1. A hull section with an outer hull and an inner hold comprising:
inner hull plating comprising an inner bottom defining a bottom of said hold and a lower inner side shell defining lower sides of said hold;
outer hull plating comprising an outer bottom defining a bottom of said outer hull and a lower outer side shell defining lower sides of said outer hull; and
a plurality of transverse girders located, in a spaced apart relationship, transversely between the inner and outer bottoms, each of said plurality of transverse girders having two associated web frames located at each end of said transverse girders and between said lower side shells;
wherein said inner hull plating and said outer hull plating are each comprised of a first metal layer and a second layer and an intermediate layer of elastomer bonded to said first and second layers so as to transfer shear forces therebetween.
2. A hull section according to claim 1, further comprising inner and outer upper side shells attached opposite said bottom hull wall to said inner and outer lower side shells respectively.
3. A hull section according to claim 2, wherein said web frames extend between said inner and outer upper side shells and said upper side shells are comprised of a first metal layer and a second metal layer and an intermediate layer of elastomer bonded to said first and second layers so as to transfer shear forces therebetween.
4. A hull section according to claim 2, wherein an intermediate layer of elastomer is bonded to said inner and outer upper side shells so as to transfer shear forces therebetween.
5. A longitudinal barge hull section according to claim 1, further comprising a longitudinal girder located longitudinally between the outer bottom and the inner bottom, said longitudinal girder being positioned substantially in the centre of the beam of the hull.
6. A longitudinal barge hull section according to claim 1, wherein said plurality of transverse girders are located at least 1 m from each other.
7. A hull section with an outer hull and an inner hold; comprising:
a bottom defining on a first side at least a part of the bottom of said outer hull and, on a second side, the bottom of said inner hold;
inner side shells defining side walls of said inner hold;
bottom outer side shells defining side walls of said outer hull and parts of the bottom of the outer hull not defined by said bottom hull wall; wherein
said bottom is comprised of a first layer, a second layer and an intermediate layer of elastomer bonded to said first and second layers so as to transfer shear forces therebetween.
8. A hull section according to claim 7, wherein said part of the bottom of said outer hull is substantially flat.
9. A hull section according to claim 7, wherein longitudinal edges of said bottom outer side shells and longitudinal edges of said inner side shells are attached adjacent longitudinal edges of said bottom.
10. A hull section according to claim 7, wherein the distance between said inner side shells and bottom outer side shells decreases as the distance from the bottom decreases.
11. A hull section according to claim 7, wherein said inner and bottom outer side shells each comprise a first layer, a second layer and an intermediate layer of an elastomer material bonded to said first and second layers so as to transfer shear forces therebetween.
12. A hull section according to claim 7, wherein the width of said at least part of the bottom of the outer hull is a constant fraction of the beam of the outer hull.
13. A hull section according to claim 7, wherein said inner hull side shells comprise an upper inner side shell and a hopper attached together.
14. A hull section according to claim 13, wherein said hopper is attached to said bottom at an angle of between 45\xb0 and 20\xb0.
15. A hull section according to claim 7, wherein a plurality of web frame plate members are attached between said inner and outer side shells joining them together.
16. A hull section according to claim 15, wherein said plurality of web frame plate members are spaced apart along the longitudinal direction of the hull at least 1000 mm apart.
17. A longitudinal barge hull section according to claim 15, wherein plates are attached to form a sheerstrake between said web frame plate members and generally parallel to said bottom of said outer hull.
18. A longitudinal barge hull section according to claim 15, wherein web frame stiffeners are attached to said web frame plate members.
19. A longitudinal barge hull section according to claim 15, wherein said web frame plate members are provided with cut-outs.
20. A longitudinal barge hull section according to claim 7, wherein a gunwale is attached at the top of the hull.
21. A longitudinal barge hull section according to claim 20, wherein said gunwale overhangs said bottom of said hold.
22. A longitudinal barge hull section according to claim 7, further comprising a connecting member comprising:
an elongate metal body of substantially constant cross-section and having at least one tapered edge formed by first and second inclined surfaces, said inclined surfaces serving as landing surfaces and weld preparations.
23. A longitudinal barge hull section according to claim 7, wherein said first layer, said second layer and said intermediate layer are of a prefabricated panel.
24. A hull section according to claim 7, wherein the first and second layers are metal.
25. A barge comprising a plurality of longitudinal barge hull sections according to claim 7 joined together.
26. (canceled)
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A method of calibrating a rotary encoder comprising a detectable portion having a plurality of detectable elements about a rotational axis of the encoder and a detector operable to detect the detectable elements and generate an output signal in response to movement of the detectable portion about the rotational axis of the encoder, the method comprising the steps of:
(a) rotating the encoder to a selected first speed then allowing the encoder to freely decelerate to a selected second speed over a spindle-free response period;
(b) during the spindle-free response period, receiving output signals from the detector, the signals including a plurality of data sets wherein each data set comprises the periods between sampling events occurring in one revolution of the encoder at an angular speed, each sampling event representing a transition between successive detectable elements of the encoder and each sampling event being spaced an angular distance apart from the preceding sampling event;
(c) selecting a data set pair from the output signals, wherein the data set pair contains two linearly uncorrelated data sets;
(d) defining a model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients and the periods between each sampling event as contained in the data set pair;
(e) solving the model for the angular distances for the data set pair using circular closure and dynamic reversal techniques; and
(f) establishing an encoder error map using the selected solved angular distances from the model.
2. A method as claimed in claim 1, wherein in the model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=Tk(\u03c90+ak+bk2)
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
a is a first order damping coefficient, and
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
is solved by circular closure and dynamic reversal techniques.
3. A method as claimed in claim 2, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
4. A method as claimed in claim 3, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
5. A method as claimed in claim 4, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
6. A method as claimed in to claim 5, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
7. A method as claimed in to claim 6, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
8. A method as claimed in to claim 7, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
9. A method as claimed in claim 8, wherein the vector pair equations are combined to cancel angular distance vector A and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
10. A method as claimed in claim 9, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
11. A method as claimed in claim 1, wherein:
in step (c), a plurality of data set pairs are selected from the output signals;
in step (e), the model is solved for the angular distances for each of the plurality of data set pairs;
step (f) comprises the following steps:
(i) calculating the repeatability of the angular distances solved for each of the plurality of data set pairs;
(ii) selecting the angular distances solved for one of the plurality of data set pairs based on the repeatability of the angular distances as compared to the repeatability of the angular distances solved for other data set pairs;
(iii) establishing an encoder error map using the selected solved angular distances from the model.
12. A method according to claim 1, wherein step (f) comprises the following steps:
(i) defining an enhanced model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients, the solution of a previously solved model, and the periods between each sampling event as contained in each data set pair;
(ii) selecting the angular distances solved for a data set pair using the previously solved model;
(iii) repeatedly solving the enhanced model for the angular distances using circular closure and dynamic reversal techniques until the variation between successive solved angular distances is below a convergence threshold; and
(iv) establishing an encoder error map using the selected solved angular distances from the enhanced model.
13. A method as claimed in claim 12, wherein in the enhanced model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=TkRk\u03c90+aPk+bPk2
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
Rk is a correction term defined by Rk=(ecTk\u22121)cTk
c is defined by c=\u2212\u0101\u03940
\u0101 is an approximation of the first order damping coefficient previously solved using the model
\u03940 is the nominal angular distance between sampling events,
a is a first order damping coefficient,
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
P is defined by
P
k
=
\u2211
i
=
1
k
\ue89e
\u0394
_
i
\u0394
0
,
and
\u0394i is an approximation of angular distance \u0394k between sampling events k and k\u22121 from the previously solved model,
is solved by circular closure and dynamic reversal techniques.
14. A method as claimed in claim 13, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
15. A method as claimed in claim 14, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
16. A method as claimed in claim 15, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
17. A method as claimed in to claim 16, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
18. A method as claimed in to claim 17, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
19. A method as claimed in to claim 18, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
20. A method as claimed in claim 19, wherein the vector pair equations are combined to cancel angular distance vector \u0394 and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
21. A method as claimed in claim 20, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
22. A system for calibrating a rotary encoder comprising a detectable portion having a plurality of detectable elements about a rotational axis of the encoder and a detector operable to detect the detectable elements and generate an output signal in response to movement of the detectable portion about the rotational axis of the encoder, the system comprising:
(a) a timer operable to receive output signals from the detector during a spindle-free response period, wherein the output signals comprise a plurality of data sets, each data set comprises the periods between sampling events occurring in one revolution of the encoder at an angular speed, each sampling event representing a transition between successive detectable elements of the encoder and each sampling event being spaced an angular distance apart from the preceding sampling event;
(b) a processor operable to receive the plurality of data sets from the timer, the processor having a memory with statements and instructions stored therein for execution by the processor of the following steps:
(i) controlling a motor to rotate the encoder to a selected first speed then to allow the encoder to freely decelerate to a selected second speed over the spindle free response period;
(ii) selecting a data set pair from the plurality of data sets, wherein the data set pair contains two linearly uncorrelated data sets;
(iii) defining a model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients and the periods between each sampling event as contained in the data set pair;
(iv) solving the model for the angular distances for the data set pair using circular closure and dynamic reversal techniques; and
(v) establishing an encoder error map using the selected solved angular distances from the model.
23. A system as claimed in claim 22, wherein in the model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=Tk(\u03c90+ak+bk2)
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
a is a first order damping coefficient, and
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
is solved by circular closure and dynamic reversal techniques.
24. A system as claimed in claim 23, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
25. A system as claimed in claim 24, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
26. A system as claimed in claim 25, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
27. A system as claimed in to claim 26, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
28. A system as claimed in to claim 27, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
29. A system as claimed in to claim 28, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
30. A system as claimed in claim 29, wherein the vector pair equations are combined to cancel angular distance vector A and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
31. A system as claimed in claim 30, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
32. A system as claimed in claim 22, wherein:
in step (ii), a plurality of data set pairs are selected from the output signals;
in step (iv), the model is solved for the angular distances for each of the plurality of data set pairs;
step (v) comprises the following steps:
(i) calculating the repeatability of the angular distances solved for each of the plurality of data set pairs;
(ii) selecting the angular distances solved for one of the plurality of data set pairs based on the repeatability of the angular distances as compared to the repeatability of the angular distances solved for other data set pairs;
(iii) establishing an encoder error map using the selected solved angular distances from the model.
33. A system as claimed in claim 22, wherein the processor is a general purpose computer.
34. A system as claimed in claim 22, wherein the timer is a field programmable gate array.
35. A system as claimed in claim 22, wherein the processor is a controller for controlling machinery having the encoder therein.
36. A system as claimed in claim 22, wherein the timer is a controller for controlling machinery having the encoder therein.
37. A system according to claim 22, wherein the memory has further statements and instructions stored therein for execution by the processor such that step (v) comprises the following steps:
(1) defining an enhanced model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients, the solution of a previously solved model, and the periods between each sampling event as contained in each data set pair;
(2) selecting the angular distances solved for a data set pair using the previously solved model;
(3) repeatedly solving the enhanced model for the angular distances using circular closure and dynamic reversal techniques until the variation between successive solved angular distances is below a convergence threshold; and
(4) establishing an encoder error map using the selected solved angular distances from the enhanced model.
38. A system as claimed in claim 37, wherein in the enhanced model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=TkRk\u03c90+aPk+bPk2
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
Rk is a correction term defined by Rk=(ecTk\u22121)cTk
c is defined by c=\u2212\u0101\u03940
\u0101 is an approximation of the first order damping coefficient previously solved using the model
\u03940 is the nominal angular distance between sampling events,
a is a first order damping coefficient,
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
P is defined by
P
k
=
\u2211
i
=
1
k
\ue89e
\u0394
_
i
\u0394
0
,
and
\u0394i is an approximation of angular distance \u0394k between sampling events k and k\u22121 from the previously solved model,
is solved by circular closure and dynamic reversal techniques.
39. A system as claimed in claim 38, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
40. A system as claimed in claim 39, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
41. A system as claimed in claim 40, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
42. A system as claimed in to claim 41, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
43. A system as claimed in to claim 42, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
44. A system as claimed in to claim 43, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
45. A system as claimed in claim 44, wherein the vector pair equations are combined to cancel angular distance vector A and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
46. A system as claimed in claim 45, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
47. A calibrated rotary encoder system comprising
(a) a rotary encoder comprising:
(i) a detectable portion having a plurality of detectable elements about the rotational axis of the encoder;
(ii) a detector operable to detect the detectable elements and generate an output signal in response to movement of the detectable portion about the rotational axis of the encoder;
(b) a controller communicative with the encoder and having a memory with an encoder error map stored thereon and statements and instructions for execution by the controller to apply the encoder error map to output signals received from the encoder to produce calibrated output signals; the encoder error map derived by:
(i) rotating the encoder to a selected first speed then allowing the encoder to freely decelerate to a selected second speed over a spindle-free response period;
(ii) during the spindle-free response period, receiving output signals from the detector, the signals including a plurality of data sets wherein each data set comprises the periods between sampling events occurring in one revolution of the encoder at an angular speed, each sampling event representing a transition between successive detectable elements of the encoder and each sampling event being spaced an angular distance apart from the preceding sampling event;
(iii) selecting a data set pair from the output signals, wherein the data set pair contains two linearly uncorrelated data sets;
(iv) defining a model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients and the periods between each sampling event as contained in the data set pair;
(v) solving the model for the angular distances for the data set pair using circular closure and dynamic reversal techniques; and
(vi) establishing the encoder error map using the selected solved angular distances from the model.
48. A system as claimed in claim 47, wherein in the model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=Tk(\u03c90+ak+bk2)
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
a is a first order damping coefficient, and
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
is solved by circular closure and dynamic reversal techniques.
49. A system as claimed in claim 48, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
50. A system as claimed in claim 49, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
51. A system as claimed in claim 50, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
52. A system as claimed in to claim 51, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
53. A system as claimed in to claim 52, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
54. A system as claimed in to claim 53, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
55. A system as claimed in claim 54, wherein the vector pair equations are combined to cancel angular distance vector A and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
56. A system as claimed in claim 55, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
57. A system as claimed in claim 47, wherein in deriving the encoder error map:
in step (iii), a plurality of data set pairs are selected from the output signals;
in step (v), the model is solved for the angular distances for each of the plurality of data set pairs;
step (vi) comprises the following steps:
(i) calculating the repeatability of the angular distances solved for each of the plurality of data set pairs;
(ii) selecting the angular distances solved for one of the plurality of data set pairs based on the repeatability of the angular distances as compared to the repeatability of the angular distances solved for other data set pairs;
(iii) establishing an encoder error map using the selected solved angular distances from the model.
58. A system according to claim 47, wherein step (vi) of deriving the encoder error map comprises:
(1) defining an enhanced model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients, the solution of a previously solved model, and the periods between each sampling event as contained in each data set pair;
(2) selecting the angular distances solved for a data set pair using the previously solved model;
(3) repeatedly solving the enhanced model for the angular distances using circular closure and dynamic reversal techniques until the variation between successive solved angular distances is below a convergence threshold; and
(4) establishing an encoder error map using the selected solved angular distances from the enhanced model.
59. A system as claimed in claim 58, wherein in the enhanced model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=TkRk\u03c90+aPk+bPk2
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
Rk is a correction term defined by Rk=(ecTk\u22121)cTk
c is defined by c=\u2212\u0101\u03940
\u0101 is an approximation of the first order damping coefficient previously solved using the model
\u03940 is the nominal angular distance between sampling events,
a is a first order damping coefficient,
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
P is defined by
P
k
=
\u2211
i
=
1
k
\ue89e
\u0394
_
i
\u0394
0
,
and
\u0394i is an approximation of angular distance \u0394k between sampling events k and k\u22121 from the previously solved model,
is solved by circular closure and dynamic reversal techniques.
60. A system as claimed in claim 59, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
61. A system as claimed in claim 60, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
62. A system as claimed in claim 61, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
63. A system as claimed in to claim 62, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
64. A system as claimed in to claim 63, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
65. A system as claimed in to claim 64, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
66. A system as claimed in claim 65, wherein the vector pair equations are combined to cancel angular distance vector A and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
67. A system as claimed in claim 66, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
68. A computer-readable medium having statements and instruction stored therein for execution by a processor for calibrating a rotary encoder comprising a detectable portion having a plurality of detectable elements about the rotational axis of the encoder and a detector operable to detect the detectable elements and generate an output signal in response to movement of the detectable portion about the rotational axis of the encoder, the processor carrying out the steps of:
(a) rotating the encoder to a selected first speed then allowing the encoder to freely decelerate to a selected second speed over a spindle-free response period;
(b) during a spindle-free response period wherein the encoder is rotated to a selected first speed then allowed to freely decelerate to a selected second speed, receiving output signals from the detector, the signals including a plurality of data sets wherein each data set comprises the periods between sampling events occurring in one revolution of the encoder at an angular speed, each sampling event representing a transition between successive detectable elements of the encoder and each sampling event being spaced an angular distance apart from the preceding sampling event;
(c) selecting a data set pair from the output signals, wherein the data set pair contains two linearly uncorrelated data sets;
(d) defining a model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients and the periods between each sampling event as contained in the data set pair;
(e) solving the model for the angular distances for the data set pair using circular closure and dynamic reversal techniques; and
(f) establishing an encoder error map using the selected solved angular distances from the model.
69. A computer-readable medium as claimed in claim 68, wherein in the model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=Tk(\u03c90+ak+bk2)
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
a is a first order damping coefficient, and
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
is solved by circular closure and dynamic reversal techniques.
70. A computer-readable medium as claimed in claim 69, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
71. A computer-readable medium as claimed in claim 70, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
72. A computer-readable medium as claimed in claim 71, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
73. A computer-readable medium as claimed in to claim 72, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
74. A computer-readable medium as claimed in to claim 73, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
75. A computer-readable medium as claimed in to claim 74, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
76. A computer-readable medium as claimed in claim 75, wherein the vector pair equations are combined to cancel angular distance vector A and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
77. A computer-readable medium as claimed in claim 76, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2
78. A system as claimed in claim 68, wherein:
in step (c), a plurality of data set pairs are selected from the output signals;
in step (e), the model is solved for the angular distances for each of the plurality of data set pairs;
step (f) comprises the following steps:
(i) calculating the repeatability of the angular distances solved for each of the plurality of data set pairs;
(ii) selecting the angular distances solved for one of the plurality of data set pairs based on the repeatability of the angular distances as compared to the repeatability of the angular distances solved for other data set pairs;
(iii) establishing an encoder error map using the selected solved angular distances from the model.
79. A computer-readable medium according to claim 68, wherein step (f) comprises the following steps:
(i) defining an enhanced model of the angular distances between each sampling event occurring in one revolution of the encoder as a function of one or more damping coefficients, the solution of a previously solved model, and the periods between each sampling event as contained in each data set pair;
(ii) selecting the angular distances solved for a data set pair using the previously solved model;
(iii) repeatedly solving the enhanced model for the angular distances using circular closure and dynamic reversal techniques until the variation between successive solved angular distances is below a convergence threshold; and
(iv) establishing an encoder error map using the selected solved angular distances from the enhanced model.
80. A computer-readable medium as claimed in claim 79, wherein in the enhanced model, an angular distance \u0394k between sampling events k and k\u22121 as defined by:
\u0394k=TkRk\u03c90+aPk+bPk2
wherein
\u03c90 is the initial angular speed,
Tk is the period between sampling events k and k\u22121,
Rk is a correction term defined by Rk=(ecTk\u22121)cTk
c is defined by c=\u2212\u0101\u03940
\u0101 is an approximation of the first order damping coefficient previously solved using the model
\u03940 is the nominal angular distance between sampling events,
a is a first order damping coefficient,
b is a second order damping coefficient, and \u03c90 is the initial spindle speed,
P is defined by
P
k
=
\u2211
i
=
1
k
\ue89e
\u0394
_
i
\u0394
0
,
and
\u0394i is an approximation of angular distance \u0394k between sampling events k and k\u22121 from the previously solved model,
is solved by circular closure and dynamic reversal techniques.
81. A computer-readable medium as claimed in claim 80, wherein the circle closure technique is applied to constrain the sum of the angular distances between sampling events to one revolution of the encoder, resulting in vector equation:
m=\u0394+aU+bV
which expresses a measurement result m as the sum of angular distance vector \u0394 and an artifact aU+bV.
82. A computer-readable medium as claimed in claim 80, wherein artifact aU+bV is solved using the dynamic reversal technique, by applying the vector equation to each data set in the data set pair to form vector pair equations:
{
m
1
=
\u0394
+
a
1
\ue89e
U
1
+
b
1
\ue89e
V
1
m
2
=
\u0394
+
a
2
\ue89e
U
2
+
b
2
\ue89e
V
2
\ue89e
\u2003
wherein m1, U1, V1, and m2, U2, V2 are known vectors calculated from the two data sets of the data set pair.
83. A computer-readable medium as claimed in claim 82, wherein vectors U1, V1, U2, and V2 are linearly uncorrelated.
84. A computer-readable medium as claimed in to claim 83, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from ten percent of the total number of sampling events occurring in one revolution of the encoder to ninety percent of the total number of sampling events occurring in one revolution of the encoder.
85. A computer-readable medium as claimed in to claim 84, wherein the vectors U1, V1, U2, and V2 are linearly uncorrelated by setting the data sets in each data set pair to be out of phase with one another by a number of sampling events within a range from twenty-five percent of the total number of sampling events occurring in one revolution of the encoder to seventy-five percent of the total number of sampling events occurring in one revolution of the encoder.
86. A computer-readable medium as claimed in to claim 85, wherein the vectors U1, V1, U2, and V2 are uncorrelated by setting the data sets in each data set pair to be out of phase with one another by half of the total number of sampling events occurring in one revolution of the encoder.
87. A computer-readable medium as claimed in claim 86, wherein the vector pair equations are combined to cancel angular distance vector \u0394 and to yield a linear equation for unknown parameters a1, b1 and a2, b2 and then approximating first order damping coefficients a1 and a2 and second order damping coefficients b1 and b2 by applying a least-square fit to the linear equation.
88. A computer-readable medium as claimed in claim 87, wherein the approximated damping coefficients a1, b1, a2, b2 are applied to the vector pair equations to provide an equation for determining angular distance between sampling intervals as:
\u0394=m1\u2212 a1U1\u2212 b1V1 or \u0394=m2\u2212 a2U2\u2212 b2V2