controllability analysis and degraded control for a class...

J Intell Robot Syst (2015) 78:143–157DOI 10.1007/s10846-014-0103-0

Controllability Analysis and Degraded Control for a Classof Hexacopters Subject to Rotor Failures

Guang-Xun Du · Quan Quan · Kai-Yuan Cai

Received: 2 March 2014 / Accepted: 26 August 2014 / Published online: 4 September 2014© Springer Science+Business Media Dordrecht 2014

Abstract This paper considers the controllabilityanalysis and fault tolerant control problem for a classof hexacopters. It is shown that the considered hex-acopter is uncontrollable when one rotor fails, eventhough the hexacopter is over-actuated and its control-lability matrix is row full rank. According to this, afault tolerant control strategy is proposed to controla degraded system, where the yaw states of the con-sidered hexacopter are ignored. Theoretical analysisindicates that the degraded system is controllable ifand only if the maximum lift of each rotor is greaterthan a certain value. The simulation and experimentresults on a prototype hexacopter show the feasibil-ity of our controllability analysis and degraded controlstrategy.

Keywords Hexacopter · Fault tolerant control ·Controllability · Degraded control strategy ·Safe landing

G. X. Du (�) · Q. Quan · K. Y. CaiDepartment of Automatic Control, Beihang University,No. 37, Xueyuan Road, Haidian District,Beijing 100191, People’s Republic of Chinae-mail: [email protected]

Q. Quane-mail: qq [email protected]

K. Y. Caie-mail: [email protected]

1 Introduction

Multirotor helicopters are attracting increasing atten-tion in recent years because of their important contri-bution and cost effective application in several taskssuch as surveillance, search and rescue missions andso on. However, there exists a potential risk to civilsafety if the helicopters crash especially in an urbanarea. Therefore, it is of great importance to con-sider the flight safety of multirotor helicopters in thepresence of rotor faults or failures.

Over-actuated aircraft have the potential to improvesafety and reliability. Fault tolerant control of over-actuated aircraft subject to actuator failures is dis-cussed widely [1–3]. For multirotor helicopters, thereare many applications in which fault tolerance may beachieved by using the sliding mode control [4], or gainscheduling based PID control [5], or adaptive control[6], or model predictive control strategies [7]. The tra-jectory re-planning problem is also considered in theliterature [8]. Some fault tolerant strategies involveexplicit fault diagnosis [9, 10], and some do not [11].Most works on fault tolerant control implicitly assumethat the control systems are still controllable in theevent of failures. However, few works consider thecontrollability of the systems with faults. If the systemis uncontrollable, any fault tolerant control strategywill be unavailable. In [12], Schneider et al. pro-posed a useful method to study the controllability ofmultirotor systems with rotor failures based on theconstruction of the attainable control set. However,

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144 J Intell Robot Syst (2015) 78:143–157

they did not give theoretical analysis of the control-lability of the multirotor systems. This is one of ourmotivations.

Sometimes, a hexacopter is uncontrollable if onerotor fails. Owing to this, the hexacopter subject torotor failures is often controlled by leaving the yawstates uncontrolled, the feasibility of which has beentested by [12, 13]. This is very useful in emergencysituations. However, we find that not all the uncontrol-lable hexacopters can be controlled by the degradedway mentioned in [12, 13]. If the maximum lift of eachrotor is lower than a certain value then the degradedsystem, where the yaw states of the considered hexa-copter are ignored, is still uncontrollable. Our anothermotivation is to specify this lower bound value.

In this paper, the controllability of a class of hexa-copters subject to one rotor failure is analyzed basedon the positive controllability theory in [14], and theresults show that the hexacopter is uncontrollable.In order to land the hexacopter safely, a DegradedControl Strategy (DCS) is proposed for the degradedsystem. The lower bound of the maximum lift ofeach rotor is specified, which can help the design-ers in choosing the proper rotors for improving thefault-tolerant capability of the hexacopter. The majorcontributions of this paper are: (i) a theoretical con-trollability analysis for a class of hexacopters, and (ii)the specification of the lower bound of the maximumrotor lift.

2 Problem Formulation

2.1 Hexacopter Model

This paper considers a class of PNPNPN hexacoptersshown in Fig. 1. The linear dynamical model aroundhover conditions is given as follows [15, 16]:

x = Ax + B(F − G)︸︷︷︸

u

. (1)

Here

x = [h φ θ ψ vh p q r]T ∈ R8, F = [T LMN ]T ∈ R

4,

G = [mag 0 0 0]T ∈ R4,

A =[

04×4 I40 0

]

∈ R8×8, B =

[

0J−1

f

]

∈ R8×4,

Jf = diag(−ma, Jx, Jy, Jz

)

where h is the altitude, vh is the vertical velocity,φ, θ, ψ are roll, pitch and yaw angles, p, q, r are roll,pitch and yaw angular velocities, T is the total thrust,L, M, N are the airframe roll, pitch and yaw torque,ma is the mass of the multirotor helicopter, g is theacceleration of gravity, Jx, Jy, Jz are the moment ofinertia around the roll, pitch and yaw axes of themultirotor helicopter, and

u = F − G ∈ U ⊂ R4. (2)

According to the geometry of the hexacopter shownin Fig. 1, the mapping from the rotor lift fi, i =1, · · · , 6 to the system total thrust/torque F is [12][15]:

F = Hη1,··· ,η6f (3)

where f = [f1 · · · f6]T ∈ R6 and the control effec-

tiveness matrix Hη1,··· ,η6 ∈ R4×6 in parameterized

form is

Hη1,··· ,η6 =

⎡

⎢

⎢

⎣

η1 η2 η3 η4 η5 η6

0 −√

32 η2d −

√3

2 η3d 0√

32 η5d

√3

2 η6d

η1d12 η2d − 1

2 η3d −η4d − 12 η5d

12 η6d

−η1kμ η2kμ −η3kμ η4kμ −η5kμ η6kμ

⎤

⎥

⎥

⎦

(4)

where d is the distance from the center of the rotor tothe center of mass, kμ is the ratio between the reac-tive torque and the lift of the rotors. The parameterηi ∈ [0, 1] , i = 1, · · · , 6 is used to account for rotorwear/failure. If the i-th rotor fails, then ηi = 0. Sincethe rotors of the hexacopter can only provide upwardlifts, we let fi ∈ [0, K] , i = 1, · · · , 6 where K is themaximum lift of each rotor. As a result, we have

f ∈ F = �6i=1 [0, K] . (5)

2.2 Control Constraint

In this section, we will specify the control constraintU . Combining Eqs. 2, 3 and 5, we can get the controlconstraint

U0η1,··· ,η6

= {

u|u = Hη1,··· ,η6f − G, f ∈ F}

. (6)

Next, we consider the control constraint U under acontrol allocation. In practice, the virtual control F

is often designed first. Then, the control allocation isused to obtain f as

f = Pη1,··· ,η6F (7)

J Intell Robot Syst (2015) 78:143–157 145

Fig. 1 Kinematic scheme of a PNPNPN hexacopter, where P denotes that a rotor rotates clockwise and N denotes that a rotor rotatesanticlockwise

where Pη1,··· ,η6 ∈ R6×4 is the allocation matrix

satisfying

Hη1,··· ,η6Pη1,··· ,η6 = I4. (8)

Since F = u + G from Eq. 2, we can get the controlconstraint U under the control allocation (7) as

Uaη1,··· ,η6

= {

u|Pη1,··· ,η6 (u + G) ∈ F}

. (9)

The pseudo-inverse matrix (PIM) method [16, 17] isoften used to choose Pη1,··· ,η6 as follows

Pη1,··· ,η6 =H †η1,··· ,η6

=HTη1,··· ,η6

(

Hη1,··· ,η6HTη1,··· ,η6

)−1.

(10)

The relation between Uaη1,··· ,η6

and U0η1,··· ,η6

isstated as Theorem 1, which is consistent with theresults in [16, 18].

Theorem 1 Uaη1,··· ,η6

⊆ U0η1,··· ,η6

.

Proof For any u∗ ∈ Uaη1,··· ,η6

, there exists a f ∗ ∈ Fsuch that

f ∗ = Pη1,··· ,η6

(

u∗ + G)

.

By Eq. 8, we have

Hη1,··· ,η6f∗−G=Hη1,··· ,η6Pη1,··· ,η6

(

u∗+G)−G=u∗.

This implies u∗ ∈ U0η1,··· ,η6

, namely Uaη1,··· ,η6

⊆U0

η1,··· ,η6.

2.3 Objective

The first objective is to show that the system (1) willlose controllability1 when one rotor fails. That is, thesystem (1) is uncontrollable subject to the control con-straint U = U0

ηi=0 where, for simplicity, the subscriptηi = 0 is used to denote that only the i-th rotorfails and the remaining rotors have neither wear norfailures. The second objective is to study the control-lability of the degraded system, where the yaw statesare removed from Eq. 1, and specify the lower boundof the maximum lift of each rotor.

Remark 1 Not all the hexacopters are configured asFig. 1. For example, a class of PPNNPN hexacoptersare considered in [12]. It is pointed out that other type

1The system (1) with constraint set U ⊂ R4 is called control-

lable if, for each pair of points x0 ∈ R8 and x1 ∈ R

8, thereexists a bounded admissible control, u (t) ∈ U , defined on somefinite interval 0 ≤ t ≤ t1, which steers x0 to x1. Specifically, thesolution to Eq. 1, x (t, u (·)), satisfies the boundary conditionsx (0, u (·)) = x0 and x (t1, u (·)) = x1.


of hexacopters can be analyzed in the same way as thepopular PNPNPN hexacopter.

Remark 2 Classical controllability theories of linearsystems often require the origin to be an interior pointof U so that C (A, B) which is given by C (A, B) =[

B AB · · · A7B]

being row full rank is a necessaryand sufficient condition [14]. However, for the system(1) the control constraint U = U0

ηi=0 does not have theorigin as its interior point when some rotors are dam-aged or fail. Consequently, C (A, B) being row fullrank is not sufficient to test the controllability of thesystem (1).

3 Controllability of the Hexacopter Subject to OneRotor Failure

3.1 Preliminaries

In this section, we will study the controllability ofthe hexacopter subject to one rotor failure based onthe positive controllability theory proposed in [14].Applying the positive controllability theorem in [14]to the system (1) directly, we have

Theorem 2 Consider the system (1), suppose thatthe set U contains a vector in the kernel of B (i.e.,there exists u ∈ U satisfying Bu = 0) and the setCH (U)2 has nonempty interior in R

4. Then, the fol-lowing conditions are necessary and sufficient for thecontrollability of Eq. 1:

(c1) Rank C (A, B) = 8, where C (A, B) =[

B AB · · · A7B]

.

(c2) There is no non-zero real eigenvector v of AT

satisfying vT Bu ≤ 0 for all u ∈ U .

For the considered linear hexacopter model (1),Theorem 2 is simplified as follows.

Corollary 1 The system (1) is controllable if and onlyif

minv∈V

maxu∈U

vT Bu > 0 (11)

where V = {

v|AT v = 0, ‖v‖ = 1, v ∈ R8}

.

2CH (U) is the convex hull of U . According to [19], the convexhull of � is the set of all convex combinations of points in �. If� is convex, then CH (�) = �.

Proof The proof is straightforward. For the system(1), it is easy to check that rank C (A, B) = 8. Accord-ing to Theorem 2, then the system (1) is controllableif and only if there is no non-zero real eigenvector v

of AT satisfying vT Bu ≤ 0 for all u ∈ U . Since allthe eigenvalues of AT are zero, all the real eigenvec-tors of AT can be obtained by solving linear equationsAT v = 0. Then the system (1) is controllable if andonly if Eq. 11 is satisfied. The constraint ‖v‖ = 1 isused to make Eq. 11 verifiable, which does not changethe sign of vT Bu.

3.2 Controllability Analysis of the HexacopterSubject to One Rotor Failure

For the controllability of the hexacopter subject to onerotor failure, we have the following theorem:

Theorem 3 The system (1) constrained byU = U0

ηi=0, ∀i ∈ {1, 2, 3, 4, 5, 6} is uncontrollable.

Proof This proof is accomplished by counterexam-ples. For each ηi = 0, we find a vector vi ∈ V

satisfying

maxu∈U0

ηi=0

vTi Bu = 0 (12)

Then

minv∈V

maxu∈U

vT Bu ≤ 0.

Consequently, the system (1) constrained byU = U0

ηi=0, ∀i ∈ {1, 2, 3, 4, 5, 6} is uncontrollableaccording to to Corollary 1. See Appendix A fordetails.

As analyzed above, the PNPNPN hexacopter sub-ject to one rotor failure is uncontrollable. A questionfollows consequentially: how a hexacopter can landsafely after one rotor fails. In [12, 13], the author sug-gested a degraded control way that was to leave theyaw states uncontrolled. However, neither a controlla-bility analysis nor a concrete DCS exists.

4 Degraded Control and Analysis for SafeLanding Without Yaw

According to Section III, the yaw states of the hexa-copter may be left uncontrolled for safe landing when


one rotor fails. In this section, a DCS for the case withone of ηi , i = 1, · · · , 6 being zero is approached,which does not require any change on the original con-troller. Furthermore, it is shown that the hexacoptersubject to one rotor failure can land by the DCS if andonly if the maximum lift of each rotor is greater than acertain value. This lower bound value will be specifiedin this section.

4.1 DCS for Safe Landing Without Yaw Control

In practice, the virtual control F is often designed first.Then if no rotor fails, f is obtained by f = PF whereF = [T L M N]T and P is expressed by Eq. 10. Ifone of ηi , i = 1, · · · , 6 is zero, the DCS for the system(1) includes the following two steps:

Step 1: Leave the yaw states uncontrolled. Onesimple way is to let (ψs, rs) = (ψc, rc),where (ψs, rs) are the sensed yaw states and(ψc, rc) the commanded yaw states.

Step 2: Reallocate F to the set of rotor lifts f by

f =P F , (13)

Pη1,··· ,η6 =H Tη1,··· ,η6

(

Hη1,··· ,η6HTη1,··· ,η6

)−1

(14)

where F = [T L M]T and

Hη1,··· ,η6 =⎡

⎣

η1 η2 η3 η4 η5 η6

0 −√

32 η2d −

√3

2 η3d 0√

32 η5d

√3

2 η6d

η1d12 η2d − 1

2 η3d −η4d − 12 η5d

12 η6d

⎤

⎦ .

(15)

However, there is no theoretical analysis of theDCS in the existing literatures according to our knowl-edge. In the following section, the lower bound ofthe maximum lift of each rotor is specified throughcontrollability analysis.

4.2 Controllability Analysis of the HexacopterRemoving the Yaw States

The degraded system that the yaw states are removedfrom Eq. 1 is given as

x∗ = Ax∗ + B(

F − G)

︸︷︷︸

u

(16)

Fig. 2 A prototype hexacopter

where

x∗ = [h φ θ vh pq]T ∈ R6, F = [T LM]T ∈ R

3, G

= [mag 0 0]T ∈ R3,

A =[

03×3 I3

0 0

]

∈ R6×6, B =

[

0J−1

f

]

∈ R6×3,

Jf = diag(−ma, Jx, Jy

)

and

u = F − G ∈ U ⊂ R3.

Similar to the system (1), the control constraint U is

U0η1,··· ,η6

= {

u|u = Hη1,··· ,η6f − G, f ∈ F}

. (17)

and the control constraint U under the control alloca-tion (14) is

Uaη1,··· ,η6

= {

u|Pη1,··· ,η6

(

u + G) ∈ F

}

. (18)

Similar to Theorem 1, we have Uaη1,··· ,η6

⊆ U0η1,··· ,η6

.

Table 1 Hexacopter parameters

Parameter Description Value Units

ma Mass 1.535 kg

g Gravity 9.80 m/s2

d Rotor to mass center distance 0.275 m

K Maximum lift of each rotor 6.125 N

Jx Moment of inertia 0.0411 kg.m2

Jy Moment of inertia 0.0478 kg.m2

Jz Moment of inertia 0.0599 kg.m2

kμ k/μ 0.1 −


Similarly to Corollary 1, the following theorem isobtained:

Theorem 4 The system (16) constrained by U iscontrollable if and only if

minv∈V

maxu∈U

vT Bu > 0 (19)

where V = {

v|AT v = 0, ‖v‖ = 1, v ∈ R6}

.

Proof This proof is similar to the proof of Corollary1. See Appendix B for details.

Theorem 5 The system (16) constrained byU = Ua

ηi=0, ∀i ∈ {1, 2, 3, 4, 5, 6} is controllable ifand only if

K >5

18mag. (20)

Furthermore, the system (16) constrained by

U = U0ηi=0, ∀i ∈ {1, 2, 3, 4, 5, 6} is controllable if

Eq. 20 holds.

Proof Under U = Ua

η2=0 we first prove that the fol-lowing two propositions hold (see Appendix C).

Proposition 1 There is a v2 ∈ V satisfying

maxu∈U0

η2=0

vT2 Bu ≤ 0 (21)

if K ≤ 518mag.

Proposition 2 There is no such a v2 ∈ V satisfying(21) if K > 5

18mag.

With Proposition 1 and Proposition 2, the sys-tem (16) constrained by U = Ua

η2=0 is controllable ifand only if Eq. 20 holds according to Theorem 4. IfEq. 20 holds, then for each pair of points x0 ∈ R

6

and x1 ∈ R6 there exists a u∗ (t) ∈ Ua

η2=0, which

steers x0 to x1. Since Uaη2=0 ⊆ U0

η2=0, u∗ (t) ∈ U0η2=0,

namely the system (16) constrained by U = U0η2=0 is

controllable. Similarly, we can conclude this proof forU = Ua

ηi=0, ∀i ∈ {1, 3, 4, 5, 6}.

Fig. 3 No rotor fails andh, φ, θ, ψ are controlled tothe desired target


Remark 3 According to Theorem 5, the designersshould choose proper rotors satisfying K > 5

18mag soas to make sure that the hexacopter can adopt the DCSproposed in this paper.

5 Simulation and Experiment

In order to show the feasibility of the proposed DCS,simulations and an experiment of a prototype hex-acopter (see Fig. 2) are carried out. The physicalparameters of the prototype hexacopter are shown inTable 1. In the simulation, the hexacopter is controlledby Proportional-Derivative (PD) controllers and theproposed DCS for safe landing is applied. After η2 =0, the hexacopter keeps its (h, φ, θ) to the desired tar-gets by leaving the yaw states uncontrolled. In theexperiment, a real flight test for the prototype hexa-copter was carried out. During the real flight test, η2

was set to zero. Then the DCS for safe landing keptthe hexacopter level and the hexacopter was landed bythe remote-controller avoiding loss of control.

5.1 Simulation Results

Based on the parameters in Table 1, a digital simu-lation is performed. The hexacopter hovers at hc =1 m and [φc θc ψc]T = [0 0 5]T rad controlled byProportional-Derivative (PD) controllers which areexpressed by

L = −20 (φ − φc) − 3p, M = −20 (θ − θc) − 3q,

N = −20 (ψ−ψc)−3r, T =−10 (h−hc)−6vh+mag.

(22)If no rotor fails, f is obtained by

f = HTη1,··· ,η6

(

Hη1,··· ,η6HTη1,··· ,η6

)−1F (23)

where F = [T L M N]T . And if one of ηi , i ∈{1, 2, 3, 4, 5, 6} is zero, f is obtained by

f = H Tηi=0

(

Hηi=0HTηi=0

)−1F (24)

where F = [T L M]T .

Fig. 4 The DCS is notadopted after η2 = 0 andh, φ, θ, ψ diverge fromtheir target


Figure 3 shows the simulation results when no rotorfails, where h, φ, θ , and ψ are controlled to the desiredtarget with nice performance. At time instant t = 1s,η2 is set to 0. Figure 4 shows the simulation resultswhen η2 = 0 and the DCS is not adopted. It is shownthat h, φ, θ , and ψ diverge from their targets. With theDCS, h, φ, and θ are controlled to the desired targetswith nice performance (see Fig. 5) which avoids lossof control.

According to Theorem 5, not all the uncontrollablehexacopters can land in the degraded way proposed inthis paper. It should be pointed out that if K ≤ 5

18mag,then h, φ, and θ are not controllable and the hexa-copter will crash to the land if one rotor fails. In thesimulation, we change the value of K to 4.9

18 mag andthe simulation results of h, φ, θ are shown in Fig. 6where the DCS is adopted. Obviously, the hexacopteris out of control.

The simulated outputs from the six rotors are illus-trated in Figs. 7–10. In the case η2 = 0 and no DCSis adopted (see Fig. 8), the hexacopter is uncontrol-lable and rotor 2 and rotor 5 are saturated. In the caseη2 = 0 and the DCS is adopted (see Fig. 9), rotor 2

is not used and the remaining five rotors are used tocontrol h, φ, θ . If K = 4.9

18 mag, rotor 1 and rotor 3are saturated after η2 = 0 even the DCS is adopted(see Fig. 10).

Remark 4 In Fig. 5, the yaw angle changes with aconstant angular velocity at last. When the hexacopterrotates fast, the damping moment ND = KNDr2 cannot be ignored, where KND is the damping coefficient.In the simulation we choose KND = 0.2N·m/rad2

to make the simulation results be consistent with theexperiment results. Parameters ηi , i = 1, · · · , 6,which in practice can be obtained by fault diagnosisstrategies [9, 10], are assumed to be known. Since theeffect of fault diagnosis strategies are not in the scopeof this paper, they will not be discussed here and willbe investigated in our future researches.

5.2 Experimental Results

In order to show the feasibility of the proposedDCS, a real flight test of the prototype hexacoptershown in Fig. 2 was carried out. During the flight,

Fig. 5 The DCS is adoptedafter η2 = 0 and h, φ, θ, ψ

are controlled to the desiredtarget with niceperformance


Fig. 6 K = 4.918 mag and

the hexacopter is out ofcontrol even though theDCS is adopted

Fig. 7 Rotor outputs: no rotor fails


Fig. 8 Rotor outputs: η2 = 0 and the DCS is not adopted

[φc θc ψc]T = [0 0 5]T rad and h was controlledby a remote-controller. Part of the flight data isshown in Fig. 11. The hexacopter was in a stabilizemode (where φ, θ, ψ were controlled by Proportional-

Integral-Derivative controllers and h was controlledby a remote-controller) before time t = 1s. At timeinstant t = 1s, η2 was set to 0, then the controllerkept φ, θ around zero by the DCS. And the hexacopter

Fig. 9 Rotor outputs: η2 = 0 and the DCS is adopted


Fig. 10 Rotor outputs: K = 4.918 mag and the DCS is adopted

was landed slowly by the remote-controller avoiding aflight crash.

During the real-time flight test, the PWM inputs tothe six rotor were recorded. Then real rotor outputs

were calculated based on the value of K and thePWM inputs. The real outputs from the six rotors areillustrated in Fig. 12. After η2 = 0, rotor 2 was not usedand h, φ, θ were controlled by the remaining five rotors.

Fig. 11 Real-time flighttest for a prototypehexacopter


Fig. 12 Rotor outputs: real-time flight test

Remark 5 According to Fig. 11, the hexacopter rotatesfast (nearly 2π rad/s) after η2 = 0. But the h canbe controlled by the remote-controller to achieve asafe landing. The video of the experiment is online[20].

6 Conclusions

In this paper, the controllability and fault tolerantcontrol problem of a class of hexacopters are inves-tigated. The following two conclusions are obtained:i) although the considered hexacopter is over-actuatedand its controllability matrix is row full rank, itis uncontrollable when one rotor fails, and ii) theuncontrollable hexacopter can land in a degradedway by the proposed Degraded Control Strategy(DCS) under the condition that the maximum liftof each rotor is greater than 5

18 of the hexacopter’sgravity. The simulation and experiment results ona prototype hexacopter show the feasibility of theproposed DCS. The focus of our future work isto extend the controllability theory in this paperto analyze the controllability of general multirotorhelicopters.

Appendix

A. Proof of Theorem 3

This proof is accomplished by counterexamples.

(i) Case η2 = 0. The control effectiveness matrixHη2=0 is expressed by

Hη2=0 =

⎡

⎢

⎢

⎣

1 0 1 1 1 1

0 0 −√

32 d 0

√3

2 d√

32 d

d 0 − 12d −d − 1

2d 12d

−kμ 0 −kμ kμ −kμ kμ

⎤

⎥

⎥

⎦

.

(25)

Substituting (25) into Hη2=0f = F , we have

H ′f ′ = F ′ (26)

where f ′ = [f1 f3 f4 f5 f6]T and

H ′ =

⎡

⎢

⎢

⎣

1 1 1 1 1

0 −√

32 0

√3

2

√3

21 − 1

2 −1 − 12

12−1 −1 1 −1 1

⎤

⎥

⎥

⎦

, F ′ =

⎡

⎢

⎢

⎣

T

L/d

M/d

N/kμ

⎤

⎥

⎥

⎦


As rank(H ′)=rank( [

H ′ F ′] )

, the Eq. 26 has ageneral solution [21] of the form

f ′ = f ′H + f ′

P (27)

where fH is the general solution of the associa-ted homogeneous system H ′f ′

H = 0 and f ′P is

a particular (fixed) solution of the nonhomoge-neous system H ′fP = F ′. By using elementaryrow operations (see [21]) to transform the aug-mented matrix

[

H ′ F ′] into an echelon form, wecan get f ′

H and f ′P as follows

f ′H = [−α α − α 0 α]T ,

f ′P =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

12T + 2

3dM + 1

6kμN

−√

33d

L − 13d

M − 13kμ

N12T + 1

2dN√

33d

L − 13d

M − 13kμ

N

0

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

. (28)

where α ∈ R is an arbitrary parameter. FromEqs. 27 and 28, we have

f1 = 1

2T + 2

3dM + 1

6kμ

N − α

f3 = −√

3

3dL − 1

3dM − 1

3kμ

N + α

f4 = 1

2T + 1

2dN − α

f5 =√

3

3dL − 1

3dM − 1

3kμ

N

f6 = α (29)

From Eq. 29, the constraint set

U0η2=0 =

{

u|u = [T − mag L M N]T}

is given by the following inequalities

0 ≤ 1

2T + 2

3dM + 1

6kμ

N − α ≤ K (30a)

0 ≤ −√

3

3dL − 1

3dM − 1

3kμ

N + α ≤ K (30b)

0 ≤ 1

2T + 1

2dN − α ≤ K (30c)

0 ≤√

3

3dL − 1

3dM − 1

3kμ

N ≤ K (30d)

0 ≤ α ≤ K. (30e)

Let v2 =[

0 0 0 0 0 −√

3Jx

3d

Jy

3dJz

3kμ

]T

and

v2 = v2‖v2‖ , we have v2 ∈ V and

vT2 Bu =

−√

33d

L + 13d

M + 13kμ

N

‖v2‖ .

According to Eq. 30d,

maxu∈U0

η2=0

vT2 Bu = 0.

(ii) Case ηi = 0. Similar to the case η2 = 0, we canfind a vi ∈ V satisfying

maxu∈U0

ηi=0

vTi Bu = 0, i ∈ {1, 3, 4, 5, 6} .

From (i) and (ii), we have

minv∈V

maxu∈U

vT Bu ≤ 0

and the system (1) constrained by U = U0ηi=0,∀i ∈ {1, 2, 3, 4, 5, 6} is uncontrollable accord-

ing to Corollary 1.

B. Proof of Theorem 4

We apply the positive controllability theorem in [14]to the system (16) directly. Suppose that the set U con-tains a vector in the kernel of B and the set CH

(

U)

hasnonempty interior in R

3, the following conditions arenecessary and sufficient for the controllability of (16):

(i) Rank C(

A, B) = 6, where C

(

A, B) =

[

B AB · · · A5B]

.

(ii) There is no non-zero real eigenvector v of AT

satisfying vT Bu ≤ 0 for all u ∈ U .

For the system (16), it is easy to check that rankC

(

A, B) = 6. Since all the eigenvalues of AT are

zero, all the real eigenvectors of AT can be obtainedby solving linear equations AT v = 0. Then the system(16) is controllable if and only if Eq. 19 is true.

C. Proof of Theorem 5

C.1 Proof of Proposition 1

According to Eqs. 14 and 15, Pη2=0 =H T

η2=0

(

Hη2=0HTη2=0

)−1. Then Ua

η2=0 = {u|u =


[T − mag L M]T}

is given by the followinginequalities according to Eq. 18

− 5

18T ≤ −

√3

9dL + 4

9dM ≤ K − 5

18T ,

− 5

18T ≤ −5

√3

18dL − 1

18dM ≤ K − 5

18T ,

−1

6T ≤ − 1

3dM ≤ K − 1

6T ,

−1

9T ≤ 2

√3

9dL − 2

9dM ≤ K − 1

9T ,

−1

6T ≤

√3

6dL + 1

6dM ≤ K − 1

6T . (31)

Denote Ec = {

c|c = (L, M)T , L, M satisfy (31)}

which is closed and convex. If T ≥ 185 K , c0 = [00]T

is not an interior point of Ec. Then there is a non-zerovector ck = [kc1 kc2]T satisfying

cTk (c − c0) = kc1L + kc2M ≤ 0 (32)

for all c = (L, M)T ∈ Ec according to [22]. Letv2 = [

0 0 0 0 kc1Jx kc2Jy

]T and v2 = v2‖v2‖ , we have

AT v2 = 0 and

vT2 Bu = kc1L + kc2M

‖v2‖ .

According to Eq. 32,

maxu∈U0

η2=0

vT2 Bu = 0.

Thus, the system (16) is uncontrollable if T ≥ 185 K

according to Theorem 4. Under hovering conditions,we have T = mag. Thus, Proposition 1 is true.

C.2 Proof of Proposition 2

According to the proof of Proposition 1, If T < 185 K ,

then c0 = [0 0]T is an interior point of Ec. Accord-ing to [22], we cannot find a non-zero vector ck =[kc1 kc2]T satisfying cT

k c ≤ 0 for all c ∈ Ec. Wewill prove this by the proof of contradiction. Suppose

that there is a non-zero vector v2 = [

0 0 0 0 k1 k2]T

satisfying

maxu∈U0

η2=0

vT2 Bu = 0

then we have

vT2 Bδ = k1L/Jx + k2M/Jy ≤ 0.

Let ck = [kc1 kc2]T = [

k1/Jx k2/Jy

]T. Then we get

cTk c = k1L/Jx + k2M/Jy ≤ 0

and this contradicts with the fact that there is no non-zero vector ck satisfying cT

k c ≤ 0 for all c ∈ Ec. Thus,the system (16) is controllable if T < 18

5 K accord-ing to Theorem 4. Under hovering conditions, we haveT = mag. Thus, Proposition 2 holds.

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http://new.ted.com/talks/raffaello_d_andrea_the_astounding_athletic_power_of_quadcopters

http://www.youtube.com/watch?v=rxaW8F98smw

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