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Contraction Based Tracking Control of Autonomous Underwater Vehicle
Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Automatic Control Proceedings of the 20th9-14, World Congress Toulouse, France, July 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 2665–2670 Contraction Based Tracking Control Contraction Contraction Based Based Tracking Tracking Control Control Autonomous Underwater Contraction Based Tracking Vehicle Control Autonomous Underwater Vehicle Autonomous Underwater Vehicle Autonomous Underwater Vehicle Majeed Mohamed and Rong Su
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Majeed Mohamed and Rong Su Majeed Mohamed and Rong Su Majeed Mohamed and Rong Department of Electrical Electrical Engineering, NTU,Su Singapore. Department of Engineering, NTU, Singapore. Department of Electrical Engineering, NTU, Singapore. (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Department of Electrical Engineering, NTU, Singapore. [email protected]). (e-mail: [email protected], (e-mail: [email protected], [email protected]). Abstract: Abstract: This This paper paper discusses discusses the the incremental incremental stability stability of of an an underwater underwater vehicle vehicle using using stability of andynamics underwater using Abstract: This theory. paper discusses the incremental vehicle the contraction Stability analysis considered in vehicle of a simple and the contraction theory. Stability analysis considered in vehicle dynamics of a simple and Abstract: This theory. paper discusses the incremental stability ofcontroller andynamics underwater using the contraction Stability analysis considered in the vehicle of avehicle simple and a more advanced model has the ability to constructing to track the desired a more advancedtheory. model Stability has the ability toconsidered constructing the controller to track the desired the contraction analysis in vehicle dynamics of a simple and atrajectory more advanced model vehicle has theposition. ability to constructing the controller to track the desired of The natural behavior of underwater of underwater underwater vehicle position. The natural contracting contracting behavior of an an the underwater atrajectory more isadvanced model has the ability model to constructing the controller to track desired trajectory of underwater vehicle position. The natural contracting behavior of an underwater vehicle ensured for a more advanced of vehicle to derive contracting exponentially vehicle is ensured for a more advanced model of vehicle to derivebehavior contracting exponentially trajectory of underwater vehicle position. The natural contracting of an underwater vehicle is ensuredThe for tuning a more parameters advanced model of vehicle to derive contracting exponentially stable of have selected analytically from stable controller. controller. The tuning parameters of the the controller controller have selected analytically from the the vehicle is ensured foranalysis a more parameters advanced model vehicleThe to derive contracting exponentially stable controller. The tuning of the of controller have selected analytically from the incremental stability using contraction theory. controller design is restricted to incremental stability analysis using contraction theory. The controller design is restricted to stable controller. The tuning parameters of the controller have selected analytically from the incremental stability analysis using contraction theory. Thebackstepping controller design is technique. restricted In to parametric-strict-feedback form to develop an incremental design parametric-strict-feedback form to develop an incremental backstepping design technique. In incremental stability analysis using contraction theory. The controller design is restricted to parametric-strict-feedback form to develop an incremental backstepping design technique. In this afford to aa controller in way it this paper, paper, proposed proposed method method afford to construct construct controllerbackstepping in aa recursive recursivedesign way and and it enforces enforces parametric-strict-feedback form to ofdevelop an incremental technique. In way and this paper, proposed method afford toan construct a controller inand a recursive it enforces incremental exponential stability underwater vehicle not just global asymptotic incremental exponential stability oftoan underwater vehicle inand not justway global asymptotic this paper, proposed method afford construct a controller a recursive and it enforces and not just global asymptotic incremental exponential stability of an underwater vehicle stability. stability. incremental exponential stability of an underwater vehicle and not just global asymptotic stability. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. stability. Keywords: Keywords: Incremental Incremental stability, stability, tracking tracking control, control, exponential exponential stability, stability, contraction contraction theory, theory, Keywords: Incremental stability, tracking control, exponential stability, contraction theory, underwater vehicle. underwaterIncremental vehicle. Keywords: stability, tracking control, exponential stability, contraction theory, underwater vehicle. underwater vehicle. 1. trajectories 1. INTRODUCTION INTRODUCTION trajectories with with respect respect to to each each other other (Lohmiller (Lohmiller (1999)). (1999)). 1. INTRODUCTION trajectories withofrespect to each other (Lohmiller (1999)). Such a property nonlinear systems is aa stronger property Such a property of nonlinear systems is stronger property 1. INTRODUCTION trajectories withofrespect to each other (1999)). Such a property nonlinear systems is (Lohmiller aastronger property than exponential convergence to single Controlling than global global exponential convergence to astronger single trajectory. trajectory. Controlling the the horizontal horizontal position position of of an an underwater underwater vehivehi- Such a property of nonlinear systems is a property than global exponential convergence to a single trajectory. Controlling the horizontal position of an underwater vehicle at of sea cle changing changingthe at the the bottom bottom of the the of sea is is of of great great imporimporIn literature, Lyapunov-based Global Expothan exponential convergenceUniform to a single trajectory. Controlling position vehiIn the theglobal literature, Lyapunov-based Uniform Global Expocle changing at horizontal the bottom of the seaanisunderwater ofaccurate great importance for precise maneuvering. Real-time positance for precise maneuvering. Real-time accurate posi- In the literature, Lyapunov-based Uniform Global Exponential Stability (UGES) of system dynamics is proven and cle changing at the bottom of the sea is of great impornential Stability (UGES) of system dynamics is proven and tance for precise maneuvering. Real-time accurate posi- In tion measurements of underwater vehicles are facilitated the literature, Lyapunov-based Uniform Global Expotion measurements of underwater vehicles accurate are facilitated nential Stability (UGES) of system dynamics isvehicle proven and extended its application to the underwater with tance for precise maneuvering. Real-time posiextended its application tosystem the underwater vehicle with tion measurements of underwater vehicles are facilitated by the latest advances in sensing technology of vehicle nential Stability (UGES) of dynamics is proven and by the latest advances in sensing vehicles technology of vehicle extended its application to the underwater vehicle with the contracting (Jouffroy and tion measurements of underwater are facilitated the nonlinear nonlinear contractingtoobserver observer (Jouffroy vehicle and Fossen Fossen by the latest advances in sensing technology of vehicle extended position and velocity (Marco and Healey (1998),Whitits application the underwater with position and velocity (Marco and Healey (1998),Whitthe nonlinear contracting observer (Jouffroy and Fossen (2004)). Likewise, the application of contraction analysis by theetlatest advances in sensing technology ofthe vehicle (2004)). Likewise, the application of(Jouffroy contraction analysis position and(1998)). velocity (Marco and Healey (1998),Whitcomb al. In order to precisely control lowthe nonlinear contracting observer and Fossen comb et al. (1998)). In order to precisely control the low(2004)). Likewise, thederive application of contraction analysis has pursued the observer by position and(1998)). velocity (Marco Healey (1998),Whithas been been Likewise, pursued to to derive the nonlinear nonlinear observeranalysis by sevsevcomb et al. Inunderwater order to and precisely control the low- (2004)). speed trajectory of vehicle, these thederive application of contraction speed et trajectory of an an underwater vehicle,control these advances advances has been pursued to the(2013),Jouffroy nonlinear observer by several authors (Majeed and Kar (2003),Joufcomb al. (1998)). In order to precisely the loweral been authors (Majeed and Kar (2013),Jouffroy (2003),Joufspeed trajectory of an underwater vehicle, these advances has in position sensing are greatly backup to the controller pursued to derive the nonlinear observer by sevin position sensing are greatly backup to these the controller eral authors (Majeed and Kar (2013),Jouffroy (2003),Jouffroy and (2002b),Jouffroy and The speed trajectory of an vehicle, advances froy authors and Lottin Lottin (2002b),Jouffroy and Slotine Slotine (2004)). (2004)). The in position sensing areunderwater greatly backup to the controller design. But, our of dynamics of eral (Majeed and Kar (2013),Jouffroy (2003),Joufdesign. But,sensing our understanding understanding of the the to dynamics of the the froy and Lottin (2002b),Jouffroy andsignificant Slotine (2004)). The contraction theory framework offers advantages in position are greatly backup the controller contraction theory framework offers significant advantages design. But, ourand understanding of the dynamics of bladed thrusters vehicle are principally limited to the froy and1)Lottin (2002b),Jouffroy andsignificant Slotine (2004)). The bladed thrusters and vehicle are principally limited to the theory framework offers advantages that the to the design. But, ourand understanding of the dynamics the contraction that it it 1) eliminates eliminates the need need to know know the equilibrium equilibrium bladed thrusters vehicle aredynamics principally limited of toused the control application while these commonly contraction theory framework offers significant advantages control application while these dynamics commonly used that it 1) eliminates the need to know the equilibrium dynamics as for an bladed thrusters andwhile vehicle aredynamics principally limited(Healey toused the point point itof of system system dynamics as it it works works for the an incremental incremental control application these commonly to dynamically positioned marine that eliminates the need to know equilibrium to actuate actuate dynamically positioned marine vehicles vehicles (Healey point of1)system dynamics asand it works fornot an require incremental convergence of trajectories, 2) does difficontrol application while these dynamics commonly used convergence of trajectories, and 2) does not require diffito actuate dynamically positioned marine vehicles (Healey et al. (1994),Whitcomb and Yoerger (1999a)). This repoint of system dynamics as it works for an incremental et actuate al. (1994),Whitcomb and Yoerger (1999a)). This re- convergence of trajectories, and 2) does not require difficult task of the selection of a suitable Lyapunov function to dynamically positioned marine vehicles (Healey cult task of the selection of a suitable Lyapunov function et al. (1994),Whitcomb anddesign Yoerger (1999a)). This re- convergence search work focuses on the of position controller of trajectories, 2) stability does notanalysis require diffisearch work focuses on the design of position controller task of the selection of aand suitable Lyapunov function for analysis. As aa result, does et al.the (1994),Whitcomb and Yoerger This re- cult for stability stability analysis. As of result, stability analysis does search work focuses on the design of (1999a)). position controller with use of a simple (Fossen (1994)) as well as more cult task of athe selection a suitable Lyapunov function with the use of a simple (Fossen (1994)) as well as more for stability analysis. As a result, stability analysis does not require skillful simplification to show negative defsearch work focuses on the design of position controller not stability require aanalysis. skillful simplification to show analysis negative does defwith the use of a simple (Fossen (1994)) as well as more for advanced models of the underwater vehicle (Whitcomb As a result, stability advanced models of the underwater vehicle (Whitcomb not require a skillful simplification to show negative definiteness of the derivative of a Lyapunov function. For with the use of a simple (1994)) as well as more initeness of the derivative of a Lyapunov function. For a a advanced models of the (Fossen underwater vehicle (Whitcomb and Yoerger (1999b)). not require a skillful simplification to show negative defand Yoerger (1999b)). ofclass the of derivative of systems, a Lyapunov function. For a particular nonlinear an integrator backadvanced models of the underwater vehicle (Whitcomb initeness particular class of nonlinear systems, an integrator backand Yoerger (1999b)). initeness ofclass the of derivative of systems, a Lyapunov function. For particular nonlinear an integrator back-a stepping in framework is The nonlinear control and (1999b)). stepping procedure procedure in contraction contraction framework is introduced introduced The Yoerger nonlinear control systems systems design design in in a a recursive recursive way way particular class of nonlinear systems, an integrator backstepping procedure in contraction framework is introduced The nonlinear control systems design in a recursive way in (Jouffroy and Lottin (2002a)). In the light of has shown for the so in (Jouffroy and Lottin (2002a)). In the light of such such has been been shown great great interest interest for the past past so many many years. years. stepping procedure in contraction framework is introduced The nonlinear systems design in anonlinear way in (Jouffroy and Lottin (2002a)). In the in light ofpaper such has been showncontrol great interest for the past sorecursive many years. framework, UGES control law is derived this The backstepping method is one of the techframework, UGES control law is derived in this paper The backstepping method is one of the nonlinear techin (Jouffroy UGES and Lottin In desired the in light ofpaper such has been shown great interest past soprocedure many years. framework, control(2002a)). law is the derived this The backstepping method is for onethe ofdesign the nonlinear techusing theory to track trajectory nique, which a in using contraction contraction theory tolaw track the desired trajectory nique,backstepping which provides provides a recursive recursive design procedure in framework, UGES control is derived in this paper The method is one of the nonlinear techusing contraction theory to track the desired trajectory nique, which provides a recursive design procedure in of vehicle position. The main contributions a manner for systems transformable into of an an underwater underwater vehicle position. The main contributions a systematic systematic manner for the the systemsdesign transformable into using contraction theoryposition. to trackThe the main desired trajectory which provides recursive procedure in of an underwater vehicle contributions anique, systematic manner fora the systems transformable into this paper are parametric strict feedback form (Khalil (1996)). In this of this paper are parametric strict feedback form (Khalil (1996)). In into this of an underwater vehicle position. The main contributions a systematic manner for the systems transformable of this paper are parametric strict Lyapunov feedback form (Khalil (1996)). In this method, suitable functions are constructed at method, suitable Lyapunov functions are(1996)). constructed at Contraction paper are theory parametric strict feedback form (Khalil In this (1)this Contraction theory based based an an incremental incremental backstepbackstepmethod, suitable Lyapunov functions areeach constructed at of(1) each stage to confirm the stability of subsystem each stage to confirm the stability of each subsystem (1) Contraction theory based to antracking incremental backstepping control law is derived the pomethod, suitable Lyapunov functions are constructed at ping control law is derived to tracking the desired desired poeach stage to confirm the stability of each subsystem (Khalil (1996)). Lyapunov stability theory is widely ex(1) Contraction theory based an incremental backstep(Khalil (1996)). Lyapunov stability theory is widely exping control law is derived to tracking the desired position of an underwater vehicle. The simple and more each stage todesign confirm the stability stability of each subsystem sition of an underwater vehicle. The simple and more (Khalil (1996)). Lyapunov theory is widely explored to the of nonlinear control system. On the ping control law isofderived to tracking theare desired poplored to(1996)). the design of nonlinear control system. On the sition of anmodels underwater vehicle. The simple and more advanced vehicles adopted (Khalil Lyapunov stability theory is widely exadvanced models of underwater underwater vehicles areand adopted plored to contraction the design of nonlinear control system. On the contrary, theory is used as latest method to sition of an underwater vehicle. The simple more contrary, contraction theory is used as latest method to the advanced models of underwater vehicles are adopted for In aa practical point this plored to contraction theanalysing design theory ofofnonlinear the for this this purpose. purpose. Inunderwater practicalvehicles point of ofareview, view, this contrary, is usedcontrol as latestsystem. methodOn toand the convergence systems (Lohmiller advanced models of adopted convergence analysingtheory of nonlinear nonlinear systems (Lohmiller and for thisis purpose. Into a be practical point of view, this result considered highly useful as it permits contrary, contraction is used as latest method to the result is considered to be highly useful as it permits convergence analysing of nonlinear systems (Lohmiller and Slotine (1998),Lohmiller and Slotine (2000)). It is treated for this purpose. In a practical point of view, this Slotine (1998),Lohmiller and Slotine (2000)). It is treated result is considereddesign to beofhighly useful exponentially as it permits for contracting convergence analysing of nonlinear systems (Lohmiller and for the the systematic systematic design ofhighly contracting exponentially Slotine (1998),Lohmiller and Slotine (2000)). It is treated as form stability since contraction theory result considereddesign to beof useful exponentially as it permits as an an incremental incremental form of of and stability since contraction theory for theis systematic contracting stable controllers. Slotine (1998),Lohmiller Slotine (2000)). It is treated stable controllers. as an incremental form of stability sinceofcontraction theory offers a stability analysis nonlinear for the controllers. systematic design of contracting exponentially offers a tool tool for for the the stability analysis ofcontraction nonlinear system system stable as an incremental form of stability sinceof theory offers a tool for the stability analysis nonlinear system stable controllers. offers a tool for the stability analysis of nonlinear system
Copyright © 2017 IFAC 2720 Copyright © 2017, 2017 IFAC 2720Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 2720 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 2720Control. 10.1016/j.ifacol.2017.08.472
Proceedings of the 20th IFAC World Congress 2666 Majeed Mohamed et al. / IFAC PapersOnLine 50-1 (2017) 2665–2670 Toulouse, France, July 9-14, 2017
(2) The selection of tuning parameters of the controller is obtained analytically as a by-product of the incremental stability analysis in contraction theory. The natural contracting behavior of an underwater vehicle is established for a simplified UGES control law in the case of a more advanced model of vehicle. (3) Contracting control law is tested through the simulation studies for the simple and more advanced model of underwater vehicle. The following Section 2 describes the design of integral backstepping position control of an underwater vehicle with its numerical simulation in section 3. Finally, the conclusions are given in section 4. 2. POSITION CONTROL OF AUTONOMOUS UNDERWATER VEHICLE We are considered in turn a simple and more advanced models of underwater vehicle to derive the UGES control law based on the contraction theory in order to track the desired trajectory of vehicle position. Contraction theory is a tool to analyze the convergence between two arbitrary system trajectories. If the trajectories of the perturbed system return to their nominal behavior with an exponential convergence rate, such a nonlinear system is said to be contracting. The main result of contraction theory is given in Lemma 1 (Lohmiller and Slotine (1998),Lohmiller and Slotine (2000),Jouffroy and Slotine (2004)). Lemma 1: For a given system x˙ = f (x, t) , there exists a scalar ξ > 0, ∀x, ∀t 0 such that ∂f ∂x −ξI < 0 or T
−ξI < 0, any trajectory which starts + ∂f ∂x in a ball of constant radius centred about a given trajectory and contained at all a time in a contraction region, remains in that ball and converges exponentially to the given trajectory. Moreover, global exponential convergence to this given trajectory is guaranteed if the whole state space region is contracting. In the case of feedback combinations of two systems, results of contraction theory are extended as follows: Consider the two systems possibly of different dimensions whose dynamics are given by x˙ 1 = f1 (x1 , x2 , t) (1) x˙ 2 = f2 (x1 , x2 , t) If these systems are connected in feedback combination, we can write virtual displacements in transformed domain as d δz1 F1 G δz1 = (2) δz2 −GT F2 dt δz2 Then the augmented system is contracting if and only if the separated plants are contracting. 1 2
∂f ∂x
2.1 A Simple Model of Underwater Vehicle A simple model to describe the speed of underwater vehicle is proposed by Healey and Marco (1992) from the nonlinear underwater vehicle equations of motion (?). The speed equations of an underwater vehicle with actuator can be written in the form of parametric-strict feedback as follows : q˙ = v η d v˙ = − |v| v + (3) m m η u η˙ = − + T T
where m is the mass of underwater vehicle in kg, d is vehicle drag coefficient in N s2 /m2 and T is the time constant of actuator. The state vector of (3) consists of vehicle position q, vehicle velocity v and vehicle torque η. Control signal u is acting as the commanded input to the actuator. In the position control of underwater vehicle, control objective is to design a feedback control law u so that the vehicle position y = q converges to a desired trajectory of yd with all other signals remaining bounded. The following lemma derives the resulting control law in contraction framework. Lemma 2: Given an underwater vehicle system (1), the actual vehicle position q(t) will track a reference trajectory yd (t) if control law u is given by −z1 (z2 + β2 ) u=T − (k1 + k2 ) z2 + m T + T (d (z1 + β1 − |z1 + β1 |) (z1 + β1 )) (4) + T [d (z1 + β1 + |z1 + β1 |) − k1 m] z2 × −e1 + + (1 − k1 ) z1 , k1 > 0, k2 > 0 m
where auxiliary variables z1 and z2 are defined as z1 = v − β1 (q) and z2 = η − β2 (q, z1 ) respectively with β1 (q) = − [q − yd ] + y˙ d and β2 (q, z1 ) = d |v| v − k1 z1 m + y¨d m.
Proof : Defining tracking error e1 = y − yd and selecting auxiliary variables as z1 = v − β(q), where β(q) the virtual control input, makes the first subsystem in (3) contracting with reference to q. Thus, the dynamics of subsystem can be written as q˙ = z1 + β1 (q) (5) The virtual displacement of this system can be represented in differential framework as δ q˙ = δz1 + J11 δq (6) where Jacobian matrix J11 is represented by ∂ [β1 (q)] (7) J11 = ∂q This Jacobian J11 is Uniformly Negative Definite (UND) in nature by the selection of β1 (q) as follows β1 (q) = − [q − yd ] + y˙ d = −e1 + y˙ d (8) Thus, z1 = v + e1 − y˙ d and hence the first subsystem can be reduced into error dynamics of vehicle position as e˙ 1 = q˙ − y˙ d = v − y˙ d (9) = −e1 + z1 By taking time derivative of z1 and using (3) and (5), we get z˙1 = v˙ + e˙ 1 − y¨d (10) η d − e1 + z1 − y¨d = − |v| v + m m Define new virtual control input β2 (q, z1 ) to make (10) contracting w.r.t. z1 . The feedback interconnection of subsystems are ensured in (5) and (10). Defining new auxiliary variable z2 = η − β2 (q, z1 ), (10) can be represented as z2 d β2 z˙1 = −e1 − |v| v + + z1 + − y¨d (11) m m m Thus, the virtual displacement in differential framework for the combination of the first two subsystems (9) and (11) can be represented as δ e˙ 1 −1 1 δe1 0 (12) = + δz2 −1 J22 1/m δ z˙1 δz1
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−1 1 Here, Jacobian matrix J = , where J22 = −1 J22 β2 (q,z1 ) ∂ d − y¨d ,which is UND in nature ∂z1 z1 − m |v| v + m by suitable selection of β2 (q, z1 ) and by considering δz2 to be bounded external input with the constant coefficient T column vector [ 0 1/m ] . For an UND of J, β2 (q, z1 ) is selected as β2 = d |v| v − k1 z1 m + y¨d m (13) Now taking time derivative of z2 and using (3) ∂β2 ∂β2 z˙2 = η˙ − q˙ − z˙1 (14) ∂q ∂z1 where ∂ ∂β2 =d v {|z1 − (q − yd ) + y˙ d |} ∂q ∂q ∂ {z1 − (q − yd ) + y˙ d } + d |v| ∂q = d (v − |v|) ∂β2 ∂ {|z1 − (q − yd ) + y˙ d |} − k1 m =d v ∂z1 ∂q ∂ + d |v| {z1 − (q − yd ) + y˙ d } ∂q = d (v + |v|) − k1 m With use of (11) and (13), the expression for (14) can be reduced into u η z˙2 = − + − d (v − |v|) v − [d (v + |v|) − k1 m] z˙1 T T z2 + (1 − k1 ) z1 z˙1 = −e1 + m (15) where η = z2 + β2 (q, z1 ) and v = z1 + β1 (q). The following selection of control u −z1 (z2 + β2 ) u=T − (k1 + k2 ) z2 + m T + T (d (z1 + β1 − |z1 + β1 |) (z1 + β1 )) (16) + T [d (z1 + β1 + |z1 + β1 |) − k1 m] z2 × −e1 + + (1 − k1 ) z1 m ensures contracting nature of final subsystem w.r.t. z2 . Thus,(15) becomes z1 z˙2 = − − (k1 + k2 ) z2 (17) m It ensures feedback combination of contracting subsystems. Now, dynamics of transformed system in differential framework can be written as −1 1 0 δe1 δ e˙ 1 δ z˙1 = −1 1 − k1 1/m δz1 (18) 0 −1/m − (k1 + k2 ) δ z˙2 δz2 The determinant of the above Jacobian matrix is given by 1 (19) det(J) = −k1 (k1 + k2 ) − 2 m
The positive values of k1 and k2 shows the UND of Jacobian matrix. The convergence of δe1 → 0, δz1 → 0, and δz2 → 0 are therefore ensured using contraction theory stated in Lemma1. Consequently, vehicle position q tracks the desired trajectory of yd as per the requirement .
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Fig. 1. Propeller geometry 2.2 A More Advanced Model of Underwater Vehicle A nonlinear function of propeller velocity and axial flow velocity can be described from a simple propeller blade liftdrag model (Whitcomb and Yoerger (1999a)). Consider a more advanced model of underwater vehicle using the detailed hydrodynamic effects on the vehicle (Bachmayer et al. (2000)). The dynamic equations of such advanced model is given by q˙ = v cD T v˙ = − v |v| m m (23) k2 Q −k1 ω˙ = ω+ u− I I I γ˙ = ω where q is the position of vehicle, v is the velocity of vehicle, ω is the angular velocity of propeller, and γ is the angular position of propeller. T represents the thrust of vehicle whose mass is m, and Q is the drag force of propeller with inertia I. The vehicle drag coefficient is denoted by cD and, k1 and k2 are designated for the backemf and gain of the motor respectively. Since the propeller represents the interface between motor dynamics and fluid dynamics, vehicle thrust and propeller drag as a function of propeller blade lift forceL and drag force D. As a result, the hydrodynamic force in body fixed T coordinates fHyd = [ T −Q ] can be computed from the lift and drag force cos θ − sin θ L (24) fHyd = sin θ cos θ D where θ is the pitch angle which is related to vthe vehicle , where velocity and propeller speed. i.e., θ = tan−1 rω r is the effective propeller radius. The geometry of the propeller is pictorially represented in Fig.1.
With effective angle of attack α = p − θ , where p is blade angle, the Lift and Drag forces in (24) reasonable 2α approximation for the α π periodic lift force and π periodic drag force are 2 2 2 L = 500 v + r ω
2
2
= 500 v + r ω
2
2
2
+ 500 v + r ω
2
2
D = 500 v + r ω
= 500 v 2 + r2 ω 2
2
2
CL max sin p cos p cos2 θ − sin2 θ
2
2
CL max sin α cos α
CL max
2
2
sin p − cos p sin θ cos θ
C0 + Ci max sin2 α
C0 + Ci max sin2 pcos2 θ − cos2 psin2 θ
+ 500 v + r ω
2
Ci max [2 sin p cos p sin θ cos θ]
(25)
(26)
with maximal lift coefficient CL max > 0 , frictional drag coefficients c0 > 0, and maximal induced drag coefficient
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Proceedings of the 20th IFAC World Congress 2668 Majeed Mohamed et al. / IFAC PapersOnLine 50-1 (2017) 2665–2670 Toulouse, France, July 9-14, 2017
∂fHyd ∂ = ∂η ∂η
−500 (v 2 + r2 ω 2 )
= −500
(v 2 + r 2 ω 2 )
c0 v 3 + (ci max + c0 − cL max )vr2 ω 2 (cL max + c0 )rωv 2 + (ci max + c0 )r3 ω 3
(v 2 + r2 ω 2 ) 2 2 2 3c0 v2 + c1 r2 ω2 v +r ω
2c2 rωv
2c1 rωv c2 v + 3c3 r2 ω 2
− ( v rω)
2
2c0 v 4 + 3c0 v 2 r 2 ω 2 + c1 r4 ω 4 (−c0 + 2c1 )v 3 rω + c1 vr 3 ω 3 c2 rωv 3 + (−c3 + 2c2 )r 3 ω 3 v c2 v 4 + 3c3 r 2 ω 2 v 2 + 2c3 r 4 ω 4
(v +r ω )
v rω , sin θ rω re(v 2 +r 2 ω 2 )
spectively. Thus, the expressions (25) and (26) are reduced into L = 500CL max r2 ω 2 − v 2 sin p cos p + 500CL max vrω sin2 p − cos2 p 2 D = 500 C0 v + r2 ω 2 + Ci max [2vrω sin p cos p] + 500Ci max r2 ω 2 sin2 p − v 2 cos2 p (27) Thus, the hydrodynamic force fHyd in (24) can be expanded as 500 rω −v b1 fHyd = v rω b2 2 2 2 (v + r ω ) b1 = CL max
r 2 ω 2 − v 2 sin p cos p + vrω sin2 p − cos2 p
b2 = C0 v 2 + r 2 ω 2 + Ci max [2vrω sin p cos p] 2
2
2
2
∂fHyd ∂η
Considering the new state vector η = [ v ω ] , can be written as in (29) . The variation of coefficient terms are c1 = ci max + c0 − cL max , c2 = cL max + c0 , and C3 = Ci max + C0 . The expansion of equation (29) becomes (30). This equation can be rewritten in terms of trigonometric expression of angle-of-attack α as shown in (31). The contraction behavior of the vehicle motion is only a function of the angle of attack . Consider now a typical underwater vehicle with mass m = 100kg, Inertia I = 1.5, extrapolated lift and drag coefficients c0 = 0.1, Ci max = 2.5 , CLmax = 1.1 and propeller radius r = ∂fHyd 0.1m. The eigenvalues of the symmetric part of ∂η √ divided by 500 v 2 + r2 ω 2 are illustrated in the Fig.2 as a function of |α|. Since both are uniformly negative, the underwater vehicle system is naturally contracting. Note that the contraction behavior increases with the energy dissipation at high angle of attack . 2.3 Controller design with a more advanced model of underwater vehicle The UGES tracking control law is developed for an advanced model of underwater vehicle (23) using contraction theory and presented in the form of following theorem. Theorem 1: Based on the result of derivation made for the natural contracting behaviour of underwater vehicle with an advanced model of (23), the actual vehicle position
cos p − sin p sin p cos p
−1
cos p − sin p sin p cos p
(30)
(31)
0
−2 −0.5
−2.5 −3 −3.5
−1
−4 −4.5 −5 −5.5 0
1
|α|
2
−1.5 0
3
1
2
3
|α|
Fig. 2. Natural contraction behavior of an underwater vehicle q(t) will track a reference trajectory if control law u is given by I k1 cD 2 u=
k2
+
(28) T
c0 v 3 + c1 vr 2 ω 2 c2 rωu2 + c3 r3 ω 3
(29)
−1.5
2
+ Ci max r ω sin p − v cos p
cos p − sin p sin p cos p
3 α + c1 cos αsin3 α (c0 −42c1 )cos α sin c2 cos α + 3c3 sin2 αcos2 α + 2c3 sin4 α
2 4 4 2 cos p sin p 2c0 cos α +33c0 cos αsin α + c31 sin α − sin p cos p −c2 sin αcos α + (c3 − 2c2 )sin α cos α cos p − sin p 2 2 2 × −500 v + r ω sin p cos p
Ci max > 0. Since the pitch angle θ = tan−1 and cos θ are replaced by √ 2 v 2 2 and √
Eigen Value− λ2
∂fHyd = ∂η
3
−500
Eigen Value− λ1
∂fHyd = ∂η
I
ω − z1 − (g1 + g2 ) z2 +
cD I (v + |v|) − g1 k2 m g1 > 0, g2 > 0
m
v − v |v|
−e1 + (1 − g1 ) z1 + z2 +
T m
,
(32)
where auxiliary variables z1 and z2 are defined as z1 = v − β1 (q, γ) and z2 = ω − β2 (q, z1 ) respectively with β1 (q) = − [q − yd ] + y˙ d − γ and β2 = cmD |v| v1 + y¨d − g1 z + γ .
Proof: Defining tracking error e1 = y − yd and selecting auxiliary variables as z1 = v − β1 (q, γ), where β1 (q, γ) the virtual control input, makes first subsystem (23) in contracting w.r.t. q. Thus, the dynamics of subsystem can be written as (33) q˙ = z1 + β1 (q, γ) For this system, virtual displacement in differential framework can be represented as (34) δ q˙ = δz1 + J11 δq where Jacobian matrix J11 is represented by ∂ J11 = [β1 (q, γ)] (35) ∂q This Jacobian J11 is UND (uniformly negative definite) in nature by the selection of β1 as follows (36) β1 (q) = − [q − yd ] + y˙ d − γ = −e1 + y˙ d − γ Thus, z1 = v + e1 − y˙ d + γ and hence the first subsystem can be reduced into error dynamics of vehicle position as e˙ 1 = q˙ − y˙ d = v − y˙ d (37) = −e1 + z1 − γ By taking time derivative of z1 with use of (23) and (33), we get
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z˙1 = v˙ + e˙ 1 − y¨d + γ˙ (38) cD T − |v| v + ω + (z1 − e1 − γ) − y¨d = m m Define new virtual control input β2 (q, z1 ) to make equation reference goes here contracting w.r.t. z1 . This ensures the feedback interconnection of subsystems in (33) and (38). Defining new auxiliary variable z2 = ω − β2 (q, z1 ), (38) can be represented as cD T z˙1 = − |v| v + z2 + β2 (q, z1 ) − e1 + z1 − γ − y¨d (39) m m So for the combination of the first two subsystems (37) and (39), differential framework based virtual displacement can be written as δ e˙ 1 −1 1 δe1 0 (40) = + δz2 −1 J22 1 δ z˙1 δz1 −1 1 Here, Jacobian matrix J = , where J22 = −1 J22 cD T ∂ ¨d + m , which is UND ∂z1 z1 − m |v| v + β2 (q, z1 ) − γ − y in nature by suitable selection of β2 (q, z1 ) and δz2 is considering as bounded external input. Since the underwater vehicle shows its naturally contracting behaviour in Fig.3, an UND of J is ensured by the selection of β2 (q, z1 ) as follows: cD |v| v1 + y¨d − g1 z + γ (41) β2 = m T As a result, (39) becomes z˙1 = −e1 + (1 − g1 ) z1 + z2 + m . Now taking time derivative of z2 and using (23) ∂β2 ∂β2 z˙2 = ω˙ − q˙ − z˙1 ∂q ∂z1 ∂β2 ∂β2 T v− = ω˙ − −e1 + (1 − g1 ) z1 + z2 + ∂q ∂z1 m (42) where ∂β2 cD = (v − |v|) (43) ∂q m cD ∂β2 (44) (v + |v|) − g1 = ∂z1 m With use of (39) and (41), the expression for (42) can be reduced into k2 Q cD −k1 ω+ u− − (v − |v|) v z˙2 = I I I m c T D − −e1 + (1 − g1 ) z1 + z2 + (v + |v|) − g1 m m (45) where ω = z2 +β2 (q, z1 ) and v = z1 +β1 (q). The following selection of control u u=
k1 cD v 2 − v |v| ω − z1 − (g1 + g2 ) z2 + I m cD I T + −e1 + (1 − g1 ) z1 + z2 + (v + |v|) − g1 k2 m m (46) I k2
ensures contracting nature of final subsystem w.r.t. z2 . Thus, (45) becomes Q z˙2 = −z1 − (g1 + g2 ) z2 − (47) I It also ensures feedback combination of contracting subsystems. Now, the dynamics of transformed system in differential framework can be written as
δ e˙ 1 δ z˙1 δ z˙2
−1
= −1 1 − g1 0
−1
T
2669
1 0
1 − (g1 + g2 )
T ∂ m + ∂z − Q I
δe1 δz1
δz2
(48)
where z = [ z1 z2 ] . The transformed matrix can be obtained from the expression of z1 = v + e1 + γ − y˙ d and z2 = ω − cmD v |v| + g1 z1 − y¨d − γ as follows 1 0 δv δz1 cD = (49) δz2 δω g1 − |v| 1 m T Therefore, z η = [ v ω ] and the Jacobian matrix F of (48) can be reduced into 0 1×2 0 T −1 1 0 1 + ∂ F = −1 1 − g1 m 0 0 −1 − (g1 + g2 ) Q ∂η −
I J1
J2
(50) From the results of natural contracting behaviour of an underwater vehicle, the jacobian of the dynamics associated to non-zero terms in J2 using the inertia as metric is given by ∂ ∂ T = (fHyd ) 0 (51) −Q ∂η ∂η The determinant of the Jacobian matrix J1 is given by (52) det(J1 ) = − [g1 (g1 + g2 ) + 1] The positive values of g1 and g2 shows the UND of Jacobian matrix F and hence the dynamics of (48) will be contracting. In other words, the contracting behaviour of overall system is ensured for the selected values u,β1 ,β2 with g1 > 0, g2 > 0 . Therefore,the exponential stability of system states is obtained since all the system states converge to each other .
Remark: In case of uncertainty in parameter k1 in (23), the approach given in (Sharma and Kar (2009)) can be used to estimate kˆ1 , the controller structure is same as proposed in (32) except that uncertain parameter k1 is replaced by kˆ1 . 3. NUMERICAL SIMULATION The performances of proposed position control laws are tested by the numerical simulation of underwater vehicle. We are primarily considered a simple model of vehicle whose mass m = 4kg, drag coefficient d = 1N s2 /m2 , actuator time constant T = 1sec. The initial values considered as [0; 0; 0]. The fourth order Runge-Kutta integrator with a fixed time step of 0.1sec is used here to integrate the control law presented in (16), and the controller gains k1 and k2 are selected as 3.79 and 11.85 respectively. The dotted line is the time history of vehicle position with proposed control law, which converges to the reference yd represented by the full line as shown in Fig. 3. The good agreement between the vehicle conbtrolled position and desired trajectory is achieved. In the case of a more advanced model of underwater vehicle, the performances of UGES control law is tested by the numerical simulation of a vehicle with initial states as q(0) = 0m, v(0) = 0.5m/sec, ω(0) = 2rad/sec., and γ(0) = 0. The control law derived in
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Position q (m) Velocity v (m/s)
0.5
REFERENCES
Desired Trajectory Controller
1 0.5 0 0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
10
20
30 time (sec)
40
50
60
0 −0.5
Control input Torque u η (N/m)
5 0 −5
50 0 −50
Fig. 3. Tracking controller responses for a simple model of underwater vehicle (46) is integrated with advanced model (23) of vehicle by using the fourth order Runge-Kutta integrator with a fixed time step of 0.01s. The dynamics of desired trajectory is tracked with use of controller gains g1 = 0.2 and g2 = 0.09. The fig. 4. shows the convergence of the vehicle tracking position to the desired trajectory. 0.18
error in q
q (m)
0.1 0.05 0 0.003 0
−0.003
ω (rad/s)
v (m/s)
0.4 0.2 0 0 −5 −10
u
γ (rad)
0 −1 −2
0 −100 −200 −300
0
0.2
0.4
0.6
0.8
1 1.2 time in sec
1.4
1.6
1.8
2
Fig. 4. Tracking controller for advanced model of vehicle 4. CONCLUSIONS A nonlinear control system was investigated via contraction theory for a simple as well as a more advanced model of an underwater vehicle. Control system was developed based on the application of 1) integral backstepping controller design technique for nonlinear system, and 2) contraction theory to ensure the incremental exponential global stability of underwater vehicle to track the desired trajectory of vehicle position. Due to the application of backstepping technique, controller design was restricted to parametric-strict-feedback form, but the selection of its tuning parameters were analytically obtained from the contraction theory based incremental stability analysis. Simulation results were presented to show the effectiveness of the design method of integral backstepping contraction control law for underwater vehicle.
Bachmayer, R., Whitcomb, and Grosenbaugh (2000). An accurate four-quadrant nonlinear dynamical model for marine thrusters: theory and experimental validation. IEEE J. of Oceanic Engineering, 25(1), 146–159. Fossen, T.I. (1994). Guidance & control of ocean vehicles. John Wiley & Sons Inc. Healey, A.J., Rock, S.M., Cody, S., Miles, D., and Brown, J.P. (1994). Toward an improved understanding of thruster dynamics for underwater vehicles. In Proceedings of the 1994 Symposium on Autonomous Underwater Vehicle Technology, 340–352. Healey, A. and Marco, D. (1992). Slow speed flight control of autonomous underwater vehicles: Experimental results with the nsp auv ii. In International Offshore and polar Engineering Conference (ISOPE), 523–532. Jouffroy, J. (2003). A relaxed criterion for contraction theory: Application to an underwater vehicle observer. In 2003 European Control Conference, 2999–3004. Jouffroy, J. and Fossen, T.I. (2004). On the combination of nonlinear contracting observers and uges controllers for output feedback. In 43rd IEEE Conference on Decision and Control, volume 5, 4933–4939. Jouffroy, J. and Lottin, J. (2002a). Integrator backstepping using contraction theory: a brief methodological note. Jouffroy, J. and Lottin, J. (2002b). On the use of contraction theory for the design of nonlinear observers for ocean vehicles. In Proceedings of the 2002 American Control Conference, volume 4, 2647–2652. Jouffroy, J. and Slotine, J.J.E. (2004). Methodological remarks on contraction theory. In 43rd IEEE Conference on Decision and Control, volume 3, 2537–2543. Khalil, H.K. (1996). Nonliner Systems. NJ: Prentice-Hall. Lohmiller and Slotine (1998). On contraction analysis for nonlinear systems. Automatica, 34(6), 683–696. Lohmiller, W. and Slotine, J.J.E. (2000). Control system design for mechanical systems using contraction theory. IEEE Trans. on Automatic Control, 45(5), 884–889. Lohmiller, W.S. (1999). Contraction Analysis of Nonlinear System. Thesis. Majeed, M. and Kar, I.N. (2013). Design and convergence analysis of stochastic frequency estimator using contraction theory. IET- Signal Processing, 7(5), 389–399. Marco, D.B. and Healey, A.J. (1998). Local area navigation using sonar feature extraction and model-based predictive control. International Journal of Systems Science, 29(10), 1123–1133. Sharma, B.B. and Kar, I.N. (2009). Contraction based adaptive control of a class of nonlinear systems. In 2009 American Control Conference, 808–813. Whitcomb, L.L. and Yoerger, D.R. (1999a). Development, comparison, and preliminary experimental validation of nonlinear dynamic thruster models. IEEE Journal of Oceanic Engineering, 24(4), 481–494. Whitcomb, L.L. and Yoerger, D.R. (1999b). Preliminary experiments in model-based thruster control for underwater vehicle positioning. IEEE Journal of Oceanic Engineering, 24(4), 495–506. Whitcomb, L.L., Yoerger, D.R., Singh, H., and Mindell, D.A. (1998). Toward precision robotic maneuvering, survey, and manipulation in unstructured undersea environments. In Robotics Research-The Eighth International Symposium, 45–54.
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IFAC PapersOnLine 50-1 (2017) 2665–2670 Contraction Based Tracking Control Contraction Contraction Based Based Tracking Tracking Control Control Autonomous Underwater Contraction Based Tracking Vehicle Control Autonomous Underwater Vehicle Autonomous Underwater Vehicle Autonomous Underwater Vehicle Majeed Mohamed and Rong Su
of of of of
Majeed Mohamed and Rong Su Majeed Mohamed and Rong Su Majeed Mohamed and Rong Department of Electrical Electrical Engineering, NTU,Su Singapore. Department of Engineering, NTU, Singapore. Department of Electrical Engineering, NTU, Singapore. (e-mail: [email protected], [email protected]). (e-mail: [email protected], [email protected]). Department of Electrical Engineering, NTU, Singapore. [email protected]). (e-mail: [email protected], (e-mail: [email protected], [email protected]). Abstract: Abstract: This This paper paper discusses discusses the the incremental incremental stability stability of of an an underwater underwater vehicle vehicle using using stability of andynamics underwater using Abstract: This theory. paper discusses the incremental vehicle the contraction Stability analysis considered in vehicle of a simple and the contraction theory. Stability analysis considered in vehicle dynamics of a simple and Abstract: This theory. paper discusses the incremental stability ofcontroller andynamics underwater using the contraction Stability analysis considered in the vehicle of avehicle simple and a more advanced model has the ability to constructing to track the desired a more advancedtheory. model Stability has the ability toconsidered constructing the controller to track the desired the contraction analysis in vehicle dynamics of a simple and atrajectory more advanced model vehicle has theposition. ability to constructing the controller to track the desired of The natural behavior of underwater of underwater underwater vehicle position. The natural contracting contracting behavior of an an the underwater atrajectory more isadvanced model has the ability model to constructing the controller to track desired trajectory of underwater vehicle position. The natural contracting behavior of an underwater vehicle ensured for a more advanced of vehicle to derive contracting exponentially vehicle is ensured for a more advanced model of vehicle to derivebehavior contracting exponentially trajectory of underwater vehicle position. The natural contracting of an underwater vehicle is ensuredThe for tuning a more parameters advanced model of vehicle to derive contracting exponentially stable of have selected analytically from stable controller. controller. The tuning parameters of the the controller controller have selected analytically from the the vehicle is ensured foranalysis a more parameters advanced model vehicleThe to derive contracting exponentially stable controller. The tuning of the of controller have selected analytically from the incremental stability using contraction theory. controller design is restricted to incremental stability analysis using contraction theory. The controller design is restricted to stable controller. The tuning parameters of the controller have selected analytically from the incremental stability analysis using contraction theory. Thebackstepping controller design is technique. restricted In to parametric-strict-feedback form to develop an incremental design parametric-strict-feedback form to develop an incremental backstepping design technique. In incremental stability analysis using contraction theory. The controller design is restricted to parametric-strict-feedback form to develop an incremental backstepping design technique. In this afford to aa controller in way it this paper, paper, proposed proposed method method afford to construct construct controllerbackstepping in aa recursive recursivedesign way and and it enforces enforces parametric-strict-feedback form to ofdevelop an incremental technique. In way and this paper, proposed method afford toan construct a controller inand a recursive it enforces incremental exponential stability underwater vehicle not just global asymptotic incremental exponential stability oftoan underwater vehicle inand not justway global asymptotic this paper, proposed method afford construct a controller a recursive and it enforces and not just global asymptotic incremental exponential stability of an underwater vehicle stability. stability. incremental exponential stability of an underwater vehicle and not just global asymptotic stability. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. stability. Keywords: Keywords: Incremental Incremental stability, stability, tracking tracking control, control, exponential exponential stability, stability, contraction contraction theory, theory, Keywords: Incremental stability, tracking control, exponential stability, contraction theory, underwater vehicle. underwaterIncremental vehicle. Keywords: stability, tracking control, exponential stability, contraction theory, underwater vehicle. underwater vehicle. 1. trajectories 1. INTRODUCTION INTRODUCTION trajectories with with respect respect to to each each other other (Lohmiller (Lohmiller (1999)). (1999)). 1. INTRODUCTION trajectories withofrespect to each other (Lohmiller (1999)). Such a property nonlinear systems is aa stronger property Such a property of nonlinear systems is stronger property 1. INTRODUCTION trajectories withofrespect to each other (1999)). Such a property nonlinear systems is (Lohmiller aastronger property than exponential convergence to single Controlling than global global exponential convergence to astronger single trajectory. trajectory. Controlling the the horizontal horizontal position position of of an an underwater underwater vehivehi- Such a property of nonlinear systems is a property than global exponential convergence to a single trajectory. Controlling the horizontal position of an underwater vehicle at of sea cle changing changingthe at the the bottom bottom of the the of sea is is of of great great imporimporIn literature, Lyapunov-based Global Expothan exponential convergenceUniform to a single trajectory. Controlling position vehiIn the theglobal literature, Lyapunov-based Uniform Global Expocle changing at horizontal the bottom of the seaanisunderwater ofaccurate great importance for precise maneuvering. Real-time positance for precise maneuvering. Real-time accurate posi- In the literature, Lyapunov-based Uniform Global Exponential Stability (UGES) of system dynamics is proven and cle changing at the bottom of the sea is of great impornential Stability (UGES) of system dynamics is proven and tance for precise maneuvering. Real-time accurate posi- In tion measurements of underwater vehicles are facilitated the literature, Lyapunov-based Uniform Global Expotion measurements of underwater vehicles accurate are facilitated nential Stability (UGES) of system dynamics isvehicle proven and extended its application to the underwater with tance for precise maneuvering. Real-time posiextended its application tosystem the underwater vehicle with tion measurements of underwater vehicles are facilitated by the latest advances in sensing technology of vehicle nential Stability (UGES) of dynamics is proven and by the latest advances in sensing vehicles technology of vehicle extended its application to the underwater vehicle with the contracting (Jouffroy and tion measurements of underwater are facilitated the nonlinear nonlinear contractingtoobserver observer (Jouffroy vehicle and Fossen Fossen by the latest advances in sensing technology of vehicle extended position and velocity (Marco and Healey (1998),Whitits application the underwater with position and velocity (Marco and Healey (1998),Whitthe nonlinear contracting observer (Jouffroy and Fossen (2004)). Likewise, the application of contraction analysis by theetlatest advances in sensing technology ofthe vehicle (2004)). Likewise, the application of(Jouffroy contraction analysis position and(1998)). velocity (Marco and Healey (1998),Whitcomb al. In order to precisely control lowthe nonlinear contracting observer and Fossen comb et al. (1998)). In order to precisely control the low(2004)). Likewise, thederive application of contraction analysis has pursued the observer by position and(1998)). velocity (Marco Healey (1998),Whithas been been Likewise, pursued to to derive the nonlinear nonlinear observeranalysis by sevsevcomb et al. Inunderwater order to and precisely control the low- (2004)). speed trajectory of vehicle, these thederive application of contraction speed et trajectory of an an underwater vehicle,control these advances advances has been pursued to the(2013),Jouffroy nonlinear observer by several authors (Majeed and Kar (2003),Joufcomb al. (1998)). In order to precisely the loweral been authors (Majeed and Kar (2013),Jouffroy (2003),Joufspeed trajectory of an underwater vehicle, these advances has in position sensing are greatly backup to the controller pursued to derive the nonlinear observer by sevin position sensing are greatly backup to these the controller eral authors (Majeed and Kar (2013),Jouffroy (2003),Jouffroy and (2002b),Jouffroy and The speed trajectory of an vehicle, advances froy authors and Lottin Lottin (2002b),Jouffroy and Slotine Slotine (2004)). (2004)). The in position sensing areunderwater greatly backup to the controller design. But, our of dynamics of eral (Majeed and Kar (2013),Jouffroy (2003),Joufdesign. But,sensing our understanding understanding of the the to dynamics of the the froy and Lottin (2002b),Jouffroy andsignificant Slotine (2004)). The contraction theory framework offers advantages in position are greatly backup the controller contraction theory framework offers significant advantages design. But, ourand understanding of the dynamics of bladed thrusters vehicle are principally limited to the froy and1)Lottin (2002b),Jouffroy andsignificant Slotine (2004)). The bladed thrusters and vehicle are principally limited to the theory framework offers advantages that the to the design. But, ourand understanding of the dynamics the contraction that it it 1) eliminates eliminates the need need to know know the equilibrium equilibrium bladed thrusters vehicle aredynamics principally limited of toused the control application while these commonly contraction theory framework offers significant advantages control application while these dynamics commonly used that it 1) eliminates the need to know the equilibrium dynamics as for an bladed thrusters andwhile vehicle aredynamics principally limited(Healey toused the point point itof of system system dynamics as it it works works for the an incremental incremental control application these commonly to dynamically positioned marine that eliminates the need to know equilibrium to actuate actuate dynamically positioned marine vehicles vehicles (Healey point of1)system dynamics asand it works fornot an require incremental convergence of trajectories, 2) does difficontrol application while these dynamics commonly used convergence of trajectories, and 2) does not require diffito actuate dynamically positioned marine vehicles (Healey et al. (1994),Whitcomb and Yoerger (1999a)). This repoint of system dynamics as it works for an incremental et actuate al. (1994),Whitcomb and Yoerger (1999a)). This re- convergence of trajectories, and 2) does not require difficult task of the selection of a suitable Lyapunov function to dynamically positioned marine vehicles (Healey cult task of the selection of a suitable Lyapunov function et al. (1994),Whitcomb anddesign Yoerger (1999a)). This re- convergence search work focuses on the of position controller of trajectories, 2) stability does notanalysis require diffisearch work focuses on the design of position controller task of the selection of aand suitable Lyapunov function for analysis. As aa result, does et al.the (1994),Whitcomb and Yoerger This re- cult for stability stability analysis. As of result, stability analysis does search work focuses on the design of (1999a)). position controller with use of a simple (Fossen (1994)) as well as more cult task of athe selection a suitable Lyapunov function with the use of a simple (Fossen (1994)) as well as more for stability analysis. As a result, stability analysis does not require skillful simplification to show negative defsearch work focuses on the design of position controller not stability require aanalysis. skillful simplification to show analysis negative does defwith the use of a simple (Fossen (1994)) as well as more for advanced models of the underwater vehicle (Whitcomb As a result, stability advanced models of the underwater vehicle (Whitcomb not require a skillful simplification to show negative definiteness of the derivative of a Lyapunov function. For with the use of a simple (1994)) as well as more initeness of the derivative of a Lyapunov function. For a a advanced models of the (Fossen underwater vehicle (Whitcomb and Yoerger (1999b)). not require a skillful simplification to show negative defand Yoerger (1999b)). ofclass the of derivative of systems, a Lyapunov function. For a particular nonlinear an integrator backadvanced models of the underwater vehicle (Whitcomb initeness particular class of nonlinear systems, an integrator backand Yoerger (1999b)). initeness ofclass the of derivative of systems, a Lyapunov function. For particular nonlinear an integrator back-a stepping in framework is The nonlinear control and (1999b)). stepping procedure procedure in contraction contraction framework is introduced introduced The Yoerger nonlinear control systems systems design design in in a a recursive recursive way way particular class of nonlinear systems, an integrator backstepping procedure in contraction framework is introduced The nonlinear control systems design in a recursive way in (Jouffroy and Lottin (2002a)). In the light of has shown for the so in (Jouffroy and Lottin (2002a)). In the light of such such has been been shown great great interest interest for the past past so many many years. years. stepping procedure in contraction framework is introduced The nonlinear systems design in anonlinear way in (Jouffroy and Lottin (2002a)). In the in light ofpaper such has been showncontrol great interest for the past sorecursive many years. framework, UGES control law is derived this The backstepping method is one of the techframework, UGES control law is derived in this paper The backstepping method is one of the nonlinear techin (Jouffroy UGES and Lottin In desired the in light ofpaper such has been shown great interest past soprocedure many years. framework, control(2002a)). law is the derived this The backstepping method is for onethe ofdesign the nonlinear techusing theory to track trajectory nique, which a in using contraction contraction theory tolaw track the desired trajectory nique,backstepping which provides provides a recursive recursive design procedure in framework, UGES control is derived in this paper The method is one of the nonlinear techusing contraction theory to track the desired trajectory nique, which provides a recursive design procedure in of vehicle position. The main contributions a manner for systems transformable into of an an underwater underwater vehicle position. The main contributions a systematic systematic manner for the the systemsdesign transformable into using contraction theoryposition. to trackThe the main desired trajectory which provides recursive procedure in of an underwater vehicle contributions anique, systematic manner fora the systems transformable into this paper are parametric strict feedback form (Khalil (1996)). In this of this paper are parametric strict feedback form (Khalil (1996)). In into this of an underwater vehicle position. The main contributions a systematic manner for the systems transformable of this paper are parametric strict Lyapunov feedback form (Khalil (1996)). In this method, suitable functions are constructed at method, suitable Lyapunov functions are(1996)). constructed at Contraction paper are theory parametric strict feedback form (Khalil In this (1)this Contraction theory based based an an incremental incremental backstepbackstepmethod, suitable Lyapunov functions areeach constructed at of(1) each stage to confirm the stability of subsystem each stage to confirm the stability of each subsystem (1) Contraction theory based to antracking incremental backstepping control law is derived the pomethod, suitable Lyapunov functions are constructed at ping control law is derived to tracking the desired desired poeach stage to confirm the stability of each subsystem (Khalil (1996)). Lyapunov stability theory is widely ex(1) Contraction theory based an incremental backstep(Khalil (1996)). Lyapunov stability theory is widely exping control law is derived to tracking the desired position of an underwater vehicle. The simple and more each stage todesign confirm the stability stability of each subsystem sition of an underwater vehicle. The simple and more (Khalil (1996)). Lyapunov theory is widely explored to the of nonlinear control system. On the ping control law isofderived to tracking theare desired poplored to(1996)). the design of nonlinear control system. On the sition of anmodels underwater vehicle. The simple and more advanced vehicles adopted (Khalil Lyapunov stability theory is widely exadvanced models of underwater underwater vehicles areand adopted plored to contraction the design of nonlinear control system. On the contrary, theory is used as latest method to sition of an underwater vehicle. The simple more contrary, contraction theory is used as latest method to the advanced models of underwater vehicles are adopted for In aa practical point this plored to contraction theanalysing design theory ofofnonlinear the for this this purpose. purpose. Inunderwater practicalvehicles point of ofareview, view, this contrary, is usedcontrol as latestsystem. methodOn toand the convergence systems (Lohmiller advanced models of adopted convergence analysingtheory of nonlinear nonlinear systems (Lohmiller and for thisis purpose. Into a be practical point of view, this result considered highly useful as it permits contrary, contraction is used as latest method to the result is considered to be highly useful as it permits convergence analysing of nonlinear systems (Lohmiller and Slotine (1998),Lohmiller and Slotine (2000)). It is treated for this purpose. In a practical point of view, this Slotine (1998),Lohmiller and Slotine (2000)). It is treated result is considereddesign to beofhighly useful exponentially as it permits for contracting convergence analysing of nonlinear systems (Lohmiller and for the the systematic systematic design ofhighly contracting exponentially Slotine (1998),Lohmiller and Slotine (2000)). It is treated as form stability since contraction theory result considereddesign to beof useful exponentially as it permits as an an incremental incremental form of of and stability since contraction theory for theis systematic contracting stable controllers. Slotine (1998),Lohmiller Slotine (2000)). It is treated stable controllers. as an incremental form of stability sinceofcontraction theory offers a stability analysis nonlinear for the controllers. systematic design of contracting exponentially offers a tool tool for for the the stability analysis ofcontraction nonlinear system system stable as an incremental form of stability sinceof theory offers a tool for the stability analysis nonlinear system stable controllers. offers a tool for the stability analysis of nonlinear system
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Proceedings of the 20th IFAC World Congress 2666 Majeed Mohamed et al. / IFAC PapersOnLine 50-1 (2017) 2665–2670 Toulouse, France, July 9-14, 2017
(2) The selection of tuning parameters of the controller is obtained analytically as a by-product of the incremental stability analysis in contraction theory. The natural contracting behavior of an underwater vehicle is established for a simplified UGES control law in the case of a more advanced model of vehicle. (3) Contracting control law is tested through the simulation studies for the simple and more advanced model of underwater vehicle. The following Section 2 describes the design of integral backstepping position control of an underwater vehicle with its numerical simulation in section 3. Finally, the conclusions are given in section 4. 2. POSITION CONTROL OF AUTONOMOUS UNDERWATER VEHICLE We are considered in turn a simple and more advanced models of underwater vehicle to derive the UGES control law based on the contraction theory in order to track the desired trajectory of vehicle position. Contraction theory is a tool to analyze the convergence between two arbitrary system trajectories. If the trajectories of the perturbed system return to their nominal behavior with an exponential convergence rate, such a nonlinear system is said to be contracting. The main result of contraction theory is given in Lemma 1 (Lohmiller and Slotine (1998),Lohmiller and Slotine (2000),Jouffroy and Slotine (2004)). Lemma 1: For a given system x˙ = f (x, t) , there exists a scalar ξ > 0, ∀x, ∀t 0 such that ∂f ∂x −ξI < 0 or T
−ξI < 0, any trajectory which starts + ∂f ∂x in a ball of constant radius centred about a given trajectory and contained at all a time in a contraction region, remains in that ball and converges exponentially to the given trajectory. Moreover, global exponential convergence to this given trajectory is guaranteed if the whole state space region is contracting. In the case of feedback combinations of two systems, results of contraction theory are extended as follows: Consider the two systems possibly of different dimensions whose dynamics are given by x˙ 1 = f1 (x1 , x2 , t) (1) x˙ 2 = f2 (x1 , x2 , t) If these systems are connected in feedback combination, we can write virtual displacements in transformed domain as d δz1 F1 G δz1 = (2) δz2 −GT F2 dt δz2 Then the augmented system is contracting if and only if the separated plants are contracting. 1 2
∂f ∂x
2.1 A Simple Model of Underwater Vehicle A simple model to describe the speed of underwater vehicle is proposed by Healey and Marco (1992) from the nonlinear underwater vehicle equations of motion (?). The speed equations of an underwater vehicle with actuator can be written in the form of parametric-strict feedback as follows : q˙ = v η d v˙ = − |v| v + (3) m m η u η˙ = − + T T
where m is the mass of underwater vehicle in kg, d is vehicle drag coefficient in N s2 /m2 and T is the time constant of actuator. The state vector of (3) consists of vehicle position q, vehicle velocity v and vehicle torque η. Control signal u is acting as the commanded input to the actuator. In the position control of underwater vehicle, control objective is to design a feedback control law u so that the vehicle position y = q converges to a desired trajectory of yd with all other signals remaining bounded. The following lemma derives the resulting control law in contraction framework. Lemma 2: Given an underwater vehicle system (1), the actual vehicle position q(t) will track a reference trajectory yd (t) if control law u is given by −z1 (z2 + β2 ) u=T − (k1 + k2 ) z2 + m T + T (d (z1 + β1 − |z1 + β1 |) (z1 + β1 )) (4) + T [d (z1 + β1 + |z1 + β1 |) − k1 m] z2 × −e1 + + (1 − k1 ) z1 , k1 > 0, k2 > 0 m
where auxiliary variables z1 and z2 are defined as z1 = v − β1 (q) and z2 = η − β2 (q, z1 ) respectively with β1 (q) = − [q − yd ] + y˙ d and β2 (q, z1 ) = d |v| v − k1 z1 m + y¨d m.
Proof : Defining tracking error e1 = y − yd and selecting auxiliary variables as z1 = v − β(q), where β(q) the virtual control input, makes the first subsystem in (3) contracting with reference to q. Thus, the dynamics of subsystem can be written as q˙ = z1 + β1 (q) (5) The virtual displacement of this system can be represented in differential framework as δ q˙ = δz1 + J11 δq (6) where Jacobian matrix J11 is represented by ∂ [β1 (q)] (7) J11 = ∂q This Jacobian J11 is Uniformly Negative Definite (UND) in nature by the selection of β1 (q) as follows β1 (q) = − [q − yd ] + y˙ d = −e1 + y˙ d (8) Thus, z1 = v + e1 − y˙ d and hence the first subsystem can be reduced into error dynamics of vehicle position as e˙ 1 = q˙ − y˙ d = v − y˙ d (9) = −e1 + z1 By taking time derivative of z1 and using (3) and (5), we get z˙1 = v˙ + e˙ 1 − y¨d (10) η d − e1 + z1 − y¨d = − |v| v + m m Define new virtual control input β2 (q, z1 ) to make (10) contracting w.r.t. z1 . The feedback interconnection of subsystems are ensured in (5) and (10). Defining new auxiliary variable z2 = η − β2 (q, z1 ), (10) can be represented as z2 d β2 z˙1 = −e1 − |v| v + + z1 + − y¨d (11) m m m Thus, the virtual displacement in differential framework for the combination of the first two subsystems (9) and (11) can be represented as δ e˙ 1 −1 1 δe1 0 (12) = + δz2 −1 J22 1/m δ z˙1 δz1
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−1 1 Here, Jacobian matrix J = , where J22 = −1 J22 β2 (q,z1 ) ∂ d − y¨d ,which is UND in nature ∂z1 z1 − m |v| v + m by suitable selection of β2 (q, z1 ) and by considering δz2 to be bounded external input with the constant coefficient T column vector [ 0 1/m ] . For an UND of J, β2 (q, z1 ) is selected as β2 = d |v| v − k1 z1 m + y¨d m (13) Now taking time derivative of z2 and using (3) ∂β2 ∂β2 z˙2 = η˙ − q˙ − z˙1 (14) ∂q ∂z1 where ∂ ∂β2 =d v {|z1 − (q − yd ) + y˙ d |} ∂q ∂q ∂ {z1 − (q − yd ) + y˙ d } + d |v| ∂q = d (v − |v|) ∂β2 ∂ {|z1 − (q − yd ) + y˙ d |} − k1 m =d v ∂z1 ∂q ∂ + d |v| {z1 − (q − yd ) + y˙ d } ∂q = d (v + |v|) − k1 m With use of (11) and (13), the expression for (14) can be reduced into u η z˙2 = − + − d (v − |v|) v − [d (v + |v|) − k1 m] z˙1 T T z2 + (1 − k1 ) z1 z˙1 = −e1 + m (15) where η = z2 + β2 (q, z1 ) and v = z1 + β1 (q). The following selection of control u −z1 (z2 + β2 ) u=T − (k1 + k2 ) z2 + m T + T (d (z1 + β1 − |z1 + β1 |) (z1 + β1 )) (16) + T [d (z1 + β1 + |z1 + β1 |) − k1 m] z2 × −e1 + + (1 − k1 ) z1 m ensures contracting nature of final subsystem w.r.t. z2 . Thus,(15) becomes z1 z˙2 = − − (k1 + k2 ) z2 (17) m It ensures feedback combination of contracting subsystems. Now, dynamics of transformed system in differential framework can be written as −1 1 0 δe1 δ e˙ 1 δ z˙1 = −1 1 − k1 1/m δz1 (18) 0 −1/m − (k1 + k2 ) δ z˙2 δz2 The determinant of the above Jacobian matrix is given by 1 (19) det(J) = −k1 (k1 + k2 ) − 2 m
The positive values of k1 and k2 shows the UND of Jacobian matrix. The convergence of δe1 → 0, δz1 → 0, and δz2 → 0 are therefore ensured using contraction theory stated in Lemma1. Consequently, vehicle position q tracks the desired trajectory of yd as per the requirement .
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Fig. 1. Propeller geometry 2.2 A More Advanced Model of Underwater Vehicle A nonlinear function of propeller velocity and axial flow velocity can be described from a simple propeller blade liftdrag model (Whitcomb and Yoerger (1999a)). Consider a more advanced model of underwater vehicle using the detailed hydrodynamic effects on the vehicle (Bachmayer et al. (2000)). The dynamic equations of such advanced model is given by q˙ = v cD T v˙ = − v |v| m m (23) k2 Q −k1 ω˙ = ω+ u− I I I γ˙ = ω where q is the position of vehicle, v is the velocity of vehicle, ω is the angular velocity of propeller, and γ is the angular position of propeller. T represents the thrust of vehicle whose mass is m, and Q is the drag force of propeller with inertia I. The vehicle drag coefficient is denoted by cD and, k1 and k2 are designated for the backemf and gain of the motor respectively. Since the propeller represents the interface between motor dynamics and fluid dynamics, vehicle thrust and propeller drag as a function of propeller blade lift forceL and drag force D. As a result, the hydrodynamic force in body fixed T coordinates fHyd = [ T −Q ] can be computed from the lift and drag force cos θ − sin θ L (24) fHyd = sin θ cos θ D where θ is the pitch angle which is related to vthe vehicle , where velocity and propeller speed. i.e., θ = tan−1 rω r is the effective propeller radius. The geometry of the propeller is pictorially represented in Fig.1.
With effective angle of attack α = p − θ , where p is blade angle, the Lift and Drag forces in (24) reasonable 2α approximation for the α π periodic lift force and π periodic drag force are 2 2 2 L = 500 v + r ω
2
2
= 500 v + r ω
2
2
2
+ 500 v + r ω
2
2
D = 500 v + r ω
= 500 v 2 + r2 ω 2
2
2
CL max sin p cos p cos2 θ − sin2 θ
2
2
CL max sin α cos α
CL max
2
2
sin p − cos p sin θ cos θ
C0 + Ci max sin2 α
C0 + Ci max sin2 pcos2 θ − cos2 psin2 θ
+ 500 v + r ω
2
Ci max [2 sin p cos p sin θ cos θ]
(25)
(26)
with maximal lift coefficient CL max > 0 , frictional drag coefficients c0 > 0, and maximal induced drag coefficient
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∂fHyd ∂ = ∂η ∂η
−500 (v 2 + r2 ω 2 )
= −500
(v 2 + r 2 ω 2 )
c0 v 3 + (ci max + c0 − cL max )vr2 ω 2 (cL max + c0 )rωv 2 + (ci max + c0 )r3 ω 3
(v 2 + r2 ω 2 ) 2 2 2 3c0 v2 + c1 r2 ω2 v +r ω
2c2 rωv
2c1 rωv c2 v + 3c3 r2 ω 2
− ( v rω)
2
2c0 v 4 + 3c0 v 2 r 2 ω 2 + c1 r4 ω 4 (−c0 + 2c1 )v 3 rω + c1 vr 3 ω 3 c2 rωv 3 + (−c3 + 2c2 )r 3 ω 3 v c2 v 4 + 3c3 r 2 ω 2 v 2 + 2c3 r 4 ω 4
(v +r ω )
v rω , sin θ rω re(v 2 +r 2 ω 2 )
spectively. Thus, the expressions (25) and (26) are reduced into L = 500CL max r2 ω 2 − v 2 sin p cos p + 500CL max vrω sin2 p − cos2 p 2 D = 500 C0 v + r2 ω 2 + Ci max [2vrω sin p cos p] + 500Ci max r2 ω 2 sin2 p − v 2 cos2 p (27) Thus, the hydrodynamic force fHyd in (24) can be expanded as 500 rω −v b1 fHyd = v rω b2 2 2 2 (v + r ω ) b1 = CL max
r 2 ω 2 − v 2 sin p cos p + vrω sin2 p − cos2 p
b2 = C0 v 2 + r 2 ω 2 + Ci max [2vrω sin p cos p] 2
2
2
2
∂fHyd ∂η
Considering the new state vector η = [ v ω ] , can be written as in (29) . The variation of coefficient terms are c1 = ci max + c0 − cL max , c2 = cL max + c0 , and C3 = Ci max + C0 . The expansion of equation (29) becomes (30). This equation can be rewritten in terms of trigonometric expression of angle-of-attack α as shown in (31). The contraction behavior of the vehicle motion is only a function of the angle of attack . Consider now a typical underwater vehicle with mass m = 100kg, Inertia I = 1.5, extrapolated lift and drag coefficients c0 = 0.1, Ci max = 2.5 , CLmax = 1.1 and propeller radius r = ∂fHyd 0.1m. The eigenvalues of the symmetric part of ∂η √ divided by 500 v 2 + r2 ω 2 are illustrated in the Fig.2 as a function of |α|. Since both are uniformly negative, the underwater vehicle system is naturally contracting. Note that the contraction behavior increases with the energy dissipation at high angle of attack . 2.3 Controller design with a more advanced model of underwater vehicle The UGES tracking control law is developed for an advanced model of underwater vehicle (23) using contraction theory and presented in the form of following theorem. Theorem 1: Based on the result of derivation made for the natural contracting behaviour of underwater vehicle with an advanced model of (23), the actual vehicle position
cos p − sin p sin p cos p
−1
cos p − sin p sin p cos p
(30)
(31)
0
−2 −0.5
−2.5 −3 −3.5
−1
−4 −4.5 −5 −5.5 0
1
|α|
2
−1.5 0
3
1
2
3
|α|
Fig. 2. Natural contraction behavior of an underwater vehicle q(t) will track a reference trajectory if control law u is given by I k1 cD 2 u=
k2
+
(28) T
c0 v 3 + c1 vr 2 ω 2 c2 rωu2 + c3 r3 ω 3
(29)
−1.5
2
+ Ci max r ω sin p − v cos p
cos p − sin p sin p cos p
3 α + c1 cos αsin3 α (c0 −42c1 )cos α sin c2 cos α + 3c3 sin2 αcos2 α + 2c3 sin4 α
2 4 4 2 cos p sin p 2c0 cos α +33c0 cos αsin α + c31 sin α − sin p cos p −c2 sin αcos α + (c3 − 2c2 )sin α cos α cos p − sin p 2 2 2 × −500 v + r ω sin p cos p
Ci max > 0. Since the pitch angle θ = tan−1 and cos θ are replaced by √ 2 v 2 2 and √
Eigen Value− λ2
∂fHyd = ∂η
3
−500
Eigen Value− λ1
∂fHyd = ∂η
I
ω − z1 − (g1 + g2 ) z2 +
cD I (v + |v|) − g1 k2 m g1 > 0, g2 > 0
m
v − v |v|
−e1 + (1 − g1 ) z1 + z2 +
T m
,
(32)
where auxiliary variables z1 and z2 are defined as z1 = v − β1 (q, γ) and z2 = ω − β2 (q, z1 ) respectively with β1 (q) = − [q − yd ] + y˙ d − γ and β2 = cmD |v| v1 + y¨d − g1 z + γ .
Proof: Defining tracking error e1 = y − yd and selecting auxiliary variables as z1 = v − β1 (q, γ), where β1 (q, γ) the virtual control input, makes first subsystem (23) in contracting w.r.t. q. Thus, the dynamics of subsystem can be written as (33) q˙ = z1 + β1 (q, γ) For this system, virtual displacement in differential framework can be represented as (34) δ q˙ = δz1 + J11 δq where Jacobian matrix J11 is represented by ∂ J11 = [β1 (q, γ)] (35) ∂q This Jacobian J11 is UND (uniformly negative definite) in nature by the selection of β1 as follows (36) β1 (q) = − [q − yd ] + y˙ d − γ = −e1 + y˙ d − γ Thus, z1 = v + e1 − y˙ d + γ and hence the first subsystem can be reduced into error dynamics of vehicle position as e˙ 1 = q˙ − y˙ d = v − y˙ d (37) = −e1 + z1 − γ By taking time derivative of z1 with use of (23) and (33), we get
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z˙1 = v˙ + e˙ 1 − y¨d + γ˙ (38) cD T − |v| v + ω + (z1 − e1 − γ) − y¨d = m m Define new virtual control input β2 (q, z1 ) to make equation reference goes here contracting w.r.t. z1 . This ensures the feedback interconnection of subsystems in (33) and (38). Defining new auxiliary variable z2 = ω − β2 (q, z1 ), (38) can be represented as cD T z˙1 = − |v| v + z2 + β2 (q, z1 ) − e1 + z1 − γ − y¨d (39) m m So for the combination of the first two subsystems (37) and (39), differential framework based virtual displacement can be written as δ e˙ 1 −1 1 δe1 0 (40) = + δz2 −1 J22 1 δ z˙1 δz1 −1 1 Here, Jacobian matrix J = , where J22 = −1 J22 cD T ∂ ¨d + m , which is UND ∂z1 z1 − m |v| v + β2 (q, z1 ) − γ − y in nature by suitable selection of β2 (q, z1 ) and δz2 is considering as bounded external input. Since the underwater vehicle shows its naturally contracting behaviour in Fig.3, an UND of J is ensured by the selection of β2 (q, z1 ) as follows: cD |v| v1 + y¨d − g1 z + γ (41) β2 = m T As a result, (39) becomes z˙1 = −e1 + (1 − g1 ) z1 + z2 + m . Now taking time derivative of z2 and using (23) ∂β2 ∂β2 z˙2 = ω˙ − q˙ − z˙1 ∂q ∂z1 ∂β2 ∂β2 T v− = ω˙ − −e1 + (1 − g1 ) z1 + z2 + ∂q ∂z1 m (42) where ∂β2 cD = (v − |v|) (43) ∂q m cD ∂β2 (44) (v + |v|) − g1 = ∂z1 m With use of (39) and (41), the expression for (42) can be reduced into k2 Q cD −k1 ω+ u− − (v − |v|) v z˙2 = I I I m c T D − −e1 + (1 − g1 ) z1 + z2 + (v + |v|) − g1 m m (45) where ω = z2 +β2 (q, z1 ) and v = z1 +β1 (q). The following selection of control u u=
k1 cD v 2 − v |v| ω − z1 − (g1 + g2 ) z2 + I m cD I T + −e1 + (1 − g1 ) z1 + z2 + (v + |v|) − g1 k2 m m (46) I k2
ensures contracting nature of final subsystem w.r.t. z2 . Thus, (45) becomes Q z˙2 = −z1 − (g1 + g2 ) z2 − (47) I It also ensures feedback combination of contracting subsystems. Now, the dynamics of transformed system in differential framework can be written as
δ e˙ 1 δ z˙1 δ z˙2
−1
= −1 1 − g1 0
−1
T
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1 0
1 − (g1 + g2 )
T ∂ m + ∂z − Q I
δe1 δz1
δz2
(48)
where z = [ z1 z2 ] . The transformed matrix can be obtained from the expression of z1 = v + e1 + γ − y˙ d and z2 = ω − cmD v |v| + g1 z1 − y¨d − γ as follows 1 0 δv δz1 cD = (49) δz2 δω g1 − |v| 1 m T Therefore, z η = [ v ω ] and the Jacobian matrix F of (48) can be reduced into 0 1×2 0 T −1 1 0 1 + ∂ F = −1 1 − g1 m 0 0 −1 − (g1 + g2 ) Q ∂η −
I J1
J2
(50) From the results of natural contracting behaviour of an underwater vehicle, the jacobian of the dynamics associated to non-zero terms in J2 using the inertia as metric is given by ∂ ∂ T = (fHyd ) 0 (51) −Q ∂η ∂η The determinant of the Jacobian matrix J1 is given by (52) det(J1 ) = − [g1 (g1 + g2 ) + 1] The positive values of g1 and g2 shows the UND of Jacobian matrix F and hence the dynamics of (48) will be contracting. In other words, the contracting behaviour of overall system is ensured for the selected values u,β1 ,β2 with g1 > 0, g2 > 0 . Therefore,the exponential stability of system states is obtained since all the system states converge to each other .
Remark: In case of uncertainty in parameter k1 in (23), the approach given in (Sharma and Kar (2009)) can be used to estimate kˆ1 , the controller structure is same as proposed in (32) except that uncertain parameter k1 is replaced by kˆ1 . 3. NUMERICAL SIMULATION The performances of proposed position control laws are tested by the numerical simulation of underwater vehicle. We are primarily considered a simple model of vehicle whose mass m = 4kg, drag coefficient d = 1N s2 /m2 , actuator time constant T = 1sec. The initial values considered as [0; 0; 0]. The fourth order Runge-Kutta integrator with a fixed time step of 0.1sec is used here to integrate the control law presented in (16), and the controller gains k1 and k2 are selected as 3.79 and 11.85 respectively. The dotted line is the time history of vehicle position with proposed control law, which converges to the reference yd represented by the full line as shown in Fig. 3. The good agreement between the vehicle conbtrolled position and desired trajectory is achieved. In the case of a more advanced model of underwater vehicle, the performances of UGES control law is tested by the numerical simulation of a vehicle with initial states as q(0) = 0m, v(0) = 0.5m/sec, ω(0) = 2rad/sec., and γ(0) = 0. The control law derived in
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Position q (m) Velocity v (m/s)
0.5
REFERENCES
Desired Trajectory Controller
1 0.5 0 0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
10
20
30 time (sec)
40
50
60
0 −0.5
Control input Torque u η (N/m)
5 0 −5
50 0 −50
Fig. 3. Tracking controller responses for a simple model of underwater vehicle (46) is integrated with advanced model (23) of vehicle by using the fourth order Runge-Kutta integrator with a fixed time step of 0.01s. The dynamics of desired trajectory is tracked with use of controller gains g1 = 0.2 and g2 = 0.09. The fig. 4. shows the convergence of the vehicle tracking position to the desired trajectory. 0.18
error in q
q (m)
0.1 0.05 0 0.003 0
−0.003
ω (rad/s)
v (m/s)
0.4 0.2 0 0 −5 −10
u
γ (rad)
0 −1 −2
0 −100 −200 −300
0
0.2
0.4
0.6
0.8
1 1.2 time in sec
1.4
1.6
1.8
2
Fig. 4. Tracking controller for advanced model of vehicle 4. CONCLUSIONS A nonlinear control system was investigated via contraction theory for a simple as well as a more advanced model of an underwater vehicle. Control system was developed based on the application of 1) integral backstepping controller design technique for nonlinear system, and 2) contraction theory to ensure the incremental exponential global stability of underwater vehicle to track the desired trajectory of vehicle position. Due to the application of backstepping technique, controller design was restricted to parametric-strict-feedback form, but the selection of its tuning parameters were analytically obtained from the contraction theory based incremental stability analysis. Simulation results were presented to show the effectiveness of the design method of integral backstepping contraction control law for underwater vehicle.
Bachmayer, R., Whitcomb, and Grosenbaugh (2000). An accurate four-quadrant nonlinear dynamical model for marine thrusters: theory and experimental validation. IEEE J. of Oceanic Engineering, 25(1), 146–159. Fossen, T.I. (1994). Guidance & control of ocean vehicles. John Wiley & Sons Inc. Healey, A.J., Rock, S.M., Cody, S., Miles, D., and Brown, J.P. (1994). Toward an improved understanding of thruster dynamics for underwater vehicles. In Proceedings of the 1994 Symposium on Autonomous Underwater Vehicle Technology, 340–352. Healey, A. and Marco, D. (1992). Slow speed flight control of autonomous underwater vehicles: Experimental results with the nsp auv ii. In International Offshore and polar Engineering Conference (ISOPE), 523–532. Jouffroy, J. (2003). A relaxed criterion for contraction theory: Application to an underwater vehicle observer. In 2003 European Control Conference, 2999–3004. Jouffroy, J. and Fossen, T.I. (2004). On the combination of nonlinear contracting observers and uges controllers for output feedback. In 43rd IEEE Conference on Decision and Control, volume 5, 4933–4939. Jouffroy, J. and Lottin, J. (2002a). Integrator backstepping using contraction theory: a brief methodological note. Jouffroy, J. and Lottin, J. (2002b). On the use of contraction theory for the design of nonlinear observers for ocean vehicles. In Proceedings of the 2002 American Control Conference, volume 4, 2647–2652. Jouffroy, J. and Slotine, J.J.E. (2004). Methodological remarks on contraction theory. In 43rd IEEE Conference on Decision and Control, volume 3, 2537–2543. Khalil, H.K. (1996). Nonliner Systems. NJ: Prentice-Hall. Lohmiller and Slotine (1998). On contraction analysis for nonlinear systems. Automatica, 34(6), 683–696. Lohmiller, W. and Slotine, J.J.E. (2000). Control system design for mechanical systems using contraction theory. IEEE Trans. on Automatic Control, 45(5), 884–889. Lohmiller, W.S. (1999). Contraction Analysis of Nonlinear System. Thesis. Majeed, M. and Kar, I.N. (2013). Design and convergence analysis of stochastic frequency estimator using contraction theory. IET- Signal Processing, 7(5), 389–399. Marco, D.B. and Healey, A.J. (1998). Local area navigation using sonar feature extraction and model-based predictive control. International Journal of Systems Science, 29(10), 1123–1133. Sharma, B.B. and Kar, I.N. (2009). Contraction based adaptive control of a class of nonlinear systems. In 2009 American Control Conference, 808–813. Whitcomb, L.L. and Yoerger, D.R. (1999a). Development, comparison, and preliminary experimental validation of nonlinear dynamic thruster models. IEEE Journal of Oceanic Engineering, 24(4), 481–494. Whitcomb, L.L. and Yoerger, D.R. (1999b). Preliminary experiments in model-based thruster control for underwater vehicle positioning. IEEE Journal of Oceanic Engineering, 24(4), 495–506. Whitcomb, L.L., Yoerger, D.R., Singh, H., and Mindell, D.A. (1998). Toward precision robotic maneuvering, survey, and manipulation in unstructured undersea environments. In Robotics Research-The Eighth International Symposium, 45–54.
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