There are a number of vehicles that tilt or bank toward the inside of turns in the manner of an airplane when it is executing a coordinated turn. In a steady turn, these vehicles tilt to an angle such that the vector combination of the acceleration of gravity and the centrifugal acceleration lies along a symmetry axis of the vehicle. Passengers in such vehicles have the sensation that there is no lateral acceleration relative to their own bodies, but they are pushed down in their seats as if the acceleration of gravity had increased somewhat. Coffee in a cup tilted with the vehicle has no tendency to spill even when the centripetal acceleration in a steady turn is high.
Cars and trucks with conventional suspensions, of course, tend to tilt toward the outside of turns. This direction of tilt is undesirable from the point of view of passenger comfort, and the tilt also shifts the center of mass position toward the outside wheels thus increasing the possibility of over- turning. The suspension springs and devices called antisway bars resist this tilting tendency, but their effectiveness is limited by the necessity providing a certain amount of suspension compliance for the sake of passenger comfort and to allow the wheels to follow roadway unevenness. Recently developed partially active suspensions are able to reduce the amount of tilt in conventional automobiles, but generally, these active suspension sys- tems do not attempt to tilt the car toward the inside of a turn.
Among ground vehicles that tilt toward the inside of turns, bicycles and motorcycles are obvious examples. In addition, a number of advanced trains tilt for passenger comfort in high-speed turns. Tall thin commuter vehicles have also been developed which tilt toward the inside of turns to reduce the chance of overturning in sharp turns.
Some tilting rains and commuter vehicles are tilted by direct action of an actuator and an automatic control system that forces the vehicle body to
tilt to the inside of a turn against its natural tendency to tilt toward the outside. This requires an active system with an energy supply, but the design of this type of tilting servomechanism is fairly straightforward.
Single-track vehicles such as motorcycles are tilted in a completely different way by action of the steering system. In this chapter, the dynamics and control of steering-controlled banking vehicles will be discussed. These vehicles are unstable in the absence of active control of the steering either by a human operator or an automatic control system.
Obviously, steering loses effectiveness as a means for balancing such vehicles at very low speeds or at rest, so another means of achieving balance must be provided. (Bicycle and motorcycle riders use their feet for this pur- pose.) Once a certain speed has been reached, the steering system not only is used to cause the vehicle to follow a desired path, but also is used to sta- bilize the vehicle and to cause it to bank at the proper angle in a steady turn. The present analysis applies not only to single-track two-wheeled vehicles, but also to three- or four-wheeled vehicles with a roll axis near the ground, a high center of gravity, and a suspension with very low roll stiffness. For such vehicles, direct tilt control using an actuator would have to be used to supplement steering control at low speeds.
There have been numerous mathematical studies of the balancing problem for bicycles and motorcycles over a period of many years. Many of the more recent studies include several degrees of freedom, geometric nonlinearities, and nonlinear tire-force models. Such mathematical models are often so complex that insight into the essential dynamics and control of the vehicles is nearly impossible. Here we will introduce a number of simplifications in order to achieve a low-order linear model that is easily understandable and yet illustrates the essential dynamics and control features of steering-controlled banking vehicles.
A. Development of the Mathematical Model
Fig. 7.1shows a number of the dimensions and variables associated with the mathematical model of a tilting vehicle. For a single-track vehicle, the sketch in Fig. 7.1 is to be imagined as existing in the ground plane.
For a multiwheeled vehicle, the two wheels represent equivalent single wheels for the front and rear axles much as was done for the ‘‘bicycle’’ model for automobiles. The ‘‘ground’’ plane would then pass through the roll center of the suspension. The dimensions a and b relate to the distances from the projection of the center of mass to the front and rear axles in the ground plane. The velocity components U and V describe the
velocity of the center of mass projection on the ground plane. (Because of the time-varying tilt angle, the center of mass has other velocity compo- nents besides these components of the projection in the ground plane.)
The coordinates x and y locate the ground plane projection of the center of mass in inertial space, and the angle /, which may be large, represents the orientation of the vehicle with respect to the y-axis. Although most vehicles use only front-wheel steering, both front and rear-steering angles, dfand dr, will be included. It will be shown that rear- wheel steering alone poses difficult control problems, but experimental vehicles have been constructed using a combination of front- and rear- wheel steering. Both steer angles will be assumed small since the model is intended for use at relatively high speeds when the turn radius, R, is large with respect to the wheelbase, (a+b).
A major simplifying assumption is that the slip angles are negligible. This may seem to be an odd assumption since in the analysis of automo- biles inChapter 6,slip angles played a major role in the stability analysis. In this chapter, however, the dynamics of tilting is prominent and the slip angle effects are not important as long as the tires do not skid. This assumption certainly breaks down at high lateral acceleration and it precludes the use of nonlinear tire characteristics, but it has the great advantage that no tire characteristics at all are involved in the model. It is certainly common experience, when riding a bicycle, that it is hard to discern any slip angle when riding at moderate lean angles. The wheels appear to roll almost exactly in the direction that they are pointed with no noticeable slip angles.
We also assume that the forward velocity U is constant. With these assumptions, motion in the ground plane is determined purely kinemati- cally. Using the small angle assumption,
tan dfidf; tandridr ð7:1Þ
simple geometric considerations result an expression for the turn radius. (Problem 7.1 involves the derivation of this relationship.)
R¼ ða þ bÞ=ðdf drÞ ð7:2Þ
It may be useful to note here that Eq. (7.2) is a generalization of the steer angle relationships encountered previously in the ‘‘bicycle model’’ of an automobile. If there is no rear steer angle and if the magnitudes of the front and rear slip angels are equal, then Eqs. (6.53) and (7.2) are identical. In addition, if the understeer coefficient should vanish, Eqs. (6.65) and (6.66) would match Eq. (7.2) in the absence of a rear steer angle. This means that if the tilting vehicle was a strictly neutral steer vehicle, the relationships would be the same as if the slip angles were actually zero (seeFig. 6.8).
The yaw rate r is given by the expression
riU=R ¼ Uðdf drÞ=ða þ bÞ ð7:3Þ
The lateral velocity in the ground plane is found again purely kinematically by considering the lateral velocities at the front and the rear.
ViUðbdfþ adrÞ=ða þ bÞ ð7:4Þ
(Problem 7.2. deals with this relationship.)
Finally the ‘‘slip angle’’ for the center of mass projection in the ground plane, b, is
If it is desired to track the location of the center of mass projection in the ground plane during computer simulation, for example, the following equations can be used:
/¼ r; y ¼ U cos/ V sin/; x ¼ U sin/ þ V cos/ ð7:6Þ
With the assumption of zero slip angles for the front and rear wheels, the motion in the ground plane is completely determined by the time histories of the front and rear steer angles. Now the dynamics of the tilting and acceleration of the vehicle body will be modeled.
B. Derivation of the Dynamic Equations
Fig. 7.2 shows the vehicle body with its center of mass a distance h above its ground plane projection and tilted at the lean angle h. The angle h functions as the single geometric degree of freedom. The principal axes of the body are assumed to be parallel to the 1-, 2-, and 3-axis system shown in the figure. The 1-axis lies along the longitudinal axis of the vehicle in the x–y plane, the 3-axis is aligned with the vehicle axis that is vertical when the lean angle is zero, and the 2-axis is perpendicular the 1- and 3-axes. The principal moments of inertia relative to the mass center are denoted I1, I2, and I3. When writing the expression for kinetic energy, one can imagine
the 1-, 2-, and 3-axes translated to the center of mass and actually being then the principal axes for the body.
The equation of motion will be derived using Lagrange’s equation much as was described inChapter 5,Eqs. (5.7)–(5.9). In this case of three- dimensional motion, the kinetic energy expression is more complicated than for plane motion and there is a potential energy term having to do with the height of the center of mass in the gravity field. Because of the zero slip angle assumption, the tire lateral forces are perpendicular to the tire velocities and thus do no work. This means that there is no need for a generalized force to represent these forces. The steer angles merely provide a prescribed kinematic motion of the ground plane axis about which the vehicle tilts. The motion variables U, which is assumed to be constant, and V, which is determined by the steer angles through Eq. (7.4), will enter the expression for the kinetic energy.
From Fig. 7.2, one can find the square of the velocity of the center of mass, noting that the U and V velocity components lie in the ground plane. The velocity of the center of mass is composed of the velocity of its projection in the ground plane with components U and V added to the components induced by the angular rotation rates r andh. All the com-. ponents of the center of mass velocity are shown translated to the center of mass location in Fig. 7.2. The square of the center of mass velocity can be written as the sum of the squares of two orthogonal components. The first component is the component in the 1-direction, (Urhsinh). The second component has to do with the vector sum of V and hh . These two vectors. are separated by the angle h. The square of the vector sum can be found using the law of cosines to be (V2+h2h.2+2Vhhcosh). The final expression. for the square of the velocity of the center of mass is
v2c¼ ðU rh sinhÞ2þ ðV2þ h2h2þ 2VhhcoshÞ ð7:7Þ The angular velocities along the 1-, 2-, and 3-directions are seen to be
x1¼ .
h; x2 ¼ r sinh; x3¼ r cosh ð7:8Þ
Then, the kinetic energy expression appropriate for three-dimensional motion (Crandall et al., 1968) is
T¼ mv2c=2 þ ðI1x21þ I2x22þ I3x23Þ=2 ð7:9Þ in which Eqs. (7.7) and (7.8) will be substituted.
The potential energy expression is just mg times the height in the gravity field,
V¼ mgh cosh ð7:10Þ
where in this case, V represents the potential energy rather than a lateral velocity.
Finally, substituting Eqs. (7.7)–(7.10) into Lagrange’s equation for h(t), which happens to be exactly Eq. (5.7), the resulting equation of motion is found to be
ðI1þ mh2Þ ¨h þ ðI3 I2 mh2Þr2cosh sinh mgh sinh
¼ mh coshðV.þ rUÞ ð7:11Þ
This equation is certainly a more complicated equation of motion than necessary for present purposes since no small-angle approximations have yet been made for the lean angle h. Only in extreme cases do bicycles or motorcycles achieve large lean angles so h will be assumed to be small enough to allow the use of the usual trigonometric small-angle approx- imations. The middle term on the left-hand side of Eq. (7.11) will be neglected since it involves the product of the small lean angle and the square of the yaw rate, which is also small for cases of practical interest. After applying the small-angle approximations for functions of h, a linearized equation results.
ðI1þ mh2Þ ¨h mghh ¼ mhð .
Vþ rUÞ ð7:12Þ
(This version of the equation can also be derived using Newton’s laws and the small-angle approximations from the beginning, but it requires the in- troduction of the tire lateral forces and then their subsequent elimination.) The terms on the right side of Eq. (7.12) are determined entirely by the time-varying steer angles using the kinematic equations derived above [Eqs. (7.3) and (7.4)]. After using these relations, the final form of the linearized equation of motion is
ðI1þ mh2Þ ¨h mghh ¼ mh ða þ bÞðb . dfUþ dfU2þ a . drU drU2Þ ð7:13Þ This equation has several interesting features. For example, if the steer angles and their rates were all zero, the equation would describe an upside- down pendulum (for small angles). The term (I1+mh2) can be recognized as the moment of inertia of the pendulum about a pivot point in the ground
plane. This should be no surprise since a motorcycle with locked steering would surely tend to fall over just as an upside-down pendulum would.
The right side of Eq. (7.13) indicates that as U!0, steering action becomes ineffective in influencing the lean angle as would be expected. What may come as a surprise is that not only do the steer angles influence the lean angle, but also the steer angle rates have an effect. In fact, for low speeds, the steer angle rates are more important than the angles themselves since as the speed decreases, the effectiveness of the rates declines only with Uwhile the effectiveness of the angles declines with U2.