© AFM, EDP Sciences 2017

1 Introduction

Compared to other types of vehicles, the two-wheeled vehicle (TWV) remains a particularly dangerous means of transport due to the inherent instability of this type of vehicles. Several works in the literature reported in [1] focus on increasing the safety and the comfort of the TWV’s rider. In this context, some of these works [2–6] deal with increasing the maneuverability and the handling performances to get a more safe and comfortable TWV. Maneuverability or handling describes the TWV’s ability to execute complex maneuvers and the facility with which the rider is able to perform them.

The evaluation of the level of handling can be done through computer simulations or through experimentation. The level of handling is obtained for any maneuver by calculating the handling indices that correlate the input and the output values of the TWV. In fact, the TWV can be assumed as a system with some control inputs such as steering, driving and braking torques and some measured outputs such as the kinematic and dynamic states with an objective to browse a well-defined path.

In the literature the majority of works [2,3] suggest that the steering torque (rider input) and the resulting roll motion (TWV output) should be the base of a good indicator for the handling evaluation.

In this context, modeling the system formed by the TWV and the rider presents the first step in order to obtain an efficient tool able to predict its behavior. One of the approaches to obtain these models is to use software tools, e.g., ADAMS/View, SimMechanics. However, these software tools do not state the equations of motion in an explicit form. To have a better insight of the physics of the system, the analytical approach is more effective. For this purpose, a new model of a TWV is proposed based on a robotic approach and including the interaction with road. The tire–road interface is modeled using the Magic Formula (MF). The developed model will be validated using the ADAMS software and will be used to evaluate the handling performance of a TWV model.

This paper is organized as follows: Section 2 gives an overview of the state of the art in modeling of the TWV and a description of the developed model. Section 3 deals with the use of the MF in modeling the tire–road interactions. Next, in Section 4, we present the developed model using the Matlab Simulink toolbox and the validation process using the ADAMS software. Section 5 deals with the evaluation of the handling performances. A description of the handling indices used as well as the performed tests maneuvers are presented. In Section 6 the developed model is used to study the influence of the geometric parameters (wheelbase and caster angle) on the handling performances. Section 7 is devoted to the conclusions and perspectives.

2 Modeling of the TWV

2.1 State of the art

In the literature, different types of modeling were presented using several approaches. The number of works dealing with the modeling of TWVs is relatively low compared to those dealing with four-wheeled vehicles. One of the first models tried to model a bicycle [7]. It was made of two rigid bodies linked via the steering mechanism. Sharp [8] was the first one who integrated the tires forces in a TWV model. Other works incorporated more parameters to model the suspension systems and for a more complex tire–road interaction models [9–11]. In order to study specific behaviors such as acceleration and braking, we will deal with model with an increasing complexity. An interesting model of a TWV was presented by Cossalter [12], which has the highest number of degrees of freedom (11 DOF).

The developed model is inspired from the works of Cossalter [12], Khalil and Kleinfinger [13] and Maakaroun [14]. This model considers the TWV as a mobile robot. The proposed TWV model has 11 DOF: the position and the orientation of the chassis (6), the front and rear suspension travel (2), the spin of the front and rear wheels (2) and the steering angle (1) (see Fig. 1).

Compared to the model presented by Maakaroun [14] where the steering axis is vertical and the trail is zero, the proposed model takes into account the caster angle ε and the trail, which have a great effect on the aligning of the front tire [1]. Another parameter is considered in our model, which is the camber angle γ (inclination angle of the wheel plan to the road) (see Fig. 2). The influence of this parameter is added to the expression of the lateral force for both tires. In fact, in the proposed model, the tires–road interaction is modeled using a TWV version of the Pacejka Magic Formula tire model [15].

The proposed model can consider the most important features of a TWV including the tires dynamics. It is easy to use for control and it can be used to predict the dynamic behavior and to analyze the handling quality and the stability of a TWV.

Fig. 1 DOF of the TWV.

Fig. 2 Geometric parameters: trail, caster and camber angle.

2.2 Geometric description

The TWV is considered as a mobile robot with a tree structure (Fig. 3). The chassis is the mobile base and the two wheels are the end effectors. Note that the rider body is considered as a part of the chassis.

• C₁ is the mobile base;

• C₃ is the handlebar and the sprung part of the fork;

• C₄ is the unsprung part of the fork;

• C₉ is the unsprung part of the rear wheel;

• C₆ and C₁₁ are the front and the rear wheels, respectively;

• C₂, C_2′, C₅, C₈, C_8′, C₁₀, C₇ and C₁₂ are virtual bodies used to define intermediate frames [10].

The Geometric model is obtained using the Modified Denavit–Hartenberg (DHM) approach developed by Khalil and Kleinfinger [16]. Each body C_j is connected to the previous one C_i with a joint (Revolute or Prismatic). According to this approach the frame R_j(O_j, x_j, y_j, z_j) attached to the body C_j is defined as follows: (note i = a(j) the index of the antecedent body of C_j and k = s(j) the index of the successor body of C_j):

• the z_j axis is along the axis of joint (j);

• the x_j axis is along the common normal between z_j and one of z_k;

• the y_j axis completes the base (x_j, y_j, z_j);

• another axis u_j is built on the common normal between z_i and z_j in the case of a tree where x_i is not perpendicular to z_i and z_j.

The geometry of the TWV is defined by the following parameters (γ_j, b_j, α_j, d_j, θ_j, r_j) defining frame (j) with respect to its antecedent frame (i) as described in Figure 4. The resulting articulated model of the TWV is shown in Figure 5.

Note that R₇ and R₁₂ are two frames attached, respectively, to the front and the rear wheel–ground contact points. The DHM geometric parameters of the tree structure of the TWV are shown in Table 1.

The transformation from the frame R_j to R_i is done using the homogenous transformation matrix defined as follows: (1)

The motion of the TWV is described by a vector (q) formed by 11 generalized coordinates in an Euler–Lagrange mixed description: (2)

• (x, y and z) and (θ, Φ and ψ) represent, respectively, the position and the orientation (roll, pitch and yaw) of the mobile base with respect to the reference frame;

• q₄ and q₉ are, respectively, the front and the rear suspension displacements;

• q₆ and q₁₁ are the angular positions of the two wheels with respect to their spin axis;

• q₃ is the steering angle.

The advantage of using the DHM technique is that it is applicable for any types of structure also for mobile robots as our case (TWV). This method allows calculating the symbolic expression of the geometric, kinematic and dynamic models with a minimum set of differential equations. In our study a simply full non-linear dynamic model of a TWV is derived using this approach. It offers the possibility of implementation of a strategy of control or other types of simulations.

Fig. 3 Multibody tree structure of the TWV.

Fig. 4 Geometric parameters according to DHM.

Fig. 5 Articulated model of the TWV.

2.3 Dynamic model of the TWV

The inverse dynamic model is obtained from two recursive equations using the algorithm of Newton–Euler [17] (see Appendix A). This model computes the joint torques as a function of the joint coordinates, speeds and accelerations as follows: (3) where [A] (11 × 11) is the system generalized inertia matrix; H (11 × 1) is a vector regrouping the centrifugal forces, the effect of gravity, the Coriolis forces and the external forces and moments; and q,  and are, respectively, the joint coordinates, speeds and accelerations including the variables of the mobile base.

In this model, for the joints (4) and (9) we add the effects of stiffness and damping of the front and rear suspension systems.

In order to model the contacts between the two wheels and the road, two constraint equations are added to the model. These constraints express the fact that vertical displacement of the contact points, relative to the ground, is null. (4) (5)

The full inverse dynamic model with constraints becomes: (6) where λ is a (2 × 1) vector formed by the normal reaction forces at wheels–road contact points and [J] is a (2 × 11) matrix corresponding to the two constraints.

3 MF modeling

3.1 State of the art

Tire modeling is an important task while investigating the TWV handling stability and dynamic behavior. Indeed, it is through the tires that the TWV interacts with the road. The forces generated between the tire and the road surface largely determine the movement of the vehicle [15]. It is essential for any study of vehicle dynamics to study these forces.

The forces in the contact area are difficult to determine and several works in the literature are still investigating the best way to model them. Several of these models are based on the physics of the contact, e.g., Dugoff [18], Gim [19], Brush [20], and Kiencke and Nielson [21]. The model presented by Pacejka [15] is an empirical one and it is based on the MF. The Pacejka model is the most commonly used model by tires and automobiles manufacturers. This model uses several parameters, which are identified based on experimental data.

3.2 Basics of the MF

MF tire models are considered as the state of the art for modeling of the tire–road interaction forces in vehicle dynamics applications [22]. This model has proven to be applicable to the TWV tires with inclination angles to the road up to 60° [22].

The inputs for the MF consist of the wheel normal load (F_z), the longitudinal and lateral slip (κ, α), and the camber angle (γ). The outputs are: the longitudinal and lateral forces (F_x, F_y) and the overturning, the rolling resistance and the aligning moments (M_x, M_y, M_z).

In this work the resulting wrench is formed by the longitudinal force, the lateral force, and the aligning moment (Fig. 6).

The longitudinal behavior of the tire is characterized by the relation between the longitudinal force and the relative speed of the tire to the road. The longitudinal slip at the tire–road interface is calculated as follows [22]: (7)

The slip angle can be calculated as follows [22]: (8)

Fig. 6 Forces and moments acting on a tire at the wheel–road contact point [22].

3.3 MF equations

In this work, the MF will be used in a pure slip condition to calculate the steady state tire forces (longitudinal and lateral) and the aligning moment. For a pure slip condition there is no coupling between the longitudinal and the lateral forces. A pure lateral slip condition occurs during cornering with a free rolling tire, whereas the longitudinal slip condition occurs when braking or driving without cornering [22].

The basic MF is used to describe the interaction forces between the tire and the road under several steady-state operating conditions. It is given under the following form: (9) where Y can be the longitudinal or the lateral force; X is the longitudinal slip κ or the slip angle α; D is the peak factor. It is the factor that determines the peak of the characteristic; C is the shape factor. It is the factor that determines the part used of the sine and, therefore, mainly influences the shape of the curve; B is the stiffness factor. It is the factor that stretches the curve; E is the curvature factor. It is the one that can modify the characteristic around the peak of the curve; and S is the horizontal or the vertical shift.

The curve produced by the sine version of the MF is shown in Figure 7. The parameters B, C, D, E and S are calculated from the tire testing data.

The longitudinal and the lateral forces are given by the following equations: (10) (11)

The aligning moment is calculated as a product of the lateral force Fy and the pneumatic trail t_r and added to the residual moment M_zr which is the moment that remains when the side force becomes equal to zero [15]: (12)

Fig. 7 Curve of the Magic Formula [15].

4 Simulation and validation

4.1 Simulation

The objective of simulation is to control a TWV to track a reference longitudinal speed profile and to stabilize the lateral dynamics when taking a turn.

The direct dynamic model of the TWV was used for the simulation. This model was deduced from the inverse dynamic model described in Section 2. It gives the joints accelerations as a function of joints positions, velocities, torques and external loads. The second type of output of the direct dynamic model is the normal reaction forces at the wheels–road contact points: (13)

All equations of the dynamic model of the TWV and the MF equations are implemented in Matlab/Simulink environment using M-files and Matlab functions. Figure 8 shows the combination of the dynamic model with the contact model.

Note that the components of the wrench applied to the chassis with respect to its DOF are null. Hence, the resulting vector of loads applied to the TWV joints can be defined as follows: (14)

The outputs of the dynamic model (state of the system such as the longitudinal and the lateral speeds of wheels at contact points and the normal forces acting on both wheels) are used as inputs for the contact model to calculate the slip quantities (longitudinal slip and slip angle). The contact model (the longitudinal and lateral forces and the aligning moment) influences the dynamic model by adding the actions of command (steering torque, propulsion torque and/or brake torque). The dynamic state of the model will be subsequently updated.

The objective of the simulation is the control of the TWV to track a reference longitudinal speed profile and to stabilize its lateral dynamics when taking a turn. In fact there is a risk of overturn of the TWV under the effect of centrifugal forces and then, the problem comes down to stabilize the roll motion (Fig. 9).

In this model, we have two controllers. The first one controls the angular velocity of the rear wheel by producing a torque applied to the rear wheel. Its function is to obtain a desired longitudinal speed. The second one controls the roll angle by generating a rider steering torque applied to the TWV handlebar.

The reference roll angle is calculated at rolling equilibrium configuration (Fig. 9) and is given by the following expression: (15)

For these two controllers a proportional one is used to calculate the needed motor torque and a proportional–integral–derivative (PID) structure is used for the steering torque. The models of the two controllers are: (16) (17)

Fig. 8 Simulation diagram of the dynamic model of the TWV with tire contact model.

Fig. 9 Equilibrium roll in turns.

4.2 Validation

To validate the developed model, an equivalent model of the TWV was carried out using ADAMS software and used in a co-simulation environment with Matlab/Simulink.

Co-simulation ADAMS/Simulink is developed according to the following steps:

• establish a multibody model of the TWV with ADAMS (Fig. 10);

• identify inputs and outputs for the control model. The studied model has two inputs (the steering torque, the motor torque) and two outputs which are the roll angle and the angular velocity of the rear wheel;

• export the ADAMS model into Simulink environment. The result is a block named adams_sub (Fig. 11) that contains the complete dynamics model of the TWV with its inputs and outputs;

• build the control scheme using the Simulink block adams_sub. The complete control scheme implemented in Simulink is shown in Figure 12;

• simulate the model and analyze the results.

Fig. 10 Multibody model of the TWV.

Fig. 11 “adams_sub” block.

Fig. 12 Control scheme in Simulink.

5 Simulation and results

To evaluate the handling performance and to understand the dynamic behavior of the developed model three tests have been performed (U turn, lane change (LC), steady turning). These maneuvers are also parts of the set of maneuvers used by the TWV manufacturers to test their produced models. For each maneuver, handling indices have been described according to the literature.

5.1 U-turn maneuver

This maneuver consists in the following steps: initially the TWV follows a straight line at a constant speed, then, the rider, at a certain distance, enters into a steady turning curve with a constant radius equal to 100 m and finally it goes back to a rectilinear trajectory as described in Figure 13. The simulation was set up at three constant velocities 60, 80 and 90 km h⁻¹ and the resulting trajectories followed by the TWV are shown in Figure 14.

We notice a slight difference in the trajectory followed by the vehicle, and this is due to the drift phenomenon that depends on many factors such as the condition of the tires, steering angle, vehicle speed and so on.

To quantify the characteristics of the TWV’s handling qualities, Koch [23] proposed the following index [Koch Index (KI)] relative to a “U” turn test: (18)

This equation relates the value of the steering torque peak () at the entry to the curved part with the roll rate peak (), normalized by the average velocity of the TWV (V_avg). This index is used to demonstrate the capacity of a TWV to enter a turn.

Figure 15 shows the evolution of the steering torque and the roll rate during a U turn for the three velocities. The peak values of steering torque and roll rate used to calculate the KI are in the transition phase.

The results of the simulation are given in Table 2. We can note a good correlation between the two developed models (Matlab and ADAMS). Values of the calculated KI are near to those given in literature for a similar type of TWV (scooter) [24].

Fig. 13 U-turn maneuver.

Fig. 14 Trajectories followed by the TWV for U-turn maneuver.

Fig. 15 Steering torque-roll rate (U turn).

5.2 LC maneuver

The LC maneuver is divided into three successive phases (Fig. 16): the first one is called the ‘entry lane’ in which the TWV travels in straight line at a constant speed for a certain distance. Then, a transient lateral displacement is accomplished to move the vehicle in the lateral direction by a predetermined distance, called the offset. In the last one, the TWV is returned to straight running in the ‘exit lane’, which is parallel to the ‘entry lane’ and offset by some lateral distance.

The overall ‘transition distance’ refers to the distance from initiation of the maneuver to the return to a straight-line trajectory.

In this study three tests are performed with a prescribed constant velocity 60, 80 and 90 km h⁻¹. The trajectory followed by the TWV is shown in Figure 17. It is clear that the transition distance increases due to the increasing velocity and the offset remains constant (1.5 m).

To quantify the efficiency with which the rider can operate the TWV during the LC maneuver, Cossalter and Sadauckas introduced a modified version of the KI, the Lane Change Roll Index (LCRI) [23]: (19)

As the maneuver consists of a double steering motion in two directions, the two peaks of the steering torque and roll rate have to be considered. Hence, we have a more complete description of the input/output relationship for a successful entry to the LC maneuver.

Figure 18 shows the evolution of steering torque and roll rate during this type of maneuver and the peaks used to calculate the LCRI. The obtained results show a good agreement both in amplitude and phase between the developed model and ADAMS model. This result is also confirmed for different speeds.

Results of the developed model of TWV and the ADAMS model tests are compared in Table 3 where the LCRI during the test is calculated. Both indices are comparable with the typical value of the LCR Index [25].

Fig. 16 Lane change (LC) maneuver.

Fig. 17 Trajectories followed by the TWV for a lane change maneuver.

Fig. 18 Steering torque-roll rate (lane change).

5.3 Steady state turn maneuver

In this scenario, the TWV is tested in a circular path (Fig. 19) at a constant speed 60 km h⁻¹ with a 3 cornering radii (40, 50 and 60 m). The trajectories followed by the TWV during these three tests are illustrated in Figure 20. The TWV has successfully tracked the desired trajectories.

In this type of tests, the handling quality is quantified using the Acceleration Index (AI) [26]. This index relates the rider action input (steering torque ) during the steady condition and the response of the TWV (lateral acceleration) as follows: (20) where is the average steering torque; V_avg is the average velocity of the TWV during the maneuver and R is the cornering radius.

The requested rider steering torques at the TWV for the three cornering radii are illustrated in Figure 21. One can note that the correlation between the developed model and ADAMS one is good. It is worth mentioning that for a given constant speed, the steering torque increases when the cornering radius decreases. By decreasing the cornering radius, the maneuver becomes more severe because the steady state value of the lateral acceleration () increases, which corresponds to an important value of the roll angle.

The developed model and ADAMS results are shown in Table 4. The AI calculated has negative values for both models because of the negative values of the steering torque. This result is a good indicator for the handling performance of the TWV. Indeed, in the case of a negative steering torque the TWV’s behavior tends to be stable. The AI value of our model is near to other AI values calculated for other TWV [27].

By comparing the results, a good correlation is shown between the two models. The results of simulations are quite similar with a small difference in the magnitude. From this comparison, it can be concluded that the proposed model is correct and accurate.

The comparison of the two models shows that there is a difference in the magnitude of the calculated indices (KI, LCRI and AI). This is because of the difference between methods used to calculate the normal force at the contact point. In fact, ADAMS uses a rheological model [22] to determine the normal load in the tire–road contact points but in our model the contact is defined as rigid one using the Lagrange multipliers.

Fig. 19 Steady state turn maneuver.

Fig. 20 Trajectory followed by the TWV for a three radius in a steady state turn maneuver.

Fig. 21 Steering torque (steady state turn).

5.4 Stability analysis

In this study, the used parameters of the PID controller are obtained through a trial-and-error process to find the best performance requirements. Therefore, to study the stability of the closed-loop system of our TWV developed model, the root locus plots under the various scenarios described above are shown in Figure 22. It can be concluded that all roots, the pink boxes marking the closed-loop pole locations, lie on the left half of the s-plane. It means that when the TWV is in motion its model is considered as stable.

Fig. 22 Root locus plots: (a) U-turn, (b) lane change, and (c) steady state turn.

5.5 Stability limits

To study the stability limits of the TWV developed model and the ADAMS one, we consider two simulations of a TWV traveling at different speeds on a curved paths. We consider two different cornering radii (20 m and 40 m) and different traveling speeds (10, 20, 30, 40, 50, 60 and 70 km h⁻¹). The limit of stability is evaluated by considering the ratio between the lateral contact force to the normal one at the frontal contact point between the wheel and the ground μ_lat. This ratio is defined as the lateral friction coefficient for the TWV and deduced from the lateral acceleration during the maneuver as follows: (21) where a_y is the lateral acceleration and g is the gravitational constant.

The acceptable maximum value of the above-defined ratio according to safety and comfort margins [28] is between 0.2 and 0.3.

Figure 23 shows the results of simulation for both models. It is found that the safety margin value is lower as the cornering radius is smaller and become more critical at higher forward speeds. In fact, this is due to the desired large value of the roll angle needed to complete the maneuver. Therefore, for a small cornering radius the forward speed should be low in order to obtain an acceptable roll angle.

By comparing the two models, it is notable that the lateral acceleration response is similar when the cornering radius and the forward speed are low. The two models start diverging from each other when the forward speed increases for a constant cornering radius.

From the results obtained for the two cornering radii simulations, the responses of the two models are similar for a roll angle value less than 17°.

According to different simulations, the TWV for both models begins to fall down when the speed reaches 50 km h⁻¹ for the cornering radius 20 m and 70 km h⁻¹ for the cornering radius 40 m which corresponds to an approximate maximum value of roll angle equal to 44°. Therefore, the stability limits are defined by the maximum value of the lateral acceleration equal to 6.4 m s⁻² for the developed model and 8.7 m s⁻² for the ADAMS model as shown in Figure 23. One can conclude that the maximum available lateral friction for our model is 0.66 and for the ADAMS model is 0.88.

Fig. 23 Stability limits of the TWV developed model and ADAMS model.

6 Sensitivity analysis of the handling

In this section, we study the effects of variation of two geometric parameters (wheelbase and caster angle) on the handling characteristics. The KI is used for the evaluation of the handling quality in U turn. This index highlights the relationship between the steering torque necessary to start the maneuver and the maximum roll rate necessary to reach the desired roll angle. The decrease of the KI corresponds to an improvement of the handling quality of the TWV. Simulations are carried out for a U turn with a radius of 60 m and at constant velocity of 60 km h⁻¹.

6.1 Influence of the wheelbase

The wheelbase (see Fig. 24) is varied around a nominal value and the KI is calculated. The objective is to determine the effect of the variation of the wheelbase on the handling quality.

The results of simulations are given in Table 5; it is easy to note that the variation of KI with respect to the initial design (e = 1.87 m) decreases by decreasing the wheelbase value. In this case, a low steering torque is needed to initiate the turn which is an indicator of a good handling. When the wheelbase increases the KI increases and the steering torque in the transition phase increases too, which affects the handling and the stability of the TWV.

Also, we can see the influence of varying the wheelbase on the steering torque during maneuver in Figure 25: when wheelbase decreases the absolute value of steering torque decreases. This means that a configuration with a low value of wheelbase is more stable which corresponds of a low value of KI.

Fig. 24 Wheelbase.

Fig. 25 Steering torque varying wheelbase.

6.2 Influence of the caster angle

The caster angle (see Fig. 26) influences stability and handling of a TWV in combination with trail by controlling the alignment of the front tire when the TWV enters in a curve. Three values of the caster angles are tested to determine the sensitivity of the KI to this parameter. The results are shown in Table 6.

The obtained results show that the KI decreases with the decreasing of the caster angle value. Moreover we can conclude from Figure 27 that by decreasing the caster angle the TWV needs a less effort to steer. This means that a configuration of our model with a low value of caster angle is more stable and presents a good handling.

Compared to the wheelbase, the caster angle does not have an appreciable influence on the KI. So we can conclude that the handling is improved essentially by decreasing the wheelbase.

Fig. 26 Caster angle.

Fig. 27 Steering torque varying caster angle.

7 Conclusions

In this work a dynamic model of a TWV was developed based on a robotic approach and a tire dynamics model, which uses the MF. The proposed model has the advantage of being simple and easy to implement to evaluate the handling performance for this type of vehicle. This evaluation is done for three maneuvers: U turn, LC and steady turning, by calculating some handling indices for each test. A validation of the proposed model was carried out using ADAMS software. A sensitivity analysis of the KI to the wheelbase distance and to the caster angle shows that the caster angle has a little effect on the KI. Therefore, one can conclude that the wheelbase distance can be used as a design parameter to improve the handling quality.

Nomenclature

AI: Acceleration Index (N s²)

a_y: lateral acceleration (m s⁻²)

[A]: generalized inertia matrix

: orientation matrix (3 × 3) of frame R_i with respect to R_j

e: wheelbase (m)

f_e: external wrenches (forces and moments) terms

F_s: Coulomb friction parameter

F_v: viscous friction parameter (N s² m⁻¹)

F_x: longitudinal force (N)

F_y: lateral force (N)

F_z: normal load (N)

g: gravitational constant (=9.81 m s⁻²)

H: vector regrouping the centrifugal forces, the effect of gravity, the Coriolis forces and the external forces and moments

: matrix corresponding to the two constraints equations

K: stiffness parameter (N m⁻¹)

K_p3: proportional gain of the steering torque controller

K_p11: proportional gain of the motor torque controller

K_i3: integral gain of the steering torque controller

K_d3: derivative gain of the steering torque controller

KI: Koch Index (N s² rad⁻¹)

LCRI: Lane Change Roll Index (N s² rad⁻¹)

M_z: aligning moment (N m)

M_zr: residual moment (N m)

: position vector of the origin O_i of frame R_i in R_j

q: vector of generalized coordinates

: vector of generalized velocities

: vector of generalized accelerations

R_c: radius of curvature of the reference trajectory (m)

R_e: effective rolling radius (m)

t_r: pneumatic trail (m)

: the homogenous transformation matrix

V: longitudinal speed (m s⁻¹)

V_x: longitudinal speed of the contact point (m s⁻¹)

V_y: lateral speed of the contact point (m s⁻¹)

: vertical velocity at the front wheel–ground contact point relative to R_f

: vertical velocity at the rear wheel–ground contact point relative to R_f

: vector of the linear velocity of the chassis

: vector of the linear acceleration of the chassis

: vector of the angular velocity of the chassis

: vector of the angular acceleration of the chassis

[x, y, z]^T: position vector of CG of chassis relative to reference frame R_f

α: slip angle (rad)

ε: caster angle (rad)

κ: longitudinal slip

γ: camber angle (rad)

Ω: angular velocity of the wheel (rad s⁻¹)

: yaw speed (rad s⁻¹)

Γ ₃: steering torque (N m)

 Γ₁₁: motor torque (N m)

: reference angular velocity for the rear wheel (rad s⁻¹)

: measured angular velocity for the rear wheel (rad s⁻¹)

θ_ref: reference roll angle (rad)

θ: roll angle (rad)

ψ: yaw angle (rad)

Φ: pitch angle (rad)

λ: vector of Lagrange multipliers

υ: vector that group all the terms of the acceleration constraint equations

μ_lat: lateral friction coefficient

Appendices

Appendix A Recursive Newton–Euler algorithm for Inverse dynamics

Forward recursion

For j = 1…n, we calculate the total forces and moments on each link (j):

This recursion is initialized by which represents the Euler-variables of the mobile base.

• σ_j a binary variable defining the joint type: σ_j = 1 if joint is a prismatic, σ_j = 0 if joint is a revolute and σ_j = 2 when joint is blocked in this case, all terms multiplied by σ_j or are eliminated;

• and are respectively the total forces and moments on the body (j);

• and are respectively the angular velocity and the angular acceleration of body (j);

• is the translational acceleration of body (j);

• M_j is the mass of body (j);

• is the vector of three components of first moment of body (j);

• are elements of the inertia matrix of body (j);

• and are, respectively, the orientation matrix of frame R_i with respect to R_j and the position vector of O_i in R_j;

• 

• 

• 

Backward recursion

For j = n…1, we calculate the forces and moments exerted on body C_j by its antecedent C_i:

This recursion is initialized by for a terminal link (j) without successor (k).

• and are the external forces and moments applied by body (j) on the environment;

• is the resulting force applied by the antecedent body on body C_j;

• is the resulting moment applied by the antecedent body on body C_j.

The inverse dynamic model is obtained by projecting or on the joint axis z_j and adding the effects of friction and elasticity if they exist:

In our study, we add for the front and rear suspensions joints this terms:

• Front suspension: and

• Rear suspension: and

Appendix B TWV parameters

L₁ = 0.88 m; L₂ = 0.55 m; R = 0.32 m; d = 0.098 m; ε = 0.42 rad; α = 0.31 rad; q₄₀ = 0.73 m; q₉₀ = 0.61 m

M₁ = 187.18 kg; XX₁ = 43.03 kg m²; YY₁ = 29.2 kg m²; ZZ₁ = 19.3 kg m²; MX₁ = MY₁ = MZ₁ = XY₁ = XZ₁ = YZ₁ = 0

M₃ = 15.42 kg; XX₃ = 0.52 kg m²; YY₃ = 0.34 kg m²; ZZ₃ = 0.3 kg m²; MX₃ = 0.04 kg m; MZ₃ = 1.53 kg m; MY₃ = XY₃ = XZ₃ = YZ₃ = 0

M₄ = 1.95 kg; XX₄ = 0.09 kg m²; YY₄ = 0.06 kg m²; ZZ₄ = 0.03 kg m²; MX₄ = 0.0067 kg m; MZ₄ = 0.39 kg m; MY₄ = XY₄ = XZ₄ = YZ₄ = 0; K₄ = 60 000 N m⁻¹; F_v4 = 35 826 N s m⁻¹

M₉ = 14.38 kg; XX₉ = 0.83 kg m²; YY₉ = 0.74 kg m²; ZZ₉ = 0.19 kg m²; MX₉ = 2.07 kg m; MZ₉ = 1.15 kg m; MY₉ = XY₉ = XZ₉ = YZ₉ = 0; K₉ = 90 000 N m⁻¹; F_v9 = 53 185 N s m⁻¹

M₆ = 9.12 kg; XX₆ = YY₆ = 0.18 kg m²; ZZ₆ = 0.3 kg m²; MX₆ = MY₆ = MZ₆ = XY₆ = XZ₆ = YZ₆ = 0

M₁₁ = 9.12 kg; XX₁₁ = YY₁₁ = 0.18 kg m²; ZZ₁₁ = 0.3 kg m²; MX₁₁ = MY₁₁ = MZ₁₁ = XY₁₁ = XZ₁₁ = YZ₁₁ = 0.

References

All Tables

All Figures

	Fig. 1 DOF of the TWV.
In the text

	Fig. 2 Geometric parameters: trail, caster and camber angle.
In the text

	Fig. 3 Multibody tree structure of the TWV.
In the text

	Fig. 4 Geometric parameters according to DHM.
In the text

	Fig. 5 Articulated model of the TWV.
In the text

	Fig. 6 Forces and moments acting on a tire at the wheel–road contact point [22].
In the text

	Fig. 7 Curve of the Magic Formula [15].
In the text

	Fig. 8 Simulation diagram of the dynamic model of the TWV with tire contact model.
In the text

	Fig. 9 Equilibrium roll in turns.
In the text

	Fig. 10 Multibody model of the TWV.
In the text

	Fig. 11 “adams_sub” block.
In the text

	Fig. 12 Control scheme in Simulink.
In the text

	Fig. 13 U-turn maneuver.
In the text

	Fig. 14 Trajectories followed by the TWV for U-turn maneuver.
In the text

	Fig. 15 Steering torque-roll rate (U turn).
In the text

	Fig. 16 Lane change (LC) maneuver.
In the text

	Fig. 17 Trajectories followed by the TWV for a lane change maneuver.
In the text

	Fig. 18 Steering torque-roll rate (lane change).
In the text

	Fig. 19 Steady state turn maneuver.
In the text

	Fig. 20 Trajectory followed by the TWV for a three radius in a steady state turn maneuver.
In the text

	Fig. 21 Steering torque (steady state turn).
In the text

	Fig. 22 Root locus plots: (a) U-turn, (b) lane change, and (c) steady state turn.
In the text

	Fig. 23 Stability limits of the TWV developed model and ADAMS model.
In the text

	Fig. 24 Wheelbase.
In the text

	Fig. 25 Steering torque varying wheelbase.
In the text

	Fig. 26 Caster angle.
In the text

	Fig. 27 Steering torque varying caster angle.
In the text