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Bodycentric modelling, identification, and acceleration tracking control of a quadrotor UAV Article
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Alkowatly, M. T., Becerra, V. M. and Holderbaum, W. (2015) Bodycentric modelling, identification, and acceleration tracking control of a quadrotor UAV. International Journal of Modelling, Identification and Control, 24 (1). pp. 2941. ISSN 17466172 doi: https://doi.org/10.1504/IJMIC.2015.071697 Available at http://centaur.reading.ac.uk/39735/
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Body-centric Modelling, Identification, andAcceleration Tracking Control of a Quadrotor UAV
Mohamad T. AlkowatlySchool of Systems Engineering,University of Reading,Reading, RG6 6AX, UKE-mail: [email protected]
Victor M. BecerraSchool of Systems Engineering,University of Reading,Reading, RG6 6AX, UKE-mail: [email protected]
William HolderbaumSchool of Systems Engineering,University of Reading,Reading, RG6 6AX, UKE-mail: [email protected]
Abstract: This paper presents the mathematical development of a body-centric nonlineardynamic model of a quadrotor UAV that is suitable for the development of biologicallyinspired navigation strategies. Analytical approximations are used to find an initial guessof the parameters of the nonlinear model, then parameter estimation methods are usedto refine the model parameters using the data obtained from onboard sensors duringflight. Due to the unstable nature of the quadrotor model, the identification process isperformed with the system in closed-loop control of attitude angles. The obtained modelparameters are validated using real unseen experimental data. Based on the identifiedmodel, a Linear-Quadratic (LQ) optimal tracker is designed to stabilize the quadrotorand facilitate its translational control by tracking body accelerations. The LQ tracker istested on an experimental quadrotor UAV and the obtained results are a further meansto validate the quality of the estimated model. The unique formulation of the controlproblem in the body frame makes the controller better suited for bio-inspired navigationand guidance strategies than conventional attitude or position based control systems thatcan be found in the existing literature.
Keywords: Quadrotor; Modelling; System Identification; Linear Quadratic TrackingControl.
Reference to this paper should be made as follows:Alkowatly, M.T., Becerra, V.M.,and Holderbaum, W. (2014) ‘Body-centric Modelling, Identification, and AccelerationTracking Control of a Quadrotor UAV’, International Journal of Modelling, Identificationand Control, Vol. x, No. x, pp.xxx–xxx.
Biographical notes: Mohamad T.Alkowatly is a current PhD student in Cyberneticsresearching visual bio-inspired aerial guidance and navigation strategies. He has an MSc inAeronautics Engineering (2009, UK) and a BSc in Informatics Engineering (2006, Syria).
Victor M. Becerra is a Professor of Automatic Control. His research interests includecomputational optimal control, nonlinear control, system identification, robotics, andartificial intelligence.
William Holderbaum is a Professor in Mathematical Engineering. His research interestsinclude boolean input systems, geometric control, rehabilitation engineering, and smartgrid systems.
Copyright c© 2014 Inderscience Enterprises Ltd.
Int. J. Modelling, Identification and Control, Vol. x, No. x, 2014 2
1 Introduction
Quadrotor Unmanned Aerial Vehicles (UAV) havegained considerable interest as a research platform[1, 2, 3, 4]. This is because, besides being affordable,they are relatively lightweight and quite manoeuvrablewhich makes them suitable for testing different flightstrategies. This manoeuvrability comes mainly from thefact that quadrotors are unstable in open-loop, and thisemphasizes the role of the control system for operatingthem. This in turn stresses the importance of developingfaithful models both for simulation and control designpurposes.
In order to obtain a particular model for a specificquadrotor, either the quadrotor dynamics have to beapproximated or the model parameters, such as theaerodynamic coefficients and moments of inertia, haveto be accurately identified. In the first approach, systemidentification has been used to identify the attitudeangular velocity transfer function of a helicopter [5],thus obtaining a linear and decoupled model of thevehicle. In [6], an Autoregressive structure with Externalinput (ARX) model of a helicopter has been identifiedby employing neural networks. The second approach,on the other hand, allows to obtain the nonlinearequations of motion of the vehicle by finding or refiningthe model parameters of a specific quadrotor UAV.It is possible to estimate most of these parametersexperimentally as discussed by Derafa et al [7], butthat requires the use of a purpose made test benchto hold the UAV and allow appropriate motion andmeasurements. Alternatively Abas et al. [8] used theunscented Kalman filter to estimate the moments ofinertia and the rotor inertia parameters of the quadrotorwhile the aerodynamic parameters were assumed tobe experimentally estimated in advance. However, noaction has been taken to validate the resulting modelparameters other than the convergence of the unscentedKalman filter, which does not imply that acceptablemodel parameters have been found. Additionally,extending this method to estimate all model parametersmight be limited by their identifiability [9]. Finally theauthors in [10] used Prediction Error Methods (PEM)to identify a linear discrete state-space model of aquadrotor UAV. Although such model can be useful forthe control design process, the linearity of the modelwill adversely affect simulation fidelity and restrict itsvalidity to small neighbourhood of the linearisationpoints.
The first take towards identifying the parametersof a nonlinear model of coaxial helicopter has beenpresented in [11] where some of the model parameters arefound experimentally or analytically while the remainingparameters are estimated using the Covariance MatrixAdaptation Evolution Strategy (CMA-ES). Note thatthe presence of the stabilizer bar as well as the tandemcounter rotating propellers in the coaxial helicopter
greatly reduces the rotational dynamics instability ifnot eliminating it all together. No such mechanicalstabilization aid is available on the quadrotor, thusthe system identification process has to be adjusted tocope with the fast unstable open loop dynamics of thequadrotor.
Many quadrotor control techniques have beenpresented in the literature. Most techniques implementseparate angular and translational control systems basedon a separate models for the rotational and translationaldynamics. Simple PID based control systems have beenpresented [12, 13, 14, 15] where the attitude dynamicsare simplified to decouple the attitude angles and thenseparate feedback loops are used to control each angle.Although the method is reported to work well, ensuringcontroller stability in all flight conditions is challenging,especially given the use of oversimplified decoupledmodels.
Sliding mode control of a quadrotor UAVs hasbeen implemented and tested by Öner et al. [16]and Bouabdullah et al. [13]. These two works reportcontrasting results, so while sliding mode controlhas been found to provide a very good trackingperformance by the former, the latter concluded thatthe switching nature and the chattering associatedwith sliding mode control is not suitable for quadrotorcontrol. Backstepping techniques have been used forposition tracking in [17], and the Linear QuadraticRegulators (LQR) has been applied for attitude controlof quadrotors in [18] where it has provided goodstability, but has shown to be slow to react possiblydue to modelling uncertainties. Another LQR strategyis demonstrated in [16] providing effective positiontracking.
Regardless of the design method used for quadrotorcontrol, all of the aforementioned methods either controlthe body attitude only or control the translationaldynamics through kinematic variables obtained in aninertial frame (position and velocities). Estimating suchkinematic variables requires special sensors and rathercomplicated calculations and filtering schemes. On theother hand, biologically observed visual navigationstrategies solve navigation problems without relying onvariables in an inertial frame [19, 20, 21, 22, 23], andrely on a body-centric frame of reference instead [24].Hence, it is more convenient, and more coherent, todevelop model based controllers in the same frame ofreference. Advantages of employing biologically inspirednavigation in UAVs are lighter onboard sensor suites andless computational power requirements [25, 26].
It should be mentioned that it is possible touse the aforementioned conventional control system toserve a bio-inspired navigation method. However, it ismore convenient, and more coherent, to develop controlsystems that do not rely on equivalent variables notnormally available in animal species. Furthermore, if thecontrol system is used to validate the applicability of a
Copyright c© 2014 Inderscience Enterprises Ltd.
Body-centric Modelling, Identification, and Acceleration Tracking Control of a Quadrotor UAV 3
bio-inspired navigation method, it is important not touse variables that the navigation strategy claims not touse. As the control system presented in this paper is usedto serve the position and velocity free bio-inspired visualnavigation method presented in [27], the control systemshould not use such variables employed conventionally.Thus, body accelerations are more suitable references forthe tracking control system. Attitude tracking controlhas been assumed to serve as acceleration trackingcontrol in [28]. However, in this research, LQ trackingcontrol design method is used to design an explicit bodyacceleration tracking control.
This paper demonstrates the modelling,identification and control of a quadrotor helicopter ina body-centric reference frame. A unique feature ofthis controller is that it provides a means to controlthe quadrotor translation by tracking body frameaccelerations, in contrast to tracking a trajectory definedin an inertial frame. The relation between the the trackedacceleration and the body attitude is analysed. To theauthors’ knowledge, this paper is the first to addressthe issues associated with numerical estimation, andvalidation of the parameters of a nonlinear quadrotormodel without requiring any special test benches.The paper provides a numerical quantification of theenhancement achieved by using the estimated modelparameters compared to the analytically approximatedparameters, and it validates the obtained modelparameters using unseen measured data. The bodyacceleration tracking control system, developed in thispaper, is used as an additional validation step for theidentified model. Finally, the model and control systempresented in this paper are used with the bio-inspirednavigation strategy demonstrated in [27].
This paper is organized as follows: The modelingof the quadrotor in a body-centric frame is presentedin section 2, estimation of model parameters usinganalytical approximations is presented in section 3,the formulation of the system identification processis presented in section 4, system identification resultsand model validation are presented in section 5, thedevelopment of a tracking low level controller basedon the proposed model is presented in section 6, andthe results of low level control experimental testing arepresented and discussed in section 7.
2 Body-Centric Quadrotor Model
The quadrotor model developed here is similar to thatdeveloped in [12]. The model formulation is brieflypresented here. Two Cartesian coordinate frames aredefined for the purpose of quadrotor modelling. Theearth surface fixed frame with axes 1e
x,1ey and 1e
z alignedwith North, East and Down (NED) local directionsrespectively, which is safely assumed to be an inertialframe for the purposes of the slow and short flight of
the quadrotor. The second frame is a body fixed framewith its origin at the body center of mass and axes 1x,1y
and 1z are aligned with forward, right (starboard), anddown body orientations. The body and earth coordinateframes, motor numbering, and their positive rotationdirection are illustrated in Figure 1. Note that the axesorientation is different from those used by [12].
Ω2 Ω4
Ω3
Ω1
1x
1y1z
1ey
1ex
1ez
Figure 1 Top view of the quadrotor showing thedefinition of used coordinates frames, motornumbering and positive motor rotation direction.Axes 1e
z and 1z are pointing into the page
2.1 Attitude and Rotation Representation
To obtain body attitude starting from the earth frame,one performs three right handed consecutive rotations(yaw, pitch, roll) with angles (ψ, θ, φ) about 1z, thenew 1y, and the new 1x axes, respectively. These threerotations define the transformation matrix Rb/e.
The quadrotor angular velocity in the earth frameωe
b/e = [φ, θ, ψ]T can be found from quadrotor angularvelocity in body frame ωb
b/e = [p, q, r]T as follows [29]:
ωeb/e =
1 tan(θ) sin(φ) tan(θ) cos(φ)0 cos(φ) − sin(φ)0 sin(φ)/ cos(θ) cos(φ)/ cos(θ)
ωbb/e (1)
2.2 Quadrotor Body Dynamics
Using Newton’s Euler formalism, the body dynamics areexpressed in the body fixed frame as:[
mI3×3 03×303×3 Iq
][V b
ωbb/e
]+
[ωb
b/e ×mVb
ωbb/e × Iqω
bb/e
]=
[Fb
τ b
](2)
4 M.T. Alkowatly et al.
where Iq is the moments of inertia tensor of thequadrotor (kg m2), V b = [u, v, w]T is the body frametranslational velocity vector (m/s), F b is the force vectoracting on the quadrotor in body frame (N), and τ b is thetorque vector acting on the quadrotor in body frame (Nm). The quadrotor is assumed to be symmetric about itsbody principal axes which are assumed to coincide withthe body frame. This assumption cancels all productsof inertia and the inertia matrix becomes the diagonalmatrix Iq = diag(Ixx, Iyy, Izz).
The external forces acting on the quadrotor body arethe weight force (mg) and the thrust forces generated bythe four propellers Ti. Each thrust forces is modelled as:
Ti = b Ω2i , i = 1, 2, 3, 4 (3)
and the total thrust force Ta = T1 + T2 + T3 + T4 isalways aligned with the body 1z axis in the negativedirection.
The total torque acting on the quadrotor iscomposed of the control torques and gyroscopic effecttorque. Control torques τx, τy that generate a positiverolling and pitching moment respectively can beexpressed as:
τx = `(T4 − T2)1x
τy = `(T1 − T3)1y
(4)
The aerodynamic drag torque Qi acting on propelleri is modeled as:
Qi = dΩ2i , i = 1, 2, 3, 4 (5)
The total drag torque that generates a positive yawingmoment is expressed as:
τz = d(Ω22 + Ω2
4 − Ω21 − Ω2
3)1z (6)
Body angular rates induce a gyroscopic effect torqueτr on each of the rotating propellers due to rotor inertiaJr (rad/s), and the total imbalance Ωr in the propellerangular velocities. τr can be expressed as:
τr = Jr(ωbb/e × 1z)Ωr =
Jr q Ωr
−Jr p Ωr
0
(7)
where
Ωr = Ω2 + Ω4 − Ω1 − Ω3 (8)
By defining the following variables:
U1 = (Ω21 + Ω2
2 + Ω23 + Ω2
4)
U2 = (Ω24 − Ω2
2)
U3 = (Ω21 − Ω2
3)
U4 = (Ω22 + Ω2
4 − Ω21 − Ω2
3)
(9)
the quadrotor equations of motion expressed in the bodyfixed coordinates frame in addition to the local earthattitude kinematics (φ, θ, ψ) can be written as follows:
p = [q r (Iyy − Izz) + Jr q Ωr + ` b U2] / Ixx
q = [p r (Izz − Ixx)− Jr p Ωr + ` b U3] / Iyy
r = [p q (Ixx − Iyy) + d U4] / Izz
u = r v − q w − g sin(θ)
v = p w − r u+ g cos(θ) sin(φ)
w = q u− p v + g cos(θ) cos(φ)− b U1 /m
φ = p+ q tan(θ) sin(φ) + r tan(θ) cos(φ)
θ = q cos(φ)− r sin(φ)
ψ = q sin(φ)/ cos(θ) + r cos(φ)/ cos(θ)
(10)
3 Initial Estimation of Quadrotor ModelParameters
3.1 Initial Estimation of Aerodynamic ModelParameters
Initial Estimation of Lift Factor. The aerodynamicforces and moments generated due to propeller rotationare derived in [30] using the blade element theory. Noassumption is made in [30] on the size of the propeller,hence the derivations presented in [30] are valid for largerpropellers like the one used in this work. If the ratio ofthe lateral quadrotor velocities to the propeller’s angularvelocity is neglected, the lift force generated by thepropeller is approximated by:
T = 2π ρ Nb c Ω2R3p
[θtw0
6−θtwf
8− λ
4
](11)
where θtw0and θtwf
are used to model the propellervariable twist angle at a point r along its radius, whichis given by:
θtw(r) = θtw0 − θtwf
(r
Rp
)(12)
Values of θtw0and θtwf
can be found from the propellergeometry using least squares fitting.
From (3) and (11) the lift factor b is expressed as:
b = 2π ρ Nb c R3p
[θtw0
6−θtwf
8− λ
4
](13)
Additionally, if a closed-loop stabilizing controller isavailable (which is the case in this work), the lift factorb can be estimated experimentally when the quadrotoris hovering, as the total thrust force should be equal tothe quadrotor weight. Thus b can also be written as:
b =mg
4Ω2h
(14)
It is important however to perform this experimentalmeasurement well above the ground to eliminate thecontribution of the ground effect.
Body-centric Modelling, Identification, and Acceleration Tracking Control of a Quadrotor UAV 5
Initial estimation of drag factor. The drag torque Q isderived in [30] under the same assumption used for liftforce, and can be approximated by:
Q = 2π ρ Nb c Ω2R4p
[Cd
16π+ λ
(θtw0
6−θtwf
8− λ
4
)](15)
Substituting (15) in (5) and solving for d results in:
d = 2π ρ Nb c R4p
[Cd
16π+ λ
(θtw0
6−θtwf
8− λ
4
)](16)
3.2 Initial Estimation of the Moments of Inertia
The estimation of the quadrotor moments of inertiaIxx, Iyy, Izz about the body axes, and the rotor inertia Jrabout its rotational axis is simplified by segmenting thequadrotor into simple shapes, working out the individualmoments of inertia, and then adding them together[12]. The quadrotor frame is approximated by two rods,the electronics are approximated by a box, the motorsare approximated by a cylinder, and the propellers areapproximated by a variable density flat cylinder.
4 System Identification for Refining ModelParameters
The Prediction Error Method (PEM) has been chosen toestimate the parameters of the nonlinear model (10) dueto its ability to provide good estimates even if the dataare collected from a closed-loop system [9]. The PEMfinds the parameter vector Θ by using an optimizationalgorithm to minimize the following error function:
E(Θ, ZT ) =1
N
N∑t=1
ε(Θ, t)TΛε(Θ, t) (17)
where ZT = (u(t), y(t)), t = 1, . . . , N is the trainingdata set, ε(Θ, t) = y(t)− y(Θ, t) is the differencebetween measured and predicted system outputs, N isthe total number of time steps in the data, and Λ isa diagonal weighting matrix used to reflect the relativeimportance of the outputs in the minimization process.To prevent the optimization algorithm from being biasedin favour variables with large mean values, matrix Λ iscalculated as shown in (18):
Λ = diag
(1
y2i
)(18)
where yi = 1N
∑Ni=1|yi| is the mean of the absolute value
of the output variable yi. This will normalize the differentvariables values for the optimization and will makethe error function E (17) unitless, which is useful forcomparing results of different data sets as well.
The parameter vector Θ to be identified is definedas:
Θ =[Ixx, Iyy, Izz, Jr, b, d
]T (19)
where the initial guess of Θ is obtained from theanalytically approximate values as described in section3. The search space is constrained within an orderof magnitude up and down the initial guess, whilethree other linear inequality constraints are added toreflect the perpendicular axis theorem constraint on themoment of inertia so that the solution must satisfy:
Ixx ≤ Iyy + Izz
Iyy ≤ Ixx + Izz
Izz ≤ Iyy + Ixx
(20)
In principle, the system identification can beperformed both in open loop and closed loopconfigurations. In an open loop configuration, the modelinputs are the motor angular velocities (Ω1 to Ω4) andthe model output is the vehicle measured state y asshown in Figure 2. In the closed loop configuration, afeedback stabilising control system is employed so thatthe inputs of system model used for identification arethe reference commands of the stabilising control (u)and the vehicle measured state y as an output. Althoughthe data is collected from a closed loop system in bothcases, the two configuration are considerably differentwhen using the prediction error method to estimatethe model parameters. The instability of the open loopmodel will cause high sensitivity of the error functionmaking the identification process very challenging. Thisis because the input data, although collected from theclosed loop system, will fail to stabilise the unstablequadrotor system in open loop due to the presence ofmodel uncertainties. Therefore the parameter estimationis performed using closed-loop data as this addressesboth issues by stabilizing the predictor and reducing thesensitivity of the error function [9]. This is possible sincein this work a prior stabilizing controller was available.
The identification experiment is performed in closed-loop (Figure 2) by choosing the input vector u to bethe remote pilot commands comprising the attitude andthrottle reference:
u =[φr, θr, ψr,Υr
]T (21)
while the output vector y is chosen to be:
y =[φ, θ, ψ, p, q, r, ax, ay, az
]T (22)(23)
where body measured accelerations ax, ay, az areincluded in the outputs to enhance the observability ofthe lift factor b. These accelerations are given by:
ax = −g sin(θ)
ay = g cos(θ) sin(φ)
az = g cos(θ) cos(φ)− b U1 /m
(24)
6 M.T. Alkowatly et al.
The pilot commands references are processed bythe on-board closed-loop feedback controller Cfb toproduce the four motor angular velocities commands[Ωr1,Ωr2,Ωr3,Ωr4] based on the measured quadrotorattitude and attitude rates. Motor commands are thensubjected to the actuator dynamics which result in theactual motor angular velocities that drive the modelpresented in (10). This scheme is illustrated in Figure 2
Quadrotor dynamics
u (ϕr, θr, ψr, Γr)
Stabilizingcontroller
(Ω1, Ω2, Ω3, Ω4)(Ωr1, Ωr2, Ωr3, Ωr4) Actuatordynamics
y(ϕ, θ, ψ, p, q, r)
Figure 2 Control scheme used for closed-loopidentification of quadrotor system
5 System Identification Results
5.1 Experimental Platform
Quadrotors are unstable systems that cannot be flownunder open loop control. In this work a UAV platform forresearch by Ascending Technologies GmbH known as thePelican has been used. The platform comes with an on-board stabilizing controller which is implemented on theon-board ARM7 embedded microcontroller and runs at1KHz. This controller allows the user to fly the quadrotorby providing attitude reference commands (roll, pitch,and yaw angles) and a throttle value by means of aremote controller. This allows the data acquisition tobe performed while flying a closed-loop stable system.Figure 3 shows a side view of the Pelican quadrotorhelicopter employed in this work.
Brushless DC motors, like those used by the Pelican,are known to have a small time constant (10ms asreported in [31]). Thus, actuator dynamics are neglectedhere for simplicity, implying [Ωr1,Ωr2,Ωr3,Ωr4] =[Ω1,Ω2,Ω3,Ω4]. The supplied stabilizing controller hasknown structure and parameters, leaving the modelparameter vector Θ in (19) to be estimated by anoptimization algorithm.
Output body rates (p, q, r) are provided by onboardMicro Electro Mechanical system (MEMS) gyroscopeswhile the attitude angles (φ, θ, ψ) are estimatedby the onboard Attitude and Heading ReferenceSystem (AHRS). The body accelerations (ax, ay, az) aremeasured by the onboard MEMS accelerometers, andcorrected for gravity using the AHRS attitude estimate.Input and output data are sampled at a rate of 50Hzwhile the quadrotor attitude and throttle is manually
Figure 3 Side view of the Pelican quadrotor helicopter
varied during the experiments. Three experimental datasets, with a duration of 500 seconds each, are collectedfor the purposes of system identification and modelvalidation.
5.2 Parameter Estimation
The initial model parameters values are obtained usingthe analytical approximations described in section 3,and shown in Table 1. These values are used as aninitial guess for the system identification process, andsubsequently refined to achieve more accurate values.
A Quasi-Newton (QN) constrained optimizationmethod from MATLAB’s optimization toolbox is usedto minimize (17) where the gradients are evaluatednumerically and the Hessian is approximated usingthe BFGS method. A refined estimate of Θ is foundusing 500 seconds of real experimental data. Smallertime segments of the experimentally measured againstmodel predicted outputs are shown in Figures 4 and5 when using the initial guess and the optimizedmodel parameters, respectively. These time segmentsof the fitted variables have been chosen to showthe fitted variables under excitation. A considerableprediction enhancement of 47.61% is achieved by theoptimization process where the error function value afterthe optimization is (E = 7.67) compared to (E = 14.64)for the initial guess. The parameters values obtained arelisted in Table 1.
5.3 Validation of The Identified Model
In order to ensure that the identified model parametersrepresent well the system dynamics, the model isvalidated against two data sets of unseen experimentalmeasurements collected from the quadrotor when itwas flown manually and all its degrees of freedomwere varied. Figure 6 shows different time segments ofmeasured outputs of the validation data set against
Body-centric Modelling, Identification, and Acceleration Tracking Control of a Quadrotor UAV 7
24 26 28 30 32 34 36 38 40−0.4−0.2
00.2
φ
156 158 160 162 164 166 168 170−0.2
00.20.4
θ
400 402 404 406 408 410 412 414 416 418 420
−0.50
0.5
ψ
155 160 165 170−4−2
02
ax
82 84 86 88 90 92 94 96 98 100 102 104−4−2
02
ay
40 45 50 55 60 65
−6−4−2
0
az
24 26 28 30 32 34 36 38 40−1
01
p
156 158 160 162 164 166 168 170−1
01
q
400 402 404 406 408 410 412 414 416 418 420−2
02
r
Time (s)
Measured Predicted
Figure 4 Comparison of model outputs using the initialparameter guesses and the training data set(segments). Angles (φ, θ, ψ) are given in radians,accelerations (ax, ay, az) are given in m/s2, andbody rates are in rad/s
24 26 28 30 32 34 36 38 40−0.4−0.2
00.2
φ
156 158 160 162 164 166 168 170−0.2
00.20.4
θ
400 402 404 406 408 410 412 414 416 418 420
−0.50
0.5
ψ
155 160 165 170−4−2
02
ax
82 84 86 88 90 92 94 96 98 100 102 104−4−2
02
ay
40 45 50 55 60 65−6−4−2
0
az
24 26 28 30 32 34 36 38 40−1
01
p
156 158 160 162 164 166 168 170−1
01
q
400 402 404 406 408 410 412 414 416 418 420−2
02
r
Time (s)
Measured Predicted
Figure 5 Quasi-Newton Optimization for model outputfitting in closed-loop using the training data set(segments). Angles (φ, θ, ψ) are given in radians,accelerations (ax, ay, az) are given in m/s2, andbody rates are in rad/s
Table 1 Initial approximations along the systemidentification refined model parameters
Parameter Initial value Refined valueIxx 1.57× 10−2 1.21× 10−2
Iyy 1.57× 10−2 1.355× 10−2
Izz 2.52× 10−2 2.179× 10−2
Jr 8.031× 10−5 1.143× 10−4
b (Analytical) 2.599× 10−6 -b (Experimental) 2.428× 10−5 2.256× 10−5
d 1.247× 10−7 3.679× 10−7
the predicted model outputs using the optimizedparameters. Additionally, the value of the error functionE evaluated when using the initial and the refined modelparameters for the training, as well as two unseen datasets are shown in Table 2. There is consistency in theimprovement of fitting quality from the initial guessacross the training and validation data sets. The modelwith the optimized parameters is shown to provide about48% better fit than the model with the initial parametersvalues for all the data sets available. This indicatesthat the model with the identified parameters capturesthe quadrotor dynamics better than the model withanalytically approximated parameters with no signs ofover fitting.
320 325 330 335 340 345 350
−0.20
0.2
φ
175 180 185 190 195 200−0.2
00.20.4
θ
50 52 54 56 58 60 62 64
−0.50
0.5
ψ
175 180 185 190 195 200
−202
ax
320 325 330 335 340 345 350
−202
ay
75 80 85 90 95−2−1
0
az
320 325 330 335 340 345 350−1
01
p
175 180 185 190 195 200−1
01
q
50 52 54 56 58 60 62 64−3−2−1
01
r
Time (s)
Measured Predicted
Figure 6 Model validation using one of the unseen datasets (segments). Angles (φ, θ, ψ) are given inradians, accelerations (ax, ay, az) are given inm/s2, and body rates are in rad/s
It is noted that the variable θ has lower fittingquality compared to other variables both using thetraining and the validation data sets shown in Figures
8 M.T. Alkowatly et al.
Table 2 Total model fitting error E using initial guessand refined model parameters showing fittingenhancement percentage for each data set
Data set Usinginitialguess
Usingrefinedparameters
Modelfittingenhancement
Trainingdata set
14.64 7.67 47.61%
Unseendata set 1
18.93 8.71 48.59%
Unseendata set 2
16.81 8.79 47.72%
5 and 6, respectively. Consequently, the variable axshows similar quality as it is directly related to θ. Thesame results have been found even when increasing theweight corresponding to those variables in the weightingmatrix Λ. This indicates that there is a trade offbetween fitting θ and ax, and fitting all other variablesin order to minimize the total error function E (17).The relative inaccuracy in fitting θ may be explained byminor violations of modelling assumptions, including forinstance the presence of an small imbalance in the centerof mass on the 1y quadrotor axis.
Nevertheless, the fitting of the corresponding bodyrate q is acceptable, although still more sluggish thanother rates, which explains the observed lag betweenthe predicted and measured values of θ. Note that theprediction error of θ is bounded for all the data setsavailable (L∞(θ − θ) < 0.58).
As a final remark, despite the good predictionaccuracy achieved by performing the identification inclosed-loop, it has been found that the best modelparameters achieved fail to predict the quadrotordynamics for more than 2 seconds if prediction isperformed in open loop. This is due to the unstablenature of the open loop system.
6 Quadrotor Low Level Control
In this section the design of a control systemis presented based on the body-centric model ofthe quadrotor dynamics (10). To fit within abiologically inspired navigation scheme, it is desired todevelop a biologically plausible low-level control. Thedefinition of a biologically plausible control system hasdifferent interpretations from different points of view(neurological or ecological). Moreover, the factors thatmake a control system biologically plausible are differentdepending on the animal species used as reference.In this work, the guidelines for biological plausibilitysuggested in [24] are followed. Hence, a biologicallyplausible control should only use the informationequivalent to that provided by sensory organs such asthe vestibular system. Additionally, the system should
use this information in the body-centric fame of referencewhich is the motive to develop the model (10) in thebody frame.
Functionally, the desired control system must beable to stabilize the fast quadrotor rotational dynamicsin addition to facilitate the control of the quadrotortranslational dynamics by a higher level controllerwithout relying on kinematic variables that are definedin an external reference, such as position or velocity. Theattitude reference, particularly the roll and pitch angles,are an exception because they are defined by the gravityvector and body shape, both of which are perceived bymany animal species.
Based on the previous guidelines, and noting thesystem dynamic equations (10), the body velocitiesu, v, w do not affect the rest of the states and can bedropped from the state vector for the purposes of controlsynthesis. The state vector x then becomes:
x =[φ, θ, ψ, p, q, r
]T (25)
the control vector u is
u =[Ω1,Ω2,Ω3,Ω4
]T (26)
and the output vector y is
y =[ψ, ax, ay, az
]T (27)
where the body accelerations ax, ay, az, which depend onthe states and the input as shown in (24), are assumedto be available through the on-board accelerometers. Onthe other hand, the heading angle ψ is only availablethrough an external reference (like the earth magneticfield or an arbitrarily chosen heading reference) and itis included in the state and output vectors only to aidtesting and flying the quadrotor by a human operator. Itis clear from the model equations (10) that dropping theheading angle dynamics ψ has no effect whatsoever onthe quadrotor dynamics in the body frame, in contrastwith the case when the model is derived in an inertialframe [13].
The dynamic equations (now a subset of (10)) arelinearised about the equilibrium point (xeq,ueq) givenby:
xeq =[0, 0, 0, 0, 0, 0
]Tueq =
[Ωh,Ωh,Ωh,Ωh
]T(28)
where Ωh = 407 rad/s for the Pelican. The linearisedequations are discretised to obtain the system matricesΦ, Γ,C,D of a linear discrete state space representationof the quadrotor system:
x(n+ 1) = Φ x(n) + Γ u(n)
y(n) = C x(n) +D u(n)(29)
The values of the system matrices Φ, Γ, C,D are shownin the appendix.
Body-centric Modelling, Identification, and Acceleration Tracking Control of a Quadrotor UAV 9
6.1 Optimal Tracker Control Design
To stabilize the rotational dynamics in addition tocontrolling the translational dynamics of the quadrotor,the following control law is used:
u(n) = −Kx(n) + Fyr(n+ 1) (30)
where matrices K,F are the state feedback andreference feed-forward gains respectively. The rotationaldynamics are stabilized due to the state feedback termin (30). To allow the tracking of desired translationalreference, the output reference yr is defined as :
yr =[ψr, axr, ayr, azr
]T (31)
The references for body accelerations axr, ayr, azr enablethe translational control of the quadrotor along the threebody axes, while the yaw angle reference ψr allows theoperator to control the quadrotor’s heading. As this low-level control is designed to be used by a high level bio-inspired navigation control system, then the low-levelcontroller is chosen to be simple as high performance isnot required from it.
Linear Quadratic (LQ) optimal control methodshave been used to obtain the values of K and F in (30).Equation (24) shows that the system has direct feed-through of the control input in the output accelerationequations (resulting in a non-zero D matrix), thus thegain matrices have been found using the formulationpresented in [32] which assumes a constant referencesignal and involves the following performance index:
J =1
2
∞∑n=0
(y(n)− yr)TQy(y(n)− yr) + u(n)TRu(n)
(32)
The obtained values of K and F are shown in theappendix. Lastly, the overall control scheme is illustratedin Figure 7.
x
Quadrotor dynamicsyr C
K
+
-
Fy
Figure 7 LQ Tracker controller structure for low-levelcontrol
6.2 Tuning and Analysis
A Simulink model is developed implementing the controlstructure shown in Figure 7 using the nonlinear model
(10). The references are provided externally by theuser via a joystick. Initial tuning of the controllerweight matrices Qy and R was performed in simulationand further tuning was performed experimentally. Thefinal chosen values of weight matrices are given in theappendix.
The LQ tracker provides sub-optimal tracking dueto the linearisation of the non-linear quadrotor dynamicsin addition to unsatisfied assumptions in the optimalityproblem, such as the constant reference value. Besidesachieving sub-optimal tracking, LQ optimal controldesign methods guarantee nominal stability under mildassumptions[33].
It is useful to study the effect of tracking accelerationcommands on the system states. When the controlleris presented with a zero reference signal to track,the feedback term in control law (30) will stabilizethe quadrotor about its original equilibrium pointxeq = 0. However, when a non-zero reference signal yris presented, the feed-forward term will displace theequilibrium point at steady state to xss. This state canbe found from the system dynamics equations (29) whenthe control law (30) is used to produce the control inputvector u. Substituting (30) in (29) and dropping the timedependency results in:
xss = Φxss + Γ(−Kxss + Fyr) (33)
solving for xss gives:
xss = Πyr (34)
where Π is given by:
Π = (I −Φ + ΓK)−1ΓF (35)
and I is the identity matrix of the same dimensions asΦ. Using the parameter values shown in the appendix,matrix (I −Φ + ΓK) is invertible and the numericalvalue of Π is shown in the appendix.
Inspecting the value of Π reveals an interestingproperty. Firstly, positive longitudinal body accelerationaxr is tracked by demanding a negative pitch anglewhile positive lateral body acceleration ayr is trackedby demanding a positive roll angle. Thus, one canlimit the maximum attitude angles (at steady state)by limiting the values of the demanded accelerationreference. Limiting axr and ayr to values in the range[−ar, ar] results in a maximum pitch and roll angles of0.102 ar rad. This maximum value is not exact if thereference is not constant and because of nonlinearitiesin the dynamics. However, ar can be set to providea safe operation of the vehicle and to prevent drivingthe attitude angles too far from the point the nonlinearmodel was linearised at. Finally, the motors cannotchange their spin direction, and hence cannot generatea negative thrust. Therefore, the maximum feasibledownward acceleration cannot exceed the value of freefall gravity acceleration of 9.805 m/s.
10 M.T. Alkowatly et al.
7 Low Level Control Experimental Resultsand Discussion
The control system presented in the previous section isimplemented on the Pelican ARM embedded processorwhich runs the control loop at a frequency of 1 KHz.The state vector x is estimated by the onboard Attitudeand Heading Reference System (AHRS) while the outputreference vector yr is provided by the remote controller.The quadrotor is flown indoors and both the commandedbody accelerations axr, ayr, azr as well as the measuredbody accelerations corrected for gravity ax, ay, az arelogged at a rate of 50 Hz. Figure 8 shows 50 secondsof the experimental data showing the reference signalsagainst measured body accelerations.
−2
0
2
4
a x (m
/s2 )
−2
0
2
4
a y (m
/s2 )
0 5 10 15 20 25 30 35 40 45 50
−2
0
2
4
a z (m
/s2 )
Time (s)
Reference Measured
Figure 8 Reference vs. actual body accelerations(segment).
The proposed controller managed to achieve its aimof stabilizing the quadrotor attitude and providing amean to control the translational dynamics. In additionto the smooth and stable flight achieved experimentally,it can be seen that the tracking performance of bodyacceleration shown in Figure 8 is very satisfactory. Theprevious sentence has to be interpreted taking intoaccount the limitations of the linearised control andthe feed forward acceleration tracking employed. Asthe quadrotor dynamics (10) and output accelerationequations (24) are linearised around zero attitude angles,then the attitude contribution to changes in the bodyaccelerations are not fully captured. This becomesevident by comparing the tracking performance of smallaccelerations in the first 10 seconds of Figure 8 comparedto simultaneous, larger, and more sustained accelerationsaround second 35 in the same figure. Another artefactof linearising the system dynamics can be seen on thevertical body acceleration az which shows a tendencyto track upward (negative) reference accelerations betterthan downward accelerations as shown in Figure 8. This
is mainly due to linearising the nonlinear relation of thethrust force with respect to the motor angular velocity(11).
In addition to the linearisation limitations, it shouldbe noted that the acceleration tracking results shownhave been achieved without an acceleration feedbackloop correcting for any observed errors. Nevertheless,the tracking performance obtained was satisfactoryespecially given that the proposed control system willact under the higher-level visual control presented in[27], where the visual variables of interest will be undera feedback control cascaded with the control systempresented here. The tracking performance presented inFigure 8 has another use in the context of this paperas the tracking performance is partly a measure ofthe quality of the linearised model employed, which isderived from the identified nonlinear model.
Another experiment is performed where individualbody accelerations have been demanded separatelyto reduce the attitude coupling effect on the outputacceleration, which in not fully captured by the linearisedmodel. Figure 9 shows 3 different time segments of theexperiment each showing one of the body accelerationsbeing excited where the reference and measured bodyaccelerations are plotted. Achieving fairly accuratetracking performance as shown in Figure 9 reflects thequality of the model parameters obtained in this paper.This can be considered an additional validation stepin addition to performing the unseen data fitting testas well as the experimental control of the quadrotorunder a model based control system. Additionally, theseresults show that despite the relative difference in thefitting quality of variables θ and ax compared to theother variables has no noticeable effects on the qualityof the tracking of longitudinal and lateral accelerationsas shown in Figures 8 and 9. Therefore, it is possibleto conclude with confidence that the proposed modelstructure and parameter estimation methods employedresulted in a nonlinear model of the quadrotor that issufficiently accurate for the purpose of control systemdesign.
8 Conclusions
In this paper, a body-centric non-linear model fora quadrotor UAV has been presented, identified,verified and used in a model based body accelerationtracking control system. The proposed approach isan attractive choice for researchers investigating bio-inspired navigation strategies for quadrotor UAVs.The parameters of the nonlinear model are initiallyapproximated by analytical methods and then furtherrefined using system identification methods. Due to theunstable nature of the quadrotor model the systemidentification is performed in closed loop to avoidthe issues associated with the integration of unstable
Body-centric Modelling, Identification, and Acceleration Tracking Control of a Quadrotor UAV 11
0
1
2
a x (m
/s2 )
−2
0
2
a y (m
/s2 )
0 2 4 6 8 10 12 14 16
−2
0
2
4
a z (m
/s2 )
Time (s)
Reference Measured
Figure 9 Reference vs. actual body accelerations when notwo accelerations are demanded simultaneously
dynamics and the high sensitivity of the error function.Performing the identification in closed loop resulted insatisfactory prediction of the quadrotor dynamics whenusing sets of unseen experimental data.
The results obtained with the proposed controlmethod demonstrate the possibility of using body-centric based control to operate the quadrotor, whichis more suitable for fulfilling bio-inspired navigationstrategies. The proposed control system has been testedexperimentally and it performed sufficiently well instabilizing the quadrotor and controlling its translation.The good tracking performance achieved via the feed-forward tracking is another direct indication of thegood quality of the model achieved via the systemidentification process carried out in this paper. Thebody-centric model and the body acceleration trackingcontrol proposed in this paper are the building blocksof the visual high-level control and state estimationpresented in [27].
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Appendix
Table A1: Discrete time linearised model and LQ trackerparameters
Φ
1 0 0 0.001 0 00 1 0 0 0.001 00 0 1 0 0 0.0010 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1
Γ 10−3 ×
1 −0.0002 0 0.0002
0.0001 0 −0.0001 00 0 0 00 −0.3305 0 0.3305
0.295 0 −0.295 0−0.0143 0.0143 −0.0143 0.0143
C
0 0 1 0 0 00 −9.805 0 0 0 0
9.805 0 0 0 0 00 0 0 0 0 0
Body-centric Modelling, Identification, and Acceleration Tracking Control of a Quadrotor UAV 13
Table A1 (continued)
D
0 0 0 00 0 0 00 0 0 0
−0.0116−0.0116−0.0116−0.0116
K
0 530.32 −211.61 0 42.4 −86.12
−529.93 0 211.61 −40.04 0 86.120 −530.32−211.61 0 −42.4−86.12
529.93 0 211.61 40.04 0 86.12
F
−211.61−54.09 0 −21.52211.61 0 −54.05−21.52−211.61 54.09 0 −21.52211.61 0 54.05 −21.52
Qy 103 ×
180 0 0 00 5 0 00 0 5 00 0 0 5
R I4
Π
0 0 0.102 00−0.102 0 01 0 0 00 0 0 00 0 0 00 0 0 0
