Algorithm 1. Spatial Topological Relation Analysis Algorithm.

Input: 3D point cloud of each object

Output: The spatial topological relations between the objects

Initialize: Create convex hull and AABB of each object from point cloud

begin

for each object A do

for each object B do

for each point $a_i$ in object A do

if $a_i$ is not in $Box\left(B\right)$

then $A\cap B°=\varnothing$, $A\cap \partial B=\varnothing$ and $A\cap B^{-}=\neg \varnothing$ continue

Compute the relative position of p by formula (4)

Check $A\cap B°$, $A\cap \partial B$ and $A\cap B^{-}$ for loop termination

end

end

end

for each object A do

for each object B do

if $A\cap B°=\neg \varnothing$ and $A°\cap B=\neg \varnothing$ then $\mathrm{R}\left(A\to B\right)=cross$

else if $A°\cap B=\varnothing$

if $A\cap B^{-}=\varnothing$ then $\mathrm{R}\left(A\to B\right)=within$

else $\mathrm{R}\left(A\to B\right)=\mathrm{partial}\mathrm{within}$

else if $A\cap B°=\varnothing$

if $A^{-}\cap B=\varnothing$ then $\mathrm{R}\left(A\to B\right)=contain$

else $\mathrm{R}\left(A\to B\right)=partial contain$

else if $A\cap B°=\varnothing$, $A\cap \partial B=\neg \varnothing$, $A°\cap B=\varnothing$ and $\partial A\cap B=\neg \varnothing$

then $\mathrm{R}\left(A\to B\right)=touch$

else $\mathrm{R}\left(A\to B\right)=disjoint$

end

end

end