Here we define the key parameters of our model and derive the formula describing how Δ⊥ depends on these parameters. Full formal derivations can be found in the Appendices.
The laser line pattern has the form of a line segment between P1 and P2 on the user's ground plane, a distinctive pattern between P2 and P3 that depends on the nature of the tripping hazard, and a line segment between P3 and P4 on the ground plane beyond the hazard. The results here apply only for the cases where the line segments P1P2 and P3P4 are long enough to be visible to the user, so we assume here the hazard is not abutting P1 or P4.
We set the point-of-view of the user as the “cyclopean eye position” (C
P), a point midway between the user's two eyes. Specifically:
\begin{equation}C_P^W = {\left[ {\begin{array}{*{20}{c}} {{x_E},}&{{y_E},}&{{z_E}} \end{array}} \right]^T} = {\left[ {\begin{array}{*{20}{c}} {0,}&{0,}&{{z_E}} \end{array}} \right]^T}\end{equation}
where x
E = y
E = 0, because we have set the origin of World Coordinates (denoted with a superscript
W) to be between the user's feet, the x-axis going to the user's right, and the y-axis pointing along the user's path as shown in
Figure 1a. The position of the laser projector, L
P, on the user's hip is related to C
P by a baseline distance described in spherical coordinates as:
\begin{equation}L_P^W = C_P^W + r\left[ {\begin{array}{@{}*{1}{c}@{}} {\cos ({\theta _{ELEV}})\cos ({\theta _{AZ}})}\\ {\cos ({\theta _{ELEV}})\sin ({\theta _{AZ}})}\\ {\sin ({\theta _{ELEV}})} \end{array}} \right]\end{equation}
where
r is the baseline distance between C
P and L
P, ϴ
ELEV is the angle the vector C
PL
P makes with the xy-plane, and ϴ
AZ is the angle this vector makes with the xz-plane (see
Appendix A: Derivation of World Coordinates for an illustration). In the analysis below, we will assume
ϴAZ = 0, putting both the laser projector and the cyclopean eye position on the xz-plane. This setting simplifies the expressions we shall derive. Simulation results in the
Discussion show that the derived results hold for small, non-zero values of
ϴAZ.
Next, we define the nearest point where the laser line intersects the tripping hazard (P
2) as located along the y-axis at a distance of
yT (“T” denotes “target”) from the user. Specifically:
\begin{equation}P_2^W = {\left[ {\begin{array}{*{20}{c}} {0,}&{{y_T},}&0 \end{array}} \right]^T}\end{equation}
Deriving an expression for Δ⊥ using these parameters requires four steps, all of which are detailed in the Appendices:
These geometric derivations in the appendices result in an expression for the
perpendicular perspective signal given in
Equation (4).
\begin{eqnarray}
{\Delta _ \bot } = - \frac{{{S_H}f\left( {y_T^2 + z_E^2} \right)\left( {{z_E}r\sin {\theta _{Laser}}\cos {\theta _{AZ}}\cos {\theta _{ELEV}} - {y_T}r\cos {\theta _{Laser}}\sin {\theta _{ELEV}}} \right)}}{{{y_T}\left( {\left( {{z_E} + r\sin {\theta _{ELEV}}} \right)\left( {y_T^2 + z_E^2 - {S_H}{z_E} + r\sin {\theta _{ELEV}}} \right) + \Sigma } \right)\sqrt {y_T^2{{\cos }^2}{\theta _{Laser}} + z_E^2} }}\end{eqnarray}
where
\(\Sigma = \left\{ {\begin{array}{@{}*{1}{c}@{}} { - {S_H}y_T^2}\\ 0 \end{array}}\right. \begin{array}{@{}*{1}{c}@{}} \textit{for descending curb}\\ \textit{for ascending curb} \end{array}\).
In
Equation (4),
f is the effective focal length of the user's eye (in simulations we assume
f to be 17 mm). Note that Δ
⊥ correctly reduces to zero as the step-height (
SH) or the baseline distance (
r) goes to zero.
Equation (4) suggests that Δ
⊥ depends strongly on ϴ
Laser. We would therefore expect human performance to exhibit a correlating dependence on ϴ
Laser.