RRT and the piano mover’s problem
Note θ max serves as a symmetric point of range of angle being blocked. If at a special state the object touches the obstacle at a rotation of α, the range of the angle being blocked by this obstacle will be (θ max - |θ max - α| , θ max + |θ max - α|).
Fig.15: plotting of function
Figure 15 shows the plot of function (1) with 𝑎 on the y- axis and θ on the x -axis. The intersection of the curve between x= 𝜋 and x= tan −1 ( 𝑤 𝑙 )+ 𝜋 2 is used in generating of 3-dimensional obstacles.
This case can expand to all four sides of the 2-dimensional obstacles. There are two main differences in other situations: 1.The transformation from θ towards angle 2 in figure 13 is different. In case shown above it is angle 2 = 𝜋−𝜃 , while for examples where the object is on the left or right of the obstacle angle 2 = 𝜋 2 −𝜃 . 2. the effect of value 𝑎 is different depending on the relative position of object and obstacle. Generally speaking, the purpose is to shift the object till it touches the obstacle. So, 𝑎 will be added to the x or y coordinate accordingly. The final program will do a simple RRT search and path-planning in a 3-dimensional space demonstrated by figure 16 which represents the continuous motion of the object in 2-dimensional search space. Note that the obstacles are much more complicated than shown on this graph. See figure 10 for more references on the shape of obstacles.
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