On a Hybrid Symbolic-Connectionist Approach for Modeling the Kinematic Robot Map - and Benchmarks for Computer Algebra

The kinematics model of a robot arm (we are considering open kinematic chains) is described by a corresponding robot map having the configuration space as its domain and the workspace as codomain. In other words, the robot map assigns to every configuration of the joint parameters a unique point of the workspace of the robot arm. We briefly discuss the general introduction of the robot map where the parameters of a translational joint are represented by points of the real line and the parameters of a rotational joint by points of the unit circle in the real plane, respectively. Thus, in general, a concrete joint configuration (point of the configuartion space) is an element of an abelian Lie group being a direct product of some copies of the real line and the unit circle. The position and orientation of the endeffector of a robot arm is represented by an element of the euclidean motion group of real 3-space. The standard problems like the direct kinematics problem, the inverse kinematics problem and the singularity problem can easily be defined. A classical method to establish the robot map is the approach by Denavit-Hartenberg. It leads to a completely symbolic description of the direct kinematics model of an arm and forms the basis for the treatment of the inverse kinematics problem. In order to represent an entire robot arm class it is of basic interest to find a completely symbolic closed form solution of the inverse kinematics problem. Using a two joint robot arm, B.Buchberger demonstrated the principle how to solve this problem with the help of a computer algebra (CA) system applying his Gröbner bases method (Buchberger Algorithm) cf. [Buc70], [Buc85], [Buc87], [BCK88]. Later we made own investigations and constructed a more complex test example (cf. [Pfa97]). We observed that very hard performance problems arose, in the corresponding CA applications, when the degree of freedom of a robot arm increases. An interesting aspect is the fact that these investigations show a natural way how to construct benchmarks for CA. A completely different method to represent the kinematic model of a robot arm is a Connectionist Network approach. The idea is to learn a robot map with the help of a suitably chosen Artificial Neural Network (ANN) using a powerful ANN simulator. The training data