Discrete modelling of pedestrian dynamics often defines the system geometry on a two-dimensional regular grid. In the simulation, we must consider that the individual pedestrians are associated with an exclusive personal space. In the traditional two-dimensional cellular automaton model (CA) and its various extensions (see , , , , , ,  etc.) this personal space is described by the cell size of the underlying regular grid. Hence, the state change on the grid can be applied to describe the system dynamics. This leads inevitably to a fixed personal space of the pedestrians. However, empirical data (e.g. ) show that in low density range, the size of this personal space varies significantly. In high density range, the size is clearly restricted by the physical size of the pedestrians. The purpose of this paper is to present a new modelling technique of pedestrian dynamics with consideration of advanced step calculation and density control on a two-dimensional regular grid.
The main contribution of the paper will be explained in the following parts.
1. In  we proposed a new method for the step calculation on a regular grid for the generalized case of local velocity larger than one grid cell size in one dimension per simulation cycle (sometimes also called multicell-step), which is necessitated by the representation of heterogeneous pedestrians. The local step choice on the underlying grid is more than a simple position change from a start position to a destination calculated by a “hardware” method, e.g. of the shortest distance, without taking into consideration the actual system dynamics. Instead, we allow small deviations on route within a local step, but with the restriction that the mathematical expectation of the possible step choices should reflect the original position transition exactly.
2. We introduce a projection mechanism to compute the intermediate steps for the position transition with consideration of local (cell) position availabilities. In the simulation cycle, we further introduce a balancing mechanism to decide the execution sequence of the simulation participants. This improvement, along with the first, enables a drastic reduction of the possible “deadlock” among the participants and at the same time gives a very reasonable explanation for the step calculation.
3. In reality, however, pedestrians’ personal space sometimes appears to be “compressible” to a certain extent in a way similar to gas particles. To our knowledge, in the CA model and its extensions, due to the nature of homogeneous rectangles (or squares) on the two-dimensional regular grid, the pedestrians’ exclusive personal space is fixed in the system construction, e.g. as 0.4m * 0.4m in . We proposed in  a solution for a flexible personal space. We observe that this exclusive personal space
varies roughly in the range of 0.3m * 0.3m (which represents the “incompressible” physical size of a pedestrian) to 1m * 1m (which corresponds to the normal, unstressed state of an average pedestrian)
in reality. Hence in the current paper, the grid cell of the regular grid is defined to be of size
0.3m * 0.3m, and the modifiable personal space for the pedestrian can be considered as a composition of exactly one grid cell to a neighbourhood of 3*3 grid cells on the regular grid. We consider two cases.
(a) Firstly, the space requirement of a single pedestrian varies empirically in correlation with local speed. That is, a higher overall speed in the system results in a lower density. This is sometimes referred
to as fundamental diagram. The fundamental diagram enables us to estimate the empirical density via the local pedestrian velocity, which can be acquired from the last simulation cycle. According to this
information, some of the eight grid cells (apart from the origin) in the corresponding 3*3-neighbourhood are to be declared as “inaccessible” for the rest of the simulation participants. This inaccessibility will be described by a transition function.
(b) Secondly, the former case considers the situation that a pedestrian behaves on his or her own behalf. In an extension of this, we distinguish the notion of “inaccessibility” under various contexts: Independent pedestrians tend to show greater “repulsive” effect to each other, whereas pedestrians which belong to the same group in the simulation environment tend to be “friendly” to each other, hence, the inaccessibility function is of a lower value.
Overall, the inaccessibility at an arbitrary cell position is always dependent on the behavioural context in the system. The density control on a local level can be achieved in this way.
We present two experimental cases to demonstrate our model. In the first experiment we simulate the group behaviours of the pedestrians, i.e. how some of these tend to stay close to each other (but the distance among them may still vary over a small range in the local step calculation), distinguishing them from other “foreign” pedestrians which exist in the simulation environment at the same time. A second experiment considers the intersection of multiple pedestrians groups with a simplification that all the participants are given pre-configured moving directions. The objective of this experiment is to demonstrate the density evolution, especially in certain critical regions (e.g. the intersection point where bottleneck phenomenom is more likely to take place), in correlation with local speed.