A methodology of modeling and analysis of planar mechanical systems is developed based on graph theoretic methods, with improvements in component models. The system model based on cutset and circuit topologies is used to derive a new hybrid cutset-circuit method of formulation of the equations of motion for planar systems. Computer-aided formulation is based on analysis of the substitution procedure mandated by the hybrid cutset-circuit formulation. A new graphical representation of the formulation process is introduced: substitution graphs. No special programming is needed for computer-aided formulation which can be achieved in a symbolic form using the off the shelf Maple symbolic mathematics system. Symbolic formulation requires only inputting the systems equations in an order and form as derived from the analysis of the hybrid formulation. An algorithm for symbolic formulation using Maple is given. A compact set of differential-algebraic equations results, which can be solved numerically. Some simple systems will result in closed-form solutions. A number of examples are given to illustrate the modeling and formulation. Numerical solutions are also given to demonstrate the effectiveness and correctness of the formulation procedure.

The problem of initial probability assignment which is consistent with the available information about a probabilistic system is called a direct problem. E.T. Jaynes' (1957) maximum entropy principle (MaxEnt) provides a method for solving direct problems when the available information is in the form of moment constraints. On the other hand, given a probability distribution, the problem of finding a set of constraints which makes the given distribution a maximum entropy distribution is called an inverse problem. A method based on the MinMax measure to solve the above inverse problem is presented. The MinMax measure of information, defined by Kapur, Baciu and Kesavan (1995), is a quantitative measure of the information contained in a given set of moment constraints. It is based on both maximum and minimum entropy. Computational issues in the determination of the MinMax measure arising from the complexity in arriving at minimum entropy probability distributions (MinEPD) are discussed. The method to solve inverse problems using the MinMax measure is illustrated by solving the problem of estimating a probability density function of a random variable based on sample data.

Kapur et al. (1995) introduced the MinMax information measure, which is based on both maximum and minimum entropy. The major obstacle for using this measure, in practice, is the difficulty in finding the minimum entropy. An analytical expression has already been developed for calculating the minimum entropy when only variance is specified. An analytical formula is obtained for calculating the minimum entropy when only mean is specified, and numerical examples are given for illustration.

The distinction between geometry and dynamic interactions is fundamental for the consistent dynamic analysis of physical systems. A unified treatment of such systems is possible when we adopt a hierarchical mathematical model with a consistent set of embedded abstractions. This new view is adopted in the general formulation strategy for obtaining a simplified dynamics model of mechanical systems. We show that there exists a consistent general extension from the model of constrained particle-mass systems (PMS) to the model of multibody systems (MBS) based entirely on graph-theoretic concepts.

With the aid of graph theory it is possible to combine the topological information of a constrained rigid-body system with the mathematical formulation of the equations of motion in a direct, systematic, and consistent procedural fashion that lends itself to a straightforward computer implementation. Such a graph-theoretical formulation for the dynamics analysis and simulation of three-dimensional constrained rigid-body systems is presented. It is shown that through the graph-theoretic approach, the topological information contributes directly to the systematic formulation procedure and also to an efficient, directly implementable computer algorithm. This algorithm can be implemented in a recursive fashion that reduces the computational cost substantially. The main components of the numerical procedure can be traced back directly to the system graphs of the formulation phase with very little effort. The resulting system of differential equations is left in its implicit form and solved numerically.>

Generalizations of the maximum entropy principle (MEP) of E.T. Jaynes (1957) and the minimum discrimination information principle (MDIP) of S. Kullback (1959) are described. The generalizations have been achieved by enunciating the entropy maximization postulate and examining its consequences. The inverse principles which are inherent in the MEP and MDIP are made quite explicit. Several examples are given to illustrate the power and scope of the generalized maximum entropy principle that follows from the entropy maximization postulate.>

The graph-theoretic models of a prismatic joint, a revolute joint, and an open kinematic chain are presented. The Denavit-Hartenberg representation of linkages is encompassed within the general framework of graph-theoretic system theory. The final mathematical model derived by this formalism is a system of differential and algebraic equations.

A set of mixed differential and algebraic equations (DAEs) which arises in the simulation of a robot manipulator is solved simultaneously using implicit integration. The dimension of the DAEs which have to be solved by LU factorization at each integration step can be reduced to the number of degrees of freedom by exploring the special structure of the Jacobian matrix of DAEs. The independent and dependent generalized coordinates are determined directly from the system topology. The simulation of a 6-R manipulator is given as an example.

The discipline of linear graph theory, which has been applied extensively to the analysis of electrical networks, is now extended to encompass general three-dimensional isolated rigid-body systems. The graph-theoretic models of a rigid body and its relevant components are discussed elaborately to pave the way for further studies in multibody systems. Utilizing the analysis based on system topology, the mathematical model of a mechanical system is derived by means of a hybrid formulation in which Euler parameters (or normalized quaternions) are incorporated to define the orientations of rigid bodies. A single body system serves as an illustrative example.