Keywords

1 Introduction

With the development of electric motor technology, the power density of electric motors has increased sharply. This helps develop a new form of power drive vehicles: distributed-drive. Distributed electric drive vehicles have a high degree of freedom in control and a high complexity in the physical modeling of the vehicle.

Vehicle state estimation, a technique for inferring the dynamic state of a vehicle from sensor data and mathematical models, provides accurate information about the vehicle state, such as vehicle position and tire force [3]. The state estimation is the foundation of the vehicle control, which directly affects the accuracy of the downstream control tasks.

Most of the tire force state estimation methods that have been studied are based on physical rule models [4]. Under the trend of increasing control complexity and more complicated feasible working conditions in distributed electric drive vehicles, the accuracy of these traditional models seems limited [7]. Therefore, establishing a more accurate model for estimating the vehicle tire force state is the starting point of this study. As the requirements for autonomous driving tasks become more complex, the limited capabilities of traditional rule models are being recognized by the industry. The attempts to use neural networks have started to increase gradually these years.

Some researchers employed a neural network for the modeling problem completely [5]. In 2019, an article published in Science Robotics used a pure neural network to achieve lateral trajectory tracking control of vehicles [8]. The other researchers did not abandon the original physical rule model. They combined the neural network model and rule-based model [1, 6]. This kind of modeling idea is a kind of affirmation of the information processing and output ability of the rule model. It will reduce the training difficulty of the neural network model theoretically.

Some articles present a theory of building neural networks which incorporates Lagrange’s theorem [2, 6]. A 2023 study used these ideas by proposing a neural network DNN. Combined with the tire magic formula, the DNN network was trained to estimate the parameters in the magic formula. After that, the tire force can be calculated [1].

The contributions of our study are as follows:

1) We proposed a combination of the two to build a tire force residual correction framework (TF-RCF) to improve the accuracy of vehicle tire force estimation.

2) In TF-RCF, we built the magic formula tire force model to have an initial estimation. The study used three neural networks, MLP, MLP-seq and MLP-mixer, to fit the residuals between the rule-based model outputs and the actual observations.

3) We tested the accuracy and the generalization ability of the TF-RCF. Results showed that the mean error is reduced from 134.9 N to 21.2 N. Meanwhile, the neural network had a good performance on the validation datasets.

2 Tire Force Residual Correct Framework

Fig. 1.
figure 1

Tire Force-Residual Correction Framework (TF-RCF)

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2.1 Tire Force Residual Correction Framework (TF-RCF)

Our study built a residual correct framework for tire force estimation. As shown in Fig. 1, the framework is constructed mainly by a rule-based tire force model and a neural network model. The rule-based model consists of a dual-mass vibration system with a magic formula. The neural network is based on the MLP, MLP-seq and MLP-mixer model. We trained the neural network to learn the residual between the real forces (derived from Carsim) and the rule-based model output forces. Finally the trained neural network can output the residual to correct the rule-based output.

In the TF-RCF, we added the 12-dimensional tire force information output from the rule-based model as the neural network model input. The experiment shows that this can reduce the training difficulty of the neural network and improve the residual estimation accuracy of the neural network.

2.2 Tire Force Model Based on Magic Formula

H.B. Pacejke tire model (magic formula) [4], is based on a large number of data summarized by the empirical tire model. Because only a uniform set of formulas is needed for the expression of tire forces, the magic formula has been adopted by a large number of tire manufacturers.

The general expression for the tire magic formula isFootnote 1:

$$\begin{aligned} Y_{x}=D-sin[C-arctan(B-x-E(B-x-arctan(B-x)))]) \end{aligned}$$
(1)

\(Y_{x}\) denotes the lateral force, longitudinal force, or backward correcting moment of the tire. The independent variable x is the lateral deflection angle or the longitudinal slip rate of the tire, respectively.

Our study used the two-mass vibration system for the wheels’ vertical loads. The vertical motion of the body, tire and ground are represented by z, \(z_{t}\) and \(z_{r}\). They satisfy the following differential equation:

$$\begin{aligned} m\ddot{z_{t}}=k_{t}(z_{r}-z_{t})+k(z-z_{t})+c(\dot{z}-\dot{z_{t}}) \end{aligned}$$
(2)

2.3 Neural Network Model Based on MLP, MLP-seq and MLP-Mixer

MLP, MLP-seq and MLP-mixer are used as the neural network model [9, 10].The detailed struture information of networks that used in the experiments is as follows: Input dimension: 26 and 38 (2 types). Output dimension: 12. Hidden dimensions: 1024. Hidden layers: 4. Sequence length: 3. MLP-mixer channels: 128. MLP-mixer layers: 4. The neural network was trained in Pytorch 1.12.1 and miniconda3. We used two loss functions: MAE (Mean Average Error) and variable weighted multi-task loss function. Adam is the optimizer. The learning rates for the weights of network and multitask loss are 1e-3 and 1e-5, respectively. The training batch size is 4 and the maximum epoch is 256.

$$\begin{aligned} Multitaskloss = \sum _{i=1}^{4}\sum _{j=x,y,z} \exp (-w_{i,j}) \cdot |\delta _{Fi,j} - \hat{\delta _{Fi,j}}| + w_{i,j} \end{aligned}$$
(3)

3 Experiments and Results

3.1 Experiment Setup

Our study used two methods to obtain dataFootnote 2:

  1. 1)

    Specific working conditions 6 (30000 sets): Design corresponding control sequences for different working conditions. These specific conditions are selected for the distributed electric drive vehicles, such as front and rear wheels rotating in the same direction and opposite directions.

  2. 2)

    Randomized control conditions 5 (21000 sets): Input random control sequences to the simulated vehicle. Our study set the random seed of the Gaussian random module for torque and angle control.The maximum frequency of the random control is 3Hz. In order to keep the vehicle stable, we set the front and rear wheels rotating in opposite direction.

We used two ways to divide the acquired data into training and validation datasets:

  1. 1)

    For data \([t,t-\varDelta t,t-2\varDelta t]\), data \([t-\varDelta t,t-2\varDelta t]\) is divided into the training set and the rest is divided into the validation set. The ratio of training and validation data is 2:1.

  2. 2)

    Classify the first 70% of the condition data as the training set and the last 30% as the validation set.

3.2 Performance Without Rule-Based Forces as Input

MLP, MLP-Seq and MLP-Mixer Performance.

The mean loss value of the original Condition 1 data is 289.10 N. The best validation loss value for MLP (256 epochs), MLP-seq (256 epochs), MLP-mixer (128 epochs) is 3.3 N, 3.8 N, 1.3 N. The accuracy of the tire force prediction is significantly improved. Also, we found that network’s performance for XYZ forces is different, which indicates the importance of using the variable weighted loss function.

We also obtained the original data of the carsim output. The network was also used to directly fit the original tire data. And the result show that the network trained with the residual dataset has better correction results than the original data in most cases. More specific results are shown in Table 1. In the Table 1, R means residual data and O means original data.

Performance with Rule-Based Forces as Input. During the experiments, we found that the MLP-mixer network is significantly overfitted in data of condition 1. To solve the overfitting problem of the network, we added the rule-based model output to the input of the neural network. Adding the rule forces to the network reduces the learning pressure of the network and the network’s difficulty in extracting the input information.

Our study set five comparison experiments for condition 5 (MLP-mixer): 1). 12d rule force+18d state+8d control, BAE: 4.3 N 2). 12d rule force+8d control, BAE: 5.4 N 3). 12d rule force+18d state, BAE: 7.2 N 4). 12d rule force, BAE: 5.3 N 5). 18d state+8d control, overfit. Results show that the neural network perform better after adding the rule force. There is no overfitting any more. Meanwhile, the comparison experiments further illustrate that the addition of rule forces as network inputs is responsible for the increase in the accuracy of the neural network.

Generalization Ability. Our study tested the generalization ability of network by random control data. Each the working condition data was divided into training and validation sets according to the first 70% and the last 30%. The sequence length was adjusted to 10. We applied the MLP-mixer network (38d). A dropout layer was also added to strengthen network’s stability. The dropout value is 0.3. Fig. 2a and 2b show the training and validation results of the network for random working conditions 1 to 5. The mean value of the original residual force in the validation set is 134.9 N and the best MAE on the validation set is 21.2 N. Due to the random control, the validation datasets are totally unseen for the network. The result can show a good generalization ability of the network.

Table 1. Detailed results for vehicle condition 1
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Fig. 2.
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MLP-mixer Performance in Random Control Conditions

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4 Conclusions

The study proposes a tire force residual correction framework (TF-RCF) consisting of a neural network model and a magic formula rule force model. The accurate tire force estimation can be used to build a more accurate dynamic model, which helps improve the accuracy in vehicle control scenarios. Our study has shortcomings: We haven’t obtain real vehicle tire force data at present. So we lack strong evidence to prove the TF-RCF can fit the data from the real car. This is a continuing direction for our study. In the acquisition of datasets, the specific working conditions and random control conditions set by our study still lack some systemic and rationality. In the future, we will improve the network architecture and obtain datasets in a quicker and more reasonable way.