Mohammad Poursina

Advances in the Application of the Divide-and-Conquer Algorithm to Multibody System Dynamics

This paper summarizes the various recent advancements achieved by utilizing the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with many aspects of modeling, designing, and simulating articulated multibody systems. This basic algorithm provides a framework to realize O(n) computational complexity for serial task scheduling. Furthermore, the framework of this algorithm easily accommodates parallel task scheduling, which results in coarse-grain O(log n) computational complexity. This is a significant increase in efficiency over forming and solving the Newton–Euler equations directly. A survey of the notable previous work accomplished, though not all inclusive, is provided to give a more complete understanding of how this algorithm has been used in this context. These advances include applying the DCA to constrained systems, flexible bodies, sensitivity analysis, contact, and hybridization with other methods. This work reproduces the basic mathematical framework for applying the DCA in each of these applications. The reader is referred to the original work for the details of the discussed methods.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

New and Extended Applications of the Divide-and-Conquer Algorithm for Multibody Dynamics

This work presents a survey of the current and ongoing research by the authors who use the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with various aspects of multibody dynamics. This work provides a brief discussion of various topics that are extensions of previous DCA-based algorithms or novel uses of this algorithm in the multibody dynamics context. These topics include constraint error stabilization, spline-based modeling of flexible bodies, model fidelity transitions for flexible-body systems, and large deformations of flexible bodies. It is assumed that the reader is familiar with the “Advances in the Application of the DCA to Multibody System Dynamics” text as the notation used in this work is explained therein and provides a summary of how the DCA has been used previously.

Extended Divide-and-Conquer Algorithm for Uncertainty Analysis of Multibody Systems in Polynomial Chaos Expansion Framework

In this paper, an advanced algorithm is presented to efficiently form and solve the equations of motion of multibody problems involving uncertainty in the system parameters and/or excitations. Uncertainty is introduced to the system through the application of polynomial chaos expansion (PCE). In this scheme, states of the system, nondeterministic parameters, and constraint loads are projected onto the space of specific orthogonal base functions through modal values. Computational complexity of traditional methods of forming and solving the resulting governing equations drastically grows as a cubic function of the size of the nondeterministic system which is significantly larger than the original deterministic multibody problem. In this paper, the divide-and-conquer algorithm (DCA) will be extended to form and solve the nondeterministic governing equations of motion avoiding the construction of the mass and Jacobian matrices of the entire system. In this strategy, stochastic governing equations of motion of each individual body as well as those associated with kinematic constraints at connecting joints are developed in terms of base functions and modal terms. Then using the divide-and-conquer scheme, the entire system is swept in the assembly and disassembly passes to recursively form and solve nondeterministic equations of motion for modal values of spatial accelerations and constraint loads. In serial and parallel implementations, computational complexity of the method is expected to, respectively, increase as a linear and logarithmic function of the size.

An improved fast multipole method for electrostatic potential calculations in a class of coarse-grained molecular simulations

This paper presents a novel algorithm to approximate the long-range electrostatic potential field in the Cartesian coordinates applicable to 3D coarse-grained simulations of biopolymers. In such models, coarse-grained clusters are formed via treating groups of atoms as rigid and/or flexible bodies connected together via kinematic joints. Therefore, multibody dynamic techniques are used to form and solve the equations of motion of such coarse-grained systems. In this article, the approximations for the potential fields due to the interaction between a highly negatively/positively charged pseudo-atom and charged particles, as well as the interaction between clusters of charged particles, are presented. These approximations are expressed in terms of physical and geometrical properties of the bodies such as the entire charge, the location of the center of charge, and the pseudo-inertia tensor about the center of charge of the clusters. Further, a novel substructuring scheme is introduced to implement the presented far-field potential evaluations in a binary tree framework as opposed to the existing quadtree and octree strategies of implementing fast multipole method. Using the presented Lagrangian grids, the electrostatic potential is recursively calculated via sweeping two passes: assembly and disassembly. In the assembly pass, adjacent charged bodies are combined together to form new clusters. Then, the potential field of each cluster due to its interaction with faraway resulting clusters is recursively calculated in the disassembly pass. The method is highly compatible with multibody dynamic schemes to model coarse-grained biopolymers. Since the proposed method takes advantage of constant physical and geometrical properties of rigid clusters, improvement in the overall computational cost is observed comparing to the tradition application of fast multipole method.

An Efficient Application of Polynomial Chaos Expansion for the Dynamic Analysis of Multibody Systems With Uncertainty

In this paper the mathematical framework of an advanced algorithm is presented to efficiently form and solve the equations of motion of a multibody system involving uncertainty in the system parameters and/or the excitations. The uncertainty is introduced to the system through the application of the polynomial chaos expansion. In this scheme, states of the system, nondeterministic parameters, and the constraint loads are expanded using modal values as well as orthogonal basis functions. Computational complexity of the application of traditional methods to solve the stochastic equations of motion of a multibody system drastically grows as a cubic function of the number of the states of the system, uncertain parameters and the maximum degree of the polynomial chosen for the basis function. The presented method forms the equation of motion of the system without forming the entire mass and Jacobian matrices. In this strategy, the stochastic governing equations of motion of each individual body as well as the one associated with the kinematic constraint at the connecting joint are developed in terms of the basis functions and modal coordinates. Then sweeping the system in two passes assembly and disassembly, one can form and solve the stochastic equations of motion. In the assembly pass the non-deterministic equations of motion of the assemblies are obtained. In the disassembly process, these equations are then recursively solved for the modal values of the spatial accelerations and the constraints loads. In the serial and parallel implementations, computational complexity of the method increases as a linear and logarithmic functions of the number of the states of the system, uncertain variables, and the maximum degree of the basis functions used in the expansion.Copyright

Forward Kinematic Analysis of Non-Deterministic Articulated Multibody Systems With Kinematically Closed-Loops in Polynomial Chaos Expansion Scheme

This paper presents the method of polynomial chaos expansion (PCE) for the forward kinematic analysis of nondeterministic multibody systems with kinematically closed-loops. The PCE provides an efficient mathematical framework to introduce uncertainty to the system. This is accomplished by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial base functions. This paper presents the detailed formulation of the kinematics of a constrained multibody system at the position, velocity, and acceleration levels in the PCE scheme. This analysis is performed by projecting the governing kinematic constraint equations of the system onto the space of appropriate polynomial base functions. Furthermore, forward kinematic analysis is conducted at the position, velocity, and acceleration levels for a non-deterministic four-bar mechanism with single and multiple uncertain parameters in the length of linkages of the system. Time efficiency and accuracy of the intrusive PCE approach are compared with the traditionally used Monte Carlo method. The results demonstrate the drastic increase in the computational time of Monte Carlo method when analyzing complex systems with a large number of uncertain parameters while the intrusive PCE provides better accuracy with much less computation complexity.Copyright

Computed torque control of articulated multibody systems in the generalized divide and conquer algorithm framework

An extension to the Generalized-Divide-and-Conquer Algorithm (GDCA) is presented in this paper in conjunction with the Computed-Torque-Control-Law (CTCL) to model and control fully actuated multibody systems. CTCL uses the inverse dynamics to provide control inputs to the system while, the dynamics of the system must be formed and solved in each iteration. Herein, the GDCA is extended to form and solve the inverse dynamics to find control torques. Further, this method is also extended to efficiently solve the equations of motion of the controlled system. This significantly reduces the complexity of modeling, simulating, and controlling the fully actuated multibody systems to O(n) or O(logn) operations in each iteration in the serial and parallel implementations, respectively.Copyright

Computed Torque Control of the Stewart platform with uncertainty for lower extremity robotic rehabilitation

Parallel manipulators play a key role in robotic rehabilitation. In reality, such systems operate under uncertainty due to the changes in the characteristics of the patients and lack of knowledge about the physical and geometrical properties of the system. In this paper, we present a robust control scheme to control a six-degree-of-freedom Stewart platform. In this application, it is aimed to follow a desired pure rotational motion required in the robotic rehabilitation of the foot for patients with diabetic neuropathy. It is assumed that uncertainty exists in the mass of the foot of the patients (the proposed approach can also be used when disturbance exists). To perform this, the method of polynomial chaos expansion (PCE) is extended and integrated with the computed torque control law (CTCL) to control the system. In PCE scheme, uncertainty is introduced to the system by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial basis functions. CTCL uses a feedback linearization technique which provides the necessary force/torque to enforce the system to follow a prescribed trajectory. This papers presents a successful implementation of the PCE-base CTCL on a Stewart platform. Finally, a comparison between the efficiency and accuracy of the Monte Carlo and PCE is conducted.

An optimal Stewart platform for lower extremity robotic rehabilitation

In this paper, two algorithms are performed to find an optimum design of a six-degree-of-freedom Stewart platform to provide a desired pure rotational motion required in the robotic rehabilitation of the foot for patients with neuropathy. To accomplish this, first, we present the kinematic and the dynamic analysis of the Stewart platform. The dynamic equations are derived by using a customized Lagrange method. Then, physically meaningful objective variables are defined such as the size of the platform, the length of the six links, the maximum stroke of the six linear actuators, the maximum actuator force, and the reachable workspace. This is followed by using two optimization methods (Genetic Algorithm and Monte-Carlo method) to study the aforementioned objective variables, resulting in the optimal solution for the desired orientation motions. Then, the detailed investigation of the effect of changes in these objective variables on the variation of the platform design variables is studied. Finally, in a numerical example, the advantages and disadvantages of using the Genetic Algorithm and the Monte-Carlo method to find the optimal design variables for a custom cost function with weighted objective variables are revealed.