We have a 3-axis accelerometer in a device. The device is placed in a
vehicle. We do not know the orientation of the device with respect to
the vehicle, or the location. We ASSUME our device is not located in
the dead centre of the vehicle, but rather offset in some way, such as
in the boot at the rear, or on the passenger seat etc.

As the vehicle drives along roads, it executes turns and hills. This
motion can be detected as changes in acceleration of our device.

We can use a compass or gyroscope to detect angular change, to assist
the algorithm, and we find the following formula may be correct

DEFINITIONS
a=Radial acceleration in m^2^2
w=Angular velocity in radians/2 (Degree of change / time in secs)
v=Velocity in m^2 (Road speed around curve)
r=radius in meters (Radius of curve)
1RAD = 180deg/PI (180 degrees =PI radians )

FORMULA
a=rw^2
v=rw
v=a/w
w=sqr(a/r)

So given ‘a’ and ‘w’ we can presumably deduce ‘v’, our road speed.
Given ‘a’ and ‘r’ we can presumably deduce ‘w’, our angular rate of
change

If we did not have a compass or gyro in the system, and instead made
the following real-world assumptions:
1.	Vehicle speed is commonly 60km/h, may be less but rarely in excess
of 110km/h
2.	Vehicle speed does not vary a lot while in motion, IE is fixed for
long periods of time
3.	Vehicle speed is relative to the radius of the curve, tighter curve
brings slower speeds, so a coefficient is in order
4.	Road curves are commonly of fixed radius, such that ‘r’ does not
change frequently during a curve, but rather there is a sequence of
curves, each with near fixed radius
5.	Our vehicle does not spend a lot of time on straight roads, IE
there is angular change to detect.
6.	Roads are built with restrictions on corners, so in a 60km/h zone a
curve may have a 20m radius, but on a highway the radius may be much
larger. In effect there is a minimum radius we can expect, as a
vehicle cannot turn on a dime, and we can assume a relationship
between likely speed and radius or curve

The question is
1.	Are the above calculations correct, and are they the best algorithm
to deduce road speed if we do not have wheel encoders or other direct
sensors but do have a compass or gyro sensor?
2.	Given the real world assumptions, how can we determine the angular
change to some degree, with no indication of either speed or compass
sensor – IE can we SOLEY use accelerometer readings to provide some
angular change estimate, given our vehicle and road assumptions? (IE
Can we determine say clockwise rotation of 0-20 degrees or perhaps
20-40 degrees?)
3. Can we determine the device offset from the vehicle, IE after some
left and right curves we may be able to deduce the device is on the
front left passenger seat or the right hand side of the boot?

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Clarification of Question by chrispeaks-ga on 26 Oct 2003 19:04 PST

This question is related to several others I have placed, perhaps you
may wish to consider them if your answere relates to more than one -
IE it may be preferable to answer two or three together by one
researcher, or use any posted answers to one question as a starting
block to others.

Below is the list of my open questions
GIS storage method / clever compression shows angle & radius for
curves?
Detecting velocity via with correction
Accelerometers to detect angular rotation or velocity 
Velocity detection via doppler effect with observer situated ON moving
object!
Compass heading determined with tilt correction algorithm (negative
pitch error)

I might suggest two separated accelerometers, one drivers side and one
passengers, in a turn they would both read a lateral moment.


there would be a discrepancy (the one on the inside of the turn would
tend to slow down and the one on the outside of the turn would tend to
speed up)  you could tell by the difference in the readings how tight
the turn is, and by the time of the turn the change in direction.

Even though the question has expired, here are some comments and questions.
What is the end use of the information to be collected? Roughness
index of the road? Center lines of each lane to create a data base for
automating the driving task. For instance to alert drivers of lane
excursion or not allowing lane change without activating turn signal?

You used "boot" instead of "trunk", so I assume you live in the mother
country. If you are looking for radius of turn, look at the anit-lock
brake sensors. Some are simple wheel rotation sensors. Tap into these
sensors to determine radius of turn. Calibrate the tires (tyres) to
compensate for differences in air pressure. Wheel count sensors can
produce the same range, resolution and repeatability as gyros and
accelerometers and at much less cost.

The Institue of Navigation and the UK counterpart have published
histories of navigation. An early device used by the Chinese used a
differential gear to keep an arrow pointing south. Instrument the
differntial to detect changes in direction.

For cost effectiveness, consider using differential GPS to map road
center lines. Consider using any of the tools designed specifically to
measure and record roughness. The simple ones detect the deflection of
the shock absorber. The more complex have a trailer with
accelerometers and differential GPS on each wheel.

The smoother the road, the less wear and tear from vertical accelerations.