angle was used as an initialization towarm start the nonlinear optimization. Although we initialized the op-timization with the kinematic steering angle, the problem tends to notbe very sensitive to the initial guess.

To obtain the steering values that are used for control, we used thesteady-state lateral velocity alongwith the current speed to find the feed-forward sideslip angle bffw as shown in Eq. 19A. Once the feedforwardsideslip command was computed, it was used in the lanekeepingfeedback control scheme, along with the computed dffw as shown inEq. 19B. The final steering command consists of the feedforwardsteering command found from solving for the network’s stationarypoint, plus an added term for path-based steering feedback. The steeringfeedback, similar to the physics-based controller, includes a feedforwardsideslip term that was calculated from solving for the optimal steady-state lateral velocity from the network.

bffw ¼ arctanðUy;ffw=UxÞ ð19AÞ

d ¼ dffw � kpðeþ xLAsinðDYþ bffwÞÞ ð19BÞ

The resulting steering command was calculated and sent to the ve-hicle over its CAN interface. This steering command was tracked via alow-level steering controller, which applies a torque on the vehicle’ssteering system to obtain a desired road-wheel angle.

Comparison of both the lookahead physics-based controller andneural network feedforward controller was implemented on an auto-mated Volkswagen GTI. Both controllers were tested at ThunderhillRaceway Park, on an oval map on the asphalt high-friction skid pad.Comparison of tracking errors occurred when both controllers weretracking a 0.9g longitudinal speed profile. A feedforward-feedback–based longitudinal controller was used in both implementations onthe vehicle. Analysis of data was completed in MATLAB 2016b.

SUPPLEMENTARY MATERIALSrobotics.sciencemag.org/cgi/content/full/4/28/eaaw1975/DC1Fig. S1. Measured and predicted sideslip from experimental control comparison.Fig. S2. Measured longitudinal speeds from experimental control comparison.Table S1. Test dataset results from simulation learning study.Table S2. Test dataset results from experimental data learning study.Movie S1. Neural network controller implemented on automated GTI, tested on oval track atThunderhill Raceway Park.

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Ux, К

βffw

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e, ΔΨData Input

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Fig. 7. Schematic of control process using a learned neural network model.

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Acknowledgments: We thank Volkswagen Group Research including P. Hochrein,B. Mennenga, and T. Drescher for their support by providing research vehicles and testfacilities, as well as fostering research collaboration and discussion. In addition, we thankthe Volkswagen Electronics Research Lab for providing vehicle support, J. Subosits andV. Laurense for support with automated GTI, J. Goh for video editing, and Thunderhill RacewayPark for providing test facilities. We would like to thank the editor and our reviewers fortheir comments and help in improving the clarity of this paper. Funding: N.A.S. is supportedby the NSF Graduate Research Fellowship and William R. and Sara Hart Kimball StanfordGraduate Fellowship. This material is based on work supported by the NSF Graduate ResearchFellowship Program under grant no. DGE-1147470. Any opinions, findings, and conclusionsor recommendations expressed in this material are those of the authors and do notnecessarily reflect the views of the NSF. Author contributions: N.A.S. conceived anddesigned data-based framework for automated vehicle control as well as coordinatedand contributed to manuscript. M.B. aided in development of optimization-based control andhelped in manuscript development. N.R.K. assisted with development and integration withphysics-based control and aided in manuscript preparation. J.C.K. designed and analyzedthe comparison between automated vehicle and human participant. J.C.G. aided inexperimental design and manuscript preparation. Competing interests: The authors declarethat they have no competing interests. Data and materials availability: We have suppliedthe relevant data and code (at https://purl.stanford.edu/zb950hd3384) to recreate theresults with the exception of the experimental low-friction vehicle data. These data wereobtained at a proprietary test facility, and at this moment, we are not free to post them.The relevant models trained on these data and results generated from these data are includedso that experimental results can be validated.

Submitted 4 December 2018Accepted 5 March 2019Published 27 March 201910.1126/scirobotics.aaw1975

Citation: N. A. Spielberg, M. Brown, N. R. Kapania, J. C. Kegelman, J. C. Gerdes, Neural networkvehicle models for high-performance automated driving. Sci. Robot. 4, eaaw1975 (2019).

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Neural network vehicle models for high-performance automated drivingNathan A. Spielberg, Matthew Brown, Nitin R. Kapania, John C. Kegelman and J. Christian Gerdes

DOI: 10.1126/scirobotics.aaw1975, eaaw1975.4Sci. Robotics

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AUTONOMOUS VEHICLES Copyright © 2019 Neural network ...€¦ · Ux(s) + + δfb δffw δ e, ΔΨ...

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