arXiv:1403.5986v3  [cs.SY]  4 Feb 2015
1
Controllability Analysis for Multirotor Helicopter Rotor Degradation and Failure
Guang-Xun Du, Quan Quan, Binxian Yang, Kai-Yuan Cai
NOMENCLATURE
h
=
altitude of the helicopter, m
φ, θ, ψ
=
roll, pitch and yaw angles of the helicopter, rad
vh
=
vertical velocity of the helicopter, m/s
p, q, r
=
roll, pitch and yaw angular velocities of the helicopter, rad/s
T
=
total thrust of the helicopter, N
L, M, N
=
airframe roll, pitch and yaw torque of the helicopter, N·m
ma
=
mass of the helicopter, kg
g
=
acceleration of gravity, kg·m/s2
Jx, Jy, Jz
=
moment of inertia around the roll, pitch and yaw axes of the
helicopter frame, kg·m2
fi
=
lift of the i-th rotor, N
Ki
=
maximum lift of the i-th rotor, N
ηi
=
efficiency parameter of the i-th rotor
ri
=
distance from the center of the i-th rotor to the center of mass, m
m
=
number of rotors
kµ
=
ratio between the reactive torque and the lift of the rotors
I. INTRODUCTION
Multirotor helicopters [1], [2], [3] are attracting increasing attention in recent years because of their
important contribution and cost effective application in several tasks such as surveillance, search and
The
authors
are
with
Department
of
Automatic
Control,
Beihang
University,
Beijing
100191,
China
(dgx@asee.buaa.edu.cn; qq buaa@buaa.edu.cn; yangbinxian@asee.buaa.edu.cn; kycai@buaa.edu.cn)
February 5, 2015
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rescue missions and so on. However, there exists a potential risk to civil safety if a mutirotor aircraft
crashes, especially in an urban area. Therefore, it is of great importance to consider the flight safety
of multirotor helicopters in the presence of rotor faults or failures [4].
Fault-Tolerant Control (FTC) [5] has the potential to improve the safety and reliability of multirotor
helicopters. FTC is the ability of a controlled system to maintain or gracefully degrade control
objectives despite the occurrence of a fault [6]. There are many applications in which fault tolerance
may be achieved by using adaptive control, reliable control, or reconfigurable control strategies [7],
[8]. Some strategies involve explicit fault diagnosis, and some do not. The reader is referred to a
recent survey paper [9] for an outline of the state of art in the field of FTC. However, only few
attempts are known that focus on the fundamental FTC property analysis, one of which is defined
as the (control) reconfigurability [6]. A faulty multirotor system with inadequate reconfigurability
cannot be made to effectively tolerate faults regardless of the feedback control strategy used [10].
The control reconfigurability can be analyzed from the intrinsic and performance-based perspectives.
The aim of this Note is to analyze the control reconfigurability for multirotor systems (4-, 6- and
8-rotor helicopters, etc.) from the controllability analysis point of view.
Classical controllability theories of linear systems are not sufficient to test the controllability of
the considered multirotor helicopters, as the rotors can only provide unidirectional lift (upward or
downward) in practice. In our previous work [11], it was shown that a hexacopter with the standard
symmetrical configuration is uncontrollable if one rotor fails, though the controllability matrix of
the hexacopter is row full rank. Thus, the reconfigurability based on the controllability Gramian
[10] is no longer applicable. Brammer in [12] proposed a necessary and sufficient condition for the
controllability of linear autonomous systems with positive constraint, which can be used to analyze
the controllability of multirotor systems. However, the theorems in [12] are not easy to use in practice.
Owing to this, the controllability of a given system is reduced to those of its subsystems with real
eigenvalues based on the Jordan canonical form in [13]. However, appropriate stable algorithms to
compute Jordan real canonical form should be used to avoid ill-conditioned calculations. Moreover,
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a step-by-step controllability test procedure is not given. To address these problems, in this Note the
theory proposed in [12] is extended and a new necessary and sufficient condition of controllability
is derived for the considered multirotor systems.
Nowadays, larger multirotor aircraft are starting to emerge and some multirotor aircraft are con-
trolled by varying the collective pitch of the blade. This work considers only the multirotor helicopters
controlled by varying the RPM (Revolutions Per Minute) of each rotor but this research can be
extended to most multirotor aircraft regardless of size whether they are controlled by varying the
collective pitch of the blade or the RPM.
The linear dynamical model of the considered multirotor helicopters around hover conditions is
derived first, and then the control constraint is specified. It is pointed out that classical controllability
theories of linear systems are not sufficient to test the controllability of the derived model (Section
II). Then the controllability of the derived model is studied based on the theory in [12], and two
conditions which are necessary and sufficient for the controllability of the derived model are given.
In order to make the two conditions easy to test in practice, an Available Control Authority Index
(ACAI) is introduced to quantify the available control authority of the considered multirotor systems.
Based on the ACAI, a new necessary and sufficient condition is given to test the controllability of the
considered multirotor systems (Section III). Furthermore, the computation of the proposed ACAI and
a step-by-step controllability test procedure is approached for practical application (Section IV). The
proposed controllability test method is used to analyze the controllability of a class of hexacopters to
show its effectiveness (Section V). The major contributions of this Note are: (i) an ACAI to quantify
the available control authority of the considered multirotor systems, (ii) a new necessary and sufficient
controllability test condition based on the proposed ACAI, and (iii) a step-by-step controllability test
procedure for the considered multirotor systems.
II. PROBLEM FORMULATION
This Note considers a class of multirotor helicopters shown in Fig.1, which are often used in
practice. From Fig.1, it can be seen that there are various types of multirotor helicopters with different
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 1.
Different configurations of multirotor helicopters (the white disc denotes that the rotor rotates clockwise and the
black disc denotes that the rotor rotates anticlockwise)
rotor numbers and different configurations. Despite the difference in type and configuration, they can
all be modeled in a general form as equation (1). In reality, the dynamical model of the multirotor
helicopters is nonlinear and there are some aerodynamic damping and stiffness. But if the multirotor
helicopter is hovering, the aerodynamic damping and stiffness is ignorable. The linear dynamical
model around hover conditions is given as [14], [15], [16]:
˙x = Ax + B(F −G)
|
{z
}
u
(1)
where
x = [h φ θ ψ vh p q r]T ∈R8, F = [T L M N]T ∈R4, G = [mag 0 0 0]T ∈R4,
A =


04×4
I4
0
0

∈R8×8, B =


0
J−1
f

∈R8×4, Jf = diag (−ma, Jx, Jy, Jz)
In practice, fi ∈[0, Ki] , i = 1, · · · m since the rotors can only provide unidirectional lift (upward
or downward). As a result, the rotor lift f is constrained by
f ∈F = Πm
i=1 [0, Ki] .
(2)
Then according to the geometry of the multirotor system shown in Fig.2, the mapping from the rotor
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1
M
2
M
m
M
1r
2r
mr
1e
2e
3e
Fig. 2.
Geometry definition for multirotor system
lift fi, i = 1, · · · m to the system total thrust/torque F is:
F = Bff
(3)
where f = [f1 · · · fm]T . The matrix Bf ∈R4×m is the control effectiveness matrix and
Bf = [b1 b2 · · · bm]
(4)
where bi = ηi¯bi, ¯bi ∈R4, i ∈{1, · · · m} is the vector of contribution factors of the i-th rotor to the
total thrust/torque F, the parameters ηi ∈[0, 1] , i = 1, · · · , 6 is used to account for rotor wear/failure.
If the i-th rotor fails, then ηi = 0. For a multirotor helicopter whose geometry is shown in Fig.2, the
control effectiveness matrix Bf in parameterized form is [16]
Bf =


η1
· · ·
ηm
−η1r1 sin (ϕ1)
· · ·
−ηmrm sin (ϕm)
η1r1 cos (ϕ1)
· · ·
ηmrm cos (ϕm)
η1w1kµ
· · ·
ηmwmkµ


(5)
where wi is defined by
wi =







1, if rotor i rotates anticlockwise
−1, if rotor i rotates clockwise
.
(6)
By (2) and (3), F is constrained by
Ω= {F|F = Bff, f ∈F} .
(7)
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Then u is constrained by
U = {u|u = F −G, F ∈Ω} .
(8)
From (2) (7) and (8), F, Ω, U, are all convex and closed.
Our major objective is to study the controllability of the system (1) under the constraint U.
Remark 1. The system (1) with constraint set U ⊂R4 is called controllable if, for each pair of
points x0 ∈R8 and x1 ∈R8, there exists a bounded admissible control, u (t) ∈U, defined on some
finite interval 0 ≤t ≤t1, which steers x0 to x1. Specifically, the solution to (1), x (t, u (·)), satisfies
the boundary conditions x (0, u (·)) = x0 and x (t1, u (·)) = x1.
Remark 2. Classical controllability theories of linear systems often require the origin to be an
interior point of U so that C (A, B) being row full rank is a necessary and sufficient condition [12].
However, the origin is not always inside control constraint U of the system (1) under rotor failures.
Consequently, C (A, B) being row full rank is not sufficient to test the controllability of the system
(1).
III. CONTROLLABILITY FOR THE MULTIROTOR SYSTEMS
In this section, the controllability of the system (1) is studied based on the positive controllability
theory proposed in [12]. Applying the positive controllability theorem in [12] to the system (1)
directly, the following theorem is obtained
Theorem 1. The following conditions are necessary and sufficient for the controllability of the
system (1):
(i) Rank C (A, B) = 8, where C (A, B) =

B AB · · · A7B

.
(ii) There is no real eigenvector v of AT satisfying vT Bu ≤0 for all u ∈U.
It is difficult to test the condition (ii) in Theorem 1, because in practice one cannot check all u
in U. In the following, an easy-to-use criterion is proposed to test the condition (ii) in Theorem 1.
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Before going further, a measure is defined as:
ρ (X, ∂Ω) ≜







min {∥X −F∥: X ∈Ω, F ∈∂Ω}
−min

∥X −F∥: X ∈ΩC, F ∈∂Ω
	
(9)
where ∂Ωis the boundary of Ωand ΩC is the complementary set of Ω. If ρ (X, ∂Ω) ≤0, then
X ∈ΩC ∪∂Ω, which means that X is not an interior point of Ω. Otherwise, X is an interior point
of Ω.
According to (9), ρ (G, ∂Ω) = min {∥G −F∥, F ∈∂Ω} which is the radius of the biggest enclosed
sphere centered at G in the attainable control set Ω. In practice, it is the maximum control thrust/torque
that can be produced in all directions. Therefore, it is an important quantity to ensure controllability
for arbitrary rotor wear/failure. Then ρ (G, ∂Ω) can be used to quantify the available control authority
of the system (1). From (8), it can be seen that all the elements in U are given by translating the all the
elements in Ωby a constant G. As translation does not change the relative position of all the elements
of Ω, the value of ρ (0, ∂U) is equal to the value of ρ (G, ∂Ω). In this Note, the Available Control
Authority Index (ACAI) of system (1) is defined by ρ (G, ∂Ω) as Ωis the attainable control set and
more intuitive than U in practice. The ACAI shows the ability as well as the control capacity of a
multirotor helicopter controlling its altitude and attitude. With this definition, the following lemma
about condition (ii) of Theorem 1 is obtained.
Lemma 1: The following three statements are equivalent for the system (1):
(i) There is no non-zero real eigenvector v of AT satisfying vT Bu ≤0 for all u ∈U or
vT B (F −G) ≤0 for all F ∈Ω.
(ii) G is an interior point of Ω.
(iii) ρ (G, ∂Ω) > 0.
Proof: See Appendix A. □
By Lemma 1, condition (ii) in Theorem 1 can be tested by the value ρ (G, ∂Ω). Now a new necessary
and sufficient condition can be derived to test the controllability of the system (1).
Theorem 2: System (1) is controllable, if and only if the following two conditions hold:
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(i) Rank C (A, B) = 8.
(ii) ρ (G, ∂Ω) > 0.
According to Lemma 1, Theorem 2 is straightforward from Theorem 1. Actually, Theorem 2 is a
corollary of Theorem 1.4 presented in [12]. To make this Note more readable and self-contained, we
extend the condition (1.6) of Theorem 1.4 presented in [12], and get the condition (ii) in Theorem 2
of this Note based on the simplified structure of (A, B) pair and the convexity of U. This extension
can enable the quantification of the controllability and also make it possible to develop a step-by-
step controllability test procedure for the multirotor systems. In the following section, a step-by-step
controllability test procedure is approached based on Theorem 2.
IV. A STEP-BY-STEP CONTROLLABILITY TEST PROCEDURE
This section will show how to obtain the value of the proposed ACAI in Section III. Furthermore,
a step-by-step controllability test procedure for the controllability of the system (1) is approached for
practical applications.
A. Available Control Authority Index Computation
First, two index matrices S1 and S2 are defined, where S1 is a matrix whose rows consist of all
possible combinations of 3 elements of M = [1 2 · · · m], and the corresponding rows of S2 are the
remaining m −3 elements of M. The matrix S1 contains sm rows and 3 columns, and the matrix S2
contains sm rows and m −3 columns, where
sm =
m!
(m −(nΩ−1))! (nΩ−1)!.
(10)
For the system in equation (1), sm is the number of the groups of parallel boundary segments in F.
For example, if m = 4, nΩ= 4, then sm = 4 and
S1 =


1
2
3
1
2
4
1
3
4
2
3
4


, S2 =


4
3
2
1


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Define B1,j and B2,j as follows:
B1,j = [bS1(j,1) bS1(j,2) bS1(j,3)] ∈R4×3
B2,j = [bS2(j,1) · · · bS2(j,m−3)] ∈R4×(m−3)
(11)
where j = 1, · · · , sm, S1 (j, k1) is the element at the j-th row and the k1-th column of S1, and
S2 (j, k2) is the element at the j-th row and the k2-th column of S2. Here k1 = 1, 2, 3 and k2 =
1, · · · , m −3.
Define a sign function sign(·) as follows: for an n dimensional vector a = [a1 · · · an] ∈R1×n,
sign (a) = [c1 · · · cn]
(12)
where ci = 1 if ai > 0, ci = 0 if ai = 0, and ci = −1 if ai < 0. Then ρ (G, ∂Ω) is obtained by the
following theorem.
Theorem 3. For the system in equation (1), if rank Bf = 4 then the ACAI ρ (G, ∂Ω) is given by
ρ (G, ∂Ω) = sign (min (d1, d2, · · · , dsm)) min (|d1| , |d2| , · · · , |dsm|) .
(13)
If rank B1,j = 3, then
dj = 1
2sign ξT
j B2,j


Λj
ξT
j B2,j

T −
ξT
j (Bffc −G)
 , j = 1, · · · , sm
(14)
where fc = 1
2[K1 K2 · · · Km]T ∈Rm and Λj ∈R(m−3)×(m−3) is given by
Λj =


KS2(j,1)
0
0
0
0
KS2(j,2)
0
0
0
0
...
0
0
0
0
KS2(j,m−3)


(15)
The vector ξj ∈R4 satisfies
ξT
j B1,j = 0, ∥ξj∥= 1
(16)
and B1,j and B2,j are given by (11). If rank B1,j < 3, dj = +∞.
Proof: The proof process is divided into 3 steps and the details can be found in Appendix B. □
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Remark 3. In practice, +∞is replaced by a sufficiently large positive number (for example, set
dj = 106). If rank Bf < 4, then Ωis not a 4 dimensional hypercube and the ACAI makes no sense
which is set to −∞. Similarly, −∞is replaced by −106 in practice). From (13), if ρ (G, ∂Ω) > 0, then
G is an interior point of Ωand ρ (G, ∂Ω) is the minimum distance from G to ∂Ω. If ρ (G, ∂Ω) < 0,
then G is not an interior point of Ωand |ρ (G, ∂Ω)| is the minimum distance from G to ∂Ω. The
ACAI ρ (G, ∂Ω) can also be used to show a degree of controllability (see [17], [18], [19]) of the
system in equation (1), but the ACAI is fundamentally different from the degree of controllability
in [17]. The degree of controllability in [17] is defined based on the minimum Euclidean norm of
the state on the boundary of the recovery region for time t. However, the ACAI is defined based on
the minimum Euclidean norm of the control force on the boundary of the attainable control set. The
degree of controllability in [17] is time-dependent, whereas the ACAI is time-independent. A very
similar multirotor failure assessment was provided in [16] by computing the radius of the biggest
circle that fits in the L-M plane with the center in the origin (L = 0, M = 0), where the L-M plane
is obtained by cuting the four-dimensional attainable control set at the nominal hovering conditions
defined with T = G and N = 0. This computation is very simple and intuitive. But the radius of
the two-dimensional L-M plane can only quantify the control authority of roll and pitch control. To
account for this, the ACAI proposed by this Note is defined by the radius of the biggest ball that fits
in the four-dimensional polytopes Ωwith the center in G.
B. Controllability Test Procedure for Multirotor Systems
From the above, the controllability of the multirotor system (1) can be analyzed by the following
procedure:
Step 1: Check the rank of C (A, B). If C (A, B) = 8, go to Step 2. If C (A, B) < 8, go to Step 9.
Step 2: Set the value of the rotor’s efficiency parameter ηi,i = 1, · · · , m to get Bf = [b1 b2 · · · bm]
as shown in (4). If rank Bf = 4, go to Step 3. If rank Bf < 4, let ρ (G, ∂Ω) = −106 and go to Step
9.
Step 3: Compute the two index matrices S1 and S2, where S1 is a matrix whose rows consist of
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1o
o
2o
Fig. 3.
(a) Standard rotor arrangement, (b) new rotor arrangement, (c) the 1-st rotor of the PNPNPN system fails, (d) the
1-st rotor of the PPNNPN system fails.
all possible combinations of the m elements of M taken 3 at a time and the rows of S2 are the
remaining (m −3) elements of M, M = [1 2 · · · m].
Step 4: j = 1.
Step 5: Compute the two matrices B1,j and B2,j according to (11).
Step 6: If rank B1,j = 3, compute dj according to (14). If rank B1,j < 3, set dj = 106.
Step 7: j = j + 1. If j ≤sm, go to Step 5. If j > sm, go to Step 8.
Step 8: Compute ρ (G, ∂Ω) according to (13).
Step 9: If C (A, B) < 8 or ρ (G, ∂Ω) ≤0, the system (1) is uncontrollable. Otherwise, the system
in equation (1) is controllable.
V. CONTROLLABILITY ANALYSIS FOR A CLASS OF HEXACOPTERS
In this section, the controllability test procedure developed in section IV is used to analyze the
controllability of a class of hexacopters shown in Fig.3, subject to rotor wear/failures, to show its
effectiveness.
The rotor arrangement of the considered hexacopter is the standard symmetrical configuration
shown in Fig.3(a). PNPNPN is used to denote the standard arrangement, where “P” denotes that
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TABLE I
HEXACOPTER PARAMETERS
Parameter
Value
Units
ma
1.535
kg
g
9.80
m/s2
ri, i = 1, · · · , 6
0.275
m
Ki, i = 1, · · · , 6
6.125
N
Jx
0.0411
kg·m2
Jy
0.0478
kg·m2
Jz
0.0599
kg·m2
kµ
0.1
-
TABLE II
HEXACOPTER (PNPNPN) CONTROLLABILITY WITH ONE ROTOR FAILED
Rotor failure
Rank of C(A, B)
ACAI
Controllability
No wear/failure
8
1.4861
controllable
η1 = 0
8
0
uncontrollable
η2 = 0
8
0
uncontrollable
η3 = 0
8
0
uncontrollable
η4 = 0
8
0
uncontrollable
η5 = 0
8
0
uncontrollable
η6 = 0
8
0
uncontrollable
a rotor rotates clockwise and “N” denotes that a rotor rotates anticlockwise. According to (4), the
control effectiveness matrix Bf of that hexacopter configuration is
Bf =


η1
η2
η3
η4
η5
η6
0
−
√
3
2 η2r2
−
√
3
2 η3r3
0
√
3
2 η5r5
√
3
2 η6r6
η1r1
1
2η2r2
−1
2η3r3
−η4r4
−1
2η5r5
1
2η6r6
−η1kµ
η2kµ
−η3kµ
η4kµ
−η5kµ
η6kµ


(17)
Using the procedure defined in Section IV, the controllability analysis results of the PNPNPN
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hexacopter subject to one rotor failure is shown in Table II. The PNPNPN hexacopter is uncontrollable
when one rotor fails, even though its controllability matrix is row full rank. A new rotor arrangement
(PPNNPN) of the hexacopter shown in Fig.3(b) is proposed in [16], which is still controllable when
one of some specific rotors stops. The controllability of the PPNNPN hexacopter subject to one rotor
failure is shown in Table III.
TABLE III
HEXACOPTER (PPNNPN) CONTROLLABILITY WITH ONE ROTOR FAILED
Rotor failure
Rank of C(A, B)
ACAI
Controllability
No wear/failure
8
1.1295
controllable
η1 = 0
8
0.7221
controllable
η2 = 0
8
0.4510
controllable
η3 = 0
8
0.4510
controllable
η4 = 0
8
0.7221
controllable
η5 = 0
8
0
uncontrollable
η6 = 0
8
0
uncontrollable
From Table II and Table III, the value of the ACAI is 1.4861 for the PNPNPN hexacopter subject
to no rotor failures, while the value of the ACAI is reduced to 1.1295 for the PPNNPN hexacopter.
It can be observed that the use of the PPNNPN configuration instead of the PNPNPN configuration
improves the fault-tolerance capabilities but also decreases the ACAI for the no failure condition.
Similar to the results in [16], changing the rotor arrangement is always a tradeoff between fault-
tolerance and control authority. That said, the PPNNPN system is not always controllable under a
failure. From Table III, it can be seen that if the 5-th rotor or the 6-th rotor fails the PPNNPN system
is uncontrollable.
The following provides some physical insight between the two configurations. For the PPNNPN
configuration, if one of the rotors (other than the 5-th and 6-th rotor) of that system fails, the remaining
rotors still comprise a basic quadrotor configuration that is symmetric about the mass center (see
Fig.3(d)). In contrast, if one rotor of the PNPNPN system fails, although the remaining rotors can
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14
1
K
2
K
5
K
1
K
2
K
1
K
5
K
5
K
2
K
(a) Controllable rotor efficiency region
(a) controllable rotor efficiency region
(b) Projection on plane
1
2
5
,
1
KK K  
(c) Projection on plane
1
5
2
,
1
KK K  
(d) Projection on plane
2
5
1
,
1
K K K  
Fig. 4.
Controllable region of different rotors’ efficiency parameter for the PNPNPN hexacopter
make up a basic quadrotor configuration, the quadrotor configuration is not symmetric about the
mass center (see Fig.3(c)). The result is that the PPNNPN system under most single rotor failures
can provide the necessary thrust and torque control, while the PNPNPN system cannot.
Therefore, it is necessary to test the controllability of the multirotor helicopters before any fault-
tolerant control strategies are employed. Moreover, the controllability test procedure approached can
also be used to test the controllability of the hexacopter with different ηi, i ∈{1, · · · , 6}. Let η1, η2,
η5 vary in [0, 1] ⊂R, namely rotor 1, rotor 2 and rotor 5 are worn; then the PNPNPN hexacopter
retains controllability while η1, η2, η5 are in the grid region (where the grid spacing is 0.04) in Fig.4.
The corresponding ACAI at the boundaries of the projections shown in Fig. 4 is zero or near to zero
(because of error in numerical calculation).
VI. CONCLUSIONS
The controllability problem of a class of multirotor helicopters was investigated. An Available Con-
trol Authority Index (ACAI) was introduced to quantify the available control authority of multirotor
February 5, 2015
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15
systems. Based on the ACAI, a new necessary and sufficient condition was given based on a positive
controllability theory. Moreover, a step-by-step procedure was developed to test the controllability of
the considered multirotor helicopters. The proposed controllability test method was used to analyze
the controllability of a class of hexacopters to show its effectiveness. Analysis results showed that the
hexacopters with different rotor configurations have different fault tolerant capabilities. It is therefore
necessary to test the controllability of the multirotor helicopters before any fault-tolerant control
strategies are employed.
APPENDIX
A. Proof of Lemma 1
In order to make this Note self-contained, the following lemma is introduced:
Lemma 3 [20]. If Ωis a nonempty convex set in R4 and F0 is not an interior point of Ω, then
there is a nonzero vector k such that kT (F −F0) ≤0 for each F ∈cl (Ω), where cl (Ω) is the
closure of Ω.
Then according to Lemma 3,
(i)⇒(ii): Suppose that (i) holds. It is easy to see that all the eigenvalues of AT are zero. By solving
the linear equation AT v = 0, all the eigenvectors of AT are expressed in the following form
v = [0 0 0 0 k1 k2 k3 k4]T
(18)
where v ̸= 0, k = [k1 k2 k3 k4]T ∈R4, and k ̸= 0. With it,
vT Bu = −k1
T −mag
ma
+ k2
L
Jx
+ k3
M
Jy
+ k4
N
Jz
.
(19)
By Lemma 3, if G is not an interior point of Ω, then u = 0 is not an interior point of U. Then, there
is a nonzero ku = [ku1 ku2 ku3 ku4]T satisfying
kT
u u = ku1 (T −mag) + ku2L + ku3M + ku4N ≤0
for all u ∈U. Let
k = [−ku1ma ku2Jx ku3Jy ku4Jz]T
(20)
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16
then vT Bu ≤0 for all u ∈U according to (19), which contradicts Theorem 1.
(ii)⇒(i): As all the eigenvectors of AT are expressed in the form expressed by equation (18), then
vT Bu = kT J−1
f u
according to equation (1) and (18) where k ̸= 0. Then there is no nonzero v ∈R8 expressed by
(18) satisfying vT Bu ≤0 for all u ∈U is equivalent to that there is no nonzero k ∈R4 satisfying
kT J−1
f u ≤0 for all u ∈U. Supposing that (ii) is valid, then u = 0 is an interior point of U. There
is a neighbourhood B (0, ur) of u = 0 belonging to U, where ur > 0 is small and constant. (ii)⇒(i)
will be proved by counterexamples.
Supposing that condition (i) does not hold, then there is a k ̸= 0 satisfying kT J−1
f u ≤0 for all
u ∈U. Without loss of generality, let k = [k1 ∗∗∗]T where k1 ̸= 0 and ∗indicates an arbitrary
real number. Let u1 = [ε 0 0 0]T and u2 = [−ε 0 0 0]T where ε > 0; then u1, u2 ∈B (0, ur) if ε
is sufficiently small. As kT J−1
f u ≤0 for all u ∈B (0, ur), then kT J−1
f u1 ≤0 and kT J−1
f u2 ≤0.
According to equation (1),
−k1ε
ma
≤0, k1ε
ma
≤0.
This implies that k1 = 0 which contradicts the fact that k1 ̸= 0.
Then, condition (i) holds.
(ii)⇔(iii): According to the definition of ρ (G, ∂Ω), if ρ (G, ∂Ω) ≤0, then G is not in the interior
of Ω, and if ρ (G, ∂Ω) > 0, then G is an interior point of Ω.
This completes the proof.
B. Proof of Theorem 3
Theorem 3 will be proved in the following 3 steps.
Step 1. Obtain the equations (25), which are the projection of parallel boundaries in F by the map
Bf.
The results in [17] are referred to in order to complete this step. First, (3) is rearranged as follows:
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17
F =

B1,j
B2,j



f1,j
f2,j


(21)
where f1,j = [fS1(j,1) fS1(j,2) fS1(j,3)]T ∈R3, f2,j = [fS2(j,1) · · · fS2(j,m−3)]T ∈Rm−3, j =
1, · · · , sm. Write (21) more simply as
F = B1,jf1,j + B2,jf2,j
(22)
If the rank of B1,j is 3, there exists a 4 dimensional vector ξj such that
ξT
j B1,j = 0, ∥ξj∥= 1.
Therefore, multiplying ξT
j on both sides of (22) results in
ξT
j F −ξT
j B2,jf2,j = 0.
(23)
According to [17], ∂Ωis a set of hyperplane segments, and each hyperplane segment in ∂Ωis the
projection of a 3 dimensional boundary hyperplane segment of F. Each 3 dimensional boundary of
the hypercube F can be characterized by fixing the values of f2,j at the boundary value, denoted by
¯f2,j, where
¯f2,j ∈Πm−3
i=1

0, KS2(j,i)
	
(24)
and allowing the values of f1,j to vary between their limits given by F, where f1,j ∈Π3
i=1

0, KS1(j,i)

.
Then for each j, if rank B1,j = 3, a group of parallel hyperplane segments ΓΩ,j =

lΩ,j,k, k = 1, · · · , 2m−3	
in Ωis obtained, and each lΩ,j,k is expressed by
lΩ,j,k = 
X|ξT
j X −ξT
j B2,j ¯f2,j = 0, X ∈Ω, ¯f2,j ∈Πm−3
i=1

0, KS2(j,i)
		
(25)
where ξj is the normal vector of the hyperplane segments.
Step 2. Compute the distances from the center Fc to all the elements of ∂Ω.
It is pointed out that, not all the hyperplane segments in ΓΩ,j specified by equations (25) belong
to ∂Ω. In fact, for each j, only two hyperplane segments specified by equations (25) belong to ∂Ω,
February 5, 2015
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18
denoted by ΓΩ,j,1 and ΓΩ,j,2, j ∈{1, · · · , sm}, which are symmetric about the center Fc of Ω. The
center of F is fc, then Fc is the projection of fc through the map Bf and is expressed as follows
Fc = Bffc
(26)
where fc = 1
2[K1 K2 · · · Km]T ∈Rm. Then the distances from Fc to the hyperplane segments given
by (25) are computed by
dΩ,j,k =
ξT
j Fc −ξT
j B2,j ¯f2,j

=
ξT
j B2,j
 ¯f2,j −fc,2


=
ξT
j B2,j¯zj

(27)
where k = 1, · · · , 2m−3, fc,2 = 1
2[KS2(j,1) KS2(j,2) · · · KS2(j,m−3)]T ∈Rm−3, ¯f2,j is specified by
(24), and ¯zj = ¯f2,j −fc,2.
Remark 4. The distances from Fc to the hyperplane segments given by (25) are defined by dΩ,j,k =
min {∥X −Fc∥, X ∈lΩ,j,k}, k = 1, · · · , 2m−3.
The distances from the center Fc to ΓΩ,j,1 and ΓΩ,j,2 are equal, which is given by
dj,max = max

dΩ,j,k, k = 1, · · · , 2m−3	
(28)
Since ¯zj ∈Z = 1
2Πm−3
i=1

−KS2(j,i), KS2(j,i)
	
, k = 1, · · · , 2m−3,
dj,max = 1
2sign
ξT
j B2,j


Λj
ξT
j B2,j

T
(29)
according to (12) (27) and (28), where Λj is given by (15).
Step 3. Compute ρ (G, ∂Ω).
As G and Fc are known, the vector FGc = Fc −G is projected along the direction ξj and the
projection is given by
dGc = ξT
j FGc.
(30)
Then if G ∈Ω, the minimum of the distances from G to both ΓΩ,j,1 and ΓΩ,j,2 is
dj = dj,max −|dGc|
(31)
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19
But if G ∈ΩC, dj specified by (31) may be negative. So the minimum of the distances from G to
both ΓΩ,j,1 and ΓΩ,j,2 is |dj|. According to (26) (29) (30) and (31),
dj = 1
2sign
ξT
j B2,j


Λj
ξT
j B2,j

T −
ξT
j (Bffc −G)
 , j = 1, · · · , sm.
But if rank B1,j < 3, the 3 dimensional hyperplane segments are planes, lines, or points in ∂Ωor Ω
and |dj| will never be the minimum in |d1|, |d2|, · · · , |dsm|. The distance dj is set to +∞if rank
B1,j < 3. The purpose of this is to exclude dj from |d1|, |d2|, · · · , |dsm|. In practice, +∞is replaced
by a sufficiently large positive number (for example, dj = 106). If min (d1, d2, · · · , dsm) ≥0, then
G ∈Ωand ρ (G, ∂Ω) = min (d1, d2, · · · , dsm) . But if min (d1, d2, · · · , dsm) < 0, which implies that
at least one of dj < 0, j ∈{1, · · · , sm}, then G ∈ΩC and ρ (G, ∂Ω) = −min (|d1| , |d2| , · · · , |dsm|)
according to (9).
Then ρ (G, ∂Ω) is computed by
ρ (G, ∂Ω) = sign (min (d1, d2, · · · , dsm)) min (|d1| , |d2| , · · · , |dsm|) .
(32)
This is consistent with the definition in (9).
VII. ACKNOWLEDGMENT
This work is supported by the National Natural Science Foundation of China (No. 61473012) and
the ”Young Elite” of High Schools in Beijing City of China (No. YETP1071).
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