An Autonomous Path Planning of a Rover Utilizing Multi-Image Shape from Shading

Abstract

For planetary exploration, path planning for a rover is one of the important missions. This paper deals with Shape from Shading (SfS) scheme for estimation of planet terrain. As a reliable reflectance model of a surface, the Hapke model is formed in the world of remote sensing. This paper proposes to utilize the Hapke model in the SfS algorithm for multiple camera images. Since the Hapke model has singularity when the gradient vector of a surface element is coincident with a certain direction, the model is modified not to show the singularity. As a result, the SfS algorithm is applicable to multiple camera images and valid regardless of the degree of albedo of the surface. Applying the proposed SfS algorithm for multiple images, an autonomous path planning based on Dynamic Programming is shown. The effectiveness of the proposed scheme is investigated in numerical simulations, and some discussion about the results is presented. INTRODUCTION For planetary exploration, considering the communication time delay between the planet and the earth, the mission is desired to be completed autonomously even in unknown environments. Estimated 3-D elevation map of planetary terrains is very efficient for path planning of rovers. Estimation from camera images is desirable from a point that the images contain terrain information over the region. ’Shape from Shading (SfS)’[1];[2] is one of the most useful techniques because of the simple equipment: a camera and signal processors. However, it has a problem to be solved, the improper convergence of the algorithm caused by Research Associate, member of AIAA y Associate Professor, member of AIAA local minimum, self-shadow, occlusion and noises on camera image. Essentially, estimation of 3-D elevation from only one 2-D camera image is an ’ill-posed’ problem and it has no unique solution. To avoid the problem, it is assumed typically for the standard SfS algorithm that the terrain surface is smooth, and the derivatives of the gradients are minimized. However, the assumption neither be always satisfiable nor improves the results drastically. To utilize multiple camera images for SfS algorithm, the Lommel-Seeliger model[3];[4] has been proposed for the reflection of a surface. However, some experimental results using real camera images are much worse than expected, while the approach can improve the results at least theoretically. One of the most influential factors for the results is improperness of the reflectance model for the albedo of the surface. On the other hand, in the world of remote sensing, the Hapke model[5] [7] is considered as the most reliable reflectance model of a surface. However, the Hapke model shows singularity when the gradient vector of a surface element is coincident with such certain direction that the reflected light angle is close to the incident light angle. Considering that the accuracy for estimated gradient of a surface element is quite low around the singularity, and that 3-D elevation map is produced to put up the surface elements one by one, the inaccuracy caused by the singularity results in unacceptable error. Therefore, this paper proposes to modify the Hapke model and apply the model to the SfS scheme for multiple camera images. First, this paper summarizes the standard SfS algorithm. Then, the explanations for the SfS utilizing multiple images and the reflectance model are followed. Making a comparison with the different models, the modified Hapke model is proposed, which is valid for 1 American Institute of Aeronautics and Astronautics AIAA/AAS Astrodynamics Specialist Conference and Exhibit 5-8 August 2002, Monterey, California AIAA 2002-4825 Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. the different degree of albedo. Based on the proposed reflectance model, a new SfS algorithm and the scheme for an autonomous path planning by Dynamic Programming are introduced. Finally, the effectiveness of the proposed scheme is investigated in numerical simulations, and some discussions about the results are followed. SHAPE FROM SHADING ALGORITHM