Vehicle crash test is considered to be the most direct and common approach to assess the vehicle crashworthiness. However, it suffers from the drawbacks of high experiment cost and huge time consumption. Therefore, the establishment of a mathematical model of vehicle crash which can simplify the analysis process is significantly attractive. In this paper, we present the application of LPVARMAX model to simulate the cartopole collision with different initial impact velocities. The parameters of the LPVARMAX are assumed to have dependence on the initial impact velocities. Instead of establishing a set of LTI models for vehicle crashes with various impact velocities, the LPVARMAX model is comparatively simple and applicable to predict the responses of new collision situations different from the ones used for identification. Finally, the comparison between the predicted response and the real test data is conducted, which shows the high fidelity of the LPVARMAX model.
In the past few decades, as one of the major concerns in the automotive industry, the vehicle crashworthiness has been attracting exceptional attention all over the world. Before appearing on roads, each car must undergo a series of crash tests to be verified whether they satisfy the safety requirements and conform to safety standards set by the road safety organizations or rate programs such as Euro NCAP or National Highway Traffic Safety Administrations (NHTSA). In terms of the crash tests, they are exceptionally costly and complicated due to the need of appropriate facilities, qualified staff, and dataacquisition system. Moreover, the proper arrangement of measuring devices (e.g., accelerometers and cameras used to record the crash event) or precise positioning of the research centers has also added the complexity of such crash tests. Thus it would be highly attractive if the overall car performance can be predicted and assessed without the need to execute numerous fullscale crash experiments, which renders the proposition of a mathematical models with accurate inputoutput behavior as the real tests to be of great interest.
Recently we can distinguish two main approaches of vehicle crash modeling: finite element method (FEM) simulations and lumped parameter modeling (LPM). FEM is considered as the most thorough computational tool with detailed insight into the vehicle crash modeling and thus has been widely employed; refer to [
On the other hand, for the modeling of complex systems, the model identification based on the data training or estimation can be an alternative to the traditional modeling by using physical laws [
A significant contribution to databased approach in modeling automobile crash was made in [
Recently, the research scope of crashworthiness has been focused on defining a dynamic vehicle crash with changing parameters according to the changeable input such as initial impact velocity [
The structure of this paper is organized as follows. The detailed introduction about the fullscale vehicletopole collision is presented in Section
In this section we present cartopole collisions with different initial impact velocities so as to test if the model proposed is capable of representing different crash scenarios.
The car tests were subjected to impact with a vertical and rigid cylinder. During the tests, the acceleration was measured in three directions, that is, longitudinal, lateral, and vertical. The yaw rate from the car’s center of gravity was also recorded. The acceleration field was 100 meters long with two anchored parallel pipelines. The pipelines have a clearance of 5 mm to the front wheel tires. The force to accelerate the test car was generated using a truck and a tackle. The release mechanism was placed 2 m before the end of the pipelines, and the distance from there to the test item was 6.5 m. The car was steered using the pipelines that were bolted to the concrete runaway. The experimental scheme is shown in Figure
Scheme of crash test.
As illustrated in Figure
Obstruction.
During the test, the acceleration at the center of gravity in three dimensions (
Cameras layout.
A 3D accelerometer was mounted on a steel bracket close to the car’s center of gravity, and it was fastened by screws to the car’s chassis. Data from the sensor was fed to an eightchannel data logger and subsequently sampled with a frequency of 10 kHz. The memory was able to store 6.5 s of data per channel. The velocity of the car was checked by an inductive monitor. It was directed towards a perforated disc mounted on a wheel on the right side of the test car. Figure
Car’s deformation.
Steps of the experiment recorded by the highspeed camera.
All the tests are central impact; thus, as already mentioned only the pulse recorded in the longitudinal direction (
Vehicle kinematics with
Vehicle kinematics with
Vehicle kinematics with
Vehicle kinematics with
There are two types of mathematical modelings of real world systems that are commonly used [
Analysis of the autoregressive model with moving average exogenous input (ARMAX) was done according to [
The ARMAX model can be simplified as
As stated in Section
In the following, we will adopt a LPVARMAX model which can be described as
The following instructions specify the steps of such identification.
Suppose that we have a set of groups of vehicle crash data, all of them are in the type of pole collision but under different impact velocities
The
From the resulting LPV model, the output
When the functions
It should be noted that the LPVARMAX model is linear with respect to
It should be noted that the recorded acceleration signal in real car crash test is actually timeseries data. Thus in the established LPVARMAX model (
In the previous literature, the most commonly used method is to establish an ARMAX model based on a group of vehicle crash data obtained under one specific impact velocity, which may not be applicable to other situations with different impact velocities. Thus it is desirable to know the models in the whole relevant volume of initial velocity instead of one (or several) specific values. Moreover, since the impact velocity can be arbitrarily valued in practice, the corresponding local model can be obtained easily with aid of the LPVARMAX model once we know the specific initial velocity, which is the advantage of our approach.
In the next section, we will present the simulation of vehicle crash to test the efficiency proposed in this paper. In the simulation, the error noise is Gaussian, and the lumped LTIARMAX models are identified via the MATLAB Identification Toolbox.
In this section, the parameters of the LPVARMAX are estimated. We have four data sets corresponding to four initial impact velocities at our disposal, of which three data sets will be used to obtain the LPVARMAX model while the rest data set will be used for the purpose of verification. If one data set under a specific initial impact velocity
With this in mind, in order to represent with a simplified model of the vehicle crash that can be applicable to the collisions with different initial impact velocity, the LPVARMAX model should be well constructed. It is assumed that the time series data of acceleration during the collision can be well represented by a LPVARMAX
Parameters of ARMAX models for different impact velocities.
Impact velocity 
Parameters of corresponding ARMAX models 












ARMAX simulation results for vehicle with
ARMAX simulation results for vehicle with
ARMAX simulation results for vehicle with
From Table
LPVARMAX perdition for vehicle with
Simulation error by LPVARMAX model for vehicle with
The simulation results show that the established LPVARMAX (2,0,2) model can not only reproduce the crash pulse signals which were utilized for the identification, but also predict the kinematic responses of crash pulses with new impact velocity which were not presented in the creation stage. Thus, it can be proved that the LPVARMAX model allows us to obtain the accurate estimations of vehicle’s acceleration signal under different initial conditions and therefore the integration of acceleration which closely resembles the original ones.
In this paper, a new method to approximate the modeling of the vehicle’s acceleration pulses under different impact velocities is investigated by using LPV approach. The LPVARMAX model allowing to approximate the behavior of vehicle crash with various initial conditions is investigated. Specifically, the parameters of the LPVARMAX model are assumed to be functions of the initial impact velocity which may produce significant influence on the acceleration pulse during the crash. By interpolating the LTIARMAX models obtained from the training data sets, the overall LPVARMAX model can be achieved which can predict accurately the kinematic responses of vehicle crashes with various impact velocities. Simulation results illustrate that the LPVARMAX model can exactly reproduce both the new vehicle crash scenarios and those used for training. Future works may include taking the masses of vehicles into account as well since the masses can affect the acceleration pulses during the collision to some extent. But this attempt may increase the complexity since it will result in the coupling of the parameters in the LPVARMAX identification process, and thus it remains further investigation.