CIRCULAR OIC/CAPUFE/TAR/008/2016

Robot controllers operate on a 0.1 s time step and are not synchronized with each other. At each time-step, the robot updates its belief about the neighboring robots and fixed obstacles, and applies one of the behaviors described in Sec- tions 3.3 and 3.4 to compute the desired heading and velocity. Robots are con-

81 3.7 Experiments

trolled with ORCA, HRVO and HL as described in Section 3.3.3, by translating the holonomic desired velocity into wheel speeds. In addition, we also consider ORCA-NH, which explicitly takes into account the non-holonomicity of the robots when computing the desired wheel velocities, as discussed in Section 3.4.1.

HL We provide HL robots with our own implementation (see Appendix B) of

the behaviors described in Section 3.3. We fixτ_{r ot} = 0.5 s and we limit the foot- bot angular speed to ωmax = 90 ° s−1 to prevent excessive slipping and camera

image blurring. Wheel speed is clipped to wmax = 30 cm/s. Human motion

characteristics are given by τ = η = 0.5 s [104]. We maintain η = 0.5 s and decreaseτ to 0.125 s to obtain a more reactive but still smooth behavior, which moves with increased caution.

HRVO We use an open source implementation[69] of the model described in

Section 3.4.1. The desired velocity is found in the velocity space through a linear optimization technique. The implementation does not have free parameters.

ORCA We use an open source implementation[126] of the model described in Section 3.4.1. The desired velocity is found in the velocity space through a linear optimization technique. The time horizon is a free parameter: a large time horizon allows the robot to anticipate crowding and avoid congestion, but at the same time penalizes it with a reduction of speed and a longer, more conservative path. In the following, we select the time horizon with the best performance for each scenario.

ORCA-NH We use the same controller as ORCA, but apply it to the effective

center and effective radius (see Section 3.4.1) of the non-holonomic robots.

3.7

Experiments

We investigate how the proposed human-like behavior compares with alternative local navigation behaviors detailed in Section 3.4. In particular, we aim to:

a) investigate how the safety margin affects the navigation safety for robots with error-free omnidirectional sensing and the trade-off between efficiency and safety for robots with realistic sensing (Section 3.7.1);

82 3.7 Experiments

c) validate simulated results by comparison with real-robot experiments per- formed in the same conditions (Section 3.7.3);

d) investigate how the navigation performance for the different behaviors scales with the number of robots (Section 3.7.4), and explore the impact of groups of heterogeneous agents implementing different navigation al- gorithms (Section 3.7.5);

e) study the emergence of macroscopic group behaviors (Section 3.7.6);

f) study how well the robots follow a prescribed smooth trajectory (Section 3.7.7). For each experiment, we compute how a given parameter affects a number of performance metrics, detailed below; for each value of the parameter, we perform R simulation runs (replicas), each lasting T seconds after a random ini- tialization. For the Cross scenario, R = 50, T = 900 s; for the Circle scenario,

R= 100, T = 100 s; for the Corridor scenario, R = 100, T = 180 s; for the Indoor

scenario, R= 100, T = 300 s.

Performance Metrics

We compute the following performance metrics, which quantify different aspects of the robots’ trajectories.

Relative throughput indicates the efficiency in navigating towards the targets.

This measure is defined for the Cross scenario as the total amount of targets that the robots were able to reach, divided by the number of targets that the robots could reach in the same time while traveling in straight lines (i.e., ignor- ing any collision). In the Circle scenario, throughput is defined as the minimal time it would take for one robot to reach the opposite side (when traveling in a straight line) divided by the actual time it took. In the Corridor scenario, the relative throughput is given by the average speed directed towards the target divided by the maximal admitted speed of the robot. The resulting quantity is a- dimensional, bounded between 0 (worst) and 1 (optimal), and is averaged over all the robots in the simulation.

Relative path length: the total length that the agents have traveled, divided by the length the agents would have covered while traveling in straight lines (i.e., ignoring any collision). This is negatively related to the energetic efficiency of the trajectories.

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Path irregularity: the amount of unnecessary turning per unit path length per-

formed by a robot; unnecessary turning corresponds to the total amount of robot rotation minus the minimum amount of rotation which would be needed to reach the same targets with the most direct path. Path irregularity is measured in

rad/m, and is averaged over all the robots in the simulation. We propose this as

an objective measure of the legibility (see Section 3.2) of the robot’s behavior. In fact, it’s difficult to infer the intention (the target) of a robot that is changing its direction often. We are currently researching the correlation of path irregular- ity with the legibility of the robot’s behavior and the subjective judgment of its friendliness by humans.

Total number of collisions: the number of collisions per robot per meter of

covered distance. Collisions are defined as discrete events, so pairs of agents repeatedly brushing against each other give rise to multiple collision events.

Safety margin violations: for a given robot r, the fraction of time during which at least one agent or obstacle penetrates the safety margin ms by more than

a given amount of space (violation length); the value is then averaged over all robots. The value is computed for every violation length between 0 and m_s. Com- pared to the number of collisions, this provides a more descriptive but less con- crete measure of safety: for example, it allows to discriminate a case in which the safety margin is frequently violated, but only by a small amount, from a case in which the safety margin is rarely violated, but with robots almost coming into contact.

Line order: a metric computed only in the Corridor scenario, where it quan- tifies, for a given moment, the segmentation of robots of two different groups (corresponding to different optimal speeds or target directions) in longitudinal lines [103]. More specifically, we divide the corridor into narrow longitudinal bands with a width of 30 cm (i.e., roughly twice the width of a foot-bot) and count the number of robots of each group n1(B), n2(B) inside a band B: the

Yamori band[152] is defined as Y (B) = |n1(B)−n2(B)|

n1(B)+n2(B). The line order OL is defined

as the average Yamori index over all bands. O_L is bounded between 0 and 1 (representing a perfect organization of the swarm classes in longitudinal lines).

Hausdorff distance: d_H(γ, γ_t) = max  max x∈γ miny∈γt |x − y|, max y∈γt min x∈γ |x − y| ‹ ,

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between the robot’s trajectoryγ and a target trajectory γ_t, quantifies deviations from the path.

Discomfort: the integrated squared magnitude of the jerk J quantifies the smoothness of the trajectory (see Section 2.3.1). Agents that would artificially reduce their speed while following a geometrical trajectory, would substantially decrease J . We get a better metric ¯J when we divide J by the fifth power of the efficiency because it become independent of the mean speed (see Equa- tion (2.3)).