# Situationally Sensitive Path Planning
## 2 Related Work
### Continuum Mechanics Models  
#### 2.3.1 Macroscopic Mechanics of Aggregate Crowd Flow  
Macro-scale models drive collective (usually group and crowd) movements by approximating global continuum functions. Traditionally, these are physics-based due to the ease of transfer between considerations of physical particles and pedestrians as point masses with velocity and rotation: both are traditions that were carried into early models of pedestrian flow and traffic flow [42; 43]. Significant success has been achieved in generating substantive patterns of crowd behavior from the mathematics of particle interactions [44; 45], and a set of case studies have shown that particle models can reproduce the statistical patterns of crowd flow and outcomes that exhibit properties of complex adaptive systems [46; 47; 48; 49]. We note, however, that most of the applications from physics are focused on phase shifts in crowd behavior, i.e., the emergent continuum or the tipping point at which a crowd begins to behave as excitable media [50]. Invariably, this draws the models into applications that focus on the extremes of crowd density. Consider, for example, how [PERSON] [51] (p. 171) outlaid the particle approach: \"A crowd of pedestrians can generally be treated as a continuum, provided the characteristic distance scale between pedestrians is much less than the characteristic distance scale of the region in which the pedestrians move.\" This brings to mind the dense packing of moving people at festivals [52] during large-volume crowd flow events such as parades and carnivals [53] and dense assemblies that are common during entertainment shows [54]. This dense-packing approach evokes crowd behavior, rather than the individual movement that one would expect through an enclosed space, i.e., continuum approaches address relatively uncommon events such as emergency egress [5], crowd panic [55], and crowd quakes [56] in which there are sudden shifts in crowd flow that block or injure participants [57]. Our interest is in more mundane and quotidian patterns of behavior. As such, the physics of excitable media are not appropriate representations. Nonetheless, a concise review is appropriate.  
Traditionally, physics-based approaches to indoor pedestrian movement make an initial assumption that pedestrian particles can be treated as moving in a continuum; they use variants of the Navier-Stokes [58] equation or assumptions of Maxwell-Boltzmann [59] distributions to consider pedestrian movement and collective crowd flow as analogous to the flow of granular media, gas, or liquids [60; 61; 62]. Examples of applications to urban design are common in the literature, and scenarios include [PERSON]'s [60] work to model crowds as fluid phenomena in a gaseous phase. Indeed, [PERSON] tested his model with an application to a featureless model passageway, but performed validation for real designs (including the footpath outside the University of Sydney's Fisher Library). [PERSON] [43; 51] partial differential equation-based models are another canonical example of pedestrian particle models, alongside a stream of physics-inspired models by [PERSON] and [PERSON] and colleagues [63; 64; 65]. Popular extensions of continuum mechanics include variants to account for excitation [44; 66], potential fields [67; 68], velocity fields [69; 70; 71], and implementations as cellular automata and finite-state machines on Turing-like [72] lattices of discretized space models [73].