On the Merits of Joint Space and Orientation Representations in Learning the Forward Kinematics in SE(3)

This paper investigates the influence of different joint space and orientation representations on the approximation of the forward kinematics. We consider all degrees of freedom in three dimensional space SE(3) and in the robot’s joint space Q. In order to approximate the forward kinematics, different shallow artificial neural networks with ReLU (rectified linear unit) activation functions are designed. The amount of weights and bias’ of each network are normalized. The results show that quaternion/vector-pairs outperform other SE(3) representations with respect to the approximation capabilities, which is demonstrated with two robot types; a Stanford Arm and a concentric tube continuum robot. For the latter, experimental measurements from a robot prototype are used as well. Regarding measured data, if quaternion/vector-pairs are used, the approximation error with respect to translation and to rotation is found to be seven times and three times more accurate, respectively. By utilizing a four-parameter orientation representation, the position tip error is less than 0.8% with respect to the robot length on measured data showing higher accuracy compared to the state-of-the-artmodeling (1.5%) for concentric tube continuum robots. Other three-parameter representations of SO(3) cannot achieve this, for instance any sets of Euler angles (in the best case 3.5% with respect to the robot length).