RRT and the piano mover’s problem
the diagonal of the object is perpendicular to the edge of the obstacle. It is then easy to work out the distance shown on the figure where L and W is the length and width of the object.
To create 3-dimensional obstacles from region 2, a function that describes the relationship between 𝑎 , the shortest distance between the obstacle and the object’s corner, and θ is needed. Note that the centre of the object keeps on the edge line of region 2. Figure 13 demonstrates an example of this.
Fig.13: mathematical model of region 2
Figure 13 also shows the function deduced by using trigonometry and the relationship between 𝑎 and θ is as follows:
𝑤 2 tan(𝜋 − 𝜃)
𝑏 =
𝑎 𝑚𝑎𝑥 +𝑤/2 sin(𝜋 − 𝜃)
𝑙 2
𝑎 sin(𝜋 − 𝜃)
= 𝑏+
+
𝑎 𝑚𝑎𝑥 +𝑤/2 sin(𝜋 − 𝜃)
𝑤 ∙ cos(𝜋 − 𝜃) 2 sin(𝜋 − 𝜃)
𝑙 2
𝑎 sin(𝜋 − 𝜃)
=
+
+
𝑤 2 ) = 𝑤 ∙ cos(𝜋 − 𝜃) + 𝑙 ∙ sin(𝜋 − 𝜃) + 2𝑎
2(𝑎 𝑚𝑎𝑥 +
√𝑙 2 +𝑤 2 −𝑤 ∙cos(𝜋−𝜃)−𝑙 ∙sin(𝜋−𝜃) 2 (1)
𝑎 =
Note that angle 2 is π – θ and size of the object is described as (length, width).
Figure 14 shows how to get θ when a = 0
Fig.14: mathematical represen tation of θ max
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