Robust Framework for the Computed Torque Control of Nondeterministic Multibody Dynamic Systems

Considering uncertainty is inarguably a crucial aspect of dynamic analysis, design, and control of a mechanical system. When it comes to multibody problems, the effect of uncertainty in the system’s parameters and excitations becomes even more significant due to the accumulation of inaccuracies. For this reason, this paper presents a detailed research on the use of the Polynomial Chaos Expansion (PCE) method for the control of nondeterministic multibody systems. PCE is essentially a way to compactly represent random variables. In this scheme, each stochastic response output and random input is projected onto the space of appropriate independent orthogonal polynomial basis functions. In the field of robotics, a required task is to force robotic arms to follow designated paths. Controlling such systems usually leads to difficulties since the dynamic equations of multibody problems are highly nonlinear. Computed Torque Control Law (CTCL) is able to overcome these difficulties by using feedback linearization to evaluate the required torque/force at any time to make the system follow a trajectory. In this paper, a mathematical framework is introduced to apply the Computed Torque Control Law to a multibody system with uncertainty. Surprisingly, it is shown that using this control scheme, uncertainty in geometry does not affect the closed-loop equations of controlled systems. Both the intrusive PCE method and the Monte Carlo approach are used to control a fully actuated two-link planar elbow arm where each link is required to follow a specified path. Lastly, a comparison of the time efficiency and accuracy between the traditionally used Monte Carlo method and the intrusive PCE is presented. The results indicate that the intrusive PCE approach can provide better accuracy with much less computation time than the Monte Carlo method.

Control Constraint of Underactuated Aerospace Systems

This paper discusses the problem of control constraint realization applied to the design of maneuvers of complex underactuated systems modeled as multibody problems. Applications of interest in the area of aerospace engineering are presented and discussed. The tangent realization of the control constraint is discussed from a theoretical point of view and is used to determine feedforward control of realistic underactuated systems. The effectiveness of the computed feedforward input is subsequently verified by applying it to more detailed models of the problems, in the presence of disturbances and uncertainties in combination with feedback control. The problems are solved using a free general-purpose multibody software that writes the constrained dynamics of multifield problems formulated as differential-algebraic equations. The equations are integrated using unconditionally stable algorithms with tunable dissipation. The essential extension to the multibody code consisted of the addition of the capability to write arbitrary constraint equations and apply the corresponding reaction multipliers to arbitrary equations of motion. The modeling capabilities of the formulation could be exploited without any undue restriction on the modeling requirements.

Parallel Linear Equation Solvers and OpenMP in the Context of Multibody System Dynamics

Computational efficiency of numerical simulations is a key issue in multibody system (MBS) dynamics, and parallel computing is one of the most promising approaches to increase the computational efficiency of MBS dynamic simulations. The present work evaluates two non-intrusive parallelization techniques for multibody system dynamics: parallel sparse linear equation solvers and OpenMP. Both techniques can be applied to existing simulation software with minimal changes in the code structure; this is a major advantage over MPI (Message Passing Interface), the de facto standard parallelization method in multibody dynamics. Both techniques have been applied to parallelize a starting sequential implementation of a global index-3 augmented Lagrangian formulation combined with the trapezoidal rule as numerical integrator, in order to solve the forward dynamics of a variable number of loops four-bar mechanism. This starting implementation represented a highly optimized code, where the overhead of parallelization would represent a considerable part of the total amount of elapsed time in calculations. Several multi-threaded solvers have been added to the original software. In addition, parallelizable regions of the code have been detected and multi-threaded via OpenMP directives. Numerical experiments have been performed to measure the efficiency of the parallelized code as a function of problem size and matrix filling ratio. Results show that the best parallel solver (Pardiso) performs better than the best sequential solver (CHOLMOD) for multibody problems of large and medium sizes leading to matrix fillings above 10 non-zeros per variable. OpenMP also proved to be advantageous even for problems of small sizes, in despite of the small percentage of parallelizable workload with respect to the total burden of the execution of the code. Both techniques delivered speedups above 70% of the maximum theoretical values for a wide range of multibody problems.

TRANSPORT IN DYNAMICAL ASTRONOMY AND MULTIBODY PROBLEMS

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.