Current research topics of Leo Dorst

Specific subjects that involve my personal research are:

• Applications of Geometric (Clifford) Algebra
  Recently (from September 1997) I have become very enamored of Geometric Algebra, and am studying it diligently to use it in geometric problems in robotics. I have done a sabbatical with Hestenes' group at Arizona State University from November 98 to April 99, and have applied it towards a solution of the collision detection problem (see below). More information here.

• Objects in Contact
  The generation of collision-free paths for a robot is commonly represented in the parameter space of the robot, better known as configuration space. Obstacles are represented as sets of forbidden states. It is an essential but unsolved problem to calculate these sets for arbitrary robots and environments. With Rein van den Boomgaard, I have developed an analytical framework for computations of objects in contact, which has now been connected to the mathematical theory of contact transformations. Using homogeneous coordinates in projective space, we expect to convert the theory into a practical computational method. Here is an overview talk in ps or in pdf.
  But using geometric algebra, one can address the issues in n-dimensional space, see here!

• Exploration
  There is a dichotomy in robot behavior, between reactive behavior and planned behavior. In an exploration task, both meet. With Ben Kröse, I try to characterize the geometry of the sensor motor configuration spaces in which the reactive behavior resides, in order to obtain a better formal understanding and characterization of this transition in behavior in exploration tasks.

• Reasoning with uncertainty in robotics
  With people from the ILLC, I am trying to get a better hold on the proper uncertainty reasoning mechanisms in robotics, notably in representations of uncertain geometry. We organized a workshop on this.

• Abstraction in path planning
  The robot path planning problem is well-understood on the lowest level, that of configuration spaces, and optimal paths may be computed using a method like wave-propagation (A*). It is an open problem how to abstract from paths generated and represented in this way to more abstract path descriptions, and corresponding planning algorithms. I am trying to approach this using geometric/topological methods (with a pinch of homology).

• Digitized straight lines
  A very long time ago, for my Ph.D. thesis, I worked on the estimation of properties of digitized straight boundaries. I solved the problem by finding a characterization of those boundaries, and the set of continuous lines that could have generated them. The transform that made the problem tractable is actually a precursor of the slope transform.