This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, to facilitate the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matix relating variations in dependent and independent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, is developed that integrates for only the independent variables, yet effectively determines dependent variables. Numerical results are presented for planar motion of two tracked vehicular systems with 13 and 24 degrees of freedom. Computational efficiency of the algorithm is shown to be an order of magnitude better than previously employed algorithms.

I. Structural Optimization FormulationD ESIGN sensitivity analysis methods for calculation of first derivatives of structural performance with respect to design parameters have become quite well developed and applied. 1>2 It is the purpose of this Note to extend the adjoint method of design sensitivity analysis to calculation of second derivatives of performance with respect to design parameters. The value of second-derivative information in design optimization is well known. Using only first-order (gradient) information, optimization algorithms must resort to onedimensional search or heuristic step-size selection methods. If second-derivative information is available, at a reasonable computing cost, much more powerful iterative optimization algorithms are possible. 3 Finite-dimensional (matrix) structural models are considered here, even though the methods presented can be extended to distributed parameter (boundary-value problem) structural models. Structural design is presumed to be described by a vector of design parameters b=[b 1 ,b 2 ,...,b k ] T . Structural response is described by a vector of nodal displacement coordinates z=[Zi,z 2 ,«.,Zn\ 7 ', called state variables.Structural behavior is modeled by a finite-element method, that yields the matrix equationwhere K(b) is an nxn symmetric, positive-definite stiffness matrix whose components are twice continuously differentiable functions of b, and Q(b) is a load vector that may depend on design. Constraints on structural performance are generally written in the form of inequality constraints Such constraints may include bounds on stress, displacement, and design variable values. The form of the function \l/ f (b,z) and the variables that appear explicitly depend on the constraint represented.

No abstract

A recursive formulation of the equations of motion of spatial constrained mechanical systems is derived, using tools of variational and vector calculus. Position, virtual displacement, velocity, and acceleration relations are developed, using relative coordinates between kinematically coupled bodies. Graph theoretic definition of connectivity of systems with tree structure is used to define computational sequences for formulation and solution of the equations of motion that is efficient and well suited for parallel computation. Using a variational form of the equations of dynamics, inertia and right-side terms are reduced from outboard bodylcentroidal reference frames to an inboard body centroidal reference frame. A recursive algorithm is developed to reduce equations of motion to a base body. A robot arm is analyzed to illustrate use and efficiency of the method. '

Numerical algorithms for mapping boundaries of manipulator workspaces are developed and illustrated. Analytical criteria for boundaries of workspaces for both manipulators having the same number of input and output coordinates and redundantly controlled manipulators with a larger number of inputs than outputs are well known, but reliable numerical methods for mapping them have not been presented. In this paper, a numerical method is first developed for finding an initial point on the boundary. From this point, a continuation method that accounts for simple and multiple bifurcation of one-dimensional solution curves is developed. Second order Taylor expansions are derived for finding tangents to solution curves at simple bifurcation points of continuation equations and for characterizing barriers to control of manipulators. A recently developed method for tangent calculation at multiple bifurcation points is employed. A planar redundantly controlled serial manipulator is analyzed, determining both the exterior boundary of the accessible output set and interior curves that represent local impediments to motion control. Using these methods, more complex planar and spatial Stewart platform manipulators are analyzed in a companion paper.