Assembly of linear accelerometers mounted on a rigid body remote from the center of mass measuring three dimensional movements

The invention provides an assembly of accelerometers for use in a system for measuring the three dimensional movements of a rigid body comprising nine linear accelerometers which are connected to said rigid body in a spacial configuration with reference to an imaginary orthogonal coordinate system. At a first position on the X-axis at a predetermined distance from the origin two accelerometers are installed, the sensitive directions thereof being parallel to the Y-axis and to the Z-axis respectively. At a second position on the Y-axis of the coordinate system at a predetermined distance from the origin two accelerometers are installed, the sensitive directions being parallel to the X-axis and to the Z-axis respectively. At a third position on the Z-axis at a predetermined distance from the origin two accelerometers are installed, the sensitive directions being parallel to the X-axis and to the Y-axis respectively. Three further accelerometers are installed midway on the edges between the first, second and third positions, the sensitive axes of these further accelerometers each coinciding with a line through the origin of the coordinate system.

BACKGROUND OF THE INVENTION 
The invention relates to an assembly of accelerometers for application in a 
system for measuring the three dimensional movements of a rigid body which 
accelerometers are connected to said rigid body in a spacial configuration 
with reference to an imaginary orthogonal coordinate system. 
In general ballistometry is the collection of techniques and algorithms 
used for the reconstruction of the motion of a free-floating body from a 
record of measurements. This includes satellite attitude reconstruction 
from onboard measurements but also the tracking of motions of a rigid body 
by any of a variety of methods. These methods may include the use of 
optical sensors, gyroscopes, accelerometers or cinematographic or 
photogrammetric observation of position an attitude of a rigid body. 
There are several reasons to opt for the use of accelerometers, one of 
which is that accelerometers and their associated electronics require the 
least mass and volume. These factors weigh heavily in space research 
programs, one of the possible application fields of this invention. 
The motion of a rigid body can always be analyzed into a linear and an 
angular velocity. The kinematic variables appear as parameters in the 
representation of the acceleration field that is related to the rigid 
body. The number of accelerometers used, their location on the rigid body 
and the relative positioning of their sensitive axes in the acceleration 
field determine the resulting number and value of the kinematic parameters 
as well as the computations necessary for the reconstruction of the motion 
of the rigid body. 
Prior art ballistometry focuses on the number of accelerometers and the 
relative positioning of their sensitive axes. In this respect it is common 
knowledge that a minimum of six linear accelerometers is required for a 
complete definition of the kinematic variables of a rigid body. Five 
linear accelerometers are required to compute all three components of 
angular acceleration about the body-fixed axes of the acceleration field 
of the rigid body. A sixth accelerometer is needed to provide, in 
addition, all three components of linear acceleration for complete 
definition of rigid body motion. 
The computation of angular acceleration of a rigid body from measured 
linear accelerations is a relatively simple procedure based on well-known 
kinematic principles. The determination of arbitrary motion with six 
sensors involves (numerical) integration or differentiation. Because of 
errors in measurement, these stepwise integration or differentiation 
procedures usually result in an accumulation of errors. These problems are 
described by PADGOANKER et al in "Measurement of Angular Acceleration of a 
Rigid Body Using Linear Accelerometers." (in Journal of Applied Mechanics, 
September 1975, pages 552-556). To solve these problems Padgoanker et.al. 
introduce the use of nine accelerometers instead of six, position them in 
a predetermined spacial configuration with their sensitive axes directed 
such that by relatively simple calculations on the accelerometer outputs 
the linear and angular acceleration components of the motion of the rigid 
body to which the accelerometers are attached can be determined. More 
specifically the prior art configuration comprises nine accelerometers, 
three of which are located at the origin of an imaginary orthogonal 
coordinate system which is fixed with respect to the rigid body of which 
the motions are measured. The sensitive axis of these three accelerometers 
are trained respectively in the direction of Z, Y and X-axis of the 
coordinate system. This prior art nine accelerometer configuration 
furthermore comprises a set of two accelerometers on each of the 
orthogonal axes of the coordinate system at a predetermined distance from 
the origin. The sensitive axes of the two accelerometers positioned on the 
X-axis of the coordinate system are trained respectively parallel to 
Y-axis and parallel to the Z-axis. The sensitive axes of the two 
accelerometers positioned on the Y-axis of the coordinate system are 
trained respectively parallel to the X-axis and parallel to the Z-axis. 
The sensitive axes of the two accelerometers positioned on the Z-axis of 
the coordinate system are trained respectively parallel to the X-axis and 
parallel to the Y-axis. 
Rotation of the rigid body can cause problems when the rotation is 
three-dimensional and there are errors in the measured linear 
accelerations. As stated before errors in measurement can result in an 
accumulation of errors when stepwise (numerical) computations are 
performed. MITAL et.al. introduce in "Computation of Rigid-Body Rotation 
in Three-Dimensional Space From Body-Fixed Linear Acceleration 
Measurements." (in Journal of Applied Mechanics, Vol. 46, December 1979, 
page 925-930) a method which generates an orthogonal transformation 
matrix, which needs to be evaluated only when it is required to transform 
a position vector from the body-fixed frame to the inertially fixed 
reference frame. 
It will be clear now that a nine sensor arrangement can allow a direct 
determination of linear acceleration (a) of the origin O of the 
acceleration field of the rigid body as well as the angular velocity 
(.omega.) and of the angular acceleration (.omega.) by algebraic 
operations on the accelerometer output. This is a stable calculation and 
leaves scope for additional extraction of parameter values from the 
comparison of angular velocity and angular acceleration by calculation and 
measurement. 
However the nine sensor arrangement according to prior art has three 
accelerometers at the origin of an imaginary orthogonal coordinate system 
which is fixed with respect to the rigid body of which the motions have to 
be measured. For no motion of the center of mass of the rigid body it is 
clear that in that case the origin of the coordinate system should be put 
at the center of mass of the rigid body and measurement of angular 
velocity is possible with three sensors. Usually a free-floating rigid 
body will show only small excursions in the center of mass location and it 
is therefore advantageous to have the origin O of the coordinate system 
near the approximate center of mass position. However in general this 
location is centrally located in the rigid body to which the sensors are 
attached and is inaccessible. These circumstances make it desirable to 
have an arrangement that has no sensors at the origin of the coordinate 
system. None of the arrangements in the quoted literature fulfil this 
requirement. 
SUMMARY OF THE INVENTION 
It is therefore the purpose of the present invention to provide a planar 
nine sensor arrangement, without sensors at the origin of the orthogonal 
coordinate system, which arrangement, with the matching computations on 
the accelerometer outputs, results in accurate definition of the motion of 
a rigid body. 
It is furthermore the purpose of the present invention to eliminate the 
laborious practice of individually mounting each accelerometer on the 
rigid body, aligning their sensitive axes and calibrating the whole rigid 
body arrangement. 
In agreement with these objects the invention provides an assembly of 
accelerometers for use in a system for measuring the three dimensional 
movements of a rigid body comprising nine linear accelerometers which are 
connected to said rigid body in a spacial configuration, with reference to 
an imaginary orthogonal coordinate system, whereby at a first position on 
the X-axis of the coordinate system at a predetermined distance from the 
origin a first and second accelerometer are installed, the sensitive 
direction of the first accelerometer being parallel to the Y-axis and the 
sensitive direction of the second accelerometer being parallel to the 
Z-axis, at a second position on the Y-axis of the coordinate system at a 
predetermined distance from the origin a third and fourth accelerometer 
are installed, the sensitive direction of the third accelerometer being 
parallel to the X-axis and the sensitive direction of the fourth 
accelerometer being parallel to the Z-axis, and at a third position on the 
Z-axis of the coordinate system at a predetermined distance from the 
origin a fifth and sixth accelerometer are installed, the sensitive 
direction of the fifth accelerometer being parallel to the X-axis and the 
sensitive direction of the sixth accelerometer being parallel to the 
Y-axis, characterised in that, a seventh accelerometer is installed at a 
fourth position midway on a line between the first and second position, 
the sensitive axis of the seventh accelerometer coinciding with a line 
through said fourth position and the origin of the coordinate system, an 
eighth accelerometer is installed at a fifth position midway on a line 
between the second and third position, the sensitive axis of the eighth 
accelerometer coinciding with a line through said eighth position and the 
origin of the coordinate system and a ninth accelerometer is installed at 
a sixth position midway on a line between the third and first position, 
the sensitive axis of the ninth accelerometer coinciding with a line 
through said sixth position and the origin of the coordinate system. 
It is preferred that the nine accelerometers are attached to a frame in the 
shape of a flat triangular base plate in an arrangement which locates two 
accelerometers on one end of each of its sides and one accelerometer 
midway along each of its sides. More preferably the triangular base plate 
consists of three elongated subassemblies the length of which corresponds 
to the length of a side of the base plate, each subassembly has two 
accelerometers at an end and one accelerometer midway between both ends.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 illustrates schematically an orthogonal reference coordinate system 
with the origin O and the three orthogonal axis X, Y and Z. Two 
accelerometers 1 and 2 are located at the position X1 on the X-axis at a 
predetermined distance from the origin O. Two accelerometers 3 and 4 are 
installed at a position Y1 on the Y-axis at a predetermined distance from 
the origin O. Two further accelerometers 5 and 6 are located at position 
Z1 having the same predetermined distance from the origin O. A further 
accelerometer 7 is installed midway on the line X1Z1 between the positions 
X1 and Z1. An other accelerometer 8 is installed midway on the line Y1Z1 
between the positions Y1 and Z1 and another accelerometer 9 is installed 
midway on the line X1Y1 between the locations X1 and Y1. It is remarked 
that in FIG. 1 the accelerometers are not illustrated as such, only the 
axis of sensitivity of the various accelerometers are indicated by means 
of the illustrated arrows. More specifically the direction of sensitivity 
of the accelerometers 3 and 6 is perpendicular to the X-axis, the 
direction of sensitivity of the accelerometers 1 and 5 is parallel to the 
Y-axis, and the direction of sensitivity of the accelerometers 2 and 4 is 
parallel to the Z-axis. Furthermore the direction of sensitivity of 
accelerometer 7 coincides with a line through the origin O, and the same 
applies to the directions of sensitivity of the two further accelerometers 
8 and 9. 
Although not necessary, per se, it is preferred that the positions X1, Y1 
and Z1 have identical distances to the origin O. In that case the 
positions X1, Y1 and Z1 determine together with the origin O a symmetrical 
tetrahedron. 
With this arrangement of nine accelerometers it is possible to calculate 
the motion of the origin O in a very accurate manner. The output from the 
accelerometers 4, 5 and 8 on the line Y1Z1 is given by: 
EQU U.sub.4 +U.sub.5 =a.sub.1 +a.sub.2 +r.sqroot.2 .omega..sub.1 .omega..sub.2 
EQU U.sub.4 -U.sub.5 =a.sub.1 +a.sub.2 +r.sqroot.2 .omega..sub.3 
EQU 2.sqroot.2 (U.sub.4 +U.sub.5)-4U.sub.8 =r{(.omega.+.omega..sub.2).sup.2 
+2.omega..sub.3.sup.2} 
where 
U.sub.4 =Output of accelerometer 4 
U.sub.5 =Output of accelerometer 5 
U.sub.8 =Output of accelerometer 8 
a.sub.1 =linear acceleration in direction of Y-axis 
a.sub.2 =linear acceleration in direction of Z-axis 
##EQU1## 
.omega..sub.1 =angular velocity about Y-axis .omega..sub.2 =angular 
velocity about Z-axis 
.omega..sub.1 =angular acceleration about Y-axis 
.omega..sub.2 =angular acceleration about Z-axis. 
Similar expressions hold for the output of the accelerometers lines X1Z1 
and Z1Y1. 
The geometrical relative arrangement of the axes of sensitivity of the 9 
accelerometers, without any accelerometers in the origin of the coordinate 
system, results in an accurate reconstruction of the motion of a rigid 
body. The physical arrangement according to this invention results in a 
pre-calibratable strap-down ballistometer, which is easy to produce, easy 
to install, and, in combination with a suitable data processor accurately 
reconstructs the motion of a rigid body. Some advantages of the inventive 
arrangement with no sensors at the origin are apparent. The origin of the 
coordinate system can now readily be put at the center of mass (or 
inertia) of the rigid body of which the motion is to be measured, since no 
physical connection, or presence of the measuring system is required at 
the center of mass. 
The electronic circuitry for processing the electric signals derived from 
the various accelerometers as well as the processor used for performing 
the necessary calculations on the obtained data in agreement with the 
above-mentioned equations are schematically illustrated in FIG. 2. As 
appears from FIG. 2 the accelerometers are in sets of three connected to 
three different conditioning circuits. The accelerometers 1, 2, 9 are 
connected to circuit 10, the accelerometers 3, 4 and 8 are connected to 
circuit 11 and the accelerometers 5, 6, 7 are connected to circuit 12. 
Each of the conditioning circuits 10, 11 and 12 receives electronic 
signals from the connected accelerometers, amplifies and shapes the 
signals, if necessary and converts the signals into digital values 
suitable for processing by a computer or digital processor 13. 
The processor 13 scans the digital values at the output of the various 
circuits 10, 11 and 12 and performs the necessary computations according 
to the above-mentioned equations to obtain the desired motion information 
which is made available at an output 14 of the processor 13. The actual 
embodiment of the circuits 10, 11 and 12 is considered to be known to a 
person skilled in the art and the same applies to the functioning and 
embodiment of the processor 13. Further details about these circuits 10, 
11, 12 and processor 13 are therefore not provided. Suitable accelerometer 
conditioning is described in "Accelerometer sensor conditioning". 
The results of the calculations performed by the processor 13, available at 
output 14 of the processor 13, can be stored for instance in a memory or 
can be transmitted, for instance, in the case of an object in space, along 
a telemetry communication path to a receiver. 
Because the nine accelerometers 1 . . . 9 are physically arranged in planar 
configuration, or zone, in a plane including the three locations X1, Y1 
and Z1 it is possible to use a planar structure support to accommodate the 
nine accelerometers. An example of such a structure is illustrated in FIG. 
3. FIG. 3 illustrates schematically a planar triangular base plate 20 onto 
which the various accelerometers 1 . . . 9 are mounted. On each corner of 
the base plate 20 a set of two accelerometers (1, 2) (3, 4) and (5, 6) is 
installed whereas midway on the edges between the three vertexes the 
further accelerometers 7, 8 and 9 are installed. 
The installation of each accelerometer is a matter of physically fixing the 
accelerometer at the preferred position in the two-dimensional plane, and 
additionally aligning the sensitive axis of the accelerometer in the 
correct direction. In fact if the physical position of the accelerometer 
is correct but the alignment of the sensitive axis is incorrect, the whole 
sensor arrangement will yield incorrect readings. 
The open space in the center of the base plate 20 can be used to 
accommodate a housing 21 in which the conditioning circuits 10, 11 and 12 
as well as the processor 13 are accommodated. For the sake of clarify the 
wires running from the various accelerometers to the housing 21 are not 
illustrated in FIG. 3. 
To improve accuracy, the edges or sides of the triangular base plate 20 
have to be selected as large as possible within the circumstances in which 
the arrangement has to be used. That implies that a relatively large base 
plate 20 has to be used which in many cases will give rise to 
difficulties. 
FIG. 4 illustrates a further embodiment of a base plate 20' which avoids at 
least part of these problems. In this embodiment the central section of 
the base plate 20' is open providing space to accommodate parts of the 
host rigid body whose motions are to be measured. In FIG. 4 only the 
accelerometers 1 . . . 9 are illustrated. It will be clear that the 
conditioning electronics and the microprocessor can be installed elsewhere 
on the rigid body of which motion is robe measured, and wires running from 
the various accelerometers to the conditioning electronics are not 
illustrated in FIG. 4. 
By subdividing the base plate 20' into three separate beams which can be 
fitted together, as illustrated in FIG. 5, the embodiment illustrated in 
FIG. 4 can be obtained. In this embodiment the housing comprises three 
beams 20a, 20b, 20c each carrying three accelerometers, two stacked at one 
and of the respective beam and one installed midway of the respective 
beam. The various accelerometers are indicated with the same reference 
numbers as used in FIG. 4. Preferably each of the beams 20a, 20b and 20c 
carries furthermore a part of the electronic circuitry necessary for 
amplifying and shaping the signals delivered by the accelerometers mounted 
on the same beam. In this way the subdivision of the base plate into three 
beams results in three identical subassemblies. This embodiment has the 
advantage that each of the subassemblies can be prefabricated, preadjusted 
and pretested before the configuration as a whole is installed on the body 
where motions are to be measured. 
The configuration of the electronic circuitry illustrated in FIG. 2 was 
developed for the embodiment of FIG. 3. It will be clear from a comparison 
between FIGS. 2 and 5 that the housing 22a may accommodate electronics for 
conditioning circuit 10, the housing 22b may accommodate electronics for 
circuit 11 and the housing 22c may accommodate electronics for circuit 12. 
Although not illustrated in FIG. 5, of each of the beams 20a, 20b and 20c 
may comprise fastening means to assemble the beams together into a 
configuration such as illustrated in FIG. 4. However, if the alignment of 
the various beams on the body, on which the accelerometers have to be 
used, will cause problems, or is taken care of in another manner, then in 
fact the beams 20a, 20b and 20c could be shorter, such that there is just 
enough space to accommodate the various sensors and electronics circuitry 
thereof. 
It is noted that the term "beam", as used in this description should be 
understood as a component or structure on which sensors can be mounted, 
can be accurately positioned with respect to each other in the prescribed 
locations, and can be aligned in the correct direction so that their 
sensitive axes are pointing in the prescribed direction. It is therefore 
not necessary that the beam is an elongated flat piece of material. The 
beam can be embodied in other forms too. 
Finally FIG. 6 illustrates an application of the inventive nine sensor 
configuration on a spacecraft. The spacecraft is as a whole indicated by 
reference number 30 and comprises a main body 31 and two solar cell panels 
32 and 33. The actual shape of the spacecraft is, however, not important 
at all. On one face of the main body 31 a triangular configuration of nine 
accelerometers according to the invention herein, indicated as a whole by 
reference number 34, is installed. Preferably, the lay-out of the 
accelerometer configuration is such that the origin O of the orthogonal 
coordinate system for which the accelerometer configuration is laid out, 
coincides with the center of mass of the spacecraft 30. 
An upper view on a preferred embodiment of the accelerometer assembly is 
shown in FIG. 7. The nine accelerometers are indicated by reference 
numbers 41 . . . 49. The accelerometers as such are considered known to a 
person skilled in the art and need no further explanation. Suitable 
accelerometers are described in CSEM "Specifications of ASMAC 01-1g" and 
P. Roussel "Solid State Microaccelerometer Experiment"; accelerometers 
have a flat pad-like generally planar configuration and the axis of 
sensitivity extends approximately perpendicular to the plane of the pad, 
centrally thereof. Each accelerometer has the shape of a rectangular box 
with longitudinal extending mounting brackets, for example 47a at opposed 
ends by means of which brackets such as 47a the accelerometers are 
fastened to a triangular frame, 50. In the central space of the triangular 
frame the coordinate axes X, Y and Z of the imaginary orthogonal 
coordinate system as well as the origin O, thereof are indicated 
schematically. 
The triangular frame 50 comprises three legs or beams identified separately 
in FIG. 7 by 50a, 50b and 50c. Each beam or leg has a number of seatings, 
for example 41b and 47b each intended to receive one of the 
accelerometers. Each seating such as 41b and 47b extends in a suitable 
direction such that after mounting the accelerometers the respective axis 
of sensitivity axis of each accelerometer is directed in the prescribed 
manner. More specifically the accelerometer 41 is mounted on seating 41b 
comprising a surface portion near one end of leg 50a which surface portion 
is oriented in relation to the imaginary orthogonal coordinate system such 
that the sensitive axis of accelerometer 41 is directed parallel to the 
Y-coordinate. The accelerometer 46 is mounted on a surface portion near 
the other end of leg 50a which surface portion is oriented in relation to 
the imaginary orthogonal coordinate system such that the sensitive axis of 
accelerometer 46 is directed parallel to the Y-coordinate. The 
accelerometer 47 is mounted on a seating 47b comprising a surface portion 
midway of leg 50a which surface portion is oriented in relation to the 
imaginary orthogonal coordinate system such that the sensitive axis of 
accelerometer 47 extends within the XZ-plane and through the origin O. It 
will be clear that the other accelerometers 42, 43, 44, 45, 48 and 49 are 
mounted on corresponding surfaces on the two other legs 50b and 50c such 
that their respective sensitive direction are oriented in the prescribed 
manner. 
The triangular frame is in FIG. 7 connected to an adjusting ring 52 by 
means of three mounting brackets 54a, 54b and 54c, respectively attached 
to the triangular frame legs 50a, 50b and 50c. When the accelerometer 
arrangement is mounted onto the body, of which the motions have to be 
measured, for example a space craft as shown in the schematic view of FIG. 
6, then first of all the adjustment ring 52 is fitted onto the body, so 
that the ring extends in the correct plane. Thereafter the triangular 
frame 50 is fixed to the adjustment ring 52 with the angular position of 
the frame 50 in relation to the ring 52 and more specifically in relation 
to the rigid body being adjusted in a suitable manner. 
It will be clear that also in this embodiment the space in the central part 
of the triangular frame 50 can be used for housing the electronics 
circuitry and the processor, necessary to process the output signals from 
the accelerometers and to perform the necessary calculations to provide 
the desired angular and linear velocity output values.