1. Signal Error Distribution
μ =0.359 σ = 0.0912 n = 488
Numberofsamples
Distance from Actual (Feet)
Iterative
μ =0.208 σ = 0.0597 n = 82 Time = .075s
Numberofsamples
Distance from Actual (Feet)
Direct
μ =0.158 σ = 0.0503 n = 79 Time = .38s
Numberofsamples
Distance from Actual (Feet)
Signal Timeline
(Not to Scale)
Processor Delay
< 1 μs Trigger Circuit Delay
Constant 17 – 25 μs
Signal Travel Time
Signal detect time
Variable, ~300 μs
Amplifier Circuit Delay
Constant 5 – 10 μs
Processor Delay
1-2 μs
Primary source of
measurement error
Combine to form
approximately
constant offset
Robust Precision n-lateration* using Distributed Ultrasound Beacons
Robustness is far more than accuracy, it
includes tolerance of error, rejection of bad
data and ability to function with missing
information. Both methods used Chauvenet’s
criterion for rejecting outliers, employed after
all position calculations were completed. In
comparing accuracy, robustness and
computational time in our two methods,
neither is a clear winner; the final choice will
be determined by ease of execution on the
microprocessor and position quality
requirements of the control algorithm. Both
calculation methods achieved an average
error substantially less than range of errors in
the original measurement, an expected
outcome from an overdetermined system.
In terms of both average error and standard
deviation, the direct method produced better
results, but not drastically so. It had more
difficulty handling erroneous readings, and is
computationally expensive. The smaller
sample size of the direct method is due
primarily to a receiver location that returned
a complex value when calculated. The
reasons behind this are not known, and will
have to be determined before the method
can be used in real time.
The iterative method was significantly faster,
computationally, although implementing
matrix math in C++ for the microprocessor
may reduce this margin. Additionally, it
handles bad or missing data better, and
requires no alteration to use more or fewer
sensor readings.
Both methods are likely to be improved with
further calibration of the sensor
measurement, especially by understanding
the linear component of error associated
with distance.
Rohan Kapoor , Chad Bieber
Department of Mechanical and Aerospace Engineering, North Carolina State University
Direct Method
Iterative Method
The Direct method uses geometry to solve for a position based on
three transmitter measurements. Creating a coordinate system on
the plane of the transmitters simplifies this drastically
In the new coordinate system, the distance to each transmitter can
be easily described.
The Iterative method uses multiple iterations of a least
squares approximation to create one solution of the
over-determined problem. It is more complicated up
front, but is capable of handling any combination of
sensors.
First, an estimated position is guessed, and the distance
equation is re-written using the difference between the
actual position and our estimated position.
eZ
K
J
r3
r2
r1
R
T1
eX
eY
xT2
T3
I
(x3 y3 z3)
(x2 y2 z2)
(x1 y1 z1)
(x y z)
Ti = (xi yi zi)
R0 = (x0 y0 z0)
R = (x y z)
ri
Dr
r0i
(Estimated location)
(Actual location)
T
1
T
2
T
3
T
4
T
5
T
6
Experimental Set Up: 6 transmitters (T) were placed in a 6 by 8
foot grid in the ceiling, angled downward towards a test area. A
single receiver (R) received the signals. Twenty-eight different
receiver locations (4 clusters of 7) were tested. The spacing
between the receiver locations in a cluster is ½ foot. Three
measurements of signal time were recorded at each receiver
location producing a total of 504 measurements (28X6X3).
2 1( )d T T= -
2 2 2
1 2
2 2 2 2
1
2 2 2 2 3
2
32 2 2 2
3 3 3
2 2 2
1
2
( )
( ) ( )
r r d
x
dr x y z
x
r x d y z y x
y
r x x y y z
z r x y
- +
=
üï= + + ïïï= - + + Þ = -ý
ïï= - + - + ïïþ
= ± - -
This result must be transformed back to the original coordinate
system (I,J,K). There are n-choose-3 different combinations of
measurements to use in this method, this specific arrangement of
transmitters results in 2 linear combinations, which do not produce a
valid location, and 18 valid combinations, which must be combined
into one composite solution
T = Transmitter
R = Receiver
Ultrasound Beacon Signal
*n-lateration is the determination of the position of a point (the receiver) from distance measurements to n ( >= 3) known points (transmitters).
Accuracy and Robustness
The signal time represents a
one-dimensional length
composed of three parts. The
part we want is the travel time
of the sound pulse. A constant
offset, composed of circuitry
and processor delays, can be
easily accounted for. The
remaining time is a variable
signal detection time, caused by
the receiver beginning to
vibrate and the digitizer
catching enough amplified
signal to see. This variable part
has an additive, random, error
as well as a linear error related
to the distance.
We conditioning this signal for
use by subtracting the mean
error to distribute the readings
around zero. In the future, we
can identify the linear
component and remove it as
well, further increasing our
accuracy.
2 2 2 2
1 0 0 0(Δ ) (Δ ) (Δ )i i ir x x x y y y z z z= + - + + - + + -
Which is the sum of three
parts:
2 2 2 2 2 2
0 0 0 0 0 0( ) ( ) ( ) 2( )Δ 2( )Δ 2( )Δ Δ Δ Δi i i i i ix x y y z z x x x y y y z z z x y z= - + - + - + - + - + - + + +
A linearized approximation of the error:
The position is approximated by ignoring the error term
and subtracting the distance to our estimated position:
[ ]2 2
1 0 0 0 0 0 0 0
Δ
2( )Δ 2( )Δ 2( )Δ 2 Δ
Δ
i i i i i i i
x
r r x x x y y y z z z x x y y z z y
z
ì üï ïï ïï ïï ï- = - + - + - = - - - í ý
ï ïï ïï ïï ïî þ
Where
Which can be written in matrix notation as
Ax b=
2 2
1 01
0 1 0 1 0 1
2 2
2 020 2 0 2 0 2
2 2
0 0 0
0
2 ,
n n n
n n
x x y y z z
x x y y z x
x x y y z z
r r
r r
r r
A b
ì üï ï-ï ïé ù- - - ï ïï ïê ú ï ïê ú -ï ï- - - ï ïê ú ï ïï ïê ú= = í ýê ú ï ïï ïê ú ï ïê ú ï ïï ïê ú ï ï- - -ê ú ï ï-ë û ï ïï ïî þ
M M M M
Overall Goal of the Project: To Develop a 3D Sensing System for the 6 Degree-of-
freedom (Orientation and Positioning) Real-time Navigation of Flocks of Flight Vehicles.
The Robustness of the Sensing System is critical. The near-term goal described here is
characterization of robustness and the accuracy of measurements in the presence of
erroneous sensor measurements, resulting from obstructions in view, out-of-range
measurements, poor directionality, and spurious signals. The distributed,
overdetermined nature of the transmitter grid creates an excellent foundation to build
an algorithm able to deliver precise, real-time information to a very large number of
flight vehicles, even with missing and invalid data.
n-lateration Method
And an error term:The distance to our estimated position:
2-D Error Geometry
Measurement
Confidence
Interval
Confidence
Area
Transmitter Location
Conclusion and beyond: Both methods show potential, though the direct
method is not capable of identifying a location if the radii measured do
not intersect. The largest gain in improving robustness will come from
identifying an individual measurement as erroneous, and (re)calculating
the position based on the remaining data. Implementing multiple
receivers to identify orientation will require modeling of the vehicle and
adapting the equations to include body coordinates. Real-time position
measurements will require an algorithm that can update position based
on one measurement at a time, using history and projected path to
provide the remaining information.
Microprocessor
40 KHz Driver Circuitry
Amplifier/Digitizer circuitry
Ultrasonic Transmitter
Receiver
Digital Output
Digital Input
A microprocessor triggers the 40KHz driver
circuit and begins timing. The driver sends a
200 μs burst of signal to the transmitter, which
emits the ultrasound pulse. When the signal
reaches the receiver, a 2-stage amplifier brings
it into the measureable realm, and a digitizer
sends a high signal back to the microprocessor.
The high input initiates an interrupt routine,
and the microprocessor records the number of
elapsed clock ticks.
Positional Accuracy begins with measurement accuracy. In
the two dimensional example, the Confidence Area can be
bounded by the Confidence Intervals of the two component
Measurements. Combining multiple measurements has the
effect of overlaying many such areas, tightening the likely
region of the actual point.
The mean error of the raw measurement is an offset from
the actual. This offset is subtracted from the raw
measurement to produce an adjusted value used to calculate
the position. In the 2-D (and 3-D) position, the measure of
error is how far the calculated position is from the actual
position, which is always positive.