G. Richard Eisler

Flexible Robot Dynamics and Controls

Contents. 1. Introduction. 2. Mathematical Preliminaries. 3. Flexible Robot Dynamic Modeling. 4. System Identification. 5. Input Shaping for Path Planning. 6. Linear Feedback Control. 7. Nonlinear Systems and Sliding Mode Control. 8. Adaptive Sliding Mode Control. Appendix A: VF02AD Optimization. Appendix B: MATLAB%% Optimization. Appendix C: Hardware & Software Support.

Effect of Very High Travel Speeds on Melting Efficiency in Laser Beam Welding

Calorimetric measurements of the net heat input to the workpiece have been made to determine the effect of very high travel speeds on laser weld melting efficiency. Very high welding speeds are required in welding applications such as automotive where lasers are now applied extensively. Travel speeds as fast as 530 mm/s for continuous wave CO2 laser welding on 304 stainless steel have been examined in this study. Melting efficiency indicates what fraction of the laser power absorbed is used to produce melting rather than undesirable base metal heating. It was found that melting efficiency initially increased then slowly decreased as fusion zone dimensions changed. Dimensionless parameter correlations for melting efficiency based on heat flow theory have been presented for both 2D and 3D heat flow geometries. The levels of melting efficiency observed are close to the maximum values that are predicted with these correlations. Determinations of the melting point isotherms and analysis of changes to the dimensionless parameters have been shown to predict the observed changes in melting efficiency. The results indicate that an enhanced melting efficiency is obtained in high speed laser welding when either the fusion zone aspect ratio or the joint geometry promote 2D heat flow.

Trajectory Matching Flight-Path Optimization of Aerospace Vehicles

A new class of six-degree-of-freedom optimization problems referred to as trajectory matching e ight-path optimization is addressed and explicitly described via an example problem. The essence of this class of e ightpath optimization problems is to force an aerospace vehicle to follow prescribed translational and rotational velocity histories or to motion-match. For this example, a lightweight, axisymmetric re-entry vehicle decoy is designed to match the motion of a typical full-weight re-entry vehicle by optimizing its mass properties. The optimization problem, which is performed in two steps, is given as the minimization of the difference between the target trajectory and the actual trajectory subject to physical constraints. The e rst step utilizes a unique stability measure to develop similarity parameters that provide excellent initial guesses for the numerical optimization routine employed in the second step. The second step utilizes a sequential quadratic programming algorithm to numerically solve the constrained minimization problem. Simulation results demonstrate the robust trajectory matching of the lightweight re-entry vehicle.

3. Introduction to Dynamic Programming

This book concerns the use of a method known as dynamic programming (DP) to solve large classes of optimization problems. We will focus on discrete optimization problems for which a set or sequence of decisions must be made to optimize (minimize or maximize) some function of the decisions. There are of course numerous methods to solve discrete optimization problems, many of which are collectively known as mathematical programming methods. Our objective here is not to compare these other mathematical programming methods with dynamic programming. Each has advantages and disadvantages, as discussed in many other places. However, we will note that the most prominent of these other methods is linear programming. As its name suggests, it has limitations associated with its linearity assumptions whereas many problems are nonlinear. Nevertheless, linear programming and its variants and extensions (some that allow nonlinearities) have been used to solve many real world problems, in part because very early in its development software tools (based on the simplex method) were made available to solve linear programming problems. On the other hand, no such tools have been available for the much more general method of dynamic programming, largely due to its very generality. One of the objectives of this book is to describe a software tool for solving dynamic programming problems that is general, practical, and easy to use, certainly relative to any of the other tools that have appeared from time to time. One reason that simplex-based tools for solving linear programming problems have been successful is that, by the nature of linear programming, problem specification is relatively easy. A basic LP problem can be specified essentially as a system or matrix of equations with a finite set of numerical variables as unknowns. That is, the input to an LP software tool can be provided in a tabular form, known as a tableaux. This also makes it easy to formulate LP problems as a spreadsheet. This led to spreadsheet system providers to include in their product an LP solver, as is the case with Excel. A software tool for solving dynamic programming problems is much more difficult to design, in part because the problem specification task in itself

Adaptive Sliding Mode Control

Traditionally, adaptive control is applied to dynamic systems that have constant or slowly-varying, uncertain or unknown parameters, such as, manipulator payloads. In the presence of changing plant dynamics, adaptive control design inherently adjusts control system parameters. Adaptive SMC is a specialized form of adaptive control algorithms that falls into the category of robust adaptive control design. A term is included in the control law development that ensures stability in the presence of disturbances, unmodeled dynamics, and modeling inaccuracies.

Linear Feedback Control

This chapter describes several linear feedback control techniques that can be used to robustly control flexible dynamic systems. As with any dynamic system, it is often difficult to accurately model the system with enough fidelity that open loop control performs as intended. Because modeling errors are often unavoidable, linear feedback is often used to compensate for these modeling uncertainty. Even though many of the flexible dynamic systems are nonlinear, their models can be adequately linearized about operating points and standard linear feedback control techniques can be applied with satisfactory results.

Flexible Robot Dynamic Modeling

Several of the control strategies for flexible link robots described in the remainder of the book rely on an accurate dynamic model of the system. Creating a dynamic model that accounts for link flexibility adds additional challenges beyond the standard rigid link robot dynamics. The most apparent complexity arises due to the additional degrees-of-freedom associated with link deformations. Although in theory this adds an infinite number of degrees-of-freedom, in practice only a finite number are used to generate a model that is sufficiently accurate for predictive simulation and control design. Another complexity (and perhaps a less obvious one) is the appearance of first-order (not negligible) dynamic effects due to second-order kinematic and force effects that at first glance appear to be negligible. For simple robot configurations, these effects can be handled in several intuitive ways. However, for complicated geometries, a systematic approach is needed to ensure that coupling effects are not inadvertently lost. Much of this chapter is devoted to describing such an approach, called the method of quadratic modes.

Nonlinear Systems and Sliding Mode Control

Many systems of practical interest are nonlinear, but sometimes it is possible to consider small motions about an operating and/or equilibrium point. In this case, a linear set of dynamic equations can be formulated, thus facilitating the use of linear analysis and design techniques. When it is inappropriate to linearize the system, the linear design tools cannot be applied and instead nonlinear analysis is required 1,2. Two of the more important analyses that are often needed are stability determination and controller design. Several examples are presented to illustrate these situations.

Input Shaping for Path Planning

Input shaping is an effective way to optimize the performance of robots, flexible structures, spacecraft, telescopes, and other systems that have vibration, control authority, tracking, and/or pointing constraints. These constraints along with the dynamics and kinematics of the system under consideration can be included in a trajectory optimization/path planning procedure to ensure that the system meets the desired performance. Input shaping is particularly useful when the closed-loop controller cannot be modified or tuned. For example, many pedestal-based robots have closed architecture control systems that restrict access to the servo-loop controls. This chapter begins with the overhead gantry robot and a vibration constraint referred to as swing-free input shaping.