In this paper, we describe an original mathematical technique for calculating the position and orientation of an electromagnetic coil from a minimum of four collinear magnetic field measurements. This problem arose in the development of a system for evaluating the efficacy of inflatable back rafts designed to mitigate complications that arise from the immobilisation of patients with suspected spinal injuries on spinal boards during transport to hospital. Electromagnetic markers are attached to points of interest on the back of an immobilised test subject. Spinal curvature is then measured by passing a magnetometer underneath the board

Burke, Ted

De Paor, Annraoi

Fitzmaurice, Ronan

English

Irish Universities

Electromagnetic measurement of spinal curvature

Arrow@TUDublin

Technological University Dublin ARROW@TU Dublin Conference Papers tPOT: People Oriented Technology 2005-07-01 Electromagnetic measurement of spinal curvature Ronan Fitzmaurice University College Dublin Ted Burke Technological University Dublin, ted.burke@tudubln.ie Annraoi De Paor University College Dublin Follow this and additional works at: https://arrow.tudublin.ie/teapotcon  Part of the Biomedical Engineering and Bioengineering Commons, and the Computer Engineering Commons Recommended Citation Fitzmaurice, Ronan, Burke, Ted, De Paor, Annraoi: Electromagnetic measurement of spinal curvature. Proceedings of IEEE EMBSS UKRI PGBIOMED'05, University of Reading, UK, July 18-20, 2005, pp.13-14. This Conference Paper is brought to you for free and open access by the tPOT: People Oriented Technology at ARROW@TU Dublin. It has been accepted for inclusion in Conference Papers by an authorized administrator of ARROW@TU Dublin. For more information, please contact yvonne.desmond@tudublin.ie, arrow.admin@tudublin.ie, brian.widdis@tudublin.ie. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License  ELECTROMAGNETIC MEASUREMENT OF SPINAL CURVATURE  R. Fitzmaurice1,2, E. Burke1,2, A. de Paor1,2 1Department of Electronic and Electrical Engineering, National University of Ireland, Dublin, Ireland 2National Rehabilitation Hospital, Dun Laoghaire, Ireland ronan.fitzmaurice@gmail.com, edward.burke@ucd.ie, annraoi.depaor@ucd.ie  Abstract - In this paper, we describe an original mathematical technique for calculating the position and orientation of an electromagnetic coil from a minimum of four collinear magnetic field measurements. This problem arose in the development of a system for evaluating the efficacy of inflatable back rafts designed to mitigate complications that arise from the immobilisation of patients with suspected spinal injuries on spinal boards during transport to hospital. Electromagnetic markers are attached to points of interest on the back of an immobilised test subject. Spinal curvature is then measured by passing a magnetometer underneath the board.  I. INTRODUCTION  It is common practice for a patient with a suspected spinal injury to be immobilised on a spinal board during transport to hospital. This minimises the risk of further spinal cord damage occurring. Typically, a rigid plastic board is used, to which the patient is tightly strapped. Prolonged immobilisation can cause serious side effects, such as pressure sores and flattening of the spine. The inflatable back raft is a cushioning device designed to alleviate these problems. In evaluating the efficacy of the raft, we designed a system to measure spinal curvature during immobilisation. Electromagnetic marker coils are attached to points of interest on the back of a subject who is immobilised on a spinal board, beneath which a magnetometer is passed. The position of each marker is calculated from the magnetic field readings. The coils are pulsed with current, one at a time, under computer control. Here, we describe the mathematical theory used to calculate the position and orientation of each coil from the magnetometer readings.  II. THEORY  Consider a circular coil of radius rcoil, with N turns, centred on a point, c, and carrying a current, I. Point p is relatively distant from c. coilr>>−=rcpr The magnetic flux density at p, due to the coil, may be approximated as that due to an equivalent magnetic dipole [1]. ( )−= 350 .34)(rmrrrmpBπµ where B is magnetic flux density, µ0 is the permeability of free space and m is the magnetic moment vector of the equivalent dipole. The direction of m is perpendicular to the plane in which the coil lies and its magnitude is defined as follows [1], 2coilrNIπ=m  The dipole axis is the line perpendicular to the plane in which the coil lies, passing through the point c. All lines tangential to the field intersect the dipole axis at some point, except those that are parallel to it. Consider a line defined by a point ),,( iii zyx=ip  and a directional vector ),,( iii nml=ih . We define the following associated values, iiiiiiiiiiiiiiixmylwzlxnvynzmu−=−=−= Two such lines intersect if and only if they satisfy the following condition [2]. 0121212212121 =+++++ nwmvlunwmvlu  At each of four points on the x-axis, the magnetic flux density, ),,()( iziyix bbb=ipB , is measured and a line, Li, tangential to the field at that point is defined. Since, in general, the four tangent lines, L1, L2, L3 and L4 are skew, there exist only two lines that intersect with all four. One is the observation line (the x-axis) and the other is the dipole axis. Using the intersection condition stated above, the following four simultaneous equations are generated from the tangent lines. 01111111 =−+++ nxbmxbwbvbub yzzyx  02222222 =−+++ nxbmxbwbvbub yzzyx  03333333 =−+++ nxbmxbwbvbub yzzyx  04444444 =−+++ nxbmxbwbvbub yzzyx  One of the two solution lines of this set of equations is the observation line and the other is the dipole axis, from which mˆ , the unit vector in the direction of the dipole moment, can readily be obtained. Now, using any one of the four tangent lines, the point c can be calculated. Here, we choose L1. First, let us rewrite the expression for the magnetic flux density at a point p as follows. ( )[ ]mrrmrmpB ˆˆˆ.ˆ34)( 30 −=πµ,  where rˆ  is a unit vector in the direction of r. Note that the vector part of this expression (contained in the square brackets) depends only on the directions of m and r (and not on their magnitudes), so lines tangential to the magnetic field at every point on any straight line passing through c are parallel. Hence, knowing the angle between the field tangent at a point and the dipole axis, we can calculate the angle between the vector r and the dipole axis. This in turn reveals the position of c. Since mˆ  and rˆ  are unit vectors, their dot product is simply the cosine of θ , the angle between them. So, the expression for the magnetic flux density may once again be rewritten: [ ]mrpB ˆˆcos3)( −= θk , where k is simply the scalar multiple from the previous form of the expression. Consider a circle lying in the plane containing both L1 and the dipole axis, with a diameter of 3 and passing through c, as shown in Figure 1. The point d is defined by mcd ˆ+=  The line L1’ is the line parallel to L1 that passes through d. b is the point of intersection of L1’ with the circle. Since α  is the angle between the vector mˆ and the field vector at p1, it can be calculated from their dot product as follows, = − )(ˆ).(cos111pBmpBα  Applying the sine rule to triangle abd gives 5.0sin5.1sin βα= , from which we obtain the following. = −3sinsin 1 αβ  And since triangles abc and abd share a common vertex angle, 2βαθ+=  The distance from b to d is βα cos5.1cos5.0 +=− db , The unit vector, rˆ , may now be expressed as follows,  −+= )()(ˆcos31ˆ11pBpBdbmrθ The perpendicular distance between the point p1 and the dipole axis may be expressed as )(ˆ 1pem −×=h , where e is any point on the axis [3]. Using this, we can calculate the magnitude of r, θcosh=r  Finally, since the magnitude of r is now known in addition to its unit vector, the point c at which the coil is located may be calculated. rpc −= 1   III. CONCLUSION  We emphasise that the validity of the method is not specific to the magnetometer used. Given exact magnetic flux density readings, this technique delivers the exact coil position. This was double-checked using simulated readings generated from the dipole field equation. The expected error in the coil position estimate depends on the precision of the specific magnetometer used as well as on the position of each of the four points relative to the coil. To ensure coil position estimates within the specification required in this application (5mm), many readings were taken for each coil, from which a large number of estimates were calculated. Outlying estimates were discarded and the average of those that remained was taken as the final estimate. The final estimates were shown, theoretically and experimentally, to satisfy the specification. The original technique described here for measuring magnetic coil position using four collinear field measurements undoubtedly has numerous potential uses in a wide variety of applications. It provides a simple and efficient means of orientation and position sensing of objects in otherwise difficult circumstances.    θ m^ α β a b  c d B(p1) 0.5 1.0 p1 3.0 L’ 1  Fig. 1: mdbcb ˆcos3cos3 −=−⇒=− θθ  REFERENCES  [1] J.R. Reitz and F.J. Milford, Foundations of Electromagnetic Theory, Addison-Wesley Publishing Company, inc., 1962, p.164. [2] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, W. Metcalfe and Son, 4th Ed., 1882, p.43. [3] E.W. Weisstein, “Point-Line Distance--3-Dimensional”, From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html 

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Electromagnetic measurement of spinal curvature

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