Underwater Robots - Gianluca Antonelli.pdf

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206 8. Interaction Control of UVMSs scheme proposed in [99], aforce control scheme isanalyzed that does not require exact dynamic compensation; however, the knowledge ofpart ofthe dynamic model can always beexploited when available. Extension of the original scheme to redundant systems is achieved via atask-priority inverse kinematics redundancy resolution algorithm [78] andsuitable secondary tasks are defined to exploit all the degrees of freedom of the system. Force control scheme. Asketch ofthe implemented scheme is provided in Figure 8.3.Inour case the Inverse Kinematics (i.e., theblock IK in the sketch) is solved at the differential level allowing to efficiently handle the system redundancy. The matrix J k ( R B I )isthe nonlinear, configuration dependent, matrix introduced in (2.58); its inverse, when resorting to the quaternion attitude representation, isalways defined. The matrix J pos( R B I , q )has been defined in (2.67). Moreover, astability analysis for the kinematically redundant case is provided. The Motion Control block can be any suitable motion control law for UVMS; thus, if apartial knowledge of the system is available, amodel based control can be applied. In Section 2.10, the mathematical model of an UVMS in contact with the environment isgiven. x p,d, x s,d ˙x c IK ζ d − 1 J k � + − η , q Fig. 8.3. External Force Control Scheme 8.4.1 Inverse Kinematics Motion Control − + J T pos UVMS +env. Let m be the number of degrees of freedom of the mission task. The system DOFs are 6+n ,where 6are the DOFs of the vehicle and n is the number ofmanipulator’s joints. When 6+n>mthe system is redundant with respect to the given task and Inverse Kinematics (IK) algorithms can beapplied to exploit such redundancy. The IK algorithm implemented is based on atask-priority approach [78] (see also Section 6.5), that allows to manage the natural redundancy of the system while avoiding the occurrence of algorithmic singularities. In this phase we can define different secondary tasks to PID + − f e f e,d
8.4 External Force Control 207 be fulfilled along with the primary task as long as they do not conflict. For example, we can ask the system not to change the vehicle orientation, or not to use the vehicle at all as long asthe manipulator is working in adexterous posture. Let us define x p as the primary task vector and x s as the secondary task vector. The frame in which they are defined depends onthe variables weare interested in: if the primary task is the position of the end-effector it should be normally defined in the inertial frame. The task priority inverse kinematics algorithm is based onthe following update law [78] ζ d = J # p [ ˙x p,d + Λ p ( x p,d − x p )] + +(I − J # p J p ) J # s [ ˙x s,d + Λ s ( x s,d − x s )] , (8.7) where J p and J s are the configuration-dependent primary and secondary task Jacobians respectively, the symbol # denotes any kind of matrix inversion (e.g. Moore-Penrose), and Λ p and Λ s are positive definite matrices. It must be noted that x p , x s and the corresponding Jacobians J p , J s are functions of η d and q d .The vector ζ d is defined in the body-fixed frame and a suitable integration must be applied to obtain the desired positions: η d , q d (see (2.58)). Let us assume that the primary task is the end-effector position that implies that J p = J pos. Tointroduce aforce control action (8.7) is modified as follow: the vector ˙x c is given as areference value to the IK algorithm and the reference system velocities are computed as: where ζ d = J # p [ ˙x p,d + ˙x c + Λ p ( x p,d − x p )] + +(I − J # p J p ) J # s [ ˙x s,d + Λ s ( x s,d − x s )] , (8.8) ˙x c = k f,p ˜ f e − k f,v ˙ f e + k f,i � t 0 ˜f e ( σ ) dσ (8.9) being ˜ f e = f e,d − f e the force error. The direction in which force control is expected are included in the primary task vector, e.g., if the task requires to exert aforce along z ,the primary task includes the z component of x . 8.4.2 Stability Analysis Pre-multiplying (8.8) by J p ∈ IR 3 × (6+n ) we obtain: J p ζ d = ˙x p,d + ˙x c + Λ p ( x p,d − x p ) . (8.10) Let us consider aregulation problem, i.e., the reference force f e,d and the desired primary task x p,d are constant. In addition, let us assume that f e,d ∈R( K ), where K is the stiffness matrix defined in (2.77), and that, as
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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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[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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Page 183 and 184: [deg] 3 2.5 2 1.5 1 0.5 7.8 Output
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Page 197 and 198: [Nm] [Nm] [Nm] 500 0 −500 50 0
Page 199 and 200: [m] [-] [deg] x 10−3 10 8 6 4 2 0
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Page 221 and 222: 8. Interaction Control of UVMSs 8.1
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Page 259 and 260: 240 A. Mathematical models ⎡ 6 .
Page 261 and 262: 242 A. Mathematical models using th
Page 263 and 264: 244 A. Mathematical models L = 2 m
Page 265 and 266: References 1. Alekseev Y.K., Kosten
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Page 275 and 276: 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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