RRT and the piano mover’s problem
The figures above show the path-finding result for simple and adaptive RRT from coordinates (0,0) to (100,100). The purple rectangles represent obstacles and red line represent the centre of the object. Note that the separation between the left two and right two blocks is 20, so it is nearly impossible for the algorithm to find the way through this separation when width=length=20.
2.22 Changes addressing the second difference
In order more conveniently to uniquely represent the state of the object in PMP, another dimension is introduced with the third axis representing the degree of rotation. Therefore, any point on this 3- dimensional space uniquely refers to one state of the object back in our 2-dimensional map, and if we manage to come up with the 3-dimensional obstacles according to 2-dimensional blocks and a 2- dimensional object. The only thing that we need to do is to perform a simple 3-dimensional RRT path searching to solve the PMP.
Fig.10: 3D obstacles (R) generated according to 2D obstacles and size of object (L) 9
Figure 10 demonstrates 3-dimensional obstacles being generated where a continuous connection from start to end point will be reflected back to a continuous motion from starting state to ending state, which is the solution to PMP.
2.3
Method of generating 3-dimensional obstacles
2.31 Categorizing of case
The whole 2-dimensional searching space can be categorized into three parts, as shown in figure 11.
9 See https://www.coursera.org/learn/robotics-motion-planning/lecture/Yh5fc/2-3-piano-movers-problem.
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