FIELD OF INVENTION The present invention relates generally to high precision machine tools. More specifically, the present invention provides a system for periodic calibration of the angular errors in a numerically controlled machine tool. BACKGROUND OF THE INVENTION

The availability of contemporary numerically controlled machine tools has increased the potential accuracy of machining operations in many industrial applications. To realize the full accuracy potential of a given machine, however, it is required that the quasi- static errors originating from loading conditions and track wear be calibrated and compensated for. The geometric relationships of the tracks and the guideways to the machine base determine the accuracy of the machine. Lack of straightness, orthogonality, and track pitch, roll, and yaw deviations, need to be initially calibrated, and periodically recalibrated over the lifetime of the machine, to insure its maximum accuracy.

In contemporary numerically controlled machine tools a work table is generally used in combination with a cutting tool. The cutting tool will generally move along a single axis (typically a vertical axis) while the work table, upon which a work piece is mounted will generally move horizontally relative to the cutting tool in order to machine the work piece as desired. In order to obtain high precision machining, the movements of the work table and cutting tool must be very precisely controlled. Furthermore, in order to provide very diverse machining capabilities it is necessary for the work table to move along two or possibly three orthogonal axes.

In order to achieve appropriate movement of the work table, the table is generally mounted on tracks and has associated therewith numerous motors or other devices capable of driving the work table. Over time, quasi static errors are generated in the

control of the work table. These errors can be generated from numerous sources including wear of the table mechanisms over time, stress induced into the systems, misalignment over time, and other load induced and thermally induced errors. As a consequence of all of these errors, angular errors develop in each of the movement axes.

As a result of all of the above mentioned errors, it is necessary to periodically calibrate the work table movement to assure precision machining capabilities. This calibration has been conventionally done using a series of instruments including autocolimators, optical straight edges, tilt meters, etc. Due to the large number of different devices traditionally used and the complexity of these devices, this calibration process is generally very time consuming and very expensive. As will be appreciated this time and expense has a negative impact on the business operations of a machining shop. Consequently it is necessary to develop a simpler and less costly method of providing periodic calibration of these machine tools.

In the field of avionics, there has always been a need to track the motions and movement of devices in inertial space. More specifically aircraft and spacecraft need to be tracked in order to maintain a current assessment of their position and orientation in inertial space. This information is then used by navigation systems and auto pilots to control the aircraft as well as numerous other systems to provide aircraft operability enhancement. As is recognized in the avionics field, these inertial navigation systems provide a highly precise measurement of position and orientation in inertial space.

Generally inertial navigation systems or inertial reference systems require the use of gyroscopes and accelerometers to track the motion of such systems. One type of gyroscope is the ring laser gyroscope as explained in US Patent 3.373,650 to Killpatrick. et al which is assigned to the assignee of the present invention.

SUMMARY OF INVENTION

The present invention uses all of the previously established capabilities of inertial reference systems to provide solutions to some of the known problems and needs

regarding precision machining. The present invention utilizes a sensor assembly which is mounted to the movable member of a machine tool to provide information regarding angular errors. Data is taken from the sensor assembly as it travels through a repeated pattern of motion on the movable member, thus allowing its actual movement to be characterized. This data is fed into a processing system to process all of the information from the sensor assembly regarding the precise angular movements of the movable member. Also provided to the processing unit is an indication of the member's position which is provided by position sensors mounted to the movable member. By collecting this data over numerous predetermined movements of the movable member, any irregularities can thus be determined. As recognized once these irregularities are determined, they can be used to compensate the precision control system which controls the movements of the member during the machining operation.

The use of inertial sensors offers some important advantages: (1) the calibration procedure is automated; (2) calibration of track pitch, roll, and yaw angular deviations, and spindle to track angular deviations, is possible with greater efficiency and accuracy; (3) the ease of use, and speed of the inertial measurement technique, make it feasible to easily recalibrate the machine in the user's environment.

It is an object of the present invention to provide a high precision system utilizing ring laser gyroscopes and other inertial sensors to measure the movements of the movable member in a precision machining tool. It is a further object of the invention to provide algorithms for high precision characterization of the angular error in these movements and consequently allow for machine compensation.

It is another object of the present invention to provide a calibration methodology, including appropriate algorithms, to provide automated calibration of a precision machine tool.

It is a further object of the present invention to provide an automated system which will perform all of the data acquisition and computation functions without requiring heavy user involvement.

BRIEF DESCRIPTION OF THE DRAWINGS

Further objects and advantages of the present invention can be seen by reading the following detailed description in conjunction with the drawings in which:

Figure 1 is a schematic of the concept which illustrates the principal elements in the invention;

Figure 2 is a diagram illustrating the source and method of processing the three informational sources to arrive at the machine angular deviations as a function of the distance along each of its tracks;

Figure 3 is a block diagram that illustrates the algorithmic approach used in convering the three informational sources into an angular deviation function;

Figure 4 is a curve that illustrates the unique type of symmetry required in the movement of an axis during the repeated machine traversals executed in the calibration process;

Figure 5 is a graphical illustration used in the derivation of the incremental and total angular deviations of a given machine axis;

Figure 6 is a graphical illustration used in the derivation of the backlash in each machine axis;

Figure 7 is a graphical illustration used in the derivation of the angular error in a machine's rotary axis.

DETAILED DESCRIPTION OF THE INVENTION

The capability to do precision machining depends on being able to compensate for the irregularities in each machine axis. Normally five parameters are used to characterize the deviations in each of the machine's axes (pitch, roll, yaw angular deviations; two cross-axis linear deviations) and, in addition, six parameters to characterize the orthogonality errors between the three machine axes - leading to a total of 21 parameters to be calibrated. In the present invention, an inertial sensor assembly is utilized in the determination of a subset of these 21 machine tool errors, which are the pitch, roll, and yaw angular errors in each axis.

A device for precision measurements of the angular errors in each axis of a machine tool is illustrated in Figure 1. Figure 1 depicts a numerically controlled machine tool 10 having a spindle 12 which moves in a vertical direction, and a movable member or table 14 which moves in two orthogonal horizontal directions. Both spindle 12 and table 14 movements are effected by irregularities that cause angular errors in the orientation of these members. These irregularities are a function of the axis position relative to a reference starting point and are characterized by pitch, roll, and yaw components. The function of the calibration technique of the present invention is to measure the machine pitch, roll, and yaw angular errors using an inertial sensor assembly.

As illustrated in Figure 1 , the basic concept is to collect data from an inertial sensor assembly 20, and a machine position encoder 16, over a complete traversal of a given axis of the machine. This information is used to derive the angular deviations of that axis. In the present embodiment, inertial sensor asembly 20 includes a triad of gyros 21 orthogonally orientated. Gyro data is collected simultaneously with the machine position data and stored in a processor or PC 30. Processing of the gyro and machine position data provides an estimate of the angular deviations (pitch, roll, yaw) of the particular axis as a function of the position along the axis of measurement. By repeating this back and forth traversal, and averaging the results, a composite result is obtained that represents the best estimate of the machine's angular deviations.

Further details of the calibration apparatus and technique are shown in Figure 2. Here it is seen that the sensor assembly 20 is interogated by the sensor electronics 22, which

also interogates the internal temperature sensors 24. This allows the gyro incremental outputs to be corrected for temperature effects using a set of gyro compensation coefficients, as provided by gyro calibration coefficient apparatus 26. Gyro compensation apparatus 28 receives the coefficients from gyro calibration coefficent apparatus 26 to produce the compensated gyro incremental angle vector Δθ . The machine position along its track, x, is measured simultaneously by sending an interogation pulse from sensor electronics 22 to machine encoder 16. This provides all of the data needed to calibrate the machine angular errors at a series of equally spaced l-y . • . Y points ' ^N along the track specified by the user.

A block diagram illustrating the basic features of the processing algorithm carried out by processor 30 is shown in Figure 3. It should be noted that, only a single axis is represented in the diagram. Consequently, identical processes will be carried out for each axis of measurement.

The diagram illustrates a number of basic functions as follows:

1. Gyro 16 as its output incremental measured over an i ^m time interval to produce a gyro incremental output 32. (Δθ _j ) This Output 32 is summed by summing apparatus 34 with all previously occuring incremental angles obtained from gyro 16, to form a cumulative angle ' 36 at the i time point.

2. The cumulative angle, ' , is filtered by a first filtering apparatus 38. This filter 38 could be a low-pass filter of first or second order, or possibly two or more such filters cascaded together. The filtering operation produces the filtered cumulative

gyro output. ' 40. at the i ^tn time point. The purpose of the filtering operation is to attenuate random contributions to the cumulative gyro output originating from quantization and other random errors in the gyro incremental output.

3. The machine position measurement, x , 44 is synchronized with the gyro data, so that the machine position corresponds to the end of the i ^m time interval. The sequence of machine position measurements is filtered by a second filtering apparatus 42. Second filter 42 is matched to first Filter 38, so that whatever filter transfer function is selected for one also applies to the other. The reason for matching the two filters is so both the gyro data and the machine position data remain synchronous. This will be true only if the filters are matched; otherwise, the two signals will become time shifted relative to one another in proportion to the time constant of the filter. The output of second filter 42 is then a filtered machine position 46.

4. The filtered cumulative gyro output 40 and filtered machine position measurement 46 are processed by the block designated "Cumulative Angle Processing and Sorting" 50. This operation breaks up the stream of cumulative gyro data and sorts it according to a predetermined segment of the track. Since the filtered cumulative gyro output occurs at a fixed spacing in time, and the filtered position output 46, an interpolation process is required to derive the desired result. After interpolation, the filtered cumulative gyro output is known for each of the predetermined positions, xk, at which the angular errors are to be determined. This is done for both the

Af forward and reverse directions of travel, leading to the filtered cumulative angle, ^k , when the machine occupied the k ^tn track position in the forward direction, and the

filtered cumulative angle, , when the machine occupied the same position on the track when travelling in the reverse direction.

5. The change in the filtered cumulative gyro output from the (k-1 ) point on the track to the k ^m point on the track is determined for both the forward and reverse directions of travel, and differenced at summing point 52. By doing so, the effect of the fixed (and generally unknown) component in the gyro output arising from gyro bias and earth rate is eliminated. At the same time, the component in the gyro output that is responsive to the change in the track angular deflection over a given track segment is reinforced (doubled). Therefore, by multiplying the difference by

1/2 in block 54, the change in the track angular deviation, ^α k 56, over the k track segment is obtained.

6. A final summing operation at block 58 leads to the cumulative track deflection change, ^α k 60, from the start of the track to its end, as a function of the track position.

The processing scheme defined above takes advantage of a knowledge of what the gyro measures as it travels down the track of a machine. The incremental gyro output over some arbitrary interval of time consists of the sum of following components:

• A fixed component due to gyro bias and the earth rate component measured by the gyro, which is constant given that the gyro orientation is fixed relative to the earth's rotational rate vector - which will always be true for a nominally straight machine track.

• A random component due to gyro quantization, gyro random walk, small extraneous motions of the gyro due to holding fixture to track angular motions, and extraneous machine high frequency angular motions that are superimposed on the angular motions due to track irregularities.

• The desired angular motions resulting from track irregularities, which are a function of the local track curvature.

The filtering applied to the cumulative gyro output is designed to severely attenuate the random components in the gyro output, while the combining of the forward and reverse cumulative angle changes is designed to eliminate the fixed (unknown) component in the gyro output. This leaves just the component that is responsive to the local track curvature, to which the track angular deviation is directly equal.

The basis for the processing scheme may be derived mathematically in the following manner. The incremental gyro filtered cumulative output over any given track interval may be expressed as follows:

θ _k -θ[., = Δα _k + Ω(tJ -tJ.,) ξt ₍₁₎

where:

= gyro filtered cumulative output at forward crossing of k ^m point on track

k = gyro filtered cumulative output at reverse crossing of k ^m point on track

^k = angle change due to track curvature over k ^tn track segment

^ ^k = random error in filtered gyro output over forward traversal of k" ¹ track segment

^ ^k = random error in filtered gyro output over reverse traversal of k ^tn track segment

^k = time associated with reaching the end of k" ¹ track segment in forward direction

^k = time associated with reaching the end of k ^tn track segment in reverse direction

W = constant angular rate measured by gyro (earth rate plus gyro bias)

and the angle change change due to track curvature is given by

where

^k = radius of curvature of track over the k ^m track segment

^Xk = distance from start point of track to the end of the k ^m track segment

As seen from (3), the angle change due to track curvature, ^k , manifests itself positively when the motion is in a forward direction through the k ^m track segment, and negatively when the motion is in the reverse direction through the same segment of the track. Therefore, this angle may be obtained by subtracting the forward and reverse cumulative angle differences, as defined above, and dividing by 1/2 or, explicitly

Δ ^α _k = ' ₂ ^ [I^V - ^θf _k -i / V - ^θ _k -ι) / (4)

The total angular deviation of the track relative to its reference starting point is determined by accumulating all of the track curvature changes as defined by (4) or, explicitly

<* _k = α -ι + Δα _k (k = l,- ",N) _{(5 )}

where ^α k is the total track angular deviation at the k ^m point on the track. Equations of the form given by (5) define the three angular errors (pitch, roll, and yaw) for each

machine axis, with each of the three gyros providing the necessary information to derive one of the three angular errors.

It should be noted from (1), (2) and (4) that the effect of the unknown angular rate, Ω , due to earth rate and gyro bias, will cancel out over the forward and reverse traversals of each segment of the track only if the following condition holds true.

_ f ^l k ^{_ t} k-l - ^{~ l} k-1 (6)

The condition stated in (6) will in turn be satisfied only if the machine's motion along the track has a symmetrical characteristic, as depicted in Figure 4. This type of machine motion, which is symmetrical about the end point of the track, may be easily commanded by the numerical controller.

The basic relationship that is used in estimating the track angular deviation from a single traversal is illustrated graphically in Figure 5. The single-traversal estimate may be significantly improved upon by processing M such sets of results, and averaging or, explicitly

1 α = — ¹ v 2. oc _k (' ) ^M "=ι (7)

where ^α k is the best estimate after M traversals, and ^α k is the angular error at the k ^tn point on the track for the i ^m traversal.

The above derivation assumes that the effect of the track curvature, ^k . is the same in both the forward and reverse direction for each segment of the track. This is not always true for all machines and, more generally, a slightly different value will be experienced when the machine is moving in the forward direction then when moving in the reverse direction. Given that this is true, the processing scheme represented by Equations ( 1 ) through (5) leads to what is referred to as the "average angular deviation".

In addition to the average angular error characterization for the machine, it is also necessary for compensation purposes to characterize the angular deviation errors for the forward and reverse directions individually. This may be accomplished using the data over complete forward reverse traversals and, by using a "closure" principle, an estimate of the constant component in the gyro output may be derived. (Closure requires that the net angular change around a closed traverse be zero). Application of the closure principle leads to the following relationship (refer to Figure 6):

where

θ ¹ = filtered cumulative angle at the start turnaround point, with the machine moving forward

θ. filtered cumulative angle at the start turnaround point, with the machine moving backward

θ ^N = filtered cumulative angle at the end turnaround point, with the machine moving forward

B filtered cumulative angle at the end turnaround point, with the machine moving backward

time at the start turnaround point, with the machine moving forward

time at the start turnaround point, with the machine moving backward

= time at the end turnaround point, with the machine moving forward

^N = time at the end turnaround point, with the machine moving backward

Ω = unknown constant component of angular rate measured by gyro

The unknown constant component in the gyro output, Ω , is then easily computed from (8), which allows the forward filtered cumulative angle data to be corrected by means of

and the reverse filtered cumulative data to be corrected by means of

Given the availability of the individual estimates, ^k and ^k , the total track angular deviation measured from the reference starting point is obtained separately for the forward direction of motion by means of

f «k = α _ ₁ + Δα _k (9)

and for the reverse direction of motion bv means of

α_ ^{= α} -ι ⁺ Δα _{( ] 0)}

As was the case for the average angular deviation, a best estimate based on M traversals may be obtained by averaging the individual results for both the forward and reverse directions.

Calibration of the Machine's Backlash Parameters

A complete characterization of a machine's angular errors must also include a measurement of the backlash in each axis. (Backlash is defined here as the change in the angular orientation of an axis that occurs almost instantaneously when the direction of axis travel is reversed.) The methodology used to determine the angular deviations of each axis of a machine also lends itself to determining the backlash in the axis, without requiring a special procedure dedicated to this purpose.

The computation of the backlash for a given axis assumes the following (refer to Figure 6):

• A stop and start occurs at each end of the track

• The backlash manifests itself exactly the same way around each end of the track, with an angular change of the same magnitude but of opposite sign experienced in the process of starting and stopping at each end of the track, such that no net angle change accumulates over a complete round trip. (This is a different manifestation of the closure condition used earlier in deriving the constant component in the gyro output.)

The gyro output has a fixed component due to earth rate and bias that is the same for the traversal around each end of the track.

• The component in the gyro output that is responsive to the track curvature will be zero around each end of the track, which will be true if the same portion of the track is used in both the forward and reverse directions at each of the two turnarounds.

The backlash may then be computed utilizing the following equations

Δθ _a = B + ΩΔt _a

(11)

Δθ _h = B+ΩΔt, (12)

in which B is the backlash angle, and the angles AH a and AH t> are the incremental filtered cumulative angle changes experienced during the turnarounds at the ends of the track, explicitly defined by

Δθ_= θf-θ.

^* r ^Λ f

Δθ _b = ΘN -Θ _N

with *a and ^Δt b being the time durations associated with the turnarounds at the two ends of the track, which are explicitly defined by

I _f _r _f where the quantities, ⁹ ! , θ, , Θ _N ,Θ _N ,t, , t, , t _N , t _N , and Ω _{are defined as} before.

From (11) and (12), the unknown constant component. W, in the gyro output is determined as

_Ω= ^Δ V ^Δθ t>

Δt _a +Δt _{b (13)}

which allows the backlash, B, to be computed as

B = Δθ _a (l - r)- rΔθ _{b ( 14)}

where r is defined by

Δt. r = ^•

Δt _a + Δt _b

Each round trip will produce an independent estimate of the backlash, with a best estimate from M such traversals being the average of the individual backlash estimates, or explicitly

1 ^M _ ^• _

B = ∑B<

M i=ι (15)

where B is the best estimate of the backlash after M machine traversals, and B ) is the backlash estimate associated with the i ^m traversal.

Calibration of the Errors in Rotarv Axes

The calibration of a rotary axis of a machine tool consists of using inertial measurements to determine the error in the indicated angle of the table, as derived from an angular encoder provided as part of the machine's output. The means by which this is accomplished parallels that used in the calibration of linear axes, with repeated clockwise/counterclockwise table rotations substituting for the forward/reverse motions used in the calibration of linear axes. The procedure assumes the following:

The table is made to rotate ±360°. with gyro data and angular encoder data being collected during the process.

• A stop and start occur at the end of each clockwise and counterclockwise rotation.

• The clockwise and counterclockwise rotations are each somewhat larger than 360°, to insure crossings of the 0° and 360° points.

• The gyro bias and scale factor are essentially constant over a ±360° table traversal.

In order to provide an accurate calibration of the angular encoder, it is necessary to simultaneously calibrate the bias and scale factor of the gyro having its input axis along the table turn axis. This may be achieved using the table angular encoder zero reference point as a convenient fiducial mark allowing the gyro bias (actually, bias plus earth rate) and scale factor to be calibrated using data collected during precise 360° clockwise and counterclockwise rotations of the table. The processing equations are derived using a dual closure condition, which leads to the two following relationships (refer to Figure 7):

θji -θf +θϊ -θ|, -Ω (t!, -t; +t - t^)=o

(16)

where W and K are the bias and scale factor of the gyro having its input axis along the table turn axis.

The gyro scale factor and bias are then determined directly from (16) and (17). which allows the error in the angular encoder to be determined by forming the difference between the corrected filtered cumulative incremental angle and its indicated value over each segment of the angular excursion of the table. This is accomplished for both the clockwise and counterclockwise motions of the table, yielding the following relationships

Δβ _k =^ _κL ^r kκ-θ _k --i "Ω(t _k -t ^r _k _-,.) _y } _j ( vx _k __ - -, _(i9)

where ^ ^k and ^■ * are the angular errors over the clockwise and counterclockwise rotations respectively, and x\_ designates the table angle at the end of the k ^m segment of its angular excursion.

The clockwise and counterclockwise differences are then averaged to provide the error for the k" ¹ segment of the table angular excursion or, explicitly

Δβ _k =l/2(Δβ^ + Δβ ^r _k

(20)

This allows the cumulative error in the angular encoder, ^ , measured from the 0° point to be determined by adding all of the incremental errors, which results in the following equation

βk=βk-ι+Δβ _{k (} 21 ₎

The estimates of the angular encoder error derived from the individual clockwise/counterclockwise rotations may be used to arrive at the best estimate from M repetitions according to

= - ∑β ^(J) M ,=, (22)

where is the best estimate after M repetitions, and ^ is the angular encoder error derived from the k ^m clockwise/counterclockwise rotation.

Detailed Description of an Alternate Embodiment

The alternative exists of using a sensor assembly 20 consisting of two accelerometer and a single gyro. This concept uses the two accelerometers to measure the two level (pitch and roll) angular deviations of the track, with the gyro being used to measure the vertical (yaw) angular deviation. This alternate technique assumes that the machine acceleration is either zero (cross-track direction), or precisely known by processing the machine's position data (along-track direction).

The processing algorithms required to convert the accelerometer outputs occurring over multiple machine traversals into angular deviations are based on the structure given in α Figure 3. The cumulative angle, ^k , may be obtained directly from the accelerometer, as opposed to accumulating the gyro Δθ '$, according to

θ, =

(23)

where

^{k ~} acceleration measured by the accelerometer at the k point on the track

* ⁼ track acceleration derived from track position at the k ^tn point on the track

g = is the acceleration due to gravity

Equation (23) is generalized so that it may be applied to either the along-track direction, or the cross-track direction (in which case, " ^~~ ).

The cumulative angle derived directly from the accelerometer is then passed through the

matched filter 38 to arrive at the filtered cumulative angle, ^k . The filtered cumulative angle is then treated in the subsequent operations illustrated in Figure 3 in a manner indistinguishable from the treatment of the same angle derived from the gyro.

The angle, ^k , appearing in (23) has a somewhat different meaning than the similarily designated cumulative gyro incremental angle, insofar as the former is a measure of the absolute angle, whereas the latter is the angle accumulated starting from a zero initial value. Since only the change in across successive segments of the track actually appears in the processing equations, the two angles may be used without distinction. In some implementations, the acceleration required in (23) may not be directly available from the accelerometer but, rather, the natural output of the instrument may be a stream of incremental velocities. When such is the case, the needed acceleration may be derived by summing the incremental velocities over some finite time interval, and then dividing by the duration of the interval.

Detailed Description of Second Alternate Embodiment

The use of measurements obtained from a sensor assembly consisting of a triad of inertial sensors, in combination with the position measurements available from the machine, is capable of producing a highly accurate calibration of the angular errors in machine tools; however, a bad calibration resulting from the use of a failed or degraded sensor is still a distinct possibility. To prevent a bad calibration from subsequenth causing a high-value part to be ruined, a sensor health monitor scheme may be employed.

The failure or degradation of a sensor in the triad may be monitored by adding a fourth gyro, whose input axis is skewed relative to the three orthogonal sensors. Assume as an example that the fourth gyro is along the diagonal of the cube formed by the input axes

of the three orthogonal sensors; then, the incremental angle sensed by the skewed gyro is related to the incremental angles occurring about the three orthogonal axes as follows:

Δθ. ^ (Λθ _χ + Δθ _y + ΔΘ-

(24)

where

ΛH ΛH ΛH ^x ' ^v ' ^z = incremental angles about the three orthogonal axes

ΛH s = incremental angle sensed by the skewed gyro

The relationship given by (24) establishes a parity relationship that provides the means of detecting a failure or degradation in any of the four sensors.

As applied to machine tool calibration, a unique use of the parity relationship may be made; that is, a parity signal may be formed from the four angular deviations computed from the individual sensors, as follows

Pk J τ = b lc_ + βk + V

(25)

where

P ^k = parity at the k ^tn point on the track (nominally zero for perfect sensors)

^α k = angular deviation at k ^m point on track determined by the along-track sensor

^ = angular deviation at k ^m point on track determined by the cross-track sensor

Y = angular deviation at k point on track determined by the vertical gyro

^ε k = angular deviation at k point on track determined by the skewed gyro

Therefore, any sensor failure or major degradation may be detected by testing the value of P ^k , and verifying that it remains within bounds that are reasonable for normally encountered fixed and random sensor errors.

In addition, the accuracy of the pitch, roll, and yaw angular deviations may be enhanced by using all four angular deviation functions. This is achieved by employing a least- squares solution that derives the three angular deviations about the orthogonal axes from the four available angular deviation functions, using the following equation

where the matrix A is defined for the example sensor configuration by

r l o o i

I o i o I ^A =

_{* * *} and ^α k ^» k' and ^ are the angular deviations (pitch, roll, yaw) about the three orthogonal machine axes, with ^ttk ' ™ ' ^ ^k and ^{£ k} being the angular deviations determined from the three orthogonal and skewed sensor outputs.