As to solve the collaborative relative navigation problem for near-circular orbiting small satellites in close-range under GNSS denied environment, a novel consensus constrained relative navigation algorithm based on the lever arm effect of the sensor offset from the spacecraft center of mass is proposed. Firstly, the orbital propagation model for the relative motion of multi-spacecraft is established based on Hill-Clohessy-Wiltshire dynamics and the line-of-sight measurement under sensor offset condition is modeled in Local Vertical Local Horizontal frame. Secondly, the consensus constraint model for the relative orbit state is constructed by introducing the geometry constraint between the spacecraft, based on which the consensus unscented Kalman filter is designed. Thirdly, the observability analysis is done and the necessary conditions of the sensor offset to make the state observable are obtained. Lastly, digital simulations are conducted to verify the proposed algorithm, where the comparison to the unconstrained case is also done. The results show that the estimated error of the relative position converges very quickly, the location error is smaller than 10 m under the condition of 10^{−3} rad level camera and 5 m offset.

Maintaining formation configuration and restructuring control is essential for cooperative spacecraft to accomplish specific missions, and formation control depends on precise relative navigation between the members [

At present, the commonly used sensors for relative measurement of spacecraft formation flying mainly include: relative GPS, microwave radar, LIDAR, visible light camera, infrared camera and laser rangefinder, etc. However, relative navigation based on GPS can be easily interfered by the environment. In addition, members in the formation will lose common star to do the relative measurement if they are far away from each other. Radio ranging navigation has the defect of mirror orbit [

Range-only measurement via radio signals and angels-only optical navigation devices are relatively simple and reliable in spacecraft formation flying, thus becoming major trend in the field, and lots of research has been conducted by scholars. Dianetti et al. [

The main contribution of this paper is to develop a novel consensus constrained relative navigation algorithm for Multi-Spacecraft Formation in Close-Range that based on the lever arm effect of the sensor offset from the spacecraft center of mass, which will avoid angles-only relative navigation algorithm from converging to the mirror orbit. Orbital maneuver and computational burden brought by complex dynamic model are avoided in the proposed method, only single optical camera is needed to realize relative navigation of close-range spacecraft formation flying.

This paper begins with a brief review of Hill-Clohessy-Wiltshire dynamics and camera offset measurement model in

The origin of a rotating local vertical local horizontal (LVLH) reference frame is collocated with the chief spacecraft center of mass. The axes of the LVLH frame are aligned with the chief spacecraft inertial position vector (x axis or radial), the normal to orbit plane (z axis or cross track), and the along-track direction (y axis completes the orthogonal set).

The position and velocity of the deputy spacecraft center of mass relative to the chief center of mass observed from the chief LVLH coordinates is denoted by

Then, under the assumptions of two-body problem and the range between the chief and deputy spacecraft is small compared to the radial distance to the center of Earth, the relative motion of the deputy with respect to the chief that is orbiting near-circular can be governed by the well-known Hill-Clohessy-Wiltshire dynamics [

It is assumed that the origin of the chief-fixed body reference frame is co-located with the chief center of mass. Without loss of generality, an optical-sensor (camera is considered in this work) offset from the chaser center of mass in the chief fixed body frame, i.e.,

Consider a formation of multiple (at least two) spacecraft, in which the inertial orbit of each spacecraft is assumed to be unknown. Furthermore, it is assumed that each spacecraft installs a directed camera used to measure the line-of-sight relative to other spacecraft and transmits its own estimation to other spacecraft by undirected broadcasting network.

As shown in

When only two spacecraft are in flight as a formation, there will be no other information except the angles-only measurements could be used to estimate the relative orbit. EKF algorithm is only suitable for the estimation of weakly nonlinear system because it expands the original system and measurement by Taylor series and retains only the linear term. Since the proposed angles-only navigation algorithm represents a nonlinear system, then the UKF introduced by Wan et al. [

Initialization

Calculate sigma points and scale weights

Time update

Measurement update

where the superscript–marks the priori estimate,

When multiple (at least three or more) spacecraft are involved in the formation, the constraint based on geometrical topology information between spacecraft may be used to improve the estimation. Then, Consensus Unscented Kalman Filter (CUKF) is a good and easy way to utilize the constraint to achieve a better estimation. The key of conducting CUKF to the orbital estimation is to construct the consensus condition. Thus, the consensus would be modeled firstly for the problem and then used in designing CUKF algorithm in the following.

As can be seen from

Since the distributed estimate strategy is considered, the relative orbit estimations are resolved in different LVLH frames of each spacecraft. Thus, after coordinate transformation, the position vector loop can be expressed as follows:

Differentiating on the both side of

Then, combing and re-organizing

So far, the physical constraint on the relative orbital vectors of the in-loop spacecraft is achieved, which can be used as the consensus condition.

Next, the distributed Consensus Unscented Kalman Filter is considered to be used to estimate the relative orbit for each spacecraft, because of the convergence characteristic of CUKF when smooth and bounded vector field of the dynamics and the measurement are given [

In this section, the Lie derivative method of the observability analysis for nonlinear systems is introduced, then theoretical observability analysis for the proposed offset camera line-of-sight measurement relative navigation system is presented.

For a general nonlinear dynamic system defined as
^{th}

It has been shown that if

The system state is a 6-dimensional vector, and the observation state is a 3-dimensional unit vector but 2 bearing angles in essence. In order to make the observability matrix potentially full rank, the Lie derivatives are required to calculated at least three times. Without loss of generality, the line-of-sight measurement is adopted to perform the observability analysis of the system. The complex observability matrix is as follows:

Consider a constant vector

If and only if

The proposed algorithm is established in MATLAB simulation environment to verify theoretical conclusions mentioned above. The spacecraft parameter settings are shown in

Spacecraft | a [km] | e | Ω [°] | i [°] | ||
---|---|---|---|---|---|---|

1 | 6800 | 0.0000 | 0 | 1.01 | 0 | 0.01 |

2 | 6800 | 0.0002 | 0 | 1.02 | 0 | 0.02 |

3 | 6800 | 0.0003 | 0 | 1.03 | 0 | 0.03 |

Parameter | Value | Parameter | Value |
---|---|---|---|

Time | 3 orbital period | Initial state error covariance matrix | diag [1 × 10^{4} m^{2}, 1 × 10^{4} m^{2}, 1 × 10^{4} m^{2}, 10 m^{2}/s^{2}, 10 m^{2}/s^{2}, 10 m^{2}/s^{2}] |

Time Steps | 30 s | ||

Initial state error std. dev | [50 m, 50 m, 50 m, 5 m/s, 5 m/s, 5m/s]^{T} |
Process noise covariance matrix | diag [0, 0, 0, 1 × 10^{−8} m^{2}/s^{4}, 1 × 10^{−8} m^{2}/s^{4}, 1 × 10^{−8} m^{2}/s^{4}] |

Consensus coefficient | 0.03 | Measurement error covariance matrix | diag [7 × 10^{−7} rad^{2}, 7 × 10^{−7} rad^{2}] |

In order to verify the effectiveness and performance of the proposed algorithm, the following two sets of simulation are performed: in the first group, offset situations are simulated to see whether the relative motion trajectory estimated by the UKF algorithm converges, and in the second group, same offset condition is set to compare convergence performance of CUKF algorithm and UKF algorithm. The statistics of 200 Monte Carlo shooting results are shown in

In order to intuitively compare the effects of different camera-offset length on the angles-only relative navigation algorithm proposed in this paper, the following statistics are defined

Applying mean error

Relative position [m] | Relative velocity [m/s] | ||||||
---|---|---|---|---|---|---|---|

x axis | y axis | z axis | x axis | y axis | z axis | ||

Mean error | 1 m | 61.9 | 127.6 | 34.7 | 0.067 | 0.139 | 0.040 |

5 m | 3.8 | 7.8 | 2.0 | 0.004 | 0.009 | 0.003 | |

10 m | 1.0 | 1.7 | 0.5 | 0.001 | 0.002 | 0.001 | |

Std error | 1 m | 60.2 | 122.9 | 33.8 | 0.065 | 0.135 | 0.039 |

5 m | 44.8 | 98.5 | 25.3 | 0.049 | 0.101 | 0.030 | |

10 m | 0.4 | 0.9 | 0.2 | 0.000 | 0.001 | 0.000 |

Suppose the camera offset of the three spacecraft are

Under such camera offset conditions, combined with the results of the previous observability analysis, we can predict:

A new angles-only cooperative relative navigation algorithm for spacecraft formation in close-range is studied in this paper. Based on the Hill-Clohessy-Wiltshire dynamics, this paper studied the convergence of UKF and CUKF when the measurement sensor (camera) is installed offset from the center of mass of the spacecraft. The research work of this paper mainly includes four aspects: (1) The relative motion model between spacecraft and the sensor measurement model with camera installed away from the center of mass are established. (2) The observability of the estimated state is analyzed by introducing the Lie derivative criterion, and the observability conditions of relative position and velocity are obtained. (3) A decentralized estimation strategy based on consistent unscented Kalman filter is designed and the consistent estimation is constructed by using multiple physical constraints. (4) The effectiveness of the algorithm is verified by standard Monte Carlo simulation, and the performance of the algorithm is tested. The results show that for the spacecraft formation with a short distance of 1–7 km, the relative navigation accuracy is within 10 m when 5 m camera offset is designed. The relative navigation algorithm proposed in this paper is based on three spacecraft. When more spacecraft participate in formation or cluster, the decentralized strategy and nonlinear estimation algorithm will be more complex, which will be the main work of future research.