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Table of Contents Modeling and identification of serial robots 48 1.1. Introduction 48 1.2. Geometric modeling 49 1.2.1. Geometric description 49 1.2.2. Direct geometric model 53 1.2.3. Inverse geometric model 55 1.2.3.1. Stating the problem 55 1.2.3.2. Principle of paul's method 57 1.3. Kinematic modeling 60 1.3.1. Direct kinematic model 60 1.3.1.1 calculation of the jacobian matrix by deviation of dgm 61 1.3.1.2. Kinematic jacobian matrix 62 1.3.1.3. Factorization of the jacobian matrix into three matrixes 65 1.3.1.4. Dimension of application working space of a robot 65 1.3.2. Inverse kinematic model 66 1.3.2.1. General form of the kinematic model 67 1.3.2.2. Inverse kinematic model for the regular case 67 1.3.2.3. Solution at the proximity of singular positions 69 1.3.2.4. Inverse kinematic model of redundant robots 70 1.4. Identification of geometric parameters 71 1.4.1. Introduction 71 1.4.2. Geometric parameters 71 1.4.2.1. Geometric parameters of the robot 72 1.4.2.2. Parameters of the robot's position in the space 72 1.4.2.3. Geometric parameters of the tool 73 1.4.3. Generalized differential model of a robot 74 1.4.4. Principle of geometric calibration 75 1.4.4.1. General form of the calibration model 75 1.4.4.2. Identifying the geometric parameters 76 1.4.4.3. Solving the identification equations 78 1.4.5. Calibration methods of geometric parameters 80 1.4.5.1. Calibration model by measuring the end-effector position 80 1.4.5.2. Autonomous calibration models 81 1.4.6. Correction of geometric parameters 83 1.5. Dynamic modeling 84 1.5.1. Lagrange formalism 86 1.5.1.1. General form of dynamic equations 87 1.5.1.2. Calculation of energy 88 1.5.1.3. Properties of the dynamic model 90 1.5.1.4. Taking into consideration the friction 90 1.5.1.5. Taking into account the inertias of actuators 91 We represent the kinetic energy of the actuator j by a term having the form iaj . The inertial parameter iaj can be written: 91 1.5.1.6. Taking into consideration the stresses entailed by the end-effector on its environment 92 1.5.2. Newton-euler formalism 93 1.5.2.1. Linear newton-euler equations in relation to the inertial parameters 93 1.5.2.2. Practical form of newton-euler equations 95 1.5.3. Determining the basic inertial parameters 96 1.6. Identification of dynamic parameters 101 1.6.1. Introduction 101 1.6.2. Identification principle of dynamic parameters 102 1.6.2.1. Solving method 102 1.6.2.2. Identifiable parameters 104 1.6.2.3. Choice of identification movements 104 1.6.2.4. Evaluation of joint coordinates 106 1.6.2.5. Evaluation of joint torques 106 1.6.3. Identification model using the dynamic model 107 1.6.4. Sequential formulation of the dynamic model 109 1.6.5. Practical considerations 110 1.6.6. Conclusion 111 1.7. Conclusion 112 1.7. Bibliography 112 2.1. Introduction 2 2.1.1. Characteristics of classic robots 2 2.1.2. Other types of robot structure 3 2.1.3. General advantages and disadvantages 6 2.1.4. Present day uses 8 2.1.4.1. Simulators and space applications 8 2.1.4.2. Industrial applications 10 2.1.4.3. Medical applications 12 2.1.4.4. Precise positioning 13 2.2. Machine types 14 2.2.1. Introduction 14 2.2.2. Plane robots with three degrees of freedom 19 2.2.3. Robots moving in space 20 2.2.3.1. Manipulators with three degrees of freedom 20 2.2.3.2. Manipulators with four or five degrees of freedom 24 2.2.3.3. Manipulators with six degrees of freedom 26 2.3. Inverse geometric and kinematic models 29 2.3.1. Inverse geometric model 29 2.3.2. Inverse kinematics 31 2.3.3. Singular configurations 34 2.3.3.1. Singularities and statics 36 2.3.3.2. State of the art 37 2.3.3.3. The geometric method 37 2.3.3.4. Maneuverability and number of conditions 40 2.3.3.4. Singularities in practice 41 2.4. Direct geometric model 41 2.4.1. Iterative method 42 2.4.2. Algebraic method 43 2.4.1.2. Reminder concerning algebraic geometry 43 2.4.2.2. Planar robots 45 2.4.2.3. Manipulators with six degrees of freedom 47 2.5. Bibliography 48 Performance analysis of robots 1 3.1. Introduction 2 3.2. Accessibility 3 3.2.1. Various levels of accessibility 3 3.2.2. Condition of accessibility 5 3.3. Workspace of a manipulator-type robot 6 3.3.1. General definition 6 3.3.2. Space of accessible positions 8 3.3.3. Primary space and secondary space 9 3.3.4. Defined orientation workspace 11 3.3.5. Free workspace 12 3.3.6. Calculation of the workspace 14 3.4. Concept of aspect 15 3.4.1. Definition 15 3.4.2. Mode of aspect calculation 16 3.4.3. Free aspects 18 3.4.4. Application of the aspects 20 3.5. Concept of browsability 22 3.5.1. Introduction 22 3.5.2. Characterization of n-browsability 24 3.5.3. Characterization of t-browsability 26 Necessary and sufficient condition 29 3.6. Local performances 32 3.6.1. Definition of dexterity 32 3.6.2. Manipulability 32 3.6.3. Isotropy index 41 3.6.4. Lowest singular value 41 3.6.5. Angles and distances of approach 42 3.7. Conclusion 43 Trajectory generation 314 4.1. Introduction 314 4.2. Point-to-point trajectory in the joint space under kinematic constraints 315 4.2.1. Fifth-order polynomial model 316 4.2.2. Trapezoidal velocity model 317 4.2.3. Smoothed trapezoidal velocity model 322 4.3. Point-to-point trajectory in the task-space under kinematic constraints 325 4.4. Trajectory generation under kinodynamic constraints 327 4.4.1. Problem statement 329 4.4.1.1. Constraints 330 4.4.1.2. Objective function 331 4.4.2. Description of the method 331 4.4.2.1. Outline 332 4.4.2.2. Construction of a random trajectory profile 333 4.4.2.4. Summary 339 4.4.3. Trapezoidal profiles 341 4.5. Examples 343 4.5.1. Case of a two dof robot 344 4.5.1.1. Optimal free motion planning problem 344 4.5.1.2. Optimal motion problem with geometric path constraint 345 4.5.2. Case of a six dof robot 346 4.5.2.1. Optimal free motion planning problem 346 4.5.2.2. Optimal motion problem with geometric path constraints 347 4.5.2.3. Optimal free motion planning problem with intermediate points 348 4.6. Conclusion 350 4.7. References 351 Basic concepts 355 The basic scheme of a stochastic technique consists of generating random trial solutions s in the search space ?. Then, a comparison between these solutions is made to retain the best. This is implemented in the following pseudo-code: 355 A) the hill climbing method 356 B) the simulated annealing method 356 Position/force control of a robot in a free or restrained space 1 5.1. Introduction 1 5.2. Free space control 2 5.2.1. Hypotheses applying to the whole chapter 2 5.2.2. Complete dynamic modeling of a manipulator robot 3 5.2.3. Ideal dynamic control in the joint space 6 5.2.4. Ideal dynamic control in an application working space 9 5.2.5. Decentralized control 10 5.2.6. Sliding mode control 11 5.2.7. Control by sliding of a higher order 14 Twisting algorithm 15 5.2.8. Adaptive control 15 5.3. Control in a constraint space 16 5.3.1. Interaction of the manipulator with the environment 16 5.3.2. Impedance control 17 5.3.3. Control during the stress entailed by a mass attached to a spring 18 5.3.4. Non-linear decoupling in a constrained space 21 5.3.5. Position/force hybrid control 22 5.3.5.1. Parallel structure 22 5.3.5.2. External structure 28 5.3.6. Specificity of the control during stress 30 5.4. Conclusion 33 5.5. Bibliography 34 chapter 6. Visual servoing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Francois chaumette 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2. Modeling visual data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.1. The interaction matrix . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.2. Mounted camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.3. Scene camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2.4. 2-d data interaction matrix . . . . . . . . . . . . . . . . . . . . . . 106 6.2.4.1. Interaction matrix of a 2-d point . . . . . . . . . . . . . . . . 106 6.2.4.2. Interaction matrix of a configurable 2-d geometric primitive 109 6.2.4.3. Interaction matrix for complex 2-d shapes . . . . . . . . . . 112 6.2.4.4. Interaction matrix by learning or estimation . . . . . . . . . . 115 6.2.5. 3-d data interaction matrix . . . . . . . . . . . . . . . . . . . . . . 116 6.2.5.1. Pose calculation . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.5.2. Interaction matrix associated with _u . . . . . . . . . . . . . 119 6.2.5.3. Interaction matrix associated with a 3-d point . . . . . . . . 120 6.2.5.4. Interaction matrix associated with a 3-d plane . . . . . . . . 122 6.3. Task function and command . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3.1. Obtaining the visual command s_ . . . . . . . . . . . . . . . . . . 123 6.3.2. Regulating the task function . . . . . . . . . . . . . . . . . . . . . . 124 6.3.2.1. Case where the dimension of s is 6 (k = 6) . . . . . . . . . . 126 6.3.2.2. Case where the dimension of s is greater than 6 (k > 6) . . . 134 6.3.3. Hybrid tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.3.3.1. Virtual links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.3.3.2. Hybrid task function . . . . . . . . . . . . . . . . . . . . . . . 141 6.3.4. Target tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.4. Other exteroceptive sensors . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9 10 modeling, performance analysis and control of robot manipulators 7.1. Introduction 173 7.2. Modeling of flexible robots 173 7.2.1. Introduction 173 7.2.2. Generalized newton-euler's model for a kinematically free elastic body 174 7.2.2.1. Definition: formalism of a dynamic model 174 7.2.2.2. Choice of formalism 176 7.2.2.3. Kinematic model of a free elastic body 177 7.2.2.4. Balance principle compatible with the mixed formalism 179 7.2.2.5. Virtual power of the field of acceleration quantities 179 7.2.2.6. Virtual power of external forces 181 7.2.2.7. Virtual power of elastic cohesion forces 183 7.2.2.8 balance of virtual powers 183 7.2.2.9. Linear rigid balance in integral form 184 7.2.2.10. Angular rigid balance in integral form 184 7.2.2.11. Elastic balances in integral form 185 7.2.2.12. Linear rigid balance in parametric form 186 7.2.2.13. Intrinsic matrix form of the generalized newton-euler model 188 Notes.- 190 7.2.3. Speed model of a simple open robotic chain 190 7.2.7. Geometric model of an open chain 195 7.2.8. Recursive calculation of the inverse and direct dynamic models for a flexible robot 197 7.2.8.1. Introduction 197 7.2.8.2. Recursive algorithm of the inverse dynamic model 197 7.2.8.3. Recursive algorithm of the direct dynamic model 201 7.2.8.4. Iterative symbolic calculation 205 7.3. Control of flexible manipulator-type robots 205 7.3.1. Introduction 205 7.3.2. Recall of notations 206 7.3.3. Control methods 207 7.3.3.1. Regulation 207 7.3.3.2. Point-to-point movement in fixed time 208 7.3.3.2.1. Control of one axis robot by operational movement planning [ben 00a, ben 03] 208 7.3.3.2.2 control of a robot with one axis through model parameterization 210 7.3.3.3. Monitoring of the movement in the joint space 212 7.3.3.3.1 the calculated torque method 212 7.3.3.3.2. Joint monitoring by retrograde integration of elastic dynamics 213 7.3.3.4. Trajectory monitoring in the operational space 215 7.4. Conclusion 219 7.5. Bibliography 220 List of Authors Index
Library of Congress Subject Headings for this publication:
Robotics.
Manipulators (Mechanism).