Consider the problem of moving the brown object (which is a union of two tetrahedra) from the configuration on the left to the one on the right. The configuration-space map for this problem is six-dimensional as the brown object has six degrees of freedom.
If a straight-line path in the 6D configuration space is taken [click here for an animation of that (600kB)] the object hits the cube as it moves; whenever the current configuration membership-tests solid in the configuration-space map the little green flag box changes to a red pyramid:
However, an algorithm by Dan Pidcock allows that straight path to be modified into a polyline in the hyperspace of the map that misses the configuration-space obstacle, allowing the brown object to move from the start to the end without colliding:
Click here for an animation of the resulting path without collisions (also 600kB).