WO2017063032A1

WO2017063032A1 - Determining elevation and bearing information of a remote point

Info

Publication number: WO2017063032A1
Application number: PCT/AU2016/050949
Authority: WO
Inventors: Jan Carlo BARCA; Kevin ACRES
Original assignee: Monash University
Priority date: 2015-10-15
Filing date: 2016-10-10
Publication date: 2017-04-20

Abstract

A method for determining elevation and bearing information of a remote point P is provided. The method includes: (a) providing a K-simplex structure having K+1 vertices and one or more transducers at each of the K+1 vertices of the structure; (b) receiving, at the one or more transducers, a remote signal associated with remote point P; (c) determining wavefront data relating to the remote signal at each of the one or more transducers (and providing said wavefront data to a receiver for processing); and (d) processing wavefront data at the receiver to determine the bearing and elevation of the remote signal.

Description

DETERMINING ELEVATION AND BEARING

INFORMATION OF A REMOTE POINT

Technical Field

[1 ] The present invention relates to a method for determining elevation and bearing information of a remote point and in particular where satellite based navigation is not available.

Background of Invention

[2] Determining location of a point in 3-D space may be achieved using satellite based navigation such as GPS or the like but there are many situations where satellite based navigation is not available or not reliable.

[3] GPS free 3D localisation has been explored and localisation methods have been applied such as traditional maximum likelihood estimation (TML), Traditional Maximum Likelihood, Closed-Form Linear Least Squares and Constrained Weighted Linear Least Squares in order to determine the location of a remote point.

[4] A common problem with these approaches is the amount of processing required. For example convergence of least squares algorithms, matrix arithmetic and the solving of a series of simultaneous equations are all reasonably complex and potentially time consuming procedures to perform via software. Complexity and high amounts of processing is not desirable in many applications where GPS free 3D localisation is required. This is because many of the applications are in very time sensitive contexts - for example a swarm of drones flying in next to each other or a plane needing to determine its location relative to a runway in real time.

[5] It would therefore be desirable to provide a method which ameliorates or at least alleviates the above-mentioned problems.

[6] The discussion of documents, acts, materials, devices, articles and the like is included in this specification solely for the purpose of providing a context for the present invention. It is not suggested or represented that any or all of these matters formed part of the prior art base or were common general knowledge in the field relevant to the present invention as it existed before the priority date of each claim of this application.

Summary of Invention

[7] According to a first aspect, the present invention provides: a method for determining elevation and bearing information of a remote point P, the method including: (a) providing a K-simplex structure having K+1 vertices and one or more transducers at each of the K+1 vertices of the structure; (b) receiving, at the one or more transducers, a remote signal associated with remote point P; (c) determining wavefront data relating to the remote signal at each of the one or more transducers and providing said wavefront data to a receiver for processing; and (d) processing wavefront data at the receiver to determine the bearing and elevation of the remote signal.

[8] Advantageously, the present invention provides an array of transducers, positioned at the vertices of a K-simplex structure, and said transducers supply time difference of arrival information to a receiver - this data is then processed without the need for satellite based navigation (such as GPS information) to determine bearing and elevation of the remote signal. Advantageously, the design is agnostic of the type of transducer used.

[9] Advantageously, the present invention derives the relative bearing and elevation information from a single signal - avoiding the need to derive an absolute location from a number of signals (as required by GPS and similar systems). GPS style similar systems require four or more transmitters and one receiver. The present invention requires only one transmitter and four receivers (or one receiver should the receiver inputs and outputs be multiplexed).

[10] Preferably, the wavefront data includes phase difference or time difference of arrival between each of the transducers.

[1 1 ] Preferably, K = 3 and the structure is a tetrahedron having vertices A, B, C and D. For a tetrahedron, processing wavefront data at step (d) includes the steps of: (i) assuming the value of remote point P to be at infinity; (ii) determining distance x to a remote point P via x = where: x is the distance from the centroid of

the structure to the remote point P; R is the radius of a circumscribed circle; b is an angle related to the remote point; and r is the distance from a vertex, of the structure, to the remote point P.

[12] The method may further reduce the effect of a "near field" signal by further including the steps of: (i) determining if a wavefront associated with remote signal from point P has curvature as it reaches the one or more transducers; (ii) determining the amount of likely error in the measurements based on the curvature. Near field in this case may be where the distance from the centroid to remote point is at a distance that matches the transducer separation distance. Advantageously, faster processing is provided but a trade-off, namely an error component, is minimised since the error component is known. In any event, measurements are being made in what is potentially a fast moving dynamic environment in which any measurement may be outdated a fraction of a second after it was made.

[13] Advantageously, by providing error correction for near field signal sources depending on the orientation of the K-simplex structure relative to the remote point, the present invention improves accuracy for bearing and elevation information for near field sources. This may apply to one or more objects in close proximity to each other. For example, heterogeneous swarm robotics where teams of Unmanned Air Vehicles (UAV) and Unmanned Ground Vehicles (UGV) interact with each other, or UAV/ Unmanned Surface Vehicle (USV) interact with each other at close proximity.

[14] Ranging information may further be provided to improve accuracy by, further including the step of determining ranging information from an out of band source. The out of band source includes one or more of altitude data or radio ranging (in the case of a sonar application - ranging information may be determined from the echo delay time).

[15] While there will be ambiguity as to whether the remote point is above or below the plane of the K-simplex structure. This ambiguity may be easily resolved by Out of band' methods, for example by comparing (or inferring) altitude information. [16] Elevation of the remote point P may be derived via the K=3 structure (i.e. a full tetrahedral structure) and bearing of the remote point P may then derived from measurements performed at an equilateral triangle face of the K=3 structure.

[17] For an equilateral triangle, processing wavefront data at step (d) includes the steps of: (i) assuming value of remote point P to be at infinity; (ii) determining distance x to a remote point P via x = 2(r₁—+r₂ + ²r₃) ≡— Kr^—rz -+—r^ r-^ and utilizing one or more of the faces of the equilateral triangle to measure the bearing to the remote point P via: r₁ + r₂ + r₃ = 0 and (r_x ² + r₂ ² + r₃ ²) = l.SR² where: I is the length of a side of a face of the equilateral triangle; R is the radius of a circumscribed circle; and r the distance from a vertex, of the structure, to the remote point P.

[18] In some situations, the signal may be degraded. The method may include at step (c), a further step of applying signal conditioning to the signal. The signal conditioning may include limiting the signal to an intermediate frequency stage. For example, in an FM receiver, the gain of the intermediate frequency stage is such that the signal may be limited to a constant level (within some constraints) irrespective of the strength of the signal being received. This may be done by the 'Intermediate Frequency' stage of a receiver.

[19] In an alternative, signal conditioning includes the steps of: (i) detecting a zero crossing or pulse in the signal; (ii) determining the longest transition time of the signal between any given pair of the one or more transducers; (iii) enabling the one or more transducers for a time period which is equal to the longest transition time; and (iv) disabling the one or more transducers until immediately prior to the next expected zero crossing or pulse. It will be appreciated that step (ii) may be pre-determined by construction dimensions of the K-simplex structure.

[20] Advantageously, signal conditioning allows the present invention to operate in an environment where there are multipath reflections.

[21 ] Further if one or more of the transducers is inoperable or in a degraded mode of operation, the method may further include the step of deriving elevation information from the remaining operable transducers according to the formula:

^{= arccos( r)}

[22] Where: S is a scaling factor; Θ is the angle of incidence of a line between the centre of the K = 2 simplex structure and the remote point P; Q is a projected point; and R is the radius of a circumscribing sphere associated with the K = 2 simplex structure. Advantageously, where the one or more transducers are inoperable or in a degraded mode of operation, the present invention may resolve ambiguity out of band. In addition, dis-ambiguated information may be inferred until such time as a crossing of the measurement plane is made by the remote signal source.

Brief Description of Drawings

[23] Figure 1 is a schematic diagram illustrating the method of the present invention;

[24] Figure 2 is a schematic diagram illustrating signal time difference of arrival at tetrahedron vertices;

[25] Figure 3 is a schematic diagram illustrating deriving range between two drones;

[26] Figure 4 is a schematic diagram illustrating the near field effect at zero degrees;

[27] Figure 5 is a schematic diagram illustrating the near field effect at 30 degrees;

[28] Figure 6 is a schematic diagram illustrating the near field effect at 60 degrees; and

[29] Figure 7 is a schematic diagram illustrating deriving positional error according to the method of the present invention.

Detailed Description [30] Figure 1 is a flow diagram 100 illustrating a method for determining elevation and bearing information of a remote point P.

[31 ] At step 105 a K-simplex structure having a K+1 vertices and one or more transducers at each of the K+1 vertices of the structure is provided. The structure is preferably a K=3 simplex structure, namely a tetrahedron or may be a K=2 or equilateral triangle structure. The structure may be applied to an object which needs to know the location of a remote point P which may be another object that may need to be avoided. The transducers may take any form. For example, they may be radio antennas, optical or audio transducers, x-ray detectors, pressure sensors etc.

[32] Control then moves to step 1 10 in which a remote signal associated with the remote point P is received at the one or more transducers, which are located at the K+1 vertices of the K-simplex structure. It will be appreciated that depending on the orientation of the K-simplex structure, the time at which the remote signal associated with remote point P reaches the one or more transducers, will be different.

[33] At step 1 15, wavefront data is determined at each of the one or more transducers. This wavefront data may include the phase difference between each of the transducers or the time difference of arrival of the signal from the remote point P between each of the transducers. As will be appreciated, this will be different for each of the vertices of the K-simplex structure. The data is then provided to a receiver for processing. The receiver may include a radio receiver, but is not limited to this. Each type of transducer may include an appropriate signal conditioner such that TDOA or phase difference may be derived.

[34] Control then moves to step 120 where the wavefront data is processed at the receiver and the bearing and elevation of the remote signal is determined thereby determining bearing and elevation of remote point P. The method may be repeated a number of times over a particular period, particularly when the remote point P is moving and it is important that this data be determined quickly.

[35] Advantageously, the present invention processes the wavefront data and determines the bearing and elevation of the remote signal with reduced processing power and with high speed. The steps involved in determining the bearing on the elevation of the remote signal by processing will be described further below. [36] In a situation where a remote point P is close to the K-simplex structure and one or more transducers, a "near field" effect may impact on determining the position of the remote point accurately. In that case, the method may further include the step of determining if a wavefront associated with the remote signal from the point P has a curvature as it reaches one or more of the transducers. While there will always be some degree of curvature in a wavefront, the curvature increases as the signal gets closer. The present invention derives the degree of curvature for accurate directional information.

[37] In the event that there is substantially no curvature or that the wavefront is substantially flat, no action needs to be taken. However, in the event the wavefront has some curvature, a calculation is performed so that it provides an offset resulting in a more accurate result being provided. It will be appreciated that the present invention may cope with signals which are as close as the transducer spacing distance. Closer signals than this would indicate a high risk of collision for example. Further processing may then be carried out to resolve the distance. Also the error factor would not be a significant factor given the angular distance covered by a very close object. This will be described further with reference to Figure 4.

[38] The method may further include the step of determining ranging information from an out of band source which be for example, one or more of altitude data, radio ranging or sonar echo.

[39] The method may further include the step of applying signal conditioning to the signal. For example, it may include limiting the signal to an intermediate frequency stage. The signal conditioning may further include the steps of (i) detecting a zero crossing or pulse in the signal; (ii) determining the longest transition time of the signal between any given pair of the one or more transducers; (iii) enabling the one or more transducers for a time period which is equal to the longest transition time; and (iv) disabling the one or more transducers until immediately prior to the next expected zero crossing or pulse is expected.

[40] In some situations, the one or more of the transducers may be inoperable, for example, due to shielding and the method may further include the step of deriving elevation information from the remaining operable transducers by calculating a scaling factor and ensuring that the scaling factor is not exceeded. This will be further described below.

[41 ] Figure 2 is a schematic diagram 200 illustrating signal time difference of arrival at vertices of a tetrahedron, namely a 3-simplex structure 205.

[42] The structure 205 includes four vertices, A, B, C and D and the structure 205 is located at a location remote from a signal denoted by 210 which originates from point P.

[43] As can be seen, the signal 210 has a wavefront which transitions on each of the tetrahedron vertices A, B, C and D at different times depending on the orientation of P with respect to the structure 205.

[44] As can be seen, the signal 210 hits vertices A before B, B before D and D before C, all of which results in a time difference of arrival. At each of the vertices, A, B, C and D there may be provided one or more transducers.

[45] The choice of a 3-simplex tetrahedron as the structure 205 may be made since the location of a point in 3D space requires that four measurements be made. This follows from the knowledge that any point in n-space requires n+1 measurement points in order to uniquely identify its location.

[46] Signals received by the transducers at vertices A, B, C and D in the structure 205 may then be communicated individually to one or more receivers, at which the time difference on arrival (TDOA) information may be extracted and ultimately processed.

[47] It will be appreciated that in an alternative, time division multiplex techniques may be used with a single receiver rather than multiple receivers. In this case, four signals may be multiplexed at a receiver input and de-multiplexed at the phase difference or time difference on arrival (comparison stage).

[48] It will be appreciated that the time difference on arrival information may include wavefront data which helps to point at the location P. From that point the mathematical properties of 2 and 3-simplexes (namely equilateral triangles and tetrahedrons), the elevation and bearing information relating to the received signal 210 may be calculated in a series of short steps.

[49] In addition, the properties of a 2-simplex may be also provided in order to provide backup elevation information in the event of degradation of one of the four signals.

[50] It will be appreciated that this arrangement can be applied across a number of technologies. For example, space based x/gamma ray source identification in which a large 3-simplex array may be constructed with gas filled x-ray detectors or spark chamber gamma ray detectors at each of the vertices of the 3-simplex structure. This would provide a cost effective adjunct to currently deployed systems.

[51 ] The present invention may also be utilised to assist with pilot guidance into an airport in the event that no other form of navigation assistance is provided (for example, in an emergency situation), where an array structure may be equipped with for example, VHF receivers. The array may be sized such that it would take advantage of, the 2.5 meter wavelength used for general aviation radio communication. This would permit an array with antenna separation of about 1 meter which would be sufficient for good direction resolution.

[52] The present invention in determining the elevation and bearing of the object may be further advanced by providing altitude information which may be given verbally by the pilot for example. This will then allow the present invention to determine both the distance, bearing and elevation of the aircraft from the airport.

[53] Also envisaged is location of submerged black boxes of aircraft in which a phased array structure of detectors may be provided for detecting the direction to submerged black box locations.

[54] Instead of searching for the strongest signal, each received ping from the black box would arrive from a known direction (as per the method of the present invention), allowing rapid completion of any search. Since sound waves travel relatively slowly (compared with radio signals), the array structure may be relatively compact but still provide acceptable resolution. [55] The present invention may also be utilised in the area of aerial swarm robotics and an example will be given in relation to this, although it will be appreciated that the invention is not necessarily limited to swarm robotics applications.

[56] In aerial swarm robotics, the present invention may be used by drones which form part of an aerial swarm. Drones by a necessity need compact low end and low energy embedded systems and the present invention provides by way of a fast calculation which need not require satellite based navigation such as GPS for example, to determine the location of one or more drones relative to another drone.

[57] The present invention can be utilised with for example, a TV transmitter on a nominated lead drone and detect the phase difference and hence the time difference of arrival (TDOA) of an unused subcarrier such as would be used for one of the two sound channels. This information can be used by other members of the swarm to help maintain their positioning within the swarms.

[58] As will be appreciated, it is important to have accurate distance measurements between the drones in the swarm and this can achieved by a number of means. For example, exchanging altitude information permits determination of distance by means of trigonometry. It will also be appreciated that low power "ultra- wideband" ranging chip sets may also be used to directly derive the distance.

[59] Sound channels associated with the module, a TV transmitter of the type commonly used for first person view (FPV) video information, are often used in radio controlled aircraft such as drones. One sound channel carrier may be used solely for vector determination whereas the other may be used as a telemetry channel for communicating altitude information. Ranging information may also be calculated from the two known values of the height difference and the angle of elevation from one drone to the other.

[60] As shown in Figure 3, two drones 305 and 310 may be provided although more drones may be included. Drone 305 may include one or more transducers associated with a k-simplex structure, preferably a 3-simplex structure, and the range 315 and height difference 320 and angle of elevation Θ may determine the range by R = h ÷ sin(0). Advantageously, the present invention may be used as a general localization device - that is, used to localize different types of UAVs or base stations as long as they transmit signals. For example, if the emitted signal has a higher frequency, then a smaller structure can be used. Base stations may be located via their up-link and UAVs via their down-link (e.g. video data transmitted back to the base). The technology can therefore be used in anti-UAV operations.

[61 ] Reduced complexity configuration provided by the present invention, may also be useful in the instance of aircraft based anti-collision aids. In this case, a k=2 simplex structure may be used and any ambiguity resolved via another means. For example, in the case of an aircraft fitted with a k = 2 simplex structure with antennas mounted at the tail and each wing tip, the present invention may now be used to unambiguously derive bearing but ambiguously derive elevation.

[62] While the elevation may be ambiguous without the use of a fourth antenna/receiver but this ambiguity may be resolved by means of transfer of altimeter data. This may be provided via telemetry on the transmitter's signal. The difference in the received altimeter data when compared with that of the receiving aircraft, would be sufficient to resolve any ambiguity in elevation.

[63] Advantageously, the present invention calculates bearing very quickly and without the need for much processing power or satellite based navigation data.

[64] It will also be appreciated that the method of the present invention may be provided in a spherical sonar system. In this instance, the spacing of the transducers in the k-simplex structure may be designed such that is greater than the pulse width of the sonar transmitter. The sonar transmitter transducer is preferably located at the centroid of the k-simplex structure and sends out periodic pulses.

[65] The power, width and spacing of the pulses may be major factors in the range and resolution of the system. Then, the receiving system rather than the determined phase differences in the incoming signal, may send incoming data to a series of four buffers in memory. These buffers may then be examined for correlation of both amplitude and time of arrival of incoming pulse reflections. A 3D map may then be created of objects that return a signal from the transmitted pulse.

Mathematical Model [66] A mathematical model may be applied to provide useful equalities for switching between spherical and edge measurements of a K-simplex. Example derivations of equations for 2-simplex (equilateral triangle) and 3-simplex (tetrahedral) are provided below.

[67] For the sake of clarity, to facilitate operations on both 2-simplex (equilateral) and 3-simplex (tetrahedral) shapes use is made of the length of an edge rather than the radius of the circumscribed sphere or circle.

[68] In the case of a 3-simplex (tetrahedron) R denotes the radius of the circumscribed sphere, 1 the length of an edge and t represents the height of the tetrahedron; then 6 ₇

t⁼ T^l

R² may be represented in terms of I

_ v 6 _ 3 _

R^{= (}T° = 8 ^l

It follows that the sum of squares of distances from the centroid to the vertices of a regular tetrahedron will then have the following equality

, 4 * 3 ,

4R²≡——I²≡ l.Sl² (1 )

In the case of an 2-simplex (equilateral triangle) R is the radius of the circumscribed circle and I the length of an edge

I

R = -j= l

V3 For an equilateral triangle an equality of the sum of squares from the centroid to the vertices may be shown as

Tetrahedron

[69] A 3-simplex shape is regular tetrahedron with 6 equal sized edges and 4 vertices. Importantly it has 12 rotational symmetries of which use is made of at least four of the symmetries. The properties of the sum of squares of distances from any given point in 3-space to each of the vertices of a regular tetrahedron are determined. The required properties are derived algebraically below. Initially each of the vertices a, b, c and d of a regular tetrahedron as a function of radius R of the circumscribing sphere are determined as shown in Table 1 below.

[70] Table 1 Vertex Placement as a Function of the Radius of a Circumscribing Sphere

An arbitrary remote point P is then provided:

where distance x from P to the centroid of the tetrahedron is

[71 ] For simplicity, an algebraic substitution for the square roots is provided: Let S = V6 and T = V2i?² , then we calculate r_a = P - a, r_b = P - b, r_c=P - c, r_d =

P-d where

r_a = P — a = [ρχ,ργ,ρζ] - r_b = P -b = [ρχ,ργ,ρζ] -

= [px

1 1 1

r_c = P — c■ = [ρχ,ργ,ρζ] - [--S.R,

1 1

= [px + -S.R,px + - Γ, ρχ + ifl]

= P - d = [ρχ,ργ,ρζ] - [0,0, R] -- = [px, py, pz— R]

The individual squares of distances are then derived:

ra ⁼ Ρ^χ2 + ^{2 ~} T.py + pz² +—R.pz +— T² +—R²

- 2 , 2 , 2 1 _ /l _ 1\ , r² px ——5. β. ρχ + py +— 7\ py + pz + -R.pz +— Γ + I— + — ) R'

l_) l_) l_) J J J

, 2 _ 2 _ 2 1 _ /l _ 1\ , r_c ² px +—S. R.px + py +— T. py + pz + -R.pz + -T + I -5 +— ) ?^z rj = px² + py² + pz²— 2R. pz + ?²

The sum of squares of distances is then

+ r_b ² + r² + r_d ²

which expands to

Substituting for S² = 6 and Γ² = 2R²

Z 12 4\

4px² + 4py² + 4pz² +-2R² + (— + -J R² = 4(px² + py² + pz²) + 4 ?^z [72] With x as the distance from the tetrahedron centroid to the remote point and s as the sum of squares of distances from the remote point to the tetrahedron vertices

X² = px² + py² + pz² s = r_a ² + r_b ² + r ² + r_d ² = 4x² + 4R² (3)

Since distance x between the centroid and point P is unknown, sum of squares Equation (3) is recreated whilst removing x from it via

(r_a - x)² + (r¾ - x)² + (r_c - x)² + (r_d - x)² which expands to

4x² - 2(r_a + r_b + r_c + r_d)x + (r_a ² + r_b ² + r ² + r_d ²) (4) However, we know from Equation (3) that Xa + r_b ² + r ² + r_d ²)≡ 4x² + 4R² (5)

By assigning r_x = r_a— x , r₂ = r_b— x , r₃ = r_c— x , r₄ = r_d— x then substituting for Equation (5) and r_1→

8x²— 2(r_x + r₂ + r₃ + r₄ + 4x)x + 4i?² = r² + r₂ ² + r₃ ² + r₄ ² (6) which expands to

2(r_x + r₂ + r₃ + r₄ + 4x)x We get

8x² + 2(r_x + r₂ + r₃ + r₄)x

Which simplifies Equation (6) to

— 2(r_x + r₂ + r₃ + r₄)x + 4R² = r² + r₂ ² + r₃ ² + r₄ ² which is equivalent to

— 2(71 + ^r2 + ^r3 + ^r4) ^{= Γ}ί + ^r2² + ^ri + ^r4 ^— 4 ?²

By multiplying both sides by -1

2(7^* ₁ + ^r ₂ + ^r ₃ + ^r ₄) = 4R²— (r² + r₂ ² + r₃ ² + r₄ ²) and solving for x

4R² - (r_x ² + r₂ ² + r₃ ² + r₄ ²)

= 2(—r_x I +—r₂ I +—r₃ 7 +— r₄) (⁷

[73] Advantageously, x can be recovered solely from the measurements from which it has been removed. Having discarded the original distance of the remote point P and as P approaches infinity the 2(r_x + r₂ + r₃ + r₄) component approaches 0. We utilize this phenomenon in a 'time difference of arrival' application by applying an equal offset to each r_k ensuring

r_k = 0 to effectively place P at infinity, whilst retaining its angular properties with respect to the centroid. (Note that time difference of arrival measurements imply that at least one of ri, r₂, r₃, r₄ = 0. Failing this, the minimum r_k first needs subtracting from all r_k). This is performed by the following procedure in which 0 is set to the required offset r_k) where

r_x = r_x— o

r₂ = r₂ - o

r₃ = r₃ - o

r₄ = r₄ - o

[74] This transform has an extra side effect insomuch as the following now holds; assuming that R is the radius of the circumscribed sphere of the tetrahedron and b an angle related to the remote point, then

b = arccos(— ) (8)

R

[75] Importantly, for any vertex the angle to the remote point P is 90 - arccos(— ) degrees relative to the plane normal to a line from the vertex to the centroid. Positive values are above the plane and negative values below (with respect to the vertex). Except for near field effects (described below with reference to Figure 4, 5 and 6), this is a direct reading relative the related plane and is not affected by changes in bearing. A corollary to this is that, in the case of a regular tetrahedron sitting with one face on the horizontal plane, that the r_x value at the upper vertex may be used directly to derive the elevation of a remote point.

[76] The closer the distance from the centroid to P the more a 'near field' effect on the vertices of the tetrahedron becomes apparent. Conversely, as P moves further away this effect decreases. For near field measurements, checks and supplementary offsets are required to avoid sine and cosine limits.

Equilateral Triangle

[77] In the case of utilizing a 2-simplex structure (i.e. equilateral triangle is a regular 2-space simplex with 3 equal sized edges and 3 vertices) the calculations are as follows: [78] The sum of squares s, assuming that I is the length of a side, x the distance from the centroid to the point p and R the radius of the circumscribed circle, may be represented thus; bearing in mind the equality at (2) above

s = 3x² + 3R²≡ 3x² + I²

[79] In a similar fashion to the 3-simplex (tetrahedron) we can derive distance x to a remote point via

3R² - fa² + r₂ ² + r₃ ²) ₌ I² - fa² + r₂ ² + r₃ ²)

2fa + r₂ + r₃) ^~ 2fa + r₂ + r₃)

[80] In contrast to the 3-simplex (tetrahedron) vertices, which are used to measure elevation, we use one, or possibly more, of the triangle faces to measure the bearing to the remote point. Not only does this require that

rl + ^r2 + ^r3 ⁼ 0 (10) but also that fa² + r₂ ² + r₃ ²) = 1.5Λ² (1 1 )

[81 ] In particular, conformance to Equation (1 1 ) ensures that near field distortion is minimized with the resulting reduction in the measurement error component.

[82] To prepare a triangle for measurement of bearing information the following steps are carried out:

1 . Subtract the smallest of r , r₂, r₃ from each of r , r₂, r₃

2. Subtract the average of r_x, r₂, r₃ from each of r_x, r₂, r₃

3. Scale values r_x,r₂, r₃ such that fa² + r₂ ² + r₃ ²) = 1.5i?²

4. Read the bearings at each r_k, a = ^{cos rfe)} [83] The scaling function in Step 3 above ensures that the triangle face becomes effectively in the same plane as the remote point. Point P = (7Ί, Γ₂, Γ₃) and unit normal N = (1,1,1) /V3, while a projected point Q is derived using

Q = P - (P. N)N (12)

Q is then scaled to derive point 0 via

Point 0 now conforms to the constraints of Equation (10) and (1 1 ), and the resulting values can be used directly for bearing derivation.

[84] Assuming S is the scaling factor and Θ the angle of incidence of a line between the centre of the triangle and the remote point, the angle of the remote point relative to the plane of the triangle can be obtained as follows:

[85] This information can be used as a check for the elevation derived from the tetrahedron as shown at Equation (8), assuming that near field effects do not cause S to drop below 1.5, in which case an elevation of 0 degrees must be assumed. This information is also useful in degraded, or intentional 'reduced complexity' modes of operation, albeit with ambiguity as to whether the remote point is above or below the plane of the triangle. This ambiguity may, however, be easily resolved by Out of band' methods, for example by comparing (or inferring) altitude information and the like.

Near and Far Fields [86] In some situations, the remote point P may be a signal source that is far enough away such that a wavefront is essentially flat as the wavefront passes through the various measurement points at the vertices of the k-simplex structure.

[87] Conversely, "near field" signal sources may have a noticeable and measureable curvature to the wavefront which in the present invention, can lead to an error (albeit a predictable error) in the measurements. Near field in this case is generally closer to the centroid than the spacing between transducers.

[88] In essence, the effect of a "near field" can cause the time difference of arrival of the signals to be either closer or further apart than expected and is largely dependent on the relative orientation of the signal and the k-simplex structure. Typically this is a 3-simplex tetrahedral structure.

[89] Figure 4 is a schematic diagram 400 illustrating where the remote signal is at zero degrees orientation relative to and on the same plane as the k-simplex structure. In this case a 3-simplex tetrahedron.

[90] The signal source P is oriented along a line 415 to the centroid and apex of the structure 410 and in this situation, the wavefront 405 travels at a longer distance to the further receivers B, C and D and then would be expected if the source point was at infinity. The result of this is that it effectively increases the time difference of arrival between the receiver at vertex A and vertices B and C.

[91 ] In Figure 4, the effect at vertices B and C is symmetric. That is, the path 420 between the remote point P is the same as that of the path between remote point P and vertices B.

[92] In contrast with Figure 5, a schematic diagram 500 is shown having a remote point P from which emanates a wavefront 505 toward a structure 510 which is at an angle of 30 degrees offset.

[93] In this instance, the delay at vertices B has the effect of making the signal source at remote point P seem further away, relative to the delay at vertices A and C. This displaces the apparent angular position of P to less than the expected 30 degrees. [94] Contrast this with Figure 6 in which a schematic diagram 600 is shown having a remote point P emanating a signal toward a structure 610 which has a bearing of 60 degrees and the wavefront 605 at vertices A and B is delayed relative to that, which would be expected for a signal from a point at infinity.

[95] The overall time difference of arrival of the signal at vertices A and B compared with C, is less than would have otherwise been expected.

[96] However as with Figure 4, because the effect of symmetric, it does not have an effect on the derived bearing.

Use of Tetrahedral Symmetries for "Best Range" Measurements

[97] In order to help maintain the best possible resolution, in elevation measurements, these are limited to ±35.5°; this being just larger than one half of the tetrahedron dihedral angle of arccos(l/₃) or « 70.53°. Irrespective of the orientation of the tetrahedron there will always be at least one vertex/centroid line that has an angle within the range of + ³ degrees from the signal source relative to its normal. This permits selection of a symmetrical orientation of the tetrahedron relative to the signal source in which the elevation is within the range±35.5°. This allows operations on shallow angles when determining the bearing component of the signal and is of great benefit when operations that rely on sines and/or cosines are carried out.

Degraded Modes of Operation

[98] The present invention also allows for contingencies for degraded modes of operation and in particular handling multipath reflections and antenna shielding.

Multipath Reflection Mitigation

[99] Multipath reflections have the potential to cause inaccurate readings. There a number of ways that this can be addressed. This present invention works entirely on phase difference or, equivalently, time difference of arrival, of a signal, at the transducers, which lends itself to the possibility of a certain amount of signal conditioning. In particular limiting at an intermediate frequency (IF) stage allows use of the 'FM capture effect'.

[100] A second method of addressing this problem is by gating the incoming signals. This can be especially useful when the antenna separation is small when compared to the wavelength of the signal being received. This works by waiting for the first event, be it zero crossing or pulse. Input from the remaining receivers is then enabled for a time equal to the longest transition time of a signal across the array. Following this time delay the receivers are all disabled until just before the next event is expected, when they are re-enabled once more.

Antenna Shielding

[101 ] It is possible that, at some orientations, one of the antennas of the array will become shielded from the remote signal. This may be, for example, as a consequence of the construction of the device around which the antenna array is built. In this instance the present invention takes advantage of Equation (14) to derive elevation information directly from the readings at the remaining three transducers; bearing in mind that this will lose resolution at higher angles and that the special case of 90 degrees needs to be handled separately.

Results of Example Simulations

[102] Simulations were carried out to derive the error component of elevation and bearing readings with respect to the distance to the signal - essentially providing a baseline for the theoretical limits of the calculations. However, in practice limits that imposed by measurement resolution and construction tolerances are likely to be less than the theoretical maximum.

[103] A regular tetrahedron with edge length of 0.5 metres was modelled. These dimensions were chosen as they would be suitable for implementation in an aerial swarm robotics context, where the size of the quad rotor drones would permit a relatively unobtrusive antenna array to be built around the main chassis of the drone.

[104] Measurements of the error components were made at 1 , 10 and 100, 1 ,000, 100,000 and 1 ,000,000 meters. This provided a sufficient selection of distances from which the behavioural properties of this system could be determined. [105] For the purposes of the simulation the angle of elevation was restricted to the range -35 to +35 degrees, which is approximately half the dihedral angle of a regular tetrahedron of « 70.5°. Doing so provides better resolution than would have been obtainable at higher angles and takes advantage of the symmetries of a tetrahedron that ensures that any given remote point will always have an angle of incidence, Θ, such that |θ | <= arccos(¹/3)/2, or about 35.26 degrees, to at least one plane centred at the centroid and normal to a vertex/centroid line.

[106] As shown in Figure 7, positional errors were calculated individually for the bearing and elevation measurements. This was achieved by assuming that the true and derived bearings form two points on an isosceles triangle, C and D respectively, and that the measurement point, A, makes the third, with oc = zCAD. The line CD is then bisected by a line AB to form two right angle triangles.

[107] The, two dimensional, error component then becomes:

CD = 2BD = 2AD sin oc/2 (15)

[108] To calculate the positional error in 3-space, the individual error components were derived, CD for the bearing measurement and CD' for the elevation measurement. The total, absolute, error was then derived by Pythagoras' theorem:

Error Component = V CD² + CD'² (16)

[109] Simulation results were plotted in three dimensions with an x axis representing degrees of bearing, from 0 to 360, z axis representing elevation from -35 through to +35 degrees and a y axis representing a suitably scaled error component.

[1 10] In each of the simulations, measurements were made of the bearing and elevation errors whilst stepping through 0 to 360 degrees of bearing and at elevations from -35 to +35 degrees, both of which were made in 5 degree increments.

Results at 1 Metre [1 1 1 ] A simulation was set up for an effective target distance of 1 metre from the centroid of the tetrahedron. Simulated measurements were then made. The bearing error was noted to have a value that approximates a sine wave of 3 times the periodicity of the target bearing as the bearing angle is increased. In this instance the error was found to be « -4sin(36), where Θ is the expected bearing, and was relatively free of any component introduced by elevation changes.

[1 12] The elevation error was found to be a little more complex with a 1.5 + 1.5cos(36) component at -35 degrees of elevation, changing to 1.5 - 1.5cos(36) at 35 degrees. In this instance there was a noticeable effect caused by changes in bearing.

[1 13] The bearing and elevation measurements for the 1 metre target distance where correlated and the absolute value of the positional errors calculated. These average out at just less than 5.4 centimetres with a maximum error of 8.4 centimetres.

Results at 10 Metres

[1 14] A simulation was set up for an effective target distance of 10 metres from the centroid of the tetrahedron. Simulated measurements were then made.

[1 15] The bearing error was once more noted to have a value that approximates a sine wave of 3 times the periodicity of the target bearing as the bearing angle is increased. In this instance the error was found to be « -0.4sin(36) and was relatively free of any component introduced by elevation changes.

[1 16] The elevation error followed a similar pattern to the 1 metre results, this time with a 0.15 + 0.15cos(36) component at -35 degrees of elevation, changing to 0.15 - 0.15cos(36) at 35 degrees. Once again there is a noticeable effect caused by changes in bearing.

[1 17] The bearing and elevation measurements for the 10 metre target distance where correlated and the absolute value of the positional errors calculated. These average out at 5.61 centimetres with a maximum error of 8.83 centimetres.

Results at 100 Metres [1 18] A simulation was set up for an effective target distance of 100 metres from the centroid of the tetrahedron. Simulated measurements were then made.

[1 19] The bearing error was once more noted to have a value that approximates a sine wave of 3 times the periodicity of the target bearing as the bearing angle is increased. In this instance the error was found to be « -0.04sin(30) and was relatively free of any component introduced by elevation changes.

[120] The elevation error, at 100 metres, followed a similar pattern to both the 1 and 10 metre results, this time with a 0.015 + 0.015cos(30) component at -35 degrees of elevation, changing to 0.015 - 0.015cos(30) at 35 degrees. Once again there is an effect caused by changes in bearing, but was now seen to be decreasing as a linear function of the distance to the target.

[121 ] The bearing and elevation measurements for the 100 metre target distance where correlated and the absolute value of the positional errors calculated. These average out at 5.61 centimetres with a maximum error of 8.84 centimetres. It was found that the positional error is effectively constant regardless of distance to the target.

Simulation Summary

[122] Positional errors were then simulated distances ranging from 1 to 1 ,000,000 metres at powers of 10 and the minimum, average and maximum error positional error components calculated. These are tabulated below.

Table 2 Error Components Compared to Distance

10 000 0.08 j 5.61 8.84

ΪΟΟ ΟΟ 0.08 : 5.61 8?84

lOOO-OOO 0 8 8.84

[123] It can be seen from the above that angular errors decrease as a linear function of distance to the target. This has the effect of producing a positional error that is unaffected by target distance. In the case of this simulation (with 50 centimetre antenna spacing) the maximum error is 8.84 centimetres.

[124] In the case where distance to a target is known then the error component can be calculated and removed from the results. However, it is considered that the amount of error is negligible for any distance greater than, say, 10 to 100 metres.

Claims

1 . A method for determining elevation and bearing information of a remote point P, the method including: a. providing a K-simplex structure having K+1 vertices and one or more transducers at each of the K+1 vertices of the structure; b. receiving, at the one or more transducers, a remote signal associated with remote point P; c. determining wavefront data relating to the remote signal at each of the one or more transducers and providing said wavefront data to a receiver for processing; and d. processing wavefront data at the receiver to determine the bearing and elevation of the remote signal.

2. The method of claim 1 , wherein wavefront data includes phase difference between each of the transducers or the time difference of arrival between each of the transducers.

3. The method of claim 1 , wherein K = 3 and the structure is a tetrahedron having vertices A, B, C and D.

4. The method of claim 3, wherein K=3 and step (d) includes the steps of: i. assuming the value of remote point P to be at infinity; ii. determining distance x to a remote point P via

where: x is the distance from the centroid of the structure to the remote point P; R is the radius of a circumscribed circle; b is an angle related to the remote point; and r is the distance from a vertex, of the structure, to the remote point P.

5. The method of claim 1 , wherein at step (d) the method further includes the steps of: i. determining if a wavefront associated with remote signal from point P has curvature as it reaches the one or more transducers; ii. determining the amount of likely error in the measurements based on the curvature.

6. The method of claim 1 , further including the step of determining ranging information from an out of band source.

7. The method of claim 6, wherein the out of band source includes one or more of altitude data, radio ranging or sonar echo.

8. The method of claim 1 , wherein K = 2 and the structure is an equilateral triangle having vertices A, B and C.

9. The method of claim 1 , wherein K=2 and step (d) includes the steps of: i. assuming the value of remote point P to be at infinity; ii. determining distance x to a remote point P via

_ 3R² -(r₁ ² +r₂ ² +r₃ ²) _ l² - r₁ ² +r₂ ²+r₃ ²)

~^~ 2(r₁+r₂ +r₃) ^~~ 2(ΤΊ+Τ·₂ +Γ₃) and utilizing one or more of the faces of the equilateral triangle to measure the bearing to the remote point P via:

r₁ + r₂ + r₃ = 0 and (r_x ² + r₂ ² + r₃ ²) = l.SR² where: I is the length of a side of a face of the equilateral triangle; R is the radius of a circumscribed circle; and r is the distance from a vertex, of the structure, to the remote point P.

10. The method of claim 1 , wherein at step (c), the method further includes the step of applying signal conditioning to the signal.

1 1 . The method of claim 10, wherein signal conditioning includes limiting the signal to an intermediate frequency stage.

12. The method of claim 10, wherein signal conditioning includes the steps of: i. detecting a zero crossing or pulse in the signal; ii. determining the longest transition time of the signal between any given pair of the one or more transducers; iii. enabling the one or more transducers for a time period which is

equal to the longest transition time; and iv. disabling the one or more transducers immediately prior to the next expected zero crossing or pulse is expected.

13. The method of claim 1 , wherein if one or more transducers is inoperable or in a degraded mode of operation, further including the step of deriving elevation

information from the remaining operable transducers according to the formula:

^{5 =} n T'^{e = arcc(,s( )}

Where: S is a scaling factor;

Θ is the angle of incidence of a line between the centre of the K = 2 simplex structure and the remote point P;

Q is a projected point; and

R is the radius of a circumscribing sphere associated with the K = 2 simplex structure.