Terrain Reconstruction Method Based on Weighted Robust Linear Estimation Theory for Small Body Exploration Zhengshi Yu, Pingyuan Cui, and Shengying Zhu yuzhengshi@gmail.com; cuipy@bit.edu.cn; zhushy@bit.edu.cn Institute of Deep Space Exploration, Beijing Institute of Technology, Beijing, People’s Republic of China; Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, Beijing, People’s Republic of China.

Contents Introduction Terrain Reconstruction Method Based on Least Square Method Terrain Reconstruction Method Based on Weighted Robust Linear Estimation Numerical Simulation and results Conclusions Thursday, April 19, 2018 Institute of Deep Space Exploration

Introduction Humans have never stop the steps to exploring the universe. 1996 NEAR 1998 Deep Space 1 1999 Stardust 2005 Deep Impact 2003 Hayabusa 2004 Rosetta Thursday, April 19, 2018 Institute of Deep Space Exploration

Small body Asteroid Comet Significance Exploring the orbital evolution, reducing potential danger of collision, Investigating the genesis and evolution of solar system, Exploiting natural resource, Testing and developing new aerospace technology. Significance Thursday, April 19, 2018 Institute of Deep Space Exploration

Terrain Reconstruction DEM generation, Physical parameter estimation, Autonomous navigation. Terrain Reconstruction Challenge Requirement VS. Long distance Severe time delay Large uncertainty Limited prior knowledge Autonomy Robustness and adaptivity Accuracy Algorithm simplicity Cost and power No terrain reconstruction algorithm is adapted in actual mission. Navigation camera & Optical information Thursday, April 19, 2018 Institute of Deep Space Exploration

Terrain Reconstruction Method Based on Least Square Method Application of Least Square Method in Terrain Reconstruction How to determine the position vector V ? Thursday, April 19, 2018 Institute of Deep Space Exploration

Nonlinear observation model Linearization The use of prior information V0 Linear observation model H is integrated weighting matrix Real position of landmark can be determined by nominal position V0.and correction Thursday, April 19, 2018 Institute of Deep Space Exploration

Robustness of Traditional Least Square Method The robustness of algorithm can be described as Influence Function and Collapse Point. Influence Function determines the influence of estimation by single observation Collapse Point describes how much bias of the model the estimated states can be tolerated Collapse Point of LSM is 0 Influence Function of LSM is infinite Robustness of least square method is very weak, algorithm based on least square method is not stable Thursday, April 19, 2018 Institute of Deep Space Exploration

Equivalent weighting matrix Terrain Reconstruction Method Based on Weighted Robust Linear Estimation The basic theory of robust linear estimation is M estimation Estimation criterion Differentiate Equivalent weighting matrix How to calculate? Thursday, April 19, 2018 Institute of Deep Space Exploration

Construction of Integrated Weighting Matrix Error Propagation Weighting Matrix Observability Weight Data Depth Weight Error Propagation Weighting Matrix Influence of different error sources to the observation model error Singular value of error propagation matrix represent the influence degree of different error sources. pl=[p l f]T Error propagation matrix Thursday, April 19, 2018 Institute of Deep Space Exploration

Integrated weighting matrix Observability Weight Observability of observation model The observability can be described by the condition number of matrix Ai . Data Depth Weight Data depth of different observation conditions Data depth describes the position of each data point in the data set. Integrated weighting matrix Thursday, April 19, 2018 Institute of Deep Space Exploration

Numerical Simulation and results 3-Demensional model of Eros 433 asteroid including 64,800 data points Simulation parameters Parameter Value Unit Position sensor accuracy 10 m Attitude sensor accuracy 0.1 deg Navigation camera imaging size 4×3 mm Navigation camera focus 0.5 Imaging error Nominal landmark position error 200 Spacecraft flies in a 36 km round orbit around the asteroid. On orbit, the navigation camera takes photos of the asteroid every 6 degrees. Scenario Thursday, April 19, 2018 Institute of Deep Space Exploration

Reconstruction errors in three axes (landmark miss match and navigation sensor fault are not considered). Weighted robust linear estimation method can improve the reconstruction accuracy. Reconstruction errors in 3 axes decrease with the increase of measurement times. When more than 6 measurements have been conducted, the reconstruction accuracy can not be improved much. Thursday, April 19, 2018 Institute of Deep Space Exploration

Reconstruction errors in three axes (landmark miss match and navigation sensor fault are considered) landmark miss match Position sensor fault Attitude sensor fault Reconstruction errors in three axes all increase due to landmark miss match and navigation sensor fault. Proposed terrain reconstruction method can improve the accuracy efficiently. Thursday, April 19, 2018 Institute of Deep Space Exploration

Conclusions A novel terrain reconstruction method is proposed based on robust linear estimation. Integrated weighting matrix is constructed with the consideration of the influence of different error sources, the observability of landmarks, and data depth of different observation conditions. The accuracy is increased compared with least square method and robust linear estimation. The robustness and adaptation of the method can be efficiently improved. Thursday, April 19, 2018 Institute of Deep Space Exploration

Thank You! Thursday, April 19, 2018 Institute of Deep Space Exploration