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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).• Influence of different joint space and orientation representations in SO(3) on learning the FK are empirically

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We propose a segment design that combines two distinct characteristics of tendon-driven continuum robots, i.e. variable length and non-straight tendon routing, into a single segment by enabling rotation of its backbone. As a result, this segment can vary its helical tendon routing and has four degrees-of-freedom, while maintaining a small-scale design with an overall outer diameter of 7 mm thanks to an extrinsic actuation principle. In simulation and on prototypes, we observe improved motion capabilities, as evidenced by position redundancy and follow-the-leader deployment along spatially tortuous paths. To demonstrate the latter on a physical prototype, a simple, yet effective area-based error measure for follow-the-leader deployment is proposed to evaluate the performance. Furthermore, we derive a static model which is used to underpin the observed motion capabilities. In summary, our segment design extends previous designs with minimal hardware overhead, while either archiving similar accuracy in position errors and planar follow-the-leader deployment, or exhibiting superior motion capabilities due to position redundancy and spatial follow-the-leader deployment.

Concentric tube continuum robots utilize nested tubes, which are subject to a set of inequalities. Current approaches to account for inequalities rely on branching methods such as if-else statements. It can introduce discontinuities, may result in a complicated decision tree, has a high wall-clock time, and cannot be vectorized. This affects the behavior and result of downstream methods in control, learning, workspace estimation, and path planning, among others.In this paper, we investigate a mapping to mitigate branching methods. We derive a lower triangular transformation matrix to disentangle the inequalities and provide proof for the unique existence. It transforms the interdependent inequalities into independent box constraints. Further investigations are made for sampling, control, and workspace estimation. Approaches utilizing the proposed mapping are at least 14 times faster (up to 176 times faster), generate always valid joint configurations, are more interpretable, and are easier to extend.