Neural Network Modeling for Steering Control of an AutonomousVehicle

Gowtham Garimella1, Joseph Funke2, Chuang Wang2, and Marin Kobilarov3

Abstract— Model-based control of dynamical systems typi-cally requires accurate domain-specific knowledge and spec-ifications of possibly proprietary system components. In thecontext of autonomous driving, steering actuator dynamicscan be difficult to model due to an integrated proprietarypower steering control module. While first-principles modelsderived from physics laws can often approximate the systembehavior, it remains generally difficult to capture non-physicallyderived behavior based on proprietary software algorithms inthe power steering system. To overcome this limitation, thiswork instead employs a recurring neural network to modelthe steering dynamics of an autonomous vehicle. The resultingmodel is then integrated into a Nonlinear Model PredictiveControl scheme to generate feedforward steering commandsfor embedded control. The proposed approach is compared totraditional first-principles steering modeling through on-vehicleexperiments and statistical data validation. As a result, it isshown that the neural network model can be automaticallygenerated with less domain-specific knowledge, can predictsteering dynamics more accurately, and perform comparably toa high-fidelity first principles model when used for controllingthe steering system of a self-driving vehicle.

I. INTRODUCTION

This paper considers the dynamical modeling of steeringsystems in passenger vehicles and the use of derived modelsfor steering control, as a basic building block for autonomousdriving. We propose an architecture consisting of a nominalphysics-based model that is embedded as a block insidea data-driven recurrent neural network (RNN). This RNNmodel serves as a transition function for model-predictive-control (MPC) to achieve desired steering behavior. The ap-proach is suitable for systems with known electromechanicalspecifications but also for black-box (e.g. third-party OEM)vehicle systems with unknown characteristics. We show thatthis method achieves marked improvement over traditionalfeed-forward techniques and study the effects of differentstrategies for combining RNNs with a simple physics-basedmodel.

The steering system can nominally be modeled basedon known physics laws from first principles. The steeringdynamics are usually modeled as a second-order forceddynamical system [24] with torque inputs induced by thesteering actuator and tire forces [15]. For modern steering

*This work was supported by and conducted at Zoox Inc.1Gowtham Garimella is with the Department of Mechanical Engineering,

Johns Hopkins University, 3400 N Charles Str, Baltimore MD 21218, USAggarime1@jhu.edu

2Joseph Funke and Chuang Wang are with Zoox Inc., Menlo Park,California, USA joe.funke|mrwang@zoox.com

3Marin Kobilarov is with Zoox Inc., Menlo Park, California, USA andwith the Johns Hopkins University, 3400 N Charles Str, Baltimore MD21218, USA marin@zoox.com

systems, a power steering motor is used as the steeringactuator and is governed by a steering controller. In thiswork, we assume that no exact knowledge of the internalcontroller logic is available and the first-principles approachis thus a very coarse approximation to the actual behavior.

Unlike the first principles models, non-parametric modelsdo not depend on strict physics-based assumptions andinstead can be derived solely from experimental input-output data. Such models can better capture complex actu-ator nonlinearities and delays and can thus provide higherpredictive accuracy with limited domain specific knowl-edge [14].Recently, RNN models have been employed inModel Predictive Control (MPC) framework to produce acontroller for robot manipulation [11], using deep layers [18]to model the robot dynamics and controls cutting taskaccurately.

A non-parametric approach can be used in conjunctionwith control policy learning [20]. Deep neural networkshave been proposed to learn a non-parametric control policybased on experience. In a reinforcement learning setting,the control policy tries to maximize a reward function bybalancing exploration and exploitation of the dynamics ofthe system [16]. Guided Policy Search Methods have beenproposed to improve the convergence of the reinforcementlearning methods [25]. While these techniques are verygeneral and could work directly with the physical system,our present work focuses on first learning an accurate androbust dynamics model only, which can then be rigorouslyvalidated and used for traditional model-based control.

Steering control has been achieved using simple firstprinciples model with robustness to disturbances [6], [9].Similarly, lateral control schemes that control the lateraldisplacement of the vehicle using first principles models havebeen developed [8], [19], [17]. The success of these methodsdepends on an accurate steering model, which is non-trivialsince the power steering dynamics is unknown.

RNN-based models have been employed for vehicle con-trols in many simulation studies [7], [22], [10], [13]. Rivalset.al [23] experimentally demonstrated neural network basedlateral control of a four-wheel-drive car. Apart from themodel-based control schemes, vision based neural networkcontrol policies have also been developed [21], [5]. However,these control policies rely heavily on the trained vision dataand when presented with new examples, can perform unex-pectedly. While the approach of mapping from sensory inputsto actions directly holds promise, for safety considerationswe consider a more traditional multi-layered model-basedapproach consisting of high-level planner providing reference

2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)September 24–28, 2017, Vancouver, BC, Canada

978-1-5386-2682-5/17/$31.00 ©2017 IEEE 2609

trajectories that are tracked by a low-level controller. Thisdecomposition allows for individual verification of each layeron a wide range of datasets.

Contributions.: In this work, a model based controlstrategy is developed for the steering dynamics of an auto-motive vehicle. An RNN model has been used to model thesteering dynamics of the vehicle. The model is augmentedwith known physics-based transition blocks to improve itspredictive capacity. The learned model is then employedin an MPC framework to produce the feed-forward controltorques for a low-level embedded steering PID controller.Experiments on Toyota Highlander vehicles have been con-ducted to compare the performance of the steering trackingbehavior using feedforward controls from different models.The limitations and advantages of several configurations ofthe proposed architecture are examined.

Organization.: The rest of the paper is organized asfollows. In section II, the steering control architecture islaid out, and the necessity for feedforward steering torque isexplained. Next, the steering models required to compute thefeedforward steering torque are explained in section III. Theprocedure to compute feedforward steering torque using themodels is specified in section IV. Finally, the effect of addingthe feedforward steering torque using different models iscompared in section V.

II. CONTROL ARCHITECTURE

Controlling the steering system requires tracking a refer-ence steering angle trajectory by applying torque inputs to thesteering system. The steering reference trajectory is gener-ated from a high-level planner as shown in Fig 1. The usageof a layered structure separates the vehicle behavior fromlow-level vehicle tracking. Following the control structure,the high-level planner produces a trajectory that is consistentwith obstacles and road rules. The MPC trajectory trackingthen produces a steering reference trajectory necessary toachieve the high-level planner trajectory. Finally, the low-level steering PID controller computes the actuator steeringtorque input required to track the steering trajectory.

Fig. 1: Control architecture showing how a desired trajectoryis converted to steering actuator commands. The focus of thepaper is to develop a feedforward steering torque to improvesteering controller.

The focus of this work is to develop an appropriatefeedforward steering torque input to improve the steeringPID controller. The steering PID controller tracks a referencesteering trajectory (δr, δr), by commanding the steering

torque τcmd as

τcmd = −kp(δ − δr)− kd(δ − δr)− ki∫

(δ − δr)dt+ τff .

Using an accurate feedforward steering torque τff iscritical as can be observed from Fig 2. The figure shows thesteering torques necessary for tracking a reference trajectoryusing a PID controller with the feedforward steering torqueturned off (τff = 0). The bulk of the control input duringthe tracking is provided by the integrator alone which leadsto poor tracking performance. The PID gains have beenchosen based on gain scheduling to track reference trajec-tories closely without oscillations. The performance of PIDcontroller is limited by the time delays inherent in the system.The PID gains used ensure stability but allows error toaccumulate over time, which is then compensated through theintegrator. The goal of introducing the feedforward steeringtorque is to improve the system response and reduce trackingerror without increasing PID gains. The parametric and non-parametric models necessary for generating the feedforwardsteering torque are investigated next.

6 8 10 12 14 16 18 20 22

Time (sec)

-30

-20

-10

0

10

20

To

rqu

e (

%)

Steering Torque

Total

Proportional

Integrator

Derivative

Fig. 2: Steering torque generated by a PID controller duringa trajectory tracking experiment. The majority of the torqueoutput is provided by the integrator indicating that thesteering system is highly nonlinear.

III. MODELING STEERING DYNAMICS

The steering model predicts the steering angle δ andthe steering rate δ based on the applied steering torqueτs. Steering dynamics are highly coupled with the lateraldynamics of the vehicle, which in turn depends on thelongitudinal velocity vx, which we regard as a recordedinput. The lateral states, lateral velocity vy and yaw rateφ, are therefore included as a part of our steering dynamicsformulation. The complete state of the steering dynamics isx = [δ, δ, φ, vy]

T and controls by u = [τs, vx]T . The stacked

states and controls are represented by z = [xT , uT ]T .The discrete steering dynamics evolve according to the

discrete-time model xi+1 = f(zi) where i defines the timeindex along the trajectory. The discrete steering dynamics

2610

predicts the next state xi+1 given the current state xi andcontrol ui. The state transition function f is an unknownnonlinear function of the previous state and controls. It canbe derived from a first principles approach or using a non-parametric model such as a neural network.

A. First principles model

The first principles model is a second order model thatintegrates the net steering torque to obtain the steering angleand its rate. The net steering torque is given by a combinationof the steering system dynamics, road wheel interactions, andpower steering torque. The steering system dynamics consistsof Coulomb friction from the steering rack, jacking torquedue to camber angle, and damping caused by rigid bodydynamics [12]. The self-aligning torque is produced due totire deformation when the steering wheel moves against thetire thread. At small slip angles, the self-aligning torque isproportional to the lateral force Fyf applied to the front tire.The resulting first principles model is

δ = kpδ︸︷︷︸jacking

+ kdδ︸︷︷︸damping

+ kaFyf︸ ︷︷ ︸Aligning

+ kcsgn(δ)︸ ︷︷ ︸Coulomb

+ g(τs, vx)τs︸ ︷︷ ︸Power Steering

,

sgn(δ) ={

1 if δ > 0−1 otherwise

.

A tire model specifies the lateral force applied to the fronttire [15]. The lateral velocity and the yaw rate of the car arepropagated using the bicycle model [24].

The power steering system applies an actuation torquebased on the torque sensor input τs and longitudinal velocityvx. The net torque from power steering system is modeledas a nonlinear gain g(τs, vx) on the torque sensor input.

g(τs, vx) = g1(τs)g2(v),

g1(τs) = γ1

[1− e−

(τs−γ2γ3

)2],

g2(v) =[γ4 + (1− γ4)e−vx/γ5

].

The power steering gain g(τs, vx) is decomposed into twomultiplicative gains. The first gain is dependent on driverinput. It increases with the driver input and saturates toa constant value γ1 as driver input becomes large. Theparameters γ2, γ3 specify the bias and the rate of change ofthe gain respectively. The second gain decreases with vehiclevelocity, where the rate of decrease is specified by γ5 andthe saturated value by γ4. The form of the gain function hasbeen chosen based on observed experimental data betweendriver input torque and assist torque. The power steeringdynamics presented here is an approximation based on thegraphs shown in Badawy et.al [2], since the true dynamicsis unknown. The unknown parameters in the first principlesmodel are obtained using standard least-squares regressionbased on the error between predicted and measured steeringangle and steering rate.

B. Neural Network Model

The neural network model approximates the nonlinearfunction xi+1 = f(zi) to predict the discrete-time steering

dynamics. The model consists of several units of neuralnetwork blocks stacked in time. Each neural network blockis divided into a known physics function and a stack offully connected layers. Figure 3 shows a block diagram ofa neural network block. The current inputs and previousoutputs are stacked together and fed into the fully connectedlayers and the physics function. The output of these layers iscombined to predict the state at the next step. These predictedvariables together with new control inputs are fed back intothe network to continue prediction for the next step.

Fig. 3: The neural network architecture used for learninglateral dynamics.

The overall discrete dynamics for a neural network modelcan be written as f(z) = fph(z) + fnn(z), where fphdenotes the physics function and fnn represents the neuralnetwork layers. The physics function incorporates basicdomain knowledge by predicting the steering angle as anintegral of the steering rate and the yaw rate based on theno-slip condition for a kinematic car model. The physicsfunction can be mathematically stated as

fph(zi) =

δi + k1δi

0vxi tan(k2δi)

0

,where the unknown parameters are the time-constant forintegration k1 and the inverse of the wheelbase of the cark2. Similarly, the neural network layers can be expressed as

fnn(zi) = gn (gn−1 (gn−2 (· · · g0 (zi)))) ,gi(x) := σ(Wix+ bi), i = 0, . . . , n− 1,

gn(x) :=Wnx+ bn.

The function gi(·) defines a single fully connected layer forthe neural network with the parameters Wi, bi. The functionσ(·) denotes the activation function used in the network. Theactivation function is chosen to be the hyperbolic tangentfunction. The last layer gn(·) is specified as a unit activationfunction since the neural network is used in a regressionproblem.

The neural network layers predict the residual dynamicsnot modeled by the physics function. The neural networklayers, therefore, capture the effects of road-tire interactions,power steering logic, and steering system dynamics. Theaddition of physics function improves the gradient flow ofthe network thereby providing better prediction performance.The increase in depth of the fully connected layers increasesthe ability of the RNN to model higher nonlinear models.At

2611

the same time, it is also harder to train deeper neural networkmodels due to the vanishing gradient problem [3]. In thiswork, the depth of the neural network layers has beenexperimentally determined to capture the unknown dynamicswell.

Training: The neural network weights and the physicsfunction parameters are optimized using time series datacollected from driving the vehicle along variable curvaturepaths with different desired longitudinal velocities. The steer-ing dynamics are controlled using a preliminary ProportionalIntegral Derivative (PID) controller during the data collectionphase.

The collected time series data is divided into fixed timehorizon segments to train the neural network model. The lossfunction during neural network training is set to minimize thedifference between propagated states and measured states foreach of the fixed time horizon segments. The loss functionalso adds an L2 regularization with a user-selected gain cr toavoid overfitting of the parameters. The goal of the trainingphase is to find the optimal parameters θ∗ that minimize thetraining cost as

θ∗ = argminθ

m∑j=1

C(x0:N , u0:N−1, θ), (1)

C(x0:N , u0:N−1, θ) =

n∑i=1

(xi − xi)TP (xi − xi) + crθT θ,

(2)s.t xi+1 = fph(zi, θ) + fnn(zi, θ), (3)zi = [xi, ui], x0 = x0, (4)θ = [W0, b0,W1, b1, · · · ,Wn, bn, k1, k2], (5)

where θ denotes the weights of the network along with theunknown physics parameters.

The cost function C measures the deviation of the prop-agated states xi from measured states xi for m sample tra-jectories. The state deviations are weighed using a diagonalmatrix P to enforce a uniform scale across the deviations.The matrix P is usually chosen as the inverse covariance ofthe sensor measurements.

The propagation of steering dynamics results in the propa-gated states significantly diverging from the measured statesfor random initialization of parameters. The optimization isthus performed in two stages: first by limiting the propaga-tion to a single step and initializing the parameters obtainedfrom the previous stage and optimizing over the entiretrajectory segment. This two-stage optimization proposed byLenz et al. [11] has been shown to perform better thanrandom initialization of parameters.

Implementation: The RNN is coded as a computationalgraph in TensorFlow [1] package. The training data isprovided by 20,000 trajectory segments with a 0.5-secondhorizon which corresponds to approximately 150 hours ofdriving data. The neural network has been trained with twofully connected layers. The optimization has been performedusing mini-batch gradient descent with a batch size of 200samples.

The optimal parameters are used to verify the performanceof the model on a test data set of 5,000 trajectory segments.The RNN model is used to propagate the lateral dynamicsusing the initial state and controls along the trajectory.Figure 4 shows the RMS error between the propagatedand measured handwheel angle along the trajectory usingdifferent trained models. The addition of a second layer to theRNN along with physics function improves the performanceof prediction significantly. Increasing the RNN layers furtherdoes not result in an improvement of performance. Hence thedepth of neural network is limited to two layers.

0 0.2 0.4 0.6 0.8 1

Time(sec)

0

5

10

15

20

25

30

35

40

45

50

Handw

heel angle

(deg)

RMS Error vs Time

First principles model

2 layer RNN no physics

1 layer RNN

2 layer RNN

3 layer RNN

Fig. 4: Averaged RMS Error between propagated and mea-sured hand wheel angle along the trajectory for all the testsegments plotted for different dynamics models.

IV. STEERING CONTROL

The steering torque used to track a steering referencetrajectory consists of a PID component and a feedforwardcomponent. The steering reference trajectory comprises of areference steering angle δr and steering rate δr. The feed-forward steering torque estimates the necessary commandto track the desired reference and thereby improves trackingperformance when compared to applying the PID componentalone. The procedure to compute feedforward steering torqueusing different models is discussed next.

A. Inverting First Principles Model

Here, the feedforward steering torque for controlling thesteering system is computed by inverting the first principlesmodel. Denoting the power steering dynamics at a givenlongitudinal velocity by P (τs) := g(τs, vx)τs, the inputtorque required to track a steering reference trajectory iscomputed as

τ∗s = P−1(−kpδr − kdδr − kaFyf

).

The desired steering angle, steering rate and forwardvelocity along the trajectory are assumed to be known duringinversion. The tire force Fyf is computed based on referencetrajectory as

Fyf = kfmvxφ,

2612

where the gain kf is the ratio of the distance betweenthe center of mass to front wheels to the wheelbase. Thefeedforward steering torque for trajectory tracking is chosento be equal to the computed input torque τff = τ∗s , sincethe computed input torque is only dependent on the referencetrajectory.

The double derivative of the reference steering angle, theCoulomb friction and the desired steering acceleration arenot accounted for when computing the feedforward steeringtorque to avoid inducing high-frequency oscillations into thesteering control.

B. Lookup table

The feedforward steering torque can also be computedusing a lookup table under steady state assumptions. Thelookup table provides the steering torque required to achievea steady state steering angle for a given velocity. The datain the lookup table is filled based on experiments in whichthe steering torque is adjusted to achieve the desired steeringangle for a given forward velocity. The feedforward steeringtorques for a given reference steering angle and forwardvelocity are then computed by interpolating the data pointsfrom the lookup table.

C. NMPC Steering Control

When using the neural network model, the feedforwardsteering torque is computed by solving an NMPC opti-mization problem. The optimization problem is set to tracka reference steering trajectory s0:N consisting of desiredsteering angle δr and desired steering rate δr using theneural network model. The longitudinal velocity is fixed tobe equal to the reference velocity from high-level plannerduring the optimization process. In contrast to the neuralnetwork optimization in Eq 5, this optimization fixes the neu-ral network parameters and optimizes over the feedforwardsteering torques to track the desired steering trajectory. Theoptimization problem for computing the feedforward steeringtorque can be written as

τ∗0:N−1 = argminτ0:N−1

N∑i=1

(si − si)TPi(si − si) + τTi−1Riτi−1,

s.t xi+1 = fph(zi) + fnn(zi),

si = [δi, δi]T , si = [δr, δr]

T

Given {vx0:N−1, x0:N}.

The first component of the optimal steering torque fromNMPC optimization is sent to the PID controller as feed-forward steering torque (τff = τ∗0 ). If there are delaysin sending the torque command, the feedforward steeringcommand can correspond to the future stamped steeringtorque as in τff = τ∗delay

Optimization Algorithm: The NMPC optimization is per-formed using a Stage-wise Newton method [4]. The al-gorithm consists of repeatedly application of two steps:backward pass to compute a quadratic relaxation of thevalue function and forward pass to update the states andcontrols that minimize the approximated value function.

This procedure is repeated until a specific time limit. Ifthe optimization did not succeed by then, the feedforwardterm is set to zero. The gradient of the cost function withrespect to the input variables can be obtained analyticallyfor the neural network model. The analytical gradients avoidusing expensive finite differencing scheme for computinggradients.

V. RESULTS

The MPC controller for steering dynamics has been testedon a passenger car equipped with a high-precision GPSsystem, and an onboard compute stack. The onboard com-pute stack consists of two computational modules: high-level computer and low-level microcontroller. The high-levelcomputer performs the high-level planning and provides asteering reference trajectory as an output. It also producesthe feedforward steering torque necessary for tracking thesteering reference trajectory using NMPC optimization de-scribed in section IV. The low-level microcontroller runs asteering PID controller that outputs a torque command basedon the input steering reference and feedforward torque.

Start

Goal

Fig. 5: Test track used for autonomous driving experiments

The goal of the experiments is to closely track the desiredwaypoint trajectory in a small test track as shown in Fig 5.The steering controller is tested at longitudinal velocitiesof 5mph and 10mph under a maximum lateral accelerationof 3.5m/ss. Large lateral acceleration and low speeds areused to test the system, as these conditions tended to causeworse steering tracking performance. The effect of addinga feedforward steering torque using different models onthe steering tracking performance is shown in Table I. Notincluding a feedforward in the controller results in the largestRMS error, as expected. Adding the lookup feedforwardmodel improves the performance only slightly, probablybecause these are more dynamic rather than steady statemaneuvers. Computing the feedforward steering torque usingthe first principles model, which incorporates these dynamiceffects, outperforms the lookup table model.

The neural network performs comparably to first principlesmodel with regards to the RMS error. It also performsbetter than the first principles model at low velocities. Thefirst principles model is not able to model the road-wheelinteractions correctly at low velocities. This is likely becauseat lower velocities, the power steering system provides largerassist, and that effect is difficult to model directly without

2613

knowing the underlying power steering software algorithm.The neural network, however, benefits from its own internalrepresentation based simply on training data.

Controller/Longitudinal velocity 5mph 10mphNo Feedforward 18.63 18.61Lookup table 16.43 14.88First principles model 12.48 9.56Neural network model 10.21 9.53

TABLE I: RMS Handwheel error (degrees) for differentmodels based on multiple trials.

The steering performances of different steering controllersare shown in Fig 6 for a single trial at 5mph and 10mphlongitudinal velocities. The initial steering angle differencesare matched across different controllers before computing theRMS error to remove the effect of various initial conditions.The PID controller without feedforward steering torque hasthe highest RMS steering error as seen in Table I, and asa result of the larger errors, the integrator performs muchof the tracking task. Adding a lookup table offers marginalimprovement, since the maneuvers are highly dynamic. Thefirst principles model and the neural network model out-perform the other methods with regards to RMS error. Themagnitude of the integrator is also small and most of thetracking performed by the feedforward steering torque.

VI. CONCLUSIONS

The use of RNN, a non-parametric model, for modelingand control of a vehicle’s steering system has been presented.The creation of the neural network model did not requiredomain specific knowledge about the power steering moduleand steering system dynamics. It used only minimal knowl-edge of the nominal car dynamics to improve the predictioncapability. The resulting neural network model outperformeda first principles model in the long-term prediction of steer-ing dynamics and operated equally well in generating afeedforward reference command for control. These resultsdemonstrate that a simple RNN, augmented with simpledynamics information but lacking domain specific knowl-edge, can be suitable for dynamical modeling and controlof a vehicle steering system. The current system is limitedto learning RNN model offline and computing feedforwardtorque online. Future work will include learning the non-parametric model online and testing the vehicle on changingroad conditions.

REFERENCES

[1] TensorFlow. https://www.tensorflow.org/, 2017.[2] Badawy Aly, Zuraski Jeff, Bolourchi Farhad, and Chandy Ashok.

Modeling and analysis of an electric power steering system. In SAETechnical Paper. Society of Automotive Engineers(SAE) International,03 1999.

[3] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependen-cies with gradient descent is difficult. IEEE Transactions on NeuralNetworks, 5(2):157–166, Mar 1994.

[4] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont,MA, 1999.

[5] Chenyi Chen, Ari Seff, Alain Kornhauser, and Jianxiong Xiao. Deep-driving: Learning affordance for direct perception in autonomousdriving. In Proceedings of the IEEE International Conference onComputer Vision, pages 2722–2730, 2015.

[6] Zhengrong Chu and C. Q. Wu. Active disturbance rejection control forautomated steering in vehicles and controller tuning. In 2016 AmericanControl Conference (ACC), pages 7567–7572, July 2016.

[7] K T R Van Ende, D Schaare, J Kaste, F Kucukay, R Henze, and F KKallmeyer. Practicability study on the suitability of artificial, neuralnetworks for the approximation of unknown steering torques. VehicleSystem Dynamics, 54(10):1362–1383, oct 2016.

[8] P. Falcone, F. Borrelli, J. Asgari, H. E. Tseng, and D. Hrovat.Predictive active steering control for autonomous vehicle systems.IEEE Transactions on Control Systems Technology, 15(3):566–580,May 2007.

[9] J. Guldner, W. Sienel, Han-Shue Tan, J. Ackermann, S. Patwardhan,and T. Bunte. Robust automatic steering control for look-downreference systems with front and rear sensors. IEEE Transactionson Control Systems Technology, 7(1):2–11, Jan 1999.

[10] A Haddoun, M E H Benbouzid, D Diallo, R Abdessemed, J Ghouili,and K Srairi. Modeling, Analysis, and Neural Network Controlof an EV Electrical Differential. IEEE Transactions on IndustrialElectronics, 55(6):2286–2294, jun 2008.

[11] Ian Lenz AND Ross Knepper AND Ashutosh Saxena. Deepmpc:Learning deep latent features for model predictive control. In Robotics:Science and Systems, Rome, Italy, 2015.

[12] Krisada Kritayakirana and J Christian Gerdes. Autonomous vehiclecontrol at the limits of handling. PhD thesis, Stanford University,2012.

[13] S Kumarawadu and T T Lee. Neuroadaptive Combined Lateral andLongitudinal Control of Highway Vehicles Using RBF Networks.IEEE Transactions on Intelligent Transportation Systems, 7(4):500–512, dec 2006.

[14] I J Leontaritis and Stephen A Billings. Input-output parametricmodels for non-linear systems Part I: deterministic non-linear systems.International journal of control, 41(2):303–328, 1985.

[15] William F Milliken and Douglas L Milliken. Race car vehicledynamics, volume 400. Society of Automotive Engineers Warrendale,1995.

[16] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves,Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. PlayingAtari with Deep Reinforcement Learning. Deep Learning Workshopat Neural Information Processing Systems NIPS, 2013.

[17] L. Ni, A. Gupta, P. Falcone, and L. Johannesson. Vehicle lateral motioncontrol with performance and safety guarantees. IFAC-PapersOnLine,49(11):285 – 290, 2016. 8th IFAC Symposium on Advances inAutomotive Control AAC 2016.

[18] Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, and Yoshua Ben-gio. How to Construct Deep Recurrent Neural Networks. InternationalConference on Learning Representations ICLR, 2014.

[19] S. Patwardhan, Han-Shue Tan, and J. Guldner. A general frameworkfor automatic steering control: system analysis. In Proceedings of the1997 American Control Conference (Cat. No.97CH36041), volume 3,pages 1598–1602 vol.3, Jun 1997.

[20] M M Polycarpou. Stable adaptive neural control scheme for nonlinearsystems. IEEE Transactions on Automatic Control, 41(3):447–451,mar 1996.

[21] Dean A Pomerleau. Alvinn: An autonomous land vehicle in a neuralnetwork. Technical report, DTIC Document, 1989.

[22] G V Puskorius, L A Feldkamp, and L I Davis. Dynamic neural networkmethods applied to on-vehicle idle speed control. Proceedings of theIEEE, 84(10):1407–1420, oct 1996.

[23] Isabelle Rivals, Lon Personnaz, Grard Dreyfus, and Daniel Canas.Real-time control of an autonomous vehicle: a neural network ap-proach to the path following problem. In 5th International Conferenceon Neural Networks and their Applications, pages 219–229. Citeseer,1993.

[24] Jihan Ryu. State and parameter estimation for vehicle dynamicscontrol using GPS. PhD thesis, Stanford University, 2004.

[25] Tianhao Zhang, Gregory Kahn, Sergey Levine, and Pieter Abbeel.Learning Deep Control Policies for Autonomous Aerial Vehicles withMPC-Guided Policy Search. arXiv preprint arXiv:1509.06791, 2015.

2614

5mph Trials 10 mph Trials

0 10 20 30 40 50 60

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 10 20 30 40 50 60

Time (sec)

-50

0

50

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 17.95 deg

0 5 10 15 20 25 30 35

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 5 10 15 20 25 30 35

Time (sec)

-50

0

50

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 19.37 deg

a) PID controller without feedforward steering torque

0 10 20 30 40 50 60

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 10 20 30 40 50 60

Time (sec)

-40

-20

0

20

40

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 15.82 deg

0 5 10 15 20 25 30 35

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 5 10 15 20 25 30 35

Time (sec)

-50

0

50

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 14.54 deg

b) Lookup feedforward steering torque

0 10 20 30 40 50 60

-400

-200

0

200

400

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 10 20 30 40 50 60

Time (sec)

-60

-40

-20

0

20

40

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 12.70 deg

0 5 10 15 20 25 30

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 5 10 15 20 25 30

Time (sec)

-50

0

50

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 9.24 deg

c) First principles feedforward steering torque

0 10 20 30 40 50 60

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 10 20 30 40 50 60

Time (sec)

-60

-40

-20

0

20

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 8.35 deg

0 5 10 15 20 25 30 35

-400

-200

0

200

Ha

nd

wh

ee

l A

ng

le(d

eg

)

Steering Angle

Actual

Ref

0 5 10 15 20 25 30 35

Time (sec)

-50

0

50

To

rqu

e (

%)

Steering Torque

Total

Integrator

Feedforward

RMS Error: 9.24 deg

d) RNN feedforward steering torque

Fig. 6: Comparison of steering performance using steering controllers with different feedforward steering torque methodsfor a single trial.

2615