Scalable Deep Kernel Gaussian Process for Vehicle Dynamics in Autonomous Racing
Authors: Jingyun Ning
Madhur Behl
Venue: CoRL 2023 Poster (2023)
Keywords: Gaussian Process
Vehicle Dynamics
Autonomous Vehicle
Deep Kernel Learning

Abstract: Autonomous racing presents a challenging environment for testing the limits of autonomous vehicle technology. Accurately modeling the vehicle dynamics (with all forces and tires) is critical for high-speed racing, but it remains a difficult task and requires an intricate balance between run-time computational demands and modeling complexity. Researchers have proposed utilizing learning-based methods such as Gaussian Process (GP) for learning vehicle dynamics. However, current approaches often oversimplify the modeling process or apply strong assumptions, leading to unrealistic results that cannot translate to real-world settings.
In this paper, we proposed DKL-SKIP method for vehicle dynamics modeling.
Our approach outperforms standard GP methods and the N4SID system identification technique in terms of prediction accuracy. 
In addition to evaluating DKL-SKIP on real-world data, we also evaluate its performance using a high-fidelity autonomous racing AutoVerse simulator.
The results highlight the potential of DKL-SKIP as a promising tool for modeling complex vehicle dynamics in both real-world and simulated environments.

Scalable Deep Kernel Gaussian Process for Vehicle Dynamics in Autonomous Racing
Jingyun Ning
Madhur Behl
Department of Electrical and Computer Engineering University of Virginia United States jn2ne@virginia. edu
Department of Computer Science University of Virginia United States madhur. behl@virginia. edu
Abstract: Autonomous racing presents challenging environment for testing the limits of autonomous vehicle technology. Accurately modeling the vehicle dy- namics (with all forces and tires) is critical for high-speed racing, but it remains a difficult task and requires an intricate balance between run-time computational demands and modeling complexity. Researchers have proposed utilizing learning- based methods such Gaussian Processes (GP) for learning vehicle dynamics. However, current approaches often oversimplify the modeling process apply strong assumptions, leading to unrealistic results that cannot translate to real- world settings. In this paper, we proposed DKL-SKIP, a method combining deep kernel learning (DKL) with SKIP-GP, for vehicle dynamics modeling. Our ap- proach outperforms standard GP methods and the Numerical algorithms for Sub- space State Space System Identification technique (N4SID) in terms of prediction accuracy. addition evaluating DKL-SKIP on worldworld data, we also evalu- ate its performance using a high-fidelity autonomous racing Auto Verse simulator. The results highlight the potential of DKL-SKIP as a promising tool for modeling complex vehicle dynamics in both real-world and simulated environments.
Keywords: Gaussian Processes, Vehicle Dynamics, Autonomous Vehicle, Deep Kernel Learning
Introduction
The rising popularity of self-driving cars has inspired the growth of autonomous racing research (Betz al., 2022). Researchers are developing algorithms for nigh-performance race vehicles which aim to operate autonomously at the edge of the vehicle's 's limits. Competitions in autonomous racing have been held not only in simulation (Hartmann et al., 2021; Babu and Behl, 2020), but also on pro- totypes ranging from 1:43 scale RC cars (Carrau et al., 2016) to 1/10 scale (O'Kelly et al., 2019), and to full-size Indy racecars (Wischnewski et al., 2022; Jung et al., 2023). To optimize racecars' perfor- mance and control for autonomous racing, is essential have a model that can precisely predict the vehicle's dynamics. Existing models (Althoff al., 2017) range from simple point mass vehicle models and single-track models complex multi-body vehicle models. However, the construction of high-fidelity vehicle dynamics models based physies is difficult due to the capture of non- linear behavior of components such as tires, suspension and steering systems. Moreover, obtaining precise parameter values for these components can be expensive and time-consuming. For example, obtaining tire parameters such as tire stiffness, slip angle, slip ratio, etc. involves combination of performing tire testing experiments on tires test rigs and building mathematical models such as the Pacejka tire model (Pacejka, 2005). Uncertain factors such as road conditions and driver inputs, along with complex subsystems like suspension and steering systems, further complicate the mod- eling process. Consequentially, researchers have shown interest in using learning- barning-based approaches to address model-output discrepancies (Xing et al., 2020; Hermansdorfer al., 2020). Specifically, Van Niekerk et al. (2017) proposed Gaussian Processes (GP) models for vehicle dynamics learning,
7th Conference Robot Learning (CoRL. 2023), Atlanta, USA.

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which has been further explored by other researchers (Jain et al., 2020; Kabzan et al., 2019; Hewing et al., 2018). However, existing work faces limitations, including scalability issues with GP models, reliance on simplified simulations, arbitrary GP kernel choices or no exploration of kernel learning, and the use of unrealistic race tracks.
This paper proposes DKL-SKIP, solution to address the mismatch between observations and state predictions obtained with an extended- -ineematti single-track model for vehicle dynamics modeling. The main contributions of our paper are stated as follows:
1. We present DKL-SKIP, new method that integrates deep kernel learning with the scalable SKIP-GP approach for vehicle dynamics modeling. This combination not only overcomes the scalability challenges associated with the traditional GP but also enhances the robust- ness to noise, and irrelevant features.
2.
We evaluated DKL-SKIP datasets collected from full-scale autonomous Indy racecar in both real-world and simulating racing environments. We demonstrate our method's 's pability to accurately capture the non-linear dynamics of a racecar, especially at the limits of its performance envelope.
2 Related Work
2.1 Gaussian processes for vehicle dynamics modeling
Hewing et al. (2018) pioneered the use of GP models to learn vehicle dynamics in a miniature autonomous racecar, demonstrating their effectiveness in capturing lateral dynamics and developing a model predictive controller. Building on this, Kabzan et al. (2019) applied GP models to learn vehicle dynamics a driverless electric racecar and developed a contouring model predictive control approach. Additionally, Jain et al. (2020) utilized exact GP models for vehicle dynamics learning in autonomous racing simulations both 1 1:43 and 1:10 scales.
However, these methods have the following limitations: (i) The conducted experiments were limited to small-scale or formula student racecar (Kabzan et al., 2019) unlike our use of a full-sized fully autonomous racecar. (ii) Lack of real-world track layouts, limiting model generalizability. Ning and Behl (2023) have demonstrated that racetrack configurations and choices highly influence the accuracy of the GP-based model. In this paper, demonstrate that our method works on data obtained from real racecar running at real racetracks. (iii) The exact GP used in Bayesrace is not scalable to the volume of woal-world data samples. In this paper, the training data is 50 times larger, rendering the Bayesrace method intractable. (iv) Previous work lacks of kernel learning exploration, and authors arbitrarily used kernel functions such as RBF or Matérn kernel, while research indicates (Ning and Behl, 2023) that the choice of the kernel can substantially impact GP model performance. Therefore, in this work, we implement a deep neural network (DNN) for kernel learning for the GP models.
2.2 Gaussian processes with deep learning
The idea of deep kernel learning, which combines deep learning with Gaussian processes, was introduced by Wilson et al. (2016), who demonstrated the potential of DKL-GP to achieve state- of-the-art results on several benchmarks. Bradshaw et al. (2017) investigated the robustness of DKL-GP models against adversarial examples and transfer learning scenarios. Although the DKL- GP method has been employed for robotics and control tasks (Lee al., 2022), to the best of our knowledge, this paper the first work of integrating GPs with deep learning approaches for vehicle dynamics modeling. Specifically, propose DKL-SKIP to leam the dynamics of full-sized, real-world racecar by incorporating deep kernel in conjunction with sparse GPs.

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Notation
Vehicle Dynamics
z.y: Vehicle position
Kinematic
Dynamic
E-Kin
Steering angle [rad]
cos v
Ug V by
Viz cos V By sin V
Heading angle [rad]
sin
U2
+ by
0 ti sin V 008
Yow [rad/s]
A8
A6
8 A6
fy. C.O.G. frontrear adle [m]
Fry
Mzz Longitudinal velocity [m//e]
= Oz
m (F= Fsu sin mygw)
022 = Oz
Fyg Lateral velocity [m/ /s]
fy
m ( (Frg Fry nos mmzw)
By
by Ty
(At 10 v2w)
ozi Longitudinal ncceleration [m//²]
(1p Pyy cos (,Fr))
sin
Fry. Fry: Tire lateral forces
FJy D sin(C arctan Baj E(Boy arctanl (B))
Fs: Longitudinal force
Fry sin(C arctan(Bo..
E Ba, arctan (Bo,)))]
Table I: Mathematical descriptions different vehicle dynamics models. The vehicle dynamics models have three components:(1) the kinematic single-track model, (2) the dynamic single-track model, and (3) the felended--knemmen model used in this paper.
Vehicle Dynamics Background
Kinematic and dynamic single-track models are widely used based on a good trade-off between simplicity and accuracy. The single-track models simplify the vehicle by lumping the front and rear wheel pairs into wheel, similar bicycle configuration (Althoff et al., 2017). The details of differential equations for the kinematic model are shown in the second column of Table 1. and y are the coordinates of the vehicle's center of gravity (C.O.G.) with respect to the inertial frame. Velocity at C.O.G. of the vehicle is denoted by v is vehicle inertial heading, and is longitudinal acceleration. 8 is the steering angle that represents the angle between the vehicle heading and the front wheel heading. The model states are {ar, y, v, 6, v) the inputs to the model are acceleration and steering rate (az. _ and the parameters of the model are the distance between front axle and vehicle C.O.G., ly. and the distance between rear-axle and vehiele 4.
Although, easier to construct, kinematic models lack accuracy at higher speeds as they neglect tire slip, thereby overlooking crucial effects like understeer and oversteer (Rajamani, 2011). On the other hand,
a dynamic single-track model (Fig. 1) takes into ac- count the complex dynamics of vehicle, including the effects of tire slip and the interaction between tire forces and vehicle motion (Hwan Jeon et al., 2011). It considers additional factors such as lateral acceleration, yaw rate, and tire forces. This model more accurate and suitable for high-speed maneu- vers and scenarios where the vehiele operates closer its physical limits. The state equations are denoted in the third column of Table 1, where F, is the lon- gitudinal force, and Ffy. Fry are the lateral forces at the front axle and rear axle, respectively. In addition to the states in kinematic models, the dynamic mod- els consider lateral speed, By, as well as the slip angle the tire, a, or a,, which defined as the angle between the heading of the wheel and the velocity vector of the wheel, and used for deriving tire lateral forces Ffy. Fry using the Pacejka equations (bottom rows in Table 1). However, the model coefficients, BCDE, expensive to obtain and need to be recalibrated every time for every new racetrack, which increases the cost of implementing such dynamic model.
JCR
w
I yl 1
Fxx
Dynamic single-track vehicle model. Reference point: C.O.G.
Figure 1:
3

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Extended kinematic model In this work, we use an extended kinematic single-track (E-kin) model, as shown in Table 1, which preserves the simplicity of a kinematic model but has identical states to a dynamic model. However, the states the E-kin model and dynamic model differ velocities and yaw rate, {ea, By, (). Specifically, unlike the dynamic models, the E-kin model uses measurable variables, such as az. 17, 6, and two parameters, 1,,4. approximate the longitudinal and lateral forces. Therefore, the E-kin model requires less effort in model calibration than a dynamic single- track model. On the other hand, these approximations cause the E-kin model to diverge from the real dynamics. Therefore, when using the E-kin model to estimate the vehicle dynamics, we employ GP models to calculate the discrepancies between the E-kin model and the racecar observations.
4 Problem Statement
In this paper, our goal is to construct extended kinematic single-track model for an autonomous racecar and use GPs to learn the mismatch between the E-kin model and the observed racecar dy- namics. In particular, we use racecar state measurements, denoted by set D = (di...,,dh d for data samples, to address the differences between the E-kin model and the observed racecar dy- namics. Every data sample, di consists of states, 8; == { Yi, Uxi, Wy; th;, 6.), and inputs, u, == {am:, }. The E-kin model is represented as fEkin, and the model output at time step is created using the identical states, and inputs from the racecar measurements noted as fEkm (St. ut). Therefore, at time the model residual r, be expressed as in Eq(1).
rs = d++ - fEkin uz)
(1)
We then use GPs for model residual learning:
e(de) = GP(()) +/2
where e(d2) is the GP function given data sample d,, and Vr ~ (O, 1) is the additive Gaussian noise. By doing so, we can construct a corrected dynamics model, fcorr, which the combination of [Ekin and GP, to approximate the racecar's observed dynamics. For instance, at time 4, the corrected model (sorr(St, Ui) can be used to approximate racecar's observed dynamics de+1 as:
feorr u+) = f'kin (8t, un) + +(d+) R de+1
As shown in Table 1, it is worth noting that the model differences between the observation and E- kin model outputs only occur in the velocity, Mr: Ty. and yaw rate states. This implies that, in this work, R = [0, 0, Eu,, Ew, 0, 0, 0, 0J. R is the set of model residuals, R == {rI, There- fore, we learn three GP models, namely Cv2, Co,, Cws to approximate model residuals in 6,2 Cu, Cw- In the next section, we describe the implementation of the DKL-SKIP method to learn these GPs.
5 DKL-SKIP
In this section, we describe the DKL-SKIP method, which is a combination of deep kernel learning and scalable GP, SKIP-GP. This combines the advantages of both DKL and SKIP-GP, and allows us to apply it to capture the non-linear relationship for vehicle dynamics modeling mismatch.
5.1 Deep kernel learning
The goal of deep kernel learning is to learn kernel function for GPs. This is achieved by applying a deep learning neural network for feature extraction before is fed into the GPs, transforming the GP, a kernel function is used to measure the similarity between inputs, which influences the predictions of GP models. One of the most commonly used kernel functions in GP is the RBF kernel, denoted in Eq( 4).
inputs into a bigher-level representation.
(-
11 2/2
k(d,, d,) = o² exp
.. E (1,..,,3
(4)

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where d, dz are the imput data sample, i.e., racecar state measurements in this paper, o² is the variance, I|d - dy || is the Euclidean distance between inputs, is the length-scale. Further details of GP can be referred to Rasmussen (2003). In this paper, we transform a kernel ked;, dz; 0) to a deep kernel function, as shown Eq( 5).
k(d;, dj; 0) - ((g(d,, w), (d)g)dd w)]0, w)
where g(d, w) is a deep learning model, such as a deep neural network. 0 is the kernel hyperpa- rameters, e.g., the length-scale (1) of RBF kernel, and the w is the DNN weight. The deep kernel hyperparameters, denoted as {0, w), can be learned jointly by maximizing the log marginal likelihood. Moreover, DKL acts as a feature extractor, which transfers each input sample (d) into lower-dimensional feature expression (d)) Hence, the transformation of kernel into a deep kernel can then be expressed in Eq(6). This enables DKL capture the most representative information from the input data well as reduce the data dimensionality.
k(d;, dz;0) - k(de, dz;7)
5.2 SKIP-GP
The kernel learning process of GPs involves using the Cholesky decomposition of the kernel ma- trix, KDD, which is matrix composed of kernel functions of data points (d1,-.., d.). In this paper, DKL transform input data into lower-dimensional features (d) leading kernel matrix, K Keee sition requires matrix-vector multiplies (MVMs), which still leads to (n²) time and (n?)) space complexity (Rasmussen, 2003). Therefore, we use SKTP-GP, proposed by Gardner al. (2018), 8), to reduce the computing complexity. There two key components in SKIP-GPP (i) Structured kernel interpolation (SKI) (Wilson and Nickisch, 2015); and (ii) Product kernel structure. First, given set of m inducing points, U = {di...., dm I. where m < n. the SKI computes kernel functions between inducing points and then approximates the true kernel functions:
deep which improves the scalability of GPs. However, computing this decompo-
DD
; k(d,, d,) Ro KUU w 1, j. E (1...,,m) m}
(7a)
Kebe ~ W WikKoW?
(7b)
where Wi the sparse matrix composed of sparse vector (wa) that contains approximation weights. And thus, Eq(7b) denotes applying the approximation for all data. This method allows SKI to achieve linear time and storage complexity. However, it leads SKI to an exponential time complexity in the dimensionality of the inputs. The SKIP-GP uses product kernel structure to ad- dress the curse of dimensionality in SKI: given the data with p dimensions, the kernel matrix of the product kernel structure can be expressed as K Kddep -(1
かD ... (P) BD where is element-wiss
bD
multiplication. And each component, K (i) bD approximated using SKI: K DD S Keo w(Orr This ensures SKIP-GP achieves linear time complexity even with high-dimensional inputs.
w
D
Combining DKL with SKIP-GP by using inducing points to approximate the deep kernel matrix, Keeep + the resulting kernel is denoted as KdUp. Then we can use this inducing kernel matrix
to approximate the deep kernel matrix: K-deey se W kelee Wp". Therefore, the DKL-SKIP achieves linear time complexity while maintaining robustness and expressiveness.
6 Experiment Setup
We validate our method using both real-world data and data obtained from Auto Verse, a high-fidelity racing simulator. All experiments were conducted on Linux- system equipped with sixteen 3.8 GHz CPU cores, 16 GB of RAM, and single NVIDIA GeForce RTX 3080 GPU.
6.1 Real-world autonomous racing setup
The real data is obtained from full-scale, fully autonomous racecar, shown in Figure 2 These racecars are engineered for the high-speed, autonomous racing competition in the Indy Autonomous
5

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Las Vegas
Motor Speedway
Cleal legl-
IAC full-sized, fully autonomous racecar, Las Vegas Motor Speedway race track; IAC racecar in AutoVerse simulator, with racce track in Texas Motor Speedway.
Figure 2: Experiment setup:
Challenge (IAC). These vehicles have been customized with sensors such LiDAR, GNSS, cam- eras, and radar, computing hardware, and other autonomy-enabling components. The data was col- lected during solo runs at speeds 80 mph at the Las Vegas Motor Speedway, Figure these runs, state measurements were obtained by the Extended Kalman filter algorithm and logged as ROS2 bag files. These bag files were processed to construct the data set, Dreat, consisting of (Va, Uys vi, Ö, 4i, an, A6, WT, uT,un] Jn which longitudinal and lateral velocities, vehicle heading, yaw rate, stecring angle, longitudinal acceleration, steering velocity, throttle command, and braking pres- sure. This dataset is recorded at 25 Hz, divided into training set encompassing 15,789 samples (631 seconds) and testing set, unseen to the model, comprising 6,432 data samples (257 seconds).
During
6.2 Simulated autonomous racing setup
For more repeatable analysis, also conduct experiments using Veroe, designed to replicate the racing environment for data collection (Autonoma, 2023). As shown in Figure 2 Auto Verse can simulate vehicle simulator for the IAC racecars. We set up single racecar scenario on the Texas Motor Speedway racetrack, shown in Figure 2 where the car utilizes a pure pursuit algorithm and runs at higher speeds, 130 mph, than those recorded during real-world data collection. This approach aims to validate our method's performance in learning vehicle dynamics the limits the vehicle's 's performance. The data set, Daims recorded 25 Hz is composed of training set consisting of 11,737 data samples (469 seconds), the testing has 7.153 data samples (286 seconds).
6.3 DKL-SKIP setup
Here, we describe the setup of deep neural network feature extractor used to define the deep kernel
of the GP model. In this work, we used fully connected DNN with [9 19-8000 800-30000000 300 300-500 5000--0--00 50--4 4| archi- tecture, where the numbers indicate the number of neurons in each layer. Specifically, the first layer has 9 inputs corresponding to 9 features in the dataset, which are (uz. Vy: w. 6, w, 0z,, Ad, ur, un). The final layer has 4 outputs, which is determined based on the experiments and means that the DNN maps to final features. These output features are then passed to the SKIP-GP ReLU acti- vation layers and dropout layers with 20% rate have been integrated into the DNN. Morcover, we consider the scalable deep kernel learning with the RBF base kernel, as shown in Eq(4), which is popular choice of base kernel. During the training process, we run 60 epochs of training using the Adam optimizer, and the learning rate is 0.02. Besides, both the hyperparameters of the deep kernel, 0, and the parameters of the DNN, w, learned using Type-II maximum likelihood estimation.

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-35
10:00
mm
-00/
-100
pei SKPP
WID
sesin
Osserved Eata
Viecteee tara
5090
Doserved Data
Cherrven Dara
M Man
ScP GP
+200
Time (secondss
(a) Real-world dataset
(b) Simulation dataset
Figure 3: Comparison of prediction performance among various models using prediction
values: (1) The SKIP-GP model is represented by blue dotted lines; (2) Red dashed lines indicate the DKL-SKIP model; and (3) The N4SID model is shown by purple dashed lines.
6.4 Data preparation
To address the mismatch between the real racecar observations and the extended kinematic model data, we use the collected data Deap, comprising of both the real-world data and the simulator data, Dezp Dreal U DDim- Given the parameters (l,,4) (,) the E-kin model fEkin are known, we can construct a dataset, Dikins to capture the model outputs when excited with the same inputs and initial conditions, D DEkin {fEknn (s;, u;)), Vi {1....,7) n), where 8,, Uy represent data sample of state measurements and control input of Deup- For GPs training and testing, the input states consist. of (Ux, uy 1. 8, w. ax A6, uT, uB }. and the outputs are Eu2 Ew,
Ew. respectively.
7 Results
In this section, we evaluate DKL-SKIP, SKIP-GP, and a system identification technique, N4SID, proposed by Van Overschee and De Moor (1994). Our comparison focuses their capabilities in predicting the mismatch between the extended kinematic single-track model and observation.
7.1 Real-world results
Figure 3a shows the comparison between DKL-SKIP, SKIP-GP and N4SID) for each error term (Eng, Fuy, ew), and we can see that DKL-SKIP is able to accurately predict the mismatch between observed states and E-kin model output. In addition, we compare these models on mean absolute error, root mean square error, normalized root mean square error, and the coefficient determination (IP)), left table of Table 2. When examining the R², DKL-SKIP outperforms N4SID and SKIP- GP methods by 99%, 62% for error ev, and 24%, 32% for error Ev., respectively. N4SID slightly outperforms DKL-SKIP in the ew eπтоr term by margin of 0.16% in terms of the cocfficient of determination. However, does poorly at predicting the other two eтrоr terms. Overall, the results on real data show DKL-SKIP superior predictive capabilities across the three error terms.
7.2 Simulation results
The comparison between the models on simulated data is shown Figure 3b and in the right side of Table 2. Once again, DKL-SKIP model consistently performs well across all error terms. This indicates the effectiveness of DKL-SKIP and its ability predict the non-linear error terms under high-speed situations. In terms of 122 DKL-SKIP outperforms N4SID and SKIP-GP methods by 99%, 2% for error E., and 63%, 34% for error Eoy, respectively. The SKIP-GP is able to make accurate predictions error terms Eu, and ew, but it is not as accurate in predicting c,, Although

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Errors in long. velocity, Ce, (real-world)
Errors in long. velocity, Ce, (Simulation)
Method
MAE
RMSE
NRMSE
R2
Method
MAE
RMSE
NRMSE
R²
N4SID
0,00527
0.006600
0.15956
0.0020
NASID
0.00754
0,00882 0.00094
0.26283
2.463
SKIP-GP
DKL-SKIP
0.00103
0.00111
0.02607
0.9733
DKL-SKIP
0,00057
0.00064
0.01874
0.9823
Errors in lateral velocity, Euy -real-world)
Errors in lateral velocity, Cryy (Simulation)
Method
MAE
RMSE
NRMSE
R²
Method
MAE
RMSE
NRMSE
N4SID
0.00609
0.00787
0.11236
0.7347
N4SID
0.00985
0.01215
0,20140
0.3230
SKIP-GP
0.00730
0.00891
0.12733
0.6590
SKIP-GP
0.00816
0.00926
0.15363
0.6141
0.00301
0.9592
Errors in yaw rate, Ew (real-world)
Errors in yaw rate, ew (Simulation)
Method
MAE
RMSE
NRMSE
12"
Method
MAE
RMSE
NRMSE
12"
NASID
4.962-6
6,78e-6
0,010001
0.9985
NASID
2.71e-06
9.470-06
0.004-18
0.9971
SKIP-GP
3.83e-5
4.51e:5
0.06757
0,8837
SKIP-GP
3.91e-05
5.18e-05
0,02456
0.9133
DKL-SKIP
5.77e-6
7.26e-6
0.01088
0,9969
DKL.-KKIP
1.95e-05
2.9e.05
0.01374
0.9829
Table 2: Evaluation of N4SID, SKIP-GP, DKL-SKIP (our method), real-world setup (left table) and simulation setup (right table): The tables show these models' performance evaluated in four metrics, such as Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Normalized Root Mean Square Error (NRMSE), and Coefficient of Determination (R²).
N4SID marginally performs better Cur by 1%, it again struggles accurately predict the other error terms, particularly Ev.. DKL-SKIP's ability to provide accurate predictions under conditions when the vehicle operates at high speeds further demonstrates its robustness and effectiveness in learning the mismatch between Ekin model and observed dynamics for autonomous racing,
8 Limitations
The DKL-SKIP model has shown promise in learning vehicle dynamics in the context of au- tonomous racing, but a few limitations are noteworthy. First, DKL-SKIP still faces computational challenges, mainly impacted by the deep learning component, which is computationally expensive, especially for large datasets. In addition, the SKIP-GP has a (dkn) space complexity which will limit GPU performance when the number of copies of dataset, k, increases. These factors limit the model's performance in real time. In this paper, without any code optimization, DKL-SKIP has an inference time ranging from 20 to 40 ms. Since it has not been verified in closed-loop evaluations, and thus, the computing time may limit its suitability for real-time prediction. This will be addressed in future work as we implement OKL-SKIP-basee model predictive control for autonomous racing. Second, as supervised learning method, the predictive accuracy of DKL-SKIP depends on the distribution of the training data. The model's performance will be impaired by unseen input data samples, such as abrupt changes in vehicle speed or direction. Finally, the states Ux, Gy, 4 in the E-kin model simply approximations. We have not explored different variable settings in their differential equations, which could impact DKL-SKIP prediction ability.
9 Conclusion and Discussion
This paper introduces DKL-SKIP, a scalable GP model combined with the deep kernel learning for learning the mismatch between an Ekin model and observed dynamics for autonomous racing. We conduct both real-world and simulation experiments to compare DKL-SKIP against SKIP-GP, and N4SID. real-world evaluation, terms of R², DKL-SKIP outperforms competitors by 99%, 62% for Ee, and 24%, 32% for Gv,, respectively. DKL-SKIP also outperforms other models on simulated data by 99%, 2% for 6c, and 63%, 34% for cu,- respectively. Besides, DKL-SKIP performs well on predicting E.. specifically, in terms of R², archives 0.9969 in real-world data, and 0.9829 in simulated data. Our future work involves implementing model predictive control and evaluation in closed-loop experiments as well as extending this methodology to multi-agent autonomous racing.

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Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant No. 2046582.
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L. Hermansdorfer, R. Trauth, J. Betz, and M. Lienkamp. End-to-end neural network for vehicle dynamics modeling. 2020 6th IEEE Congress on Information Science and Technology (CiSt), pages 407-412, 2020. 10.:10.1 109/CiS14939992022 9357196.
B. Van Niekerk, A. Damianou, and B. Rosman. Online constrained model-based reinforcement learning. Conf. Uncertainty in Artificial Intelligence, 2017.
A. Jain, M. O'Kelly, Chaudhari, and M. Morari. Bayesrace: Learning to race autonomously using prior experience. arXiv preprint 2xxv:2005.04755 2020.
J. Kabzan, L. Hewing, A. Liniger, and M. N. Zeilinger. Learning- based model predictive control for autonomous racing. IEEE Robotics and Automation Letters, 4(4):3363-3370, 2019.
L. Hewing, A. Liniger, and M. N. Zeilinger. Cautious nmpe with gaussian process dynamics for autonomous miniature race cars. In 2018 European Control Conference (ECC), pages 1341- 1348. IEEE, 2018.

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J. Ning and M. Behl. Vehicle dynamics modeling for autonomous racing using gaussian processes. arXiv preprint arXiv:23000033005, 2023.
A. G. Wilson, Z. Hu, R. Salakhutdinov, and E. P. Xing. Deep kernel learning. In Artificial intelli- gence and statistics, pages 370-378. PMLR, 2016.
Bradshaw, G. G. Matthews, and Ghahramani. Adversarial examples, uncertainty, and transfer testing robustness in gaussian process hybrid deep networks. arXiv preprint 2017.
Lee, J. Feng, M. Humt, M. G. Müller, and R. Triebel. Trust your robots! predictive uncertainty estimation of neural networks with sparse gaussian processes. Conference on Robot Learning, pages 1168-1179, PMLR, 2022.
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J. Hwan Jeon, S. Karaman, and E. Frazzoli. Anytime computation of time-optimal off-road vehicle maneuvers using the rrt. In 2011 50th IEEE Conference Decision and Control and European Control Conference, pages 3276-3282. IEEE, 2011.
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Gardner, G. Pleiss, R. Wu, K. Weinberger, and A. Wilson. Product kernel interpolation for scalable gaussian processes. In International Conference on Artificial Intelligence and Statistics, pages 1407-1416. PMLR, 2018.
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(202)) Autoverse simulator of autonoma lab, 2023. URL http:///attonmmaaass.oom Accessed: May 5, 2023.
Van Overschee and B. De Moor. N4sid: Subspace algorithms for the identification of combined Seterministic-stochastt systems, Automatica, 30(1):75-995 1994.
10

---

In this supplementary material, we present a generalized overview of the N4SID system identifica- tion technique. In addition, we include the details of the ablation studies that were undertaken the development of DKL-SKIP.
A N4SID System Identification Algorithm
In this paper, we implement an alternative technique known as N4SID, a data-driven approach designed for system identification (Van Overschee and De Moor, 1994). This method utilizes a subspace-basee strategy, which segregates the data into deterministic and stochastic elements by projecting them onto distinct orthogonal subspaces. The algorithm calculates the system's 's state se- quence, -tatettransition matrid, input matrid, and output matrix from these subspaces, offering concise and effective depiction of the inherent dynamical system. Essentially, N4SID is capable of directly approximating system behavior using input-output data in state-space format. A descrip- tion of the algorithm is shown below:
Algorithm N4SID Algorithm
1: procedure N4SID
2: Normalize input-output data.
3: Estimate a covariance matrix by applying QR decomposition.
4: Compute singular value decomposition (SVD) of the covariance matrix.
5:
Divide the SVD output into observable and unobservable subspaces. From the observable subspace, compute the system matrices A, B, C, and D.
6:
7: Perform the balancing transformation and reduction of the state-space model.
8: Return the state-space model.
9:
end procedure
In this paper, considering the collected input-output data set u(k), ( with u(k) E R" repre- senting the input data sample, (Vzk, Pyk, 1ik, Wk, ark; Aōk, urk, uBB and y(k) TR denoting the output vector, E S Swx at time k, N4SID can effectively approximate the system's dynam- ies using state-space representation, as demonstrated
xt + (k)) + Bu(k) (v(1) = Cx((k)) + Du(k)
(8)
B DKL-SKIP setup
B.1 Architecture of neural network
The architecture of DNN was inspired by the work on Deep Kernel Learning by Wilson al. (2016). While adopted their layered approach, made modifications make it suitable for tasks.
One key adaptation the configuration of the first layer, which comprises nine neurons, mirroring the nine input features in our dataset. The output layer dimensionality decided through experimen- tation. Through our tests, we identified an optimal configuration for the output layer that balances maintaining prediction accuracy and simplifying the computational task. Specifically, the output dimensionality to four neurons. This design choice aids the SKIP-GP model in producing more accurate predictions by reducing the complexity it needs to handle. This is also the reason our DKL-SKIP performs better than SKIP-GP in terms of prediction accuracy.
reduced
The number of hidden layers (network capacity) was decided based on the experiment. Below, we compare the R2 score as model performance across different architectures, and the results shown in Table 3. As shown in the results, it is clear to that the capacity of the DNN is augmented, there's s typical trend of enhancement in our model's performance. This improvement is attributed
11

---

Architecture
R2 scores (v²)
R2 scores (vy)
R2 scores (w)
9 50->4
0.96095
0.95242
0.89471
2-+300-+50-11
0.97435
0.96716
0,99369
0.97489
0.96347
0.99522
9 10000880003000055000000
0,95730
0.93647
0.91790
Table 3: R2 scores for different architectures. As DNN capacity increases, performance generally improves, but it can decline if capacity becomes excessive.
R2 Score vs Learning Rate
0.980
0.078
R2 Score
0.076
0.074
0.072
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Learning Rate
Figure 4: Performance comparison across different learning rates (0.001 to 0.035) with the optimal R2 score of 0.9806 achieved at rate of 0.02.
to the increased ability of the DKI. to capture complex patterns and relationships in the dynamics data. However, it's essential to strike a balance. If the DNN's capacity becomes overly large, it may start to overfit the training data, which eventually degrades the overall performance of our model.
In addition, we also carried out an ablation study determine the optimal learning rate for tuning the DNN and the Gaussian Process (GP) kernel function. The results shown in Figure 4. In our study, we executed range of experiments, testing learning rates between 0.001 and 0.035, We used the R2 score our benchmark for model performance. Our findings revealed that learning rate of 0.02 yielded the highest performance, achieving an R2 score of 0.9806. Moreover, this rate ensured consistent stability during training.
B.2 Input data features
In this paper, the DKL-SKIP model was not exposed to the testing data during the training process. The training and testing data were collected during distinct runs on the racetrack. The variations in throttle/braking and steering between these datasets resulted in differing vehicle dynamics. To offer clearer understanding, we've depicted each input data in Figure 5.
The input features of our data sets consisting of {ez, Uy. 1/, S,w, Ad, ur, uB } representing lon- gitudinal and lateral velocities, vehicle heading, yaw rate, steering angle, longitudinal acceleration, steering velocity, throttle command, and braking pressure, respectively.
These features serve as the initial input for the DKL feature extractor. Within the DKL framework, series of processes occur: First, the dimensionality of this input data is significantly reduced to simplify the data structure. This condensed data is then transformed, mapping it to more abstract
12

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Taining set
Testing set
RASSA
Figure 5: Distinct racetrack for training and testing data collection, highlighting differences in vehicle dynamics due to variations in throttle/braking and steering.
feature space. Out of this transformed data, DKL identifies and selects the top four most significant and representative features. These features are then forwarded the input for the GP model, which responsible for making the final target predictions.
13


Reviews:
[{"review_id": "avMG7ipwC6E", "reviewer": "Reviewer_AEJt", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The authors paper is located in the field of mobile ground robotics, especially autonomous cars. The special focus of the authors work is on autonomous race cars, cars that drive on high speed and high acceleration in dynamic environment completely autonomously. A big importance in this field is modeling and determining the correct vehicle dynamics - especially from a nonlinear point of view e.g. spring/damper dynamics and tyre dynamics. The authors propose an approach called \"Deep Learning kernel\" in order to model/detect the vehicle dynamics correctly without modeling them with explicit ODEs. The authors use a real-world race car (Indy Autonomous Challenge) which provides a rich dataset and apply their approach to the this data. Ultimately, the approach is applied to a simulation environment an compared to other approach e.g. Gaussian process for Vehicle dynamics estimation.", "summary_of_recommendation": "The paper presents the application of a Deep Kernel Learning approach in order to estimate the vehicle dynamic behavior of an autonomous racecar. The authors used real-world data from a racecar to train the DNN and then apply it on the real car and in simulation. The authors demonstrate that their approach can capture the nonlinear behavior of the vehicle much better than other approaches. Unfortunately, the authors miss to have some special evaluation scenarios that clearly demonstrate that their approach can be generalized through many various scenarios. Nevertheless, a solid paper that is addressing questions in the vehicle dynamics and autonomous car community.", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "The experimental setup section is a litte bit poor. I understand that the Indy Autonomous Challenge car is driving on an oval only, but the vehicle dynamics are not explored here thoroughly since the car is driving on similar speeds and jus steering to the left side. The authors need to integrate an experimental setup that shows more dynamic behavior e.g. left/right steering, cornering, braking and steering -> Only this captures the nonlinear dynamics more thoroughly and it is questioned why a simple track like the Las Vegas Motor Speedway is used.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Quality:\n- The paper provides a thorough state of the art and is covering many interesting and important papers in the field of autonomous racing and vehicle dynamics modeling. The authors show clearly which papers adress which method so far e.g. the Gaussian Process papers. \n- Overall, the paper is of good merit but lacks some information, structural refinements, and some additional explanations in the method section. The method section is quite short, the additional explanation of the vehicle dynamics backgound (section 3) and Table 1 is a litte bit redundant\n\nClarity:\n-The paper's content is presented in a clear and concise manner, making it easy for readers to comprehend the research objectives, methods, and findings. The language used is precise, avoiding unnecessary jargon, and effectively conveying complex ideas without sacrificing understanding. Moreover, the logical flow of information is well-maintained, with each section naturally leading to the next, enhancing the overall coherence of the paper\n\nOriginality:\n- The paper is addressing the problem of vehicle dynamics estimation, which is a very well know problem in the ground robotics community and has been done many times. The news value and originality value of the paper lies in the application of the DKL-SKIP method, which is a combination of deep kernel learning and a scalable GP, SKIP-GP. From my view the paper offers therefore a good originality since this approach combines the advantages of both DKL and SKIP-GP, and allows to apply it to capture the non-linear relationship for vehicle dynamics modelling mismatch. If this approach is done right the nonlinearities of spring/damper and tyres can be capture more precisely.\n- By displaying the current SoTA in the field, the authors have introduced innovative approaches and setting their work apart from previous research clearly.\n- Unfortunately, right now the authors are missing to display their main conributions in order to display the reader more clearly what was done in the paper. I recommend entering the main contributions.\n\nSignificance:\n- While acknowledging the paper's merits in presenting a nice approach, it is essential to recognize that its significance may be relatively moderate when compared to some existing well-established works in the field. Especially classic modelling of the vehicle dynamics is still state of the art. The parameter identification is here the crucical part which takes time. If the authors can address a litte bit more the real-world value e.g. saving time and still being precise enough. It would be helpful.\n- While the paper contributes valuable insights and offers an innovative perspective, it might not drastically revolutionize the current state of research. The approach may resonate with specific audiences in the field of autonomous driving and mobile ground robots or research contexts, making it relevant in those particular domains only. \n\nStrength:\n- The approach of Deep Kernel Learning was firstly used in this paper to estimate the vehicle dynamics of a mobile ground vehicle with ackerman steering (car)\n- The paper is well written and structure and has a clarity in the reading\n- Comprehensive Literature Review: The paper showcases a comprehensive and up-to-date literature review that demonstrates the authors' in-depth understanding of the subject.\n- The paper is using real world data from a real vehicle in order to estimate the vehicle dynamics. Many authors are using simulation environments only but here real data is used\n\nWeaknesses:\n- The papers method is not explained thoroughly e.g. training setup and DNN setup. The DNN architecture is describe in one sentence which does not explained thoroughly how the method actually works. It is recommended to enter a clear picture of the process of using real-world data, training the DKL, and applying the DKL\n-You are missing your discussion section completely. From a scientific paper point of view, you need to enter a critical and reflective discussion. Explain to the reader what is good in your approach and what is bad. Be honest here about what needs to be done in addition to enhancing your system further. In addition, you can draw here conclusions from other authors. \n- The result section is quite poor only display three values of the estimation (velocity an yawrate). It makes sense to display the other values too and highlight the best values in Table 2", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "LJRwtCxbsU", "reviewer": "Reviewer_Fjkf", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper combined deep kernel learning with Gaussian Process to develop a learning-based residual dynamics model for autonomous racing. The proposed method is evaluated on offline racing data collected on a real-world race track with a full-scale racecar. ", "summary_of_recommendation": "Given the limited novelty and insufficient experimental study, I recommend rejecting this paper in its current form. It could potentially be turned into a valuable contribution to the community, if the authors could validate the proposed method by combining it with a model-based controller and evaluating it in a closed-loop fashion on the real-world racing platform. ", "technical_quality": "Fair: Significant additional evidence required to support the claims.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Highly relevant to robotics but no hardware experiments", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "It seems that the GP model is only used to estimate the mean of the model residual and the randomness is injected with a unit Gaussian noise. Then why can't the authors simply replace the GP model with a neural network (e.g., MLP or RNN), especially considering that the proposed method already incorporates a neural network for kernel learning? ", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Strengths: The experiment is based on data collected on a real-world track with a full-scale autonomous racecar, which could potentially provide exceptionally valuable insights into advancing real-world autonomous racing algorithms. \n\nWeakness:\n1. This work has very limited novelty and contribution in methodology. The proposed model is a straightforward combination of existing deep kernel learning with sparse GP, and GP for racing dynamics model learning has also been widely studied in the literature. The uniqueness of this work mainly lies in its experimental platform, built on a real-world track with a full-scale autonomous racecar. \n2. Unfortunately, the current experimental study is very limited and insufficient to validate the effectiveness of the proposed method. The proposed DKL-SKIP is evaluated on offline collected data. The data distribution highly depends on the controller and the reference path/speed adopted to drive the racecar. More importantly, its distribution could be significantly different from the trajectory distribution under closed-loop control (i.e., controlling the racecar using the learned GP model). Thus, the evaluation results on offline data are insufficient to validate the robustness of the proposed method. ", "extra_scores": {"impact": "2: Unsure, difficult to assess."}, "extra_text": {}}, {"review_id": "9DznjgY1fd", "reviewer": "Reviewer_nK5o", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The authors proposed to model the dynamics of an autonomous racecar by learning the residuals between an extended kinematic model and the real observations with a Gaussian Process (GP). To overcome the limitations of previous GP-based approaches, this work combines two existing techniques for more flexible and scalable GPs modeling, that are deep kernel learning and one variant of scalable GPs called SKIP-GP. Besides, in conrast to previous work, the proposed method is evaluated on the real data of a full-size racecar and in a high-fidelity simulator. Benefits have been shown against the standard GP and a traditional method via empirical experiments.", "summary_of_recommendation": "The idea including the method and its evaluation is sound and of value to the community of GP-based dynamics modeling. Though the technical novelty is limited, I would vote for a weak accept for this work.", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Highly relevant to robotics but no hardware experiments", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "I would suggest the authors to refine the writing accordingly and address the concerns in the review above.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The idea proposed by this work is generally convincing. The authors try to solve a problem highly relevant to mobile robots and autonomous driving. Though the technical novelty is limited, i.e. combining two existing approaches and there are not much texts on describing the technical difficulties, I think the straightforward combination is valuable in this application scenario. Because it addresses the limitations in previous works, i.e. a learnable kernel matrix and the scalability issue of GPs. More importantly, the validation of the proposed idea has been performed on real data of a full-size racecar, more realistic than test data of only small scale racecar in other related work. Besides, there is another data set in simulation used in the experiements. On the other hand, the presentation is generally clear and concise, I don't have much difficulty on understanding the key ideas. However, there are some suggestions that might be helpful on further improving the manuscript:\n\n### Writing\n- Minor: \"Gaussian Process\" -> \"Gaussian Processes\";\n\n- The DKL and SKIP-GP might be better to be put into a (sub)section called background or preliminaries. Then the authors can elaborate the proposed combination in a more detailed and formal way followed by the following optimization process. A discussion on the complexity comparison would be more informative for the readers that are interested in this method.\n\n### Methods\n- Since there is missing description on SKIP-GP (see above), why does SKIP-GP lead to an exponential time complexity in the dimensionality of the inputs? This should be the another motivation for using DKL on top of SKIP-GP. But from the texts, it is not straightforward to understand. \n\n- Why are the letters for inducing points the same as those for features from DKL? They are both $\\tilde{d}_i$.\n\n- As mentioned in the limitation, for real world deployment, out-of-distribution data might occur and cause unexpected behaviors of the model. Since GPs are also good at providing reliable uncertainty estimates. To evaluate the ability of detecting these data can offer more safety guarantee for the autonomous racecar. Have the authors consider this aspect. \n", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}, {"review_id": "7n439WRAPXg", "reviewer": "Reviewer_c6Hr", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The paper proposes a combination of GP regression with learned kernel functions for the task of residual dynamics learning for autonomous racing. The proposed method, termed DKL-SKIP, is shown to achieve better predictive performance on both real-world as well as simulated data. ", "summary_of_recommendation": "My recommendation is based on :\n- the promising results reported in the paper\n- the missing interpretation of the results\n- missing details about the evaluation procedure (are results obtained on seen or unseen data, sampling frequency, choice of hyperparamters)\n", "technical_quality": "Fair: Significant additional evidence required to support the claims.", "clarity_of_presentation": "Fair: Major revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Relevant but unlikely to deploy to hardware in near future", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "It would be interesting to understand if the superior performance of DKL-SKIP is across both the training and testing set or only on the testing set. I.e. is the model generalizing better or does it in general have better approximation capacity? The paper does not clearly state if the stated results have been achieved on seen or unseen data. In general, the paper would benefit from a more in-depth analysis of the results: why does the proposed method perform so much better than the RBF-based GPs? Previous research has achieved convincing results without doing deep kernel learning. What is different about the task at hand in this paper that results in such a substantial drop in performance of the published baselines?\n\n", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Strengths\n============\n- The paper shows experiments on both simulated and real-world data.\n- The related work section cites relevant work on Gaussian Process-based residual dynamics identification for autonomous race cars.\n\nConstructive Criticism\n==================\n- The abstract of the paper introduces the abbreviation GP as Gaussian Processes, but does not provide the context on DKL, SKIP, or N4SID. The readability of the paper could be greatly improved if abbreviations were introduced before they are used.\n\n- The limitation reported in L64 is not an algorithmic limitation. Just because the method presented in this paper has been deployed on a full-scale car does not make the approach better. If there are objective and measurable effects that can only be observed on the full-scale race car, then please list them instead of making such a sweeping statement.\n\n- The related work mentions that previous approaches that identified residual dynamics models from vehicle data used simple GPs with RBF or Matern Kernels, and argues that this choice of kernel functions is suboptimal. It would be interesting to compare with these approaches and report their performance for reference as well. Note that these approaches did not use SKIP-GP.\n\n- It would be interesting to understand if the superior performance of DKL-SKIP is across both the training and testing set or only on the testing set. I.e. is the model generalizing better or does it in general have better approximation capacity? The paper does not clearly state if the stated results have been achieved on seen or unseen data. In general, the paper would benefit from a more in-depth analysis of the results: why does the proposed method perform so much better than the RBF-based GPs? Previous research has achieved convincing results without doing deep kernel learning. What is different about the task at hand in this paper that results in such a substantial drop in performance of the published baselines?\n\n- Figure 3: replace the x-axis with time instead of sample count, add a grid on the plots.\n\n- L204: at which rate is this data recorded? I.e. how much track expose time do these sample numbers correspond to?\n\n- The size of the MLP used to do kernel learning has a very specific size. How was this network size found? What would happen if the network capacity increased/decreased? The learning rate of 0.02 is surprisingly high btw, wondering if this hyperparameter has been ablated or not.\n\n- The writing of the paper could be improved, here is a list of the most prominent issues:\n  - L43: minor stylistic feedback: a paper should not call its own method 'innovative'\n  - L50: the method is not trained on the car, it's trained on data collected from the car\n  - L266: an inference time should be reported in a unit of time, not a unit of frequency (i.e. 20-40ms inference time)\n", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors present an extended kinematic single track model for an autonomous race car. A GP model is used to learn the discrepancy between the extended kinematic model and observed dynamics in real and simulated environments. The proposed DKL-SKIP approach  is a deep kernel learning + SKIP-GP combination and is shown to outperform baselines in real and simulated experiments.

The reviewers recommendations where split with 3 weak accept and 1 strong reject recommendation.

Strengths of the proposed work include
- the paper was found to be well written
- valuable review of related work
- the inclusion of real world experiments
- relevance to autonomous driving community
- improved performance reported over state of the art

Areas of potential improvement:
- more challenging race tracks could be considered instead of oval tracks
- additional closed-loop evaluation was requested by a reviewer
- additional exploration and interpretation of the results was requested
- the degree of novelty of the approach was questioned and significance relative to existing
  and classical methods was an area of concern voiced
- additional critical reflection about the method was requested

In summary, I believe the work to be of interest to the CoRL community and the experimental evaluation is of value in particular, 
but the lack of strong accept recommendations and mixed opinions by the reviewers and remaining concerns also indicate that it may be valuable to incorporate additional improvements. In particular, the inclusion of an additional focus on the qualitative interpretation of the results, its limitations etc could be strengthening the work further. This submission may therefore currently be just near the acceptance threshold.
Decision: Accept (Poster)