Planning paths around dynamic agents means they must be calculated quickly, and in order to evaluate different potential paths they must also be diverse. In crowded environments, paths that belong to different homotopy classes may still move in very similar directions. Therefore, processing time would be wasted on planning paths that do not offer any diversity between them. Also, the dynamic environment results in any paths planned too far in the future becoming unreliable, due to escalating prediction errors in the pedestrian model as T increases.
Both of these issues are now addressed in order to ensure the selected paths move in different enough directions from one another. The number of paths will
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(a) (b)
Fig. 5.11 Visualisation of how paths (arrows pointing from the AMR’s centre) are chosen, in order to aide exploration of the model environment. A maximum of three paths (highlighted with green arrowheads) are selected, with only one occurring in the left and right (dark blue), and the front (light blue) quadrants of the circle. This path diversification will force the CPP to explore the environment, allowing for considerate paths to be discovered, and thus preventing three similar paths all occurring around the same point. The yellow has highlights the extension of the FoV beyond the path selection area. (a) An example of a FoV with a viewing angle of 270o, with 3 separate 90o quadrants. (b) An example of a FoV with a viewing angle of 135o, with 3 separate 45o quadrants.
also be limited, preventing too many potential paths being calculated, whilst also restricting the number to choose related to at what t along T , the CPP is planning paths.
Ensuring Path Diversity
After an initial reduction in the repetition of paths, a more explicit method is used to ensure the paths chosen are also divergent. Depending on the population density of the crowd surrounding the AMR the number of potential paths may be very large. Therefore, the maximum number of paths which can be chosen is limited to three, reducing the algorithm’s processing speed and path redundancy. However, the three most desirable paths may be very similar, e.g. three paths that move left, in between, and right of two ellipses directly ahead. The three chosen paths are forced to diverge by ensuring they are selected to the left, the right, and ahead of the search’s start point at t, by positioning three circular quadrants at these locations around the AMR, Fig. 5.11. Only the most desirable path is chosen from each quadrant, which will force the CPP to move to different parts of the model environment at t + 1.
The three paths selected will aide in producing globally diverse paths, however the further into the prediction time-horizon, T , the CPP goes, the more unreliable and repetitive the overall paths may become. To prevent predicting multiple paths with less reliable data, the number of paths predicted at each time-step gradually reduced. Tis equally divided into thirds so that for the first third of T three paths are selected; for the second third of T two paths are selected; and for the last third of T only one path is selected. The number of paths at each time-step, pt, is dependent upon the current time-step as well as the overall T value, which satisfies the following inequality, pt= 3, if ∆ < 1 3 1, if ∆ > 2 3 2, otherwise (5.8) where ∆ = t T− 1
T. The maximum number of local path segments, Pt, is a summation series of nested product series:
Pt= T X t=1 pt= p1+ (p1× p2) + · · · + T −1 Y n=1 pn+ T Y n=1 pn (5.9)
The maximum number of PT paths, which occur by connecting consecutive Ptpaths together for the length of T , is a simple product series, seen as the nested product series in Eq. (5.9): PT = T Y t=1 pt= p1× p2× · · · × p(T −1)× pT (5.10) The maximum number of searches required to create all the potential PT paths is calculated using the difference between the maximum Pt and PT paths:
f(p) = 1 + Pt− PT = 1 +h p1+ p2+ · · · + p(T −1)+ pT i − " p1+ (p1× p2) + · · · + T −1 Y n=1 pn+ T Y n=1 pn # (5.11)
Fig. 5.12a shows the number of Pt paths, PT paths, and separate searches required when increasing T from 1 to 12; Fig. 5.12b shows an visual example using a tree structure when T = 4.
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(a)
(b)
Fig. 5.12 Example of the maximum number of FGPs that can be calculated for various prediction time-horizon lengths, T . (a) Each coloured box represents the maximum number of branches that can be created at that time-step (Green: 3, Yellow: 2, Red: 1). The numbers within the boxes represent the maximum possible accumulating calculations required at each time-step.(b) Example of tree when T = 4.
Completing the Prediction Time-Horizon
By removing paths that share similar ellipses, limiting the number of Pt paths to a maximum of three, and ensuring that of those three paths they diverge to the left, the right, and forward from the search start, the CPP will be able to create multiple unique, diverse, and considerate PT paths the length of T .
As can be seen in the CNS flow diagram, Fig. 5.1, if t < T the pedestrian model will update the uncertainty ellipses according to the size and position they should be at t + 1. The AMR’s FoV that limits what portion of the environment can be searched remains the same, as the AMR has not yet moved. The VD-VG roadmap is recalculated using the updated ellipses, and remains limited to within the FoV as before. However, the searches that find the next Ptpaths at t + 1 begin at the end of the Pt paths selected at t. The search repeats for all Pt paths until t = T , along with the same path filtering techniques, e.g. Fig. 5.13. Once all possible PT paths have been calculated, as defined by Eq. (5.9), they can be connected to one another to create a global path along the entire prediction time-horizon,
PT(n) = P1(n) + P2(n) + · · · + P(T −1)(n) + PT(n) (5.12) where N is the number of potential PT paths defined by Eq. (5.10), and {n ∈ Z | 0 ≤ n ≤ N }. The most desirable global PT path can then be determined by the CPP for the AMR to move along, e.g. Fig. 5.14.