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Hybrid modeling of multibody vehicles with partially known physics: discovering complex behaviors of tires

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Abstract

There are multibody systems whose physics are partially known owing to their complexity and nonlinearity. Therefore, motion equations are not utterly available to be utilized for the prediction, control, design, and monitoring of these systems. To alleviate this issue, this study aims at developing a hybrid modeling procedure to discover respective unidentified physics and, subsequently, provide a holistic governing model of the original mechanism. For approach development, a vehicle with unmodeled tires is thoroughly considered in this research work. Tires profoundly impact the dynamics of vehicles, influencing their handling, drivability, and ride comfort. Advanced chassis control systems used to improve vehicles’ safety, performance, and reliability also require knowledge of tire behavior. Nevertheless, tires are very challenging to model as they are very complex and nonlinear components. Although simplified models are often employed, they are incapable of fully capturing tire behaviors. Using neural networks, i.e., black-box models, of the tire represents a common alternative. However, these approaches do not work outside the training data distribution, and they need costly and hard-to-measure experimental data for training purposes. Thus, this research study proposes a hybrid method by combining partially known physics of vehicle dynamics and a neural network to compensate for the unknown physics of tires. The developed approach learns the tire dynamics automatically from vehicle responses without requiring costly measured tire forces but solely relying on signals from an inertial measuring unit. The suggested methodology is validated experimentally, providing accurate and stable results. The time-depending behaviors of tires during cornering are also discovered and reported. The developed model is generic and can handle either linear or nonlinear physics-based models. However, the linear tire model integrated into the hybrid procedure in this study limits the simulation to stationary trajectories and cannot address the physics of tires when a vehicle undergoes nonstationary maneuvers.

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Data Availability

The data and materials presented in this study are available on request from the corresponding author.

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Acknowledgements

The authors gratefully acknowledge the support of VLAIO (Flemish Innovation & Entrepreneurship) through the O&O project AI4Test (HBC.2022.0005).

Funding

This project is financed through the European Recovery and Resilience Facility (RRF).

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Authors and Affiliations

  1. Dept. Electromechanical, Systems and Metal Engineering, Ghent University, B-9052, Ghent, Belgium

    Ehsan Askari & Guillaume Crevecoeur

  2. MIRO Core Lab, Flanders Make, 3920, Lommel, Belgium

    Ehsan Askari & Guillaume Crevecoeur

  3. Siemens Digital Industries Software, Interleuvenlaan 68, 3001, Leuven, Belgium

    Davide Gorgoretti

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  1. Ehsan Askari
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Contributions

E.A. wrote and reviewed the manuscript, developed the method, validated it, wrote scripts, and D.G. wrote the manuscript and provided experiments data, and G.C. reviewed the manuscript and mentored. All authors reviewed the manuscript.

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Correspondence to Ehsan Askari.

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Appendices

Appendix A

In this appendix, the formulations associated with RNN are given.

$$\begin{aligned} \begin{aligned} &V_{t}^{\left (1\right )} = w_{1} u_{t},\qquad h_{t}^{\left (1\right )} = \varphi _{1} \left ( V_{t}^{\left (1\right )} \right )\\ & V_{t}^{(2)} = w_{2} \left ( h_{t}^{\left (1\right )} + \psi _{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) \right ),\qquad h_{t}^{(2)} = \varphi _{2} \left ( V_{t}^{\left (2\right )} \right )\\ & V_{t}^{(3)} = w_{3} \left ( h_{t}^{\left (2\right )} + \psi _{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) \right ),\qquad y_{t} = \varphi _{3} \left ( V_{t}^{\left (3\right )} \right )\\ & S_{t}^{\left (1\right )} = h_{t}^{\left (1\right )} + \psi _{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right )\\ & S_{t}^{\left (2\right )} = h_{t}^{\left (2\right )} + \psi _{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) \end{aligned} \end{aligned}$$
(A.1)

The gradient of network output with respect to weights, computing the weight correction at each training iteration, Eq. (6), is given as follows:

$$\begin{aligned} \begin{aligned} &\frac{d z_{t}}{d w_{3}} = \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) h_{t}^{\left (2\right )}\\ & \frac{d z_{t}}{d w_{2}} = \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) \left ( h_{t}^{\left (1\right )} + \psi _{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) \right )\\ &\phantom{\frac{d z_{t}}{d w_{2}} = }{} + \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \\ &\frac{d S_{t -1}^{\left (2\right )}}{d w_{2}} \frac{d z_{t}}{d w_{1}} = \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) w_{2} \dot{\varphi}_{1} \left ( V_{t}^{\left (1\right )} \right ) u_{t} \\ &\phantom{\frac{d S_{t -1}^{\left (2\right )}}{d w_{2}} \frac{d z_{t}}{d w_{1}} = }{}+ \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \frac{d S_{t -1}^{(2)}}{d w_{1}} \\ &\phantom{\frac{d S_{t -1}^{\left (2\right )}}{d w_{2}} \frac{d z_{t}}{d w_{1}} = }{}+ \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) w_{2} \dot{\psi}_{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) w_{hh}^{\left (1\right )} \frac{d S_{t -1}^{(1)}}{d w_{1}} \end{aligned} \end{aligned}$$
(A.2)

in which

$$\begin{aligned} \begin{aligned} \frac{d S_{t}^{(2)}}{d w_{2}} ={}& \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) \left ( h_{t}^{\left (1\right )} + \psi _{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) \right ) + \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \frac{d S_{t -1}^{(2)}}{d w_{2}} \\ \frac{d S_{t}^{(2)}}{d w_{1}} = {}&\dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) w_{2} \dot{\psi}_{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) w_{hh}^{\left (1\right )} \frac{d S_{t -1}^{(1)}}{d w_{1}} + \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) w_{2} \dot{\varphi}_{1} \left ( V_{t}^{\left (1\right )} \right ) u_{t} \\ &{}+ \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \frac{d S_{t -1}^{(2)}}{d w_{1}} \\ \frac{d S_{t}^{(1)}}{d w_{1}} ={}& \dot{\varphi}_{1} \left ( V_{t}^{\left (1\right )} \right ) u_{t} + \dot{\psi}_{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) w_{hh}^{\left (1\right )} \frac{d S_{t -1}^{(1)}}{d w_{1}} \end{aligned} \end{aligned}$$
(A.3)

The gradient of network output with respect to recurrent weights is, in turn, determined as follows:

$$\begin{aligned} \begin{aligned} \frac{d z_{t}}{d w_{hh}^{2}} ={}& \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) S_{t -1}^{(2)}\\ &{} + \dot{\varphi}_{3} \left ( V_{t}^{\left (3\right )} \right ) w_{3} \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \frac{d S_{t -1}^{(2)}}{d w_{hh}^{2}} \\ \frac{d z_{t}}{d w_{hh}^{1}} ={}& \dot{\varphi}_{3} \left ( V_{t}^{\left ( 3 \right )} \right ) w_{3} \dot{\varphi}_{2} \left ( V_{t}^{\left ( 2 \right )} \right ) w_{2} \dot{\psi}_{1} \left ( w_{hh}^{\left ( 1 \right )} S_{t -1}^{\left ( 1 \right )} \right ) S_{t -1}^{\left ( 1 \right )}\\ &{} + \dot{\varphi}_{3} \left ( V_{t}^{\left ( 3 \right )} \right ) w_{3} \dot{\psi}_{2} \left ( w_{hh}^{\left ( 2 \right )} S_{t -1}^{\left ( 2 \right )} \right ) w_{hh}^{\left ( 2 \right )} \frac{d S_{t -1}^{\left ( 2 \right )}}{d w_{hh}^{1}} \\ &{}+ \dot{\varphi}_{3} \left ( V_{t}^{\left ( 3 \right )} \right ) w_{3} \dot{\varphi}_{2} \left ( V_{t}^{\left ( 2 \right )} \right ) w_{2} \dot{\psi}_{1} \left ( w_{hh}^{\left ( 1 \right )} S_{t -1}^{\left ( 1 \right )} \right ) w_{hh}^{\left ( 1 \right )} \frac{d S_{t -1}^{\left ( 1 \right )}}{d w_{1}}, \end{aligned} \end{aligned}$$
(A.4)

where

$$\begin{aligned} \begin{aligned} \frac{d S_{t}^{\left (2\right )}}{d w_{hh}^{2}} ={}& \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) S_{t -1}^{\left (2\right )} + \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \frac{d S_{t -1}^{\left (2\right )}}{d w_{hh}^{2}}\\ \frac{d S_{t}^{(2)}}{d w_{hh}^{1}} ={}& \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) w_{2} \dot{\psi}_{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) w_{hh}^{\left (1\right )} \frac{d S_{t -1}^{(1)}}{d w_{hh}^{1}}\\ &{} + \dot{\varphi}_{2} \left ( V_{t}^{\left (2\right )} \right ) w_{2} \dot{\psi}_{1} \left ( w_{hh}^{\left (1\right )} S_{t -1}^{\left (1\right )} \right ) S_{t -1}^{(1)} \\ &{}+ \dot{\psi}_{2} \left ( w_{hh}^{\left (2\right )} S_{t -1}^{\left (2\right )} \right ) w_{hh}^{\left (2\right )} \frac{d S_{t -1}^{(2)}}{d w_{hh}^{1}} \\ \frac{d S_{t}^{\left ( 1 \right )}}{d w_{hh}^{1}} = {}&\dot{\psi}_{1} \left ( w_{hh}^{\left ( 1 \right )} S_{t -1}^{\left ( 1 \right )} \right ) S_{t -1}^{\left ( 1 \right )} + \dot{\psi}_{1} \left ( w_{hh}^{\left ( 1 \right )} S_{t -1}^{\left ( 1 \right )} \right ) w_{hh}^{\left ( 1 \right )} \frac{d S_{t -1}^{\left ( 1 \right )}}{d w_{hh}^{1}}. \end{aligned} \end{aligned}$$
(A.5)

Appendix B

The differentiation of the hybrid model’s output with respect to the output of the neural network for both hybrid models of S1 and S2 is given in this appendix.

The derivatives associated with the S1 hybrid model are as follows:

$$ \frac{d \mathbf{o}}{d z_{1}} = \left [ \textstyle\begin{array}{c} \left ( - \frac{1}{m} \frac{x_{3}}{v_{x}} - \frac{l_{f}}{m} \frac{x_{4}}{v_{x}} + \frac{\delta}{m} \right ) + \left ( - \frac{l_{f} x_{3}}{I_{zz}} - \frac{l_{f}^{2} x_{4}}{I_{zz}} + \frac{l_{f} v_{x}}{I_{zz}} \delta \right ) dt\\ \left ( - \frac{l_{f}}{I_{zz}} \frac{x_{3}}{v_{x}} - \frac{l_{f}^{2}}{I_{zz}} \frac{x_{4}}{v_{x}} + \frac{l_{f}}{I_{zz}} \delta \right ) dt \\ \textstyle\begin{array}{c} \left ( - \frac{l_{f}}{I_{zz}} \frac{x_{3}}{v_{x}} - \frac{l_{f}^{2}}{I_{zz}} \frac{x_{4}}{v_{x}} + \frac{l_{f}}{I_{zz}} \delta \right )\\ \left ( - \frac{1}{m} \frac{x_{3}}{v_{x}} - \frac{l_{f}}{m} \frac{x_{4}}{v_{x}} + \frac{\delta}{m} \right ) dt \end{array}\displaystyle \end{array}\displaystyle \right ] $$
(B.1)

and

$$ \frac{d \mathbf{o}}{d z_{2}} = \left [ \textstyle\begin{array}{c} \left ( - \frac{1}{m} \frac{x_{3}}{v_{x}} + \frac{l_{r}}{m} \frac{x_{4}}{v_{x}} \right ) + \left ( \frac{l_{r} x_{3}}{I_{zz}} - \frac{l_{r}^{2} x_{4}}{I_{zz}} \right ) dt \\ \left ( \frac{l_{r}}{I_{zz}} \frac{x_{3}}{v_{x}} - \frac{l_{r}^{2}}{I_{zz}} \frac{x_{4}}{v_{x}} \right ) dt \\ \textstyle\begin{array}{c} \left ( \frac{l_{r}}{I_{zz}} \frac{x_{3}}{v_{x}} - \frac{l_{r}^{2}}{I_{zz}} \frac{x_{4}}{v_{x}} \right )\\ \left ( - \frac{1}{m} \frac{x_{3}}{v_{x}} + \frac{l_{r}}{m} \frac{x_{4}}{v_{x}} \right ) dt \end{array}\displaystyle \end{array}\displaystyle \right ]. $$
(B.2)

In addition, those for S2 hybrid model can be written by

$$ \frac{d \mathbf{o}}{d z_{1}}^{T} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \frac{\cos \delta}{m} & \frac{l_{f} \cos \delta}{I_{zz}} dt & \textstyle\begin{array}{c@{\quad}c} 0 & \frac{l_{f} \cos \delta}{I_{zz}} \end{array}\displaystyle \end{array}\displaystyle \right ] $$
(B.3)

and

$$ \frac{d \mathbf{o}}{d z_{2}}^{T} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \frac{1}{m} & - \frac{l_{r}}{I_{zz}} dt & \textstyle\begin{array}{c@{\quad}c} 0 & - \frac{l_{r}}{I_{zz}} \end{array}\displaystyle \end{array}\displaystyle \right ]. $$
(B.4)

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Askari, E., Gorgoretti, D. & Crevecoeur, G. Hybrid modeling of multibody vehicles with partially known physics: discovering complex behaviors of tires. Multibody Syst Dyn (2024). https://doi.org/10.1007/s11044-024-09983-3

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