Journal of Systems Engineering and Electronics Vol. 24, No. 1, February 2013, pp.108–117
New celestial assisted INS initial alignment method for lunar explorer Weiren Wu1,2 , Xiaolin Ning2,3,* , and Lingling Liu2,3 1. Lunar Exploration and Aerospace Engineering Center, Beijing 100037, China; 2. School of Instrument Science and Opto-electronics Engineering, Beihang University, Beijing 100191, China; 3. Science and Technology on Inertial Laboratory Fundamental Science on Novel Inertial Instrument and Navigation System Technology Laboratory, Beijing 100191, China
Abstract: In the future lunar exploration programs of China, soft landing, sampling and returning will be realized. For lunar explorers such as Rovers, Landers and Ascenders, the inertial navigation system (INS) will be used to obtain high-precision navigation information. INS propagates position, velocity and attitude by integration of sensed accelerations, so initial alignment is needed before INS can work properly. However, traditional ground-based initial alignment methods cannot work well on the lunar surface because of its low rotation rate (0.55◦ /h). For solving this problem, a new autonomous INS initial alignment method assisted by celestial observations is proposed, which uses star observations to help INS estimate its attitude, gyroscopes drifts and accelerometer biases. Simulations show that this new method can not only speed up alignment, but also improve the alignment accuracy. Furthermore, the impact factors such as initial conditions, accuracy of INS sensors, and accuracy of star sensor on alignment accuracy are analyzed in details, which provide guidance for the engineering applications of this method. This method could be a promising and attractive solution for lunar explorer’s initial alignment.
Keywords: lunar exploration, initial alignment, inertial navigation, celestial navigation. DOI: 10.1109/JSEE.2013.00014
1. Introduction In the future lunar exploration programs of China, the inertial navigation system (INS) will be used on explorers such as Rovers, Landers and Ascenders. According to installation mode, INS can be classified into two types: platform inertial navigation system and strap-down inertial navigation system (SINS) [1]. The SINS is used in this study because of low cost, small size and good reliability. The Manuscript received March 27, 2012. *Corresponding author. This work was supported by the National Natural Science Foundation of China (61233005), the Program for New Century Excellent Talents in University (NCET-11-0771), and the Aerospace Science and Technology Innovation Fund (10300002012117003).
navigation information parameters of INS such as position, velocity, and attitude are obtained by integration, therefore the first step is to obtain initial position, velocity, and attitude of the explorer before INS begins to work. When a lunar explorer is at rest, its initial velocity is zero and its initial position can be obtained accurately by ground tracking system. The major task is to determine the initial attitude of the explorer, namely, the coordinate transformation from the explorer body frame to the navigation frame. This procedure is the so-called initial alignment. Traditional ground-based INS initial alignment methods usually use the earth’s gravity vector to obtain horizontal information, and use the earth’s rotation rate vector to obtain azimuth information when the object is at rest [2–4]. Because the lunar rotation rate is only 0.55◦/h (about 1/27 of that of the Earth), it cannot be used to obtain azimuth information when the accuracy of the gyroscopes is not higher than 0.001◦/h. Thus, other means are demanded to assist INS alignment of lunar explorer. A celestial navigation system (CNS) will be a good choice for INS alignment on lunar surface. INS/CNS integrated navigation methods have been widely used and studied for ballistic missiles [5–7], airplanes [8–10] and spacecrafts [11,12]. These methods mainly focus on eliminating the accumulated errors of the INS for a moving object by using CNS output. However, there are few researches on the initial alignment method of the INS assisted by CNS, especially for lunar explorer. Though the basic principles of these methods are similar, the mathematical models and impact factors are different because of the differences in objects, measurements, coordinates and applications. A new autonomous INS initial alignment method assisted by celestial observations for lunar explorer is pro-
Weiren Wu et al.: New celestial assisted INS initial alignment method for lunar explorer
posed, which uses star observations to help INS estimate its attitude, gyroscope drifts and accelerometer biases. Moreover, the impacts of some factors on alignment accuracy are analyzed at the end of this paper, providing guidance for its engineering applications.
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⎤ − sin λ − sin L cos λ cos L cos λ n T ⎣ cos λ − sin L sin λ cos L sin λ ⎦ Rm n = (Rm ) = 0 cos L sin L (1) where L and λ are the latitude and longitude of the explorer, respectively. ⎡
2. System model Many reference frames [13] used in this celestial assisted INS initial alignment method are described in this section. Then, system models and a filter method are introduced. 2.1 Reference frames (i) The moon inertial frame and the moon fixed frame As shown in Fig. 1, the moon inertial frame om xi yi zi has its origin at the center of the moon. Its z-axis is normal to the equatorial plane, x-axis is in the equatorial plane and points to the vernal equinox, and the y-axis completes a right-handed orthogonal frame. The moon fixed frame om xm ym zm has the same origin and z-axis as om xi yi zi . Its x-axis is in the equatorial plane and points to the prime meridian (0◦ longitude), and its y-axis completes a righthanded orthogonal frame. (ii) The navigation frame The navigation frame oxn yn zn is a local vertical frame and its origin is at the position of the explorer. Its x-axis points to the east, the y-axis points to the north, and the z-axis points upward. The transformation matrix from the navigation frame to the moon fixed frame Rm n can be defined as ⎡
cos θ cos ψ − sin θ sin ϕ sin ψ Rnb = (Rbn )T = ⎣ cos θ sin ψ + sin θ sin ϕ cos ψ − sin θ cos ϕ where ϕ, θ, ψ are the pitch, roll, and yaw angles, respectively. 2.2 System model and filter method Traditional ground-based INS alignment methods usually use a linear INS error equation as the state equation, and the horizontal velocity errors of INS as measurements [14,15]. The observability of this method is relatively low. While in this new method, celestial observations are added as measurements, and the nonlinear INS error equation based on large initial azimuth misalignment angle is used as the state equation. Its observability is greatly improved compared with the traditional one. The state equation and measurement equation of this celestial assisted INS initial
Fig. 1
Reference frames
(iii) The explorer body frame The explorer body frame oxb yb zb is rigidly attached to the explorer and has its origin at the center of the mass of the explorer. Its x-axis is in the symmetry plane of the body and points to the direction along which the rover moves. Its z-axis is perpendicular with the symmetry plane of the body, and the y-axis completes a right-handed orthogonal frame. Rnb is the transformation matrix from the explorer body frame to the navigation frame, which can be expressed as − cos ϕ sin ψ cos ϕ cos ψ sin ϕ
⎤ sin θ cos ψ + cos θ sin ϕ sin ψ sin θ sin ψ − cos θ sin ϕ cos ψ ⎦ . cos θ cos ϕ
(2)
alignment method is given below. 2.2.1 State equation On the earth, initial attitude angles can be roughly calculated by accelerometers and gyroscopes before refined initial alignment. When the gyroscope drift is 0.1◦ /h, the accelerometer bias is 10µg, and the latitude of the object is 3◦ , the accuracy of three misalignment angles are about 2 , 2 and 0.38◦, respectively according to (3) [16]. However, on the moon, the accuracy of three misalignment angles decrease to about 12 , 12 , and 10◦ respectively under the same conditions because of its smaller gravity and much lower rotation rate. The horizontal attitude angles are acceptable with this precision, but the error of the yaw angle
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is too large. If the accuracy of the yaw angle can reach the same level as that on the earth, the accuracy of gyroscopes must be higher than 10−3◦ /h. It is impossible for realization in lunar missions now. ⎧ ⎪ ⎨ δφE = ∇N /g δφN = −∇E /g (3) ⎪ ⎩ δφU = −∇E tan L/g − εE /(Ω cos L)
where δφE , δφN and δφU are the errors of the three misalignment angles, ∇E and ∇N are the horizontal accelerometer biases, εE is the east gyroscope drift, Ω is the rotation rate, and g is the gravity. For these reasons mentioned above, a large initial azimuth misalignment angle may exist. The state equation of this new method using the nonlinear INS error equation based on large initial azimuth misalignment angle is expressed as follows:
⎧ φ˙ E = −(sin φU )ωim cos L + φN ωim sin L − δVN /R + εE ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ φ˙ N = (1 − cos φU )ωim cos L − φE ωim sin L + δVE /R + εN ⎪ ⎪ ⎪ ⎪ ⎪ φ˙ U = (−φN sin φU + φE cos φU )ωim cos L + δVE tan L/R + εU ⎪ ⎪ ⎪ ⎪ ⎪ δ V˙ E = (1 − cos φU )fE + (sin φU )fN − gm (φN cos φN + φE sin φU ) + 2ωim sin LδVN + ∇E ⎪ ⎪ ⎪ ⎨ ˙ δ VN = −(sin φU )fE + (1 − cos φU )fN + gm (φE cos φU − φN sin φU ) − 2ωim sin LδVE + ∇N ⎪ ε˙E = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ε˙N = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ε˙U = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ˙E =0 ∇ ⎪ ⎪ ⎪ ⎩ ˙ ∇N = 0 where φE , φN and φU are the three misalignment angles, εN and εU are the north and upward gyroscope drifts, R is the mean of the moon radius and has the value of 1 738 km, ωim is the rotation rate of the moon, gm is the gravity of the moon and has the value of 1.618 m/s2 , and fE , fN , and fU are the projections of the outputs of accelerometers in the navigation frame. Assume the state vector X = [φE , φN , φU , δVE , δVN , εE , εN , εU , ∇E , ∇N ]T , the state equation of this method can be simplified as ˙ = f (X) + W X (5)
(4)
[xs , ys , zs ]T in the star sensor frame can be computed. At the same time, the direction vector of the star in the moon inertial frame can be obtained from the astronomical almanac, which is given as follows: ⎤ ⎤ ⎡ xi cos δ cos RA si = ⎣ yi ⎦ = ⎣ cos δ sin RA ⎦ zi sin δ ⎡
(7)
where δ and RA are the declination and right ascension of the star. The relation between ss and si is
where W is the process noise.
ss = Rsb · Rbn · Rnm · Rm i · si
2.2.2 Measurement equation
where Rsb is an installment matrix of the star sensor, and Rm i is a transformation matrix from the moon inertial frame to the moon fixed frame. According to (6) and (8), we can obtain the following measurement equation of this method.
Traditional INS alignment methods usually use INS horizontal velocity errors as measurements. While in this new method, starlight vectors of CNS as well as horizontal velocity errors of INS are used as measurements. (i) INS horizontal velocity errors As the explorer is at rest, the horizontal velocities calculated from INS are velocity errors, and its measurement equation is given as δV = [δVE , δVN ].
(6)
(ii) Starlight vectors Given the two-dimensional (2D) star centroid from the threshold star image, a 3D starlight unit vector ss =
Z = [δV, ss ] = h(X) + V
(8)
(9)
where V is measurement noise. 2.2.3 Filter method The Kalman filter (KF) is usually applied to traditional INS initial alignment with the traditional linear INS error equation. However, state equation and measurement equation
Weiren Wu et al.: New celestial assisted INS initial alignment method for lunar explorer
in this new method are nonlinear. Extended Kalman filter (EKF) and unscented Kalman filter (UKF) are two methods usually utilized in this nonlinear system. The comparison of alignment performances of the three filter methods is presented in Section 3.2.
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ters. The drift of each gyroscope is 0.05◦/h (1σ) and the bias of each accelerometer is 10 µg (1σ) with an updated rate of 100 Hz. The measurement errors of these sensors are shown in Fig. 3.
3. Simulation and analysis In this section, simulations are used to test the capability and feasibility of this celestial assisted INS alignment method. 3.1 Simulation conditions In simulations, supposing the lunar explorer is at rest, its true position is 3◦ S and 336◦66 E, and its initial yaw, roll, and pitch angles are 20◦ , 0◦ , and 0◦ , respectively. The astronomical almanac used is the JPL DE405 planetary ephemeris. The star sensor is technically installed parallel to the explorer body frame, with its optical axis pointing to the zenith as shown in Fig. 2. The accuracy of the star sensor is 3 (1σ) and its updated rate is 5 Hz. The inertial measurement unit (IMU), fixed parallel to the explorer body frame precisely, is composed of three optical fiber gyroscopes and three quartz pendulum accelerome-
Fig. 3
Fig. 2
Installation of star sensor and IMU
Initial position errors of latitude and longitude are assumed to be zero. Meanwhile, initial attitude errors are 1 in pitch, roll, and yaw angles. The filter period is 0.2 s and the entire simulation time is 5 min.
Measurement errors of sensors
3.2 Simulation results This section presents simulation results to demonstrate the performance of this celestial assisted INS alignment method. Fig. 4 shows the estimation results of attitude, gyroscope drifts and accelerometers biases of this method using EKF. After the filter convergence period, the estimated values of the three attitude angles quickly converge to 0◦ , 0◦ , and 20◦ . The estimations of the east, north, and upward gyroscope drifts are −0.082◦/h, −0.013 5◦ /h and 0.083 5◦ /h, respectively. The estimations of the east and north accelerometer biases are 3.671µg and 30.411µg, respectively. Fig. 5 shows the estimation errors of the attitude, gyro-
scope drifts and accelerometer biases of this method. From these results, it can be concluded that this celestial assisted INS alignment method can meet the requirements of the high-precision initial alignment for a stationary lunar explorer. The alignment performances of KF, EKF, and UKF are compared, when the initial errors of the yaw angle are 3 , 1 000 and 1 800 , respectively. The state equation used in KF is the traditional linear INS error equation. The state equation used in EKF and UKF is the nonlinear INS error equation. The simulation results are presented in Table 1. Under the condition of small initial error of the yaw angle, the alignment performances of KF, EKF and UKF are similar. However, the nonlinear INS error equation demon-
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strates its obvious advantages under the condition of the
large initial azimuth misalignment angle.
Fig. 4 Estimation of attitude, gyroscope drifts and accelerometers biases
Fig. 5
Estimation errors of attitude, gyroscope drifts, and accelerometer biases
It can be seen from Table 1 that the performance of UKF is slightly better than that of EKF. As UKF costs much Table 1 Model Linear model (KF) Nonlinear model (EKF) Nonlinear model (UKF)
Initial errors of yaw angle/( ) 3 1 000 1 800 3 1 000 1 800 3 1 000 1 800
more computation time than EKF, EKF is still used in the following analysis.
Performance of different filter methods
Estimation errors of attitude/( ) East −0.090 1 1.066 3 12.974 0 −0.089 6 0.796 9 2.552 1 −0.088 8 −0.443 8 0.889 7
North −0.404 7 2.728 5 4.550 8 −0.380 7 −0.063 5 0.743 6 −0.375 2 −0.083 9 −0.613 2
Upward −0.035 8 31.104 1 42.462 7 −0.029 4 −0.589 4 −4.070 8 −0.027 1 −0.370 5 3.867 4
Estimation errors of gyroscope drifts/(◦ /h) East North Upward 0.008 8 0.010 8 0.000 2 −0.027 2 0.353 4 −1.560 9 0.019 3 0.476 8 −2.122 1 0.008 7 0.010 7 0.000 0 −0.044 1 −0.019 4 0.031 7 −0.131 2 −0.091 0 0.218 4 0.008 6 0.010 7 −0.000 1 0.008 0 −0.002 3 0.021 2 −0.054 6 0.070 8 −0.205 1
Estimation errors of accelerometer biases/µg East North −0.075 1 0.039 5 22.801 4 18.076 5 37.281 5 41.791 7 −0.062 1 0.010 4 −3.995 0 7.466 9 −6.972 5 9.510 6 −0.154 6 0.136 0 −0.562 0 0.439 5 4.110 5 2.589 8
4. Analysis of impact factors
4.1 Initial conditions
Besides the state equation and filter method, the alignment accuracy of this method mainly rests on initial conditions and the obtained accuracy of these measurements, which mainly lies on the accuracy of INS sensors and the star sensor, as shown in Fig. 6.
Since the navigation information of INS is obtained by integration, the initial accuracy of position and attitude has great impacts on the accuracy of INS measurements. Besides, the position information is included in Rnm and attitude information is included in Rbn , both of them will
Weiren Wu et al.: New celestial assisted INS initial alignment method for lunar explorer
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affect the obtained accuracy of star observations.
angle gradually increases from 1 to 1 200 and the initial
4.1.1 Initial attitude errors
errors of other two angles are kept 1 . Other conditions are
Fig. 7 shows the results when the initial error of a yaw
the same as described above. Table 2 shows the details.
Fig. 6
Impact factors on alignment accuracy
From these results, we can see that the initial error of the yaw angle has some impact on alignment accuracy but not obviously. And the greater the initial error of a yaw angle
is, the worse the alignment accuracy is. The relationship between the initial error of a yaw angle and alignment accuracy is nonlinear, as shown in Fig. 7.
Fig. 7 Influence of initial attitude errors Table 2 Initial errors of yaw angle /( ) 1 10 60 600 1 200
Estimation errors of attitude/( ) East −0.025 4 −0.020 1 0.037 6 0.198 0 1.278 6
North −0.032 6 −0.031 4 0.066 0 0.542 1 1.470 1
Upward 0.054 0 0.047 2 0.012 8 −0.028 8 −0.892 1
Influence of initial attitude errors
Estimation errors of gyroscope drifts/(◦ /h) East North Upward −1.079 3 × 10−5 −6.415 9 × 10−4 −8.931 0 × 10−4 −0.000 1 −0.000 8 0.000 1 −0.007 4 −0.002 1 0.000 5 −0.017 7 −0.016 4 0.004 7 −0.070 3 −0.070 6 0.066 3
4.1.2 Initial position errors Fig. 8 shows the results when initial position errors of latitude and longitude gradually increase from 0 km to 1 km, and initial attitude errors are kept 1 . Other conditions are the same as described above. Table 3 shows the details. It can be concluded from Table 3 that initial position er-
Estimation errors of accelerometer biases/µg East North 0.029 1 −0.064 6 −0.038 0 0.024 5 −0.128 4 0.148 6 −1.689 8 6.148 2 −6.151 3 8.875 1
rors have a great impact on the estimation accuracy of attitude and horizontal accelerometer biases. Alignment accuracy decreases as the initial position errors increase. There is a linear relationship between initial position errors and the alignment accuracy as shown in Fig. 8. When initial position errors are increased by 0.1 km, the estimation er-
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rors of the east, north and upward attitude are increased by about 7.13 , 15.17, and 0.67, respectively. And the es-
timation errors of the east and north accelerometer biases are increased by about 12.19 µg and 5.61 µg, respectively.
Fig. 8 Influence of initial position errors Table 3 Initial position errors/km 0 0.01 0.05 0.1 0.5 1
Influence of initial position errors
Estimation errors of attitude/( ) East −0.025 4 −0.735 5 −3.575 8 −7.126 2 −35.528 7 −71.030 5
North −0.032 6 1.487 3 7.567 0 15.166 7 75.964 4 151.962 8
Upward 0.054 0 0.116 0 0.364 2 0.674 4 3.154 4 6.250 8
East −1.079 3 × 10−5 −1.114 5 × 10−5 −1.255 3 × 10−5 −1.431 3 × 10−5 −2.839 3 × 10−5 −4.598 9 × 10−5
4.2 Accuracy of INS sensors 4.2.1 Gyroscope drifts Fig. 9 shows the results when the drift of each gyroscope gradually increases from 0.01 ◦ /h to 10 ◦ /h, with other conditions unchanged. Table 4 shows the details. Generally speaking, the gyroscope drifts have little im-
Estimation errors of gyroscope drifts/(◦ /h) North −6.415 9 × 10−4 −6.402 0 × 10−4 −6.346 4 × 10−4 −6.276 8 × 10−4 −5.720 5 × 10−4 −5.025 4 × 10−4
Upward −8.931 0 × 10−4 −8.933 2 × 10−4 −8.942 1 × 10−4 −8.953 3 × 10−4 −9.042 7 × 10−4 −9.154 7 × 10−4
Estimation errors of accelerometer biases/µg East North 0.029 1 −0.064 6 1.245 6 0.503 2 6.111 5 2.774 4 12.193 9 5.613 5 60.853 1 28.326 2 121.677 4 56.717 8
pact on the alignment accuracy compared with other impact factors. It can be seen from Table 4 and Fig. 9 that the greater the gyroscope drifts are, the worse the alignment accuracy is. Data from Table 4 illustrate that attitude, gyroscope drifts and horizontal accelerometer biases can be well estimated, but the estimation accuracy decreases when the gyroscope drifts are large.
Fig. 9 Influence of gyroscope drifts
Weiren Wu et al.: New celestial assisted INS initial alignment method for lunar explorer Table 4 Gyroscope drifts/(◦ /h) 0.01 0.05 0.1 0.5 1 5 10
Estimation errors of attitude/( ) East −0.026 4 −0.026 0 −0.025 4 −0.021 0 −0.015 4 0.034 7 0.110 2
North −0.032 7 −0.032 7 −0.032 6 −0.032 6 −0.032 7 −0.044 9 −0.089 1
Upward 0.054 5 0.054 3 0.054 0 0.051 4 0.048 1 0.014 3 −0.046 5
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Influence of gyroscope drifts
Estimation errors of gyroscope drifts/(◦ /h) East North 3.068 5 × 10−4 2.474 7 × 10−4 −1.456 7 × 10−4 1.636 6 × 10−4 −1.079 3 × 10−5 −1.456 7 × 10−4 −0.001 2 −0.004 8 −0.002 3 −0.010 4 0.007 2 −0.073 6 0.063 5 −0.198 6
4.2.2 Accelerometer biases Fig. 10 shows the results when the bias of each accelerometer gradually increases from 1 µg to 1 mg, when other conditions remain unchanged. Table 5 shows the details. From these results, we can see that the accelerometer
Upward −9.448 0 × 10−4 −9.448 0 × 10−4 −8.931 0 × 10−4 −8.124 8 × 10−4 −0.001 1 −0.016 5 −0.068 9
Estimation errors of accelerometer biases/µg East North 0.028 6 −0.063 8 0.028 8 −0.064 2 0.029 1 −0.064 6 0.031 5 −0.068 2 0.034 9 −0.072 5 0.089 8 −0.109 5 0.284 6 −0.189 4
biases have little impact on the estimation of attitude and gyroscope drifts, but they have a great impact on that of accelerometer biases. The relationship between the accelerometer biases and the estimation errors of accelerometer biases is obviously linear, as shown in Fig. 10.
Fig. 10 Influence of accelerometer biases Table 5 Accelerometer biases /µg 1 5 10 50 100 500 1 000
Influence of accelerometer biases
Estimation errors of attitude/( ) East −0.026 2 −0.025 9 −0.025 4 −0.021 9 −0.017 5 0.017 5 0.061 5
North −0.034 9 −0.033 9 −0.032 6 −0.022 8 −0.010 6 0.087 5 0.209 9
Upward 0.053 3 0.053 6 0.054 0 0.056 9 0.060 7 0.089 9 0.125 4
Esat 1.360 7 × 10−5 2.762 9 × 10−6 −1.079 3 × 10−5 −1.192 7 × 10−4 −2.549 5 × 10−4 −0.001 3 −0.002 7
4.3 Accuracy of star sensor 4.3.1 Star sensor accuracy Fig. 11 shows the results when the star sensor accuracy gradually increases from 0.1 to 60 , with other conditions unchanged. Table 6 shows the details.
Estimation errors of gyroscope drifts/(◦ /h) North Upward −4.673 7 × 10−4 −8.394 3 × 10−4 −5.448 0 × 10−4 −8.632 8 × 10−4 −6.415 9 × 10−4 −8.931 0 × 10−4 −0.001 4 −0.001 1 −0.002 4 −0.001 4 −0.010 1 −0.003 8 −0.019 9 −0.006 8
Estimation errors of accelerometer biases/µg East North −0.030 9 0.020 8 −0.004 2 −0.017 2 0.029 1 −0.064 6 0.295 9 −0.444 2 0.629 4 −0.918 7 3.297 2 −4.714 4 6.632 2 −9.458 3
It can be seen from Table 6 that the star sensor accuracy has some impact on alignment accuracy but not obviously. That is because the error of star sensor is Gaussian noise, which can be reduced effectively by EKF. From Fig. 11, we can see that the relationship between the star sensor accuracy and the estimation errors of attitude and accelerometer biases is obviously linear.
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Star sensor accuracy/( ) 0.1 0.5 1 5 10 30 60
Fig. 11
Influence of star sensor precision
Table 6
Influence of star sensor acuracy
Estimation errors of attitude/( ) East 0.001 0 −0.002 6 −0.007 2 −0.043 7 −0.089 3 −0.271 9 −0.546 0
North 0.001 3 −0.003 4 −0.009 2 −0.056 1 −0.114 7 −0.349 0 −0.700 4
Upward 0.001 9 0.009 1 0.018 1 0.089 8 0.179 5 0.538 3 1.076 5
Estimation errors of gyroscope drifts/(◦ /h) East North Upward −3.830 7 × 10−4 −0.001 2 −3.173 6 × 10−5 −0.001 1 −1.505 4 × 10−4 −3.317 2 × 10−4 −2.675 4 × 10−4 −0.001 0 −2.777 0 × 10−4 −0.001 5 2.4595 × 10−4 −4 6.320 2 × 10−4 −0.003 0 8.878 4 × 10 0.003 5 0.004 3 −0.008 9 0.007 3 0.009 7 −0.017 8
4.3.2 Installation errors of star sensor Fig. 12 shows the results when the installation errors of star sensor are gradually increased from 0 to 1 000 , with other conditions unchanged. Table 7 shows that the installation errors of star sensor have a great impact on alignment accuracy. In a word, the bigger the installation errors of star sensor are, the worse the alignment accuracy is. This is because the installation
Fig. 12
Estimation errors of accelerometer biases/µg East North 0.065 1 −0.094 0 0.060 2 −0.089 9 0.054 0 −0.084 9 0.004 3 −0.044 4 −0.057 8 0.006 1 −0.306 1 0.208 3 −0.678 6 0.511 6
errors of star sensor are constant, and they cannot be eliminated or reduced by EKF. The relationship between the installation errors of star sensor and the alignment accuracy is obviously linear as shown in Fig. 12. When the installation errors of star sensor are increased by 10 , the estimation errors of the east, north and upward attitude are raised by the same level, and the estimation errors of east and north accelerometer biases are increased by about 7.97 µg and 7.94 µg respectively.
Influence of installation errors of star sensor
Weiren Wu et al.: New celestial assisted INS initial alignment method for lunar explorer Table 7 Installation errors of star sensor/( ) 0 5 10 50 100 500 100 0
Influence of installation errors of star sensor
Estimation errors of attitude/( ) East −0.025 4 −5.025 3 −10.025 5 −50.035 5 −100.069 8 −501.213 9 −100 4.808 7
North −0.032 6 −5.032 0 −10.031 0 −50.014 9 −99.972 8 −498.764 1 −995.071 6
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Upward 0.054 0 5.054 1 10.053 9 50.044 0 100.009 9 498.864 0 995.250 4
Estimation errors of gyroscope drifts/(◦ /h) East North Upward −1.079 3 × 10−5 −6.415 9 × 10−4 −8.931 0 × 10−4 −2.866 1 × 10−5 −6.377 9 × 10−4 −9.041 0 × 10−4 −4.653 0 × 10−5 −6.339 8 × 10−4 −9.151 1 × 10−4 −0.001 0 −1.894 9 × 10−4 −6.035 7 × 10−4 −0.001 1 −3.681 9 × 10−4 −5.656 0 × 10−4 −0.002 0 −0.001 8 −2.631 9 × 10−4 −0.003 6 1.113 1 × 10−5 −0.003 1
The influence of the main impact factors on the alignment accuracy is analyzed in this section. From these results, we can conclude that the initial position errors and installation errors of star sensor have a significant impact on the alignment accuracy, which should be paid more attention during the alignment procedure.
5. Conclusions To solve the problem of autonomous INS initial alignment of lunar explorer in the future lunar exploration programs of China, a new celestial assisted INS initial alignment method is studied. Simulations show that this method can not only speed up alignment, but also lead to higher alignment accuracy. This method provides a promising and attractive scheme for explorers on lunar surface. Furthermore, an analysis of impact factors shows that the initial position errors and the installation errors of star sensor have a significant impact on the alignment accuracy, which should be focused on during the alignment process in the future.
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Estimation errors of accelerometer biases/µg East North 0.029 1 −0.064 6 −3.971 1 3.938 4 −7.971 2 7.941 5 −39.972 6 39.966 0 −79.974 4 79.996 7 −399.988 1 400.242 4 −800.003 8 800.550 3
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Biographies Weiren Wu was born in 1953. He received the Ph.D. degree in control science and technology from Huazhong University of Science and Technology. He is mainly working on mission design of deep space, tracking and control technology, and navigation technology. E-mail:
[email protected] Xiaolin Ning was born in 1979. She received the B.E. degree in computer science from Shandong Teachers University in 2001, and the Ph.D. degree in mechanical engineering from Beihang University in 2008. She is mainly working on the guidance, navigation, and control system of spacecraft and autonomous navigation of deep space explorers. E-mail:
[email protected] Lingling Liu was born in 1989. She received the B.S. degree in measurement & control technology and instrument from Henan University of Science and Technology, in 2011, and now is a postgraduate student in Beihang University. She is mainly working on autonomous navigation of deep space explorers. E-mail:
[email protected] 
