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.

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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

Robot arm: One linear and one rotary joint. Robot arm: Two rotary joints. Robot arm: Two rotary joints.
One linear and one rotary joint. Two rotary joints. Two rotary joints.

Three Degrees of Freedom

Robot arm: One linear and two rotary joints. Robot arm: Three rotary joints. Robot arm: Three rotary joints. Robot arm: Three rotary joints.
One linear and two rotary joints. Three rotary joints. Three rotary joints. Three rotary joints.

 


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.

 

One robot in a 2D world. Two robots in a 2D world.
One robot in a 2D world. The contours of the value function are shown. Two robots in a 2D world. The blue robot's goal is on the right and the red robot's goal is on the left.

 

Two robots in a 2D world. Two robots in a 2D world. Two robots in a 2D world.
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.