Range Space Method Research Articles

Lie Algebraic Control for the Stabilization of Nonlinear Multibody System Dynamics Simulation

It is well known that when equations of motion are formulated using Lagrange multipliers for multibody dynamic systems, one obtains a redundant set of differential algebraic equations. Numerical integration of these equations can lead to numerical difficulties associated with constraint violation drift. One approach that has been explored to alleviate this difficulty has been contraint stabilization methods. In this paper, a family of stabilization methods are considered as partial feedback linearizing controllers. Several stabilization methods including the range space method, null space method, Baumgarte's method, and the damping and stiffness penalty methods are examined. Each can be construed as a particular partial feedback linearizing controller. The paper closes by comparing several of these constraint stabilization methods to another method suggested by construction: the variable structure sliding (VSS) control. The VSS method is found to be the most efficient, stable, and robust in the presence of singularities.

Constraint violation in concurrent range space methods for transient dynamic analysis

Despite the fact that many authors have dismissed redundant coordinate formulations as being of order N 3, and hence less attractive than recursive formulations, this paper presents recent research that demonstrates that a class of redundant, nonrecursive multibody formulations consistently achieve order N computational cost for systems of rigid and/or flexible bodies. This formulation is based upon the rapidly convergent preconditioned range space formulation for nonlinear multibody dynamics. The method can be traced to its foundation in equality constrained quadratic optimization and incorporates some aspects of subdomain decomposition techniques used in variational boundary value problems. Until recently, however, this approach had not been investigated in the context of multibody simulation, and presents theoretical questions unique to nonlinear dynamics. Nonrecursive methods have additional advantages with respect to recursive order N methods: (1) the formalisms retain the highly desirable order N computational cost; (2) the techniques are amenable to concurrent simulation strategies; (3) the approaches do not depend upon system topology to induce concurrency; (4) the methods can be derived to balance the computational load automatically on concurrent multiprocessors. However, the preconditioned range space formulation can suffer from degradation in accuracy due to constraint violation drift. This effect becomes even more pronounced when one induces concurrency via domain decomposition. A central theme of this paper is the derivation and development of concurrent constraint stabilization methods that retain the order N computational cost of the underlying preconditioned range space formulation. It is demonstrated that variants of the penalty and augmented Lagrangian methods for nonlinear multibody dynamics are well suited for concurrent constraint stabilization. Moreover, this paper presents new theoretical results regarding the rate of convergence of order N constraint stabilization schemes associated with the newly introduced class of methods.

Numerical Continuation and Singularity Detection Methods for Parametric Nonlinear Programming

Numerical methods are developed for continuation, solution-type determination, and singularity detection in the parametric nonlinear programming problem. This problem is first converted to a closed, “active set” system of equations $\bar F ( z,\alpha ) = 0$, which includes a nonstandard normalization of the multipliers. A framework is then developed for combining various numerical continuation methods with a large number of null and range space methods from constrained optimization. By exploiting the special structure in the parametric optimization problem, solution-type classification and singularity detection are shown to require minimal additional expense beyond that involved in the continuation procedure itself. Due to the special structure of these problems, singularity detection methods are more comprehensive than those for general nonlinear equations. In this development, the Schur complement and related results play an important and unifying role. As an illustration, these methods are used to produce a “global” parametric analysis for a model problem from design optimization. This example exhibits an extensive number of solution paths, each of the basic types of singularities, multiple optima, regions of sensitivity, and jump phenomena.

Sub-domain decomposition methods and computational controls for multibody dynamical systems

This paper presents a concurrent methodology to simulate the dynamics of flexible multibody systems with a large number of degrees of freedom. A general class of open-loop structures is treated and a redundant coordinate formulation is adopted. A range space method is used in which the constraint forces are calculated using a preconditioned conjugate gradient method. By using a preconditioner motivated by the regular ordering of the directed graph of the structures, it is shown that the method is order N in the total number of coordinates of the system. The overall formulation has the advantage that it permits fine parallelization and does not rely on system topology to induce concurrency. It can be efficiently implemented on the present generation of parallel computers with a large number of processors. Validation of the method is presented via numerical simulations of space structures incorporating large number of flexible degrees of freedom.