# A Robust Planning Algorithm for Groups of Entities in Discrete Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Planning for Multiple Autonomous Beings

## 3. Definition of the Planning Problem

#### 3.1. Formal Model of the Considered Problem

## 4. A-Star and Wavefront Algorithms for Multi-Entity Planning

#### 4.1. Important Assumptions

#### 4.2. The Complexity of Multi-Dimensional Planning

#### 4.3. Multi-Entity A-Star

- $V=\{{v}_{1},{v}_{2},\dots ,{v}_{n}\}$ is a set of vertices,
- $E=\{{e}_{1},{e}_{2},\dots ,{e}_{m}\}$ is a set of edges, each indicating a pair of vertices $v\in V$: ${e}_{i}=({v}_{j},{v}_{k})$,
- $c:E\to {\mathbb{R}}^{+}\cup \left\{0\right\}$ is a function that assigns a weight to every edge $e\in E$.

- states are represented by vertices $v\in V$ (each state has $n\ast k$ components),
- actions are represented by edges $e\in E$: if an action of changing state from state ${v}_{j}$ to state ${v}_{k}$ is possible, ${e}_{i}=({v}_{j},{v}_{k})\in E$.

- Ƒ denotes the fringe, i.e., the set of vertices, which are the possible candidates for the next move;
- V denotes the set of vertices, which have already been visited.

- ${\u01a4}_{E}$ is a sequence of edges, indicating the optimal path from start to goal;
- ${\u01a4}_{V}$ is a sequence of vertices corresponding to ${\u01a4}_{E}$.

- (1)
- Initially assume: $V\leftarrow \varnothing $, $\u0191\leftarrow \left\{{v}_{s}\right\}$, $g\left({v}_{s}\right)=0$.
- (2)
- Choose the node ${v}^{\prime}\in \u0191$ to be expanded:$$\underset{{v}^{\prime}}{arg\; min}f\left({v}^{\prime}\right)=g\left({v}^{\prime}\right)+h\left({v}^{\prime}\right)$$
- (3)
- If ${v}^{\prime}={v}_{g}$, go to Step 5.Otherwise, expand node ${v}^{\prime}$. $\forall {v}^{\u2033}\in V:\phantom{\rule{0.166667em}{0ex}}{e}^{\u2033}=({v}^{\prime},{v}^{\u2033})\in E{,}^{\prime},{v}^{\u2033}\notin V$, and do the following:
- calculate the value of function g: $g\left({v}^{\u2033}\right)=g\left({v}^{\prime}\right)+c\left({e}^{\u2033}\right)$,
- add ${v}^{\u2033}$ to the fringe: $\u0191=\u0191\cup \left\{{v}^{\u2033}\right\}$.

- (4)
- Remove ${v}^{\prime}$ from the fringe after expansion and move it to the visited set:$$\u0191=\u0191\setminus \left\{{v}^{\prime}\right\},\phantom{\rule{0.222222em}{0ex}}V=V\cup \left\{{v}^{\prime}\right\}.$$
- (5)
- Reconstruct the optimal path:
- (a)
- Let ${e}_{p}=\pi \left({v}_{g}\right)$.
- (b)
- Let ${\u01a4}_{E}={(}^{\prime},),{\u01a4}_{V}={(}^{\prime},)$ (empty sequences).
- (c)
- Append ${e}_{p}$ to the beginning of ${\u01a4}_{E}$: ${\u01a4}_{E}=({e}_{p},{\u01a4}_{\varepsilon})$.Append ${p}_{v}$ to the beginning of ${\u01a4}_{V}$: ${\u01a4}_{V}=({v}_{p},{\u01a4}_{V})$.
- (d)
- Let ${v}_{\pi}=v:\pi \left({v}_{p}\right)=(v,{v}_{p})$.
- (e)
- Let ${e}_{p}=\pi \left({v}_{\pi}\right)$.
- (f)
- If ${e}_{p}\ne 0$, go to Step 5c.Otherwise, STOP.

#### 4.4. Multi-Entity Wavefront

- (1)
- vertex labeling;
- (2)
- selecting a path.

## 5. A-Star-Wavefront Hybrid Algorithm

- there is a global plan provided by the A-star algorithm;
- sub-spaces of a possible collision in the search space are identified; they called collision sub-spaces;
- for the above sub-spaces, the wavefront algorithm is used to give the optimal path to exit the sub-space, thus to return to the original A-star-based path.

- (1)
- the number of neighborhoods (N) that might result in collisions, either among agents or obstacles: ${d}_{c}$;
- (2)
- the size of the collision sub-space (S): ${r}_{c}$.

## 6. Experimental Results

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Wojnicki, I.; Ernst, S.; Turek, W.
A Robust Planning Algorithm for Groups of Entities in Discrete Spaces. *Entropy* **2015**, *17*, 5422-5436.
https://doi.org/10.3390/e17085422

**AMA Style**

Wojnicki I, Ernst S, Turek W.
A Robust Planning Algorithm for Groups of Entities in Discrete Spaces. *Entropy*. 2015; 17(8):5422-5436.
https://doi.org/10.3390/e17085422

**Chicago/Turabian Style**

Wojnicki, Igor, Sebastian Ernst, and Wojciech Turek.
2015. "A Robust Planning Algorithm for Groups of Entities in Discrete Spaces" *Entropy* 17, no. 8: 5422-5436.
https://doi.org/10.3390/e17085422