Code developed by Enrico Eberhard.
📝 Please read our paper: Herbst, E.C.*, Eberhard, E.A.*, Hutchinson, J.R. & Richards, C.T. (2022) Spherical frame projections for visualising joint range of motion, and a complementary method to capture mobility data. Journal of Anatomy. * shared first authorship
If you use this method, please cite our paper (Herbst et al. 2022) 
Spherical frame projection (SFP) was conceived as a convenient visualisation for orientational range of motion (ROM) data. The ROM of a three-dimensional rotational joint, such as a ball joint, is a subset of the group SO(3). If ROM is defined as the boundary between feasible and infeasible orientation (pose), then it is the surface of a four-dimensional volume. This kind of thing is rather difficult to represent visually or understand intuitively.
In most cases, Euler angles are used to reduce the dimensionality of the orientation problem to three dimensions. However, a bounded ROM volume generated from a sample of Euler angles may not accurately represent or intuitively inform the reachability of joint due to inherent distortions in the mapping, particularly near the poles.
The SFP approach projects the frame of a pose onto the surface of a unit sphere. The frame is represented by a set of three orthogonal basis vectors, commonly labelled X, Y and Z, expressed in some base coordinate system. If a pose is represented by a 3x3 homogeneous transformation matrix, then the three projected frame points of that pose are just the three columns of the matrix.
In a given sampling of ROM, a set of poses can be projected. The result is a distribution of points on the surface of the sphere which are divided into a triplet of regions; one for each frame axis. The reachable area of each frame is represented by the respective points, and a 2D surface boundary can be drawn around these points.
Each frame axis can then be imagined to be constrained by its respective spherical surface boundary. Combining the constraints of each 2D surface is enough to define and constrain the full rotational range.
This repository contains MATLAB code to generate and visualize an SFP mapping from a given dataset of orientation samples.