We will design an extended interacting multiple models adaptive estimator (EIMMAE) for attitude determination of a stereoimagery satellite. This algorithm is based on interacting multiple models (IMM) extended kalman filters (EKF) using star sensor and gyroscope data. In this method, the motion of satellite during stereoimaging manoeuvres is partitioned into two different modes: “manoeuvring motion” mode and “uniform motion” mode. The proposed algorithm will select the suitable Kalman filter structure to estimate gyro errors accurately in order to maintain the peak attitude estimation error less enough at the beginning of manoeuvres while the satellite is in “manoeuvring motion” mode and then will select the suitable star sensor measurement noise level at the end of manoeuvres while the satellite is in “uniform motion” mode to reduce attitude estimation errors. It will be shown that using the proposed algorithm, the attitude estimation accuracy of stereoimagery satellite will be increased considerably. The effectiveness of the proposed algorithm will be examined and compared with the previous proposed methods through numerical simulations.

Data collection in stereo mode is the simplest and most convenient way for 3D topographic data acquisition to produce new and revision of old inaccurate databases and maps which has been matured over the 100 years [

In [

Stereoimaging Scenario.

To achieve the above-mentioned scenario, an accurate and fast attitude determination system is crucial to provide precise attitude knowledge for successful satellite operations. So, we consider two basic modes of flight for achieving high accuracy attitude determination task during each of stereoimaging maneuvers: “uniform motion” mode with constant angular velocity while the satellite reaches the end of stereoimaging maneuvers and “maneuvering motion” mode with high angular velocity at the beginning of stereoimaging maneuvers [

Difficulties associated with high-accuracy attitude estimation techniques to satisfy the above-mentioned requirements are due to the inherent nonlinearities of satellite dynamic model [

In order to deal with the above-mentioned problems, several approaches have been proposed [

To circumvent the aforementioned problems in order to maintain the peak attitude estimation errors less enough in the “maneuvering motion” mode and to reduce the attitude estimation errors in the “uniform motion” mode, in this paper, we will design a new adaptive attitude estimation structure based on interacting multiple models (IMM) using star sensor and gyroscope measurements data. The proposed structure consists of two different IMM estimator structures which are called: “IMM_CT” and “IMM_L”. The “IMM_CT” structure will be selected automatically based on the model conditional likelihood functions to estimate the gyro errors in order to maintain the peak attitude estimation errors less enough while the satellite is in “maneuvering motion” mode. The “IMM_L” structure will be selected automatically to identify the suitable star sensor measurement noise level to reduce attitude estimation errors while the satellite is in “uniform motion” mode. It will be shown that by using the proposed adaptive attitude estimation method, the attitude estimation requirements of stereoimagery satellite will be satisfied simultaneously. The effectiveness of the proposed algorithm method will be examined and compared with previous proposed methods through numerical simulations.

The satellite being studied here is assumed to be rigid and equipped with a three-axis gyro, a star sensor, and four reaction wheels.

The nonlinear kinematic equations of a satellite can be written as follows [

We formulate a simplified gyro model as described in [

At the end of stereoimaging maneuvers, the effect of scale factor and misalignment on the estimator is minimal. So, we can formulate a simplified gyro model while the satellite is in “uniform motion” mode as follows [

The stars are assumed to be inertially fixed neglecting the effects of proper motion and velocity aberration. It is established in that the photograph image plane coordinates of the

In summary, the computed orientation angles using the star vectors (

Star sensor model.

According to Figure

This adaptive estimator structure contains a bank of Kalman filters run in parallel at every time, each based on a particular model, to obtain model-based estimates and check the status of the operation system; the overall state estimate is a kind of combination of those model-based estimates. Therefore, each cycle of IMM algorithm consists of four major steps: (i) model conditional reinitialization (interacting or mixing of the estimates), in which the input to the filter matched to a certain mode is obtained by mixing the estimates of all filters at the previous time under the assumption that this particular mode is in effect at the present time; (ii) model-conditional filtering, performed in parallel for each mode; (iii) mode probability update, based on the model conditional likelihood functions; (iv) estimate combination, which yields the overall state estimate as the probabilistically weighted sum of the updated state estimates of all filters. The probability of a mode being in effect plays a key role in determining the weights in the combination of state estimates and covariance matrices. The dynamics of the plant for designing the Kalman filters for each mode are described as follows:

General structure of the IMM method [

Steps from

Interaction/mixing of the estimates (for

Predicted mode probability:

Mixing estimate:

Mixing covariance:

Model-condition filtering (for

Predicted state:

Predicted covariance:

Measurement residual:

Residual covariance:

Filter gain:

Updated state:

Updated covariance:

Mode probability update (for

Likelihood function:

Mode probability:

Combination of estimates

Overall state estimate:

Overall covariance:

In this section, the IMM baseline algorithm will be adopted to design an attitude estimator while the satellite is in “maneuvering motion” mode. This estimator consists of three EKFs (

IMM_CT attitude estimator structure.

Using (

Residual update (for

IRU compensated rate:

Steps 1 to

Attitude, angular rate, and IRU compensation parameters update

In Steps from

As we mentioned in Section

Here, the IMM baseline algorithm will be adopted to (

IMM_L attitude estimator structure.

Residual update (for

IRU compensated rate:

Steps

Attitude and IRU compensation parameters update

In this structure, we have considered that

In this section, we will design an extended interacting multiple models adaptive estimator (EIMMAE) to maintain the peak attitude estimation errors less enough while the satellite is in “maneuvering motion” mode and to reduce attitude estimation errors while the satellite is in “uniform motion” mode simultaneously. The proposed algorithm will select the suitable Kalman filter structure while the satellite is in “maneuvering motion” mode and then will select the suitable star sensor measurement noise level while the satellite placed in “uniform motion” mode. The proposed EIMMAE attitude estimator structure is shown in Figures

EIMMAE structure.

IMM_L estimator block.

The IMM_L estimator block in Figure

In “interaction/mixing of the estimates IMM_L” block, we have supposed that the probability of transition between different EKF structures of IMM_L estimator is zero (

Residual update IMM_CT (for

IRU compensated rate:

Steps

Residual update IMM_L (for

PE_1 block (for

Interaction/mixing of the estimates IMM_L ( for

Mixing estimate:

Mixing covariance:

Steps

Combination of estimates

Overall state estimate:

Overall covariance:

Overall residual:

Residual covariance:

PE_2 block:

Residual of tertiary model:

Steps

Attitude, angular rate, and IRU compensation parameters update

Angular rate estimation (for

Overall angular rate estimate:

A simulation is done to show effectiveness and efficiency of the EIMMAE method for attitude estimation of a stereoimagery satellite with

Mission sequence for stereoimaging.

Maneuver | Attitude description | {Roll, Pitch, Yaw} Degree | |

From | to | ||

Initial Slew | Slew from Initial condition to nadir pointing during 100 sec | ||

Time: | |||

Slew 1 | Slew from nadir pointing to point 1 during 100 sec | ||

Time: | |||

Slew 2 | Slew from point 1 to nadir pointing (point 2) during 100 sec | ||

Time: | |||

Slew 3 | Slew from point 2 to point 3 during 100 sec | ||

Time: | |||

Slew 4 | Slew from point 3 to nadir pointing during 100 sec | ||

Time: |

Stereoimaging Scenario.

In this scenario, a typical star sensor data has been considered based on produced data in [

(a) Number of identified stars (b) Star sensor measurement errors and boundaries.

According to Figure ^{3/2} and ^{1/2}. The initial bias of the gyro for each axis is given by 0.01 deg/hr. The initial covariance of the gyro for the attitude error is set to 0.28 deg^{2}, and the initial covariance for the gyro drift is set to 0.22 (deg/hr)^{2}. The sampling interval for the star sensor and gyroscope is considered

In this simulation, we have supposed that the satellite will achieve a number of large angle maneuvers in a few seconds (100 seconds for each maneuver) to estimate the gyro bias, misalignments, and scale factor errors accurately, by applying this method according to Steps from

Posterior probability of the EKFs structures in IMM_CT method.

The gyro errors estimations are shown in Figures

Gyro bias estimation.

Scale factor estimation.

Upper misalignment error estimation.

Lower misalignment error estimation.

These figures verify the effectiveness of IMM_CT method to estimate gyro errors accurately and to maintain the peak error of attitude estimation less enough while the satellite is in the “maneuvering motion” mode as mentioned in [

According to Figure

Standard deviation of the star sensor measurement noise levels around each axis.

Levels | |||

Roll | Pitch | Yaw | |

(Low) | |||

(High) |

Therefore, we should consider at least

Different model for star sensor measurement noise levels.

Number of models ( | Star sensor measurements noise | ||

Covariance ( | |||

Roll | Pitch | Yaw | |

Model #1 | High | High | High |

Model #2 | High | High | Low |

Model #3 | High | Low | High |

Model #4 | High | Low | Low |

Model #5 | Low | High | High |

Model #6 | Low | High | Low |

Model #7 | Low | Low | High |

Model #8 | Low | Low | Low |

It has been shown in Figure

Number of correct model in

According to Figure

This section clearly shows the effectiveness of the proposed EIMMAE structure for attitude estimation of the stereoimagery satellite. By applying this method during stereoimaging scenario (according to Table

Posterior probability of the EKFs structures in EIMMAE structure.

The posterior probability of the models for star sensor measurements noise level (Table

Posterior probability of the models for the star sensor measurements noise level.

The attitude estimation errors using the EIMMAE structure will be compared with IMM_L and IMM_CT method in Figures

Estimation errors using IMM_L and EIMMAE.

Estimation errors using IMM_CT and EIMMAE.

The comparison results between EIMMAE structure with IMM_L and IMM_CT for attitude estimation at the end of maneuvers (

Comparing results of EIMMAE structure with IMM_L and IMM_CT methods.

Method | Average RMS error for 3 maneuvers (degree) | ||

_{0}=_{n}= | |||

Roll | Pitch | Yaw | |

IMM_CT | |||

IMM_L | |||

EIMMAE |

In order to show effectiveness and advantages of the proposed EIMMAE method to estimate the attitude of the satellite at the imaging moment, we will compare the obtained results with the other previous attitude estimation methods. For this purpose, we have summarized 1 sigma estimation errors of different attitude estimation methods in Table

Comparison of attitude estimation methods.

Method | Gyro noise characteristics | Star sensor noise characteristics | 1 sigma estimation error | ||

Roll | Pitch | Yaw | |||

EKF [ | ^{3/2 }^{1/2} | (_{x,}_{y}) = 3.6 to 6 arcsec_{z} = 70.8 to 180 arcsec | 0.00066 | 0.00072 | 0.00499 |

EKF [ | ^{3/2 }^{1/2} | (_{x}, _{y}) = 10 arcsec_{z} = 20 arcsec | 0.00172 | 0.00155 | 0.00172 |

MMAE [ | ^{3/2 }^{1/2} | (_{x}, _{y,}_{z}) = 3.6 arcsec | 0.0016 | 0.0015 | 0.0015 |

IMMAE [ | ^{3/2 }^{1/2} | (_{x,}_{y,}_{z}) = 3.6 arcsec | 0.0013 | 0.0012 | 0.0013 |

EIMMAE* | ^{3/2 }^{1/2} | (_{x,}_{y}) = 3.6 to 6 arcsec_{z} = 70.8 to 180 arcsec | 0.00032 | 0.00031 | 0.0027 |

This table approves that the proposed EIMMAE algorithm improves the steady-state attitude estimation accuracy of the satellite considerably around roll and pitch axes at the end of stereoimaging maneuvers. According to this table, attitude estimation error around the yaw axis associated with EIMMAE method is more than the attitude estimation error around yaw axis associated with the other methods. The reason is that we have used only one star sensor with lower accuracy than the star sensors which have been used in the other mentioned methods. Therefore, in the similar situation, by using star sensor with the same characteristics being used in the other methods, we can provide more accurate attitude knowledge by applying the EIMMAE algorithm while imaging takes place. So, fulfillment of the EIMMAE algorithm is more effective than the other previous mentioned methods to reduce the peak attitude estimation errors at the start of the maneuvers and to increase accuracy of attitude estimation at the end of stereoimaging maneuvers.

In this paper, we have designed a new adaptive attitude estimator (EIMMAE) based on interacting multiple models (IMM) extended Kalman filters (EKF) for attitude determination of a stereoimagery satellite. The proposed algorithm consists of two different IMM estimator structures: “IMM_CT” structure to select the suitable Kalman filter structure to maintain the peak attitude estimation errors less enough at the start of maneuvers and “IMM_L” structure to select the suitable star sensor measurement noise level to reduce attitude estimation errors while the satellite is in “uniform motion” mode. We have shown in Table