Integrated non-contacting torque and absolute position sensor for steering applications

[0001] This application claims the benefit of the U.S. Provisional Patent Application Ser. No. 60/542,511, filed on Feb. 6, 2004, which is hereby incorporated by reference.

[0002] The present invention relates to an integrated torque and position sensor for detecting the relative rotational angular displacement between two rotatable shafts joined by a torsion bar and for detecting the angular position of a rotatable shaft. The present invention also provides a method of calibrating and calculating the angular position of shaft.

[0003] The need to measure both the torque and the angular position of a steering shaft is important to automotive applications utilizing an electric power steering system (EPS). A number of standalone sensors have been developed related to either determining torque or calculating the angular position of a steering wheel. Generally, it is necessary to calculate the torque in order to determine the amount of electrical assist to apply when a driver turns a steering wheel. In determining torque, the driver typically turns the steering wheel which is connected to an input shaft. The input shaft is coupled to an output shaft which is connected to a steering mechanism. The input and output shafts are generally coupled together with a torsion bar and the torsion bar allows for relative rotation between the shafts. The input shaft may rotate with respect to the output shaft by a predetermined number of degrees, e.g. +/−12 degrees. An example of a torque sensor is disclosed in U.S. Patent Application Publication No. 2004/0250631 filed on Jul. 13, 2004 which is assigned to the assignee of the present invention and is hereby incorporated as a reference. An example of a position sensor is disclosed in U.S. Pat. No. 6,720,763 which is assigned to the assignee of the present invention and is hereby incorporated as a reference.

[0004] Position sensors are used for determining the angular position of the shaft as a user turns a steering wheel. Examples of position sensors are disclosed in U.S. Pat. No. 5,930,905 (the '905 patent) to Zabler et al., and in U.S. Pat. No. 6,630,823 (the '823 patent) to Tateishi et al. The '905 patent discloses a gear having a plurality of teeth coupled to a shaft. The teeth engage a plurality of additional teeth disposed on a second gear and a third gear. A pair of absolute sensors are positioned in proximity to the second gear and the third gear for generating an output that corresponds to the angular positions of the gears. As long as the number of teeth on each of the gears are known, it is possible to calculate the angular position of the shaft based on the outputs of the absolute sensors.

[0005] The '823 patent discloses a gear having a plurality of teeth coupled to a shaft. As the shaft rotates, the teeth on the gear engages with additional gear teeth disposed on a second gear. A bevel gear disposed on a different plane from the second gear engages a change gear. A first sensing element is positioned in proximity with the change gear for outputting a first detection signal which repeats continuously. A screw is positioned through the change gear and rotates with the change gear as the change gear rotates. A driven body is coupled with the screw and is axially displaced in response to rotating the change gear. A second sensing element is positioned in proximity to the driven body and outputs a second detection signal which gradually increases or decreases. A detection circuit determines the angular position of the shaft based on the first and the second detection circuits.

[0006] Although these prior art position sensors are useful, an opportunity exists for a position sensor that minimizes the number of gear teeth disposed on a gear while providing the angle position of the shaft with accuracy. By reducing the number of gear teeth on a gear, the sensor is not susceptible to losing accuracy as a result of the teeth wearing down over time. Additionally, an opportunity exists for combining the torque and position sensing capabilities into an integrated sensor to occupy less package space in a vehicle. Finally, by combining the functionality of a sensor which calculates both the torque and angular position of a shaft, common parts may be utilized within the sensor for providing additional cost savings.

[0007] It is the object of the present invention to provide an integrated torque and position sensor for measuring the relative rotation between an input and an output shaft and for measuring the angular position of the output shaft. The integrated torque and position sensor includes a support housing that supports the output shaft about an axis and an input shaft that is axially aligned with the output shaft for rotation about an axis. A torsion bar interconnects the input shaft and the output shaft for allowing relative rotation between the shafts in response to a torque being applied to the input shaft. A wheel is coupled to the output shaft for rotation therewith. A torque sensing mechanism is disposed about the shafts for measuring the relative rotation between the input and the output shaft. An incremental sensing mechanism generates an incremental output which is indicative of the angular position of the output shaft and a segment sensing mechanism provides a segment output indicative of the revolution in which the output shaft is disposed. A sensor casing is supported by the support housing and the sensor casing supports portions of the torque, incremental, and segment sensing mechanisms.

[0008] It is a further object of this invention to provide a sensor assembly for measuring the angular position of a shaft. A wheel is coupled with the shaft for incremental angular rotation and through a plurality of revolutions. An incremental sensing mechanism detects the rotation of the wheel and provides an incremental output which is indicative of the incremental angular rotation of the wheel. A change gear is supported by the sensor casing for rotation through predetermined angular segments. A segment sensing mechanism responds to the change gear for providing a segment output indicative of the angular segment in which the wheel is disposed. An actuation mechanism interconnects the wheel and the change gear for rotating the change gear through each of the predetermined angular segments for each occurrence of rotation of the wheel completely through a predetermined number of increments of angular rotation and to prevent rotation of the change gear during rotation of the wheel through each of the predetermined number of increments of rotation.

[0009] The subject invention also provides a method of calibrating a position sensor of a type that measures the incremental angular rotation of a shaft and the segment in which the shaft is disposed to produce the final angular position of shaft. The method includes the steps of rotating a shaft, generating a first incremental output having a first incremental amplitude and an incremental phase angle which is indicative of the incremental angular rotation of the shaft, generating a second incremental output having a second incremental amplitude and the incremental phase angle which is indicative of the incremental angular rotation the shaft, generating a first segment output having a first segment amplitude and a segment phase angle which is indicative of the angular segment in which the shaft is disposed, and generating a second segment output having a second segment amplitude and the segment phase angle phase which is indicative of the angular segment in which the shaft is disposed.

[0010] The method further includes performing a Fourier analysis on the incremental and segment outputs from the sensor to produce incremental and segment dc components, incremental and segment fundamental outputs and incremental and segment harmonic outputs. A compensation equation is used to provide a common amplitude between the first and the second incremental outputs. A first incremental final output corresponds to the compensated first incremental output and a second incremental final output corresponds to the compensated second incremental output wherein the compensated factors also correct the phase shift errors between the first and the second incremental outputs. Likewise, a compensation equation is used to provide a common amplitude between the first and the second segment outputs. A first segment final output corresponds to the compensated first segment output and a second segment final output corresponds to the compensated second segment output wherein the compensated factors also correct the phase shift errors between the first and the second incremental outputs.

[0011] Accordingly, the integrated torque and position sensor sets forth a reduction in the amount of teeth necessary to calculate the angle position which over time ensures greater accuracy in calculating the angle position of a shaft. At the same time, the integrated torque and position sensor utilizes common components for performing both functions of calculating the angle position and the torque of the shaft.

[0047] Referring to the FIGS. 1–2, wherein like numerals indicate like parts throughout all of the figures, an integrated torque and position sensor (integrated sensor hereafter) is generally shown at 10 and provides the torque and the angular position of a shaft 12. A support housing 14 supports the integrated sensor 10 and the shaft 12. The shaft 12 includes an input shaft 16, an output shaft 18 and a torsion bar 20 positioned therebetween for allowing relative rotation between the input shaft 16 and the output shaft 18. The input shaft 16 is coupled to a steering wheel (not shown). The output shaft 18 is coupled to a steering system (not shown), for example, a rack and pinion gear mechanism.

[0048] Referring to FIGS. 1–2, the integrated sensor 10 utilizes a wheel 24 and a torque sensing mechanism 39 for measuring the relative rotation between the input shaft 16 and the output shaft 18. The wheel 24 includes a first side 26, a second side 28 and a middle portion 30 disposed therebetween. A coupling ring 31 is coupled to the wheel 24 for securing the wheel 24 to the output shaft 18.

[0049] The torque sensing mechanism further includes a torque rotor assembly 37 supported by said input shaft 16. The torque rotor assembly 37 includes a back iron 32 which is made of a soft magnetic material, such as a nickel iron alloy. The back iron 32 may be constructed by a stamping process or produced from a powdered metal using a sintering process or through a machining process.

[0050] Referring to FIGS. 2–4, the back iron 32 includes an inner surface 34 and an outer surface 36 and is centered on an axis (A). The input shaft 16 includes an enlarged portion 22 and the inner surface 34 of the back iron 32 is press fit over the enlarged portion 22 for non-rotatably coupling the back iron 32 to the input shaft 16. The output shaft 18 is coupled to the second side 28 of the wheel 24.

[0051] Referring to FIGS. 2 and 4–5, the back iron 32 defines a plurality of pockets 38 disposed on the outer surface 36. The torque rotor assembly 37 includes a plurality of magnets 40 disposed in the pockets 38. The magnets 40 may be permanently affixed within the pockets 38 by either crimping, injection molding or by adding an adhesive. The back iron 32 includes a plurality of support structures 42 integrally formed between the pockets 38 on the outer surface 36. The torque rotor assembly 37 includes a retaining ring 44 that is placed over the back iron 32 and is radially spaced from the pockets 38 for applying an inward force against the magnets 40. The retaining ring 44 may not be necessary if the magnets 40 are affixed to the pockets 38 by adhesive or other methods.

[0052] Referring to FIG. 5, an arrangement of six magnets 40 are disposed circumferentially in six pockets 38 on the outer surface 36 of the back iron 32. The north and south pole arrangements for the magnets 40 may take on any number of different embodiments. In one embodiment, the magnets 40 are magnetized with all north poles facing outwardly and south poles facing inwardly.

[0053] Referring to FIGS. 2–3 and 6, the wheel 24 includes a first torque stator 46 and a second torque stator 56. The first torque stator 46 and the second torque stator 56 are centered on the axis (A) and are axially spaced and parallel to each other. The first torque stator 46 includes a first upper surface 48 and a first lower surface 50. Likewise, the second torque stator 56 includes a second upper surface 58 and a second lower surface 60. The first torque stator 46 and the second torque stator 56 are disposed within the integrated sensor 10 whereby the first lower surface 50 of the first torque stator 46 faces the second upper surface 58 of the second torque stator 56. The first torque stator 46 and the second torque stator 56 may be manufactured using a stamping process or may be made from a powdered metal using a sintering process or may be made using a machining process.

[0054] Referring to FIGS. 2 and 7–8, the first torque stator 46 includes a circular base 52 and a plurality of teeth 54 extending inwardly from the circular base 52 in a radial direction. Additionally, the second torque stator 56 includes a circular base 62 and a plurality of teeth 64 extending inwardly from the circular base 62 in a radial direction. As shown in FIG. 8, the teeth 54, 64 of the first torque stator 46 and the second torque stator 56 are angularly offset or out of phase with each other. Alternative embodiments may include teeth 54, 64 from the first torque stator 46 and the second torque stator 56 as being angularly aligned or in phase with each other.

[0055] Referring to FIGS. 1 and 3, the wheel 24 is made of a non-magnetic material such as plastic and is over molded over the first torque stator 46 and the second torque stator 56. As stated above, the first torque stator 46 and the second torque stator 56 are disposed within the wheel 24 and the wheel 24 fixes the out of phase relationship between the extending teeth 54, 64 of the first torque stator 46 and the second torque stator 56 and the axial displacement between the first torque stator 46 and the second torque stator 56 after the over molding process is performed. The axial displacement between the first torque stator 46 and the second torque stator 56 defines a gap (G) for establishing a measurement zone through which the magnetic flux from the plurality of magnets 40 passes.

[0056] Referring to FIGS. 2, 13 and 14, a portion of the torque sensing mechanism 39 includes at least one torque magneto-sensitive element 66 for measuring the magnetic flux generated by the magnets 40. As discussed above, the teeth 54, 64 of the first torque stator 46 and the second torque stator 56 are offset or out of phase with each other. A magnetic flux is measured by the torque magneto-sensitive elements 66 and varies depending on the alignment of the magnets 40 to the wheel 24 and the teeth 54, 64 of the first torque stator 46 and the second torque stator 56.

[0057] The magnetic flux generated by each magnet 40 flows through the stators 46, 56, the support structures 42 of the back iron and through the gap (G). The magnetic circuit that is formed by the magnets 40 has mainly two regions, an upper magnetic zone that is formed between the magnets 40 and the first torque stator 46, and a lower magnetic zone is formed between the magnets 40 and the second torque stator 56. The differential magnetic flux between these two zones flows through the gap (G) and is sensed by the torque magneto-sensitive elements 66. The differential magnetic flux is indicative of the relative rotation between the input shaft 16 and the output shaft 18. At a no load torque condition, both of the zones produce the same amount of flux resulting in a zero differential flux.

[0058] As shown in FIG. 9, a plot of the flux density of the magnets 40 versus the relative rotational displacement of the shafts 16, 18 is generally shown at 45. The relative rotation between the input shaft 16 and the output shaft 18 may vary between an angle of +/−θ with an associated +/−G Gauss. In one embodiment, the relative rotation between the input shaft 16 and the output shaft 18 may vary between an angle of +/−8 degrees. Depending on relative rotational displacement between the input shaft 16 and the output shaft 18, the differential flux either flows up or down in the gap (G).

[0059] Referring to FIGS. 1 and 10–14, the integrated sensor 10 includes a circuit board 70 and a sensor casing 78. The sensor casing 78 supports the circuit board 70. The circuit board 70 includes a torque leg 72, a position leg 74 and a flexible portion 76 positioned therebetween. The position leg 74 will be discussed in more detail below. The torque magneto-sensitive elements 66 are disposed on the torque leg 72 and positioned within the gap (G) for sensing a change in the magnetic flux. As shown in FIG. 1, the sensor casing 78 is supported by the sensor housing 14 which is mounted to a stationary member (not shown) and serves to retain the sensor casing 78 in proximal distance with the wheel 24 and the input shaft 16 and the output shaft 18. As shown in FIGS. 1–2, a support board 80 is positioned underneath the torque leg 72 for preventing the torque leg 72 from flexing. A wire harness (not shown) delivers signals to and from a microcontroller to the integrated sensor 10.

[0060] The angle position sensing capability of the integrated sensor 10 measures the angle position of the output shaft 18, or shaft 18. The integrated sensor 10 includes an incremental sensing mechanism 68 that detects the angular position of the shaft 18 and is indicative of the incremental angular rotation or the angular position of the shaft 18 between any angle of 0 and 360 degrees.

[0061] Referring to FIG. 2, the incremental sensing mechanism 68 includes a ring magnet 82. The ring magnet 82 further includes a first side 84 and a second side 86. The first side 84 of the ring magnet 82 is coupled to the second side 28 of the wheel 24 and is axially spaced from the first torque stator 46 and the second torque stator 56. Accordingly, the ring magnet 82 rotates with the wheel 24 for generating a magnetic flux which corresponds to the incremental phase angle of the shaft 18.

[0062] Referring to FIG. 15, the ring magnet 82 is diametrically magnetized wherein one half of the ring magnet 82 includes north poles and the other half of the ring magnet 82 includes south poles. The flux lines flow radially and axially from the north to the south poles.

[0063] Referring to FIGS. 2, 12 and 14, a portion of the incremental sensing mechanism 68 includes at least one incremental magneto-sensitive element 94 for detecting the magnetic flux of the ring magnet 82. As stated above, the circuit board 70 includes the torque leg 72 and the position leg 74 wherein the torque leg 72 and the position leg 74 are axially spaced from each other. The position leg 74 includes a pair of circumferentially extending arms 96 positioned adjacent the wheel 24. The sensor casing 78 further includes a pair of casing arms 79 that support the extending arms 96 of the circuit board 70. The incremental magneto-sensitive elements 94 are circumferentially positioned from each other and disposed on the extending arms 96.

[0064] Referring to FIGS. 10–13, the sensor casing 78 defines a gear cavity 98. The integrated sensor 10 further includes a segment sensing mechanism 97. The segment sensing mechanism 97 includes a change gear 100 having change teeth 101 rotatably received and supported by the gear cavity 98.

[0065] Referring to FIGS. 11–14, the segment sensing mechanism 97 further includes a segment magnet 106. In the illustrated embodiment, the segment magnet is in the shape of a disc, however the segment magnet 106 may be in the form of a ring, rhombus or square. A portion of the segment sensing mechanism 97 includes at least one segment magneto-sensitive element 108. The segment magnet 106 is coupled to the change gear 100 wherein both the change gear 100 and segment magnet 106 are rotatably received and supported by the gear cavity 98. The segment magnet 106 may be coupled to the change gear 100 by injection molding. The segment magneto-sensitive elements 108 are disposed adjacent the segment magnet 106 on the position leg 74 of the circuit board 70. As shown in FIG. 16, the segment magnet 106 is diametrically magnetized wherein one half of the segment magnet 106 includes north poles and the other half of the segment magnet 106 includes south poles. The flux lines flow radially and axially from the north to the south poles. As the change gear 100 rotates, the segment magnet 106 rotates, resulting in a change in the magnetic flux which is detected by the segment magneto-sensitive elements 108. As stated above, the position of the change gear 100 is indicative of the revolution in which the shaft 18 is disposed.

[0066] Referring to FIGS. 2, 13 and 17, the change teeth 101 includes a first plurality of change teeth 102 disposed on a first plane (H1) and a second plurality of change teeth 104 disposed on a second plane (H2). The first plurality of change teeth 102 and the second plurality of change teeth 104 are parallel and axially spaced from each other. Additionally, the first plurality of change teeth 102 are out of phase or not angularly aligned with the second plurality of change teeth 104. As shown in FIG. 17, the first plurality of change teeth 102 and the second plurality of change teeth 104 each include five teeth. The change gear 100 may be configured to include any number of change teeth and may be disposed on any number of planes. An engagement surface 110 is positioned between each of the five change teeth for the first plurality of change teeth 102 and for the second plurality of change teeth 104.

[0067] Referring to FIGS. 18 and 20, drive teeth 111 are disposed on the middle portion 30 of the wheel 24. The drive teeth 111 includes a first pair of drive teeth 112. The first pair of drive teeth includes a first drive tooth 114 positioned on the first plane (H1) on the middle portion 30 of the wheel 24 and a second drive tooth 116 positioned on the second plane (H2) on the middle portion 30 of the wheel 24.

[0068] Referring to FIG. 19, a second pair of drive teeth 118 is disposed on the wheel 24 one hundred and eighty degrees from the first pair of drive teeth 112. Likewise, the second pair of drive teeth 118 includes a first drive tooth 120 disposed on the first plane and a second drive tooth 122 disposed on the second plane. The wheel 24 may be configured to include more drive teeth and the pairs of drive teeth 112, 118 may be positioned at any angle from each other.

[0069] Referring to FIGS. 19–20, a plurality of blocking cams 124 are positioned on the middle portion 30 of the wheel 24. More specifically, a first blocking cam 126 is positioned on the first plane of the middle portion 30 between the first drive teeth 114, 120 for the first pair of drive teeth 112 and for the second pair of drive teeth 118. The first blocking cam 126 extends through one hundred and eighty degrees on one diametrical side of the wheel 24. Additionally, a second blocking cam 128 is positioned on the second plane of the middle portion 30 between the second drive teeth 116, 122 for each of the first pair of drive teeth 112 and the second pair of drive teeth 118. The second blocking cam 128 extends through the other one hundred and eighty degrees on the other diametrical side of the wheel 24. The first pair of drive teeth 112 and the second pair of drive teeth 118 define a plurality of ends of the blocking cams 124. The integrated sensor 10 further includes an actuation mechanism 129. The actuation mechanism 129 includes the first pair of drive teeth 112 and the second pair of drive teeth 118, the blocking cams 124, and the first plurality of change teeth 102 and the second plurality of change teeth 104.

[0070] Referring to FIGS. 21 and 23–24, the actuation mechanism 129 interconnects the wheel 24 and the change gear 100 when the wheel 24 rotates completely through a predetermined number of increments. Meaning, that for every one hundred and eighty degrees of rotation of the wheel 24, either the first pair of drive teeth 112 engages with either one change tooth 103 in the first plurality of change teeth 102 or one change tooth 105 in the second plurality of change teeth 104, or the second pair of drive teeth 118 engages with either one change tooth 103 in the first plurality of change teeth 102 or one change tooth 105 in the second plurality of change teeth 104 for rotating the change gear 100 through a predetermined angle of thirty six degrees. One skilled in the art will recognize that a change gear 100 may be provided with any number of change teeth. For example, ten change teeth may be used for the first plurality of change teeth 102 and the second plurality of change teeth 104 resulting in a total of twenty teeth and rotating the change gear 100 eighteen degrees upon each occurrence of the drive teeth 112 engaging one of the change teeth in either the first plurality of change teeth 102 or the second plurality of change teeth 104.

[0071] As shown in FIGS. 19–20, the first blocking cam 126 and the second blocking cam 128 are axially spaced and parallel to each other. A first channel 130 is disposed below the first blocking cam 126, and a second channel 132 is disposed above the second blocking cam 128.

[0072] Referring to FIGS. 21–22, (assuming that this is the initial condition in which the shaft 18 is disposed in terms of the angular position of the shaft 18), a first change tooth 105 from the second plurality of change teeth 104 is positioned in the first channel 130 below the first blocking cam 126. The engagement surface 110 disposed between a first change tooth 103 and a second change tooth 107 on the first plurality of change teeth 102 engages the first blocking cam 126 for preventing rotation of the wheel 24. Referring to FIGS. 19, 21–22, after rotating the shaft 18 counterclockwise through an angle in excess of one hundred and eighty degrees and below an angle of three hundred and sixty degrees, the change gear 100 rotates thirty six degrees when the second drive tooth 122 of the second pair of drive teeth 118 engages with the change tooth 105 disposed in the second plurality of change teeth 104.

[0073] Referring to FIGS. 23–24, after the change gear 100 rotates thirty six degrees, the second change tooth 107 in the first plurality of change teeth 102 is rotated into the second channel 132 above the second blocking cam 128 and the engagement surface 110 between the first change tooth 105 and the second change tooth 109 disposed in the second plurality of change teeth 104 engages with the second blocking 128 cam for preventing rotation of the change gear 100 until the wheel 24 rotates through another one hundred and eighty degrees of rotation.

[0074] Referring to FIGS. 20–21, in response to rotating the wheel 24 to an angle less than one hundred and eighty degrees in the clockwise direction, the first drive tooth 116 of the first pair of drive teeth 112 would engage the first change tooth 105 of the second plurality of change teeth 104 and result in rotating the change gear 100 an additional thirty six degrees.

[0075] Whenever the shaft 18 is rotated to any angle which is not equal to a multiple of one hundred and eighty degrees the either the first blocking cam 126 engages with the engagement surface 110 disposed on the first plurality of change teeth 102 or the second blocking cam 128 engages with the engagement surface 110 disposed on the second plurality of change teeth 104.

[0076] Referring to FIGS. 13–14, and as stated above, the incremental magneto-sensitive elements 94 are disposed on the extending arms 96 of the position leg 74 and are spaced both axially and radially from the ring magnet 82. The incremental magneto-sensitive elements 94 includes a first incremental magneto-sensitive element 134 and a second incremental magneto-sensitive element 136. The incremental magneto-sensitive elements 134, 136 output an incremental output that correlates to an angle of between 0 and 360 degrees. The incremental output includes a first incremental output, P₁, and a second incremental output, P₂. The first incremental output, P₁, is generated from the first incremental magneto-sensitive element 134, and the second incremental output, P₂, is generated from the second incremental magneto-sensitive element 136.

[0077] Referring to FIG. 25, the incremental and segment outputs are generally shown at 133, the first incremental output, P₁, is defined by a sinusoidal voltage output and the second incremental output, P₂, is defined by a cosinusoidal voltage output. The incremental outputs P₁ and P₂ are defined by following equations:
P₁=A₁ sin θ₁  (a)
P₂=B₂ cos(θ₁+δ_1,2)  (b)
The variables A₁ and B₂ are defined as a first incremental amplitude and a second incremental amplitude, respectively. The angle θ₁ is the incremental phase angle of the shaft and δ_1,2 is defined as an incremental quadrature error angle. Ideally, P₁ and P₂ are shifted ninety degrees from each other and the incremental quadrature error angle is equal to zero.

[0078] As stated above, the segment magneto-sensitive elements 108 are disposed on the torque leg 72 of the circuit board 70. A first segment magneto-sensitive element 138 and a second segment magneto-sensitive element 140 output a segment output which correlates to an angle of between 0 and 360 degrees. The segment output will be used to provide the segment or revolution in which the shaft 18 or the wheel 24 is disposed. The segment output includes a first segment output, P₃, which is an output produced from the first segment magneto-sensitive element 138 and a second segment output, P₄, which is an output produced from the second segment magneto-sensitive element 140.

[0079] Referring to FIG. 25, the first segment output, P₃, is defined by a stepped sinusoidal voltage output and the second segment output, P₄, is defined by a stepped cosinusoidal voltage output. The segment outputs, P₃ and P₄, are defined by the following equations:
P₃=A₃ sin θ₂  (c)
P₄=B₄ cos(θ₂+δ_3,4)  (d)
The variables A₃ and B₄ are defined as a first segment amplitude and a second segment amplitude, respectively. The angle θ₂ is the segment phase angle is the angle position of the shaft and δ_3,4 is defined as a segment quadrature error angle. Preferably, P₃ and P₄ are shifted ninety degrees from each other and the segment quadrature error angle is equal to zero.

[0080] As shown in FIG. 26, a solver equation uses the outputs P₁ and P₂ to calculate and produce a final incremental angle position of 0 to 360 degrees for the shaft 18. Likewise, the solver equation uses outputs P₃ and P₄ to calculate and produce a final segment angle position which comprises actual segment angles between 0 and 360 degrees. This interaction between the final incremental angle position and the final segment angle position will be discussed in more detail after the derivation for the solver equation is shown. The solver equation is defined as:

[0081] $\begin{array}{cc}{\phi }_{\mathrm{old}}={\phi }_{\mathrm{old}}-\left(\frac{f\left(\theta ,\phi \right)}{\frac{df\left(\theta ,\phi \right)}{d\phi }}\right)& \left(e\right)\end{array}$

[0082] To show the derivation of the solver equation, the amplitudes A₁, A₃ will be defined by A_m and the amplitudes B₂, B₄ will be defined by B_m. The angles θ₁ and θ₂ are defined by the angle θ_m. Finally, the quadrature error angles δ_1,2 and δ_3,4 are defined as δ.

[0083] The equations (a), (b), (c), and (d) may be rewritten as:
P₁(or P₃)=A_m sin θ_m  (f)
P₂(or P₄)=B_m cos(θ_m+δ)  (g)
It is important to note that P₁ and P₃ are not equal to each other. Likewise, P₂ and P₄ are not equal to each other. The purpose of writing equations (f) and (g) is to show that the same derivation for establishing the application of the solver equation using the outputs P₁ and P₂ also applies to the outputs P₃ and P₄.

[0084] Due to tolerances created in the manufacture of various components used in the integrated sensor, it is necessary to generate an equation where A_m=B_m. Additionally, it is necessary to ensure that a phase shift of ninety degrees exists between the sinusoidal and cosinusodial outputs.

[0085] A gain factor of

[0086] $\frac{A_m}{B_m}$
is multiplied to equation (g) for generating the following equation:

[0087] $\begin{array}{cc}\begin{array}{c}\frac{A_m}{B_m}{P}_2\left(\mathrm{or}{P}_4\right)=\frac{A_m}{B_m}\left(\cos {\theta }_m\cos \delta -\sin {\theta }_m\sin \delta \right)B_m\\ =A_m\cos {\theta }_m\cos \delta -A\sin {\theta }_m\sin \delta \\ \frac{A_m}{B_m}{P}_2\left(\mathrm{or}{P}_4\right)=A_m\cos {\theta }_m\cos \delta -P_1\left(\mathrm{or}{P}_3\right)\sin \delta \end{array}& \left(h\right)\end{array}$
Equation (h) may be re-written using equations (f) and (g):

[0088] $\begin{array}{cc}{A}_m\cos {\theta }_m\cos \delta =\frac{A_m}{B_m}{P}_2\left(\mathrm{or}{P}_4\right)+P_1\left(\mathrm{or}{P}_3\right)\sin \delta & \left(i\right)\end{array}$
In solving for A_m cos θ_m in equation (i), equation (j) is produced resulting in an equation that represents the different amplitude B_m and the error angle δ in the form of a common amplitude A_m.

[0089] $\begin{array}{cc}{A}_m\cos {\theta }_m=\frac{\left(\frac{A_m}{B_m}{P}_2\left(\mathrm{or}{P}_4\right)+P_1\left(\mathrm{or}{P}_3\right)\sin \delta \right)}{\cos \delta }& \left(j\right)\end{array}$

[0090] By arbitrarily setting P′₂(or P′₄) to A_m cos θ_m, it is possible to define the numerator of the solver equation as follows:
f(θ_m, φ)=P₁(or P₃)cos φ−P′₂(or P′₄)sin φ=A_m sin(θ_m−φ)=0  (k)

• • [0091] equation (k) is rewritten as;
    f(θ_m, φ)=A_m sin θ_m cos φ−A_m cos θ_m sin φ=A_m sin(θ_m−φ)=0  (l)
    The angle φ is selected to set equation (l) to zero, accordingly by setting angle φ=θ_m, this satisfies the condition that f (θ_m, φ)=0.

[0092] The denominator for the solver equation is defined as the following:

[0093] $\begin{array}{cc}\frac{df}{d\phi }\left({\theta }_m,\phi \right)=-P_1\left(\mathrm{or}{P}_3\right)\sin \phi -P_2^{\prime }\left(\mathrm{or}{P}_4^{\prime }\right)\cos \phi & \left(m\right)\end{array}$
The solver equation is shown as the following after substituting equations (l) and (m) into equation (e):

[0094] $\begin{array}{cc}{\phi }_{\mathrm{new}}={\phi }_{\mathrm{old}}-\left(\frac{P_1\left(\mathrm{or}{P}_3\right)\cos \phi -P_2^{\prime }\left(\mathrm{or}{P}_4^{\prime }\right)\sin \phi }{-P_1\left(\mathrm{or}{P}_3\right)\sin \phi -P_2^{\prime }\left(\mathrm{or}{P}_4^{\prime }\right)\cos \phi }\right)& \left(n\right)\end{array}$
Accordingly, φ_old is an initial angle value which is arbitrarily set to any angle between 0 and 360 degrees, after φ_old is inserted into the solver equation (n), an angle φ_new is produced, if the angle φ_new is not a pre-determined angle, angle φ_new is set to φ_old, and equation (n) is solved again generating a new angle φ_new. Depending on which of the outputs P₁, P₂, P₃, P₄ were inserted into the solver equation (n), the converged angle φ_new is assigned to either a final incremental angle position or a final segment angle. If the outputs P₁, P₂ were inserted into the solver equation (n), the converged angle φ_new is set to the final incremental angle and the output is any angle between 0 and 360 degrees. If P₃, P₄ was inserted in the solver equation (n), then the converged angle φ_new is set to the actual segment angle.

[0095] A total of five actual segment angles correspond to the five change teeth of the first plurality of change teeth 102. Likewise, five actual segment angles correspond to the five change teeth of the second plurality of change teeth 104. As either the first pair of drive teeth 112 or the second pair of drive teeth 118 engages with any change tooth 103 of the first plurality of change teeth 102 or any change tooth 105 of the second plurality of change teeth 104, the change gear 100 and the segment magnet 106 is rotated a total of thirty six degrees in either direction. In response to the segment magnet 106 being rotated by thirty six degrees, new values for P₃ and P₄ are outputted from the segment magneto-sensitive elements 138, 140 and inserted into the solver equation (n) for producing the actual segment angle. As the user rotates the shaft 18 over the full range of shaft travel of +/−900 degrees, the wheel 24 engages the change teeth 102,104 every one hundred and eighty degrees or ten times (total angular rotation of shaft/# of change teeth) resulting in a total of ten final segment angles. The selection of the number of change teeth is calculated by taking the total angular travel of the shaft 18 divided by the angle between the first pair of drive teeth 112 and the second pair of drive teeth 118. Accordingly, the selection of ten teeth dictates the number of steps represented in the sinusoidal and cosinusodial outputs of P₃ and P₄ respectively. Every two turns of the segment magnet 106 is indicative of a new revolution in which the shaft 18 is disposed. As stated earlier, one skilled in the art will recognize that a change gear 100 with five change teeth would result in the drive teeth engaging the change teeth every three hundred and sixty degrees. As a result, a total of five final segment angles would be utilized.

[0096] Upon rotating the shaft 18, the ring magnet 82 rotates in the same direction generating a new set of output values P₁ and P₂. The incremental outputs P₁ and P₂ are then applied to the solver equation (n) for producing the final incremental angle.

[0097] An algorithm uses the actual segment angle to produce the final segment angle (FinalSegAng) and a final angle position equation determines a final angle position (FinalPos). The algorithm and the final angle position equation are defined by equations (o) and (p) respectively:
Actual Segment Angle−Δθ≦FinalSegAng≦Actual Segment Angle+Δθ  (o)
FinalPos=FinalIncAng+n·360 degrees  (p)

• • [0098] for n=+(0, 1, 2, 3)
    The FinalSegAng includes ten angles and each angle corresponds to an integer n. The FinalSeqAng and corresponding integer n are shown in Table 1:

[0099]

	TABLE 1
	FinalSegAng	Integer(n)
	0	0
	36	0
	72	1
	108	1
	144	2
	180	−3
	216	−2
	252	−2
	288	−1
	324	−1

[0100] It should be noted that anyone skilled in the art may derive different final segment angles (FinalSeqAng) other than those defined in Table 1 and may also assign different integers to the FinalSegAng. The actual segment angle which is produced from the solver equation (n) will ultimately converge on any of the final segment angles with the tolerance Δθ. After the final segment angle has been determined by matching the actual segment angle, the integer n is determined, and n and the FinalIncAngle produced by the solver equation (n) using outputs P₁, and P₂ are inserted into the final angle position equation (p) to produce the final position angle (FinalPos). Ideally, each final segment angle is configured to represent each change tooth on the change gear 106. As the number of change teeth utilized on the change gear either increase or decrease, the number of final segment angles used will decrease or increase. Accordingly, the final position angle (FinalPos) is an angle between +/−900 degrees and is indicative of the angle in which the shaft 18 is disposed. These values depend on the initial position of the shaft 18 and the change gear 100. The calculation of the final position angle of between +/−900 may extend to a larger angle and may also be smaller than +/−900. The limit of the final angle position will depend on the number of change teeth utilized on the change gear 106.

[0101] Each of the calculated final segment angles generated by the solver equation (n) has a tolerance of +/−18 degrees, which is equivalent to the angle rotation of the change gear at thirty six degrees. The calculation of the final incremental angle, the actual segment angle, the final segment angle and the final angle position will be performed on a device not included with the integrated sensor 10. The device may include a controller electrically coupled to the integrated sensor 10 which receives the outputs P₁, P₂, P₃, P₄. Referring to FIG. 25, the raw data, as generated from the incremental magneto-sensitive elements 94 and the segment magneto-sensitive elements 108 P₁, P₂, P₃, P₄ is shown.

[0102] Referring to FIG. 26, each of the outputs P₁, P₂, P₃, P₄ is inserted into the solver equation and the resulting final incremental angle 142 and the final segment angle 144 are shown over the full angular rotation of the shaft 18. The final position angle 146 is generated after executing the algorithm and after solving the final position angle equation and is defined as any angle between +/−900 degrees.

Method of Calibrating and Compensating the Outputs of the Integrated Sensor

[0103] Due to tolerance issues or design variations of the various components utilized in the integrated sensor 10, different amplitudes may exist between A₁ and B₂ for the incremental outputs P₁ and P₂; and A₃ and B₄ for the segment outputs P₃ and P₄. Further, a ninety degrees phase shift between the angles may not exist between the incremental outputs P₁ and P₂ and the segment outputs P₃ and P₄. Other errors may lead to unbalanced amplitudes, positive and negative cycles may not be spanned over 180 degrees, and higher order harmonics may be present in the outputs P₁, P₂, P₃ and P₄. The sources of these errors or non-idealities may include, magnet anisotropies, cocking of either the ring magnet 82 or segment magnet 106 after assembly, variations between the incremental and the segment magneto-sensitive element's sensitivity (mV/G), and a voltage offset (V_OQ) variation. The voltage offset (V_OQ) is a produced by the magneto-sensitive elements.

[0104] Referring to FIG. 27, the incremental uncompensated waveforms 148 correspond to the outputs of the measured first incremental output P₁ and the measured second incremental output P₂. It illustrates how non-idealities may affect the resulting output waveforms 148. The amplitudes of the incremental outputs P₁ and P₂ each have different peak values, P₁ and P₂ are not phase shifted at ninety degrees and the waveforms 148 are not symmetrical.

[0105] Referring to FIG. 33, the segment uncompensated waveforms 150 correspond to the outputs of the measured first segment output P₃ and the measured second segment output P₄. It illustrates how non-idealities may affect the resulting output waveforms 150. The amplitudes of the segment outputs P₃ and P₄ each have different peak values and P₃ and P₄ may not be phase shifted to ninety degrees.

[0106] Referring to FIG. 29, a method for calibrating the incremental outputs P₁ and P₂ is generally shown at 200. In step 202, the calibration process requires that the shaft 18 is rotated. In step 204, the first incremental magneto-sensitive element 134 outputs P₁ in response to detecting the changing magnetic flux of the ring magnet 82. In step 206, the second incremental magneto-sensitive element 136 outputs P₂ in response to detecting the changing magnetic flux of the ring magnet 82. In step 208, a zero crossing is created for the P₁ output in response to rotating the shaft 18.

[0107] In step 210, the calibration method samples and collects the incremental outputs P₁ and P₂ from zero to three hundred and sixty degrees. In step 212, the voltage offset (V_OQ) is subtracted from the collected incremental outputs P₁ and P₂ as produced by the incremental magneto-sensitive elements 134, 136. In step 214, a Fourier analysis is performed on P₁ to produce a first incremental dc component DC_P₁, a plurality of first incremental harmonic outputs, and a first incremental fundamental output. It should be noted that the first incremental dc component DC_P₁ corresponds to a harmonic order of zero after performing the Fourier analysis. The plurality of first incremental harmonic outputs include a plurality first incremental harmonic amplitudes AN_P₁ and a plurality of first incremental harmonic phase angles PhN_P₁, where N is the order or the number of harmonics generated by the Fourier analysis. In the illustrated embodiment, the desired waveform is the first incremental fundamental output which corresponds to the first order harmonic (N=1) after performing the Fourier analysis. The remaining plurality of first incremental harmonic outputs corresponds to errors generally present in P₁ and are defined as first incremental errors. It should be noted that additional or different harmonic outputs may be selected as the desired waveform or the waveform that represents errors in P₁ and the selection of the harmonic outputs which comprise errors or are selected as the desired waveform are not limited by the illustrated embodiment. The first incremental fundamental output includes a first incremental fundamental amplitude A1_P₁ and a first incremental fundamental phase angle Ph1_P₁. Accordingly, DC_P₁, AN_P₁, PhN_P₁, A1_P₁, and Ph1_P₁ are defined as the first incremental calibration parameters.

[0108] In step 216, the Fourier analysis is also performed on P₂ to produce a second incremental dc component DC_P₂, a plurality of second incremental harmonic outputs, and a second incremental fundamental output. The second incremental dc component DC_P₂ corresponds to a harmonic order of zero after performing the Fourier analysis. The plurality of second incremental harmonic outputs include a plurality of second incremental harmonic amplitudes AN_P₂ and a plurality of second incremental harmonic phase angle PhN_P₂, where N is an integer and is the order or the number of harmonics generated by the Fourier analysis. The desired waveform is the second incremental fundamental output which corresponds to the first order harmonic after performing the Fourier analysis. The remaining plurality of second incremental harmonic outputs corresponds to the errors generally present in P₂ and are defined as first incremental errors. It should be noted that additional or different harmonic outputs may be selected as the desired waveform or the waveform that represents errors in P₂ and the selection of those harmonic outputs are not limited by the illustrated embodiment. The second incremental fundamental output includes a second incremental fundamental amplitude A1_P₂ and a second incremental fundamental phase angle Ph1_P₂. Accordingly, DC_P₂, AN_P₂, PhN_P₂, A1_P₂, and Ph1_P₂ are defined as the second incremental calibration parameters.

[0109] Referring to FIG. 30, a block diagram for the method of compensating the incremental outputs P₁ and P₂ is generally shown at 250. In the illustrated embodiment, it is necessary to use instantaneous values for the first and the second incremental outputs. These values change for every angle of rotation of the shaft 18. The compensation process produces a first incremental final output and a second incremental final output for every angle of rotation of the shaft.

[0110] A first incremental adder 252 and a second incremental adder 254 is used to eliminate the voltage offset (V_OQ) as generated from the incremental magneto-sensitive elements 134,136. A third incremental adder 256 subtracts DC_P₁ from the P₁ to create a first incremental adjusted output P′₁. Likewise, a fourth incremental adder 258 subtracts DC_P₂ from P₂ to create a second incremental adjusted output P′₂.

[0111] After producing P′₁, a previous position of the shaft (prevpos), AN_P₁, and PhN_P₁ are inserted into a first incremental compensation block 260 wherein a first incremental compensation equation eliminates the first incremental harmonic outputs. This in effect eliminates most of the non-idealities in the P₁ output and produces the first incremental final output P′₁. The first incremental compensation equation is defined as:
P″₁=P′₁−AN_—P₁*sin [(prevpos+PhN_—P₁*N)]  (q)
It should be noted that if the previous position (prevpos) of the shaft was not known, an initialization routine will produce the (prevpos) value of the shaft. The first incremental final output P″₁ is the compensated value for P₁.

[0112] After producing P′₂ from the fourth incremental adder 258, the previous position of the shaft (prevpos), AN_P₂, and PhN_P₂ are inserted into a second incremental compensation block 262 wherein a second incremental compensation equation eliminates the second incremental harmonic outputs. This in effect eliminates most of the non-idealities in the P₂ output and produces a second incremental corrected output P″₂. The second incremental compensation equation is defined as:
P″₂=P′₂−AN_—P₂*sin [(prevpos+PhN_—P₂*N)]  (r)
The output of the incremental compensation equation converges on a single value to produce the second incremental corrected output P″₂.

[0113] After generating P″₂, P″₂ is multiplied to a first gain block 264 wherein the gain block includes an incremental gain factor which is defined by the ratio of fundamental amplitudes:
A1_—P₁/A1_—P₂  (s)
By multiplying P″₂ with the incremental gain factor, this produces a second incremental normalized output P″₂. This step is performed such that the amplitudes of P₁ and P₂ are equal.

[0114] By establishing a common amplitude between P₁ and P₂, the next step is to calculate the incremental quadrature error angle δ_1,2 between the outputs P₁ and P₂, ideally a ninety degrees phase shift is required between the outputs P₁ and P₂. To calculate δ_1,2, a fifth incremental adder 268 adds ninety degrees to Ph1_P₁, and Ph1_P₂ is subtracted from the sum of ninety degrees and Ph1_P₁ to calculate the incremental quadrature error angle δ_1,2. After solving for δ_1,2, an incremental correction block 266 produces a second incremental final output {tilde over (P)}₂. The incremental correction block 266 includes an incremental correction equation and is defined by the following equation:
{tilde over (P)}₂=[(P′″₂+P″₁ sin δ_1,2)/cos δ_1,2]  (t)
Accordingly, the final incremental output {tilde over (P)}₂ is the compensated value for P₂.

[0115] By employing the method as described above, the objective of providing a common amplitude between the incremental outputs P₁ and P₂ and achieving a phase shift of substantially ninety degrees between the angles of the incremental outputs P₁ and P₂ is satisfied. As shown in FIG. 28, the compensated incremental waveforms 152 of the final incremental outputs P″₁ and {tilde over (P)}₂ illustrates that the amplitudes are similar and the resulting phase shift between the compensated waveforms 152 are substantially ninety degrees. The first final incremental output P″₁ and the second final incremental output {tilde over (P)}₂ generated for each angle of rotation of the shaft 18 is stored on a controller.

[0116] It is now possible to employ the solver equation to produce the final incremental angle of the shaft 18. As stated above, the final incremental angle is any angle between 0 and 360 degrees. Referring to FIG. 30, the first and the second incremental final outputs are inserted into a solver equation block 270 where the solver equation provides the final incremental angle of the shaft 18. The solver equation (n) from above is re-written as:

[0117] $\begin{array}{cc}{\phi }_{\mathrm{new}}={\phi }_{\mathrm{old}}-\left(\frac{P_1^{\mathrm{\prime \prime }}\cos \phi -{\tilde{P}}_2\sin \phi }{-P_1^{\mathrm{\prime \prime }}\sin \phi -{\tilde{P}}_2\cos \phi }\right)& \left(u\right)\end{array}$
The variable φ_new will converge on a value of between 0 and 360 degrees and is defined as the final incremental angle. As stated above, the converged angle φ_new is assigned to the final incremental angle.

[0118] Referring to FIG. 31, a method for calibrating the segment outputs P₃ and P₄ is, generally shown at 300. In step 302, the calibration process requires that the shaft 18 is rotated to a maximum number of rotations or a total of eighteen hundred degrees. In step 304, the first segment magneto-sensitive element 138 outputs P₃ in response to detecting the changing magnetic flux of the segment magnet 106. In step 306, the second segment magneto-sensitive element 140 outputs P₄ in response to detecting the changing magnetic flux of the segment magnet 106. In step 308, a controller stores P₃ and P₄ against a plurality of master encoder readings.

[0119] In step 310, the voltage offset (V_OQ) is subtracted from the segment outputs P₃ and P₄ as produced by the segment magneto-sensitive elements 138, 140. In step 312, a Fourier analysis is performed on P₃ to produce a first segment dc component DC_P₃ and a first segment fundamental output. It should be noted that the first segment dc component DC_P₃ corresponds to a harmonic order of zero after performing the Fourier analysis. The first segment fundamental output includes a first segment fundamental amplitude A1_P₃ and a first segment fundamental phase angle Ph1_P₃. Accordingly, DC_P₃, A1_P₃, and Ph1_P₃ are defined as the first segment calibration parameters.

[0120] In step 314, the Fourier analysis is also performed on P₄ to produce a second segment dc component DC_P₄ and a second segment fundamental output. The second segment dc component DC_P₄ corresponds to harmonic order of zero after performing the Fourier analysis. The second segment fundamental output includes a second segment fundamental amplitude A1_P₄ and a second segment fundamental phase angle Ph1_P₄. Accordingly, DC_P₄, A1_P₄, and Ph1_P₄ are defined as the second segment calibration parameters.

[0121] Referring to FIG. 32, a block diagram for the method of compensating the segment outputs P₃ and P₄ is generally shown at 350. In the illustrated embodiment, it is necessary to use instantaneous values for the first and the second segment outputs. The value changes for every angle of rotation of the change gear 100. The compensation process produces a first segment final output and a second segment final output for every angle of rotation of the shaft, and each first segment final output and second segment final output is stored in a controller.

[0122] A first segment adder 352 and a second segment adder 354 are used to eliminate the voltage offset (V_OQ) generated from the segment magneto-sensitive elements 138,140 and the adders 352, 354. A third segment adder 356 subtracts DC_P₃ from the P₃ output to create a first segment final output P′₃. Accordingly, P′₃ is the compensated value for P₃. Likewise, a fourth segment adder 358 subtracts DC_P₄ from the P₄ output to create a second segment adjusted output P′₄.

[0123] After generating P′₄, P′₄ is multiplied to a segment gain block 360 wherein the gain block includes a segment gain factor which is defined by the ratio of fundamental amplitudes:
A1_—P₃/A1_—P₄  (v)
By multiplying P′₄ with the segment gain factor, this produces a second incremental normalized output P″₄. This step is performed such that the amplitudes of P₃ and P₄ are equal.

[0124] By establishing a common amplitude between P₃ and P₄, the next step is to calculate the segment quadrature error angle δ_3,4 between the outputs P₃ and P₄, ideally a ninety degrees phase shift is required. To calculate δ_3,4, a fifth incremental adder 362 adds ninety degrees to Ph1_P₃, and Ph1_P₄ is subtracted from the sum of ninety degrees and Ph1_P₃, to calculateδ_3,4. After solving forδ_3,4, a segment correction block 364 produces a second segment final output {tilde over (P)}₄. The segment correction block 364 includes a segment correction equation and is defined by the following equation:
{tilde over (P)}₄=[(P″₄+P′₃ sin δ_3,4)/cos δ_3,4]  (w)
Accordingly, the second segment final output {tilde over (P)}₄ is the compensated value for P₄.

[0125] By employing the method as described above, the objective of providing a common amplitude between the segment outputs P₃ and P₄ and achieving a phase shift of substantially ninety degrees between the angles of the segment outputs P₃ and P₄ is satisfied. As shown in FIG. 34, the compensated segment waveforms 154 of the final segment outputs P′₃ and {tilde over (P)}₄ illustrates that the amplitudes are similar and the resulting phase shift between the compensated waveforms 154 are substantially ninety degrees. The first segment final output P′₃ and the second segment final output {tilde over (P)}₄ generated for each angle of rotation of the shaft 18 is stored on a controller.

[0126] It is now possible to employ the solver equation to produce the final segment angle of the shaft 18. The first and the second segment final outputs are inserted into a solver equation block 366 where the solver equation provides the final segment angle of the shaft 18. The solver equation (n) from above is re-written as:

[0127] $\begin{array}{cc}{\phi }_{\mathrm{new}}={\phi }_{\mathrm{old}}-\left(\frac{P_3^{\prime }\cos \phi -{\tilde{P}}_4\sin \phi }{-P_3^{\prime }\sin \phi -{\tilde{P}}_4\cos \phi }\right)& \left(x\right)\end{array}$
The variable φ_new will converge to the actual segment angle which is a value between 0 and 360 degrees.

[0128] After calculating the final incremental angle and the actual segment angle, the actual segment angle is compared to the final segment angle and corresponding integer (n) as listed in Table 1 above. In response to comparing the actual segment angle to the final segment angle, the final position equation associated with the final segment angle takes the final incremental angle as calculated from equation (u) and inserts the final incremental angle and the corresponding integer (n) into the final angle position equation (p) to produce the final angle position of the shaft 18 which may be any angle between +/−900 degrees.

[0129] Obviously, many modifications and variations of the present invention are possible in light of the above teachings. The invention may be practiced otherwise than as specifically described within the scope of the appended claims.