Non-adaptive velocity tracking controller for a class of

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 65, No. 4, 2017 DOI: 10.1515/bpasts-2017-0051

Non-adaptive velocity tracking controller for a class of vehicles P. HERMAN* and W. ADAMSKI Poznan University of Technology, 3A Piotrowo St., 60-965 Poznan, Poland

Abstract. A non-adaptive controller for a class of vehicles is proposed in this paper. The velocity tracking controller is expressed in terms of the transformed equations of motion in which the obtained inertia matrix is diagonal. The control algorithm takes into account the dynamics of the system, which is included into the velocity gain matrix, and it can be applied for fully actuated vehicles. The considered class of systems includes underwater vehicles, fully actuated hovercrafts, and indoor airship moving with low velocity (below 3 m/s) and under assumption that the external disturbances are weak. The stability of the system under the designed controller is demonstrated by means of a Lyapunov-based argument. Some advantages arising from the use of the controller as well as the robustness to parameters uncertainty are also considered. The performance of the proposed controller is validated via simulation on a 6 DOF robotic indoor airship as well as for underwater vehicle model. Key words: marine vehicle, fully actuated hovercraft, indoor airship, non-adaptive control, quasi-velocities, diagonal inertia matrix.

1. Introduction The use of robotic marine vehicles, ground vehicles and air vehicles for different applications has been growing in the last decades. One of their advantages is low cost as compared to full scale and fully manned vessels. The setpoint, trajectory tracking and path following control strategies for robotic vehicles have received increased attention of researchers. Numerous control algorithms have been proposed for the class of vehicles considered here. Tracking strategies related to underwater vehicles are presented, e.g., in [1–4]. Some control strategies for surface vessel including ships and hovercrafts can be found in [5–15]. Tracking controllers useful for marine vessels are described also in [16–20]. Referring to the airship trajectory tracking problem control algorithms are shown, e.g., in [21–24]. There are vehicles which, due to their construction, can be regarded as fully actuated. One may refer to the following works concerning underwater vehicles [1, 3, 25–29], surface vehicles [10, 30, 31, 15], hovercrafts [32, 33], and airships [34–36]. Velocity tracking control algorithms are rather rarely presented in the robotic literature as far as marine or aerial vehicles are concerned. However, tracking control strategies suitable for underwater vehicles or surface ships are shown, e.g., in [18, 37]. The velocity controllers related to velocity tracking for this class of vehicles are given in [38, 39]. The same type of controller for underwater vehicles are presented in [40]. The sliding mode control based approaches for velocity tracking for unmanned surface vessels are considered, e.g., in [6, 41, 42]. Moreover, velocity control algorithms are useful for other mechanical systems as unmanned helicopters [43] or quad-rotors [44]. Velocities

*e-mail: [email protected] Manuscript submitted 2016-09-01, revised 2016-12-22, initially accepted for publication 2016-12-27, published in August 2017.

are applied also for mobile platform control [45]. The tracking control algorithms for aerial indoor blimp using velocity-based controller are introduced in [21, 22]. In this paper a velocity control algorithm realized in the body-fixed frame for a class of fully actuated vehicles is introduced and discussed. The considered method is recommended for control of underwater vehicles, hovercrafts, or indoor airship moving slowly (with linear velocity below 3 m/s) and under assumption that the external disturbances are weak (if they can be omitted in equations of motion). It means that the submarine moves in calm water or the airship flies inside the hall. This velocity tracking controller is based on a velocity variables transformation arising from the inertia matrix decomposition. Therefore, the vehicle equations of motion are transformed into a velocity space. As a result, the obtained differential equations allow us to track the moving object velocity. It is noticeable that after the system inertia matrix decomposition the obtained controller includes the dynamic parameters set of the vehicle. Additionally, each rate is regulated separately in a sense (dynamical couplings are shifted to the appropriate velocity variable). One of benefits relies on that the system response of the system is fast and the velocity error convergence is quick. Consequently, the desired trajectory is reached in short time. Another advantage arising from the use of the controller is that the gain matrix related to the velocity error includes the system parameters highly dependent on the vehicle dynamics (for various vehicles the control coefficients can be different). The novelty of the presented strategy is that the nonlinear controller is realized after transformation of equations of motion arising from the inertia matrix decomposition. Moreover, the proposed controller is universal in the following sense. First, it is suitable both for 6 DOF and for vehicles moving in the horizontal or vertical plane. Second, it can be applied in the same general (or reduced to 3 DOF) form for fully actuated marine vehicles, hovercrafts, and indoor airships. Third, some

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Bull. Pol. Ac.: Tech. 65(4) 2017 Unauthenticated Download Date | 8/25/17 12:47 PM

P. Herman and W. Adamski

symmetric, positive definite and their elements are known. As symmetric, positive definite and their elements are known. As it arises from the literature [22] forEuler a class of marine vehicle 6 controllers which have simpler form can be deduced from it it η 2 R the vector of positions angles, ν 2 R6 is arisesis from the literature [22]and for a class of marine vehicle models such approximation is reasonable. Similarly conclu(particular cases of the controller). The stability of the vehicle models the vector of body-fixed linear is and angular velocity composuch approximation reasonable. Similarly conclu6 for indoor airships moving with low velocity. sion can be made under the controller is shown based on the Lyapunov argument. sion nents,can andbe τ 2 R the indoor control airships vector. The components of two madeisfor moving with low velocity. T T Remarknamely 2. Noteν = [u, v, w, p, q, r] that various moving can be described Additionally, the robustness to the system parameter changes vectors, andsystems η = [x, y, z, ϕ, θ, ψ] Remark 2. Note that various moving systems can be described is considered. are related to the motion variable in of surge, roll,for unusing Eqs.(1) and (2). Equations thissway, type heave, are used Eqs.(1) (2). Equations of this typea 6£6 are used for unThe mathematical model describing the dynamics and ki- using pitch, and yaw, and respectively. Additionally, J(η) block derwater vehicles [3, 13, 26, 34, 46] and forissurface vessels [8, derwater vehicles [3, 13, 26, 34, 46] and for surface vessels [8, nematics of the class of vehicles is introduced in Section 2. diagonal transformation between the body-fixed 12, 14, 15, 17, 33, 35].matrix However, hovercrafts [2, 18,frame 30, 40] as 12, 14, 15, 17, 33, 35]. However, hovercrafts [2, 18, 30, 40] as The proposed velocity tracking controller is presented and con- to theas inertial reference matrix well indoor airshipsframe [43, (usually 44, 63, the 64]earth). can beThe also described as indoor 44, 63, 64] can be also described sidered in Section 3. Simulation results for an airship as well well J(η) depends on airships the Euler [43, angles. by these equations. as for a underwater vehicle model are contained in Section 4. by these equations. Introducing now a transformation ofMrates in the form: Section 5 offers conclusions. Remark 1. Recall, the inertia matrixof is constant, symIntroducing nowthat a transformation rates in the form: metric, and in general, non-diagonal, i.e. it contains off-diagonal ν = ϒξ , (3) ν = ϒξwith , a diagonal inertia (3) elements. In order to obtain equations matrix allows us diagonal, to design a decoupled controller in the where which ϒ is an upper invertible matrix with constant 2. Dynamical model of vehicle in terms where ϒ iseach an upper invertible matrix with constant sense that rate candiagonal, be separately the matrix M three elements it is possible toregulated decompose the matrix M into of generalized velocity components elements itdecomposed. is possible togeneral decompose the matrix Minertia into three −T −1 should be In components of the matrices, i.e. M = ϒ−T Nϒ−1 . The obtained matrix N is diagoFig. 1. Coordinate system forof6the DOF vehicle . The N is diagomatrices, i.e. M ϒ Nϒ fluid The six dynamical modelfor considered on= geometry, flowobtained rates andmatrix other uncernal anddepend it contains constant elements on the diagonal. Fig.DOF 1. Coordinate system 6 DOF vehicle here class matrix and it containsthe constant elements on the are diagonal. of vehicles (Fig. 1) is expressed in the body-fixed reference nal tainties. Moreover, added mass coefficients often estiCalculating the time derivative of ν we have ν˙ = ϒξ˙˙ . Taking frame by [18]: mated using experimental studies and empirical relations which ˙ Calculating the time derivative of ν we have ν =ϒ ξ . Taking by [22]: the above into account and inserting (3) into (1), and next preby [22]: are above not quite accurate. a result, it should be stated that if prethe into accountAsand inserting (3) into (1), and next T (as in [32]) we can write: multiplying both sides by ϒ appears to by be ϒ non-symmetric it cannot M ν˙ +C(ν)ν + D(ν)ν + g(η) = τ,,(1) (1) the matrix Mboth T (as in [32])then we can write:be sides M ν˙ +C(ν)ν + D(ν)ν + g(η) = τ, (1) multiplying decomposed into a diagonal form and the proposed approach ˙ η˙ = J(η)ν, (2) Mϒξ˙ +C(ν)ϒξ + D(ν)ϒξ + g(η) = τ, (4) is not valid. Thus, the of the matrix M (4) η˙ = J(η)ν,,(2) (2) Mϒ ξ +C(ν)ϒξ +decomposition D(ν)ϒξT+ g(η) = τ,inertia T T T T ˙ ϒisT possible Mϒξ + ϒ C(ν)ϒξ + ϒ D(ν)ϒξ + ϒ g(η) = ϒ τ. (5) is assumed the matrix symmetric, where (1) is the vehicle dynamics and (2) is the kinematics. In (1)ϒ Mϒξ˙ + ϒifTitC(ν)ϒξ + ϒTthat D(ν)ϒξ + ϒTisg(η) = ϒT τ. (5) where where (1) is the vehicle dynamics and (2) is the kinematics. In 6×6 symmetric, positive definite andknown. theirare elements As (1) isMthe dynamics (2) ismatrix the kinematics. In positive definite and their elements are Asknown. it arisesare from symmetric, positive definite and their elements Asknown. these equations ∈ vehicle R means theand inertia including Grouping now thedefinite terms of the equation, theare transformed equasymmetric, symmetric, positive positive definite and and their their elements elements are known. known. As As these equations M ∈M 2 R R6×66×6 means the inertia matrixincluding including these equations means the inertia matrix the the literature [18] for a class of marine vehicle models such symmetric, positive definite and their elements are known. As it arises from the literature [22] for a class of marine vehicle Grouping now the terms of the equation, the transformed equait arises from the literature [22] for a class of marine vehicle the rigid body inertia matrix and the added mass matrix. More- itit arises symmetric, positive definite their elements are known. A tions from of motion can be written in athe following form: arises from the theisliterature literature [22] [22] for for aand class class of of marine marine vehicle the rigid body inertia and the added addedmass mass matrix. Morerigid body inertiamatrix matrix approximation Similarly can be vehicle made T > it arisestions from the literature [22] for ainclass of marine vehicle models suchreasonable. approximation istheconclusion reasonable. Similarly conclu˙ =matrix. ofapproximation motion can be following form: models such iswritten reasonable. Similarly conclu0  and M 0 are Moreover, fulfilled. over, two conditions M =T Mand T it arises from the literature [22] for a class of marine vehic models models such such approximation approximation is is reasonable. reasonable. Similarly Similarly concluconclu̇ ˙ two conditions  > 0 > and0 Mand  = 0M are=fulfilled. However, for indoor airships moving with low velocity. 0 are fulfilled. over, two conditionsM = M M=M ˙ models approximation Similarly conclucan for indoor with Nmade ξ˙ +Cairships )ξ +moving Dairships (ξ )ξwith +moving glow = π, low velocity. (6) sion can such besion made forbesuch indoor velocity. However, we must take into account that vehicles are differξis(ξreasonable. ξmoving ξ (η) models approximation is reasonable. Similarly concl sion sion can can be be made made for for indoor indoor airships airships moving with with low low velocity. velocity. NNote ξ +C )ξ +moving Dξmoving (ξ )ξwith + glow =can π, be described (6) we must take into account that vehicles different.are Therefore, However, we must take into account thatare vehicles differξ (ξvarious ξ (η)velocity. sion can2. beRemark madethat for indoor airships ˙ = 0 must 2.various that Remark Note moving cansystems be described ent. Therefore, the M condition be carefully checked ˙that η = J(η)ϒξ , systems (7)  ̇  = 0 mustM sion be made for indoor airships moving with low veloci the condition checked for each con-Remark Remark 2. can Note various moving systems can be described Remark 2. 2. Note Note that that various various moving moving systems systems can can be be described described ent. Therefore, the condition M˙be=carefully 0 must be carefully checked η˙ Equations =and J(η)ϒξ , this (7) Remark 2. using Note that various moving systems can be described Eqs.(1) (2). Equations of this type are used for un6×6 Eqs.(1) using and (2). of type are used for unfor each considered object whether it is acceptable. Besides, sidered object whether it is acceptable. Besides, C(ν) 2 R using Eqs. (1) and (2).Equations Equations of this this type for Remark 2.(2). Note that various moving systems can be describ using using Eqs.(1) Eqs.(1) and and (2). Equations of of this type typeare are areused used used for forunununfor each considered object whether it is acceptable. Besides, where the appropriate matrices and vectors are given as fol6×6 using Eqs.(1) and[3, (2). of26, this type are used forvessels underwater vehicles [3,matrices 13, 34, 46] andsurface for surface [8, derwater vehicles 13,Equations 26,and 34, 46] and for surface vessels [8, vessels C(ν) ∈is Rthe matrix is theofmatrix Coriolis and terms centrifugal terms Coriolisofand centrifugal (that satisfies derwater 48] and where thevehicles appropriate and vectors are given as using Eqs.(1) (2). Equations ofsurface this type are used for u derwater derwater vehicles vehicles [3, [3,[46, 13, 13, 2, 26, 26,25, 34, 34,47, 46] 46] and for forfor surface vessels vessels [8, [8, folC(ν) ∈ R6×6 is Tthe matrix terms lows: T 6×6 6 of Coriolis 6×66and centrifugal derwater vehicles [3, 13, 26, 34, 46] and for surface vessels [8, 12, 14, 15, 17, 33, 35]. However, hovercrafts [2, 18, 30, 40] as 15, 17, 33,8,35]. However, [2, 18, 30, 40] as ∀ν ∈ R6 ), D(ν) ∈ Rof is12, the14, (that satisfies C(ν) −CT (ν), C(ν) = –C (ν),=8ν 2 R ), D(ν) 2 R is the matrix hydrody[49, 50, 51,vehicles 11, 12]. However, [5,18, 52,30, 32,40] 53]as 6×6 derwater [3,hovercrafts 13,hovercrafts 26,hovercrafts 34, 46] and for surface 12, 12,lows: 14, 14, 7, 15, 15, 17, 17, 33, 33, 35]. 35]. However, However, hovercrafts [2, [2, 18, 30, 40] asvessels [ (ν), ∀ν ∈ R ), D(ν) ∈ R is the (that satisfies C(ν) = −C 6 66 14, 12, 15, 17, 33, 35]. However, hovercrafts [2, 18, 30, 40] as well as indoor airships [43, 44, 63, 64] can be also described T can damping termsdamping (D(ν) > 0, 8ν 2 R , ν 6=>  0), g(η) 2 R well as indoor [43, airships also by (8) as as indoor 44, [54–57] 63, 64] can be behovercrafts alsodescribed described matrixnamic of hydrodynamic terms (D(ν) 0, ∀ν ∈well R is, well = ϒHowever, 12,airships 14,airships 15, 17, [43, 33, [2, 18, 30, well as as indoor indoor airships [43,N35]. 44, 44, 63, 64] 64] can can be be also also described described T Mϒ, matrix the of vector hydrodynamic damping terms (D(ν) > and 0, ∀ν ∈ by R6 ,these = ϒ63, Mϒ, (8)40] well asthese indoor airships [43, 44,N 63, 64] can be also described of6 gravitational and buoyancy forces moments, equations. by these equations. equations. is the vector of gravitational and buoyancy ν = 0), g(η) ∈R T 6 well as indoor airships [43, 44, 63, 64] can be also describ by by these these equations. equations. C(ν)ϒ, (9) of gravitational and buoyancythese ν = 0), g(η) ∈ R is the vector ξ (ξ ) = ϒT Introducing now now a transformation ofC(ν)ϒ, rates the form: equations. Introducing aCξtransformation ofinrates in the form:(9) Introducing now a transformation in the form: Euforces and moments, η ∈ R66 is the vector of positions and by C (ξ )of =rates ϒof by these equations. Introducing Introducing now now a a transformation transformation of rates rates in in the the form: form: T forces and moments, η ∈ R is the vector of positions and Eua transformation in the form: Dξ (ξ )of=rates ϒ D(ν)ϒ, (10) ler angles, ν ∈ R6 is the vector of body-fixed linear and angu-Introducing now Introducing now transformation the form: ν T=D(ν)ϒ, ϒξ , of rates in(3) ),,(3) =ϒ (10) (3) νD=ξa (ξ ϒξ ler angles, ν ∈ R6 is the vector of body-fixed linear and angu6 T ν ν = = ϒξ ϒξ , , (3) (3) lar velocity components, and τ ∈ R6 is the control vector. The νg= ϒξ ,= ϒT g(η), (3) (11) ξ (η) lar velocity components, and τ ∈ R is the control vector.T The ginvertible (η) = ϒmatrix g(η), (11) ( ν = ϒξ , ϒ is an upper invertible matrix with constant ξdiagonal, and ϒwhere components of two vectors, namely ν = [u, v, w, p, q, r]T where iswhere anϒupper diagonal, with constant is upperdiagonal, diagonal, invertible matrix with constant T where where ϒϒ is is an anan upper upper diagonal, invertible invertible matrix matrix with with constant constant components of two vectors, namely ν = [u, v, w, p, q, r] and π = ϒ τ. (12) T where is an upper invertible with constant itdiagonal, is possible toϒTdecompose the matrix M into three in ϒelements η = [x, y, z, φ , θ , ψ] are related to the motion variableelements itelements is possible the matrix matrix MM into three it possible todecompose the matrix into three πdecompose =diagonal, τ. (12) ϒ−T istoan–Tdecompose upper invertible matrix with consta elementswhere itit is is is possible possible to to decompose the the matrix matrix M M into into three three in elements η = [x, y, z, φ , θ , ψ]T are related to the motion variableelements −T −1 −1 –1 it is possible to decompose the matrix M into three Nϒ . The obtained matrix N is diagomatrices, i.e. M = ϒ surge, sway, heave, roll, pitch, and yaw, respectively. AddiNϒ .ϒ−1 The obtained matrix NN is diagomatrices, i.e. elements M i.e. =(6) ϒ and matrices, Ntogether obtained matrix isN diagonal −T −1...The Equations (3) with (7) describe the motion of a Fig. 1. roll, Coordinate for 6respectively. DOF vehicle Addiit is(3) possible towith decompose the matrix M into Nϒ Nϒ The The obtained obtained matrix matrix N is ismotion diagodiagomatrices, matrices, i.e. i.e. M MM = ϒ = = ϒϒ−T Fig.sway, 1. Coordinate system for 6system DOFyaw, vehicle surge, heave, pitch, and −1 Equations and together (7)diagonal. describe of a thr Nϒ .diagonal The obtained matrix isthe diagomatrices, i.e. Mand =(6) ϒ−T nal it contains constant elements onitN the diagonal. Fig. Fig. 1. 1.isCoordinate Coordinate system system for for 66 DOF DOF vehicle vehicle and it contains constant elements on the diagonal. tionally, J(η) a 6 × 6 block diagonal transformation matrix nal and it contains constant elements on the −T −1 vehicle, where N is a matrix. As was mentioned indiag Fig. 1.J(η) Coordinate system for diagonal 6 DOF vehicle Nϒmatrix. . The obtained matrix ˙N isin i.e. Ma diagonal =ϒ nal nal and and ititmatrices, contains contains constant constant elements elements on on the theAs diagonal. diagonal. tionally, is Fig. a 6× block transformation matrix  ̇ mentioned where N isthe itν ̇ϒ was ˙ nalCalculating andvehicle, it contains constant elements on the diagonal. Calculating the time derivative of ν we have  = ϒξ . Taking ˙ 1. 6Coordinate for 6 DOF vehicle Calculating time derivative of ν we have ν = ϒ ξ . Taking between the body-fixed frame tosystem the inertial reference frame ˙ the time derivative of ν we have ν = ξ . Taking [31] there are various possible decomposition methods. Hownal and contains constant ˙˙(1), Calculating Calculating the theitvarious time time derivative derivative of ofdecomposition ννelements we we have have νon ν = =˙the ϒϒξξ˙˙diagonal. .. Taking Taking the body-fixed frame to the inertial reference frameCalculating by [22]: [31] there are possible methods. Howtheinto above into account and inserting (3)(1), next 22]:between ˙(3) the time derivative of ν(3) we have νinto = ϒ ξnext . and Taking the above into account and inserting into (1),preand next pre(usually the earth). The matrix J(η) depends on the Eulerthe an-above account and inserting into and ever, in this work the Loduha-Ravani method which is related by by(usually [22]: [22]: the T T (3) Calculating the time derivative of (1), ν we have ν˙is= ϒξ˙ . Taki the theever, above above into into account account and and inserting inserting (3) into into (1), and and next next prepreearth). The matrix J(η) depends on the Euler the an- above 22]:gles. pre-multiplying both sides by ϒ (as in [58]) we can write: in this work the Loduha-Ravani method which related T into account and inserting (3) into (1), and next pre[32]) we caniswrite: multiplying both sides by ϒ (as (asTT in [32]) weincan write: multiplying both sides by ϒaccount ˙ +C(ν)ν M ν+ D(ν)ν + g(η) =(1) τ, (1) to the generalized velocity components (GVC) [36] used (the M ν˙ +C(ν)ν D(ν)ν ++g(η) = τ, the above into and inserting (3)write: into (1), and(the next pr (as in in [32]) [32]) we we can can write: multiplying multiplying both both sides sides by by ϒϒ (as gles. by [22]: T (as ˙˙ +C(ν)ν M Mνν +C(ν)ν + + D(ν)ν D(ν)νmatrix + +g(η) g(η) = =isτ, τ,constant,(1) (1) to the generalized velocity components (GVC) [36] is used in [32]) we can write: multiplying both sides by ϒ Remark 1. Recall, that the inertia M sym˙ M ν +C(ν)ν + D(ν)ν + g(η) = τ, (1) T obtained matrix ϒ is only an upper triangular matrix containing ˙ ˙ ˙ ηM =ν (2)ξobtained in [32]) we can multiplying both sides by=ϒτ, +C(ν)ϒξ ,(4) Mϒmatrix ξ+ +C(ν)ϒξ + D(ν)ϒξ +(as g(η) = τ, η˙ = J(η)ν, (2)τ, sym-Mϒ Remark 1. Recall, that the inertia matrix M isg(η) constant, D(ν)ϒξ g(η) (4)write: (4) ˙J(η)ν, +C(ν)ν + D(ν)ν = ϒ only an matrix containing ˙˙ = ηη = J(η)ν, J(η)ν, (2) (2)off- (1) metric, and in general, non-diagonal, i.e. + it contains Mϒ Mϒξξ˙˙ on +C(ν)ϒξ +C(ν)ϒξ + +is D(ν)ϒξ D(ν)ϒξ + +upper g(η) g(η) triangular = = τ, τ, (4) (4) ˙ones the diagonal). η˙ =in J(η)ν, (2) offT T + g(η) =Tτ, T Mϒ +C(ν)ϒξ + D(ν)ϒξ (4) ˙ T ξones T T metric, and general, non-diagonal, i.e. it contains ˙ Mϒdiagonal). + ϒ+ C(ν)ϒξ + ϒ+ D(ν)ϒξ +=ϒτ,τ. g(η) = ϒT τ. (5)( onϒ˙˙ the Mϒ C(ν)ϒξ ϒ D(ν)ϒξ ϒ +g(η) (1) is the vehicle dynamics and (2) is thewith kinematics. = J(η)ν, (2) TTξ + Tξ T˙ξ+C(ν)ϒξ TT D(ν)ϒξ TT = TT (5) Mϒ + g(η) diagonal elements. Inη˙ order to(2) obtain equations re (1) iswhere the vehicle dynamics and is the kinematics. In a diag-ϒ TIn ϒϒ Mϒ Mϒξξ + +ϒϒ C(ν)ϒξ C(ν)ϒξ + +ϒϒ D(ν)ϒξ D(ν)ϒξ + +ϒϒ g(η) g(η) = = ϒϒ τ. τ..(5)(5) (5) where where (1) (1) is iselements. the the vehicle vehicle dynamics and and (2) (2) equations is is the the kinematics. kinematics. In In ϒ Mϒ diagonal Indynamics tomeans obtain withIna including diag6×6 ξ˙ + ϒT C(ν)ϒξ + ϒTT D(ν)ϒξ + ϒT T g(η) = ϒT τ. 6×6 (1) isthese the M vehicle dynamics and (2) istothe kinematics. T T (5) equations M ∈order Rallows the inertia matrix ˙+ onal inertia matrix which us design a decoupled conereequations ∈ R means the inertia matrix including 6×6 6×6 ϒ Mϒ ξ ϒ C(ν)ϒξ + ϒ D(ν)ϒξ + ϒ g(η) = ϒTequaτ. ( Grouping now the terms of the equation, the transformed Grouping now the terms of the equation, the transformed equawhere matrix (1)M dynamics andmatrix (2) is the kinematics. In 3. Design of decoupled non-adaptive velocity these these equations equations Mis ∈ ∈the RR vehicle means means the the inertia inertia matrix including including onalthe inertia which allows us to design aincluding decoupled con6×6 Grouping Grouping now now the the terms terms of of the the equation, equation, the the transformed transformed equaequaeigid equations M ∈ R means the inertia matrix rigid body inertia matrix and the added mass matrix. More3. Design of decoupled non-adaptive velocity troller in the sense that each rate can be regulated separately bodythese inertia matrix and the added mass matrix. MoreGrouping now the terms of the equation, the transformed 6×6 Grouping now the terms of the equation, the transformed equations of motion can be written in the following form: tions of motion can be controller written in the following form: the transformed equ M ∈and R rate the inertia matrix including the the rigid rigid body body inertia inertia matrix matrix and the themeans added added mass mass matrix. matrix. MoreMoretroller in theequations sense that regulated tracking T each Grouping now thebe terms ofin the tions tions of oftracking motion motion can cancontroller be becan written written in in the the following following form: form: igid body inertia and the added matrix. More>=0be = 0separately are fulfilled. over, two conditions M0TT= M Tcan equations of motion written theequation, following the matrix M matrix should be decomposed. In general components > and M˙mass 0˙and areM˙fulfilled. two conditions M = inertia M tions of motion can be written in the following form: form: ˙ the rigid body matrix and the added mass matrix. More> > 0 0 and and M M = = 0 0 are are fulfilled. fulfilled. over, over, two two conditions conditions M M = = M M ˙ the matrix M should be decomposed. In general components T ˙ ˙ tions of motion can be written in the following form: ξ+ +C (ξ )ξ=+ gdecoupled π, in the (6) > 0into and MTvehicles = 0 that are ,ever, two conditions M into =M However, we must take account vehicles areand differ- In this N ξ +C˙˙ξthe (ξN )ξ D ξ (ξ )ξ + gDξξ(η) π, (6) section general controller which is= ξ (η) = of the inertia matrix depend on geometry, fluid flow rates we musttwo take account that arefulfilled. ˙differN N ξ ξ +C +C (ξ (ξ )ξ )ξ + + D D (ξ (ξ )ξ )ξ + + g g (η) (η) = π, π, (6) (6) > 0 and M = 0 are fulfilled. over, conditions M = M However, However, we we must must take take into into account account that that vehicles vehicles are are differdiffer˙ In this section the general controller which is decoupled in the ξ ξ ξ ξ ξ ξ of the inertia matrix depend on geometry, fluid flow rates and ˙ , (6) N ξ +C (ξ )ξ + D (ξ )ξ + g (η) = π, (6) ˙ wever, we must take into account that vehicles are different. Therefore, the condition M = 0 must be carefully checked ξ ξ ξ sense of the vector of the transformed variables ξ is presented. ˙ other However, uncertainties. Moreover, the added mass coefficients are Therefore, the condition M = 0 must be carefully checked ˙ η = J(η)ϒξ , (7)( ˙ η = J(η)ϒξ , (7) ˙ ˙ N ξ +C (ξ )ξ + D (ξ )ξ + g (η) = π, must take into account that vehicles are different. ent. Therefore, Therefore, the thewe condition condition M M = = 0 0 must must be be carefully carefully checked checked ξ ξ ξ sense of the vector of the transformed variables ξ is presented. ˙ ˙ other uncertainties. Moreover, the added mass coefficients are = = J(η)ϒξ J(η)ϒξ,, (7) (7) ˙ object Therefore, the condition M = 0 must be˙studies carefully checked forestimated each considered whether it isand acceptable. Besides, ˙ =ηηJ(η)ϒξ η , (7) using experimental empirical relaachoften considered object whether it is acceptable. Besides, ent. Therefore, the condition 06must be carefully checked for for each each considered considered object object whether whether ititMstudies is is= acceptable. acceptable. Besides, Besides, Fig. Coordinate system for DOF vehicle where ,,(7) η˙ = J(η)ϒξ often estimated using experimental and empirical relathe appropriate matrices and arefolgiven as fol-( 6×6 where the matrices and vectors arevectors given as 6×6 each object whether it is acceptable. C(ν) ∈ Rare is 1.the matrix ofand Coriolis andBesides, centrifugal terms which not quite accurate. As a result, it should be 3.1.appropriate Control algorithm Theand controller be used ) ∈tions Rconsidered is the matrix of Coriolis centrifugal terms where where the the appropriate appropriate matrices matrices and vectors vectors can are are given given as asfor folfol-fully 6×6 6×6 considered object whether it is acceptable. Besides, C(ν) C(ν) ∈ ∈for RR each is is the the matrix matrix of of Coriolis Coriolis and and centrifugal centrifugal terms terms where the appropriate matrices and vectors are given as foltions which are not quite accurate. As a result, it should be 3.1. Control algorithm The controller can be used for fully 6×6 lows: T 6 6×6 lows: T 6 6×6 ) satisfies ∈stated R (that is the matrix of Coriolis and centrifugal terms (ν), ∀ν ∈ R ), D(ν) ∈ R is the satisfies C(ν) = −C that the matrix appears non-symmetric actuated underwater vehicles, hovercrafts or indoor (ν), ∈ R ),ofto D(ν) ∈ R andis the then itterms C(ν)if∈= where the appropriate matrices and vectors areairships. given as fo 6×6 lows: lows: TM T∀ν 6be 6×6 C(ν) R−C the matrix centrifugal (ν), (ν), ∀ν ∀ν6∈ ∈toRR6Coriolis ),), non-symmetric D(ν) D(ν)6×6 ∈ ∈ RR6×6 isthen the thelows: (that (that satisfies satisfies C(ν) C(ν) = =T is −C −C stated that if the matrix M appears be it R6 ,actuated underwater vehicles, hovercrafts or indoor airships. 6is T (ν), ∀ν ∈ R ), D(ν) ∈ R is the C(ν) = −C T that matrix of hydrodynamic damping terms (D(ν) > 0, ∀ν ∈ cannot be decomposed into a diagonal form and the proposed Moreover, it is assumed the vehicle moves, i.e. the , ixsatisfies of hydrodynamic damping terms (D(ν) > 0, ∀ν ∈ R lows: T 6 6×6 N = ϒ Mϒ, 6 6 N = ϒ Mϒ, Pol. Ac.: Tech. 65(4) (8) 2017airship (8) TT theBull. (ν), ∀ν ∈ Rand ), 0, D(ν) (that satisfies C(ν) = −C matrix matrix of of460 hydrodynamic hydrodynamic damping damping terms terms (D(ν) (D(ν) > > 0, ∀ν ∀νproposed ∈ RR ,, is the be decomposed into avector diagonal form the Moreover, it is assumed moves, i.e. the airship 6∈ 6 is N Nthe = = ϒϒ Mϒ, Mϒ,vehicle (8) (8) 6is T that , ix cannot of hydrodynamic damping terms (D(ν) > 0, ∀ν ∈ R the of gravitational and buoyancy ν = 0), g(η) ∈ R approach not valid. Thus, the decomposition of the ineris in flight phase or marine vehicle flows. Other T 0), g(η) ∈ R is the vector of gravitational and buoyancy N= ϒ Mϒ, (8) motion (9) 6, T (ξ ) =Unauthenticated Tϒ C(ν)ϒ, C ofRR66hydrodynamic damping terms (D(ν) > 0, ∀ν ∈ R is is the the vector vector of of gravitational gravitational and and buoyancy buoyancy νν approach = = 0), 0),matrix g(η) g(η)is ∈ ∈not C (ξ ) = ϒ C(ν)ϒ, (9) ξ T T valid. Thus, the decomposition of the ineris in flight phase or the marine vehicle flows. Other motion 6 6 N = ϒ Mϒ, ( ξ C 6 is the 0),and g(η) ∈ Rand of gravitational and that buoyancy C (ξ)consideration. )= = ϒϒ C(ν)ϒ, C(ν)ϒ, (9) (9) the vector ofand positions andisEu- phases are not taken forces moments, η ∈vector T Date tia matrix Misg(η) (1) ifRvector it isisof assumed the matrix into positions Eues moments, ηthe ∈isvector R∈possible Download | 8/25/17 12:47 PM ξ)ξ(ξ 6 is 66 is C (ξ = ϒ C(ν)ϒ, (9) the of gravitational and buoyancy ν =  0), R T is the the vector vector of of positions positions and and EuEuforces forces and and moments, moments, η η ∈ ∈ R R ξ T T tia matrix M (1) if it is of assumed that the matrix is phases are not taken 6 is the vector )= (10)( (ξ ) =ϒ ϒD(ν)ϒ, C(ν)ϒ, (10) positions and Eues and ην ∈is Rpossible Dξ (ξinto ) =consideration. ϒDD(ν)ϒ, ler thebody-fixed vector of body-fixed linear and anguξC(ξ ngles, νmoments, ∈angles, R6 isand the vector of linear and 6 is D Dξξ(ξ (ξ)) = = ϒξTTD(ν)ϒ, D(ν)ϒ, (10) (10) T ϒ the vector ofangupositions forces moments, η6 of ∈ ler ler angles, angles, νν6 ∈ ∈ RR66 is is the the vector vector ofRbody-fixed body-fixed linear linear and and anguangu- and Eu6 D (ξ ) = ϒ D(ν)ϒ, (10) T ngles, ν ∈ R is the vector of body-fixed linear and angu-

z ξ=ξ − dσ ,ξ(15) νd(ν − ν+Λz), isΛz), velocity erro s ν(σ ssthe − ϒΛz), = =(14) ξξrr − −xξξwhere = = (16) ϒϒT rξ==)ξ ϒ + + ϒ ϒ g(η) r d + Λz), s = T C(ν)ϒξ rξ ξ(5) Grouping now the terms of the equation,+ the transformed equaϒand − ξν˜ν=Λ = += (mν˜+ (16) (16 = [s r =ξ ϒ (ν r ξ0˙ ξs= r(ν ϒT Mϒξ Mϒξ˙now +ϒ ϒthe C(ν)ϒξ +the ϒT D(ν)ϒξ D(ν)ϒξ +the ϒT transformed g(η) = =ϒ ϒT τ. τ.equa(5) ˙˜ (+ ξ ϒ−1 {A} >ξξ(17) 0 (λ the mξ[[ ˜ϒis λrelated s˙ξ =ξ ξ˙r − ν), m is the xxcan = −1 Grouping terms of equation, = ˙ ˙ −1 ˙ ˙ ˙ sym ˙ index d to the desired ˜ ˜ s ˙ = ξ − ξ = ϒ ( ν + Λ ν), (17) −1 −1 ˜ ˜ Grouping now the terms of the equation, the transformed equa˙ ˙ ˙ ˙ ˙ ˙ s ˙ = ξ − ξ = ϒ ( ν + Λ ν), (17) r ˙ tions of motion can be written in the following form: −1 ξ r x = ξ −1 −1 ˜ ˜ ˜ ˜ ˜ and ν = ν − ν is the velocity error vector (the quantity with s ˙ s s ˙ ˙ = ξ − ξ = ϒ ( ν + Λ ν), = = ξ ξ − − ξ ξ = = (17) ϒ ϒ−− ˙r ϒ− ˙(ν=quantity ˙rΛz), ˙˜ = − ξ =dϒerror (νs˙ vector += (16) and ν = νd −equaν sisξ = theξrvelocity (the with ˙˜ + find upper bound rξ rr Grouping now the terms of the equation, the transformed ξrΛz), = ξs˙ξ ξ˜ξan(15) ˜tocan d+ ξ ϒ − ( ξ ν + Λ ϒ ν), ( ν Λ ν), (17) (17 tions of motion can be written in the following form: {A} λ ξ ξ Grouping now the terms of the equation, the transformed equam the index the actual velocity), k and ν˜ index = νthe νisisrelated the velocity velocity error vector (the xquantity tions of motion can be written in the following form: T , zwith T ]Twithout d− −1 dd˙ desired to=the desired velocity whereas index dπ,is related to whereas without ˙ +C = [s one can wr tions of motion can be written in the following form: −1 ˜ and ν = ν − ν is the velocity error vector (the quantity with ˙ ˜ ˜ T s ξ − ξ = ϒ ( ν + Λz), (16) and ν = ν − ν is the velocity error vector (the quantity with ˙ can fin N ξ (ξ )ξ + D (ξ )ξ + g (η) = (6) fo r for all t ≥ 0 and x ∈ d ξ ˜ ˜ ξ s ˙ = ξ − ξ = ϒ ( ν + Λ ν), (17) ˜ ˜ ˜ tions of motionN can beξ (ξ written inξthe following form: and νν˜ν==isνtoν and and = νν> − the velocity − − ννand is is(the the theisvelocity quantity velocity error erro with 0, N aquantity sT Λ vector =ννTerror Λ= ξ dν dTdvector T diagonal Tisvelocity), index isvelocity), related the desired velocity ˜ r= and νand − −νthe velocity νkis the velocity error (the vector wit ξ˙˙ +C + D )ξ + ggξξthe (η)index = π, (6) dactual > for 0,x =on > quantity 0, without kI(the kwith k0, the toa class the kkDI ˙= >whereas > 0,error totracking the actual kdthe ξ (ξ )ξ ξ (ξ all I = Dto Non-adaptive velocity controller of=vehicles index ddindex is related velocity whereas without D [s −1 T=, z(17) TI ]Tbased D desired N + D )ξ + = π, (6) index isdfor related to the desired velocity whereas without ˙rto− ˙ k= T Therefore, tt T T ξ (ξ )ξ ξ (ξ ξ (η) ˜ ˜ index d is related the index index desired d d is is velocity related related whereas to to the the desired desired without for a = 0 is gl equilibrium point [s s ˙ = ξ ξ ϒ ( ν + Λ ν), η˙ ξξ˙˙=+C J(η)ϒξ , (7) for N +C )ξ + D (ξ )ξ + g (η) = π, (6) ˙ T ξthe > 0,positive kI = kImatrix. > 0, The =velocity kwith thethe index the actual velocity), index is index related dactual istoN related to kthe desired whereas velocity without withou ξ (ξ )ξ, + Dξ (ξ )ξ + ξwhereas Dmatrix. TT > Laaξ DkkT TThe > 0, and is a desired diagonal strictly =adto Λ and ν˜==Λ − 0, ν is velocity error vector (the quantity and NΛ is diagonal strictly positive N˙˙ ξ=+C gξξΛ(η) =νTdπ,> (6) Therefo η J(η)ϒξ (7) for 0, = > 0, k = k the index to the velocity), k T T ξ ξ > 0, > 0, k = k = the index to the actual velocity), k II Tactual D ckD D index the state IIT that T > Tk3. TTher D η J(η)ϒξ ,, (7) >>k0, = kquantity > 0, k> kIkwill the index actual velocity), the the index to the the actual velocity), velocity), Ther IkI= D Remark we as Tactual Tthe T 0, and N istoto a[sthe diagonal positive matrix. = ΛTequilibrium D 0, 0 0,= kclusion 0,with = = kDpositive ksimplicity the index to the index actual velocity), kto ˙˙ = η = J(η)ϒξ (7) ID=>kIThe D DFor and ν˜the = −velocity νN is the velocity vector (the ,velocity), zexponentially ]T =strictly 0error iskwithout globally exponentially stable. point index d is related to[s= the whereas D > I Ther ,Λ zTTT ]desired = 0νand globally stable. equilibrium point dis 2N clusion where the appropriate matrices and vectors are given asΛfol> 0, N is a diagonal strictly matrix. The Λ T T T > 0, and is a diagonal strictly positive matrix. The Λ = ξ η = J(η)ϒξ , (7) ξΛ w for all t ≥ 0 and x ∈ R with the controller (1 T T T T T > 0, and N is a diagonal strictly > > 0, 0, and and positive N N is is a a matrix. diagonal diagonal The str st Λ = Λ Λ Λ = = Λ Λ clusi clusi constant and diagonal. Note also th where the appropriate matrices and vectors are given as folT T , z ] = 0 is globally exponentially stable. equilibrium point [s > N is ato and diagonal isI will aglobally strictly diagonal positive strictly The matrix. Th Λ= Λ > Λd0, =will ΛkFor index isand related the desired velocity whereas the appropriatematrices matrices and vectors given asthe follows: Remark 3. we that kDk, kDmatrix. ΛTTare TT T 0, Ipositive 0,N = k,assume >Λ 0, =Tsimplicity theare index to3. actual velocity), where the appropriate and vectors are given as fol,, and k without ,Tstable. and Λ are with th Remark 3. For simplicity we assume that lows:where Dξassume kequilibrium are Remark For simplicity we that clus for all = exponentially equilibrium point [s T,will T T0 Iand D IT ,kzzDTTT]]> = 0,kzis is exponentially stable. equilibrium point [s Therefore, based the where the appropriate matrices and vectors are given as folTthe ξξT,point TNote T T]globally = 00that isthe globally exponentially ,,analozzare ]]TT0,= =error 00on stable. is is glo glo equilibrium [s equilibrium point point [s [sisTξITξvelocity with with gous to virtual ve lows: T > = k > 0, k = k the index to the actual velocity), k constant and diagonal. also that quantity s , z ] = 0 , is z globally ] = is exponentially globally exponentially stable. stabl equilibrium equilibrium point [s point [s ξ where the appropriate matrices and vectors are given as fol, k , and Λ Remark 3. For simplicity we will assume k D I ξ I D D I > 0, and N is a diagonal strictly positive matrix. The Λ = Λ lows: Theref ξ ξ is analoconstant and diagonal. Note also that the quantity s is analoconstant and diagonal. Note also that the quantity s with Λ are Remark 3. simplicity we will assume kFor ξ T that state spa kthat , and and are Remark 3.ΛtoFor For simplicity we will assume that kclusion D ,, ksimplicity Imatrix. lows: Iξ,velocity ,ΛThe kthe ,vector and ΛΛ are 3.and simplicity we Remark Remark will assume 3. 3. For kξsimilar we we will will ass as >globally 0, Nvelocity issimplicity awe diagonal strictly positive Λ = virtual error vector s,that whereas N = ϒTT Mϒ,,(8) (8) to the reference defi T ]gous T and TRemark D Ikare lows: isr, is constant Note also thatassume the quantity and ar 3. Remark For simplicity 3.For For will we will that kDDs,simplicity kanaloclusion , zvelocity =Remark 0diagonal. isthe exponentially stable. equilibrium point [sconstant ξksIthat DΛ,similar I , and gous to the virtual velocity error vector sassume whereas ξanalois gous to the virtual error vector s whereas ξ is similar N = ϒ (8) rcontroller T Mϒ, r T T T is and diagonal. Note also that the quantity ξ with the (13): is analoconstant and diagonal. Note also that the quantity s ξ to the reference velocity vector defined by Slotine and Li [61]. , z ] = 0 is globally exponentially stable. equilibrium point [s T ξ N = ϒ Mϒ, (8) T is analoconstant and diagonal. Note constant constant also that and and the diagonal. diagonal. quantity Note Note s also also tha ththt However, because of the presence ξ ξ gous to the virtual velocity error vector s whereas ξ is similar with C (ξ ) = ϒ C(ν)ϒ, (9) is analois analo constant and constant diagonal. and diagonal. Note also Note that the also quantity that the s quantity s r N = ϒ Mϒ, (8) T ξ T toHowever, the reference velocity vector defined by Slotine and Li [52]. , kI ,vector and ΛLisare Remark For simplicity we will assume that kSlotine to the3.reference velocity vector defined and [52]. D gous to the virtual velocity error whereas ξξIrr,ξalso is similar Nξξ= ϒ) =Mϒ, (8) gous to the virtual velocity error vector sto whereas is similar ofbythe presence the matrix take here C (ξ ϒ T C(ν)ϒ,,(9) kwhereas and Λdynamics are Remark 3.velocity For simplicity we will assume that kϒ gous tobecause the virtual velocity gous gous error to vector the the virtual virtual s, swe velocity velocity ξrξexponenti error error vec vec is similar Dand to consideration ofis is globally C (ξ )) = ϒ (9) to (9) the reference vector defined by Slotine Li [52]. TTC(ν)ϒ, gous to the gous virtual to the velocity virtual error velocity vector error s whereas vector ξ whereas is similar isin simila ξ r r However, because of the presence the matrix ϒ we take here is analoconstant and diagonal. Note also that the quantity s However, because of the presence the matrix ϒ we take here in C (ξ = ϒ C(ν)ϒ, (9) TT D(ν)ϒ, to the reference velocity vector defined by Slotine and Li [52]. ξ that D (ξ ) = ϒ (10) in to consideration also dynamics of the system. Moreover, for to the reference velocity vector defined by Slotine and Li [52]. ξ is analoconstant and diagonal. Note also the quantity s is glob ξ ξvehicle C )) = ϒϒTC(ν)ϒ, (9) totothe reference velocity vector to toeach the theby defined reference reference by velocity velocity Slotine vector vector Li [52]. defin defin considered we should ξξ(ξ because of the presence matrix ϒSlotine we take here inand D (ξ D(ν)ϒ, (10) glo to the reference the reference velocity vector velocity defined vector by and Slotine Li [52]. andfor Liis [52 is gl toeach consideration also dynamics of thesdefined system. Moreover, gous to the virtualHowever, velocity error vector sthe whereas ξthe similar consideration also dynamics of the system. Moreover, for ,to (10) considered vehicle we error should take into values of D (ξ )) = = ϒ (10) r is However, because of presence the matrix ϒ we take here in gous to the virtual velocity vector whereas ξtake similar TT D(ν)ϒ, However, because of the presence the matrix ϒaccount we3.2. here in here ξ(η) r is is gl D (ξ = ϒ D(ν)ϒ, (10) However, because of the presence However, However, the because because matrix of of ϒ we the the take presence presence th in th controlling forces and force momen T g = ϒ g(η), (11) Robustness iss to consideration also dynamics of the system. Moreover, for ξξ (ξ ) = ϒT D(ν)ϒ, However, However, because of because the presence of the the presence matrix the ϒ we matrix take ϒ here we take in here D (10) each considered vehicle we should take account of R3i to the reference velocity vector defined bydynamics Slotine and Li [52]. controlling forces and force moments. Thus, is necessary to 3.2. each considered vehicle we should take into account values ofit Moreover, ggξξ(η) = ϒ (11) to the reference velocity vector defined by into Slotine and Li values [52]. T g(η), to consideration also of the system. for to consideration also dynamics of the system. Moreover, for (η) = ϒ g(η), (11) to consideration also dynamics to to consideration consideration of the system. also also dynamics Moreover, dynamics of of for tt check these for the vehicle. T ta isvalues globally exponentially tainty we must glob each considered vehicle we should take account values of 3.2. tothe consideration tothese consideration also dynamics also dynamics of the system. the Moreover, system. Moreover, for fo 3.2. ggπξξ= (η) (11) ,(11) check values for the vehicle. T g(η), ϒTT= τ.ϒ (12) controlling forces and force moments. Thus, it is values necessary toisconsi However, because of presence the matrix ϒ we here inof controlling forces and force moments. Thus, it take is necessary to However, because of the presence theinto matrix ϒ we take here in each considered vehicle we should take into account of each considered vehicle we should take into account values of (η) = ϒ g(η), (11) tainty 3.2. ξ= ϒT τ. each considered vehicle we each each should considered considered into vehicle vehicle account we we should should of π (12) Proof. closed-loop system gtw gorithm. The sensiti controlling forces and force moments. Thus, ittake isThe necessary tofor tainty each considered each considered vehicle we vehicle should we should into account take into values account ofvalues values ot taint π = ϒ (12) to consideration also dynamics oftake the system. Moreover, T τ. check these values for the vehicle. to consideration also dynamics of the system. Moreover, for check these values for the vehicle. controlling forces and force moments. Thus, it is necessary to controlling forces and force moments. Thus, it is necessary to π = ϒ τ. (12) gorithm Twith (7) describe the motion of a taint 3.2. R Equations (6) and (3) together controlling forces and force controlling controlling moments. forces forces Thus, and and it is force force necessary momen momen to controller (13) can be written as fol ti 3.2. Robustness issue tionships tht Proof. The closed-loop system (6,take 7) together the conπ = ϒ with τ..(12) (12) check values forand the vehicle. controlling controlling forces forces force and moments. force moments. Thus, it together iswith Thus, necessary it with is between necessary tothegorit each considered vehicle we should into account values of gorit each considered we should take into account values of Proof. The closed-loop system (6), (7) Proof. Thevehicle closed-loop system (6), (7) together with the Equations (6) and (3) together (7) describe the motion of aa these check these values for the vehicle. check these values for the vehicle. tionshi tainty gori Equations (6) and (3) together with (7) describe the motion of vehicle, where N is a diagonal matrix. As it was mentioned in check these values for the check check vehicle. these these values values for for the the vehicle. vehicle. troller (13) can be written as follows: tainty we must consider Taking into accou Proof. The closed-loop system (6), (7) together with the controlling forces and force moments. Thus, it is necessary to tions check these check values these for values the vehicle. for the vehicle. ˙ Equations (6) and (3) with (7) describe the motion of tions controlling forces and force moments. Thus, it system is necessary to controller (13) can be written as (6), follows: controller (13) can be written asclosed-loop follows: vehicle, where N is diagonal matrix. As it was mentioned inaaProof. N ξ (7) +Ctogether (ξ )ξ + with D )ξ + ggorithm (η) The Equations (6) (3)aa together together with (7)(7) describe the motion of ξtogether ξ (ξthe ξTaki Proof. The closed-loop system (6), (7) with tions vehicle, where N is diagonal matrix. As it was mentioned Equations (6) and (3) together with describe the motion ofin [31] there are and various possible decomposition methods. HowProof. The system Proof. Proof. (6), The The (7) closed-loop closed-loop together with system system the (1 gorithm. Thethe sensitivity (14)-(17) (note that τ check these values forclosed-loop theas vehicle. controller (13) can be written follows: Ta Proof. The Proof. closed-loop The closed-loop system (6), system (7) together (6), (7) with together the with th vehicle, where N is a diagonal matrix. As it was mentioned in Ta check these values for the vehicle. [31] there are various possible decomposition methods. Howcontroller (13) can be written as follows: tionshi vehicle, where N isthe aN diagonal matrix. AsAsititwas mentioned inD N ˙r(13) ˙)ξ controller (13) can beD written as follows: ˙ +C (14)-(1 a vehicle, where is a diagonal matrix. was mentioned in Ta [31] there are various possible decomposition methods. However, in this work Loduha-Ravani method which is related = N +C (ξ )ξ + D (ξ )ξ + g ξ +C (ξ )ξ + (ξ )ξ + g (η) ξ controller (13) can be written controller controller as follows: (13) can can be be written written as as foll foll Proof. The closed-loop system (6), (7) together with the N ξ (ξ )ξ + (ξ + g (η) re tionships between the va asξ follows: r r ξrewritten ξ ξ can ξ can ξwith as ξis related ξ (14)controller controller (13) be (13) written beaswritten follows: [31] there are various possible decomposition methods. How(14)Proof. The closed-loop system (6), (7) together thefollows: ever, in this the Loduha-Ravani method which Taki [31] there arework various possible decomposition methods. HowN ξ˙ +C (ξ )ξ + D (ξ )ξ + g (η) [59] there are various possible decomposition methods. How(18) rewritte (14) ever, in this work the Loduha-Ravani method which is related ξ ξ ξ ˙ controller (13) can be written as follows: to the generalized velocity components (GVC) [36] is used (the Tinto account T g˙˙ (η) + k Taking N ξξ˙˙+ +C (ξ )ξ + D (ξ )ξ gg(ξ (η) ˙r is rewri Nwritten +C (ξ )ξ + D (ξ + (η) ever, in this work the Loduha-Ravani method which related rewr ξξ(ξ = ξ˙Nrξ˙)ξ +C + D )ξ + ϒ k)ξ zν˙ d+ (18) controller can be)ξ as follows: = N(13) ξis +C DξN + gD (η) +++ kD sξξξ(ξ +)ξ (18) what leads to the generalized velocity components (GVC) [36] used (the N N NgkξξξgIξz+C +C (ξ )ξ + (η) (ξto: )ξ )ξ ++ D= (ξ ++gg(14)-(1 (η) ˙+C ˜ + r )ξ rϒ D sTξ+ I)ξ Λ ν) τD M( rξ +C r+C D ξg ever, in this work the Loduha-Ravani method which is related ξ (ξ ξξξ)ξ ever, in this work the Loduha-Ravani method which is related ξξ(ξ ξξ(ξ ξξ(η) NN ξ)ξ +ξξ(ξ (ξ)ξ )ξ + + (η) to the generalized velocity components (GVC) [36] is used (the ˙ obtained matrix ϒ is only an upper triangular matrix containing ξ (ξ ξ (ξ ξD ξ+ ξ (η) ξk ˙ (14)-(17) (note that τrewr = = ξ +C (ξ )ξ + (ξ )ξ g (η) + s + ϒ k z (18) T τ = r r r D I N ξ +C (ξ )ξ + D (ξ )ξ + g (η) to the generalized velocity components (GVC) [36] is used (the ξ ξ ξ ξ ˙ rewritt T ˙ ξ ξ ξ obtained ϒ is only an upper triangular thematrix generalized velocity components (GVC) [60] iscontaining used (the +C (ξ (ξ + + kkD ssξξ + ϒϒk)ξ kksII+ z+ (18) T T+g(η) to theto generalized velocity components (GVC) [36] used (the(ξ= ˙r)ξ ˙ ˙ =)ξN Nξ+ ξ˙rleads +C (ξ )ξ +D D (ξr)ξ )ξ +gg(ξ (η) + + z (18) τ ˙matrix rr + rrD ξξN ξξ)ξ ξξ(η) τ)gg= rg D + T obtained matrix ϒ is only an upper triangular matrix containing what to: = ξ = = N N ξ ξ +C (ξ + )ξ + g +C +C (η) (ξ + (ξ )ξ + D D ϒ (ξ (ξ k )ξ )ξ z (18) + + ones on the diagonal). ˙ N ξ +C (ξ )ξ + D (η) what leads to: N s ˙ + [C (ξ ) + D (ξ rewritten as follows: r r+ rrkξgD s(η) ξ ξg(ξ ξξ+ ξξϒ IkIξrzr (18 ξ containing ξ =which ξleads ξ+ N ξ=r N =ξ+C Nto: ξ)ξ +C (ξ +C +ξ ξD (ξ (ξ )ξ(ξ )ξ D (η) )ξ + ϒTDkkrkDr Iξzszξ(18) + (18) obtained matrix ϒ is only an upper triangular matrix τ+ ˙ r(ξ r )ξ r+ r+ ξ ξ ξ ξ obtained matrix ϒ is only an upper triangular matrix containing ones on the diagonal). ξ + D )ξ + g (η) + k s + ϒ obtained matrix ϒ is only an upper triangular matrix containing what leads to: r τ= r r D ξ I ξ ξ ξ ones on the diagonal). T T what leads to: what leads to: ones on on diagonal). the diagonal). =D ν now = N ξ˙r +CξN(ξs˙ξ)ξ+ (ξ )ξ gsξ˙)ξ(η) +]s s)ξ+ kwhat Nleads [C ) 0. +(18) kLyapunov ]sTξ to: + ϒτDenoting kfunction =ν˙0. (19) to: what leads to: ˜νr+C( [CD (ξ )what + D (ξ ++ kto: ϒ+TDϒ kξI z(ξ = Λν) = M( r+ r+ Iz As aleads D(19) Iz ξleads D ξk(ξ ones ones on the the diagonal). d +candidate ξ what ξleads ξD+ what what to: leads to: Denoti 3. Design of decoupled non-adaptive velocity N sN (19) ˙ξ + [Cξ (ξ ) + Dξ (ξ ) + kproposed: 0..able (19) a are to rewrite th D ]sξ + ϒ kTT Iz = Deno Deno ))+ + kkDDthe ]s + ϒϒNTTsks˙]s 0. 3. Design of decoupled non-adaptive velocity Ns˙s˙ξξ + +[C [C (ξ +D D (ξ))expression )+ +D ]sξξ)following +is kIIzz+ =[C 0.T(ξ (19) + ϒ ξξ(ξ ξ (ξ TkI(19) what to: As a Lyapunov function candidate expression N s(ξ ˙ξfollowing += + [C ϒ )I +g(η) = +=D D 0.1are (ξ (ξ (19) )(19 )+ + abl Asleads a Lyapunov function candidate the Den 3. Design decoupled non-adaptive velocity ξξξ(ξ ξD ξϒ ξ (ξ ξis ξDenoti N s˙ξN+s˙ξ[C N s˙ξξ+ )(ξ+[C D )+ (ξ + +kϒN kD˙kξξ]s = + 0. kz)(19) z+ 0. + [C ) [C + D(ξ (ξ ) +kD k(ξ ]sξξ)+ + ϒD 0. (19) trackingof controller D]s Iξzξ= ξ ξ are aa 3. Design of decoupled non-adaptive velocity ξ are T As a Lyapunov function candidate the following expression is ˙ + τ = M ν tracking controller 3. Design of decoupled non-adaptive velocity r , z)isis = sξare L (sξnow As a Lyapunov function candidate thefollowing following expression proposed: are abl 3. Designcontroller of decoupled non-adaptive velocity a aproposed: function candidate the expression tracking As a Lyapunov Lyapunov function candidate the following expression isν = νNs + Denoting N s˙ξ +in [CAs + aD ) Lyapunov + kDfunction ]sξ1+function ϒTcandidate kI z1=candidate 0. As a aLyapunov As As aa(19) Lyapunov Lyapunov the following function function expression candidate candidate ttξ ξ (ξa)proposed: ξ (ξ tracking controller As Lyapunov the following expression isisr 2 d is In this section the general controller which is decoupled the proposed: As Lyapunov As function candidate function candidate the following the following expression expression 1 1 tracking controller tracking controller T T T T proposed: In this section the general controller which is decoupled in the proposed: to rewrite ab Ns z kare z. able (20)the , z)proposed: =proposed: sξ NsξL+(sξz,1z) kIT= z. sξproposed: L (sξproposed: proposed: Ithe 1 ξ T+ (20) In this the general controller which decoupled in the sense ofsection the vector of the transformed variables ξ is presented. Calculating time (20) derivative of th proposed: 2 2 As is a Lyapunov function candidate the following expression is 2 2 1 1 In this section the general controller which is decoupled in the 1 1 , z) = s Ns + z k z. L (s T T sense of the vector of the transformed variables ξ is presented. I ξT + 121T1z1zTTkkI z. 11Note ξ1ss1TNs In this section the general controller which is decoupled in the N that (20) L TT sense the vector of the transformed variables ξξinis presented. Inof this section, the general controller decoupled the sense of (sLξξ,,(sz) z)ξL = z.z..(20) (20) Lξ(s 2 (s= τleads =comparing M11to: ν˙ssrTT+C( ξξ + ,= z) sNs Ns + zT ξkto: kI + (20) ,ξξTz) = s2ξ2sTNs z) = = zL kIk(20) z.I z. (20) Ns L L (s (s Iz. proposed: sense of the vector of the transformed variables is presented. ξ+ 1th ξ(20) ξL ξξ,,z) ξξ 2 Calculating the time derivative of the function ξξ Ns Calculating the time derivative of the function leads , z) = s , z) = Ns z Ns + z (20) (20 L (s L (s Note 2 T I ξ ξ ξ ξ 2 2 sense of the vector of the transformed variables ξ is presented. ˙ ξ ξ 2 2 2 2 vector algorithm of the transformed variables ξcan is presented. N s˙ϒs ++ϒ L 3.1. the Control The controller be used for Calculating fully Note 2of the function 2 2 L (20) 2 (sleads ξ , z) = ξNote 1the 1 T Note th time derivative to:ssξ = Tthe ξ ,2 Note 3.1. Control algorithm The controller can be used for fully Calculating time derivative of the function L (20) leads to: , z) = s Ns + z k z. (20) L (s Calculating the time derivative of the function L (20) leads to: of the function L Calculating the time derivative (20) leads to: 1 I 1 ξ ξ ξ 3.1. Control algorithm The controller can be used for fully Calculating the time derivative Calculating Calculating of the function the the time time L derivative derivative (20) leads of of the to: th TTderivative T of T actuated underwater vehicles, hovercrafts or indoor airships. TCalculating T ˙ ˙ ˙ ˙ Calculating the time derivative the time of the function the L function (20) leads L (20) to: leads to 2 2 , z) =+sν˜ξ Nk1s˙Iξz.T+Because s Ns ν˜matrices kI z. (21) 3.1. Control algorithm The be used for fully sξξ Ns (21) Lfully (sξ , z) = sξ N s˙ξ +L (s ξ +Note actuated underwater vehicles, hovercrafts or indoor airships. 3.1. Control algorithm. Thecontroller controllercan can be used for M,reformulate and ϒ (13 ha T Tξ 3.1. Control The can be used for fully that(21) comparing Thus, we ˙ (sξ , z) 1˙TT2Tξ ξ˙+ ν1˜ T the 2˙ = actuated underwater vehicles, hovercrafts or indoor airships. Moreover, it isalgorithm assumed that thecontroller vehicle Calculating moves, i.e. the airship 1ξleads sz) N= k Ns Lof T+ s1 Tz. T TTs˙L T I ξ T 11alw ˙ d ξ actuated underwater vehicles, hovercrafts or indoor airships. ˙ ˙ ˜ the time derivative the function (20) to: ˙ , s N s ˙ + s + ν k z. (21) Ns L (s Thus, ˜ ˙ actuated underwater vehicles, hovercrafts or indoor indoor airships. . (21) , z) = s N s ˙ + s + ν k z. (21) Ns L (s T T T T T Moreover, it is assumed that the vehicle moves, i.e. the airship I ˜ 1 1 , z) = s N s ˙ + s + ν k z. (21) Ns L (s ξ ξ ξ 2 thus N = (ϒ Mϒ) = 0. Using I ˙ ˙ ˙ ξ ξ ˙ ξ ξ ξ lo actuated underwater vehicles, hovercrafts or airships. I lows: ξξξTM, ξ ξ= s2ϒTN Ts˙ξ˙ + TL ,constant z) +ξ,ν,˜z) z) = s= sξξ N Nss˙˙ξξThus, + (21) s LL (s L (s ξN Thus ˙ (smatrices ˙s(s Moreover, is assumed the vehicle moves, i.e. the airship Iksz. Thus is in Moreover, flight it phase or the that marine vehicle flows. Other motion Because constant elements, Because theairship matrices M, andthe have only , sz) ˙and = ξ2sξ2sξhave sξ ξξ+only +sξνs˜dtTξNs k˙Iξ(s z. ν˜ Tk= z. (21 Ns Ns Lϒ ,+ s2s= ϒs I(21) ξ , z) = ξ+ ξ ξ+ Moreover, it is assumed that the vehicle moves, i.e. the airship ξ ξN ξelements, it is assumed that the vehicle moves, i.e. the ξ 2 2 lows: Thu is in flight phase or the marine vehicle flows. Other motion gets: lows: 2 2 Moreover, it is assumed that the vehicle moves, i.e. the airship d 1 T d T Because the matrices M, and ϒ have only constant elements, 1 lows ˙ is in flight phase or the marine vehicle flows. Other motion T 0. T0.and phases areflight not taken into consideration. Because thesTmatrices matrices M, ϒhave have only constant elements, thus Nthe =+matrices Using also the relationship (19) one lows thus N˙Other =L (ϒ = also the relationship (19) one T ˙Mϒ) Because M, and ϒϒ have only constant elements, is in phase or the marine vehicle flows. Other Because the ϒ have only constant elements, is in phase or the vehicle motion , Mϒ) z) = sthe ˙ξUsing += ν˜M, kand (21) Ns (s Because M, and only constant elements, dtd(ϒ dt˙motion I z. N Land = Tmatrices ξthus ξ0. phases are not taken consideration. ξ NdBecause ξthe matrices M, Because and Because ϒ have the the only matrices matrices M, M, and elements, ϒϒslows hav ˙ s= T ξhav is in flight flight phase or into the marine marine vehicle flows. flows. Other motion Thus, we reformulate th ˙d ̇d = N (ϒ Mϒ) Using also the relationship (19) one Tconstant Because the Because matrices the M, matrices and ϒ M, have and only ϒ have constant only elements, constant element T= 2 thus N (ϒ Mϒ) = 0. Using also the relationship (19) one d phases are not taken into consideration. ˙ T 2 T gets: ˙ dt gets: L (s [−C (ξ )s − D , z) = s dt ˙ phases are taken not taken into consideration. thus N  =  (ϒ M ϒ) = 0. Using also the relationship (19) one thus N = (ϒ Mϒ) = 0. Using also the relationship (19) one d d d T T T phases are not into consideration. ξ ξ ξ ξ( thus N = (ϒ Mϒ) = 0. Using also the relationship (19) one ξ Non-adaptive velocity controller ˙ ˙ ˙ dtT˙= tracking dt d d T thus N thus thus N N = = (ϒ Mϒ) = 0. Using also the (ϒ (ϒ relationship Mϒ) Mϒ) = = 0. 0. Using Using (19) one also als dt ˙ phases are not taken into consideration. Because the matrices lows: gets: thus Nand = thus NMϒ) =dtonly = (ϒ0. Mϒ) Using = 0. also Using the relationship also relationship (19) one (19) on dt dt theIts M,gets: have constant elements, It Bull. Pol. Ac.: Tech. XX(Y)gets: 2016 dtϒ(ϒ time Tdt T derivative ha gets: T Tgets: T (ξ )s ˙gets: ˙ (sdξXX(Y) ˜ L (s , z) = s [−C (ξ )s − D − k s − ϒ k z] gets: gets: + ν k z. L , z) = s [−C (ξ )s − D (ξ )s − k s − ϒ k z] Bull. Pol. Ac.: Tech. 2016 D I I D I Itstime tim T ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ 1 Its gets: gets: T T ξ Non-adaptive velocity tracking controller ˙ TheoremTheorem 1. Consider the vehicle dynamic model (6), the kiT T ˙ Bull. Pol. Ac.: Tech. XX(Y) 2016 thus N = (ϒ Mϒ) = 0. Using also the relationship (19) one T ˙ L (s , z) = s [−C (ξ )s − D (ξ )s − k s − ϒ k z] theL f 1. ConsiderBull. the vehicle model Its L (6), (s˙˙ξ ,the z) T=kine(ξ )sabove (ξ )s − ξkDcan sξD−write z]I L ξsξ [−C ξ −D ξ result ξ ξ we ξ ϒ kTT dt XX(Y) Itssξ˙tim ti+ + =˙following ssξTNs Pol. Ac.: dynamic Tech. 2016 Iin ξξThe M (s (ξ )s − k(ξ s)s − ϒϒ z) = ssTξTξT[−C Tξ)s TT TkkIIz] T= Ac.: Tech. 2016 T L (s z)TkL = [−C (ξ )s −ξD Dξξ)s (ξ )s − khowever sξξ= −− z]−(ξ nematic relationship (7), and theBull. transformation (3) L Non-adaptive velocity tracking controller D ˙ (s Its ti) ξξν ξξ(ξ ξξ − ξξ˙˙D 2(22) Dz) ˜,,,(3) T[−C Tk + z. gets: + ν˜ T kXX(Y) (22) L L (s (s (ξ − k [−C [−C s (ξ ϒ )s )s − z] − D D (ξ (ξ , z) = s , , z) = s s Recall (9), that s C (ξ I ˙ ˙ Non-adaptive velocity tracking controller I z. D I (10)): matic relationship (7), velocity andPol. the velocity transformation toξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ L (s L (s z) = s [−C , z) = (ξ s )s [−C − D (ξ (ξ )s )s − − D k (ξ s )s − − ϒ k k s z] − ϒ k z] T + DI (22) ξ in ξthe ξ following ξ ξ ξ ξ form ξ D ξ(using ξ (22) ξ (22) ξI ξ ν˜ kcontroller ξ I z. ξwrite . Considertogether the vehicle dynamic model (6), the Non-adaptive kine- The above velocity with the following controller: +result ν˜ tracking TT I z.we can T C(ν)s A Non-adaptive velocity tracking controller time derivative has) th Assuming for simpli  (assuming =ϒs ˜˜TT =Its ˜ν˜k)s gether with the following controller: + ν k z. (22) TT T T TCk s T C(ν)(ϒs T T + k z. (22) T I 0 s(using ˙ I T ˜ ξ b Recall however (9), that s (ξ )s = (ϒs ) =Assum + ν + + ν k z. ν k k z. z. (22) L (s , z) = s [−C (ξ − D (ξ )s − k s − ϒ z] TξC(ν)(ϒs Recall however (9), s C (ξ =Dthat (ϒs The above result we can write inξ the the form Theorem 1. Consider Consider the vehicle vehicle dynamic(3) model (6), the kinekineIk)s Ifollowing I= onship (7), and the velocity transformation to- (6), ξI )s ξ= ξ ) Assum ξ+ ξ ξs)Twrite ξ (10)): ξ(9), ξ s)(ϒs ˜ ξkabove ξ νthat +ξξν˜result (22) (22 ξCC(ν)(ϒs Recall (ξ ) ) = I z.ξhowever Iξz. 0 A s T T ξ ξ ξ The we can in following form (using Theorem 1. the dynamic model the n a and using (29) we re 11 ξ ξ usT Ta skew-s Recall however (9), thatwesL C )sξ=s= (ϒs )T C(ν)(ϒs )˙ = =iss(using and (the matrix C(ν) ∈ R Assu ˙ TξC(ξ T above T ξξfollowing ξ= ˙r +C TC(ν)s ,(ξ z) − The result can in form Theorem 1. Consider vehicle dynamic model (6), the s˙Assu + ν˜ L (ϒs   (13) (10)): matic relationship (7),Nthe and the velocity transformation (3) toT C(ν)s the following ξξwrite π (7), ξand = )ξ )ξ gkξz.(η) + ktosRecall ϒ ksRecall z, however (9), )s C(ν)(ϒs = s(10)): = = ϒs )because because s))T= 0))all for T Tall and T  that s+TrνC(ν)s =(3) 0kine(assuming = ϒs )(assuming because ssssTTξs(s C(ν)s = 0= for all however (9), that (ξ )s =the (ϒs C(ν)(ϒs =ssTM r + Dtransformation DRecall Iabove Assu The we ξ(9), can write the (using Theorem 1. controller: Consider the vehicle dynamic model the kineξ (ξ ξ (ξ ξ+ ξTξin ξξC(ν)s ξC(ν)(ϒs sC(ν)s =0ξ0result (assuming =C )ξRecall sfollowing = 0form for ˜+T(6), ξϒs ξA matic relationship the velocity (22) T T T T however that Recall s however however (9), (9), that that C (ξ )s (ϒs ) C C (ξ (ξ )usi = )s )u T T ξ ξ I z z 0 T T and ξ ξ ξ ξ ξ 2 fore, taking into account (16) we and Recall however Recall however (9), that (9), s that C (ξ )s C = (ξ (ϒs )s ) = C(ν)(ϒs (ϒs ) C(ν)(ϒs ) = ) ξ ξ ξ (13) Recall, however (9), that s C (ξ)s  = (ϒs ) C(ν)(ϒs ) = 0 A s s n n s C(ν)s = 0 (assuming s = ϒs ) because s C(ν)s = 0 for all ˙ ξ ξ ξ ξ ξ ξ ξ ξ M (10)): matic relationship (7), and the velocity transformation (3) toξ ξ ξ ξ       n M ν = τ −Cν − Dν 11 ξ ˙ ξmatrix ξ ThereMand ν ha == TTC(ν)s TTC(ν)s getherrelationship with the the following following controller: ξone) skew-symmetric one) [22]. Theres(10)): ∈ T))because matrix C(ν) a(the skew-symmetric [22]. s ∈ R (the(3) ˙to(the matrix C(ν) C(ν) isisξϒs aaskew-symmetric one) [22]. Theres=∈Ris R= T transformation matic (7), and the velocity        s 0 (assuming s = s = 0 for all T T T T T T with controller: , (24) , z) − L (s s C(ν)s = 0 (assuming s = ϒs because s C(ν)s = 0 for all Assuming for simplifica T ξ n , ξ +Cgether (ξ )ξ + D (ξ )ξ + g (η) + k s + ϒ k z, (13) ξ ˙ T T where T T T T M ν = s s s C(ν)s = 0 (assuming s = C(ν)s C(ν)s ϒs ) because = = 0 0 (assuming (assuming s C(ν)s s s = = = 0 ϒs ϒs for ) ) all be be = s C(ν)s = 0 (assuming s = ϒs ) because s C(ν)s = 0 for all r r D I ξ ξ ξ ξ 0 A s s ξ       ξ ξ ξ (the matrix C(ν) is a skew-symmetric one) [22]. Theres ∈ R T gether with the following controller: Recall however (9), that staking (ξ(assuming )s = )ξskew-symmetric C(ν)(ϒs )L 11 sC C(ν)s = = 0(ϒs (assuming ϒs ) sbecause =have: sbecause C(ν)s s 0C(ν)s for (ξ all =) + 0Hence for ξ= ˙)0(s z11 0is A ν˙H ξC(ν) ξϒs ATheren (the fore, taking into account have: ξwe ξ= into account (16) we 2T A s[22]. ˙ξ0C(ν)s have:  = issξaas= M ν˙a ξ(16) ,and z) [D kM n matrix gether the (ξ following ˙ +C [22]. sssaccount ∈ RRns 2 R ξ(16) ξ−s  t+ kDfore, D ]s ,There(24) z) = − L (sznwe T ξis aawe matrix C(ν) skew-symmetric one) ∈fore, using (29) receiv Hence, recalling that ξ,There˙(the ξn(the π= =N Nwith )ξr + Dξcontroller: (ξ )ξr + gξ (η) (η) +taking ϒTT kkI z, z,into (13) matrix C(ν) a skew-symmetric one) n (the matrix C(ν) isisssa∈ skew-symmetric (the matrix matrix C(ν) C(ν) one) is [22]. skew-sy skew-sy TheresL ∈∈ R ∈ RRnn (the 00 ξone) A ss[18]. ssis M ν˙ (24) ,, z) = − (s ξ    11 T ξ ξ fore, taking into account (16) we have: ξ π ξξ˙˙rr +C +sTkC(ν)s ssξξ + ϒ (13) (the matrix (the C(ν) matrix is a C(ν) skew-symmetric a skew-symmetric one) [22]. one) There[22]. There s ∈ R s R z z 0 A D I A ˙ ξ (ξ )ξr + Dξ (ξ )ξr + g ξ T Hence, = 0k (assuming staking = ϒs ) ξbecause C(ν)s = 0T]s all Tfor TTk .T 11 T TAT2T + ξ(16) ξTand ,,into z) = − L (s ˙˙(s TsT[D T,, A22 =(24) fore, account (16) we have: ξinto ˜ kD s)ξdσ TITzk+ z 0 T taking TzA z = ν(σ , (14) fore, into account (16) we have: π = N ξ +C (ξ )ξ + D (ξ )ξ + g (η) + + ϒ z, (13) T ˙ ˜ where fore, taking account we have: where = ϒ D(ν)ϒ Λ 2 ˙ L , z) = −s (ξ ) + k − s ϒ k ν k z ˙ r r r I Henc T A D I 11 (24) z) = − L (s D I ˜ ξ ξ ξ ˜ L (s , z) = −s [D (ξ ) + k ]s − s ϒ k z + ν k z ξ ξ ξ fore, taking into account (16) fore, fore, we taking taking have: into into account account (16) (16) we we hav ha Henc = kI D˙ ]s z s˙Λ Lϒ (skξI z, , z)(13) = −s )+ ]sξ −into sξξξ ϒ(16) +νhave: z ξ −s [D Dk0T π = N ξr +C sξn + DT ξ ξktaking ξzkI zwe ξ (ξfore, z=) τ+−Cν MITξν˙(ξ − M +g L =ξ− s−Dν fore, taking into account account (16) we ξ [D 2 ξ TA t ξ (ξ )ξr + Dξ (ξ )ξr + gξ (η)0 + where (the matrix C(ν) Theres k∈DR Hen 0I ξ ATIξhave: ˜ kzTTI z definite. Thus 2z + L˙is (s˙˙ξaT ,skew-symmetric z) =T−s )one) +z kD[22]. ]sξTT−matrix νpositive where symmetric isz+ TTthe T T ξ [D TTsTξ ϒ ATϒkTA ξ) (ξ  −1  ˜ ˜ = −s [D (ξ + k ]s − z Λ k z. (23) T T T T L (s , z) = −s [D (ξ ) + k ]s − s k ν k z T T T T T T z = ν(σ ) dσ , (14) A (s ,− z) = [D (ξ )]sξ+ kD z s+= ν˜−s kIII zz[D DA where A11)= D(ν)ϒ + and = kIkIsL .z. Note that where ˙ (s ˙˙ϒ(s ξkk  z) ξξ −s ξξ k ξξΛ ξ[D D]sΛ Tξξξk,I− Tϒ TTk = )[D + ξr = ϒ fore, (νd taking += Λz), (15) (16) we ˜(ξ  tt −sinto = +L kϒ ]s zhave: (23) (M L (s]s ,ξD = −s )is− ]s −s (ξ )s)of z+ kkDDν˙]s ξT D−s IξAξT ˙−s D˙ (s Iz) rTξξ+ ξξ(ξ ξT[D Dϒ Ik+ where ξ (ξ ξDz(ξ ξξν C s]sm ˜eigenvalue ξ [Daccount ξs= {A} > 0− minimal L L)Λ (s −s ,Tz. z) (ξ )ξT22 + [D kT(λ )+ −the +ksL kξ,IIz) − zHence, + ϒ kξkξIrecalling zI[D z++ ν˜ Tk(23) zthe 0 T ξξ , z)ξ = m m(ξ Ithat ξk ξ=− ξ]s ξBull. ξ+ ξTTν ξλ ξkAD ˜˜ )) dσ APol. Ac.: Tech. XX(Y) 2016 zz = ν(σ dσ ,,,(14) =  t ν(σ (14) = −s [D (ξ + ]s z Λ k z. (23) where A = ϒ D(ν)ϒ k and = Λ k . Note that the symmetric matrix A is positive definite. Thus, assuming T T T T D I −1 D I 11 22 ξ ξ T T T ξ (23) (14) where A −1 = ϒ D(ν)ϒ + k and A = Λ k . Note that t = −s [D (ξ ) + k ]s − z Λ k z. (23) T T T T T T T T T T T T T ˜ Non-adaptive velocity tracking controller sξ = ξr −(15) ξL =˙ ϒ +(14) Λz), (16) D 11 22Tk of = )ϒ + kϒ ]skξξI)2016 − zkν˜TDupper Λ k− (23) D II z.z can an bound the De TXX(Y) T ξ]s Dfind Tkk ξξTk[D ξ (ξ T T ξr = ϒ (νd + Λz),0 ν(σ = −s =TΛ −s −s [D ]sof z.[D [D (ξ (ξ )II).L + +˙kkderivative. ]s − −that Λ (23) (sξλ, (m z)ν{A} = −s [D (ξis−s )symmetric + − smatrix zA + k]s ˜˜ )) dσ zz = Bull. Pol. Ac.: Tech. D D]s D A = + and A kktime Note ξD(ξ ξI zξkthe ξξone ξwhere T[D T s˙zz+Λ ν˜(23 k sTξξ M the is positive definite. Thus, assuming ξ= D 11 22 ξIk −s = −s [D )ξϒ+ kD(ν)ϒ (ξ )+ − + z Tkkkpositive Λ z.= zmatrix Λ z.Λ (23) >ξ0XX(Y) (λ the minimal eigenvalue D I− IA) m = 00 −1 ν(σ dσ ,, (14) ξ=ξ(ξ ξ]s where A Tξ2016 T+ = D(ν)ϒ and A = Λ . = Note that ξ Ac.: ξxξXX(Y) the symmetric matrix A is definite. Thus, assuming Bull. Pol. Tech. D I 11 22 Bull. Pol. Ac.: Tech. 2016 = [s , z ] one can write: −1 −1 ˙ ˙ ξ = ϒ (ν + Λz), (15) −1 ˙ r d 0 ξ ˜ ˜ T T T T the symmetric matrix A is positive definite. Thus, assuming ˜ s ˙ = ξ − ξ = ϒ ( ν + Λ ν), (17) s = ξ − ξ = ϒ ( ν + Λz), (16) {A} >Λ00result (λ is2016 thecan minimal eigenvalue of now the matrix A) one λ]smAc.: upper bound time Denoting ξrthe = ϒvehicle Λz), (15) r r ξ ,(15) Bull. Tech. XX(Y) ξ 1. Consider −1(νd + = −scan =assuming s(using [Dξfind (ξ )an + kPol. − z> kof z.mmthe (23) (Mone ν˙ r +C the symmetric matrix Aderivative. is positive definite. Thus, The above we write in the following form Theorem dynamic D IXX(Y) (λ is the minimal eigenvalue of the matrix A) λ mξ{A} Bull. Pol. Ac.: Tech. 2016 ξξs r = ϒ Λz), (15) −1 (νd + −1 ˜ model (6), ξthe kinePol. Ac.: Tech. XX(Y) 2016 T TBull. T one {A} > 0 (λ is the minimal eigenvalue of the matrix A) one λ Bull. Pol. Ac.: Tech. XX(Y) 2016 Bull. Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. XX(Y) XX(Y) 2016 2016 = ϒ (ν + Λz), (15) The above result we can write in the following form (using (10)): ξ − ξ = ϒ ( ν + Λz), (16) −1 2 can find an upper bound of the time derivative. Denoting now m m x = [s , z ] can write: r −1 r d ξ ˙ ˙ ˙ Bull. Ac.: Bull. Ac.: 2016 XX(Y) 2016 ˜ + Λz), >Tech. 0Pol. (λXX(Y) the minimal eigenvalue of ≤ the−λ matrix A)now one λPol. (10)): matic relationship and (3) (16) − = νtransformation , L (t, x) ξto˜isϒthe can find an upper bound of the time derivative. Denoting m {A} m is Tech. s˙ξ = ξr − and ξ (7), =sνξ˜ϒ= (ξνν˜˙rdthe + ν), (17) −ξξΛ νvelocity error vector (the quantity m {A}||x|| −1(velocity T ,with T ]Tupper ˜ ssξ = ξ − = ϒ ( ν + Λz), (16) , (16) −1 can find an bound of the time derivative. Denoting now r x = [s z one can write: −1 T T T ˙ ˙ ˜ ˙ = ξ − ξ = ϒ ( ν + Λz), (16)       can find an upper bound of the time derivative. Denoting now gether with the following controller: r ξ ˜ ˜ x = [s z T˙ ] T one can write: = − ξ˙ = =ϒ ϒ−1 ν˙˜ + +Bull. Λν), ν), Tech.(17) XX(Y) 2016 index d ξξis related to (the desired velocity whereas without ˙rr − ξT ,,L ˜ Pol. Ac.: ss˙˙ξξξ = Λ (17) ≤can −λ for allsmwrite: t{A}||x|| ≥ 0T andA2 ,11 x ∈ R02N . (25) xx = [s zzT(t, ]]T x) one −1(ν − ν is the velocity error vector quantity with ˙r − ξξ˙˙(the s ˙˜˙velocity), TT T(17) ξ˙ ,> ˜ ξ = [s one can write: s ˙ = ξ = ϒ ( ν + Λ ν), −1 ˙ 2 ξ ,(24) ξ 0, = k > 0, k = k the index to the actual k T ˙r +C s˙+ ξ(ξ = (ν˜+ (17) I LIξ (s2N (17) + ν), ,direct method (24) − L˙˙ (t, x) Dwith r− {A}||x|| (25) [5 ≤ −λ −λ 2, ξ = ξ , z) =Therefore, π = ν˜N ξ= (ξ )ξ D )ξξr + gϒξ (η) kΛ s˜ξ ,+ ϒD kI z, (13) based onmthe Lyapunov ˜ and ν − ν is velocity error vector quantity related to the desired velocity whereas without rthe D(the ξ ξ d {A}||x|| (25) L (t, x) ≤ T and ν = νd − νΛis=the velocity error vector (the quantity with z ˙ (t, x) ≤ 0−λm A for allpositive t ≥ 0 and x ∈ RThe. 2,z 2 > 0, and N is a diagonal strictly matrix. Λ T T {A}||x|| , (25) L 2 ˜ and ν˜ = = νisdvelocity), − ν is error vector (the with clusion space origin ˙that the index related tokDvelocity the desired velocity whereas without 0, quantity 0, Tkvelocity = kI > = kdesired o where the actual 2N  ≤state m the {A}||x|| , of the system (25)(6), L −λ and ν ν νequilibrium is the the velocity (the quantity with m D >error index ddand isdν˜− related to the without for all all tthe ≥ Lyapunov 0 and and xx ∈ ∈R Rdirect Therefore, onstable. method [52], con2N(t,.. x)  = ν the velocity quantity withbased ,Ivector zT ]Tvector 0whereas is(the globally point [sξerror d ¡ νtois the T exponentially T=> for t ≥ 0 index d is related desired velocity whereas without with the controller (13): A 2N > 0, = k 0, k = k the index to the actual velocity), k 0, the and N is a diagonal strictly positive matrix. The T T I > that D >whereas index dindex istorelated tothe desired ..system (6), for all ≥ 00based and xxon ∈ R 0,k ,the 0,whereas kII = kwithout =velocity kwe index the actual velocity), kDDvelocity Therefore, the Lyapunov direct method [52], the the concont toFor clusion state space origin theLyapunov (3) together d is related thesimplicity desired ITwithout D T > for all ttΛ ≥are and ∈of R2N Remark 3. will assume that T [52], T to T = TT, z> based on the direct method D kTherefore, I , and > 0, = k 0, k = k the index the actual velocity), k T T T T T T T 0, and N is a diagonal strictly positive matrix. The Λ = Λ D I ] 0 is globally exponentially stable. point [s T I D ˜ z = ν(σ ) dσ , (14) > 0, = k > 0, k = k the=index actual velocity), k Therefore, based on the Lyapunov direct method [52], the conwhere A = ϒ D(ν)ϒ + k and A = Λ k . Note that index to the actual velocity), k  = k  > 0, k  = k  > 0, where A  = ϒ D(ν)ϒ + k and A  = Λ k . Note that the sym>to 0, the and N is a diagonal strictly positive matrix. The Λ ΛξTthe D I clusion that the state space origin of the system (6), (3) together with the controller (13): D I 11 22 D I 11 D 22 I D I I D (t) s Therefore, on the Lyapunov method the conconstant Note also that thestable. quantity sξ is analoT T = diagonal. Tis ξ [52], clusion thatbased the state space origin ofdirect the system (6), (3) together > N a0]and diagonal strictly positive The Λ = Tand iskglobally globally exponentially equilibrium point [s Tis= = 0, lim Thus, , kI strictly , andexponentially Λpositive are matrix. Forequilibrium simplicity will that state  > 0, a diagonal matrix. The the metric matrix A is positive Thus, assuming λ m{ A } > 0 > 0, 0,we and NTξTand is positive matrix. The Λ =Λ ΛTΛ = Λ clusion that the space origin of the system (6), (3) together Dstrictly symmetric matrix A isdefinite. positive definite. assuming ξassume ,,tozzTTaN ]diagonal 00 is stable. point [s with the controller (13): T t→∞ −1 clusion that the state space origin of the system (6), (3) together gous the virtual velocity error vector s whereas ξ is similar with the controller (13): z(t) r T = T0 T is , z ] globally exponentially stable. equilibrium point [s T T T ξ = ϒ (ν + Λz), (15) (t) s rξthat point , z=quantity ]will is globally exponentially (λm isthe thecontroller eigenvalue ofeigenvalue the matrix can find A) an one andstable. Λstable. are λwith Remarkequilibrium 3.Note For simplicity we assume that  A)  analod diagonal. also sξ isthat ξ (13): z [s]the 0d = 0 isvelocity globally exponentially equilibrium point ξwe > 0minimal (λm is the minimal ofone the matrix m {A} ,, kkII ,, and are Remark 3. For simplicity will assume kkDDdefined   ξ , reference = 0, (26) lim with the controller (13): to [s the vector byΛ Slotine and Li [52]. −1ξ ˜the (t) s , k , and Λ are Remark 3. For simplicity we will assume that k   t→∞ is analoconstant and diagonal. Note also that quantity s D I virtual velocity error vector s whereas is similar ξ s = ξ − ξ = ϒ ( ν + Λz), (16) z(t) rthe that globally can be made. find an in upperisbound of the now r we also sξ (t)derivative. Λ are Remark 3. Fordiagonal. simplicity will assume kD , kthe  =convergent  time ξ is analoconstant and Note that quantity sI ,ξξand 0, Denoting (26) limexponentially However, because of the presence ϒ wecan take here ssz(t) = 0, (26) lim T , zT ]T one can write: is analoconstant and diagonal. Note also that the quantity ssrξ ismatrix t→∞ ξ (t) gous to the virtual velocity error vector s whereas ξ similar ence velocity vector defined by Slotine and Li [52]. (t) x = [s −1 is analoconstant and diagonal. Note also that the quantity ˙ ˙ t→∞ ξ = lim gous toBull. the virtual vector s whereas isglobally similar z(t) ξ rξis ˜ of ξξthe ˙velocity = ξr −error ξ2017 = also ϒ (dynamics ν˜˙ + Λν), (17) exponentially to sconsideration system. Moreover, forconvergent can Pol. Ac.: 461 (26) = 0, 0, (26) limbe made. ξTech. t→∞ gous to the virtual velocity error vector ss whereas similar z(t) to the theof reference velocity vector defined by Slotine and Li [52]. r is ecause the 65(4) matrix ϒ we take here in ξand 3.2. Robustness issue In case of vehicle parame t→∞ gous toreference the presence virtual velocity error vector whereas is similar z(t) to velocity vector defined by Slotine Li [52]. r each considered vehicle we should take into account values of 2 is globally exponentially convergent can be made. Unauthenticated ˙ to the reference velocity vector defined by Slotine and Li [52]. {A}||x|| , (25) L (t, x) ≤ −λ is globally exponentially convergent can be made. However, because of the presence the matrix ϒ we take here in ˜ ation also dynamics the system. Moreover, for and ν = ν − ν is the velocity error vector (the quantity with m tainty we convergent mustDate consider robustness of the proposed dbecause to the reference velocity defined by Slotine Li [52]. However, of the vector presence the matrix ϒ we and take here init is necessary | 8/25/17 12:47 PM is exponentially can be made. controlling forces and force moments. Thus, 3.2. Robustness issue toIn case ofDownload vehicle parameters unceris globally globally exponentially convergent can analysis be made. However, because of the presence the matrix ϒ we take here in to consideration also dynamics of the system. Moreover, for ered vehicle we should take into account values of index d is related to the desired velocity whereas without 2N gorithm. The sensitivity will be done usi However, because of the presence the matrix ϒ we take here in to considerationcheck also dynamics of for thethe system. Moreover, formust for all Robustness t ≥ 0 robustness and x ∈issue R of .In these values 3.2. case of vehicle vehicle parameters uncertainty consider thecase proposed control alT >we T vehicle. to consideration also dynamics of the system. Moreover, for 3.2. Robustness issue In of parameters uncereach considered vehicle we should take into account values of forces and force moments. Thus, it is necessary to > 0, k = k 0, = k the index to the actual velocity), k tionships between the variables in the given below I D to consideration also dynamics of the system. Moreover, for I D

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(10)): to- (10)): 11T0  A 11 ξ z  ,ξ  (24)    sξwrite ,˙z)(s =, z) − ξwe Labove (s ˙ (s -ation zTcan The The result result we can in inAthe the following form form (using 2 following (24) , (using (24) ,toz)ξL = − − L 0 form 11 ξabove ξ can= (10)): (3)result (10)): --13) ξwrite above we write in the following z z, sξ(using 0 A ˙    2 sA 0 A A szξ ,  s s (24) , z) = − L (s   11 11 z z zξ0 0 A 0 T ξ ξ ξ z, (13)     2 2 ˙ ˙ T (10)): (10)): -)): (24) = (s −ξ , z) =−  szT  A  ,      , z sξ (24) 011 13) L (sξ , z)L T2 AA sξ z0 , s (24) sξ zξ0  A z02A11 A11 211s 000 A ,= z)z− =− L˙ (s A ˙ s s ξ 13) T T ξ ξ ξ ξ , (24) , z) L (s       A  0 A A T ˙ (sξξ, z)T= − T z, z)kz=  A (24) −AA11 L˙T(ssξsξ+ , (24) , L11 + kI z, (13) 0A 0= )) ϒwhere A = ϒTs D(ν)ϒ Λ2 k0sIsξ.zξ zANote that 011Az22 ξA11D and 0 s ˙ ˙ Herman W. Adamski z and z zTΛ 0 A ξ ξ22T ΛT k where A = ϒ D(ν)ϒ + k and A . T kNote that T T   INote A A= 2.P.that , , (24) (24) , , z) z) = = − − L L (s (s D 11 22 ˙ ξ ξ 14) )here T    A where A = ϒ D(ν)ϒ = ϒ + D(ν)ϒ k and + k A and = Λ A k . = Note that , (24) , z) = − L (s I Note 22 (14)symmetric the is positive definite. Thus, Amatrix D(ν)ϒ + kDD22 Λzz kI . assuming ξ11 where 11 11 = ϒ A zDzis 00and I=Thus,  that  AAAAz2222 z Aϒmatrix 0A Adefinite. the symmetric matrix positive assuming T D(ν)ϒ TA TThus, TThus, 2 definite. A Λdefinite. 14) ehere symmetric the symmetric matrix is positive is positive assuming assuming A where A = ϒ = + D(ν)ϒ k and + k A and = A k . = Note Λ k . that Note the symmetric matrix A is positive definite. Thus, assuming       A A D D I I 11 11 22 22 {A} > 0 (λ is the minimal eigenvalue of the matrix A) one that λ m m  eigenvalue  Tminimal T k . A)Note 15) {A} > 0 (λ is the of the matrix one λ (15) 14) m m T T where A = ϒ D(ν)ϒ + k and A = Λ thatLet now define the control input in the following form: A A T T Tone D I 11 22 > 0mupper (λ is > the 0 (λ minimal is the minimal eigenvalue eigenvalue of the matrix of the A) matrix one A) λthe {A} > 0 (λ is the minimal eigenvalue of the matrix A) λ{A} symmetric matrix A matrix is positive A is definite. positive definite. Thus, assuming Thus, assuming msymmetric m m)e{A} m m where A = ϒ D(ν)ϒ + k and A = Λ k . Note that can find an upper bound of the time derivative. Denoting now TDD(ν)ϒ + Tone A Tof T now the time derivative. Denoting x = [s ] . Let IA 11bound 22k= and (14) )15) ξ , z where A = ϒ = Λ k Note where A = ϒ D(ν)ϒ + k and A Λ k . Note that nowthat define the control input in the following form: D I 11 22 D I 11 22 can find an upper bound of the time derivative. Denoting now (16) the symmetric matrix is the positive definite. Thus, assuming T0 T T can find upper bound time derivative. Denoting now T Tm Tof TThus, n{A} can upper find an bound upper of bound the time of the derivative. time derivative. Denoting Denoting now > {A} is > the 0anϒ (λ minimal is the minimal eigenvalue eigenvalue of the matrix matrix one A)now one λ the symmetric matrix A+ isAkmatrix positive definite. assuming can write: m,one xfind = [san z(λ one can write: )m16) T22k where where AAm]]T11 = = ϒ D(ν)ϒ D(ν)ϒ + kpositive and and Adefinite. A = = Λ ΛNote kkIthe .definite. . A) Note Note that that T T T D D I 11 22 ξ Let now define in following form: the symmetric A is positive Thus, assuming the symmetric matrix A is Thus, assuming 15) T T T ere A = ϒ D(ν)ϒ + k and A = Λ . that x = [s , z one can write: T T T T T T D I 11xλξ= Letthe now input in th = ,(λ z0] upper ]write: can write: )16) (λ is the minimal eigenvalue of the matrix A) one τ = Mˆ ν˙ r the ˆ −T ˆcontrol ˆ input [s [s ]mx{A} one one write: an find an can upper find an bound ofcan bound the time of22the derivative. time derivative. Denoting Denoting now now .(34)(34) + Cν kDdefine kI z.control ϒˆ −1 s +the ξzcan (17) {A} >ξ[s0,0> ismone the minimal eigenvalue of the matrix A) one λ,mz{A} r + Dν r + gˆ + ϒ (15) )= m{A} the the symmetric symmetric matrix matrix A A is is positive positive definite. definite. Thus, Thus, assuming assuming ξλ 17) > 0 (λ is the minimal eigenvalue of the matrix A) one λ > (λ is the minimal eigenvalue of the matrix A) one symmetric matrix A is positive definite. Thus, assuming 2 −T −1 m m m m ˙ T T T T T T 16) ˆ ˆ ˆ ˆ ˆ can find an upper bound of the time derivative. Denoting now ) [sλλξcan (t, x) ≤the −λ (25) Lcan Let now define the input therinput following Let Let now define define the input in the following form: Cν +inDν gˆˆ+inˆϒ kform: kˆform: z. ϒ Mˆ sν˙+ τ =control M ν˙ r +the ˆ− ˆform: x (16) = [s , z{A} ] > one z(λ can ]m is write: one write: ˆ (34) m {A}||x|| 2 matrix Letnow now define thecontrol control input inthe the following 2 ,,(25) rcontrol D−T ICν find an upper bound of time derivative. Denoting now ˙ (t, + Dν τϒˆfollowing = −T ˙an > 00,the (λ is the the minimal minimal eigenvalue eigenvalue of of the the matrix A) A) one one r+ rare r + gˆ + ϒ ξT ˆ+ {A}||x|| , 2A) (25) L x) ≤ −λ m m0{A} m 17) 2 the ))= {A}||x|| , (25) L (t, x) ≤ −λ y with m can find upper bound of time derivative. Denoting now can find an upper bound of the time derivative. Denoting now ϒ known. where the parameters in M, C, D, g, ˆ –T , and –T T T ˙ ˙ A} > (λ is minimal eigenvalue of the matrix one m   ̂   ̂   ̂   ̂   ̂ m {A}||x|| {A}||x|| , , (25) (25) L (t, x) ≤ L −λ (t, x) ≤ −λ T,]zT one where the parameters −T in M , C−1 , D, −T g ̂ , ϒ−T , −1 and ϒ −Tare known. =TT [s ] one can write: with m m −1 2N xall =xfind zan can ξT and )17) Tbound Twrite: −T −1 ˆDν ˆττ= ˆg, ˆˆ−T ˆ,ϒϒ ˆˆand find upper upper bound of of the the time time derivative. derivative. Denoting now ˆ+ ˆCν ˆˆgˆr+ ˆreceive: ˆDν ˆˆM, ˆ(34) ˆrkr+ ˙ rInserting without ˙ˆν(34) ˙˙define forcan t[sfor ≥ 0all x0L ∈ R,2N ξ ,,zan 2one +Inserting Cν +ˆM + ϒ kˆkin z. (34) ϒ τ =now Mˆ where ν(25) now control the following Cν + + gˆˆggˆ+ + ϒ s s+ + kkIkz. (34) ϒ τLet = M νˆDν 2N areform: known. the parameters in C, D, ˆ+ ϒ 2 Denoting 2Denoting x≥one = [s zxx 2 R ].T≤ can write: xcan = [s ]Ttt ¸ 0 can write: ˆ D, into (33) we ˆ(34) ˆ g, r= Ik find an upper bound the time derivative. now rν r+ r+ D IIz. + Cν + Dν + ˆˆinput +sϒ ϒ kDthe sϒ z. (34) M ˙of ˙−λ where parameters in M, C, ˆ rr+ rthe rD D . into (33) we receive: and ∈ R ξ {A}||x|| {A}||x|| , , (25) (t, x) L (t, x) ≤ −λ ξ (17) ))out with . for all t ≥ 0 and x ∈ R m m 2N 2N T T T T T T T all . . r t ≥ for 0 all and t x ≥ ∈ 0 R and x ∈ R T T T x x = = [s [s , , z z ] ] one one can can write: write: 2 > 0, ˙ Therefore, based on the Lyapunov direct method [52], the conInserting (34) into (33) we receive: I 2 ξ ξ [s , z ] one can write: −T −1 {A}||x|| , (25) L (t, x) ≤ −λ −T −T ) Therefore, based on the Lyapunov direct method [61], the con[52], −T −T −T −T T ˙ Inserting (34) into (33) we receive: ˆ with m ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ −T −T ˆ ˆ ˆ ˆ ˙ ˆ ˆ ˆ ξ > ˆϒ ˙ r−+ {A}||x|| (25) L (t, x)direct ≤ −λ −λ ˆC)ν ,ˆD, and known. where the parameters in D, ˆin hout 2the Therefore, based the Lyapunov direct method the conCν + Dν + ˆϒ kIknown. z.known. τparameters =M, M νC, 2 , [52], ,(D ,and ϒϒˆ s r+ are are where where the the parameters ing, in M, C, D, g,gg, ˆˆ+ ϒϒ 2N 2N [(M M) ν˙rM, +ˆ C, (C +are −ϒϒ D)ν L =(25) sparameters hx.0, r− D ˙m(t, , kand and are known.(34) Fi where the M, C, D, g, ˙the rϒ rϒ The herefore, Therefore, on the Lyapunov Lyapunov method direct method the [52], concon{A}||x|| ,(25) L x)system ≤[52], −λ {A}||x|| , m(6), L (t, x) .xon .≤ all t≥ for 0based all and tthe x≥ ∈state 0based Ron and ∈ R quantity with hor m clusion that space origin of the system (6), (3) together clusion that the state space origin of the (3) clusion together T 2 2 vi Let now define the control input in the following form: hout ˙ ˆ r ˆ+−1 ˆ(33) ˆ r− M) 2N ˙˙space TD)ν Inserting (34) into (33) weinto receive: > 0, clusion that state space origin of the (6), (3) together Inserting Inserting (34) into (33) we receive: [(M −−M) ν˙ rwe + − C)ν (D − L = s(34) 2system ˆFig. tstable. {A}||x|| {A}||x|| ,, together (25) (25) L L (t, (t, x) x) ≤ ≤ −λ −λ Inserting (34) into (33) we(C receive: ˆˆ−T 2N .{A}||x|| for all tthe ≥ 0based xon ∈2N R ˙and ˙z.rare hThe m m + (C − C)ν = s−T ˆkDD, usion that clusion the state that space state origin of origin the system of the(6), system (3) (6), (3) together ˆL˙g, +g gˆ − Ds −receive: ϒ sϒˆ− k[(M + ν˜ϒˆTˆ−T kIν(35) (35) ϒ herefore, Therefore, based on the Lyapunov the Lyapunov direct method direct [52], method the [52], conthe conr , (25) L (t, x) ≤ −λ . for all t ≥ 0 and x ∈ R I,z]and reas without t with the controller (13): known. where the parameters in M, C, ˆ with the controller (13): m 2N viro > 0, all 0the and x∈ R ˆ ˙ .direct .Lyapunov forTthe all tcontroller ≥ 0 for and x(13): ∈t ≥ R −T −1 T T −T −1 T ble. The with ˆ ˆ ˆ ˆ ˆ ,usion ˆ T −T Therefore, based on method [52], the cond Λ are   ˜ tk,ith Fig. 2. Diagram  Fig. Fig + Cν + Dν + g ˆ + ϒ k s + k z. (34) ϒ τ = M ν +g − g ˆ − Ds − ϒ k s − k z] + ν k z. (35) ϒ ˙ T 2N 2N ˆ ˙ ˙ the controller with the controller (13): (13): ˆ ˆ ˆ ˆ ˆ ˆ ˆinto ˆC)ν ˆ+g ˆˆ ˆ r − clusion space the state origin space origin the system ofr the(6), system (6), (3) r (3) together r L DL I[(M D I D)ν ˙= ˆ Crˆ r+ Therefore, based on the Lyapunov direct method [52], the=together conˆν˙M > tthe k that [(M − M) + (C +(C (D D)ν s [52], Inserting (34) we receive: −M − M) M) νrν + + (C −C − C)ν C)ν (D − L = sνs˙sr[(M gˆI−g˜Ds kDFig ϒ I = .. ofon for all all t0,≥ ≥xstate 0∈ 0that and and xxon ∈ R R rr D, [(M − M) (C − C)ν +(D (DD − D)ν L =con˜− ˜− ˆ˙˙r(33) ˜− rr− rr+  The Denoting now = M, = −C, =D)ν D− = g− ˆ −ϒg, for Therefore, based Lyapunov direct method the Therefore, based the Lyapunov method [52], the con.∈ all t for ≥I 0clusion and R2N (t) sξdirect ble. are  the = (13):  analo(t) sξorigin vironment th that the state space of the system (6), (3) together viro vir vir ith the controller with the controller (13): −T −1 T −T −T −1 −1 T T −T −T clusion that the state space origin of the system (6), (3) together matrix. The 0, (26) lim ee,ble. −T ˆ− ˆˆM, ˆϒ ˆ˜ˆ= ˆ+ ˆˆ (the ˆ˙Ds ˆ+g ˆ the ˆD− ˆconˆ(26) ˜ϒ ˆDDs ˜− ˆ+(35) Therefore, Therefore, based based on onthat the the Lyapunov Lyapunov direct direct method [52], [52], the conconsξ (t) ˆz] T− = system 0,method lim ˜s s− ˜obtained gˆ − − k− − k(C ν kD z.= ϒ +g +g gˆknown. ggˆ− −− kIϒs kD ksignals kInow + ν˜˜νD, kTkIgk˜z. ϒ ,−and where the parameters M, C, D, g, ˆ(6), ϒ Denoting now M = C −C, − = gˆ(35) −(35) g,C˜ = Cˆ − ˆ−1 from and using the expression ˜ˆ= clusion the state ofin the system (3)Denoting together the space of (6), (3) together I+ IID IrIˆz. +g − ˆM − Ds − ϒ kDenoting sD kz] + ν z. ̂ − (35) ϒ refore, based on thestate Lyapunov direct method the (t) (t) sξorigin sz(t) t→∞ [(M M) ν˙sϒrsC + C)ν (D − D)ν Lϒ = sare M M M, ξϒ ˜=  − ̂ Ds ˜  = C  D ̂  ¡ C,  z] ̂ z] alo.are clusion ξ the r˜− space   with the controller (13): thethat (26) now M  = M  ¡ M, C  = D  ¡ D, g˜ = g  ¡ g, t→∞ lim =origin 0,,[52], (26) e.similar z(t) with controller (13): ntially stable. = 0, = 0, (26) (26) lim lim ˆ clusion clusion that that the the state state space space origin origin of of the the system system (6), (6), (3) (3) together together t→∞ Inserting (34) into (33)together we receive: and (the obtained from thetheexpression s= ϒsˆand ˆ ξ ( the controller) get: with controller (13): withthe thestate controller (13): Li [52]. −T −1 the T the ion that spacethe origin of the system (6), are  (3)  ξC (t) sz(t) sz(t) t→∞ expression sg,= ϒs ˆsignals ˜= ˜using ˜C˜˜= ˆ−C, and expression s = ϒs (the signals ˜we ˆ− ˆˆˆ−g− ˜D ˆgD ˆˆ− aloξt→∞ ξ (t) ˆ− ˜= ξ ˆD can z(t) Denoting M =using Mˆnow − M, C = C D,ϒ gD ˜˜== ˆD g,D, .ilar +g ˆ−C, −M, Ds −= kusing s=obtained − k− z] +g˜from (35) Denoting Denoting now M˜M M M C˜D C C −C, D = D gνg˜= g, D I D, Denoting now M = M −M, M, =ϒ C −C, − D, ˜= =kgˆIggˆ− ˆz.− − g,Conseq is exponentially be 0, made. = 0, = (26) now (26) lim lim khere , and Λthe areglobally ehealo  with with the controller controller (13): (13):convergent  convergent Ithe the controller) we get: in (t) s is globally exponentially can be made. controller (13): T ξ [(M − M) controller) Twe Tobtained T controller) we get: Fig. 2.(the Diagram the control strategy in in ˆ +ξ Cν 52]. ˆϒs ˆˆrξthe t→∞ t→∞ ˆ rthe ˆ the ilar ˙ D)ν ˜ sr s+ ˜get: ˜ϒs z(t) from using expression s(expression = (the signals signals from fromis reduc and and using using the = ν˙ r +and (C − C)ν + (D − L˙ss= sz(t) ξ (t) M ν˙ϒs Dν g) ˜obtained (signals ν˜ Tof + zobtained Λ L = −s made.  =sˆbe (26) lim globally exponentially be made. (the obtained and using the expression s=signals = r expression ξξ = rM I zg (t) ˜ =(the ˜tested ˆ airship is analoyever, sis --globally ξ0, ξ (t)can = 0, (26) limt→∞  convergent is globally exponentially exponentially convergent convergent can belim can made. Denoting now Mˆ − M,ϒs C˜+ Cˆ− −C, D = D −)kD, ˜ from = gˆ − g, ξ for is globally exponentially convergent can be made. T T T T ilar vironment for the z(t) = 0, (26) = 0, (26) lim ˙ e52]. in ˜ ˜ ˜ t→∞In s −T uncer˙ rget: rlues (t)oft→∞ sξz(t) the we get: −T 3.2. Robustness issue case vehicle parameters the the controller) controller) we we get: M νz. + Cν Dν)sr + g) ˜ ˆTL − ν˜ −s +Tz(M Λ L −s ˜ r + Dν ˜ ν˙ r)k ˜ r + g) the controller) I zCν ˆget: ξ(t) + ˜ −0, gˆz(t) − − ϒˆ controller) kD(26) s− k= + ν˜TT((D kthe (35) ϒˆ −1 (t) slim t→∞ −s + ϒ krD+ kξ˙I(z= ϒˆ −1 ξglobally isofsimilar r-52]. I z] using Iwe z(t) ξlim r3.2. (the signals obtained and expression s+ =ν˜ϒs is tainty globally exponentially exponentially convergent convergent can be+g made. beDs made. = =can 0, (26) (b) from The ma Robustness issue In case of vehicle parameters uncerfor = 0, (26) lim e in we must consider robustness of the proposed control alT −T −1 T .r.sary t→∞ t→∞ ˆ ˆ 3.2. Robustness issue In case of vehicle parameters uncerT T T T T T T T T T T T T −T −1 T z(t) z(t) ˜ T −T −1 to −s (D + ϒ k )s + ν k z ϒ ˙ T T T T ˜ ˙ ˙ is globally exponentially convergent can be made. ˆ ˆ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ t→∞ and Li [52]. D I− ˜g, ˙= 3.2.must Robustness Inof case ofof vehicle parameters uncertainty Robustness 3.2. Robustness issue Inissue. issue case In vehicle case of parameters vehicle parameters unceruncer˜g) ˙˜ν= ˙ν˙˙r+ ˜˜I z(D ˜˜r r+ z(t) (M ν˙= + Cν + + (Dν νDν z˜g) Λ the controller) get: (((M + Cν Cν + Dν g) ˜g) ()k ν(˜(s−s νν ++ z+ zzΛ L −s (D + + ν˜ k ˜be ˆ−s = −sD, Cν + + − ϒˆΛ kI zDIIkzzϒˆD ϒ )s)splified ˆ− ˆ −s is globally globally exponentially convergent can be made. r= r(M tainty we consider robustness the proposed control alrν r r+ M Cν + Dν + g) ˜˜− − + Λϒ)k)k )k L = −s efor in r+ rrrwe rr− r+ Denoting now M˜proposed =done M M, CL == CrelaD˜L D − gM ˜ν˜Dν g+ ˆ+ − gorithm. The sensitivity analysis will be using the s.2. of isconsider globally exponentially convergent can made. is exponentially convergent can be made. Consequently, impact of dyn tainty we must consider robustness of the control al-−C, T −1 T −T the −1 ˆ ˆ we must robustness of the proposed control algorithm. ˜ take here in ˜ ˜ neninty T −T T T T −T −T −1 −1 T T T we tainty must we consider must consider robustness robustness of the proposed of the proposed control alcontrol al˙ 2. Robustness 3.2. Robustness issue In issue case of In vehicle case of parameters vehicle parameters unceruncerT T T T = −s ( M ν + Cν + Dν + g) ˜ − s (D + ϒ k )s ϒ T −T −1 T ˆ ˆ ˆ ˆ ˆ ˆ ˆrela˜+ν˙g) ˜ + g) gorithm. Theexponentially sensitivity analysis will be done usingsway. the Tϒ rϒ D Cν ˜zIIzz= ˜r + ˜(ν˜(rTM for ˜r˜νν ˜+ tionships between the variables inthe the given below is is globally globally exponentially convergent convergent can can be be made. made. −s (D signals + ϒ−s +˜from kϒˆCν z˜ )s ϒ −s (D + kν˙˜kDrkD+ + kIk+ Dν ˜ and using expression = ϒs ˙kobtained ylobally to ˜ krCν −z (r)s M ν˙ν + ˜rzT+ Λ(D k+ zrϒ −s D −s (D + )s + sith of ξ (the r˜ +−s I)s (ϒˆ= M ν + Dν g) −Dν Λ˜T )kr + L = −s D is reduced, too. exponentially convergent can be made. the ξ Iϒ I z τr = M norithm. gorithm. The sensitivity analysis will be done using the relaThe sensitivity analysis will be done using the relationships 3.2. Robustness issue In case of vehicle parameters uncerMoreover, forbetween rsrinty T TTT T ˆ T T˜ ˙ gorithm. The sensitivity The sensitivity analysis analysis will be done will using be done the using relathe relawe tainty must we consider must consider robustness robustness of the proposed of the proposed control alcontrol al3.2. Robustness issue In case of vehicle parameters uncer˜ ˜ Taking into account inversion of the relationship (12) and tionships the variables in the given below way. T −T −1 T T −T −T −1 −1 T T T T of the controller) we below get: (Dν νr−1 + +)s −z −s ϒr+ Tk TDν −1 ˆ r+ ˜Λ ˆg) ˆˆ (36) ˜ˆξ˜g) ˜M ˜−1 yf to Tz˜ν Ts r˜s+ IM ˆ˜z−T ˙˜(D ˙˙r+ 3.2. Robustness issue In of vehicle parameters uncer3.2.tainty Robustness issue Inin case of vehicle parameters uncer˜Dν ˜ϒˆ(D T −s −T = −s (alM˜ ν˙= + Cν + + ˜r−T sD kTskD(D ϒ −s −s (T(˜M ν= Cν + g) ˜Cν g) ˜ϒ − (D ϒ kkDk−s ϒˆϒϒ (M˜ ν˙ r one. +4 C˜ kˆϒ = ϒˆ)s)s runt r= rˆ(M between the variables variables the given below way. tionships between the variables in the given way. rν rr+ D Iϒ = −s + Cν + Dν g) ˜ˆ−The − )s(36) ˆ˜Cν ˆ+ ˆDν ϒ )s ν zTΛ+ −T rr+ we must consider robustness ofcase the proposed −s kr− )r + ϒs z−z Λ k+ (b) matrix values of forithm. I(D onships tionships between between the the variables inϒcase the given in the below given way. below way. Dkϒ I z. M isDa ξdiagonal gorithm. The sensitivity The sensitivity analysis analysis will be done will using be done the using relathe relaξ+− (14)-(17) (note that τ In = π) the input forces vector τcontrol can be tainty we must consider robustness of the proposed control alξTϒ T(D + ϒ becaus Taking into account inversion of the relationship (12) and y to 3.2. 3.2. Robustness Robustness issue issue In case of of vehicle vehicle parameters parameters unceruncerthe T T T T T T T T T T T T T T T T −T −1 T T tainty we must consider robustness of the proposed tainty we must consider robustness of proposed control Taking into account inversion of the relationship (12) and Robustness issue In case vehicle parameters uncerTϒ Tˆ˜ϒ ˙= ˆˆϒˆr˜(TM ˜ (12) ˆ˜r−s ˆal˜Cν ˜˜g) ˜the ˜and ˜ν˜)Dν ˜Dν ˜k˜ϒ Taking into account inversion of the relationship (12) and T TDν T −T −1 ˆ 4. ˙−s ˜z = ˙˜νϒs ˙ˆ˙r+ Λrelak(control −s νM + + ˜plified (the M ν˙given + Cν Dν + g) ˜al− + ΛϒˆkT(D L −s ((˜M Cν + + ˜g) ˜+ −z −z kIk)k z(IIzTM = ϒ −s + ϒ − zr r+ z. ˆϒˆ−T gorithm. The analysis will done the −Tof as follows: necessary to(note oofonships Iν rbe r +using r−z I˜z( rν r+ −z +Dν +g) g) ˜is z= = −s rewritten as follows: Taking into Taking account into inversion account inversion of the relationship of relationship (12) and ˙ξr kξCν ξ+ = −s ν+ + + Dν + ˜Λ − sIˆrform kkDD(36) )s ϒˆϒˆ−1 ξΛ −s tionships between between the variables variables in the given in the below way. below way. (D +ϒ )ϒs − r˜r+ rg) r(D ξzΛ gorithm. The sensitivity analysis will be done using the relarM rCν –T (14)-(17) that τsensitivity = the ϒ π) the input forces vector τcontrol can be ξDϒCν tainty tainty we we must must consider consider robustness robustness of of the the proposed proposed control alalξpositive −T the (14–17) (note that τ = ϒ π) the input forces vector τ can be Based on [52] we can find the strictly constants βi ξ 4. gorithm. The sensitivity analysis will be done using the relagorithm. The sensitivity analysis will be done using the relawe must consider robustness of the proposed control aloty (14)-(17) (note that τ = ϒ π) the input forces vector τ can be −T −T Tin the −T −1 be T tionships between the variables given below way. ˆ ˆ T T −T −1 T T T T T T T T T T −T −T −1 −1 T T T T 4)-(17) (14)-(17) (note that (note τ = ϒ that τ = ϒ π) the input π) the forces input vector forces τ can vector τ can be ˜relaTaking into Taking account into inversion account inversion of the relationship ofgiven relationship (12) and (12) −s (D +the ϒ kDusing )s +−s νrelakˆIand z(D + ϒ+ Λz) Tˆ−z T(D −1 TCν ˆϒin ˆΛ ˆthe ˜Λ ˆˆIM ˆkˆ1,D(D ˆˆ Λ ˜ξ− ˜(36) tionships between the variables inwill the below way. rewritten as follows: 4.1. zthe (18) ˆz −T sug ˜ variables ˙ d follows: gorithm. gorithm. The The sensitivity analysis analysis will be be done done the the ϒˆiξ= z. ϒ (ϒs ν˙− + Dν +ofg) ˜ the = −s + k− kDfind )ϒs )ˆ)kensure ϒs Λ kTr kI[52] (36) (36) ϒϒ ϒˆzϒϒ 4. Simulati +analysis Λ ν) +C(ν)(ν Λz) + D(ν)(ν τ= M( νsensitivity 4. rewritten as τzzconvergence = M on can strictly positive βither4. +C(ν)ν D(ν)ν + Iϒ r z+ r constants D IIrz. −s k−s Λ kν˙z. z. (36) din+ dusing ξorder 4. where .(D .we .+ ,+6k)ϒˆϒs to trackξBased tionships the variables in the given below way.−s tionships between the the given below way. r+ ξ ϒBased ξϒ D ξϒ thm. The sensitivity will be done using the relaξ− on we can find stri rewritten as follows: ξ[52] −Tbetween −T Taking into account inversion the relationship (12) and her with the eewritten T of T −T −1 rewritten as follows: as follows: 14)-(17) (14)-(17) (note that (note τ = ϒ that τ = ϒ π) the input π) the forces input vector forces τ can vector be τ can be ˆ ˆ ˜ −T −1 ˜ ˜ Taking into account inversion of the relationship (12) and T sult of tionships tionships between the the variables variables in in the the given below below way. ˜ to = (given M ν˙relationship Cν +way. Dν g) ˜ where −ing s (D ϒ1, )s ˜ the ˜D(ν)(ν i+ =and . zero. .Tk.ϒ 6Tϒ(D inTherefore, order r+ r +(27) D −Tto ensure T1,|[ +g(η) ϒthe kgiven ϒ−s νthe + Λz) + k+I z.rΛz) ≥ M ν˙order +trackCν error to choosing βpositive Taking into account the relationship (12) Taking into inversion of (12) −T D theΛ variables in below way. ˆ find ˆconvergence ˆ, we ˆ −1 r4.1. ˜account where i− =ipositive .T.ϒˆk. I,βobtain 6(of in ensure ai + ν) +C(ν)(ν + Λz) +(inversion τ between = M(follows: ν˙ dbetween 4.1. 4.1 eships −s + ϒ kthe )ϒs zpositive Λwe z. ϒ 4.1 dϒ d of −T (14)-(17) (note that τ+ =−T π) the input forces vector τ and can bewe Based on [52] can find the strictly Based Based on on [52] [52] can can find the strictly strictly constants βIrβiβ+ Dpositive because ϒ˜ =Indoor (the ide ξconstants iconstants ii(36) Based on [52] we can find the strictly constants ξ we written rewritten as as follows: T ˜ ˙ (14)-(17) (note that τ = ϒ π) the input forces vector τ can be T + Λ ν) +C(ν)(ν + Λz) + D(ν)(ν + Λz) τ = M( ν of T T T T ˆ −T ˜ sia Taking Taking into into account account inversion inversion of of the the relationship relationship (12) (12) and and ˜ d d d ˙ ˆ 18) ˜ ˜ ˜ ϒkI as (Msymmetric νr + Cνrregarding +or ing error toibe Therefore, choosing β ≥to ˜ (14)-(17) ˜= ˙+C(ν)(ν ˜= (assuming and Λ|[zero. + ΛM( ν) + Λνν) +C(ν)(ν Λz) + D(ν)(ν + Λz) + Λz) + Λz) τ =(14)-(17) M( ν˙ τdDenoting = νas (note that = ϒ vector τ+ can (note that = π) the input forces vector τν˙ rcan be −z (M + g) ˜|zero. Λ zπ) =D(ν)(ν −s ϒswhere aking into account inversion of the (12) −T −1 r6 dnow dνϒd+ dΛ d=and ingkDpositive error Therefore, choosin sults sult sul r.Dν rg)] I+ ˙ dk+ ˜kthe + Λz, = νΛz) ν, and ν˜forces + Λz we su ξinput rewritten follows: i+ =Cν 1, .+.Based ,Dν order ensure convergence ofi the trackwhere where i in i= 1,1, .we ....,.to 6we ,receive 66in in order order to to ensure ensure convergence convergence ofof the the tracktrack+g(η) +τττrϒ= k−T ϒ (ν˙τdνr˜relationship + + (27) . (19) −T where i = 1, . , in order to ensure convergence of the trackD I z. (27) on [61] can find the strictly constants β , where −T −1 T rewritten as follows: sim i ˜ −T (14)-(17) (14)-(17) (note (note that that = ϒ ϒ π) π) the the input input forces forces vector vector τ τ can can be be Based on receive [52] we(assuming can find Dν strictly constants βofkiof6Da ˜forces −1 −1 g)] ˜ i | matrices): we k˜the symmetric or AS50 18) +g(η) + k+ ϒ (dϒ νΛz) +Λz) + kI −T z. (27) Dν diagonal ˜τrewritten ˜ϒ ˙= follows: as follows: r+ Iˆ˜positive Λ ν) +C(ν)(ν + Λν) +C(ν)(ν Λz) + + + D(ν)(ν + Λz) Λz) τ =rewritten M( ν˙are τd + = M( ν D−T )-(17) (note that ϒϒ−T π) the input vector beerror TˆT( + g)] ˜˙Λ ||[kwe receive (assuming of airship ˆDTrand d rewrite dϒ dΛz) dˆ + able to the above in the form: ˆas TD(ν)(ν T −1 T T Therefore, ˜β ˜Cν ˜˜r r+ T ˜M ˜ν˜˙νrν +g(η) +g(η) + ϒ + kas (ν˜k+ (Λz) + ν˜(D k+ z. +τ kkcan z. (27) (27) i+ ˙ ˆ ˆ ˆ D Dequation I+ Iing ≥ |[ ϒ ( M ν Cν + to zero. choosing β ≥ ≥ |[ ϒ ϒ M ( + + Cν + ing ing error error to to zero. zero. Therefore, Therefore, choosing choosing β i = 1, …, 6 in order to ensure convergence of the tracking error ˙ i r r i i r ≥ |[ ϒ ( M + Cν + ing error to zero. Therefore, choosing β −s ϒ ) ϒs − z Λ k z. (36) ϒ ϒ 4. Simulation results i r r D I 6of D ξ diagonal 18) co rewritten as as follows: follows: ˜+ +C(ν)(ν ξd+ +ν) Λ ν) Λz) +and D(ν)(ν + Λz) sion isas τ M( = M( ν˙rdΛ ˙−T ˜D(ν)(ν where i 6= 1, . . . , 6 in orderTtodiagonal ensure convergence of the trackmatrices): = ν Λz, = ν + Λ ν, s = ν Λz we Denoting now ν −T −1ν −1 d˙Λz) d˜˜Λz) ) r Trewritten T d ritten follows: ˜ ˙ T T + +C(ν)(ν + + + τ = ν matrices): simulations we sim sim   ̂   ̂ ˜   ̂ T ˜ ˜ ˜ ˜ d d d +g(η) + ϒ + ϒ+ϒνΛ (r˙ν kr+ + Λz) (+ + νν, k+ z. Λz) k++ z.Λz) (27) (27) sim zero. choosing (M νΛ  ̇ rΛ  + C νras  + D ˜)]ijoror ˜Λz) ˙−T ˜D(ν)(ν + g)] ˜+ we (assuming k(assuming andT βΛi ¸ j[ϒ kkIkDkas symmetric orsymmetric + ϒDenoting kI zτDenoting (18) + + g)] ˜g)] ˜˜ i |ii|Therefore, we we receive receive (assuming and and kkIkIIas symmetric Dν Dν )enoting =+g(η) Λz, ν˙˙ϒ + Λ and s+ = νΛz + Λz we now D+ I+ IDν ˜d= ˙+C(ν)ν ˜ννM ˙ d=ν ˙krνD r we id| + D ν) +C(ν)(ν + Λz) D(ν)(ν = M( rreceive rto D + Λ ν) Λz) = M( ν + g)] | we receive (assuming and Λ as symmetric or Dν g(η) = ν+ rντ d+C(ν)(ν dD(ν)ν Tν(r + g r D d d r d d ˆ ˙ ˜ ˙ ˜ ˜ ˜ ˜ com ˜ + Λz, = ν + = Λz, ν + ν Λ = ν, ν and + s Λ = ν, ν and + s = ν Λz we now ν now −1 [1 T −T −1 T T ˙ r r r r d d d ≥ |[ ϒ M ν + Cν + ing error to zero. Therefore, choosing β are able to rewrite the above equation in the form: 4.1.Λ+TkIndoor ˙≤ ˆ model ˆairship rIn rsectio −T −1 + ϒequation k+ ϒ (d+ ν˜the + k+ z. (27) L − β sξki Dϒˆ and (D ϒˆI as ksymmetric z 6ΛDOF kIthis z. model Based on [52] weΛz) can find the strictly positive )19) ˜form: ˜  + Λz 6∑constants 666DD ˜ν) ˜ ν+C(ν)(ν D Idiagonal Denoting now  = ν  + Λz, ν ̇rΛz)  = ν  −T ̇+  + Λν ,+ and s = ν wediagonal we receive (assuming i |sξ i | − β D ϒ 6 i )ϒsξor−diagonal + + Λ Λ+g(η) ν) +C(ν)(ν + Λz) D(ν)(ν D(ν)(ν + Λz) Λz) τντ˙ = = M( M( −1 matrices): +g(η) + ϒ k−T (νthe ν˜ν, Λz) k(+ (27) rabove d−T diagonal matrices): matrices): d+ dϒ d˜z. −1 are to the D Id˜ν diagonal matrices): ˜ν˙ν˙rddto T k as + Λrewrite ν) +C(ν)(ν + Λz) D(ν)(ν + Λz) =able M(able +ϒ kD−1 ϒ sin + z. (28) [1]. ˜ − dnow dthe dk+ T ˆ T (assuming −T kDˆ and −1 the T symmetric TT proposed ˜ eenoting are rewrite able the rewrite above equation above equation in form: in the form: ˙ ˙ ˙ ˜ ˙ ˜ ˜ +g(η) + ϒ ϒ + Λz) + k z. (27) sy +g(η) + ϒ k ϒ ( + Λz) + k z. (27) ID = ν + Λz, = ν + = Λz, ν + ν Λ = ν and s Λ = ν, ν and + Λz s = we ν + Λz we Denoting ν now ν ˙ ˆ ˆ + g)] ˜ | we receive Λ or Dν I D I r r r sults regarding use of the ct d d d d T −T 19) i=1 r i I (20) to L ≤ β |s | − s (D + ϒ k ) ϒs − z Λ k z. ϒ ϒ ˆcom where i−1 = 1, .+ .Λz) .g(η) ,in6 + in˜kkorder ofi the areτable toν˙rewrite the above equation the form: to ensure convergence matrices): −T −T −1 D − ∑ βiξ|sξ i | −coming ∑ ξ i trackL˙ ≤ sξ ϒIˆ (Dfrom +com ϒ co ξ +C(ν)ν + D(ν)ν = M ne19) isable to ˜ ˜ −T −1 +g(η) +g(η) + + ϒ ϒ k k ϒ ϒ ( ( ν ν + + Λz) + z. z. (27) (27) r r r ˙ ˙ ˜ = ν + Λz, ν = ν + Λ ν, and s = ν + Λz we Denoting now ν 6 D D I I 6 6 sys r r (37) d d T ˜ 6 are rewrite able to the rewrite above the equation above equation in the form: in the form: +g(η) + ϒ k ϒ ( ν + Λz) + k z. (27) ˙ ˙ ˜ ˜ T ˙ diagonal matrices): = ν + Λz, ν = ν Λ ν, and s = ν + Λz we Denoting now ν m +C(ν)ν + D(ν)ν + g(η) τ = M ν i=1 of airship AS500 (assuming indoor test w ˆ D I ˜ ˜ r= Note that comparing (13) we have the drzero. r νerror |[ ϒ we (TM ν + Therefore, choosing βL i=1 T˙ r + CνˆrT−T T Tˆ ˆˆ [1]. [1] TTˆ ˆTˆTT ˆ −1 ˆ ˆ−T −T −1 T TT [1]. T TT The maxi ˙+ ˜+≤and ˜˙˙≥ ˙= ˙+ +ing D(ν)ν + + g(η) τ(19) =able Mnow ν˙to τ+C(ν)ν ν˙νrddthe ν+ Λz, ν˜in = ν˙form: Λν˜˙ν, s =L +≤ Λz now =+ϒ + Λz, νD(ν)ν + Λrelationships: =L Λz we Denoting ν=rrr M [1] −T −1 −T −1 ˆβiβ|s ˙νi|s rDenoting r+C(ν)ν requation rν, ˆzϒ ˆ−1 rν r and rr s dto ds + dg(η) = 0. are ))nzads ˆ ˆ is to: − β | s (D + ϒ k ) ϒs − Λ k z. ϒ ϒ ≤ − − β |s | − | − s s (D (D + + ϒ ϒ k k ) ϒs ) ϒs − − z z Λ Λ k k z. z. ϒ ϒ ϒ rewrite above the i D I (37) k ϒ k z. (28) i D D I I L ≤ − |s | − s (D + ϒ k ) ϒs − z Λ k z. ϒ ϒ T ∑ ξ i ξ ∑ ∑ ξ ξ i i ξ ξ This condition leads us to conclusion that the tracking error fo ξ D I i D I ξ ξ −T −1 ma T ξ i ξ are able able to the above in the form: simulations were done in system MATLAB/Sim ξ ˙rr− ˙ν|above ˙D(ν)ν ˜+ ˜−1 = ννν˙to + + Λz, ν˙νequation Λ Λsνν, and and ssΛz = =inwe ν˜ν˜the + +Λz Λz we we Denoting now now ννs+ −T −1 6 + g)] ˜= receive (assuming k(29) and Λ kI i=1 as symmetric or Dν rable r== drν d˙,rabove dkdwe were +ϒ s= + z. syst sys )sn isDenoting ˙are equation form: are rewrite the in rϒ iν D ˜+ ˜g(η) = M +C(ν)ν + D(ν)ν + g(η) τ= M νrewrite D Is+ sya = νr τd+C(ν)ν Λz, = ν˙−T ν, and νform: noting now νrto ˙˜ν, ˙+ skΛz, = ν, s+ ˙+ = − ν. ϒs i=1 i=1 rD rIϒ rI= rrewrite dsνequation +ϒ +ϒ ϒ + kΛ z. kthe (28)(28) (28) rkthe rz. ξk i=1 isleads D This condition us| − tosconclusion that the Tfor TDOF −T −1tracking Terror Tinputs. for convergence guaranteed t → ∞ if the vehicle dynamics is TT ˙ ˆ ˆ ˆ ˆ (28) 20) This condition leads us to conclusi 6 model with six signal is sexpression are are able able to to rewrite rewrite the the above above equation equation in in the the form: form: L ≤ − β |s (D + ϒ k ) ϒs − z Λ k z. ϒ ϒ ˙ + D(ν)ν + g(η) τ above =ν˙M νequation diagonal matrices): (37) (37) i ξi D (37) ξ Tmax Note that comparing (13) we the relationships: T −T −1have −T in −1 ∑ r +C(ν)ν rD(ν)ν rg(η) (37) ξ x,y,z I= T27, max m T able to rewrite the the form: (21) +C(ν)ν + τ = M m r r r +ϒ +ϒ k ϒ s + k k ϒ z. s + k z. (28) (28) Note that comparing (13) we have the relationships: D D I I convergence is guaranteed for t → ∞ if the vehicle dynamics is The ˙ r D(ν)ν Thus, reformulate Lyapunov candidate as fol- not exactly known. (37) sote +C(ν)ν + D(ν)ν =the M ν+ +r relationships: g(η) τwe = M ν˙ r +C(ν)ν C r + g(η) r the rfunction i=1 convergence is guaranteed for t → ∞ coming from the report [9], were also 20)thatNote comparing that comparing (13) weτhave (13) we have relationships: the −T −1 This condition leads us to conclusion that the tracking error This This condition condition leads leads us us to to conclusion conclusion that that the the tracking tracking error error for the airship A for for t This condition leads us to conclusion that the tracking error −T + −1 for +ϒ k ϒ s + k z. (28) 6 ˙ ˙ D I +C(ν)ν +C(ν)ν + D(ν)ν D(ν)ν + + g(η) g(η) τ τ = = M M ν ν ˙ −1 s + (37)Cas +ϒ ϒ s= +−T z. relationships: (28) rrν, rrI− to: thatτNote , rr srwe = νD(ν)ν s+ϒ ˙sg(η) ν˙krkthe ν. (29) = ϒs not exactly known. −T 20) Dthe r −kk na +ν +have =lows: Mssthat ν˙= ξ(13) not exactly known. [1]. The maximal forces and torques a ments, r +C(ν)ν r −1 T ˆconvergence TI z. −Tconvergence −1 ˆ(28) Tguaranteed T k ϒ k +ϒ ϒ + z. (28) ))ote comparing comparing have (13) we relationships: ˙ ˆ ˆ ˙ ˙ is guaranteed for t → ∞ if the vehicle dynamics is D D I convergence is is guaranteed for for t t → → ∞ ∞ if if the the vehicle vehicle dynamics dynamics is is , s = − ν, s ˙ = ν − ν. (29) ϒs The diagram o The Th isz guaranteed t conclusion → ∞ if the vehicle iserror r= Lν1we − β1˙ri |s (D + ϒ (29) kconvergence )ϒscondition Λ kleads Th ξs = (20) ˙ss∑ ˙i |r − sν. D ϒThis I z. and −T −1 −1 , −−T sν, s˙≤ − νν, −the = (29) sNote = ϒs scomparing = ϒs1νξ+ϒ ξν ξ −properties that (13) have r+ϒ rϒ= r+ usfor toadvantages thatdecoupled thedynamics tracking 1˙ξ ϒ ξ ,comparing s to:one Note nal −T 3.3. Some of conkk+ ϒ + kksν. z. z.Trelationships: (28) (28) re that (13) the relationships: 19) Tϒ−1 Tz.have T (28) D Dwe IIrelationships: system were assumed as follows: F ma +ϒ kthe s k Note that comparing (13) we have the not exactly known. Thus, we reformulate Lyapunov function candidate as folNs + z k z = s Ms + z k z. (30) not not exactly exactly known. known. s L = i=1 )s21) Case 1 set of D I Cas Ca I I not exactly known. ξ Ca ξr,comparing ˙function ˙ relationships: that (13) we Notewe comparing relationships: to: ,s = s= −Lyapunov sν,we = 2νshave ˙r = −ν ν, −2sν. ˙ = have ν˙ rcandidate − 2the ν. (29) (29) sthat =reformulate ϒs ϒs convergence is guaranteed for t which →=∞27, ifof vehicle dynamics isreli Thus, the as folrthe SomeThe properties and advantages decoupled con2ν ξNote ξ(13) troller proposed controller, isthe non-interacting properties and advant (37) T 267, 27 Nm. Thein values hus, we Thus, reformulate reformulate the Lyapunov the Lyapunov function function candidate candidate as folas3.3. folmax3.3. x,y,z Some lows: (20) leads to: we nal parameters :e: that nal nal Note Note that that comparing comparing (13) (13) we we have have the the relationships: relationships: ˙ ˙ na = ϒs , s = ν − ν, s ˙ = − ν. (29) s . (29) This condition leads us to conclusion that the tracking error ν r r ξ comparing (13) we have the relationships: ˙ us to ˙conclusion shas =This νϒs − ν, s˙= =νν˙ν˙function − ν. (29) = ϒsthe 21) not exactly known. lows: troller The proposed controller, which is AS800 non-interacting in have Its time derivative the form: the sense of the quasi-acceleration vector, gives somecontroller, useful r− r− ξ,, Lyapunov troller The proposed w condition leads that the tracking error :nts, for the airship (both airships ˙ ˙ ws: lows: , s = − ν, s ˙ = ν − ν. (29) s = s = ν ν, s ˙ ν. (29) ss = ϒs hus, we Thus, reformulate we reformulate the Lyapunov function candidate candidate as folas folr r z] r r 3.3. Some properties and advantages of decoupled con3.3. 3.3. Some Some properties properties and and advantages advantages of of decoupled decoupled concon1 reliesconon tracki relie rel convergence is guaranteed foradvantages t !  if theof vehicle dynamics 3.3. Some properties and decoupled 1 T ξ 1 T ˙˙ 1˙˙ T 1 Tξ rel 21) the sense of the quasi-acceleration vector, gives some useful ))ws: advantages. Consider the practical interest of the controller. T + Trk Ts˙s˙s= Tfor t → , , s s = = ν ν − − ν, ν, = ν ν − − ν. ν. (29) (29) s s = = ϒs ϒs 1 1 1 1 Thus, we reformulate the Lyapunov function candidate as folNs z z = Ms + z k z. (30) L = s ˙ r r r the sense of the quasi-acceleration convergence is guaranteed ∞ if the vehicle dynamics is ξ ξ The diagram of the control strategy is I I ˜ ˙ ˙ ˜ ˙ ˙ T T T T lows: ξ M s ˙ + ν k z = s (M ν − M ν) + ν k z. (31) L = s , s = ν − ν, s ˙ = ν − ν. (29) s = ϒs . Thus, (21) 1r1+ Thus, we reformulate Lyapunov function candidate asThe folξ 1 r1T+Tthezthe Thus, we reformulate function candidate folis not exactly known.controller, troller proposed controller, which is non-interacting in troller troller The The proposed proposed controller, which which non-interacting non-interacting inin I z1= r 2the troller The proposed controller, whichisis isof non-interacting incon- pr one TThus, TT T T 2 zcandidate T I as k1ILyapunov s= Ms k1function (30) s= L1ξreformulate = nts, 2kI zLyapunov 2Ns I z. we reformulate Lyapunov candidate as folwe the function as folξ zNs ξξ Ns 3.3. Some properties and advantages of decoupled + + = z s k Ms z + s z Ms k z. + z (30) k z. (30) L = s L s advantages. Consider the practical interest the controller. I I I lows: ξ )nts,(22) advantages. Consider the practical not exactly known. −T ξ ξ 2 2 2 2 Case 1 set of nominal parameters. Inνin td lows:lows: of the quasi-acceleration gives some useful the the sense sense of of the quasi-acceleration vector, vector, gives gives some some useful 121TCT = C(ν), = ϒ useful kD ϒ−1 in 1. From (27) wequasi-acceleration observevector, that the gain matrix KDnon-interacting 21the 2function 2121 TDT =candidate 21the 21Lyapunov Thus, Thus, we we21reformulate reformulate the Lyapunov Lyapunov function function candidate as asg(η) folfolthe sense ofthe the quasi-acceleration vector, gives some useful one Tlows: T the T form: T gsense Assuming for simplification D(ν), = ,s, lows: we reformulate the candidate as folIts time derivative has troller The proposed controller, which is Ns + z Ns k z + = z s k Ms z = + s z Ms k z. + z (30) k z. (30) L = s L = s ϒs )elements, = ntξIts I I I I ,one −T −1 ξ 3.3. Some properties and advantages of decoupled parameters set of= the is taken 1 equation: 1advantages.1. T ξ 1ξhas ξT form: fo time derivative the Consider the practical interest of the controller. advantages. advantages. Consider Consider the the practical practical interest interest ofof the the controller. controller. ϒ airship kconϒ inFrom (27) we observe that the matrix K includes also dynamics ofnal the system. Consequently, lows: lows: 11 T21form: 1=1below advantages. Consider the practical interest of the controller. D Dthe (29) receive the 2Tgiven 2using 21we s time derivative Its and time derivative has the has the 1.gain From (27) we observe that the gai T22sT Ms T2 zT kI z. 1 (30) L s2form: the sense of theconquasi-acceleration vector, gives some useful 11+ I z 1properties ξ 1+ T 1=sTT Ns TξNs TkzI zk= T1 ξ . (30) troller. The proposed controller, which is non-interacting in L˙ = + z s Ms + z k z. (30) = L (19) one T T T T ee,s:hip T T T 3.3. Some and advantages of decoupled for all I relies on tracking the velocity trajectory foll ˜ ˙ ˙ ˜ + νform: k+ z 12=z s2form: (M ν+ −s M ν) += νzT 2kskIIz.z. (31) s= ξ2Ns 2zMs alsoτ dynamics of the system. Consequently, the inr1 I= zT+ Ms + z kincludes (30) kT Ns z= (30) L TM Tξhas signal is strictly related not only toalso also −1 I+ I z.put −T −T −1 −1 ξν 1212ν 1212ss˙˙the 2− includes dynamics of the of syst time derivative Its time derivative set Ts −T ˜TzL ˙TM ˙kν− T= Tg2T= T− g. M k+ zνs˜12the M ν) kk− (31) L = s= ˙TL esThereT = ϒ−T kkinematics From (27) observe that the gain matrix K advantages. Consider thethe practical interest of the controller. s) D(ν (32) M =L τ˙˙has −Cν −ν Dν − −C(ν = ϒϒϒ kbut kDkThe ϒϒ 1.1. From From (27) (27) we observe observe that that the gain gain matrix matrix KDK ˙= r1 I z. Itroller 1Ns 1TsIkξzkντI˙TII(M 1T− 1˙sνL the sense of the quasi-acceleration gives some useful Dvector, rs˙2 DϒD D 2TrT1. 2(30) = ϒ−1 1.we From (27)τwe we observe that the gain matrix K 2(M ˜sTξξTξ+ ˙Ms ˜+ ˙˜zz. ˜s) s + ν k M s ˙ = + (M k = − s ν) ν + M k ν) + ν (31) k z. (31) L = D= D The proposed controller, which is non-interacting in TM T− Ns + z z z z = = s Ms + z k z. z. (30) = s r r I I I I I ξ ξ put signal is strictly related not only to kinematics but also to the vehicle dynamics. This means that the matrix k is Its time derivative has the form: Ns + z k z = s Ms + z k z. (30) L = s 22) D I I put signal τ is strictly related not o ξ follows: = [sin(π/20 · t) + 2, 0, ν follo fol ξ 2 2 2 2 2 2 2 2 Its time derivative has the form: Assuming for = C(ν), = T simplification TTC2form: T ˙2D˙ =˜D(ν), T ˙ T g(η) also advantages. Consider theofof practical interest ofdthe the incontroller. dynamics of theuseful system. Consequently, includes includes also also dynamics dynamics the system. Consequently, Consequently, the the inin˙ Its ˙˙ + includes also dynamics ofthe thesystem. system. Consequently, the in−T −1fol ˙= ˜(31) the quasi-acceleration vector, gives some the form: time derivative the form: M sderivative ν˜sT2has kMIderivative zs˙has = +the s˜Tthe (M ksense zνhas − sof M (M ν) + −νD(ν), M kν) +ggνincludes kg(η) (31) =2stime LIts =time Cs 0rC(ν), [22] we Hence, recalling that sν T ItsL for simplification C DνD(ν), = = robtain: I z.= I z.= g(η) I= = ϒ k ϒ 1. From (27) we observe that the gain matrix K to the vehicle dynamics. This means that the matrix k is –T –1(e.g chosen according to dynamics of the controlled plant D D D ssuming Assuming for simplification for simplification C = C(ν), C = D C(ν), = D = g D(ν), g(η) g ϒAssuming k z] 22) T T T T to the vehicle dynamics. This me I using 1. strictly From (27) we observe that matrix Kalso kbut ϒalso ˙we and (29) the given below equation: τput is related not related only to the kinematics but put signal signal τcontroller. ττisis strictly strictly related not notgain only only toto kinematics also D = ϒ D0.1 Its Its time time derivative derivative has the form: T receive T s Tν ˙ ν˜T+ ˜ z.ksignal 0, 0,but · cos(pi/1 s˙the +Tadvantages. ν˜form: k= (M ν˙equation: −ν) Mthe ν) (31) L = s has put signal isalso strictly related only tokinematics kinematics but also )ime = rM I z. (31) I z s= Consider practical interest of the k derivative the form: ˙ν˙r below T(M M sM ˙M + − + kput L =receive T˙ν using (29) we receive the below T Tsgiven T includes dynamics ofnot the system. Consequently, the in- of ˙(29) IkzTzIgiven chosen according to dynamics ofchosen the Consequently, controlled plant (e.g 22) ˙+ ˙˙has for a heavy vehicle the control coefficients can be different ˜kkTIs= ˙T˙D(M s˙receive ν zC(ν), = (M −D(ν), Msν) + ν˜ν˜˙= k− zIIz. L = sTwe ˜= ˙g = ˜ Tto ˜ν˜T= d and using and (29) using we the given the below equation: equation: according to dynamics M s ˙ + ν k z = M ν) + ν k z. (31) L = ssfor M s ˙ + ν = s (M ν M + k (31) L for . (31) includes also dynamics of the system. the inAssuming simplification simplification C C D C(ν), = = g D(ν), g(η) g(η) r− Ito ))ssuming r I I r the vehicle dynamics. This means that the matrix k is to the the vehicle vehicle dynamics. dynamics. This This means means that that the the matrix matrix k k is is D D D to the vehicle dynamics. This means that the matrix k is T T T T T T T T (22) D )all =M ν˙˙ = ˙˙ = s) − D(ν −ν+ − g.z.(32) τ T−Cν − Dν − g˙receive −C(ν −1 related ˜ν˜s= ˙ν) ˜ν˜s) TM Tkkτ Tν) put signal τ kstrictly isvehicle). strictly related not only tocan kinematics but alsonon TIIz. M swe ˙s˙− + += z−C(ν = = ss− (M (M − − + (31) (31) L L ss= forthan aput heavy vehicle the control coefficients begains different r− r+ k The set selected for the Assuming for C = C(ν), DI z. = D(ν), ggain = g(η) a light ifonly the system parameters are rwe signal is not to kinematics but also = ϒτ−T 1. (27) observe that the matrix Kfor for a of heavy vehicle the control co ˜simplification ˙+ ˙ν˙ν˙s) ˜M ˜kk− nd using and (29) using we receive (29) the given the given equation: below equation: M sτ˙= + ν z= (M M ν) νM k˙= L = skTsimplification +Cν Dν + gr+ − Ds − ν(31) kgz. (33) Daccording Dtoϒ Assuming for C = D D(ν), g(η) − − D(ν −− s) g. (32) M ν˙τ = τsν˙ −Cν − Dν gνν τIIgzFrom r= rν rC(ν), I= rbelow I(M )) ν˙=(23) chosen according to dynamics of the Even controlled plant (e.g chosen chosen according to dynamics dynamics ofof the the controlled controlled plant plant (e.g (e.g rC(ν), rτ) chosen according to dynamics of the controlled plant (e.g − s) − D(ν − s) − − s) D(ν g. − (32) s) − g. (32) M = −Cν M = − Dν −Cν − g − Dν τ −C(ν − = τ −C(ν Assuming for simplification C = C(ν), D = D(ν), g = g(η) Assuming for simplification C = D = D(ν), g = g(η) r r r r = ereto the vehicle dynamics. This means that the matrix k is all light vehicle). Even ifmeans the system parameters are for simplification C = C(ν), D = D(ν), g = g(η) and than tofor thea vehicle dynamics. This thataϒ,the matrix kD iserror follows: and (29) we receive the given below equation: not exactly known, thanks to the matrix the velocity T Cs D ifΛ C(ν)(ϒs )Assuming =using includes also dynamics ofggthe system. Consequently, the in= than for light vehicle). Even T and (29) we receive given below equation: =the 0we [22] we obtain: Hence, recalling that s= Fi ξ using a− heavy vehicle thevehicle control coefficients can be different for for aaaheavy heavy vehicle the the control control coefficients coefficients can can be be different Assuming Assuming for simplification simplification C Crgiven = =−C(ν C(ν), C(ν), D Drs) = =− D(ν), D(ν), = = g(η) g(η) TDν for heavy vehicle thedynamics control coefficients can bedifferent different and receive the given below equation: and using we receive below equation: s) − D(ν − − s) D(ν − g.r for − (32) s) g. (32) ν˙Hence, = τfor −Cν M ν˙simplification =−(29) τfor Dν −Cν −Tusing g− τ(29) −C(ν − gthe = τ− rereall uming C = C(ν), D = D(ν), g = g(η) Cs = 0 [22] we obtain: recalling that s r T using (29) we receive the given below equation: chosen according to of the controlled plant (e.g chosen according to dynamics of the controlled plant (e.g =M not exactly known, thanks to the matrix ϒ, the velocity error decreases quickly. l Cs = 0 [22] Cs = we 0 obtain: [22] we obtain: ence, recalling Hence, recalling that s that s put signal τ is strictly related not only to kinematics but also 3 not exactly known, thanks to the m Fig du )s = 0 for all lerethan for a light vehicle). Even if the system parameters are than than for for aaalight light vehicle). vehicle). Even Even ififif the the system system parameters are are and and using (29) (29) we we receive receive the the given given below below equation: equation: kD can =parameters diag{95, 95, 95, 60, 6T − s) − −− s)g. −(32) g. (32) M τT −Cν − − gsτTbelow =−C(ν τ −C(ν thanfor for lightvehicle vehicle). Even thecoefficients system parameters are using (29) receive given desired ˙using ˜ τTTDν ˙−r equation: ˙ dynamics. a heavy the control coefficients becan different s) − D(ν s) M =˙˙ν˙we τ=−Cν −Cν Dν = M ˙ν− + νthe k− = (M νvehicle − Mrs) ν) + ν˜ TTD(ν kr I− zrs) L = srecalling TDν T for a quickly. heavy vehicle the control beThe different r− Iz decreases T 2.not The inertia matrix Nthe gives information about the ˙sT+ = −Cν − Dν = τν) −C(ν D(ν s) −exactly g.diagonal (32) Mss˜− to This means that the matrix kthanks is − D(ν −exactly g. (32) M ν˙νL = − g0g˜Ts= τTthe −C(ν decreases quickly. Cs = [22] Cs = we0ν [22] we obtain: recalling that that ˜ ˙ ˙ ˜ dur Tτ T= Tobtain: T Tnot r− Dthanks rg˙− rk− M ˙ ν k z s (M − M + ν z = s Nb [22].L There-l-ence, known, thanks to the matrix ϒ, velocity error ˙Hence, ˙ not exactly known, known, to to the the matrix matrix ϒ, ϒ, the the velocity velocity error error I r I not exactly known, thanks to the matrix ϒ, the velocity error ˙ ˙ ˜ ˙ ˜ s˙ + kMI zs˙that = + ν s s= kCs = s0M (M ν) νs) + − ν−M kν) νs) k− == Lrecalling =ν− s˙ Dν =sττM than for a light vehicle). Even if theifsystem parameters are3 T(M T T(32) Fig. a) 60, and kIsystem =(without diag{85, 85, 85, 60 Iτzν r− rwe I z r+ I zg. Fig. Fig ˙ ˙ Fig than for a light vehicle). Even the parameters are − − s) − D(ν D(ν − − s) − g. (32) (32) M M ν ν −Cν −Cν − Dν − − g g = τ −C(ν −C(ν 2. The diagonal inertia matrix N gives information about the T = [22] obtain: Hence, ˜ inertia related to each quasi-acceleration dynamr r r = s (M ν +Cν + Dν + g − Ds − τ) + k z. (33) ν -ν˙ = τHence, chosen according to dynamics of the controlled plant (e.g 2. The diagonal inertia matrix N giv r r r I T T − s) − D(ν − s) − g. (32) −Cν − Dν − g = τ −C(ν Nex Cs = 0 [22] we obtain: recalling that s T T T T T T T se r r T T decreases quickly. decreases decreases quickly. quickly. ˙ ˙ ˜+ T decreases quickly. s sν˙˙Hence, (M +Cν − − τ) + z. not exactly known, thanks to the matrix ϒ, theϒ, velocity error the ˙Ds Cs = [22] weνk˜ Iobtain: recalling sM = 0ν [22] we recalling s+ M + kM sν˙+ νs˜+r(M kDν −sgDs (M ν) νg˙τ) + −+ ν0˜Ds M L ==ssT= L =sν˜νs˙Trthat rCs during airs duri du Irz Irzthat r+ robtain: I˙zkν ˙= = 23) Hence, (M (M +Cν + + g˙ra= − Dν − − ν˜kν) − τ) (33) ν˜kTI zk(33) (33) du not exactly thanks toinertia thequasi-velocity matrix the velocity error inertia related toknown, each quasi-acceleration (without dynamΛ = diag{0.35, 0.35, 0.35, r +Cν rTTDν r+ I z.+ I z.coefficients ical couplings). Moreover, each ξ is indei T T T T T for heavy vehicle the control can be different related to each quasi-acce sen ˙ af 2. inertia matrix N gives information about the about Cs = [22] wewe Hence, Hence, recalling that s˜s+ TCs T[22] Tν 2.2. The The diagonal diagonal inertia inertia matrix matrix NN gives information information about the the ˙T ν˜+ ˜ The Hence, obtain: decreases quickly. s˙that νk˜sITzCs = 0 k= z00s= s[18] (M ν˙obtain: − ν) kITz diagonal L sTthat The diagonal inertia matrix Ngives gives information about the Trecalling Iwe robtain: = [22] obtain: nce, that sTTM ˙we ˙Mν˜+ Next, in Fig. 4 Nex Ne s(M ˙M + = (M − M ν) k+ z(33) L =νrecalling s= Tr˙ν T T2. ˜ T krecalling ˙s= ˙0rT=Dν Ne r− IzM decreases quickly. ν23) sCs +Cν + gT+− Dν Ds + gτ) −s+ Ds −kν˜˙τ) ν˜ ν) kI+ z. (33) ˙ν+ ical couplings). Moreover, each quasi-velocity ξ is inde˙˙(M pendent from other quasi-velocities and allows one to deI z = sL rT+Cν rν rthan rss˙+ IrTz.− i ˜ ˙ ˜ ˜ ˙ M ν k z = (M ν k z L s M s ˙ + k z = (M ν − M ν) + k = for a light vehicle). Even if the system parameters are ical couplings). Moreover, each 3 I I afte I r I an 2. to Theeach diagonal inertia matrix N(without gives information about the velociti T inertia related quasi-acceleration dynaminertia inertia related related toto each each quasi-acceleration quasi-acceleration (without (without dynamdynamThe desired linear and angular inertia related to each quasi-acceleration (without dynamTTTsT (M˜ν TrTT+Cνr + TT Dν˙r˙ + g − TT + T ˙ ˜ ˙ ˙ 3 = 23) Ds − τ) ν k z. (33) sented. The er sent sen ˜ ˙ ˙ ˜ ˜ I T T sen The diagonal inertia matrix N gives information M Mks˙s˙zν + +r ν+Cν νs kk(M zz = = sDν s− (M (M − − Mν) ν) + + k˜ zzktoI z. L νν+kν pendent from other quasi-velocities and allows one quasi-velociti toξabout 3the termine the kinetic energy reduced the variable , i.e.theang = (M + +ν) − τ) (33) T32. ider+ ˙ννggrr− ˜Ds known, thanks matrix the velocity error other M s= ˙= +sssνT˜ (M ν˙ rDν M + νM k− s L related to each quasi-acceleration (without Iν I rzτ) er ˙˙r= (33) = sIITrrnot (M ν˙exactly Dν + g− − τ)(33) + ν˜ical kIϒ, z.inertia (33) )) L˙ = (23) +Cν + Ds + ν˜ITITTDs kI z. ical couplings). Moreover, each quasi-velocity ξby isfrom ical couplings). couplings). Moreover, Moreover, each each quasi-velocity quasi-velocity ξξiξdynamis indeindeFig. 31pendent a) respectively. Note afte that r r+Cν r+ i b), iiis ical couplings). Moreover, each quasi-velocity is inde1 each 1 related 6and T Mν T Nξ 2indeafter about 30 aft T T afts inertia to quasi-acceleration (without dynamtermine the kinetic energy reduced by the variable ξ , i.e. K(ξ ) = ν = ξ = N ξ . These indepeni 3 3 ∑ ˙ ˜ ˜ ˙ T T i ical couplings). Moreover, each quasi-velocity ξ is inde= = s s ) (M (M ν ν +Cν +Cν + + Dν Dν + + g g − − Ds Ds − − τ) τ) + + ν ν k k z. z. (33) (33) decreases quickly. termine the kinetic energy reduce i=1 i erro r r r r r r I I i 2 2 2 af pendent from otherfrom quasi-velocities and allows one dependent pendent from other other quasi-velocities quasi-velocities and allows allows one one toaccording todede= s (M ν˙ r +Cνr + Dνr + g − Ds − τ) + ν˜ kI z. (33) during the airship motion to pendent from other one de1couplings). 6 and T Nξ 21to angular velocit ang 1 ξto 1ang 3information T Mν = an Moreover, each quasi-velocity indeν T Mν = 12the ξquasi-velocities = 12 ∑ Nand .allows These indepenK(ξ )ical =quasi-velocities dent are used in the decoupled iT is i ξproposed 3 2. The diagonal inertia matrix N gives about i=1 i ν ξ Nξ = K(ξ ) = 2 ∑ afte ar termine the kinetic the energy reduced byNext, the variable i.e. linear termine the kinetic energy energy reduced reduced by by the variable ξ2ξiξ,ii,velocity i.e. 2ξ 2err in Fig. 4the a) each er i ,variable 3 kinetic 3 termine termine the kinetic energy reduced byproposed the variable ,i.e. i.e. error reduction erro err pendent other quasi-velocities and allows one to dequasi-velocities are used in the decoupled controller. 111 TTTTfrom 133 dent 6 1 1 1 1 1 6 6 T 2 T T 2 2 inertia related to each quasi-acceleration (without dynamdent quasi-velocities are used in 1These 6 NNξ TNξ aris Mν) ))= == Nξ Nsented. K(ξ Mν = Nξ = ξiξi.i2..These These indepenindepenK(ξ ∑ The decreases, as it was i ξ= i indepenierror K(ξ =222ν2ξνν Mν Mν= =2 2ξ12∑ξξi=1 Nξ = N These indepen3 ) = 2 νK(ξ ∑ i 2. 22∑ i=1 i=1 iAc.: after about 20 afte aft i=1 462 Bull. Pol. 65(4) 2017 aft termine the kinetic energy reduced by Tech. the variable ξdecontroller. 3. Some particular cases of the presented controller can be i , i.e. ical couplings). Moreover, each quasi-velocity ξ is indecontroller. into i an dent quasi-velocities are1 used inare the proposed decoupled dent dent quasi-velocities quasi-velocities are used in in the the proposed decoupled decoupled after about 30 second the error is close t dent quasi-velocities are used in proposed decoupled 1 used 1the 6proposed T T 2 arises from the aris ari ari Unauthenticated ν Mν = ξ Nξ = N ξ . These indepenK(ξ ) = 3. Some particular cases of the presented controller can be de∑ duced. We can point at two cases: i i i=1 pendent from other quasi-velocities and allows one to de3. Some particular cases of the prese 2 2 2 ang cl controller. controller. controller. angular velocity errors for angular vari controller. Date | 8/25/17 12:47 PM into account in into int int dent quasi-velocities are used in the proposed decoupled duced. We can point atDownload two cases: termine the kinetic energy reduced by the variable ξcases duced. Webe can point at two cases: clos i , i.e. (a) For a symmetric vehicle in thecan xy-plane we get ydeab 3. Some particular cases of thecases presented controller de3. 3. Some Some particular particular of of the the presented presented controller controller can can be be deg = error reduction is not so fast as the lin 3. Some particular cases of the presented controller can be deangular velocit ang ang 2 . These indepenan controller. K(ξ ) = 21 ν T Mν = 12 ξ T Nξ = 12 We N ∑6i=1duced. (a) aascan symmetric vehicle inabout the we yg = areable i ξpoint 0;We a results controller is xy-plane simplified and reduced. be i For duced. can at two cases: duced. We can point point atthe attwo two cases: cases: after 20 second allget signals sig

Desired angular velocity [rad/s]

Desired linear velocity [m/s]

Consequently, impact of dynamic couplings effect ˜ = Dˆ − (thesignals signals obtained obtained using usingthe theexpression expression ss== ϒsξC oting now M˜˜ = Mˆˆ − M, C˜˜ϒs = −C, D D, g˜ = gfrom ˆfrom − g, ξˆ (the isis reduced, reduced,too. too.the Consequently, the impact of dynamic couplings effect ˆC ˜ = Dˆ −obtained (the D signals using the we expression s= ϒs oting now M =get: M − M, C= D, g˜ = gˆfrom − g, ξˆ −C, is reduced, too. Consequently, the impact of dynamic couplings effect ˆ ontroller) ontroller) we get: signals obtained from using the expression s = ϒs is reduced, too. ˆ ξξ (the (b) (b) The The matrix matrix M M is is a a diagonal diagonalone. one. ItItsuch suchcase casethe thesimsimontroller) we get: (the signals obtained from using the expression s = ϒs is reduced, too. TT ˜we TT TT TT get: (b) The matrix M is a diagonal one. It such case the sim˙ontroller) ˙==−s ˜ ˜ ˜ ˜ ˜ ˙ ˙ ˜ ˜ ( ( M M ν ν + + Cν Cν + + Dν Dν + + g) ˜ g) ˜ − − ( ( ν ν + + z z Λ Λ )k )k z z L −s plified form formis isas as controller) we (b) plified The matrix M is follows: afollows: diagonal one. It such case the simT T T II ˙ = −sTT (M ˜ r rr + Dν ˜ ν˙r rrget: ˜ r rr + g) ˜ + Cν ˜ − ( ν + z Λ )k z L plified form is as (b)controller The matrix M is afollows: diagonal one. It such case the simTT (M −T −1 −1Dν ˜Dϒˆϒ ˜ϒ ˜ ++ for a class of vehicles νˆ˙ϒˆr−T +kCν + ˜kkI Iz−z (ν˜ TT + Non-adaptive zTT ΛTT )kII z velocity tracking L˙˙ = −s ˜ν˜TTg) −s −s (D + kD )s plified form is as follows: rˆ + rν T + ˜ T(D −T −1)s T ˜ ˜ ˙ ˙ τ τ = = M M ν ν +C(ν)ν +C(ν)ν + +D(ν)ν D(ν)νr r++g(η) g(η)++kkDDss++kkI Izz (38) (38) ˙ ˜ ˆ ˆ ( M ν + Cν + Dν + g) ˜ − ( ν + z Λ )k z L =−s −s r r r r plified form is as follows: ˜ r r r I )s + ν T kI z TT T (D˜˜+ ϒ −T˜˜kD ϒ −T ˆˆ−1 −1 ˆ+ τ = M ν˙ r +C(ν)νr + D(ν)νr + g(η) + kD s + kI z (38) ˆϒˆ−T ˜Dν ˜)sr r+ ˜˜g) kDr r+ k− ϒˆˆ+−1 ==−s −sTTTT(D ((M M+ ν˙ν˙rϒ Cν Cν Dν ++νg) ˜ − (D + + ϒ k k )s )s ϒ ϒ I zssT(D −T −1 T rˆ+ D D ˙ τ = M νwe +== D(ν)ν + g(η) + kD s + kI z (38) r +C(ν)ν rϒ ˆ −T ˆ −1 ˜kDrϒ+ Dν ˜)s ++ν˜g) (DM˜+ν˙ϒ I z sT (D + ϒ because because obtain obtainϒ II(the (therridentity identity matrix). matrix). =−s −s ˜ k− )s r + Cν τ = M ν˙ rwe +C(ν)ν + D(ν)ν + g(η) + kD s + kI z (38) −T kD ϒ T Dν T TTT( rϒ ˆ˜g) ˆ −1 ˜TTϒˆϒ ˜zν ˆT+ ˜Cν ˜− ˜˜ν˙ν˙rrr++Cν ˜Dν ˜ r r+ ˙ = T−s ( M + Cν g) s (D ϒ k )s ϒ because we obtain = I (the identity matrix). −z −z Λ Λ k k z = = −s −s ( ( M M + + Dν + + g) ˜ r r D T T −T −1 I I r r Interpreted ξ˜ξT ˆ + ˆ ˆ T T T(M ˜ ˜ ˙ ˜ ˜ ˜ = −s ν + Cν Dν + g) ˜ − s (D + ϒ k )s ϒ because we obtain ϒ = I (the identity ˙rr + Cν r −sT r T (M ν −zT ΛT kI z = Dν ˜ D ϒ Airship_Model simout matrix). r + g) ˆ ˆTˆ−1 ˜ TrrT+ Fcn because we obtain ϒ = I (the identity matrix). TTT kI z =ˆˆ−T −TξξT ϒ TDν T˜ r +MATLAB −zTTTϒˆϒ (−1 M˜˜))ϒs νˆ˙ϒs Cν + g) ˜ Λ −s r +− ˆ ˆ −s −s (D (D + + ϒ ϒ k k − z z Λ Λ k k z. z. (36) (36) ϒ ϒ ˆ ˜ 4. 4. Simulation Simulation results results ˜ D D I I ˙ ξ ξ −z M ν + Cν ˜ ξξT Λ T kI z = ˆ−s −Tξ ϒ ˆ(−1 T+ Dν T r + g) r r ˆ ˆ −sTξ ϒˆ T (D + ϒˆ −T kD ϒˆ −1 )ϒs z Λ kI z. (36) 4. Simulation results Clock ˆ ξξ − Generator S-Function To Workspace −sξT ϒˆ T (D + ϒˆ −T kD ϒˆ −1 )ϒs − zTT ΛTT kTrajectory (36) 4.1. 4. Simulation results I z. constants 4.1. Indoor Indoor airship airship model model In Inthis thissection section we wepresent presentsome somerereˆ ed d −s on onξ[52] [52] we we can can find find the the strictly strictly positive positive constants β β (36) ϒ (D + ϒ kD ϒ )ϒsξ − z Λ kI z. ii 4. Simulation results 4.1. Indoor airship model In this section we present some reed on [52] we can find the strictly positive constants β sults regarding regarding the theuse use of ofthe theproposed proposed controller controller for forthe the model model 4.1. Indoor airship model In this section we present some rere e i ion ==1,1, . . . ., ,6we 6inincan order order totoensure ensure convergence convergence of of the thetracktracked [52] find the strictly positive constants βii in sults Fig. 2. Diagram of the control strategy in MATLAB/Simulink environment for the tested airship sults regarding the use of the proposed controller for the model 4.1. Indoor airship model In this section we present some rere i = 1, . . . , 6 in order to ensure convergence of the trackT T ed on [52] we can find the strictly positive constants β of of airship airship AS500 AS500 (assuming (assuming indoor indoor test test with with low low velocity). velocity). The The ˆ ˆ ˜ ˜ ˜ ˜ i regarding the use of the proposed controller for the model Mof ν˙ν˙r rthe ++Cν Cν + sults error rror toto1, zero. zero. Therefore, choosing ββi i≥≥|[|[ϒϒT((M r r+ re i = . . . , 6Therefore, in order tochoosing ensure convergence trackof airship AS500 (assuming indoor test with low velocity). The ˆ ˜ ˜ sults regarding the usein of the proposed controller for the model ≥ Mof ν˙ the + Cν + simulations error to zero. βΛ TT |[ϒ re i˜ = 1,we . .receive .receive , 6Therefore, in order tochoosing ensure convergence trackwere were done done inMATLAB/Simulink MATLAB/Simulink environment environment for for T( of airship AS500 (assuming indoor test with low velocity). The ˆas ˜ rror + +g)] ˜g)] (assuming (assuming kkDDand and as symmetric or i |to i |we (symmetric M˜˜ ν˙ rr + Cν + simulations error zero. Therefore, choosing βii Λ≥Tkk|[I Iϒ T simulations were done in MATLAB/Simulink environment for of airship AS500 (assuming indoor test with low velocity). The ˆ ˜ + g)] ˜ | we receive (assuming k and Λ k as symmetric or ˙ ≥ |[ ϒ ( M ν + Cν + error to zero. Therefore, choosing β i D I 6 6 DOF DOF model model with with six six signal signal inputs. inputs. The The blimp blimp parameters parameters i r r T simulations were done in MATLAB/Simulink environment for m: onal nalg)] matrices): + ˜ matrices): i | we receive (assuming kD and ΛT kI as symmetric or 6 DOF model with six signal inputs. The blimp parameters simulations were done in MATLAB/Simulink environment for onal + g)] ˜ matrices): (assuming k and Λ k as symmetric or coming coming from from the the report report [9], [9], were were also also exploited exploited in in reference reference i | we receive D I pendent from other quasi-velocities and allows one to6deforcessix andsignal torquesinputs. applied by theblimp controlparameters system were DOF maximal model with The onal matrices): 6(34) 6 coming from the report [9], were also exploited in reference 6, i.e. DOF model with six signal inputs. The blimp parameters onal matrices): The The maximal maximal forces forces and and torques torques applied by byin the control control −T −1 kinetic energy reduced ξ[1]. assumed as report follows: F Tthe 6 ββ|s|stermine coming from the [9], were alsoapplied exploited reference ˆ−T ˆϒs ˆ −−by i[1]. max x, y, z = 107, 13, 40 N, max x, y, z = 27, ≤≤−−∑ −ssTξTξT1ϒˆthe (D ++ϒˆϒ kk ϒˆϒˆ−1 ))ϒs zzTTthe ΛΛTTkvariable kz.z. ϒˆTTTT(D ∑ ξi |i |− [1]. The maximal forces and torques applied by the control 6 i i ξK(ξ) =  T DD ˆ −1 6ˆ ξξ 2 T T I I −T 1 1 coming from the report [9], were also exploited in reference ˆ ˆ ν Mν =  ξ Nξ =  ∑ N ξ . These independent 267, 27 Nm. The values were taken from [64] for the airship ≤ − β |s | − s (D + ϒ k ) ϒs − z Λ k z. ϒ ϒ system system were were assumed assumed as as follows: follows: F F = = 107, 107, 13, 13, 40 40 N, N, i i i=1 i D I max max x,y,z x,y,z [1]. The maximal forces and torques applied by the control ∑ ξ i ξ re 6 T i=1 2ˆ −T k ϒ 2 )ϒs ξ2 ˆ T (D + ϒ ˆ ξ − zTT ΛTT kI z. ˆ −1 ≤ known. −i=1 βi |squasi-velocities system were assumed as follows: Fmax =taken 107, 13, 40[7] N, D in ∑ ξ i | − sξT ϒ [1]. The maximal forces and torques applied by the control T −T −1 x,y,z are used the proposed decoupled conAS800 (both airships have similar construction). The diagram ˆ ˆ ˆ ˆ (37) (37) T T = = 27, 27, 267, 267, 27 27 Nm. Nm. The The values values were were taken from from [7] ≤ − i=1 β |s | − s (D + ϒ k ) ϒs − z Λ k z. ϒ ϒ max maxx,y,z x,y,zwere assumed as follows: Fmax x,y,z = 107, 13, 40 N, system D I ∑ i troller. ξi ξ ξ i=1 (37) T = 27, 267, 27 Nm. The values were taken from [7] system were assumed as follows: Fmax = construction). 107, 13, 40 N, of the control strategy is presented insimilar Fig. 2. x,y,z x,y,z i=1 leads condition condition leads us us toto conclusion conclusion that that the the tracking tracking error error for for the the airship airship AS800 AS800 (both (both airships have have similar construction). (37) Tmax = 27, 267, 27 Nm.airships The values were taken from [7] max x,y,z Fig. us 2. Diagram of the control strategy in inerror MATLAB/Simulink en-AS800 condition leads to conclusion that the tracking for the airship (both airships have similar construction). 3. Some particular cases of the presented controller can be de(37) T = 27, 267, 27 Nm. The values were taken from max x,y,z ergence ergence isisguaranteed guaranteed for for t t → → ∞ ∞ if if the the vehicle vehicle dynamics dynamics is is The The diagram diagram of of the the control control strategy strategy is is presented presented in in Fig. Fig. 2.2. [7] condition leads us to conclusion that the tracking error for the airship (both airships have similar construction). vironment for the tested airship duced. Wetocan point at iftwo cases: Case 1of –AS800 set ofcontrol nominal parameters. In this example the nomergence is guaranteed for t → ∞ the vehicle dynamics is The diagram the strategy is presented in Fig. 2.nomicondition leads us conclusion that the tracking error m of the control strategy in in MATLAB/Simulink enfor the airship AS800 (both airships have similar construction). (35) exactly xactly known. known. Case 11--set setofofof nominal nominal parameters. parameters. In this thisexample example the thenomiI z. ergence is guaranteed for t → ∞vehicle if the in vehicle dynamics is y Case The diagram the control strategy isIn presented in Fig. 2. (a)  F or a symmetric the xy-plane we get  = 0; inal parameters set of the airship is taken into account. The task g exactly known. he tested airship Case 1 set of nominal parameters. In this example the nomiergence is guaranteed for t → ∞ if the vehicle dynamics is nal Theparameters diagram ofset the strategy is presented in Fig. 2.task nal parameters set of ofcontrol the thethe airship airship isistaken taken into intoexample account. account. The The task exactly known. Case 1 relies - set of nominal parameters. In this the nomias a results the controller is simplified and reduced. on tracking velocity trajectory described by: g ˜ = g ˆ − g, nal parameters setthe of velocity the airship is taken into account. The task Consequently, the of dynamic effect exactly known. Case 1on-tracking set of nominal parameters. In this example the nomiSome Some properties properties and and advantages advantages of ofofimpact decoupled decoupled concon- couplings relies relies on tracking velocity trajectory trajectory described by: by: nal parameters setthe of the airship is takendescribed into account. The task Consequently, the impact dynamic couplings effect ined from Some properties and advantages of decoupled conrelies on tracking the velocity trajectory described by: is reduced, too. nal parameters set of the airship is taken into account. The task equently, the impact of dynamic couplings effect er rSome The Theproperties proposed proposed controller, controller, which isisof non-interacting non-interacting inin relies on tracking the velocity trajectory described by: is reduced, too.which and advantages decoupled con= = [sin(π/20 [sin(π/20 · · t) t) + + 2, 2, 0, 0, sin(π/25 sin(π/25 · · t), t), ν ν er The proposed controller, which is non-interacting in d d Some and decoupled conrelies the velocity trajectory described by:(39) uced, too. sense ense of ofproperties the the(b)  quasi-acceleration quasi-acceleration vector, vector, gives some useful useful (b) Theadvantages matrix M isisof agives diagonal Itthe case on the simThe matrix M is which a diagonal one. Itsome suchone. case simpli tracking er The proposed controller, non-interacting insuch · t), νd = [sin(π/20 · t) + 2, 0, sin(π/25 sense of the quasi-acceleration vector, gives some useful er The proposed controller, which is non-interacting in = [sin(π/20 + 2, 0, sin(π/25 ν 0,0, 0,0, 0.1 0.1··t) ·cos(pi/15 cos(pi/15 · ·t)] t)]TTT. . · t), (39) (39) d k z fied form is as follows: ntages. ntages. Consider Consider the the practical practical interest interest of of the the controller. controller. plified form is as follows: I sense of the quasi-acceleration vector, gives some useful + 2, 0, sin(π/25 νd = [sin(π/20 matrix MConsider is a quasi-acceleration diagonal one. It such case the 0, 0, 0.1· t) · cos(pi/15 · t)]T . · t), (39) ntages. the practical interest ofgives thesimcontroller. sense of the vector, some useful 0,gains 0, 0.1 · cos(pi/15 · t)]Tcontroller .controllerisis(39) ntages. Consider the practical interest of the controller. −T −T −1 −1 The The set set of of selected selected gains for for the the nonlinear nonlinear as as d form is as follows: ==controller. ϒϒ−Trk+ kDDϒg(η) ϒ−1 + kD s + kI z (38) om om (27) (27)we weobserve observe that the the matrix matrix ˙ r interest 0, 0.1 · cos(pi/15 · t)] . controller(39) τ= Mgain νgain +C(ν)ν + DDD(ν)ν ntages. Consider thethat practical ofrKK the The set of 0, selected gains for nonlinear the nonlinear isasas The set of selected gains for the controller is −1 = ϒ k ϒ om (27) we observe that the gain matrix K (38) D D ˆ follows: follows: −1 )s ϒ The set of selected gains for the nonlinear controller is as cludes ludes also also dynamics of ofthe the system. system. the ininDν ϒ−T kthe om (27) wedynamics observe that the gain K ˙ r +C(ν)ν Dϒ M g(η) + matrix kDConsequently, sConsequently, + kϒ (38) −T identity −1 matrix). follows: r + D(ν)ν r+ I zD The follows: set of selected gains for the nonlinear controller is as because we obtain == I (the cludes also dynamics of the system. Consequently, the in= ϒ k ϒ om (27) we observe that the gain matrix K D D follows: tsignal signalalso ττisisstrictly strictlyrelated related not only onlytoto kinematics kinematicsbut but also also cludes dynamics of thenot system. Consequently, the in- follows: kkDD==diag{95, diag{95,95, 95,95, 95,60, 60,60, 60,60}, 60}, (40) (40) tthe signal τ isdynamics strictly related not only to kinematics but also cludes also ofwe theobtain system. Consequently, the inse we obtain ϒ= I (the identity matrix). because ϒ = I (the identity matrix). kD = diag{95, 95, 95, 60, 60, 60},,(40) (40) the vehicle vehicle dynamics. This This means means that that the the matrix matrix kkDD isis t signal τ isdynamics. strictly related not only to kinematics but also =diag{85, diag{95,85, 95,85, 95,60, 60,60, 60,60}, 60}, (40) kkI ID== diag{85, 85, 85, 60, 60, 60}, (41) (41) vehicle dynamics. This means the matrix is (36) tthe signal τ is strictly related not only tothat kinematics butk(e.g also 4. Simulation results D osen osen according according to to dynamics dynamics of of the the controlled controlled plant plant (e.g k = diag{95, 95, 95, 60, 60, 60}, (40) the vehicle dynamics. This means that the matrix kD is (41) ID= diag{85, 85, 85, 60, 60, 60}, osen according to dynamics of the controlled plant (e.g the vehicle dynamics. This means that the matrix k is k = diag{85, 85, 85, 60, 60, 60}, (41) , (41) Λ Λ = = diag{0.35, diag{0.35, 0.35, 0.35, 0.35, 0.35, 1.0, 1.0, 1.0, 1.0, 1.0}, 1.0}, (42) (42) D I rosen aaheavy heavy vehicle the the control control coefficients coefficients can can be beplant different different ion results 4.1.to Indoor airship model In this section we present some re-Λ according dynamics of the controlled (e.g kI = = diag{0.35, diag{85, 85, 85, 60, 60,1.0, 60}, (41) nstants βvehicle ivehicle 0.35, 0.35, 1.0, 1.0}, (42) rnosen afor heavy control coefficients can beplant different according tothe dynamics of the controlled (e.g 4. Simulation results = diag{0.35, 0.35, 0.35, 1.0, 1.0, 1.0}, (42) an for a a light light vehicle). vehicle). Even Even if if the the system system parameters parameters are are sults regarding the use of the proposed controller for the modelΛ r a heavy vehicle the control coefficients can be different the trackThe The desired desired linear linear and and angular angular velocities velocities profiles profiles are are given given inin airship model In this section we present some reΛ = diag{0.35, 0.35, 0.35, 1.0, 1.0, 1.0},,(42) (42) an for a light vehicle). Even ifcoefficients the system parameters are rtexactly aexactly heavy vehicle the control can be test different The desired linear and angular velocities profiles are change given in known, known, thanks thanks to to the the matrix matrix ϒ, ϒ, the the velocity velocity error error of airship AS500 (assuming indoor with low velocity). The ˜ for a light vehicle). Even if the system parameters are ν˙an + Cν + Fig. Fig. 3 3 a) a) and and b), b), respectively. respectively. Note Note that that three three profiles profiles change g the use of the proposed controller for the model r exactly r light 4.1. Indoor airship model. Insystem this section we present some reThe desired linear and angular velocities profiles are given in t known, thanks to the matrix ϒ, the velocity error an for a vehicle). Even if the parameters are Fig. 3 a) and b), respectively. Note that three profiles change creases creases quickly. quickly. The desired linear and angular velocities profiles are given in simulations were done inϒ,MATLAB/Simulink environment fordesired t exactly known, thanks towith theoflow matrix the controller velocity error mmetric orsults regarding the use thevelocity). proposed for the model The linear and angular velocities profiles are given during during airship airship motion motion according according toto sinusoidal sinusoidal functions. functions. 500 (assuming indoor test The Fig. 3 the a)theand b), respectively. Note that three profiles change creases quickly. t exactly known, thanks toN the matrix ϒ, the velocity error during the airship motion according to sinusoidal functions. he e diagonal diagonal inertia inertia matrix matrix N gives gives information information about about the the Fig. 3 a) and b), respectively. Note that three profiles change 6 DOF model with six signal inputs. The blimp parameters creases of AS500 (assuming indoor testfor with low velocity). Fig. 3 and b), respectively. three profiles Next, ininin Fig. Fig. 44a)a)each each linear linear velocity velocityNote error error time time history history isischange preprewere donequickly. in airship MATLAB/Simulink environment during the airship motion according to that sinusoidal functions. he diagonal inertia matrix N gives information about the Next, creases quickly. Next, induring Fig.airship 4the a) airship each linear velocity error history is and preertia rtia related related toto each each quasi-acceleration quasi-acceleration (without (without dynamdynamduring the motion according toexpected, sinusoidal functions. coming from report [9], were about also exploited in reference The simulations were done in information MATLAB/Simulink environment motion according totime sinusoidal functions. he diagonal inertia matrix Nthe gives the sented. sented. The The error error decreases, decreases, as as it it was was expected, quickly quickly and l with six signal inputs. The blimp parameters Next, in Fig. 4 a) each linear velocity error time history is preertia related to each quasi-acceleration (without dynamhe diagonal inertia matrix N gives information about the sented. The error decreases, as it was expected, quickly and al lTthe couplings). couplings). Moreover, Moreover, each each quasi-velocity quasi-velocity ξ ξ is is indeindefor[9], 6to DOF model with six signal inputs. The blimp parameNext, in Fig. 3 a) each linear velocity error time history is preNext, in Fig. 4 a) each linear velocity error time history is pre[1]. The maximal forces and torques applied by the control T report i i ertia related each quasi-acceleration (without dynamafter after about about 30 30error second second the theerror erroras isisclose totoexpected, zero. zero.In InFig. Fig. 44b) b)the the were also exploited in reference Λcouplings). krelated sented. The decreases, itclose was quickly and I z. al Moreover, each quasi-velocity ξapplied indeertia to each quasi-acceleration (without dynami is ters coming from the report [62] were also in [63]. The sented. The error decreases, as it was expected, quickly and after about 30 second the error is close to zero. In Fig. 4 b) the ndent ndent from from other other quasi-velocities quasi-velocities and and allows allows one one to to dedesented. The error decreases, as it was expected, quickly and system were assumed as follows: F = 107, 13, 40 N, al couplings). Moreover, each quasi-velocity ξi ismax index,y,z angular angular velocity velocity errors errorsthe for forerror angular angular variables variables are are shown. The ximal forces and torques applied by theallows control after about 30 second is close to zero. Inshown. Fig. 4 b)The the ndent from other quasi-velocities and one to deal couplings). Moreover, each quasi-velocity ξ is indei angular velocity errors for angular variables are shown. The rmine mine the the kinetic kinetic energy energy reduced reduced by by the the variable variable ξ ξ , , i.e. i.e. (37) after about 30 second the error is close to zero. In Fig. 4 b) the Tmax x,y,z = 27,x,y,z 267, Nm. TheN, values were taken from [7] isis not ito i dendent from other quasi-velocities and allows one error reduction reduction not so sofor fast fast as as the thevariables linear linear velocity velocity error error but but assumed as follows: Fmax =27 107, 13, 40 angular velocity errors angular are shown. The rmine kinetic reduced the variable ξtoi , dei.e. error 11energy 11 66 by 1the 1from TTMν TTNξ 22.allows ndent other quasi-velocities and one error reduction is not so fast as the linear velocity error but (ξ ξ ) ) = = ν ν Mν = = ξ ξ Nξ = = N N ξ ξ . These These indepenindepenking error angular velocity errors for angular variables are shown. The ∑ ∑ for the airship AS800 (both airships have similar construction). i i rmine the kinetic energy reduced by the variable ξ , i.e. i=1 i=1 i i 21 TNm. The 221 6 taken 2from [7] i T Nξ =were after after about 20 20second second all signals aresignificantly significantly reduced. reduced. ItIt 7, 267, values errorabout reduction is notall sosignals fast as are the linear velocity error but (ξ )quasi-velocities =227 Mν =22112energy ξare Nproposed These indepen∑the rmine the reduced by variable ξpresented i ξthe i , i.e. 1 1 νisTkinetic 6 i=1 i2 . strategy T 2 2 after about 20 second all signals are significantly reduced. It nt nt quasi-velocities are used used in in the proposed decoupled decoupled ynamics error reduction is not so fast as the linear velocity error but The diagram of the control is in Fig. 2. a) b) (ξAS800 ) = 21 ν(both Mνairships = 21 ξ T Nξ =similar N ξ . These indepen∑ i arises arises from from the the fact fact that that part part of of dynamical dynamical couplings couplings is is taken taken have construction). 1 6 i=1 i T 2 2in theNproposed after about 20 second all signals are significantly reduced. It nt) quasi-velocities used decoupled (ξ = 2 ν MνCase = 2 ξ1are Nξ = ξ . These indepen∑ i i=1 i arises from the fact that part of dynamical couplings is taken 2 ntroller. ntroller. afteraccount about 20 all signals are significantly reduced.the It - is setused of nominal parameters. In this example the nomint the quasi-velocities are in thein proposed 3 0.1 into into account ininsecond the thecontrol control algorithm. algorithm. However, However, because because the of control strategy presented Fig. 2. decoupled arises from the fact that part of dynamical couplings is taken ntroller. nt quasi-velocities are used in ofthe proposed decoupled into account in the control algorithm. However, because the ome me particular particular cases cases of of the the presented presented controller controller can can be be dedearises from the fact that part of dynamical couplings is taken nal parameters set the airship is taken into account. The task ntroller. 0.08 angular velocity velocity trajectory changes sinusoidal sinusoidal the theerror error isisonly only fntroller. nominal parameters. In this example the nomiinto account in trajectory the controlchanges algorithm. However, because the 2.5 ome particular cases of the presented controller can be de- angular angular velocity trajectory changes sinusoidal thevelocity error isvarionly ced. ced. We We can can point point at aton two two cases: cases: pled conintoby: account in the control algorithm. However, because the relies tracking the velocity trajectory ome particular cases of the presented controller can be described de- close 0.06 close to to zero. zero. The The control control signals signals related related to to linear linear velocity varis set of the airship is taken into account. The task angular velocity trajectory changes sinusoidal the error is only ced. We can point at two cases: ome particular cases of the presented controller can be de2 close to zero. The control signals related to linear velocity variracting in angular velocity trajectory changes sinusoidal the error is only ced. We can point atvehicle two cases: a) ) For For aavelocity symmetric symmetric inin the xy-plane xy-plane we get get0,yyggsin(π/25 == ables ables are arezero. reported reported inFig. Fig.55signals a). a).Their Their values values after short short time time are are king the trajectory by: · t)we 0.04 close to Theincontrol related to after linear velocity vari=the [sin(π/20 + 2, · t), νdescribed ced. We can point at vehicle two cases: d in a) For a symmetric vehicle the xy-plane we get y = ables are reported in Fig. 5 a). Their values after short time are me useful close to zero. The control signals related to linear velocity varig 1.5 0;0; as asaasymmetric results resultsthe thecontroller controller isis simplified simplified and and reduced. reduced. below below 20 20 N. N. From From Fig. Fig. 5 5 b) b) we we see see that that the the applied applied torque torque TTyy 0.02 T a) For vehicle in the xy-plane we get y = ables are reported in Fig. 5 a). Their values after short time are g · t)] . 0, cos(pi/15 (39) [sin(π/20 · t)the + 2, 0, sin(π/25 ·0,t),0.1 · and νntroller. asaasymmetric results controller is simplified reduced. below 20 N. From Fig. 5 b) we see that the applied torque T a)d =0; For vehicle in the xy-plane we get y = ables are reported in Fig. 5 a). Their values after short time are g 1 the controller is simplified and reduced. 0 0; as a results below 20 N. From Fig. 5 b) we see that the applied torque Tyy T 0; as a resultsThe controller and below 20 is N.as From Fig. 5 b) we see that the applied torque Ty · the cos(pi/15 · t)]is. simplified set of selected gains for(39) thereduced. nonlinear controller Bull. Bull.Pol. Pol.Ac.: Ac.:Tech. Tech.XX(Y) XX(Y)2016 2016 –0.02 ϒ−T kD ϒ0,−10, 0.1 0.5 Bull. Pol. Ac.: Tech. XX(Y) 2016 follows: tly, the in-gains for the nonlinear controller is as –0.04 selected Bull. Pol. Ac.: Tech. XX(Y) 2016 0 Bull. Pol. Ac.: Tech. XX(Y) 2016 cs but also –0.06 kD = diag{95, 95, 95, 60, 60, 60}, (40) atrix kD is –0.5 –0.08 = diag{85, 85,(40) 85, 60, 60, 60}, (41) kplant 95, 95, 60, 60,kI60}, D = diag{95, (e.g –0.1 –1 30 0.35, 40 0.35, 1.0, 50 1.0,60 10 20 30 40 50 60 =20diag{0.35, 1.0}, (42) 0 = diag{85, 85,0 85, 60,1060,Λ60}, (41) ekIdifferent t [s] t [s] meters are Λ = diag{0.35, 0.35, 0.35, 1.0, 1.0,and 1.0}, (42) The desired linear angular velocities profiles are given in ocity error Fig. 3 a) andFig. 3.respectively. Case 1 are and given CaseNote 2:in a) desired linearprofiles velocitieschange ud, vd, wd, b) desired angular velocities pd, qd, rd b),profiles that three d linear and angular velocities during the airship motion according to sinusoidal functions. b), respectively. Note that three profiles change about the Next, in Fig. a) each linear velocity error time history is prership motion according to 4sinusoidal functions. ut dynam-Bull. Pol. Ac.: Tech. 65(4) 2017 463 sented. The error decreases, it was expected, quickly and velocity error time history isaspreξ4i a)is each inde-linear Unauthenticated after second the error is close to zero. In Fig. 4 b) the errortodecreases, as about it was30expected, quickly and Download Date | 8/25/17 12:47 PM one de- angular velocity errors for angular variables are shown. The 0ble second the error is close to zero. In Fig. 4 b) the ξi , i.e. reduction is are not shown. so fast as the linear velocity error but ity errors for error angular variables The

P. Herman and W. Adamski

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Fig. 5. Case 1 – control signals: a) applied forces fx, fy, fz, b) applied torques Tx, Ty, Tz

after about 30 second the error is close to zero. In Fig. 4 b) the angular velocity errors for angular variables are shown. The error reduction is not so fast as the linear velocity error but after about 20 second all signals are significantly reduced. It arises from the fact that part of dynamical couplings is taken

into account in the control algorithm. However, because the angular velocity trajectory changes sinusoidal the error is only close to zero. The control signals related to linear velocity variables are reported in Fig. 5 a). Their values after short time are below 20 N. From Fig. 5 b) we see that the applied torque Ty

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Fig. 6. Case 2 – 50% weight reduction (∆ν ´ ν˜ ): a) linear velocity errors ∆u, ∆v, ∆w, b) angular velocity errors ∆p, ∆q, ∆r

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Non-adaptive velocity tracking controller for a class of vehicles

a)

4.2. Underwater vehicle model. The simulations were done for 6 DOF model of underwater vehicle which parameters can be found in [58].

Control signal for linear variables [N]

120 100

Case 3 – set of nominal parameters. In this example the nominal parameters are used and the velocity tracking trajectory is described by (39). The aim is to show that the proposed control scheme is appropriate also for marine vehicles. The control gains selected for velocity tracking task are as follows:

80 60 40 20 0 –20

0

10

20

30

40

50

60

t [s]

kD = diag{10, 10, 10, 10, 10, 10},(43)



kI = diag{10, 10, 10, 10, 10, 10},(44)



Λ = diag{1.5, 1.5, 1.5, 1.5, 1.5, 1.5},(45)

They are different than for the airship because the dynamics of these two vehicles is quite different. Moreover, the control gains are directly related to the system dynamics. In Fig. 8 a) three linear velocity errors are shown. We observe that the ∆u tends to zero after about 7 s, while the others are close to zero at the beginning of the movement. From Fig. 8

150 100 50 0

a)

–50 –100 –150 –200

0

10

20

30

40

50

60

t [s]

Fig. 7. Case 2 – 50% weight reduction, control signals: a) applied forces fx, fy, fz, b) applied torques Tx, Ty, Tz

Case 2 (robustness test) – 50% weight reduction. In order to investigate sensitivity to the parameter changes of the controller the robustness test was done. It was assumed that the blimp weight has been reduced to 50%. Such situation may result from loss of gas in the blimp or if not all parameters are known exactly. The desired velocity was the same as for the nominal parameters set. The linear velocity errors are shown in Fig. 6 a) whereas the angular velocity errors in Fig. 6 b). Similarly, as previously the decreasing of the initial errors is great. In spite of the fact that their values after about 30 s are slightly bigger than for the case of nominal parameters, the controller works still correctly with acceptable performance. From Fig. 7 a) and Fig. 7 b), it arises that the forces and torques have comparable values as in Case 1. These observations lead us to conclusion that the proposed control algorithm is robust to parameter changes.

1.5 1 0.5 0 –0.5

0

5

10

15

20

15

20

t [s]

b)

0.1

Angular velocity error [rad/s]

has during the motion maximal value over 100 Nm. It can be concluded that the dynamical couplings affect the movement in this direction.

2

Linear velocity error [m/s]

Control signal for angular variables [Nm]

b)



0.08 0.06 0.04 0.02 0

–0.02

0

5

10

t [s]

Fig. 8. Case 3 (∆ν ´ ν˜ ): a) linear velocity errors ∆u, ∆v, ∆w, b) angular velocity errors ∆p, ∆q, ∆r

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P. Herman and W. Adamski

Control signals for linear variables [N]

a)

various fully actuated vehicles, namely marine (underwater) vehicles, hovercrafts or indoor airships moving with low velocity. Its robustness was discussed and formally proven. It was also mentioned that simpler controllers can be deduced from the controller discussed. Simulation results for both 6 DOF airship and underwater vehicle model show effectiveness of the proposed methodology.

1400 1200 1000 800 600

Acknowledgments. The work was funded by the National Science Centre of Poland by decision DEC-2011/03/B/ST7/02524 and it was supported by the University Grant No. 09/93/ DSPB/0611.

400 200 0 –200

0

5

10

15

20

t [s]

Control signals for angular variables [Nm]

b) 200 0 –200 –400 –600 –800 –1000

0

5

10

15

20

t [s]

Fig. 9. Case 3 – control signals: a) applied forces fx, fy, fz, b) applied torques Tx, Ty, Tz

b) we see that all angular velocity errors are reduced to zero after about 5 s. We can note that the dynamical coupling causes that at the start all variables are actuated (in spite of that only is r tracked). The velocity tracking without overshoot can be explained by the strong mechanical couplings and great mass of the vehicle (m = 250 kg [58], whereas m = 18.375 kg [62] for the airship). Observing the control signals time history related to linear velocity variables given in Fig. 9 a) we see the greatest force changes for the fx. It arises from the fact that in the initial point the velocity is +2 [m/s] and next it is reduced. From Fig. 9 b) it is noticeable that the applied torque Ty has the greatest values. This fact can be explained by strong dynamical couplings which act in this direction.

5. Conclusions A velocity tracking controller based on Lyapunov techniques has been derived in this work. The controller can be used for

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Non-adaptive velocity tracking controller for a class of vehicles

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