Adaptive Control with Neural Networks-based Disturbance Observer for a Spherical UAV

20th 20th IFAC IFAC Symposium Symposium on on Automatic Automatic Control Control in in Aerospace Aerospace August 21-25, 2016. Quebec, Canada 20th IFAC Symposium on Automatic Control in Aerospace August 21-25, 2016. Sherbrooke, Sherbrooke, Quebec, Canada 20th IFAC Symposium on Automatic Control in Aerospace Available online at www.sciencedirect.com August 21-25, 2016. Sherbrooke, Quebec, Canada August 21-25, 2016. Sherbrooke, Quebec, Canada

ScienceDirect IFAC-PapersOnLine 49-17 (2016) 308–313

Adaptive Control with Neural Adaptive Control with Neural Adaptive Control with Neural Adaptive Control with Neural Networks-based Disturbance Observer Networks-based Disturbance Observer Networks-based Disturbance Observer Networks-based Disturbance Observer Spherical UAV Spherical UAV Spherical UAV Spherical UAV

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∗ ∗∗ ∗∗∗ Matassini Antonios Matassini ∗∗ Hyo-Sang Hyo-Sang Shin Shin ∗∗ Antonios Tsourdos Tsourdos ∗∗∗ ∗∗ ∗∗∗ ∗∗∗∗ Matassini Hyo-Sang Shin Antonios ∗∗∗∗ Innocenti Mario Innocenti Matassini ∗Mario Hyo-Sang Shin ∗∗ Antonios Tsourdos Tsourdos ∗∗∗ ∗∗∗∗ Mario Innocenti Mario Innocenti ∗∗∗∗ ∗ ∗ Universit` a di Pisa, Italy, (e-mail: Universit` a di Pisa, Italy, (e-mail: [email protected]) [email protected]) ∗ ∗∗ a di Pisa, Italy, (e-mail: [email protected]) ∗∗ Cranfield University, UK (e-mail: [email protected]) ∗ Universit` Cranfield University, UK (e-mail: [email protected]) Universit` a di Pisa, Italy, (e-mail: [email protected]) ∗∗∗∗∗ Cranfield University, UK (e-mail: [email protected]) ∗∗∗∗∗ Cranfield University, UK (e-mail: [email protected]) Cranfield University, UK (e-mail: [email protected]) Cranfield University, UK (e-mail: [email protected]) ∗∗∗ ∗∗∗∗ Cranfield University, UK (e-mail: [email protected]) ∗∗∗∗ Universit` a di Pisa, Italy, (e-mail: [email protected]) ∗∗∗ Universit` a di Pisa, Italy, (e-mail:[email protected]) [email protected]) Cranfield University, UK (e-mail: ∗∗∗∗ a ∗∗∗∗ Universit` Universit` a di di Pisa, Pisa, Italy, Italy, (e-mail: (e-mail: [email protected]) [email protected]) Abstract: This This paper paper develops develops a a control control scheme scheme for for aa Spherical Spherical Unmanned Unmanned Aerial Aerial Vehicle Vehicle Abstract: Abstract: paper a scheme aa Spherical Unmanned Aerial (UAV) can used in scenarios where traditional navigation and (UAV) which whichThis can be be useddevelops in complex complex scenarios where for traditional navigation and communications communications Abstract: This paper develops a control control scheme for Spherical Unmanned Aerial Vehicle Vehicle (UAV) can be used scenarios where and systems would The is the control systemswhich would not succeed. The proposed proposed scheme is based based on onnavigation the nonlinear nonlinear control theory theory (UAV) which cannot be succeed. used in in complex complex scenariosscheme where traditional traditional navigation and communications communications systems would not succeed. The proposed scheme is based on the nonlinear control theory combined with Adaptive Neural-Networks Disturbance Observer (NN-DOB) and controls the combinedwould with Adaptive Neural-Networks Observer (NN-DOB) controls the systems not succeed. The proposedDisturbance scheme is based on the nonlinearand control theory combined with Adaptive Neural-Networks Disturbance Observer (NN-DOB) and controls the attitude and altitude of the UAV in presence of model uncertainties and external disturbances. attitude and altitude of the UAV in presence of model uncertainties and external disturbances. combined with Adaptive Neural-Networks Disturbance Observer (NN-DOB) and controls the attitude and of in of external The can estimate the without knowledge of their The NN-DOB NN-DOB can effectively effectively estimate the uncertainties uncertainties without the the and knowledge of disturbances. their bounds bounds attitude and altitude altitude of the the UAV UAV in presence presence of model model uncertainties uncertainties and external disturbances. The NN-DOB can effectively estimate the uncertainties without the knowledge of their bounds and the control system stability is proven using Lyapunov’s stability theorems. Numerical and NN-DOB the control stability is proven using Lyapunov’s theorems. Numerical The cansystem effectively estimate the uncertainties without stability the knowledge of their bounds and the stability proven Lyapunov’s stability theorems. Numerical simulation results demonstrate the the method on UAV model simulation resultssystem demonstrate theis validity ofusing the proposed proposed method on the the UAV under under model and the control control system stability isvalidity provenof using Lyapunov’s stability theorems. Numerical simulation results demonstrate the uncertainties and external external disturbances. uncertainties and disturbances. simulation results demonstrate the validity validity of of the the proposed proposed method method on on the the UAV UAV under under model model uncertainties and external disturbances. uncertainties and external disturbances. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Spherical Spherical UAV, UAV, model model uncertainties uncertainties and and external external disturbances, disturbances, disturbance disturbance Keywords: Spherical UAV, model uncertainties and external disturbances, observer, adaptive control, neural networks. observer, adaptive control, networks. Keywords: Spherical UAV, neural model uncertainties and external disturbances, disturbance disturbance observer, observer, adaptive adaptive control, control, neural neural networks. networks. 1. INTRODUCTION Disturbance 1. INTRODUCTION Disturbance Observer. Observer. The The proposed proposed approach approach adjusts adjusts rorotor speed and flap angles to control attitude and altitude 1. INTRODUCTION Disturbance Observer. The proposed approach adjusts rotor speed andObserver. flap angles control attitude altitude 1. INTRODUCTION Disturbance Thetoproposed approachand adjusts roof spherical UAV. tor speed and angles of the the spherical UAV. speed and flap flap angles to to control control attitude attitude and and altitude altitude Recent Recent researches researches (Gui (Gui et et al. al. [2015],Mizutani [2015],Mizutani et et al. al. [2015]) [2015]) tor of the spherical UAV. of the spherical UAV. Recent researches (Gui et al. [2015],Mizutani et al. [2015]) have proposed the design of ’spherical-like’ UAVs. In this The basic control system is designed based have proposed the design of ’spherical-like’ UAVs. In this Recent researches (Gui et al. [2015],Mizutani et al. [2015]) The basic control system is designed based on on the the article the is coaxial, flap spherical have design ’spherical-like’ UAVs. In Lyapunov control theory. NN-DOB adaptively estimates basic control system is designed based on articleproposed the focus focusthe is about about a of coaxial, flap actuated, actuated, spherical have proposed the designa of ’spherical-like’ UAVs. In this this The Lyapunov NN-DOB adaptively The basic control controltheory. system is designed basedestimates on the the helicopter, developed University the article the is coaxial, actuated, spherical control NN-DOB adaptively estimates model and disturbances these helicopter, developed inaa Cranfield Cranfield University with the Lyapunov article the focus focus is about aboutin coaxial, flap flap actuated, with spherical model uncertainties uncertainties and external external disturbances and these Lyapunov control theory. theory. NN-DOB adaptively and estimates specific purpose of exploration and exploitation of complex helicopter, developed in Cranfield University with the estimates are incorporated in the original control commodel uncertainties and external disturbances and these specific purpose of exploration and exploitation complex helicopter, developed in Cranfield Universityofwith the model estimates are incorporated in thedisturbances original control comuncertainties and external and these specific purpose of and environments (Dixon and Fernandez Fernandez [2013]). of mands obtained from the Lyapunov control theory. The are incorporated in the original control comenvironments (Dixon and [2013]). specific purpose of exploration exploration and exploitation exploitation of complex complex estimates mands obtained from the Lyapunov controlcontrol theory. comThe estimates are incorporated in the original environments (Dixon and Fernandez [2013]). stability of the entire system is analytically investigated. mands obtained from the Lyapunov control theory. The environments (Dixon and Fernandez [2013]). stability of the entire system is analytically investigated. mands obtained from the Lyapunov control theory. The The spherical frame provides protection to the inner comThe spherical frame provides protection to the inner com- stability of entire system analytically investigated. The the control The analysis analysis results confirm that the proposed proposed control stability of the theresults entire confirm system is isthat analytically investigated. The spherical protection the inner components of UAV and to along floor if ponents of the theframe UAVprovides and allows allows to roll roll to along the floor if The The spherical frame provides protection to the the inner comcontrol system guarantees the stability of entire system under the analysis results confirm that the proposed system guarantees the stability of entire system under the The analysis results confirm that the proposed control if the environment permits. The coaxial motors provide as ponents of the UAV and allows to roll along the floor the environment permits. The coaxial provide ponents of the UAV and allows to rollmotors along the floor as if system presence of uncertainties and disturbances, even without guarantees the stability of entire system under the presence of uncertainties and disturbances, even without system guarantees the stability of entire system under the much thrust as possible in the small volume of the sphere the environment permits. The coaxial motors provide as much thrust as possible in The the small volume of provide the sphere the environment permits. coaxial motors as presence knowledge their and knowledgeof ofuncertainties their bounds. bounds. presence ofof uncertainties and disturbances, disturbances, even even without without and yaw through propeller speed. much thrust possible in small of and allow allow yawas control through differential propeller speed. knowledge much thrust ascontrol possible in the the differential small volume volume of the the sphere sphere of their bounds. knowledge of their bounds. and allow yaw control through differential propeller speed. The flaps, placed below the propellers, allow a decoupled main advantage of the proposed control method The allow flaps, yaw placed below the propellers, allow a decoupled and control through differential propeller speed. The The main advantage of the proposed control method is is the the roll pitch control a vectoring The The flaps, placed below propellers, allow aa decoupled applicability to every rigid (DoF) body in which forces and main advantage of the proposed control method is the roll and and pitch control inthe a thrust thrust vectoring manner. The The The flaps, placed belowin the propellers, allowmanner. decoupled applicability to every rigid (DoF) body in which forces The main advantage of the proposed control method is and the final result of this is compact, roll pitch in vectoring The torques to every finaland result of control this design design is a a well-protected, well-protected, compact, roll and pitch control in aa thrust thrust vectoring manner. manner. The applicability torques can can be be applied. applicability to applied. every rigid rigid (DoF) (DoF) body body in in which which forces forces and and of this design is a well-protected, compact, easily controlled, flexible and agile UAV for operations in final result torques can be applied. easilyresult controlled, flexible UAV for operations in torques final of this designand is agile a well-protected, compact, can be applied. This paper paper is is organised organised as as follows: follows: Section Section 22 introduces introduces the the complex environments. This easily controlled, flexible complex environments. easily controlled, flexible and and agile agile UAV UAV for for operations operations in in This notations used throughout this paper and describes paper is organised as follows: Section 22 introduces the notations used throughout this paper and describes complex environments. This paper is organised as follows: Section introduces the complex environments. As dynamic of 3 used this paper and the As in in the the spherical spherical UAV, UAV, system system identification identification is is quite quite notations dynamic model model of the the spherical spherical UAV. Sections 3 explains explains notations used throughout throughout this UAV. paper Sections and describes describes the is quite challenging in airframes. As in the UAV, system the proposed control method for a single integrator system model of the spherical UAV. Sections 3 explains challenging in small small and and unconventional airframes. ModAs in the spherical spherical UAV,unconventional system identification identification is Modquite dynamic the proposed control for UAV. a singleSections integrator system dynamic model of themethod spherical 3 explains elled of an often uncertainchallenging in unconventional airframes. Modfor single system and proposed its most mostcontrol useful method extensions. Section describes the elled dynamics dynamics of such suchand an airframe airframe often contain contain uncertainchallenging in small small and unconventional airframes. Mod- the and its useful extensions. 44 describes the the proposed control method for aaSection single integrator integrator system ties. Moreover, small external disturbances could become elled dynamics of such an airframe often contain uncertainapplication of the proposed control method on the and its most useful extensions. Section 4 describes the ties. Moreover, disturbances could become and elled dynamics ofsmall suchexternal an airframe often contain uncertainapplication of the proposed control method the spherispheriits most useful extensions. Section 4 on describes the severeMoreover, since they theysmall might be relatively relatively strong to to suchbecome plat- application ties. external disturbances could cal performance of proposed method is of control method on spherisevere since might be strong such aa platties. Moreover, small external disturbances could become cal UAV. UAV. Control Control performance of the the proposed method is application of the the proposed proposed control method on the the spheriform, to and severe since might relatively strong aa platdemonstrated via numerical simulations in Section 5 and, UAV. Control performance of the proposed method is form, unlike unlike to relatively relatively large and conventional conventional aircraft. severe since they they might be be large relatively strong to to such suchaircraft. plat- cal demonstrated via numerical simulations in Section 5 and, cal UAV. Control performance of the proposed method is Therefore, when designing a control system small form, unlike to and conventional aircraft. numerical in finally, Section Sectionvia presents thesimulations conclusions. Therefore, when designing large a flight flight control system for for small demonstrated form, unlike to relatively relatively large and conventional aircraft. finally, 66 presents the conclusions. demonstrated via numerical simulations in Section Section 5 5 and, and, and/or unconventional UAVs, incorporating uncertainties Therefore, when designing a flight control system for small finally, Section 6 presents the conclusions. and/or unconventional UAVs, incorporating uncertainties Therefore, when designing a flight control system for small finally, Section 6 presents the conclusions. and/or unconventional UAVs, incorporating and disturbances disturbances is of of great great importance. and is importance. and/or unconventional UAVs, incorporating uncertainties uncertainties 2. 2. MODELING MODELING and disturbances is of great importance. and disturbances is develop of greataimportance. 2. MODELING This paper aims to control system for the spher2. MODELING This paper aims to develop a control system for the spherThis paper aims a the ical UAV, UAV, which is develop able to to cope cope withsystem model for uncertainties The final final design design of of the the spherical spherical UAV UAV is is shown shown in in Fig. Fig. 1; 1; ical is able with model uncertainties This paperwhich aims to to develop a control control system for the spherspher- The is able to cope with model uncertainties is located in the and external disturbances. In order to achieve this the = ical UAV, which final of is shown in Fig. 1; b ,, z b }, and UAV, external disturbances. order to model achieveuncertainties this aim, aim, we we The the body-fixed body-fixed frame Bspherical = {x {xbb ,, yyUAV z }, is located in the ical which is able to In cope with The final design designframe of the theB spherical UAV is shown in Fig. 1; b b yyb ,,the zzb }, is in Body Axis (BAC), geometric centre of propose an control with Neural and external disturbances. order achieve this aim, frame B {x propose an adaptive adaptive controlIn system with Neural NetworkBody Axis Centre Centre (BAC), which is geometric centre of and external disturbances. Insystem order to to achieve thisNetworkaim, we we the the body-fixed body-fixed frame B = =which {xbb ,, is is located located in the the b the b }, Body Axis Centre (BAC), which is the geometric centre propose an adaptive control system with Neural Networkpropose an adaptive control system with Neural Network- Body Axis Centre (BAC), which is the geometric centre of of

Tommaso Tommaso Tommaso Tommaso

Copyright 2016 IFAC 308 Copyright©© ©2016, 2016IFAC IFAC (International Federation of Automatic Control) 308Hosting by Elsevier Ltd. All rights reserved. 2405-8963 Copyright © 2016 IFAC 308Control. Peer review ©under of International Federation of Automatic Copyright 2016responsibility IFAC 308 10.1016/j.ifacol.2016.09.053

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309

From the Newton-Euler equations it holds:         X u˙ p u Y = m v˙ + m q × v (6) Z w˙ r w         p˙ p L p M = IB q˙ + q × IB q (7) r˙ r N r where m is the UAV mass, IB is the Inertia matrix in the T body frame B, vB = [u v w] is the linear velocity in B, T T ω = [p q r] is the angular velocity in B, F = [X Y Z] T are the applied forces in B and T = [L M N ] are the applied torques in B. F andT include gravity contribution and external disturbances. Fig. 1. UAV Prototype developed in Cranfield University. symmetry; the axes directions are shown in the left bottom corner. 2.1 Actuation principles The actuation system is composed of two contra-rotating motors along z-axis and provides a force F and a torque τ ; the relation with the PWMs is found with a test rig. 1 (1) F = f1 (PWM1 , PWM2 ) 1 + τm s 1 τ = f2 (PWM1 , PWM2 ) (2) 1 + τm s where τm represents a first order time constant and f1 and f2 are nonlinear function of both PWM1 and PWM2 . T

T

Define [X Y Z]Act as the applied forces and [L M N ]Act as the applied torques in the body frame B. The two flaps make the UAV behave as a thrust vectored nozzle; the force F produced by the motors is steered in three dimensions due to the flap angles α1 and α2 :       Fx −cα1 sα2 X 1 +sα1 cα2 F Y = Fy =  2 2 (3) cα1 sα2 + c2α2 −cα cα Fz Z Act 1 2 From now on we define cx = cos x, sx = sin x and tx = tan x for the simplicity of the notation. L and M torques are generated because the forces X and Y are applied in a point on the z-axis,dV below the BAC.           0 −dV Fy 0 Fx L M (4) = 0 × Fy + 0 = +dV Fx τ Fz τ dV N Act Merging (3) and (4), the actuators action is shown in (5):     X Fx Y   Fy      1 Z   Fz  (5) = L    −dV Fy  τm s + 1 M  +dV Fx  N Act τ 2.2 Dynamics

Consider the motion of the body-fixed frame B about an Earth-fixed reference frame E = {xe , ye , ze }. There are two assumptions: first, the body is assumed to be rigid; second, the Earth is flat and E is considered inertial. The frame E is selected with the North-East-Down (NED) configuration. 309

The relation between the body-fixed angular velocity ω  T and the Euler angles derivatives, η˙ = φ˙ θ˙ ψ˙ , is determined writing the body rates components into the inertial frame:   1 sφ tθ cφ tθ  φ (8) η˙ = 0 scφφ −s cφ  ω = Jω 0 cθ cθ Therefore, using the Direct Cosine Matrix (DCM), the linear velocities in the reference frame are computed as:   d PN PE = DCM T vB vE = (9) dt P D The integration of (8) and (9) with the proper initial condition makes the position and attitude of the UAV known. For more details, see Stevens et al. [2015]. 3. ADAPTIVE CONTROL LAW DESIGN The proposed control system considers the system dynamics as a single integrator system: x˙ = k · u + D(x, u, t) (10) where x is the state to be controlled, k is a known constant and D(x, u, t) denotes a disturbance. Note that model uncertainties, external disturbances and/or neglected dynamics are incorporated in the disturbance, D(x, u, t). The aim of the control law is to make x follow a desired command xC . Define the tracking error as (11) e = xC − x Define a positive definite Lyapunov function 1 (12) V1 = e2 2 The derivative of V1 is ˙ = e(x˙ C − k · u − D(x, u, t)) (13) V˙ 1 = ee˙ = e(x˙ C − x) This mean that, if we could choose 1 u∗ = (x˙ C − D(x, u, t) + αe), α > 0 (14) k We could obtain V˙ 1∗ = −αe2 (15) Which is negative definite. In practical application the uncertainty D(x, u, t) is (obviously) unknown, and it is ¯ that achieves also difficult to know an upper bound D ¯ D(x, u, t) < D ∀x, u, t For that reason an online adaptive observer is proposed for the estimation of D. The proposed control law achieves the

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tracking of the command xC along with the estimation of the uncertainty. 3.1 Disturbance Estimation The NN-DOB is implemented with a Radial Basis Function Neural Network (RBF-NN), which has an efficient capacity for approximating nonlinear dynamics; see Lavretsky and Wise [2012] for more details. The NN-DOB estimator has an input layer, a hidden layer and an output layer, as shown in Fig. 2.

Fig. 2. RBF-NN structure. The hidden layer is composed of N neurons, each one with a centre, µi , and a width, σi . Each hidden node contains a RBF, a nonlinear function satisfying φ(x) = φ(x). The output depends only on the distance from the center, scaled by the width: e − µi φi (e) = f ( ) (16) σi The most used RBF are gaussian, logarithmic and multiquadric functions. The output layer is computed using the following weighted sum: Dφ =

N 

Wi φi (e) = WT Φ

(17)

i=1

where Wi is the weight between the ith hidden neuron and T T the output, W = [W1 · · · WN ] and Φ = [φ1 · · · φN ] . Define δ as the minimum estimation error between the real disturbance and Dφ and W∗ as the weight vector that achieves the minimum δ: (18) W∗ = arg min(D − Dφ ) W

So that D(x, u, t) = Dφ (W∗ ) + δ = W∗ T Φ + δ (19) The estimation error δ and the optimal weight W∗ are unknown, so the total disturbance D is estimated as ˆ = WT Φ(e) + δˆ (20) D Where W and δˆ are calculated through an adaptive law. 3.2 Control Law Define the Lyapunov function 1 1 ˆ 2 (21) V2 = V1 + (W∗ − W)T (W∗ − W) + (δ − δ) 2η1 2η2 310

This function V2 can be rewritten as V2 = 12 z T P z, where     e 1 1 ∗ z = W − W , P = diag 1, IN ×N , (22) η1 η2 δ − δˆ

Note that V2 is positive definite with respect to the variable z, and not only w.r.t the error e. The derivative of V2 is 1 1 ˆ δ− ˙ δ) ˆ˙ (23) ˙ ∗ − W)+ ˙ ˙ (W∗ −W)T (W (δ− δ)( V˙ 2 = ee+ η1 η2 For every disturbance the couple (W∗ , δ) is fixed, so we can assume ˙ ∗=0 (24) W ˙δ = 0 (25) Substituting (13), (24) and (25) into (23) yields 1 1 ˆ δˆ˙ (26) ˙ (δ− δ) V˙ 2 = e(x˙ C −k·u−D)− (W∗ −W)T W− η1 η2 Design the adaptive law as ˙ = −η1 Φ e W (27)

˙ (28) δˆ = −η2 e Design the control action as 1 ˆ u = (αe + x˙ C − D) (29) k ˆ is calculated from (20) and α, η1 , η2 are positive where D tuning parameters. Substituting (27), (28) and (29) into (26) yields ˆ ˆ − D) + (W∗ − W)T Φe + (δ − δ)e V˙ 2 = e(−αe + D 2 = −αe (30) This control action makes the derivative of V2 only negative semi definite, and for that reason the stability is checked with the Barbalat’s Lemma. Define V¨2 as: ˆ − D) (31) V¨2 = −2αee˙ = −2αe(−αe + D ˆ ¨ Since e, u, D, D are all bounded, then V2 is bounded, hence V˙ 2 is uniformly continuous. Combined with the fact that V2 is bounded from below and V˙ 2 is negative semi-definite, then it infers that V˙ 2 → 0 as t → ∞. 3.3 Estimation Direction The disturbance equation (19) can be rewritten as    T  W∗ ∗T D = W Φ(e) + δ = Φ 1 = pT L∗ (32) δ  ∗   W Where L∗ = and pT = ΦT 1 is the linear map δ between L∗ and D. Note that pT ∈ R1×(N +1) , so its kernel has rank (33) rank(ker(pT )) = N All L who get the correct disturbance estimation are in L: L = {L|L = L∗ + Lker , Lker = span ker(pT )} (34) The estimation is perfect if L reaches the set L. The adaptive law (27) and (28) can be written as   η Φ e, L(0) = L0 (35) L˙ = − 1 η2 Note that if η1 = η2 = η, the estimation action is L˙ = −ηep, L(0) = L0

(36)

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From the basic algebra theorem,

311

1.8

1.8

η = 0.2 η=2 η=6

1.6

The estimation update is perpendicular to L, so it is made in the fastest direction and the set of weights reachable from L0 is Lnearest , as shown in Fig. 3.

r (rad/s)

(37)

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p

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η = 0.2 η=2 η=6

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Im(p) ⊥ ker(pT )

0 0

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Fig. 4. Step response, r channel. Left: α = 6, Right: α = 15. L

4.1 Single Integrator and Non Linear Control

W∗

Fig. 3. The variation of the disturbance estimation is perpendicular to L.

3.4 Advantages of the proposed control action As written in (29), the control action is 1 ˆ u = (x˙ C + αe − D) k It is straightforward to understand that the control action is composed by three terms: a proportional action, a Feed Forward and an estimation of the disturbance. Moreover, this kind of control doesn’t act like a Sliding Mode Control, so there is no chattering in the control action. This characteristic generalizes the control structure, as it can be applied to any system that acts as a single integrator.

The linear control law (29) can be applied to the motion equations (6), (7) because the inputs are directly integrated. From (6), the relation between F and vB is similar to a single integrator. 1 d vB = F − ω × v B ≈ k · u + D (41) dt m Likewise, from (7) the relation between T and ω is similar to a single integrator. d −1 −1 −1 ω = IB (T − ω ˆ IB ω) = IB T − IB ω ˆ IB ω ≈ k · u + D dt (42) For example, the x-axis law (43) X = mv˙ x + m(−r · vy + q · vz ) Is equal to (10) with the change of variables x = vx , k = u = X and D = r · vy − q · vz .

1 m,

Similarly, the Non Linear control law (40) can be applied to the UAV when the input variables are F and T and the regulated variables are the linear and angular velocities in the Earth frame vB and η. ˙ The demonstration that these systems are NL-MIMO-AC is made in Appendix A and B.

3.5 Non-Linear MIMO Extension The control laws (27), (28) and (29) can be extended to every Non-Linear, Multi Input Multi Output affine-in-thecontrol (NL-MIMO-AC) system whose input-to-state map g(x) has rank bigger than the state dimension. Consider the following dynamic system: x˙ = f (x) + g(x)u + D

(38)

x ∈ RN , D ∈ RN , u ∈ RM where M ≥ N ; if the map g(x) : RM → RN has always rank ≥ N there exist an inverse function for every state x. rank g(x) ≥ N → ∀x ∃ g −1 (x)

(39)

Note that if M > N the pseudo-inverse function g(x)† = (g(x)T g(x))−1 g(x)T can be used instead of the inverse. Rather, given the system (38) and condition (39) holds, the control action that makes e = 0 a stable equilibrium is ˆ u = g −1 (x)(−f (x) + x˙ C + αe − D) (40) 4. SPHERICAL UAV APPLICATION The proposed control law is applied to the Spherical UAV by computing the desired control effort with (29),(40) and resolving the control allocation problem linearizing (5). 311

4.2 Commanded Variables and Control Allocation The chosen control logic in this paper is the control of UAV altitude and attitude, so the selected variables, depending of the data available from the sensors, are xInt or xN L . The plant can be considered as a ’Single-Integrator’-like system if the state variables are xInt , while it can be seen as a NL-MIMO-AC when the state variables are xN L . In both case, the commanded variables are uC , as shown in (44).       P˙D Z w  φ˙  L p  uC =   , xInt =   , xN L =  (44)  θ˙  M q N r ψ˙ Given uC , the control allocation problem consists in findT ing the set of inputs uR = [F τ α1 α2 ] that reach the closest desired command (in norm). This is done linearizing and inverting equation (5), f (uR ), around the current set of real input uR0 . ∂f |u (uR − uR0 ) (45) uC = f (uR ) ≈ f (uR0 ) + ∂uR R0 Inverting (45) yields  ∂f −1 | uR 0 (uC − f (uR0 )) (46) u R = uR 0 + ∂uR

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(a) Nominal Plant

(b) Torque Disturbance

(c) Uncertainty and Disturbance

Fig. 5. System response for three simulation scenarios. Top: Adaptive Control. Bottom: Baseline PID Controller. The black dashed lines represent the commands. Note that the Jacobian is always full rank due to the constraints in uR shown in Table 1. 4.3 External Loop The proposed control method is applied when the controlled variables are (generalised) velocities. In practical applications the command is in position or attitude, so an outer loop is needed. Assuming that the internal loop

∆F − ∆m Fˆ

x x m ˆ Where Dunc = represents the uncertainties m+∆m ˆ due to ∆m and ∆Fx , Df lex deals with the flexibility terms, Dcoupling considers the channels interaction and Dext is an external acceleration due to wind gusts or interactions with the external environment.

5. SIMULATION RESULTS The effectiveness of the proposed control system is demonstrated through non-linear simulations. The UAV parameters used in simulations are taken from the prototype and shown in Table 1. The controlled variables are the Euler Table 1. UAV Data

Fig. 6. Double Loop Control Scheme. behaves like a controlled 2nd order, as shown in Fig. 4, the outer controller can be synthesised with the linear control theory: a P or PI could be sufficient, root locus, LQR and so on. Note that the control actions given in (29) and (40) requires the derivative of the command x˙ C , which can be easily given by the outer controller. 4.4 Disturbance Sources The disturbance sources can be divided in uncertainties and external actions; the former deal with the approximated knowledge of the system, while the latter are due to the environment. For example, re-write (43) supposing that only an estimation of the mass, m ˆ and of the control action, Fˆx is known:  d 1    vx = Fx + (r · vy − q · vz ) dt m (47) m=m ˆ + ∆m   F = Fˆ + ∆F x x x

The dynamic system is rewritten again as: d 1 vx = (Fˆx + ∆Fx ) + (r · vy − q · vz ) dt m ˆ + ∆m 1 ˆ = Fx + Dunc + Df lex + Dcoupling + Dext m ˆ

312

Parameter Mass [Kg]



Inertia Kg ·

Symbol m m2



Offset distance [m] Force range [N ] Torque range [N · m] Angle range [rad] Motors constant [s]

IB dV

τm

Value

3393.00.590 6.0



−6.9 10 6.0 3918.3 47.1 −6.9 47.1 2745.9 8.572 · 10−3 F ∈ [0.6818 7.6542] τ ∈ [−0.0352  0.0352]  −6

α1 , α 2 ∈ − π 4

π 4

0.175

T

angles η with initial condition η0 = [0 0 0] . Moreover, a null-loop on P˙ D is built in order to concentrate the effort on the attitude. The desired commands is the following:   5 deg t ≤ 12 10 deg t ≤ 6 , ψC = φC = 10 deg, θC = 0 deg

t > 12

0 deg

t>6

The control parameters are tuned as follows; the adaptive coefficients  are all set as η1 = η2 = 2, and the proportional gain are αP˙D αφ αθ αψ = [5 6 6 6]. For all channels the outer loop is a simple P controller with gain [kφ kθ kψ ] = [2 2 4]. The number of neurons used is 9, the RBF centers µi are evenly spaced in [−1 1], and the widths σi are all 0.25. The performances are compared with a standard PI controller for the velocity, while the outer loop is the same.

IFAC ACA 2016 August 21-25, 2016. Quebec, Canada Tommaso Matassini et al. / IFAC-PapersOnLine 49-17 (2016) 308–313

313

Mizutani, S. et al. (2015). Proposal and experimental validation of a design strategy for a uav with a passive rotating spherical shell. In IEEE/RSJ IROS 2015, 1271– 1278. Stevens, B., Lewis, F., and Johnson, E. (2015). Aircraft Control and Simulation: Dynamics, Controls Design, and Autonomous Systems. Wiley. Appendix A. NL-MIMO-AC, POSITION Fig. 7. Torque disturbance. 5.1 Numerical results The performance of the controller with the nominal plant is shown in Fig. 5a. As shown, the adaptive control system is much more decoupled and stable than the PI controller in the nominal case. In the second simulation, to check the disturbance rejection, a time-varying torque disturbance is added in the attitude channels. The total disturbance is shown in Fig. 7 and it is composed by a fixed term plus a time-varying disturbance. Fig. 5b shows that the angles error based on the proposed method is much more limited, about ±0.4 degrees than the PI method which has an error of about ±2 degrees. The simulation results confirm that the proposed approach effectively copes with the external disturbances and model uncertainties. Moreover, the performance comparison shows that the proposed outperforms over the standard PI controller.

In the third simulation a fault simulation is proposed: at t = 12 the UAV Inertia matrix becomes the 30% of the original one and a huge torque disturbance is added to the yaw channel: from Fig. 5c the proposed control method works way more better that the PI control and it can recover easily the steady state condition after the fault. 6. CONCLUSION In this paper, an on-line adaptive control scheme is developed for a Spherical UAV in the presence of model uncertainties and external disturbances. The disturbance observer is designed based on a RBF-NN and the stability is proven using the Lyapunov control theory. Numerical simulation results confirms that the proposed control method is simple, easy to implement and quite effective. Further extensions include real test simulations, the application of the proposed control scheme on different platforms and the demonstration of the proposed control law in case of time-varying disturbances. REFERENCES Dixon, R. and Fernandez, Y. (2013). Design of a Spherical UAV for Operations in Complex Environments. Master’s thesis, Cranfield University. Gui, H. et al. (2015). Attitude control of spherical unmanned aerial vehicle based on active disturbance rejection control. In 34th Chinese Control Conference, 1191–1195. Lavretsky, E. and Wise, K. (2012). Robust and Adaptive Control: With Aerospace Applications. Advanced Textbooks in Control and Signal Processing. Springer. 313

The relation between vB and vE is (9). Deriving the inverse of (9) it holds ˙ vE + DCM v˙ E (A.1) v˙ B = DCM Substituting (A.1) in (41) and expliciting v˙ E yields d 1 ˙ vE + . . . vE = DCM T F − DCM T DCM dt m T T −1 −DCM (J η) ˙ × (DCM vE ) − [0 0 g] (A.2) Equation (A.2) is in form (38), where ˙ vE − DCM T (J −1 η) f = −DCM T DCM ˙ × (DCM vE )

(A.3)

D = − [0 0 g] 1 has always rank = 3. And the map g = DCM T m

(A.4)

T

Appendix B. NL-MIMO-AC, ATTITUDE The relation between ω and η is (8). Deriving the inverse of (8) it holds d (B.1) ω˙ = ( J −1 )η˙ + J −1 η¨ dt Where   0 0 −cθ θ˙ d −1  (B.2) J = 0 −sφ φ˙ cφ cθ φ˙ − sφ sθ θ˙  dt 0 −cφ φ˙ −sφ cθ φ˙ − cφ sθ θ˙ Substituting (B.1) and (8) in (42) and expliciting η¨ yields     d d −1 −1 −1 (J −1 η)×(I η˙ = JIB T −J ( J −1 )η˙ +IB ˙ η) ˙ BJ dt dt (B.3) Equation (B.3) is in the form (38), where     d −1 −1 f (η, η) ˙ = −J ( J )η˙ + IB ˙ × (IB J −1 η) ˙ (J −1 η) dt (B.4) −1 ⇒ g −1 = IB J −1 (B.5) g(η, η) ˙ = JIB Note that rank g −1 = 3 is ensured if θ = ± π2 and the inertia matrix IB is diagonal; the condition must be checked in all the other cases.