RRT and the piano mover’s problem
Fig.11: categorizing space around an obstacle
The blue block represents an obstacle and the three categories are:
1. Within the blue block. If the centre of the object is generated within this region, whatever the rotation is, it will definitely be blocked.
2. The red area, where the outer red line represents the maximum distance from centre point of the object to the edge of obstacle when both are still in contact with each other. If the centre of the object is generated within this region, we need to wo rk out in what range of θ the object will get blocked. 3. The outer region. When the centre of the object is generated within this region it will definitely not get blocked. Therefore, it is fairly easy to build up 3-dimensional obstacles for region 1 and 3. The only thing we are left with is region 2.
2.32 A mathematical approach for region 2
In order to perform any further calculation, the maximum distance that can be reached between the side of the obstacle and centre point of the object needs to be calculated (which is the distance between edge of the obstacle to the outer red line in region 2).
Figure 12 shows this extreme situation, where the black horizontal line represents one edge of the obstacle and the red line represents edge of region 2. The distance between these two lines occurs when
Fig.12: demonstration of extreme situation
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