The Kalman Filter: Navigation's Integration Workhorse
INNOVATION
noisy measurements to estimate the desired The Kalman Filter: signals. The estimates are statistically opti mal in the sense that they minimize the mean Navigation's square estimation error. This has been shown to be a very general criterion in that many Integration other reasonable criteria (the mean of any monotonically increasing, symmetric error Workhorse function such as the absolute value) would yield the same estimator. The Kalman filter was a dramatic improvement over its mini Larry J. Levy mum mean square error predecessor, in vented by Norbert Wiener in the 1940s, The Johns Hopkins University which was primarily confined to scalar sig Applied Physics laboratory nals in noise with stationary statistics. Figure 2 illustrates the Kalman filter algo rithm itself. Because the state (or signal) is typically a vector of scalar random variables "Innovation" is a regular column featuring (rather than a single variable), the state ·discussions about recent advances in GPS uncertainty estimate is a variance-covariance technology and its applications as well as the matJ.ix- or simply, covariance matrix. Each fundamentals ofGPS positioning. The column diagonal term of the matrix is the variance of is coordinated by Richard Langley of the a scalar random variable - a description of Department of Geodesy and Geomatics its uncertainty. The term is the variable's Since its introduction in 1960, the Kalman Engineering at the University of New mean squared deviation from its mean, and filter has become an integral component in Brunswick, who appreciates receiving your its square root is its standard deviation. The thousands of military and civilian navigation comments as well as topic suggestions for matrix's off-diagonal terms are the covari systems. This deceptively simple, recursive future columns. To contact him, see the ances that describe any correlation between digital algorithm has been an early-on favorite "Columnists" section on page 4 of this issue. pairs of variables. for conveniently integrating (or fusing) The multiple measurements (at each time navigation sensor data to achieve optimal When Rudolf Kalman formally introduced point) are also vectors that a recursive algo overall system performance. To provide the Kalman filter in 1960, the algorithm was rithm processes sequentially in time. This current estimates of the system variables - well received: The digital computer had suf means that the algorithm iteratively repeats such as position coordinates- the filter uses ficiently matured, many pressing needs itself for each new measurement vector, statistical models to properly weight each new existed (for example, aided inertial naviga using only values stored from the previous measurement relative to past information. It tion), and the algorithm was deceptively sim cycle. This procedure distinguishes itself also determines up-to-date uncertainties of the ple in form. Engineers soon recognized, from batch-processing algorithms, which estimates for real-time quality assessments or though, that practical applications of the must save all past measurements. for off-line system design studies. Because of algorithm would require careful attention to Starting with an initial predicted state esti its optimum performance, versatility, and ease adequate statistical modeling and numerical mate (as shown in Figure 2) and its associ of implementation, the Kalman filter has been precision. With these considerations at the ated covariance obtained from past infor especially popular in GPS/inertial and GPS forefront, they subsequently developed thou mation, the filter calculates the weights to be stand-alone devices. In this month's column, sands of ways to use the filter in navigation, Larry Levy will introduce us to the Kalman surveying, vehicle tracking (aircraft, space ------~ filter and outline its application in GPS craft, missiles), geology, oceanography, fluid I 1 Multiple navigation. dynamics, steel/paper/power industries, and 1 noise Dr. Levy is chief scientist of the Strategic demographic estimation, to mention just a inputs Systems Department of The Johns Hopkins few of the myriad application areas.
University Applied Physics Laboratory. He Multiple I received his Ph.D. in electrical engineering EQUATION-FREE DESCRIPTION noises Multiple noisy 1 -from Iowa State University in 1971. Levy has The Kalman filter is a multiple-input, multi ___outputs______]1 worked on applied Kalman filtering for more ple-output digital filter that can optimally than 30 years, codeveloped the GPS translator estimate, in real time, the states of a system concept in SATRACK (a GPS-based missile based on its noisy outputs (see Figure 1). tracking system), and was instrumental in These states are all the variables needed to developing the end-to-end methodology for completely describe the system behavior as a evaluating Trident II accuracy. He conducts function of time (such as position, velocity, graduate courses in Kalman filtering and voltage levels, and so forth). In fact, one can Figure 1. The purpose of a Kalman filter system identification at The Johns Hopkins think of the multiple noisy outputs as a multi is to optimally estimate the values of University Whiting School of Engineering and dimensional signal plus noise, with the sys variables describing the state of a teaches Navtech Seminars's Kalman Filtering tem states being the desired unknown system from a multidimensional signal short course. signals. The Kalman filter then filters the contaminated by noise.
September 1997 GPS WORlD 65 I N NO VA TION
used when combining this estimate with the estimate follows only the deterministic trans (if all states are observable) because of the first measurement vector to obtain an formation, because the actual noise value is accumulated information from the measure updated "best" estimate. If the measurement unknown. The covariance prediction ac ments. However, because information is lost noise covariance is much smaller than that of counts for both, because the random noise's (or uncertainty increases) in the prediction the predicted state estimate, the measure uncertainty is known. Therefore, the predic step, the uncertainty will reach a steady state ment's weight will be high and the predicted tion uncertainty will increase, as the state when the amount of uncertainty increase in state estimate's will be low. estimate prediction cannot account for the the prediction step is balanced by the uncer Also, the relative weighting between the added random noise. This last step completes tainty decrease in the update step. If no ran scalar states will be a function of how the Kalman filter's cycle. dom noise exists in the actual model when "observable" they are in the measurement. One can see that as the measurement vec the state evolves to the next step, then the Readily visible states in the measurement tors are recursively processed, the state esti uncertainty will eventually approach zero. will receive the higher weights . Because the mate's uncertainty should generally decrease Because the state estimate uncertainty filter calculates an updated state estimate using the new measurement, the state esti mate covariance must also be changed to Pred;ctw ;o;:~ Compute weights from predicted state reflect the information just added, resulting state estimate easurements covariance & measurement noise covariance ~:Now in a reduced uncertainty. The updated state & covariance each cycle estimates and their associated covariances t form the Kalman filter outputs. IPredict state estimate & covariance I Update state estimates as weighted Finally, to prepare for the next measure to next t1me step linear blend of predicted state estimate ment vector, the filter must project the ~ & "~ mooomomoo< updated state estimate and its associated covariance to the next measurement time. Compute new covariance of updated state estimate Y Upd:ted The actual system state vector is assumed to state estimate change with time according to a deterministic Figure 2. The Kalman filter is a recursive, linear filter. At each cycle, the state linear transformation plus an independent estimate is updated by combining new measurements with the predicted state random noise. Therefore, the predicted state estimate from previous measurements.
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66 GI'S WORLD Seplember 1997 Circle 54 changes with time, so too will the weights. Generally speaking, the Kalman filter is a digital filter with time-varying gains. Inter ested readers should consult "The Mathe matics of Kalman Filtering" sidebar for a summary of the algorithm. If the state of a system is constant, the Kalman filter reduces to a sequential form of deterministic, classical least squares with a weight matrix equal to the inverse of the measurement noise covariance matrix. In other words, the Kalman filter is essentially a recursive solution of the least-squares prob lem. Carl Friedrich Gauss first solved the problem in 1795 and published his results in 1809 in his Theoria Motus, where he applied the least-squares method to finding the orbits of celestial bodies (see the "What Gauss Said" sidebar). All of Gauss's statements on the effectiveness_ of least squares in analyzing measurements apply equally well to the Kalman filter.
A SIMPLE EXAMPLE A simple hypothetical example may help clar ify the concepts in the preceding section. Consider the problem of determining the actual resistance of a nominal 100-ohrn resis tor by making repeated ohmmeter measure ments and processing them in a Kalman filter. First, one must determine the appropriate statistical models of the state and measure-
What Gauss Said
If the astronomical observations and other quan tities, on which the computation of orbits is based, were absolutely correct, the elements also, whether deduced from three or four obser vations, would be strictly accurate (so far indeed as the motion is supposed to take place exactly according to the laws of Kepler), and, therefore, if other observations were used, they might be confirmed, but not corrected. But since all our measurements and observations are nothing more than approximations to the truth, the same must be true of all calculations resting upon them, and the highest aim of all computations made concerning concrete phenomena must be to approximate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requi site for the determination of the unknown quanti ties. This problem can only be properly undertak en when an approximate knowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most accurate manner possible. - From Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, a translation by C.H. Davis of C. F. Gauss's 1809 Theoria Matus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. Davis's 1857 translation was republished by Dover Publications, Inc., New York, in 1963.
Circle Sl September 1997 GPS WORLD 67 Recruitment INNOVATION Director Avionics Engineering Center Ohio University 104.------.------.------4 The School of Engineering and * Computer Science has reopened its 103 '1n * E 3 search for director of the Avionics Ul * .r:. true' E. _§ * * * Q) Engineering Center (AEC). The AEC (.) E. 102 * "' c is a recognized leader in the research, ro 2 • Predicted g +! - ~ development, and field engineering > <'ll101 0 • Updated ·u; (.) support of landing and navigation Q) Q) * measurement a: 1ii systems in the National Airspace + estimate E 100 ~ System (NAS). The Center has a staff w • of 2 8 full-time technical and • • I administrative personnel and 3 0 99 0 0 • 5 10 15 0 2 3 4 5 student interns. Staff and students are Time steps Time steps funded by grants and contracts from state and local governments, federal Figure 3. The individual resistance Figure 4. Initially, the predicted uncer sponsors such as FAA, NASA, DOD, measurements of a nominally 100-ohm tainty (variance) of the resistor's value 2 and industrial firms. Research facilities resistor are scattered about the true is 4 ohms and is simply based on the include extensive laboratories and value of slightly less than 102 ohms. manufacturer-provided tolerance value. single and multi-engine aircraft. The Kalman filter estimate gradually However, after six measurements, the The Director is responsible for all converges to this value. estimated variance drops to below 2 aspects of AEC operations including 0.2 ohms . technical, administrative, and ment processes so that the filter can compute ·promotional activities. In addition, the the proper Kalman weights (or gains). Here, where x010 denotes the best estimate at time 0, Director is expected to advise graduate only one state variable- the resistance, x based on the measurement at time 0. Note students on thesis and dissertation is unknown but assumed to be constant. So that the measurement receives a relatively activities. To facilitate these academic the state process evolves with time as high weight because it is much more precise duties, appointment as a part-time (less uncertain) than the initial state estimate. [1] faculty member in the School of The associated uncertainty or variance of the Electrical Engineering and Computer Note that no random noise corrupts the updated estimate is Science at the rank of Professor is state process as it evolves with time. Now, anticipated. the color code on a resistor indicates its pre PolO= (1 - KolPot- cision, or tolerance, from which one can Minimum qualifications: earned =(4l1)4=l [5] Ph.D. in electrical engineering or deduce - assuming that the population of closely related discipline; 10 years resistors has a Gaussian or normal histogram Also note that just one good measurement direct experience in research, -that the uncertainty (variance) of the 100- decreases the state estimate variance from 4 engineering, and project management ohrn value is, say, (2 ohrnf So our best esti to 4/5. According to equation [1], the actual in electrical engineering and aviation mate of x, with no measurements, is x01_ = state projects identically to time 1, so the esti fields; demonstrated capability for 100 with an uncertainty of P01_ = 4. Repeated mate projection and variance projection for effective interaction with government ohmmeter measurements, the next measurement at time 1 is and industry leaders at all levels; broad zk xk + vk , [2] xllo xoto; Plio= · [6] knowledge of civil aviation operations, = = PolO=~ the NAS infrastructure, and the federal directly yield the resistance value with some Repeating the cycle over again, the new gain agencies involved in developing and measurement noise, vk (measurement errors is operating the NAS. It is desirable that from tum-on to turn-on are assumed uncorre Plio _ 4/5 the Director be a rated pilot with lated). The ohmmeter manufacturer indicates [7] Plio+ R - 4/5 + 1 instrument certificate. Salary will be the measurement noise uncertainty to be Rk = 1 commensurate with background and (1 ohrn)2 with an average value of zero about and the new update vari.ance is experience. the true resistance. P111 = (1- K1)Pllo Send resume, letter of application, Starting the Kalman filter at k = 0, with the [8] and names and contact information of initial estimate of 100 and uncertainty of 4, -( 1 \1_4 -415+1}5_9. three professional references to Dr. the weight for updating with the first mea Dennis Irwin, Chair, School of surement is Figure 3 represents a simulation of this Electrical Engineering and Computer process with the estimate converging toward Science, Ohio University, Athens, OH K - Pot- - 4 [3] the true value. The estimation uncertainty for o-po,_+Ro -4+1 45701. Screening of applications will this problem, which the Kalman filter pro begin on September 2, 1997 and with the updated state estimate as vides, appears in Figure 4. One can see that continue until the position is filled. the uncertainty will eventually converge to Ohio University is an affirmative action Xoto = (1-Ko)xot- + KoZo zero. [4] and equal opportunity employer. A New Set of Conditions. Let's now change the =(4l1)100 +(4! 1)zo problem by assuming that the measurements
68 GPS WORlD September 1997 INNOVATION
astic use in aided inertial applications . 4 Integrating GPS with an inertial naviga tion system (INS) and a Kalman filter pro VJ vides improved overall navigation perfor ~ 3 .3. mance. Essentially, the INS supplies virtually Q) () noiseless outputs that slowly drift off with c time. GPS has minimal drift but much more -~"' 2 • Predicted > 0 • Updated noise. The Kalman filter, using statistical () Q) models of both systems, can take advantage iii E 1 • Best of their different error characteristics to opti ~ total mally minimize their deleterious traits . w • • • estimates • • • • As shown in the "The Mathematics of 0 l ______l 0 2 3 4 5 Kalman Filtering" sidebar, the Kalman filter Possible resets Time steps (years) is a linear algorithm and assumes that the process generating the measurements is also Figure s. The true resistance of a Figure 6. An integrated GPS receiver linear. Because most systems and processes resistor in an environment with a widely and inertial navigator use a Kalman (including GPS and INS) are nonlinear, a varying temperature is not quite con filter to improve overall navigation method of linearizing the process about some stant. If this is modeled assuming a performance. known reference process is needed. Figure 6 variation with an average of zero and a illustrates the approach for integrating GPS 2 variance of 0.25 ohms , the Kalman GPS/INS INTEGRATION and inertial navigators. Note that the true val filter estimate of the resistance variance We can see that the Kalman filter provides a ues of each system cancel out in the measure converges to 0.4 ohms2 after only a few simple algorithm that can easily lend itself to ment into the Kalman filter so that only the measurements. integrated systems and requires only ade GPS and inertial errors need be modeled. quate statistical models of the state variables The reference trajectory, one hopes, is suffi and associated noises for its optimal perfor ciently close to the truth so that the error are taken one year apart with the resistor mance. This fact led to its wide and enthusi- models are linear and the Kalman filter is placed in extreme environmental conditions so that the true resistance changes a small !--;::======~ amount. The manufacturer indicates that the small change is independent from year to year, with an average of zero and a variance of 114 ohms2. Now the state process will evolve with time as [9] where the random noise, wk, has a variance of Qk = 114. In the previous case, the variance prediction from time 0 to time 1 was constant as in equation [6]. Here, because of the ran dom noise in equation [9], the variance pre diction is Puo=Poto+Qo=~+!=l.05 · [10] Now the gain and update variance calcula tions proceed on as in equations [7] and [8] but with larger values for the predicted vari ance. This will be repeated every cycle so that the measurement update will decrease the variance while the prediction step will increase the variance. Figure 5 illustrates this tendency. Eventually, the filter reaches a steady state when the variance increase in the Contact: The Embedded Projects Manager prediction step matches the variance decrease in the measurement update step, with Pk+ Ilk= Test Equipment Division 0.65 and Pklk = 0.4. The Qk represents a very Dynamics Research Corporation important part of the Kalman filter model 93 Border Street, West Newton, MA 02165 because it tells the filter how far back in time to weight the measurements. An incorrect Tel: (800) 522-4321 Ext #15# value of this parameter may dramatically Loc: (617) 965-1346 I Fax.: (617) 244-7726 affect performance.
Circle 53 September 1997 GPS WORLD 69 INNO V ATION
The Mathematits of Kalman Filtering GPS·ONLY NAVIGATION In some applications, an INS is not desired or The Kalman filter assumes that the system state time tk, based on measurements up to ti, and Pklj may not be available, as in a stand-alone GPS vector, xk, evolves with time as is the corresponding "optimal" estimation error receiver. In such cases, the Kalman filter . xk+1 =
Update covariance PRACTICAL DESIGNS Regardless of an application's equipment Pklk = ~ - KkHJ Pk/k- 1 x klk be it GPS, INS, or other devices- develop ing a practical Kalman filter-based naviga Figure 7. The Kalman filter algorithm involves four steps: gain computa tion system requires attention to a variety of tion, state estimate update, covariance update, and prediction. design considerations. The filter's covariance analysis portion optimal. For most GPS applications this is pled arrangement when the receiver outputs (not requiring real data; see Figures 2 and 7) the case. provide position without raw measurements. uses predetermined error models of potential So, even though the overall systems are The open-loop correction approach of Fig systems (GPS, inertial, and so forth) to pre nonlinear, the Kalman filter still operates in ure 6 is termed linearized Kalman filtering. dict the particular configuration's perfor the linear domain. Of course, the state vari An alternate approach in which the algo mance. The filter designer repeats this for ables for the Kalman filter must adequately rithm feeds the estimates back to the inertial different potential equipment (models) until model all error variables from both systems. system to keep the reference trajectory close the requirements are satisfie~ . In some cases, GPS errors could include receiver clock, to the truth is an example of extended Kal one must implement the Kalman filter in a selective availability, ionospheric, tropos man filtering. "small" computer with only a few states to pheric, multipath, and satellite ephemeris and clock errors. Inertial inaccuracies, on the Further Reading For a comprehensive selection of reprints of other hand, could include position, velocity, Kalman filter theory and application papers, including orientation, gyro, accelerometer, and gravity The literature on Kalman filtering abounds, with appli some of the germinal ones from the 1960s and those cations ranging from spacecraft navigation to the from the IEEE Transactions on Automatic Control errors. The equipment quality and the appli demographics of the French beef cattle herd. To ease special issue, see cation requirements will determine how you into it, here are a few suggestions. • Kalman Filtering: Theory and Application, edited extensive the error models must be. For the seminal introduction of the Kalman filter by H.W. Sorenson, published by IEEE Press, New algorithm, see If the GPS outputs are user position, one York, 1985. • "A New Approach to Linear Filtering and For a discussion about speCial covariance analy terms the integration architecture as loosely Prediction Problems," by R.E. Kalman in the Journal sis and numerically robust algorithms, see the lecture coupled. A tightly coupled architecture of Basic Engineering, the Transactions of the notes depicts one in which the GPS outputs are American Society of Mechanical Engineers, Series D, • Applied Kalman Filtering, Navtech Seminars, pseudoranges (and possibly carrier phases) Vol. 83, No. 1, pp. 35-45, March 1960. Course 457, presented by L.J. Levy, July 1997. For an excellent, comprehensive introduction to For an introductory discussion about GPS and and the reference trajectory is used (along Kalman filtering, including a GPS case study, see inertial navigation integration, see with the GPS ephemeris from the receiver) to • Introduction to Random Signals and Applied • "Inertial Navigation and GPS," by M.B. May, in predict the GPS measurements . In the tightly Kalman Filtering (with Matlab exercises and solutions), GPS World, Vol. 4, No.9, September 1993, pp. 3d edition, by R.G. Brown and P.Y.C. Hwang, pub 56-66. coupled system, the measurement errors lished by John Wiley & Sons, Inc., New York, 1997. Several good Web sites devoted to Kalman filter would be in the range domain rather than the For discussions about various Kalman filter appli ing exist, including position domain. Usually, the tightly coupled cations, see • "The Kalman Filter," a site maintained by G. arrangement is preferred because it is less • IEEE Transactions on Automatic Control, Special Welch and G. Bishop of the University of North issue on applications of Kalman filtering, Vol. AC-28, Carolina at Chapel Hill's Department of Computer sensitive to satellite dropouts, and adequate No. 3 published by the Institute of Electrical and Science:
70 GPS WOR lD September 1997 INNOVATION
model the process. This suboptimal filter of choice in GPS-based navigation systems. optimal integrated combination of large must be evaluated by special covariance It requires sufficiently accurate multidimen scale systems. The recursive nature of the fil analysis algorithms that recognize the differ sional statistical models of all variables and ter allows for efficient real-time processing. ences in the real-world model producing the noises to properly weight noisy measurement Off-line covariance studies enable the inte measurements and the implemented filter data. These models enable the filter to grated system performance to be predicted model. Finally, once the filter meets all per account for the disparate character of the before development, providing a convenient formance requirements, a few simulations of errors in different systems, providing for an and easy-to-use system design tool. • all processes should be run to evaluate the adequacy of the linearization approach and search for numerical computational errors. Corredion Table 4. Defining parameters of the PZ·90 datum In most cases, the extended Kalman filter Some of the Parametry Zemli 1990 (PZ- (with resets after every cycle) will ameliorate 90) datum parameter values listed in Parameter Value any linearization errors. Numeric computa Table 4 of this column's July 1997 article, 72.921 15 x 10·6 radians s·1 tional errors caused by finite machine word "GLONASS: Review and Update," are Earth rotation rate 9 length manifest themselves in the covariance incorrect. Although these values are iden Gravitational constant 398 600.44 x 10 m3s-2 matrices, which become nonsymmetric or tified as pertaining to PZ-90 in the latest Gravitational constant of atmosphere 0.35 x 109 m3s·2 have negative diagonal elements, causing version of the GLONASS Interface Control Document (dated October 4, Speed of light 299 792 458 ms·1 potentially disastrous performance. This 1995), they actually pertain to its prede Second zonal harmonic problem can be alleviated by increasing the cessor, the Soviet Geodetic System 1985. of the geopotential 1082.6257 x w-s computational precision or by employing a The correct PZ-90 parameter values, to be Ellipsoid semimajor axis 6378136m theoretically eq-uivalent but more numeri used with GLONASS ephemerides, are Ellipsoid flattening 1/298.257 84 listed to the right. The inverse ellipsoid cally robust algorithm. Equatorial acceleration flattening is a rounding of the full preci of gravity 978 032.8 mgal sion value of 298.257 839 303. Note also Correction to acceleration CONCLUSIONS that the sign of the geopotential second Because of its deceptively simple and easily of gravity at sea level zonal harmonic, 12, is given here as posi because of atmosphere --0.9 mgal programmed optimal algorithm, the Kalman tive in keeping with convention. filter continues to be the integration method
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