# Fast Motion Model of Road Vehicles with Artificial Neural Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Outlook

#### 1.2. Motivation

## 2. Nonlinear Single Track Vehicle Model

#### 2.1. Model Components

#### 2.2. Dynamics of the Chassis

#### 2.3. Dynamics of the Wheels

#### 2.4. Steering Actuation

#### 2.5. Closed Loop Control

#### 2.6. Simulation of Model

## 3. Random Trajectory Planning

^{13}, which is very difficult to handle. Instead of the parameter sweeping, learning data are generated based on simulated scenarios. When defining the driving maneuvers, our goal is to create a wide variety of dynamic situations. Regarding longitudinal dynamics, sections with intensive acceleration and braking and smaller variations around a constant traveling velocity are necessary. Considering lateral dynamics, mild curves, as well as sharp turns, are essential to reach an extensive range of lateral acceleration. Combinations of curves with acceleration and braking are also desirable. Definition of these simulation maneuvers manually would be, on the one hand, a massive effort. On the other hand, it would probably also not provide the required diversity. Thus, a randomized motion planning approach is applied to generate the reference data for vehicle dynamics simulations.

#### 3.1. Motion Planning Based on Piecewise Linear Curvature and Travel Velocity Functions

#### 3.2. Random Planning

- The knot points ${\dot{x}}_{p}^{i}$ of the traveling velocity profile are chosen randomly between allowed lower ${\dot{x}}_{p,min}$ (10 m/s) and upper ${\dot{x}}_{p,max}$ (30 m/s) limits.
- Minimal allowed radii are calculated for each section based on maximal allowed lateral acceleration ${\ddot{y}}_{p,max}$ (5 m/s
^{2}).$${r}_{p,min}^{i}=\frac{{\left({\dot{x}}_{p}^{i}\right)}^{2}}{{\ddot{y}}_{p,max}}$$ - Radii ${r}_{p}^{i}$ of each section are chosen randomly in such a way that the values are between ${r}_{p,min}^{i}$ and ${m}_{{r}_{min}}{r}_{p,min}^{i}$. The factor ${m}_{{r}_{min}}$ (10) is a planning parameter.
- Lengths $\mathrm{\Delta}{\sigma}_{p}^{i}$ of each section are chosen randomly in such a way that the values are between ${p}_{{c}_{min}}2\pi {r}_{p}^{i}$ and ${p}_{{c}_{max}}2\pi {r}_{p}^{i}$. The factors ${p}_{{c}_{min}}$ (0.1) and ${p}_{{c}_{max}}$ (0.2) are planning parameters.
- Curvature values ${\kappa}_{p}^{i}=\frac{1}{{r}_{p}^{i}}$ are calculated, and half of them are inverted to provide left and right turns with equal probability.
- A ${p}_{s}$ (0.35) proportion of the curvature values ${\kappa}_{p}^{i}$ is nulled out to provide straight segments in a way that neighboring straight segments are not allowed.
- Transitions are calculated between each of the previous sections in such way that the proportion of their lengths to the segments lengths ${p}_{t}$ (0.4) is given.
- Arc length knot points ${\sigma}_{p}^{i}$ are calculated by a cumulative sum of segment lengths $\mathrm{\Delta}{\sigma}_{p}^{i}$.
- The curvature and travel velocity profile calculated in 1–8 is then provided to the planner described in Section 3.1 to get the random reference trajectory.

## 4. Neural Network Based Vehicle Model

`tensorflow`,

`keras`and

`scikit-learn`. The training is performed on a desktop computer with an Intel

^{®}Core™ i5-7600 CPU, 32 GB of RAM, 500 GB of NVME SSD storage, and an NVIDIA

^{®}GeForce GTX 1050 Ti GPU.

#### 4.1. Input–Output Concept

#### 4.2. Learning Sample Generation

`keras`’

`Model.fit`method so that the training algorithm asks for the samples in increasing order. By knowing the number of samples per binary file from the metadata, it is easy to decide if a new file must be loaded. The generator algorithm also takes care of the shuffling of samples per buffer file by creating random indices for samples.

#### 4.3. Neural Network Architecture

#### 4.4. Training Process

#### 4.5. Motion Prediction in Feedback Loop

- Input vector ${X}_{nv}\left({t}_{nv}^{j}\right)$ is applied to the neural network to compute output ${Y}_{nv}\left({t}_{nv}^{j}\right)$. In practice, input and output vectors must be scaled with the scaling vectors in Equations (59) and (60), but this is not reflected to in the equations for the sake of simplicity.
- Vehicle state variables for which estimations $\mathrm{\Delta}\xi \left({t}_{nv}^{j}\right)$ are available in the neural network output ${Y}_{nv}\left({t}_{nv}^{j}\right)$ are calculated by$$\xi \left({t}_{nv}^{j+1}\right)=\xi \left({t}_{nv}^{j}\right)+\mathrm{\Delta}\xi \left({t}_{nv}^{j}\right)$$
- For variant V0, yaw angle is calculated by numerical integration as$${\psi}_{nv}\left({t}_{nv}^{j+1}\right)=\frac{\mathrm{\Delta}{\dot{\psi}}_{nv}\left({t}_{nv}^{j}\right)+\mathrm{\Delta}{\dot{\psi}}_{nv}\left({t}_{nv}^{j+1}\right)}{2}\mathrm{\Delta}{t}_{nv}.$$
- For variants V0–2, position coordinates are calculated by numerical integration as$$\begin{array}{cc}\hfill {x}_{nv}^{G}\left({t}_{nv}^{j+1}\right)& =\frac{{\dot{x}}_{nv}^{G}\left({t}_{nv}^{j}\right)+{\dot{x}}_{nv}^{G}\left({t}_{nv}^{j+1}\right)}{2}\mathrm{\Delta}{t}_{nv},\hfill \end{array}$$$$\begin{array}{cc}\hfill {y}_{nv}^{G}\left({t}_{nv}^{j+1}\right)& =\frac{{\dot{y}}_{nv}^{G}\left({t}_{nv}^{j}\right)+{\dot{y}}_{nv}^{G}\left({t}_{nv}^{j+1}\right)}{2}\mathrm{\Delta}{t}_{nv},\hfill \end{array}$$
- Vehicle state derivative variables for which estimations $\mathrm{\Delta}\xi \left({t}_{nv}^{j}\right)$ are available in the neural network output ${Y}_{nv}\left({t}_{nv}^{j}\right)$ are estimated directly with differences$$\dot{\xi}\left({t}_{nv}^{j+1}\right)\simeq \frac{\mathrm{\Delta}\xi \left({t}_{nv}^{j}\right)}{\mathrm{\Delta}{t}_{nv}}$$
- Inertial accelerations in momentaneous vehicle frame are evaluated as$$\begin{array}{cc}\hfill {\ddot{x}}_{nv,I}^{V}\left({t}_{nv}^{j+1}\right)& =\frac{\mathrm{\Delta}{\dot{x}}_{nv}^{V}\left({t}_{nv}^{j}\right)}{\mathrm{\Delta}{t}_{nv}}-{\dot{y}}_{nv}^{V}\left({t}_{nv}^{j}\right){\dot{\psi}}_{nv}\left({t}_{nv}^{j}\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\ddot{y}}_{nv,I}^{V}\left({t}_{nv}^{j+1}\right)& =\frac{\mathrm{\Delta}{\dot{y}}_{nv}^{V}\left({t}_{nv}^{j}\right)}{\mathrm{\Delta}{t}_{nv}}+{\dot{x}}_{nv}^{V}\left({t}_{nv}^{j}\right){\dot{\psi}}_{nv}\left({t}_{nv}^{j}\right),\hfill \end{array}$$
- Input vector of next step, ${X}_{nv}\left({t}_{nv}^{j+1}\right)$ is assembled from vehicle inputs and results of Equation (62).
- Steps 1–7 are repeated until simulation is finished.

## 5. Simulation Results

`keras`. First, an example of the regression fit of the standalone neural network is presented and analyzed. Then, a closed-loop vehicle motion simulation example is shown to give a picture of the performance of the proposed algorithm in terms of output signals. Finally, the performance of different variants is evaluated.

#### 5.1. Regression Fit

_{MAX}stands for maximal absolute error, while MAE (Mean Absolute Error) is used conventionally. We can see the biggest estimation errors in the case of changes in wheel speeds and longitudinal slip values. This behavior is logical, since the piecewise linear travel velocity profile that is used for the input sample generation results in a square-shaped longitudinal acceleration (and wheel angular acceleration) signal with jumps (this can be seen in Figure 6 as well). Learning these jumps is complicated for the neural network. While E

_{MAX}values are very high for these variables, they are only reached for a minimal number of samples. As MAE values show, average estimation error remains very small even in these cases. In comparison, changes in lateral slip values are much slower since the clothoid transition segments between the circular arcs and straight sections enable lateral tire forces (and wheel slips) to build up gradually. Accordingly, corresponding E

_{MAX}values are much slower. In general, we can say that while the regression provided by the neural network is not perfect, it is sufficient to be used for short-term vehicle simulations presented in Section 5.2.

#### 5.2. Prediction of Vehicle Motion

^{2}and lateral acceleration exceeding 2 m/s

^{2}. The output of the neural network based model is able to nicely reproduce the output of the original dynamic model. The maximum error of position estimation is below 60 cm, and the maximal yaw (heading) angle estimation error remains under 2${}^{\circ}$. Prediction error of longitudinal velocity does not exceed 0.03 m/s, while yaw rate prediction error is below 0.6${}^{\circ}$/s. Inertial accelerations are also calculated with a maximal error of 0.2 m/s

^{2}. The biggest error can be observed in the case of lateral velocity, where the neural-network-based solution occasionally has a large offset of 0.05 m/s. However, the importance of this state variable from the motion planning point of view is not as high as that of the other ones listed above. The results show that with the application of appropriate safety boundaries applied to the position of the vehicle, the output of the neural-network-based model can be used for motion prediction in the online optimization loop of trajectory planner algorithms.

#### 5.3. Evaluation of Input–Output Concept

_{MAX}(maximal absolute error) values calculated for the whole test dataset and all vehicle simulation output quantities in case of the different variants V0–3 considering a 3 s long simulation time. Figure 9 shows the same information considering 10 s long simulations. The values labeled with ODE represent the estimation error that is present even if we assume perfect neural network performance. For these data, the worst case, considering the V0 variant with the maximal number of variables obtained by numerical integration, is selected. Please note that a logarithmic scale is used for the axis of error values.

## 6. Discussion

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DDPG | Deep Deterministic Policy Gradient |

I/O | Input–Output |

JSON | JavaScript Object Notation |

MCTS | Monte Carlo Tree Search |

MAE | Mean Absolute Error |

MPC | Model Predictive Control |

NWU | North West Up |

ODE | Ordinary Differential Equation |

PBD | Position Based Dynamics |

PCA | Principal Component Analysis |

RBFNN | Radial Basis Function Neural Network |

ReLU | Rectified Linear Unit |

SELU | Scaled Exponential Linear Unit |

SMC | Sliding Mode Control |

Nomenclature | |

${\gamma}_{[x/y]}^{G}$ | Dynamic quantity in inertial earth-fixed north-west-up coordinate system |

${\gamma}_{[x/y]}^{V}$ | Dynamic quantity in rotating vehicle-fixed forward-left-up coordinate system |

${\gamma}_{[f/r],[x/y]}^{W}$ | Dynamic quantity in rotating front or rear wheel-fixed forward-left-up coordinate system |

${M}_{d}$ | Driving torque (non-negative) [Nm] |

${M}_{b}$ | Braking torque (non-negative) [Nm] |

${\delta}_{sw}$ | Steering wheel angle [rad] |

${x}_{v}^{G}$, ${y}_{v}^{G}$ | Position of vehicle center of gravity in x (north), and y (west) directions in earth-fixed coordinate system [m] |

${\dot{x}}_{v}^{G}$, ${\dot{y}}_{v}^{G}$ | Velocity of vehicle center of gravity in x (north), and y (west) directions in earth-fixed coordinate system [m/s] |

${\ddot{x}}_{v}^{G}$, ${\ddot{y}}_{v}^{G}$ | Acceleration of vehicle center of gravity in x (north), and y (west) directions in earth-fixed coordinate system [m/s^{2}] |

${\dot{x}}_{v}^{V}$, ${\dot{y}}_{v}^{V}$ | Velocity of vehicle center of gravity in x (forward) and y (left) directions in vehicle-fixed coordinate system [m/s] |

${\ddot{x}}_{v,I}^{V}$, ${\ddot{y}}_{v,I}^{V}$ | Inertial acceleration of vehicle center of gravity in x (forward) and y (left) directions in instantaneous vehicle-fixed coordinate system [m/s^{2}] |

${\psi}_{v}$ | Yaw angle (rotation angle around up axis) in earth-fixed coordinate system [rad] |

${\dot{\psi}}_{v}$ | Yaw rate (angular velocity in z (up) direction) of vehicle in vehicle-fixed coordinate system [rad/s] |

${\ddot{\psi}}_{v}$ | Yaw acceleration (angular acceleration in z (up) direction) of vehicle in vehicle-fixed coordinate system [rad/s^{2}] |

${\rho}_{[f/r]}$ | Turning angle of front and rear wheels [rad] |

${\dot{\rho}}_{[f/r]}$ | Angular velocity of front and rear wheels [rad/s] |

${\ddot{\rho}}_{[f/r]}$ | Angular acceleration of front and rear wheels [rad/s^{2}] |

${s}_{[f/r],[x/y]}$ | Slips of front and rear wheels in x (longitudinal) and y (lateral) direction |

${\dot{s}}_{[f/r],[x/y]}$ | Slip derivatives of front and rear wheels in longitudinal (x) and lateral (y) direction [1/s] |

${\delta}_{f}$ | Steering angle of front wheel [rad] |

${\dot{\delta}}_{f}$ | Steering angle derivative of front wheel [rad/s] |

${F}_{[f/r]a,[x/y]}^{G}$ | Acting tire forces of front and rear wheels in x (north) and y (west) directions in earth-fixed coordinate system [N] |

${F}_{[f/r]a,[x/y]}^{V}$ | Acting tire forces of front and rear wheels in x (forward) and y (left) directions in vehicle-fixed coordinate system [N] |

${F}_{[f/r]a,[x/y]}^{W}$ | Acting tire forces of front and rear wheels in x (longitudinal), and y (lateral) directions in wheel-fixed coordinate systems [N] |

${F}_{[f/r]n,[x/y]}^{W}$ | Tire forces of front and rear wheels in x (longitudinal), and y (lateral) directions in wheel-fixed coordinate systems in case of pure longitudinal or lateral slip [N] |

${F}_{[f/r]c,[x/y]}^{W}$ | Tire forces of front and rear wheels in x (longitudinal), and y (lateral) directions in wheel-fixed coordinate systems in case of combined slip [N] |

${F}_{d,[x/y]}^{G}$ | Aerodynamic drag forces in x (north) and y (west) directions in earth-fixed coordinate system [N] |

${F}_{d,[x/y]}^{V}$ | Aerodynamic drag forces in x (forward) and y (left) directions in vehicle-fixed coordinate system [N] |

${F}_{[f/r],z}^{V}$ | Tire load forces in z (up) direction in vehicle-fixed coordinate system [N] |

${M}_{[f/r],d}$ | Driving torques of front and rear wheels [Nm] |

${M}_{[f/r],b}$,${M}_{[f/r],ba}$ | Intended and acting braking torques of front and rear wheels [Nm] |

${M}_{[f/r],rr}$,${M}_{[f/r],rra}$ | Calculated and acting rolling resistance torques of front and rear wheels [Nm] |

${c}_{[f/r],M}$ | Torque distribution factor to front and rear wheels |

${\dot{x}}_{[f/r]}^{W}$, ${\dot{y}}_{[f/r]}^{W}$ | Center point velocities of front and rear wheels in x (longitudinal) and y (lateral) directions in wheel-fixed coordinate system [m/s] |

${\dot{x}}_{[f/r]}^{V}$, ${\dot{y}}_{[f/r]}^{V}$ | Center point velocities of front and rear wheels in x (forward) and y (left) directions in vehicle-fixed coordinate system [m/s] |

${v}_{[f/r],r}$ | Rolling velocity of front and rear wheels [m/s] |

${s}_{[f/r]d,[x/y]}$ | Damped slips of front and rear wheels in x (longitudinal) and y (lateral) direction |

${K}_{[f/r],[x/y]}$ | Slip stiffness of front and rear wheels in x (longitudinal) and y (lateral) direction |

${k}_{[f/r]d,x}$ | Actual longitudinal slip damping factor |

${l}_{[f/r]a,[x/y]}$ | Slip dependent actual relaxation lengths of front and rear tires in x (longitudinal) and y (lateral) direction [m] |

${X}_{v}$ | State vector of vehicle |

${\dot{X}}_{v}$ | Derivative of vehicle state vector |

${m}_{v}$ | Total mass of vehicle [kg] |

${\theta}_{v,z}$ | Moment of inertia of the vehicle around z (up) axis [kgm^{2}] |

${h}_{v}$ | Center of gravity height of the vehicle [m] |

${l}_{v,[f/r]}$ | Horizontal distance of vehicle center of gravity and the front and rear axes [m] |

${A}_{v,f}$ | Frontal area of the vehicle [m^{2}] |

${c}_{v,d}$ | Aerodynamic drag coefficient of the vehicle |

${\rho}_{a}$ | Density of air [kg/m^{3}] |

${\theta}_{[f/r]}$ | Moment of inertia of the front and rear wheels [kgm^{2}] |

${r}_{[f/r]}$ | Radii of the front and rear wheels [m] |

${v}_{ba,0}$, ${v}_{ba}$ | Minimal and actual rolling velocity at which braking torque shall be fully applied [m/s] |

${k}_{{v}_{ba}}$ | Braking torque dependent factor for ${v}_{ba}$ [1/Ns] |

${A}_{[f/r],rr}$,${B}_{[f/r],rr}$,${C}_{[f/r],rr}$ | Rolling resistance coefficients [1], [s/m], [s^{2}/m^{2}] |

${v}_{rra}$ | Rolling velocity at which rolling resistance torque shall be fully applied [m/s] |

${D}_{[f/r],[x/y]}$ | Maximum values of Magic Formula for front and rear wheels in x (longitudinal) and y (lateral) direction |

${C}_{[f/r],[x/y]}$ | Shape factors of Magic Formula for front and rear wheels in x (longitudinal) and y (lateral) direction |

${B}_{[f/r],[x/y]}$ | Stiffness factors of Magic Formula for front and rear wheels in x (longitudinal) and y (lateral) direction |

${E}_{[f/r],[x/y]}$ | Curvature factors of Magic Formula for front and rear wheels in x (longitudinal) and y (lateral) direction |

${\mu}_{[f/r]}$ | Coefficient of friction at front and rear wheels |

${k}_{[f/r],x}$ | Initial longitudinal slip damping factor |

${v}_{sd}$ | Rolling velocity at which slip damping should switch off |

${s}_{da}$ | Minimal value of wheels slips at which superposition of forces shall first be considered |

${l}_{[f/r],[x/y]}$,${l}_{[f/r]m,[x/y]}$ | Initial and minimal relaxation lengths of front and rear tires in x (longitudinal) and y (lateral) direction [m] |

${k}_{s}$ | Steering ratio |

${T}_{s}$ | Settling time of steering mechanism |

$\mathrm{\Delta}{t}_{v}$ | Sample time of vehicle model solution [s] |

${t}_{v}$ | Time of vehicle motion simulation [s] |

${\sigma}_{p}$, ${\sigma}_{p}^{i}$ | Arc length, arc length knot points [m] |

${\kappa}_{p}$, ${\kappa}_{p}^{i}$ | Curvature of path, curvature profile knot points [1/m] |

${\dot{x}}_{p}$, ${\dot{x}}_{p}^{i}$ | Travel speed along path, travel speed profile knot points [m/s] |

${N}_{p}^{i}$ | Number of knot points specified for curvature and travel speed profile |

$\mathrm{\Delta}{t}_{p}^{i}$ | Time needed to travel along path section i [s] |

$\mathrm{\Delta}{\sigma}_{p}^{i}$ | Length of path section i [m] |

${\tilde{\dot{x}}}_{p}^{i}$ | Average travel speed of path section i [m/s] |

${t}_{p}$, ${t}_{p}^{i}$ | Travel time along path, time needed to reach end of path section i [s] |

${\ddot{x}}_{p}$, ${\ddot{x}}_{p}^{i}$ | Longitudinal acceleration along path, and at path section i [m/s^{2}] |

${\dot{\psi}}_{p}$, ${\dot{\psi}}_{p}^{j}$ | Yaw rate (angular velocity in z direction) along path, yaw rate output samples [rad/s] |

${\ddot{y}}_{p}$ | Centripetal acceleration along path [m/s^{2}] |

${\psi}_{p}$, ${\psi}_{p,0}$, ${\psi}_{p}^{j}$ | Yaw angle (rotation angle around up axis) along path and initially, yaw angle output samples [rad] |

${x}_{p}$, ${x}_{p,0}$, ${x}_{p}^{j}$${y}_{p}$, ${y}_{p,0}$, ${y}_{p}^{j}$ | Position in x (north), and y (west) directions in earth-fixed coordinate system along path and initially, position output samples [m] |

${\sigma}_{p,s}$ | Arc length resolution for numeric calculations [m] |

${N}_{p}^{j}$ | Number of reference trajectory points |

${N}_{r}$ | Number of road segments |

${\dot{x}}_{p,[min/max]}$ | Maximal and minimal allowed travel speed [m/s] |

${\ddot{y}}_{p,max}$ | Maximal allowed centripetal acceleration [m/s^{2}] |

${r}_{p}^{i}$ | Radius of path section i [m] |

${r}_{p,min}^{i}$ | Minimal allowed radius for path section i [m] |

${c}_{r,min}$ | Multiplier factor for minimal allowed radii |

${c}_{c,[min/max]}$ | Multiplier factor path section length in proportion to circumference |

${c}_{s}$ | Proportion of straight sections compared to curved sections |

${c}_{t}$ | Proportion of transition section length to normal section length |

$\mathrm{\Delta}{t}_{nv}$ | Prediction time of neural network based vehicle model [s] |

$\xi $ | General state variable |

$\mathrm{\Delta}\xi $ | Change of state variable in $\mathrm{\Delta}{t}_{nv}$ time |

${X}_{nv}^{VAR}$ | Input vector of neural network variant VAR |

${Y}_{nv}^{VAR}$, ${Y}_{nv}^{i}$ | Output vector of neural network variant VAR. Element of output vector |

${I}_{nv}^{tr}$, ${I}_{nv}^{tt}$ | Matrices of training and testing input samples |

${O}_{nv}^{tr}$, ${O}_{nv}^{tt}$ | Matrices of training and testing output samples |

${N}_{v}^{tr}$ | Total number of vehicle simulation samples generated for training |

${c}_{{X}_{nv}}$ | Vector of input scales (maximum absolute value of each variable in ${X}_{nv}$) |

${c}_{{Y}_{nv}}$ | Vector of output scales (maximum absolute value of each variable in ${Y}_{nv}$) |

${L}_{nv}$ | Training loss |

${n}_{{Y}_{nv}}$ | Number of elements of output vector ${Y}_{nv}$ |

${w}_{{L}_{nv}}$, ${w}_{{L}_{nv}}^{i}$ | Weighting vector for squared error in estimation of ${Y}_{nv}$. Element of weighting vector |

${\widehat{Y}}_{nv}$, ${\widehat{Y}}_{nv}^{i}$ | Estimation of ${Y}_{nv}$ by the neural network. Element of estimation vector |

${t}_{nv}$, ${t}_{nv}^{j}$ | Time of neural network based vehicle simulation. Time at simulation step j [s] |

${\psi}_{nv}$ | Yaw angle (rotation around up axis) in earth-fixed coordinate system. Computed by neural network based vehicle model [rad] |

${x}_{nv}^{G}$, ${y}_{nv}^{G}$ | Position of vehicle center of gravity in x (north), and y (west) directions in earth-fixed coordinate system [m]. Computed by neural network based vehicle model |

${\ddot{x}}_{nv,I}^{V}$, ${\ddot{y}}_{nv,I}^{V}$ | Inertial acceleration of vehicle center of gravity in x (forward) and y (left) directions in instantaneous vehicle-fixed coordinate system [m/s^{2}]. Computed by neural network based vehicle model |

${E}_{MAX}$ | Maximum absolute estimation error of neural network |

${\tau}_{r}$ | Run time as factor to real-time speed |

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Name | Layout |
---|---|

n256l3v1 | [64, 128, 64] |

n256l3v2 | [32, 192, 32] |

n256l4v1 | [32, 96, 96, 32] |

n256l4v2 | [64, 128, 128, 64] |

Variable | E_{MAX} | MAE | Variable | E_{MAX} | MAE |
---|---|---|---|---|---|

$\times {\mathbf{10}}^{-\mathbf{2}}$ | $\times {\mathbf{10}}^{-\mathbf{2}}$ | ||||

$\mathrm{\Delta}{\dot{x}}_{v}^{V}\left(t\right)$ | 0.17043 | 0.08441 | $\mathrm{\Delta}{s}_{f,x}\left(t\right)$ | 0.93073 | 0.09424 |

$\mathrm{\Delta}{\dot{y}}_{v}^{V}\left(t\right)$ | 0.16793 | 0.07394 | $\mathrm{\Delta}{s}_{f,y}\left(t\right)$ | 0.16374 | 0.08827 |

$\mathrm{\Delta}{\dot{\psi}}_{v}\left(t\right)$ | 0.24512 | 0.08250 | $\mathrm{\Delta}{s}_{r,x}\left(t\right)$ | 0.42943 | 0.08377 |

$\mathrm{\Delta}{\dot{\rho}}_{f}\left(t\right)$ | 0.82452 | 0.15126 | $\mathrm{\Delta}{s}_{r,y}\left(t\right)$ | 0.13317 | 0.05913 |

$\mathrm{\Delta}{\dot{\rho}}_{r}\left(t\right)$ | 0.51242 | 0.14104 | $\mathrm{\Delta}{\delta}_{f}\left(t\right)$ | 0.15871 | 0.04865 |

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## Share and Cite

**MDPI and ACS Style**

Hegedüs, F.; Gáspár, P.; Bécsi, T.
Fast Motion Model of Road Vehicles with Artificial Neural Networks. *Electronics* **2021**, *10*, 928.
https://doi.org/10.3390/electronics10080928

**AMA Style**

Hegedüs F, Gáspár P, Bécsi T.
Fast Motion Model of Road Vehicles with Artificial Neural Networks. *Electronics*. 2021; 10(8):928.
https://doi.org/10.3390/electronics10080928

**Chicago/Turabian Style**

Hegedüs, Ferenc, Péter Gáspár, and Tamás Bécsi.
2021. "Fast Motion Model of Road Vehicles with Artificial Neural Networks" *Electronics* 10, no. 8: 928.
https://doi.org/10.3390/electronics10080928