ModQuad: The Flying Modular Structure that Self-Assembles in Midair

David Saldana1∗, Bruno Gabrich1∗, Guanrui Li1, Mark Yim1, and Vijay Kumar1.

Abstract— We introduce ModQuad, a novel flying modularrobotic structure that is able to self-assemble in midair andcooperatively fly. The structure is composed by agile flying mod-ules that can easily move in a three dimensional environment.The module is based on a quadrotor platform within a cuboidframe which allows it to attach to other modules by matchingvertical faces. Using this mechanism, a ModQuad swarm isable to rapidly assemble flying structures in midair using therobot bodies as building units. In this paper, we focus on twoimportant tasks for modular flying structures. First, we proposea decentralized modular attitude controller to allow a team ofphysically connected modules to fly cooperatively. Second, wedevelop a docking method that drives pairs of structures to beattached in midair. Our method precisely aligns, and correctsmotion errors during the docking process. In our experiments,we tested and analyzed the performance of the cooperativeflying method for multiple configurations. We also tested thedocking method with successful results.

I. INTRODUCTION

In biological systems such as ant or bee colonies, collec-tive effort can solve difficult problems such as exploring,transporting food and building massive structures. Someant species are able to build living bridges by clinging toone another spanning the gaps in the foraging trail. Thiscapability allows them to rapidly connect disjoint areas inorder to transport food and resources to their colonies.

Recent works in robotics have been focusing on applyingswarm behaviors to solve collective tasks such as construc-tion and transportation. There are two main approaches toassemble structures using autonomous mobile robots. In thefirst approach, robots transport building units to assemblestructures. In [1] is presented an algorithm to control a swarmof robotic termites that can transport, assemble and climbover building blocks. A quadrotor swarm can also be used totransport structure parts, like bricks and beams with magnets,in order to assemble three dimensional structures [2], [3]. Inthe second approach, robots use their own bodies as buildingunits [4], [5], [6]. A cubic block module was introducedin [7] which is able to attach and to detach other modules

* Equal contributors1 D. Saldana, B. Gabrich, G. Li, M. Yim and V. Kumar are

with the GRASP Laboratory, University of Pennsylvania, Philadel-phia, PA, USA: dsaldana, brunot, lguanrui, yim,kumar@seas.upenn.edu.

The authors acknowledge the help and support of Michael Whitzer duringthe intial state of the project. We also acknowledge Justin Thomas for hisintroduction to quadrotor control and multi-UAV systems.

The authors gratefully acknowledge the support of the Brazilian agencyCAPES. The support of DARPA grant HR00111520020, ONR grantsN00014-15-1-2115 and N00014-14-1-0510, ARL grant W911NF-08-2-0004, NSF grant IIS-1426840, NSF grant 1138847 and TerraSwarm, oneof six centers of STARnet, a Semiconductor Research Corporation programsponsored by MARCO and DARPA.

Fig. 1. Flying Modular Structures. A square shape with four modules onthe left and a line shape with three modules on the right.

forming three dimensional structures. A self-assemble andself-reconfigurable robot system was developed in [8], whererobots are coordinated at the same time to dock to formplanar structures. Recent algorithms have been focusing onassembling large scale structures using hundreds of robots[8], [9]. Modular boats with rectangular [10] and squareshapes [6] have been proposed to self-assemble and to self-reconfigure planar structures on the water. These systemsrequire algorithms that avoid deadlocks during the assemblyprocess. In recent work, we speed up the assembly processby parallelizing docking actions [11].

In [12] is proposed a scalable multi-rotor aircraft whichcan be manually extended depending on the payload re-quirement. Oung et al. [13] present a reconfigurable aerialplatform based on wheeled hexagonal modules. Each modulecontains a single propeller that is not able to fly by itself,but it can move on the ground to dock other modules andcooperatively fly. This modular system can also increase itspayload capability by scaling the number of docked modules.In addition, it can self-assemble on the ground using omni-directional wheels. A cooperative multi-quadrotor system fortransportation purposes is presented in [14]. The authors usemultiple quadrotors forming different configurations to graspand transport different shaped objects.

Flying structures that can self-assemble in midair has notbeen shown in the literature. This paper introduces Mod-Quad, the self-assembly structure that can cooperatively flybased on autonomous modules (See Figure 1). Using a largenumber of robots, it is possible to assemble structures suchas bridges, rectangular platforms, among others. In contrastto related works, instead of assembling on the ground or onthe water, we propose a faster way to assemble structures.Docking modules in midair offers a relevant advantage interms of speed during the assembly process. The individual

modules are small and agile so they can rapidly movethrough environments with obstacles. There are scenarioswhere the time-response is crucial to save human lives.For example, in a burning building, the individual modulescan rapidly navigate through cluttered environments from abase-station to the target building. Then, they can assemblebridges, external staircases or platforms near windows tooffer alternative exits.

The contributions of this paper are threefold. i) We in-troduce ModQuad, a novel flying modular robotic structurethat is able to self-assemble in midair and cooperativelyfly. ii) ModQuad is the first modular system that is able toself-assemble in midair. We present a docking method thataccurately aligns and attaches pairs of flying structures inmidair. iii) We present a stable decentralized modular attitudecontroller to allow a set of attached modules to cooperativelyfly. Our controller generates the required moments minimiz-ing motor saturation.

II. MODQUAD DESIGN

Our design is focused on developing a modular robot withthe ability to dock in midair. In this way, we extend thecapabilities of the regular aerial flying vehicles. Our modularrobot, ModQuad, is propelled by a quadrotor within a light-weight cuboid frame with a passive docking mechanism. Itsmain components are described as follows.

A. Flying Vehicle

The ModQuad is propelled by a quadrotor platform. Inthis case, we use the Crazyflie 2.0. This robot has been usedin swarm applications with large numbers of robots [15].It is open-source and open-hardware. The vehicle weighs27g, having a maximum payload of 15g. Its dimensions are92x92x29mm and its battery lasts around five minutes. Itslow-cost and total payload gives an acceptable scenario fora large number of modules.

B. Modular Frame

The quadrotor is enclosed and attached to a cuboid frameas it can be seen in Figure 2. The dimensions of thiscuboid are 116x116x48mm. Light-weight carbon fiber rodsconnected by eight 3-D printed ABS connectors form theframe. The frame weight is important due to tight payloadconstraints. Our current frame design weights 7g , about halfthe payload capability.

C. Docking Mechanism

To enable rigid attachments between modules, we in-clude a docking mechanism in the modular frame. We usedNeodymium Iron Boron (NdFeB) magnets as passive actua-tors due to their large strength-to-weight ratio. The magnetsare redsquares square, 6.35x6.35x0.79mm. Two magnets arelocated at each of the eight 3-D printed connectors in theframe (See Figure 2). These magnets enable our modulesto have connections on the four vertical faces. When twomodules are connected face-to-face, the four magnets areable to provide a bonding force equivalent to a 1kg, which

magnet -

modular frame

quadrotor

magnet +

magnet +

magnet -

3-D printed connector

Fig. 2. A Flying Modular Robot. It is based on a quadrotor within acuboid frame. It is equipped with a docking mechanism, made up of squarepermanent magnets located in the corners of the frame. This enables themodules to rigidly attach to each other.

is a strong force in comparison to the module mass (40g)The strength-to-weight ratio is approximately 25:1. Althougha module has six faces, we focus on planar structuresonly, where vertical faces perform horizontal dockings. Onceattached, undocking is a difficult task and is left to futurework.

III. MODQUAD MODEL

In this section, we describe a general model for ModQuadand state the two main problems that we solve in this paper.This system is able to assemble structures from modulesdefined as follows.

Definition 1 (Module). A module is a flying robot thatcan move by itself in a three dimensional environment andhorizontally dock to other modules.

This module is based on a quadrotor platform within acuboid modular frame. The lower and upper faces of thecuboid have an area w×w and the frame has a height h. Adocking mechanism allows the module to horizontally attachto other modules. The modular robot has a mass m, includingthe quadrotor, the frame and the docking mechanism.

We consider a team of N modules, which are indexedby the set M = 1, ..., N. All modules are homogeneous,including shape, mass, inertia, and actuators. We define a setof connected modules as a structure.

Definition 2 (Structure). A flying structure, S ⊆ M,is a non-empty set of rigidly connected modular robotsthat behaves as a single rigid body. These modules arehorizontally connected by docking along the sides so theresulting shape has the same height h.

A. Coordinate frames

We set three different coordinate frames to define themodule and the structure pose:

1) The world coordinate frame, W : or inertial frame isfixed and has its z-axis pointing upwards. We denote thelocation of the center of mass of the ith module in the worldframe W by xi ∈ R3. The robot attitude is represented bythe Euler angles Θi = [φi, θi, ψi]

> for roll φi, pitch θi, and

φ1 roll

θ1 pitch

ψ1 yaw

f21 f24 f11 f14

f22 f23 f12 f13

Fig. 3. Representation of an structure with two modules. The black arrowsrepresent the rotor forces, and the curved gray arrows represent the rotormoments. The blue, green, and red axis represent the world, module andstructure coordinate frames respectively.

yaw ψi. We can see the attitude angles with respect to theworld frame for module 1 in Figure 3.

2) The module coordinate frame, Ri: is defined foreach robot. The origin is attached to the center of massand the x-axis is aligned to the front of the module. Theangular velocities in the module frame are denoted by Ωi =[pi, qi, ri]

>.3) The structure coordinate frame, S: is defined for a set

of attached modules S. The origin is attached to the center ofmass of the structure. We assume that all modules point in thesame direction. Thus, the x-axis of the structure coordinateframe is parallel to the x-axis of all module coordinateframes in the structure. Figure 3 illustrates two modules andtheir associated coordinate frames.

We assume that the modules have approximately the sameyaw orientation ψ1 = ... = ψN . Without loss of generality,we assume that this orientation is ψi = 0, for all i ∈M.

B. Robot actuators

Each module i has four vertical rotors in a square config-uration, indexed by j = 1, ..., 4, each with an angular speedωij that generates vertical forces

fij = kF ω2ij ,

and momentsMij = ±kM ω2

ij ,

where kF and kM are motor constants that can be obtainedexperimentally. The sign of the moment depends on thedirection that the motor spins. It is either positive forcounterclockwise or negative for clockwise.

The coordinates of the rotors 1 to 4 in the module frameRi are (d,−d, 0), (−d,−d, 0), (−d, d, 0) , and (d, d, 0)respectively, where d is the distance from the rotor to therespective x- or y-axis of the module coordinate frame.

C. Robot sensors

Each module i is able to measure its pose, its location inthe world coordinate frame xi, its angles for roll φi, pitch θi,and yaw ψi, and its respective angular velocities pi, qi, and

ri. We identify if a module i is connected to another modulej using the location sensor. We check if ‖xi − xj‖ = w issatisfied, where ‖·‖ denotes the Euclidean norm.

D. Dynamics of the modular rigid body

A set of connected modules – structure S – forms a singlerigid body. We denote the number of robots in the structure Sby n = |S| (|S| is the cardinality of S). All modules in thestructure share the same plane and have the same orientation.We denote the location of the ith module in the structurecoordinate frame by (xi, yi, zi).

The thrust and attitude of the flying structure depends onthe forces and moments produced by each rotor. The totalthrust F and the roll, pitch and yaw moments, denoted byMx, My, and Mz respectively, are computed as the result ofall the rotor forces in the structure

FMx

My

Mz

=∑i

1 1 1 1yi1 yi2 yi3 yi4−xi1 −xi2 −xi3 −xi4kMkF

−kMkFkMkF

−kMkF

fi1fi2fi3fi4

(1)

where (xij , yij) denotes the location of the rotor j = 1, ..., 4that belongs to the ith module with respect to S. In this dy-namical system, we can control the force of each individualactuator by the input vector

[fi1, fi2, fi3, fi4]> = ui. (2)

The resultant force and moments generate translational androtational accelerations.

nm xS =

00

−nmg

+ RWS

00∑ij fij

,where g is the gravity constant, RW

S ∈ R3×3 is the rotationmatrix that transforms from the structure coordinate frame Sto the world coordinate frame W .

We assume that the each module is symmetric andits inertia tensor is a diagonal matrix, denoted by I =Diag(Ix, Iy, Iz). We can describe the rotational accelerationsusing the linearized model

IS

pqr

=

Mx

My

Mz

, (3)

where IS is the mass moment of inertia of the structure S.We can compute the structure inertia, denoted by IS , based

on the inertia matrix of a single module I, using the parallelaxis theorem

IS = nI +m

∑i y2i 0 0

0∑i x

2i 0

0 0∑i x

2i + y2i

. (4)

E. Objective

In this paper, we want to control two of the main actionsthat the team of modular robots can perform. These actionsare cooperative flying and docking in midair.

Problem 1. Given a desired attitude Θ∗ and thrust F fora flying structure S, the problem is to find a control inputui, for all i ∈ S, such that the modular flying structure isdriven to the attitude Θ∗ while generating the thrust F .

Problem 2. Given a pair of structures S1 and S2, find thecontrol inputs ui, for all i ∈ S1 and uj for all j ∈ S2, suchthat the structure S1 docks to the structure S2 in the threedimensional space.

We approach Problem 1 and 2 in Section IV and Vrespectively.

IV. CONTROL OF THE FLYING MODULAR STRUCTURE

In order to allow the flying structure to navigate in a threedimensional environment, we control thrust and attitude togenerate vertical and horizontal translations, and rotation inthe yaw angle. We assume that the robots know the shapeof the structure as well as their location xSi = [xi, yi, zi]

>

in the structure coordinate frame S.The total thrust can be computed by dividing the total

thrust F among all rotors. The contribution for each rotor isgiven by

fij =F

4n.

The controller first generates moments for the structure, thenwe formulate a modular attitude controller to satisfy thosemoments.

A. Controlling moments

We want to generate a desired moment M = [Mx, My ,Mz]

> on the structure S using the rotor forces fij . Toachieve this, each rotor ij can contribute to the total momentof the structure with Mij = [Mxij

,Myij ,Mzij ]>. The local

moments Mij must satisfy

M =∑ij

Mij .

Since the system is redundant, this equation has infinitesolutions. We might distribute the total moment evenlyamong all actuators, but it would overload the robots thatare closer to the center of mass because a high force wouldbe required given the small moment arm. In [14], the authorscompute the pseudo inverse of the dynamics. Although thisapproach minimizes the squared sum of the local momentsand forces, it overloads some of the actuators. This isespecially problematic when there are many small rotors thatcan easily saturate. In a similar way, the authors in [13]distribute the forces linearly with respect to the center ofmass, overloading and saturating the rotors that are far fromthe center of rotation. Additionally, the batteries of thosemodules drain faster than the rotors in the middle. The mainproblem in these cooperative systems is that if one module

fails, the whole structure fails. For these reasons, we want todevelop a method that given thrust and moments [F,M]>,it minimizes the maximum rotor force as:

u∗S = argminuS

‖uS‖∞, (5)

subject to (1), where uS is the vector of all control inputsin the structure and ‖·‖∞ denotes the infinity norm, alsoknown as Chebyshev norm, which returns the maximumabsolute value in the input vector. In this way, it is possibleto generate the maximum performance of the system withoutsaturating the motors. A solution that satisfies the structuresdynamics (1) is

fij =F

4n+Mxij

yij+Myij

xij+ (−1)j+1 kF

kMMzij . (6)

The solution for (5) evenly distributes the forces among allrotors. To achieve this, we proportionally distribute the totalmoments Mx and My based on the distance of the actuatorto the respective axis. Then, we have

Mxij =|yij |∑ij |yij |

Mx, (7)

Myij =|xij |∑ij |xij |

My. (8)

In the case of Mz , all the rotors are in the same plane. Thenwe can evenly distribute the yaw moment among all rotorsas

Mzij =Mz

4n. (9)

Substituting (7), (8), and (9), in (6), we obtain

fij =F

4n+χ(yij)Mx∑

ij |yij |+χ(xij)My∑

ij |xij |+ (−1)j+1 kFMz

nkM,

where χ denotes the signum function χ(x) := x/|x|. Usingthe location of the rotor in the robot coordinate frame, wecan define the constants of the structure

Cx =∑ij

|yij |= 2∑i

|yi + d|+|yi − d|

Cy =∑ij

|xij |= 2∑i

|xi + d|+|xi − d|

Cz =4nkMkF

.

Then, we can write the control input for robot i, in a compactform

ui = Pi C

[FM

], (10)

where the matrix

Pi =

1 χ(yi − d) χ(xi + d) 11 χ(yi − d) χ(xi − d) −11 χ(yi + d) χ(xi − d) 11 χ(yi + d) χ(xi + d) −1

has a form 1,−14×4 that depends on the locationof the ith module (xi, yi) in the structure coordinateframe and the distance d. The structure matrix C =

Diag([1/4n,C−1x , C−1y , C−1z ]>) contains the constants of thestructure, which is the same for all robots.

We can see in (10) that the rotor forces only depend on thequadrant in the structure coordinate frame. In approaches like[13] and [14], the rotors with higher |xij | and |yij | saturatefirst without maximizing the use of the other rotors. In ourapproach, the rotors in the same quadrant generate the sameforce. Therefore, instead of saturating a few rotors, all rotorsin the quadrant saturate at the same time. This approach isvery efficient for low-cost quadrotors where the force interval[fmin, fmax] is narrow. However, this approach is not energyefficient and drain the batteries faster than [13] and [14].

B. Decentralized Modular Attitude control

We want to drive a structure S to a desired attitude Θ∗ =[φ∗, θ∗, ψ∗]> and thrust F . In our decentralized setup, thecentral trajectory planner sends Θ∗ and F to the structure.Then, each module i ∈ S has to independently computeits control input ui based on its location in the structurecoordinate frame and its local inertial sensor. We use aproportional-derivative controller for a single quadrotor:

φ∗

= Kp,φ (φ∗ − φ) +Kd,φ (p

∗ − p),θ∗

= Kp,θ (θ∗ − θ) +Kd,θ (q

∗ − q),ψ∗

= Kp,ψ (ψ∗ − ψ) +Kd,ψ (r∗ − r),

where Kp,·Kd,· > 0 are gain constants for a single robot.We can write this in a compact form,

Ω∗= Kp(Θ

∗ −Θ) + Kd(Ω∗ −Ω) (11)

where Kp and Kd are diagonal matrices that include thegain constants. In this controller, we can set the desiredangular velocities to zero Ω∗ = 0. Without re-tuning thegain constants, we want to be able to fly the structure for anyconfiguration and any number of robots. Since all modulesin a structure are on the same plane and pointing towardsthe same direction, each robot is able to use its local sensorto estimate the attitude Θ and angular velocities Ω of thewhole structure S. In this way, we can say that Θ1 = ... =Θn = Θ and Ω1 = ... = Ωn = Ω. Thus, each modulei ∈ S is able to independently compute the desired angularaccelerations Ω

∗using its local measurements Θi,Ωi in

(11). Maintaining the same angular desired accelerations ofa single robot for the multi-robot structure is hard to achievebecause inertia grows rapidly with the number of robots(from (4)). Additionally, agility is reduced with the size ofthe rigid body as it is studied in [16], [17]. Therefore, wecan define the desired angular acceleration of the structureΩ∗S as a function of the desired acceleration of a single

module Ω∗. We propose a function that increases the angular

accelerations proportionally to the location of the rotors and

inversely proportional to the inertia of the structure,

pS =IxISx

Cx4dp

qS =IyISy

Cy4dq

rS =IzISzCz r.

In this equation, Cx and Cy are related to the locations of allrobots in the structure, while 4d is related to the distance ofthe four rotors in a single module. We can rewrite the newdesired acceleration of the structure S in a compact form

Ω∗S = I−1S IDΩ

∗(12)

where the matrix D = Diag([Cx/4d, Cy/4d, Cz]>) con-tains the constants of the structure. From (10), the controlinput that satisfies the new desired angular accelerations isdescribed by

ui = Pi C

[F

ISΩ∗S

].

Using (12), we can write the control input as a functionof the desired angular acceleration of a single module Ω

∗:

ui = Pi E

[F

IΩ∗

], (13)

where the matrix E reformulates C to satisfy the new angularaccelerations as E = Diag([1/4n, 4d, 4d, 4]>). We highlightthat in our modular attitude controller (from (13)), it is notnecessary to tune the attitude gains for each configuration ofthe structure. As a summary, given a control input [F,Θ∗]>,we use (11) to obtain the desired acceleration of a singlerobot Ω

∗. We use the computed Ω

∗and the desired input

thrust F to obtain the desired input ui based on (13). When anew structure is assembled, each module only needs to knowthe number of robots in the structure n and its location inthe structure to compute Pi.

V. HOVERING AND DOCKING IN MIDAIR

In this section, we describe our method to dock a struc-ture S1 to a structure S2, resulting a new structure S3 =S1 ∪ S2. In order to achieve this goal, we initially describe

how to control the linear velocity of the structure using themodular attitude controller from previous section. Then, weuse the velocity controller to hover in a desired waypointand to dock pair of structures in midair.

A. Controlling linear velocity

We can control a structure S as a single quadrotor [14],[18]. All robots have the same desired yaw angle andwithout loss of generality, we assume that this angle iszero. The desired linear acceleration x∗S = [x∗S , y

∗S , z∗S ]> is

transformed to the attitude controller [F,Θ∗]> using

[FΘ∗

]=

mg +m z∗S−y∗S/gx∗S/g0

. (14)

Robot 1

IMU 1

1/n

Vel. Sensor 1

+

+

x∗S

...

x∗

Eq. (11) Eq. (13)

Robot n

IMU n Vel. Sensor n

-+

Module 1

Module n

u1

un

Eq.(14)

Eq.(15)

xS

x1

xnEq. (11) Eq. (13)

Θ,Ω

Θ,Ω

Θ1,Ω1

Θn,Ωn

Fig. 4. Block diagram that illustrates the interaction between the velocitycontroller and the distributed modular attitude controller for the flyingstructure.

We can move the robot with a desired velocity x∗S using

x∗S = Ka(x∗S − xS). (15)

Then, we can control the linear velocities by considering thedesired velocity as the control input

x∗S = wS . (16)

The velocity controller and its interaction with the modularattitude controller is illustrated in Figure 4. It starts with thevelocity controller, from (16). Its output x∗S is transformedinto a desired attitude [F,Θ∗]> for the whole structure using(14). In a decentralized manner, each module i receivesthe desired attitude and combines it with feedback Θi,Ωi

from its inertial measurement unit (IMU) based on (11).Using the desired angular acceleration of the structure andthe desired thrust, [F, Ω]>, each module computes its owncontrol input ui using (13). In order to close the loop for thevelocity controller, we compute the velocity of the structureby averaging the velocity of all modules.

Now, we proceed to dock pairs of flying structures inmidair. In our approach, one flying structure waits in ahovering action and the other structure performs the dockingaction. It reduces the motion errors and disturbances from thehovering structure, so the other structure can precisely alignand dock. Both the hovering and the docking actions use thevelocity controller as described in the following subsections.

B. Hovering

In the hovering state, we want to maintain the roboticstructure in a desired point x∗. We can drive it by using thevelocity controller, from (15), as

wS = Kv(x∗S − xS). (17)

C. Docking action

In order to dock structure S1 to S2, we take as reference apair of modules (i, j), such that i ∈ S1 and j ∈ S2. Assum-ing that module i docks to module j, and it docks throughthe x-axis of Rj . The location of module i in the coordinateframe of module j, Rj , is denoted by (x

(j)i , y

(j)i , z

(j)i ).

Our docking method is based on a gradient function ∇f ,which is followed by the velocity controller as

wS = ∇f(x(j)i , y

(j)i , z

(j)i ).

0.0 0.1 0.2 0.3 0.4 0.5

0.2

z ‖∇f‖

x

0.1

0.0

0.1

0.2

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

Fig. 5. Gradient function in the xz−plane. It is symetric and depicts thesame for the yz−plane.The arrows represent the direction of the gradienton a given point and the background color represents the magnitude of thegradient.

Our docking function defines what we call approachingtunnel, which is a cylinder with radius r aligned with thex−axis of Rj . The radius r is related to the motion error thatis tolerable during docking, e.g. a small misalignment error(< 25mm) is fixed by the attraction force of the magneticdocking mechanism. If the module i is inside the tunnel,it will move towards module j with constant velocity v.Otherwise, meaning that module i is not aligned to module j,the module will rapidly be attracted towards the tunnel beforeapproaching to the other module. If the module is outsidethe tunnel and close to the yz-plane, the gradient will pushit backwards to correct the alignment before approaching.

Our gradient function for docking is given by

∇f(x, y, z) = −

g(x)x/|x|k1 yk2 z

,where the rejection/approaching tunnel function is

g(x, y, z) =

−k3 e−k4|x| if y2 + z2 ≥ r,

v otherwise.(18)

Using this function, the module in any arbitrary locationwill be driven towards the x-axis of Rj (by the proportionalcontroller in the y− and z−axis). When the module is closeto the x−axis, with a distance less than r, within the tunnel,it will be approaching the origin with constant speed v.When the module is outside the tunnel, it is rejected of theyz−plane. Using the method, we guarantee that the moduleis aligned for the docking action, since the module canonly approach to the origin if it is within the tunnel. Wedepict the gradient function ∇f(x, y, z) for the xz−plane inFigure 5. Since the function is symmetric, the vector field inthe xy−plane is the same as the vector field in the xz−plane.We can see that the only way to approach to the zero is by thepurple region, where the module slowly moves and maintainsthe alignment during the docking action. Theoretically, thisgradient-following approach guarantees a successful dockingaction, since the function f has a single minimum and theonly way to reach it is by a proper alignment.

VI. EXPERIMENTS

We performed three different experiments to study: the dy-namics performance of the modular system; the behavior ofthe docking action; and the interaction of the flying controllerand the docking method for multiple configurations.

In our experimental testbed, we used the Crazyflie-ROSnode [19] to control the robots, and to determine the robotpose in space as well as relative locations for docking. Weare using a motion capture system (VICON) operating at100 Hz. We are able to measure the angular velocities of therobots using their IMU sensors. On the computer, we runthe velocity (15) and attitude (14) controllers as ROS nodes.The same attitude command is broadcasted to all robots inthe structure via 2.4GHz radio. The original firmware ofthe Crazyflie robot was modified to implement the forcedistribution from (13). The PD controller from (11) is alreadyimplemented in the firmware.

A. Dynamics performance

In this experiment, we evaluate the dynamics performancefor multiple modules in a line configuration. We used ourmodular attitude controller sharing same gains for all mod-ules. The modules are distributed along the x-axis of thestructure in such a way that the x−axis of the structure andx−axis of the modules are aligned. In our setup, we attachedthe robotic structure to a mechanical system that constrainsthe system motion allowing only rotation in the pitch angle.We tested the angular velocity in pitch θ for n = 1, 3, 5, 7modules using the same gains. We set the desired pitch angleto zero, θ∗ = 0, and an initial angle θ = π/4. From theinitial angle we released the structure allowing it to achieveits desired angle. Figure 6(a) shows the angular velocitiesof the structure meanwhile it drives its error to zero. It ispossible to observe how the time response is reduced withrespect to the number of modules. Using the obtained data,we estimated the angular accelerations of the structure. Basedon (12), we can obtain that the angular acceleration in theline configuration is reduced with order O(1/n), since theinertia grows cubically and Cy grows quadratically in theline configuration. It matches the results of our experiment,since the maximum estimated angular accelerations weremax(q) = 38.93, 16.52, 9.61, 8.55 (rad/s2) for n =1, 3, 5, 7 respectively. A similar test were conducted for theyaw angle case (see Figure 6(b)). In comparison to the pitchcase, we can see that the increment of modules generates ahigher impact in the reduction of the angular velocity. Theangular acceleration in yaw is reduced with order O(1/n2)(see (12)) because the constant Cz for the yaw angle growsslower than Cx. For this reason, the yaw angle is the mostaffected when the number of modules in the structure isincreased.

We can see that all robots try to converge to the steadystate, but the angular acceleration is reduced when we in-crease the number of robots. A small number of modules arevery agile and converge to zero in a faster way. In contrast,a large number of robots present a smoother behavior.

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Fig. 6. Angular velocities in the y− and z−axis for n = 1, 3, 5, 7robots. The modules are arranged in a line configuration. In this experimentthe robot structure is released from an angle φ = π/4 and (13) drives it toφ = 0.

B. Docking action

In our second experiment, we want to test the behaviorof the docking action from multiple initial points. We seta static and a moving module. The module, that performsthe docking action, follows the gradient function, from (18),with constants k1 = 3.0, k2 = 3.0, k3 = 1.0, k4 = 10.0 andv = 0.1. We present multiple trajectories that the moduleperforms during the docking action from different initiallocations (see Figure 7). The docking action is successfullycompleted even in the presence of motion errors. We can seethat the module follows the gradient and always executes theapproaching procedure satisfying the alignment requirement.We can also see that the robots are overshooting on thedesired z and have to perform an additional curve. However,this behavior and motion errors are always compensated bythe docking function. In the experiments we observed thatthe magnetic field created by the docking mechanism wasable to compensate and correct misalignments between twostructures generating successful docking even with motionerrors.

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Fig. 7. Robot trajectories during the docking action. The initial locationsof the the docking robot are represented by the blue rectangle. This robotmoves towards a static module (red rectangle) and docks where the dashedgreen rectangle is located.

C. Assembling Structures

In our third experiment, we want to test the integrationof the flying modular attitude controller and the dockingmethod. We performed the docking experiment for multi-ple robots and multiple configurations. We implemented aROS-node that automatically detects when two or multiplerobots are attached by using the robot locations from theVICON System. For each detected structure, the matrix Pi

in (13) is uploaded to every module in the structure usingradio communication. Once all modules receive the updatedparameters, the structure recovers its stability.

In general, we tested our controller for structures withmultiple configurations. The angular acceleration in rolland pitch angles is reduced when the number of robots isincreased but they are still stable. However, in the case of theyaw angle, due to the low-powerful motors of the Crazyflie,we experimented motor saturations. For this reason, the gainsin yaw should be maintained as low as possible. Althoughthe controller maintains the modules stable, the scalability ofthe system is mainly constrained to the yaw moment that therotors can offer without saturation. We also observed that theshear stress between connected modules was not an issue dueto the high strength-to-weight ratio the docking mechanismprovides.

VII. CONCLUSIONS AND FUTURE WORK

This paper introduces ModQuad, a novel flying modularrobotic structure that is able to self-assemble in midair andcooperatively fly. One of the main challenges was developinga modular attitude controller in which each module has touse its own inertia sensor to compute its local control input.We implemented this controller in low-cost robots. Althoughour approach is not energy efficient, it allows us to use low-cost motors with a low maximum thrust. In our approach, thegain constants in our controller do not need to be re-tunedas the configurations change, but the angular acceleration isreduced with the addition of modules.

Using a velocity controller, we are able to dock multiplerobots in midair in order to assemble arbitrary structures.The docking system and control has been validated throughmultiple experiments.

Future work includes examining how the system scales asmore modules are attached and rigorously determining theeffects of motors saturation on the performance and stabilityof the structure. In addition, methods and mechanisms forundocking will be explored.

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