# Persistent multi-robot formations with redundancy

## Summary (2 min read)

### 1 Introduction

- For applications such as collective transport, multi-robot formations need to maintain a global shape.
- Since this may increase sensing and communication costs, a “persistence theory” for directed distance constraints between points was proposed by Hendrickx et al. [5] (see also [4]), effectively cutting costs in half by assigning one of the two agents to be responsible for sensing and maintaining a distance.
- Redundancy is wellunderstood in rigidity theory, and the associated objects form the foundation for the main contribution of this paper: an approach for constructing persistent leader-follower formations with redundancy.
- Restricting to acyclic formations implies that such special conditions do not impact their construction, and the authors present simulation results (Section 5) verifying their approach.

### 2 Preliminaries

- For a given graph, almost all associated embeddings, called generic embeddings, share the same rigidity properties (see, e.g., [19]).
- One can interpret each of the vertices as representing an autonomous agent with out-going edges specifying distance constraints to neighbors that it is responsible for satisfying.
- While the formations in Figures 2(a) and 2(b) have the same underlying undirected graph, only one is persistent.
- The authors consider a specific type of persistent formations called leader-follower formations.

### 3 Redundancy for persistence theory

- Communication links and sensors can fail, motivating the need for redundancy in a multi-robot system.
- Refer back to the formation in Figure 2(d).
- While its underlying undirected graph is redundantly rigid, the formation is not redundantly persistent; without the edge −→ 32, the resulting formation ) is no longer persistent.
- Furthermore, the authors can use “pebble collection” moves in the pebble game to find an orientation H where exactly one vertex vL has out-degree 0, another vertex vC incident to vL, has out-degree 1 and all other vertices have out-degree 2.
- Proposition 2 gives a recursive approach for constructing persistent leader-follower formations with any desired number of vertices whose out-edge sets each contain redundancy.

### 4 Special geometric conditions

- In Section 3, the authors presented approaches for constructing generically persistent graphs, applying to situations where agents are positioned with a generic embedding in the plane.
- As noted in Section 2, for particular geometric configurations, a generically rigid graph may become flexible.
- The authors illustrate these two types of special positions with some simple examples.
- Observe that the persistent formations of Fig. 5 and 6 are acyclic, while the non-persistent formations are not.

### 5 Simulation

- Each vertex in the graph is implemented by a robot that is equipped with a generic emitter, and each directed edge by equipping the source robot with a receiver that listens to the target robot’s emitter.
- Every follower computes and moves to a goal position based on its assigned distance constraints: with two constraints, the closest point of the 2 intersection points of 2 circles, and, with three constraints, the average of 3 points computed for each pair of constraints.
- The expected position of the follower robots is computed using the position of the leader and co-leader positions; since the co-leader is on the same robot as the leader, its distance constraint is always satisfied.
- Data for each simulation was then computed as the mean of the mt , µt and Mt across all simulation steps.
- The results confirm the expected behavior of the formations; the simulations testing redundancy performed comparably to the control with all edges present.

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### Additional excerpts

...distributed systems [2], [7], [11], [9], [10], [3], [18], [8]....

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##### References

1,068 citations

394 citations

### "Persistent multi-robot formations w..." refers background in this paper

...While rigidity theory has been applied to the construction and analysis of formations of autonomous agents [11, 3], the approach assumes undirected constraints, leading to a model where both agents would be responsible for the constraint....

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379 citations

213 citations

### "Persistent multi-robot formations w..." refers background or methods or result in this paper

...We can now state the technical definitions from [5] for persistence....

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...[5] (see also [4]), effectively cutting costs in half by assigning one of the two agents to be responsible for sensing and maintaining a distance....

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...Then, by Propositions 1 and 3 in [5], we obtain an acyclic generically minimally persistent graph G....

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...We rely on the following result of [5]: Theorem 1 (Theorem 3 of [5])....

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...However, if a persistent graph is acyclic, then there exists an ordering of the vertices such that (1) the first vertex has out-degree 0, (2) the second out-degree 1, and (3) every other vertex has ≥ 2 out-edges to vertices earlier in the ordering [5]....

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209 citations

### "Persistent multi-robot formations w..." refers background or methods in this paper

...Graphs of this type are a generalization of Henneberg I graphs, given the use of the “vertex addition” step for the followers that was first described by Henneberg for minimally rigid graphs [18, 19]....

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..., [19]) and is well-understood, with a quadratic algorithm for determining the bar-and-joint rigidity properties [8]....

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##### Frequently Asked Questions (2)

###### Q2. What are the contributions in "Persistent multi-robot formations with redundancy" ?

To distribute control, the authors construct leader-follow formations in the plane that are persistent: designated “ leader ” robots control the movement of the entire formation, while the remaining “ follower ” robots maintain directed local links sensing data to other robots in such a way that the entire formation retains its overall shape. In this paper, the authors present an approach based on rigidity theory for constructing persistent leader-follower formations with redundancy ; specified robots may experience sensor link failure without losing the persistence of the formation. Within this model, the authors consider the impact of special positions due to certain geometric conditions and provide simulation results confirming the expected behavior.