Relative Navigation of Long-Range Non-Cooperative Targets Based on Monocular Sequence Images

DONG Tianshu,WANG Dayi,HUANG Meili,HOU Bowen

Beijing Institute of Spacecraft System Engineering,Beijing 100094

Abstract:Along with the increase of the number of failed satellites,plus space debris,year by year,it will take considerable manpower and resources if we rely just on ground surveillance and early warning.An alternative effective way would be to use autonomous long-range non-cooperative target relative navigation to solve this problem.For longrange non-cooperative targets,the stereo cameras or lidars that are commonly used would not be applicable.This paper studies a relative navigation method for long-range relative motion estimation of non-cooperative targets using only a monocular camera.Firstly,the paper provides the nonlinear relative orbit dynamics equations and then derives the discrete recursive form of the dynamics equations.An EKF filter is then designed to implement the relative navigation estimation.After that,the relative“locally weakly observability”theory for nonlinear systems is used to analyze the observability of monocular sequence images.The analysis results show that by relying only on monocular sequence images it has the possibility of deducing the relative navigation for long-range non-cooperative targets.Finally,numerical simulations show that the method given in this paper can achieve a complete estimation of the relative motion of longrange non-cooperative targets without conducting orbital maneuvers.

Key words:relative navigation,monocular sequence images,EKF filter

1 INTRODUCTION

Along with the increase in the number of failed satellites,space debris increases year by year,relying on ground surveillance and early warning will consume a lot more manpower and resources.Autonomous relative navigation of spacecraft for long-range non-cooperative targets can solve this problem.In this paper,the situation where a chaser spacecraft is navigating with respect to a non-cooperative target at large separations(dozens of kilometers)is studied.Equipped with a single optical camera,the relative navigation system has the advantages of simplicity,cheapness,compactness,and low power consumption.Moreover,comparing with other sensors such as lidar,radar or stereo-cameras,the simple camera,which has the inherent characteristics of being passive,robust,with a high-dynamic range capability[1],is most suitable for long-range non-cooperative targets relative navigation.

Historically monocular vision was difficult for measurement of the relative distance between moving objects directly,and the monocular vision relative navigation system was generally considered as an unobservable system.This problem has been explored in several research studies.Nardone and Aidala[2]analyzed the observability requirements for target motion with 2-D bearings-only,and their paper shows that for certain types of maneuvers the estimation process could remain unobservable.Hammel[3]extended the observability requirements previously established for bearing-only tracking in 2-D to 3-D.Fogel and Gavish[4]presented the observability of Nth-order nonlinear dynamics.However,the target’s Nth-order dynamic is too simple for orbital dynamics,hence cannot be applied to the issues covered in the article.Woffinden and Geller[5]derived an analytical expression of the observability criteria for angles-only navigation,and the influence of the initial orbital maneuver on the observability of the system was analyzed.In the literature[5],the problem of observability of the relative navigation system is converted into whether the systematic observation and the initial value of the system are uniquely mapped.The criteria shows that with a calibrated thrust maneuver,observability can be guaranteed for all possible relative trajectories.Gaias and Amico[6]addressed the design and the implementation of a relative navigation tool that makes use of camera-based angles-only measurements,and the maneuver required to achieve full observability.More recently,Ardaens and Gaias[7]solved the problem of the angles-only navigation by modeling the relative motion with relative orbital elements,where the initial state is obtained by orbital maneuver.It can be noted that the previous articles have maintained the possibility of improving the observability by use of an orbital maneuver.This article analyzes the impact of the orbital manifold on the observability and implements the relative motion estimation without a maneuver.

The remainder of the paper is organized as follows:Section 2 provides the model of the relative orbit dynamics and the discrete form.The model of monocular sequence images is obtained in Section 3,and the design of the EKF relative navigation filter is provided in Section 4.Section 5 illustrates the possibility of monocular sequence images to obtain complete relative navigation information.Finally,the simulation is covered in Section 6 to verify the effectiveness of the algorithm.

2 MODELING SPACECRAFT RELATIVE MOTION

In the following,the relative orbit dynamics equation based on Keplerian orbit is provided.Until now,numerous models of relative orbital dynamics have been studied,such as the CW equation[8],the separated orbit parameter equation[9],the curve coordinate equation[10],etc..Those relative orbit dynamic models can be divided into two types:nonlinear and linear.The linear relative orbit dynamics model has the advantage of small calculation amount and convenient sequential calculation,while the non-linear model retains a more manifold information.To describe the relative motion between chaser and target realistically,a non-linear model[11]based on the Cartesian frame is introduced:

whereX∈R6is the relative state described in the LVLH frame of the chaser;fchaserandRchaserare the true anomaly and the geocentric distance of the chaser.Since the chaser and target are assumed to be in a Keplerian orbit,perturbation effects and orbital maneuvers are not considered in the equation (1).

The continuous relative model cannot be used for estimating the relative motion directly.Therefore,a discrete dynamics model is given as follows:

whereFkis determined by the first order of Taylor series of Eq.(1);F1andF2are given by

3 MEASUREMENT MODEL

As the relative navigation system is equipped with a monocular sensor,the only effective measurement is monocular sequential images.Monocular sequential images could be expressed as a series of image space points.However,image space points are uneasy to explain the relationship between observations and relative states.Therefore,angle measurements will be used to take the place of image space points in this paper.Figure 1 shows the observed geometric relationship.

Figure 1 Observed geometric relationship

The measurement model is supplied as:

whereαandβare measurement angles;is the observation noises vector;xC,yC,zCis the vectorrin the frame of monocular sensor;ρ=||(xC,yC,zC)||2=||(x,y,z)||2is defined as the absolute distance between the chaser and the target;xC,yC,zCandx,y,zhave the relationship as follows:

4 EKF FILTER OF MONOCULAR SEQUENCE IMAGES

In this section,the design of an EKF filter based on the non-linear relative orbit model and monocular sequence images will be introduced.For the problem of non-linear system estimation,use of the EKF filter is a common method.Compared with UKF,EKF has a smaller calculation amount and is more suitable for satellite orbit applications.The EKF algorithms are given as:

State update:

whereFk ,k-1is provided in equation (3).

Measurement update:

whereHkis the Jacobian matrix of the measurement equation(5).

5 OBSERVABILITY ANALYSIS

For the observability criteria of non-linear systems,Hermann and Krener[13]provided an observable rank criterion based on the method of“locally weakly observability”.In this section,the theory is chosen to analyze the observability of the relative navigation of monocular sequence images.The system’s Lie derivative is defined as follows:

whereXis the state vector of the system;fandhare equations of states and observations respectively.Define the system observability matrixNAas:

If the system observability matrixNAis nonsingular,the relative navigation system will satisfy the locally weak observability.In this time,the relative navigation system could estimate the relative motion based on monocular sequence images only.For calculating the rank ofNA,analytical expressions of,are provided.his given as:

According to the analytical expression ofNAfrom (13)to(18),the observability of monocular sequence images could be analyzed.Two spacecraft formation orbits shown in Table 1 are taken as examples to show the potential observability of monocular sequence images.

Since the relative motion is a bounded orbit in space and is consistent with the period of the host star,the simulation timeTuses the period of the chaser.The trajectory of the relative trajectory in the LVLH frame of the host star and its projection in the viewing plane (X-Yplane)can be seen in Figure 2 and Figure 4.

Figure 3 and Figure 5 show that the rank number of the observability matrix meets the observability condition.Different from previous research conclusions[5-7],the results show that the monocular sequence images approach has the possibility to achieve a complete estimation of relative motion.Moreover,it is unnecessary for the chaser to conduct a maneuver or model the effective orbital perturbation to improve observability.

6 SIMULATION

To verify the effectiveness of the monocular sequence images with the EKF filter design in this paper,along with two spacecraft formation orbits as shown in Table 1 are taken as examples and simulated in a MATLAB environment.It is assumed that the host star does not perform any active orbit maneuvers and does not predict any prior information.The initial state error is set as 10%·Xreal(t0),and the measurement noise is set as 80''(3δ).The estimated results of the relative trajectory and the true data of the state are shown in Figure 6 and Figure 7.

Table 1 Relative orbital parameters

Figure 2 Fly-by orbit in the LVLH frame of the host star and its projection in the X-Y plane

Figure 5 The rank number of the observability matrix NA of mix orbit

It can be seen from Figure 6 and Figure 7 that the estimated results approximately coincide with the relative motion trajectory,which indicates that the filter designed in this paper is effective.Furthermore,the results also show the effectiveness of monocular sequence images in obtaining the complete relative motion.

Figure 6 Real trajectories and estimates of fly-by

Figure 7 Real trajectories and estimates of mix

7 CONCLUSIONS

A relative navigation method based on monocular sequence images is presented in this paper to solve the problem of longrange non-cooperative targets motion estimation.Different from the previous research,this method could estimate the complete relative motion of the long-range target without performing an orbital maneuver.The non-linear model of the relative orbit dynamics is provided,and its discrete form is presented for the foundation of the filter design.At the same time,a mathematical expression for monocular sequence images is presented.EKF filter is designed to solve the problem of non-linear system estimation.According to the analytical form of the observability matrix,the observability analysis of the monocular sequence images shows the possibility of achieving a complete estimation of relative motion.The final numerical simulation verifies the correctness of the filtering algorithm.