This invention relates to inertial instruments such as gyroscopes and accelerometers, and more specifically to the detection of errors in real time for inertial instruments providing the ability to correct for such errors while the inertial instruments remain in operation.
The ability to independently reverse the sign of the scale factor (SF) terms for gyros is described in U.S. Pat. No. 7,103,477. A solution is described for scale factored input rates for two gyros A and B involving equations with scale factors (SFA and SFB), and bias terms (BiasA and BiasB). However, if SFA and/or SFB are in disagreement, the results of the equations are adversely affected.
An exemplary inertial measurement apparatus in accordance with the present invention has first and second inertial instruments that are oriented to have parallel sense axes and that produce respective first and second sensed output signals representative of an inertial attribute to be measured. Respective first and second scale factors are used in producing the first and second sensed output signals. A substitute scale factor is determined to be an equivalent of the second scale factor and is based on the first scale factor and a difference between the first and second scale factors. Differences in the first and second scale factors are calculated based on the first scale factor and the substitute scale factor during first and second time intervals where a sign of one of first and second scale factors changes from one state during the first time interval to the other state during the second time interval. First and second corrected output signals are generated based on the respective first and second sensed output signals and correction of said second scale factor error.
A method for implementing error corrections is a further embodiment of the present invention.
Features of exemplary implementations of the invention will become apparent from the description, the claims, and the accompanying drawings in which:
One aspect of the present invention resides in the recognition of the difficulties associated with inertial instrument errors in the scale factors, especially in a real time environment in which it is desirable maintain the inertial instruments in continuing operation while minimizing such errors.
Alternatively, the functions and calculations can be implemented in an application specific integrated circuit or other form of hardware implementation. In addition to the functionality and calculations made by the instrument 15, additional functionality provided by the two or more gyros could be incorporated into a single device.
The bias errors may be directly observable if the sense axes of two instruments, gyroscopes (gyros) in this exemplary embodiment, are located along the same axis relative to the attribute being sensed and are sequenced as described. Both gyros sense rotations about the same axis. The measurements MeasA and MeasB made by gyros A and B during each ith measurement interval are:
MeasA(i)=SFA*dTheta_in+BiasA*Ti (Eq1)
MeasB(i)=SFB*dTheta_in+BiasB*Ti (Eq2)
where:
MeasA and MeasB are the measurement of incremental angle made by two gyros A and B, respectively;
dTheta_in is the true input angle displaced by the two gyros A and B having parallel sense axes;
SFA and SFB are the scale factor coefficients of gyros A and B respectively which relate the physical output of the gyros to input angle;
Ti is the time interval over which the two gyros are angularly displaced.
Independently reversing the sign of the scale factor terms of each gyro yields equations:
MeasA(i)=KmodeA*SFA*dTheta_in+BiasA*Ti (Eq3)
MeasB(i)=KmodeB*SFB*dTheta_in+BiasB*Ti (Eq4)
where:
KmodeA and KmodeB independently take on the values of +1 or −1 to provide a sequence of measurements in which the scale factor terms are reversed.
The 8 equations for the combination of Kmode values are:
MeasA(1)=+1 *SFA*dTheta_in+BiasA*Ti (Eq5)
MeasB(1)=+1 *SFB*dTheta_in+BiasB*Ti (Eq6)
MeasA(2)=+1 *SFA*dTheta_in+BiasA*Ti (Eq7)
MeasB(2)=−1 *SFB*dTheta_in+BiasB*Ti (Eq8)
MeasA(3)=−1 *SFA*dTheta_in+BiasA*Ti (Eq9)
MeasB(3)=−1 *SFB*dTheta_in+BiasB*Ti (Eq10)
MeasA(4)=−1 *SFA*dTheta_in+BiasA*Ti (Eq11)
MeasB(4)=+1 *SFB*dTheta_in+BiasB*Ti (Eq12)
These 8 equations provide a solution for the four scale factored input rates SFA*dTheta_in through SFA*dTheta_in4, and the two bias terms BiasA and BiasB.
In this embodiment the term SFB is modeled as:
SFB=SFA*(1+dSFB) (Eq13)
where dSFB is the difference between SFA and SFB.
Substituting the equivalent of SFB from Eq13 into Eq3 and Eq4 yields:
MeasA(i)=KmodeA*SFA*dTheta_in+BiasA*Ti (Eq14)
MeasB(i)=KmodeB*(SFA+SFA*dSFB)*dTheta_in+BiasB*Ti (Eq15)
If the measurements MeasA and MeasB over successive intervals are summed or differenced, the product of SFA*dTheata_in(i) can be made to drop out, leaving a remainder containing elements of dSFB*SFA*dTheta_in, BiasA and BiasB.
The differencing of the A and B gyro measurements may be selected to provide observability of the bias errors and scale factor imbalance. A sequence of KmodeA, KmodeB and summing or differencing of equations Eq14 and Eq15 can be found that makes the errors dSFB, BiasA and BiasB observable as frequency and phase multiplexed signals in the combined measurement from gyroscopes A and B; this is explained in more detail below.
MeasA(i)=KA*(KmodeA*SFA*dTheta_in+BiasA*Ti) (Eq16)
MeasB(i)=KB*(KmodeB*(SFA+SFA*dSFB)*dTheta_in+BiasB*Ti) (Eq17)
In the above equations Eq16 and Eq17 coefficients KmodeA and KmodeB represent the sign of the A and B gyro scale factors, respectively. The coefficients KA and KB are also given the values of +/−1. Values of KA, KB, KmodeA and KmodeB are selected to force the following:
(KA*KmodeA+KB*KmodeB)*dTheta_in(i)=0 (Eq18)
KA and KB serve to control whether Eq16 and Eq17 are summed or differenced, and to control the order of difference: Eq16−Eq17 or Eq17−Eq16.
The summation of Eq16 and Eq17, assuming Eq18 is satisfied, has only terms containing the errors desired. Neglecting the product containing dSFB*SFA*dTheta_in for the moment, by sequentially alternating the signs of KA and KB, the effect of BiasA and BiasB on Eq18a changes sign, effectively “modulating” the error caused by these terms in the summation Eq18a. Waveforms 40 and 42 represent this effect from BiasB and BiasA, respectively.
The effect of dSFB on Eq18a is affected only by KmodeB. Changing the sign of the scale factor of GyroB in
Once a sequence for KA, KB and KmodeB are selected, KmodeA, the sign of the scale factor of Gyro A, can be selected to force the condition prescribed by Eq18. The result is that BiasA, BiasB, and dSFB appear in the summation of Eq16 and Eq17 as square waves correlated with
Exemplary instrument 10 implements a Kalman filter to demodulate the error signals shown in
The Kalman filter H matrix for each observation is:
H(i)=[KB(i)*KmodeB(i)*SFA*dTheta_in(i),KA(i)*Ti, KB(i)*Ti] (Eq20)
Those skilled in the art will understand the operation of the Kalman Filter with regard to Z(i) and H(i).
All of the elements within the H matrix are known at each interval: KA, KB, KmodeA, KmodeB and the raw (uncorrected) measurement from gyro A, i.e. SFA*dTheta_in(i). Hence, the above process can be applied recursively by instrument 1 to determine the real time measurement of the bias errors of gyros A and B, and the imbalance (if any) between the scale factors of gyros A and B. The output dTheta_out of instrument 1 consists of a constantly updated angle measurement based on the raw output from gyros A and B, and as corrected to compensate for any bias and/or scale factor errors determined as explained above. Thus, correction of such errors is implemented while the gyros continuously operate in a dynamic environment, i.e. dynamic updates are performed during times where the gyro inputs are subject to change.
With regard to the illustrative embodiment, a digital implementation will contain counts, i.e. numerical values, corresponding to inputs and outputs of the inertial instrument measurements from two parallel instrument inputs/channels Win(A) and Win(B). The instrument 10 preferably processes the required inputs and generates outputs in substantially real time.
Although exemplary implementations of the invention have been depicted and described in detail herein, it will be apparent to those skilled in the art that various modifications, additions, substitutions, and the like can be made without departing from the spirit of the invention. The above implementation, described in terms of a gyroscope, is equally applicable to a pair of parallel accelerometers having controllable reversibility of the sign of the scale factors.
The scope of the invention is defined in the following claims.
Number | Name | Date | Kind |
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7103477 | Lee | Sep 2006 | B1 |
20060055584 | Waite et al. | Mar 2006 | A1 |
Number | Date | Country | |
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20090292497 A1 | Nov 2009 | US |