Simulation and performance evaluation of race ...

Simulation and performance evaluation of race motorcycle dynamics based on parts of real circuit 987012 V. COSSALTER, M. DA LIO, R. LOT

99

Simulation and performance evaluation of race motorcycle dynamics based on parts of real circuit 987012

V. COSSALTER, M. DA LIO, R. LOT Department of Mechanical Engineering, University of Padova, Padova Italy

L. FABBRI Aprilia S.p.A., Racing Department, Noale, Italy

Abstract – This paper focuses on the evaluation of race motorcycle performance by means of simulation. There are two sections. The first deals with the evaluation of the trajectory to be followed for maximum efficiency. This problem is here solved with a new approach, called “the optimal manoeuvre method” which consists of solving a two-point optimal control problem. The optimal manoeuvre, which corresponds to the minimum time for moving between given initial and final positions, is found and the time needed to complete the manoeuvre is taken as a measure of vehicle manoeuvrability. This particular solution is thus the fastest way for a motorcycle to complete the given circuit part. When finding the solution, all physical constraints like tyre traction, lane borders, and available thrust and braking forces are taken into account and satisfied. The second section of the paper concerns the study of the behaviour of the vehicle in its symmetry plane. The aim is to evaluate the suspension parameters and set them. This study is carried out by integrating a system of equations, which describe the dynamics of the vehicle in its symmetry plane in two dimensions. To carry out this study, the braking and thrust forces found in the previous step (the optimal manoeuvre) and the apparent gravity due to the curvature of the trajectory and camber are used. The comparison of the results of these simulations with data gathered with telemetry show significant agreement.

Introduction This paper presents the evaluation of the performance of race motorcycles carried out in two steps by simulations. First the gross motion of the whole vehicle is studied and the manoeuvre which requires the minimum time for moving between given initial and final positions on a circuit is found using a new method called “the optimal manoeuvre”, which is an application of optimal control. 99

100 The motions in the motorcycle plane of the wheels and frames are subsequently studied relative to a reference frame which moves according to the gross motion defined above. This makes it possible to study the operation of suspensions while the vehicle moves along the path required by the optimal manoeuvre.

The optimal manoeuvre method This method is described in detail in papers [1,2]. The basic idea underlying the method is that of using optimal control to determine the driver’s actions, i.e. control inputs, that make a vehicle complete a given “manoeuvre” in the “most efficient way”. In this paper only “manoeuvres” that consist of following a portion of a given circuit1 are considered (Fig.1), and the words “most efficient” here are always interpreted as “complete the manoeuvre in

Finish

the minimum time”2 . The resulting mo-

Final boundary conditions

tion must also comply with boundary conditions at the beginning and at end of the given manoeuvre and with trajectory

Start

constraints that express physical limits that cannot be exceeded. The boundary conditions and trajectory constraints used for

Optimal path

this paper are listed in table 1. The manoeuvre that is found using this method represents the “maximum” per-

Initial boundary conditions

formance that a given vehicle can produce, and corresponds to the motion that would

Fig. 1 - The optimal manoeuvre corresponds to minimum time

result if the driver were perfect; i.e. if he or

for moving between initial and final positions on a circuit.

she applied the optimal control input. Besides allowing unambiguous evaluation of vehicle performance (in this case measured by the time a motorcycle takes to travel the portion of the given track) this method also overcomes another crucial problem in vehicle handling studies: namely the modelling of the pilot, which is especially critical in the case of unstable vehicles, as indeed motorcycles are. The solution of the optimal control problem requires some manipulation of the equations of motion. As explained in papers [1-5], a penalty, which defines the optimality criterion, is minimised subject to the equality constraints given by the equations of motions. The constrained minimisation yields a set of ordinary differential equations formed by the original equations of motion plus an equal number of co-equations for the lagrange multipliers. The

1

Represented by a plane strip that the vehicle has to stay on.

2

Different possible meaning of “most efficient”, as well as different kinds of “manoeuvres”, are also discussed in papers [1,2].

100

101 set of all these equations is solved as a boundary value problem, since conditions are imposed both at the beginning and at the end of the manoeuvre [1-5].

Initial boundary conditions

assigned: position, velocity, tyre loads (consistent with thrust), tyre lateral forces (consistent with velocities).

free: thrust.

Final boundary conditions

assigned: roll angle, lateral, roll and yaw velocities, tyre lateral forces (consistent with velocities) and exit direction.

free: lateral position, longitudinal velocity, tyre loads, thrust.

Trajectory constraints

Tyre forces must stay within traction limits. Vehicle must stay within lane borders. Thrust cannot exceed the maximum available. High rates of change of thrust/braking forces and steering force are penalised.

Table 1 - Manoeuvre boundary conditions and trajectory constraints. The penalty function used here is somewhat different from that presented in [2]. The difference is that here what is minimised is the final time, whereas in [2] the average velocity and the final space were maximised. This reformulation requires a change of the independent variable from time to space, and the introduction of an additional state variable and corresponding equation [3-5]. The mathematical model of the motorcycle is the same one given in [2] (but with a few improvements). It focuses on gross motion only, and does not include a detailed analysis of steering angle and includes no analysis of steering torque at all. The control inputs here are the lateral force on the front tyre (as a result of steering) and thrust and braking forces. This model makes it possible to determine only the gross motion of the vehicle and to find the manoeuvre, which, according to available driving forces, uses the minimum time.

In-plane analysis Once the gross motion is known the small motions of the various bodies forming the motorcycle and their interactions may be studied. This makes it possible to analyse the working of the suspensions, pitching and heaving of the vehicle as well as the differences of the tyre forces with respect to the ideal manoeuvre induced by these small motions. For this analysis, the motorcycle is modelled as a system of four rigid bodies in its plane of symmetry, as described in appendix 1, and the relative equations of motion are derived by means of the natural coordinates approach [6]. The equations shown in appendix 1 depend on the apparent gravity g *, which is the sum of the gravity acceleration and the acceleration of the moving reference frame3 . With this approach the study of the suspension system is not coupled with gross motion, and can be carried out as a two dimensional analysis, with real gravity replaced by an apparent gravity which includes the contribution of the moving reference frame.

3

Apparent torque induced by the moving reference frame is neglected.

101

102

Case studied Optimal manoeuvre First the optimal manoeuvre of a race motorcycle, whose parameters4 are given in appendix 2, is found. The road to be followed is the simple circuit shown in Fig.2. B

A

0

C D E

F

-100

-200

G -300 -400

-300

-200

-100

0

100

200

300

Fig.2 – Simple circuit. The three curves on the right represent those of S.Donato, Luco and Poggio Secco of the Mugello circuit in Italy. 80 70

A

B

Velocity

[m/s]

60

telemetry

50

C

40

F

30

E D

20 10 0

G

simulation 0

100

200

300

400

500

600

700

800

900

Curvilinear abscissa [m] Fig.3 – Comparison of velocity between telemetry and simulation. Although it is not a real circuit when considered as a whole, the three curves on the right are the curves of S.Donato, Luco and Poggio Secco of the Mugello circuit in Italy. The figure shows the optimal path, as resulting 4

Only those parameters that are relevant for the optimal manoeuvre and the mathematical model used are shown in appendix 2.

102

103 from the minimum time optimal manoeuvre. The points marked A, B, C, D, E, F are 100 meters apart from one another. Fig.3 shows a comparison between the simulated velocity and the telemetry on the real curves. Fig.4 compares the roll angle. As can be seen, comparisons show a good agreement. Fig.5 shows the longitudinal force on front and rear wheels in the simulation.

simulation

D

1

[rad] Roll angle

E

C

G

B 0

A telemetry

-1

F 0

100

200

300

400

500

600

700

800

900

Curvilinear abscissa [m] Fig.4 – Comparison of roll angle between telemetry and simulation. 1000

A

longitudina force [N]

rear wheel

E

500

0

G A

B

D C

F E F

-500

-1000

B -1500 0

100

front wheel

C 200

300

400

500

600

700

800

900

Curvilinear abscissa [m] Fig.5 – Longitudinal force on front (dashed line) and rear wheels (simulation). Positive values mean thrust. According to the optimal manoeuvre (solid lines), the motorcycle initially (A,B) proceeds straight (Fig.2). At about 50 meters, in the middle between A and B, it starts braking (Fig.3 and Fig.5). Here Fig.5 shows a sudden passage 103

104 from a state in which the rear wheel is applying thrust (A) to a state in which both wheels are braking (B). The front wheel brakes more than the rear wheel because it has a greater load (in fact the braking force is optimally divided). Braking causes a decrease in velocity, as shown in Fig.3 which lasts until D. Fig.4 shows that the roll angle gradually increases from A to D. To do this, the vehicle initially turns slightly to the left (A,B) which causes rolling to the right, then curves to the right (B,C,D) and reaches the inner border of the S. Donato curve just past D. Thereafter it begins to accelerate (D,E,F). As shown in Fig.5, at this point the rear wheel is applying a thrust and the velocity increases (Fig.3). During acceleration (longitudinal force positive) only the rear wheel pushes; the slightly negative value of the front wheel longitudinal force (E on dashed line in Fig.5) is needed for wheel angular acceleration.

Steering angle

[rad]

0.02

D

peak E

0.01

0

A B

G

C

F

-0.01

dip -0.02

0

100

200

300

400

500

600

700

800

900

Curvilinear abscissa [m] Fig.6 – Steering angle (simulation). The motorcycle stays rolled until E where it suddenly inverts its roll angle (E,F). This manoeuvre is accomplished by steering towards the inner side of the curve, which corresponds to the first peak in E, Fig.6. Similar manoeuvres are also shown at about 600 m (the dip which corresponds to another inversion of the roll angle between the Luco and Poggio Secco curves) and at 800 m (which corresponds to bringing the motorcycle vertical for the straight line G).

Parameter analysis A parameter analysis was carried out to estimate the potential benefits of certain changes in the motorcycle design. The variations that were considered were: 1) displacement of the overall centre of mass 10% higher than the initial height, 2) displacement of the overall centre of mass 10% further back than the initial position, 3) wheelbase increase of 10%, 4) overall mass decrease of 10%, 5) rear tyre sideslip stiffness increase of 10%, 6) front tyre sideslip stiffness increase of 10%, 7) rear tyre camber stiffness increase of 10%, 8) front tyre camber stiffness increase of 10%. In all the cases the changes only affect the listed parameter, the remaining parameters are not modified. 104

105 This means, for example, that the increase in wheelbase, case 4, must be interpreted as not being accompanied by an increase in mass and/or moment of inertia (unlike what would occur in reality). Table one reports the difference in the time for one lap. As can be seen, all changes except case 7 produce a beneficial effect. Displacing the mass centre upwards (without increasing the moments of inertia), case 1, is beneficial because it gives faster rolling. A wheelbase increase, case 3, and displacement of overall mass centre backwards, case 2, (again with no collateral effect) gives faster yawing. Reduced overall mass, case 4, is obviously beneficial because faster acceleration is possible (the thrust limit is unchanged). Increased tyre sideslip stiffness, both front and rear wheel, cases 5 and 6, is clearly beneficial as it reduces the slip angles.

Variation

Time difference (ref. motorcycle time lap 35.561 s)

1) displacement of overall mass centre 10% upwards (h +10%)

- 0.010 s

2) displacement of overall mass centre 10% backwards (b -10%)

-0.086 s

3) wheelbase increased by 10% (p +10%)

-0.081 s

4) overall mass decreased by 10% (m –10%)

-0.187 s

5) rear tyre sideslip stiffness increased by 10% ( C1r +10%)

-0.018 s

6) front tyre sideslip stiffness increased by 10% ( C1f +10%)

-0.035 s

7) rear tyre camber stiffness increased by 10% ( C 2r +10%)

+0.008 s

8) front tyre camber stiffness increased by 10% ( C 2f +10%)

-0.011 s

Table 2 – Effect of variation of some parameter. As can be seen from table 2, in cases 7 and 8, increasing rear camber stiffness worsens performance, whereas increasing front camber stiffness improves it. This has something to do with front/rear balance: in fact, the motorcycle of the reference case has approximately the same slip angles on the front and rear wheels in steady state turning. Understeering results from increasing the camber stiffness on the rear wheel, and this worsens performance, conversely moderate oversteering (which is obtained in case 8) is beneficial.

In-plane analysis Figures 7 and 8 compare the suspension strokes resulting from simulation and telemetry. At the beginning, during the braking phase (B,C), the rear suspension elongates and the front one shortens. In the curve (D), when the motion is almost steady circular, both suspensions are compressed (although the front one less than during the previous braking phase). At the exit of the curve, during acceleration (E), the front suspension elongates. The rear sus-

105

106 pension also elongates because at this point, despite acceleration, the trajectory is straight and the centrifugal forces that compressed both suspensions in D are missing. The pitching of the vehicle is shown in Fig.9 for the reference case and for some alternative ones. As is shown, an anti-dive system on the front wheel (dotted line) produces its effect mainly during the braking phase, where the front suspension stroke and pitching are reduced. A progressive spring on the rear wheel also reduces the pitching in the braking phase (dashed line). A completely different effect is conversely obtained by displacing the pinion downwards. This effect occurs only when the chain is pulled, i.e. during acceleration. The different direction of the chain that can be obtained by displacing the sprocket results in a different elongation of the rear suspension. 0.345

0.30

telemetry 0.330

E

A

G 0.315

0.20

0.15

0.100

F

D

0

100

200

300

400

500

0.300

600

700

800

0.285 900

Curvilinear abscissa [m] Fig.7 – Rear suspension stroke. 0

E G simulation

-0.05

A D F

-0.1

telemetry

B C

-0.15

0

100

200

300

400

500

600

Curvilinear abscissa [m] Fig.8 – Front suspension stroke. 106

700

800

900

spring stroke [m]

simulation

C

0.25

front spring stroke [m]

rear arm angle [rad]

B

107

picht angle [rad]

A

anti-dive front susp. progressive rear spring

0

E

pinion position 40 mm down

D F

reference case anti-dive front suspension progressive rear spring pinion position 40 mm down

B C

-0.1 0

100

200

300

400

500

600

700

800

900

Curvilinear abscissa [m] Fig.9 – Motorcycle pitch and the effect of some design variations.

Conclusions This paper has shown an approach to the analysis of motorcycle performance which is founded on two steps. First there is a gross motion analysis, which is carried out using the “optimal manoeuvre method” and which yields the optimal path and acceleration/braking that carries the motorcycle from a given initial condition to a given final conditions in the minimum time possible. Then there is an analysis of the relative motions of the various bodies which form the motorcycle in the neighbourhood of the previously found gross motion. This analysis gives insight into the operation of the suspension systems. A case of a typical race motorcycle was considered and it was shown that a fairly good accordance exists between simulations and telemetric data, concerning both the overall motion (velocity and roll angle) and the suspension strokes. A parametric analysis considering different physical properties for the motorcycle and the tyres was done. As a result of this analysis the potential benefits of modified designs were clearly evaluated in terms of the time saved in one lap. A similar parameter analysis was done for the suspension system showing the changes in the pitching that may result from adopting anti-dive front suspension, progressive rear springs or a different pinion position.

107

108 FSf

Symbol list m1 I1 x1 , y1 θ1 rr Nr Sr Mr m4 I4 x4 , y4 θ4 rf Nf Sf Mf m2 I2 x2 , y2 c2 , s2 I12 l12 MSr MFr

MFf

rear wheel mass. rear wheel moment of inertia. coordinates of rear wheel centre G1. rear wheel rotation. rear wheel radius. rear wheel load. rear wheel longitudinal force. rear wheel rolling resistance. front wheel mass. front wheel moment of inertia. coordinates of front wheel centre G4 . front wheel rotation. front wheel radius.

Fa TC

θ3 rp rc a, b

a1 , b1

front wheel load. front wheel longitudinal force.

a3 , b3

front wheel rolling resistance.

a l ε λ1 , λ2 , λ3

frame/driver mass. frame/driver moment of inertia. coordinates of frame point P2 . director cosines of unit vector u 2 , fixed to the frame. rear fork moment of inertia (mass is divided between G1 and P2 ). rear fork length (distance between G1 and P2 ). torque of suspension on rear fork (acts between frame and fork). rear braking torque (acts between fork and rear wheel).

g* g u v ϕ ψ

force of suspension on front wheel (acts between frame and wheel). front braking torque (acts between frame and front wheel). aerodynamic drag. chain tension (acts between frame and rear wheel). chain angle (derived from known positions of point G1and sprocket centre). sprocket radius. rear sprocket radius. coordinates of frame/driver centre of mass in a reference frame with origin in P2 and x axis u2 . coordinates of aerodynamic drag centre in a reference frame with origin in P2 and x axis u2 . coordinates of sprocket centre in a reference frame with origin in P2 and x axis u 2 . front suspension sliding axis. distance between P2 and axis a . inclination of sliding axis a . Lagrange multipliers. apparent gravity (includes acceleration of moving reference frame). acceleration of gravity. forward velocity. lateral velocity. roll angle. yaw angle.

Appendix 1 The mathematical model for the in-plane analysis considers the motorcycle made-up of the 4 bodies shown in Fig.A1 below. They are: 1) the rear wheel, 2) the frame plus driver considered as rigidly connected, 3) the rear fork, 4) the front wheel.

108

109 aerodynamic drag

mg

*

G2 u1 θ1 G1(x1,y1)

u12

(x2,y l 2)

P2(x2,y2) u2

a

ε m4 g*

m1 g*

u4

θ4 G4(x4,y4)

front tyre forces

rear tyre forces

Fig. A1 This system is described by means of 10 natural coordinates, namely: the rear wheel centre x1 , y1 , the rear wheel rotation5 θ1 , the position x2 , y2 of point P2 (shared between rear fork and frame), the direction cosines s2 , c2 of unit vector u 2 fixed to the frame, the front wheel centre x4 , y4 and rotation θ 4 . Since there are only 7 degrees of freedom (horizontal and vertical displacements of frame, frame pitch, strokes of the two suspensions, rotations of the two wheels), three coordinates are redundant. The way in which the redundant coordinates are dependent on one another is expressed by the constraint equations listed below; the first expresses the fact that vector u 2 must have a unit length, the second that the distance between points P2 and G1 must be constant and equal to rear fork length, the third that the front wheel centre G4 must stay on line a fixed to the frame which represents the front suspension sliding axis. c2 2 + s2 2 = 1

( x2 − x1 )2 + ( y2 − y1 )2 = l12 2

(A1.1)

( x4 − x2 )[c2c(ε ) − s2 s(ε )] + ( y4 − y2 )[s2c(ε ) + c2 s(ε )] = l Pseudo-gravity forces ( mg* , m1g* , m4 g* ) act on the frame and the two wheels (not on rear fork because its mass is divided between points G1 and P2 ) which depend on the apparent gravity g * that accounts for the acceleration of the reference frame. For the centre of mass of the frame it is:

[

]

g* = g cos(ϕ ) + uψ˙ s(ϕ ) − h ϕ˙ 2 + ψ˙ 2 s 2 (ϕ ) + (v˙ + bψ˙˙ )s(ϕ )

(A1.2)

where ϕ is the roll angle, ψ the yaw, u the longitudinal velocity, v the lateral velocity of the moving reference frame as derived from the optimal manoeuvre.

5

To be rigorous angle θ1 should not be considered a “natural” coordinate. In this special case nevertheless, it is a cyclic coordinate and never appears as θ1 but only as θ˙1 . This latter can be considered as a new natural coordinate: in fact trigonometric functions of θ1 never appears.

109

110 6

Tyre loads ( Nr , N f ) are modelled as functions of tyre deformation with a simple linear model . Tyre longitudinal forces ( Sr , S f ) are modelled with a Pacejika-like formula as functions of longitudinal slip. Rolling resistance ( Mr , M f ) and aerodynamic drag ( Fa ) are also included. The internal forces (not shown in figure) are: the forces due to suspensions ( MSr acting between frame and rear fork, FSf acting between frame and front wheel); the force TC exchanged between rear wheel and frame by means of the chain and due to the motor; the forces due to the brakes and acting between front wheel and frame ( MFf ) or between rear wheel and fork ( MFr ), respectively. This leads to the 10 ordinary differential equations listed below, which, with the three constraints equations above, form a system of 13 differential algebraic equations in 13 unknowns: the 10 natural coordinates plus the three lagrange multipliers λ1 , λ2 , λ3 .  I12  I12 y1 − y2 ( MFr − MSr )  m1 + 2  x˙˙1 − 2 x˙˙2 + 2λ2 ( x1 − x2 ) = Tc c(θ 3 ) + Sr + l12  l12 l12 2 

(A1.3)

 I12  I x −x y1 − 122 ˙˙ y2 + 2λ2 ( y1 − y2 ) = − g* m1 + Tc s(θ 3 ) + Nr − 1 2 2 ( MFr − MSr )  m1 + 2  ˙˙ l12  l12 l12  I1θ˙˙1 = Mr + MFr + rr Sr − rc Tc −

−

 I12 I12  ˙˙ x m s2 ) − 2λ2 ( x1 − x2 ) + λ3 [s2 s(ε ) − c2 c(ε )] = + + 1 2   x˙˙2 + m2 ( a c˙˙2 − b ˙˙ l12 2 l12 2   y −y = − Tc c(θ 3 ) − 1 2 2 ( MFr − MSr ) − [s2 c(ε ) + c2 s(ε )]FSf − Fa l12  I12 I  ˙˙ y +  m2 + 122  ˙˙ y2 + m2 (b c˙˙2 + a ˙˙ s2 ) − 2λ2 ( y1 − y2 ) − λ3 [s2 c(ε ) + c2 s(ε )] = 2 1 l12 l12   x −x = − m2 g* − Tc s(θ 3 ) + 1 2 2 ( MFr − MSr ) l12

[

]

[

]

[

]

m2 ( a x˙˙2 + b ˙˙ y2 ) + m2 ( a 2 + b 2 ) + I2 c˙˙2 + 2λ1c2 + λ3 ( x4 − x2 )c(ε ) + ( y4 − y2 )s(ε ) =

[

] (

[

]

m2 ( a ˙˙ y2 − b x˙˙2 ) + m2 ( a 2 + b 2 ) + I2 ˙˙ s2 + 2λ1s2 + λ3 ( y4 − y2 )c(ε ) − ( x4 − x2 )s(ε ) =

[

] (

)

= − m2 g a + TC c(θ 3 )b3 − s(θ 3 )a3 + rp c2 − MSr + MFf c2 + s2 ( y2 − y4 ) + l[2 s2 2 c(ε ) + 2 s2 c2 s(ε ) − c(ε )] + b1 Fa + FSf s2 s(ε ) − c2 c(ε ) *

m4 x˙˙2 − λ3 [s2 s(ε ) − c2 c(ε )] = S f + FSf [s2 c(ε ) + c2 s(ε )] m4 ˙˙ y2 + λ3 [s2 c(ε ) + c2 s(ε )] = − m4 g* + N f + FSf [s2 s(ε ) − c2 c(ε )]

6

)

= − m2 g b − TC c(θ 3 )a3 + s(θ 3 )b3 + rp s2 + MSr + MFf s2 + c2 ( y2 − y4 ) + l[2c2 2 s(ε ) + 2 s2 c2 c(ε ) − s(ε )] − a1 Fa + FSf s2 s(ε ) − c2 c(ε ) *

Detailed expressions for forces are not given for brevity.

110

111 I4θ˙˙4 = M f + MFf + rf S f This system is solved using the penalty formulation [6]. Chain tension TC and braking torques MFf and MFr are control inputs that are defined by a simple control algorithm (PD) in order to reproduce the forward velocity u of the optimal manoeuvre.

Appendix 2 Reference motorcycle parameters. Parameter

Value (SI units)

Motorcycle mass* m (Kg)

171.9

Roll moment of inertia* I x (Kg m2)

9.0

Pitch moment of inertia* I y (Kg m2)

25.4

Yaw moment of inertia* I z (Kg m2)

20.0

Roll-Yaw product of inertia* I xz (Kg m2)

-2.5

Wheelbase p (m)

1.345

Horizontal position of mass centre b - see Fig.1 (m)

0.626

Height of mass centre h - see Fig.1 (m)

0.642

Parameter for wheels/flywheel gyroscopic effect I e (Kg m)

3.049

Average toroidal radius rt (m)

0.065

Rear wheel toroidal radius rt r (m)

0.08

Front wheel toroidal radius rt f (m)

0.05

Tyre** sideslip stiffness C1 (1/rad)

20

Tyre camber stiffness C2 (1/rad)

0.8

Tyre relaxation length σ r (m)

0.2

Coefficient for the adherence in the penalty function f a

1.3

* driver included

σr ˙ F + F = (C1λ + C2ϕ ) N u N for the vertical loads.

** supposed equal for both rear and front tyres (forces are evaluated with equation: stands for the lateral force (both front and rear),

λ and ϕ

for sideslip and roll angles,

where

F

References 1. Da Lio, M.: Analisi della Manovrabilità dei Veicoli. Un Approccio Basato sul Controllo Ottimo. ATA Journal 50-1 (1997), pp. 35-42 (in Italian). 2. Cossalter, V, Da Lio, M, Lot, R, Fabbri, L.: A General Method for the Evaluation of Vehicle Manoeuvrability with Special Emphasis on Motorcycles, Vehicle System Dynamics, to be published.

111

112 3. Cossalter, V., Da Lio, M., Biral, F., Fabbri, L.: Evaluation of Motorcycle Manoeuvrability with the Optimal Manoeuvre Method, SAE Meeting: 1998 Motorsports Engineering Conference&Exposition, Dearborn, Michigan, 16-19 november 1998 4. Noton, A.R.M.: Variational Methods in Control Engineering. Pergamon, 1965. 5. Bryson, A.E., Ho, Y.: Applied Optimal Control. Hemisphere Publishing, 1975. 6. Lewis, F.L., Symros, V.L.: Optimal Control. Wiley, 1995. 7. Jalón, J.C., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems, Springer Verlag, 1994.

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