Underwater Robots - Gianluca Antonelli.pdf

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130 6. Kinematic Control of UVMSs • Manipulability .Since the fuzzy approach tries to move the manipulator alone, its manipulability has to be checked. Acomputationally limited measure of the manipulability can be obtained by checking the minimum singular value of the Jacobian [195, 77]. Since the manipulability function is strongly non-linear it is possible to adopt the following approach: when close to asingular configuration, the system tries to reconfigure itself in a dexterous configuration. The task, thus, is anominal manipulator configuration whose Jacobian is given by: J s 1 =[O 6 × 6 I 6 ]; • Mechanical limits. Due to the mechanical structure, each joint has a limited allowed range. In case ofareal-time trajectory, avoidance of such limits is crucial. Forthis reason the minimum distance from amechanical limit is considered as another secondary task. Notice that the Jacobian is the same of to the previous task: J s 2 =[O 6 × 6 I 6 ]; • Vehicle attitude. Asfor the previous cases, the vehicle attitude (roll and pitch angles) has tobekept null when possible. The Jacobian is then given by J s 3 = � � 0 0 0 1 0 0 0 0 0 0 0 0 . 0 0 0 0 1 0 0 0 0 0 0 0 Due to the simple structure ofthe matrices, the pseudoinversion of the secondary tasks is trivial: J † s 1 = J T s 1 , J † s 2 = J T s 2 , J † s 3 = J T s 3 . As shown in Section 6.6, the above tasks are activated by fuzzy variables. The fuzzy inference system has 3inputs, namely: ameasure of the robot manipulability, ameasure of the distance from the joints limits and ameasure of the vehicle attitude. Hence, 3linguistic variables can be defined that can take the values: manipulability = { singular, not singular} joint limits = { close, not close} vehicle attitude = { small, not small} The output isgiven by the linguistic variables β and the 3 α i ’s. The latter can take the following values: α i = { high, low } . The linguistic variable β ,named motion can take the following values: motion = { vehicle, manipulator } . As an example, the membership function ofthe linguistic variable joint limits is reported inFigure 6.18. The FIS outputs are considered at two different levels. The variable motion ( β in (6.15)) can be considered at ahigher level with respect to the
[-] 1 0.8 0.6 0.4 0.2 0 close 0 20 40 [deg] 60 80 6.6 Fuzzy Inverse Kinematics 131 Fig. 6.18. Membership function of the linguistic variable: joint limits used in the fuzzy singularity-robust task priority technique variables α i ’s. Roughly speaking, the motion has to be normally assigned to the manipulator, i.e., if all the α i ’s are null. Hence, the value of β is related to the value of the α i ’s and the first set of rules to be designed concerns the 3linguistic variables α 1 , α 2 and α 3 .Acomplete and consistent set of fuzzy rules for 3linguistic variables each ofthem defined intwo fuzzy sets requires 2 3 rules for every output leading to32total rules. The rules have been arranged in ahierarchical structure that gives higher priority tothe kinematic singularity ofthe manipulator and lower tothe vehicle attitude. Hence, the following 8rules have been used: 1. if (manipulator is singular) then ( α 1 is high); 2. if (manipulator is not singular) then ( α 1 is low); 3. if (manipulator is singular) then ( α 2 is low); 4. if (joint limits is not close) then ( α 2 is low); 5. if (manipulator is not singular) and (joint limits is close) then ( α 2 is high); 6. if (manipulator is singular) or (joint limits is close) then ( α 3 is low); 7. if (vehicle attitude is small) then ( α 3 is low); 8. if (manipulator is not singular) and (joint limits is not close) and (vehicle attitude is not small) then ( α 3 is high). In detail, the rules are developed as follows: • The first two rules concern the primary task. Inthis case, the manipulator singularity isofconcern and the variable α 1 is activated when the manipulator isclose to asingularity.
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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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Page 99 and 100: 3.9 Numerical Comparison Among the
Page 101 and 102: 80 4. Fault Detection/Tolerance Str
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Page 115 and 116: 5.3 Experiments of Dynamic Control
Page 117 and 118: [m] [m] 5 4 3 2 1 0 0 100 200 300 4
Page 119 and 120: 5.3 Experiments of Dynamic Control
Page 121 and 122: 5.4 Experiments of Fault Tolerance
Page 123 and 124: [deg] 120 100 80 60 40 20 0 −20 5
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 155 and 156: joint positions 1-3 [deg] joint pos
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
Page 163 and 164: 7.2 Feedforward Decoupling Control
Page 165 and 166: 7.2 Feedforward Decoupling Control
Page 167 and 168: 7.4 Nonlinear Control for UVMSs wit
Page 169 and 170: 7.5 Non-regressor-Based Adaptive Co
Page 171 and 172: 7.6 Sliding Mode Control 7.6 Slidin
Page 173 and 174: 7.6 Sliding Mode Control 153 u = B
Page 175 and 176: 7.6 Sliding Mode Control 155 Practi
Page 177 and 178: 7.7 Adaptive Control 157 of gravity
Page 179 and 180: Let us consider the scalar function
Page 181 and 182: 7.7 Adaptive Control 161 The vehicl
Page 183 and 184: [deg] 3 2.5 2 1.5 1 0.5 7.8 Output
Page 185 and 186: 7.8 Output Feedback Control 165 It
Page 187 and 188: 7.8 Output Feedback Control 167 whe
Page 189 and 190: 7.8 Output Feedback Control 169 C T
Page 191 and 192: 7.8 Output Feedback Control 171 wit
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Page 195 and 196: [m] [-] [deg] x 10−3 10 8 6 4 2 0
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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[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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