Ackermann steering

(trapezoidal

geometry)

(Erasmus Darwin 1758, Rudolph Ackermann 1810) 𝑙= 𝑓 𝑟𝑝𝑖𝑛 ; = −𝑓 −𝑟𝑝𝑖𝑛 ; 𝑙 Steering Knuckle Rack Tie Rod Steering Axis 𝑙 𝑟𝑝𝑖𝑛 𝑆𝐴 𝑆𝐴𝑦 𝑟𝑝𝑖𝑛 𝛾 sym m et ry w he n = 0 𝛾

View from top and rear:

Eq [2.62] shows the relation between steering angles for rack steering, with 𝑇 =Tie Rod Length, 𝑆𝐴 =Steering Arm Length, 𝑆𝐴=Rack to Steering Axis lengths.

𝑇 2= ( 𝑆𝐴 − 𝑆𝐴 cos(𝛾 + 𝑙))2+ ( 𝑆𝐴𝑦− 𝑆𝐴 sin(𝛾 + 𝑙) + 𝑟𝑝𝑖𝑛∙ ) 2 ; 𝑇 2_{= ( 𝑆𝐴 − 𝑆𝐴 cos(𝛾 − ))}2_{+ ( 𝑆𝐴} 𝑦− 𝑆𝐴 sin(𝛾 − ) − 𝑟𝑝𝑖𝑛∙ )2; [2.62] The 𝑇 determines a toe-in. To get _toe-in, design (or adjust) the 𝑇 to:

𝑇 = √( 𝑆𝐴 − 𝑆𝐴 cos(𝛾 − toe-in))2+ ( 𝑆𝐴𝑦− 𝑆𝐴 sin(𝛾 − toe-in))2;.

2.6.2 Steering System Forces

(This section has large connection with 2.2.4.6Other Forces and Moments in Lateral.)

The steering wheel torque, 𝑇 , 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 sys- tem, typically that assists the steering wheel torque with a factor varying between 1 and 10, but less for small 𝑇 (highway driving) than large 𝑇 (parking), see Figure 2-81. Here, the variation in assis- tance is assumed to be hydraulic and follows a so-called boost curve. At 𝑇 = 0, the assistance is ≈0.45/0.55≈1 and for 𝑇 = 4 Nm, it is ≈0.9/0.1≈10. Assisted Rack Force SteWhlTq UnAssisted Assistance

Figure 2-81: 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

wheel torque. From Reference (Rösth, 2007). Right: Unassisted and assisted steering wheel torque.

2.7 Environment Sensing System

This subsystem has to be mentioned since it is maybe the most important new enabler for today’s de- velopment of automated driving. The technology to sense (radar, camera, lidar, GPS, etc) is not typi- cally part of vehicle dynamics, but many vehicle dynamics control functions can be invented or im- proved through usage of the information from the subsystem. Some typically available information is listed in 3.5.1 and 4.6.1. Another vehicle dynamics aspect is that some sensor fusion, but primarily some predictions, can be made using vehicle dynamics models.

2.8 Vehicle Aerodynamics

The flow of air around the vehicle body produces different external forces and moments acting on the vehicle. The fluid mechanics will not be covered in this course. However, practical first order models for aerodynamic forces have been established and are presented here.

2.8.1 Longitudinal Relative Wind Velocity

The most relevant aerodynamic force of interest in this course is the resistance force to forward mo- tion, _𝑎𝑖 , which is proportional to the square of the longitudinal component of the wind speed rela- tive to the vehicle, _𝑒𝑙. For aerodynamic loads resisting forward motion of the vehicle, the Equation [2.63] can be used. The parameters 𝜌 and 𝐴 𝑛𝑡 represent the drag coefficient, the air density and a

reference area of the vehicle, respectively. The 𝐴 _𝑛𝑡 is the area of the vehicle projected on a vehicle transversal plane.

− _𝑎𝑖 =

2∙ ∙ 𝜌 ∙ 𝐴 𝑛𝑡∙ 𝑒𝑙

2_{; [2.63]}

Typical values of drag coefficients ( ) for cars can be found from sources such as: (Robert Bosch GmbH, 2004), (Barnard, 2010), (Hucho, 1998), and (Schuetz, 2015). These coefficients are derived from coast down tests, wind tunnel tests or CFD (Computational Fluid Dynamics) calculations. The air resistance can often be neglected for city speeds, but not at highway speeds.

Since a car structure moving through the air is not unlike an aircraft wing, there are also an aerody- namic lift force and pitch moment. This affects the vertical forces on front and rear axle, and conse- quently the tyre to road grip. Hence, it affects the lateral stability.

𝑎𝑖 = 1 2∙ 𝑙∙ 𝜌 ∙ 𝐴 𝑛𝑡∙ 𝑒𝑙 2_; 𝑀𝑎𝑖 𝑦=₂∙ 𝑝 ∙ ∙ 𝜌 ∙ 𝐴 𝑛𝑡∙ 𝑒𝑙2; [2.64] The coefficient 𝑙 represents the lift characteristics of the vehicle. For extreme vehicle, such as racing

cars, one can achieve negative _𝑙, but often by sacrificing with higher . The forces _𝑎𝑖 and _𝑎𝑖 are assumed to act through the same reference point, often centre of gravity (CoG), which defines 𝑀𝑎𝑖 𝑦.

One can replace [ 𝑎𝑖 𝑎𝑖 𝑀𝑎𝑖 𝑦] with equivalent [ 𝑎𝑖 𝑎𝑖 𝑎𝑖 ] or

[ 𝑎𝑖 𝑎𝑖 𝑎𝑖 ], Figure 2-82 and Eq [2.65]. − 𝑎𝑖 𝑀𝑎𝑖 𝑦 𝑎𝑖 𝑎𝑖 𝑎𝑖 − _𝑎𝑖 𝑎𝑖 𝑎𝑖 − _𝑎𝑖 𝑎𝑖 = 0 𝑎𝑖 ≈ aerodynamic reference point, often ≈CoG

Figure 2-82: Force-equivalent ways to model longitudinal-wind aerodynamic forces in 𝑥 -plane. 𝑎𝑖 and

𝑎𝑖 differs between mid and right figure but 𝑎𝑖 is the same for all 3 figures.

𝑎𝑖 =₂∙ 𝑙 ∙ 𝜌 ∙ 𝐴 𝑛𝑡∙ 𝑒𝑙2; 𝑎𝑖 =₂∙ 𝑙 ∙ 𝜌 ∙ 𝐴 𝑛𝑡∙ 𝑒𝑙2;

For a reference height _𝑎𝑖 , often CoG heigth.

[2.65]

2.8.2 Lateral Relative Wind Velocity

When the wind comes from the side, there can be direct influences on the vehicle lateral dynamics. Es- pecially sensitive are long but light vehicles (such as buses or vehicles with unloaded trailers). The problem can be emphasized by sudden winds (e.g. on bridges or exiting a forested area). Besides direct effects on the vehicle lateral motion, side-winds can also disturb the driver through disturbances in the steering wheel feel.

Similar expressions to the longitudinal loads are derived for lateral forces and from side-winds. _{𝑎𝑖 𝑦}=₂∙ ∙ 𝜌 ∙ 𝐴 ∙ 𝑦 𝑒𝑙2;

𝑀𝑎𝑖 =₂ 𝑦 ∙ 𝜌 ∙ 𝐴 ∙ ∙ 𝑦 𝑒𝑙2;

The speed _{𝑦 𝑒𝑙} is the lateral component of the vehicle velocity relative to the wind. Note that 𝐴 and may now have other interpretations and values than in Equations [2.63]-[2.65], e.g. 𝐴 𝑛𝑡 or 𝐴 𝑖 𝑒.

2.9 Driving and Transport Application

The driver drives and experiences the vehicle in the short time scale through pedals, steering wheel and seat. But the drivers’/users’ choice of load (cargo weight and position) and choice of route is im- portant on the longer time scale. One can differ between, Ref (Pettersson, 2019):

• Transport Application is how the vehicle is used by one user/owner during its lifetime. It can be commuting 2 × 15 km/day, 5 days/week (e.g. for passenger car) or loading and transport- ing timber on 10 km forest road + 300 km high way, twice per day (e.g. for a truck).

• Transport Operation is how the vehicle is used along a specific route. It is typically 10 min to 10 h driving. Driving Cycles, (𝑡), mentioned in 3.3.1.1 have the purpose to describe approxi- mately the same.

• Transport Mission is the purpose of one Transport Operation, such as where to stop and load/unload a certain payload.

2.9.1 Mission, Road and Traffic

This section is kept very short, but it is included for completeness, beside 2.9.2. See more in 3.3.1.

2.9.2 Driver

To study how different vehicle designs work in a vehicle operation a driver model is needed. In its eas- iest form, a driver model can be steering wheel angle _𝑡 0;. Another extreme interpretation of what can be called a driver model is an implicit/inverse statement, like “driver will push accelerator pedal so that speed 20 [ 𝑠⁄ ] during the manouvre”, which leads to that accelerator pedal posi- tion becomes an output, as opposed to input, to the vehicle model. Beyond those very simple driver models, there is often need for a driver model which react on vehicle states in relation to an environ- ment or traffic. In this section, driver models are primarily thought of as models of the driver of the subject vehicle, but when modelling surrounding traffic carefully, each object vehicle can also use a driver model.

The driver interacts with the vehicle mainly through steering wheel, accelerator pedal and brake pe- dal. In addition to these, there are clutch pedal, gear stick/gear selector, and various buttons, etc., see Figure 2-83, but we focus here on the first 3 mentioned.

•Accelerator Pedal Position, •Brake Pedal Force, •Steering Wheel Angle

•Accelerator Pedal Force, •Brake Pedal Position, •Steering Wheel Torque,

•Seat motion,

•View of environment relative to vehicle

Other:

• Clutch Pedal and Gear Stick or Gear Selector • Parking Brake • Direction indicators • HMI (buttons, lamps,

text, sounds, …) • …

Figure 2-83: Interface of driver, and some commonly assumed causality.

Driver’s control of vehicle dynamics, or vehicle motion including position, can be discussed in longitu- dinal (mainly pedals) and lateral (mainly steering wheel).

Driver reacts on several stimuli, such as motion (mainly through seat), sounds, and optical. Among mo- tion, it is primarily the accelerations (and their time derivative, jerk) that is sensed by the driver, but also rotational velocity in yaw can be sensed by human. Among the optical there is looming (optical expansion of an object in the driver’s field of view [ 𝑒 𝑠]) is often used as a cause for how driver uses the pedals. The optical flow (the pattern of apparent motion of objects, surfaces, and edges in the

Driver models are here discussed as models of the “human driver” for use in vehicle verification simu- lations. However, driver models can also be understood as models of “virtual driver”, and then they are actually implemented as algorithms in the vehicle product, e.g. as prediction algorithms or automated driving controllers. In the first context, it is often important to vary the driver model (at least its pa- rameters, maybe even its equations) for robustness, as mentioned in Figure 1-3. In the latter context, the driver model of the subject vehicle is rather varied for optimization/satisficing, see Figure 1-3 again.

An important aspect of driver modelling is how the user (driver or occupant) experiences the vehicle. This is often referred to as subjective evaluation, but for some cases one can establish methods to ob- jectively calculate a measure of how good or bad the experience is. The measure can sometimes be a physical quantity but often it has to be a rating or grading without unit. Examples are “driveability [rating 0-10, high is good]”, “steering effort [deg/s, low is good]” and “ride comfort [m/s2_{, low is} good]”.

2.9.2.1 Driver Modelling

As in all modelling, it is important to model, or select model, after what the model should be used for. Driver modelling for “verification of vehicle functions” and for implementation in “driving automa- tion functions” are similar in that they should react on the vehicle’s environment, but there are also differences. Driver models for verification of vehicle functions should be as human-like as possible. They should also judge feedback to driver, such as assessing steering effort. Driver models for use in driving automation functions should also be human-like to facilitate cooperation between human and automated driving, such as hand-over/take-back or simultaneous control. However, there are also rea- sons to not mimic all aspects from a human driver, such as the human’s inability to watch in several directions simultaneously.

A categorization of modelling concept is whether the model uses equations that reflect the biological processes human’s perception, cognition and neuro-muscular or equations from a vehicle model. The first concept (exemplified in 2.9.2.3.3) would rather use angle to obstacle as opposed to distance to ob- stacle, since humans rather see angles than distances. The latter concept (exemplified in 2.9.2.2.1) as- sumes that driver has adapted to the specific vehicle and (subconsciously) operates the vehicle in a good way; a kind of inverse model thinking. Overall, both concepts can reflect approximately the same driving, but they are differently parameterized; typically, in biological parameters and vehicle parame- ters, respectively.

Driver can be modelled in 2 parts: Strategic and Operative. A division in Longitudinal and Lateral is also relevant. One can think of different ways of arranging these dimensions on each other; one possi- ble way is shown in Figure text.

Strategic Longitudinal Strategic Lateral Operative Longitudinal Operative Lateral Pedals, ForwardOrReverse, CruiseButtons SteeringWheel

Requested longitudinal motion of subject vehicle, e.g. 𝑅𝑒

Longitudinal motion relative to other road users/obstacles and to road grades and curves, road unevennesses, lane widths, legal speed, ...

Lateral motion relative to other road users/obstacles and to road edges.

Actual longitudinal motion of subject vehicle

Actual lateral motion of subject vehicle. Requested lateral motion of

subject vehicle, e.g. 𝑅𝑒

Driver

In document Vehicle Dynamics Compendium (Page 124-129)