Perturbation Techniques for Flexible Manipulators

A manipulator, or 'robot', consists of a series of bodies (links) connected by joints to form a spatial mechanism. Usually the links are connected serially to form an open chain. The joints are either revolute (rotary) or prismatic (telescopic), various combinations of the two giving a wide va­ riety of possible configurations. Motive power is provided by pneumatic, hydraulic or electrical actuation of the joints. The robot arm is distinguished from other active spatial mechanisms by its reprogrammability. Therefore, the controller is integral to any de­ scription of the arm. In contrast with many other controlled processes (e. g. batch reactors), it is possible to model the dynamics of a ma­ nipulator very accurately. Unfortunately, for practical arm designs, the resulting models are complex and a considerable amount of research ef­ fort has gone into improving their numerical efficiency with a view to real time solution [32,41,51,61,77,87,91]. In recent years, improvements in electric motor technology coupled with new designs, such as direct-drive arms, have led to a rapid increase in the speed and load-carrying capabilities of manipulators. However, this has meant that the flexibility of the nominally rigid links has become increasingly significant. Present generation manipulators are limited to a load-carrying capacity of typically 5-10% of their own weight by the requirement of rigidity. For example, the Cincinatti-Milicron T3R3 robot weighs more than 1800 kg but has a maximum payload capacity of 23 kg.

Contenu

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.