Firdaus E. Udwadia

A New Perspective on Constrained Motion

The explicit general equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of lagrangian mechanics.

An efficient QR based method for the computation of Lyapunov exponents

Abstract An efficient and numerically stable method to determine all the Lyapunov characteristic exponents of a dynamical system is developed. Numerical experiments are presented highlighting some aspects of convergence, accuracy and efficiency in the computation of the Lyapunov characteristic exponents.

Simultaneous optimization of controlled structures

A formulation is presented for finding the combined optimal design of a structural system and its control by defining a composite objective function as a linear combination of two components; a structural objective and a control objective. When the structural objective is a function of the structural design variables only, and when the control objective is represented by the quadratic functional of the response and control energy, it is possible to analytically express the optimal control in terms of any set of “admissible” structural design variables. Such expression for the optimal control is used recursively in an iterative Newton-Raphson search scheme, the goal of which is to determine the corresponding optimal set of structural design variables that minimize the combined objective function. A numerical example is given to illustrate the computational procedure. The results indicate that significant improvement of the combined optimal design can be achieved over the traditional separate optimization.

A unifying framework for robot control with redundant DOFs

Abstract Recently, Udwadia (Proc. R. Soc. Lond. A 2003:1783–1800, 2003) suggested to derive tracking controllers for mechanical systems with redundant degrees-of-freedom (DOFs) using a generalization of Gauss’ principle of least constraint. This method allows reformulating control problems as a special class of optimal controllers. In this paper, we take this line of reasoning one step further and demonstrate that several well-known and also novel nonlinear robot control laws can be derived from this generic methodology. We show experimental verifications on a Sarcos Master Arm robot for some of the derived controllers. The suggested approach offers a promising unification and simplification of nonlinear control law design for robots obeying rigid body dynamics equations, both with or without external constraints, with over-actuation or underactuation, as well as open-chain and closed-chain kinematics.

A new perspective on the tracking control of nonlinear structural and mechanical systems

Based on recent results from analytical dynamics, this paper develops a class of tracking controllers for controlling general, nonlinear, structural and mechanical systems. Unlike most control methods that perform some kind of linearization and/or nonlinear cancellation, the methodology developed herein views the nonlinear control problem from a different perspective. This leads to a simple and new control methodology that is capable of ‘exactly’ maintaining the nonlinear system along a certain trajectory, which, in general, may be described by a set of differential equations in the observations/measurements. The approach requires very little computation compared with standard approaches. It is therefore useful for online real–time control of nonlinear systems. The methodology is illustrated with two examples.

Uniqueness of Damping and Stiffness Distributions in the Identification of Soil and Structural Systems

As the interest in the seismic design of structures has increased considerably over the past few years, accurate predictions of the dynamic responses of soil and structural systems has become necessary. Such predictions require a knowledge of the dynamic properties of the systems under consideration. This paper is concerned with the uniqueness of the results in the identification of such properties. More specifically, the damping and stiffness distributions, which are of importance in the linear range of response, have been investigated. An N-storied structure or an N-layered soil medium is modeled as a coupled, N-degree-of-freedom, lumped system consisting of masses, springs, and dampers. Then, assuming the mass distribution to be known, the problem of identification consists of determining the stiffness and damping distributions from the knowledge of the base excitation and the resulting response at any one mass level. It is shown that if the response of the mass immediately above the base is known, the stiffness and damping distributions can be uniquely determined. Following this, some nonuniqueness problems have been discussed in relation to the commonly used ideas of system reduction in the study of layered soil media. A numerical example is provided to verify some of these concepts and the nature of nonuniqueness of identification is indicated by showing how two very different (yet physically reasonable) systems could yield identical excitation-response pairs. Errors in the calculation of the dynamic forces, due to erroneous identification have also been illustrated thus making the results of the present study useful from the practical standpoint of the safe design of structures to ground shaking.

Optimal tracking control of nonlinear dynamical systems

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

Creativity and innovation in organizations: Two models and managerial implications

Abstract Technological innovation is emerging as the single most important factor to influence business success in todays intensely competitive and dynamic environment. Accordingly, scholars as well as practitioners are contributing to a rapidly growing body of knowledge for the effective management of innovation. However, surprisingly, very little attention is being paid to the organizational and managerial issues pertaining to creativity , which is the most basic and the most critical element in the process of innovation. This paper highlights creativity as the central issue in management of innovation, and presents two models to further our understanding of the dynamics of creativity in organizational settings, and the place of creativity in the innovation process. For a comprehensive understanding of creative behavior and performance in organizations this paper develops a Multiple Perspective Model. This model includes three perspectives, the Individual , the Technical , and the Organizational , which focus respectively on the distinctive individual characteristics associated with creativity, the needed technical resources—material as well as human—for creativity, and the organizational practices and manegerial actions that aid or stifle creativity. The exposition of this model is followed by an analysis of its implications for the management of creativity. Next, a model of the innovation process is proposed, in which innovation is shown as being contingent on a cascade of creative efforts in various functional areas and across different fields of specialization. These two models are expected to be useful to both researchers and practitioners in ferreting out the issues of primary significance, nurturing creativity and enhancing innovation in organizations.

Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

Some Uniqueness Results Related to Building Structural Identification

This paper studies the nature of uniqueness in the identification of building structural systems subjected to strong ground shaking. Characterization of the stiffness distribution in the structure from a knowledge of response of one of the floors to a base excitation is investigated. It is shown that uniqueness of the results, in the inverse problem, can be established by proper sensor location. At sensor locations where nonunique solutions are present, an upper bound on the number of such solutions has been presented. The degree of nonuniqueness is found to monotonically increase with increasing height of sensor in the building system from at most one, for a sensor located at the first floor level, to at most N! for a sensor located at the Nth floor of an N story structure.