ROTARY ENCODER
This invention relates to rotary encoders, for example for sensing rotary position in
robot joints.
Rotary encoders are widely used for sensing the position of rotatable elements such
as shafts. Examples of their application include robot arm joints, automotive drive
shafts and control wheels or knobs.
10 One common type of position sensor is the Hall effect magnetic sensor. These
sensors have a ring around which is arranged a set of alternating magnetic poles. A
sensor interacts with the ring, and is located so that the magnetic poles move past the
sensor as the rotation that is desired to be sensed takes place. For example, the ring
could be attached about a shaft and the sensor could be attached to a housing within
15 which the shaft rotates. The sensor detects changes in magnetic polarity as the poles
move past the sensor. By counting the number of changes in polarity the amount of
rotation from a reference position can be sensed. To sense the direction of rotation
two such pairs of rings and sensors can be provided, and arranged so that one sensor
detects magnetic transitions of its ring at rotation positions that are offset from the
20 positions where the other sensor detects magnetic transitions of its ring. By
considering the relative timing of transitions detected by each sensor the direction of
rotation can be sensed.
Similar properties can be got from other forms of two-state rotation sensing devices,
25 for example optical sensors that sense transitions from black to white on a rotating
disc, or eddy current sensors that detect the presence or absence of a tooth on a
toothed wheel rotating past a sensor.
An enhancement of the approach discussed above is to measure the position of the
30 poles relative to the sensors to multi-bit accuracy, and to arrange the rings of poles
such that each position of the shaft within a 360° range is associated with a unique set
of outputs from the sensors. This may be achieved by providing different numbers of
poles on each ring and making the numbers of poles the rings co-prime to each other.
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A problem with sensors of this nature is that they can only detect relative position, or
if they can detect absolute position it is only within a range of 360°. The state detected
by the sensor(s) is independent of the number of whole revolutions made by the shaft.
This is immaterial for certain applications, but for other applications it requires
5 additional steps to be taken in order to form a measurement of absolute position. One
example of an application where absolute position is important is in robotics. Some
robot joints may be capable of rotating more than 360°, and when the robot is operating
it is important to know how many rotations have been undergone from a reference
position. That information might be necessary to avoid excessive twisting of internal
10 cables due to driving the joint too far in one direction, or to provide reassurance that if
the robot is reset part-way through a procedure any parts the robot was holding when
it was reset can be restored to their original condition. In some applications the shaft
whose motion is being sensed is connected to another mechanism that provides a
limit to the shaft's travel after some number of rotations. In that situation it is common
15 for the encoder to be calibrated by rotating the shaft until the limit is reached and then
resetting the count on the encoder. A count can then be maintained of the net number
of transitions detected since the shaft was at the limit, the count being incremented or
decremented depending on the direction of rotation. The number of whole rotations
undergone since the shaft was at the limit can be determined by dividing the count by
20 the number of transitions expected in a full rotation. One problem with this is that the
shaft must be turned to its limit in order to perform the calibration. That may be
undesirable in some situations, for example if the shaft is holding an instrument that is
inserted into an object that could be damaged by large amounts of rotation of the
instrument.
25
It is desirable to have an improved or alternative way of allowing the position of a
rotating object to be sensed.
According to the present invention there is provided a device for sensing the relative
30 rotary position of first and second parts about a rotation axis, the device comprising a
follower constrained to move on a first track fast with the first part and on a second
track fast with the second part, the first track being linear and the second track
comprising a plurality of circular arcs and at least one transition section connecting
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one of the circular arcs to another, the tracks being arranged so as to convert relative
rotation of the parts into linear motion of the follower.
The second track may be generally spiral. Each circular arc may be of constant radius
5 about the rotation axis. The first track may be perpendicular to the rotation axis.
10
15
20
25
Each circular arc may be of a different radius from the other(s).
All the circular arcs may lie in a single plane perpendicular to the rotation axis.
The second track may be generally helical. Each circular arc may lie in a single plane
perpendicular to the rotation axis. The first track may be parallel to the rotation axis.
All the circular arcs may be of the same radius.
Each of the circular arcs may lie in a different plane from the other(s) perpendicular to
the rotation axis.
Each circular arc may occupy more than 270° of a circle.
The device may further comprise a sensor for sensing the position of the follower in
the first track.
The sensor may be a switch providing a single bit output.
The device may comprise a second sensing mechanism for sensing the absolute or
relative rotary position of the first and second parts about the rotation axis over a range
not greater than 360°.
30 The second sensing mechanism may be a magnetic sensing mechanism.
The first and second tracks may be defined by channels. The follower may be located
in both channels.
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The follower may comprise a rigid linear element located in both channels.
The second track may comprise a passage open to the radial exterior and
communicating with the outermost one of the circular arcs whereby the follower can
5 be introduced into the outermost arc of the second track.
According to a second aspect of the present invention there is provided a robot arm
comprising a device as set out above. The device may be arranged for sensing
rotation about a joint of the arm. The joint may be a revolute joint arranged so that its
10 rotation axis extends longitudinally with respect to the limbs of the arm between which
it is located.
15
20
The present invention will now be described by way of example with reference to the
accompanying drawings.
In the drawings:
Figure 1 is a general representation of a shaft equipped with a position encoder
mechanism.
Figure 2 shows a portion of the periphery of a disc 5 of figure 1, illustrating a generally
spiral track of the position encoder in more detail.
In the arrangement to be described below, an encoder for the relative rotational
25 position of two objects is capable of absolute determination of the relative rotational
position in whole rotations. (By "absolute determination" is meant that the position can
be determined directly from the output of the sensor(s) without, for example, the need
to count up the amount of motion since the relative rotational position was in a
reference configuration). The absolute determination is preferably over a range
30 greater than 360°. This is achieved by a follower constrained to run (a) in a generally
spiral or helical path about the rotation axis and (b) in a linear path along or transverse
to the rotation axis. The major portions of the spiral or helical path are circular, so that
for the majority of the relative rotational travel of the two objects there is no motion of
the follower along the linear path.
5
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Figure 1 shows a shaft 1 equipped with a position encoder. The position encoder is
capable of detecting motion of the shaft, the direction of that motion and the absolute
number of rotations of the shaft from a limit position.
The shaft is configured to rotate about a rotation axis 2. Three discs are attached to
the shaft so as to rotate with it. Discs 3 and 4 allow for magnetic encoding of rotary
position. Disc 5 allows for mechanical encoding of rotary position.
10 Discs 3 and 4 carry a number of permanent magnets defining magnetic poles 6. On
each disc the poles are arranged in a circle having the rotation axis 2 of shaft 1 as its
axis. On each disc the magnets are arranged so that around the circle of poles the
poles exposed at the sensing surfaces 7, 8 alternate between north and south poles.
Magnetic sensors 9, 10 are disposed adjacent to the sensing surfaces 7, 8 and aligned
15 with the rings of magnetic poles 6. The magnetic sensors are fast with the body
relative to which shaft 1 rotates: for example by being fixed to a housing for shaft 1.
As a result, when the shaft rotates, together with discs 7, 8, the rings of magnetic poles
6 revolve past the sensors 9, 1 0. The sensors are capable of detecting transitions
between north and south poles in the ring of poles as such transitions move past the
20 sensors. The sensors could, for example be Hall effect sensors, reed sensors,
magnetoresistive sensors or inductive sensors. For relative position sensing each
sensor 9, 10 is arranged so that when a transition from a north pole to a south pole
passes the sensor the output of the sensor goes from high to low, and when a
transition from a south pole to a north pole passes the sensor the output of the sensor
25 goes from low to high. For absolute position sensing within a range of 360° each
sensor is arranged to provide a multi-bit output representing the relative position of the
neighbouring poles to it and the rings of poles are arranged such that each position of
the shaft within a 360° range is associated with a unique set of outputs from the
sensors. This may be achieved by providing different numbers of poles on each ring
30 and making the numbers of poles the rings co-prime to each other. The outputs from
the sensors pass to a processing unit 11 .
The circumferential positions of the sensors 9, 1 0 and the rotational positions of the
disc 7, 8 about axis 2 are chosen so that the transitions between the poles on disc 7
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as sensed by sensor 9 occur for different rotational positions of the shaft from the
transitions between the poles on disc 8 as sensed by sensor 10. This allows the
direction of rotation of the shaft to be inferred from the relative order of high/low and
low/high transitions as sensed by each sensor. The rings and sensors allow a relative
5 position of the shaft to be determined.
The number of magnetic poles around the discs can be selected based on the
application; but there could, for example, be around 30 to 40 pairs of north/south poles.
For absolute position sensing within a 360° range using the technique described above
10 the numbers of pairs on the rings should be co-prime.
The mechanical encoder comprising disc 5 will now be described.
Disc 5 is fast with the shaft 1 so as to rotate with the shaft. Disc 5 has a series of
15 formations 20 which have an extent along the direction of the shaft. The formations
define a generally spiral path 21 in the plane of the disc 5, i.e. in a plane perpendicular
to the axis of the shaft. A follower 30 is guided by the formations in a radial direction
(i.e. in a direction perpendicular to the axis of the shaft), so that rotation of the shaft
can cause motion of the follower in a radial direction. As will be described below, the
20 radial position of the follower can be detected in order to establish the absolute position
of the shaft 1 in whole rotations from a reference position.
In more detail, disc 5 has a formation of ridges 20 which extend in an axial direction:
i.e. along the axis 2 of the shaft 1. The ridges are configured so as to define a generally
25 spiral groove 21 between them. For the majority of its path the spiral groove is of
constant radius about the rotation axis 2, for example as indicated in figure 2 at 22 and
23. The region 22 is of a first radius, and the region 23, which is within region 22, is
of a second radius smaller than the first radius. In a transition region 24 the ridges are
configured so as to define a smooth change in radius of the groove between the first
30 radius and the second radius, as indicated at 25. The interior track 23 of the groove
terminates in an end wall 26. The exterior track 23 of the groove terminates in a radial
passageway 27 which extends radially outwardly and opens to the circumference of
the disc 5.
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A mechanical sensing device 28 is located adjacent to disc 5. The mechanical sensing
device is fast with the body relative to which shaft 1 rotates: for example by being fixed
to a housing for shaft 1. As a result, when the shaft rotates the disc 5, which is fast
with the shaft, rotates past the sensing device 28. The sensing device has a track 29
5 within which follower 30 is constrained to move. The track 29 is directed perpendicular
to axis 2. The follower is a pin extending along the axis 2. The pin runs snugly in the
groove 21 and also in the track 29. The mechanical sensing device is arranged to
maintain the orientation of the pin parallel to the axis 2, for example by constraining a
flat head 31 of the pin (shown with a D-shape in figure 2) to prevent rotation of the
10 head about axes perpendicular to axis 2.
The interaction of the track 29, the groove 21 and the follower 30 is such that when
the disc 5 is rotated the position of the follower in a radial direction within the groove
is controlled by its running in groove 21. When one of the regions of the groove 22,
15 23 that have constant cross-section are aligned with the track 29 the disc can rotate
without the follower 30 moving in track 29. The follower will remain in either a radially
outward position aligned with groove portion 22 or a radially inward position aligned
with groove portion 23. When the shaft is rotated so that the transition region 24
revolves past the track 29 the follower is forced from a radially inward position to a
20 radially outward position (when the shaft is turned anti-clockwise as viewed in figure
2) or from a radially outward position to a radially inward position (when the shaft is
turned clockwise as viewed in figure 2).
A microswitch 32 is positioned so as to detect when the follower 30 is in its radially
25 outward position: i.e. aligned with groove portion 22. The output of the microswitch
forms the output of the mechanical sensor device 28, and is passed to the processing
unit 11. The radial position of the follower 30 in track 29 could be detected in other
ways. For example, its presence could be detected at the radially inner position rather
than the radially outer position; and the detector could be a single bit (on/off) switch
30 (e.g. a mechanical, magnetic or optical switch) or could provide a more detailed
indication of position along the length of the track 29. The output of the sensor 28 is
an absolute position signal indicating the number of revolutions of the shaft from a
reference point.
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The outputs of sensors 9, 1 0, 28 pass to the processing unit 11. The processing unit
comprises a processor device 40, which could be hard coded to interpret the signals
from the sensors 9, 1 0, 28 or could be a general purpose processor configured to
execute software code stored in a non-transient way in memory 41. The processor
5 device combines the signals from the sensors to form an integrated output signal at
42.
10
The data from the sensors may be used by the processor device 40 in a number of
ways.
1. The mechanical encoder arrangement 5, 28 could be implemented (either by itself
or together with another position encoder such as that provided by discs and sensors
3, 4, 9, 1 0) to provide a simple output representing the number of whole revolutions of
the shaft 1 from a reference location. If the reference location is taken to be the end
15 26 of the path 21 then non-detection of the follower 30 by microswitch 32 could indicate
zero revolutions and detection of the follower 30 by microswitch 32 could indicate one
revolution.
2. The mechanical encoder arrangement 5, 28 could be used to indicate a reference
20 location for resetting the relative position count associated with the relative position
measurement system. When the count is desired to be reset the output of the
mechanical position encoder is known. The shaft 1 is then rotated in a direction
selected in dependence on that output so as to move the transition zone 24 towards
the follower. With reference to figure 2, if the output of the mechanical position
25 encoder indicates that the follower is in outer track region 22 then the shaft is rotated
clockwise and if the output of the mechanical position encoder indicates that the
follower is in inner track region 23 then the shaft is rotated anti-clockwise. This
determination may be made by the processing unit 11 in response to a signal to reset
the count, and signalled to a drive unit (e.g. a motor) for driving the shaft in the
30 appropriate direction. When the processing unit subsequently detects a transition of
the output of the mechanical position encoder it knows that the transition zone is
aligned with the sensor 28. At that point the count can be reset. This approach has
the advantage that it avoids the need to move the shaft to an extreme position to reset
the count at a position where both the rotational position and the number of rotations
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of the shaft from a predetermined reference position are known. It may be that the
precise rotational position of the shaft at which the output of the mechanical switch
transitions when the follower is in the transition region 28 is different depending on the
rotation direction of the shaft. In that case, the same procedure as above can be
5 followed, but in one direction of the shaft when the transition is detected in the output
from sensor 28 the shaft is then turned in the opposite direction and the subsequent
transition used to indicate resetting of the count.
3. In the example described above, the discs and sensors 7, 8, 9, 10 are capable of
10 sensing relative position. In an alternative arrangement they could be capable of
sensing absolute position within a 360° revolution of the shaft. This can be done in a
number of ways. For example, the sensors 9, 10 could be capable of sensing their
relative position between magnetic poles 6 and outputting an analogue or multi-bit
representation of that relative position, and the numbers of poles on the discs 7, 8
15 could be selected so that in combination the sensors 9, 10 yield a value uniquely
representing the position of the shaft within a 360° revolution. In another example, the
poles could be located on the discs in a binary encoded fashion, so that in combination
the sensors yield a digital output uniquely representing the position of the shaft within
a 360° revolution. In combination with the mechanical encoding arrangement 5, 24
20 this approach allows the absolute position of the shaft both within a 360° revolution
and in whole revolutions from an end point to be immediately determined without the
need for movement of the shaft. This is useful in that it allows the position of the shaft
to be fully determined immediately on start-up of the system, without the requirement
for a calibration step as discussed at 2 above.
25
In the example discussed above, the groove 21 varies in radius about axis 2 and the
radial position of the follower 30 indicates the absolute position of the shaft. In an
alternative arrangement the grove could be a generally helical groove, and the follower
could move axially in the groove to indicate absolute shaft position. Over the majority
30 of its length such a helical groove would be of constant position along axis 2, and there
would be a transition zone in which its axial position varies with radial position of the
shaft. The track 29 would be disposed in an axial direction. The sensor 32 would
sense axial motion of the follower.
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The part circular arcs 22, 23 or their helical equivalents preferably extend over more
than half a circle, more preferably over more than 270°, more preferably over more
than 300°. The transition zone 25 preferably extends at 50° or less to the arcs where
it connects to them, as shown in figure 2. The transition zone preferably includes a
5 linear path section.
Instead of running in a groove 21/29 the follower 30 could be guided in another way,
for example by riding on a raised track which it overlaps on either side.
10 Slot 27 which extends to the periphery of the disc 5 can be used to help assemble the
mechanical position encoder. In the example of figures 1 and 2, during assembly the
sensor 28 can be introduced radially to the disc and the follower inserted into the spiral
channel 21 through the slot 27.
15 In the example of figures 1 and 2 the mechanical sensor can distinguish only between
zero and one whole rotations from the end stop 26. The spiral groove could be
extended to cover more revolutions, with a zone of constant radius for each rotation
covered by the groove and a transition zone between each pair of adjacent constant
radius zones. The transition zones would be located at a common circumferential
20 position about the rotation axis 2. Similarly, in the case of a helical groove the groove
could be extended to cover more revolutions, with a zone of constant axial position for
each rotation covered by the groove and a transition zone between each pair of
adjacent constant axial position zones. In this case the transition zones would again
be located at a common circumferential position about the rotation axis 2. The sensor
25 device 28 would be adapted so that it can determine which of the constant radial/axial
position zones the follower is aligned with. For example, the sensor device could have
multiple microswitches, one aligned with each of the constant radial/axial position
zones, or with all but one of those zones.
30 In the example above, the discs and sensors 3, 4, 9, 1 0 sense relative position by
means of magnetic interaction between the discs and the sensors. They could sense
motion in other ways. For example the sensors could be optical sensors that sense
transitions from one colour or reflectivity to another on a rotating disc, or the sensors
could be eddy current or other electrical sensors that detect the presence or absence
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of a tooth on a toothed wheel rotating past a sensor. If the sensors 3, 4 are magnetic
sensors they could be of any suitable type, for example Hall effect or reed switch
sensors.
5 The control mechanism for the shaft could be arranged so as to automatically prevent
over-rotation of the shaft past the end positions 26, 27 in dependence on position data
output by the processing unit 11.
Instead of being attached to a shaft the discs could be attached to any other part that
10 rotates relative to another part. In the case of a shaft, one or more of the discs could
be attached to the housing of the shaft and the sensors could rotate with the shaft.
The discs could be replaced by members having the same function but different
shapes, e.g. they could be of the form of a cylinder, an annulus or a cuboid.
15 Some applications of the arrangement are as follows. The arrangement could be used
in a robot arm in which rigid members forming the limbs of the arm are coupled by
revolute joints. The shaft could be fast with one limb of the arm and the housing could
be formed by the adjoining limb of the arm, revolution of the shaft relative to that
housing representing relative rotation of the two limbs. The position encoding
20 arrangement can then be used to establish the relative position of the arms. In another
example, the shaft could be the shaft extending from a vehicle's steering wheel and
the housing could be the housing for that shaft, which is part of the main body of the
vehicle. Motion of the steering shaft could be sensed to control an electrical power
steering system or simply to establish the steering demand (e.g. so as to help control
25 the vehicle's stability control system).
The applicant hereby discloses in isolation each individual feature described herein
and any combination of two or more such features, to the extent that such features or
combinations are capable of being carried out based on the present specification as a
30 whole in the light of the common general knowledge of a person skilled in the art,
irrespective of whether such features or combinations of features solve any problems
disclosed herein, and without limitation to the scope of the claims. The applicant
indicates that aspects of the present invention may consist of any such individual
feature or combination of features. In view of the foregoing description it will be evident
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to a person skilled in the art that various modifications may be made within the scope
of the invention.

CLAIMS
1 . A device for sensing the relative rotary position of first and second parts about a
rotation axis, the device comprising a follower constrained to move on a first track fast
with the first part and on a second track fast with the second part, the first track being
linear and the second track comprising a plurality of circular arcs and at least one
transition section connecting one of the circular arcs to another, the tracks being
arranged so as to convert relative rotation of the parts into linear motion of the follower,
wherein the second track is generally spiral, each circular arc is of constant radius
about the rotation axis and the first track is perpendicular to the rotation axis.
2. A device as claimed in claim 1, wherein each circular arc is of a different radius
from the other(s).
3. A device as claimed in claim 1 or 2, wherein all the circular arcs lie in a single plane
perpendicular to the rotation axis.
4. A device for sensing the relative rotary position of first and second parts about a
rotation axis, the device comprising a follower constrained to move on a first track fast
with the first part and on a second track fast with the second part, the first track being
linear and the second track comprising a plurality of circular arcs and at least one
transition section connecting one of the circular arcs to another, the tracks being
arranged so as to convert relative rotation of the parts into linear motion of the follower,
wherein the second track is generally helical, each circular arc lies in a single plane
perpendicular to the rotation axis and the first track is parallel to the rotation axis.
5. A device as claimed in claim 4, wherein all the circular arcs are of the same radius.
6. A device as claimed in claim 4 or 5, wherein each of the circular arcs lies in a
different plane from the other(s) perpendicular to the rotation axis.
7. A device as claimed in any preceding claim, wherein each circular arc occupies
more than 270° of a circle.
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8. A device as claimed in any preceding claim, further comprising a sensor for sensing
the position of the follower in the first track.
9. A device as claimed in claim 8, wherein the sensor is a switch providing a single
bit output.
10. A device as claimed in any preceding claim, comprising a second sensing
mechanism for sensing the absolute or relative rotary position of the first and second
parts about the rotation axis over a range not greater than 360°.
11. A device as claimed in claim 1 0, wherein the second sensing mechanism is a
magnetic sensing mechanism.
12. A device as claimed in any preceding claim, wherein the first and second tracks
are defined by channels and the follower is located in both channels.
13. A device as claimed in claim 12, wherein the follower comprises a rigid linear
element located in both channels.
14. A device as claimed in claim 12 or 13 as dependent on claim 1, wherein the
second track comprises a passage open to the radial exterior and communicating with
the outermost one of the circular arcs whereby the follower can be introduced into the
outermost arc of the second track.
15. A device substantially as herein described with reference to the accompanying
drawings.
16. A robot arm comprising a device as claimed in any preceding claim, arranged for
sensing rotation about a joint of the arm.
17. A robot arm as claimed in claim 16, wherein the joint is a revolute joint arranged
so that its rotation axis extends longitudinally with respect to the limbs of the arm
between which it is located.
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AMENDED CLAIMS
received by the International Bureau on 22 November 2016 (22.11.2016)
1. A device for sensing. the relative rotary position of first and second parts about a
rotation axis. the device comprising a follower constrained to move on a first track fast
with the first part and on a second track fast with the second part, the frrst track being
linear and the second traCk comprising a plurality of circular arcs and at feast one
transition section connecting one of the circular arcs to another. the tracks being
arranged so as to convert relative rotation of the parts into linear motion of the follower.
wherein the second track is generafly spiral, each circular arc is of constant radius
about the rotation axis and the first track is perpendicular to the rotation axis.
2. A device as claimed in claim 1, wherein each circular arc is of a different radius
from the other(s).
3. A device as claimed in claim 1 or 2, wherein all the circular arcs lie in a single plane
perpendicular to the rotation axis.
4. A device for sensing the relative rotary position of first and second parts about a
rotation axis, the device comprising a foltower constrained to move on a first track fast
with the first part and on a second track fast with the second part, the first track being
linear and the second track. comprising a plura.lity of circular arcs and at least one
transition section connecting one of the circular arcs to another. the tracks being
arranged so as to convert relative rotation of the parts into linear motk>n of the foUower.
wherein the second track is generally helical, each circular arc lies in a single plane
perpendicular to the rotatton axis and the first track is parallel to the rotation axis.
5. A device as claimed in claim 4r wherein all the circutar arcs are of the same radius.
6. A device as clsdmed in ctatm 4 or 5.. wherein each of the circular arcs lies in a
different plane from the other{s) perpendicular to the rotation axis.
7. A device as claimed in any preceding claim, wherein ea.ch circular arc occupies
more than 270° of a circle.
AMENDED SHEET (ARTICLE 19)
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8. A device as claimed in any preceding claim, further comprising a sensor for sensing
the position of the follower in the first track.
9. A device as claimed in claim 8, wherein the sensor is a switch providing a single
bit output.
1 o. A device as claimed in any preceding claim, comprising a second sensing
mechanism for sensing the absOlute or relative rotary position of the first and second
parts about the rotation axis over a range not greater than 3601!).
11. A device as claimed in claim 10, wherein the second sensing mechanism is a
magnetic sensing mechanism.
12. A device as ctaimed in any preceding claim, wherein th.e first and second tracks
are defined by channels and the follower is located in both channels.
13. A device as claimed in ctaim 121 wherein the follower comprises a rigid linear
element located in both channelS.
14. A device as claimed in claim 12 or 13 as dependent on claim 1. wherein the
second track comprises a passage open to the radial exterior and communicating with
the outermost one of the circular arcs whereby the follower can be introduced into the
outermost arc of the second track.
15. A robot arm comprising a device as claimed in any preceding cfaim, arranged for
sensing rotation about a joint of the arm.
16. A robot arm as claimed In claim 15, wherein the joint is a revolute joint arranged
so that its rotation axis extends tongituOinany with respect to the fimbs of the arm
between which it is located ..