With so many extant models, some multiple decades old, the obvious question is, “Why base a pedestrian model on a robot motion planning technique?” We have previously dismissed macroscopic models and cellular automata approaches. The former place significant limits on crowd heterogeneity and the latter’s discrete artifacts lead to dissatisfying simulation results. So, for heterogeneous agents planning in a continuous domain, the choice comes down to velocity obstacles or social forces. Certainly, it seems that velocity obstacles, which inherently encode position and velocity
to predict future state, seems a more sophisticated model than a simple social force model which merely generates forces based on current relative positions. However, this putative advantage is not unique to velocity-obstacle-based methods. Various social force derivatives have introduced terms in the force definition which include relative velocity (Johansson et al., 2007; Chraibi et al., 2010) and prediction (Karamouzas et al., 2009; Zanlungo et al., 2011). However, we feel that when comparing each model’s pros and cons in other respects, the velocity obstacle approach offers the greatest net benefit.
Implementation: Social forces are very simple in principle. A force is generated for each agent pair and the final agent response arises from the superpositioning of the individual responses. In contrast, implementing velocity obstacles can be very complex. Generally, computing a truncated velocity obstacle for entities witharbitraryshape and optimizing with respect to the union of a number of such velocity obstacles may prove infeasible. However, with the assumption of disk agents, the ORCA formulation provides a relatively straight-forward mathematical basis for defining the velocity obstacles, and, assuming the optimization function is convex, the algorithm for optimizing the response velocity is relatively simple – albeit still far more complex than implementing a straight-forward social force model (such as (Helbing et al., 2000)).
Efficiency: Depending on the social force model, computing the net force acting on an agent can be quite efficient8. However, the ORCA formulation defines a problem that can be solved very efficiently. So, in many ways, the difference in cost of a single time step between the two approaches can be considered negligible. However, as previously indicated, the agent-agent repulsion functions typically have a large slope at close distances. This requires a small time step, much smaller than is required for ORCA (as we show in Chapter 5). For a given amount of simulation time, more time steps need to be taken with a social force model, leading to a lower efficiency.
Convergence: The choice of time step can also affect the simulation outcome. As the timestep size goes to zero, the simulation should converge to a consistent result. However, when the time step is large, changes to the time step can easily lead to significantly different outcomes. This is largely due to the standard practice of using a low-order explicit integration scheme. These integration schemes are simple to implement but converge to a consistent solution slowly as the time step
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This is particularly true for (Helbing et al., 2000), as the models become more sophisticated the cost can increase significantly (Johansson et al., 2007; Karamouzas et al., 2009; Chraibi et al., 2010; Zanlungo et al., 2011).
decreases. Both models will suffer from artifacts in integrating the velocity to compute position for the next time step. However, where the velocity obstacle model is afirst ordermodel, the social force model is asecond ordermodel. A second order model will be less compatible with low-order explicit integration leading to additional convergence issues for social force models.
This can be understood intuitively. It comes down to the fact simple fact that social force models require two integrations per time step and velocity obstacles only require one. In social forces, a force and its resultant acceleration are computed. To get position, the acceleration must be integrated twice. In fact, the new velocity the agent takes is not just a function of the simulator state, but also the time step. In contrast, a velocity-obstacle-based model computes the new velocity directly from the simulator state. So, it only need be integrated once to get a new position. More significantly, the velocity is independent of the simulation time step.
Although the connection between model stability and this property has not been formally proven, empirical evidence suggests that this is what allows crowd simulators based on velocity obstacles to take large time steps and still produce consistent results (e.g., Chapter 5).
Behavior: Finally, the two models operate in a fundamentally different paradigm for resolving
multipleagent interactions. With social forces, the repulsive force represents a response to a single agent. When there are multiple agent interactions, the individual responses are linearly combined. The implication of this is that aspects of individual responses can cancel each other out. Velocity obstacles work through constraints. Rather than combining specific responses, each agent contributes to the constrained space. Then the solution is computed based on the feasible velocity space. Figure 2.5 illustrates the impact of these differences.
With social forces, the blue and red agents, independently, cause the grey planning agent to slow and turn a small amount (see Figure 2.5 (a) and (b)). When they are combined, the two turning effects cancel each other out and, instead, the agent is slowed significantly (see Figure 2.5 (c)). The slowing is not necessarily the best strategy; it does not reduce the risk of future collision. With velocity obstacles, each agent alone has a similar effect on the planning agent; the agent slows and turns slightly9. However, when combined, the agent recognizes that the only way to avoid inevitable collision is to speed up, passing between the red and blue agents while there is still space available.
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The amount the agent turns and slows could be made almost identical between the velocity obstacle and social force illustrations if the force coefficients were modified.
(a) (b) (c)
(d) (e) (f)
Figure 2.5: A comparison of how social force models and velocity space models handle multiple agent-agent interactions. The grey agent is computing its new velocity. In all figures, the dashed grey arrow represents the planning agent’s previous velocity. The planning agent’s new velocity is the solid black arrow. (a) The resultant velocity due to the acceleration caused by the repulsive force from the blue agent. (b) The resultant velocity due to the acceleration caused by the repulsive force from the red agent. (c) The resultant velocity due to the superpositioning of the red and blue agents’ forces. (d) The resultant velocity due to the half-plane constraint implied by the blue agent. (e) The resultant velocity due to the half-plane constraint implied by the red agent. (f) The resultant velocity due to both half-plane constraints from the red and blue agents. In (d), (e), and (f), the truncated cone velocity obstacleandthe corresponding half-plane constraint are shown.
The difference in behavior is purely a function of how the influence of pair-wise agent interactions are combined. We feel the effect produced by taking the union of velocity obstacles yields more robust behavior than the linear combination of forces.
CHAPTER 3: COMPOSITE AGENTS