Perturbation Techniques for Flexible Manipulators

1 Introduction.- 1.1 Objectives.- 1.2 Book outline.- 2 Present trends in the dynamics and control of flexible manipulators.- 2.1 Introduction.- 2.2 Space Structures.- 2.2.1 Dynamics.- 2.2.2 Control design.- 2.3 Dynamics of flexible manipulators.- 2.3.1 Lagrange's equation and modal expansion.- 2.3.2 Lagrange's equation and finite elements.- 2.3.3 Newton-Euler equation and modal expansion.- 2.3.4 Newton-Euler equation and finite elements.- 2.3.5 Other methods.- 2.4 Control of flexible manipulators.- 2.4.1 Non-adaptive control.- 2.4.2 Adaptive control.- 2.5 Conclusions.- 3 Dynamic model of a single-link flexible manipulator.- 3.1 Introduction.- 3.2 Analytic model of a single-link arm.- 3.2.1 Energy terms.- 3.2.2 Derivation of the dynamic equation.- 3.2.3 Solution of the differential equation.- 3.3 Dynamic model based on natural modes.- 3.3.1 Truncation of the modal model.- 3.3.2 Combined modal and finite element models.- 3.4 Dynamic models based on assumed modes.- 3.4.1 Model based on cantilever modes.- 3.4.2 Model based on pinned-pinned modes.- 3.4.3 Choice of assumed modes.- 3.5 Models for control design.- 3.5.1 Variation of poles and zeros with loading.- 3.5.2 Truncation revisited-exact residues versus exact zeros.- 3.6 Conclusions.- 4 An experimental single-link flexible arm.- 4.1 Introduction.- 4.2 Experimental apparatus.- 4.2.1 Overview.- 4.2.2 The flexible arm.- 4.2.3 Actuator and power amplifier.- 4.2.4 Sensors.- 4.2.5 Computer for control law implementation.- 4.2.6 The controlled system.- 4.3 Experimental identification of the system model.- 4.3.1 D.C. gains.- 4.3.2 Poles and zeros.- 4.3.3 Damping.- 4.4 Verification of the arm model.- 4.5 Conclusions.- 5 Control design for the single-link arm.- 5.1 Introduction.- 5.2 Digital control design.- 5.2.1 Pole placement control.- 5.3 Experimental flexible arm control.- 5.3.1 Hub angle control.- 5.3.2 Tip position control.- 5.4 Conclusions.- 6 Multi-link flexible arm dynamics.- 6.1 Introduction.- 6.2 Dynamic model of a planar two-link flexible arm.- 6.2.1 Kinetic energy.- 6.2.2 Potential energy.- 6.2.3 External work.- 6.2.4 System Lagrangian and final dynamic equations.- 6.3 Linearisation of the dynamic model.- 6.4 Properties of the linearised model.- 6.5 Variation of the linearised model with loading and configuration.- 6.5.1 Convergence properties of the modal model.- 6.5.2 Effect of changing loading and arm configuration.- 6.5.3 Effect of joint locking.- 6.6 Gravitational effects.- 6.6.1 Potential energy due to gravity.- 6.6.2 Modification of the dynamic model to include gravitational effects.- 6.6.3 Effect of changing load and orientation in the presence of gravity.- 6.7 Control of multi-link manipulators.- 6.8 Conclusions.- 7 A perturbation approach to changing dynamics.- 7.1 Introduction.- 7.2 Canonical perturbation theory.- 7.2.1 Hamiltonian mechanics.- 7.2.2 Canonical transformations.- 7.2.3 Separation of variables.- 7.2.4 Time-dependent perturbation theory.- 7.2.5 Action-angle variables.- 7.3 Extension of canonical perturbation theory to flexible systems.- 7.4 Perturbation analysis based on orthogonality of modes.- 7.4.1 Orthogonality properties of natural modes.- 7.4.2 Eigenvalue expansion theorem.- 7.4.3 Perturbation analysis.- 7.5 Assumed modes formulation.- 7.5.1 Perturbations in the boundary conditions.- 7.5.2 Modified Euler method.- 7.5.3 Complexity and accuracy.- 7.5.4 Application of the perturbation analysis to damped systems.- 7.6 Conclusions.- 8 Extended perturbation techniques.- 8.1 Introduction.- 8.2 Extended perturbation analysis.- 8.2.1 State-space expansion.- 8.2.2 Symmetric expansion.- 8.3 Discrete-time perturbation analysis.- 8.4 Corrective control design.- 8.5 Explicit control correction.- 8.5.1 Experimental demonstration of corrective control design.- 8.6 Direct update of controller gains.- 8.6.1 Reconfiguration of a pole placement regulator.- 8.6.2 Compensation for system perturbations.- 8.7 Investigation of corrective control on the two-link arm model.- 8.8 A general perturbation-based control scheme.- 8.9 Conclusions.- 9 Looking to the future - high performance control.- 9.1 Introduction.- 9.2 Controller Design.- 9.3 The Provision of Feedforward.- 9.3.1 Demand Filtering by Impulse Convolution.- 9.3.2 Inverse Dynamics.- 9.4 Feedback control.- A Symbols and Nomenclature.- B Experimental determination of arm parameters.- C Modal model of the single-link arm.- D Mass and stiffness terms for the two-link arm.- E Derivation of the assumed modes perturbation analysis.- F Transformation from a state-space to an input/output model.