Most algorithms for closed form solutions of inverse kinematic of serial manipulators deal with 6R manipulators. It is often hypothesized that some of these algorithms can be extended to 6R/P manipulators, though the extension of such algorithms for mixed rotational and prismatic joints are not trivial. Our first objective in this project is to develop an inverse kinematic algorithm that provides analytic solutions for general 6R/P serial manipulators. We would like to mainly focus on the HUPF algorithm which is known to work well with 6R manipulators. We prefer this algorithm because some of the participants (H-P Schröcker and M. Pfurner) in this project were the originators of the HUPF algorithm and we have a very good understanding of it. Understanding the kinematic mapping of 6R/P manipulators will allow us to approximate the collision-free configuration and Cartesian spaces of these manipulators as semi-algebraic sets. The next objective would be to plan path within these spaces. It is natural to believe that an algebraic path-planning algorithm is most suited in these spaces with good algebraic structure. Our next objective will be to develop a new algebraic algorithm and compare them with commonly known path-planning algorithms (such as path-planners based on random-sampling) in terms of performance (i.e. time complexity). One new algorithm that we wish to develop is based on the medial axes of these semi-algebraic spaces. There exists efficient algorithms for developing medial axes of semi-algebraic sets (e.g. by the extension of the powercrust method developed by Nina Amenta). We would like to take advantage of these efficient algorithms for medial axes and skeleton to develop new roadmap for a planner. Specifically, we would like to investigate the use of minimum medial axis for planning path in a specific semi-algebraic set that approximates the collision-free configuration or Cartesian space of the robot.