The need to find alternative urban mobility solutions for delivery and transport has led mobility companies to devote enormous resources for research-based solutions to increase vehicle safety. This paper documents a virtual approach to investigate the influences of different load states to the vehicle dynamic of light electric vehicle. A model basing on a three-dimensional
multibody system was used, which consists of five bodies. By applying methods of multibody modelling the generalized equations of motion were generated. To include the behavior within the contact point between road and vehicle a simplified tire models was added. The implementation of the equations allowed a first validation of the model via simulations. In a final modeling step the simulation results were interpreted in respect of plausibility. Afterwards, the model was simulated numerically to investigate different load states of the vehicle, by applying constant steering stimuli and variable velocities. In sum, the investigated model approach is useful to identify safety relevant parameters and shows the effects of load states to the vehicle dynamics. Furthermore, it behaves plausibly regarding general vehicle dynamics. These results prove the general usability of the model for the development controllers and estimators in driver assistances systems.
Vehicle Dynamics Multibody System Tricycle Rigid Model Numerical Simulation1. Introduction
In the recent past, electric light vehicles have regained importance as urban mobility solutions, what is confirmed by increasing unit sales in Europe [1] [2]. This kind of vehicle is especially interesting for the lightweight transportation sector because of the small amounts of space needed for parking and maneuvering operations.
This causes the need of intelligent, traffic-safety increasing systems to improve urban traffic. By just increasing mechanical stability and driving force, higher loads could be transported, but without respect to the vehicle safety and the vehicle dynamics.
Model based approaches offer a cost-effective opportunity for the development of overall light electric vehicle solutions. Such an approach must ensure a high degree of flexibility.
For instance, the positions, in which physical quantities are calculated within the model, should be adaptable to variable positions of sensors within the real prototype. This is due to measured quantities like the centrifugal acceleration varies by changing the sensors position within a vehicle. The modelling approach presented below allows adding any number of coordinate systems. Therefore, physical quantities acting in the points of origin of a system dedicated to a specific sensor can be calculated for an appropriate comparison with measurement data [3] [4]. Because the field of research engaged in dedicated modelling is still narrow it is our aim to broaden the foundation for further model-based developments of specific assistance systems.
The main objectives of this study are the development of a vehicle model, which enables quantity variation. Additionally, the specific goals are:
· A mathematical model for the description of the dynamics of a representative light electric vehicle.
· The model-verification by means of selected investigation maneuvers.
· The analysis of additional masses to the vehicle behavior.
2. Conceptual Reviews/State of Research2.1. Configuration of Three-Lane Vehicle System
Three-track small vehicles can have a wide variety of configurations. They can have two wheels in the front and one wheel in the rear (2F1R) or, conversely, one wheel in the front and two wheels in the rear (1F2R). In the case of three-track vehicles for cargo transportation, the area of an additional load can be located in the front area or in the rear area. The latter is the most common. Also, in many three-lane vehicles there is a tendency to lean in the direction of the curve, similar to a bicycle. This paper is not focused on such three-lane vehicles. However, Figure 1 shows the topology of a three-lane vehicle on which this paper focuses.
2.2. Review of Empirical Literature
The literature review below is divided into two main areas. On the one hand, an overview of the modeling approaches and the analysis focus of three-lane vehicles is given. On the other hand, a more general overview of modeling approaches for the investigation of the load states to the driving behavior is given.
The objective of modeling the dynamics of three-lane vehicles has a wide field of application with different focal points of investigation. In [5] a simulation model is described with the aim of investigating the structure vibrations with respect to the ride comfort. The simulation model based on the method of multi-body modeling. Due to the investigation aspect, however, the model has only one longitudinal degree of freedom. For this reason, this model variant cannot be used for an investigation of lateral movements. The same applies to [6]. Here, too, only the longitudinal degree of freedom is taken into account, with the focus of investigation on the electric drive train in terms of efficiency. The editors of the paper [7] focuses within his investigation to the influence of geometric parameters on the stability of various three-lane vehicle topologies. The roll over threshold is used for evaluation. The basis of the investigation is a highly simplified model that is based on static equations. Based on the lower stability of three-lane vehicles, some three-wheeled vehicles have a more complex mechanical tilt mechanism. The investigation of the stability on such vehicles is carried out in [8] and [9], among others. Here, models according to the Newton-Euler method are applied. Modeling focus is the representation of the inclination during cornering. The overarching goal of these works is to develop tilt controllers. A first general modeling approach to describe longitudinal and lateral dynamics for three-lane vehicles is provided by [10]. Here, a model for a 2F1R vehicle is developed from several simplified models using Newton’s and Euler’s equations. The tire model is represented by maps that generate different longitudinal and lateral forces depending on different ground conditions. Finally, some simulations are performed for plausibility investigations, focusing on different ground conditions. The model offers the possibility to shift the center of mass arbitrarily, which is generally not discussed further. A similar model is built in [11]. Based on the same modeling method, a simulation model is derived and developed here for the 1F2R vehicle. The objective of the development of this model was on the one hand the investigation of the tilting stability and on the other hand the model was used for the development of a torque controller for an electric motor. In relation to four-wheeled vehicles, in which there are model variants with a wide variety of complexities and levels of detail, the amount of modeling approaches or simulation models is relatively small, despite the potential to participate in the development of driver assistance systems. To complete the literature review with respect to the objective of the paper, some papers dealing with the influence of the loading condition will be presented below.
In [12], the influence of the load on the braking behavior of light transport vehicles is investigated. Methodically, simple static equations and several measurement experiments are used. The investigation of the influence of the height changes of the entire center of gravity, which can result from additional loading, is presented in [13] and [14]. Here again simple basic static equations are discussed. Due to the influence of the loading condition on the driving behavior, [15] and [16] already deal with methods to estimate the changed vehicle mass or center of gravity height. To summarize, the model in this paper differs from the other models in the way that this model has a high flexibility with the possibility of analyzing the effect of different loads on the behavior of different drive systems.
3. Research Methodology
This chapter describes a stepwise modelling procedure in which the successive steps cover different fields of mechanics. Foundation for the modelling is a physical model scheme as simplified description of the tricycles mechanics. It is designed as multibody system with discrete components for springs and dampers, which represent their ideal physical behavior [3]. Each body of the system has its own coordinate system. Based on this scheme the multibody systems kinematics of positions are formulated. Thereby, the absolute and relative positions of all bodies are described using vectors and rotational matrices. The kinematics of velocity and acceleration are successively calculated by deviation. Meanwhile, Jacobian matrices are generated, which map generalized quantities of the multibody system onto actual quantities of all bodies [3] [4]. For this kind of description, tools like coordinate transformations and formulas of relative kinematics are used [4] [17]. After the systems kinematics are known a description of the kinetics is derived. A free-body system is generated by the methods of sections as foundation for the kinetic description [17]. Hereinafter, the word “forces” is used summing up the word “forces” and “torques” as kinetic quantities. The free-body system contains all inner and outer forces per body as well as all constrained forces. Due to the known kinematic behavior, those forces can be expressed as function of kinematic quantities. For instance, a damper force depends on relative velocities of bodies connected by the damper. Afterwards, these forces are summed up in addition of inertial, centrifugal and coriolis forces producing newton and euler equations per body [3]. Merging the newton and euler equations of all bodies within a vector generates the newton-euler equation of the multibody system [3]. In ordinary multi-body systems with holonomic constraints the Newton-Euler-Equation can be transformed to the generalized equations of motion of the Multi-body system using jacobian matrices. The structure of the generalized equations of motion is comparable to the structure of Lagrange’s equations of the first kind [4]. They contain the degrees of freedom of the multibody system and their derivatives as variables [3]. To display the systems behavior, the generalized equations of motion can be solved numerically within an appropriate software environment.
3.1. Model Introduction
Foundation of the modelling procedure is an abstracted representation of the vehicle shown in Figure 2. It visualizes a multibody system with several characteristics defined below. The multibody system consists of five bodies. Each wheel is represented by one body. The bodies of the frame and the loading are represented by one mass point each. The indices m and J symbolize the mass and the mass moment of inertia per body. Each body includes a Cartesian coordinate system in its center of gravity. Furthermore, an inertial coordinate system is positioned at any place in space.
In this case the term “position” includes the term “orientation” as well. Besides these coordinate systems, several auxiliary coordinate systems are used. Subsequent to these basic specifications, the mechanical degrees of freedom are defined, which are marked by violet arrows and are listed in Table 1. The sense of rotation for the rotary degrees of freedom is clockwise in direction of the double-arrows.
To ensure clarity, the discrete components mentioned in the previous chapter and all input variables are shown in Figure 3. Several assumptions are made to generate this abstract model of the tricycle. For instance, the geometrical parameters are neglect. The Rolling motion refers to an artificial rolling axis, which is located at road level. The pitching motion is neglected due to a focus on lateral dynamics. The masses of the wheels are assumed to contain the masses of motor, braking system and tire components. The frame mass sums up the masses of all vehicle components fixed to the frame. Furthermore, the frame and the wheels are assumed to be rigid. Moreover, the model is valid only for continuous road-contact. A loss of contact can be modelled additionally.
3.2. Kinematics
In this section the description of the models kinematics is given. It consists of the kinematics of position, the kinematics of velocity and the kinematics of acceleration. These three areas of kinematics are deduced successively. They are exemplified by means of the vehicle-fixed coordinate system V. This is because the matrices of other bodies reach impractical large sizes and the mathematical procedures are similar.
In the first instance the kinematics of position are described. For that purpose absolute and relative positions vector and rotational matrices are defined for each body respectively coordinate system, which represent functions of the degrees of freedom. From a mathematical point of view, the degrees of freedom represent a set of generalized coordinates, which are now summarized in the following vector of the generalized coordinates [17].
q _ = [ X V Y V Ψ Z b o d y δ φ γ F γ R L γ R R ] T (4.1)
Degrees of Freedom of the multibody system
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