Incremental nonlinear control for attitude tracking of a fixed-wing

UAV - simulation and practical implementation

Afonso D’Alberto Goncalves Canas Mendesafonsocmendes@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

November 2017

Abstract

In this work, two well-studied nonlinear control techniques - Nonlinear Dynamics Inversion andBackstepping - are taken into consideration. In particular, they are complemented with an increasinglypopular robust control technique that considers the incremental dynamics of the vehicle in orderto capture changes in its aerodynamics through angular acceleration feedback instead of complexmodelling. The approach yields increased robustness to model uncertainties and improved trackingperformance, at the expense of additional sensor information and fast control actuation. Additionally,command filters are considered in order to protect the design against constraining actuator dynamics.Both robust approaches are properly tuned, tested and compared in closed-loop simulations. Robust-ness tests for one selected technique are performed to validate its integration into the open-sourceArdupilot c© real-time software. Simulation results using the JSBSIM flight simulator are obtained tofurther certify the integrated solution for practical experimentation.Keywords: Nonlinear Dynamics Inversion, Backstepping, Incremental Control, Command Filtering,Ardupilot

1. Introduction

The inherently compact structure, high manoeuvra-bility and low-cost production of unmanned aerialvehicles (UAVs) represent only some of the reasonswhy these systems have become increasingly pop-ular in recent years. As the corresponding mar-ket expands, it is expected that regulatory author-ities establish tighter bounds on the performanceof these systems to mostly ensure public safety butalso long-term integration into the civil airspace en-vironment. In turn, flight safety is closely related tothe performance of the automatic flight controllers.

The most commonly used approach by the in-dustry to design flight controllers is to use linearmodels of the aircraft and employ the various toolsfrom linear control theory to obtain a linear con-troller. Since most real systems are inherently non-linear, linear descriptions are approximations withlimited validity and so are the controllers which aredeveloped using those descriptions.

On the other hand, advances in computer andmicroprocessor technology have rendered the im-plementation of nonlinear control systems a simplematter. Since this type of control techniques con-cerns generic nonlinear descriptions of the system tobe controlled, it is likely to be more efficient thanlinear control and valid for a larger set of conditions.

Nonlinear Dynamics Inversion (NDI) is a con-trol methodology that started being developed inthe late 1970s by independent authors who first ad-dressed the problem of linearising a nonlinear sys-tem through state feedback [6].

Backstepping (BKS) is a systematic, Lyapunov-based control design method for a broad class ofnonlinear systems [6]. Its origins can be traced backto the early 1990s, when its development experi-enced a rapid acceleration due to the ease that thetechnique showed to have in dealing with unknownparameters and external disturbances.

A common drawback to NDI and BKS designs isthe dependency on model knowledge. The nominalcharacter of these techniques makes them, in theirsimplest forms, sensitive to model uncertainties.

One particular nonlinear and robust control so-lution that has been continuously gaining momen-tum in recent years is incremental control. Devel-opment of nonlinear control theory based on in-crements of nonlinear control action can be tracedback to the late 1990s by the papers of [9] and oth-ers. It has been studied in recent years for atmo-spheric flight control of fixed-wing aircraft [7] andhelicopters [8] and also for space applications [1].It exploits the advancements in sensor technology(and general hardware) to reduce the control syn-

1

thesis dependency on nominal aircraft model dataand therefore enhance its flexibility and robustnessproperties, while minimizing the necessity for adap-tive elements in the flight control system.

Applications of Incremental BKS (IBKS) can befound in [11] for an F-16 aircraft, in [3] for missilecontrol and in [2] for spacecraft attitude control.

Incremental NDI (INDI) [9] has also found prac-tical applications both for aircraft [7, 8] and space-craft [1] flight control.

This project fits in the context presented this faras it concerns the study of INDI and IBKS and theirapplicability to the flight control problem of fixed-wing aircraft. It includes a comparison between theapproaches based on simulation results and the pro-motion of one selected technique for integration inthe real-time software Ardupilot c© .

This paper is structured as follows: in Section2, standard NDI and BKS control laws are re-viewed, followed by a short presentation on incre-mental dynamics and command filtering. Section3 details the fixed-wing UAV rotational model andthe derivation of the command filtered INDI andIBKS control laws for attitude control. In Section 4,MATLAB/Simulink® implementation results arediscussed and in Section 5 the integration of oneincremental controller into the Ardupilot c© frame-work is briefly commented. Section 6 provides themain conclusions and some final remarks.

2. Nonlinear Control approaches2.1. Nonlinear Dynamics Inversion

Consider the following affine-in-control, square,multivariable, generic, nonlinear, n-th order sys-tem:

x = f (x) +G (x)u , y = h (x) (1)

where x ∈ Rn is the state vector, u ∈ Rm is theinput vector, y ∈ Rm is the output vector, f and hare smooth vector fields and G is an n×m matrixwhose column are smooth vector fields in Rn. TheNDI control design procedure consists in finding adirect relation between the output y and the inputu and invert it. By differentiating the output withrespect to time, one obtains:

y =dh (x)

dxx = ∇h (x)

[f (x) +G (x)u

]= Lfh (x) + Lgh (x)u (2)

where Lfh denotes the first-order Lie derivative ofh in the direction of f and is used to simplify thenotation. If Lgh (x) is invertible, the control law:

u = Lgh (x)−1 [

ν − Lfh (x)]

(3)

linearises the nonlinear system from the virtual con-trol ν to y, i.e., ν = y, which can be used to ren-der the closed-loop asymptotically stable with lin-ear control, for example. If the reference for y is

denoted by yd

and the tracking error by e = yd− y,

then the linear control law with feedforward termν = Ke+ y

dyields:

e+Ke = 0 (4)

If K is a positive definite matrix, then the errordynamics is asymptotically stable.

If Lgh (x) is not invertible, consecutive time-differentiations are performed until an explicit re-lation between every output yi ∈ y and the input uis found. If the total number of differentiations isnot equal to n, then the system possesses so-calledinternal dynamics, which cannot be controlled bythe input vector and their behaviour must be con-sidered in order to guarantee closed-loop stability.

For higher-order, cascaded systems, a commonapproach to simplify NDI design is to study the ex-istence of a time-scale separation between the dif-ferent subsystems and exploit that separation to de-sign NDI control laws individually, as explained in[7]. This simplification reduces the number of dif-ferentiations that need to be performed to obtainfeedback linearisation, facilitating the design.

2.2. Backstepping

For the sake of simplicity, consider the followinggeneric, strict-feedback, nonlinear system:

x1 = f1

(x1) +G1 (x1)x2

...xi = f

i(x1, · · · , xi) +Gi (x1, · · · , xi)xi+1...

xk = fk

(x1, · · · , xk) +Gk (x1, · · · , xk)u

(5)

where it is assumed that every xi-subsystem is ofthe same order n and that every state in each sub-system has a relative degree of one. Then, usingLyapunov stability concepts, it can be shown [6]that for a candidate Lyapunov function of the form:

Vk =1

2

k∑i=1

z>i zi (6)

where zi are the tracking tracking errors, given by:

zi = xi − αi−1 (7)

the system in Eq.(5) can achieve global asymptotictracking of a smooth and bounded reference sig-nal, with the first n time-derivatives known andbounded, xd for x1, if the following stabilizing con-trol laws are used:

α1 = G−11

[−C1z1 − f1 + xd

]αi = G−1i

[−Cizi − f i + αi−1 −G

>i−1zi−1

] (8)

for i = 2, · · · , k, with each Ci and C1 being diag-onal positive definite matrices and where the finalBackstepping control law is given by u = αk.

2

2.3. Incremental control

Consider now a generic, n-th order, nonlinear sys-tem given by:

x = f (x, u) , y = x (9)

Take the first-order Taylor series expansion ofEq.(9) around the current augmented state (x0, u0):

x ≈ x0 +∂f

∂x

∣∣∣∣∣(x0,u0)

(x− x0) +∂f

∂u

∣∣∣∣∣(x0,u0)

(u− u0)

= x0 + F x (x− x0) + Fu (u− u0) (10)

For high sampling frequencies and fast control actu-ation, the state variation can be neglected and theformer result reduces to:

x ≈ x0 + Fu (u− u0) (11)

assuming that it is possible to obtain the statederivative feedback x0 and that the actuator state isobservable. This results allows modifying the NDIcontrol law of equation Eq.(3) to:

u = u0 + F−1u [ν − x0] (12)

and the final Backstepping control law of Eq.(8) to:

u = u0 + F−1u

[−Ckzk − x0 + αk−1 −G

>k−1zk−1

](13)

where x0 is used to designate the current observa-tion of the state xk.

It is noticeable that the effect of consideringincremental dynamics removes the necessity of amodel for the state-dependent dynamics of theplant, f (x), both in the NDI and in the BKS controllaws, which results in increased robustness againstmodel mismatch. This simplification is achieved atthe expense of requirements of fast control actua-tion, high sampling frequency and increased stateobservability.

2.4. Command filtering

Since the end goal of this work is practical imple-mentation of an incremental NDI (INDI) or incre-mental BKS (IBKS) controller, actuator dynamicsare an important aspect to consider. In reality, theycan include magnitude bounds, rate limitations andother phenomena that deviates them from an ideal,instant response behaviour. Therefore, commandfilters are considered now to take those limitationsinto account and protect the closed-loop systemagainst reference signals that force the system be-yond its capabilities.

2.4.1 Command filtered Backstepping

Command filtered (CF) Backstepping is formallypresented in [5] and consists, basically, in definingmodified tracking errors

zi = zi − χi (14)

and guaranteeing global asymptotic regulation ofthose modified errors instead. The variables χi areused as estimates of the effect of the command fil-ters and are obtained from stable linear filters:

χi

= −Ciχi +Gi

[xi+1,r − αi + χ

i+1

](15)

where xi,r is the output of the i-th CF and rep-resents a filtered estimation of the reference signalfor the xi state, αi−1, that takes into account theknown limitations of the system. The reader is re-ferred to [5] for the proof of how the control laws:

α1 = G−11

[−C1z1 − f1 + xd

]αi = G−1i

[−Cizi − f i + xi,r −G

>i−1zi−1

] (16)

for i = 2, · · · , k can be used in combination withthe CFs to obtain a globally asymptotically stableclosed -loop system in terms of the modified track-ing errors. The CFs are also useful to provide thederivatives of these control laws.

2.4.2 Command filtered NDI

The approach of command filtering for NDI is sim-ilar to that of BKS and is even more relevant formulti-loop designs that exploit the time-scale sep-aration principle because then the CFs can also beused to obtain the feedforward terms for the linearcontrol laws. For the system in equation Eq.(1),the control signal obtained from the dynamics in-version in Eq.(3), here denoted by u0, is filtered bythe CF before being directed to the nonlinear sys-tem. Then, the stable linear filter:

χ = −Kχ+ Lgh (x)(u− u0

)(17)

is used to estimate the loss of authority of the con-troller due to constraints imposed by the CF andthe tracking error is hedged with χ to obtain a mod-ified tracking error that is then used by the linearcontrol law. It can be shown that the dynamics ofthis modified error e = e− χ becomes:

˙e = −Ke+Kχ (18)

revealing that it converges asymptotically to zerowhen the CF does not impose any constraints onthe control signal and is bounded otherwise.

3. Fixed-wing attitude control3.1. Rotational model

In vector notation, the Euler equations of rotationalmotion for a rigid body with constant inertial masscan be expressed by:

M = Jω + ω × Jω (19)

where M is the total external moment acting onthe system, ω = [p , q , r]> is its angular velocity inthe body frame and J is the inertia tensor. Theexternal moment is assumed to be decomposed in

3

contributions from an aerodynamic model, Maero,and the propulsion system, Mprop. In turn, theaerodynamic moments are given by:

Maero = Ma + rac × F a,b (20)

where ra,c is the relative position of the mean aero-dynamic centre of the aircraft with respect to itsCG, Ma are pure aerodynamic torques and F a,b arethe aerodynamic forces written in the body frame,which can be obtained from the wind-axes forcesF a,w through a simple rotation matrix [4]:

F a,b = T bw (α, β)F a,w (21)

with α being the angle of attack and β the sideslipangle. The propulsion moment has a similar de-composition. Defining the control input as u =[δa , δe , δr]

>, with δa, δe and δr being the aileron, el-

evator and rudder deflections, respectively, and as-suming a linear relation between the aerodynamicforces and moments and u, as in:

Ma = M0a+Muu and F a,w = F 0

a,w+Fuu (22)

the rotational dynamics equation becomes:

ω = J−1[M0

a + RacT bwF0a,w +Mprop − ΩJω

]+ J−1

[Mu + RacT bwFu

](23)

with Rac denoting (rac×) and Ω denoting (ω×). Inthis work, the attitude vector will be considered toinclude the pitch θ and roll φ angles and the sideslipangle instead of the yaw angle. Using kinematicrelations [4, 11], the final rotational model is givenby:

x1 = f1

(x1) +G1 (x1)x2 (24a)

x2 = f2

(x1, x2) +G2 (x1, x2)u (24b)

where the state vectors are x1 = [φ, θ, β]>

= ϕβ

and x2 = ω and:

f1≡[0 0 fβ

]>G1 ≡

1 sφtθ cφtθ0 cφ −sφw√

u2 + w20

−u√u2 + w2

f2≡ J−1

[M0

a + RacT bwF0a,w +Mprop − ΩJω

]G2 ≡ J

−1[Mu + RacT bwFu

](25)

where u, v and w are the body velocity vector com-ponents and where sφ and cφ is used to denote sineand cosine and where:

fβ =1√

u2 + w2

(fx + fy + fz

)(26a)

fx = − uvV 2t

[ax − gsθ] (26b)

fy =

[1−

(v

Vt

)2] [ay + gsφcθ

](26c)

fz = −vwV 2t

[az + gcφcθ

](26d)

with Vt being the total velocity√u2 + v2 + w2 and

ax, ay and az the specific forces in the body frame.

3.2. NDI control lawsBased on the time-scale separation principle [8], thesystem can be controlled separately by an NDI lawfor the dynamics equation and another for the kine-matics. Using standard NDI with command filter-ing, the rate control problem can then be solvedby:

u0 = T−1u J

[ν2 − J

−1[T 0 − ΩJω

]](27a)

ν2 = K2 (ωd − ω + χ2) + ωd (27b)

χ2 = −K2

χ2 + TuJ

−1(u− u0

)(27c)

Tu = Mu + RacT bwFu (27d)

T 0 = M0a + RacT bwF

0a,w +Mprop (27e)

where K2 is the proportional gain of the rate con-trol loop (the inner loop), the feedforward term ωdis provided by the outer loop and where the actualcontrol signal u is obtained at the output of theCF. It is clear that model dependency results in acomplex control law. Under the assumptions men-tioned before, a simplification can be obtained withthe employment of incremental dynamics:

ω ≈ ω0 + J−1[Mu + RacT bwFu

](28)

which is simpler than Eq.(23), resulting in the fol-lowing INDI control law:

u0 = u0 + T−1u J[K2 (ω − ωd + χ

2) + ωd − ω0

](29)

For the (outer) control loop, incremental controlis not considered since it is not expected to bringabout any clear advantage. The command filteredNDI control law for the outer loop is then given by:

ω0d = G−11

[ν1 − f1

](30a)

ν1 = K1

(ϕβ,d − ϕβ + χ

1

)+ ϕ

β,d (30b)

χ1 = −K1

χ1 +G1

(ωd − ω0

d

)(30c)

where the feedforward term ϕβ,d is assumed to be

provided externally.

3.3. BKS control lawsSince the dynamic system represented by Eq.(24)and Eq.(25) is in strict-feedback form, the BKSdesign procedure can be readily applied to solvethe attitude control problem. The adaptation ofthe generic laws in Eq.(16) is straightforward andthe command filtered BKS control laws for attitudecontrol read:

α1 = G−11

[C1

(ϕβ,d − ϕβ

)+ ϕ

β,d − f1

](31a)

4

χ1 = −C1

χ1 +G1 (ωd − α1 + χ

2) (31b)

u0 = T−1u J

[C2 (ωd − ω) + ωd − J

−1[T 0 − ΩJω

]+ G>1

(ϕβ − ϕβ,d − χ1

)](31c)

χ2 = −C2

χ2 + TuJ

−1(u− u0

)(31d)

where ϕβ,d is assumed to be provided externally, ωdand ωd are the outputs of the CF of the outer loopand the final control input u is obtained by filteringthe signal u0 with the CF of the inner loop. As forINDI, the expression for the latter can be simplifiedby considering the incremental dynamics:

u0 = T−1u J[C2 (ωd − ω) + ωd − ω0

+ G>1

(ϕβ − ϕβ,d − χ1

)](32)

4. MATLAB/Simulink® implementation

4.1. Aerosonde® model

The Aerosonde® is a small, inverted V-tail, singlepropeller, fixed-wing UAV. The Simulink® modelused herein is based on the Aerosim Blockset li-brary and includes linear aerodynamics based oncomponent build-up and dimensionless equations,from which the terms in Eq.(22) can be readily ob-tained. In particular, the control derivatives usedby the incremental controllers are given by:

Mu = qSCδ and Fu = qS [C1 + C2Su] (33)

where q is the dynamic pressure, S is the wing areaand where the following definitions have been used:

Cδ =

bClδa 0 bClδr0 cCmδe 0

bCnδa 0 bCnδr

(34a)

C1 =

0 0 0CYδa 0 CYδr

0 −CLδe 0

(34b)

C2 =

−CDδa −CDδe −CDδr0 0 00 0 0

(34c)

with b the wing span, c the mean aerodynamicchord and Su is a diagonal matrix with the signs ofthe elements of u in its diagonal.

4.2. Preliminary results

Preliminary results are now presented to the readerin Figure 1 regarding an attitude tracking task withboth INDI and IBKS controllers from the previ-ous sections, using the simplification Tu = Mu,but without command filters. This simplificationis valid, since it was verified that more than 95%of the control effectiveness is due to the Mu termalone at all times. The reference signals consist of10 deg doublets with the duration of 3 seconds for

pitch and roll and constant zero sideslip. A 100Hz control frequency was used. The gains were de-signed the same for every channel, being k2 = 10for the inner loop and k1 = 2.5 for the outer loop.

Time [s]0 5

[?;3

][d

eg]

-10

0

10

? 3 -

Time [s]0 5

-[d

eg]

-0.2

0

0.2

Figure 1: Attitude tracking results with INDI (solidlines) and IBKS (dash-dotted lines) controllers.The black lines represent signal references.

The results in Figure 1 are satisfactory as thereferences are tracked efficiently in the pitch and rollchannels with minor sideslip peaks due to adverseyaw.

4.3. Introduction of actuator dynamics

After achieving successful attitude tracking in nom-inal conditions, the simulated system was aug-mented with an actuators model, incorporating auniform time-delay of 30 ms, magnitude bounds|u| ∈ ±[30 , 30 , 40]> deg, rate limits of 1 rad/s anda lowpass frequency response with a 13 rad/s cut-off frequency. In turn, this motivates the introduc-tion of command filtering in the control framework.For both loops, second-order CFs were used [10],as they allow to impose both magnitude and ratelimits. The design of the CF of the inner loop isdriven by the the actuators modelling, whereas forthe CF of the outer loop the bandwidth is imposedby the bandwidth of the controller in the inner loop.In order to illustrate the advantages of using theCFs, a magnitude limit of 0.7 deg was forced onthe aileron deflection and a 5 deg step signal wasused as reference for the roll angle, whilst keepingthe pitch and sideslip at zero. The results are de-picted in Figure 2 and one can notice that the CFprevents continued aileron saturation and exagger-ated overshooting without compromising the set-tling time. The differences between the results forINDI and IBKS are justified by the fact that thetwo command filtered designs are slightly differentand therefore equal gains are not expected to yieldthe same results.

4.4. Angular acceleration feedback

In reality, angular acceleration sensors, often costly,are not commonly integrated in the sensing sys-tems. The majority of the literature on incremen-tal attitude control has therefore tackled the issue

5

Time [s]0 2 4 6 8 10

?[d

eg]

0

2

4

IBKS INDI

Time [s]0 2 4 6 8 10

/ a[d

eg]

-0.8

-0.6

-0.4

-0.2

0

0.2

Figure 2: Effect of aileron saturation on roll track-ing. The black lines represents the roll reference.The solid lines correspond to the case with theCF included in the simulation and the dashed lineswithout.

of obtaining ω0 by using estimation/filtering meth-ods on IMU data. An intuitive solution is to usebackward finite differences on the measured angu-lar rates. This approach is relevant for this worksince the end goal is practical implementation. In[7], the authors implemented a predictive filter thatis designed by offline parameter estimation basedon ordinary least squares optimization. Since thecontrollers developed in this chapter are intendedfor experimental implementation, this approach isnot adequate, as the filter design should not dependon the availability of a good simulation model. An-other option is the use of a first-order highpass (HP)filter. By inspection of its generic transfer function:

HHP (s) =a1s

1 + τs(35)

with a1 ∈ R and τ ∈ R, one may notice that forlow frequencies, s 1/τ , the denominator is dom-inated by the constant term and the highpass filterbecomes a low-frequency derivator. Furthermore,the validity of the approximation increases with adecreasing value for the time constant of the filter,τ . However, decreasing τ results in an increase ofthe high frequency gain, given by a1/τ . This factmight be undesirable if the input signal presentshigh-frequency noise. For that reason, a secondorder bandpass (BP) filter is also proposed. Thetransfer function of such a filter can be representedby:

HBP (s) =a1s

s2 + 2ζωns+ ω2n

(36)

where ωn is the natural frequency and ζ is thedamping factor. At low frequencies, the filter ap-proximates a derivator:

HBP ≈a1ω2n

s (37)

if a1 is designed as a1 = ω2n.

Results were obtained for the angular acceler-ation vector estimate through numerical differen-tiation of a noisy angular rate signal. The sen-sor model was obtained from the datasheet of acommercial MEMS-based sensor with an integratedIMU and GNSS receiver. First and second or-der finite differences, FD1 and FD2, respectively,were implemented with a uniform sampling timeof 0.01 seconds. The first-order highpass filter,HP1, and the second-order bandpass filter, BP2,were discretized at 100 Hz using the same finitedifferences, with the BP filter requiring, addition-ally, the second-order FD for the second-order time-derivative. A 10 Hz cut-off frequency was used forboth filters. The results for the roll channel for ashort transient of time are depicted in Figure 3. Theresults for the BP2 filter include three different val-ues of ζ to better illustrate its effect on performance.In Figure 3(b), a bar plot of the root-mean-squareerror (RMSE) of each method is presented to aid incomparing performance.

Time [s]1 1.05 1.1 1.15 1.2 1.25 1.3

_p[d

eg/s

2]

-50

0

50

100

(a) Roll acceleration. The solid and dash-dotted red linesconcern the FD1 and FD2 approaches. The blue lines re-gard the BP2 filter - the solid line was used for ζ = 1.5,the dashed line for ζ = 0.5 and the dash-dotted line forζ = 0.05. The HP1 filter is represented by the green line.

MethodFD1 FD2 HP1 BP1=0:05BP1=0:5 BP1=1:5

Rel

ativ

e R

MS

E [%

]

0

50

100

(b) Relative RMSEs for the different methods with respectto the maximum of the set

Figure 3: Roll acceleration results using finite dif-ferences and linear filters.

6

From Figure 3(a) it is possible to conclude thatthe second order FD is more sensitive to noise thanfirst order, which is confirmed by a 40% perfor-mance decrease in terms of RMSE between the twoFDs. The filters clearly produced better results.Regarding the BP2 filter in specific, inspection ofFigure 3 shows how decreasing ζ implies more over-shoot (higher passband peak gain). The HP1 filterpresents very similar behaviour when compared tothe BP2 filter for ζ = 1.5. This could be expectedsince the width of the passband grows with ζ. How-ever, if very high-frequency noise plays a more rele-vant role, the benefits of the BP2 filter become moreclear. In sum, the fact that finite differences are sosensitive to sensor noise renders them unfit for im-plementation. The linear filters present roughly thesame performance and, although the HP1 filter isa simpler design, the BP2 filter is expected to offeradditional high-frequency advantages in a practicalimplementation.

Recall that the core principle behind incremen-tal control is that the desired angular accelerationis obtained one time step ahead. If delay exists inthe measurement, the incremental step assumptionis violated and degradation deteriorates. Since thefilter does indeed induce a time delay, this problemmust be taken into consideration. One intuitive so-lution is to achieve synchronization between ω0 andω, as seen from the controller, by using a lowpassfilter on the angular rates measurement with thesame characteristic polynomial as the BP2 filter.This way, the virtual control signal and the angularacceleration vector that are available at any giventime for the computations of the inner loop corre-spond to the same (delayed) time step and there-fore the controller does not overshoot/undershootits output signal.

The sampling frequency was verified to have aminimum acceptable value of 15 Hz. The design forthe bandpass and the lowpass filter was then set toa damping factor ζ = 2, a cut-off frequency of 10Hz and a sampling frequency of 20 Hz.

4.5. Calibration

In this section, sensor noise is added to the simula-tion. In order to justify the choice of the controllerarchitecture to promote for practical implementa-tion, four different command filtered, incrementalcontrollers are briefly compared:

The PI-INDI controller that consists of stan-dard NDI in the outer loop and INDI in theinner loop, both robustified by integral action;

The R-INDI controller that consists of stan-dard NDI in the outer loop and INDI in theinner loop, both robustified by discontinuouscontrol, as in [12];

The PI-IBKS controller that consists of the

IBKS controller with integral action in bothloops, as shown in [10];

The R-IBKS controller that consists of theIBKS controller with discontinuous terms inboth loops.

Since the different controllers will compete forperformance, it is adequate for the tuning of theirparameters to be performed in the same systematicway. In short, the method employed consists in set-ting intervals for the gains of each controller andthen recording some performance variables for eachsimulation run with a different combination of pa-rameter values inside those intervals. In order tosimplify the method, each controller was assumedto have four parameters only - one proportional gainand one robust control gain for each loop. For sim-plicity, the gains of the inner and outer loops weretested separately. For each controller, after obtain-ing data from every simulation, an nv-by-ns perfor-mance matrix D is built, with nv being the numberof performance variables and ns the number of sim-ulations. Then, a cost function is quantified, foreach channel (c ∈ [φ, θ, β]), and for each pair s ofgains, as:

Jcs =1

‖w‖1

nv∑i=1

wiDis

Di

(38)

Di = maxs=1,··· ,ns

Dis

with Dis denoting the element (i, s) of D, Di be-ing the maximum element in each row of D and wbeing a vector weighing each performance variable.The optimal set of gains is found, for each loop,by finding the minimum of the unified performanceindex:

J∗ = mins=1,··· ,ns

Js = mins=1,··· ,ns

Jφs + Jθs + Jβs (39)

It can be argued that the actual optimal gains canbe different for each channel. For that reason, theperformance index is also minimized separately foreach control channel and two additional simulationsare performed to verify if using the different gainsin each channel results in performance increase.

After defining the optimal gains for each loop,the search intervals are refined and so are the fixedvalues that are used for one loop as the other isbeing tested. The procedure is repeated until theperformance obtained with the extreme values ofeach interval becomes nearly identical.

The performance variables used in all channels in-clude settling time, overshoot, tracking error RMS,control vector RMS (control effort) and controlmoving standard deviation (MSTD).

4.6. Position Control

Since the selected controller will be integrated inthe open-source autopilot software Ardupilot c© , itis adequate to choose the controller that seems to

7

respond better to the position controllers of thatframework. Therefore, the L1 and TECS controllerswere implemented in MATLAB/Simulink® and in-troduced in the simulation, as shown in Figure 4.The former is responsible for lateral navigation andthe latter by coupled height and speed control. Themanoeuvre selected for this comparison consists ofa lateral/directional “zig-zag” with two alternatingreference altitudes at a constant speed of 20 m/s.It was designed to excite the couplings between lat-eral and longitudinal motions. The results obtainedwith the four incremental controllers are depicted inFigure 5. Results obtained with the baseline linearcontroller from Ardupilot c© are also included to pro-vide a comparison point. It is noticeable in Figure5 that the altitude tracking of the TECS controllerdoes not present satisfactory results, and the lateralposition controller presents steady-state cross-trackerror. This can be explained by the fact that thetwo position controllers are decoupled and thereforethey are both inefficient when aggressive manoeu-vres are being simultaneously executed in the lon-gitudinal and lateral planes. Nevertheless, it canbe concluded that the four incremental controllersappear to have the same performance. Moreover, itcan be observed that the nonlinear control solutionsare more efficient in dealing with the motion cou-plings - the linear controller clearly loses more alti-tude when banking after completion of waypoints 3and 5. In order to perform also a brief quantitativeanalysis, the attitude tracking performance of eachcontroller is evaluated based on the tracking errorRMS in Table 1, where it is evident that the nonlin-ear, incremental controllers yield roughly the sameresults. It is arguable that the small performancedifferences in attitude tracking error RMS couldbe mitigated with a more optimal scheme to tunethe controllers. It is possible to conclude that themain advantage of the nonlinear control techniquesover the baseline linear control scheme proved tobe a combination of both performance increase andalso the ease of performing fine tuning. In sum,only at the expense of requiring a model for thecontrol dependent dynamics, the incremental con-trollers showed to be a simpler solution for attitudecontrol with better closed-loop performance.

Regarding the choice of the nonlinear controllerto be promoted for practical implementation, thePI-INDI controller was selected. The reason behindthis choice is related to the fact that NDI is slightlyconceptually simpler than Backstepping and pro-vides equally satisfying results. Furthermore, it isthe nonlinear approach which is most inline with thearchitecture of the baseline linear controller, whichsuggests that its implementation in the Ardupilot c©

software should be taken as a first step before en-gaging on more elaborate control approaches.

Table 1: Waypoint navigation performance metrics.

ControllerTracking RMSE [deg]

φ θ β

PI-INDI 0.27 0.23 0.37× 10−4

R-INDI 0.10 0.14 3.38× 10−4

PI-IBKS 0.28 0.20 1.02× 10−4

R-IBKS 0.28 0.22 9.38× 10−4

Linear 0.37 0.48 21.48× 10−4

4.7. Robustness

Some robustness tests were performed on the PI-INDI controller before engaging on its practical im-plementation. These allowed concluding that thecontroller is relatively insensitive to uncertainties inthe control derivatives up to 100%, with the pitchresponse being more susceptible to degradation; italso showed to be robust against inertia tensor mis-matches as performance was not seriously compro-mised for deviations of ±50%. Given the designassumptions of the incremental controller, its sensi-tivity to actuator time-delay was studied and it wasverified to achieve satisfactory results with delays ofup to 150 ms, which is admissible for practical inte-gration. It was also confirmed that the incrementalcontroller is dependant on high sampling frequency,with similar results for 50 Hz and 100 Hz. The di-rectional channel showed to be the most sensitiveto low control frequencies (< 40 Hz). Finally, ro-bustness against external disturbances was analysedwith the introduction of simulated wind, leading tothe conclusion that it has a serious effect on perfor-mance with wind speeds of up to 10 m/s.

5. Ardupilot c© integration

This final section concerns the pre-experimental im-plementation and testing of the nonlinear control al-gorithm in the Ardupilot c© framework. Fortunately,the software includes a native software-in-the-loop(SITL) simulator that allows testing the code underdevelopment without the need for any additionalhardware and it can interface with a number ofdifferent vehicle simulators to obtain sensor data,either the ones built in the Ardupilot c© code or sev-eral external alternatives. In this project, the JS-BSIM simulator was used. As for the fixed-wingmodel, the already available model for the Rascal110 fixed-wing aircraft was used, in order to takeadvantage of the fact that the control derivativesused by JSBSIM can be easily extracted from theunderlying code, which in turn allows designing theINDI controller for nominal conditions.

The discretization of the bandpass filter (fromwhich the angular accelerations are obtained), thelowpass filter (used to delay the angular rates sam-pling) and the command filters must take into ac-count the fact that, in a real-time system, the sam-

8

Waypoints L1

TECS

Incrementalcontroller Actuators

Aerosonde®

model

Lowpass Filter

Bandpass Filter Sensors

φd

ω

ω0

Position control

θd

Figure 4: Block diagram of the final Simulink® control architecture.

North position [m]400 600 800 1000 1200 1400 1600 1800

Eas

t pos

ition

[m]

-100

-50

0

50

100

WP1

WP2

WP3

WP4

WP5

WP6

WP7

PI-INDIR-INDIPI-IBSR-IBSLinear

North position [m]400 600 800 1000 1200 1400 1600 1800

Alti

tude

[m]

190

200

210

WP1

WP2

WP3

WP4

WP5

WP6

WP7

Figure 5: Position control results for controller comparison.

pling time may vary with time. Therefore, its im-plementation in Ardupilot c© was changed from theone in MATLAB/Simulink® by considering finitedifferences with non-uniform grid spacing.

The Ardupilot c© code includes sideslip compen-sation by means of lateral acceleration feedback.However, for many small UAVs, this setup maybe inefficient if the aircraft does not have sufficientfuselage side area to produce a measurable lateralacceleration. For that reason, active sideslip controlis optional in Ardupilot c© control and the ruddercan then be used to simply maintain coordinatedflight which, in theory, occurs at zero sideslip aswell. To retain this design (and operational) flex-ibility, the INDI controller was adapted in a waysuch that when active control of the sideslip angleis not desired, the outer loop performs dynamicsinversion for the roll and pitch rates only, whilstthe yaw rate is fed directly to the inner loop tokeep the flight coordinated. The main results ob-tained with JSBSIM are depicted in Figure 6 andcorrespond to the position control of the fixed-wingaircraft using both the PI-INDI and baseline linearattitude controllers, without active sideslip regula-tion. The 8-shaped manoeuvre was selected because

it allows studying the response of the aircraft toaggressive manoeuvring, its ability to keep a coor-dinated flight and also its response to a roll refer-ence doublet. Inspection of Figure 6 suggests thatthe controllers have very similar performance andthe lateral position tracking appears to be achievedefficiently. A deeper look into the quantitative re-sults of Table 2 allows concluding that the attitudetracking presents roughly the same quality in bothcases, with the most significant difference regard-ing the sideslip regulation task. This suggests, asexpected, that the nonlinear controller is more ef-ficient at keeping the flight coordinated. Addition-ally, the altitude RMS is marginally smaller withINDI, as the coupled nonlinear controller is moreefficient at maintaining pitch when banking, as seenalso in the MATLAB/Simulink® simulations.

6. Conclusions

This work addressed the practical applicability ofincremental control to the attitude control prob-lem of fixed-wing aircraft. The first results wereobtained from a MATLAB/Simulink® implemen-tation with a nonlinear model for the Aerosonde®

UAV, for which both IBKS and INDI approaches

9

East position [m]-400 -300 -200 -100 0

Nor

th p

ositi

on [m

]

-200

-100

0

100

200

300

400

500

600

INDI Linear

Figure 6: JSBSIM simulation results without activesideslip control.

Table 2: Performance metrics for INDI and linearcontrollers in the JSBSIM simulation.

ControllerAltitude tracking Tracking RMS [deg]

RMS [m] φ θ β

INDI 1.461 4.98 0.84 7.94

Linear 1.726 4.85 1.11 11.9

yielded satisfactory results when command filterswere integrated in each framework and the actua-tors dynamics were included in the simulation. Thesuitability of the bandpass filter to obtain the an-gular acceleration measurement was verified. Then,two INDI and two IBKS controllers, with either in-tegral or discontinuous control in both loops, wereconsidered for practical implementation. A sim-ple optimization method was developed to tune thecontrollers under a common criterion. The posi-tion and attitude controllers from Ardupilot c© wereintegrated in the simulation environment in orderto assess which nonlinear technique maximizes theperformance of the integrated design and the re-sults endorsed the expected conclusion that perfor-mance can be increased with the adoption of nonlin-ear control, which presents more intuitive tuning, atthe expense of knowledge on the control derivatives.The PI-INDI controller was then selected for prac-tical implementation. Robustness tests were per-formed to appraise some limitations of the chosencontroller, validating important robustness prop-erties for practical implementation, including therequirements imposed by incremental control de-sign. Simulation results using the JSBSIM sim-ulator and the Ardupilot c© implementation of the

PI-INDI controller, without active sideslip control,were very satisfactory as the controller showed toachieve better performance than its linear counter-part at keeping the flight coordinated and maintain-ing altitude. Unfortunately, due to calendar incom-patibilities, the experimental flight tests could notbe performed before the submission deadline of thisproject and are left out as future work.

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