Symbolic Linearized Equations for Nonholonomic Multibody Systems With Closed-Loop Kinematics

2018 ◽  
Author(s):  
Márton Kuslits ◽  
Dieter Bestle
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Linearized Equations ◽  
Nonlinear Equations Of Motion ◽  
Loop Constraints

Multibody systems and associated equations of motion may be distinguished in many ways: holonomic and nonholonomic, linear and nonlinear, tree-structured and closed-loop kinematics, symbolic and numeric equations of motion. The present paper deals with a symbolic derivation of nonlinear equations of motion for nonholonomic multibody systems with closed-loop kinematics, where any generalized coordinates and velocities may be used for describing their kinematics. Loop constraints are taken into account by algebraic equations and Lagrange multipliers. The paper then focuses on the derivation of the corresponding linear equations of motion by eliminating the Lagrange multipliers and applying a computationally efficient symbolic linearization procedure. As demonstration example, a vehicle model with differential steering is used where validity of the approach is shown by comparing the behavior of the linearized equations with their nonlinear counterpart via simulations.

A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

2022 ◽  
Author(s):  
Alfonso García-Agúndez Blanco ◽  
Daniel García Vallejo ◽  
Emilio Freire ◽  
Aki Mikkola
Keyword(s):  
Multibody Systems ◽  
Jacobian Matrix ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Constrained Multibody Systems ◽  
Multibody Model ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, a human propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels, is analysed. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a Differential-Algebraic system of equations (DAE system), are linearized along the steady forward motion resorting to a recently validated linearization procedure, which allows the maximum possible reduction of the linearized equations of motion of constrained multibody systems. The approach enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of spurious null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

Null Space Method of Differential Equation Type for Motion Analysis of Multibody Systems

2017 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yusaku Yamashita
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the constraint Jacobian. In previous reports, one of the authors presented methods which use the null space matrix. In the procedure to obtain the null space matrix, the inverse of a matrix whose regularity may not be always guaranteed. In this report, a new method is proposed in which the null space matrix is obtained by solving differential equations that can be always defined by using the QR decomposition, even if the constraints are redundant. Examples of numerical analysis are shown to validate the proposed method.

Dynamic Modeling and Simulation of Parallel Mechanisms Using the Virtual Spring Approach

2000 ◽  
Author(s):  
Jiegao Wang ◽  
Clément M. Gosselin ◽  
Li Cheng
Keyword(s):  
Closed Loop ◽  
Differential Algebraic Equations ◽  
Parallel Mechanisms ◽  
Algebraic Equations ◽  
New Approach ◽  
Flexible Links ◽  
Simulation Results ◽  
Loop Constraints ◽  
Virtual Springs ◽  
Good Agreement

Abstract A new approach for the dynamic simulation of parallel mechanisms or mechanical systems is presented in this paper. This approach uses virtual springs and dampers to include the closed-loop constraints thereby avoiding the solution of differential-algebraic equations. Examples illustrating the approach are given and include the four-bar mechanism with both rigid and flexible links as well as the 6-dof Gough-Stewart platform. Simulation results are given for the four-bar linkages and the 6-dof manipulator. The results achieve a good agreement with the results obtained from other conventional approaches.

Direct Linearization of Continuous and Hybrid Dynamical Systems

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado ◽  
Andrew J. Sinclair
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equations ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Hybrid Dynamical Systems ◽  
Linearized Equations ◽  
Direct Linearization ◽  
Taylor Series Expansions ◽  
Nonlinear Equations Of Motion ◽  
And Control

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

A Geometric Derivation of the Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Elias Paraskevopoulos
Keyword(s):  
Numerical Methods ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Main Idea ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Motion Constraints ◽  
Stiffness Properties

A new set of equations of motion is presented for a class of mechanical systems subjected to equality motion constraints. Specifically, the systems examined satisfy a set of holonomic and/or nonholonomic scleronomic constraints. The main idea is to consider the equations describing the action of the constraints as an integral part of the overall process leading to the equations of motion. The constraints are incorporated one by one, in a process analogous to that used for setting up the equations of motion. This proves to be equivalent to assigning appropriate inertia, damping and stiffness properties to each constraint equation and leads to a system of second order ordinary differential equations for both the coordinates and the Lagrange multipliers associated to the motion constraints automatically. This brings considerable advantages, avoiding problems related to systems of differential-algebraic equations or penalty formulations. Apart from its theoretical value, this set of equations is well-suited for developing new robust and accurate numerical methods.

Constraint Wrench Formulation for Closed-Loop Systems Using Two-Level Recursions

2007 ◽  
Vol 129 (12) ◽  
pp. 1234-1242 ◽  
Author(s):  
Himanshu Chaudhary ◽  
Subir Kumar Saha
Keyword(s):  
Lagrange Multipliers ◽  
Spanning Tree ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Minimal Set ◽  
Closed Loops ◽  
Closed Loop Systems ◽  
Subsystem Level ◽  
Body Level

In order to compute the constraint moments and forces, together referred here as wrenches, in closed-loop mechanical systems, it is necessary to formulate a dynamics problem in a suitable manner so that the wrenches can be computed efficiently. A new constraint wrench formulation for closed-loop systems is presented in this paper using two-level recursions, namely, subsystem level and body level. A subsystem is referred here as the serial- or tree-type branches of a spanning tree obtained by cutting the appropriate joints of the closed loops of the system at hand. For each subsystem, unconstrained Newton–Euler equations of motion are systematically reduced to a minimal set in terms of the Lagrange multipliers representing the constraint wrenches at the cut joints and the driving torques/forces provided by the actuators. The set of unknown Lagrange multipliers and the driving torques/forces associated to all subsystems are solved in a recursive fashion using the concepts of a determinate subsystem. Next, the constraint forces and moments at the uncut joints of each subsystem are calculated recursively from one body to another. Effectiveness of the proposed algorithm is illustrated using a multiloop planar carpet scraping machine and the spatial RSSR (where R and S stand for revolute and spherical, respectively) mechanism.

Efficient Parallel Computer Simulation of the Motion Behaviors of Closed-Loop Multibody Systems

2007 ◽  
Author(s):  
Shanzhong Duan ◽  
Andrew Ries
Keyword(s):  
Parallel Computing ◽  
Multibody Systems ◽  
Closed Loop ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Parallel Computer ◽  
Constraint Forces ◽  
Computing Systems ◽  
Closed Loops

This paper presents an efficient parallelizable algorithm for the computer-aided simulation and numerical analysis of motion behaviors of multibody systems with closed-loops. The method is based on cutting certain user-defined system interbody joints so that a system of independent multibody subchains is formed. These subchains interact with one another through associated unknown constraint forces fc at the cut joints. The increased parallelism is obtainable through cutting joints and the explicit determination of associated constraint forces combined with a sequential O(n) method. Consequently, the sequential O(n) procedure is carried out within each subchain to form and solve the equations of motion while parallel strategies are performed between the subchains to form and solve constraint equations concurrently. For multibody systems with closed-loops, joint separations play both a role of creation of parallelism for computing load distribution and a role of opening a closed-loop for use of the O(n) algorithm. Joint separation strategies provide the flexibility for use of the algorithm so that it can easily accommodate the available number of processors while maintaining high efficiency. The algorithm gives the best performance for the application scenarios for n>>1 and n>>m, where n and m are number of degree of freedom and number of constraints of a multibody system with closed-loops respectively. The algorithm can be applied to both distributed-memory parallel computing systems and shared-memory parallel computing systems.

Automatic Differentiation in Automatic Generation of the Linearized Equations of Motion

2021 ◽  
Author(s):  
Bruce Minaker ◽  
Francisco González
Keyword(s):  
Multibody Systems ◽  
Automatic Differentiation ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Order Reduction ◽  
Automatic Generation ◽  
Analysis Model ◽  
Direct Evaluation ◽  
Base Function ◽  
Linearized Equations

Abstract In the ongoing search for mathematically efficient methods of predicting the motion of vehicle and other multibody systems, and presenting the associated results, one of the avenues of continued interest is the linearization of the equations of motion. While linearization can potentially result in reduced fidelity in the model, the benefits in computational speed often make it the pragmatic choice. Linearization techniques are also useful in modal and stability analysis, model order reduction, and state and input estimation. This paper explores the application of automatic differentiation to the generation of the linearized equations of motion. Automatic differentiation allows one to numerically evaluate the derivative of any function, with no prior knowledge of the differential relationship to other functions. It exploits the fact that every computer program must evaluate every function using only elementary arithmetic operations. Using automatic differentiation, derivatives of arbitrary order can be computed, accurately to working precision, with minimal additional computational cost over the evaluation of the base function. There are several freely available software libraries that implement automatic differentiation in modern computing languages. In the paper, several example multibody systems are analyzed, and the computation times of the stiffness matrix are compared using direct evaluation and automatic differentiation. The results show that automatic differentiation can be surprisingly competitive in terms of computational efficiency.

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