U.S. patent application number 10/605414 was filed with the patent office on 2004-12-23 for sporting equipment provided with a motion detecting arrangement.
This patent application is currently assigned to IMEGO AB. Invention is credited to Storek, David.
Application Number | 20040259651 10/605414 |
Document ID | / |
Family ID | 33519670 |
Filed Date | 2004-12-23 |
United States Patent
Application |
20040259651 |
Kind Code |
A1 |
Storek, David |
December 23, 2004 |
SPORTING EQUIPMENT PROVIDED WITH A MOTION DETECTING ARRANGEMENT
Abstract
Method and arrangement for detecting movement-parameters in a
moving object. The parameters include acceleration and angular
velocity of the object, the arrangement includes an Inertial
Navigation System (INS) having at least one gyroscope and
accelerometer for measuring an acceleration, angular velocity and
effect of attraction of gravity on the sporting equipment.
Inventors: |
Storek, David; (Goteborg,
SE) |
Correspondence
Address: |
TRACY W. DRUCE, ESQ.
1496 EVANS FARM DR
MCLEAN
VA
22101
US
|
Assignee: |
IMEGO AB
S-411 33
Goteborg
SE
|
Family ID: |
33519670 |
Appl. No.: |
10/605414 |
Filed: |
September 29, 2003 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60319580 |
Sep 27, 2002 |
|
|
|
Current U.S.
Class: |
473/131 |
Current CPC
Class: |
A63B 69/0024 20130101;
A63B 69/3632 20130101; G01C 21/16 20130101; A63B 2024/0012
20130101; G01P 15/00 20130101; A63B 2225/50 20130101 |
Class at
Publication: |
473/131 |
International
Class: |
A63B 057/00 |
Foreign Application Data
Date |
Code |
Application Number |
Sep 27, 2002 |
SE |
0202873-6 |
Claims
1. An arrangement for detecting movement-parameters in a moving
object, said parameters comprising acceleration and angular
velocity of said object, said arrangement comprising an Inertial
Navigation System (INS), comprising at least one gyroscope and
accelerometer for measuring an acceleration, angular velocity and
effect of attraction of gravity on said object.
2. A system for detecting and analyzing motion data comprising
acceleration and angular velocity, the system comprising: an
arrangement for detecting movement-parameters of an equipment, said
arrangement comprising: an Inertial Navigation System (INS),
comprising a number of sensors: at least one gyroscope and
accelerometer for measuring acceleration and angular velocity and
effect of attraction of gravity on said equipment, a computer unit
communicating with said arrangement and comprising processor for
processing data received from said arrangement and compensating for
said effect of attraction of gravity on said equipment.
3. A golf club comprising an arrangement for detecting
movement-parameters of said golf club, said arrangement comprising
an Inertial Navigation System (INS), wherein said arrangement
further comprises a number of sensors for measuring a acceleration,
angular velocity and effect of attraction of gravity on said golf
club.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] The present application claims the benefit of U.S.
Provisional Application Ser. No. 60/319,580 filed Sep. 27, 2002 and
claims priority to Swedish Application No. 0202873-6 filed Sep. 27,
2002. Said provisional application is expressly incorporation
herein by reference in its entirety.
BACKGROUND OF INVENTION
[0002] 1. Technical Field
[0003] The present invention relates to sporting equipment provided
with an arrangement for detecting, tracking and analyzing motion
parameters of the equipment.
[0004] 2. Background
[0005] A general Inertial Navigation System (INS) consists of both
hardware and software. The hardware is a well-defined collection of
sensors that detect all possible accelerations and rotational
velocities, in all relevant room coordinates. Usually, the sensors
are accelerometers and gyroscopes. The task of the software is to
compute the movement-parameters, like position and orientation of
the hardware, while also carefully keeping track of coordinate
transformations during the motion. It is this coordinate
transformation property that distinguishes an INS from a sensor
system that only integrates accelerometer and gyro signals.
[0006] The basic movement-parameters are the velocity, position,
angular acceleration and orientation in a specific coordinate
system. To be useful, this coordinate system must be fixed with
respect to the earth for reasons that will become more obvious
after studying the example in a later section. For a golf club,
with an INS mounted to the butt of the shaft, the shaft can then
tracked with respect to, for example, the associated golf ball. It
is also reasonably straight forward to understand that other parts
of the club may be tracked, through a pure mathematical
translation, as long as the response of the club material between
the mounting point and the tracking point is known. If the club is
assumed to be a rigid body, then any part of the club can easily be
tracked. Furthermore, the movement-parameters must be resolved in
the three room (the x-y-z) coordinates and also resolved in time.
The basic movement-parameters can now easily be recomputed to any
other interesting parameters such as player hand speed, various
angles and speeds with respect to the swing plane, or anything that
mathematical combination or translation will allow from the basic
movement-parameters and the originally measured parameters.
[0007] One of the objects of the present invention is to provide an
arrangement including such an INS for utilization in the play of
sports, including automotive sports, leisure sports, toys and the
like. Most especially, the present invention relates to an
arrangement for sporting devices such as golf clubs, but other
arrangements such as tennis rackets, hockey sticks, baseball bats
and the like may also utilize and benefit from the teachings of the
invention. Consequently, the invention includes quantifying and
storing: movement information as variable values, computed results
and movement-parameters from the INS sensors. Accordingly, it is
possible to use the variables in real-time applications, or store
the variables for later analysis.
[0008] It is useful to consider a simplified system to obtain an
intuitive feeling for what an INS does with raw sensor signals and
hence the difference between a collection of sensors and an INS. A
simplified system can consist of, for example a ship that starts
moving, sails straight, makes a U-turn, and comes back on a path
parallel to the original straight course. While doing all this,
even through the U-turn, the ship accelerates with a constant
acceleration.
[0009] An accelerometer rigidly mounted to the ship, in the
direction of travel, will produce a signal looking like a straight
horizontal line. Integrated once, this signal will yield the
velocity. By integrating the signal again, the position of the
accelerometer is obtained. It is impossible to see any turning of
the boat from this position trace, in fact, nothing but a
constantly accelerating motion can be deduced from any of these
traces. If the boat would have omitted the U-turn and kept sailing
absolutely straight forward, with the same acceleration as in the
first case, all the curves would have appeared identical to the
original case.
[0010] However, the result is not an error; the accelerometer shows
a signal relevant to its own co-ordinate frame. The U-turn is only
relevant for an observer that can relate the boat to an outside
object that does not turn, like the water or a shoreline. The
accelerometer is simply not "smart" enough to know, in that it has
no capability for determining that it has turned.
[0011] The accelerometer mounted to the deck, measures acceleration
in the sensor system. The task is then to convert this measurement
to a fixed system outside the boat, the navigation system. Unless
this conversion is done properly, the answer expressed in the
navigation system, which is the useful system to an observer, will
be wrong.
[0012] The remedy is to equip the boat with additional sensors. In
the illustrative case of the boat, only one more accelerometer
(accelerometer y), and perhaps one more gyroscope need be added.
Generally speaking, the boat can never sail in the z-direction
(up-down) and it can only turn around the z-axis (on the water
surface). By further adding INS software and hence completing the
INS, the INS can keep track of turns and therefore correctly
transform sensor signals from the boat co-ordinate system to a
global coordinate system that views the boat relative to the water
or land and which will be generally termed the navigation system.
Of course, the velocities in the x and y-directions are also
readily available.
[0013] This simplified system does not consider any wave action,
however. When a wave hits the boat, the entire INS can move up and
down. A movement strictly in the z-direction will not affect the
INS because it is irrelevant if the boat travels exactly on the
water surface or one meter above it, as long as all the original
maneuvering stays the same. If the boat rocks sideways, however,
then the INS will tilt with respect to the water surface or at
least with respect to the coastline. The x-axis accelerometer, for
example, will then point in a new and unknown direction with
respect to the coastline. Hence, the INS will be unable to
correctly compute its position and orientation with respect to
land. The best way to get around this problem is to add two more
gyroscopes and one more accelerometer to the system. Now, the boat
can rock and roll and in principle all the movement-parameters will
be "accurately" measured.
[0014] The INS is usually used in aircrafts. For example, according
to U.S. Pat. No. 6,285,954, strap down inertial navigation systems
are frequently used in missiles and aircraft. Physically isolated
and stabilized apparatus, such as a gimballed platform that is
physically angularly stabilized relative to the local vertical
direction, require precise and mechanically complex angle
positioning apparatus, and are being systematically replaced by
systems of the strap down type.
[0015] A state-of-the-art strap down inertial navigation system has
three rotation sensors or gyros and three accelerometers rigidly
attached to a supporting vehicle. The rotation sensors are each
positioned and oriented to sense angular displacement about one of
three defined orthogonal axes attached to the vehicle body and
known as the body coordinate system. The accelerometers are each
positioned and oriented in a fixed direction relative to the
vehicle in order to sense velocity changes (incremental velocities)
along three different orthogonal axes in the body system. In a
strap down system, the accelerometer axes are not angularly
stabilized. Hence, all inertial sensors output a raw signal
meaningful in the body coordinate system.
[0016] Typically, the accelerometers are constantly changing
direction relative to gravity, navigation velocities cannot be
computed by directly integrating the accelerometer signals.
Instead, a stable computational frame or analytic navigation
coordinate system is continually computed. The output signals from
the rotation sensors are used by an attitude integration apparatus
to calculate the directions of local vertical, together with two
other axes orthogonal to the local vertical direction.
[0017] Sensed angle changes and accelerations (incremental
velocities) are continually rotated through the calculated angles
from the vehicle body axes to the calculated navigation axes. Angle
signals from the rotation sensors are used to update the
computer-stored angular position and incremental velocity data for
both the angle sensors and accelerometers relative to the
navigation coordinate system.
[0018] The rotation sensors and accelerometers have fixed relative
directions in the body coordinate system. An angular transformation
matrix of direction cosines is computed in an attitude integration
apparatus. The accelerometer signals, which are incremental changes
in velocity, in the strap down body coordinate system are converted
in a coordinate transformation computer from that system into
corresponding signals in the stabilized navigation coordinate
system.
[0019] After transformation into the navigation coordinate system,
the incremental velocity signals are integrated or summed to form
updated velocity signals. The rotation sensor and accelerometer
signals are sampled, and the sampled signals are delivered to a
computer which is programmed to accept the signals and to calculate
both velocities along the three axes in the stabilized navigation
coordinate system and attitude angles relative to this system.
[0020] In U.S. Pat. No. 4,303,978, a plurality of inertial
measuring unit (IMU) modules, each comprising (including, but not
limited to) gyros and accelerometers for sensing inertial
information along two orthogonal axes, are strap-down-mounted in an
aircraft, preferably such that the sense axes of the IMUs are
skewed with respect to one another. Inertial and temperature
signals produced by the IMU modules, plus pressure signals produced
by a plurality of pressure transducer modules and air temperature
signals produced by total air temperature sensors are applied to
redundant signal processors. The signal processors convert the raw
analogue information signals into digital form, error compensate
the incoming raw digital data and, then, manipulate the compensated
digital data to produce signals suitable for use by the automatic
flight control, pilot display and navigation systems of the
aircraft. The signal processors include: an interface system
comprising a gyro subsystem, an accelerometer and air calibration
data subsystem and an air data and temperature subsystem; a
computer; an instruction decoder; and, a clock. During computer
interrupt intervals, raw digital data is fed to the computer by the
interface subsystems under the control of the instruction decoder.
The computer includes a central processing unit that compensates
raw digital gyro and accelerometer data to eliminate bias, scale
factor, dynamic and temperature errors, as necessary. The central
processing unit also modifies the gyro and accelerometer data to
compensate for relative misalignment between the sense axes of the
gyros and accelerometers and for the skewed orientation of these
sense axes relative to the yaw, roll and pitch axes of the
aircraft. Further, accelerometer data is transformed from body
coordinate form to navigational coordinate form and the result used
to determine the velocity and position of the aircraft. Finally,
the central processing unit develops initializing alignment signals
and develops altitude, speed and corrected temperature and pressure
signals.
[0021] WO 00/69528 relates to an instrumented golf club system
having an instrumented golf club, an interface means and a
computing means is disclosed herein. The instrumented golf club
includes a plurality of sensors, an internal power supply, an
angular rate sensor and an internal ring buffer memory for
capturing data relating to a golf swing. The interface means is
capable of transferring data from the instrumented golf club to the
computing means for processing the data and presenting the data in
a useful and informative format. The data may be used to assist a
golfer's swing, or to design an appropriate golf club for a
specific type of golfer.
[0022] However, although the invention points out a need for high
precision and includes a rate sensor, the initial system
orientation with respect to gravity is never considered. This
collection of sensors fails all of the three points for a
successful INS, particularly for an INS useful for adapting golf
clubs.
[0023] WO 02/38184 is directed to systems and methods for analyzing
the motion of sporting equipment, such as a golf club, a baseball
bat, a hockey stick, a football or a tennis racquet, for example.
The systems comprise a motion sensing system in communications with
the sporting equipment to measure motion parameters wherein the
motion sensing system has at least one accelerometer or at least
one gyroscope, and a command station having a data acquisition
system to process the measured motion parameters and produce data.
The motion sensing system may be located on the sporting equipment
or, optionally, within the sporting equipment. The systems and
methods described herein can be used to determine the impact
location of the sporting equipment with another object, the
experienced forces, the velocity of the sporting equipment and/or
angular orientation of the sporting equipment during a motion.
[0024] U.S. Pat. No. 6,157,898 describes a device for measuring a
movable object, such as a baseball, football, hockey puck, soccer
ball, tennis ball, bowling ball, or a golf ball. Part of the
device, called the object unit, is embedded, secured, or attached
to the movable object of interest, and consists of an accelerometer
network, electronic processor circuit, and a radio transmitter. The
other part of the device called the monitor unit is held or worn by
the user and serves as the user interface for the device. The
monitor unit has a radio receiver, a processor, an input keypad,
and an output display that shows the various measured motion
characteristics of the movable object, such as the distance, time
of flight, speed, trajectory height, spin rate, or curve of the
movable object, and allows the user to input data to the
device.
[0025] According to the disclosed invention of U.S. Pat. No.
6,157,898, only one accelerometer network is used to measure the
acceleration in distance/time and thus the system provides poor
precision. Moreover, the invention of U.S. Pat. No. 6,157,898 is to
be integrated within a ball or similar object and not likely for a
golf ball and the like.
SUMMARY OF INVENTION
[0026] One of the objects of the present invention is to provide an
arrangement comprising an INS, rate sensors, inclinometers and
temperature sensors for sports equipment, leisure equipment, toys
and the like. Most especially, the present invention relates to an
arrangement for sporting equipment such as golf clubs, but other
equipment pieces such as tennis rockets, hockey sticks, and the
like can also employ the teachings of the invention.
[0027] Thus, an object of the invention is to use motion parameters
to provide better practicing, playing and competition ability.
Preferably, information received from the sensors is resolved, for
example, into three room-coordinates (x-y-z), and even with respect
to time. Consequently, an "all in one" quantifying and storing of a
movement is possible by computing results and variables from the
sensors. Accordingly, it is possible to use variables in real-time
applications or for later analysis.
[0028] The invention has many advantageous applications. One
example is its capability to quantify the repetitive motion or
motion patterns experienced when learning to play a sport which can
be used to expose errors and shortages in motion and store a motion
pattern or variables, comparing motion pattern and variables with
stored data, to compare the data with results or performances
(something accomplished).
[0029] Using the stored data can help the development of better
tools and equipment that improve the movement patterns of the
user.
[0030] Using the arrangement of the invention, for example in a
golf club, permits a teacher, when satisfied with a pupil's swing,
to store the motion pattern. Then, the pupil can, at anytime, train
his or her swing by comparison with the stored data. Errors and
defects thereby become obvious.
[0031] The pupil and the teacher can compare the results of the
swings, for example hooks or slices, with the quantified movements
and or conclusions with it conclusions quantify the gesture and
pull development. A classical kind of feed back.
[0032] The player obtains feedback directly in the quantified
motion about various news in the swing or the equipment.
[0033] This information can be a different range between the leg
and the boll, a different power in the swing, angle of the wrist
before the forward motion, a new grip of the handle, a new club,
different shows or different ground.
[0034] The feedback can be obtained after follow-up analysis, or
also through a signal before, during or after the motion.
[0035] These signals can be triggered by differences in computed
parameters in the stored motion and actual motion. This can be used
to practice driving position, part of movement or body orientation
during the swing.
[0036] A user can compare his swing with an expert swing.
[0037] Characteristics of a swing can be translated to quantified
parameters, thus allowing other judgment possibilities.
[0038] Another object of the invention is to compensate for the
ambient temperature of the equipment using the arrangement of the
invention.
[0039] Therefore, in arrangement described above, parameters
regarding acceleration and angular velocity of the object are
deduced. The arrangement includes an Inertial Navigation System
(INS) having at least one gyroscope and accelerometer for measuring
an acceleration, angular velocity and effect of attraction of
gravity on said object. The arrangement comprises means for
communicating with a computer unit for receiving and storing
relevant data. The arrangement comprises means for communicating
with a computer unit for compensating for said effect of attraction
of gravity.
[0040] Most preferably, the data quantifies a movement by computing
results and variables from said sensors.
[0041] In a preferred embodiment, a minimum sensor set up comprises
at least one accelerometer, temperature sensor, gyroscope,
amplifiers and filters. The accelerometer and gyroscope form a time
dependent sensor inputs to said INS.
[0042] According to a preferred embodiment, the signals from
sensors are generally filtered in filters and amplified in
amplifiers before they form inputs to an INS filter. At least one
of the filters is an extended Kalman filter, comprising a sensor
model, a measurement noise model, a processor for dynamics and a
processor for noise model. Calibration data is used regarding
offsets, scale factors and the directions of sensitivity of
physical sensors. Preferably, estimated output parameters,
represented by a vector matrix, are one or several of orientations,
angular velocities, angular accelerations, positions, velocities,
and accelerations respectively, which variables represent time
dependent results from the INS.
[0043] Most preferably, the object is sporting equipment such as a
golf club.
[0044] Preferably, three coordinate systems are used: a navigation
system, a club system and a sensor system. The navigation system
defines a position of the club, typically as an angle of the club
head with respect to a ball. The sensor is arranged in at least one
of a handle, shaft or head of the golf club.
[0045] The computer unit provides for at least one of: computing
and storing movement data about the equipment, or comparison
between new and stored movement data.
[0046] The invention also relates to a system for detecting and
analyzing motion data comprising acceleration and angular velocity.
The system comprises an arrangement for detecting
movement-parameters of an equipment piece. The arrangement
comprising an Inertial Navigation System (INS) that has a number of
sensors: at least one gyroscope and accelerometer for measuring
acceleration and angular velocity and effect of attraction of
gravity on the equipment, a computer unit communicating with the
arrangement and comprising processor for processing data received
from the arrangement and compensating for the effect of attraction
of gravity on the equipment.
[0047] Preferably, the computer unit comprises an audio and a video
output arrangement for communication with a user of the equipment.
The computer unit comprises storing means for storing data from the
arrangement and providing the data for education or training of the
user. The equipment (piece) is exemplarily one of a golf club,
ice-hockey stick, baseball bat, tennis/badminton/table tennis
racket, and/or a fishing rod.
[0048] Preferably, the sensor system has a specific coordinate
system, which is transformed to a fixed coordinate system with
respect to a fixed point in an environment of the equipment.
[0049] In one embodiment, the invention also relates to a golf club
comprising an arrangement for detecting movement-parameters of the
golf club. The arrangement comprises an Inertial Navigation System
(INS). The arrangement further comprises a number of sensors for
measuring an acceleration, angular velocity and effect of
attraction of gravity on the golf club. The club also comprises
means for communication with a computer unit for receiving and
storing relevant data. The club also comprises means for
communication with a computer unit for compensating for the effect
of attraction of gravity. Preferably, the data quantifies a
movement by computing results and variables from the sensors. A
minimum sensor set up comprises at least one accelerometer,
temperature sensor, gyroscope, amplifiers and filters. The
accelerometer and gyroscope form time dependent sensor inputs to
the INS. The actual sensors are generally filtered in filters and
amplified in amplifiers before they form inputs to an INS filter.
The filter is an extended Kalman filter, comprising a sensor model,
a measurement noise model, a processor for dynamics and a processor
for noise model. Preferably, the club uses calibration data
comprising offsets, scale factors and directions of sensitivity of
physical sensors. The estimated output parameters, represented by a
vector matrix, are orientations, angular velocities, angular
accelerations, positions, velocities, and accelerations
respectively, which variables represent the time dependent results
from the INS. Most preferably, three coordinate systems are used: a
navigation system, a club system and a sensor system. The
navigation system defines a position of the club an angle of a club
head with respect to a ball. The sensor is arranged in at least one
of a handle, shaft or head of the golf club.
[0050] Preferably, when tracking at an end of a shaft, a position
is about .ltoreq.10 mm, preferably within 5 mm throughout an entire
swing. All angles at the end of the swing are within 2.degree., and
preferably within 0.5 to 1.degree.. Linear and angular velocities
of the club are within 2%, preferably within 1% of its maximum
values. A sample rate exceeds 120 samples/sec., preferably 250
samples/second to resolve a final part of the motion. When tracking
at the club head, the position is within 10 mm, preferably within
2-5 mm. A loft/lie angle is within .+-.1.degree., preferably within
.+-.0.5.degree.. An open/close angle is resolved to within
.+-.0.5.degree., preferably within .+-.0.1.degree.. Angular rates
are, within about 2% preferably within 1% of their maximum values.
A linear velocity is within 1 mph, preferably 0.5 mph.
[0051] The invention also relates to a method of analyzing a
movement of a sporting equipment piece comprising, the steps of:
collecting movement-parameters of the equipment by means of an
Inertial Navigation System (INS), by measuring acceleration, an
angular velocity and effect of attraction of gravity on the
equipment. The method further comprises the step of compensating
for the effect of attraction of gravity. Most preferably, the
equipment is a golf club, comprising gyro sensors and accelerometer
sensors.
[0052] The method comprises the steps of: prior to a swing,
initializing the INS by holding the club substantially absolutely
still in a well-defined position, removing offsets of the gyros in
the club by keeping them still, and removing offsets of the
accelerometers by comparison to the known effect of gravity on each
sensor.
[0053] The method comprises the further steps of: immediately prior
to a swing, aiming at a ball in a direction of a flag for the
sensors to define a direction toward the flag, continuously
activating the sensors in a back swing and measuring accelerations
and rotational velocities. The method also comprises the step of
measuring temperatures.
[0054] The method further comprises the steps of: collecting data
unit the data is discriticizes, performing rudimentary signal
processing, and storing the data in an internal memory.
[0055] Preferably, all stored data is transferred to a computer
unit after a performed swing. A three dimensional acceleration and
angular velocity are measured.
[0056] The invention also relates to a golf club comprising, an
arrangement for detecting movement-parameters, such as acceleration
and angular velocity of the golf club, the arrangement comprising:
an Inertial Navigation System (INS), having a number of sensors, at
least one being a gyroscope for measuring an acceleration, angular
velocity and effect of attraction of gravity on the sporting
equipment, communication arrangement for communicating with a
computer unit for receiving and storing relevant data and provided
for calculating compensation for the effect of attraction of
gravity and quantifying the motion of the club, at least one
accelerometer, temperature sensor, gyroscope, amplifiers and
filters, the accelerometer and gyroscope form a time dependent
sensor inputs to the INS, filters for filtering signals from
sensors and amplified in amplifiers before they form inputs to an
INS filter, wherein at least one of the filters is an extended
Kalman filter, comprising a sensor model, a measurement noise
model, a processor for dynamics and noise model.
[0057] The estimated output parameters, represented by a vector
matrix, are orientations, angular velocities, angular
accelerations, positions, velocities, and accelerations
respectively, which variables represent the time dependent results
from the INS.
BRIEF DESCRIPTION OF DRAWINGS
[0058] In the following, the invention will be further described in
a non-limiting way under reference to the accompanying drawings in
which:
[0059] FIG. 1 is a block diagram representing the minimum sensor
set up in a general INS,
[0060] FIG. 2 is a block diagram of a generic INS filter, with
lower level corrections included,
[0061] FIG. 3 represents in a schematic way, the grip of a golf
club with an attached sensor module,
[0062] FIG. 4 represents in a schematic way, a golf club with
attached sensor modules according to the invention, and
[0063] FIG. 5 represents in a schematic way, a golf club used as a
control device, according to another aspect of the invention.
DETAILED DESCRIPTION
[0064] As mentioned above, there is an enormous difference between
a collection of sensors and an INS. Not any collection of sensors
can be "upgraded" to a useful INS, particularly for sporting
equipments and especially golf where the above specifications make
the problem extremely hard. In general, there are three points to
obtain a successful Inertial Navigation System, for achieving the
objects of the present invention.
[0065] A sufficient number of inertial sensors must be present to
measure and account for all relevant accelerations and rotations of
the system. In this case three-dimensional accelerations and
angular velocities. This enables the correct transformation of
coordinates from the sensor system to the fixed navigation system,
while continuously computing the movement-parameters.
[0066] All sensors must measure their respective acceleration or
rotation sufficiently close to the truth, not to render the
subsequent computations and transformations useless. This normally
means that a host of unwanted effects in the raw sensor signals
first must be identified, and compensated for, before the
information can be further processed. The unwanted effects must
include effects due to temperature. Normally, imperfections of the
sensors themselves, like non-linearities, poorly known direction of
sensitivity and electric drifts, must be accounted for.
Furthermore, imperfections in the sensor system assembly, such as
imperfect mounting, will give rise to poorly known directions of
sensitivity and sensor positions and perhaps to an imperfect
transmission of forces through the assembly. Finally, it is
imperative to know the location of the sensors as well as possible
to be able to include the effects of angular accelerations and
Coriolis forces that always occur in distributed sensor
systems.
[0067] All sensors must be initialized properly prior to actual
measurement and tracking of movement parameters. This
initialization serves to remove as much as possible of the unwanted
but ever-present offset. Assuming an absolutely still INS prior to
measurement, the gyro offset is partly due to electrical offset and
partly to the Earth's rotation. In the application for golf, the
Earth's rotation can be ignored and hence this offset level can
simply be subtracted during the later tracking. The offset of the
accelerometer, however, is partly due to electrical offset and
partly to the Earth's gravitation. The gravitation is large and can
almost never be ignored, particularly not for golf. To separate the
two offsets, the orientation of the INS versus the direction of
gravity, prior to tracking, must be known or the accelerometer must
have a sufficiently predictable electrical offset. A predictable
electric offset can be compensated for and removed and hence the
direction of gravity can be computed and compensated for. In
practice, the accelerometers that are suitable for precise
measurements of a fast changing acceleration seldom have a
predictable electric offset for long periods of time. In the
application for golf the electrical offset must be predictable over
time periods of minutes, if not for hours. The solution is often to
incorporate a third type of sensor called an inclinometer. The
inclinometer has a very predictable offset over longer periods of
time so it can measure the orientation of the INS versus the
g-vector prior to tracking. The inclinometers generally rely on
extreme low-pass filtering and a narrow acceleration range to
perform well. Hence, they are usually less suitable for later
tracking. Another way to overcome the offset problem is to fix the
INS in a well-defined orientation, with respect to gravity, shortly
prior to tracking.
[0068] The block diagram of FIG. 1 represents the minimum sensor
set up 10 in a general INS. It comprises at least one accelerometer
input 11, temperature sensor input 12 and gyroscope input 13. The
accelerometers and gyroscopes form the analogue time dependent
sensor inputs to an INS. The actual sensors are generally filtered
in low-pass filters 14a-14c, amplified in amplifiers 15a-15c and
digitized in analogue to digital converters before they form inputs
to an INS filter. Furthermore, the signals are perhaps stored in a
memory (not shown).
[0069] FIG. 2 illustrates the block diagram of a generic INS filter
20, with lower level corrections included. The filter, e.g. build
as an extended Kalman filter, comprises a sensor model 21, a
measurement noise model 22, a processor for dynamics 21 and a
processor for noise model 24. The time dependent sensor signals
arrive directly from the hardware part of the INS, as illustrated
in FIG. 1. The calibration data are typically offsets, scale
factors and the directions of sensitivity of the physical sensors.
The estimated output parameters, represented by a vector matrix 25,
are the orientations, angular velocities, angular accelerations,
positions, velocities, and accelerations respectively. These
variables represent the time dependent results from the INS. It is
obvious that the Kalman filter is given as an example and other
filter types can be used to achieve same results.
[0070] One feature of the INS of the invention is its ability to
convert measured quantities between different coordinate
systems.
[0071] In case of a golf club, for example, in principle there can
be three types of coordinate systems that need to be covered in
this context: the navigation system, i.e. the center of the golf
ball, the club system, i.e. the head or grip, and a representative
sensor system. The sensor coordinate system defines the axis of
sensitivity and the location of a given sensor, thus there is one
sensor system for each sensor used in the model.
[0072] It should be appreciated that number of sensors used can be
varied to cover the needs of the measurement, e.g. if two motion
parameters are needed to be measured then only a setup of sensors
sufficient for measuring two parameters (e.g. in two directions)
are needed. Thus, one or several sensor setups can be used.
However, in the examples given, references are usually made to a
threedimensional (three coordinates) system.
[0073] FIG. 3 represents the grip 30 of a golf club 31 with an
attached sensor module 32. The sensor module, in this case shaped
as a ball, can obviously be located in other places on a club, e.g.
inside the grip, inside the handle or the club head. Several
modules can also be provided simultaneously. The three typical
coordinate systems are displayed.
[0074] As depicted in FIG. 3, the coordinate systems mentioned
above are hereafter referred to as: O.sub.NXYZ (the navigation
system), a O.sub.CSjxyz (the club system) and
O.sub.Six.sub.iy.sub.iz.sub.i (the sensor system). In FIG. 3, the
club frame is represented by the grip coordinate system, CS1, but
in principle could also represent the head system, CS2. The sensors
are enumerated by the index i and are rigidly fixed close to the
handle in an attached sensor module.
[0075] However, one is not interested in the results that are
expressed in the sensor system but only interested in what happens
or has happened in the navigation system, i.e. where the club and
what the angle of the club head are, with respect to the ball. This
is a transformation from the sensor system to a fixed system, e.g.,
with respect to the boll, flag or any other fixed object in the
environment of the player.
[0076] An advantages application for INS according to the present
invention is implementation in a golf club, thus allowing analyze
of a golf stroke, such as a swing. The main reason is that the
motion has a short duration, is well defined, is easily repeated on
command, and occurs within relative narrow tolerances of
motion.
[0077] The short duration is important because the interesting
coordinates, position and angle, are computed through a series of
integrations.
[0078] The ever-present noise then propagates an error in position
that grows as the time squared. Furthermore, any other sensor
error, such as various offsets and sensor imperfections, contribute
to the same errors.
[0079] Long-term tracking is the very difficult problem in INS and
is normally solved by periodic coordinate updates from other
measurement sources.
[0080] A golf swing always begins with the aiming at the ball, just
before the back motion (swing) begins. There forward motion is
always a downward motion in the direction of the ball as well as a
hit. The motion always finishes with a smooth follow-through. The
motion is always smooth because the human body is bad at generating
fast discontinuities once in motion. All these features, and
several more, make it simpler to select time periods in which the
sensor can be initialized, i.e. when offsets can be cancelled and
initial positions and orientations can be established.
[0081] The trimming of the entire INS is immensely simplified by
the repletion factor in golf. One can basically keep repeating an
almost identical motion until the many parts of the INS are
optimized.
[0082] The relatively good knowledge of what the swing will look
like, before the actual swing occurs, provides the Kalman filter
with powerful information that increases the precision of the final
result. A golf swing is much easier to track than a perfectly
random motion. In addition, the sensor ranges, bias points and
orientations can be carefully trimmed to optimize their
golf-performance. This, of course, precludes measuring a different
motion with the same system.
[0083] The hardware part 10 of the INS is mounted on or most
preferably inside the golf club 11 shaft 12 or inside the handle 13
or the club head 14, as illustrated in FIG. 4. Thus the sensors can
be distributed within the golf club (or other sporting equipment).
The entire device is then calibrated on an accurate rate table and,
if necessary, on a shaker table or linear acceleration stage
according to IEEE (Institute of Electrical and Electronics
Engineers) standards.
[0084] A computer unit 14, residing nearby (or integrated inside
the club, not shown), holds much of the signal processing power and
substantial part of the INS software algorithms (Kalman filter) and
communicates with the hardware, either remotely or by electric
connection. The computer unit can be a conventional PC, laptop,
handheld computer or any other type, comprising processing unit,
memory, I/O unit, and storing unit for receiving and processing
data from the INS. In case the sensors are distributed within the
golf club, the information is processed centrally in the computer.
A memory unit can be arranged in the club to store data in the club
and transmit it to the computer in a later stage. The communication
between the club and the computer is achieved through RF, IR,
etc.
[0085] In the case of a golf application, the data can be used to
simulate a golfer and the computer can be used to quantify for
repetitively learning a motion pattern or a motion (when sporting)
can be used to expose errors and shortages in motion and store a
motion pattern or variables, comparing motion pattern and variables
with stored data, to compare the data with results or performances
(something accomplished). Using the stored data can help developing
better tools and equipments with respect to the movement pattern of
the user. When a teacher is satisfied with a pupils swing, the
motion pattern can be stored. Then, the pupil can at anytime train
the swings and compare it with the stored data. Errors and defects
become the obvious. The pupil and the teacher can compare the
results of the swings, for example hooks or slices, with the
quantified movements and or conclusions with it conclusions
quantify the gesture and pull development. The player can obtain
feedback directly in the quantified motion about various news
(changes) in the swing or the equipment. These news can be a
different range between the leg and the boll, a different power in
the swing, angle of the wrist before the forward motion, a new grip
of the handle, a new club, different shows or different ground. The
feedback can be obtained after followup analyses or also through a
signal before, during or after the motion. These signals can be
triggered by differences in computed parameters in the stored
motion and actual motion. This can be used to practice driving
position, part of movement or body orientation during the swing. A
user can compare his swing with an expert swing. Characteristics of
a swing can be translated to quantified parameters, thus allowing
otherjudgment possibilities.
[0086] Prior to swing, the INS is initialized for a few seconds by
holding the club absolutely still in a well-defined position. The
offsets of the gyros are here simply removed by keeping them still,
as the effect of Earth's rotation (gravity force) is negligible.
The offsets of the accelerometers are removed by comparison to the
known effect of gravity on each sensor. A number of inclinometers
or low-g accelerometers, especially suitable to resolve effects of
gravity, in the INS may aid in this initialization.
[0087] Immediately prior to a swing, the player normally aims at
the ball in the direction of the flag for a few seconds, which is a
way for the sensors to define the direction toward the flag. Now,
the back swing starts and all the sensors are continuously active
and measuring accelerations and rotational velocities. In addition,
auxiliary data such as temperature is also measured. The data
collection unit discriticizes the data, performs rudimentary signal
processing, and finally stores the data in an internal memory.
Directly after the completed swing, all stored data is transferred
to the computer.
[0088] The digital signal processing is perfected in the computer,
several sensor software compensations are performed, and suitable
transformations of coordinate frame are performed. Finally, the
data from all the sensors is filtered together in the main INS
algorithm, the Kalman filter. Golf specific information, such as
maximum velocity at a point or other restrictions known from the
study of many golf swings, may also enter the filter at this time.
Furthermore, data from external sensors, like magnetic coils near
the ball or radar speeds, may also provide the filter with useful
information. This filter then returns computed values for velocity,
position, angular acceleration, and angle in the navigation frame.
All these values are discrete, for each time step, and are
available in the three space coordinates, both for the
accelerometers and for the gyroscopes. All is stored on the
computer and can conveniently be graphed on demand.
[0089] This information in the computer can now be called a
complete record of one golf swing. This record is unique and
depends on the player and ambient conditions such as equipment, the
course, and the weather.
[0090] The information from the INS, just like from any other
measuring equipment, will always contain an error. The precision
and accuracy of the INS will ultimately set a limit to the measured
and computed results.
[0091] By keeping all of the ambient conditions constant a player
can study his golf swing by comparison of the exhaustive records
from several swings. This study may indicate inconsistencies in the
swing or deviations compared to swings where a coach has provided
human quality information.
[0092] By keeping the swing as constant as possible, such as
professional players are skilled in doing, various ambient
conditions can be studied through comparison of records. The club
or the golf shoes may have changed. Only the imagination sets the
limit as to how the INS can improve performance.
[0093] The temperature measurement is used to compensate the values
obtained from the sensors. Both ambient and sensor temperatures can
be measured. A temperature sensor and compensation can be
integrated within the sensor by the sensor manufacturer, by the
system supplier, who adds a measurement/compensation device or by
the user of the equipment who adds a digital or analogue device for
measurement, e.g. in form of digital thermometer or a sticker on
the sensors. In the latter case the user must measure, compensate
and calibrate the sensors for the temperature by him.
[0094] The temperature is measured actively substantially on all
sensors or the entire system (filters, processors etc.). The system
can thus be calibrated fixed for a certain temperature, independent
of the ambient temperature or selfheating. Thus, much higher
precision is obtained.
[0095] The temperature compensation can be obtained in two ways:
The entire system is rotated on, e.g. a rotation table in a
climatic chamber,
[0096] Three signals are obtained: The through value from the
rotation table (rotation velocity), Output from first sensor (S1),
The temperature (T) of S1.
[0097] Then a graph for each coordinate system for each temperature
with a speed of rotation on e.g. x-axis, and the output of S1 on,
e.g. y-axis is provided. If output of S1 varies modestly with
respect to the speed of the rotation (for a fixed T), the speed of
the rotation is described as a function: RS(S1). If the output
varies unreasonably, a table can be generated. Steps 3-5 are
repeated for different temperatures. The sensor is then integrated
into the system (e.g. a golf club). When using the system in the
real environment, two signals S1.sub.R and T.sub.R are obtained.
T.sub.R decides which table or function to be used and S1R points
out the speed of the rotation in the table or the function. This is
the calibrated rotation speed. Experiments have shown that there is
a connection between the temperature and S1. This means that the
temperature measurement must be conducted very close to the
sensors, i.e. in reality a temperature measurement for each sensor
in the system. However, it is possible to thermally connect the
sensors and thus obtain one value.
[0098] Another example of an application is illustrated in FIG. 5,
in which a golf club 12 comprising the arrangement 10 of the
invention, integrated inside the club head 14, is used as a cursor
control device on a computer display 15. The cursor, here
illustrated as a club icon 16, is controlled by communicating the
position, velocity, angular velocity etc., of the club to a
processing interface of the computer, which translates the club
data to a set of position data to be displayed. This arrangement
(not limited to golf clubs) can be used both for gaming and
practice, e.g. in a golf simulator.
[0099] In a preferred embodiment, when tracking at the end of the
shaft, the position should be known to within 10 mm, preferably 5
mm throughout the entire swing. All angles at the end of the swing
should be known to within 2.degree., preferably 0.5-1.degree.. The
linear and angular velocity of the club should be known to within
2%, preferably 1% of its maximum values. The system sample rate
must exceed 120 samples/sec., preferably 250 samples/second to
resolve the interesting final part of the motion. When tracking at
the club head the position should be known to within 10 mm,
preferably 2-5 mm to usefully resolve the impact of the ball. The
loft/lie angle should be resolved to within .+-.1.degree.,
preferably .+-.0.5.degree.. The open/close angle should be resolved
to within .+-.0.5.degree., preferably .+-.0.1.degree.. The angular
rates should be known to within 2%, preferably 1% of their maximum
values. The linear velocity should be known to within lmph,
preferably 0.5 mph.
[0100] Even though a golf club is exemplified herein, it is obvious
that the arrangement and method of the invention can be implemented
in other sporting equipments such as icehockey stick, baseball bat,
tennis/badminton/table tennis rackets, etc., or any other equipment
in which a motion analyses is needed. Moreover, the invention can
be used in any other moving objects such as a vehicle, airplane
etc.
[0101] The invention is not limited the shown embodiments but can
be varied in a number of ways without departing from the scope of
the appended claims and the arrangement and the method can be
implemented in various ways depending on application, functional
units, needs and requirements etc.
[0102] The quantifications of the invention(s) disclosed herein, as
well as the protection being sought therefore, in terms of scope
and breadth, can be characterized in a plurality of ways. Examples
are included below in the following exemplary claim sets:
Set One
[0103] 1. An arrangement for detecting movement-parameters in a
moving object, said parameters comprising acceleration and angular
velocity of said object, said arrangement comprising an Inertial
Navigation System (INS), comprising at least one gyroscope and
accelerometer for measuring an acceleration, angular velocity and
effect of attraction of gravity on said object.
[0104] 2. The arrangement of claim 1, comprising means for
communicating with a computer unit for receiving and storing
relevant data.
[0105] 3. The arrangement of claim 1, comprising means for
communicating with a computer unit for compensating for said effect
of attraction of gravity.
[0106] 4. The arrangement of claim 2, wherein said data quantifies
a movement by computing results and variables from said
sensors.
[0107] 5. The arrangement of claim 1, wherein a minimum sensor set
up comprises at least one accelerometer, temperature sensor,
gyroscope, amplifiers and filters.
[0108] 6. The arrangement of claim 5, wherein said accelerometer
and gyroscope form a time dependent sensor inputs to said INS.
[0109] 7. The arrangement of claim 1, wherein signals from sensors
are generally filtered in filters and amplified in amplifiers
before they form inputs to an INS filter.
[0110] 8. The arrangement of claim 7, wherein at least one of said
filters is an extended Kalman filter, comprising a sensor model, a
measurement noise model, a processor for dynamics and a processor
for noise model.
[0111] 9. The arrangement of claim 8, wherein calibration data are
used comprises on or several of offsets, scale factors and the
directions of sensitivity of physical sensors.
[0112] 10. The arrangement of claim 9, wherein estimated output
parameters, represented by a vector matrix, are one or several of
orientations, angular velocities, angular accelerations, positions,
velocities, and accelerations respectively, which variables
represent time dependent results from the INS.
[0113] 11. The arrangement of claim 1, wherein said object is
sporting equipment.
[0114] 12. The arrangement of claim 1, wherein said equipment is a
golf club.
[0115] 13. The arrangement of claim 12, wherein three coordinate
systems are used: a navigation system, a club system and a sensor
system.
[0116] 14. The arrangement of claim 12, wherein said navigation
system defines a position of the club an angle of the club head
with respect to a ball.
[0117] 15. The arrangement of claim 12, wherein said sensor is
arranged in at least one of a handle, shaft or head of said golf
club.
[0118] 16. The arrangement of claim 2 or 3, wherein said computer
unit provides for at least one of:--computing and storing movement
data about said object, or--comparison between new and stored
movement data.
Set Two
[0119] 17. A system for detecting and analyzing motion data
comprising acceleration and angular velocity, the system
comprising: an arrangement for detecting movement-parameters of an
equipment, said arrangement comprising: an Inertial Navigation
System (INS), comprising a number of sensors: at least one
gyroscope and accelerometer for measuring acceleration and angular
velocity and effect of attraction of gravity on said equipment, a
computer unit communicating with said arrangement and comprising
processor for processing data received from said arrangement and
compensating for said effect of attraction of gravity on said
equipment.
[0120] 18. The system of claim 17, wherein said computer unit
comprises audio and video output arrangement for communication with
a user of said equipment.
[0121] 19. The system of claim 17, wherein said computer unit
comprises storing means for storing means for storing data from
said arrangement and providing said data for education or training
of said user.
[0122] 20. The system of claim 17, wherein said equipment is one of
a golf club, ice-hockey stick, baseball bat, tennis/badminton/table
tennis rackets, fishing rod.
[0123] 21. The system according to claims 18, wherein said sensor
system has a specific coordinate system, which is transformed to a
fixed coordinate system with respect to a fixed point in an
environment of said equipment.
Set Three
[0124] 22. A golf club comprising an arrangement for detecting
movement-parameters of said golf club, said arrangement comprising
an Inertial Navigation System (INS), wherein said arrangement
further comprises a number of sensors for measuring a acceleration,
angular velocity and effect of attraction of gravity on said golf
club.
[0125] 23. The golf club of claim 22, comprising means for
communication with a computer unit for receiving and storing
relevant data.
[0126] 24. The golf club of claim 22, comprising means for
communication with a computer unit for compensating for said effect
of attraction of gravity.
[0127] 25. The golf club of claim 23, wherein said data quantifies
a movement by computing results and variables from said
sensors.
[0128] 26. The golf club of claim 22, wherein a minimum sensor set
up comprises at least one accelerometer, temperature sensor,
gyroscope, amplifiers and filters.
[0129] 27. The golf club of claim 26, wherein said accelerometer
and gyroscope form a time dependent sensor inputs to said INS.
[0130] 28. The golf club of claim 22, wherein the actual sensors
are generally filtered in filters and amplified in amplifiers
before they form inputs to an INS filter.
[0131] 29. The golf club of claim 28, wherein said filter is an
extended Kalman filter, comprising a sensor model, a measurement
noise model, a processor for dynamics and a processor for noise
model.
[0132] 30. The golf club of claim 29, using calibration data
comprising offsets, scale factors and directions of sensitivity of
physical sensors.
[0133] 31. The golf club of claim 30, wherein estimated output
parameters, represented by a vector matrix, are orientations,
angular velocities, angular accelerations, positions, velocities,
and accelerations respectively, which variables represent the time
dependent results from the INS.
[0134] 32. The golf club of claim 31, wherein three coordinate
systems are used: a navigation system, a club system and a sensor
system.
[0135] 33. The golf club of claim 32, wherein the navigation system
defines a position of the club an angle of a club head with respect
to a ball.
[0136] 34. The golf club of claim 32, wherein said sensor is
arranged in at least one of a handle, shaft or head of said golf
club.
[0137] 35. The golf club of claim 34, wherein when tracking at an
end of a shaft, a position is about .ltoreq.10 mm, preferably
within 5 mm throughout an entire swing.
[0138] 36. The golf club of claim 35, wherein all angles at the end
of the swing are within 2.degree., preferably within 0.5 to
1.degree..
[0139] 37. The golf club of claim 34, wherein linear and angular
velocities of the club are within 2%, preferably within 1% of its
maximum values.
[0140] 38. The golf club of claim 34, wherein a sample rate exceeds
120 samples/sec., preferably 250 samples/second to resolve a final
part of the motion.
[0141] 39. The golf club of claim 34, wherein when tracking at the
club head the position is within 10 mm, preferably within 2-5
mm.
[0142] 40. The golf club of claim 34, wherein a loft/lie angle is
within .+-.1.degree., preferablywithin .+-.0.5.degree..
[0143] 41. The golf club of claim 34, wherein an open/close angle
is resolved to within .+-.0.5.degree., preferably within
.+-.0.1.degree..
[0144] 42. The golf club of claim 34, wherein angular rates are,
within about 2% preferably within 1% of their maximum values.
[0145] 43. The golf club of claim 34, wherein a linear velocity is
within 1 mph, preferably 0.5 mph.
Set Four
[0146] 44. A method of analyzing a movement of a sporting equipment
comprising, the steps of: collecting movement-parameters of said
equipment by means of an Inertial Navigation System (INS), by
measuring acceleration, an angular velocity and effect of
attraction of gravity on said equipment.
[0147] 45. The method of claim 44, comprising further step of
compensating for said effect of attraction of gravity.
[0148] 46. The method of claim 44, wherein said equipment is a golf
club, comprising gyro sensors and accelerometer sensors.
[0149] 47. The method of claim 46, comprising the steps of:--prior
to a swing initializing said INS by holding said club substantially
absolutely still in a well-defined position,--removing offsets of
said gyros in said club by keeping them still,--removing offsets of
the accelerometers by comparison to the known effect of gravity on
each sensor.
[0150] 48. The method of claim 46, comprising the steps
of:--immediately prior to a swing, aiming at a ball in a direction
of a flag for the sensors to define a direction toward the
flag,--continuously activating said sensors in a back swing and
measuring accelerations and rotational velocities.
[0151] 49. The method of claim 48, comprising the step of measuring
temperatures.
[0152] 50. The method of claim 48, comprising the steps
of:--collecting data unit the data is discriticizes,--performing
rudimentary signal processing, and--storing the data in an internal
memory.
[0153] 51. The method of claim 50, wherein all stored data is
transferred to a computer unit after a performed swing.
[0154] 52. The method of claim 44, wherein a three dimensional
acceleration and angular velocity are measured.
Set Five
[0155] 53. A golf club comprising, an arrangement for detecting
movement-parameters, such as acceleration and angular velocity of
the golf club, said arrangement comprising: an Inertial Navigation
System (INS), having a number of sensors, at least one being a
gyroscope for measuring an acceleration, angular velocity and
effect of attraction of gravity on said sporting equipment,
communication arrangement for communicating with a computer unit
for receiving and storing relevant data and provided for
calculating compensation for said effect of attraction of gravity
and quantifying the motion of said club, at least one
accelerometer, temperature sensor, gyroscope, amplifiers and
filters, said accelerometer and gyroscope form a time dependent
sensor inputs to said INS, filters for filtering signals from
sensors and amplified in amplifiers before they form inputs to an
INS filter, wherein at least one of said filters is an extended
Kalman filter, comprising a sensor model, a measurement noise
model, a processor for dynamics and noise model.
[0156] 54. The club of claim 53, wherein estimated output
parameters, represented by a vector matrix, are orientations,
angular velocities, angular accelerations, positions, velocities,
and accelerations respectively, which variables represent the time
dependent results from the INS.
* * * * *