Object-Oriented Modelling and Simulation of an ATV F. Donida, G. Ferretti, G. Magnani, M. Zampini Politecnico di Milano, Dipartimento di Elettronica e Informazione, P.zza L. Da Vinci, 32, 20133, Milano, Italy e-mail:
[email protected]
Abstract As a basic step toward the design of a remotely (wireless) guided robotic system, a dynamic model of an All-Terrain Vehicle (ATV) has been developed, and it is described in this paper. The modelling approach is based on the object-oriented modelling paradigm defined in Modelica, within the Dymola environment, allowing a fully modular approach to physical system modelling and featuring advanced symbolic manipulation techniques and numerical solvers for nonlinear differential-algebraic equations. The crucial steps of the model development were the description of the steering system and of the suspensions, and the modelling of the interaction among tires and terrain. A first virtual rider model has been also implemented, allowing to track both a steer angle and a target speed profile during different maneuvers. Moreover, as an essential feature of the rider model, the displacements of the rider have been also taken into account.
1
Introduction
As a basic step toward the design of a remotely (wireless) guided robotic system, a dynamic model of an All-Terrain Vehicle (ATV) has been developed, and it is described in this paper. The vehicle, equipped with a robot arm, should be capable of travelling over small and medium distances, scan the surrounding soil seeking things like mines or reactive shells, pick up objects, soil samples. Even when teleoperated, a feedback controller (autopilot or virtual rider) is needed to ensure the vehicle stability during motion in uneven and sloping terrains ([1], [2]). A main problem in stabilizing and controlling the motion of the overall system is the dynamic interaction between the motion of the arm and that of the supporting vehicle. The dynamic model described in this paper has been developed just to support the design of the autopilot and of the underlying servoactuators of the steering, brake and throttle commands. Several ATV models have been reported in the literature, see for example [3], [4], [5], [6]. The model described in this paper is based on the object-oriented modelling paradigm defined in Modelica, within the Dymola environment. Dymola is a state-of-the-art simulation tool, allowing a fully modular approach to physical system modelling and featuring advanced symbolic manipulation techniques and numerical solvers for nonlinear differential-algebraic equations (DAE) systems ([7],[8],[9]). The modelling of the steering system and of the suspensions has been first dealt with, requiring a careful definition of the joints in order to ensure mobility. The models have been easily assembled based on the Modelica Multibody library [10] and above all on the symbolic manipulation capabilities of the Dymola environment, which allow an efficient management of kinematic closed loops. As expected, another crucial point has been the modelling of the interaction among tires and terrain, which has a major impact on ride and stability properties of the vehicle [11], [12], [13]. A first virtual rider model has been also implemented, allowing to track both a steer angle and a target speed profile during different maneuvers. Moreover, as an essential feature of the rider model, the displacements
Figure 1: Degrees of freedom of the vehicle model. of the rider have been also taken into account. In the case of an ATV, these displacements seem to be better modelled as translational motions of the rider’s center of mass, as also reported on some ATV user manuals. The paper is organized as follows. Section 2 illustrates the modular approach adopted, based on the Modelica/Dymola environment, highlighting the crucial part of the model, constituted by model of the steering and suspensions systems and by the tire-road interaction model. Section 3 introduces the virtual rider closed-loop control systems and the description of the rider displacements. Finally, Section 4 ends the paper with some concluding remarks.
2
The ATV Model
The model is characterized by 15 d.o.f.: the chassis has the 6 d.o.f. of a floating body, 3 rotational d.o.f. are introduced for the steer and for the two front hubs, 2 translational d.o.f. for the front suspensions, 4 d.o.f. for the rear suspensions and for the rear hubs (see Figure 1). The suspensions are modeled as spring-damper systems, with non-linear characteristics: two for the front suspensions and two for the rear suspensions. The exogenous variables (inputs) of the model are the torque applied to the steer, the torque applied to the central differential, the variable selecting the traction method (2WD, 4WD and 4WD locked) and the forces applied to the four brakes. It must be pointed out that spreading the model over local force balances, thanks to the concept of acausal mechanical connector, saves the explicit assembly of the overall model and allows to easily account for gyroscopic and inertial effects.
Figure 2: Front arms and shock absorber assemblies (left) and the steer arm and EPS unit (right). From the modelling point of view the main difficulties were related to the modelling of the steering system and the tire-road interaction model, described in the following.
2.1
The Steering System
The steering system is equipped with an Electric Power Steering (EPS) unit, an electric motor acting at the steering column according to the steering input. The EPS is connected on the top to the handle bars through the steering stem and on the bottom to the left and right steering knuckles through the pitman arm and the left and right tie-rods. To model the steer-suspensions coupling it is crucial to consider the front arms and shock absorber assemblies. In Fig. 2 half of the front structure is shown, which includes 11 joints, 6 revolute (R1, R2, R3, R4, R5 and R6), 4 spherical (S1, S2, S3 and S4)) and 1 translational (the suspension). Note that 4 closed loops are formed: 2 are relevant to the suspension (R1-R4-R3) and the arms (R2-S1-S2-R3) and 2 are relevant to the steer: (R2-S1-S3-S4-R6 and R3-S2-S3-S4-R6). Two revolute joints (R1 and R4) connect the upper and lower side of the suspension, two other revolute joints (R2 and R3) connect the upper and lower arms to the chassis, one revolute joint (R6) connect the pitman arm to the lower flange of the EPS unit and one revolute joint (R5) is the hub bearing. The spherical joints S1 and S2 connect the upper and lower arms to the steering knuckles, the spherical joint S3 connects the steering knuckle to the tie-rod, while the spherical joint S4 connects the tie-rod to the pitman arm. The tree structure of the assembly has therefore 18 d.o.f. while the closed loop structure has only 3 d.o.f., one allowing the displacement of the spring-damper system, another transmitting the handlebar motion while the last simply allows the rotation of the wheel. The Dymola implementation of the structure is reported in Fig. 3. The left suspension loop block (top left) is connected to the handelbar joint R6 and to the rotational (steer) joint S1 z, representing the rotational d.o.f. of S1 around the vertical axis. For the sake of precision, the rotation axis of joint S1 z is not parallel to the vertical axis, the caster angle is 5 deg. To remove redundancies just one spherical joint S3 has been introduced, while only two axes of motion, specifically S4 x and S4 z, have been considered for joint S4.
Figure 3: Dymola models of the kinematic chain (left) and the suspension and arms loops (right).
Figure 4: Rear suspension assembly (left), rear view of the ATV (center) and Dymola implementation (right). For the same reason, neglecting the rotation of the knuckle, the suspension structure can be simplified as a planar one, with only 6 rotational joints with parallel axes, closing two planar loops: R2-S1 x-S2 x-R3 and R1-R4-R3. In Fig. 4 (left) a scheme of the rear suspension is shown. The assemblies are very similar to those on the front side, without steer and with no need of spherical joints, replaced with two single d.o.f. rotational joints (R5 and R6). As a consequence, the rear mechanical chain has only two loops: one (R1-R3-R4) includes the suspension and the other (R2-R3-R5-R6) the knuckle, as in the case of the front structure. The Dymola model is exactly the same of the previous subsystem, see Fig. 4 (right). The front arms and shock absorber assemblies are designed to ensure the best performance on all terrains by making the stroke of the suspensions and the transmission quite independent from the roll angle of the vehicle. This a very common peculiarity concerning the majority of the off-road vehicles. Thanks to this particular system, when changing the stroke, the wheels roll angle maintains nearly constant. Adopting the multibody approach and according to the data available in the service manual [14], each component has been singularly modeled, leading to a very accurate overall model. As a result, some detailed simulations have been performed, obtaining significant and specific results on the dynamic behavior of the ATV. The main simulation experiments performed on the ATV model were made to investigate three main aspects
Figure 5: Variation of the front structure at varying suspension stroke. related to the influence of the suspension stroke on the steer angle 1 , the steer-handlebar angles coupling, the distance between the wheels of the same axis (front or rear). In addition, some simulations have been performed in order to investigate the handling characteristics of the vehicle. The experiments have been first made using a model of the half structure and then considering the model of the whole ATV. Finally, for some of the previous simulations, the results have been compared with the experimental data. In a first experiment the behavior of the kinematic chain of the front structure has been investigated, in Fig. 5 some screenshots of the 3D animation are reported. A force has been applied to the wheel axis (not shown in the picture) causing a variation of the length of the left suspension. Various configurations have been reached, starting from totally compressed (top left), getting to normal load and finally with fully extended (bottom right). Even if the steer angle is maintained in neutral position, the wheel’s toe-in changes in a quite apparent way, while the camber angle is unchanged. Figure 6 shows the steer angle as a function of the handlebar angle for different working point of the suspension. On the graph the static measurements have been reported, the steer angle has been measured turning clockwise the handlebar from 0 to 55 deg. Comparing the different lines on the plot, some points have to be outlined. First, the relationship is clearly nonlinear and at least a third order polynomial is required to fit the curve. Second, for a null angular displacement of the bottom flange of the EPS unit there is an offset in the steer angle, corresponding to the “toe-in” (see Fig. 6). Third, considering that the static measurement revealed a linear relation between the steer angle and the handlebar angle, with a proportional coefficient of 0.8, the results of the simulations fit very well the experimental data. Figure 7 shows the results obtained from a set of simulations, where the ATV follows a sinusoidal trajectory at different speeds, see Fig. 8. The left and right steer angles of the front wheels are plotted as a function of the steer angle. Note that when turning left/right, the right/left suspension is subject to compression while the other (left/right) to extension. However, the variation of the amplitude of the two suspensions is different under compression and extension. For example, when turning right (negative steer angle) the left suspension is slightly compressed, maintaining a quite linear characteristic, while the right suspension extends considerably, showing a nonlinear behavior, according to the curves of Fig. 6. 1
In this paper, the steer angle is defined as the angle between the plane of the wheels and the plane of symmetry of the vehicle.
Figure 6: Steer angle as function of the handlebar angle for different suspension strokes.
Figure 7: Left and right wheel angles as a function of the steer angle, during a sinusoidal trajectory. “Without vehicle” refers to the linearized design characteristic considering half of the front structure.
Figure 8: Screenshot of the 3D animation of the previous experiment.
Figure 9: Neutral (dotted) and simulated (solid) steer angle with respect to the lateral acceleration (left) and understeering coefficient with respect to the speed (right). Finally, some simulations were performed in order to investigate the understeering/oversteering characteristics of the vehicle. In all this set of simulations the vehicle is driven at a constant forward speed V at various turning radii. The understeering coefficient Kus has been calculated along the simulations, as a function of the steer angle δ and the lateral acceleration ay [15]:
Kus =
dδ gL − 2 d(ay /g) V
(1)
where g is the gravity acceleration and L the wheelbase of the vehicle. The results are reported in Fig. 9. Note that the dependence of the simulated steer angles (left) with respect to the lateral acceleration is nearly linear and close to the neutral steer (Kus = 0), this means that the ATV maintains nearly the same handling characteristic for the same turn, even at different speeds. Moreover, considering the medium value of the derivative of the steer angle with respect to the lateral acceleration (right), one could calculate the medium handling characteristic depending only on the speed. The plot reveals that the ATV is slightly understeering for low speeds (Kus > 0) and becomes neutral by increasing the speed, see Fig. 9 (right).
2.2
The powertrain
The drivetrain is an important component to be considered for a correct modelization of the vehicle moving in various scenarios. The ATV considered in this work is a 4WD vehicle, it is endowed with a central differential distributing power between the front and the rear axles, and select between different transmission modes. When the vehicle is at rest, the rider can choose to drive the ATV as a 2WD, as a limited-slip 4WD or as fully locked differential 4WD-all vehicle, by pushing a button. The front and rear axles are equipped with open differentials allowing each of the driving wheels to rotate at different speeds, while supplying equal torque to each of them. The model of the powertrain is then made up of four main sub-components: the engine (input torque), the gear (introducing a gear-dependent ratio), the central differential and the front and rear axles differentials. All the differentials, central, front and rear, are supposed to be ideal and modeled with the following system of equations:
(ϑr + ϑl )ρ 2 = τr
ϑeng = τl
ρτeng = τl + τr
(2) (3) (4)
where ϑeng and τeng are the engine input pinion angle and torque, while ϑl/r and τl/r are angle and torque at the axles of the left/right wheel respectively.
2.3
The tire-road interaction model
The road model is essentially defined by a surface of equation f (x, y, z) = 0, from which the relevant normal unit vector can be computed as n = ∇f /|∇f |, and by the friction characteristics, which can be associated to the friction coefficient (see e.g., [16], [17], [18]) µ.
Figure 10: Reference frames for the wheel–road interaction. The tire-road contact point frame is defined by introducing four frames, shown in Figure 10: the hub frame, the wheel frame, the ideal contact point frame and the real contact point frame (these last frames are nearly overlapped in the figure).
The hub frame is attached to the wheel hub, with its origin in the middle of the hub, and rotates with the wheel around the yh axis. The origin of the wheel frame is the same of the hub frame, and also the unit vector yw coincides with yh , while xw is the unit vector associated with the rotational equivalent wheel velocity direction, obtained as the intersection between the plane orthogonal to yw and containing the center of the wheel frame and the road tangent plane at the ideal contact point. The unit vector zw completes the frame: yw = yh yw × n xw = |yw × n| zw = xw × yw The ideal contact point frame has the origin laying on the road plane, along the direction identified by zw , the unit vector zi coincides with the road normal unit vector n, the unit vector xi coincides with xw and yi simply completes the frame: zi = n xi = xw yi = zi × xi The location ri of the origin of the ideal contact frame is computed as: ri = rh − ∆z zw with rh being the location of the origin of the hub frame and with ∆z being the distance between the hub origin and the road plane in the zw direction. The ideal and real contact frames are aligned with each other and differ only in the origin position; the location rr of the real contact frame is displaced from ri (see e.g., [16], [18]) by the so-called tire trail, hence δx rr = ri + R r δ y 0 where Rr is the rotation matrix from the real contact frame to the absolute frame. In turn, the displacements δx , δy and δz have been computed as in [16]: δx = fw R sgn(vx ) δy = (∆z − R) sin(ϕ) δz = (∆z − R) cos(ϕ) where R is the wheel radius, vx is the forward wheel ground contact point velocity and fw is the rolling friction coefficient, which can be computed as: fw = A +
B C 2 + vx , p p
with p being the tire pressure and vx the wheel forward velocity, i.e. vx vy = RrT vr = RrT drr , dt vz
(again, note that the velocity has not been explicitly computed: the statement der() has been simply invoked). The quantities A, B, C are suitable non-negative parameters, and the wheel roll angle ϕ is given by ϕ = − arctan
Ty zw r Tz zw r
The forces arising from the tire-road interaction can be decomposed into a vertical force Fz , a longitudinal force Fx and a lateral force Fy . The normal force Fz can be computed as: dδz Fz = − Kel δz + Del zr dt with Kel and Del being the elastic and damping constants describing the tire elasticity properties. The longitudinal and lateral forces have been computed according to the Pacejka model [18], as highly nonlinear functions of the normal force Fz , of the longitudinal wheel slip λ, of the tire sideslip angle α and of the wheel roll angle ϕ: Fx = Fx (Fz , λ, α, ϕ) Fy = Fy (Fz , λ, α, ϕ) where 0 1 0 RhT ωh − vx Rω − vx λ = = vx vx vy α = − arctan vx R
with ωh being the angular velocity of the hub frame and ω its component along the hub axis. The Pacejka formulas are based on a semi-empirical model, and depend on a huge number of parameters estimated from data. The model has been implemented according to the results proposed in [19], where extensive tests on different tires have been carried out, so as to estimate the needed parameters via numerical optimization. An alternative simplified linear model has been also implemented, where the longitudinal force Fx is computed as a linear function of the longitudinal wheel slip λ, and the lateral force Fy is computed as a linear function of the tire sideslip angle α and of the roll angle ϕ. The tire relaxation dynamics has been also taken into account, according to [19], thus the sideslip angle has been filtered through a time-variant time constant: d˜ α dt τ
1 (α − α ˜) τ a + bvx + cvx2 = Kyα vx =
where Kyα is a coefficient derived from the Pacejka model and a, b, c are suitable parameters. At the beginning of this section, the longitudinal and lateral friction coefficients µx and µy have been introduced. As a matter of fact, these coefficients are employed in the model so as to describe different possible road surfaces, such as dry asphalt road, wet road, icy road and the like. According to the Pacejka-based force model given in [19], where this kind of description is not explicitly modeled, we used the friction coefficients as a scaling factor applied to F˜x and F˜y . Hence, the final expression of the longitudinal and lateral forces is
Fˆy = µy F˜y , Fˆx = µx F˜x . The friction coefficients values lie in the set µi ∈ [0.2, 1], where the value 1 represents high-grip asphalt road and the value 0.2 icy road (note that the friction coefficients can be defined as a function of the position: µ = µ(x, y)). It must be pointed out that this modelling choice cannot fully capture all the effects of road surface changes. As a matter of fact (see also [17]), the coefficient of friction does not act as a simple scaling factor on the longitudinal and lateral forces, but, for example, it also causes a shift of the peak of the longitudinal force as function of the wheel longitudinal slip. A force model which captures this behavior is the so-called Burckhardt model (see [17], [20]). On the other hand, though, this model is not rich enough to describe the complex dependency of Fx and Fy on the other variables, such as side slip angle and roll angle which are indeed crucial for motorcycle ATV motion. This is why we decided to adopt a very accurate force description and resorted to represent the road surface changes as a scaling factor. Finally, the self-alignment moment is also taken into account, which acts at the real contact point. This moment is computed as Mal = Mal (Fz , α, ϕ, λ), hence as a nonlinear function of the vertical load Fz , the wheel side slip angle α, the roll angle ϕ and the longitudinal wheel slip λ. This moment accounts for the fact that the lateral force, when acting at the real contact point between tire and road, generates a moment that rotates the tire in the direction of decreasing side slip angle. To complete the model and transfer the ground load to the ATV the balance of the forces at the hub frame must be set: Fh + Fˆx + Fˆy + Fz = 0 τh + (rr − rh ) × (Fˆx + Fˆy + Fz ) + Mal = 0.
2.4
Aerodynamics
As far as aerodynamic forces are concerned, the proposed vehicle model takes into account • the Drag Force, which is directed along the longitudinal axis; • the Lift Force, which is directed along the vertical axis. These forces are applied in a specified point called pressure center which is generally shifted forward with respect to the chassis center of gravity (see [16]). The drag force affects the maximum speed and acceleration values and it is proportional to the square of the forward speed. It is computed as 1 Fd = ρCx A˜ v2, 2
(5)
where ρ is the air density, Cx is the drag coefficient, A is the section of the motorbike area in the forward direction and v˜ is the vehicle center of gravity forward velocity. The force Fd is directed along the chassis longitudinal axis. The lift force (also proportional to the square of the forward speed) reduces the vertical load on the front and rear tire. It is computed as 1 Fl = ρCl A˜ v2, (6) 2 where Cl is the lift coefficient and all the other parameters have the same meaning as in (5). The force Fl is directed along the chassis vertical axis.
3
The virtual rider
In this section the virtual rider model is introduced. Two main control loops have been implemented, in order to follow a reference trajectory by specifying a given handlebar angle, and to track a speed profile along a specified maneuver. When driving an ATV, the rider has also to lean both laterally and longitudinally, moving the overall center of gravity. This makes driving an ATV similar to driving a motorcycle, even if, differently from the twowheeled vehicles, the ATV does not need any counter steering maneuver. More in detail, when entering a turn, the rider has to apply a torque to the handlebar and move the upper body toward the inside of the turn while keeping his weight on the outer footrest handlebar and leaning in order to ensure stability. This aspect becomes more important at high speed/turn radius ratio. In the same way, when riding on a climb or a hill, the rider has to move his center of gravity forward or backward. This behavior of the rider has been emulated through a moving mass, translating laterally and longitudinally with respect to the saddle along two translational joints.
3.1
Target trajectory and speed tracking
A PID control law is adopted to control the handlebar angle δh , using the torque τh applied to the handlebar as the control variable: Z h ¯ h h dδh (t) τh (t) = KP (δh (t) − δh (t)) + KI (δ¯h (σ) − δh (σ)) dσ + KD , dt h have empirically chosen. This where δ¯h is the reference handlebar angle, and the gains KPh , KIh and KD control loop essentially compensates for the handlebar torque disturbances arising from the reaction forces at the wheels.
A closed loop speed control has been also implemented through a PI control law: Z τV (t) = KPV (V¯ (t) − V (t)) + KIV (V¯ (σ) − V (σ)) dσ where τV is the traction (applied to the flange of the central differential) or braking torque (which can be either applied to the front wheels only or to both front and rear wheels after being properly split - see also [21]) needed to follow the target speed value V¯ .
Figure 11: Simulation scheme of the rider displacements.
Figure 12: Screenshot of the 3D animation of the vehicle making the same turn at the same speed with (right) and without (left) riders displacements.
3.2
Driver displacements
Differently from previous works ([9],[22]), where rotational joints were adopted, in this work the motions of the rider with respect to the vehicle have been taken into account by means of a couple of translational joints, along the vehicle longitudinal and lateral direction. In fact, in general the rider of an ATV has to move considerably with respect to the saddle in order to maintain the stability of the vehicle, and his motions are more similar to a couple of translations than to a couple of rotations. Of course, the new degrees of freedom need to be actuated in order to model the rider motions during a maneuver along a curve. In particular, the displacements of the translating mass along the lateral direction, CGy , and longitudinal direction, CGx , have been related to the steer angle δ, the forward velocity V and the forward acceleration ax as shown in Fig. 11. The lateral displacement CGy is defined by the product of a velocity-dependent gain K1 (V ) and the steer angle δ. The block F represents a low-pass filter on the steer angle, introduced to filter out fast steer angle dynamics. The longitudinal displacement CGx depends both on the steer angle and on the forward speed, through the gains K3 and K2 (V ), this latter being also velocity-dependent. The values of K3 and the functions K1 (v) and K2 (v) have been empirically chosen in order to avoid rollover, and could be designed to model different rider behaviors. Figure 12 shows a comparison between two different riders, making the same turn with the same speed, with (right) and without (left) rider displacements. The two screenshots show that the vertical forces at the
internal wheels during the turn with a rigid rider are smaller than the vertical forces obtained by applying the scheme in 11. This aspect could be also analytically assessed considering the rollover indicator ROI, which is an index of the rollover risk [23]: ROI =
F zl − F zr F zl + F zr
(7)
where F zl /r is the sum of the left/right normal forces (front and rear) at the points of contact of the wheel with the soil. This index can assume values from -1 to 1 and can be used to estimate the right/left rollover probability. As it can be observed in Figure 12, adding the d.o.f. of the (rider) mass improves the stability of the vehicle in making a curve. Specifically, the rollover indicator relevant to the screenshot on the left is about 0.8, while it is about 0.5 for the screenshot on the right. Thus, the rider translational degrees of freedom and the adoption of the scheme in Fig. 11 can improve the vehicle stability.
4
Conclusion
In this paper, the dynamic model of an ATV has been outlined, based on the object-oriented modelling paradigm defined in Modelica, within the Dymola environment, and some simulation results have been discussed. The model is currently being endowed with the model of the automatic steering system, based on a servomotor directly actuating the steer, and of its relevant control loops (current, velocity and position). In first preliminary results, the model allowed to assess the influence of the motion of the supported robot arm on the vehicle stability. Even at medium velocities it is necessary to coordinate the motion of the robot arm with the riding manoeuvres, in order to maintain stability and avoid overturning during trajectory changes. By comparison with experimental data gathered on a real ATV some discrepancies were observed with reference to the classical Pacejka model, even assuming a tire-asphalt contact. The main motivations are currently under investigation but, at a first glance, it appears to be necessary to account for a larger contact area between tire and soil and for the tire tread. Moreover the tire-soil contact needs to be accurately investigated, based on some experimental measurements and on a more specific tire-soil interaction model, such as the one considered in [24], where the soil compression has been taken into account.
References [1] W. El Messoussi, O. Pages, and A. El Hajjaji, “Four-wheel steering vehicle control using takagi-sugeno fuzzy models,” Fuzzy Systems Conference, 2007. FUZZ-IEEE 2007. IEEE International, pp. 1–6, 23-26 July 2007. [2] J. Boot, “Atv control regulating a 4wd/4ws autonomous guided vehicle,” Master’s thesis, Eindhoven University of Technology Department of Mechanical Engineering, 2004-2005. [3] F. Mancosu and D. Arosio, “Vehicle, road, tyre and electronic control systems interaction: Increasing vehicle active safety by means of a fully integrated model for behaviour prediction in potentially dangerous situations,” Industrial Electronics, 2005. ISIE 2005. Proceedings of the IEEE International Symposium on, vol. 1, pp. 337–342, June 20-23, 2005. [4] A. Chevrefils and S. Filizadeh, “Modeling and transient simulation of an all-electric all-terrain vehicle (atv),” Electrical and Computer Engineering, 2007. CCECE 2007. Canadian Conference on, pp. 401– 404, 22-26 April 2007.
[5] T. Bruns and E. Schafer, “Modelling and identification of an all-terrain vehicle,” International Journal of Vehicle Systems Modelling and Testing, vol. 2, no. 3, pp. 276–295, 2007. [6] K. A. A. El-Gawwad, D. A. Crolla, A. M. A. Soliman, and F. M. El-Sayed, “Off-road tyre modelling ii: effect of camber on tyre performance,” Journal of Terramechanics, vol. 36, pp. 25–28, January 1999. [7] S. Mattsson, H. Elmqvist, and M. Otter, “Physical system modeling with modelica,” Control Engineering Practice, vol. 6, pp. 501–510, 1998. [8] M. Otter, H. Elmqvist, and S. Mattsson, “The new modelica multibody library,” in 3rd International Modelica Conference, Link¨oping, Sweden, 2003, pp. 311–330. [9] F. Donida, G. Ferretti, S. Savaresi, and M. Tanelli, “Object-oriented modeling and simulation of a motorcycle,” Mathematical and Computer Modelling of Dynamical Systems, vol. 14, no. 2, pp. 79– 100, 2008. [10] M. Otter, H. Elmqvist, and S. Mattsson, “The new modelica multibody library,” in Proc. 3rd International Modelica Conference, Link¨oping, Sweden, November 3-4, 2003, pp. 311–330. [11] Miszklevitz and S. L. Defense, “A dynamic simulation of soil-wheel interaction,” Technical Information Center (DTIC), Tech. Rep., 1973. [12] B. Heimann, N. Bouzid, and A. Trabelsi, “Road-wheel interaction in vehicles: A mechatronic view of friction,” Mechatronics, 2006 IEEE International Conference on, pp. 137–143, July 2006. [13] N. Bouton, R. Lenain, B. Thuilot, and P. Martinet, “Backstepping observer dedicated to tire cornering stiffness estimation: Application to an all terrain vehicle and a farm tractor,” Proc. on IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1763–1768, 2007. [14] “YAMAHA YFM7FGPW SERVICE MANUAL,” Yamaha Motor Corporation, U.S.A., Tech. Rep., 2006. [15] J. Y. Wong, Theory of Ground Vehicles. [16] V. Cossalter, Motorcycle Dynamics.
New York: Wiley, 1991.
Milwaukee, USA: Race Dynamics, 2002.
[17] U. Kiencke and L. Nielsen, Automotive Control Systems. [18] H. Pacejka, Tyre and Vehicle Dynamics.
Springer-Verlag, Berlin, 2000.
Oxford: Buttherworth Heinemann, 2002.
[19] R. S. Sharp, S. Evangelou, and D. J. N. Limebeer, “Advances in the modelling of motorcycle dynamics,” Multibody System Dynamics, vol. 12, pp. 251–283, 2004. [20] M. Burckhardt, Fahrwerktechnik: Radschlupf-Regelsysteme.
Vogel Verlag, W¨urzburg, 1993.
[21] D. J. N. Limebeer, R. S. Sharp, and S. Evangelou, “The stability of motorcycles under acceleration and braking,” Proc. I. Mech. E., Part C, Journal of Mechanical Engineering Science, vol. 215, pp. 1095–1109, 2001. [22] G. Ferretti, S. Savaresi, F. Schiavo, and M. Tanelli, “Modelling and simulation of motorcycle dynamics for Active Control Systems Prototyping,” in Proceedings of the 5th MATHMOD Conference, Vienna, Austria, 2006. [23] N. Bouton, R. Lenain, B. Thuilot, and J.-C. Fauroux, “A rollover indicator based on the prediction of the load transfer in presence of sliding: application to an all terrain vehicle,” Robotics and Automation, 2007 IEEE International Conference on, pp. 1158–1163, 10-14 April 2007. [24] G. Ferretti and R. Girelli, “Modelling and simulation of an agricultural tracked vehicle,” Journal of Terramechanics, vol. 36, no. 3, pp. 139–158, June 1999.
