Signal Error Distributionμ =0.359 σ = 0.0912 n = 488

Num

ber o

f sam

ples

Distance from Actual (Feet)

Iterativeμ =0.208 σ = 0.0597 n = 82 Time = .075s

Num

ber o

f sam

ples

Distance from Actual (Feet)

Directμ =0.158 σ = 0.0503 n = 79 Time = .38s

Num

ber o

f sam

ples

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 timeVariable, ~300 μs

Amplifier Circuit DelayConstant 5 – 10 μs

Processor Delay1-2 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 BieberDepartment 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

Jr3

r2r1

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)

riDr

r0i (Estimated location)

(Actual location)

T1

T2

T3

T4

T5

T6

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 21 2

2 2 2 21

2 2 2 2 32

32 2 2 23 3 3

2 2 21

2

( )( ) ( )

r r dxdr x y zxr x d y z y xy

r x x y y zz r x y

- +=üï= + + ïïï= - + + Þ =-ýïï= - + - + 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 21 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 20 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 21 0 0 0 0 0 0 0

Δ2( )Δ 2( )Δ 2( )Δ 2 Δ

Δi i i i i i i

xr 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 ýï ïï ïï ïï ïî þ

Where

Which can be written in matrix notation as

Ax b=

2 2

1 010 1 0 1 0 1

2 2

2 020 2 0 2 0 2

2 20 0 0 0

2 ,

n n n n n

x x y y z zx x y y z x

x x y y z z

r rr 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

ReceiverDigital Output

Digital Input

40 KHz Signal

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.