The orientation of the IMUs is estimated by sensor fusion of the ACC, GYR, and MAG data. Since the system is likely to be used in indoor environments and near ferromagnetic materials, we employ an algorithm that reduces the influence of non-homogeneous magnetic fields by assuring that the inclination (roll and pitch) portion of the orientation remains unaffected [6].

The attachment of the individual sensors to the body is predetermined by the *sensor strip* layout and by the layout of the *base unit*. The coordinate systems of ACC, GYR, and MAG are therefore transformed to follow the ISB recommendations for the hand and forearm [5]. The estimated IMU orientations coincide with the finger segment orientations as defined by the ISB and are represented as quaternions ^{Segi}**q**. They describe the segment orientation relative to a common reference frame defined by gravity and magnetic north.

To obtain the joint angle *α*_{A} between Segment 1 and 2, the relative quaternion ${}_{\mathrm{Seg1}}{}^{\mathrm{Seg2}}\mathbf{q}_{A}$ is calculated from the two segment orientations surrounding the joint *A* by

$${}_{\mathrm{Seg1}}{}^{\mathrm{Seg2}}\mathbf{q}_{A}={}^{\mathrm{Seg1}}\mathbf{q}_{A}{}^{-1}\otimes {\mathrm{Seg2}}^{}{\mathbf{q}}_{A},$$(1)

where ${}^{\mathrm{Seg1}}\mathbf{q}_{A}$ is the orientation of the proximal segment and ${}^{\mathrm{Seg2}}\mathbf{q}_{A}$ is the orientation of the distal segment. This relative quaternion can be decomposed into Euler angles given a sequence of rotation axes. For the hand and wrist, intrinsic *z*-*x*’-*y*” Euler angles are used to obtain the joint angles specified by the ISB.

Writing the quaternion ${}_{\mathrm{Seg1}}{}^{\mathrm{Seg2}}\mathbf{q}$ as

$${}_{\mathrm{Seg1}}{}^{\mathrm{Seg2}}\mathbf{q}={q}_{w}+\mathbf{i}{q}_{x}+\mathbf{j}{q}_{y}+\mathbf{k}{q}_{z}$$(2)

yields the following *z*-*x*’-*y*” Euler angles:

$$\begin{array}{ccccc}\alpha \hfill & =atan2(2\left({q}_{z}{q}_{w}-{q}_{y}{q}_{x}\right),{q}_{w}^{2}+{q}_{y}^{2}-{q}_{x}^{2}-{q}_{z}^{2})\hfill & & & \\ \beta \hfill & =\mathrm{arcsin}\left(2\left({q}_{x}{q}_{w}+{q}_{y}{q}_{z}\right)\right)\hfill & & & \\ \gamma \hfill & =atan2(2\left({q}_{y}{q}_{w}-{q}_{x}{q}_{z}\right),{q}_{w}^{2}-{q}_{y}^{2}-{q}_{x}^{2}+{q}_{z}^{2})\hfill & & & \end{array}$$(3)

with

*α*: flexion (pos.), extension (neg.),

*β*: adduction (pos.), abduction (neg.),

*γ*: pronation (pos.), supination (neg.).

For the approximate 1 D joints (DIP, PIP, and TIP), the angles *β* and *γ* are negligible, and *γ* is close to zero for the approximate 2 D joints (MCP).

The accuracy of the estimated joint angles is influenced by three major factors: the calibration of ACC, GYR, AND MAG; the accuracy of the orientation estimation algorithm; the precision with which the sensors are attached to the finger segments. While algorithms exist that allow to reduce the latter [7], no such methods are employed at the current state.

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