Four Wheel Steering Comparison with two wheel steering

Kensell Kyle Corominas Hife

SA105X – Kandidatarbete med fördjupning i Fordonsteknik

VT 2014

Abstract

What are the differences between two wheel steering and four wheel steering? The aim of

this project is to compare these two in terms of advantages gained from four wheel steering.

A simulation is conducted on ADAMS Cars platform based on the RCV (Research Concept

Vehicle) model, developed at KTH in Stockholm.

Sammanfattning

Vad är skillnaden mellan tvåhjulsstyrning och fyrhjulsstyrning? Syftet med detta projekt är

att jämföra dessa två med hänsyn till vilka fördelar man får ut från fyrhjulsstyrning. En

simulering utfördes på ADAMS Cars plattform baserad på RCV (Research Concept Vehicle)

modellen som utvecklats på KTH i Stockholm.

Contents

1 Vehicle dynamics …………………………………………………………………1

3.1 Single track model …………………………………………………..........1

3.2 Two track model ………………………………………………………….3

2 RCV Model ………….……………………………………….…………………….4

4.1 Ackermann geometry………………………………………………..……..4

4.2 Camber …………………………………………….……………………... 5

3 Simulation ……………………………………………………………………….... 5

4 Results ………………………………………………………………………..…… 6

5 Conclusion …………………………………………………………………...…… 10

6 Reference …………………………………………………………….…………… 11

1

1 Vehicle Dynamics

A vehicle has 6 degrees of freedom as shown in Figure 1.1. The vertical motion, lateral

motion, longitudinal motion and the 3 rotational motions around X, Y, Z axis (roll, pitch and

yaw).

1.1 Single track model

A single track model, also called bicycle model, assumes the front wheels as one unit as well

as the rear wheels. The free body diagram is shown in Figure 1.2 for two wheel steering, the

same principle applies for four wheel steering. Three different co-ordinate systems are

implemented with the prefix 𝑋𝐶𝑜𝐺 , 𝑥𝑤 and 𝑥𝑓𝑖𝑥 which follows the vehicle’s center of gravity,

the wheel center and a fixed point on the ground. Roll and pitch motion are neglected as well

as the load transfers between wheels during cornering and linear tire characteristics is

assumed.

The wheel velocity (𝑣𝑤𝑓 , 𝑣𝑤𝑟) is a superposition of the vehicles velocity (𝑣𝐶𝑜𝐺) and the

rotational velocity around the center of gravity with yaw angle velocity �̇�. The yaw angle 𝜓

is defined as the angle between the longitudinal axis 𝑋𝐶𝑜𝐺 of the vehicles co-ordinate system

and the longitudinal axis 𝑥𝑓𝑖𝑥 of a fixed co-ordinate system.

Figure 1.1 Vehicle co-ordinate system, [5].

2

The angle between the wheels longitudinal axis (𝑥𝑤) and its velocity vector is called wheel

slip angle (𝛿𝑓 , 𝛿𝑟) while the angle between wheels longitudinal axis (𝑥𝑤) the vehicles

longitudinal axis (𝑋𝐶𝑜𝐺) is the steering angle (𝜃𝑓). Vehicle side slip angle 𝛽, is the angle

between the vehicles velocity vector and its longitudinal axis.

Due to caster, contact forces between wheel and road acts towards the rear of the contact area

center point. Caster is the angle between the steering axis and the vertical axis of the wheel

view from the side. At positive caster, the steering axis is tilted back with respect to the

vertical axis, which give stability and the realigns the wheels after a turn [3].

This model is only valid for lateral acceleration below 0.4𝑔, above it the load transfers can

no longer be neglected [1]. The model can be expressed as a state-space system [1], from

which the transfer function for the vehicle side slip can be obtained. A constant ratio K is

assumed for the front and rear steer angle. Solving for steady-state gain equals to zero yields:

𝐾 =−𝑏+

𝑚𝑎

𝐶𝑓(𝑎−𝑏)𝑣𝐶𝑜𝐺

𝑎+𝑚𝑏

𝐶𝑟(𝑎−𝑏)𝑣𝐶𝑜𝐺

(1)

𝐶𝑓 and 𝐶𝑟 are the corner stiffness of the front and rear axle respectively while 𝑎 and 𝑏 are the

distance from the center of gravity to the front axes and rear axle respectively. From equation

(1) the front and rear wheels are steered differently for various speeds, at low speeds the rear

Figure 1.2 Single track geometry for two wheel steering.

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wheels are steered at opposite direction from the front while at high speeds the rear wheels

are steered in same direction. This results in high maneuverability at low speeds as well as

increased safety in higher speeds.

1.2 Two track model

The single track model has its limitations namely for higher lateral forces it is no longer valid.

A two track model is shown in Figure 1.3, applying newton’s second law for the vehicles

plane motion yields:

𝑚𝑎𝑥 = 𝐹𝑑 + 𝐹𝜇 + ∑ 𝐹𝑥𝑖6𝑖=1 (2)

𝑚𝑎𝑦 = ∑ 𝐹𝑦𝑘6𝑘=1 (3)

𝐼𝑧�̇� = 𝑀𝑓𝑙 + 𝑀𝑓𝑟 + 𝑀𝑟𝑙 + 𝑀𝑟𝑟 + 𝑎 ∑ 𝐹𝑦𝑘6𝑘=1 +

𝑐

2∑ 𝐹𝑥𝑖

6𝑖=1 (4)

𝐹𝑑 and 𝐹𝜇 are drag resistance and rolling resistance respectively. 𝐹𝑓𝑙𝑥,𝑦, 𝐹𝑓𝑙𝑥,𝑦, 𝐹𝑓𝑟𝑥,𝑦, 𝐹𝑟𝑟𝑥,𝑦

are tire longitudinal and lateral forces respectively. 𝑀𝑓𝑙,𝑟𝑀𝑟𝑙,𝑟 are the moments at each wheel.

Using equations (1 - 4) the vehicles velocity and acceleration can be calculated.

Figure 1.3 Two track geometry.

4

2 RCV Modell

In reference to [4], the RCV was created in collaboration with several departments at KTH,

Stockholm. It is almost a full steer by wire vehicle, except for the friction brakes, with the

ability to independently control the torque, camber angle and steering angle of all four

wheels. Ackermann steering is used, with speed dependent steering of the rear wheels, since

high maneuverability is preferred. The ratio between the rear and front wheel steering angle

was derived thru tests since most of the variables in equation (1) are unknown.

2.1 Ackermann geometry

The Ackermann geometry takes into consideration that the outer wheel has to travel longer

than the inner wheel during a curve. Each wheel are turning at the same center of rotation as

shown in Figure 2.1. The inner wheel steering angle is greater than the outer wheel which

allows the vehicle to negotiate a turn without wheel slip angle. The model however is suitable

for low speeds since the centrifugal forces can be set to zero [4].

The relation between the inner wheel steering angle and outer wheel steering angle is derived

as:

𝜃𝑓𝑙 = cot−1 (cot(𝜃𝑓𝑟) −𝑐

𝑎+𝑏) (5)

With the track 𝑐, 𝜃𝑓𝑙 the inner wheel steering angle and 𝜃𝑓𝑟 the outer wheel steering angle.

Figure 2.1 Ackerman steering geometry.

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2.2 Camber

Camber is the wheel angle relative to the vertical axis 𝑍𝐶𝑜𝐺 as shown in Figure 2.2. Positive

camber is defined when the top of the wheel is tilted outwards while negative camber is when

the top is tilted inwards.

When entering a corner most of the cars weight shifts towards the outer wheels. With neutral

camber the outer wheels will gain a positive camber while the inner wheels gain a negative

camber, when the suspension compresses during cornering resulting in reduced contact area

between wheel and ground. To counteract this a negative and positive camber can be

implemented to the outer and inner wheel. During cornering the wheels will gain neutral

camber creating maximum contact area. On the other hand having negative camber during a

straight road will wear the tires unevenly as well as reduce road grip.

3 Simulation

A co-simulation between Simulink and ADAMS/Cars is used for this simulation. The RCV

simulation model can be set in two modes, two wheel steering and four wheel steering.

Different parameters, such as camber gain, simulation speed and three different input signals

can be set as well. Two signal inputs are used, ramp and sine input, which simulates curve as

well as slalom driving. The ramp input is used for two different speeds namely

30 𝑘𝑚 ℎ⁄ and 60 𝑘𝑚 ℎ⁄ with and without camber gain. As for the sine input a speed of 30 𝑘𝑚 ℎ⁄ is used.

Figure 2.2 Positive and negative camber, [6].

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4 Results

Figures 4.1 and 4.2 shows the vehicle paths for the two modes at both studied speeds, Figures

4.3 – 4.6 shows the lateral accelerations and normal forces respectively, while Figures 4.7

and 4.8 shows the vehicle side slip for the ramp input. Figures 4.9 - 4.11 shows the lateral

acceleration, normal forces and vehicle side slip for the sine input.

Four wheel steering without camber adjustments has the lowest lateral acceleration at low

speeds, during higher speeds though camber adjustments are needed. Four and two wheel

steering has almost identical lateral acceleration at higher speeds when the curve radius is

constant. On the other hand there are small differences between the two modes when it comes

to normal forces at low speeds, at higher speeds the differences becomes marginal.

The vehicle slide slip angles are negligible for the two modes at low speeds, however the slip

angles increases with the speed, the two wheel steering mode without camber has the highest

slip angle.

During the sine input simulation, four wheel steering mode has smaller lateral acceleration.

Normal forces acting on the front axles are somewhat the same with the exception that the

four wheel steering mode has quicker response time. Normal forces on the rear axles however

has noticeable differences. When it comes to vehicle side slip angles the four wheel steering

mode has much lower slip angles.

Figure 4.1 Difference in vehicle path at 30 𝑘𝑚 ℎ⁄ .

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Figure 4.2 Difference in vehicle path at 60 𝑘𝑚 ℎ⁄ .

Figure 4.3 Lateral acceleration at 30 𝑘𝑚 ℎ⁄ .

Figure 4.4 Lateral acceleration at 60 𝑘𝑚 ℎ⁄ .

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Figure 4.5 Normal forces on each wheel at 30 𝑘𝑚 ℎ⁄ .

Figure 4.6 Normal forces on each wheel at 60 𝑘𝑚 ℎ⁄ .

Figure 4.7 Vehicle slide slip at 30 𝑘𝑚 ℎ⁄ .

9

Figure 4.8 Vehicle side slip at 60 𝑘𝑚 ℎ⁄ .

Figure 4.9 Lateral acceleration at 30 𝑘𝑚 ℎ⁄ .

Figure 4.10 Normal forces on front and rear axles at 30 𝑘𝑚 ℎ⁄ .

10

5 Conclusion

The aim of the work was to compare the difference between two wheel steering and four

wheel steering. There is a general difference between the two modes whether it be lower or

higher speeds. At lower speeds the difference isn’t much notable, however as the speed

increases it becomes more considerable. Most notably the lateral forces which affects how a

vehicle “feels”, for example, driving a vehicle with high center of gravity the lateral forces

generated during cornering can somewhat be disturbing if the forces are high enough.

Understanding these differences can be a great tool for enthusiasts who seeks the best out of

the two.

Figure 4.11 Vehicle side slip at 30 𝑘𝑚 ℎ⁄ .

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6 References

[1] C. O’Kane and S. Timoney, “Investigation of Four-wheel Steering Algorithms for a

Formula SAE Car”, in SAE 2004 World Congress and Exhibition, Detroit, Michigan, Dublin,

2004

[2] Z. Tianjun and Z. Changfu, “Advanced Integrated Control Strategy Based on SBW and

4WS”, in Circuits, Communications and Systems, 2009, Pacific-Asia Conference, Chengdu,

Handan, China, 2009.

[3] U. Kiencke and L. Nielsen, “Automotive Control Systems for Engine, Driveline and

Vehicle”. 2nd edition, 2005, pp. 305-318.

[4] M. Engman, P. Falk, C. Hommel, M. Ljungberg and G. Spiric, “RCV Dynamics Control”,

KTH, Stockholm, 2013.

[5] https://law.resource.org/pub/us/cfr/ibr/005/sae.j1733.1994.html

[6] http://www.ictworkshopsolutions.com/2011/06/wheel-alignment-3/