Outline. 1 Introduction. 2 Problem and solution. 3 Bayesian tracking model of group debris. 4 Simulation results. 5 Conclusions

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2 Outline 1 Introduction 2 Problem and solution 3 Bayesian tracking model of group debris 4 Simulation results 5 Conclusions

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5 Problem The limited capability of the radar can t always satisfy the detection and resolution of the small space debris with high density, which usually lead to the loss of individual observation information. This paper particularly focuses on the scenario with a low detection probability.

6 Solution Due to the bad observation for individuals, the objects information must be fully utilized without any lose. The space debris cloud exhibits an overall "bulk" evolution, which can be tracked in group. And it is relatively easy to realize. The group characteristics should be incorporated to improve the tracking performance(e.g. group center, group configuration).

7 The incorporation of the group characteristics To make up the information loss of observation, we incorporate the group center into the Bayesian tracking model The group center approximately obey the two-body motion equation, and can indicate the evolution of the group debris. By using the relationship between the group center and individual trajectories, the objects number and individual trajectory can be estimated. The group center L t 1 is set as the average value of the objects states 1 L t 1 = i X t 1,i e t 1,i e t 1,i =1

8 Bayesian group tracking model with the group center Observation: the position vectors Z 1:t Estimation: the group center L t, the state existence variable e t, and the individual state X t By assuming one step Markov Chain state transition, the standard Bayesian filtering prediction and update steps are: p(l t, X t, e t Z 1:t ) = p(z t X t, e t, L t ) p(l t, X t, e t Z 1:t 1 ) p(z t Z 1:t 1 ) p(l t, X t, e t Z 1:t 1 ) p(lt, X t, e t L t 1, X t 1, e t 1 ) (2) = dx t 1 de t 1 dl t 1 p(l t 1, X t 1, e t 1 Z 1:t 1 ) (1)

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10 The solution details Here, we introduce the calculation procedure of two models as example. The calculation of other models in the solution flow can be found in the reference 1. The prediction model of the group center p(l t L t 1, X t 1, e t 1 ). The association probability between the group center and individuals, i.e MRF model ψ(x t, e t, L t ). 1 J. Huang, W. D. Hu, L. F. Zhang, Bayesian Group Tracking Method For Space Debris, In Proceedings of Sixth European Conference on Space Debris, 2013.

11 The prediction model of the group center Based on the hypothesis of normal distribution of prediction state, we directly predict the current group center according to the previous individual states, i.e.: p(l t L t 1, X t 1, e t 1 ) p(l t X t 1, e t 1 ) p(e t 1 ) N ( F X t 1, Q L ) p(et 1 ) Where F is the state transfer matrix and obtained from Kepler s equation, N(a, b) is the Gauss distribution, Q L is the prediction covariance matrix.

12 Association probability between group center and individuals The observation of the group center is taken as the mean value of the multiple objects current states L t = 1 X t,i. N Υe i Υ e The association probability between the group center and the individuals can be calculated with JPDA filtering as follows: ψ(x t, e t, L t ) = ( Lt L t,ω, Q L, p fa, p d,λ) p JPDA (Ω) Ω p JPDA Ω denotes the validation matrix of joint association event, Q L is the observation covariance, p fa indicates the probability of false alarm, p d signifies the detection probability, and λ is the expected number of false alarm in the observation area.

13 Calculation of Bayesian tracking based on MCMC-Particle algorithm The probability calculation for the aforementioned Bayesian tracking problem is nonlinear and intractable, and it is impossible to derive a closed-form solution. We have used the improved MCMC-Particle (Markov Chain Monte Carlo-Particle) algorithm 2 to solve the Bayesian tracking problem. 2 Pang, S. K., Li, J., Godsill, S. J. Detection and tracking of coordinated groups. IEEE Trans. Aerosp. Electron. Syst., 2011, 47,

14 Simulation parameters The detection probability of individual is a(km) e i( ) Ω( ) ω( ) E( ) Object Object Object Object Object 1 Object 2 Object 3 Object Observations of space objects X (km) Y (km) Z (km) Real trajectories of space debris 4 X (km) Y (km) Z (km) The observations 4

15 Simulation results 7427 Estimated trajectory of object 1 Estimated trajectory of object 2 Estimated trajectory of object 3 Estimated trajectory of object 4 X (km) Y (km) Z (km) The original state particles The trajectories With the group center Estimated trajectory of object 1 Estimated trajectory of object 2 Estimated trajectory of object 3 Estimated trajectory of object X (km) Y (km) The original state particles The trajectories Without the group center 2 Z (km)

16 Simulation results Number of detected objects With group center Without group center Time step (s) RMSE of position estimation (m) RMSE of velocity estimation (m/s) With group center Without group center Time step (s) Time step (s) Average number of detected objects RMSE of the estimation Results of 30 Mont Carlo runs Result analysis By incorporating the group center, the group evolution can be exhibited clearly. In the case of low detection probability, the group tracking method can greatly improve the accuracy of the detected number and states of the objects.

17 Conclusions We have proposed a novel group tracking method, which reveals the overall evolution of the group objects. And the trajectories of individual objects are simultaneously reconstructed explicitly. The group tracking provides a new method for cataloguing the high dense space debris within a "bulk". This paper only focused on the improvement of the transition state distribution p(x t X t 1 ). In the future, the likelihood function of observations p(z t X t ) will be constructed to utilize the measurement information more efficiently.

18 T HANKS!