Migrate vel model desc

README.md

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+# How to work on the site
+
+## Local Dependencies
+
+Install latest `hugo` with the package manager of your choice, e.g.
+
+```txt
+brew install hugo
+```
+
+## Local Preview
+
+You can start a local preview webserver with:
+
+Note: pages will autoregenerated and reloaded on save on the content.
+
+```txt
+hugo server --buildDrafts --disableFastRender
+```

archetypes/default.md

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+title = '{{ replace .File.ContentBaseName "-" " " | title }}'
+date = {{ .Date }}
+draft = true
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content/_index.md

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+---
+title: Pedestrian Dynamics
+toc: true
+cascade:
+  type: docs
+---
+
+Insert Motivation here...
+
+## Explore
+
+{{< cards >}}
+  {{< card link="models" title="Models" icon="book-open" >}}
+  {{< card link="about" title="About" icon="user" >}}
+{{< /cards >}}
+

content/about.md

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+---
+layout: page
+title: About
+permalink: /about/
+---
+
+## Who we are / Why we do this

content/models/_index.md

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+---
+title: Pedestrian Models
+weight: 1
+toc: false
+---
+
+{{< cards >}}
+  {{< card link="collision_free_speed_model" title="Collision Free Speed" icon="plus-circle" >}}
+  {{< card link="generalized_centrifugal_force_model" title="Generalized Centrifugal Force" icon="plus-circle" >}}
+{{< /cards >}}

content/models/collision_free_speed_model/index.md

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+---
+title: Collision Free Speed Model
+math: true
+---
+
+## Introduction to collision-free speed model
+
+The collision-free speed model [[1]](#Tordeux2015) is a
+mathematical approach designed for pedestrian dynamics, emphasizing the
+prevention of collisions among agents.
+
+The direction in which an agent moves is determined through an isotropic
+combination of exponential repulsion from nearby agents. The strength of this
+repulsion is influenced by the proximity to others within their surroundings,
+treating all directions equally in terms of influence.
+
+Agents adjust their speed according to the nearest neighbor in their headway,
+allowing them to navigate through congested areas without overlapping or
+obstructing each other. The collision-free speed model takes into account the
+length of the agent, which determines the required space for movement, and the
+maximum achievable speed of the agent.
+
+This simplified and computationally efficient model aims to mirror real-world
+pedestrian behaviors while maintaining smooth movement dynamics.
+
+## Mathematical description
+
+Models that establish a relationship between speed and spacing, known as the OV
+function, were originally introduced in traffic flow studies. They have since
+been adapted for pedestrian modeling, offering a straightforward way to control
+the fundamental diagram. The collision-free speed model is mathematically
+represented as a derivative equation for the velocity of each pedestrian.
+Typically, this can be expressed as
+
+$$\dot{\mathbf{x}}_i=V_i\big(s_i(\mathbf{x}_i,\mathbf{x}_j,\ldots)\big)\times\mathbf e_i(\mathbf{x}_i,\mathbf{x}_j,\ldots)$$
+
+where $x_i$ represents the position of pedestrian $i$ and $V_i$ represents
+their speed.
+
+The speed function $V_i$ regulates the overall speed of the pedestrian, while
+the direction function $\textbf{e}_i$ determines the direction in which the
+pedestrian moves.
+
+### Direction function
+
+The direction function is governed by a weighted sum of exponential repulsion
+from neighboring pedestrians, which is calibrated by the repulsion rate and
+distance.
+
+{{< figure src="figure1.png" caption="Calculation of the movement direction" >}}
+
+This mathematical representation ensures that the pedestrians are able to
+adjust their speed and direction based on their interactions with neighboring
+agents, ultimately resulting in a collision-free movement.
+
+$$\mathbf e_i(\mathbf x_i,\mathbf x_j,\ldots)=\frac{1}{N}\left(\mathbf e_0+\sum_j R(s_{i,j})\right)$$
+
+with $\mathbf e_0$ the desired direction, $N$ a normalization constant such
+that $\|\mathbf e_i\|=1$ and $R(s)=a\,\exp\big((l-s)/D\big)$ the repulsion
+function calibrated by the coefficient $a>0$ and distance $D>0$.
+
+{{< figure src="figure2.png" caption="Repulsive influence in the direction" >}}
+
+### Speed function
+
+The velocity is calculated by multiplying two functions: A speed function $V_i$
+and a direction function $\textbf{e}_i$.
+
+In the following, the OV function is the piecewise linear
+
+$$V(s)=\min\\{v_0,\max\\{0,(s-l)/T\\}\\}$$
+
+satisfies
+
+$$\begin{align*}V(s)&\gt0\quad\forall s\gt l\\\\ V(s)&=0\quad\forall s\le\ell\end{align*}$$
+
+{{< figure src="figure3.png" caption="OV speed function vs fundamental diagram" >}}
+
+The spacing is calculated along the direction of motion and is defined as the
+spacing to the nearest neighbor that may collide with the agent. See following
+picture:
+
+{{< figure src="figure4.png" caption="Calculation of the minimal speed in the direction of motion" >}}
+
+### Parameters
+
+
+The collision-free speed model depends on five parameters:
+- Pedestrian diameter ($l$)
+- Desired speed ($v_0$)
+- Time gap ($T$)
+- Repulsion rate and distance ($a$ and $D$)
+
+## Limitations of the collision-free speed model
+
+Despite its ability to simulate pedestrian dynamics and replicate real-world
+phenomena, the collision-free speed model has some limitations that should be
+acknowledged.
+
+- Initially, the model operates on basic assumptions and might not encompass
+  the intricate nuances of real pedestrian actions, particularly with its
+  representation of agents as circles.
+- It overlooks elements like response time and visual interpretation.
+- The model may not represent stop-and-go dynamics in crowded regions or
+  gridlocks aptly, unless manifested in confined bottlenecks with a circular
+  form.
+- The model does not take into account other influencing variables like
+  obstacles or environmental conditions that could impact pedestrian movement.
+
+Numerous studies have presented different enhancements to the collision-free
+speed-based pedestrian model with the aim of addressing its limitations and
+improving the accuracy of pedestrian simulations. For instance, Xu
+[[2]](#Xu2019) proposed a comprehensive velocity model that takes into
+account wall influence and incorporates velocity-based ellipses for accurate
+distance calculations. Moreover, improvements were made to the direction
+function in order to ensure seamless changes in pedestrian directions during
+simulation. Similarly, other researchers such as [[3]](#Rzezonka2022),
+[[4]](#Zhang2021), and [[5]](#Xu2021) introduced additional refinements to
+enhance the direction function further.
+
+
+## Challenges in Implementing Collision Free Speed Models
+
+Numerical solution of the first-order ordinary differential equation defined
+by the model is solved as follows:
+
+{{< figure src="figure5.png" caption="Update algorithm" >}}
+
+The implementation of collision-free speed models presents several challenges.
+One challenge is the lack of definition for agent-wall interactions in the
+original model. However, this issue has been tackled by proposed enhancements
+to the model, such as Xu's generalized velocity model. Another challenge lies
+in accurately calibrating the parameters in both the speed and direction
+models. In certain symmetrical scenarios, determining a well-defined direction
+function can be difficult.
+
+### Isotropical direction influence
+
+The direction model is uniform, meaning it does not differentiate between
+various directions of influence. The model treats all directions equally and
+does not consider specific pedestrian preferences or biases in their movement.
+This may lead to certain unrealistic situations where the agent's direction is
+influenced by agents from behind them.
+
+### Balancing Collision Avoidance with Performance: Selecting the Appropriate Time-Step
+
+The continuous definition of the model is proved to be collision-free in any
+situation. However, the discretisation of the model, as often required for
+computational efficiency, can introduce potential collision problems. While the
+speed model has a fast and efficient implementation, it is crucial to select a
+sufficiently small time step when solving the ordinary differential equation
+with Euler scheme in order to ensure that the model remains collision-free.
+Therefore, the model is collision-free in discrete time if
+
+$$\delta t \le \min\left\\{\frac T2,\frac{l(\sqrt2-1)}{v_0\sqrt2}\right\\}$$
+
+The condition for collision-free dynamics is determined solely by the
+parameters of the speed model. For exampl, if we use parameter values of $T=1$
+s, $v_0=1.2$ m/s and $l$, with a smallness condition on the time step
+approximate to $\delta t \le0.072$ s for explicit Euler schemes and circular
+pedestrian shape.
+
+### Parameter calibration
+
+Additionally, accurately calibrating the repulsion rate and distance in the
+direction model can prove challenging due to variation based on specific
+environmental conditions and crowd dynamics.
+
+
+## The collision-free speed model in the literature
+
+The following map shows works from the literature connected to the
+collision-free speed model.
+
+{{< figure src="figure6.png" caption="Citation map of the Collision Free Speed Model">}}
+
+## References
+
+- <a name="Tordeux2015"></a>[1] Tordeux, A., Chraibi, M., Seyfried, A. (2016).
+  Collision-Free Speed Model for Pedestrian Dynamics. In: Knoop, V., Daamen, W.
+  (eds) Traffic and Granular Flow '15.
+  <br/>https://doi.org/10.1007/978-3-319-33482-0_29
+
+- <a name="Xu2019"></a>[2] Xu, Q., Chraibi, M., Tordeux, M., Zhang (2019).
+  Generalized collision-free velocity model for pedestrian dynamics. Physica A:
+  Statistical Mechanics and its Applications, Volume 535.
+  <br/>https://doi.org/10.1016/j.physa.2019.122521
+
+- <a name="Rzezonka2022"></a>[3] Rzezonka, J., Chraibi, M., Seyfried, A., Hein,
+  B., Schadschneider, A. (2022). An attempt to distinguish physical and
+  socio-psychological influences on pedestrian bottleneck. Royal Society Open
+  Science. <br/>https://royalsocietypublishing.org/doi/10.1098/rsos.211822
+
+- <a name="Zhang2021"></a>[4] Zhang, S., Zhang, J., Chraibi, M., Song, W.
+  (2021). A speed-based model for crowd simulation considering walking
+  preferences. Communications in Nonlinear Science and Numerical Simulation,
+  Volume 95. <br/>https://doi.org/10.1016/j.cnsns.2020.105624
+
+- <a name="Xu2021"></a>[5] Xu, Q., Chraibi, M., Seyfried, A. (2021).
+  Anticipation in a velocity-based model for pedestrian dynamics.
+  Transportation Research Part C: Emerging Technologies, Volume 133.
+  <br/>https://doi.org/10.1016/j.trc.2021.103464
+
+- <a name="Xu2021"></a>[6] Tordeux, A. talk in TGF15, Delft.
+<br/>[Slides.](https://www.vzu.uni-wuppertal.de/fileadmin/site/vzu/Pres_1st_order_models.pdf)