Quaternions as a solution to determining the angular kinematics of human movement

Abstract

The three-dimensional description of rigid body kinematics is a key step in many studies in biomechanics. There are several options for describing rigid body orientation including Cardan angles, Euler angles, and quaternions; the utility of quaternions will be reviewed and elaborated. The orientation of a rigid body or a joint between rigid bodies can be described by a quaternion which consists of four variables compared with Cardan or Euler angles (which require three variables). A quaternion, q = (q 0 , q 1 , q 2 , q 3), can be considered a rotation (Ω = 2 cos −1 (q 0)), about an axis defined by a unit direction vector ðq 1 = sinð Ω 2 Þ; q 2 = sinð Ω 2 Þ; q 3 = sinð Ω 2 ÞÞ. The quaternion, compared with Cardan and Euler angles, does not suffer from singularities or Codman's paradox. Three-dimensional angular kinematics are defined on the surface of a unit hypersphere which means numerical procedures for orientation averaging and interpolation must take account of the shape of this surface rather than assuming that Euclidean geometry based procedures are appropriate. Numerical simulations demonstrate the utility of quaternions for averaging three-dimensional orientations. In addition the use of quaternions for the interpolation of three-dimensional orientations, and for determining three-dimensional orientation derivatives is reviewed. The unambiguous nature of defining rigid body orientation in three-dimensions using a quaternion, and its simple averaging and interpolation gives it great utility for the kinematic analysis of human movement.

A Multiclass Classification Model for Tooth Removal Procedures

“…Six-dimensional time series from the force/torque sensor consist of 3 dimensions (“XYZ”) for both forces and torques. A further 7-dimensional time series is gathered from the robot arm—3 dimensions for the position of the end-effector (“XYZ”) and a 4-dimensional representation of the orientation of the end-effector in quaternions ( Challis 2020 ). For data analysis, Python was used (Python Language Reference, version 3.9; Python Software Foundation) ( van Rossum and Drake 1995 ) and the Scikit-learn 1.0.1 module ( Pedregosa et al 2011 ).…”

Design and Validation of an IMU Based Full Hand Kinematic Measurement System

“…In this work, many linear Euclidian operations such as computing RMSE, performing linear dimensionality reduction using PCA etc., are performed. However, since quaternions are defined on a non-linear manifold, the linear operations just mentioned cannot be directly applied on them [29,31]. To perform these linear operations, the hemispherized relative quaternions were first linearized using logarithmic mapping and then the linear operations were performed on them.…”

“…Following this step, the linearized orientations were converted back to quaternions using exponential mapping [31]. For more details on the applicability of quaternions for biomechanical analysis, refer to [29], and for more information on logarithmic and exponential mapping, refer to [30][31]. A short primer on logarithmic and exponential maps is presented in Appendix II.…”

Recreating the Motion Trajectory of a System of Articulated Rigid Bodies on the Basis of Incomplete Measurement Information and Unsupervised Learning

“…The result of the calculations is to obtain, among others, angles between individual members and the location of individual points in a given coordinate system. The description of the model can also be realized using the Denavit-Hartenberg notation together with the Newton-Euler equations [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ] or the algebra of quaternions and dual quaternions [ 21 , 22 , 23 , 24 , 25 ].…”

“…There are two sub-problems in the problem of human movement acquisition: the first concerns the acquisition of human movement in a controlled laboratory environment using, for example, MoCap technology or pressure platforms [ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 ], the second in uncontrolled conditions of everyday life, e.g., with the use of IMU sensors and shoes pressure inserts [ 25 , 34 , 35 , 36 , 37 ]. The selection of the test environment does not affect the defined model of the rigid body system, while the measurement technique and the time of the measurements are changed.…”