Additional refers to low speed. The gradient is defined at certain high-speed steady state cornering conditions, in-
cluding straight-line driving. Steer angle can be either road wheel angle or steering wheel angle.
The first term in Eq [4.9], , can be seen as a reference steer angle 𝐴, which is the Ackermann steer angle. The (ISO 8855) defines 𝐴 as the steer angle which would be needed to give same instantaneous centre of rotation if the vehicle would have had two axles, perfect Ackermann steering and no tyre side slip. So, 𝐴 corresponds to L/R for a two-axle vehicle. An extended definition of 𝐴 is the steering angle
needed for a certain curvature 1 𝑝 at low speed. The understeer gradient is then understood as a measure of how this is changed with increasing speed .
The understeering gradient, 𝐾 , is normally positive, which means that most vehicles require more steer angle for a given curve, the higher the speed is. Depending on the sign of 𝐾 a vehicle is said to be oversteered (if 𝐾 < 0), understeered (if 𝐾 > 0) and neutral steered ( 𝑓 𝐾 = 0). In practice, all vehi- cles are designed as understeered, because over steered vehicle would become unstable and difficult to control.
The 𝐾 in Eq [4.9] is called “understeer gradient” and has hence the unit rad/N or 1/N. Sometimes one can see slightly other definitions of what to include in definition of understeer gradient, which have different units, see 𝐾 2 and 𝐾 in Eq [4.13].
=𝐿 𝑅+ 𝑟∙𝑙𝑟− 𝑓∙𝑙𝑓 𝑓∙ 𝑟∙𝐿 ∙ ∙𝑔∙𝑣𝑥2 𝑔∙𝑅 = {𝐾 2= ∙ ∙ 𝑟∙𝑙𝑟− 𝑓∙𝑙𝑓 𝑓∙ 𝑟∙𝐿 [1 𝑜𝑟 𝑟 ]} = 𝐿 𝑅+ 𝐾 2∙ 𝑣𝑥2 𝑔∙𝑅; =𝐿 𝑅+ 𝑟∙𝑙𝑟− 𝑓∙𝑙𝑓 𝑓∙ 𝑟∙𝐿 ∙ ∙𝑣𝑥2 𝑅 = {𝐾 = ∙ 𝑟∙𝑙𝑟− 𝑓∙𝑙𝑓 𝑓∙ 𝑟∙𝐿 [ ⁄ 2𝑜𝑟 𝑎 ⁄ 2]} = 𝐿 𝑅+ 𝐾 ∙ 𝑣𝑥2 𝑅 ; [4.13] 𝐾 is the definition used in (ISO 8855). For 𝐾 , one can sometimes see the unit “rad/g” used, which
present compendium recommended to not use.
If vertical loads on axles are only due to gravity ( 𝑖 = ( − 𝑙𝑖) ⁄ ) and tyres linear with vertical
load (𝐶𝑖𝑦 = 𝐶𝐶𝑖𝑦 𝑖 ) we can express 𝐾 2= 1 𝐶𝐶⁄ − 1 𝐶𝐶⁄ ;.
4.3.3.1 Understeering as a Fix Built-In Measure
The understeering gradient 𝐾 can be understood as how much additionally to the reference steer an- gle one has to steer, to reach a certain centrifugal force, = ∙ 2⁄ (or, if using 𝐾 , a certain accel-
eration = 2⁄ ): = 𝐴+ 𝐾 ∙ = 𝐴+ 𝐾 ∙ ∙ 2 ; ⇒ 𝐾 = − 𝐴= ; 𝑜𝑟 𝐾 =
Understeering is a steady state property and does depend on which axle is steered, see Figure 4-19. 1. If small (Low speed)
2a: …𝐶 𝑦≪ 𝐶 𝑦⇒ curve radius increases, which indicates under-steering
2b: …𝐶 𝑦≪ 𝐶 𝑦⇒ curve radius increases, which indicates over-steering 2. If increases, increases and… 𝐶 𝑦 𝐶 𝑦 𝑦 𝑦
Figure 4-19: Under- and over-steering for a two-axle vehicle with 𝑙 = 𝑙 . It does not depend on which axle is steered, but which axle is first in the direction of motion. Figure drawn for vehicle driving forward.
4.3.3.2 Understeer Gradient as Varying with Steady State Lat-
eral Acceleration
So far, the understeering gradient is presented as a fix vehicle parameter. There is nothing that says that a real vehicle behaves linear, so in order to get a well-defined value of 𝐾 , the and the should be small. However, if we accept that 𝐾 can change with , 𝐾 can be defined as a differential quantity. 𝐾 can also be understood as how much the additional steer angle, , has to increase per increased centrifugal force, , or per centrifugal acceleration, :
𝐾 =𝜕( ) 𝜕 = 𝜕 𝜕 ( − 𝐴) = 𝜕 𝜕 ; 𝑜𝑟 𝐾 = 𝜕( ) 𝜕 = 𝜕 𝜕 ; [4.14] Equation [4.14] shows the understeering gradient as a function of , rather than a scalar parameter. But it is still fix and built-in in the vehicle. If assessing understeering for a lateral forces up to near road friction limit, Equation [4.14] is more relevant than Equation [4.9], because it reflects that understeer- ing gradient changes.
4.3.3.3 Understeering as a Varying Quantity during a Transient
Manoeuvre
A third understanding of the word understeering is quite different and less strictly defined. It is to see the understeering as a variable during a transient manoeuvre. For instance, a vehicle can be said to un- dersteer if tyre side slip is larger on front axle than on rear axle, | | > | |, and over-steer if opposite, | | > | |. This way of defining understeering and oversteering is not built-in in vehicle but varies over time through a (transient) manoeuvre. E.g., when braking in a curve a vehicle loses grip on rear axle due to temporary load transfer from rear to front. Then the rear axles can slide outwards signifi- cantly, and the vehicle can be referred to as over-steering at this time instant, although the built-in un- dersteering gradient is >0. This “instantaneous” under-/over-steering (binary, not an understeer gra- dient) can be approximately found from log data with this simple approximation:
𝑛𝑒 𝑡 𝑎𝑙= ≈ { ≈ ∙ } ≈
∙
≈ { 𝑦≈ ∙ } ≈
∙ 𝑦
2 ; [4.15]
If the actual vehicle has | | < | 𝑛𝑒 𝑡 𝑎𝑙| the vehicle oversteers, and vice versa. This is often very prac-
tical since it only requires simply logged data, and 𝑦. Note that when and 𝑛𝑒 𝑡 𝑎𝑙 have differ-
ent signs, neither understeer or oversteers is suitable as classification, but it can sometimes be called “counter-steer”. An example of applying Eq [4.15] is shown in Figure 4-20, where one also see that the ESC system does not follow the Eq [4.15] when deciding ESC interventions; ESC has more advanced “reference models”, see 4.6.2.1.
A second look at Equation [4.9] tells us that we have to assume absence of propulsion and braking on front axle, = 0, to get the relatively simple final expression. When propulsion on front axle ( > 0), the required steer angle, , will be smaller; the front propulsion pulls in the front end of
the vehicle. When braking on front axle ( < 0), the required steer angle, , will be larger; the front braking hinders the front end to turn in. To keep constant, which is required within definition of steady state, one have to propel the vehicle because there will always be some driving resistance to overcome. Driving fast on a small radius is a situation where the driving resistance from tyre lateral forces becomes significant, which is a part of driving resistance which was only briefly mentioned in 3.2.
Figure 4-20: Log data from passenger car with ESC in a double lane change. Upper: Vehicle motion. Middle:
𝑛𝑒 𝑡 𝑎𝑙 from Eq [4.15] used to define “instantaneous under-/over-steering” (US/OS). Lower: Pressure to
each wheel brake.
4.3.3.4 Neutral Steering Point
An alternative measure to understeering coefficient is the longitudinal position of the neutral steering point. The point is defined for lateral force disturbance during steady state straight-ahead driving, as opposed to steady state cornering without lateral force disturbance. The point is where a vehicle-ex- ternal lateral force, such as wind or impact, can be applied on the vehicle without causing a yaw veloc- ity ( = 0), i.e. only causing lateral velocity ( 𝑦≠ 0). From this definition, we can derive a formula for
calculating the position of the neutral steering point, see Figure 4-21. The result is con-
densed in Eq [4.16]. 𝑙 = 𝐶 ∙ 𝑙 − 𝐶 ∙ 𝑙 𝐶 + 𝐶 = 𝐾 ∙ 𝐶 ∙ 𝐶 𝐶 + 𝐶 ; 𝑒𝑟𝑒 𝐾 = 𝐶 ∙ 𝑙 − 𝐶 ∙ 𝑙 𝐶 ∙ 𝐶 ∙ ; [4.16] We can see that the understeer gradient from steady state cornering model appears also in the for- mula for neutral steering point position, 𝑙 . Since 𝐶 𝐶 𝑛 are positive, the neutral steering point is behind of CoG for understeered (two-axle) vehicles, and in front of CoG for oversteered (two-axle) ve- hicles. This is why 𝑙 and 𝐾 can be said to be alternative measures for the same vehicle function/char- acter, the yaw balance.
4.3.4 Required Steer Angle
A fundamental property of the vehicle is what steer angle that is required to negotiate a certain curva- ture (=1/path radius = 1 𝑝). This value can vary with longitudinal speed and it can be normalized with wheel base, . From Equation [4.9], we can conclude:
𝑵𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒆𝒅 𝒓𝒆𝒒𝒖𝒊𝒓𝒆𝒅 𝒔𝒕𝒆𝒆𝒓𝒊𝒏𝒈 𝒂𝒏𝒈𝒍𝒆 = ∙ 𝑝= 1 + 𝐾 ∙ ∙
2
; [4.17] The normalized required steer angle is plotted for different understeering gradients Figure 4-22. It is the same as the inverted and normalized curvature gain, see 4.3.5.2.
L
F
fyF
ry = 0b
Mathematical model:
Equilibrium: 0 = 𝑦+ 𝑦+ 𝑒; 0 = 𝑦 𝑙 − 𝑦 𝑙 − 𝑒 𝑙 ; Constitution: 𝑦 = −𝐶 𝑠 𝑦; and 𝑦= −𝐶 𝑠 𝑦; Compatibility: 𝑠 𝑦 = 𝑠 𝑦 = 𝑣𝑦 𝑣𝑥; Eliminate 𝑦, 𝑦 𝑠 𝑦 𝑠 𝑦 𝑒yields: 𝑙 = 𝑟 𝑙𝑟− 𝑓 𝑙𝑓 𝑓+ 𝑟 ;Identify understeering gradient, 𝐾 = 𝑟 𝑙𝑟− 𝑓 𝑙𝑓
𝑓 𝑟 𝐿 Then:
𝑙
= 𝐾
𝐶
𝐶
𝐶
+ 𝐶
;
Physical model:
• Steady state ( = 𝑦= = 0) • Straight ahead driving ( = 0) • No steering• Small tyre and vehicle side slip. Then, angle=sin(angle)=tan(angle).
F
ey𝑙
𝑙
𝑙
Figure 4-21: Model for definition and calculation of neutral steering point.
0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 Ku = 2.525e-6 [1/N] Ku = 0e-6 [1/N] Ku = -1.794e-6 [1/N] vx [m/s] re q u ir e d s te e ri n g , d f* R /L [ ra d ] Understeered Oversteered Neutral steered
Critical speed (for oversteered vehicle) Characteristic speed (for
understeered vehicle)
Figure 4-22: Normalized steer angle ( ∙ ⁄ ) for Steady State Cornering