. 2022 Jul 28;19:83. doi: 10.1186/s12984-022-01053-z

Table 3.

Mathematical notation

Equation	Description
$p_{j}^{t} = {[x_{j}^{t} y_{j}^{t}]}^{'}$	${[x_{j}^{t} y_{j}^{t}]}^{'}$ denotes the transposed vector of 2D coordinates in the image of a body joint j from a set of joints J (Fig. 1); t denotes the frame number
$P^{t} (j_{1}, j_{2}) = p_{j_{2}}^{t} - p_{j_{1}}^{t} = {[x_{j_{2}}^{t} - x_{j_{1}}^{t} y_{j_{2}}^{t} - y_{j_{1}}^{t}]}^{'}$	Vector directed from joint $j_{1}$ to joint $j_{2}$
$‖ P^{t} (j_{1}, j_{2}) ‖$ $d^{t} (j_{1}, j_{2}) = ‖ p_{j_{1}}^{t} - p_{j_{2}}^{t} ‖$	$‖ P^{t} (j_{1}, j_{2}) ‖$ is the euclidean norm of vector $P^{t} (j_{1}, j_{2})$ and, alternatively, $d^{t} (j_{1}, j_{2})$ is the euclidean distance between two selected joints, $j_{1}$ and $j_{2}$
$Δ x^{t} (j_{1}, j_{2}) = x_{j_{1}}^{t} - x_{j_{2}}^{t}$ $Δ y^{t} (j_{1}, j_{2}) = y_{j_{1}}^{t} - y_{j_{2}}^{t}$	Displacement between two selected joints, $j_{1}$ and $j_{2}$ , in the X ( $Δ x$ ) and Y ( $Δ y$ ) axis
$a^{t} (j_{1}, j_{2}, j_{3}) = arccos (\frac{P^{t} (j_{2}, j_{1}) \cdot P^{t} (j_{2}, j_{3})}{‖ P^{t} (j_{2}, j_{1}) ‖ \cdot ‖ P^{t} (j_{2}, j_{3}) ‖})$	Angle between two vectors, $P^{t} (j_{2}, j_{1})$ and $P^{t} (j_{2}, j_{3})$ , defined by two points, $j_{2}$ to $j_{1}$ and $j_{2}$ to $j_{3}$