Underwater Robots - Gianluca Antonelli.pdf

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48 3. Dynamic Control of 6-DOF AUVs Let now consider the positive semi-definite function: V = 1 2 s T M � v s > 0 , ∀ s �= α � 0 Q � , α ∈ IR (3.5) after straightforward calculation, its time derivative isgiven by: ˙V = − s T [ Λ + D � RB] s < 0 , ∀ s �= 0 (3.6) Since V is only positive semi-definite and the system is non-autonomous the stability can not bederived by applying the Lyapunov’s theorem. By further assuming that ˙p r is twice differentiable, then ¨ V is bounded and ˙ V is uniformly continuous. Hence, application of the Barbălat’s Lemma allows to prove global convergence of s → 0 as t →∞.Due to the definition of the vector s ,its convergence tozero also implies convergence of ˜p to the null value. In case of perfect knowledge ofthe dynamic model, moreover, the convergence ofthe error tozero can be demonstrated even for K I = O ,i.e., without integral action. Compensation of the persistent effects. If areduced version of the controller is implemented, e.g., by neglecting the model-based terms in (3.4), the restoring moment is not compensated efficiently. Infact, let consider avehicle in the two static postures shown in Figure 3.1 and let suppose that thevehicle, starting from the left configuration, is driven to the right configuration and, after awhile, back tothe left configuration. Inthe left configuration the integral action in(3.3) does not give any contribution to the control moment as expected, because the vectors of gravity and buoyancy are aligned. Furthermore, in the right configuration the integral action will compensate exactly at the steady state for the moment generated by the misalignment between gravity and buoyancy. When the vehicle is driven back to the left configuration, anull steady-state compensation error ispossible after the integral action is discharged; this poses asevere limitation tothe control bandwidth that can be achieved. Asimilar argument holds inthe typical practical situation in which the compensation implemented through the vector g � RB is not exact. On the other hand, since the error variables are defined inthe earth-fixed frame, the controller is appropriate to counteract the current effect. This point will be clarified in next Subsections, when discussing the drawbacks of the controllers C and D with respect to the current compensation. Finally, anadaptive version of this controller is not straightforward. In fact, since the dynamic model (2.53) does not depend on the absolute vehicle position, asteady null linear velocity ofthe vehicle with anon-null position error would not excite acorrective adaptive control action. As aresult, null position error at rest cannot be guaranteed in presenceofocean current. From the theoretical point ofview, this drawback can be avoided by defining the velocity error using the current measurement. However, from the practical point ofview, this approach cannot achieve fine positioning ofthe vehicle since local vortices can make current measurement too noisy.
o x z o b z b x b 3.3 Earth-Fixed-Frame-Based, Non-model-Based Controller 49 f b f b M r r b r b o b x b z b r g r g f g f g Fig. 3.1. Planar view parallel to the xz earth-fixed frame of two different configurations and corresponding restoring forces and moments: r G ( r B )isthe center of gravity (buoyancy), f G ( f B )isthe gravity (buoyancy) force, and M R is the y component ofthe restoring moment Reduced Controller. According tothe motivation given in the introduction of this Section, the following reduced version of the control law isconsidered g I τ � v = ˆg � RB( R I B )+ ˙p r , (3.7) where ˆg � RB( R I B )isamodel-based estimate of the restoring generalized force acting on the vehicle. Itmight beuseful tosubstituting the definition of ˙p r yielding τ � v = ˆg � RB( R I B )+K D ˙˜p d + K P ˜p + K I that has aclear interpretation. � t 0 θ ˜p ( τ ) dτ (3.8) 3.3 Earth-Fixed-Frame-Based, Non-model-Based Controller In 1999, J.Yuh proposes an earth-fixed-frame-based, non-model-based controller [320]. The control input is given by:
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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
Page 19 and 20: XXIV Notation ˜x error variable de
Page 21 and 22: XXVI Contents 3. Dynamic Control of
Page 23 and 24: XXVIIIContents A. Mathematical mode
Page 25 and 26: 2 1. Introduction Maybe the first i
Page 27 and 28: 4 1. Introduction (Massachusetts, U
Page 29 and 30: 6 1. Introduction Fig. 1.4. Coordin
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Page 33 and 34: 10 1. Introduction An example of cr
Page 35 and 36: 12 1. Introduction Fig. 1.8. Romeo
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Page 39 and 40: 2.2 Rigid Body’s Kinematics 17 Th
Page 41 and 42: J T k,oqJ k,oq = 1 4 I 3 , that all
Page 43 and 44: 5. compute the quaternion Q by the
Page 45 and 46: 2.3 Rigid Body’s Dynamics 23 Let
Page 47 and 48: 2.4 Hydrodynamic Effects 25 τ 1 =
Page 49 and 50: C A ( ν )=− C T A ( ν ) ∀ ν
Page 51 and 52: 2.4 Hydrodynamic Effects 29 effects
Page 53 and 54: 2.5 Gravity and Buoyancy 2.5 Gravit
Page 55 and 56: 2.6 Thrusters’ Dynamics 33 Fig. 2
Page 57 and 58: 2.7 Underwater Vehicles’ Dynamics
Page 59 and 60: y z 2.8 Kinematics of Manipulators
Page 61 and 62: 2.9 Dynamics ofUnderwater Vehicle-M
Page 63 and 64: 2.9 Dynamics ofUnderwater Vehicle-M
Page 65 and 66: 2.11 Identification 43 f e = K ( x
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Page 75 and 76: o o b y x y b x b 3.5 Model-Based C
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Page 81 and 82: 3.8 Comparison Among Controllers 59
Page 83 and 84: 3.9 Numerical Comparison Among the
Page 87 and 88: force [N] moment [Nm] 20 10 0 −10
Page 101 and 102: 80 4. Fault Detection/Tolerance Str
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Page 115 and 116: 5.3 Experiments of Dynamic Control
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Page 119 and 120: 5.3 Experiments of Dynamic Control
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5.4 Experiments of Fault Tolerance
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[deg] 120 100 80 60 40 20 0 −20 5
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6. Kinematic Control of UVMSs “.
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6.2 Kinematic Control 107 UVMS. Thi
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6.2 Kinematic Control 109 singulari
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6.2 Kinematic Control 111 case of t
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6.5 Singularity-Robust Task Priorit
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6.6 Fuzzy Inverse Kinematics 121 It
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Simulations 6.6 Fuzzy Inverse Kinem
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6.6 Fuzzy Inverse Kinematics 125 Fi
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vehicle attitude [deg] 20 10 0 −1
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6.6 Fuzzy Inverse Kinematics 129 It
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[-] 1 0.8 0.6 0.4 0.2 0 close 0 20
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joint positions 1-3 [deg] joint pos
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e.e. position error [m] 0.02 0.01 0
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7. Dynamic Control of UVMSs 7.1 Int
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7.2 Feedforward Decoupling Control
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7.2 Feedforward Decoupling Control
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7.4 Nonlinear Control for UVMSs wit
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7.5 Non-regressor-Based Adaptive Co
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7.6 Sliding Mode Control 7.6 Slidin
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7.6 Sliding Mode Control 153 u = B
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7.6 Sliding Mode Control 155 Practi
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7.7 Adaptive Control 157 of gravity
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Let us consider the scalar function
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7.7 Adaptive Control 161 The vehicl
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[deg] 3 2.5 2 1.5 1 0.5 7.8 Output
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7.8 Output Feedback Control 165 It
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7.8 Output Feedback Control 167 whe
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7.8 Output Feedback Control 169 C T
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7.8 Output Feedback Control 171 wit
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[m] [deg] [deg] 0.1 0.05 0 −0.05
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[m] [-] [deg] x 10−3 10 8 6 4 2 0
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[Nm] [Nm] [Nm] 500 0 −500 50 0
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[m] [-] [deg] x 10−3 10 8 6 4 2 0
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[N] [N] [N] 20 0 −20 40 20 0 −2
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[Nm] [Nm] [Nm] 50 0 −50 350 300 2
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7.9 Virtual Decomposition Based Con
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joint torques [Nm] 200 0 −200 −
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e.e. orientation error [deg] 2 1 0
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vehicle position [m] 0.1 0.05 0 −
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8. Interaction Control of UVMSs 8.1
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8.3 Impedance Control 203 8.2 Dexte
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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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