## Abstract

We study the problem of tracking multiple moving targets using a team of mobile robots. Each robot has a set of motion primitives to choose from in order to collectively maximize the number of targets tracked or the total quality of tracking. Our focus is on scenarios where communication is limited and the robots have limited time to share information with their neighbors. As a result, we seek distributed algorithms that can find solutions in a bounded amount of time. We present two algorithms: (1) a greedy algorithm that is guaranteed to find a 2-approximation to the optimal (centralized) solution but requiring |*R*| communication rounds in the worst case, where |*R*| denotes the number of robots, and (2) a *local* algorithm that finds a \(\mathcal {O}\left( (1+\epsilon )(1+1/h)\right) \)—approximation algorithm in \(\mathcal {O}(h\log 1/\epsilon )\) communication rounds. Here, *h* and \(\epsilon \) are parameters that allow the user to trade-off the solution quality with communication time. In addition to theoretical results, we present empirical evaluation including comparisons with centralized optimal solutions.

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## Notes

- 1.
An algorithm is called a \(\mathcal {O}(x)\) approximation to a maximization problem if it guarantees a solution whose value is at least \(\frac{c}{x}\) of the optimal value, where

*c*is some constant. - 2.
After these assumptions, we omit the time index (i.e.,

*k*) for notational convenience. - 3.
If all \(x_{m}^{i}=0\) for a robot

*i*, then it can choose any motion primitives since the objective value will remain the same. - 4.
- 5.
Although we model linear motion for the targets, more sophisticated models for the prediction of target states can also be employed.

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## Acknowledgements

The authors would like to thank Dr. Jukka Suomela from Aalto University for fruitful discussion.

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This material is based upon work supported by the National Science Foundation under Grant No. 1637915.

This is one of the several papers published in *Autonomous Robots* comprising Special Issue on Robot Communication Challenges: Real-World Problems, Systems, and Methods.

## Appendices

### Proof of lemma 1

Equation (5) of a max-min linear program is equivalent to the following max-min problem if the scalar variable *w* which represents the inner minimization is eliminated:

From Eqs. (5) and (7), the following relationship is satisfied:

Since Eq. (2) does not require \(x_{m}^{i}\) to be a linear value, Eq. (2) is equivalent to Eq. (5) with additional integer constraints.

### Proof of lemma 2

Considering \(c_{i,m}^j\), which is a weight between *m*-th motion primitive of *i*-th robot and *j*-th target on graph \(\mathcal {G}_S\), a quality of tracking (\(w(\mathbf t _j)\)) for *j*-th target can be defined as follows:

Therefore, the sum of quality of tracking over all targets is:

Equation (10) is obtained by taking into account the conditional term of the first equation explicitly. The last equation follows from the property that \(y_i^j\) chooses the maximum value of \(\sum _{m\in P^i}c_{i,m}^jx_{m}^{i}\) among all robots, which is shown in lines 10–14 of Algorithm 2. Therefore, the last equation is equal to the inner term of Eq. (4).

### Greedy performs poorly for the Bottleneck variant

We present an example of instance that shows an arbitrary poor performance of the greedy algorithm when applied to the Bottleneck variant. Consider the following case where there are two robots (\(\mathbf r _i\)) having two motion primitives (\(\mathbf p _m^i\)) for each and two targets. The realization of the communication and sensing graphs are as in the following table. The tracking quality in this example corresponds to the number of targets being tracked.

\(\mathbf p _1^1\), \(\mathbf p _1^2\) | \(\mathbf p _2^1\), \(\mathbf p _2^2\) | |
---|---|---|

\(\mathbf r _1\) | \(\mathbf t _1\) | \(\varnothing \) |

\(\mathbf r _2\) | \(\varnothing \) | \(\mathbf t _2\) |

Let’s apply the Bottleneck version of greedy algorithm to this case. Since the objective of the Bottleneck variant is to maximize the minimum tracking quality, the robot 1 (\(\mathbf r _1\)) chooses motion primitive 2 (\(\mathbf p _2^1\)) because choosing motion primitive 1 (\(\mathbf p _1^1\)) gives the value of 1 while choosing motion primitive 2 (\(\mathbf p _2^1\)) gives the value of 0. For the same reason, the robot 2 (\(\mathbf r _2\)) chooses motion primitive 1 (\(\mathbf p _1^2\)). This gives the total value of 0, whereas the optimal solution is 2 as the first robot and second robot choose motion primitive 1 (\(\mathbf p _1^1\)) and motion primitive 2 (\(\mathbf p _2^2\)), respectively. The similar case is reproducible with a larger number of robots, motion primitives, and targets. Thus, the simple greedy performs arbitrarily badly for the Bottleneck variant.

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Sung, Y., Budhiraja, A.K., Williams, R.K. *et al.* Distributed assignment with limited communication for multi-robot multi-target tracking.
*Auton Robot* **44, **57–73 (2020). https://doi.org/10.1007/s10514-019-09856-1

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### Keywords

- Multi-robot system
- Task assignment
- Distributed algorithm