RRT and the piano mover’s problem
Fig.16: RRT search in 3D space
3. Conclusion
This essay has discussed the implementation of the RRT algorithm to solve the piano mover’s problem, introducing a higher-level search space in order to make a point in this space uniquely identify a state in the original space. Therefore, basic RRT search can be used for path-planning; this transforms the problem by creating new higher-level obstacles according to original ones. To solve this question, original 2-dimensional space is split into three areas, where generating higher-level obstacles for two areas is relatively simple. Then, we focus on the third and most important area. We successfully formulate the relationship between the distance away from edge of the obstacle and the angle of rotation. Therefore, it is possible to build up the 3-dimensional obstacles according to all this information. However, things get trickier on the four quarter-circles on vertices. I did not manage to find a function like the one in 2.32 on these regions. We can approximate them using rectangles without losing too much accuracy, but I believe there should be a nicer solution than that. In addition, when the object is approaching a vertex on the obstacle the function will be no longer applicable. Such case is shown in figure 17.
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