## Abstract

We propose a new paradigm for construction in which teams of quadrotor helicopters assemble 2.5-D structures from simple structural nodes and members equipped with magnets. The structures, called Special Cubic Structures (SCS), are a class of 2.5-D truss-like structures free of overhangs and holes. Quadrotors equipped with grippers pick up, transport, and assemble the structural elements. The design of the nodes and members imposes constraints on assembly, which are incorporated into the design of the algorithms used for assembly. We show that any SCS can be built using only the feasible assembly modes for individual structural elements and present simulation and experimental results for a team of quadrotors performing automated assembly. The paper includes a theoretical analysis of the SCS construction algorithm, the rationale for the design of the structural nodes, members and quadrotor gripper, a description of the quadrotor control methods for part pickup, transport and assembly, and an empirical analysis of system performance.

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Ascending Technologies, GmbH. http://www.asctec.de.

Vicon Motion Systems. http://www.vicon.com.

ROS-Matlab Bridge. http://github.com/nmichael/ipc-bridge.

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## Acknowledgements

The authors would like to acknowledge Yash Mulgaonkar for assistance with the experiments and the fabrication of the parts and Professor Nathan Michael for help with the software interfaces and infrastructure.

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## Appendix: Proof of Lemma 2

### Appendix: Proof of Lemma 2

In order to prove Lemma 2 we will consider constructing an individual square during the WFR algorithm in space 0 during wave *k* as shown in Fig. 21(a). There are 8 immediate neighbors of space 0 so there are 256 possible situations that can occur. We will show that many of the 256 possible situations are impossible and that the only ones remaining can be constructed using the construction primitives P1–P4 shown in Fig. 4.

We introduce *Inward L-shapes* as illustrated in Fig. 19(a) and define them with the aid of Fig. 19(b). We define an *Inward Pointed L-shape* as a situation where square A was built during wave *k* and squares B, C, and D were built during wave *k*−1. We define an *Inward Pointless L-shape* as a situation where square A was built during wave *k* and squares B and C were built during wave *k*−1 and space D is to be left empty. Note that *Inward L-shapes* can be rotated versions (90^{∘}, 180^{∘}, and 270^{∘}) of the ones shown in Fig. 19(a). We first show that *Inward L-shapes* imply locally disconnected regions of squares during a previous construction wave.

### Lemma 3

*For SCSs the Inward Pointless L*-*shape shown in Fig*. 19(a) *implies that spaces E*, *F*, *G*, *I*, *J*, *and K must contain a set of disconnected squares built during wave*
*k*−2.

### Proof

No squares were built in E, F, G, I, J, or K prior to wave *k*−2 because then B or C would have been built before wave *k*−1. The fact that square B (C) was built during wave *k*−1 and A during wave *k* implies that one or more squares were built in spaces I, J, and K (E, F, and G) during wave *k*−2. Spaces G and I cannot both contain squares because this would imply a hole in the original structure at D. Therefore, E, F, G, I, J, and K must contain a set of disconnected squares built during wave *k*−2. □

### Lemma 4

*For SCSs the Inward Pointed L*-*shape shown in Fig*. 19(a) *implies that for some wave number*
*w*
*there exists a set of squares of the shape of the region E–K that are disconnected and filled during wave*
*w*.

### Proof

We can use the same logic as in the proof of Lemma 3 to show that if G and I are not both occupied then E–K are disconnected and *w*=*k*−2. If G and I are occupied and H is not then Lemma 3 tells us that during wave *k*−3 we must have a set of nodes shaped like E–K that are disconnected, meaning *w*=*k*−3. The only other possibility is that G, H, and I are all filled. In this situation we again have an *Inward Pointed L-shape*. So, if we repeat the logic we eventually result in a disconnected region shaped like E–K or a region shaped like G–I with all squares filled. However, we cannot continue to have three squares built at every wave because the WFR algorithm must start at wave 1 with a single built square. Therefore, for some wave number, *w*, we must have a region shaped like E–K that is disconnected. □

We now show that *Inward L-shapes* imply globally disconnected regions.

### Lemma 5

*For SCSs the WFR algorithm cannot result in the Inward Pointless L*-*shape*.

### Proof

From Lemma 3 we know that spaces E, F, G, I, J, K contain two disconnected regions. As shown in Fig 20(a), we can draw a curve outward from square A that must separate the two regions of squares (Region 1 and 2). If they are completely disconnected this contradicts Lemma 1 so they must be connected in some way along the *Connection Route*. The Connection Route implies at least three rotated *Inward L-shapes* must exist. However, theses *Inward L-shapes* must also eventually result in locally disconnected regions like that shown in Fig. 20(a). Consider any one of the *Inward L-shapes*, it must eventually separate Regions 2 and 3 as shown in Fig. 20(b). Region 2 is then completely disconnected from region 1 and 3 which contradicts Lemma 1. Therefore, the WFR algorithm cannot result in an *Inward Pointless L-shape*.

□

### Lemma 6

*For SCSs the WFR algorithm cannot result in the Inward Pointed L*-*shape*.

### Proof

According to Lemma 4 we must eventually result in two disconnected regions in a shape like E–K. Then the same logic used to prove Lemma 5 can be used to prove this lemma. □

Now we consider building a square in space 0 during wave *k* as shown in Fig. 21(a) and the 256 possible situations that can occur. Here we note that any of the spaces 4, 5, 6, and 7 that are filled at the time of construction may have been filled during wave *k* or *k*−1 and any of them not filled at this point will never be filled. Blanks in spaces 1, 2, 3, or 8 may or may not be filled during wave *k*. Any squares built in spaces 1, 2, 3, or 8 must have been built during wave *k*−1.

### Lemma 7

*At no point during the WFR algorithm for a SCS can a hole exist in the constructed structure*.

### Proof

Since the original structure does not contain holes, any hole in the structure must contain spaces that are to be filled. Any hole that is to be filled must contain a *Inward L-shape* in the top right corner. Since *Inward L-shapes* are impossible the WFR algorithm cannot result in a structure with holes. □

We can now eliminate all of the impossible situations and prove Lemma 2.

### Proof of Lemma 2

We first eliminate any situation that contains squares at both spaces 1 and 3 like shown in Fig. 21(b). These squares must have been filled during wave *k*−1 meaning these situations are *Inward L-shapes* which are impossible according to Lemmas 5 and 6. This eliminates 64 situations.

We next consider situations where prior to construction we have (at least) two disconnected regions in spaces 1–8 like the example shown in Fig. 21(c). These may be connected outside of the spaces 1–8 but if this is true then building square 0 results in a hole in the structure. This contradicts Lemma 7 and we can dismiss 115 more situations as impossible.

The next situation to consider is one with squares built in spaces 3, 5, and 7 and not at 1, see Fig. 21(d) as an example. Square 3 must have been built during wave *k*−1. If Square 5 was built during wave *k*−1 then we would have an *Inward L-shape* so it must have been built during wave *k*. Square 7 must have been built during wave *k* or wave *k*−1. If it was built at wave *k* then one or more of the spaces directly below 6, 7, and 8 must have been filled during wave *k*−1. Either way square 3 must be connected to squares at the bottom. If they are connected around space 1 then the structure contains a hole, which is impossible. If they are connected around space 5 at least two *Inward L-shapes* are required which is impossible due to Lemmas 5 and 6. We can then dismiss 16 more situations.

The final situation to consider is one with squares in spaces 1, 5, and 7 and not in 3, see Fig. 21(e) as an example. The same logic used above can be used to show that blocks on the left must somehow be connected to blocks on the right which implies that this situation is impossible. We can then dismiss 16 more situations.

The remaining 45 situations can all be built using the constructions primitives P1–P4. □

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Lindsey, Q., Mellinger, D. & Kumar, V. Construction with quadrotor teams.
*Auton Robot* **33**, 323–336 (2012). https://doi.org/10.1007/s10514-012-9305-0

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DOI: https://doi.org/10.1007/s10514-012-9305-0