On-Line MEMS Gyroscope Bias Compensation Technique Using Scale Factor Nulling

Transcription

1 On-Line MEMS Gyroscope Bias Compensation Technique Using Scale Factor Nulling Matthew J. Schultheis Penn State Great Valley School of Graduate Professional Studies 30 East Swedesford Road, Malvern, PA 19355, USA Abstract Two methods of estimating and compensating for the bias of MEMS gyroscopes when two gyroscopes are mounted input axis parallel are explored. One method is a general bias compensation technique that can be used with gyroscopes of any technology. The second method presented uses a unique characteristic of MEMS gyroscopes that allows for nulling the rate sensitive component of the gyroscope output. This method can lead to a decrease of the errors in gyroscope performance due to bias instabilities by over an order of magnitude. Keywords MEMS, Coriolis, Gyroscope, Bias Stability, Bias Compensation, Bias Estimation Nomenclature Ω Rotation Rate [deg/hr] B Gyroscope Bias [deg/hr] Ω Gyroscope Output [deg/hr] SF Gyroscope Scale Factor Error [ppm] 1 Introduction If one wishes to know the position, velocity, and attitude of a platform, be it a submarine, tank, missile, et cetera, one way of accomplishing this goal is with an inertial navigation system (INS). An INS is initialized with the platforms current location, velocity, and attitude. The INS then receives inertial space measurements of specific forces from accelerometers, and angular rotations from gyroscopes (Note: This paper will refer to all devices capable of sensing DC angular rate a gyroscopes, irrespective of whether or not the device in question s functionality is dependent on gyroscopic precession). Traditionally, while gyroscopes have a long history of use in many applications such as ship and aircraft navigation, they have been too expensive to use in many cost-sensitive applications, such a automobiles, or artillery shells. The advent of Micro-Electro-Mechanical Systems (MEMS) has allowed for gyroscope production costs to drop significantly. However, these new MEMS gyroscopes do not have performance on par with more expensive mechanical or optical gyroscopes. To make up for this performance deficiency, it is possible to combine the output of multiple gyroscopes. One way to do this is due to a unique capability of MEMS Coriolis gyroscope is to zero the rate dependent portion of the output, which allows for a measurement of the gyroscope bias. The rate sensitivity is then re-enabled, and the estimated bias can then be subtracted from the gyroscope output, allowing for a more accurate rate measurement. Matthew J. Schultheis, 2007: I grant the Pennsylvania State University the non exclusive right to use this work for the University's own purposes and to make single copies of the work available to the public on a not-for-profit basis if copies are not otherwise available.

2 2 Gyroscope Model The underlying physics that allows MEMS gyroscopes to function is the Coriolis force, also known as a fictitious force. A proof mass in the gyroscope which is driven (vibrated) in the X-axis, when rotated about the Z-axis, will experience a force along the Y-axis. The force experienced is given by the equation: F = 2 m*( v Ω ) (1) cor F This force cor is proportional to the input angular rate, Ω, that to which the gyroscope is subjected, as well as the mass of the proof mass, m, and its velocity, v. A block diagram of such a gyroscope is shown below in Figure 1. Figure 1 A Simple Coriolis Gyroscope (Nunzi et.al, 2006.) Current MEMS gyroscopes are more complex than this, and feature coupled oscillations between both the drive (X) axis and the sense (Y) axis. A couple of examples of coupled oscillator gyroscopes currently commercially available are shown below in Figure 2 and Figure 3.

3 Figure 2 Analog Devices ADXRS MEMS Gyroscope Figure 3 Astrise Coriolis Vibratory Gyroscope The Coriolis gyroscopes shown above, as well as many other current designs can be represented by the generalized model shown below in Figure 4 (Roszhart, 2006). V yo R A Demod. V out A c G c C dc k y, c y Y (coriolis) axis m Z (rate) axis X (dither) axis A d, G d R V xd cos(ω d t) k x, c x V xo Figure 4 Generic, Coupled Mode MEMS Gyro with Electrostatic Drive and Pickoff Mechanisms

4 Using the model shown above, the gyroscope s total rate sensitive response can be written as (Roszhart, 2006): V out 2 2 ε Ac Ad Qx Q 0 y = A ( Vxo Vxd Vyo ) Ω m x Gc G ω d ω y z (2) To remove the rate sensitive component of the gyroscope output, must be set zero. This can be achieved by setting the term ( V V V to zero, which can be accomplished by setting eitherv xo (X-axis bias voltage), or xd xo xd yo ) V (X-axis dither voltage), or V yo V out (Y-axis bias voltage) to zero. Once the rate sensitive component of the gyroscope output is removed, any remaining output that is left is bias error and noise. The bias that remains, an error source, can then be estimated, and removed mathematically from the gyroscope output once the rate sensitivity is restored. In general, the gyroscope output contains both the input to the gyroscope as well as systematic and random errors (Titteron and Weston, 2004). Where: Ω = (1 + SF ) * Ω + M Ω + M Ω + B + B a + B a + B a a + ε (3) x x x y y z z fx gx x gz z gxz x z x Ω x : The total gyroscope output (rate sensitive output plus bias plus noise) Ω, Ω, Ω z : The angular rotation rotate about the x, y, and z axis x y SF x : The x-axis scale factor (the difference between M y, M : y- and z- axis cross axis sensitivities z SF x and one is the scale factor error) B fx : The g-insensitive bias Bgx, Bgz, B gxz : Bias g sensitivities ε x : Zero mean measurement noise (angle random walk) While the gyroscope model above encompasses many of the error terms that affect gyroscope performance, the equation can be simplified by ignoring small errors. Also, error terms whose contributions neither affect nor are affected by the bias compensation process are also ignored, without loss of generality. One example of such an error source is the acceleration sensitive errors which are not included in the reduced model, as the bias compensation technique will only estimate and compensate for biases in the electronics. Acceleration sensitivities are caused by actual forces acting on the proof mass, not an electrical error. The reduced gyroscope model is then: ( ) % (4) Ω= Bt () SFt () * Ω+ ε

5 Both the bias and scale factor terms above are shown as a function of time. This is because both the bias and scale factor terms include stationary and time varying errors. One example of a time varying error source, and a major contributor to MEMS gyroscope performance errors, is temperature variation (Leland, 2005). This temperature variation can often be modeled as a Gauss-Markov process, with the correlation time being a function of the temperature profile that to which the gyroscope is expected to be subjected. The Gauss-Markov model also works for many other environmental sensitivity terms, such as magnetic, humidity, and pressure. For the purposes of this investigation, a Gauss-Markov process will be included in each the bias and the scale factor, where appropriate. Thus the gyroscope model becomes: % (5) Ω= ( B () t + B () t + B ()) t + (1 + SF () t + SF ())* t Ω+ ε const env markov gyro markov const markov A block diagram of the gyroscope model to be used is shown below in Figure 5. Figure 5 Block Diagram of Gyroscope Error Model To make a measurement of bias using either of the techniques described below, the output of the gyroscope needs to be measured for some amount of time. The measurement time required to get an adequate estimate of bias is related to the angle random walk of the gyroscope. Angle random walk is a white noise and is a dominate noise source (Niu et. al, 2006). The standard deviation in gyroscope rated data due to angle random walk can be computed by: ARW σ arw = (6) dt The uncertainty in the bias estimate due to ARW (at the 1-sigma level) is: σ arw δ = (7) N Where N is the number of samples taken in a bias estimate. To determine the amount of time needed to estimate the bias, a sampling time of 1/1024 seconds was assumed. This sampling time is used throughout the following simulations. The percent of maximum error due to ARW was plotted against averaging time, and is shown in Figure 6 below.

6 Figure 6 Bias Estimation Error Due to ARW vs. Averaging Time An averaging time of 0.5 seconds (512 samples) was chosen, as this provided for a reduction of the error due to ARW to less than 5% of the maximum. Also, further averaging leads to little gain in bias estimation accuracy. 3 Conventional Bias Compensation A conventional bias compensation scheme that works with gyroscopes of any technology is to use the output from one of the gyroscopes to estimate the bias of the other gyroscope. This process is then reversed to estimate the bias of the first gyroscope. This methodology is then repeated in time to update the bias estimate. The two gyroscopes can initially be modeled as: ( ) % (8) Ω= B () t SF() t * Ω+ ε ( ) % (9) Ω = B () t SF () t * Ω+ ε B () t 2 If one wishes to measure the bias of the second gyroscope ( ), one can use the output of the first gyroscope as a reference signal. The input rate can be estimated as (using only the first gyroscope): Ω Ω= Ω % B () t 1 1 ( 1 + SF ( t) ) 1 (10) The estimate of the bias of the second gyroscope is then: Ω % 1 B1() t B2() t =Ω% 2 ( 1 + SF2() t ) 1 + SF1 ( t) (11)

7 Likewise, the estimate of the bias of the first gyroscope is: Ω % 2 B2() t B1() t =Ω% 1 ( 1 + SF1() t ) 1 + SF2 ( t) (12) It should be noted this method requires a priori knowledge of the bias of one of the gyroscopes to work. To determine the initial biases, one may use a method such as gyro compassing, or some other method of independent bias measurement. The error in this initial bias estimate will also carry through and affect the performance of the system. The gyroscope error terms that will be used in this paper come from the Common Guidance, Common Sense program, a United States Army program to develop a common inertial measurement unit (containing 3 accelerometer and 3 gyroscopes) that can be produced cheaply for many different programs. The error budget for the conventional compensation is shown in Table 1. Table 1 Variable Scale Factor Gyroscope Error Budget Parameter Value Units Model ARW 0.2 deg/rt-hr White Noise (over DC to 512 Hz) Environmentally Driven Bias Gyroscope Specific Bias Environmentally Driven Scale Factor 100 deg/hr Gauss-Markov, with various correlation times 10 deg/hr Gauss-Markov, 30 second correlation 100 parts per million (PPM) time Gauss-Markov, with various correlation times Rate Profile 100 Deg/hr Gauss-Markov with 20 minute correlation time An example time plot of the raw bias and compensated bias is shown below. Note that as time increases, the validity of the bias estimate decreases. This is due to the dependence of the bias estimate for one gyroscope on knowledge of the correct scale factor of both gyroscopes, as well as the error in the bias estimate from the other gyroscope.

8 Figure 7 Example Output of One Monte Carlo Variable Scale Bias Compensation Run Like the binary mode analysis, various combinations of simulation run length, environmental signal correlation time, and bias update time were simulated: Table 2 Variable Scale Factor Simulation Parameters Parameter Values Modeled Simulation Length 60, 300, 600, 1800 seconds Environmental Markov Correlation Time 5, 15, 30, 60, 300, 900, 1200 seconds Bias Estimate Update Interval 1, 5, 10, 15, 30, 60, 150, 300 seconds The various parameters were use to determine the best bias estimate update interval to use depending on the environmental signal correlation time. Updating the bias estimates more frequently than necessary would remove the square-root of 2 noise reduction benefit of just averaging the 2 gyroscope outputs. Updating the bias estimates less frequently than necessary would leave an invalid bias estimate being compensated for, which could lead to decreased performance. To evaluate the performance of each simulation setup, for each run, a root-mean-square (RMS) error was calculated for each of the each of the raw gyroscope outputs, an arithmetic average of gyroscope 1 and gyroscope 2, and the

9 compensated output of the gyroscope. The values for these parameters form each of the 30 runs were then RMSed together, to create one data point for each combination of run length, environmental signal correlation time, and bias estimate update time. Shown below in Figure 8 are contour plots of the ratio of RMS error computed using the averaging methodology divided by the RMS error computed using the bias compensation methodology. The higher the contour value, the better the compensation methodology works as compared to the averaging methodology. A value of one indicates that the two methodologies work equally as well. A negative value indicates that the averaging methodology performed better than the bias compensation methodology. Note that although scale factor errors were modeled, as required to estimate bias, the effects of the error in scale factor are not included in the results. Figure 8 Contour Plot of Results from Conventional Bias Compensation Simulations The results from the conventional compensation methodology show, as expected that at longer correlation times for the environmental signal correlation time, the better the bias compensation methodology performs. Also, the shorter the run time, the better the performance of the compensation methodology, going from 5X improvement at 60 seconds under the best conditions, to nearly identical performance under the best conditions at 1800 seconds. 4 Nulled Scale Factor Model and Simulation Another method of bias compensation to be considered is specific to MEMS Coriolis gyroscopes having a scale factor that can be nulled (zeroed) out. This nulling removes the rate sensitivity of the gyroscope, leaving only the bias and noise in the output signal

10 Due to the fact that, in the nulled scale factor mode bias compensation methodology, errors in scale factor play no part in the bias estimation process, the scale factor error term is dropped from the gyroscope error model. Also, the constant portion of the bias is dropped from the model, as it will be completely estimated in the first, and each subsequent, bias estimate. The reduced error therefore then becomes: % (13) Ω= B () t + B () t + ε env markov gyro markov The gyroscope errors used in the following simulations are: Table 3 Binary Scale Factor Gyroscope Error Budget Parameter Value Units Model ARW 0.2 deg/rt-hr White Noise (over DC to 512 Hz) Environmentally Driven Bias Gyroscope Specific Bias 100 deg/hr Gauss-Markov, with various correlation times 10 deg/hr Gauss-Markov, 30 second correlation time The simulation was set up with varying parameters, to better measure the effects of different bias estimate update intervals when the environmental signal properties vary. The parameters used in the simulation, and their values are: Table 4 Binary Scale Factor Simulation Parameters Parameter Values Modeled Simulation Length 60, 300, 600, 1800 seconds Environmental Markov Correlation Time 5, 10, 15, 30, 60, 120, 300, 600, 900, 1200 seconds Bias Estimate Update Interval 1, 2, 5, 10, 15, 30, 60, 120, 300 seconds A Monte-Carlo analysis of 30 runs was performed for each combination shown above. The sample output from one of the runs is shown below. The blue line shows the raw gyroscope error from one of the gyroscopes, the green line shows the estimated bias value for the gyroscope, and the red line shows the error in the compensated gyroscope output. The reduction in gyroscope error can clearly be seen in Figure 9.

11 Figure 9 Example Output of One Monte Carlo Nulled Scale Bias Compensation Run

12 Contour plots of the simulation results are shown below in Figure 10. Figure 10 Contour Plot of Results from Nulled Scale Factor Simulations The simulation results show, as expected, that the longer the correlation time, the better the improvement in performance, at all simulation run lengths. This is due to the fact that at long correlation times, the previous bias estimate remains valid for a longer period of time. The performance also improves relative to the averaging method as the bias estimate is updated more frequently. To achieve optimum performance using this method, the environmental model should be slow changing, and the bias estimate updates should be made frequently. Unlike the conventional bias compensation methodology, which at best is a 5X improvement in performance (at short run times), the nulled scale factor compensation methodology performs best at longer run times with improvement better than 25X that possible from averaging the output of two sensors. Also, unlike the conventional bias estimation method, the nulled scale factor method is very dependent on the bias estimate update time. The faster the bias estimate is updated, the better the performance. 5 Conclusions The analysis and simulations above show that it is feasible to use two input-axis parallel MEMS gyroscopes, having scale factors that can be nulled, to perform real time bias estimation and compensation in addition to the independent noise reduction that is possible. The magnitude of the performance gain is heavily dependent on a couple of factors. The first factor is the rate of change of the gyroscope bias, usually due to environmental change. The slower the rate

13 of change in the bias, the longer that the bias estimate remains valid. The other factor in performance is the rate at which the bias estimate is updated. Continuously updating the bias estimate provides the best performance, at a factor of 10X to 25X that performance gain achieved just averaging the output of the two gyroscopes. The possibility of a 10X to 25X improvement in gyroscope performance while using the nulled scale factor methodology indicates that the extra cost in both the monetary sense, as well as size, power consumption, and calculations would be worth the performance gain in certain applications. The simulation results also indicated that for maximum benefit from the compensation method, a good understanding of the bias processes, including environmental signals in the application / mission, is required. Another benefit of both of the bias compensation methodology is the inherent redundancy of having two input axis parallel gyroscopes. This method could also be used as a status monitor to aid in the identification of faulty gyroscopes based on large biases. 6 Further Investigation One area of possible further investigation is a better model of the gyroscope error process. Errors in scale factor and the ability to null it will lead to errors in the bias estimates. Including these terms in the analysis could lead to more realistic results, including better knowledge of the optimum bias estimate update time. The typical inertial navigation system has 3 orthogonal gyroscopes to measure angular rate. In such a configuration, it may be possible to implement the bias compensation methodology with the addition of only one more gyroscope in an umbrella angle configuration (the fourth 45 degrees from each of the three orthogonal axes). Such a bias estimation methodology would be more computationally intensive, but would limit the excess manufacturing costs, as well as power and size requirements. However, due to the way MEMS devices are manufactured, it may be advantageous to stick with the IA parallel mounting scheme. Many navigation and other systems requiring gyroscopes often times have external references, such as GPS. When using GPS, it is possible to estimate the gyroscope biases using estimation techniques including Kalman filtering. Inertial navigation systems are generally good at providing high frequency position, velocity, and attitude data, but are inferior to external reference systems such as GPS at low frequencies (Maybeck). 7 Acknowledgements I would like to acknowledge the following persons for their assistance in the preparation of this paper: Dr. Michael Piovoso, Professor of Electrical Engineering, Penn State University Great Valley, for serving as my advisor Mr. Terry Roszhart, Research Engineer, Penn State University Applied Research Laboratory Navigation Research and Development Center, for the initial idea of scale factor nulling to estimate bias Mr. Phil Franco, Research Engineer, Penn State University Applied Research Laboratory Navigation Research and Development Center, for verifying the validity of my analysis assumptions and techniques 8 References Leland, R. P., (2005), Mechanical-Thermal Noise in MEMS Gyroscope, IEEE Sensors Journal, Vol. 5, No. 3, June 2005 Maybeck, P. S., (1994), Stochastic Models, Estimation and Control, Arlington: Navtech Book and Software Store Niu, X., Goodall, C., Nassar, S. El-Sheimy, N., (2006), An Efficient Method for Evaluation the Performance of MEMS IMUs, Position, Location, and Navigation Symposium, San Diego, CA, 4/25/2006 4/27/2006 Nunzi, E., Antonello, P., Carbone, P., Oboe, R., Lasalandra, E., Spinola, G., Prandi, L., Rizzo, A., (2006) A Demodulation Technique for the Sensing Circuit of a MEMS Gyroscope, IMTC 2005 Instrumentation and Measurement Technology Conference, 4/24/2006-4/27/2006, Sorrento, Italy.

14 Titterton, D.H., Weston, J. L., (2004), Strapdown Inertial Navigation Technology, Lexington, Massachusetts: MIT Lincoln Laboratory Roszhart, T., (2006), Real Time IMU Bias Compensation without External Reference Using Commutation MEMS Gyros, Proposal to United States Army Research, Development and Engineering Command CERDEC, Fort Monmouth, NJ United States Army, Research Development and Engineering Command, Armament Research, Development and Engineering Center, (Unknown), Common Guidance IMU Performance Specifications 9 Biography Matthew Schultheis is an Assistant Research Engineer at Penn State University Applied Research Laboratory Navigation Research and Development Center. His work includes research of the error characteristics of inertial sensors, as well as the modeling of many inertial navigation systems, including the most accurate inertial navigation systems in the world. He is currently working towards a Master s degree in Systems Engineering at Penn State University Great Valley.