Diagnosis and influence of dynamic parameters on the motion of the electrical kart NASSER Habib(1), NAAMANE Aziz(1), Nacer Kouider M’SIRDI(1) (1) LSIS ; Laboratoire des Sciences de l’information et des Systèmes UMR CNRS 6168. Domaine Universitaire St Jérôme, Av. Escadrille Normandie-Niemen; 13397. Marseille cedex 20. France www.lsis.org . -

Abstract: In this article we study the dynamics of an electric kart in different situations. Even we test the effects of variations of parameters such as longitudinal velocity and the added mass by the driver and the bicycle model can be used in Bond Graph. Then we will develop a control system to follow a path. This has to avoid over or understeering. Finally, we include some diagnosis procedures based on filters and estimators. Stability criterion will be used to detect critic turning situations.

Keywords: Dynamics sensitivity, Bond Graph, bicycle model, filters and estimators, stability criterion, over or understeering detection. 1. Introduction Road safety is a crucial point in the functional description of vehicles and requires more and more incorporation of intelligent, modular and communicating systems. The main goals are: driver assistance, the decision in difficult cases, and alarms to improve the vigilance of driver. To meet this need, the whole world made several projects in order to decrease the large number of road accidents (SmartSenior in Germany 2009) [1] and search for other renewable and inexhaustible energy (Toyota Prius 2009) [2]. The exploitation of modern tools of nonlinear system, the development of new methods for diagnosis, the enormous progress of electronic components and the quality of information from onboard observers, make the problem of driving assistance largely feasible. In order to model the motion of the car, we can use either 4wheels model or bicycle model in this paper we will focus on the latest one because it is less complicated and very useful in control. Certainly, there are many critical situations which need a specific study such as brutal braking, oversteering and understeering. In this paper we observe the influence of parameters variations on the model, also we will develop a control system. Finally, to detect over and understeering situations, we present two ways for diagnosis using residues [3] and stability criterion. Application will consider the prototype we have developed (an electrical kart) in LSIS [4]. 2. Electrical kart modelling: 2.1 vehicle description: The electrical kart is a cross-country car model equipped with 4 batteries en serial connection with brushless motor. The transmission of movement is carried out with a system pulley belt. The steering system adopted for this vehicle is rather simple, as well as a traditional brake. 2.2 kart modelling: Assumptions: The aerodynamic forces are neglected. The rear wheels can not rotate around the z-axis.

The longitudinal speed Vx is considered constant and very large compared with constant lateral velocity Vy and the yaw rate ωz. We consider only small steering angles. To simplify the analysis, we neglect rotation around x-axis. The bicycle model is a simple system of 2 d.o.f. developed by [5]. It will be used to symbolize the transverse velocity and the yaw rotation of the kart (figure 2.1). As shown in figure 2.1 in the kart dynamics the front and rear pair of wheel can be replaced by one producing the average effect.

Figure 2.1: bicycle model In this section we will try to rewrite all the equations describing the model above, and then we start by applying the Newton-Euler equation to the kart: (1) (2) Drift angle:

(3)

Transverse forces: (4) The radius of curvature is: (5) For the pneumatic motion we can use Pacejka (Magic formula) as in [6]: (6)

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Where i= {av, ar} and a4, a3 are identified on dry ground. The normal forces and resistances are:

The bicycle model gives a good representation for the transverse behaviour of the vehicle at low speeds, and it is especially easy to implement.

(7) 3. Influence of parameters variations: Let us analyse the sensitivity of the dynamics when changing some parameters like added mass and Vx. In the following tests the input confined to the steering angle of front wheels is chosen, as a half period of sine with amplitude 0.0348 rad or 2 °, and duration 3.08s

(8) (9) (10) By following the procedure proposed by [7] we obtain the bicycle model on Bond Graphs (figure 2.2). The transverse efforts are generated by the rolling resistances Rav and Rar. Bond Graph is a graphical description that has been made in order to obtain a mathematical model of our dynamic system. It shows the energy flow between components of the kart. I J

1 wz

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Figure 2.4: Input for trial bicycle model

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3.1 Influence of longitudinal velocity: Due to actions of the pilot, slope and wind action, the longitudinal velocity can change. We stabilize all the others parameters in our test. Each simulation uses a different and constant Vx= 5, 7, 9 and 12 m/s respectively for tests 1, 2, 3 and 4.

AV

Fav

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Figure 2.2: bicycle model on Bond Graph 2.3 Simulation results on 20-Sim: As first simulation based on the dynamic and geometric

As shown in figure 3.1 below, we propose that the rear steering angle was neglected in our case.

parameters specific to kart we took as effort of the front wheel is about MSe = -

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This force generated by a second order model. The figure (2.3) shows that we are in the case of a low turn around δ=2° and purely circular and part will be useful in diagnosis based on filters and estimators.

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Figure 3.1: bicycle model with null rear steering angle The equation that describes the transverse acceleration of center of gravity can be written: (11) Figure 2.3: simulation of model

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We notice that more the longitudinal velocity increases, more the transverse acceleration becomes important when turning (fig 3.2).

Time (s)

Figure 3.5: variation of yaw angle Time (s)

Figure 3.2: transverse acceleration The velocity (Vy) from the center of gravity is represented by the following integral [8]: (12)

3.2 influence of added mass: Using the module designed for virtual reality applications in CAD/CAM Software Catia, we analyze the variation of the position of center of gravity in terms of the mass added by drivers who do not have the same weight (Each time one driver). These data are classified in the following table:

From figure 3.3 we observe that the transverse velocity starts to decline then it becomes negative in varying Vx. Also, the increase of the longitudinal velocity has a great influence on the trajectory of the model, it causes a larger variations.

Throughout this study we keep Vx = 5m/s. As the mass added by the weight of the driver is significant regard to the mass of the kart (unlike a normal car), the increase in mass causes a longer course and the kart takes a turn looser (figure 3.6).

Figure 3.3: transverse velocity The yaw acceleration in Figure 3.4 increases with the longitudinal velocity, which implies a tighter turning. In the same way, the yaw velocity increases.

Figure 3.6: displacement

Figure 3.4: yaw acceleration In figure 3.5 we find that Vx has a significant influence on the yaw angle of the kart that can explain the advantage of the vehicle to turn. Finally, the variation of longitudinal velocity may distort the motion of vehicle mainly in turning which causes over or understeering.

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The centre of gravity moves backward when increasing the mass, then the transverse velocity decreases. For the same reason, the lateral acceleration decreases slightly with the extra mass (Figure 3.7 zoom). Practically, the addition of mass and the variation of center of gravity influence only few the yaw rate (Figure 3.8) of the electric kart especially the center of gravity moves backward. Once the turn is completed, Vy goes to zero, the yaw angle remains constant and the trajectory becomes a strait. Secondly the added mass only affects the final value of the yaw angle.

Vx= cos (δ) Vx (G) – sin (δ) Vy (G) Vy= sin (δ) Vx (G) +cos (δ) Vy (G)

(13) (14)

To control this type of avoidance, an input (steering angle) of the following form is applied (see figure 4.2) sum of two periods sinus inverted and the amplitude equal to 1° either 0.0174 rad).

Time (s)

Figure 3.7: lateral acceleration

Time (s)

Figure 3.8: yaw rate Figure 4.1: path obstacle avoidance Finally, looking at the influence of lateral forces we can say that variations in the position of center of gravity and in the added mass have a small influence on the front lateral force. As the position of center of gravity recedes by adding mass, we note that the rear lateral force of the kart increases and hence has important influence (fig 3.9).

In this context a model developed in Matlab / Simulink is used to calculate the transverse velocity, the yaw angle, angular velocity and the transverse displacement as a function of longitudinal displacement which is the trajectory of the kart in real time, according to the form of input. As model we used the bicycle model developed in section 2.

Time (s)

Figure 3.9: rear forces The weight of the driver in our case has a big influence in the behaviour of the electrical kart thus we must take it into account in order to obtain motion as real as possible and to avoid risks. 4. Obstacle avoidance. To avoid an obstacle, we need a tracking trajectory and a control system able to produce a torque used to adjust the yaw rate [9]. The choice of such adjustment is influenced by the fact that in vehicle, a PI corrector is implemented to follow the trajectory using different features. 4.1 Proposed control: Trajectories: Path obstacle avoidance in the form below (figure 4.1) is derived from the following equations giving the desired velocities:

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Time (s)

Figure 4.2: input to avoid obstacle

The controller and estimation of drift angle: An important step therefore is to apply a tracking path to purposes of driving assistance. The following closed loop describes a regulation of the trajectory (Figure 4.3) which allows the system to follow a bicycle path reference (wz_réf) given by the bicycle model. From the control path, we tried to propose a

control system is active safety equipment designed to improve the trajectory control of vehicle.

To ensure efficacy, necessary we detect many parameters and state of the electrical kart which is the main goal of the next section. 5. Diagnosis: The reasons why the industry was interested in the safety of systems are security (human and environmental) and the need to increase productivity and reliability. This resulted in the appearance of surveillance methods whose purpose is the detection and isolation of faults (or Fault Detection and Isolation (FDI)). As a first step we developed observers based on filtering estimators in order to make diagnosis on the driving situation then we try to detect over or understeering by some stability criterion.

Figure 4.3: A control System A control system generally contains two parts (Fig 4.3) PI controller is already processed and another part for compensation. As the latter case, the input of our system is the yaw rate of the kart and the compensation is based on the estimation of the drift angle. One way to calculate this drift angle is to rewrite the problem as follows figure 4.4: d= arctan (Vy/Vx) (15)

5.1 filter estimator for bicycle model: The basic principle of the residual generation using observers is to make an estimate of output from the quantities accessible to measurement. The residual vector is then constructed as the difference between the estimated output and the measured output. The residues generated are generally not zero in reality, because of noise and uncertainties [10]. According to the Bond Graph model [11] the state vector is written as follows taking into account that the integral causalities:

(16)

So the two variables that can be measurable in our bicycle model are the transverse velocity and yaw rate of the kart (figure 5.1). Time (s)

Figure 4.4: estimation of drift angle We observe that drift angle follows the form of steering angle. 4.3 Simulation and results: We found (Figure 4.5) in paths that despite the proposed control there are still areas tracking where problems remain. We shall develop a compensation more advanced than that proposed above (based on proportional representation). Figure 5.1: Block diagram of the estimators As for the simulation of these filters, we have developed blocks that generate the two desired residues as shown in figure 5.2. We notice that without faults the two residues converge quickly to zero else if we detect a gap between measured outputs and estimated one, we detect abnormal operation of process.

Figure 4.5: trajectories

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From this parameter we will be (in certain limits) able of detecting the condition of the state of turning: tends to infinity (equal powers of directors If front and rear axles as ) we are in normal case; If If

> 0 then a case of understeering;