steering and turning vehicles 1

8/9/2019 Steering and Turning Vehicles 1

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ME 379M/397 Prof. R.G. LongoriaCyber Vehicle Systems

Department of Mechanical EngineeringThe University of Texas at Austin

Steering and Turning VehiclesKinematic models of 2D steering and turning

Prof. R.G. Longoria

Spring 2014


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Overview

Steering mechanisms

Forces and motion of wheels/tires

Differential steering of single-axle vehicle

Kinematic, or Ackerman, steering

Two-axle vehicle steering and turning

Simulation and animation of 2D turning


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5th wheel

steering

Likely developed by the Romans, and

preceded only by a 2 wheel cart.

Consumes spacePoor performance unstable

Longitudinal disturbance forces have large

moment arms

'hand wheel' angle =H

Articulated-vehicle steering

Tractors, heavy industrial

vehicles

turntable steering

Classical steering mechanisms


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Differential steer Synchro-drive Tricycle

and some systems also employ Ackermann-type.What is minimum

# of actuators?

Yaw on purpose: common steering mechanisms


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Differential steering is very common. Why?

Simple mechanism Does not take up a lot of space (e.g., used even

for some larger, full-scale vehicles)

There are disadvantages (tear up the terrain,wear on system, tires, etc.)

For robotics, very common:

Sliding pivot

Realized as a caster?Recall:

Table 2.1 of Siegwart, et

al (2011) provides a nice

overview of different

wheel configurations used

in some robotic vehicles.


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First, tire force/moment concepts

Gillespie (1992)

Wong (2001)

Three forces and three moments are induced at

the tire-surface interface, with origin at center of

contact, O.

Insight into this general view of wheel forces is

essential for higher speed and for larger robotic

vehicles and autonomous passenger vehicles.


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ME 379M/397 Prof. R.G. LongoriaCyber Vehicle Systems Department of Mechanical EngineeringThe University of Texas at Austin

Fig. 1.15 (Wong)

1max( , )x xd

x

r v vs r v r

= = Longitudinal slip is defined,

Longitudinal slip refers to the deficit in distance traveled

compared to free rolling.

Tractive effort is related to slip. For typical tires,the trend is as shown below.

(Wong)

NormalTractive EffortLoadCoefficient

( )x d z

F s F=

slip velocity

slip

region

Remember, real wheels/tires can slip or skid


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There is also lateral slip, inducing lateral or cornering forces on wheels/tires

sF

O

Plane of

motionD

Plane of motion

with applied side

force, Fs

A slip angle, , defines the difference between thewheel plane and the direction of motion, whichmay arise due to induced motion or because of an

applied side force, Fs.

A cornering force, Fya, is induced in the lateraldirection between the tire and ground, and it is

found to be applied along an axis off the wheel

axis. You should think of this as a frictional force.

Pneumatic

trailThe couple Ta acting on the wheel tends

to turn it so its plane coincides with the

direction of motion. Steering andsuspension systems must constrain the

wheel if it is to stay, say, in the plane

OA.

a y pT F t=p

ts

F

sF

yF

sF

Equivalent force-

torque system


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The slip angle, , is shown here as the anglebetween direction of heading and direction of travelof the wheel (OA).

The lateral force, Fy, (camber angle of the wheel

is zero), is generated at a tire-surface interface, and

may not be collinear with the applied force at the

wheel center. The distance between these two

applied forces is called the pneumatic trail.

The induced self-aligning torque helps a steered

wheel return to its original position after a turn.

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 Wongs book, Chapter 1 (Section 1.4).

Wong

side slip is due to the

lateral elasticity of the

tire.

sF

yF

sF

yFpt

More concepts related to tire cornering effects and induced forces


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For the simple vehicle model shown to the left,

there are negligible forces at pointA. Thiscould be a pivot, caster, or some other omni-

directional type wheel..

Assume the vehicle has constant forward

velocity, U.

Maybe we are interested in the yaw stability of

the vehicle about its CG.

Assume the wheels roll without slip and cannot slip laterally. Designate the right wheel 1

and the left 2. What are the velocities in a body-fixed frame?

U

A

( )

1 1

2 2

1 11 2 1 22 2

Velocities at

each wheel

( )

x w

x w

x w

v R

v R

v v v R

=

=

= + = +

Yaw rate is ( )1 2w

z

R

B =

Since the velocity along the rear axle is

constrained to be zero, you can show that,2y z

v l=

Differential steering of a single-axle vehicle in planar, turning motion

Y

X


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Derivation of the CG velocities

i C Civ v r= +

You can apply where i is any point of interest.

First, relate the velocity of the left wheel to the CG by,

( ) ( )2 2 2 2 2 2 2

C C x y z x z y z

B Bv v i v r v i v j k l i j v i v l j

= = + = + + + = +

2

2

x x z

Bv v =

2 2 0y y zv v l= =We get the two relations:

The second is the lateral constraint and tells us, 2y zv l=

Now for the velocity of the right wheel,

( ) ( )1 1 1 2 2 2 2

C C x y z x z y z

B Bv v i v r v i v j k l i j v i v l j

= = + = + + = + +

12

x x z

Bv v = +So,

Adding this relation and the one above yields,

( ) ( )

1 2

1 2 1 2

2

1

2 2

x x x

w

x x x

v v v

R

v v v

+ =

= + = +

&

( ) ( ) ( )1 1 1 2 1 2 1 22 2 1 1 1

2 2

wz x x x x x x x

Rv v v v v v v

B B B B

= = = =

Finally, the yaw rate can be found,


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X

Lets define the vehicles kinematic state in the inertial frame by,

y

Y

I

X

Y

=

q

Velocities in the local (body-fixed)

reference frame are transformed

into the inertial frame by the

rotation matrix,

cos sin 0

( ) sin cos 0

0 0 1

=

R

Inverting, we arrive at the velocities in the global reference

frame,

( ) I= q R q

cos sin 0

( ) sin cos 0

0 0 1

x

I y

z

X U v

Y V v

= = = =

q q

or, specifically,

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

the velocities in the inertial frame in

terms of the wheel velocities are,

X

21 2 1 2

21 2 1 2

1 2

( ) cos ( ) sin2

( )sin ( ) cos2

( )

w w

w wI

w

R l R

BXR l R

YB

R

B

+

= = + +

q

Position and velocity in inertial frame

1l

2l

B


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X

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

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

xy

Y

the velocities in the global reference frame are,

track widthB =

1 2

1 2

1 2

( ) cos

2( )sin

2

( )

w

wI

w

R

XR

Y

R

B

+

= = +

q

1

2 0

l L

l

=

=

Example: Differentially-driven single-axle vehicle with CG on axle

Note: this defaults to the common mobile robot model seen throughout robotics

literature.


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-1 -0.5 0 0.5 1-0.5

0

0.5

1

1.5

X

Y

1. Specify and plot initial location

and orientation of the vehicle CG.2. Initiate some handle graphicsfunctions for defining the body.

3. Perform a fixed wheel speedsimulation loop to find state, q.

4. The state of the robot is used todefine the position and orientationof the vehicle over time.

5. A simple routine is used to

animate 2D motion of the vehicle

by progressive plotting of thebody/wheel positions.

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 onthe model results.

The key elements of this code are:

Simulation of differentially-steered vehicle with 2D animation


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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, , are given by:

The average value (small angles)

is the Ackerman angle,

B

L

io

R Turn Center

Ref. Wong, Ch. 5

cot cot

steer angle of outside wheel

steer angle of inside wheel

track

wheelbase

o i

o

i

BL

B

L

=

=

=

=

=

Ackermann steering geometry2 2

o i

L L

B BR R <

+

Ackermann

L

R =

Simple relationship betweenheading and steering wheel angle.

Turning at low speed and kinematic (or Ackerman) steering


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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 rodsfor independent

suspension.

knuckle

Lankensperger/Ackermann-type steering*

*Lankensperger was the inventor,

Ackermann the patent agent.http://en.wikipedia.org/wiki/Georg_Lankensperger


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At low speed the wheels primarily 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 theextent 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 L2/(2R).

Additional notes/comments on Ackermann steering


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B

L

i

o

Can you pass the vehicle

through a given position?

Using the basic geometry of Ackermann steering this concept give you an idea

of the workspace this steering can give you

1. Assume low-speed turning

2. Project along rear-axle

3. Define R = L/max4. Project from CG5. Project ideal turning path


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

actually given a steer angle, , as shown. A kinematic model for the steered basic vehiclein the inertial frame is given by the equations,

cos cos

sin sin

tan

w

w

X v R

Y v R

v

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=Rw, andsteer angle, .

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

These kinematic equations can be readily simulated.

Y

X

( )1 21

2v v v= +

X

x

y

L

wheel baseL =

tricycle

Note:

tan

tz

t

v

L

vv

= =

=

Example: Single-axle vehicle with front-steered wheel; rolling rear wheels


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Y

X

( )1 21

2v v v= +

X

y

L

wheel baseL =

Derivation of equations

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,

cos

vv

=

tanz

v

L =

This means that the lateral velocity at the front

steered wheel must be,

sin tant

v v v = =

Now we can find the angular velocity about the CG, which is located at thecenter of the rear axle as,


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

% 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];

Example: simulation and animation of steered tricycle kinematic model% 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

= 0.33 rad = 19.1 deg


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The models introduced here provide a review of fundamentalkinematics principles and how they can be applied to vehicle

systems.

The concepts of differential and Ackermann steering are

demonstrated through simulations.

These kinematic models are commonly used in mobile robot

applications for path planning, estimation, and control.

If you know where you want to go, these steering mechanismscan be used to estimate the required control (steer angle) as

long as you are at low speed (no slip).

These kinematic models cannot tell you anything about the

effect of forces or stability. For that insight, we need dynamic

models.

Summary of differential and kinematic vehicle turning


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References1. 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. (or any later edition).

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

5. Karnopp, D.C., and D.L. Margolis, Engineering Applications of Dynamics, John Wiley &

Sons, New York, 2008. An excellent intermediate engineering dynamics text focused on

building mathematical models. Good overview of fundamental material.

6. Rocard, Y., Linstabilite en Mecanique, Masson et Cie, Paris, 1954.

7. Segel, L., Theoretical Prediction and Experimental Substantiation of the Response of the

Automobile to Steering Control, The Institution of Mechanical Engineers, Proceedings ofthe Automobile Division, No. 7, pp. 310-330, 1956-7.

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

9. Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc., New York, 2001 (3rd

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

Good overview of vehicle dynamics concepts.

steering and turning vehicles 1

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