# Optimal Path Planning under Different Norms in Continuous State Spaces

Optimal path planning under full state and map knowledge is often
accomplished using some variant of Dijkstra's algorithm, despite the
fact that it represents the path domain as a discrete graph rather
than as a continuous space. We compare Dijkstra's
discrete algorithm with a variant (often called the Fast Marching
Method) which more accurately treats the underlying continuous space.
Analytically, both generate a value function free of local minima, so
that optimal path generation merely requires gradient descent. We
also investigate the use of optimality metrics other than Euclidean
distance for both algorithms. These different norms better represent
optimal paths for some types of problems, as demonstrated by planning
optimal collision-free paths for multiple robot scenarios.
More details can be found in our ICRA 2006 paper and our
SINUM paper.

Click on the images or links below to view videos.

## Robot Arms

The objective is to efficiently move the arm to touch the end effector to the goal region indicated by a green circle. In the problems with two degrees of freedom, the contours of the value function are shown.
### Two Degrees of Freedom

### Three Degrees of Freedom

## Holonomic Robots

The objective is to efficiently reach a goal region. In the case of the two-robot problems, the joint-objective is for each robot to reach its own goal region.

Two robots in a 2D world. The blue robot is constrained to a circular path, while the red robot may move anywhere within the obstacle-free space. The blue robot's goal is on the right and the red robot's goal is on the left.