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Nonlinearities in spacecraft attitude determination problem has been studied intensively during the past decades. Traditionally, multiplicative extended Kalman filter_MEKF_algorithm has been a good solution for most nominal space missions. But in recent years, advances in space missions deserve a revisit of the issue. Though there exist a variety of advanced nonlinear filtering algorithms, most of them are prohibited for actual onboard implementation because of their overload computational complexity. In this paper, we address this difficulty by developing a new algorithm framework based on the marginal filtering principle, which requires only 4 sigma points to give a complete 6-state attitude and angular rate estimation. Moreover, a new strategy for sigma point construction is also developed to further increase the efficiency and numerical accuracy. Incorporating the presented framework and novel sigma points, we proposed a new, nonlinear attitude and rate estimator, namely, the Marginal Geometric Sigma Point Filter. The new algorithm is of the same precision as traditional unscented Kalman filters, while keeping a significantly lower computational complexity, even when compared to the reduced sigma point algorithms. In fact, it has truly rivaled the efficiency of MEKF, even when simple closed-form solutions are involved in the latter.

Nonlinearities in spacecraft attitude determination problem have been studied intensively during the past decades. Since the early 1980’s, multiplicative extended Kalman filtering (MEKF) algorithm [

In recent years, advances in space missions, such as the greater agility demand and the application of lower cost sensors, deserve a revisit of the nonlinearity issue. Although a variety of advanced nonlinear filtering algorithms exist, only few of them are close to the restrict numerical expense requirements of actual onboard implementations. In the existing methods, the well-known sigma point Kalman filters (SPKFs) [

In spite of being efficient among nonlinear filters, baseline SPKF still seems computational costly for engineering implementation. If we denote

Anyhow, applications of simplex sets have reduced the required sigma points to 50% of the traditional nonaugmented algorithms and have made a significant improvement in numerical efficiency. In fact, the numerical efficiency of simplex SPKF algorithm is able to rival or even exceed its EKF counterpart if general formed Riccati equations and numerical integration process are involved in the latter. However, for typical quaternion-based spacecraft attitude problems, since simple analytical solutions and sparse matrices exist for MEKF’ covariance propagation and measurement updates, general simplex SPKF algorithm still compares unfavorably with MEKF in efficiency.

Clearly, to develop a competitive algorithm alternative to the MEKF for practical applications, we need a still further reduction of

The main contributions of this article include two parts. First, we have derived in detail a marginal version of SPKF algorithm for typical 6-state attitude determination system. The new algorithm uses merely 4 sigma points to give a complete attitude and angular rate estimation; hence it is able to achieve a high numerical efficiency that truly rivals the MEKF. Second, we have proposed a new set of simplex sigma points for Euclidean Geometric space, named the Geometric Simplex sigma point set. The new set has a symmetric structure, a lower computational expense and is numerically more accurate. It would be of use in a variety of 3-dimensional modeled dynamic problems.

The organization of this paper proceeded as follows. First, we established a general 6-state stellar-inertial spacecraft attitude kinemics and measurement model and analyzed the partially linear structure in the system. Then a marginal SPKF estimator is derived in detail. Next, we looked into the asymmetrical properties of existing sigma point construction algorithms and proposed the Geometric simplex set. Finally, we incorporated the proposed sigma point set into the marginal filtering framework to configure a complete attitude estimator, named the Marginal Geometric Sigma Point Filter, and inspected its performance in simulation with comparisons to the traditional MEKF and Spherical Simplex SPKF.

For spacecraft attitude estimation, quaternion has been the most widely used attitude parameterization. The quaternion is given by a 4-dimension vector defined as

From (

In actual calculation process, (

To cope with the unit-length constraint of quaternion, local error states (also known as the local disturbance states) are introduced into filter design. We describe the local attitude error as a 3-dimensional rotation vector

After all generality, the observation model in this article is established as an automatic star sensor with quaternion measurements

General problem would involve numerical integration of Riccati equations to evaluate the

We consider now the application of SPKF to the system discussed above. A set of

The SPKF attitude estimator is as follows. At the beginning time of each filtering step

Next, propagate the

Next we compute the covariance predictions. If we take

Above is the framework of an SPKF version attitude estimator. A blocked-form procedure summary is listed in Table

Now we look into some special structures of the above estimator. First, noting (

Then noting (

The above discussions leads to the following conclusion: as long as we can construct a set of sigma points that matches the given mean estimations of

So now our goal becomes to construct a minimum set of sigma points, which is able to fully capture all the available information given as (a) unbiased mean estimation, that is,

To match the unbiased mean, we have

To further reduce the computational expense and make a better symmetry property, assign equal weights

To match

Looking into (

To avoid state augmentation, we would like to have the propagation noise terms incorporated into the filter with trapezoidal approximation. However, as

Construction of the base set

However, both sets lack numerical accuracy, and they are complex to compute, as irrational numbers

In order to make a better symmetry property, we propose a new base set of sigma points here as in the 3rd row of Table

Simplex base sets for 3-dimensional space.

Sigma set | |
---|---|

Spherical simplex | |

Schmidt orthogonal | |

Geometric simplex | |

Residues of numerical mean estimation.

Mean | SS | |||

SO | ||||

GS | 0 (precise) | 0 (precise) | 0 (precise) | |

Covariance | SS | 3.878960 | ||

SO | 4.249187 | |||

GS | 1.734723 | 0 (precise) |

SS: Spherical simplex set. SO: Schmidt orthogonal set. GS: Geometric simplex set.

Comparisons of total arithmetic operations.

Algorithm | Phase | Multiplies | Adds | Square |

roots | ||||

MEKF | Propagation | 350 | 270 | 2 |

Measurement update | 260 | 185 | 1 | |

One full filter step* | ||||

One observation circle | ||||

MGSPF | Propagation | 350 | 285 | 5 |

Measurement update | 380 | 280 | 1 | |

One full filter step | ||||

One observation circle | ||||

SSUKF | Propagation | 810 | 565 | 8 |

Measurement update | 455 | 370 | 1 | |

One full filter step | ||||

One observation circle |

Pseudocode and computational expense evaluation of the MEKF with closed-form solutions.

Algorithm | ||||
---|---|---|---|---|

Initialize | ||||

Propagation | 6 | 5 | 1 | |

6 | 2 | |||

3 | ||||

49 | 30 | |||

51 | 30 | |||

108 | 81 | |||

54 | 57 | |||

54 | 48 | |||

3 | ||||

22 | 15 | 1 | ||

Measurement update | 3 | |||

114 | 58 | |||

16 | 12 | |||

5 | 4 | |||

18 | 9 | |||

27 | 27 | |||

27 | 27 | |||

27 | 27 | |||

22 | 15 | 1 | ||

3 | ||||

Pseudocode and computational expense evaluation of the MGSPF algorithm.

Algorithm | ||||||
---|---|---|---|---|---|---|

Initialize | ||||||

Propagation | 25 | 13 | 3 | |||

0 | 54 | |||||

6 | 5 | 1 | ||||

6 | 2 | |||||

3 | 3 | |||||

6 | 2 | |||||

6 | 2 | |||||

32 | 24 | |||||

5 | 1 | |||||

3 | 3( | |||||

3 | ||||||

6( | ||||||

9( | ||||||

22 | 15 | 1 | ||||

For | ||||||

Measurement update | 3 | 3( | ||||

3 | ||||||

9 | ||||||

9 | ||||||

9 | ||||||

114 | 58 | |||||

16 | 12 | |||||

5 | 4 | |||||

18 | 12 | |||||

27 | 27 | |||||

27 | 27 | |||||

27 | 27 | |||||

22 | 15 | 1 | ||||

3 | ||||||

For |

Geometry simplex sigma points in 3-dimensional Euclidean space.

(i) The new set is more intuitive to comprehend and apply, especially for the 3-dimensional Euclidean space, the true space where we are, and the true space in which a variety of dynamical problems as guidance, navigation, and so on take place.

(ii) Lower computation expense and better round-off error behavior. The new set is free from calculating any irrational numbers. Furthermore, as it is only constituted of

(iii) The Geometric simplex set has a better symmetrical structure, which would help to further increase the numerical accuracy, including (a) single dimension symmetry completely fulfilled (or in matrix language, each row of

Suppose that we have already obtained a 3-dimensional unbiased state

The numerical experiment is programmed with double precision float numbers in MATLAB, and some typical results are listed in Table

Incorporating the Geometric Simplex sigma point set into the Marginal SPKF framework, we would have a new nonlinear SPKF estimator for attitude estimation, namely, the Marginal Geometric Sigma Point Filter (MGSPF) algorithm, summarized in Table

For the computing effort of the propagation phase, there is little difference between MGSPF and MEKF, both are about half of the SSUKFs. For measurement update phase, the MGSPF takes some more arithmetic operations, but still only 80% of the SSUFK. In fact, if we take into account that in most actual implementations, there exist more propagation steps than observation steps, the total computational expense of MEKF and MGSPF would be very close. It is clear that the MGSPF has achieved a truly rivalizing efficiency as the MEKF, even when simple analytical closed-form solutions are included in the MEKF, and they are almost 50% of the SSUKF.

In this section we apply the proposed Marginal Geometric Sigma Point Filter (MGSPF) algorithm to the typical stellar-inertial spacecraft attitude determination system with numerical simulations. To give a comparison, the multiple extended Kalman filter (MEKF) and a nonaugmented spherical simplex unscented Kalman filter (SSUKF) with trapezoidal approximation of the propagation noise are also simulated.

Parameters of the simulated model are set as follows. The spacecraft’s initiation attitude is

The initial states of all filters are set equivalently as

The simulation results of attitude estimation error and and gyro bias estimation error are, respectively, illustrated in Figures

Estimation error history of the attitude.

Estimation error history of the gyroscope bias.

We now address the issue of tuning. The main difference between MGSPF and traditional sigma point filters in parameter selection is mainly reflected in the usage of

Influence of the bias noise parameter in MGSPF.

A new, minimum sigma points algorithm for spacecraft attitude and angular rate estimation has been developed. By marginalizing out the linear substructures within the random walk gyro bias model and the attitude involving, only observation model, the new algorithm needs only 4 sigma points to give a complete 6-state attitude and angular rate estimation. The algorithm’s computational expense is only 50% of the traditional SSUKF algorithm. It has truly rivaled the MEKF algorithm’s computing speed even when simple analytical closed-form solutions are included. Yet it is still able to achieve the same accuracy as traditional unscented Kalman filters.

A new, symmetrical, and numerically more efficient simplex sigma set has been presented. The new set is completely free from irrational numbers and is free from any multiplication operations during sigma point construction. The new set introduces almost none of round-off error for mean reference and smaller error for covariance reference. It would be of use for the implementation in a variety of 3-dimensional Euclidean space involving dynamical problems such as positioning and attitude estimation problems.

With the remarkable reduction in computational expense, the sigma point Kalman filter would gain a significant upgrading in its competitiveness as a candidate algorithm for actual onboard implementation.