04 Steering And Turning Vehicles - Department of Mechanical ...

Steering and Turning Vehicles 

ME 360/390 – Prof. R.G. Longoria 

Vehicle System Dynamics and Control 

Prof. R.G. Longoria 

Spring 2012 

Department of Mechanical Engineering 

The University of Texas at Austin

• Steering mechanisms 



Overview 

• Differentially-steered vehicle – kinematic 

• Ackermann steered vehicle – kinematic 

• Simulation examples 



Classical steering mechanisms 

δ = 

H 

5 th wheel 

steering 

'hand wheel' angle 

‘turntable steering’ 

•Likely developed by the Romans, and preceded only by a 2 

wheel cart. 

•Consumes space 

•Poor performance – unstable 

•Longitudinal disturbance forces have large moment arms 



•Articulated-vehicle steering 

•Tractors, heavy industrial 

vehicles 



Common Steering Mechanisms 

Differential steer Synchro-drive Tricycle 

What is minimum 

# of actuators? 



…and some systems also employ Ackermann-type. 



Kinematic model of single-axle turning vehicle 

Y 

Recall the simple 2D turning vehicle with kinematic state quantified by, 

y 



x 

ψ 

X 

X 

[ ] † 

qI 

= X Y ψ 

The velocities in the local (body-fixed) reference frame 

are transformed into a global frame by the rotation 

matrix, 

⎡ cosψ sinψ 0⎤ 

R( 

ψ ) = 

⎢ 

−sinψ 

cosψ 0 

⎥ 

⎢ ⎥ 

⎢⎣ 0 0 1⎥⎦ 

and the velocities are then, 

Inverting, we arrive at the velocities in the global reference frame, 

qɺ = R( ψ ) ⋅qɺ 

I 

qɺ = Ψ( ψ ) ⋅qɺ 

where, 

⎡cosψ −sinψ 

0⎤ 

Ψ( 

ψ ) = 

⎢ 

sinψ cosψ 0 

⎥ 

⎢ ⎥ 

⎢⎣ 0 0 1⎥⎦ 

Let’s apply these relations to the case of a single-axis vehicle that has two wheels 

differentially driven with controlled speed. 


The University of Texas at Austin 

I

Y 

Recall: position and velocity in inertial frame 

We defined the vehicle’s kinematic state in the inertial frame by, 

l2 

y 

B 

l1 

x 

ψ 

So, for our simple (single-axle) vehicle, 

the velocities in the inertial frame in 

terms of the wheel velocities are, 



X 

X 

Velocities in the local (body-fixed) 

reference frame are transformed 

into the inertial frame by the 

rotation matrix, 

or, specifically, qɺ = R( ψ ) ⋅qɺ 

I 

q 

I 

⎡ X ⎤ 

= 

⎢ 

Y 

⎥ 

⎢ ⎥ 

⎢⎣ ψ ⎥⎦ 

⎡ cosψ sinψ 0⎤ 

R( 

ψ ) = 

⎢ 

−sinψ 

cosψ 0 

⎥ 

⎢ ⎥ 

⎢⎣ 0 0 1⎥⎦ 

Inverting, we arrive at the velocities in the global reference 

frame, 

⎡ Xɺ ⎤ ⎡U ⎤ ⎡cosψ ⎢ ⎥ 

qɺ I = Yɺ ⎢ 

V 

⎥ 

( ψ ) 

⎢ 

⎢ ⎥ = 

⎢ ⎥ 

= Ψ ⋅ qɺ 

= 

⎢ 

sinψ ⎢ψ ⎥ 

⎣ 

ɺ 

⎦ 

⎢⎣ Ω⎥⎦ 

⎢⎣ 0 

−sinψ 

cosψ 0 

0⎤ 

⎡ vx 

⎤ 

0 

⎥ ⎢ 

v 

⎥ 

⎥ ⎢ y ⎥ 

1⎥⎦ 

⎢⎣ ω ⎥ z ⎦ 

qɺ 

I 

⎡ Rw l2Rw ⎤ 

⎢ ( ω1 + ω2)cos ψ − ( ω1 −ω2 

)sinψ 

2 

B 

⎥ 

⎡Xɺ ⎤ ⎢ ⎥ 

⎢ ⎥ Rw l2Rw = Yɺ 

⎢ ( ω1 ω2)sin ψ ( ω1 ω2)cos ψ ⎥ 

⎢ ⎥ = + + − 

⎢ 2 

B 

⎥ 

⎢ψ ⎥ 

⎣ 

ɺ 

⎦ ⎢ ⎥ 

⎢ 

Rw 

( ω1 −ω2 

) 

⎥ 

⎢⎣ B 

⎥⎦ 



Example: Differentially-driven basic vehicle 

For a kinematic model of a differentially-driven vehicle, we assume there is no slip, and 

that the wheels have controllable speeds, ω 1 and ω 2 . If the CG is on the rear axle, 

the velocities in the global reference frame are, 



Y 

y 

X 

x 

ψ 

B = track width 

qɺ 

I 

l = L 

l 

1 

2 

= 

0 

⎡ Rw 

⎤ 

⎢ ( ω1 + ω2) cosψ 

2 

⎥ 

⎡Xɺ ⎤ ⎢ ⎥ 

⎢ ⎥ Rw 

= Yɺ 

⎢ ( ω1 ω2)sin ψ ⎥ 

⎢ ⎥ = + 

⎢ 2 

⎥ 

⎢ψ ⎥ 

⎣ 

ɺ 

⎦ ⎢ ⎥ 

⎢ 

Rw 

( ω1 −ω2 

) ⎥ 

⎢⎣ B 

⎥⎦ 



Simulation of the differentially-steered single-axle vehicle trajectory 

A simple code in Matlab to compute and plot out the vehicle trajectory is given below. 

% Differentially-steered kinematic vehicle model 

% Requires right (#1) and left (#2) wheel velocities, omegaw1 and omegaw2, 

% as controlled inputs for single axle, to be passed as global parameters 

% Wheel radius, R_w, and axle track width, B, are also required 

% Updated 2/20/12 RGL 

function Xidot = DS_vehicle(t,Xi) 

global R_w B omegaw1 omegaw2 

X = Xi(1); Y = Xi(2); psi = Xi(3); 

% NOTE: these are global coordinates 

% These equations assume CG on single axle 

Xdot = 0.5*cos(psi)*R_w*(omegaw1+omegaw2); 

Ydot = 0.5*sin(psi)*R_w*(omegaw1+omegaw2); 

psidot = R_w*(omegaw1-omegaw2)/B; 

Xidot=[Xdot;Ydot;psidot]; 

% test_DS_vehicle.m 

clear all 

global R_w B omegaw1 omegaw2 

% Rw = wheel radius, B = track width 

% omegaw1 = right wheel speed 

R_w = 0.05; B = 0.18; 

omegaw1 = 4; omegaw2 = 2; 

Xi0=[0,0,0]; 

[t,Xi] = ode45(@DS_vehicle,[0 10],Xi0); 

N = length(t); 

figure(1) 

plot(Xi(:,1),Xi(:,2)), axis([-1.0 1.0 -0.5 1.5]), axis('square') 

xlabel('X'), ylabel('Y') 



Vehicle trajectory in XY 



Simulation of differentially-steered vehicle with 2D animation 

A code in Matlab to plot out the vehicle trajectory including a simple graphing animation of 

the vehicle body/orientation is provided on the course log. This provides visual feedback on 

the model results. 

The key elements of this code are: 

Y 

1.5 

1 

0.5 

0 

-0.5 

-1 -0.5 0 

X 

0.5 1 



1. Specify and plot initial location 

and orientation of the vehicle CG. 

2. Initiate some ‘handle graphics’ 

functions for defining the ‘body’. 

3. Perform a fixed wheel speed 

simulation loop to find state, q. 

4. The state of the robot is used to 

define the position and orientation 

of the vehicle over time. 

5. A simple routine is used to 

animate 2D motion of the vehicle 

by progressive plotting of the 

body/wheel positions. 



Lankensperger/‘Ackermann’-type steering 

‘knuckle’ 

1. Steering arm 

2. Drag link 

3. Idler arm 

4. Tie rod/rack 

5. Steering wheel 

6. Steering shaft 

7. Steering box 

8. Pitman arm 



Rigid axle with 

kingpin 

Divided track rods 

for independent 

suspension. 



Turning at low (Ackermann) speed 

• What is low-speed? 

– Negligible centrifugal forces 

– Tires need not develop lateral 

forces 

• Pure rolling, no lateral sliding 

(minimum tire scrub). 

• For proper geometry in the turn, 

the steer angles, δ, are given by: 

L L 

δo ≅ < δi 

≅ 

R + B R − B 

2 2 

• The average value (small angles) 

is the Ackerman angle, 

δ = 

Ackermann 

L 

R 



L 

δ o 

B 

δ i 

R 

Simple relationship between 

heading and steering wheel angle. 

Ref. Wong, Ch. 5 

Turn Center 

“Ackermann steering geometry” 

cotδ o − cotδ 

i = B 

L 

δ = steer angle of outside wheel 

o 

δ = steer angle of inside wheel 

i 

B = track 

L = wheelbase 



Wheel slip angle definition 

A positive slip angle is said to give a 

negative force on the wheel (to the left). 

Gillespie (1992) 

*To align the wheel with heading direction 

(vector), you would have to go –α. 

The lateral force should simply always 

oppose the motion – the lateral force is a 

dissipative force! 



Fy = Cα ⋅α 

The slip angle is the angle between the 

wheel’s direction of heading (wheel plane) 

and its direction of travel. So, to compute 

the slip angle, need to track tire velocity 

components, as shown below. 

Excerpt from Liljedahl, et al (1996) 



Additional notes/comments on Ackermann steering 

• At low speed the wheels will roll without slip angle. 

• If the rear wheels have no slip angle, the center of the turn lies 

on the projection of the rear axle. Each front steered wheel has 

a normal to the wheel plane that passes through the same center 

of the turn. This is what Ackermann geometry dictates. 

• Correct Ackermann reduces tire wear and is easy on terrain. 

• Ackermann steering geometry leads to steering torques that 

increase with steer angle. The driver gets feedback about the 

extent to which wheels are turned. With parallel steer, the trend 

is different, becoming negative (not desirable in a steering 

system – positive feedback). 

• Off-tracking of the rear wheels, ∆, is related to this geometry. 

The ‘∆’ is R[1-cos(L/R)], or approximately L 2 /(2R). 





Using the basic geometry of Ackermann steering 

L 

δ o 

B 

δ i 



Can you pass the vehicle 

through a given position? 

1. Assume low-speed turning 

2. Project along rear-axle 

3. Define R = L/δ max 

4. Project from CG 

5. Project ideal turning path 



Example: 2D vehicle with front-steered wheel 

A wheeled vehicle is said to have kinematic (or Ackermann) steering when a wheel is 

actually given a steer angle, δ, as shown. A kinematic model for the steered basic vehicle 

in the inertial frame is given by the equations, 

‘tricycle’ 

Xɺ = v cosψ = Rwω 

cosψ 

Y 

δ 

Yɺ x 

= vsin = R sin 

ψ ω ψ 

v 

ψɺ = tanδ 

L 

where it is assumed that the wheels do not slip, so we 

can control the rotational speed and thus velocity at each 

wheel-ground contact. 

So, the input ‘control variables’ are velocity, v=R w ω, 

and steer angle, δ. 

In this example, the CG is located on the rear axle. 

These kinematic equations can be readily simulated. 



w 

y 

Note: 

L 

1 

v = v + v 

2 

vt 

ωz = ψɺ 

= 

L 

vt 

tanδ 

= 

v 

( ) 

1 2 



ψ 

L = wheel base 

X 

X

Derivation of equations 

Y 

y 

L 

ψ 

δ 

1 

v = v + v 

2 

( ) 

1 2 



x 

L = wheel base 

X 

X 

Note that the forward velocity at the front wheel 

is simply, v, but because of kinematic steering 

the velocity along the path of the wheel must be, 

v 

ωz = tanδ 

L 

v δ 

= 

v 

cosδ 

This means that the lateral velocity at the front 

steered wheel must be, 

v = vδ sinδ = v tanδ 

Now we can find the angular velocity about the CG, which is located at the 

center of the rear axle as, 

t 



Example: simulation and animation of steered tricycle kinematic model 

% ----------------------------------------------------- 

% tricycle_model.m 

% revised 2/21/12 rgl 

% ----------------------------------------------------function 

qdot = tricycle_model(t,q) 

global L vc delta_radc delta_max_deg R_w 

% L is length between the front wheel axis and rear wheel axis [m] 

% vc is speed command 

% delta_radc is the steering angle command 

% State variables 

x = q(1); y = q(2); psi = q(3); 

% Control variables 

v = vc; 

delta = delta_radc; 

% kinematic model 

xdot = v*cos(psi); 

ydot = v*sin(psi); 

psidot = v*tan(delta)/L; 

qdot = [xdot;ydot;psidot]; 



% Physical parameters of the tricycle 

L = 2.040; % Length between the front wheel axis and rear wheel axis [m] 

B = 1.164; % Distance between the rear wheels [m] 

m_max_rpm = 8000; % Motor max speed [rpm] 

gratio = 20; % Gear ratio 

R_w = 13/39.37; % Radius of wheel [m] 

% desired turn radius 

R_turn = 3*L; 

delta_max_rad = L/R_turn; % Maximum steering angle [deg] 

R = 6.12 m 

δ = 0.33 rad = 19.1 deg 



Summary of kinematic vehicle turning 

• The models introduced here provide additional review of 

fundamental kinematics principles and how they can be applied 

to vehicle systems. 

• The concepts of differential and Ackermann steering are 

demonstrated through simulations. 

• The kinematic models are commonly used in mobile robot 

applications for path planning, estimation, and control. 

• These kinematic models cannot tell you anything about the 

effect of forces or stability. 





Vehicle directional stability 

Directional stability refers to a vehicle’s ability to 

stabilize its direction of motion against disturbances. 

So, we are typically concerned with: 

•Unforced transient operations (natural response) 

•Forced response – aperiodic inputs (step, ramp, etc.) 

•Forced response – periodic forcing (sine) 

•Steady-state directional response 

We’ll discuss both lateral and longitudinal stability. 





2D vehicle of Rocard 



One of the earliest analyses of vehicle stability was 

conducted by Y. Rocard (1954) (Steeds, 1960). 

The vehicle was simplified as a rigid rectangular 

frame with a wheel at each corner and the plane of 

each wheel is vertical and parallel to the frame. 

The steering force is assumed to be directly related to 

slip angle, 

F = Kα 

and the contact forces are assumed not to be affected 

by vehicle motion. 

There is no steering angle in this model. 

First, let’s review the lateral forces using modern 

notation. 



Lateral tire forces induced by slip angles 

The slip angle, α, is derived using body-fixed variables. 

α δ 

− 1 y 1 z 

f = f − tan ⎜ ⎟ 

vx 

α 

⎛ l Ω − v ⎞ 

−1 

2 z y 

r = tan ⎜ ⎟ 

vx 

⎝ ⎠ 



Wong (2001) 

⎛ v + l Ω ⎞ 

⎝ ⎠ 

From Liljedahl, et al (1996) 

v 1 y l 

− ⎛ + 1Ωz 

⎞ 

− α f = tan ⎜ ⎟ −δ 

f 

⎝ vx 

⎠ 

v 1 y l 

− ⎛ − 2Ω 

z ⎞ 

− α r = tan ⎜ ⎟ − 

� 

δ r 

⎝ vx 

⎠ 

rear steer 



Tire cornering forces 

• The slip angle, α, is the angle between direction of 

heading and direction of travel of a wheel (OA). 

• A lateral force, F yα , also referred to as a cornering 

force (camber angle of the wheel is zero), is 

generated at a tire-surface interface, but this may not 

be collinear with the applied force at the wheel 

center. 

• A torque is induced, referred to as a self-aligning 

torque. This torque helps a steered wheel return to 

its original position after a turn. 

• The distance between these two applied forces is 

called the pneumatic trail. 

• The self-aligning torque is given by the product of 

the cornering force and the pneumatic trail. 

• For more on self-aligning moment and pneumatic 

trail, see Wong, Section 1.4. 



t p 

Fs 

Fyα “…side slip is due to the 

lateral elasticity of the 

tire.” 

Fs 

Fyα Wong 



Typical data on tire cornering forces 

“linear region” 

Fig. 1.23 from Wong 



α 

Maximum cornering forces: 

•passenger car tires: 18 degrees 

•racing car tires: 6 degrees 

(Wong) 


Ratio to normal load 

α 

Variables that impact cornering force: 

•Normal load 

•Inflation pressure 

•Lateral load transfer 

•Size 



Cornering stiffness and coefficient 

• The cornering stiffness will depend on tire 

properties such as: 

– tire size and type (e.g., radial, bias-ply, etc.), 

– number of plies, 

– cord angles, 

– wheel width, and 

– tread. 

• Dependence on load is taken into account 

through the cornering coefficient, 

where Fz is the vertical load. 



CC 

α = 

C 

α 

F 



z

Cornering stiffness and coefficients 

Cornering stiffness for car, light truck 

and heavy truck tires 




Effect of normal load 

on the cornering 

coefficient 




Now, Rocard’s linearized model 

Rocard derived two 2nd order ODEs for this 

problem*, and for linear approximations found 

the characteristic equation, 

( ) 

2 2 

s + Rs + S s = 0 

2 2 

⎡ ⎤ 

2 ⎛ a ⎞ ⎛ b ⎞ 

R = ⎢K1 ⎜1+ K 2 ⎟ + 3 ⎜1+ 2 ⎟⎥ 

MV ⎣ ⎝ k ⎠ ⎝ k ⎠⎦ 

2 

4 K1K3 ( a + b) 2( K1a − K3b) S = − 2 2 

Mk 

( MkV ) 

A critical speed 

is defined by: 

V 

= 

2 1 3 

c 

1 − 3 



2 

2 K K ( a + b) 

M ( K a K b) 

*Refer to Steeds handout for details; we’ll 

come back to this later 

Stable for all speeds if: 

2 

I = Mk 

K b > K a 

3 1 

•choose the position of the mass center of the 

vehicle and the steering force characteristics for 

the front and rear tires 

•If the steering force are equal, then stability is 

assured if b is greater than a, or putting the CG in 

front of the midpoint of the wheelbase. 



Directional Stability per Rocard 

Let’s match this up with modern notation: 

V 

2 

2 

2 2 K1K3 ( a + b) 

f α rα 

c = = 

M K1a − K3b M C f α L1 − Crα L2 



K = C 

2C 

C L 

( ) ( ) 

1 

K = C 

R factor = Cr L2 − C f L1 

> 0 (for stability at all speeds) 

α α 

So, only compute a critical speed if R factor is less than or equal to zero. 

A critical speed is defined by, 

3 

f α 

rα 

(Steeds, 1960) 



The ‘Bicycle’ or Single-Track Model 

Adapted from Wong 

How would you add a disturbance? 

Need to determine the ‘external forces’. 



•Assume: symmetric vehicle, no roll or pitch 

•Represent the two wheels on the front and rear 

axles by a single equivalent wheel. 

•The bicycle model will have at least three states: 

–forward translational momentum or velocity of 

the CG 

–lateral translational momentum or velocity of the 

CG 

–yaw angular momentum or velocity about the CG 

Refer to Wong, Chapter 5, Eqs. 5.25 – 5.27: 

( ɺ ) � 

m vx −VyΩ z = Fxf cos( δ f ) + Fxr − Fyf 

sin( δ f ) 

�� 

( ɺ ) 

front drive rear drive lateral force effect 

m v + V Ω = F + F cos( δ ) + F sin( δ ) 

y x z yr yf f xf f 

I Ω ɺ = l F cos( δ ) − l F + l F sin( δ ) 

z z 1 yf f 2 yr 1 xf f 



Z 

tan 

−1 

⎛ y 2 z ⎞ 

⎜ ⎟ = δ r −α 

r 

vx 

α r 

Y 



x t 

v − l Ω 

⎝ ⎠ 

l 2 

vy 

l 1 

y 

y t 

A more general ‘bicycle’ model schematic 

ωz 

x 

β 

vx 

α f δf 



X 

v −1 ⎛ y ⎞ 

β = sideslip = tan ⎜ ⎟ 

⎝ vx 

⎠ 

tan 

ψ 

v + l Ω 

− 1 ⎛ y 1 z ⎞ 

⎜ ⎟ = δ f −α 

f 

vx 

⎝ ⎠

Reduced bicycle model: 2 DOF by assuming V x = constant = V 



v = V 

x 

δ = δ = small steering angle 

f 

( ɺ ) � 

m vx −VyΩ z = Fxf cos( δ f ) + Fxr − Fyf 

sin( δ f ) 

�� 

mvɺ = F + F + F δ − mv Ω 

y yr yf xf f x z � 

∼0 

I zΩ ɺ 

z = l1Fyf − l2Fyr + l1Fxf δ f 

�� 

Note how δ falls out, but it enters again in the lateral forces, since, 

v 1 y l1 z vy l 

− ⎛ + Ω ⎞ ⎛ + 1Ωz 

⎞ 

α f = δ f − tan ⎜ ⎟ ≈ δ f − ⎜ ⎟ 

⎝ vx ⎠ ⎝ vx 

⎠ 

front drive rear drive lateral force effect 

∼0 

l 1 2Ω z − vy l2Ω z − v 

− y 

⎛ ⎞ ⎛ ⎞ 

αr 

= tan ⎜ ⎟ ≈ ⎜ ⎟ 

⎝ vx ⎠ ⎝ vx 

⎠ 



As before, we can calculate vehicle trajectory calculations 

Recall, these models solve for forward and lateral velocity and yaw 

velocity, resulting from input steer angles, δ, relative to the bodyfixed 

axes. 

To find the trajectory of the CG in the Earth-based coordinates, we 

use transformation equations for basic 2-D trajectory simulations, 



Xɺ = v cos( ψ ) − v sin( ψ ) 

x y 

Yɺ = v sin( ψ ) + v cos( ψ ) 

ψɺ 

= Ω 

x y 

z 



Example: directional stability with zero steer + disturbance 

• We examine directional stability, reviewing Steeds’ form of 

Rocard’s (linearized) model (see class handout), which is 

essentially the bicycle model with steer angle, δ = 0. 

• Determine if this vehicle with baseline parameter data given 

(next slide) is stable, and compare with results from a 

simulation of the bicycle model subjected to a force 

‘perturbation’ at the front wheel applied in the +y direction. 

• Additional ‘car specification’ data is provided as Appendix B. 





Example: baseline vehicle parameter data (from simple_bicycle.m) 

(In Matlab script form) 

% Baseline values 

g = 9.81; 

L = 3.075; L1 = 1.568; L2 = L-L1; 

% Inertia parameters 

m = 1945; % total mass, kg 

W = m*g; % weight, N 

iyaw = 0.992; % yaw dynamic index 

% the following is a defined relation between yaw dynamic index and 

% the yaw moment of inertia (see Dixon reference and table of car specs) 

Iz = iyaw*m*L1*L2; % moment of inertia about z, kg-m^2 

Wf = L2*W/L; % static weight on front axle 

Wr = L1*W/L; % static weight on rear axle 

% Refer to Wong, Section 1.4 for guide to the following parameters 

CCf = 0.171*180/pi; % front corning stiffness coefficient, /rad 

CCr = 0.5*0.181*180/pi; % rear corning stiffness coefficient, /rad 

Cf = CCf*Wf/2; % corning stiffness per tire, N/rad (front) 

Cr = CCr*Wr/2; % rear cornering stiffness per tire, N/rad (rear) 





Disturbance force application 

if (t>=tdon & t

For Nominal Case, Vehicle Stabilizes 

Given case: 

R factor = 4200.4 

V c = non-existent 

v x =V= 5L/sec 

A lateral pulse disturbance is 

applied at front wheel. 

Afterwards, the vehicle 

stabilizes (lateral and yaw 

velocities go to zero). 



4 x 10-3 Lateral Velocity 

2 

0 

-2 

-4 

-6 

0 5 10 15 

50 

0 

-50 

Position 

0 50 100 150 200 

6 x 10-3 Yaw Velocity 

4 

2 

0 

-2 

0 5 10 15 

x 10-3 Yaw 

3 

2 

1 

0 

0 5 10 15 

Used: simple_bicycle.m 



Change rear ‘lateral stiffness’ (i.e., the cornering stiffness values of tires) 

In this case, the rear lateral 

stiffness was cut in half. 

R factor = −33814. 

V c = 18.2 m/s 

Let (since vehicle is now 

unstable for V > Vc) 

V = 1.2* V c =21.8 

After the pulse disturbance, 

the vehicle does not stabilize. 



50 

0 

-50 

Lateral Velocity 

-100 

0 5 10 15 

100 

50 

0 

-50 

Position 

0 100 200 

15 

10 

5 

Yaw Velocity 

0 

0 5 10 15 

15 

10 

5 

Yaw 

0 

0 5 10 15 

Used: simple_bicycle.m 



Example: a double lane change using simple bicycle model 

Using the stable case for the 

‘simple_bicycle’ model, an 

‘open loop’ lane change is 

achieved by: 

(put right into function file): 

% double lane change 

if (t=1 & t=2 & t=3 & t=4 & t=5 & t=6) deltaf = 0; end; 



1 

0.5 

0 

-0.5 

Lateral Velocity 

-1 

0 2 4 6 8 

6 

4 

2 

0 

Position 

-2 

0 50 100 150 

0.4 

0.2 

0 

-0.2 

Yaw Velocity 

-0.4 

0 2 4 6 8 

0.4 

0.2 

0 

-0.2 

Yaw 

-0.4 

0 2 4 6 8 

Used: simple_bicycle_lanechange.m 



Bicycle model summary 

• The classic bicycle (or single-track) model forms the basis for 

understanding steering and steering control. 

• Lateral forces induced by tires depend on wheel slip angle and 

play a key role in lateral stability. See also the Rocard handout. 

• We saw earlier that longitudinal traction (driving or braking) 

influences lateral forces, so driven bicycle models (where there 

are traction/braking forces) require us to model the coupling 

between longitudinal and lateral tire forces (friction ellipse). 

• In extreme maneuvers, for example, it would be possible to 

predict yaw instability if lateral forces were significantly 

reduced. 







References 

1. Den Hartog, J.P., Mechanics, Dover edition. 

2. Dixon, J.C., Tires, Suspension and Handling (2nd ed.), SAE, Warrendale, PA, 

1996. 

3. Greenwood, D.T., Principles of Dynamics, Prentice-Hall, 1965. 

4. Gillespie, T.D., Fundamentals of Vehicle Dynamics, SAE, Warrendale, PA, 1992. 

5. Hibbeler, Engineering Mechanics: Dynamics, 9th ed., Prentice-Hall. 

6. Meriam, J.L. and L.G. Kraige, Engineering Mechanics: Dynamics (4th ed.), Wiley 

and Sons, Inc., NY, 1997. 

7. Rocard, Y., “L’instabilite en Mecanique”, Masson et Cie, Paris, 1954. 

8. Segel, L., “Theoretical Prediction and Experimental Substantiation of the 

Response of the Automobile to Steering Control,” The Institution of Mechanical 

Engineers, Proceedings of the Automobile Division, No. 7, pp. 310-330, 1956-7. 

9. Steeds, W., Mechanics of Road Vehicles, Iliffe and Sons, Ltd., London, 1960. 

10. Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc., New York, 

2001 (3rd ed.). 





APPENDIX 

Example Car Specifications* 

*Table C.1 from J.C. Dixon, “Tires, Suspension and Handling” (2 nd ed.), SAE, Warrendale, PA, 1996. 





Appendix: 

More on Steering Mechanisms 



DaNI: 4-wheeled, differentially driven 

•Motor: Tetrix DC (handout) 

•Gear Ratio: 2:1 

•Shaft Diameter: 4.73 mm 

•Wheel Diameter: 100 mm 

•Wheel base: 133 mm. 



Under what conditions would you need to 

use a dynamic model? 



General Steering System Requirements 

• A steering system should be insensitive to disturbances from the ground/road 

while providing the driver/controller with essential ‘feedback’ as needed to 

maintain stability. 

• The steering system should achieve the required turning geometry. For 

example, it may be required to satisfy the Ackermann condition. 

• The vehicle should be responsive to steering corrections. 

• The orientation of the steered wheels with respect to the vehicle should be 

maintained in a stable fashion. For example, passenger vehicles require that 

the steered wheels automatically return to a straight-ahead stable equilibrium 

position. 

• It should be possible to achieve reasonable handling without excessive 

control input (e.g., a minimum of steering wheel turns from one locked 

position to the other). 





Passenger Steering Requirements 

• Driver should alter steering wheel angle to keep deviation from course low. 

• Correlation between steering wheel and driving direction is not linear due to: 

a) turns of the steering wheel, b) steered wheel alterations, c) lateral tire 

loads, and d) alteration of driving direction. 

• Driver must steer to account for compliance in steering system, chassis, etc., 

as well as need to change directions. 

• Driver uses visual as well as ‘haptic’ feedback. For example, roll inclination 

of vehicle body, vibration, and feedback through the steering wheel (effect of 

self-centering torque on wheels). 

• It is believed that the feedback from the steering torque coming back up 

through the steering system from the wheels is the most important 

information used by many drivers. 





Ref. Milliken & Milliken 

Impact of Steering Geometry 



B = 0.56 

L 

Ref. Wong 



Ref. Wong 



Steering Results and 

Error in Ackermann 

B = 0.56 

L 

“Steering error curves” for a front beam axle 



Impact on Steering Geometry 

of using Ackermann in High Speed 

A car with a steering geometry chosen for low-speed (Ackermann) 

will not be as effective at higher vehicle speeds. 

In a high speed turn, the inside tire will have a lower normal force, 

which means it could achieve the same amount of lateral cornering 

force with a smaller slip angle. 

Using Ackermann will result in more instances in which the inside 

tire is dragged along at too high a slip angle, unnecessarily raising 

the temperature and slowing the vehicle down with excessive slip 

induced drag. 





1. Steering arm 

2. Drag link 

3. Idler arm 

4. Tie rod/rack 

5. Steering wheel 

6. Steering shaft 

7. Steering box 

8. Pitman arm 

Steering Systems – Rigid Axle 



Rack and pinion is not suitable for steering wheels on rigid 

front axles, as the axles tend to move in the longitudinal 

direction. This movement between the wheels and steering 

system can induce unintended steering action. 

Only steering gears (1) with rotational movement are used. 

Design must minimize effects from motion. 

7 

8 

2 

1 

You may see some older model 

Toyota Land Cruisers that use 

this type of steering design. 



Heisler (1999) 

Axle-Beam Steering Linkage 






Axle-Beam Steering Linkage 





Front Axle and Tie Rod Assembly 



From: J.W. Durstine, “The Truck Steering System From 

Hand Wheel to Road Wheel”, SAE, SP-374, 1973 (L. Ray 

Bukendale Lecture). 



Steering Gearbox and Ratio 

The gearbox provides the primary means for reducing the rotational input from 

the steering wheel and the steering axis. 

Steering wheel to road wheel ratios may vary 

with angle, but have values of 15:1 in passenger 

cars and may go as high as 36:1 for heavy 

trucks. 

Rack and pinions are commonly designed to 

have a variable gear ratio depending on steer 

angle. 

The actual steer ratio can be influenced by 

steering system effects, such as compliance. 

The plot here from Gillespie shows how much 

this can change. 



Experimentally measured steering 

ratio on a truck. 

From Gillespie (Fig. 8.18). 



3 Steering arms 

7 Tie rod joints 

8 Steering rack 



Rack-and-Pinion – 1 

Used on most passenger cars and some light trucks, 

as well as on some heavier and high speed vehicles. 

Used on vehicles with independent suspensions. 

Steering ratio is ratio of pinion revolutions to rack travel. 

Some advantages: simple, manufacturing ease, 

efficient, minimal backlash, tie rods can be joined 

to rack, minimal compliance, compact, eliminate 

idler arm and intermediate rod 

Some disadvantages: sensitive to impacts, greater 

stress in tie rod, since it is efficient you feel 

disturbances, size of steering angle depends on rack 

travel so you have short steering arms and higher 

forces throughout, cannot be used on rigid axles 






Rack-and-Pinion – 2 



Split track-rod with relay-rod 








Summary 

• Steering mechanisms were reviewed briefly 

• Steering geometry – how do you select steering 

system? 

• Steering control – how does it translate into 

steering mechanism design? 

• There is a complex relationship to suspension 

(see Segel, Gillespie)

04 Steering And Turning Vehicles - Department of Mechanical ...

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