The psi-angle model and the equivalent tilt(ET) model have been widely used for in-flight alignment (IFA) to align and to calibrate a strapdown inertial navigation ...
I. INTRODUCTION
Comparison of SDINS In-Flight Alignment Using Equivalent Error Models MYEONG-JONG YU JANG GYU LEE, Member, IEEE Seoul National University HEUNG-WON PARK, Member, IEEE Agency for Defense Development
The psi-angle model and the equivalent tilt(ET) model have been widely used for in-flight alignment (IFA) to align and to calibrate a strapdown inertial navigation system (SDINS) on a moving base. However, these models are not effective for a system with large attitude errors because the neglected error terms in the models degrade the performance of a designed filter. In this paper, with an odometer as an external aid, a velocity-aided SDINS is designed for IFA. And equivalent error models applicable to IFA with large attitude errors are derived in terms of rotation vector error and additive and multiplicative quaternion errors. It is found that error models in terms of additive quaternion error (AQE) become linear. Thus the proposed error models reduce unmodeled error terms for a linear filter. From a number of van tests, it is shown that the proposed error models effectively improve the performance of IFA.
Manuscript received September 17, 1998; revised January 24, 1999. IEEE Log No. T-AES/35/3/06419. Authors’ current addresses: M.-J. Yu, Automation and Systems Research Institute & School of Electrical Engineering, Seoul National University, SAN 56-1, Shinrim-Dong, Kwanak-Ku, Seoul, 151-742, Korea; J. G. Lee, Automatic Control Research Center & School of Electrical Engineering, Seoul National University, Seoul, 151-742, Korea; H.-W. Park, Agency for Defense Development, Taejon, 305-600, Korea.
c 1999 IEEE 0018-9251/99/$10.00 ° 1046
The alignment of a strapdown inertial navigation system (SDINS) determines the transformation matrix between a body frame to a navigation frame in the local-level frame. The stationary initial alignment which consists of a coarse alignment and a fine alignment is usually performed when a vehicle is at rest. In this case, if low-grade sensors are used for cost reduction, it is virtually impossible to detect small attitude errors because its accuracy heavily depends on inertial sensors employed in alignments. For some applications, the coarse alignment is only performed or the initial attitude is directly obtained from other sources such as a stored attitude or a master inertial navigation system (INS) in order to reduce the initial alignment time. Especially, when a coarse alignment is done during a flight, the attitude cannot be accurately determined due to an uncertainty in an angular body rate and an acceleration. In cases mentioned above, the initial attitude errors may be very large. Large attitude errors do not guarantee the accuracy and reliability of a system after beginning a navigation mode. In-flight alignment (IFA) has been developed for not only estimating the navigation errors but also calibrating the error sources of the inertial sensors on a moving base. When an aided SDINS is designed for IFA, one of the main concerns is the SDINS error model because it plays an important role in analyzing the characteristics of the navigation error propagation and in implementing an optimal filter. Therefore, in recent years, a considerable amount of effort has been devoted to developing effective error models [1—8]. According to the attitude error states utilized in an error model, linearized SDINS error models can be classified into the psi-angle, the phi-angle, equivalent tilt (ET), or additive quaternion error (AQE) models [1—8]. Especially, in SDINS using the quaternion for attitude computation, the error models based on the relationship between ET and AQE have been developed in previous papers [6—8]. In addition, it has been shown that ET model and AQE model are equivalent to each other. So, ET model rather than AQE model have generally been employed to reduce computations [6—8]. But they are valid only for the case where attitude errors are small enough [9]. And this condition is a constraint that limits applying these models. That is, SDINS error models presented, so far, are not effective in implementing an optimal filter for IFA in the case with large attitude errors because the neglected terms in error models can significantly degrade the designed filter. In order to improve the performance of IFA, new error models are required. An IFA method for a system with large attitude errors is proposed here. With an odometer as an external aid, the velocity-aided SDINS is designed.
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New error models applicable to IFA for the system in the presence of large attitude errors are derived. To obtain equivalent error models, the quaternion error (QE) is divided into two types. One is the AQE, and the other is the multiplicative QE (MQE). The rotation vector error (RVE) is introduced as a new variable. Exact relationships between these attitude errors are also derived. Then equivalent attitude and velocity error models are obtained in terms of RVE, AQE, and MQE, respectively. And the most useful model in the obtained error models is proposed. A comprehensive study reveals that the AQE model gives the least modeling error. It is confirmed by van tests. In the next section, an IFA is designed. In Section III, equivalent error models are derived and a linearized error model suitable for IFA in case of the presence of large attitude errors is proposed. Van tests are explained in Section IV. Finally, conclusions are presented. II.
IN-FLIGHT ALIGNMENT
The IFA is a procedure of aligning and calibrating SDINS on a moving base. In the aided SDINS designed for IFA, in order to effectively estimate the large attitude errors, velocity measurements rather than position measurements are frequently employed since velocity errors contain more information about attitude errors than position errors. When the SDINS error model and the measurement model for IFA are obtained, it is important to select an effective attitude error model and a velocity error model because the performance of IFA depends on the selected error models. Useful error models are obtained in the next section. In this section, an odometer-aided SDINS is designed. An odometer measures a forward velocity with its main error being a scale factor. Two-boresight angles related to an odometer is also inevitable due to the difficulties in mounting SDINS on a vehicle. The SDINS error model augmented with the biases of inertial sensors and errors associated with the odometer can be written as follows x_ = Fx + G! · ¸ · ¸ F11 F12 G x_ = ! x+ 0 0 0
(1) (2)
where F and G denote a dynamic matrix and an input noise matrix, respectively. x is the error states defined by x = [V, A, rba , rbg , rbr ]T : (3) The basic error states consist of velocity errors (V), attitude errors (A), accelerometer random biases (rba ), and gyro random biases (rbg ). The odometer scale factor error and two-boresight angles are modeled as random biases and are augmented to the error model
as additional error states (rbr ) together with basic error states. A measurement model is obtained by subtracting an SDINS-indicated velocity from a reference velocity provided by an odometer. That is, the measurement model is written by ˜n ¡C ˜ nV z=V b odometer
˜ n ±V = ±Vn ¡ ¢Cbn Vodometer ¡ C b odometer = Hx + v
(4)
where subscripts n and b denote a navigation and a ˜ n is the SDINS-indicated body frames, respectively. V velocity and Vodometer is the odometer forward velocity. ˜ n and ¢C n represent a transformation matrix and C b b a transformation matrix error, respectively. ±Vn is velocity error states. ±Vodometer is the velocity error related to the odometer and is composed of the odometer forward velocity and additional error states. H is the measurement matrix. Depending on the formulation of velocity error models and attitude error models, the various dynamic matrices, input noise matrices, and measurement matrices in (2) and (4) can be obtained. This is elaborated in the next section. III. EQUIVALENT ERROR MODELS The psi-angle model has been generally used for IFA [2]. But the psi-angle model or ET model is not effective in case of large attitude errors because the neglected terms in an error model can degrade the efficiency of the designed filter [9]. Therefore, new SDINS error models are required to improve IFA. In this section, equivalent error models which can be applied to a system with large attitude errors are derived. Then the most useful linearized error model for IFA is selected. A. New Relationship Between Attitude Errors In SDINS, the attitude of a vehicle is computed using an effective quaternion method. In this case, the rotation vector and the quaternion are used as parameters to compute the attitude. Therefore, the attitude error state can be expressed by RVE or QE. Until now, the attitude error has been expressed using ET or AQE [6—8]. However, for a system with large attitude errors, it is difficult to express an exact relationship between attitude errors using ET and AQE [9]. In this work, RVE, AQE, and MQE are introduced to derive new relation equations between attitude errors. First, ¢©n (´ [¢ÁN , ¢ÁE , ¢ÁD ]T ) is defined as RVE between a true navigation frame and an indicated navigation frame in the local-level frame, where ¢ÁN , ¢ÁE , and ¢ÁD are the components of RVE in N,
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˜ n ¡ Qn ) E, and D frame, respectively. Next, ±Q(´ Q b b ˜ n are the true is defined as AQE, which Qbn and Q b quaternion and the indicated quaternion, respectively. Finally, Qnn0 (´ [q0n , q1n , q2n , q3n ]T ) is defined to be MQE given in terms of RVE, where n0 represents an indicated navigation frame containing errors. The four elements of MQE are defined as Qnn0
= [q0n , q1n , q2n , q3n ]
T T
= [1 + cn , sn ¢ÁN , sn ¢ÁE , sn ¢ÁD ] , where
µ ¶ ¶ ©0n ©0n 1 cn = cos sin ¡ 1, sn = , 2 ©0n 2 q ©0n = (¢ÁN )2 + (¢ÁE )2 + (¢ÁD )2 :
(5)
µ
(6)
When it is assumed that the attitude errors are small enough, RVE becomes equivalent to ET. Equations (12) and (13) are also simplified as (14) and (15), respectively ¢©n = ¡2YT ±Q
(14)
˜ nT ˜ nQ ±Q = ¡ 12 Y ¢©n + Q b b ±Q:
(15)
Equations (14) and (15) have the same form as the relationship between ET and AQE presented previously in [6—8]. Compared with the previous results, newly proposed relation equations, (10), (12), and (13), represent the exact conversion equations between attitude errors. In a similar manner, the relationship between AQE and MQE can be obtained as follows
Then the new relationship between RVE and AQE can be derived as follows. The indicated quaternion can be decomposed into
˜ n ¡ Yqn ±Q = (1 ¡ q0n )Q b
(16)
qn = ¡YT ±Q
(17)
˜ n ´ Q n0 = Q n0 Q n Q b n b b
˜ nT ˜ nQ ±Q = ¡Yqn + Q b b ±Q
(18)
qn = [q1n , q2n , q3n ]T :
(19)
0
(7)
where Qnn is composed of the four elements of MQE. Without loss of generality, (7) can be converted into (8) ˜ n: Qbn = Qnn0 Q (8) b And (8) can be written as (9) based on the quaternion multiplication in [1] 32 ˜ 3 2 3 2 q0n ¡q1n ¡q2n ¡q3n q0 q0 7 6 6q 7 6q q0n ¡q3n q2n 7 6 q˜ 1 7 7 6 7 6 Qbn = 6 1 7 = 6 1n 76 7 4 q2 5 4 q2n q3n q0n ¡q1n 5 4 q˜ 2 5 q3 q3n ¡q2n q1n q0n q˜ 3 (9) ˜ n . From where q˜ 0 , q˜ 1 , q˜ 2 , and q˜ 3 are the elements of Q b (9) the relationship between AQE and RVE can be obtained as (10) ˜ n ¡ s Y(Q ˜ n ) ¢©n ±Q = ¡cn Q b b n
q˜ 2
¡q˜ 2 q˜ 3 q˜ 0 ¡q˜ 1
3 ¡q˜ 3 ¡q˜ 2 7 7 7: q˜ 1 5
(11)
q˜ 0
1 T Y ±Q: sn
(12)
1048
(20)
According to previous papers [6—8], it is known that this term cn can be neglected by normalizing the quaternion. But the equations presented above, (6) and (20), show that it cannot be neglected in spite of the normalization procedure for a system with large attitude errors. Therefore, it must be considered as an important factor when analyzing and deriving new error models.
In this section, equivalent attitude error models for SDINS are derived in terms of RVE and MQE from the attitude error model in terms of AQE, respectively. The obtained error models are compared with the conventional ET model [6—8]. The quaternion differential equation is defined to be (21) in [1 and 7] ˜_ n = 1 [Q ˜ n ]! b : Q b b nb 2
(21)
From (21), the attitude error model in terms of AQE is equal to (22) in [1, 7, 8, 10]
Then, multiplying both sides of (12) by Y, another AQE can be derived as follows ˜ nT ˜ nQ ±Q = ¡sn Y ¢©n + Q b b ±Q:
= ¡(q˜ 0 ±q0 + q˜ 1 ±q1 + q˜ 2 ±q2 + q˜ 3 ±q3 ):
B. Equivalent Attitude Error Model
Multiplying both sides of (10) by the transpose matrix of Y, RVE is given by ¢©n = ¡
From (5), (6), (10), (13), (16), and (18), we can obtain cn as µ ¶ ©0n ˜ nT ±Q cn = cos ¡ 1 = q0n ¡ 1 = ¡Q b 2
(10)
where 2 ˜ ¡q1 6 q˜ 0 ˜ n) ´ Y = 6 Y(Q 6 b 4 ¡q˜ 3
where
(13)
_ = M ±Q + 1 (U ±! b ¡ Y ±! n ) ±Q ib in 2
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b where ±!ib denotes the gyro error model and M, U are represented by 2 3 0 ¡!X ¡!Y ¡!Z 0 !Z ¡!Y 7 16 6 !X 7 M´ 6 7 2 4 !Y ¡!Z 0 ¡!X 5
!Z
2
¡
0
16 6 !N 6 2 4 !E
!D 2 ˜ ¡q1 6 q˜ 0 ˜ n) ´ U = 6 U(Q 6 b 4 q˜ 3 ¡q˜ 2
!Y
¡!N 0
!D ¡!E ¡q˜ 2 ¡q˜ 3 q˜ 0 q˜ 1
¡!X
0
¡!E
¡!D 0
!N 3 ¡q˜ 3 q˜ 2 7 7 7: ¡q˜ 1 5
¡!D
3
!E 7 7 7 (23) ¡!N 5 0
(24)
q˜ 0
The RVE model equivalent to AQE model can be obtained by using the relation equations, (10), (12), (21), and (22). Differentiating both sides of (12), we get µ ¶ 1 T _ 1 _T d 1 n _ ¢© = ¡ Y ± Q ¡ Y ±Q ¡ YT ±Q: sn sn dt sn (25) Inserting (10) and (22) into (25) results in µ ¶ _ n = ¡ 1 YT ± Q _ ¡ 1 Y_ T ±Q ¡ d 1 YT ±Q ¢© sn sn dt sn · ¸ µ ¶ d 1 T T T _ = Y MY + Y Y + s Y Y ¢©n dt sn n · ¸ µ ¶ c ˜ n + d 1 s YT Q ˜n ˜ n + Y_ T Q + n YT M Q b b b sn dt sn n ¡
1 T b n Y [U ±!ib ¡ Y ±!in ]: 2sn
(26)
Substituting the elements of (21) into the elements of Y_ T , and then using the matrices Y and U, the following relationships can be obtained
d dt
µ
1 sn
¶
YT Y = I
(27)
˜n YT U = C b
(28)
_n sn ¢©n = c ¢©n (¢©n )T ¢© 1 c= 2 ©0n
_ n = [I ¡ c ¢©n (¢©n )T ]¡1 ¢© · ¸ 1 ˜n b n n £ ¡!in X ¢©n ¡ [Cb ±!ib ¡ ±!in ] : 2sn (34) Inserting the first-order term of [I ¡ c ¢©n (¢©n )T ]¡1 into (34) after the binomial expansion, (34) can be n readily simplified as (35) because (¢©n )T !in X ¢©n is zero _ n = ¡! n X ¢©n ¡ ¢© in
1 [I + c ¢©n (¢©n )T ] 2sn
˜ n ±! b ¡ ±! n ]: £ [C b ib in
(35)
Assuming very small attitude errors and neglecting high-order terms in the error product, (35) is simplified as follows _ n = ¡! n X¢©n ¡ [C ˜ n ±! b ¡ ±! n ] ¢© in b ib in
(36)
which shows that RVE model can be reduced to the conventional ET model presented in [6—8]. From (22), (34), and (36), it should be noted that AQE model is equivalent to RVE model. The equivalency holds whether attitude errors are small or large. It has been commonly accepted that AQE model is equivalent to ET model, which holds only when attitude errors are small [6—8]. But this assumption is no longer valid if attitude errors are quite large. Large attitude errors are generated when we have to use a low-grade inertial measurement unit or when we solve IFA problems from a stored attitude which is frequently done for tactical missiles. In a similar manner, the attitude error model with respect to MQE can be easily derived from AQE model as follows. Differentiating both sides of (17) and rearranging each term on the right hand side of (37) by using (22), (27), (28), (31), and (32), an MQE model is obtained by
(29)
µ µ ¶¶ ©0n ©0n 1¡ cot (30) 2 2
n X YT MY + Y_ T Y = ¡!in
˜ n is an indicated transformation matrix from where C b n the body frame to the navigation frame and !in X n denotes a skew-symmetric matrix of the !in . Inserting (27)—(33) into (26) results in the following RVE model
(31)
˜n=0 ˜ n + Y_ T Q YT M Q b b
(32)
˜n=0 YT Q b
(33)
_ ¡ Y_ T ±Q q_ n = ¡YT ± Q b n ¡ Y ±!in ] = ¡[YT M + Y_ T ] ±Q ¡ 12 YT [U ±!ib
˜ n + [YT MY + Y_ T Y]qn = (q0n ¡ 1)[YT M + Y_ T ]Q b b n ¡ 12 YT [U ±!ib ¡ Y ±!in ] n ˜ n ±! b ¡ ±! n ]: Xqn ¡ 12 [C = ¡!in b ib in
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Differentiating (20), q_ 0n and the approximated c_ n are also given by 1 nT ˜ b b n q [Cn ±!ib ¡ ±!in ], (38) q_ 0n = 2q0n ˜ b ±! b ¡ ±! n ]: c_ n = 14 ¢©nT [C n ib in
(39)
It is shown in this section that AQE model is equivalent to RVE model and MQE model. The above results show that AQE model is very useful as an attitude error model because AQE model is linear and does not leave unmodeled error terms for IFA with the linear filter. In the next section velocity error model is formulated using AQE, RVE, and MQE. And the most useful velocity error model for IFA is proposed. C. Equivalent Velocity Error Model The conventional velocity equation for SDINS is represented by n )XVn + g n : V_ n = Cbn f b ¡ (2!ien + !en
(40)
The velocity error model (41) can be obtained by perturbing the above equation [1, 7—9]
˜ n ± f˜ b + V ˜ n X(2± !˜ n + ± !˜ n ) + ±g n : (41) +C b ie en
In the velocity error model, ¢Cbn f˜ b is strongly related to attitude errors. It is important to compute ¢Cbn f˜ b exactly to analyze a correlation between attitude and velocity errors. Moreover, one of the problems that arises in implementing IFA with a linear filter is the problem of linearizing error models. In the case where the system has only large attitude errors, ¢Cbn f˜ b in (41) is nonlinear and the remaining terms are linear. So, the linearization of velocity error model is only related with how to linearize the transformation matrix error, ¢Cbn . And the transformation matrix error is also associated with the measurement model, (4). Therefore, it is important to obtain the exact transformation matrix error. The transformation matrix error is defined as the difference between a true transformation matrix, Cbn and an indicated ˜ n. transformation matrix, C b In the sequel, it is shown that different transformation matrix errors can be obtained based on which attitude error is employed among AQE, RVE, and MQE. First, we derive this term with respect ˜ n ¡ ±Q) into elements to AQE. Substituting Qbn (´ Q b of true transformation matrix given by (42), (43) is obtained. ˜ n ¡ ¢C n Cbn = C b b 0
1
2
= 4 2(q0 q3 + q1 q2 ) 2(q1 q3 ¡ q0 q2 )
3
2(q1 q2 ¡ q0 q3 )
2(q1 q3 + q0 q2 )
q20 ¡ q21 + q22 ¡ q23
2(q2 q3 ¡ q0 q1 )
2(q0 q1 + q2 q3 )
¢C12 = 2(q˜ 1 ±q2 + q˜ 2 ±q1 ¡ q˜ 0 ±q3 ¡ q˜ 3 ±q0 ¡ ±q1 ±q2 + ±q0 ±q3 ) ¢C13 = 2(q˜ 1 ±q3 + q˜ 3 ±q1 + q˜ 0 ±q2 + q˜ 2 ±q0 ¡ ±q1 ±q3 ¡ ±q0 ±q2 ) ¢C21 = 2(q˜ 1 ±q2 + q˜ 2 ±q1 + q˜ 0 ±q3 + q˜ 3 ±q0 ¡ ±q1 ±q2 ¡ ±q0 ±q3 ) ¢C22 = 2(q˜ 0 ±q0 ¡ q˜ 1 ±q1 + q˜ 2 ±q2 ¡ q˜ 3 ±q3 ) ¡ ±q20 + ±q21 ¡ ±q22 + ±q23 ¢C23 = 2(q˜ 2 ±q3 + q˜ 3 ±q2 ¡ q˜ 0 ±q1 ¡ q˜ 1 ±q0 ¡ ±q2 ±q3 + ±q0 ±q1 )
q20
¡ q21
¡ q22
+ q23
3 5
¢C32 = 2(q˜ 0 ±q1 + q˜ 1 ±q0 + q˜ 2 ±q3 + q˜ 3 ±q2 ¡ ±q0 ±q1 ¡ ±q2 ±q3 ) ¢C33 = 2(q˜ 0 ±q0 ¡ q˜ 1 ±q1 ¡ q˜ 2 ±q2 + q˜ 3 ±q3 ) ¡ ±q20 + ±q21 + ±q22 ¡ ±q23
where ¢Cij is the component of ¢Cbn . Next, the equivalent matrix, (44) can be derived in terms of RVE by inserting (10) into (43). We get ˜n ¢Cbn = ¡2[cn RR + RR + RR2 ]C (44) b where RR is defined to be 2 0 ¡sn ¢ÁD 6 RR = 4 sn ¢ÁD 0
sn ¢ÁE
3
7 ¡sn ¢ÁN 5 :
sn ¢ÁN
(45)
0
Finally, the transformation matrix error with MQE, (46) is directly obtained from (44) ˜n ¢Cbn = ¡2[q0n I + RQ ][RQ ]C (46) b where I is the identity matrix, RQ is defined as 3 2 0 ¡q3n q2n 7 6 RQ = 4 q3n 0 ¡q1n 5 : ¡q2n
q1n
(47)
0
The above results show that (43), (44), and (46) are equivalent to each other. Equivalent velocity error models in terms of AQE, RVE, or MQE are obtained by substituting (43), (44), or (46) into the transformation matrix error of (41), respectively. However, in the case where a system has large attitude errors, the obtained velocity error models are nonlinear because of the transformation matrix error. Therefore, in IFA with a linear filter, how to linearize the transformation matrix error in order to reduce the modeling error is important. Linearized velocity error models are as follows. Let ¢Cbn (±q) be the transformation matrix error obtained by eliminating the second order terms of the error product in (43). On using ¢Cbn (±q) and U, ¢Cbn f˜ b is also given by ¢Cbn f˜ b = ¢Cbn (±q)f˜ b 2 u2 f˜ b ¡u1 f˜ b 6 ˜b ˜b = 26 4 u3 f ¡u4 f u f˜ b u f˜ b 4
3
(42) 1050
(43)
¢C31 = 2(q˜ 1 ±q3 + q˜ 3 ±q1 ¡ q˜ 0 ±q2 ¡ q˜ 2 ±q0 ¡ ±q1 ±q3 + ±q0 ±q2 )
¡sn ¢ÁE
n )X ±Vn ± V_ n = ¢Cbn f˜ b ¡ (2!˜ ien + !˜ en
2 q2 + q2 ¡ q2 ¡ q2
¢C11 = 2(q˜ 0 ±q0 + q˜ 1 ±q1 ¡ q˜ 2 ±q2 ¡ q˜ 3 ±q3 ) ¡ ±q20 ¡ ±q21 + ±q22 + ±q23
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u4 f˜ b ¡u f˜ b 1
¡u2 f˜ b
3 ¡u3 f˜ b 7 u2 f˜ b 7 5 ±Q ¡u f˜ b 1
VOL. 35, NO. 3
(48)
JULY 1999
where ui is the ith column of matrix U. The linearized velocity error model in terms of AQE is derived by substituting (48) into (41). The linearized velocity error model in terms of RVE simplified from (41) and (44) is similar to ET model. It is shown in the last part of this section. Neglecting high-order terms in the error product, (46) is simplified as ˜ n: ¢Cbn = ¡2RQ C b
(49)
From (49), ¢Cbn f˜ b is given by ˜ n f˜ b ]Xqn : ¢Cbn f˜ b = 2[C b
(50)
Linearized velocity error model in terms of MQE is obtained by inserting (50) into (41). Generalized linear velocity error models can be obtained as follows. In (44), compared with elements of RR or diagonal elements of RR2 , elements of cn RR and remaining elements of RR2 are ignored since they are relatively small and cannot be linearized. Using (6), the transformation matrix error obtained by considering the diagonal elements of RR2 is given by 3 2 a1 cn ¡sn ¢ÁD sn ¢ÁE 7 ˜n 6 : ¢C n = ¡2 4 s ¢Á a c ¡s ¢Á 5 C b
n
D
2 n
¡sn ¢ÁE
n
sn ¢ÁN
N
b
a3 cn
(51)
Equation (51) cannot be used for implementing IFA because of its nonlinearity. But using (5), (17), and (20), we can transform the nonlinear error model into a linear one in terms of AQE as shown in (52) ˜ n f˜ b ]XYT ±Q ¢Cbn f˜ b = ¡2[C b 2 3 a1 0 0 6 7 ˜ n ˜ b ˜ nT + 2 4 0 a2 0 5 C b f Qb ±Q: 0
0
(52)
a3
This is a very useful velocity error model for IFA which is linear in terms of ±Q. Now, we want to show how to find a1 , a2 , and a3 in (52). The difference between the diagonal elements of (44) and those of (51) is approximately given by 2
6 (¢Cbn )ED = 2 4
(a1 ¡ 2)cn ¡ 1=4(¢ÁN )2 0
0
model obtained in (51) when a1 , a2 , and a3 are 2, 2, and 0, respectively [9]. And all the elements of (53) become approximately 0. This completes the development of error models to be used to design an alignment filter even if attitude errors are large. In van tests, filters designed by new models and by conventional model are compared. In the remaining of this section, we examine the relationship between the two models. Eliminating high-order terms in the error product, (51) is simplified as (54) which is the conventional error model 3 2 0 ¡¢ÁD ¢ÁE 7 ˜n 6 ¢C n = ¡ 4 ¢Á : (54) 0 ¡¢Á 5 C b
D
¡¢ÁE
N
¢ÁN
In this case, (55) can be obtained by using (54) and (14) ˜ n f˜ b ]X ¢©n : (55) ¢Cbn f˜ b = [C b The simplified (54) and (55) in terms of RVE have the same form as ET model obtained in the case where a system has the sufficiently small attitude error in [6—8]. On comparing the new model (51) and (52) with the conventional model (54) and (55), we can see that the conventional model produces a large modeling error in a system with large attitude errors. IV. VAN TESTS AND RESULTS To verify the validity of the proposed error model in a system with large attitude errors, equivalent error models are applied to van tests for IFA. A filter used for van tests is the linear Kalman filter. The linear Kalman filter uses (2) as the state equation and (4) as the measurement equation. It has been described in Section III that (2) and (4) employ linear error models based on AQE. According to the used attitude and velocity error model, each van test is performed in three different cases. In case A, the conventional ET model is considered. From (36), (41), (54), and (55), the conventional ET model is summarized as follows
0
0
(a2 ¡ 2)cn ¡ 1=4(¢ÁE )2
0
Each ai must be chosen to minimize (53). Depending on applications, ai can be chosen under the conditions that each ai is a positive real value and the sum of ai ’s belongs to closed interval [0, 4] since all the diagonal elements of RR2 are negative real values and the trace of RR2 is approximately equal to 4cn . For example, in case of having large ¢ÁD and very small ¢ÁN and ¢ÁE , (44) is approximately simplified as the
b
0
0
(a3 ¡ 2)cn ¡ 1=4(¢ÁD )
3 2
7 ˜n 5 Cb :
_ n = ¡! n X ¢©n ¡ [C ˜ n ±! b ¡ ±! n ] ¢© in b ib in
(53)
(56)
˜ n f˜ b ]X ¢©n ¡ (2!˜ n + !˜ n )X ±Vn + C ˜ n ± f˜ b ± V_ n = [C b ie en b ˜ n X(2± !˜ n + ± !˜ n ) + ±g n +V ie en
n ˜ nV ˜n z = ±Vn ¡ [C b odometer ]X¢© ¡ Cb ±Vodometer :
YU ET AL.: COMPARISON OF SDINS IN-FLIGHT ALIGNMENT USING EQUIVALENT ERROR MODELS
(57)
(58) 1051
In case B, the AQE model equivalent to the linear MQE model is considered. In the velocity error model, we use the error model obtained in (52) when a1 , a2 , and a3 are 0. This error model is equivalent to (50). From (17), (22), (41), (49), and (52), the case B model is as follows _ = M ±Q + 1 (U ±! b ¡ Y ±! n ) ±Q (59) ib
2
˜ n ± f˜ b + V ˜ n X(2 ± !˜ n + ± !˜ n ) + ±g n +C b ie en
(60)
T ˜ nV ˜n z = ±Vn + 2[C b odometer ]XY ±Q ¡ Cb ±Vodometer : (61)
In the case C, the new AQE model is considered. We use the error model obtained in (52) when a1 , a2 , and a3 are 1. From ¢Cbn (±q) and (10), this error model is equal to (48). From (22), (41), and (52), the case C model is as follows _ = M ±Q + 1 (U ±! b ¡ Y ±! n ) (62) ±Q ib
in
˜ nT ±Q ˜ n f˜ b ]XYT ±Q + 2C ˜ n f˜ b Q ± V_ n = ¡2[C b b b n n ˜ n ± f˜ b ¡ (2!˜ ie + !˜ en )X ±Vn + C b
˜ n X(2± !˜ n + ± !˜ n ) + ±g n +V ie en
(63)
T ˜ nV z = ±Vn + 2[C b odometer ]XY ±Q
˜ nT ˜n ˜ nV ¡ 2C b odometer Qb ±Q ¡ Cb ±Vodometer :
(64)
In the velocity-aided SDINS, measurements to the Kalman filter operating with 3 s sample period are obtained by the difference between the SDINS-indicated velocity and the odometer-indicated velocity. That is, during IFA an external information source that aids an INS is velocity. The external velocity data are measured by an odometer at every 20 ms. A linear prediction filter which minimizes mean-square errors is designed to decrease the influence of measurement errors [11]. Finally, the velocity information for updating the INS are obtained from the linear prediction filter every 3 s. When it is assumed that the height of a trajectory does not vary much, error states can be partitioned into horizontal states and vertical states in order to reduce the amount of computation since vertical states are only weakly coupled to horizontal states. So, for van tests, two conventional linear Kalman filters are designed for horizontal states and vertical states, respectively. Horizontal states consist of velocity errors (north, east), attitude errors (3-ET state or 4-AQE state), accelerometer random biases (x, y axis), gyro random biases (x, y, z axis), the scale factor of the odometer, and the boresight angle (z axis). And vertical states consist of the velocity error (down), the accelerometer random bias (z axis), and the boresight angle (y axis). The indirect feedback configuration is used in each Kalman filter. This has an advantage in maintaining 1052
Bias Scale Factor
Gyro
Accelerometer
3 deg/h 500 ppm
500 ¹g 500 ppm
in
˜ n f˜ b ]XYT ±Q ¡ (2!˜ n + !˜ n )X ±Vn ± V_ n = ¡2[C b ie en
2
TABLE I Specification of Inertial Sensors
an adequacy of the linearized SDINS error model because the estimates are fed back into the states to correct navigation errors and to calibrate inertial sensors [12]. When two Kalman filters are practically implemented, the U=D covariance factorization filter is used. That is, the time update and the measurement update are calculated using a modified weighted Gram-Schmidt method and scalar measurement update, respectively [12]. And the Kalman filter estimates are used to correct INS data and external velocity data every 3 s. For van tests, equipment including includes inertial sensors, an odometer, and an INS as a reference system are mounted on a van. Ten tests were conducted to obtain a data base on a highway. The chosen road does not have sharp turns or bumps. Wind gusts are not considered. The flight distance, the flight time, and the average velocity of the vehicle are 8 km, 10 min, and 70—80 km/h, respectively. During the van tests if track slip or other conditions occur, odometer data may be corrupted. The INS Kalman filter automatically rejects these data not to degrade the system. Table I shows the specification of inertial sensors used for the test. In this paper, before the IFA is entered, initial attitude errors are assumed to be large. Therefore, the ground alignment which consists of a coarse and a fine alignment is not performed since the large initial errors are taken care of in the IFA filter. The alignment procedure is the following. When the IFA mode is selected at rest, the position, velocity, and attitude measured by a reference INS are transferred to the designed SDINS. Initial attitude errors are also automatically added to the attitude data. That is, the transferred data are used as the initial position, velocity, and attitude for the designed SDINS. In other words, the IFA mode begins at the starting point, i.e., t = 0. The initial roll, pitch, and heading errors are assumed 2 deg, 2 deg, and 3 deg, respectively. Each test is carried out under the same conditions. The INS data used for reference has an accuracy of better than 0.14 mil. Resulting differences between the designed SDINS and the reference INS are considered as the errors of the designed SDINS. Table II shows the van test results after 10 minutes (t = 600 s). Test results show that case C is the most efficient model in the three cases. When the performance of case A is compared with that of case B, the good performance of case B is readily explained with the fact that case B can reduce more
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TABLE II Simulation Result CEP (%)
Heading Error (1 sigma)
1.12 1.09 0.90
0.685 deg 0.455 deg 0.317 deg
Case A Case B Case C
of AQE is the most efficient model among the three cases. It shows also that the performance of SDINS can be significantly improved using the newly obtained error models and relation equations. REFERENCES [1]
modeling errors than case A in the attitude error model. Comparing the performance of case B with that of case C, we conclude that it is important to derive exact error terms associated with attitude errors in the velocity error model to improve SDINS performance. From these test results, we can conclude that the newly obtained error models and relation equations can be used to improve the performance of SDINS.
[2]
[3]
[4]
V.
CONCLUSIONS
In this paper some SDINS error models which can be used to design an alignment filter for IFA with large attitude errors are presented. The RVE, MQE, and AQE are introduced to model attitude error states. It is shown how these attitude errors are related each other. Using the relationship, equivalent error models in terms of the RVE, MQE, and AQE are derived, respectively. Based on the equivalent error models, effective linearized models in terms of AQE are proposed. It is shown that the AQE model is very useful as an attitude error model in the sense that this model is linear and does not leave unmodeled error terms. It is also proposed how the nonlinear transformation matrix error is linearized from the relationship between RVE, MQE, and AQE. The linearized transformation matrix error reduces a modeling error in SDINS. An effective linearized velocity error and a measurement model are also presented. It is numerically proved that the proposed error models have the least modeling error in designing an IFA filter. In order to verify the efficiency of the proposed error models, an odometer-aided SDINS is designed for IFA. Three different Kalman filters are designed based on the conventional model and the new models. Several van tests are done on a highway. Van test results show that the proposed error model in terms
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Siouris, G. M. (1993) Aerospace Avionics Systems: A Modern Synthesis. New York: Academic Press, 1993. Goshen-Meskin, D., and Bar-Itzhack, I. Y. (1992) Observability analysis of piece-wise constant system, Part II: Application to inertial navigation in-flight alignment. IEEE Transactions on Aerospace and Electronic Systems, 28, 3 (Oct. 1992), 1068—1075. Weinreb, A., and Bar-Itzhack, I. Y. (1978) The psi-angle error equation in strapdown inertial navigation systems. IEEE Transactions on Aerospace and Electronic Systems, AES-14, 3 (May 1978), 539—542. Britting, K. R. (1971) Inertial Navigation Systems Analysis. New York: Wiley Interscience, 1971. Goshen-Meskin, D., and Bar-Itzhack, I. Y. (1992) Unified approach to inertial navigation system error modeling. Journal of Guidance, Control, and Dynamics, 15, 3 (May—June 1992), 648—653. Friedland, B. (1978) Analysis strapdown navigation using quaternions. IEEE Transactions on Aerospace and Electronic Systems, AES-14, 5 (Sept. 1978), 764—768. Shibata, M. (1986) Error analysis strapdown inertial navigation using quaternions. Journal of Guidance, 9, 3 (May—June 1986), 379—381. Chung, D., Lee, J. G., Park, C. G., and Park, H. W. (1996) Strapdown INS error model for multiposition alignment. IEEE Transactions on Aerospace and Electronic Systems, 32, 4 (Oct. 1996), 1362—1366. Yu, M. J., Park, H. W., and Jeon, C. B. (1997) Equivalent nonlinear error models of strapdown inertial navigation system. In Proceedings of the AIAA 1997 GNC Conference, Aug. 1997; AIAA Paper 97-3563. Vathsal, S. (1991) Derivation of the relative quaternion differential equation. Journal of Guidance, 14, 5 (Sept.—Oct. 1991), 1061—1064. Davenport, W. B., Jr. (1970) Probability and Random Processes. New York: Mcgraw-Hill, 1970. Maybeck, P. S. (1979) Stochastic Models, Estimation, and Control, Vol. 1. New York: Academic Press, 1979.
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Myeong-Jong Yu was born in Korea in 1964. He received the B.S. and M.S. degrees in electronics engineering from Kyungpook National University, Korea in 1987 and 1990, respectively and is currently pursuing the Ph.D. degree at the School of Electrical Engineering, Seoul National University, Seoul, Korea. From 1990 to 1997, he worked as a Senior Research Engineer in the guidance and control systems department at the Agency for Defense Development, Korea. Since 1998 he has been a Research Assistant at the Automation and Systems Research Institute, Seoul National University. His research interests include inertial navigation systems and robust estimation theory. Jang Gyu Lee (S’75–M’77) received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea in 1971, and the M.S. and Ph.D. degrees from the University of Pittsburgh in 1974 and 1977, respectively. In 1977, he was employed by The Analytic Sciences Corporation (TASC), Reading, MA, doing research in the areas of missile parameter identification, missile guidance, and security assessment of power plants. From 1981 to 1982, he was with the Charles Stark Draper Laboratory, Cambridge, MA, where he worked in the areas of inertial navigation systems and optimal control of underwater vehicle. In 1982, Dr. Lee joined the faculty of Seoul National University, College of Engineering, School of Electrical Engineering, as an Assistant Professor, and since 1992 he is Professor at the same school where he is engaged in both teaching and research. He is also responsible for the graduate course sequences in inertial navigation theory, and optimal control and estimation theory. He is currently a visiting professor with the Bradley Department of Electrical and Computer Engineering and the Center for the Study of Science in Society, Virginia Polytechnic Institute and State University in Blacksburg, VA. Dr. Lee is the author of more than 100 journal papers and 250 conference papers in inertial navigation, optimal control, and estimation theory. In addition to his academic duties at Seoul National University, he has been the director of the Automatic Control Research Center since its inception in Dec. 1994, which is one of the academic research centers in Korea sponsored and supported by the Ministry of Defense. His current research interests include theory and applications of inertial navigation systems, vehicle parameter identification, automated guided vehicles, and target tracking systems. Dr. Lee is a member of National Academy of Engineering in Korea, a council member of KIEE (Korean Institute of Electrical Engineers), a senior member of AIAA, and a member of Sigma Xi, IEEE, IFAC Technical Committee on Aerospace, and IFAC Technical Committee on Education.
Heung Won Park (S’92–M’95) was born in Seoul, Korea, in 1956. He received the B.S. degree in mechanical design and production engineering and the M.S. and Ph.D. degrees in control and instrumentation engineering from the Seoul National University, Seoul, Korea, in 1979, 1988, and 1995, respectively. Since 1979, he has worked as a Principal Research Engineer in the Guidance and Control Systems Department at the Agency for Defense Development, Korea. From 1991 to 1995, he was a Research Assistant at the Automation and Systems Research Institute, Seoul National University. His research interests include inertial navigation system analysis and design, estimation theory and application, and large-scale systems. 1054
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