4.1 Introduction
The lateral motion of a vehicle is needed to follow the roads’ curves and select route in intersections as well as to laterally avoid obstacles that appear. The vehicle needs to be steerable. With some
simplification, one can say that lateral dynamics is about how steerable the vehicle is for different given longitudinal speeds. Vehicle steering is studied mainly through the vehicle degrees of freedom: yaw rotation and lateral translation.
A vehicle can be steered in different ways:
• Applying steering angles on, at least one, road wheel. Normally both of front wheels are steered with approximately same angle.
• Applying longitudinal forces on road wheels; directly by unsymmetrical between left and right side of vehicle, e.g. one sided braking, or indirectly by deliberately use up much friction
longitudinally on one axle in a curve, so that that axle loses lateral force.
• Articulated steering, where the axles are fixed mounted on the vehicle body, but the vehicle itself can bend.
The chapter is organised with one group of functions in each section as follows: • 4.2 Low speed manoeuvers
• 4.3 Steady state cornering at high speed • 4.4 Stationary oscillating steering • 4.5 Transient handling
• 4.6 Lateral Control Functions
Most of the functions in “4.6 Lateral Control Functions”, but not all, could be parts of ”4.5 Transient handling”. However, they are collected in one own section, since they are special in that they partly rely on software algorithms.
The lateral dynamics of vehicles is often experienced as the most challenging for the new automotive engineer. Longitudinal dynamics is essentially motion in one plane and rectilinear. Vertical dynamics may be 3 dimensional, but normally the displacements are small and in this compendium the vertical dynamics is mainly studied in one plane as rectilinear. However, lateral dynamics involves motion in the vehicle coordinate system which introduces curvilinear motion since the coordinate system is rotating as the vehicle yaws.
The turning manoeuvres of vehicles encompass two concepts. Handling is the driver’s perception of the vehicle’s response to the steering input. Cornering is usually used to describe the physical response (open-loop) of the vehicle independent of how it influences the driver.
4.1.1 References for this chapter
• “Chapter 25 Steering System” in Reference (Ploechl, 2013).• “Chapter 27 Basics of Longitudinal and Lateral Vehicle Dynamics” in Reference (Ploechl, 2013). • “Chapter 8: Electronic Stability Control” in Reference (Rajamani, 2012)
4.2 Low speed manoeuvers
This section is about operating vehicles in low speeds, including stand-still and reverse. Specific for low speed is that inertial effects can be neglected, i.e. one can assume that left hand side in motion
LATERAL DYNAMICS
In low speed, one often needs to find the path with orientation and understand the steering system and how tyres can be modelled to track ideally. This, and the resulting one-track model for low speeds, is described in Sections 4.2.1, …,4.2.6.
4.2.1 Path with orientation
The path and path with orientation was introduced in Section 1.5. The path, in global coordinate system, is related to vehicle speeds, in vehicle fix coordinates, as given in Figure 4-1 and Equation [4.1].Knowing ( (𝑡) (𝑡) 𝜔 (𝑡)), we can determine “path with orientation” (𝑥(𝑡) 𝑦(𝑡) (𝑡)), by time integration of the right hand side of the equation. Hence, the positions are typically “state variables” in lateral dynamics models.
x y
𝜔
Figure 4-1: Model for connecting “path with orientation” to speeds in vehicle coordinate
system.
[𝑥̇𝑦̇] = [cos( ) sin( )
sin( ) cos( ) ] ∙ [ ] ; ̇ = 𝜔 ;
[4.1] It should be noted that in some problems, typically manoeuvring at low speed, the real time scale is of less interest. Then, the problem can be treated as time independent, e.g. by introducing a coordinate, s, along the path, as in Equation [4.2].
𝑥 = 𝑠̇ ∙ cos( ) 𝑠̇ ∙ sin( ) ; 𝑦 = 𝑠̇ ∙ cos( ) + 𝑠̇ ∙ sin( ) ; =𝜔 𝑠̇ ; 𝑤ℎ 𝑟 𝑝𝑟 𝑛𝑜𝑡 𝑠 𝑟 𝑛𝑡 𝑡 𝑜𝑛 𝑤 𝑡ℎ 𝑟 𝑠𝑝 𝑡 𝑡𝑜 𝑠 [4.2]
Here, 𝑠̇ can be thought of like an arbitrary time scale, with which all speeds are scaled. One can
typically chose 𝑠̇ = 1 [ 𝑠⁄ ]. However, in this compendium we will keep notation t and the dot notation for derivative.
4.2.2 Vehicle and wheel orientations
For steered wheels, there are often reason to translate forces and velocities between vehicle coordinate system and wheel coordinate system, see Figure 4-2 and Equation [4.3].
Transformation from wheel coordinates to vehicle coordinates: [ ] = [ cos( ) sin( ) sin( ) cos( ) ] ∙ [ 𝑤 𝑤] ; 𝑛 [ ] = [ cos( ) sin( ) sin( ) cos( )] ∙ [ 𝑤 𝑤] ; Transformation from vehicle coordinates to wheel coordinates:
[ 𝑤 𝑤] = [ cos( ) sin( ) sin( ) cos( )] ∙ [ ] ; 𝑛 [ 𝑤 𝑤] = [ cos( ) sin( ) sin( ) cos( )] ∙ [ ] ; [4.3]
𝒗
𝒙𝒚𝒗
𝒙𝒗𝒗
𝒚𝒗𝒗
𝒙𝒘𝒗
𝒚𝒘 Components in vehicle coordinates Components in wheel coordinates Velocity of body,above this wheel
𝑭
𝒙𝒚𝑭
𝒙𝒗𝑭
𝒚𝒗𝑭
𝒙𝒘𝑭
𝒚𝒘 Components in vehicle coordinates Components in wheel coordinates Force in road plane,acting on this wheel
𝒗
𝒙𝒚𝒗
𝒙𝒚𝑭
𝒙𝒚𝑭
𝒙𝒚Figure 4-2: Transformation between forces and velocities in vehicle coordinate system and wheel coordinate system.
4.2.3 Steering System
The steering system is here referred to the link between steering wheel and the road wheel’s steering, on the steered axle. It is normally the front axle that is steered. Driver’s interaction is two-folded, both steering wheel angle and torque, which is introduced in Section 2.9. In present section, we will focus on how wheel steering angles are distributed between the wheels.
4.2.3.1 Chassis steering geometry
The most basic intuitive relation between the wheels steering angles is probably that all wheels rotation axes always intersect in one point. This is called Ackermann geometry and is shown in Figure 4-3. The condition for having Ackermann geometry is, for the front axle steered vehicle that:
1 tan( )= 𝑅 𝑤 ⁄ ; 1 tan( 𝑜) =𝑅 + 𝑤 ⁄ ; } ⇒ 1 tan( 𝑜) = 1 tan( )+ 𝑤 ; [4.4]
The alternative to Ackermann steering geometry is parallel steering geometry, which is simply that = 𝑜. Note that Ackermann geometry is defined for a vehicle, while parallel steering is defined for an axle. This means that, for a vehicle with 2 axles, each axle can be parallel steered, which means that the vehicle is non-Ackermann steered. However, the vehicle can still be seen as Ackermann steered with respect to mean steering angles at each axle.
For low-speed, Ackermann gives best manoeuvrability and lowest tyre wear. For high-speed, Parallel is better in both aspects. This is because vehicles generally corner with a drift outwards in curves, which means that the instantaneous centre is further away than Ackermann geometry assumes, i.e. more towards optimal for parallel. Hence the chosen geometry is normally somewhere between Ackermann and parallell.
LATERAL DYNAMICS
Practical arrangement to design the steering geometry is shown in Figure 4-7. The design of linkage will also make the transmission from steering wheel angle to road wheel steering angle non-linear. This can lead to different degrees of Ackerman steering for small and large steering wheel angles.
d
id
oL
w
Common intersection of all wheels’ axes of rotation
L
w
i o)1tan(
)
tan(
1
d
d
1 turn centrea Ackermann error, front. b Ackermann error, rear.
Rr
Figure 4-3: Ackermann steering geometry. Left: One axle steered. Right: Both axles steered and including “Ackermann errors”. From (ISO 8855).
In traditional steering systems, the steering wheel angle has a monotonically increasing function of the steering angle of the two front axle road wheels. This relation is approximately linear with a typical ratio of 15..17 for passenger cars. For trucks the steering ratio is typically 18..22. In some advanced solutions, steering on other axles is also influenced (multiple-axle steering, often rear axle steering). There are also solutions for dynamically adding steering angle through a planetary gear and electric angle-controlled motor on the steering shaft, so called Active Front Steering (AFS).
In reference (Tagesson, 2017), there is a good descriptive chapter about steering systems for heavy vehicles.
4.2.3.2 Steering system forces
(This section has large connection with Section 2.4.6.2 Tyre aligning moment.)
The steering wheel torque, 𝑇𝑠𝑤, should basically be a function of the tyre/road forces, mainly the wheel-lateral forces. This gives the driver a haptic feedback of what state the vehicle is in. The torque/force transmission involves a servo actuator, which helps the driver to turn the steering system, typically that assists the steering wheel torque with a factor varying between 1 and 10, but less for small 𝑇𝑠𝑤 (highway driving) than large 𝑇𝑠𝑤 (parking), see Figure 4-4. Here, the variation in assistance is assumed to be hydraulic and follows a so-called boost curve. At 𝑇𝑠𝑤= , the assistance is ≈0.45/0.55≈1 and for 𝑇𝑠𝑤= 4 Nm, it is ≈0.9/0.1≈10. Assisted Rack Force SteWhlTq UnAssisted Assistance
Figure 4-4: Left: Boost Curve with different working areas depending on the driving envelope. Middle: Torque distribution between manual torque, FM, and assisting torque, FA, depending on applied steering
For vehicle dynamics, one important effect of a steered axle, is that the lateral force on the axle tries to align the steering in the direction that the body (over the steered axle) moves, i.e. towards a zero tyre side slip. This is designed in via the sign of the caster trail, see Figure 4-5. Also, asymmetry in
longitudinal tyre forces (wheel shaft torques and/or brake torques) affects the steering wheel torque. This is analysed in the following.
Steering axis offset at ground, s
(drawn positive)
(sometimes called Scrub radius)
forward
Steering-axis inclination angle,
KPI