263-0007-00L-Advanced-Systems-Lab/docs/Overview_Formulas_Variables.md

Overview — Formulas & Variables

As a reference the paper was used (9805244v1.pdf)

1. Variables

Per pedestrian \alpha

Paper	Code	Type	Meaning	Ref
r_\alpha	curr_pos	Vector2d	position
v_\alpha	curr_vel	Vector2d	actual velocity (capped)
w_\alpha	w	Vector2d	preferred velocity (integrates force)
e_\alpha	e	Vector2d	desired direction, unit vector
r_\alpha^k	destination	Vector2d	next destination
v_\alpha^0	desired_v0	double	desired speed
v_{\max,\alpha}	v_max	double	maximum speed = 1.3 \cdot v_\alpha^0
F_\alpha	f_i	Vector2d	total force accumulator

Global / scene

Paper	Code	Meaning	Ref
n_p	n_p	number of pedestrians (CLI arg 1)
n_{\text{iter}}	n_iter	number of timesteps (CLI arg 2)
—	BOARD_LENGTH	walkway length, x-axis (50 m)
—	BOARD_WIDTH	walkway width, y-axis (10 m)
B	Obstacle	line segment (start, end)

Constants

Paper	Code	Value	Unit	Ref
\tau_\alpha	tau_alpha	0.5	s
V_{\alpha\beta}^0	V0_AB	2.1	m²/s²
\sigma	SIGMA	0.3	m
U_{\alpha B}^0	U0_AB	10.0	m²/s²
R	R	0.2	m
c	c	0.5	—
\varphi	phi	45	degree (2 \varphi = 90)
\Delta t	delta_t	2	s (look-ahead, used in b)
—	dt	0.2	s (Euler integration step)

2. Formulas

2.1 Desired direction — Eq. 1 (p. 4)

e_\alpha = \frac{r_\alpha^k - r_\alpha}{\| r_\alpha^k - r_\alpha \|}

2.2 Acceleration toward desired velocity — Eq. 2 (p. 4)

F_\alpha^0 = \frac{v_\alpha^0 \cdot e_\alpha - v_\alpha}{\tau_\alpha}

Uses actual velocity v_\alpha.

2.3 Pedestrian–pedestrian potential & force — Eq. 3 (p. 4), Eq. 13 (p. 7)

V_{\alpha\beta}(b) = V_{\alpha\beta}^0 \cdot \exp\!\left(-\frac{b}{\sigma}\right)

f_{\alpha\beta} = -\nabla_{r_{\alpha\beta}} V_{\alpha\beta}(b)
= \frac{V_{\alpha\beta}^0}{\sigma} \cdot \exp\!\left(-\frac{b}{\sigma}\right) \cdot \nabla b

2.4 Elliptical b — Eq. 4 (p. 5)

r_{\alpha\beta} = r_\alpha - r_\beta

s_\beta = v_\beta \cdot \Delta t

2b = \sqrt{\left(\| r_{\alpha\beta} \| + \| r_{\alpha\beta} - s_\beta \cdot e_\beta \|\right)^2 - s_\beta^2}

Gradient (derived from Eq. 4, not given explicitly in paper):

\nabla b = \frac{\| r_{\alpha\beta} \| + \| r_{\alpha\beta} - s_\beta \, e_\beta \|}{4b}
\left(
  \frac{r_{\alpha\beta}}{\| r_{\alpha\beta} \|}
  + \frac{r_{\alpha\beta} - s_\beta \, e_\beta}{\| r_{\alpha\beta} - s_\beta \, e_\beta \|}
\right)

2.5 Border potential & force — Eq. 5 (p. 5), Eq. 13 (p. 7)

U_{\alpha B}\!\left(\| r_{\alpha B} \|\right) = U_{\alpha B}^0 \cdot \exp\!\left(-\frac{\| r_{\alpha B} \|}{R}\right)

f_{\alpha B} = \frac{U_{\alpha B}^0}{R} \cdot \exp\!\left(-\frac{\| r_{\alpha B} \|}{R}\right) \cdot \frac{r_{\alpha B}}{\| r_{\alpha B} \|}

where r_{\alpha B} = r_\alpha - r_B^\alpha, and r_B^\alpha is the nearest point of border B to \alpha.

2.6 Perception weight — Eq. 7 (p. 6)

w(e,\, f) =
\begin{cases}
1 & \text{if } e \cdot f \geq \| f \| \cos\varphi \\
c & \text{otherwise}
\end{cases}

2.7 Weighted force terms — Eq. 8 (p. 6) + team convention

F_{\alpha\beta} = w(e_\alpha,\; -f_{\alpha\beta}) \cdot f_{\alpha\beta}

F_{\alpha B} = w(e_\alpha,\; -f_{\alpha B}) \cdot f_{\alpha B}
\qquad \text{(team convention; not explicit in paper)}

Note: Paper Eq. 8 only defines w weighting for ped–ped F_{\alpha\beta}; applying it to border F_{\alpha B} is our extension. (Reason see page 6, top: "However, the formulas above for attractive and repulsive effects only hold for situations that are perceived in the desired direction e_alpha(t) of motion.")

2.8 Total force — Eq. 9 (p. 6), no attractors (Eq. 6) / fluctuations

F_\alpha = F_\alpha^0 + \sum_\beta F_{\alpha\beta} + \sum_B F_{\alpha B}

Paper Eq. 9 also includes attractor term \sum_i F_{\alpha i} (Eq. 6, p. 5) and Eq. 10 adds fluctuations; both omitted here per team convention.

2.9 Speed cap — Eq. 12 (p. 7)

g\!\left(\frac{v_{\max}}{\| w \|}\right) =
\begin{cases}
1 & \text{if } \| w \| \leq v_{\max} \\[4pt]
\dfrac{v_{\max}}{\| w \|} & \text{otherwise}
\end{cases}

2.10 Equations of motion — Eq. 10 (p. 6), Eq. 11 (p. 7)

\frac{dw_\alpha}{dt} = F_\alpha

v_\alpha = w_\alpha \cdot g\!\left(\frac{v_{\max}}{\| w_\alpha \|}\right)

\frac{dr_\alpha}{dt} = v_\alpha

3. One Euler timestep (per pedestrian)

Discretisation of Eq. 10–11; not given explicitly in paper.

F_\alpha \leftarrow F_\alpha^0(v_\alpha,\, e_\alpha) + \sum_\beta F_{\alpha\beta} + \sum_B F_{\alpha B}

w_\alpha^{\text{new}} \leftarrow w_\alpha + F_\alpha \cdot dt

v_\alpha^{\text{new}} \leftarrow w_\alpha^{\text{new}} \cdot g\!\left(\frac{v_{\max}}{\| w_\alpha^{\text{new}} \|}\right)

r_\alpha^{\text{new}} \leftarrow r_\alpha + v_\alpha^{\text{new}} \cdot dt

e_\alpha^{\text{new}} \leftarrow \frac{r_\alpha^k - r_\alpha^{\text{new}}}{\| r_\alpha^k - r_\alpha^{\text{new}} \|}

Convention: force pass over all \alpha first, integration pass second.