Underwater Robots - Gianluca Antonelli.pdf

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154 7. Dynamic Control of UVMSs Let consider the Lyapunov function candidate V = ˜ε T ˜ε . (7.19) The time derivative of V is: ˙V =2˜ε T ˙˜ε = − ˜ε T ˜η ν 2 − ˜ε T S ( ˜ε ) ν 2 . (7.20) Plugging (7.17) into (7.20) and taking Λ o = λ o I 3 with λ o > 0, gives ˙V = − ˜ηλo ˜ε T ˜ε − λ o ˜ε T S ( ˜ε ) ˜ε = − ˜ηλo ˜ε T ˜ε . which isnegative semidefinite with ˜η ≥ 0. It must be noted that, in view of (7.18), ˜η is anot-decreasing function oftime and thus it stays positive when starting from apositive initial value. The set R of all points ˜ε where ˙ V =0is given by R = { ˜ε = 0 , ˜ε :˜η =0} ; from (2.7), however, it can berecognized that ˜η =0 ⇒ �˜ε � =1 and thus ˙˜η>0inview of (7.18). Therefore, the largest invariant set in R is M = { ˜ε = 0 } and the invariant set theorem ensures asymptotic convergence tothe origin. Manipulator joint error dynamics. The manipulator joint error dynamics on the sliding manifold is described bythe equation − ˙q + Λ q ˜q = 0 whose convergence to ˜q = 0 is evident taking Λ q > O . 7.6.2 Simulations Dynamic simulations have been performed inorder to show the effectiveness of the discussed control law. The UVMS simulator was developed in the Matlab c� / Simulink c� environment. Forthis simulations, the vehicle data are taken from [145]; they refer to the experimental Autonomous <strong>Underwater</strong> Vehicle NPS Phoenix. Atwo-link manipulator with rotational joints has been considered which ismounted under the vehicle body with the joint axes parallel to the fore-aft direction; since the vehicle inertia along that axis is minimum, this choice increases dynamic coupling between the vehicle and the manipulator. The length of eachlink is 1m,the center of gravityiscoincidentwith the center of buoyancy and itissupposed tobeinthe geometrical center of the link; each link is not neutrally buoyant. Dry and viscous joint frictions are also taken into account. As for the control law, implementation of(7.10) was considered; however, it is well known that the sign function would lead to chattering inthe system.
7.6 Sliding Mode Control 155 Practical implementation of (7.10), therefore, requiresreplacementofthe sign function, e.g., with the sat function u = B † [ K D s + ˆg ( q , R B I )+K S sat(s ,ε)] , (7.21) where the sat(x ,ε)isthe vector function whose i -th component is ⎧ ⎨ 1 ifxi >ε sat(x ,ε) i = − 1 ⎩ x i ε ifxi < − ε otherwise. Convergence tothe equilibrium of the UVMS under this different control law can be easily demonstrated starting from (7.12) following the guidelines in [268]. In detail, it is obtained that ˙ V < 0inthe region characterized by � s � ≥ ε ,while the sign of ˙ V is undetermined in the boundary layer characterized by � s �
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Springer Tracts in Advanced Robotic
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Gianluca Antonelli Underwater Robot
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Editorial Advisory Board EUROPE Her
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Elalocomotiva sembrava fosse un mos
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Acknowledgements The contributions
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Preface to the Second Edition The p
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XVIII Preface to the First Edition
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XX Preface tothe First Edition mani
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XXII Notation η q =[η T 1 ε T η
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XXIV Notation ˜x error variable de
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XXVI Contents 3. Dynamic Control of
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XXVIIIContents A. Mathematical mode
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2 1. Introduction Maybe the first i
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4 1. Introduction (Massachusetts, U
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6 1. Introduction Fig. 1.4. Coordin
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8 1. Introduction Fig. 1.5. Pig, ma
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10 1. Introduction An example of cr
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12 1. Introduction Fig. 1.8. Romeo
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2. Modelling ofUnderwater Robots
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2.2 Rigid Body’s Kinematics 17 Th
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J T k,oqJ k,oq = 1 4 I 3 , that all
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5. compute the quaternion Q by the
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2.3 Rigid Body’s Dynamics 23 Let
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2.4 Hydrodynamic Effects 25 τ 1 =
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C A ( ν )=− C T A ( ν ) ∀ ν
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2.4 Hydrodynamic Effects 29 effects
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2.5 Gravity and Buoyancy 2.5 Gravit
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2.6 Thrusters’ Dynamics 33 Fig. 2
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2.7 Underwater Vehicles’ Dynamics
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y z 2.8 Kinematics of Manipulators
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2.9 Dynamics ofUnderwater Vehicle-M
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2.9 Dynamics ofUnderwater Vehicle-M
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2.11 Identification 43 f e = K ( x
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3. Dynamic Control of 6-DOF AUVs 3.
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3.2 Earth-Fixed-Frame-Based, Model-
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o x z o b z b x b 3.3 Earth-Fixed-F
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3.4 Vehicle-Fixed-Frame-Based, Mode
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o o b y x y b x b 3.5 Model-Based C
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3.6 Mixed Earth/Vehicle-Fixed-Frame
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3.7 Jacobian-Transpose-Based Contro
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3.8 Comparison Among Controllers 59
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3.9 Numerical Comparison Among the
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force [N] moment [Nm] 20 10 0 −10
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80 4. Fault Detection/Tolerance Str
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5. Experiments of Dynamic Control o
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5.3 Experiments of Dynamic Control
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[m] [m] 5 4 3 2 1 0 0 100 200 300 4
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5.3 Experiments of Dynamic Control
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5.4 Experiments of Fault Tolerance
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Page 125 and 126: 6. Kinematic Control of UVMSs “.
Page 127 and 128: 6.2 Kinematic Control 107 UVMS. Thi
Page 129 and 130: 6.2 Kinematic Control 109 singulari
Page 131 and 132: 6.2 Kinematic Control 111 case of t
Page 133 and 134: 6.5 Singularity-Robust Task Priorit
Page 141 and 142: 6.6 Fuzzy Inverse Kinematics 121 It
Page 143 and 144: Simulations 6.6 Fuzzy Inverse Kinem
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Page 147 and 148: vehicle attitude [deg] 20 10 0 −1
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Page 159 and 160: e.e. position error [m] 0.02 0.01 0
Page 161 and 162: 7. Dynamic Control of UVMSs 7.1 Int
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Page 171 and 172: 7.6 Sliding Mode Control 7.6 Slidin
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Page 177 and 178: 7.7 Adaptive Control 157 of gravity
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Page 181 and 182: 7.7 Adaptive Control 161 The vehicl
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8.4 External Force Control 205 The
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8.4 External Force Control 207 be f
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8.4 External Force Control 209 that
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8.4 External Force Control 211 UVMS
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[N] [m] 350 300 250 200 150 100 50
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[deg] [deg] 10 5 0 −5 −10 50 40
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8.5 Explicit Force Control 217 wher
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8.5 Explicit Force Control 219 wher
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[m] [deg] 0.2 0.1 0 −0.1 8.5 Expl
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[m] [deg] 0.2 0.1 0 −0.1 −0.2 0
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226 9. Coordinated Control of Plato
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238 10. Concluding Remarks control
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240 A. Mathematical models ⎡ 6 .
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242 A. Mathematical models using th
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244 A. Mathematical models L = 2 m
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References 1. Alekseev Y.K., Kosten
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References 249 28. Antonelli G.and
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References 251 62. Caccia M., Indiv
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References 253 96. Cui Y. and Sarka
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References 255 134. Garcia R., Puig
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References 257 168. Kato N. and Lan
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References 259 mera. In: IEEE Inter
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References 261 235. Podder T.K. and
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References 263 273. Solvang B., Den
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References 265 310. Yoerger D.R., S
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