IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005

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Observability of Error States in GPS/INS Integration Sinpyo Hong, Man Hyung Lee, Senior Member, IEEE, Ho-Hwan Chun, Sun-Hong Kwon, and Jason L. Speyer, Fellow, IEEE

Abstract—Observability properties of errors in an integrated navigation system are studied with a control-theoretic approach in this paper. A navigation system with a low-grade inertial measurement unit and an accurate single-antenna Global Positioning System (GPS) measurement system is considered for observability analysis. Uncertainties in attitude, gyro bias, and GPS antenna lever arm were shown to determine unobservable errors in the position, velocity, and accelerometer bias. It was proved that all the errors can be made observable by maneuvering. Acceleration changes improve the estimates of attitude and gyro bias. Changes in angular velocity enhance the lever arm estimate. However, both the motions of translation and constant angular velocity have no influence on the estimation of the lever arm. A covariance simulation with an extended Kalman filter was performed to confirm the observability analysis. Index Terms—Global Positioning System, GPS/INS, inertial measurement unit (IMU), inertial navigation, lever arm, observability.

I. INTRODUCTION

T

HIS paper considers the integration of an accurate singleantenna Global Positioning System (GPS) measurement system with a low-grade inertial measurement unit (IMU). The positioning accuracy of the GPS measurement is assumed to be centimeter level. This type of accuracy can be obtained from carrier phase differential GPS (CDGPS). The errors in IMU are assumed to be so large that gyro error is much greater than the rotational speed of earth and accelerometer error is much greater than the rotational speed of earth multiplied by the velocity estimation error. Low-cost inertial sensors made with the current microelectromechanical systems (MEMS) technologies usually have these noise levels. There are a couple of important problems in this type of integration. Lever arm uncertainty can be a serious problem in accurate navigation systems. While the GPS antenna is mounted on the outside surface of a vehicle, IMU is usually placed inside the vehicle. Thus, the direct measurement of the distance

Manuscript received March 3, 2004; revised July 20, 2004. This work was supported by the Korea Science and Engineering Foundation through the Advanced Ship Engineering Research Center at Pusan Nation University. The review of this paper was coordinated by Dr. R. Klukas. S. Hong is with the Advanced Ship Engineering Research Center, Pusan National University, Busan 609-735, Korea (e-mail: [email protected]). M. H. Lee is with the School of Mechanical Engineering, Pusan National University, Busan 609-735, Korea. H.-H. Chun is with the Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, Korea. S. H. Kwon is with the Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, Korea. He is also presently a visiting scholar at the Department of Civil Engineering, Texas A&M University, College Station, TX 77843–3136 USA. J. L. Speyer is with the Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095-1597 USA. Digital Object Identifier 10.1109/TVT.2004.841540

between the GPS antenna and IMU is often quite difficult. The error in the estimated value of the lever arm, the relative position of the GPS antenna with respect to IMU, can be of great concern in accurate navigation systems [1]. The lever arm error in large vehicles can be much greater than the centimeter-level error in CDGPS measurements. In addition, the lever arm error can increase errors in the estimates of position, attitude, and inertial sensor biases [2], [3]. For a vehicle that moves with slow changes in attitude and acceleration, the attitude of the vehicle is unobservable with a single-antenna GPS measurement system [4]. Moreover, as will be shown later in this paper, the component of gyro bias in the direction of specific force is not observable if the gyro has a large error. Thus, yaw error can increase significantly fast with time. These unobservable modes increase errors in the estimates of position, velocity, and accelerometer bias. Especially, position error can be considerable with large attitude error if the lever arm is long. One approach to dealing with the above problems is designing an estimator for the uncertain values such as the lever arm, attitude, and biases of IMU. This approach can be particularly useful when there is no other instrument to measure the uncertain values. It is well known that motions of a vehicle can improve observability of the states of inertial navigation systems (INS) and inertial sensor errors. The effect of maneuvering on the observability during in-flight alignment (IFA) was investigated in [5]–[9]. A control-theoretic approach to the observability study on IFA was first introduced in [9] using a piecewise constant system modeling [10]. The research on the observability enhancements was mainly concerned with the effect of the translatory motions such as changes in acceleration. In this paper, a control-theoretic framework was introduced for the observability analysis of errors in the integrated systems of GPS and INS. Changes in attitude as well as acceleration were considered in the observability analysis. The errors in the observability analysis were described in the earth-centered earth-fixed (ECEF) frame and the body frame. Errors in position, velocity, IMU attitude, biases in gyro and accelerometer, and lever arm were considered in the observability analysis. The terms associated with position and velocity errors in the velocity error propagation equation were neglected because they were relatively very small compared with other error terms in the integration of low-grade IMU and accurate GPS measurements. This simplification made the observability analysis uncomplicated. Among the inertial sensor errors such as biases, scale-factor errors, and alignment errors, biases are the most unpredictable and dominant in low-grade sensors. Since the period of testing for the error estimation is relatively short compared with the time-constant of bias drifts, the biases in in-

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ertial sensors ae considered to be constant in this paper. Time derivatives of GPS measurements were used for observability test. This approach changes the observability test on the 18 error states to that on at most nine error states; errors in attitude, gyro bias, and lever arm. Compared with the analysis based upon the null space test of observability matrices [3], [4], this approach makes the observability analysis simpler. In this paper it is shown that the time-invariant error dynamics model for the integrated navigation system has seven unobservable modes when the position of a vehicle is measured with a single antenna GPS measurement system. Errors in the lever arm, attitude, and component of gyro bias in the direction of gravity are not observable if the vehicle moves with constant attitude and acceleration. Both the errors in attitude and lever arm contribute to the position error. It was shown in previous research that if the lever arm error is neglected, the time-invariant error dynamics model with two horizontal channels has three unobservable modes [11], [12]. Later, in [4] and [9], the error model for the three channels was also shown to have three unobservable modes. The addition of the unobservable gyro bias in this paper for the three channels results from the omission of the terms associated with the position and velocity errors in the velocity error propagation equation. It is shown in this paper that the unobservable errors in the time-invariant system can be made observable through maneuvering of the vehicle. Based on the observability analysis of time-varying systems, all of the above seven unobservable modes are shown to be made observable if both the directions of angular velocity and acceleration change. Acceleration changes are shown to improve the estimation of attitude and gyro bias. The components of errors in attitude and gyro bias that are orthogonal to the direction of the acceleration change are made observable. This result is consistent with that obtained from the piecewise constant system modeling [9]. Changes in angular velocity decrease the lever arm error. The components of the lever arm error that are orthogonal to the axis of rotation are made observable. The motion with a constant angular velocity does not have any effect on the estimation of the lever arm in the GPS/INS systems in which low-grade inertial sensors are employed. Covariance simulation results were given to support the observability analysis results. One of the main contributions of this paper is a control-theoretic approach to the observability analysis on the time-varying error dynamics model of GPS/INS. Even though it is recently known that rotational motions improve the estimation of the lever arm, rigorous research on the effect of rotational motions on observability was rarely published [3], [13]. This paper extends the application of control-theoretic approaches to more general time-varying error models. With this approach, the effect of motions of rotation as well as translation on the estimation of errors in navigation systems can be studied. The second contribution is that the relationship between the motions of the vehicle and the observability of errors in the low-grade INS aided by a single antenna GPS measurement system is explicitly given. One useful application of the observability property is the measurement of the relative distance between two objects, one of which is inside of a vehicle and the other of which is on the

outside surface of the vehicle. Rapid changes in acceleration and angular rate are desirable for the observability enhancement by maneuvering. Small vehicles are usually capable of the motion changes required for time-varying observability with their own power. Large vehicles may need external forces such as air wind, water waves, or gravity for the rapid motion changes. Many notations in this paper follow those in [14]. For a vector , is the vector decomposed in a coordinate frame . denotes the rotation matrix from a frame to a frame . denotes the column vector of the angular velocity of a frame relative to a frame , decomposed in a frame . denotes the screw-symmetric cross-product matrix of . , , , and denote the estimate, estimation error, time derivative, and denotes the absolute transpose of a matrix , respectively. and denote the cross-product value of a vector . and dot (scalar) product of vectors and , respectively. denotes the th time derivative of a matrix . denotes an identity matrix. “0” denotes a zero matrix with an appropriate dimension. , , , , and used for coordinate frames denote the earth-centered inertial (ECI) frame, ECEF frame, earth-fixed tangential frame (east, north, up), body-fixed navigation frame (north, east, down), and body frame (forward, right, down), respectively. II. NAVIGATION ERROR MODEL A navigation error propagation model is introduced in this section. The errors in the estimates of position, velocity, attitude, biases in the inertial sensors, and lever arm from their true values are considered in the error propagation equations. Velocity error propagation equation is made simple by neglecting error terms associated with errors in position and velocity. These error terms are relatively small in the integration of low-grade inertial sensors with accurate GPS measurements. The simplified velocity error propagation equation makes the observability analysis in Section III uncomplicated. Reference frames for error states are chosen in such a way that the observability analysis in the next section is more convenient. Inertial sensor biases and errors in attitude and lever arm are represented in the body frame. Position and velocity vectors are represented in the ECEF frame. The navigation equations in the ECEF frame are [14], [15] (1) (2) (3) where , , and are the position, velocity, and gravity, respectively, and is the specific force. Let the errors in the mechanization of the navigation equations be modeled as (4) (5) (6) (7) (8) where is the position error, is the velocity error, is the attitude error, is the cross-product matrix of ,

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is the accelerometer measurement, is the accelerometer bias, is the accelerometer noise, is the gyro measurement, is the gyro bias, and is the gyro noise. Attitude error in INS error analysis has usually been represented in the navigation where is the reference frame such that frame of INS [14]–[16]. The relation between and is for small . Relatively simple observability analysis will be obtained with in the next section. The linearized error propagation equations are [14], [15] (9) (10) (11) (12) (13) where ; , , and are the cross-product , and , respectively. The error propagamatrices of , tions for and are related with each other such that . The state augmentation technique [11], [17], [18] for errors in the inertial sensors is used in this paper. Since biases are the major error sources in the low-grade inertial sensors [19], other errors such as scale factor error and misalignment errors are neglected in this paper. As in [9] and [12], biases are assumed to be constant because the test time for the error estimation is relatively short compared with the time constant of the bias drift. The maximum singular value of is in the order of 10 . The and are in the orders of 10 and 10, remagnitudes of , , and are in the orders spectively. The magnitudes of of 0.1, 0.2, and 0.01, respectively, in carrier-phase differential GPS. The magnitudes of and are assumed to be in the orders of 0.1 and 0.001. Thus, in this paper, the gravity gradient and rotational motion of earth can be considered less important. Instead of (10) and (11), the following equations are used in the following sections to simplify the observability analysis:

Fig. 1.

GPS/INS measurement system.

servability properties are investigated by examining the properties of time derivatives of the GPS position measurement. Investigation of the time derivatives of the measurement changes the observability test on the 18 error states to that on at most nine error states; errors in attitude, gyro bias, and lever arm. These error states are represented in the same reference frame, the body frame. Comparing with the attitude error represented in the navigation reference frame, the attitude error given the same reference frame of the inertial sensor biases and lever arm error makes time derivatives of measurement estimation error simpler. These nine error states together with the simplified velocity error propagation model make the observability test straightforward. In addition, test of the time derivatives of the measurement makes the physical interpretation of the observability analysis easy. A. Observability Definitions Before the main part of this section is given, the definitions of observability of linear systems used in this paper are introduced. Consider the linear system

(14) (15) A single antenna GPS measurement system is given in Fig. 1 where is the lever arm between the GPS antenna and IMU. Lever arm uncertainty is modeled such that (16) (17) where is the lever arm error in the body frame. Then, the GPS measurement estimation error can be written as [3], [13] (18) is the cross-product matrix of the lever arm where is the error in the GPS measurement.

where and are, respectively, the and matrices whose entries are continuous functions of time defined over ( , ). Definition 3.1: The dynamic equation is observable at if such that for any state at time , there exists a finite over the time interval the knowledge of the output suffices to determine the state . Define a sequence of observability matrices by the equation

and

III. OBSERVABILITY PROPERTIES In this section, observability analyses are made for both the time-invariant and time-varying error dynamics models. Ob-

Suppose and in the system of . Then, the time-varying system

are analytic functions is observable

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at time if there exists a finite time of the matrix

.. .

such that the rank

Then, the INS mechanization error equations and measurement estimation error can be written as (24) (25)

(19)

and in the system are constant. is [20]. Suppose Then, the time-invariant linear system is observable if and only if the rank of the matrix

where and include the first-order approximation errors and sensor noises. For simplicity of expression, the symbol of dependency of matrices and vectors on time is and omitted in the rest of this section. For example, are replaced with and , respectively. Suppose is an unob. Then servable mode of the system (26)

.. .

(27) (28) (29)

is . If the linear time-invariant system is observable, then it is observable at every initial time, and the determination of the initial state can be achieved in any nonzero time interval [20]. The above observability tests are the same as finding a state such that vector

(30) .. .

.. .

(31) where

for a time . If there is no nonzero state that satisfies the above conditions, then the system is observable at . If the system is time-invariant, the system’s observability does not depend upon time. In the rest of this section, observability properties of errors in the integration of GPS and INS are provided. The proofs of them are given in Appendix A.

(32) with (33) (34) (35)

B. Observability Condition In this section a general observability condition for the integrated system of GPS/INS is introduced. By the application of observability condition to the time derivatives of GPS measurement estimation error, an observability test on the 18 error states is transformed to that on at most nine error states. For the analysis on the time-varying system, it is assumed in this paper that and are 1 times continuously differentiable . where is the number of error states. In this paper Let

(20) (21)

is a full-rank matrix, Since (28)–(31) implies that

in

(36) with (37) Let

Since

and

are constant, and

(38) , we have (39) (40) (41)

(22) where (23)

,

. Define (42)

HONG et al.: OBSERVABILITY OF ERROR STATES IN GPS/INS INTEGRATION

.. .

Then, the condition

.. .

,

.. .

(43)

735

for with . Thus (44) can be satisfied with arbitrary nonzero elements of , , and the component of in the direction of . Therefore, the system has seven unobservable modes. Let be the matrix whose columns represent the unobservable modes such that

is the same as (47)

(44) The remaining conditions with the following relations:

,

is always satisfied (45) (46)

and (37). Thus, the observability of the system depends on the rank of . If it has a full column rank, then the system is observable. Since the null space of in the nine-dimensional real vector space determines the unobservable subspace of the , a basis of the null space of will be given in system the following observability analysis. in (32) is equal to Remark 3.1: . This is the same as . This shows is the sum of errors in the estimates of specific force that and tangential and centrifugal accelerations of antenna 1. These in errors are induced by the attitude estimation error . (32) is the same as . Thus it is the error in the estimate of the centrifugal acceleration of antenna . 1. This error is induced by the uncertainty in the gyro bias is the same as . Thus it is the sum of errors in the estimates of tangential acceleration and centrifugal acceleration. The estimation errors are caused by the . lever arm error Remark 3.2: Equations (37) and (44)–(46) state that the unobservable modes are determined by the nine error states , , and that make all the time derivatives of the estimation error of the measurement acceleration zero vectors. If there are no such nonzero states, then the system is observable. Remark 3.3: Note that the reference frame of is the same as that of and . It can be easily seen that if the attitude error was represented with another reference frame, time derivatives of would be more complicated. Remark 3.4: From (33)–(35) and (39)–(41), it can be seen , , and their time derivatives. that is determined by , Thus, the observability properties depend on the specific force and angular velocity of the vehicle. If the speed of the vehicle is not very fast, the following relations hold: . Thus, the observability properties of navigation errors are expected to be independent of the reference frame of the INS mechanization equations such as ECEF, local geographic frame, or (local fixed) tangential frame. C. Time-Invariant Systems In this section observability property is presented for the timeinvariant system in which acceleration and attitude are constant. and in (22) and (23) are constant. Then Suppose

Then the results of the observability analysis is summarized with the following property. Property 3.1: The time-invariant system has seven unobservable modes. The set of seven columns of is a basis of the unobservable subspace. These unobservable modes induce errors in other estimates such that (48) (49) (50) Remark 3.5: An important observability property of the integrated navigation system of a low-grade IMU with a single-antenna GPS measurement system is that the component of gyro bias in the direction of the specific force is unobservable if a vehicle moves with constant attitude and acceleration. Consider the case in which a vehicle moves on a straight horizontal line with a constant velocity. Then, the yaw estimation error increases as time passes. The error increase rate is proportional to the vertical component of the gyro bias estimation error. However, in the navigation system with an accurate IMU that can detect the rotational motion of earth, the increase rate of the yaw estimation error can be negligible because gyro bias is observable [4], [9]. D. Systems With Time-Varying Acceleration This section investigates the effect of acceleration changes on the observability of GPS/INS systems. It is assumed that the vehicle moves with a constant attitude. For the convenience of expression, a group of matrices are introduced in the following. Each column in the matrices represents an unobservable mode that satisfies (44) for various motion conditions given in Properties 3.2–3.6. Let

(51) where ; and are defined in the folconsists of lowing Properties 3.2 and 3.4, respectively. seven modes. Any three-dimensional with the constraint can be expressed as a linear combination of the

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first three columns. The component of in the direction of can be expressed with the fourth column. Any three-dimencan be expressed as a linear combination of sional vector last three columns. consists of five modes: It represents and the components of and in the three-dimensional consists of five modes. The component of direction of . in the direction of can be expressed with the first column. The second column represents the component of in the with the constraint that is in the direction of direction of . The ratio of to is the same as that of to . The last consists of three columns represent three-dimensional . five modes. The first column represents the component of in with the constraint that is in the direction the direction of . The ratio of to is the same of that of to . The second column represents the of in the direction of . The last three columns component of represent three-dimensional . consists of four modes: in the direction of and It represents the component of three-dimensional . Observability properties for time-varying accelerations are given below. is constant and and Property 3.2: Suppose that all the time derivatives of it have the same direction such that

, where are real numbers. If

, then the system has seven unobservable modes. The set of seven column is a basis of the unobservable subspace. Othervectors in has five unobservable modes. The set wise, the system is a basis of the unobservable of five column vectors in subspace. is constant, that and Property 3.3: Suppose that have different directions, and that is constant. Then, the has five unobservable modes. The set of the five system column vectors in is a basis of the unobservable subspace. is constant, that Property 3.4: Suppose that and have different directions, and that all of the have the same direction such that time derivatives of , where

are real numbers. If

, then the system has five unobservable is a basis of modes. The set of the five column vectors in has the unobservable subspace. Otherwise, the system four unobservable modes. The set of the four column vectors in is a basis of the unobservable subspace. is constant and , , and Property 3.5: Suppose that are linearly independent. Then the system has three unobservable modes . The set of the last three column vectors is a basis of the unobservable subspace. in From Properties 3.2–3.5, the following property can be obtained. is constant and is timeProperty 3.6: Suppose that varying. If any three vectors in are linearly independent, then the system has three unobservhas at least one additional unable modes . Otherwise, observable mode.

Remark 3.6: Property 3.4 states that a change in acceleration direction makes the components of attitude error that are perpendicular to the acceleration change observable except for a very rare case. This result is in agreement with the observability analysis made by the piecewise constant modeling in [9]. E. General Time-Varying Systems Next, observability properties are presented for the case in which both the acceleration and attitude of a vehicle change. In the following, roll, pitch, and yaw are denoted by , , and , respectively. As in the previous subsections a group of matrices are introduced in the following for Properties 3.7–3.12:

(52) , , where , and is the position vector from the vehicle to the center of rotation, represented in the body frame. represents a compo. consists of six modes: Any nent of in the direction of three-dimensional with the constraint can be represented with a linear combination of the first three columns. The last three columns represent the three-dimensional . represents the component of in the direction of . represents the component of in the direction of . consists of two modes. The first column represents the component of in the direction of . The second column represents the comin the direction of with the constraint that ponent of is in the direction of . The ratio of to is . Then the observability the same as that of to properties for the general time-varying system are given below. is constant, then the system has Property 3.7: If three unobservable modes . The set of the last three column is a basis of the unobservable subspace. vectors in and all orders of the time derivatives Property 3.8: If has an of it have the same direction, then the system unobservable mode. is a basis of the unobservable subspace. and are constant, then the system Property 3.9: If has six unobservable modes. The set of the six column vectors in is a basis of the unobservable subspace. Property 3.10: If is constant and both and are linear has an unobservable functions of time, then the system is a basis of the unobservable subspace. mode. and both and are linear functions Property 3.11: If of time, then the system has an unobservable mode. is a basis of the unobservable subspace. Property 3.12: If a vehicle rotates on the horizontal plane and a constant angular acceleration with a constant radius , then the system has two unobservable modes. The is a basis of the unobservset of the two column vectors in able subspace. depends upon the rank The observability of the system of . However, is very complicated for general motions, and corresponding analytic observability conditions are not easy to

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TABLE I ACCELERATIONS AND ATTITUDES IN THE NUMERICAL OBSERVABILITY TESTS

,

,

,

TABLE II DESCRIPTIONS OF NUMERICAL TESTS

,

TABLE III DIMENSION OF THE NULL SPACE OF

IN THE

NUMERICAL TESTS

, ,

,

.

obtain. In the rest of this section, results of a numerical observability test are given for the simple motions that can represent many types of typical vehicle motions. Let

.. .

.. .

.. .

(53)

Note that in (43) is the same as . If has a full column rank for a certain motion with , then also has a full column rank. Thus the system is observable with that motion. , , and for are very complicated, is Since chosen for the numerical test of observability in the following. To test the rank of , singular values of it were investigated. is positive, then the matrix If the smallest singular value of has a full column rank. The lever arm in the numerical tests was . Table I shows accelerations and attitudes of the vehicle in the numerical tests. The descriptions of the above tests are given in Table II. Unobservable modes in the above numerical tests were obtained between seconds 1 and 1 at 0.1 s intervals. The number of unobservable modes for each test was found to be constant for the time interval in the tests. The test results are given in Table III. Since the above observability test is based on the null space that is a part of the full observability matrix , the test on dimensions of null spaces in Table III are actually upper bounds of the number of unobservable modes. The table shows that upper bounds of the numbers of unobservable modes in Tests 7 and 8 are zeros. Since the number of unobservable modes is nonnegative, it can be inferred that all the errors are observable in Tests 7 and 8. On the other hand, the properties given in this section provide lower bounds of the numbers of unobservable modes for Tests 1 through 6. Properties 3.3, 3.5, 3.9, 3.10, 3.11, and 3.12 state that the lower bounds of the numbers of unobservable modes in Tests 1–6 are 5, 3, 6, 1, 1, and 2, respectively.

As can be seen in Table III, the lower bound of the numbers of unobservable modes in each of the tests is the same as the upper bound of the numbers of them for the same test. Thus, the null space dimensions in Table III are actually the numbers of unobservable modes of the tests. The numerical test on the observability matrix shows that the lever arm error can be found with rotational motion, not with translatory motion. Changes in both the magnitude and direction of angular velocity make the system observable. The same numerical test results on the observability were obtained for the following lever arms: [10 1 1] , [1 10 1] , and [1 1 10] . In the observability analysis given in this section, process and measurement noises are neglected. Obviously, the presence of these noises can be harmful to the performance of state estimators. In many cases, especially in the inertial navigation systems, Kalman filtering techniques are often employed to evaluate the estimator behaviors in the presence of sensor noises. Error covariance matrix in the Kalman filter can be used to inspect the performance degradation of estimator due to the noises. Even though the effect of the noises is not considered in the observability analysis in this section, the analysis can be helpful in understanding the limits of estimator performance. If a state is unobservable, the state cannot be estimated even in the most favorable situation in which no noise is present in the system. If the unobservable mode is unstable, estimation error can grow without limit. Low-cost MEMS inertial sensors usually have a lot of broadband noises. Thus, for the state estimator design, proper modeling of sensor noises is required. Allan variance technique [21] can be a useful means to investigate the time-domain characteristics of random signals. Allan variance tests showed that noises in the low-cost MEMS inertial sensors can usually be modeled as the sum of random constant bias and white noise for the time period of one or two minutes [22], [23]. For longer time periods, the noises can be modeled as exponentially correlated processes with time constant longer than 100 s. When the time required

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for the maneuvering is less than or equal to 100 s, noise modeling of random constant bias and white noise can be justified for low-cost MEMS inertial sensors.

IV. SIMULATION RESULTS The observability analysis given in the previous section does not provide sufficient information on the performance of error estimators. It only decides whether a measurement system is observable or not. For the analysis of the degree of observability [24], covariance analysis is employed in this section. It can also be considered as an efficient means of producing statistical information on the behavior of state estimators [25]. Assuming all the errors in the process and sensor to be Gaussian white, numerical simulation results on the standard deviation (STD) of errors are provided in the following. In the simulation INS mechanization, differential equations were integrated with low-grade IMU data at 100 Hz. The solutions to the mechanization differential equations were corrected with an extended Kalman filter at 1 Hz using the measurement from the GPS antenna. The statistics of sensor noises in the numerical simulation are as follows: all the noises in measurements from GPS and IMU are assumed to be Gaussian white. The STD of the GPS measurement noise in the tangential frame was [0.06 0.06 0.08] in meters. Bias and the STD of the accelerometer noise were [ 0.04 0.05 0.1] and 0.01 in m/s , respectively. Bias and the , STD of the gyro noise were [ 0.09 0.05 0.1] and 0.1 in respectively. The lever arm of GPS antenna was [1.0 1.0 1.0] in meters in the body frame. The motion of the vehicle in the simulation is given in Figs. 2–7. The vehicle was motionless for five seconds after the start of the numerical simulation. It accelerated and moved northward with a constant velocity for five seconds, respectively. The vehicle changed its roll angle for five seconds. Then, it moved without changing its velocity and attitude for five seconds. The vehicle moved up and down, changing its pitch angle for five seconds. Once again, it moved without changing its velocity and attitude for five seconds. Finally, it moved eastward and back westward, changing its yaw angle for ten seconds. Thus, the vehicle experienced a motion of a full six degrees of freedom in the simulation. Figs. 8 –12 show errors in the estimates of INS states. The STDs of roll and pitch estimation errors in Fig. 8 decreased significantly at the beginning. This is because the horizontal components of static unobservable attitude error can be approximated to the accelerometer bias divided by gravity, which is about from 0.3 to 0.4 degrees [13]. The figure shows that the STD of yaw error decreased when the horizontal components of the specific force changed. The STD of yaw error during the first 5 s given in Fig. 8 can be misleading. Even though the STD of yaw estimation error remained constant while the vehicle was motionless in the beginning, the yaw estimation error in Fig. 9 was shown to increase in the beginning. This is the case when the signs of the initial yaw error and gyro bias were the same. If the sign of gyro bias had been different from that of the initial yaw error, the absolute value of yaw error would have decreased in the beginning.

Fig. 2. Vehicle trajectory.

Fig. 3. Vehicle velocity.

Fig. 4. Vehicle acceleration.

In Fig. 10, the horizontal components of gyro bias error were continuously reduced from the beginning. The figure confirms that these errors are observable without maneuvering. The figure also shows that the vertical component of gyro bias error remained almost constant until the twenty-fifth second. It was reduced by large amounts when the specific force and attitude changed simultaneously between seconds 25 through 30 and 35 through 45. The figure shows that the change in the horizontal acceleration is not effective in the estimation of the vertical component of gyro bias. In Fig. 12, it can be seen that both right-hand and downward components of errors in the lever arm estimate began to

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Fig. 5. Vehicle specific force.

Fig. 9.

Fig. 6. Vehicle attitude.

Fig. 10. Standard deviation of gyro bias estimation error.

Fig. 7.

Vehicle attitude rate.

Fig. 11. Standard deviation of accelerometer bias estimation error.

Fig. 8.

Standard deviation of attitude estimation error.

be reduced significantly after the vehicle began to experience a change in roll rate at the fifteenth second. The figure also show that forward and downward components of the lever arm error were reduced noticeably when pitch angle started to change at the twenty-fifth second. Thus, Property 3.8 was numerically demonstrated. The simulation results showed that acceleration changes were effective to improve attitude estimation. They also showed that changes in angular velocity reduced lever arm estimation error. However, changes in both acceleration and attitude were necessary for the estimation of the vertical component of gyro bias.

Error in the estimate of yaw angle.

Fig. 12. Standard deviation of lever arm estimation error.

V. CONCLUSIONS In this paper the observability of errors in the integration of a low-grade IMU with an accurate single antenna GPS measurement system was studied. A control-theoretic approach was adopted for the observability analysis of a time-varying error dynamics model. The effects of the motions of both translation and rotation on the error estimation were investigated. Errors in the estimates of position, velocity, attitude, biases of inertial sensors, and GPS antenna lever arm from their true values were considered in the observability analysis.

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Velocity error propagation equation was made simple by neglecting error terms associated with errors in position and velocity since these error terms are relatively small. The reference frames of errors were chosen in such a way that the observability analysis was less complicated. Inertial sensor biases and errors in attitude and lever arm were represented in the body frame. Position and velocity vectors are represented in the ECEF frame. Observability properties were investigated by examining the properties of the time derivatives of the GPS position measurement. Analysis with the time derivatives of the measurement changed the observability test on the 18 error states to that on at most nine error states: errors in attitude, gyro bias, and lever arm. These error states were represented in the same reference frame, the body frame. These nine error states together with the simplified velocity error propagation model made the observability test straightforward. Observability analysis results showed that the time-invariant error dynamics model has seven unobservable modes: attitude, lever arm, and the component of gyro bias in the direction of the specific force. Due to the unobservable gyro bias, yaw error can increase quite fast if the change in the acceleration of the vehicle is small. It was also shown that all the unobservable errors can be made observable throughthe maneuvering.Acceleration changes improve the estimates of attitude and gyro bias. The changes in angular rate enhance the estimate of the lever arm between the GPS antenna and IMU. However, the motion with a constant angular velocity has no influence on the lever arm estimation. A covariance simulation on the error state was given to verify the results of observability analysis. The numerical simulation results showed that acceleration changes were effective to improve attitude estimation. They also showed that changes in angular velocity reduced lever arm estimation error. Changes in both acceleration and attitude were shown to be effective for the estimation of the vertical component of gyro bias. Even though the numerical simulation on the error covariance can be considered as a useful means for the statistical analysis of the degree of observability, it has limitations on the simulation of the behavior of state estimators for real measurement systems. Precise mathematical descriptions of vehicle trajectory, low-frequency components of errors in the inertial sensors, and multipath error in GPS measurements are not easy to obtain. Field tests may be necessary as a separate work to obtain experimental evaluation of the effect of maneuvering on the error estimation.

Thus, if , then (54)–(56) are satisfied with the for any value of following three cases. First, with . Second, and is in the direction and is arbitrary. Otherwise, of . Third, the above equations are satisfied with the following three cases. and is in the direction of . The First, second and third cases are the same as the above second and third cases. This completes the proof of Property 3.2. B. Proof of Property 3.3 From (44), it follows that (57) (58) Obviously, the following three sets of , , and satisfy (57) and (58); First, with in the direction of . Second, with and where is an arbitrary real number. Third, an arbitrary value of with . This completes the proof of Property 3.3. C. Proof of Property 3.4 , it follows that

From (36) with

(59) Decompose

and

such that (60) (61)

where , , , , from (59), it follows that

, and

are real numbers. Then,

(62) Since that

,

, and

are linearly independent, it follows (63) (64) (65)

However, (63) and (64) imply that

APPENDIX A

(66)

A. Proof of Property 3.2 Note that . Thus , , and for . These relations also hold in the proofs of Properties 3.3–3.6. From (44), it follows that

Since

is not parallel with , and . From (36) with , it follows that

(54)

. Thus,

(67)

(55)

(68) .. .

.. . (56)

(69)

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Thus, if , then (67)–(69) are . Otherwise, . satisfied with Obviously, (44) holds with any nonzero . This completes the proof of Property 3.4. D. Proof of Property 3.5

for completes the proof Property 3.9.

. This

H. Proof of Property 3.10 , , , where , , and Let are constant initial angles, is a constant pitch rate, is a constant yaw rate, and is time. Then, it follows that

Decompose and into the forms of (60) and (61), into the form respectively, and decompose (70) where , , and are real numbers. The proof of is satisfied with Property 3.4 showed that (36) with and . For , (36) implies that

(71) Since , , and not zero, it follows that

are linearly independent, and

(75) , , where Then, the proof follows from the relation

,

.

is

(72) (73) (74) Equations (72) and (73) imply that . With this . Thus, . Indeed result, (74) implies that . This completes the proof (44) is satisfied with any nonzero of Property 3.5.

(76) with

,

, and This completes proof of Property 3.10.

.

I. Proof of Property 3.11 Let , , where , , and are constant initial angles, is a constant roll rate, is a constant yaw rate, and is time. Then, it follows that

E. Proof of Property 3.7 The property can be easily derived from the relations and , for , because is constant. This completes the proof of Property 3.7. F. Proof of Property 3.8 Since

(77)

and all orders of the time derivatives of it have

the same direction, for all proof of Property 3.8.

,

. Thus, . This completes the

Thus, we have (78) shown at the bottom of the page. If then

(79)

G. Proof of Property 3.9 Property 3.9 can be proved with the following relations: , , , and

,

Otherwise, is a full rank matrix if proof of Property 3.11.

. This completes

(78)

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J. Proof of Property 3.12 Let , , and where , , and are initial angles, is a constant yaw angular acceleration, and is time. Then, it follows that

(80)

(81) and , where is a real Let number. The observability test condition implies that . and This implies that where is a real number. Thus, the system has two unobservable modes in (52). This completes the proof of Property 3.12. REFERENCES [1] T. Bell, “Error analysis of attitude measurement in robotic ground vehicle position determination,” Navigation, vol. 47, no. 4, pp. 289–296, Winter 2000–2001. [2] X. He and L. Jianye, “Analysis of lever arm effects in GPS/IMU integration system,” Trans. Nanjing Univ. Aeronaut. Astronaut., vol. 19, no. 1, pp. 59–64, Jun. 2002. [3] S. Hong, Y. S. Chang, S. K. Ha, and M. H. Lee, “Estimation of alignment errors in GPS/INS integration,” in Proc. Inst. Navigation GPS 2002, Portland, OR, pp. 527–534. [4] S. Hong, M. H. Lee, J. A. Rios, and J. L. Speyer, “Observability analysis of INS with a GPS multi-antenna system,” Korean Soc. Mech. Eng. Int. J., vol. 16, no. 11, pp. 1367–1378, 2002. [5] A. A. Sutherland Jr., “The Kalman filter in transfer alignment of inertial guigance systems,” J. Spacecraft Rockets, vol. 5, pp. 1175–1180, 1968. [6] J. Baziw and C. T. Leondes, “In-flight alignment and calibration of inertial measurement units-part I: general formulation,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-8, pp. 440–449, 1972. [7] I. Y. Bar-Itzhack and B. Porat, “Azimuth observability enhancement during inertial navigation system in-flight alignment,” J. Guidance Contr., vol. 3, pp. 337–344, 1981. [8] B. Porat and I. Y. Bar-Itzhack, “Effect of acceleration switching during INS in-flight alignment,” J. Guidance Contr., vol. 4, pp. 385–389, 1981. [9] D. Goshen-Meskin and I. Y. Bar-Itzhack, “Observability analysis of piece-wise constant systems-part II: application to inertial navigation in-flight alignment,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 4, pp. 1068–1075, 1992. [10] , “Observability analysis of piece-wise constant systems-part I: theory,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 4, pp. 1056–1067, 1992. [11] I. Y. Bar-Itzhack and N. Berman, “Control theoretic approach to inertial navigation systems,” J. Guidance Contr., vol. 11, no. 3, pp. 237–245, 1988. [12] Y. F. Jiang and Y. P. Lin, “Error estimation of INS ground alignment through observability analysis,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 92–97, 1992. [13] S. Hong, M. H. Lee, S. H. Kwon, and H. H. Chun, “A car test for the estimation of GPS/INS alignment errors,” IEEE Trans. Intell. Transp. Syst., vol. 5, no. 3, pp. 208–218, 2004. [14] K. R. Britting, Inertial Navigation System Analysis. New York: WileyInterscience, 1971.

[15] M. Wei and K. P. Schwarz, “A strapdawn inertial algorithm using an earth-fixed cartesian frame,” Navigation, vol. 37, no. 2, pp. 153–167, 1990. [16] D. Goshen-Meskin and I. Y. Bar-Itzhack, “Unified approach to inertial navigation system error modeling,” J. Guidance Contr. Dyn., vol. 15, no. 3, pp. 648–653, 1992. [17] A. Gelb, Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974. [18] J. A. Farrell and M. Barth, The Global Positioning System & Inertial Navigation. New York: McGraw-Hill, 1999. [19] J. A. Farrell, T. D. Givargis, and M. J. Barth, “Real-time differential carrier phase GPS-aided INS,” IEEE Trans. Contr. Syst. Technol., vol. 8, no. 4, pp. 709–721, 2000. [20] C. T. Chen, Linear System Theory and Design. New York: Holt, Rinehart and Winston, 1984. [21] D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE, vol. 54, no. 2, pp. 221–230, Feb. 1966. [22] D. Gebre-Egziabher, R. C. Hayward, and J. D. Powell, “A low-cost GPS/inertial attitude heading reference system (AHRS) for general aviation applications,” in Proc. 1998 IEEE Position, Location Navigation Symp., Palm Springs, CA, 1998, pp. 518–525. [23] H. Hou and N. El-Sheimy, “Inertial sensors errors modeling using Allan variance,” in Proc. ION GPS/GNSS 2003, Portland, OR, pp. 2860–2867. [24] F. M. Ham and R. G. Brown, “Observability, eigenvalues, and Kalman filtering,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 2, pp. 269–273, 1983. [25] P. S. Maybeck, Stochastic Models, Estimation, and Control. New York: Academic Press, 1979, vol. I.

Sinpyo Hong received the B.A. degree from Pusan National University, Busan, Korea, in 1982, the M.S. degree from Korea Advanced Institute of Science and Technology, Seoul, Korea, in 1985, and the Ph.D. degree from the University of California, Los Angeles (UCLA), in 1993, all in mechanical engineering. From 1985 to 1986, he was a Design Engineer with Korea Heavy Industries and Construction Co. From 1986 to 1990, he was a Research Engineer for the development of a robot with Samsung Advanced Institute of Technology, Giheung, Korea. From 1993 to 1997, he was involved in the development of a navigation system for a unmanned aerial vehicle at UCLA. Since 2000 he has been working on the development of an unmanned land vehicle at Pusan National University, where he currently is a Research Professor with the Advanced Ship Engineering Research Center. His current research interests include design and analysis of integrated navigation systems and control systems. Dr. Hong is a member of the Institute of Navigation.

Man Hyung Lee (S’79–M’83–SM’01) was born in Korea in 1946. He received the B.S. and M.S. degrees in electrical engineering from Pusan National University, Pusan, Korea, in 1969 and 1971, respectively, and the Ph.D. degree in electrical and computer engineering from Oregon State University, Corvallis, in 1983. From 1971 to 1974, he was an Instructor in the Department of Electronics Engineering, Korea Military Academy, Seoul. He was an Assistant Professor in the Department of Mechanical Engineering, Pusan National University, from 1974 to 1978. From 1978 to 1983, he was a Teaching Assistant, Research Assistant, and Postdoctoral Fellow at Oregon State University. Since 1983, he has been a Professor in the College of Engineering, Pusan National University, where he was a Pohang Iron and Steel Co., Pohang, Korea (POSCO) Chair Professor in the School of Mechanical Engineering from 1997 to 2003 and was Dean of the College of Engineering from March 2002 to February 2004. His research interests are estimation, identification, stochastic processes, bilinear systems, mechatronics, micromachine automation, and robotics. He is the author of more than 750 technical papers and was Program Chair of the 1998 4th ICASE Annual Conference. Dr. Lee is a member of the American Society of Mechanical Engineers, Society for Industrial and Applied Mathematics, and Society of Photo-Optical Instrumentation Engineers. He was General Cochairman of the 2001 IEEE International Symposium on Industrial Electronics (ISIE) and is a General Cochairman of The 30th Annual Conference of the IEEE Industrial Electronics Society (IECON ’04).

HONG et al.: OBSERVABILITY OF ERROR STATES IN GPS/INS INTEGRATION

Ho-Hwan Chun received the B.Sc. and M.Sc. degrees in ship and ocean engineering from Pusan National University, Korea, in 1983 and 1985, respectively. He received the Ph.D. degree from the Department of Naval and Ocean Engineering, Glasgow University, U.K., in 1988. He was Yard Research Fellow at Glasgow University from 1988 to1990 and a Principal Researcher with the Hyundai Maritime Research Institute, Ulsan, Korea, from 1991 to 1993. He has been an Associate Professor in the Department of Naval Architecture and Ocean Engineering, Pusan National University, since 1994. In 2002, he became Director of the Advanced Ship Engineering Research Center (ASERC), designated by the Ministry of Science and Technology, Korea. His main research area is hydrodynamics such as hull form design, hull and structure interactions with waves, WIG and drag reduction, etc.

Sun-Hong Kwon received the B.S. degree in naval architecture from Pusan National University, Busan, Korea, in 1978, the M.E. degree in ocean engineering from Stevens Institute of Technology, Hoboken, NJ, in 1983, and the Ph.D degree in aerospace and ocean engineering from Virginia Polytechnic Institute and State University, Blacksburg, in 1986. He joined Pusan National University in 1986 as a Faculty Member in the Department of Naval Architecture and Ocean Engineering. His current research interests are the application of wavelet analysis to ocean engineering and development of sea wave monitoring system by using radar images.

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Jason L. Speyer (M’71–SM’82–F’85) received the S.B. degree in aeronautics and astronautics from the Massachusetts Institute of Technology, Cambridge, in 1960 and the Ph.D. degree in applied mathematics from Harvard University, Cambridge, MA, in 1968. His industrial experience includes research at Boeing, Raytheon, Analytical Mechanics Associated, and the Charles Draper Laboratory. He was the Harry H. Power Professor in Aerospace Engineering at the University of Texas, Austin, and is currently a Professor in the Mechanical and Aerospace Engineering Department, University of California, Los Angeles. He spent a research leave as a Lady Davis Visiting Professor at The Technion—Israel Institute of Technology, Haifa, in 1983. He was the 1990 Jerome C. Hunsaker Visiting Professor of Aeronautics and Astronautics at the Massachusetts Institute of Technology. Dr. Speyer is a Fellow of the American Institute of Aeronautics and Astronautics. He has twice been an elected member of the Board of Governors of the IEEE Control Systems Society. He has been an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and Chairman of the Technical Committee on Aerospace Control. From October 1987 to October 1991 and from October 1997 to October 2001, he was a member of the USAF Scientific Advisory Board. He received the Mechanics and Control of Flight Award and Dryden Lectureship in Research from the American Institute of Aeronautics and Astronautics in 1985 and 1995, respectively. He received an Air Force Exceptional Civilian Decoration in 1991 and 2001 and the IEEE Third Millennium Medal in 2000.

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Observability of Error States in GPS/INS Integration Sinpyo Hong, Man Hyung Lee, Senior Member, IEEE, Ho-Hwan Chun, Sun-Hong Kwon, and Jason L. Speyer, Fellow, IEEE

Abstract—Observability properties of errors in an integrated navigation system are studied with a control-theoretic approach in this paper. A navigation system with a low-grade inertial measurement unit and an accurate single-antenna Global Positioning System (GPS) measurement system is considered for observability analysis. Uncertainties in attitude, gyro bias, and GPS antenna lever arm were shown to determine unobservable errors in the position, velocity, and accelerometer bias. It was proved that all the errors can be made observable by maneuvering. Acceleration changes improve the estimates of attitude and gyro bias. Changes in angular velocity enhance the lever arm estimate. However, both the motions of translation and constant angular velocity have no influence on the estimation of the lever arm. A covariance simulation with an extended Kalman filter was performed to confirm the observability analysis. Index Terms—Global Positioning System, GPS/INS, inertial measurement unit (IMU), inertial navigation, lever arm, observability.

I. INTRODUCTION

T

HIS paper considers the integration of an accurate singleantenna Global Positioning System (GPS) measurement system with a low-grade inertial measurement unit (IMU). The positioning accuracy of the GPS measurement is assumed to be centimeter level. This type of accuracy can be obtained from carrier phase differential GPS (CDGPS). The errors in IMU are assumed to be so large that gyro error is much greater than the rotational speed of earth and accelerometer error is much greater than the rotational speed of earth multiplied by the velocity estimation error. Low-cost inertial sensors made with the current microelectromechanical systems (MEMS) technologies usually have these noise levels. There are a couple of important problems in this type of integration. Lever arm uncertainty can be a serious problem in accurate navigation systems. While the GPS antenna is mounted on the outside surface of a vehicle, IMU is usually placed inside the vehicle. Thus, the direct measurement of the distance

Manuscript received March 3, 2004; revised July 20, 2004. This work was supported by the Korea Science and Engineering Foundation through the Advanced Ship Engineering Research Center at Pusan Nation University. The review of this paper was coordinated by Dr. R. Klukas. S. Hong is with the Advanced Ship Engineering Research Center, Pusan National University, Busan 609-735, Korea (e-mail: [email protected]). M. H. Lee is with the School of Mechanical Engineering, Pusan National University, Busan 609-735, Korea. H.-H. Chun is with the Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, Korea. S. H. Kwon is with the Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, Korea. He is also presently a visiting scholar at the Department of Civil Engineering, Texas A&M University, College Station, TX 77843–3136 USA. J. L. Speyer is with the Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095-1597 USA. Digital Object Identifier 10.1109/TVT.2004.841540

between the GPS antenna and IMU is often quite difficult. The error in the estimated value of the lever arm, the relative position of the GPS antenna with respect to IMU, can be of great concern in accurate navigation systems [1]. The lever arm error in large vehicles can be much greater than the centimeter-level error in CDGPS measurements. In addition, the lever arm error can increase errors in the estimates of position, attitude, and inertial sensor biases [2], [3]. For a vehicle that moves with slow changes in attitude and acceleration, the attitude of the vehicle is unobservable with a single-antenna GPS measurement system [4]. Moreover, as will be shown later in this paper, the component of gyro bias in the direction of specific force is not observable if the gyro has a large error. Thus, yaw error can increase significantly fast with time. These unobservable modes increase errors in the estimates of position, velocity, and accelerometer bias. Especially, position error can be considerable with large attitude error if the lever arm is long. One approach to dealing with the above problems is designing an estimator for the uncertain values such as the lever arm, attitude, and biases of IMU. This approach can be particularly useful when there is no other instrument to measure the uncertain values. It is well known that motions of a vehicle can improve observability of the states of inertial navigation systems (INS) and inertial sensor errors. The effect of maneuvering on the observability during in-flight alignment (IFA) was investigated in [5]–[9]. A control-theoretic approach to the observability study on IFA was first introduced in [9] using a piecewise constant system modeling [10]. The research on the observability enhancements was mainly concerned with the effect of the translatory motions such as changes in acceleration. In this paper, a control-theoretic framework was introduced for the observability analysis of errors in the integrated systems of GPS and INS. Changes in attitude as well as acceleration were considered in the observability analysis. The errors in the observability analysis were described in the earth-centered earth-fixed (ECEF) frame and the body frame. Errors in position, velocity, IMU attitude, biases in gyro and accelerometer, and lever arm were considered in the observability analysis. The terms associated with position and velocity errors in the velocity error propagation equation were neglected because they were relatively very small compared with other error terms in the integration of low-grade IMU and accurate GPS measurements. This simplification made the observability analysis uncomplicated. Among the inertial sensor errors such as biases, scale-factor errors, and alignment errors, biases are the most unpredictable and dominant in low-grade sensors. Since the period of testing for the error estimation is relatively short compared with the time-constant of bias drifts, the biases in in-

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ertial sensors ae considered to be constant in this paper. Time derivatives of GPS measurements were used for observability test. This approach changes the observability test on the 18 error states to that on at most nine error states; errors in attitude, gyro bias, and lever arm. Compared with the analysis based upon the null space test of observability matrices [3], [4], this approach makes the observability analysis simpler. In this paper it is shown that the time-invariant error dynamics model for the integrated navigation system has seven unobservable modes when the position of a vehicle is measured with a single antenna GPS measurement system. Errors in the lever arm, attitude, and component of gyro bias in the direction of gravity are not observable if the vehicle moves with constant attitude and acceleration. Both the errors in attitude and lever arm contribute to the position error. It was shown in previous research that if the lever arm error is neglected, the time-invariant error dynamics model with two horizontal channels has three unobservable modes [11], [12]. Later, in [4] and [9], the error model for the three channels was also shown to have three unobservable modes. The addition of the unobservable gyro bias in this paper for the three channels results from the omission of the terms associated with the position and velocity errors in the velocity error propagation equation. It is shown in this paper that the unobservable errors in the time-invariant system can be made observable through maneuvering of the vehicle. Based on the observability analysis of time-varying systems, all of the above seven unobservable modes are shown to be made observable if both the directions of angular velocity and acceleration change. Acceleration changes are shown to improve the estimation of attitude and gyro bias. The components of errors in attitude and gyro bias that are orthogonal to the direction of the acceleration change are made observable. This result is consistent with that obtained from the piecewise constant system modeling [9]. Changes in angular velocity decrease the lever arm error. The components of the lever arm error that are orthogonal to the axis of rotation are made observable. The motion with a constant angular velocity does not have any effect on the estimation of the lever arm in the GPS/INS systems in which low-grade inertial sensors are employed. Covariance simulation results were given to support the observability analysis results. One of the main contributions of this paper is a control-theoretic approach to the observability analysis on the time-varying error dynamics model of GPS/INS. Even though it is recently known that rotational motions improve the estimation of the lever arm, rigorous research on the effect of rotational motions on observability was rarely published [3], [13]. This paper extends the application of control-theoretic approaches to more general time-varying error models. With this approach, the effect of motions of rotation as well as translation on the estimation of errors in navigation systems can be studied. The second contribution is that the relationship between the motions of the vehicle and the observability of errors in the low-grade INS aided by a single antenna GPS measurement system is explicitly given. One useful application of the observability property is the measurement of the relative distance between two objects, one of which is inside of a vehicle and the other of which is on the

outside surface of the vehicle. Rapid changes in acceleration and angular rate are desirable for the observability enhancement by maneuvering. Small vehicles are usually capable of the motion changes required for time-varying observability with their own power. Large vehicles may need external forces such as air wind, water waves, or gravity for the rapid motion changes. Many notations in this paper follow those in [14]. For a vector , is the vector decomposed in a coordinate frame . denotes the rotation matrix from a frame to a frame . denotes the column vector of the angular velocity of a frame relative to a frame , decomposed in a frame . denotes the screw-symmetric cross-product matrix of . , , , and denote the estimate, estimation error, time derivative, and denotes the absolute transpose of a matrix , respectively. and denote the cross-product value of a vector . and dot (scalar) product of vectors and , respectively. denotes the th time derivative of a matrix . denotes an identity matrix. “0” denotes a zero matrix with an appropriate dimension. , , , , and used for coordinate frames denote the earth-centered inertial (ECI) frame, ECEF frame, earth-fixed tangential frame (east, north, up), body-fixed navigation frame (north, east, down), and body frame (forward, right, down), respectively. II. NAVIGATION ERROR MODEL A navigation error propagation model is introduced in this section. The errors in the estimates of position, velocity, attitude, biases in the inertial sensors, and lever arm from their true values are considered in the error propagation equations. Velocity error propagation equation is made simple by neglecting error terms associated with errors in position and velocity. These error terms are relatively small in the integration of low-grade inertial sensors with accurate GPS measurements. The simplified velocity error propagation equation makes the observability analysis in Section III uncomplicated. Reference frames for error states are chosen in such a way that the observability analysis in the next section is more convenient. Inertial sensor biases and errors in attitude and lever arm are represented in the body frame. Position and velocity vectors are represented in the ECEF frame. The navigation equations in the ECEF frame are [14], [15] (1) (2) (3) where , , and are the position, velocity, and gravity, respectively, and is the specific force. Let the errors in the mechanization of the navigation equations be modeled as (4) (5) (6) (7) (8) where is the position error, is the velocity error, is the attitude error, is the cross-product matrix of ,

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is the accelerometer measurement, is the accelerometer bias, is the accelerometer noise, is the gyro measurement, is the gyro bias, and is the gyro noise. Attitude error in INS error analysis has usually been represented in the navigation where is the reference frame such that frame of INS [14]–[16]. The relation between and is for small . Relatively simple observability analysis will be obtained with in the next section. The linearized error propagation equations are [14], [15] (9) (10) (11) (12) (13) where ; , , and are the cross-product , and , respectively. The error propagamatrices of , tions for and are related with each other such that . The state augmentation technique [11], [17], [18] for errors in the inertial sensors is used in this paper. Since biases are the major error sources in the low-grade inertial sensors [19], other errors such as scale factor error and misalignment errors are neglected in this paper. As in [9] and [12], biases are assumed to be constant because the test time for the error estimation is relatively short compared with the time constant of the bias drift. The maximum singular value of is in the order of 10 . The and are in the orders of 10 and 10, remagnitudes of , , and are in the orders spectively. The magnitudes of of 0.1, 0.2, and 0.01, respectively, in carrier-phase differential GPS. The magnitudes of and are assumed to be in the orders of 0.1 and 0.001. Thus, in this paper, the gravity gradient and rotational motion of earth can be considered less important. Instead of (10) and (11), the following equations are used in the following sections to simplify the observability analysis:

Fig. 1.

GPS/INS measurement system.

servability properties are investigated by examining the properties of time derivatives of the GPS position measurement. Investigation of the time derivatives of the measurement changes the observability test on the 18 error states to that on at most nine error states; errors in attitude, gyro bias, and lever arm. These error states are represented in the same reference frame, the body frame. Comparing with the attitude error represented in the navigation reference frame, the attitude error given the same reference frame of the inertial sensor biases and lever arm error makes time derivatives of measurement estimation error simpler. These nine error states together with the simplified velocity error propagation model make the observability test straightforward. In addition, test of the time derivatives of the measurement makes the physical interpretation of the observability analysis easy. A. Observability Definitions Before the main part of this section is given, the definitions of observability of linear systems used in this paper are introduced. Consider the linear system

(14) (15) A single antenna GPS measurement system is given in Fig. 1 where is the lever arm between the GPS antenna and IMU. Lever arm uncertainty is modeled such that (16) (17) where is the lever arm error in the body frame. Then, the GPS measurement estimation error can be written as [3], [13] (18) is the cross-product matrix of the lever arm where is the error in the GPS measurement.

where and are, respectively, the and matrices whose entries are continuous functions of time defined over ( , ). Definition 3.1: The dynamic equation is observable at if such that for any state at time , there exists a finite over the time interval the knowledge of the output suffices to determine the state . Define a sequence of observability matrices by the equation

and

III. OBSERVABILITY PROPERTIES In this section, observability analyses are made for both the time-invariant and time-varying error dynamics models. Ob-

Suppose and in the system of . Then, the time-varying system

are analytic functions is observable

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at time if there exists a finite time of the matrix

.. .

such that the rank

Then, the INS mechanization error equations and measurement estimation error can be written as (24) (25)

(19)

and in the system are constant. is [20]. Suppose Then, the time-invariant linear system is observable if and only if the rank of the matrix

where and include the first-order approximation errors and sensor noises. For simplicity of expression, the symbol of dependency of matrices and vectors on time is and omitted in the rest of this section. For example, are replaced with and , respectively. Suppose is an unob. Then servable mode of the system (26)

.. .

(27) (28) (29)

is . If the linear time-invariant system is observable, then it is observable at every initial time, and the determination of the initial state can be achieved in any nonzero time interval [20]. The above observability tests are the same as finding a state such that vector

(30) .. .

.. .

(31) where

for a time . If there is no nonzero state that satisfies the above conditions, then the system is observable at . If the system is time-invariant, the system’s observability does not depend upon time. In the rest of this section, observability properties of errors in the integration of GPS and INS are provided. The proofs of them are given in Appendix A.

(32) with (33) (34) (35)

B. Observability Condition In this section a general observability condition for the integrated system of GPS/INS is introduced. By the application of observability condition to the time derivatives of GPS measurement estimation error, an observability test on the 18 error states is transformed to that on at most nine error states. For the analysis on the time-varying system, it is assumed in this paper that and are 1 times continuously differentiable . where is the number of error states. In this paper Let

(20) (21)

is a full-rank matrix, Since (28)–(31) implies that

in

(36) with (37) Let

Since

and

are constant, and

(38) , we have (39) (40) (41)

(22) where (23)

,

. Define (42)

HONG et al.: OBSERVABILITY OF ERROR STATES IN GPS/INS INTEGRATION

.. .

Then, the condition

.. .

,

.. .

(43)

735

for with . Thus (44) can be satisfied with arbitrary nonzero elements of , , and the component of in the direction of . Therefore, the system has seven unobservable modes. Let be the matrix whose columns represent the unobservable modes such that

is the same as (47)

(44) The remaining conditions with the following relations:

,

is always satisfied (45) (46)

and (37). Thus, the observability of the system depends on the rank of . If it has a full column rank, then the system is observable. Since the null space of in the nine-dimensional real vector space determines the unobservable subspace of the , a basis of the null space of will be given in system the following observability analysis. in (32) is equal to Remark 3.1: . This is the same as . This shows is the sum of errors in the estimates of specific force that and tangential and centrifugal accelerations of antenna 1. These in errors are induced by the attitude estimation error . (32) is the same as . Thus it is the error in the estimate of the centrifugal acceleration of antenna . 1. This error is induced by the uncertainty in the gyro bias is the same as . Thus it is the sum of errors in the estimates of tangential acceleration and centrifugal acceleration. The estimation errors are caused by the . lever arm error Remark 3.2: Equations (37) and (44)–(46) state that the unobservable modes are determined by the nine error states , , and that make all the time derivatives of the estimation error of the measurement acceleration zero vectors. If there are no such nonzero states, then the system is observable. Remark 3.3: Note that the reference frame of is the same as that of and . It can be easily seen that if the attitude error was represented with another reference frame, time derivatives of would be more complicated. Remark 3.4: From (33)–(35) and (39)–(41), it can be seen , , and their time derivatives. that is determined by , Thus, the observability properties depend on the specific force and angular velocity of the vehicle. If the speed of the vehicle is not very fast, the following relations hold: . Thus, the observability properties of navigation errors are expected to be independent of the reference frame of the INS mechanization equations such as ECEF, local geographic frame, or (local fixed) tangential frame. C. Time-Invariant Systems In this section observability property is presented for the timeinvariant system in which acceleration and attitude are constant. and in (22) and (23) are constant. Then Suppose

Then the results of the observability analysis is summarized with the following property. Property 3.1: The time-invariant system has seven unobservable modes. The set of seven columns of is a basis of the unobservable subspace. These unobservable modes induce errors in other estimates such that (48) (49) (50) Remark 3.5: An important observability property of the integrated navigation system of a low-grade IMU with a single-antenna GPS measurement system is that the component of gyro bias in the direction of the specific force is unobservable if a vehicle moves with constant attitude and acceleration. Consider the case in which a vehicle moves on a straight horizontal line with a constant velocity. Then, the yaw estimation error increases as time passes. The error increase rate is proportional to the vertical component of the gyro bias estimation error. However, in the navigation system with an accurate IMU that can detect the rotational motion of earth, the increase rate of the yaw estimation error can be negligible because gyro bias is observable [4], [9]. D. Systems With Time-Varying Acceleration This section investigates the effect of acceleration changes on the observability of GPS/INS systems. It is assumed that the vehicle moves with a constant attitude. For the convenience of expression, a group of matrices are introduced in the following. Each column in the matrices represents an unobservable mode that satisfies (44) for various motion conditions given in Properties 3.2–3.6. Let

(51) where ; and are defined in the folconsists of lowing Properties 3.2 and 3.4, respectively. seven modes. Any three-dimensional with the constraint can be expressed as a linear combination of the

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first three columns. The component of in the direction of can be expressed with the fourth column. Any three-dimencan be expressed as a linear combination of sional vector last three columns. consists of five modes: It represents and the components of and in the three-dimensional consists of five modes. The component of direction of . in the direction of can be expressed with the first column. The second column represents the component of in the with the constraint that is in the direction of direction of . The ratio of to is the same as that of to . The last consists of three columns represent three-dimensional . five modes. The first column represents the component of in with the constraint that is in the direction the direction of . The ratio of to is the same of that of to . The second column represents the of in the direction of . The last three columns component of represent three-dimensional . consists of four modes: in the direction of and It represents the component of three-dimensional . Observability properties for time-varying accelerations are given below. is constant and and Property 3.2: Suppose that all the time derivatives of it have the same direction such that

, where are real numbers. If

, then the system has seven unobservable modes. The set of seven column is a basis of the unobservable subspace. Othervectors in has five unobservable modes. The set wise, the system is a basis of the unobservable of five column vectors in subspace. is constant, that and Property 3.3: Suppose that have different directions, and that is constant. Then, the has five unobservable modes. The set of the five system column vectors in is a basis of the unobservable subspace. is constant, that Property 3.4: Suppose that and have different directions, and that all of the have the same direction such that time derivatives of , where

are real numbers. If

, then the system has five unobservable is a basis of modes. The set of the five column vectors in has the unobservable subspace. Otherwise, the system four unobservable modes. The set of the four column vectors in is a basis of the unobservable subspace. is constant and , , and Property 3.5: Suppose that are linearly independent. Then the system has three unobservable modes . The set of the last three column vectors is a basis of the unobservable subspace. in From Properties 3.2–3.5, the following property can be obtained. is constant and is timeProperty 3.6: Suppose that varying. If any three vectors in are linearly independent, then the system has three unobservhas at least one additional unable modes . Otherwise, observable mode.

Remark 3.6: Property 3.4 states that a change in acceleration direction makes the components of attitude error that are perpendicular to the acceleration change observable except for a very rare case. This result is in agreement with the observability analysis made by the piecewise constant modeling in [9]. E. General Time-Varying Systems Next, observability properties are presented for the case in which both the acceleration and attitude of a vehicle change. In the following, roll, pitch, and yaw are denoted by , , and , respectively. As in the previous subsections a group of matrices are introduced in the following for Properties 3.7–3.12:

(52) , , where , and is the position vector from the vehicle to the center of rotation, represented in the body frame. represents a compo. consists of six modes: Any nent of in the direction of three-dimensional with the constraint can be represented with a linear combination of the first three columns. The last three columns represent the three-dimensional . represents the component of in the direction of . represents the component of in the direction of . consists of two modes. The first column represents the component of in the direction of . The second column represents the comin the direction of with the constraint that ponent of is in the direction of . The ratio of to is . Then the observability the same as that of to properties for the general time-varying system are given below. is constant, then the system has Property 3.7: If three unobservable modes . The set of the last three column is a basis of the unobservable subspace. vectors in and all orders of the time derivatives Property 3.8: If has an of it have the same direction, then the system unobservable mode. is a basis of the unobservable subspace. and are constant, then the system Property 3.9: If has six unobservable modes. The set of the six column vectors in is a basis of the unobservable subspace. Property 3.10: If is constant and both and are linear has an unobservable functions of time, then the system is a basis of the unobservable subspace. mode. and both and are linear functions Property 3.11: If of time, then the system has an unobservable mode. is a basis of the unobservable subspace. Property 3.12: If a vehicle rotates on the horizontal plane and a constant angular acceleration with a constant radius , then the system has two unobservable modes. The is a basis of the unobservset of the two column vectors in able subspace. depends upon the rank The observability of the system of . However, is very complicated for general motions, and corresponding analytic observability conditions are not easy to

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TABLE I ACCELERATIONS AND ATTITUDES IN THE NUMERICAL OBSERVABILITY TESTS

,

,

,

TABLE II DESCRIPTIONS OF NUMERICAL TESTS

,

TABLE III DIMENSION OF THE NULL SPACE OF

IN THE

NUMERICAL TESTS

, ,

,

.

obtain. In the rest of this section, results of a numerical observability test are given for the simple motions that can represent many types of typical vehicle motions. Let

.. .

.. .

.. .

(53)

Note that in (43) is the same as . If has a full column rank for a certain motion with , then also has a full column rank. Thus the system is observable with that motion. , , and for are very complicated, is Since chosen for the numerical test of observability in the following. To test the rank of , singular values of it were investigated. is positive, then the matrix If the smallest singular value of has a full column rank. The lever arm in the numerical tests was . Table I shows accelerations and attitudes of the vehicle in the numerical tests. The descriptions of the above tests are given in Table II. Unobservable modes in the above numerical tests were obtained between seconds 1 and 1 at 0.1 s intervals. The number of unobservable modes for each test was found to be constant for the time interval in the tests. The test results are given in Table III. Since the above observability test is based on the null space that is a part of the full observability matrix , the test on dimensions of null spaces in Table III are actually upper bounds of the number of unobservable modes. The table shows that upper bounds of the numbers of unobservable modes in Tests 7 and 8 are zeros. Since the number of unobservable modes is nonnegative, it can be inferred that all the errors are observable in Tests 7 and 8. On the other hand, the properties given in this section provide lower bounds of the numbers of unobservable modes for Tests 1 through 6. Properties 3.3, 3.5, 3.9, 3.10, 3.11, and 3.12 state that the lower bounds of the numbers of unobservable modes in Tests 1–6 are 5, 3, 6, 1, 1, and 2, respectively.

As can be seen in Table III, the lower bound of the numbers of unobservable modes in each of the tests is the same as the upper bound of the numbers of them for the same test. Thus, the null space dimensions in Table III are actually the numbers of unobservable modes of the tests. The numerical test on the observability matrix shows that the lever arm error can be found with rotational motion, not with translatory motion. Changes in both the magnitude and direction of angular velocity make the system observable. The same numerical test results on the observability were obtained for the following lever arms: [10 1 1] , [1 10 1] , and [1 1 10] . In the observability analysis given in this section, process and measurement noises are neglected. Obviously, the presence of these noises can be harmful to the performance of state estimators. In many cases, especially in the inertial navigation systems, Kalman filtering techniques are often employed to evaluate the estimator behaviors in the presence of sensor noises. Error covariance matrix in the Kalman filter can be used to inspect the performance degradation of estimator due to the noises. Even though the effect of the noises is not considered in the observability analysis in this section, the analysis can be helpful in understanding the limits of estimator performance. If a state is unobservable, the state cannot be estimated even in the most favorable situation in which no noise is present in the system. If the unobservable mode is unstable, estimation error can grow without limit. Low-cost MEMS inertial sensors usually have a lot of broadband noises. Thus, for the state estimator design, proper modeling of sensor noises is required. Allan variance technique [21] can be a useful means to investigate the time-domain characteristics of random signals. Allan variance tests showed that noises in the low-cost MEMS inertial sensors can usually be modeled as the sum of random constant bias and white noise for the time period of one or two minutes [22], [23]. For longer time periods, the noises can be modeled as exponentially correlated processes with time constant longer than 100 s. When the time required

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for the maneuvering is less than or equal to 100 s, noise modeling of random constant bias and white noise can be justified for low-cost MEMS inertial sensors.

IV. SIMULATION RESULTS The observability analysis given in the previous section does not provide sufficient information on the performance of error estimators. It only decides whether a measurement system is observable or not. For the analysis of the degree of observability [24], covariance analysis is employed in this section. It can also be considered as an efficient means of producing statistical information on the behavior of state estimators [25]. Assuming all the errors in the process and sensor to be Gaussian white, numerical simulation results on the standard deviation (STD) of errors are provided in the following. In the simulation INS mechanization, differential equations were integrated with low-grade IMU data at 100 Hz. The solutions to the mechanization differential equations were corrected with an extended Kalman filter at 1 Hz using the measurement from the GPS antenna. The statistics of sensor noises in the numerical simulation are as follows: all the noises in measurements from GPS and IMU are assumed to be Gaussian white. The STD of the GPS measurement noise in the tangential frame was [0.06 0.06 0.08] in meters. Bias and the STD of the accelerometer noise were [ 0.04 0.05 0.1] and 0.01 in m/s , respectively. Bias and the , STD of the gyro noise were [ 0.09 0.05 0.1] and 0.1 in respectively. The lever arm of GPS antenna was [1.0 1.0 1.0] in meters in the body frame. The motion of the vehicle in the simulation is given in Figs. 2–7. The vehicle was motionless for five seconds after the start of the numerical simulation. It accelerated and moved northward with a constant velocity for five seconds, respectively. The vehicle changed its roll angle for five seconds. Then, it moved without changing its velocity and attitude for five seconds. The vehicle moved up and down, changing its pitch angle for five seconds. Once again, it moved without changing its velocity and attitude for five seconds. Finally, it moved eastward and back westward, changing its yaw angle for ten seconds. Thus, the vehicle experienced a motion of a full six degrees of freedom in the simulation. Figs. 8 –12 show errors in the estimates of INS states. The STDs of roll and pitch estimation errors in Fig. 8 decreased significantly at the beginning. This is because the horizontal components of static unobservable attitude error can be approximated to the accelerometer bias divided by gravity, which is about from 0.3 to 0.4 degrees [13]. The figure shows that the STD of yaw error decreased when the horizontal components of the specific force changed. The STD of yaw error during the first 5 s given in Fig. 8 can be misleading. Even though the STD of yaw estimation error remained constant while the vehicle was motionless in the beginning, the yaw estimation error in Fig. 9 was shown to increase in the beginning. This is the case when the signs of the initial yaw error and gyro bias were the same. If the sign of gyro bias had been different from that of the initial yaw error, the absolute value of yaw error would have decreased in the beginning.

Fig. 2. Vehicle trajectory.

Fig. 3. Vehicle velocity.

Fig. 4. Vehicle acceleration.

In Fig. 10, the horizontal components of gyro bias error were continuously reduced from the beginning. The figure confirms that these errors are observable without maneuvering. The figure also shows that the vertical component of gyro bias error remained almost constant until the twenty-fifth second. It was reduced by large amounts when the specific force and attitude changed simultaneously between seconds 25 through 30 and 35 through 45. The figure shows that the change in the horizontal acceleration is not effective in the estimation of the vertical component of gyro bias. In Fig. 12, it can be seen that both right-hand and downward components of errors in the lever arm estimate began to

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Fig. 5. Vehicle specific force.

Fig. 9.

Fig. 6. Vehicle attitude.

Fig. 10. Standard deviation of gyro bias estimation error.

Fig. 7.

Vehicle attitude rate.

Fig. 11. Standard deviation of accelerometer bias estimation error.

Fig. 8.

Standard deviation of attitude estimation error.

be reduced significantly after the vehicle began to experience a change in roll rate at the fifteenth second. The figure also show that forward and downward components of the lever arm error were reduced noticeably when pitch angle started to change at the twenty-fifth second. Thus, Property 3.8 was numerically demonstrated. The simulation results showed that acceleration changes were effective to improve attitude estimation. They also showed that changes in angular velocity reduced lever arm estimation error. However, changes in both acceleration and attitude were necessary for the estimation of the vertical component of gyro bias.

Error in the estimate of yaw angle.

Fig. 12. Standard deviation of lever arm estimation error.

V. CONCLUSIONS In this paper the observability of errors in the integration of a low-grade IMU with an accurate single antenna GPS measurement system was studied. A control-theoretic approach was adopted for the observability analysis of a time-varying error dynamics model. The effects of the motions of both translation and rotation on the error estimation were investigated. Errors in the estimates of position, velocity, attitude, biases of inertial sensors, and GPS antenna lever arm from their true values were considered in the observability analysis.

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Velocity error propagation equation was made simple by neglecting error terms associated with errors in position and velocity since these error terms are relatively small. The reference frames of errors were chosen in such a way that the observability analysis was less complicated. Inertial sensor biases and errors in attitude and lever arm were represented in the body frame. Position and velocity vectors are represented in the ECEF frame. Observability properties were investigated by examining the properties of the time derivatives of the GPS position measurement. Analysis with the time derivatives of the measurement changed the observability test on the 18 error states to that on at most nine error states: errors in attitude, gyro bias, and lever arm. These error states were represented in the same reference frame, the body frame. These nine error states together with the simplified velocity error propagation model made the observability test straightforward. Observability analysis results showed that the time-invariant error dynamics model has seven unobservable modes: attitude, lever arm, and the component of gyro bias in the direction of the specific force. Due to the unobservable gyro bias, yaw error can increase quite fast if the change in the acceleration of the vehicle is small. It was also shown that all the unobservable errors can be made observable throughthe maneuvering.Acceleration changes improve the estimates of attitude and gyro bias. The changes in angular rate enhance the estimate of the lever arm between the GPS antenna and IMU. However, the motion with a constant angular velocity has no influence on the lever arm estimation. A covariance simulation on the error state was given to verify the results of observability analysis. The numerical simulation results showed that acceleration changes were effective to improve attitude estimation. They also showed that changes in angular velocity reduced lever arm estimation error. Changes in both acceleration and attitude were shown to be effective for the estimation of the vertical component of gyro bias. Even though the numerical simulation on the error covariance can be considered as a useful means for the statistical analysis of the degree of observability, it has limitations on the simulation of the behavior of state estimators for real measurement systems. Precise mathematical descriptions of vehicle trajectory, low-frequency components of errors in the inertial sensors, and multipath error in GPS measurements are not easy to obtain. Field tests may be necessary as a separate work to obtain experimental evaluation of the effect of maneuvering on the error estimation.

Thus, if , then (54)–(56) are satisfied with the for any value of following three cases. First, with . Second, and is in the direction and is arbitrary. Otherwise, of . Third, the above equations are satisfied with the following three cases. and is in the direction of . The First, second and third cases are the same as the above second and third cases. This completes the proof of Property 3.2. B. Proof of Property 3.3 From (44), it follows that (57) (58) Obviously, the following three sets of , , and satisfy (57) and (58); First, with in the direction of . Second, with and where is an arbitrary real number. Third, an arbitrary value of with . This completes the proof of Property 3.3. C. Proof of Property 3.4 , it follows that

From (36) with

(59) Decompose

and

such that (60) (61)

where , , , , from (59), it follows that

, and

are real numbers. Then,

(62) Since that

,

, and

are linearly independent, it follows (63) (64) (65)

However, (63) and (64) imply that

APPENDIX A

(66)

A. Proof of Property 3.2 Note that . Thus , , and for . These relations also hold in the proofs of Properties 3.3–3.6. From (44), it follows that

Since

is not parallel with , and . From (36) with , it follows that

(54)

. Thus,

(67)

(55)

(68) .. .

.. . (56)

(69)

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Thus, if , then (67)–(69) are . Otherwise, . satisfied with Obviously, (44) holds with any nonzero . This completes the proof of Property 3.4. D. Proof of Property 3.5

for completes the proof Property 3.9.

. This

H. Proof of Property 3.10 , , , where , , and Let are constant initial angles, is a constant pitch rate, is a constant yaw rate, and is time. Then, it follows that

Decompose and into the forms of (60) and (61), into the form respectively, and decompose (70) where , , and are real numbers. The proof of is satisfied with Property 3.4 showed that (36) with and . For , (36) implies that

(71) Since , , and not zero, it follows that

are linearly independent, and

(75) , , where Then, the proof follows from the relation

,

.

is

(72) (73) (74) Equations (72) and (73) imply that . With this . Thus, . Indeed result, (74) implies that . This completes the proof (44) is satisfied with any nonzero of Property 3.5.

(76) with

,

, and This completes proof of Property 3.10.

.

I. Proof of Property 3.11 Let , , where , , and are constant initial angles, is a constant roll rate, is a constant yaw rate, and is time. Then, it follows that

E. Proof of Property 3.7 The property can be easily derived from the relations and , for , because is constant. This completes the proof of Property 3.7. F. Proof of Property 3.8 Since

(77)

and all orders of the time derivatives of it have

the same direction, for all proof of Property 3.8.

,

. Thus, . This completes the

Thus, we have (78) shown at the bottom of the page. If then

(79)

G. Proof of Property 3.9 Property 3.9 can be proved with the following relations: , , , and

,

Otherwise, is a full rank matrix if proof of Property 3.11.

. This completes

(78)

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J. Proof of Property 3.12 Let , , and where , , and are initial angles, is a constant yaw angular acceleration, and is time. Then, it follows that

(80)

(81) and , where is a real Let number. The observability test condition implies that . and This implies that where is a real number. Thus, the system has two unobservable modes in (52). This completes the proof of Property 3.12. REFERENCES [1] T. Bell, “Error analysis of attitude measurement in robotic ground vehicle position determination,” Navigation, vol. 47, no. 4, pp. 289–296, Winter 2000–2001. [2] X. He and L. Jianye, “Analysis of lever arm effects in GPS/IMU integration system,” Trans. Nanjing Univ. Aeronaut. Astronaut., vol. 19, no. 1, pp. 59–64, Jun. 2002. [3] S. Hong, Y. S. Chang, S. K. Ha, and M. H. Lee, “Estimation of alignment errors in GPS/INS integration,” in Proc. Inst. Navigation GPS 2002, Portland, OR, pp. 527–534. [4] S. Hong, M. H. Lee, J. A. Rios, and J. L. Speyer, “Observability analysis of INS with a GPS multi-antenna system,” Korean Soc. Mech. Eng. Int. J., vol. 16, no. 11, pp. 1367–1378, 2002. [5] A. A. Sutherland Jr., “The Kalman filter in transfer alignment of inertial guigance systems,” J. Spacecraft Rockets, vol. 5, pp. 1175–1180, 1968. [6] J. Baziw and C. T. Leondes, “In-flight alignment and calibration of inertial measurement units-part I: general formulation,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-8, pp. 440–449, 1972. [7] I. Y. Bar-Itzhack and B. Porat, “Azimuth observability enhancement during inertial navigation system in-flight alignment,” J. Guidance Contr., vol. 3, pp. 337–344, 1981. [8] B. Porat and I. Y. Bar-Itzhack, “Effect of acceleration switching during INS in-flight alignment,” J. Guidance Contr., vol. 4, pp. 385–389, 1981. [9] D. Goshen-Meskin and I. Y. Bar-Itzhack, “Observability analysis of piece-wise constant systems-part II: application to inertial navigation in-flight alignment,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 4, pp. 1068–1075, 1992. [10] , “Observability analysis of piece-wise constant systems-part I: theory,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 4, pp. 1056–1067, 1992. [11] I. Y. Bar-Itzhack and N. Berman, “Control theoretic approach to inertial navigation systems,” J. Guidance Contr., vol. 11, no. 3, pp. 237–245, 1988. [12] Y. F. Jiang and Y. P. Lin, “Error estimation of INS ground alignment through observability analysis,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 92–97, 1992. [13] S. Hong, M. H. Lee, S. H. Kwon, and H. H. Chun, “A car test for the estimation of GPS/INS alignment errors,” IEEE Trans. Intell. Transp. Syst., vol. 5, no. 3, pp. 208–218, 2004. [14] K. R. Britting, Inertial Navigation System Analysis. New York: WileyInterscience, 1971.

[15] M. Wei and K. P. Schwarz, “A strapdawn inertial algorithm using an earth-fixed cartesian frame,” Navigation, vol. 37, no. 2, pp. 153–167, 1990. [16] D. Goshen-Meskin and I. Y. Bar-Itzhack, “Unified approach to inertial navigation system error modeling,” J. Guidance Contr. Dyn., vol. 15, no. 3, pp. 648–653, 1992. [17] A. Gelb, Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974. [18] J. A. Farrell and M. Barth, The Global Positioning System & Inertial Navigation. New York: McGraw-Hill, 1999. [19] J. A. Farrell, T. D. Givargis, and M. J. Barth, “Real-time differential carrier phase GPS-aided INS,” IEEE Trans. Contr. Syst. Technol., vol. 8, no. 4, pp. 709–721, 2000. [20] C. T. Chen, Linear System Theory and Design. New York: Holt, Rinehart and Winston, 1984. [21] D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE, vol. 54, no. 2, pp. 221–230, Feb. 1966. [22] D. Gebre-Egziabher, R. C. Hayward, and J. D. Powell, “A low-cost GPS/inertial attitude heading reference system (AHRS) for general aviation applications,” in Proc. 1998 IEEE Position, Location Navigation Symp., Palm Springs, CA, 1998, pp. 518–525. [23] H. Hou and N. El-Sheimy, “Inertial sensors errors modeling using Allan variance,” in Proc. ION GPS/GNSS 2003, Portland, OR, pp. 2860–2867. [24] F. M. Ham and R. G. Brown, “Observability, eigenvalues, and Kalman filtering,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 2, pp. 269–273, 1983. [25] P. S. Maybeck, Stochastic Models, Estimation, and Control. New York: Academic Press, 1979, vol. I.

Sinpyo Hong received the B.A. degree from Pusan National University, Busan, Korea, in 1982, the M.S. degree from Korea Advanced Institute of Science and Technology, Seoul, Korea, in 1985, and the Ph.D. degree from the University of California, Los Angeles (UCLA), in 1993, all in mechanical engineering. From 1985 to 1986, he was a Design Engineer with Korea Heavy Industries and Construction Co. From 1986 to 1990, he was a Research Engineer for the development of a robot with Samsung Advanced Institute of Technology, Giheung, Korea. From 1993 to 1997, he was involved in the development of a navigation system for a unmanned aerial vehicle at UCLA. Since 2000 he has been working on the development of an unmanned land vehicle at Pusan National University, where he currently is a Research Professor with the Advanced Ship Engineering Research Center. His current research interests include design and analysis of integrated navigation systems and control systems. Dr. Hong is a member of the Institute of Navigation.

Man Hyung Lee (S’79–M’83–SM’01) was born in Korea in 1946. He received the B.S. and M.S. degrees in electrical engineering from Pusan National University, Pusan, Korea, in 1969 and 1971, respectively, and the Ph.D. degree in electrical and computer engineering from Oregon State University, Corvallis, in 1983. From 1971 to 1974, he was an Instructor in the Department of Electronics Engineering, Korea Military Academy, Seoul. He was an Assistant Professor in the Department of Mechanical Engineering, Pusan National University, from 1974 to 1978. From 1978 to 1983, he was a Teaching Assistant, Research Assistant, and Postdoctoral Fellow at Oregon State University. Since 1983, he has been a Professor in the College of Engineering, Pusan National University, where he was a Pohang Iron and Steel Co., Pohang, Korea (POSCO) Chair Professor in the School of Mechanical Engineering from 1997 to 2003 and was Dean of the College of Engineering from March 2002 to February 2004. His research interests are estimation, identification, stochastic processes, bilinear systems, mechatronics, micromachine automation, and robotics. He is the author of more than 750 technical papers and was Program Chair of the 1998 4th ICASE Annual Conference. Dr. Lee is a member of the American Society of Mechanical Engineers, Society for Industrial and Applied Mathematics, and Society of Photo-Optical Instrumentation Engineers. He was General Cochairman of the 2001 IEEE International Symposium on Industrial Electronics (ISIE) and is a General Cochairman of The 30th Annual Conference of the IEEE Industrial Electronics Society (IECON ’04).

HONG et al.: OBSERVABILITY OF ERROR STATES IN GPS/INS INTEGRATION

Ho-Hwan Chun received the B.Sc. and M.Sc. degrees in ship and ocean engineering from Pusan National University, Korea, in 1983 and 1985, respectively. He received the Ph.D. degree from the Department of Naval and Ocean Engineering, Glasgow University, U.K., in 1988. He was Yard Research Fellow at Glasgow University from 1988 to1990 and a Principal Researcher with the Hyundai Maritime Research Institute, Ulsan, Korea, from 1991 to 1993. He has been an Associate Professor in the Department of Naval Architecture and Ocean Engineering, Pusan National University, since 1994. In 2002, he became Director of the Advanced Ship Engineering Research Center (ASERC), designated by the Ministry of Science and Technology, Korea. His main research area is hydrodynamics such as hull form design, hull and structure interactions with waves, WIG and drag reduction, etc.

Sun-Hong Kwon received the B.S. degree in naval architecture from Pusan National University, Busan, Korea, in 1978, the M.E. degree in ocean engineering from Stevens Institute of Technology, Hoboken, NJ, in 1983, and the Ph.D degree in aerospace and ocean engineering from Virginia Polytechnic Institute and State University, Blacksburg, in 1986. He joined Pusan National University in 1986 as a Faculty Member in the Department of Naval Architecture and Ocean Engineering. His current research interests are the application of wavelet analysis to ocean engineering and development of sea wave monitoring system by using radar images.

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Jason L. Speyer (M’71–SM’82–F’85) received the S.B. degree in aeronautics and astronautics from the Massachusetts Institute of Technology, Cambridge, in 1960 and the Ph.D. degree in applied mathematics from Harvard University, Cambridge, MA, in 1968. His industrial experience includes research at Boeing, Raytheon, Analytical Mechanics Associated, and the Charles Draper Laboratory. He was the Harry H. Power Professor in Aerospace Engineering at the University of Texas, Austin, and is currently a Professor in the Mechanical and Aerospace Engineering Department, University of California, Los Angeles. He spent a research leave as a Lady Davis Visiting Professor at The Technion—Israel Institute of Technology, Haifa, in 1983. He was the 1990 Jerome C. Hunsaker Visiting Professor of Aeronautics and Astronautics at the Massachusetts Institute of Technology. Dr. Speyer is a Fellow of the American Institute of Aeronautics and Astronautics. He has twice been an elected member of the Board of Governors of the IEEE Control Systems Society. He has been an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and Chairman of the Technical Committee on Aerospace Control. From October 1987 to October 1991 and from October 1997 to October 2001, he was a member of the USAF Scientific Advisory Board. He received the Mechanics and Control of Flight Award and Dryden Lectureship in Research from the American Institute of Aeronautics and Astronautics in 1985 and 1995, respectively. He received an Air Force Exceptional Civilian Decoration in 1991 and 2001 and the IEEE Third Millennium Medal in 2000.