Motorcycle modelling and roll motion stabilization by rider leaning and ...

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dynamic model for the rider–motorcycle system and some relevant characteristics ... It was noticed that in 75% of the accidents, the rider attempts a braking or an.
Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada, August 28-31, 2005

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Motorcycle modelling and roll motion stabilization by rider leaning and steering torque Sa¨ıd Mammar, St´ephane Espi´e, Christophe Honvo Abstract— This paper is devoted to the modelling and the stabilization of the roll motion of a motorcycle. The proposed model processes a nonlinear tire-road interaction forces, and includes the rider leaning movement for stabilization. The embedded rider-motorcycle model is then stabilized using H∞ control. A prefilter is finally added in order to ensure reference model tracking. Simulation results show the effectiveness of the approach. Index Terms— Motorcycle dynamics, Roll dynamics, Rider assistance, Simulator.

I. I NTRODUCTION Since 2002, the French government decided to make fight against the road accidents one of the three large programs of its five-year period. In spite of encouraging results obtained since the installation of speed control systems, the motorcycle remains a particularly dangerous mode of transport: the number of death is still very high, and if one takes account of the number of travelled kilometers, the risk of death for a motorcycle rider is 21 times higher than that of other drivers. During the last twenty years, passengers cars experience several advances in passive and active safety systems. Almost today cars are all equipped with one or several airbags and ABS systems. More powerful systems such as ESP, brake assist systems and traction control systems or belt tensioners are becoming quickly democratized. On the other hand, during the same period, the backwardness taken by motorcycles is continually growing. As an example, the ABS, which exists since nearly 15 years, is still reserved to few top range models. The braking distributor is less expensive but still remains marginal on the whole of the market. Motorcycle airbags, whose development seems difficult, does not come to be imposed. The recent braking amplifier seems to be promising, but its diffusion remains for the moment very restricted. The French project SUMOTORI, was motivated by this fact, it aims to overcome partly these deficiencies. Its objective relates to the three following aspects • •

To acquire knowledge on the behavior of the twowheeled vehicles in situation of accident. To prove the feasibility of an onboard system which is able to detect critical situations.

This work was supported by SUMOTORI PREDIT project. ´ S. Mammar and C. Honvo are with Universit´e d’Evry val d’Essonne, France. LSC/CNRS-FRE 2494, 40 rue du Pelvoux CE1455, 91025, Evry, Cedex, France, (e-mail: [email protected]). S. Espi´e is with INRETS- MSI, 4 avenue du G´en´eral Malleret Joinville, 94114, Arcueil, France, e-mail: ([email protected]).

0-7803-9354-6/05/$20.00 ©2005 IEEE



To design a demonstrator of this onboard system in order to warn the rider and/or to activate passive safety systems. The purpose of this paper is to develop a motorcycle model which will be used as a simulator. It combines a stabilization control module which allows the stabilization of the motorcycle roll motion either by acting on steering torque or by acting on rider lean angle. The synthesis approach ensures stability of the two following configurations : feedback only on steering torque and feedback on both lean angle and steering torque. This paper is organized as follows: in section 2, the results of an accidentology study are provided, some relevant cases of control loss are detailed. In section 3, a description of the dynamic model for the rider–motorcycle system and some relevant characteristics are presented. Tire road interaction model is highlighted. In section 4, the simultaneous stabilization approach is presented. Controllers are synthesized using an H∞ performance index. Robust stability of the system according to forward speed variations is evaluated. Finally a speed scheduling procedure is proposed. Simulation results are carried out in section 5. Maneuvers are performed both in straight-line and in cornering motion. Section 6 is devoted to the conclusions, and the major parameters used in this paper are listed. II. R ELEVANT ACCIDENTOLOGY A recent accidentology study revealed that more than 70% of urban and sub-urban motorcycle accidents occur at speed lower than 50 km/h [10],[9]. It is also shown that 15% of the accidents are accident alone. When another vehicle is involved during the accident, the rider is responsible in 37% of the cases and the adverse vehicle in 50% of the cases. The remaining 13% are due to weather conditions or infrastructure failure. The study of 200 reported motorcycle accidents leads to the isolation of 47 cases of control loss due to the incompatibility between the rider and the machine leading to control loss. This incompatibility is originated either by the overestimation of the rider or motorcycle performance or by bad interpretation of the environment. It was noticed that in 75% of the accidents, the rider attempts a braking or an avoidance emergency maneuver. III. M OTORCYCLE MODEL Since the two rigid bodies model proposed in [1], up to seven bodies and more complex models have been studied [2],[4]. The complexity was generally made necessary for the

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study of some specific behavior during acceleration, braking or steering [3],[5],[6]. The model presented in this paper has four joined rigid bodies and is inspired from those presented in [7] and [8]. It includes the driver upper body lean motion. As in [7], the four rigid bodies of this driver-motorcycle are • Front fork assembly: front fork, handle bar, front wheel and shock absorber. • Rear frame: engine rotating in transverse axis, chassis, the lower body and legs of the rider. • Rear wheel. • Rider upper body: the upper body of the rider including hands an head. This model can be reduced to only three rigid bodies by putting the rear wheel body in the rear frame. In this case, it is no more possible to model the twisting movement at the rear swinging arm. Table 1. Mass and inertia Parameter value description m 165kg Total mass: rider and motorcycle 15kg Front fork mass mf mr 130kg Rear part mass 20kg Upper body mass mp The caster angle of the front fork is equal to 21◦ . The distances are shown on figure 1, other model data, including mass, inertia, spring and damper coefficient are shown in Table 1. The obtained nonlinear model has 8 state space variables given by (1) x = [v, r, p, φ , δ˙ f , δ f , f f , fr ]T where v is the lateral speed, r is the yaw rate, (p, φ ) are respectively the roll rate and the roll angle, (δ˙ f , δ f ) are respectively the steering angle rate and angle value. The forces f f and fr are the front and rear tires lateral forces. Equations of motion are obtained by gathering the lateral motion equation, the yaw equation, the roll equation, the front fork hinge equation and finally the two front and rear tire forces relaxation equations. The model presents 4 separate inputs represented by the rider control  torque T ,  ˙ ¨ the rider lean motion represented by φ , φ , φ r r r . The whole   control vector is T, φr , φ˙r , φ¨r . A. Tire road interaction The front and rear tire lateral forces are function of the sideslip angle αi at the tire-road contact location and the

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Fig. 2. Front lateral force as a function of the sideslip angle and the camber angle.

camber angle γi . Here, the index i stands for f (front) or r (rear). In this paper, the magic formula of Pacejka is used for each tire in order to determine the lateral forces [18].    bi (γi ) (1 − ei ) αi −1 fi (αi , γi ) = di (γi ) sin ci tan +ei tan−1 (bi (γi )αi ) (2) The coefficients bi , c f , di , ei depend on the tire characteristics, on the road conditions, and on the vehicle operational conditions. Parameter values, when road friction is high and camber angle is zero, are given in Table 2. Table 2. Tire model parameters on high friction road Tire bi ci di ei Front 14.2574 1.2509 1010.0 -1.661 Rear 13.9481 1.1009 1835.8 -1.542 Let µ be a common road adhesion coefficient with µ = 0.2 for icy road and µ = 1 for nominal road adhesion. The effect of the road adhesion on the lateral forces is incorporated in5 theµ magic formula by changing bi to (2 − µ ) bi , ci to 4 − 4 ci and di to µ di . In addition, the influence of the camber angle is handled by adapting the formulas for bi and di   2 |γi | 2 µ di −→ µ di 1 − π   |γi | (2 − µ )bi −→ (2 − µ )bi 1 − 4π Figure 2 provides a 3D mesh plot of the front tire lateral force as a function of the sideslip angle and the camber angle. The increase of the camber angle leads to a reduction of the lateral forces. Figure 3 shows the roll angle response during a straight running at a speed of 23 m/s. A front steering torque input of 10N.m is added during 0.5s. At this forward speed, the motorcycle is stable. However, this motorcycle model is unstable for a large range of forward speed. The stability is only obtained around the speed of 20m/s. This can be verified from the behavior of the real parts of the poles which are plotted on figure 4. The

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actual motorcycle roll angle. Notice that this first control action may achieve stabilization or not according to the forward speed. The final motorcycle-rider model is obtained by embedding the motorcycle model with the rider leaning movement model. The control input is the steering angle torque which will be set in part by the rider according to the desired roll angle. The controller is designed in the next section.

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IV. C ONTROL DESIGN

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The control philosophy processes in a two degree of freedom control approach. The steering torque T is computed as follows (figure 6)   T = W1 C f b φ +C f f φdes + Tr

The previous model is now completed with a rider leaning model for roll motion stabilization (Figure 5). It is adapted for driver model for lane keeping and rider leaning movement. This type of rider control assumes that the rider tries to stabilize the motorcycle by inclining left and right his upper body [17]. This model contains several components [11], [8]. The first one is called structural model. It is constituted by a time delay and a filter representing inherent human processing time and neuromotor dynamics. This component represents the high frequency driver compensation component, the dead time is of about τ p = 0.2s and the filter is a second order low pass filter with damping factor ξn = 0.707 and natural frequency ωn = 10rad.s−1 . The second component corresponds to the rider lead and predictive actions. It is modelled by a first order lead filter, where the time constant τL is representative of the rider mental load. The third component is a simple gain representing the proportional action of the rider face to the perceived motorcycle roll angle relative to the desired roll angle. It must be noted that all these parameters are not constant and are only valid for restricted motorcycle and rider configurations. This model is controlled by a feedback action, proportional to the difference between the desired roll angle φdes and the

where Tr is the torque applied by the rider. The real torque which is thus applied to the motorcycle is (Ta + Tr ). The feedback controller C f b ensures robust stability of the feedback loop with guaranteed damping enhancement of the roll motion. It has also in charge the disturbance rejection problem and the rider incoherent inputs. Controller C f f acts as a prefilter of the reference signal by adding a feedforward action. This controller is synthesized such that to make the motorcycle roll angle response to robustly follow as close as possible the response of a chosen reference model. This fact constitutes robust model matching [14]. In the following, a two stages approach is adopted for the synthesis of the feedback and the feedforward components. At the first stage the feedback part Cb f of the controller is computed using the H∞ coprime based loop shaping method of [16]. Afterwards, the new motorcycle model which incorporates the feedback controller is computed, thus the feedforward part C f f is synthesized by a second H∞ optimization. All the controllers are synthesized on a nominal linear system, at fixed forward speed and full road adhesion. The forward speed is considered as a varying parameter. The robustness domain of the controller is one of the main interest.

linear models are obtained by linearization around constant forward speeds.

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We consider first the sub-system Gφ T which maps the front-fork steering torque to the roll angle. In order to ensure zero steady state error and to reject possible perturbation on the steering torque or coming from the wind force, a weighting compensator is added on the input of the system. The compensator is chosen as a combination of a PI and lead filters.

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0.8s + 1 4s + 1 (3) s 0.02s + 1 The gain k is a function of the forward speed. For 20m/s, the gain is set equal to 10. Let now Gs be the shaped plant (Gs = Gφ T W1 ). According to the gap-metric, with this choice of W1 , the stability of the motorcycle for admissible parameters variations is guaranteed. The stabilizing H∞ feedback controller C f b is computed using the non-iterative procedure in [12] with a relaxed value of the maximal stability margin. The relaxed stability margin for this speed is γ1 = 1.76. The controller is implemented as shown in figure 5. W1 (s) = k(v)

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The weighting compensators are left at the system input. Let G f f be the mapping from the reference input φre f to the output φ . We seek now a single input, single output feedforward controller C f f . Let M0 be the desired transfer function between φdes and φ . This reference model is chosen 1 . The settling time is about 1.5 sec, this model as M0 = 0.5s+1 avoids overshot on motorcycle response. The feedforward controller C f f is synthesized in order to keep the error signal z small in H∞ sense for the class of perturbed systems according to road adhesion variations [13] The error signal z is computed from (z = φ − M0 φdes ) .When  including the controller C f f , the error signal is thus z = G f f C f f − M0 φdes , it can be written in LFT form which is suitable for H∞ optimization    −M0 G f f ,C f f φdes (4) z = l ft I 0

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V. S IMULATION RESULTS A first set of simulations addresses disturbance rejection, while the second set concerns reference signal tracking. A. Disturbance rejection The motorcycle is supposed to be on straight road section, at nominal speed and full road adhesion with zero roll angle. It is subject to a step disturbance in torque front fork. This torque disturbance input appears at time t1 = 1 sec and disappears at t2 = 1.5 sec. It is assumed that the rider doesn’t react to this disturbance. In this case, only controller C f b is in action. One can note from Figure 7 that the roll angle does not exceeds 2 degree and returns to zero in steady state. The response is well damped. The corresponding steering angle is limited to 2deg. In figure 8, we examine the robustness of the synthesized controller when the road adhesion is reduced by 20%, while the speed is maintained to 20m/s. The responses remain acceptable even if the degradations are notable.

B. Cornering maneuver First of all, The previous closed system is simulated for a 10deg step reference on the roll angle without the prefilter part C f f . The rider leaning motion is included. Figure 9 shows that the roll angle response exhibits a overshot of more than 20%. The settling time is of about 4sec. However, the steering angle is too large. The same maneuver is now simulated with the prefilter part. Results are shown in figure 10. As expected, the roll angle response follows the response of the reference model. The settling time is respected. In order to test the flexibility of the synthesis procedure, the time constant of the reference model is varied from 0.3sec to 15sec by steps of 0.2sec. The prefilter part of the controller is synthesized for each value of the time constant time and the previous maneuver is simulated. The obtained roll angle

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responses are shown in figure 11. One can notice that the responses change accordingly. Finally the reference model is changed to a second order 1 . As expected, the roll angle transfer function M0 = s2 +s+1 response follows closely the reference model (Figure 12).

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In this paper some general aspects of a motorcycle model are presented. Tire-road contact forces are provided and stability aspects are discussed. This model is completed by a rider body leaning motion model. This leaning control may or may not stabilize the motorcycle. A control synthesis procedure is developed for the stabilization of the roll motion by the use of the front fork steering torque. The controller performs as feedback and feedforward control. An H∞ measure index is used for robust stability. The feedforward controller

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[10] S´ecurit´e routi`ere, la s´ecurit´e routi`ere en France, bilan de l’ann´ee 2002, la documentation franc¸aise, Paris, 2003. [11] Modjtahedzadeh, A., and R. A. Hess, A model of driver steering control behavior for use in assessing vehicle handling qualities, 15, 456-464, 1993. [12] McFarlane, D., and K. Glover. A loop shaping design procedure using H∞ synthesis. IEEE Transactions on Automatic Control, 37, 759-769, 1992. [13] Giusto, A., and F. Paganini , “Robust synthesis of feedforward compensators”, IEEE Transactions on Automatic Control, 44, 15781582, 1999. [14] Kasenally, E. M., D. J. N. Limebeer and D. Perkins, On the design of robust two degree of freedom controllers., 29.1, 157-168, 1993. [15] Kamata, Y., Nishimura, H., System identification and attitude control of motorcycle by computer-aided dynamics analysis. JSAE Review, 24, 411-416, 2003. [16] S. Mammar and D. Koenig, Vehicle Handling improvement by Active Steering, Vehicle System Dynamics Journal, vol 38, No3, 211-242, 2002. [17] Miyagishi, S., Kageyama, I., Takama K., Baba, M., Uchiyama, H., Study on construction of a rider robot for two-wheeled vehicle, JSAE Review, 24, 321-326, 2003. [18] Tezuka, Y., Ishii, H., Kiyota, S., Application of the magic formula tire model to motorcycle maneuverability analysis, JSAE Review 22, 305-310, 2001.

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N UMERICAL VALUES

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a = L − b = 0.87m, ar = 0.14m, L = 1.4m, b = 0.53m, c = 0.04m, c1 = 0.12m, e = 0.37m, f = 0.70m, h = 0.65m, h = 0.82m, hb = 0.92m, hr = 0.79m, hv = 0.62m, h1 = 0.82m, h1 = 0.7m, k = −c = −0.04m, ε = 21 deg.

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is synthesized for reference model matching. The simulation results show that the synthesized controller ensures both good disturbance rejection and cornering maneuvers. Results show also that performances remain acceptable in case of road adhesion reduction. However, is seems that a forward speed scheduling procedure is necessary in order to handle with guaranteed performance forward speed variations. R EFERENCES [1] Sharp, R.S, The stability and control of motorcycles. Jour. Mech. Eng. Sci. 13(5), 316–329, 1971. [2] Sharp, R.S. and D.J.N. Limebeer, A motorcycle model for stability and control analysis. Multibody System Dynamics 6(2), 123–142, 2001. [3] Sharp, R.S., Stability, control and steering responses of motorcycles. Vehicle System Dynamics 35(4–5), 291–318. and Zeitlinger, Lisse. Vienna. 334–342, 2001. [4] Sharp, R.S., S. Evangelou and D.J.N. Limebeer. Improvements in the modelling of motorcycle dynamics. In: ECCOMAS Thematic Conference on Advances in Computational Multibody Dynamics (J.A.C. Ambr´osio, Ed.). Lisbon. MB2003-029 (CD-ROM), 2003. [5] Sharp, R.S., The stability of motorcycles in acceleration and deceleration. In: Inst. Mech. Eng. Conference Proceedings on “Braking of Road Vehicles”. MEP. London, 45–50; 1976. [6] Sharp, R.S., Variable geometry active rear suspension for motorcycles. In: Proc. of the 5th International Symposium on Automotive Control (AVEC 2000). Ann Arbor MI., 85–592, 2000. [7] Sharp, R.S., Vibrational modes of motorcycles and their design parameter sensitivities. In: Vehicle NVH and Refinement, Proc Int Conf.. Mech. Eng. Publications, London. Birmingham, 107–121, 1994. [8] Lai H. C., Liu, J.S., Lee, D.T., Wang, L.S., Design parameters study on the stability and perception of riding comfort of the electrical motorcycles under rider leaning, Mechatronics, 13, 49-76, 2003. [9] The National Highway Traffic Safety Administration, Motorcycle Safety Program, Janvier 2003.

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