This paper presents an overview and comparison of minimal and complete rigid body motion trajectory descriptors, usable in applications like motion recognition and programming by demonstration. Motion trajectory descriptors are able to deal with potentially unwanted variations acting on the motion trajectory such as changes in the execution time, the motion's starting position, or the viewpoint from which the motion is observed. A suitable rigid body motion trajectory descriptor retains only the trajectory information relevant to the application. This paper compares different trajectory descriptors for rigid body motion and validates their usefulness for dealing with motion variation in a motion recognition experiment. Furthermore, a new type of invariant trajectory descriptor is introduced based on the Frenet-Serret formulas.

The instantaneous (ISA) and average (ASA) screw axes are techniques commonly adopted in motion analysis to functionally locate the rotation axis and center of rotation of a joint. Several approaches for calculating such axes were proposed in literature and the main limitations were identified as the need for using a threshold on angular displacements or velocities for handling the cases where the ISA is ill-defined and the need for a method for reliably estimating the center or rotation in limit cases, such as a purely rotational motion in the three-dimensional space. Furthermore, in many applications, such as in biomechanics, it is useful to statistically estimate the dispersion or variation of the ISA with respect to the ASA. In this paper we propose a novel method for estimating an ASA. Our method represents an improvement over previous methods as it: (i) exploits an optimization procedure based on the instantaneous differential kinematics (screw twist); (ii) removes the need for a threshold by introducing a weighting based on the norm of angular velocity; (iii) handles the singular cases where the position of the ASA is ill-defined by proposing a regularization term in the optimization. In addition, we proposed a method for estimating the uncertainty in the ASA calculation. The same quantities serve as a measure of the dispersion of the ISAs with respect to the ASA. The method was tested on real data and surrogate data: (i) a human gait analysis trial representing the motion of a knee, (ii) the experimental recording of the free swing motion of a mechanical hinge and (iii) synthetically generated motion data of a purely rotational (cylindrical) motion. The results showed that the new method had a lower sensitivity to noise, was able to reasonably handle the singular cases and provide a detailed analysis of ISA dispersion.

In learning by demonstration, the generalization of motion trajectories far away from the set of demonstrations is often limited by the dependency of the learned models on arbitrary coordinate references. Trajectory shape descriptors have the potential to remove these dependencies by representing demonstrated trajectories in a coordinate-free way. This paper proposes a constraint-based optimization framework to generalize demonstrated rigid-body motion trajectories to new situations starting from the shape descriptor of the demonstration. Experimental results indicate excellent generalization capabilities showing how, starting from only a single demonstration, new trajectories are easily generalized to novel situations anywhere in task space, such as new initial or target positions and orientations, while preserving similarity with the demonstration. The results encourage the use of trajectory shape descriptors in learning by demonstration to reduce the number of required demonstrations.

The motion of a rigid body can be represented by the instantaneous screw axis (ISA, also known as the helical axis). Recently, an invariant representation of motion based on the ISA, namely, the screw axis invariant descriptor (SAID), was proposed in the literature. The SAID consists of six scalar features that are independent from the coordinate system chosen to represent the motion. This method proved its usefulness in robotics; however, a high sensitivity to noise was observed. This paper aims to explore the performance of inertial sensors for the estimation of the ISA and the SAID for a simple experimental setup based on a hinge joint. The free swing motion of the mechanical hinge was concurrently recorded by a marker-based optoelectronic system (OS) and two magnetic inertial measurement units (MIMUs). The ISA estimated by the MIMU was more precise, while the OS was more accurate. The mean angular error was ≈2.2° for the OS and was ≈4.4° for the MIMU, while the mean standard deviation was ≈2.3° for the OS and was ≈0.2° for the MIMU. The SAID features based on angular velocity were better estimated by the MIMU, while the features based on translational velocity were better estimated by the OS. Therefore, a combination of both measurements systems is recommended to accurately estimate the complete SAID.