WO2002084218A1

WO2002084218A1 - Inclinometer based north finder

Info

Publication number: WO2002084218A1
Application number: PCT/IB2001/000583
Authority: WO
Inventors: Ahmet Yalcin
Original assignee: Ahmet Yalcin
Priority date: 2001-04-10
Filing date: 2001-04-10
Publication date: 2002-10-24

Abstract

Inclinometer Based North Finder is a new concept that allows reaching direction information from inclination. It uses specially defined 'Yalcin Curve' which is unique for each latitude of the measurement point on the Earth. The 'Yalcin Curve' is simply a measure of the amount of deformation of gravity acceleration caused by centripetal acceleration in the measurement point.The North finder can be implemented using inclinometers placed on a plane in a predefined order. The device is also equipped with a high speed microprocessor unit which calculates the 'Yalcin Curve' values depending on the Latitude of the measurement point and compares them with the measured 'Yalcin Curve' values obtained from the outputs of the inclinometers used. Inclinometer Based Direction Finder has many advantages over today's other North Finding Gyros. It provides higher accuracy and real time output in a lower cost and lighter package.

Description

INCLINOMETER BASED NORTH FINDER

INTRODUCTION

The work outlined here describes how to design a north finder using traditional inclinometers. "From inclination to direction" is completely a new concept and to be able to achieve it, very high precise inclinometers are necessary. This work outlines step by step implementation procedures how to find direction and how to improve direction accuracy by increasing the number of inclinometers used.

The work also describes why it is not possible to observe perfect planes using inclinometers and why we have been observing a deformed sky instead of a uniform one.

The work defines a special curve, which is unique for each latitude of the measurement point on the Earth. I will call this curve as a "Yalcin" curve. The direction is found by fitting the calculated "Yalcin" curve and the measured "Yalcin" curve in the measurement point on the Earth.

North- finding devices, in general, use Magnetic North and true North as direction reference.

Electronic and magnetic compasses are based on the magnetic field of the Earth and use the Magnetic Poles as direction reference. Those direction finders are low cost devices but do not give precise directions and very much influenced by the magnetic materials close the devices.

The Mechanical Gyros, Laser Ring Gyros and Fibre Optic Gyros are based on the effect of the rotation of the Earth and use the Geographic Poles as the direction reference. Those devices give better accuracy and are independent of magnetic disturbances. But they are very expensive and heavy devices and do not give real time outputs.

GPS based direction finders are also new devices appeared on the market recently. But they are also costly and lion real-time devices. Furthermore, they are outdoor devices and GPS antennas should be located on the measurement plane in a specified order, which may not be practical for most of the applications.

On the other hand, Inclinometer based North finders described here, also uses the effect of rotation of Earth, but it is a low cost, high accurate, small and light device giving real time response.

THE INCLINOMETER

An inclinometer is an electronic slope measurement device, which may measures the slope in a single axis direction or two axes perpendicular to each other. In this work, we will work with single axis inclinometers as shown in Figure 1. Although Figure 1 shows a typical liquid based tilt sensors the whole work can be applicable to any kind of inclinometers.

As it is seen from Figure 1, a single axis tilt sensor has one balancing axis and one sensitive axis. The liquid surface inside of the tilt sensor is always perpendicular to the centre of gravity. If the centre of gravity (gravity acceleration vector direction), which will be called in this work as "the balancing vector direction", is parallel to balancing axis, than the liquid level of the sensor will be parallel to the sensitive axis. In this case the inclinometer will be in null position and the output of the inclinometer will be zero.

The tilt sensor has also a cross axis, which is perpendicular to both the sensitive axis and the balancing axis. Ideally, a single axis inclinometer is not sensitive to the angle changes in the cross axis direction. This means that, the output of the inclinometers will remain unchanged if we turn it around its sensitive axis.

WHAT DOES AN INCLINOMETER MEASURE?

Let x-y be a flat plane perpendicular to the centre of gravity (Figure 2). Let P be another plane having y-axis as an intersection line with x-y plane with a maximum slope of a_m- Let place an inclinometer on the P plane of which sensitive axis remains in the same direction with x axis. Here k is the normal unit vector of x-y plane. Let n_P be the normal vector of the P plane and up_m be the unit vector on the P plane in the maximum slope direction. It is clear that the inclinometer in Figure 2 will measure the α_m angle, which is an angle between the sensitive axis of the inclinometer and projection of it (x-axis) on the x-y plane. α_m is also the angle between the balancing axis of the inclinometer and the balancing vector effective on the inclinometer or the angle between the normal unit vectors of P-plane (where the inclinometer stands on) and x-y plane (perpendicular to the balancing vector).

So far, we assumed that the inclinometer is under the influence a single gravitational acceleration only. But in real world it is not. Let us now see what really happens in the real world.

EFFECT OF THE ROTATION OF EARTH

Since the Earth is rotating in its North-South pole axis with 24 hours/day angular speed, every point on the Earth under the influence of a centripetal acceleration. The magnitude of this centripetal acceleration is:

1 a = (2π/T)². R_L where: a: magnitude of centripetal acceleration

T: period of the rotation

R : the distance between the measurement point on the Earth and the rotation axis.

If the radius of the Earth is R_e than R = R_e. Cos L, where L is the latitude angle of the measurement point. Than equation 1 becomes (Figure 3):

2 a = (2π/T)². R_e. Cos L

Using the values T= 24 hours = 8,6 .10⁴ s and R_e = 6378137 m (WSG 84 datum) and ignoring the flattening we obtain

3 a = 0,03373 . Cos L m/s²

Equation 3 says that the centripetal acceleration is maximum at the Equator (a_e = 0,03373 m/s² ) but zero at the North and South Poles.

Since the effective acceleration will be the sum of gravitational and centripetal accelerations, we can say that, the centripetal acceleration will bend the gravitational acceleration towards the equator.

Let us use the coordinate system showing in Figure 4 where a_L is the centripetal acceleration vector at latitude L, which is towards the Equator (south direction at Northern Hemisphere). We can now, calculate the bend amount on the gravitational acceleration in terms of latitude. From Figure 4, the following equations are valid:

4 a_L = a_L. (Sin L i + Cos L k)

5 g = - g. k Where a_L : centripetal acceleration vector at latitude L

&L '■ magnitude of a g : gravitational acceleration vector having magnitude g From equations 4 and 5 we obtain the resultant vector g_R as the sum of a and g:

6 R = a_L. Sin L i + (a . Cos L - g). k We can get the bend amount δ as the angle between g and resultant vector g_R by a dot product of those vectors:

7 g • gR = g • gR • Cos δ

Knowing that, aL = ae . Cos L, from equations 4, 5, 6 and 7 we obtain: 8

USING THE INCLINOMETER IN REAL WORLD

Let us now see what we measure with the inclinometer in real World.

The x-y plane in Figure 5 is a horizontal plane perpendicular to the gravitational acceleration. Let us place an inclinometer on this horizontal plane of. which sensitive axis in west-east direction (y-axis). The balancing axis of the inclinometer will be in the same direction with the gravitational acceleration because it is placed on a horizontal plane perpendicular to it. In this case the centripetal acceleration vector will be in a perpendicular plane to the sensitive axis of the inclinometer and will have no effect on it. The normal unit vector of the x-y plane (k) will be parallel to the balancing axis and the inclinometer will give zero output.

Now, let us turn the inclinometer by 90° to north-south direction. The balancing axis is still in gravitational acceleration direction but the inclinometer is under the influence of centripetal acceleration as well. Hence the apparent gravitational acceleration (balancing vector) for the inclinometer will now be the resultant vector g_R. Since the balancing axis is still in g direction the inclinometer will measure an angle other than zero. Now, we can say that, when we turn the inclinometer from west-east direction to south-north direction the balancing vector for the inclinometer will move from the gravity acceleration g to the resultant vector gR. Hence, g will be the apparent gravity acceleration vector for the inclinometer in south or north direction. Now, we can say that:

• There is no a single apparent gravitational acceleration (balancing vector) for the inclinometers in real world,

• There is no a single absolute flat surface (horizontal plane) for inclinometers,

If we turn the inclinometer around its balancing axis on any plane, the direction of the balancing axis will remain unchanged and will be in parallel with the plane normal. On the other hand when we turn the inclinometer on the plane around its balancing axis, the balancing vector will be changed due to the centripetal acceleration. Since the inclinometer simply measures the angle between its balancing axis and balancing vector (apparent gravitational acceleration vector), we may expect a varying output from inclinometer when we turn it even on a horizontal plane. Moreover, we can say that there is no any horizontal plane for the inclinometers on which we can measure zero angle in each direction. This is valid for on all rotating planets in Universe.

To realise this, our inclinometer should be sensitive enough. Fortunately, with today's technologies it is possible to manufacture far better inclinometers.

Let us now look at more closely what we measure with inclinometers on various planes. Figure 6 shows the effects of the centripetal acceleration on various directions in particular latitude. Here we will consider a plane A described by the a_L centripetal acceleration and east-west line. On this A plane the centripetal acceleration will have maximum effect on the inclinometers in a direction and zero effect in its perpendicular east-west direction. The effect of the centripetal acceleration, λ angle apart from north-south direction on A plane is proportional to ax = a_L. Cos λ. From Figure 6 we can write:

9 ax = a_λ . (Cos μ . Cos φ .i + Cos μ . Sin φ. j + Sin μ .k) where, μ : angle between aχ and x-y plane φ : projection of λ angle on x-y plane Let n_A be the normal unit vector of A-plane. (It is perpendicular to both a_L and j so UA = -Cos L i + Sin L k ). Since λ is the angle between a_L and aχ and aχ J_ n_A from formulas 4 and 9 we obtain: 10 μ = Atan (Cos φ . Cot L) and

11 λ = Acos (Sin μ / Cos L)

Hence we can find aχin terms of φ angle.

The sum of g and χ (g_φ) will be the effective balancing vector for the inclinometer placed on x-y plane and φ angle apart from south-north direction. 12 g_φ = a_λ .Cos μ . Cos φ .i + a_λ .Cos μ . Sin φ. j + (a_λ .Sin μ -g) ,k

Since we can obtain angle μ in terms of φ (Eq. 10) g_φ is a function of φ. Equation 12 gives the balancing vector bunch (apparent gravity acceleration vector) for the inclinometer in φ direction. Each direction on x-y plane corresponds to a different balancing vector, hence different horizontal plane. The normal unit vectors for those bunches of horizontal planes are opposite directions of g_φ vector bunch. If n_φ is normal unit vector bunch of apparent horizontal planes, than we can write the following equations:

13 n_φ = (-1/g_φ ) . [a_λ .Cos μ .Cos φ A + a_λ .Cos μ . Sin φ. j + (a_λ .Sin μ -g) .k] If δ is the angle between g and g_φ, it can be expressed as:

14 δ = Acos [-(a_λ .Sin μ -g) /g_φ ] Figure 7, shows δ-φ graphics for various latitude values. Figure 7 gives very interesting results:

• The balancing vectors (apparent gravity acceleration vectors) for the inclinometers will vary depending on direction and latitude.

• As we approach to the Equator, the maximum angle deviation moves from south direction to east and west directions.

• As we approach to the poles the maximum value of δ angle gets smaller.

INCLINOMETER ON ANY PLANE

It is clear that a plane perpendicular to g or g_R is a theoretical plane only. It is not possible practically implement such planes with very precise angle tolerance. Hence we have to investigate the situation on any plane. Let us consider a plane N of which normal unit vector is n (Figure 8).

15 n = a i + b j + c k where a² + b² + c² =1

Now, we have to find the unit vector u_φ on this plane in the same direction of the sensitive axis of the inclinometer. From Figure 8:

16 u_φ = Cos p . Cos φ. i + Cos p . Sin φ. j + Sin p. k where p is the angle between sensitive axis of the inclinometer and x-y plane. Since u_φ ± n, u_φ . n =0, than:

17 p = Atan [(-1 / c) . (a. Cos φ + b. Sin φ)] We assume that the N plane perpendicular to n is a known plane hence a, b and c are known coefficients. Hence, p can be calculated in terms of φ angle. We know that if the inclinometer will measure angle ψ, it should be:

18 ψ = π/2 - Acos (u_φ . n) hence:

19 ψ = Asin [(-1/gφ) . (a_λ . Cos μ. Cos p + (a_λ . Sin μ -g). Sin p )] Using relevant equations we can find ψ angle in terms of φ, and obtain ψ-φ graphics as seen in Figure 9. Since there may be infinite number of planes on which we can place the inclinometer, we put 3 different graphics each with different plane having maximum slope in different direction. We also kept the maximum slope of the plane very small to distinguish the differences of the graphics from a cosine graphic. Figure 9 shows the inclinometer outputs in various directions, (south, south-east, north) for various latitudes. They look like a cosine function with various phase angles modulated by another graphics. It is also clear that ψ_φ = - ψφ_+π- The shape of the graphics highly depend on the latitude and we get a shaip change in east-west or west east direction toward the Equator and the graphics look like a cosine function as we move to the Poles.

From the above discussion we get following results:

1. An inclinometer giving extremely precise outputs provides also direction information, (i.e. its output is a function of the slope of the plane in its sensitive axis direction plus the effect of the centripetal acceleration in this direction) 2. A Precise elevation measurement depends on the direction of the elevation. Equal elevation angles in different directions measured by the inclinometer are not equal. An inclinometer measures true elevation angles in exact east-west or west-east direction but an elevation in south direction is not equal the same angle in the north direction. Hence, if the observatory is not at the Poles, the astronomer in the observatory will view a slightly bended sky depending on where the observer is on the Earth. In general, we can say that the universe seems to be deformed to the observer on a rotating planet. Hence for a precise elevation measurement, the angle should be corrected by taking into the consideration the direction of the elevation.

Using those above characteristics of the inclinometers we can design a direction finder which may have very useful specifications.

INCLINOMETER BASED NORTH FINDER

Assuming that we are measuring the slope accurately enough, it is possible to construct a direction finder using inclinometers.

Figure 9 in the previous section, shows what we measure with inclinometer on any plane, while we turn it on the plane around its balancing axis or plane normal. We may use equivalent sensors rather than turning the single sensor on the plane. Let us assume that, we use infinite number of sensors fitted in every direction on the plane. If we read the outputs of them we can get similar graphics shown in Figure 9.

We know that the graphics shown in Figure 9 has an information about the slope angle of the plane and also direction information (varying effect of centripetal acceleration according to direction). Each graphics in Figure 9 define a slightly bended surface. Hence, we may filter the effect of the centripetal acceleration by defining a new pseud plane.

Let us now, define a pseudo plane represented by two unit vectors defined by the angles measured by two different inclinometers on the plane. We can define infinitive number of pseudo planes depending on the inclinometers chosen. The pseudo plane is a pure plane passing through the origin of the specified coordinate system, hence, it will intersect the apparent deformed plane specified by the inclinometers outputs shown in Figure 9.

Let us choose the pseudo plane with the angles measured by the inclinometers placed the first one to south direction and the second one perpendicular to the first one as shown in Figure 10. If the angles we measure by the inclinometers chosen are α and β respectively, we can define following unit vectors:

21 u₂ = Cos β. Cos γ i + Cos β . Sin γ j + Sin β k where: Ui : unit vector from north to south direction defined by first inclinometer u₂ : second unit vector perpendicular to Ui defined by the second inclinometer γ : projection of perpendicular angle on the x-y plane. The unit vector u₂ is not exactly in east direction because the pseudo plane defined by Ui and u₂ is not parallel to x-y plane. Since Ui 1 U₂, ui . u₂ = 0, than:

22 γ = Acos (-Tan α . Tan β)

From formulas 20 and 21 we can calculate the normal unit vector of the pseudo plane nι₂ as:

23 nπ = -Sin α . Cos β. Sin γ i - Cos β/ Cos α j + Cos α . Cos β. Sin γ k If we call the components of nπ in the Equation 23 as a, b and c respectively, we can calculate the slope angle η in φ direction of the pseudo plane as in Equation 17. Hence:

24 η = Atan [(-1 / c) . (a. Cos φ + b. Sin φ)]

Figure 11 shows the graphics of inclinometer outputs (Lat 45), η pseudo plane slopes (psd sip) and the difference (diff). Figure 11 shows the graphics at latitude 45° only to avoid too many curves. I will call the difference graphics as pseudo difference graphics. The slope change of the pseudo plane looks like a pure cosine function and intersects the graphics of Figure 9 in four points. The first two points are where the selected inclinometers stand (in our example 0° and 90° or south and east directions), the other two points are the other end of the intersection lines (180° and 270° or north and west direction).

From Figure 11, we can say that, the inclinometer output is a sum of pure sine graphics

(pseudo plane) and centripetal acceleration effect (difference). Taking into consideration the scale factor in all graphics in Figure 11, we notice that the effect of centripetal acceleration is unique. The pseudo difference graphic is not depend on the plane where the inclinometers stand. Both inclinometer outputs and pseudo plane slope angles vary depending on the plane we work on, but the difference is fixed.

We have reached to this fixed graphics which is independent of the working plane using a specially defined pseudo plane. Let us now, shift the both selected inclinometers by a certain amount of angle and define a new pseudo plane. Figures 12 gives two pseudo planes on the same working plane. The first pseudo plane is defined by the inclinometers in south and east directions (0° and 90° from south) and the second pseudo plane is defined by the inclinometers at 45° and 135° angles from the south direction.

From Figures 11 and 12, it can be seen that if we shift the pseudo plane by shifting the representative inclinometers we obtain different shifted pseudo difference graphic. Hence we can say that, the effect of the centripetal acceleration does not depend on the plane where the inclinometers stand, but the pseudo difference angle graphics depend on the phase angle of the pseudo plane selected. We now filtered the effect of the centripetal acceleration from the apparent bended plane but the shape of it depends on the pseudo plane selected.

As a new step, let us define more pseudo planes using more inclinometer twins on a single measurement plane, such that they will span all directions homogenously i.e. the phase angles between the pseudo planes selected should be equal. We know that, a pseudo plane defined by α and α + π/2 is the same plane defined by α + π and α + 3π/2. Hence if we choose two different pseudo planes and if the first one is chosen with the angles α and α + π/2, the second one should be defined by the angles α + π/4 and α + 3π/4. With this configuration we can assume that we have 8 inclinometers on the plane with π/4 angles between them. Similarly, choosing three different pseudo planes which means 6 different inclinometers on the plane with 30° apart, allows us to span all directions at 12 points. Choosing 8, 10, 12, .. inclinometers with 22.5°, 18°, 15°, .. apart allow us to span all directions at 16, 20, 24, ... points respectively. Figure 13 shows how to place inclinometers to span all directions homogeneously with 3, 4, 5, ... pseudo planes.

Each pseudo plane allows us to filter the centripetal acceleration effect and to obtain a single pseudo difference angle graphics. Figure 14 gives pseudo difference angle graphics for 3, 4, and 6 pseudo planes selected. Here the graphics are given for the latitude 45° only. We know that the pseudo difference angle graphics depend on the phase angles of the pseudo plane selected. This is why we have different pseudo difference angle graphic for each case.

Now, let us get the sum of the pseudo difference angles in each case as Figure 15, which shows the sum of the pseudo difference angles for 4^th, 5^th and 6^th order setup in the same graphics. We reach to interesting new graphics look like each other in each case, no matter how many pseudo planes are selected. If we shift the selected pseudo planes by shifting the representing inclinometers on the plane, we observe that we will have no noticeable difference because they are homogenously distributed on the plane and total effect of shifting the inclinometers will be zero. We will call those unique graphics as Yalcin Curves. Depending on the number of pseudo planes used, Yalcin Curves can be 2^nd order, 3^rd order, 4^th order ...etc. Figure 16 show the Yalcin curves for various latitude angles. It can be seen that, Yalcin Curves are almost same for all orders except we get bigger amplitudes as we use more pseudo planes.

The Yalcin Curves have the following specifications:

1. They are purely symmetric according to south north direction.

2. They have exactly null value in west-east and east-west directions.

3. They are independent of actual plane maximum slope angle and direction. 4. Their amplitude depend on the number of pseudo planes used and the latitude of the measurement point.

5. The curves have positive maximum slope in east direction and negative maximum slope in west direction. Those maximum slopes go to infinity as we approach to the Equator.

6. The north direction corresponds to a minimum value, and the south direction corresponds to a maximum value on the curves.

7. As we increase the latitude, the curves approach to a sine graphics having smaller amplitude.

8. As we approach to the Equator, the curve approaches to zero except a discontinuity at east and west directions.

Using Yalcin curves it is very easy to find the direction of any inclinometer used to create pseudo planes. The software developed for this purpose should implement the following steps:

1. Accept latitude of the measurement point and depending on how many pseudo planes are used, calculate the Yalcin Curve with desired directional intervals. For instance if you require 1 miliradian accuracy, you need to calculate curve points with one miliradian intervals (6283 points).

2. Read the outputs of the inclinometers organised on a plane as specified above.

3. Calculate directional unit vectors for each inclinometer. 4. Determine the normal unit vectors for each pseudo plane selected.

5. Calculate the slopes on the pseudo planes corresponding the directions of the other inclinometers.

6. Get the difference between the actual inclinometer value and pseudo plane slope values obtained with step 5.

7. Sum all the differences for each inclinometer and get the measured Yalcin Curve values in selected inclinometer directions.

8. Compare the measured Yalcin Curve values and actual values calculated in step one and find the direction. 9. There may be several methods for an efficient comparison in step 8. For instance, it seems the best area for comparison to be east or west direction vicinities because of rapid and relatively linear change. It can be find easily any two succeeding points, which are close to east or west direction. Any two adjacent points in measured Yalcin Curve values having maximum difference between, are the ones close to east or west directions. The angle between the inclinometers and the angle of the maximum and minimum points close to east and west directions should be considered. Comparison can be done for all inclinometers used. If the direction finder is n^th order (n-pseudo planes), than we have 2n inclinometers to make comparison at 2n points. With this method it is not necessary to calculate all the points of the actual Yalcin Curve in step one. Figure 13 shows that no matter whatever the order is, all inclinometers are in π radians. Hence the actual Yalcin

Curve values in π radians are sufficient for comparison (3142 points).

10. Theoretically, it is possible to find the direction using minimum two pseudo planes only (second order direction finder). But we may have trouble getting comparison points close to each or west directions. It will even get harder if the measurement is done in higher latitudes. Hence using more pseudo planes (3^rd, 4^th, 5^lh,.. order direction finder) will allow us to find better comparison points and get better accuracy.

11. The above steps may take less than a second with today's microprocessor technologies hence we can say that our direction finder will give a continuous output.

Claims

THE CLAIMS

1. A method to reach direction information, referenced true North, from inclination using the "Yalcin Curve" of the measurement point on the Earth having the following steps to implement:

• Select the order of "Yalcin Curve" depending on the accuracy required, using a higher order of "Yalcin Curve" give better accuracy.

• Use 2n units of high precise inclinometers, if n is the order of "Yalcin Curve" selected, • Place the n twins of inclinometers on a plane counter clockwise or reverse such that:

• A certain amount of angle should be present between each twin of inclinometers, any angle can be used because any two crossing lines define a plane, but 90° is the best to define a plane,

• The first inclinometer of the first twin should be assumed as a reference inclinometer,

• A 180°/2n angle should be present between the first inclinometers of each twin to be able to scan all directions homogeneously,

• Some inclinometers can be placed with 180° phase angle to balance the weight on the plane, • Calculate the n order "Yalcin Curve" as follow:

• Assuming the measurement plane is a plane perpendicular to gravity acceleration vector, and taking into consideration the centripetal acceleration vector of the measurement point on the earth, the angles between the gravity acceleration vector and apparent balancing vector (the sum of gravity acceleration vector and centripetal acceleration vector effect in the calculated direction) are calculated with, at least, required accuracy intervals.

• Those angles are the outputs of inclinometers on the plane perpendicular to gravity acceleration vector in the calculated direction.

• The pseudo planes are found for each inclinometer twins on the measurement plane by using above calculated angle values,

• The pseudo difference angles which are the angles between the pseudo planes and the outputs of the other inclinometers are calculated, • The "Yalcin Curve" is obtained by summing the pseudo different angles calculated above. This is "Calculated Yalcin Curve".

• Read the outputs of the inclinometers on the measurement plane,

• Calculate the measured "Yalcin Curve" values using 2n inclinometer outputs for 2n points with the same steps outlined above to find "Calculated Yalcin Curve".

• Find the direction of the referenced inclinometer by matching 2n "Measured Yalcin Curve" values by "Calculated Yalcin Curve" values.

2. A method according to patent claim 1 characterised in that, same concept implemented by integrated components in a single small housing rather than discrete inclinometers.

3. A method according to patent claim 1 characterised in that, converting apparent deformed sky to uniform sky (sky normahser) by correcting the elevation angle of the star under observation using the angle between gravity acceleration vector and apparent balancing vector.