110a of the first aircraft module 100a with two points of a face of the dodecahedral cage structure 110b of the second aircraft module 100b.

As shown in FIG. 6C-6D, in any of the embodiments disclosed herein, the aircraft can further include a third aircraft module 100c and a fourth aircraft module 100d. The third aircraft module 100c can include a polyhedral cage structure 110c, a propeller 120 disposed in an interior 111c of the polyhedral cage structure 110c, and a motor 130 disposed in the interior 111c of the polyhedral cage structure 110c and configured to drive the propeller 120. The fourth aircraft module 100d can include a polyhedral cage structure 110d, a propeller 120 disposed in an interior 111d of the polyhedral cage structure 110d, and a motor 130 disposed in the interior 111d of the polyhedral cage structure 110d and configured to drive the propeller 120.

As shown in FIGS. 6A-B, the modular aircraft 200 can include a fifth 100e and sixth 100f aircraft module configured similarly to the third 100c and fourth 100d aircraft modules described above.

In any of the embodiments disclosed herein, the aircraft modules can be configured such that an axis of rotation of the propeller 120 of the first aircraft module 100a and an axis of rotation of the propeller 120 of the second aircraft module 100b are parallel and an axis of rotation of the propeller 120 of the third aircraft module 100c and an axis of rotation of the propeller 120 of the fourth aircraft module 100d are parallel, and the propeller 120 of the first aircraft module 100a and the propeller 120 of the third aircraft module 100c rotate in a direction opposite that of the propeller 120 of the second aircraft module 100b and the propeller 120 of the fourth aircraft module 100d as shown in FIG. 5B, in contrast to FIG. 5A.

In any of the embodiments disclosed herein, the first 100a, second 100b, third 100c, and fourth 100d aircraft modules lie in a substantially flat configuration 200a, as in any of FIGS. 5A-6C.

As shown in FIG. 6D, in any of the embodiments disclosed herein, the propeller 120 of the first aircraft module 100a and the propeller 120 of the third aircraft module 100c rotate in a direction opposite that of the propeller 120 of the second aircraft module 100b and the propeller 120 of the fourth aircraft module 100d, wherein the first 100a, second 100b, third 100c, and fourth 100d aircraft modules lie in a tetrahedral configuration 200b.

As shown in FIG. 7, an exemplary embodiment of the present disclosure provides a method of using a modular aircraft. The method can include coupling 701 a plurality of aircraft modules to form the modular aircraft, wherein each aircraft module can include a polyhedral cage structure, a motor, and wherein coupling the plurality of aircraft modules can include attaching each polyhedral cage structure of an aircraft module in the plurality of aircraft modules to at least one polyhedral cage structure of another aircraft module in the plurality of aircraft modules.

In any of the embodiments disclosed herein, the method can further include attaching 702 a payload to the modular aircraft at a first point, controlling 703 the motor of at least one of the plurality of aircraft modules to compensate for a moment induced by the payload to maintain an orientation of the modular aircraft, and arranging 704 the plurality of aircraft modules such that control of the modular aircraft is maintained upon a failure of at least one motor of the plurality of aircraft modules.

In any of the embodiments disclosed herein, the method can further include attaching 707 a second payload to a second portion of the modular aircraft and splitting 708 the modular aircraft into two or more smaller modular aircrafts.

In any of the embodiments disclosed herein, the method can further include remotely piloting 705 the modular aircraft to a target site, and employing 706 an end effector at the target site. The payload can include the end effector, and the end effector can be selected from the group including: an RGB camera, a hyperspectral camera, an infrared camera, a gas detector, a pellet spreader, a cargo container, a passenger transport pod. a LIDAR sensor, an ultrasonic sensor, a sprayer, a grasper, and a bucket.

In any of the embodiments disclosed herein, the plurality of aircraft modules can include four aircraft modules, and coupling the plurality of aircraft modules can further include arranging the plurality of aircraft modules in a tetrahedral configuration.

In any of the embodiments disclosed herein, coupling the plurality of aircraft modules can further include arranging the plurality of aircraft modules in a flat configuration.

In any of the embodiments disclosed herein, the coupling the plurality of aircraft modules can further include arranging the plurality of aircraft modules in a configuration such that for at least six aircraft modules the respective axis of rotation of the propeller of each of the six modules is not parallel with any of the other axes of rotation of the six aircraft modules. This results in the ability to vector the thrust of the aircraft in three dimensions while generating moments in arbitrary directions by adjusting the control of the propellers of the aircraft modules.

In any of the embodiments disclosed herein, the method can further include navigating the modular aircraft in three dimensions along a path while keeping an attitude of the aircraft fixed and changing the attitude of the modular aircraft while maintaining a fixed position.

In any of the embodiments disclosed herein, the navigated path and attitude can be chosen independently.

In any of the embodiments disclosed herein, the method can further include attaching a payload to the modular aircraft wherein an orientation of the payload is fixed with respect to the aircraft and controlling the orientation of the payload with respect to a fixed reference frame by adjusting the attitude of the modular aircraft.

The following examples further illustrate aspects of the present disclosure. However, they are in no way a limitation of the teachings or disclosure of the present disclosure as set forth herein.

Examples

The following table contains notations that are used throughout this section.

TABLE 1

Notations

i; j
Set of integers from i to j

J ._TK
Matrix transpose operation

In
Identity matrix of size n × n

0_m×n
Matrix of size m × n with entries 0

1_m×n
Matrix of size m × n with entries 1

0_n
Vector of size n with entries 0

1_n
Vector of size n with entries 1

Im(.)
Image of a matrix

⊥
Orthogonal complement of a linear subspace of Rⁿequipped

with the inner product

×
Cross-product in R³

SO(3)
3D rotation group

so(3)
Set of skew-symmetric matrices of size 3 × 3

[.]×
Application from R³to so(3) such that for all u, v ∈ R³,

[u] × v = u × v

int(.)
Interior of a set

N(μ, Σ)
Multivariate normal distribution of mean μ ∈ Rⁿand covariance

Σ
∈ Rn × n

;n − 1
;n − 1

J K

Modular robotic systems employ a group of compatible autonomous components, which, through interactions between components, hardware redundancy, and reconfiguration capabilities, promise to offer increased versatility and robustness over other robotic systems. These promises and the rapid development of Unmanned Air System (UAS) technologies have triggered investigations regarding the feasibility of autonomous modular aerial vehicles.

Current applications for UAS include mapping, surveying, inspection, delivery, filming, photography, spraying, and seeding. Operators for these applications can have large fleets of different UAS with a variety of payloads that span a range of weights and dimensions.

Many current UAS require the UAS to perform translational motion and position stabilization by tilting its airframe, which makes them ill-suited for carrying/delivering payloads which need to remain at a constant orientation.

For rotorcraft systems, the most straightforward advantage of modularity is the ability to scale the number of rotors to the weight of the payload.

Modular systems could allow operators to use a unified system of compatible UAS which could be configured, manually or automatically, to adapt to specific payload and mission requirements. With self-contained modules and overactuated vehicles, redundancy of hardware ensures robustness and tolerance to failures, such as that of a rotor, flight computer, or inertial measurement unit. Besides, the use of a fleet of similar modules can also simplify maintenance and replacement schedules for UAS operators.

A potential benefit of reconfigurable modular UAS is their ability to be configured in fully actuated or omnidirectional aerial vehicles. Fully actuated aerial vehicles can independently change their thrust and torque around hovering conditions in all directions. Omnidirectional aerial vehicles are fully actuated aerial vehicles that are able to hover in any attitude.

Fully actuated aerial vehicles can perform translational motion and position stabilization without the need for tilting their airframe, but simply by changing the direction of their thrust vector. This makes them suitable for carrying payloads which need to remain at a constant orientation and for achieving a more reactive and efficient position control in adverse weather conditions than their underactuated counterparts. The ability of fully actuated aerial vehicles to remain at a fixed position while compensating for external torque and force disturbances also makes them good candidates for operating robotic. Omnidirectional aerial vehicles, since able to achieve arbitrary poses, can track six-DoF trajectories robustly. This presents an advantage for three-dimensional mapping, sensing, and object manipulation.

A crucial point of modular robotic designs is the type of docking mechanism for modules to connect. A robust and general docking mechanism allows a fleet of modules to be extended over time with other modules dedicated to specific tasks, such as sensing, actuation, or computing. For modular UAS, these added modules can be for instance battery packs, robotic arms, or electrooptical sensors. Modular rotorcraft UAS can therefore constitute the main component of a heterogeneous but compatible fleet of modules by providing the propulsion system, while being completed with specialized modules.

Many current efforts to develop modular rotorcraft UAS have focused on vehicles that form coplanar configurations, that is with vehicles assembling in a horizontal plane. Three-dimensional configurations of modular rotorcraft have seldom been explored, except for coaxial rotorcraft systems.

Despite the fact that three-dimensional configurations may incur more pronounced wake interactions between rotors, reducing their overall efficiency, they also benefit from some advantages. These are increased rigidity, more compact designs, and, if modules exhibit the appropriate symmetries, the ability to tilt them such as to create fully actuated or omnidirectional vehicles. With a careful positioning of the rotors in a three-dimensional configuration, it is possible to mitigate the effects of wake interactions.

The modular aircraft disclosed herein is a rotorcraft module designed to exploit all the aforementioned advantages. It can include a rigid unit with a regular dodecahedron frame, a fixed-pitch propeller, a slot for holding lithium-polymer batteries, and the electronics required for autonomous sensing, control of the propeller, and communication with a ground station. By adding an appropriate connection mechanism at the vertices of the dodecahedron frame, modules can be rigidly joined together. The symmetries of the regular dodecahedron allow a variety of geometrical configurations to be achieved. Coplanar assemblies are possible, but other configurations, with three-dimensional positioning and orienting of the modules, are also enabled. This opens to way to the creation of entirely new rotor configurations, while also allowing to replicate existing and proven configurations.

Described below is the modular aircraft disclosed herein in terms of module design and achievable configurations. An emphasis is placed on the versatility offered by the regular dodecahedron shape. A solution to the problem of allocating control inputs for overactuated systems of many modules is disclosed herein. Different objectives for optimal control allocation are introduced and compared, such as power consumption or control authority. Building upon the different metrics introduced for implementing control allocation strategies, a Mixed-Integer Programming (MIP) formulation is provided to describe feasible modular configurations some of their properties. This formulation is shown to be well approximated with second-order cone constraints, thereby allowing the use of solvers specialized to tackle these problems in order to study optimal configurations.

The building block of the presented modular UAS consists of a self-contained unit with a frame in the shape of a regular dodecahedron. Each module uses a fixed-pitch propeller for propulsion which can either be rotating clockwise (CW) or counter-clockwise (CCW). The propeller axis intersects two opposed vertices of the dodecahedron frame, and the propeller plane coincides with its center.

The polytope D is characterized by V^D, E^D, and F^D, which are respectively: the vertices of D, a subset of R³, the edges of D, unordered pairs of V^D, and the faces of D, unordered quintuples of V^D.

D corresponds to a regular dodecahedron centered in 0₃, of inradius 1, and with two opposed vertices on the vertical axis. For the mathematical description of modular configurations, the Modular aircraft disclosed herein can be assimilated without loss of generality to D, regardless of its actual size. The order of S^Dis 60.

H^D⊂R³is the closed convex hull of V^D.

The rotation group of D is S^D:={R∈SO(3)}|RV^D=V^D.

To form a rotorcraft vehicle capable of performing stable flight, modules can be combined together in different configurations. The set of possible configurations is determined by the two different methods of joining two modules together. Both methods consist in connecting a face of the first module frame with a face of the second module frame such that both faces are in the same plane. The difference between the two methods resides in the alignment of the two regular pentagon faces. Formally, both these connection methods can be defined as the composition between a rotation and a translation.

For R∈SO(3) and v∈R³, the isometry c_v,Rof R³is given by ∀x∈R³, C_v,R(x):=v+Rx.

With the first connection, both faces have a pair of non-adjacent vertices coinciding and are symmetrically opposed around the axis formed by these vertices. The second dodecahedron can be obtained from the first one by a single rotation of c radians around that axis or alternatively with a single translation. Because it can be achieved by connecting two opposing faces of two modules in the same orientation, this connection is given the name opposed face matching. Any rotation preserving D can be combined with this method to obtain different orientations of the resulting module. The set of all opposed face matching connections is denoted C^ofm.

For R∈S^D, v_i∈V^D, and v_j∈V^Dwith {v_i,v_j}∈/E^Dand {v_i,v_j}⊂f for some f in F^D, the opposed face matching connection c_R,{i,j}^ofmis given by c_R,{i,j}^ofm:=c_v_i_+v_j, R

The second method consists in having the two faces coinciding exactly. A translation followed by a rotation can be used to deduce the position of the second dodecahedron from the first one. The vertices of both matching faces coincide exactly. It can also be preceded by any rotation preserving D.

The set of all exact face matching connections is denoted C^efm. It is assumed that such formed connections are rigid.

For f={v_i0,v_i1,v_i2,v_i3,v_i4}∈F^D, the exact face matching connection c^efm_R,fis given by c_R,f^efm:=c_v, RR_v,π where

v
=

2
5

⁢

(

v

i
0

+

v

i
1

+

v

i
2

+

v

i
3

+

v

i
4

)

and R_v,π is the rotation of angle π around v.

Though perhaps less intuitive, the opposed face matching connection method is the most useful as it allows to form triangles and tetrahedra with respectively three and four modules where each pair of modules is connected. Such connections can be used to form a variety of structurally efficient configurations. In fact, several common rotor configurations can be recreated by using this connection method. Besides, since all connections via opposed face matching can be described with translations only, modules resulting from successions of these connections all have the same frame orientation (not accounting for rotor orientation). For these reasons, from now on, all connections are assumed to be based on the opposed face matching method without explicitly referring it.

A configuration C of size n∈N is a triple containing n positions, n orientations, and n rotor spinning directions: C={(p_i,R_i,ϵi), i∈0;n−1}. p_i∈R³is the position of module i in the vehicle coordinate frame. R_i∈SO(3) is the orientation matrix of module i, where the natural orientation for R_i=I₃is the one shown in FIG. 1. ϵ_i∈{−1,1} indicates the spinning direction of the rotor of module i, with ϵ_i=1 for clockwise a rotor and ϵ_i=−1 for counterclockwise a rotor. Without loss of generality, it can be assumed that p₀=0₃and R₀∈S^D.

For two modules i and j with positions p_i∈R³and p_j∈R³and orientations R_i∈SO(3) and R_j∈SO(3), (p_i,R_i) and (p_j,R_j) are said to be compatible if cpi,

Ri
⁢

int

H
D

⋂

C

p
j

,

R
j

int

(

H
D

)

=

∅
.

By extension, modules i and j are said to be compatible.

For p_iand p_jas above with R_i∈S^Dand R_j∈S^D, (p_i,R_i) and (p_j,R_j) are compatible if and only if

p
i

-

p
j

2

∉

int

(

H
D

)

.

By definition of S^D, cpi, R_iint^(H^D⁾=p_i+R_iint(^H^D)=p_i+ int^(H^D⁾.

The same is true when substituting i for j, such that cpi,

Ri
⁢

int
HD

⋂

c

p

j
,

R
j

⁢

int
⁡
(

H
D

)

=

∅
⇔

p
i

-

p
j

+

int

(

H
D

)

⋂

i
⁢

int

(

H
D

)

=

∅
⇔

{

p
i

-

p
j

}

⋂

2
⁢

int

(

H
D

)

=

∅
⇔

p
i

-

p
j

2

∉

int

(

H
D

)

,

where the fact that H^Dis a convex set was used.

Given a finite sequence of n connections (c_p_o_,R_o_{. . .}c_p_n−1_,R_n−1, the succession of these connections is given by C_p0,R0⊗ . . . ⊗Cp_n−1,R_n−1:=c_p0+ . . . +p_n−1,R_n−1. . . R₀.

Note that this definition is different from the composition of isometries. This is due to the fact that the translations are always done in the coordinate frame of the vehicle.

Given a set of connections C⊂C^ofm, for a module with position p and orientation R, (p,R) is said feasible with respect to C if there exists a finite sequence of connections c_i∈C, 0≤i≤n, such that c₀⊗ . . . ⊗c_n=c_p,R.

By extension, this module is said feasible with respect to C.

Given C above, a configuration C={(p_i,R_i, ϵ_i), i∈0;n−1} is feasible with respect to C if for every i∈0;n−1, (p_i,R_i) is feasible with respect to C and for every i∈0;n−1 and j∈0;n−1 with i [Image Omitted] j, (p_i,R_i) and (p_j,R_j) are compatible.

Assuming that all opposed face matching connections are allowed, the feasibility of every module can be determined by studying its position and orientation independently. Its position must be a finite sum of translation vectors as herein and its orientation must belong to SD.

In the rest of this subsection, feasible configurations and their geometries are explored in more details.

Since it has been shown that compatibility between modules and feasibility of position with respect to C^ofmdepends only on the set of translation vectors used for connections, all connections are grouped by equal translations. C_vis the subset of C^ofmdefined by C_v:={C_u,R∈C^ofm|u=v}. C⁻ is defined by C⁻:={C_u+v|{u, v}∉E^D, {u,v}⊂F^D}.

The set C is a partition of C^ofmand it has 60 elements.

The fact that C is a partition of C^ofmis immediately deduced from the above. According to these same definitions, the number of elements of C^ofmis equal to the number of faces of the regular dodecahedron, that is 12, multiplied by the number of pairs of non-adjacent vertices of a given face, that is 5, which gives 60.

C⁻ is isomorphic to the set of translations used to make connections. Therefore, in a slight abuse of notation, elements of C⁻ are sometimes directly assimilated to the vector their represent in the following. This set of vectors determines the feasible positions for modules in a configuration, defined below by taking the combinations of possible translations with integer coefficients.

The set P is defined as the additive subgroup of R³generated by the translation vectors of all possible connections: P:=^XvZ; Cv∈C⁻.

For any open subset of R³, there is an infinite number of feasible positions for a module in that subset. Fact 2.5 shows that for a given pair of connected dodecahedra, there are only 45 possible positions that a third dodecahedron can take to connect with the first one while being compatible with the second one. Among these 45 configurations, 4 also form a connection with the second dodecahedron. As more dodecahedral modules are added, the constraints represented by the requirement of not intersecting with other dodecahedra increase and make the enumeration of possible configurations difficult. P is dense in R³. For C_u∈C⁻, there are 45 elements C_vof C⁻ such that u and v are compatible. _v∈C⁻, u−2v∈/.

This can be proven by checking whether for every C int ( [Image Omitted] ).

A restricted set of possible connections can be used, such that the set of feasible positions induced by these connections is a three-dimensional lattice, therefore parameterizable by 3 integer variables, with the guarantee that different positions from that lattice are compatible. This restricted set of connections allows only one possible connection per face for a total of 12 different induced relative positions between two dodecahedra. Only 8 vertices of the dodecahedron are used for these connections, with each of these vertices involved in 3 different possible connections. These vertices form a cube and can be chosen arbitrarily as long as they satisfy this requirement, though one of the vertices must be the top or bottom one to recreate classic configurations.

Note that in practice, since modules may be rotated according to all symmetries of the regular dodecahedron, all vertices of their frame need to be able to accommodate connections with other modules. However, once the orientation of a module is fixed, only a subset of these vertices can be used to form connections.

There exists a subset V^cubeof V^Dthat defines a cube. The set of unordered pairs of V ^cubecorresponding to the edges of this cube are denoted E^cube. Any subset of V^Dverifying the same condition can be obtained from V^cubewith a rotation in S^D. For {u,v}∈E^cube, {u,v}∈/E^Dand there is a face f∈F^Dsuch that {u,v}⊂f, implying that C_u+v∈C. C^cube={C_u+v|{u,v}∈E^cube}. The vectors of C^cubeform a cuboctahedron. By choosing an appropriate basis of R³, the coordinates of the vertices of V^cubeare given by (±½,±½,±½). In this same basis, the coordinates of the vectors in {u+v,(u,v)∈E^cube} are (±1,±1,0), (±1,0,±1), and (0,±1,±1), which correspond to the vertices of a cuboctahedron.

The set of positions feasible with respect to C^cubeis defined by P^cube=X vZ; C_v∈C^cube.

P^cubeis a three-dimensional lattice for which a basis can be chosen among C^cube. Every point of p^cubecan therefore be associated with a unique triple of three coordinates and every pair of distinct coordinates corresponds to compatible positions.

P^cubeis a three-dimensional lattice of R³and a basis of it can be found among the vectors of C^cube. p^cubeis generated by (1,1,0), (1,0,1), and (0,1,1), which are three linearly independent vectors, thus forming the basis of a lattice.

For a given position of a module in a configuration, there are as many possible orientations of that module as there are elements of the symmetry group of the regular dodecahedron, which is of order 60. In practice, these 60 orientations can be partitioned in 20 groups of 3 equivalent orientations that correspond to identical orientations of the normal of the module's rotor. This partition is defined: for v∈V^D, S_vis the subset of S^Ddefined by

S
v

:=

{

R
∈

S
v

❘

R
[

0

0

1

]

T

=

v


v


.

S⁻ is defined by S⁻:={S_v|v∈V^D.

Since elements from a same group S, are equivalent in term of module's orientation, from now on, the orientation R_i∈S_vof a module i in a configuration is directly described by

η
i

=

v


v


,

the unit-vector in the direction of the thrust produced by the module in the configuration frame. A configuration C of n modules is therefore written C={(p_i,η_i,ϵ_i), i∈0;n−1}.

By further grouping the different feasible orientations by the angle formed between the rotor axis and the vertical axis, named tilt angle, six groups are obtained. The three group with angles inferior to π/2 radians, corresponding to a positive vertical thrust, are represented in FIG. 4A. The first of these three angles is equal to 0° and corresponds to a vertical orientation, and the two others are approximately equal to 41.81° and 70.53°.

Configurations with modules at different tilt angles have a lower thrust to weight ratio than configurations with parallel, upward pointed modules. Nonetheless, such configurations have a couple of advantages over standard configurations that are detailed herein.

Tilted modules generate a thrust with a horizontal component, therefore allowing the total thrust of the modular vehicle to reside in a two- or three-dimensional space. This can be used to reject force disturbances on the airframe directly with a thrust in the opposite direction, rather than changing the attitude of the vehicle to align its vertical axis against the disturbance. This ability can lead to a better stability in position and also allows the vehicle to keep its orientation constant while maintaining its position, a desirable property when carrying a payload that requires to remain in a fixed orientation, such as a camera. Payloads that operate by physical contact with the environment, such as robotic manipulators and contact probes, also benefit from configurations with at least six modules whose orientations allow the generation of decoupled three-dimensional thrusts and torques. Such configurations indeed allow to maintain a fixed position and orientation while compensating for arbitrary force and moment disturbances induced by the payload interacting with its environment.

For vehicle configurations having modules whose rotor axis does not intersect the vertical axis of the vehicle body frame, these modules can be used to generate yawing moments with their positive lever arm around that axis, instead of relying on the rotors' counteracting torques. The advantage of using lever arms for moment generation comes from the fact that lever arms increase linearly with the distance of the module to the axis of the moment of interest, whereas the magnitude of available counteracting torques remains constant irrespectively of a module's position. This can be an issue for effectively controlling the yaw of a vehicle. Indeed, when a module is the subject of a force perturbation incurring a yawing moment on the vehicle, this yawing moment also scales linearly with the lever arm of the module. This disproportionate increase between moments from force disturbances and available counteracting torques as vehicles employ more modules therefore leads to a vanishing yaw authority of coplanar configurations.

The dynamics of a configuration C can be described by considering rigid body kinematics and forces and moments induced by each module's rotor.

The body-fixed frame B, which is the frame of the configuration already used to describe feasible positions and orientations, is attached to the assembled vehicle to describe its motion with respect to an inertial frame I. The notation p_i∈R³is reused to describe the position of module i∈0;n−1. However p₀=[0 0 0]{circumflex over ( )}T, instead of having, the origin of B is chosen such that Σ_i=0ⁿ⁻1=0. Since all modules are assumed to have the same mass and have their center of gravity coinciding with the center of the dodecahedron frame, the origin of B is also the center of mass of the vehicle.

The position and orientation of the vehicle in I are given by the vector x∈R³and the attitude matrix R∈SO(3). The angular velocity of the vehicle in B is given by the vector Ω∈R³. Each module is assumed to generate a thrust and torque both proportional to the square of the rotation speed of its rotor. The total thrust and torque induced on the vehicle in B are respectively written MC,Tu and MC,τu. MC,T∈R3×n and MC,τ∈R3×n are the thrust and torque matrices. They depend on the rotors' thrust and drag coefficients, their directions of rotation, and the modules' positions and orientations. u∈U is the control vector, which corresponds to the rotors' squared rotation speeds. The admissible control set is defined by: U:={u∈Rⁿ|u_min≤u_i≤u_max,∀i∈0;n−1}, where u_minand u_maxare positive real values. Alternatively, the matrix A∈R^2n×nand vector b∈R²n are defined to represent U with U={u ∈Rⁿ|Au≤b}

The thrust generated by the rotors leads to the following translational dynamics: m_cx″=RM_C,Tu+m_CG. m_Cis the mass of the vehicle and G is the gravity vector. As to rotational dynamics, they are given by J_CΩ=−Ω×J_CΩ+τ_g+M_C,τ u, where J_Cis the inertia tensor of the vehicle in B and τ_gis a term containing the gyroscopic moments incurred by the rotation of the rotors. The kinematic equation relating the derivative of the attitude matrix to the angular velocities is R=R[Ω]×

where [.]×: R³7→so(3) denotes the cross-product from the left.

The mass mc and inertia tensor J_Cof C can be determined from the mass mM and inertia tensor J_Mof a module by adding the modules' masses and by using the parallel axis theorem:

m_c=nm_M,J_c=_Σi=0ⁿ⁻1J_M=m_M(p_i^Tp_iI₃−p_ip_i^T)

The thrust and torques matrices of C are given by:

M_C,T=k_T[η₀. . . η_n−1],

M_C,τ=k_τ[ϵ₀η₀. . . ϵ_n−1η_n−1]+k_T[p₀×η₀. . . p_n−1×η_n−1

Multiple aerodynamics effects have not been included in the dynamics of the vehicle. Typically, they are considered as disturbances which are handled by the control system, for instance with the use of the integral terms of PID control loops or by directly measuring force disturbances and using them in control algorithms.

Although no formal guarantee is usually provided regarding the stability of the control system of UAS with respect to unmodeled dynamics, it is important to know how unmodeled effects scale as the number of modules in a configuration increases to form an intuition on the behavior of large modular vehicles. If indeed, disturbance terms have a larger marginal increase as modular vehicles grow than modeled terms to which they add, then typical control systems may not be able to handle them properly. Some of these effects and how they scale with large configurations are discussed in the following.

In the definition of the control vector given in this section, it is assumed that the rotation speed of each module's rotor is reached instantly, that the thrust and torque produced by the rotor are linear in its squared rotation speed, and that there is no drag induced in forward flight. These assumptions are not exactly true, although they are used for designing control systems of most multirotors with satisfactory performance.

In practice, for each change of desired rotor speed, there is a transitional regime during which the rotor speed increases or decreases to the given setpoint. During that regime, for each rotor, a reaction torque of value −J_rω is incurred around the axis of that rotor, where J_ris the inertia of the rotor and w its rotation speed. This reaction torque is therefore expected to remain a small disturbance as more modules are added airspeed with respect to a rotor as the vehicle moves means that produced thrust and torque are not static for a constant rotor speed and that drag is also created. A formulation of rotor thrust, torque, and lateral drag has been given previously using blade element theory and blade momentum element theory. These three values' dominant terms scale quadratically with the airstream velocity, as it can be expected from standard formulations of lift and drag. It is directly seen that a problem might arise for modules far away from a configuration's center of mass when it rotates, as these will experience higher velocities than modules close to the center of mass, leading to an inaccurate modeling of the thrust and torque they create. This must be taken in account when controlling large configurations. For instance, when using a cascaded control architecture where angular velocity setpoints are used to reach a prescribed attitude, the magnitude of these setpoints should be roughly proportional to the squared inverse of the size of the configuration. The slower achieved attitude changes for large configurations are another reason to favor tilted rotors configurations, since a position control relying on changes of attitude rather than three-dimensional thrust may be unreliable.

Another aerodynamic effect that is not included in the model of the dynamics used in this work is the drag on the airframe of the modular vehicle. For multirotors, airframe drag is usually modeled to be linear, or quadratic in the velocity of the vehicle. Therefore, as long as typical angular velocities are scaled down as configurations grow in size for reasons explained in the previous paragraph, typical drag terms per module will also remain limited.

Control strategies for multirotor vehicles are usually designed such that the resulting control vector consists of a collective thrust and three torque values. The actual motor inputs are then deduced from this vector via a matrix multiplication. This remain true of different types of control architectures and objectives, for instance PID controllers for position and attitude stabilization or differential flatness-based methods for trajectory tracking. In some works, instead of outputting torques, the controller outputs angular velocities, as it assumed that a low-level high-bandwidth controller is able to track angular velocity setpoints accurately. For fully actuated multirotor vehicles, the collective thrust is replaced by a three-dimensional thrust resulting in a six-dimensional control vector.

When controlling a quad-rotorcraft, the transformation from desired collective thrust and torque to motor inputs relies on a four-by-four matrix inversion and is therefore one-to-one. So is the transformation between thrust and torque input to motor inputs for a fully-actuated hexa-rotorcraft. However, for overactuated vehicles, which large modular configurations are, there are multiple motor inputs leading to same thrust and torque values. The problem of control allocation consists in finding optimal motor inputs that yield the desired thrust and torque setpoints. This problem is crucial for configurations involving many modules and is therefore the subject of the first part of this section. Higherorder control methods that output the desired thrust and torque setpoints are not studied for modular configurations since specific methods for the type of vehicle assembled can be used.

For a configuration C of n modules, a dynamics matrix Mthat bundles the thrust and torque matrices together is defined:

M
:=

[

M

C
,
T

M

C
,
τ

]

∈

ℝ

6
×
n

For readability, the subscript C is not repeated for the dynamics matrix as there is no ambiguity.

The rank of M is written p≤6. The set of achievable thrust and torque values S is defined as the image of the admissible control set by M: S:=MU.

Assuming that U is a polytope, S is a polytope too and is included in a linear subspace of R⁶of dimension p.

In general, the problem of control allocation is a constrained minimization problem that can be formulated by: minimize f(u) subject to u∈U, Mu=s, wherefis a cost function and s∈S is a control setpoint.

A desirable objective for the control allocation problem is to minimize the total power consumption of the modular configuration.

The power induced by a single rotor is a function of multiple variables such as the angular speed of the rotor, the relative velocity of the inflow of air, and the electrical parameters of the motor. This makes the determination of the power consumption a complex problem in general. A simplified model based on momentum theory can be used to derive an ideal power consumption P for a rotor generating a thrust T. This model assumes that a rotor is a perfect actuator disk of surface area a inducing a constant velocity of the airflow along its surface. For a stationary rotor, the induced power according to this model is given by:

P
=

T
3

2
⁢
ρ
⁢
a

where p is the density of air.

Since the control inputs are proportional to the thrusts output by each module, the problem of minimizing total power consumption for a given control setpoint can be rewritten: minimize

∑

i
=
0

n
-
1

⁢

u
i

3
2

subject to u∈U and Mu=s.

Although this optimization problem does not have a closed-form solution, the objective function is convex while the constraints are linear, making it solvable in practice with efficient numerical methods such as interior-point methods.

Minimizing total power consumption is the most energy-efficient strategy at the scale of the whole configuration, but it will tend to incur larger control inputs for modules further from the center of gravity of the configuration because of the greater torque they can achieve through their larger lever arm. As modules are designed to be self-contained and to rely on their own energy supply, those modules with larger control inputs will deplete their energy source faster. For this reason, it might be beneficial to ensure that overall, modules contribute the same amount of energy to the control of the configuration. Two different objectives are suggested to express that goal. With the first objective the goal is to find the control inputs that minimize the maximum power consumption required from a module, as formulated by: minimize max

0
≤
i
≤

n
-

1
⁢

u
i

3
2

subject to u∈U and Mu=s, which is equivalent to: minimize max 0≤i≤n−1, subject to u∈U and Mu=s, and finally to minimize λ subject to u∈U, Mu=s, u₀≤λ, and u_n−1≤λ.

This minimization problem is therefore a linear program, which makes it solvable efficiently in practice.

A second objective that is suggested to achieve an overall similar power consumption between modules is to minimize the mean squared deviation of the estimated power consumption. This can by formulated by the problem: Minimize

∑

i
=
0

n
-
1

⁢

(

u
i

3
2

-

1
n

⁢

∑

i
=
0

n
-
1

⁢

u
i

3
3

)

subject to u∈U and Mu=s, which is not a convex program.

This problem can be changed to minimize the mean squared deviation of the control inputs, which in turn is a convex, second-order cone-representable, program: minimize Σ_i=0ⁿ⁻1(u_i−1/n Σ_i=0ⁿ⁻1u_i) subject to u∈U and Mu=s.

Even though the optimization problems introduced for control allocation can be numbers solved in practice, the computation time required to find a solution might still be too high for the requirements of high-frequency flight controllers, for which update rates of 400 Hz are common. This fact is particularly true of vehicles with many propellers. For this reason, it might be desirable to use a fixed allocation matrix C∈R^n×6such that MCs=s, ∀s∈S.

Such a matrix always exists since, by definition, S⊂Im(M). The constraint Ct=0, ∀t ∈S^⊥ can also be added on the matrix C to reduce the number of variables involved in searching for such a matrix. This method has the advantage of only requiring a matrix multiplication for the computation of the motor inputs u=Cs given the control setpoint s. Note however that there is no guarantee that the control input vector obtained by this method is feasible when the target s is reachable. A new set of reachable thrust and torque setpoints S^Cusing the allocation matrix C is therefore defined by S^C={s∈S|Cs∈U}.

For configurations with redundant degrees of actuation, that is when n>p, multiple choices of the matrix C exist. In the following, several optimization problems are suggested to find specific allocation matrices corresponding to different objectives that may be desired.

As just mentioned, using a fixed allocation matrix C reduces the set of achievable thrust and torque vectors and therefore the control authority of the configuration. Nonetheless, it is possible to choose C such that the resulting control authority is maximized. A way to do this is to maximize the radius of a sphere enclosed in the space of achievable thrust and torque pairs under the constraints that the resulting control inputs are feasible and that the resulting allocation matrix is a pseudo-inverse of M as defined above. Formally, it results in the following problem: maximize r subject to MC(T0+s)=T0+s, ∀s ∈S Ct=0, ∀t ∈S⊥s∈S,∥s∥2≤r=⇒AC(T0+s)≤b.

T₀is the value at which the zero-dynamics are achieved, that is the thrust and torque vector required for hovering (typically[0 mg 0 0 0]^T).

MC(T₀+s)=T₀+s and AC(T₀+s)≤b for s∈Im(M) automatically imply that T₀+s ∈S by definition of S as given in eq. (12). Therefore the constraint T₀+s∈S, ∥s∥₂≤r=⇒AC(T₀+s)≤b can be replaced by s∈Im(M), ∥s∥₂≤r=⇒AC(T₀+s)≤b.

By introducing the matrix H∈R^6×6which is the orthogonal projection on Im(M) and by noticing that sup{A_iCs|s∈Im(M),∥s∥₂≤r}=sup {A_iCHx|x∈ [Image Omitted] ⁶, ∥Hx∥₂≤r=r∥HC^TA_i^T∥₂

where A_iis the i-th row of A, this can be rewritten: maximize r subject to MCs=s, ∀s ∈S Ct=0, ∀t ∈S^⊥ r∥HC^TA^T_i∥₂b_i−A_iCT₀, ∀i∈0;n−1. This problem is similar to the one of finding the center of the sphere of maximum radius enclosed in a polytope, known as the Chebyshev center. Unlike in the classic problem, the polytope involved here is not fixed and the last inequality is non-convex. For this reason, the constraint CT₀=u^eqwith u^eq∈Rⁿis added, which is not excessively restrictive. It may indeed be desired that u^eqis preset such that all modules have the same commanded input at hover. With this relaxation and by defining

z
=

1
r

,

the problem immediately above can be transformed in a convex optimization problem: minimize z subject to MC=H, CH=C, CT0=u_eq, and ∥HC^TA^Ti∥₂≤biz− Aiu^eqz, i∈0; n−1, the fact that: MCs=s ∀s ∈S⇔MCH=H⇔MC=H and Ct=0, ∀t∈S^⊥⇔CH=C⇔CH=C is used to reformulate the equality constraints in problem (26). Problem (26) is a second-order cone program, making the use of interior points methods specific to this type of program possible.

The allocation problem for minimizing the power consumption of a configuration can be adapted to the case of a fixed allocation matrix. To do so, a random vector X˜N(T₀,Σ) is introduced, where T₀is the hovering thrust and torque vector and Σ∈S₆⁺ a covariance matrix. X corresponds to a thrust and torque vector to be transformed in a set of control inputs and is assumed to belong to Im(M), which is guaranteed by T₀∈Im(M) and Im(Σ)⊂Im(M). With this characterization of X, the requirement that X∈S is omitted and it is assumed that Σ is chosen such that most of the values taken X are actually in S.

Under these assumptions, the control input for module i is a random variable given by u_i˜N(C_iT₀, C_iΣC_i^T.

Since the power consumption of that module is proportional to |u_i|^3/2, where the absolute value is to ensure validity of the expression, the average power consumption P_iof module i is given by the absolute moment of order:

3
2

⁢

of
⁢

u
i

:

P
i

∝

E
(

❘
"[LeftBracketingBar]"

u
i

❘
"[RightBracketingBar]"

3
2

)

This absolute moment is given by the confluent hypergeometric function of the first kind, namely ₁F₁(a,b,z), with

a
=

-

3
4

,

b
=

1
2

,

and
⁢

z

=

-

(

C
i

⁢

T
0

)

2

2
⁢

C
i

⁢

∑

C
i
T

.

By once again assuming that CT₀=u^eq, the total average power P consumption of the configuration is then given by

P
∝

∑

i
-
0

n
-
1

⁢

F
1

(

-

3
4

,

1
2

,

-

u
i
eq

2
⁢

C
i

⁢

∑

C
i
T

)

.

The problem: minimize

∑

i
-
0

n
-
1

⁢

F
1

(

-

3
4

,

1
2

,

-

(

C
i

⁢

T
o

)

2

2
⁢

C
i

⁢

∑

C
i
T

)

subject to MC=H and C=CH therefore belongs to the class of convex programs.

Similarly to what was done previously to find a control input vector that yields specific thrust and torques while minimizing the maximum module power consumption, it if possible to formulate a convex program to find an allocation matrix that minimizes the maximum average power consumption: minimize λ subject to MC=H, CH=C, CT₀=u^eq, and

1

F
1

⁢

(

-

3
4

,

1
2

,

-

C
i

⁢

T
0
2

2
⁢

C
i

⁢

∑

C
i
t

)

≤
λ

,

∀i∈0; n−1

By reusing the same notations, one can formulate a problem whose objective is to minimize the average mean squared deviation of the control vector. For i∈0;n−1, the mean deviation of the control u_iis a normally distributed random variable: u_i−x u_j˜N(V_iu^eq,V_iCΣC^TV_i^T).

V_iis defined as the row vector of size n with entries equal to

-

1
n

everywhere except for the i-th entry which is equal to

n
-
1

n

.

The average squared deviation of u_iis therefore equal to:

E(|u_i−Σ_j=0ⁿ⁻1u_j|²=(V_iu^eq)²+V_iCΣC^TV_i^T.

The minimization of the average mean squared deviation of the control vector can then be formulated as the following second-order cone representable program: minimize Σ_j=0ⁿ⁻1V_iCΣC^TV_i^Tsubject to MC=H, CH=C, and CT0=ueq.

Since all the constraints in this program are equality constraints, it is possible to reduce it to a single linear equation. To see that, the above is reformulated by vectorizing the variable C: minimize vec(C^T)^T(W⊗Σ)vec(C^T) subject to: (I_n⊗H−1_6n)vec(C^T)=0; (M⊗I₆)vec(C^T)=vec(H);(I_n⊗T₀^T)vec(C^T)=u^eq, where W:=Σ_j=0ⁿ⁻1V_i^T

V
i

=

I
n

-

1
n

⁢

1

n
×
n

.