M. Vidyasagar

Robot Modeling and Control

Preface. 1. Introduction. 2. Rigid Motions and Homogeneous Transformations. 3. Forward and Inverse Kinematics. 4. Velocity Kinematics-The Jacobian. 5. Path and Trajectory Planning. 6. Independent Joint Control. 7. Dynamics. 8. Multivariable Control. 9. Force Control. 10. Geometric Nonlinear Control. 11. Computer Vision. 12. Vision-Based Control. Appendix A: Trigonometry. Appendix B: Linear Algebra. Appendix C: Dynamical Systems. Appendix D: Lyapunov Stability. Index.

Nonlinear Systems Analysis

Introduction. Non-linear Differential Equations. Second-Order Systems. Approximate Analysis Methods. Lyapunov Stability. Input-Output Stability. Differential Geometric Methods. Appendices: Prevalence of Differential Equations with Unique Solutions, Proof of the Kalman-Yacubovitch Lemma and Proof of the Frobenius Theorem.

Robust controllers for uncertain linear multivariable systems

Abstract This paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems. In the first part, it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties, and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty. In the second part, in contrast with the first part, it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known. The type of plant uncertainty is the so-called “stable-factor” uncertainty, and necessary and sufficient conditions are given for robust stabilization. In the third part, the model of uncertainty is a ball in the space of rational matrices metrized by the so-called graph metric, and sufficient conditions for robust stabilization are derived.

Paper: Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems

In this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system. These methods are based on the concept of a maximal Lyapunov function, which is introduced in this paper. A partial differential equation characterizing a maximal Lyapunov function is derived, and the relationships of this equation as compared to Zubovs partial differential equation are discussed. An iterative procedure is given for solving the new partial differential equation. This procedure yields Lyapunov function candidates that are rational functions rather than polynomials. The method is applied to four two-dimensional examples and one three-dimensional example, and it is shown that the estimates obtained using this method are, in many cases, substantially better than those obtained using known methods.

Robust linear compensator design for nonlinear robotic control

The motion control of robotic manipulators is investigated using a recently developed approach to linear multivariable control known as the stable factorization approach. Given a nominal model of the manipulator dynamics, the control scheme consists of an approximate feedback linearizing control followed by a linear compensator design based on the stable factorization approach. Using a multiloop version of the small gain theorem, robust trajectory tracking is shown under the assumption that the deviation of the model from the true system satisfies certain norm inequalities. In turn, these norm inequalities lead to quantifiable bounds on the tracking error.

On the stabilization of nonlinear systems using state detection

In this paper, we study the problem of stabilizing a nonlinear control system by means of a feedback control law, in cases where the entire state of the system is not available for measurement. The proposed method of stabilization consists of three parts: 1) determine a stabilizing control law based on state feedback, assuming the state vector x(t) can be measured; 2) construct a state detection mechanism, which generates a vector z(t) such that z(t)-x(t)\rightarrow 0 as t\rightarrow \infty and 3) apply the previously determined control law to z(t) . This scheme is well established for linear time-invariant systems, and its global convergence has previously been studied in the case of nonlinear systems. Hence, the contribution of this paper is in showing that such a scheme works in the absence of any linearity assumptions, and in studying both local asymptotic stability and global asymptotic stability.

Bad Data Rejection Properties of Weighted Least Absolute Value Techniques Applied to Static State Estimation

A new method for obtaining an estimate of the state of a power system using weighted least absolute value (WLAV) techniques is presented. This paper shows how the WLAV estimator simultaneously detects and rejects bad data while obtaining an accurate estimate of the state. The test results presented show that the WLAV estimates, obtained with or without bad data present, are comparable to the estimates obtained by a weighted least squares (WLS) estimator using only good data.

Transfer functions for a single flexible link

The authors examine some issues in the transfer function modeling of a single flexible link. Using the assumed-modes approach, it is possible to find the transfer function between the torque input and the net tip deflection. It is shown that when the number of modes is increased for more accurate modeling, the relative degree of the transfer function becomes ill-defined. This can greatly affect the performance of a controller designed using the model. In addition, any attempt to identify the transfer function is also affected. An alternative approach that uses the rigid body deformations minus the elastic deformations as the output is proposed. This solves the above problems and results in a transfer function with a well-defined relative degree of two. Even if three or four vibration modes are used, the leading coefficient in the numerator is much larger with the proposed new output.<<ETX>>

NEW ALGORITHMS FOR CONSTRAINED MINIMAX OPTIMIZATION

A constrained minimax problem is converted to minimization of a sequence of unconstrained and continuously differentiable functions in a manner similar to Morrisons method for constrained optimization. One can thus apply any efficient gradient minimization technique to do the unconstrained minimization at each step of the sequence. Based on this approach, two algorithms are proposed, where the first one is simpler to program, and the second one is faster in general. To show the efficiency of the algorithms even for unconstrained problems, examples are taken to compare the two algorithms with recent methods in the literature. It is found that the second algorithm converges faster with respect to the other methods. Several constrained examples are also tried and the results are presented.

A Lagrangian formulation of the dynamic model for flexible manipulator systems

This paper presents a procedure for deriving dynamic equations for manipulators containing both rigid and flexible links. The equations are derived using Hamiltons principle, and are nonlinear integro-differential equations. The formulation is based on expressing the kinetic and potential energies of the manipulator system in terms of generalized coordinates. In the case of flexible links, the mass distribution and flexibility are taken into account. The approach is a natural extension of the well-known Lagrangian method for rigid manipulators. Properties of the dynamic matrices, which lead to a less computation, are shown. Boundary-value problems of continuous systems are briefly described. A two-link manipulator with one rigid link and one flexible link is analyzed to illustrate the procedure.