U.S. patent number 6,188,355 [Application Number 09/210,541] was granted by the patent office on 2001-02-13 for wireless six-degree-of-freedom locator.
This patent grant is currently assigned to Super Dimension Ltd.. Invention is credited to Pinhas Gilboa.
United States Patent |
6,188,355 |
Gilboa |
February 13, 2001 |
Wireless six-degree-of-freedom locator
Abstract
A method for determining the position and orientation of an
object with respect to a reference frame, made up of providing the
object with three independent transmitters of electromagnetic
radiation, providing three independent receivers of the
electromagnetic radiation, each of the receivers having a fixed
position in the reference frame, transmitting the electromagnetic
radiation, using the transmitters, a first of the transmitters
transmitting the electromagnetic radiation including at least a
first frequency, a second of the transmitters transmitting the
electromagnetic radiation including at least a second frequency
different from the first frequency, and a third of the transmitters
transmitting the electromagnetic radiation including at least a
third frequency different from the first frequency, receiving
signals corresponding to the electromagnetic radiation, at all
three of the receivers, at a plurality of times, each of the
signals including components of at least one of the three
frequencies, for each of the receivers, forming a first function of
the components including the components of the signal received by
the each receiver from the first transmitter at the first
frequency, a function of the components including the components of
the signal received by the each receiver from the second
transmitter at the second frequency, and a function of the
components including the components of the signal received by the
each transmitter from the third transmitter at the third frequency,
the functions being independent of a time delay between the
transmitters and the receivers and inferring the position and the
orientation of the object from the functions.
Inventors: |
Gilboa; Pinhas (Haifa,
IL) |
Assignee: |
Super Dimension Ltd. (Herzelia,
IL)
|
Family
ID: |
11070965 |
Appl.
No.: |
09/210,541 |
Filed: |
December 14, 1998 |
Foreign Application Priority Data
Current U.S.
Class: |
342/448;
324/207.17 |
Current CPC
Class: |
G01S
5/0205 (20130101); G01S 5/0247 (20130101); G01S
5/06 (20130101) |
Current International
Class: |
G01S
5/02 (20060101); H04B 007/185 (); G01S
005/02 () |
Field of
Search: |
;342/448
;324/207.17,207.22,244,245 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
Primary Examiner: Blum; Theodore M.
Attorney, Agent or Firm: Friedman; Mark M.
Claims
What is claimed is:
1. A method for determining the position and orientation of an
object with respect to a reference frame, comprising the steps
of:
(a) providing the object with three independent transmitters of
electromagnetic radiation;
(b) providing three independent receivers of said electromagnetic
radiation, each of said receivers having a fixed position in the
reference frame;
(c) transmitting said electromagnetic radiation, using said
transmitters, a first of said transmitters transmitting said
electromagnetic radiation including at least a first frequency, a
second of said transmitters transmitting said electromagnetic
radiation including at least a second frequency different from said
first frequency, and a third of said transmitters transmitting said
electromagnetic radiation including at least a third frequency
different from said first frequency;
(d) receiving signals corresponding to said electromagnetic
radiation, at all three of said receivers, at a plurality of times,
each of said signals including components of at least one of said
three frequencies;
(e) for each of said receivers, forming a first function of said
components including said components of said signal received by
said each receiver from said first transmitter at said first
frequency, a second function of said components including said
components of said signal received by said each receiver from said
second transmitter at said second frequency, and a third function
of said components including said components of said signal
received by said each transmitter from said third transmitter at
said third frequency, said functions being independent of a time
delay between said transmitters and said receivers; and
(f) inferring the position and the orientation of the object from
said functions.
2. The method of claim 1, wherein said third frequency is different
from said second frequency.
3. The method of claim 1, wherein, for each of said receivers, said
first function includes said components of a strongest of said
signals received by any of said receivers at said first frequency,
said second function includes components of a strongest of said
signals received by any of said receivers at said second frequency,
and said third function includes components of a strongest of said
signals received by any of said receivers at said second
frequency.
4. The method of claim 1, wherein said second frequency and said
third frequency are even multiples of said first frequency.
5. The method of claim 4, wherein said second frequency and said
third frequency are equal.
6. The method of claim 4, wherein, for each of said receivers, all
three of said functions include said components of a strongest of
said signals received by any of said receivers at said first
frequency.
7. A method for determining the position and orientation of an
object with respect to a reference frame, comprising the steps
of:
(a) providing the object with three independent transmitters of
electromagnetic radiation;
(b) providing three independent receivers of said electromagnetic
radiation, each of said receivers having a fixed position in the
reference frame, at least one of said receivers being spatially
extended;
(c) transmitting said electromagnetic radiation, using said
transmitters, a first of said transmitters transmitting said
electromagnetic radiation including at least a first frequency, a
second of said transmitters transmitting said electromagnetic
radiation including at least a second frequency different from said
first frequency, and a third of said transmitters transmitting said
electromagnetic radiation including at least a third frequency
different from said first frequency;
(d) receiving signals corresponding to said electromagnetic
radiation, at all three of said receivers, at a plurality of times;
and
(e) inferring the position and the orientation of the object
noniteratively from said signals.
8. The method of claim 7, wherein said third frequency is different
from said second frequency.
9. The method of claim 7, wherein each of said signals including
components of at least one of said three frequencies, the method
further comprising the step of:
(f) for each of said receivers, forming a first function of said
components including said components of said signal received by
said each receiver from said first transmitter at said first
frequency, a second function of said components including said
components of said signal received by said each receiver from said
second transmitter at said second frequency, and a third function
of said components including said components of said signal
received by said each transmitter from said third transmitter at
said third frequency, said functions being independent of a time
delay between said transmitters and said receivers;
said position and orientation of the object being inferred from
said functions.
10. The method of claim 7, further comprising the step of:
(f) calibrating said inferring of the position and orientation of
the object.
11. The method of claim 10, wherein said calibrating includes
predicting said signals at a number of calibration positions and a
number of calibration orientations.
12. The method of claim 11, wherein said number of calibration
positions is at least 36 and said number of calibration
orientations are at least 36.
13. The method of claim 10, wherein said calibrating includes
measuring said signals at a number of calibration positions and a
number of calibration orientations.
14. The method of claim 13, wherein said number of calibration
positions is at least 36 and said number of calibration
orientations are at least 36.
Description
FIELD AND BACKGROUND OF THE INVENTION
The present invention relates to a method for monitoring the
position and orientation of a moving object, of the type in which
the moving object transmits electromagnetic signals, representative
of the position and orientation thereof, to a fixed receiver. More
particularly, the present invention relates to an open loop method
in which either the transmitter or the receiver may be spatially
extended and in which both the position and the orientation of the
moving object are computed noniteratively.
It is known to track the position and orientation of a moving
object with respect to a fixed frame of reference, by equipping the
moving object with a transmitting apparatus that transmits
electromagnetic radiation, placing a receiving apparatus in a known
and fixed position in the fixed frame of reference, and inferring
the continuously changing position and orientation of the object
from signals transmitted by the transmitting apparatus and received
by the receiving apparatus. Typically, the transmitting apparatus
includes three orthogonal magnetic dipole transmitters; the
receiving apparatus includes three orthogonal magnetic dipole
receivers; and the object is close enough to the receiving
apparatus, and the frequencies of the signals are sufficiently low,
that the signals are near field signals. Also typically, the system
used is a closed loop system: the receiving apparatus is hardwired
to, and explicitly synchronized with, the transmitting apparatus.
Representative prior art patents in this field include U.S. Pat.
No. 4,287,809 and U.S. Pat. No. 4,394,831, to Egli et al.; U.S.
Pat. No. 4,737,794, to Jones; U.S. Pat. No. 4,742,356, to Kuipers;
U.S. Pat. No. 4,849,692, to Blood; and U.S. Pat. No. 5,347,289, to
Elhardt. Several of the prior art patents, notably Jones, present
non-iterative algorithms for computing the position and orientation
of magnetic dipole transmitters with respect to magnetic dipole
receivers.
Of particular note are U.S. Pat. No. 4,054,881, to Raab, and U.S.
Pat. No. 5,600,330, to Blood. Raab purports to teach an open loop
system. Raab's system is "open loop" only in the sense that there
is no communication from the receiving apparatus to the
transmitting apparatus; but it still is necessary to synchronize
the transmitting apparatus and the receiving apparatus explicitly.
Raab provides several methods for synchronizing the receiving
apparatus with the transmitting apparatus, for example a phase
locked loop in the case of frequency domain multiplexing, and code
timing signals, in the case of spread spectrum multiplexing. In all
cases, however, Raab's system requires that the receiver generate a
reference signal that is mixed with the received signal, both for
the purpose of synchronization and for the purpose of resolving
sign ambiguities in all three independent coordinates of the space
in which the object moves. In Blood's system, the transmitters are
fixed in the fixed reference frame, and the receivers are attached
to the moving object; but by reciprocity, this is equivalent to the
situation in which the receivers are fixed and the transmitters
move. Blood's transmitters are spatially extended, and so cannot be
treated as point sources. Blood also presents an algorithm which
allows the orientation, but not the position, of the receivers
relative to the transmitters to be calculated non-iteratively.
It thus is apparent that there is further room for simplification
of the art of tracking a moving object using near field
electromagnetic signals. The explicit synchronization required by
Raab demands additional hardware and/or signal processing that
would not be necessary if explicit synchronization were not
required. Blood's iterative calculation of position adds complexity
and processing time, to systems with spatially extended
transmitters or receivers, that are absent from systems with point
sources and point receivers. It would be highly advantageous to
have a noniterative method of inferring both the position and the
orientation of a transmitting apparatus relative to a spatially
extended receiving antenna without explicit synchronization of the
transmitters and the receivers.
SUMMARY OF THE INVENTION
According to the present invention there is provided a method for
determining the position and orientation of an object with respect
to a reference frame, including the steps of: (a) providing the
object with three independent transmitters of electromagnetic
radiation; (b) providing three independent receivers of the
electromagnetic radiation, each of the receivers having a fixed
position in the reference frame; (c) transmitting the
electromagnetic radiation, using the transmitters, a first of the
transmitters transmitting the electromagnetic radiation including
at least a first frequency, a second of the transmitters
transmitting the electromagnetic radiation including at least a
second frequency different from the first frequency, and a third of
the transmitters transmitting the electromagnetic radiation
including at least a third frequency different from the first
frequency; (d) receiving signals corresponding to the
electromagnetic radiation, at all three of the receivers, at a
plurality of times, each of the signals including components of at
least one of the three frequencies; (e) for each of the receivers,
forming a first function of the components including the components
of the signal received by the each receiver from the first
transmitter at the first frequency, a function of the components
including the components of the signal received by the each
receiver from the second transmitter at the second frequency, and a
function of the components including the components of the signal
received by the each transmitter from the third transmitter at the
third frequency, the functions being independent of a time delay
between the transmitters and the receivers; and (f) inferring the
position and the orientation of the object from the functions.
According to the present invention there is provided a method for
determining the position and orientation of an object with respect
to a reference frame, including the steps of: (a) providing the
object with three independent transmitters of electromagnetic
radiation; (b) providing three independent receivers of the
electromagnetic radiation, each of the receivers having a fixed
position in the reference frame, at least one of the receivers
being spatially extended; (c) transmitting the electromagnetic
radiation, using the transmitters, a first of the transmitters
transmitting the electromagnetic radiation including at least a
first frequency, a second of the transmitters transmitting the
electromagnetic radiation including at least a second frequency
different from the first frequency, and a third of the transmitters
transmitting the electromagnetic radiation including at least a
third frequency different from the first frequency; (d) receiving
signals corresponding to the electromagnetic radiation, at all
three of the receivers, at a plurality of times; and (e) inferring
the position and the orientation of the object noniteratively from
the signals.
FIG. 1 shows schematically the hardware of the present invention. A
moving object 10 is provided with three independent magnetic dipole
transmitter coils 12, 14 and 16 that are powered by transmission
circuitry 18. Fixed within the reference frame with respect to
which object 10 moves are three independent, spatially extended
receiver antennas 20, 22 and 24, electrically coupled to reception
circuitry 26. As defined herein, "independent" means that the time
varying magnetic fields created by one of coils 12, 14 or 16 cannot
be expressed as a linear combination of the time varying magnetic
fields created by the other two coils, and that the time varying
signals received by one of antennas 20, 22 or 24 cannot be
expressed as a linear combination of the signals received by the
other two antennas. Preferably, coils 12, 14 and 16 are mutually
orthogonal, as shown in FIG. 1. In the most preferred embodiments
of the present invention, antennas 20, 22 and 24 are coplanar, as
shown in FIG. 1.
Although only one tracked object 10 is illustrated in FIG. 1, it
will be readily appreciated from the description below that the
present invention is easily adapted to the simultaneous tracking of
several objects 10.
Transmission circuitry 18 and reception circuitry 26 need not be
explicitly synchronized, as long as the clocks of transmission
circuitry 18 and reception circuitry 26 do not drift with respect
to each other during one measurement of the position and
orientation of object 10 with respect to antennas 20, 22 and 24.
This requirement is easily achieved using clocks based on modern
crystal oscillators. Two algorithms are presented below whereby
signals received by antennas 20, 22 and 24 at a plurality of
reception times are transformed into a 3.times.3 matrix M that is
independent of any unknown time shift .DELTA. between the clock of
transmission circuitry 18 and reception circuitry 26. One of these
algorithms requires synchronization of transmission circuitry 18
and reception circuitry 26 at the beginning of a sampling cycle in
order to resolve a phase ambiguity in the matrix M; the other
algorithm needs no such synchronization, and resolves the phase
ambiguity using the phases of the received signals. A third
algorithm is presented below whereby a rotationally invariant
3.times.3 position matrix W and a 3.times.3 rotation matrix T are
inferred noniteratively from the matrix M. The Euler angles that
represent the orientation of object 10 relative to the fixed frame
of reference are calculated noniterativly from the elements of T,
and the Cartesian coordinates of object 10 relative to the fixed
frame of reference are calculated from the elements of W. A
preliminary calibration of the system, either by explicitly
measuring the signals received by antennas 20, 22 and 24 at a
succession of positions and orientations of object 10, or by
theoretically predicting these signals at the successive positions
and orientations of object 10, is used to determine polynomial
coefficients that are used in the noniterative calculation of the
Euler angles and the Cartesian coordinates.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention is herein described, by way of example only, with
reference to the accompanying drawings, wherein:
FIG. 1 is a schematic overall depiction, partly in perspective, of
the hardware of the present invention;
FIG. 2 is a schematic diagram of a preferred embodiment of the
transmission circuitry of the present invention;
FIG. 3 is a schematic diagram of a simpler preferred embodiment of
the transmission circuitry of the present invention;
FIG. 4 is a schematic diagram of a preferred embodiment of the
reception circuitry of the present invention;
FIG. 5 is a schematic diagram of a simpler preferred embodiment of
the reception circuitry of the present invention.
FIG. 6 shows the three component antennas of a set of coextensive
linearly independent receiving antennas.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention is of a method for determining the position
and orientation of a target with respect to a reference coordinate
frame, by transmitting electromagnetic signals from the target to a
receiver that is fixed in the reference coordinate frame, with
simpler synchronization of the transmitter and the receiver than in
the prior art, or even without explicit synchronization of the
transmitter and the receiver, in a system in which the transmitter
and receiver are not hardwired. The scope of the present invention
also includes a noniterative determination of both position and
orientation with respect to spatially extended receiver
antennas.
The principles and operation of target tracking according to the
present invention may be better understood with reference to the
drawings and the accompanying description.
Most generally, the signals transmitted by coils 12, 14 and 16 must
be temporally independent, in the sense that the signal supplied to
any one of coils 12, 14 and 16 by circuitry 18 is not a linear
combination of the signals supplied to the other two coils by
circuitry 18. This is achieved most simply and most preferably by
transmitting from each coil at a different frequency. For
definiteness, the angular frequency of the transmissions from coil
12 is designated herein as .omega..sub.1, the angular frequency of
the transmissions from coil 14 is designated herein as
.omega..sub.2, and the angular frequency of the transmissions from
coil 16 is designated herein as .omega..sub.3.
The transmitted signals induce received signals in antennas 20, 22
and 24. Reception circuitry 26 is operative to digitize the
received signals at a sequence of times t.sub.m which are
preferably but not necessarily equally spaced. It should be noted
that this spacing need not be synchronous with the transmission
frequencies. Conceptually, reception circuitry 26 consists of three
receivers, each coupled to a different antenna, and computational
means for inferring the position and orientation of target 10 from
the signals received by the three receivers. The received signals
may be organized in a matrix s of three rows, one row for each
receiver, and as many columns as there are times t.sub.m, one
column for each time. Each element of s can be written as:
c.sub.i1, c.sub.i3 and c.sub.i5 are the in-phase components of the
signals received by receiver i from coils 12, 14 and 16,
respectively. c.sub.i2, c.sub.i4 and c.sub.i6 are the quadrature
components of the signals received by receiver i from coils 12, 14
and 16, respectively. Note that components c.sub.i1 and c.sub.i2
refer to frequency .omega..sub.3, components c.sub.i3 and c.sub.i4
refer to frequency .omega..sub.2, and components c.sub.i5 and
c.sub.i6 refer to frequency .omega..sub.3. The components c.sub.ij
can themselves be arranged in a matrix c of three rows and six
columns. The matrices s and c are related by a matrix A of six rows
and as many columns as there are in matrix S:
Because the transmission frequencies and the reception times are
known, matrix A is known. Equation (2) is solved by
right-multiplying both sides by a right inverse of matrix A: a
matrix, denoted as A.sup.-1, such that AA.sup.-1 =I, where I is the
6.times.6 identity matrix. Right inverse matrix A.sup.-1 is not
unique. A particular right inverse matrix A.sup.-1 may be selected
by criteria that are well known in the art. For example, A.sup.-1
may be the right inverse of A of smallest L.sup.2 norm.
Alternatively, matrix c is determined as the generalized inverse of
equation (2):
where the superscript "T" means "transpose". The generalized
inverse has the advantage of being an implicit least squares
solution of equation (2).
Whether the right inverse A.sup.-1 or the generalized inverse
A.sup.T (AA.sup.T).sup.-1 is used to solve equation (2), the
right-multiplication of matrix s constitutes a digital filtering
operation that returns the amplitudes and phases of the received
frequency components in the form of the elements of matrix c. In
general, the elements of matrix c must be corrected for amplitude
and phase distortions introduced, for example, by reception
circuitry 26. This can be done easily given the transfer functions
of reception circuitry 26. In the following discussion, it is
assumed that such corrections have been made.
The present invention includes two preferred algorithms for forming
the matrix M from the matrix c. In the first algorithm, each column
of M is formed from components corresponding to the same frequency.
Let receiver i' be the receiver with the largest signal magnitude
at frequency .omega..sub.j. In other words, at one particular value
of j,
is largest for i=i'. Then
and the other two elements (i.noteq.i') of the j-th column of M
are
The reference to the receiver with the largest signal magnitude
tends to suppress noise. The fact that the two matrix elements of
the j-th column of M for i.noteq.i' are projections of their
signals onto the signal of largest magnitude tends to suppress eddy
current noise, which tends to be 90.degree. out of phase with the
signal.
In the second algorithm, frequencies .omega..sub.2 and
.omega..sub.3 are chosen to be even multiples of frequency
.omega..sub.1, and all matrix elements are referred to the
strongest signal at frequency .omega..sub.1. The first column of M
is as in the first algorithm. Let .omega..sub.2 =.xi..omega..sub.1
and .omega..sub.3 =.zeta..omega..sub.1, where .xi. and .zeta. are
even integers. Note that the matrix ##EQU1##
is a rotation matrix. The elements of the second column of M are
the first elements of the column matrices obtained by the operation
##EQU2##
for i=1,2,3. The elements of the third column of M are the first
elements of the column matrices obtained by the operation
##EQU3##
for I=1,2,3.
It is straightforward to show that as long as the transmitter clock
and the receiver clock do not drift relative to each other, these
expressions for the elements of matrix M are independent of any
time shift .DELTA. between the transmitters and the receivers:
substituting t.sub.m +.DELTA. for t.sub.m in equation (1) does not
change the value of the matrix elements of M. If not for the sign
ambiguity of the square roots, there would be no need to
synchronize the receivers with the transmitters. In fact, under the
first algorithm for forming M, it is necessary to synchronize the
receivers with the transmitters, as described below, to resolve the
sign ambiguity of each M.sub.i'j. Under the second algorithm for
forming M, the remaining ambiguity is resolved as described
below.
Note that under the second algorithm, there is only one sign
ambiguity. This can be explained as follows: Assign the signal of
frequency .omega..sub.1 an arbitrary phase .phi.. Then, if all
three transmissions are synchronized, under the near field
approximation, the phase of the signal of frequency .omega..sub.2
=.xi..omega..sub.1 is .xi..phi. and the phase of the signal of
frequency .omega..sub.3 =.zeta..omega..sub.1 is .zeta..phi.. The
sign ambiguity of (c.sup.2.sub.i',1 +c.sup.2.sub.i',2).sup.1/2 is
equivalent to an ambiguity of .pi. radians in .phi.. But then the
phase of the signal of frequency .omega..sub.2 is unambiguously
.xi..phi.+.xi..pi.=.xi..phi. modulo 2.pi. and the phase of the
signal of frequency .omega..sub.3 is unambiguously
.zeta..phi.+.zeta..pi.=.zeta..phi. modulo 2.pi., because .xi. and
.eta. are even integers.
Alternatively, frequencies .omega..sub.2 and .omega..sub.3 may be
the same even multiple .xi. of .omega..sub.1. Coil 12 transmits a
signal proportional to sin .omega..sub.1 t. Coil 14 transmits a
signal proportional to sin .xi..omega..sub.1 t. Coil 16 transmits a
signal proportional to cos .xi..omega..sub.1 t. Matrix A has only
four rows, two for frequency .omega..sub.1 and two for frequency
.xi..omega..sub.1. The second column of M is formed as above. The
elements of the third column of M are the second elements of the
column matrices obtained by the operation ##EQU4##
for i=1,2,3.
Let T be the orthonormal matrix that defines the rotation of object
10 relative to the reference frame of antennas 20, 22 and 24. Write
M in the following form:
where T.sub.0 is an orthogonal matrix and E is in general a
nonorthogonal matrix. In general, T.sub.0 and E are functions of
the position of object 10 relative to the reference frame of
antennas 20, 22 and 24. Let
W.sup.2 is real and symmetric, and so can be written as W.sup.2
=Pd.sup.2 P.sup.T =(PdP.sup.T).sup.2, where d.sup.2 is a diagonal
matrix whose diagonal elements are the (real and positive)
eigenvalues of W.sup.2 and where P is a matrix whose columns are
the corresponding eigenvectors of W.sup.2. Then W=PdP.sup.T =E also
is symmetric. Substituting in equation (10) gives:
so that
If T.sub.0 is known, then T, and hence the orientation of object 10
with respect to the reference frame of antennas 20, 22 and 24, can
be computed using equation (13).
The orthogonal rotation matrix T is used to resolve the
above-described residual sign ambiguity in the second algorithm for
forming M. Specifically, the first column of T should be the cross
product of the second and third columns of T. If, after following
the above procedure for forming T, the first column thereof comes
out as the negative of the cross product of the second and third
columns, then the sign of (c.sup.2.sub.i',1 +c.sub.2i',2).sup.1/2
must be reversed.
For any particular configuration of antennas 20, 22 and 24, T.sub.0
may be determined by either of two different calibration
procedures.
In the experimental calibration procedure, object 10 is oriented so
that T is a unit matrix, object 10 is moved to a succession of
positions relative to antennas 20, 22 and 24, and M is measured at
each position. The equation
gives T.sub.0 at each of those calibration positions.
There are two variants of the theoretical calibration procedure. In
the first variant, coils 12, 14 and 16 are modeled as point
sources, including as many terms in their multipole expansions as
are necessary for accuracy, and their transmitted magnetic fields
in the planes of antennas 20, 22 and 24 are calculated at a
succession of positions relative thereto, also with object 10
oriented so that T is a unit matrix. The EMF induced in antennas
20, 22 and 24 by these time-varying magnetic fields is calculated
using Faraday's law. The transfer function of reception circuitry
26 then is used to compute M at each calibration position, and
equation (14) gives T.sub.0 at each calibration position. The
second variant exploits the principle of reciprocity and treats
antennas 20, 22 and 24 as transmitters and coils 12, 14 and 16 as
point receivers. The magnetic field generated by each antenna at
the three frequencies .omega..sub.1, .omega..sub.2 and
.omega..sub.3 is modeled using the Biot-Savart law. Note that each
frequency corresponds to a different coil. The signal received at
each coil is proportional to the projection of the magnetic field
on the axis of the coil when object 10 is oriented so that T is a
unit matrix. This gives the corresponding column of M up to a
multiplicative constant and up to a correction based on the
transfer function of reception circuitry 26.
To interpolate T.sub.0 at other positions, a functional expression
for T.sub.0 is fitted to the measured values of T.sub.0.
Preferably, this functional expression is a polynomial. In the
special case of the "coextensive" preferred embodiment of spatially
extended antennas 20, 22 and 24 described below, it has been found
most preferable to express the Euler angles .alpha., .beta. and
.gamma. that define T.sub.0 as the following 36-term polynomials.
The arguments of these polynomials are not direct functions of
Cartesian coordinates x, y and z, but are combinations of certain
elements of matrix W.sup.2 that resemble x, y and z, specifically,
a=W.sup.2.sub.13 /(W.sup.2.sub.11 +W.sup.2.sub.33), which resembles
x; b=W.sup.2.sub.23 /(W.sup.2.sub.22 +W.sup.2.sub.33), which
resembles y, and c=1/W.sup.2.sub.33, which resembles z. Using a
direct product notation, the 36-term polynomials can be expressed
as:
where AZcoe, ELcoe and RLcoe are 36-component vectors of the
azimuth coefficients, elevation coefficients and roll coefficients
that are fitted to the measured or calculated values of the Euler
angles. Note that to fit these 36-component vectors, the
calibration procedure must be carried out at at least 36
calibration positions. At each calibration position, W.sup.2 is
computed from M using equation (11), and the position-like
variables a, b and c are computed from W.sup.2 as above.
Similarly, the Cartesian coordinates x, y and z of target 10
relative to the reference frame of antennas 20, 22 and 24 may be
expressed as polynomials. In the special case of the "coextensive"
preferred embodiment of spatially extended antennas 20, 22 and 24
described below, it has been found most preferable to express x, y
and z as the following 36-term polynomials:
where Xcoe, Ycoe and Zcoe are 36-component vectors of the
x-coefficients, the y-coefficients, and the z-coefficients,
respectively; and d=log(c). As in the case of the Euler angles,
these position coordinate coefficients are determined by either
measuring or computing M at at least 36 calibration positions and
fitting the resulting values of a, b and c to the known calibration
values of x, y and z. Equations (15) through (20) may be used
subsequently to infer the Cartesian coordinates and Euler angles of
moving and rotating object 10 noniteratively from measured values
of M.
Referring again to the drawings, FIG. 2 is a schematic diagram of a
preferred embodiment of transmission circuitry 18. Transmission
circuitry 18 is based on a control unit 30 that receives inputs
from one or more external input ports 32 (three are shown) and from
a counter 34, and produces four different outputs: transmitted
signals TX0, TX1 and TX2 directed to coils 12, 14 and 16, and a
reset signal directed to counter 34 and a set reset flip flop
(SR-FF) 36. The identity and functionality of the other components
of transmission circuitry 18 will be clear from the following
description of the operation of transmission circuitry 18.
Transmission circuitry 18 operates in two modes, reception mode and
transmission mode. On startup, transmission circuitry 18 is in
reception mode: a T/R line 38 from SR-FF 36 sets analog multiplexer
switches 46 so that coils 12, 14 and 16 are used as receiving
antennas, the outputs of which are fed into the inputs of a trigger
circuit 42. Transmission circuitry 18 remains in reception mode
until one of coils 12, 14 or 16 receives a trigger signal. The
trigger signal may be simply a short pulse which is higher than a
pre-set threshold level. This is appropriate to tracking either a
single object or multiple objects transmitting at separate sets of
frequencies. Alternatively, if multiple objects are tracked, then
each object, or each subgroup of objects, may be assigned its own
modulation sequence, such as a unique digital code, to serve as a
trigger signal. In this way, for example in an application to a
three dimensional game, only the game pieces in play are activated.
Upon receipt of such a signal, trigger circuit 42 changes the state
of SR-FF 36, changing the operational mode of transmission
circuitry 18 to transmission mode. In transmission mode, SR-FF 36
sets T/R line 38 so that analog multiplexer switches 46 connect
coils 12, 14 and 16 to drivers 44. The output of an oscillator 40
is fed via a gate 45 to counter 34. Counter 34 is a "divide by N"
counter with N outputs. Counter 34 counts up from zero, and the N
outputs of counter 34 are fed into control unit 30.
The signals generated by control unit 30 at outputs TX0, TX1 and
TX2 are periodic signals with three different fundamental
frequencies, set via input ports 32. For example, to enable the
second algorithm for forming matrix M, a signal with a fundamental
frequency of 1000 Hz may be supplied via output TX0, a signal with
a fundamental frequency of 2000 Hz may be supplied via output TX1,
and a signal with a fundamental frequency of 4000 Hz may be
supplied via output TX2. (To enable the alternative version of the
second algorithm, in which, for example, the signals supplied via
outputs TX1 and TX2 both have the same fundamental frequency, but
with a difference in relative phase, the relative phase also is set
via input ports 32.) The signals may be pure sinusoids, square
waves, or periodic signals of any other convenient waveform.
Solving equation (2) for the matrix c is equivalent to performing a
Fourier analysis of the received signals, to recover the
fundamental sinusoids. After these signals have been generated for
a sufficiently long time to allow reception circuitry 26 to compute
the position and orientation of target 10, control unit 30 sends a
Reset signal to counter 34 and SR-FF 36 to put transmission
circuitry 18 back into reception mode.
Most preferably, control unit 30 includes switches that can be
operated by a user to change outputs TX0, TX1 and TX2 dynamically.
For example, the signal supplied to output TX1 or TX2 can be
changed, from a signal whose frequency is one multiple of the
frequency of the signal supplied to output TX0, to a signal whose
frequency is another multiple of the frequency of the signal
supplied to output TX0.
FIG. 3 is a schematic diagram of a simpler preferred embodiment of
transmission circuitry 18, suitable for use with the second
algorithm for forming the matrix M. This embodiment operates only
in the transmission mode. The output of an oscillator 140 is fed
directly to a counter 134. Like counter 34, counter 134 is a
"divide by N" counter with N outputs. Counter 134 counts up from
zero, and the N outputs are fed into a control unit 130 which
produces three transmitted signals TX0, TX1 and TX2 that are
directed to coils 12, 14 and 16 via drivers 144.
FIG. 4 is a schematic diagram of a preferred embodiment of
reception circuitry 26. Antennas 20, 22 and 24 are alternately and
successively connected to a control/processing unit 50 via an
analog selector switch 52, an amplifier/filter 54 and an A/D
converter 56. The timing of control/processing unit 50 and A/D
converter 56 is controlled by an oscillator 62. Note that
oscillators 40 and 62 need not be synchronized. Most preferably,
control/processing unit 50 includes switches similar to the
switches of control unit 30 described above. Also connected to, and
controlled by, control/processing unit 50 is a driver 58 and an
excitation antenna 60.
To start an acquisition cycle, control/processing unit 50 sends an
excitation trigger signal to excitation antenna 60 via driver 58.
Note that this form of explicit synchronization is considerably
simpler than the synchronization of Raab, which requires the mixing
of the received signals with a reference signal at the receiver.
The trigger signal transmitted by excitation antenna 60 is received
by transmission circuitry 18 of FIG. 2 and causes transmission
circuitry 18 of FIG. 2 to flip from reception mode to transmission
mode. Using analog selector switch 52, control/processing unit 50
selects one antenna 20, 22 or 24 at a time, thereby sampling the
signals of all three antennas, at a rate sufficiently high to meet
the Nyquist sampling criterion. Control/processing unit 50 computes
the position and orientation of target 10, using one of the two
algorithms described above.
FIG. 5 is a schematic diagram of a simpler preferred embodiment of
reception circuitry 26, for use with the embodiment of transmission
circuitry 18 illustrated in FIG. 3. This embodiment lacks driver 58
and excitation antenna 60. Correspondingly, control/processing unit
150 which controls this embodiment lacks an "excite" output port.
Otherwise, the embodiment of FIG. 5 is identical in construction
and operation to the embodiment of FIG. 4.
In a most preferred embodiment of the present invention, all three
spatially extended antennas 20, 22 and 24 are coextensive, i.e.,
they occupy substantially the same volume of space, without losing
their linear independence, as taught in PCT Publication No. WO
9603188, entitled "Computerized Game Board", which is incorporated
by reference for all purposes as if fully set forth herein.
Particular reference is made to FIGS. 13A through 14E of that
publication, and the accompanying description. FIG. 13A shows two
flat rectangular antennas 500 and 550 that, when overlapping,
respond differently to transmissions from a game piece (or
equivalently from target 10), as illustrated in FIGS. 14A and 14B,
despite the occupancy by the two antennas 500 and 550 of
substantially the same volume of space. FIG. 6 shows three antennas
20', 22' and 24, in the style of antennas 500 and 550 of WO
9603188, that, when superposed spatially, constitute a set of three
coextensive linear independent antennas suitable for use with the
present invention.
While the invention has been described with respect to a limited
number of embodiments, it will be appreciated that many variations,
modifications and other applications of the invention may be
made.
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