Automated Symbolic Derivation of Dynamic Equations of Motion for Robotic Manipulators

1986 ◽  
Vol 108 (3) ◽  
pp. 172-179 ◽  
Author(s):  
M. C. Leu ◽  
N. Hemati
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Robotic Manipulators ◽  
Dynamic Equations ◽  
Computer Memory ◽  
Symbolic Language ◽  
Symbolic Form ◽  
General Computer Program ◽  
Lagrange Formalism ◽  
General Computer

A general computer program for deriving the dynamic equations of motion for robotic manipulators using the symbolic language MACSYMA has been developed. The program, developed based on the Lagrange formalism, is applicable to manipulators of any number of degrees of freedom. Examples are given to illustrate how to use this program for dynamic equation generation. Advantages of expanding the dynamic equations into symbolic form are presented. Techniques for improving efficiency of equation generation, overcoming computer memory limitation, and approximating manipulator dynamics are discussed.

Analysis of Varying Natural Frequencies and Damping Ratios of a Sample Parallel Manipulator Throughout its Workspace Using Linearized Equations of Motion

2001 ◽  
Author(s):  
Kris Kozak ◽  
Imme Ebert-Uphoff ◽  
William Singhose
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Parallel Architecture ◽  
Robotic Manipulators ◽  
Parallel Manipulators ◽  
Dynamic Equations ◽  
Extreme Variation

Abstract This article investigates the dynamic properties of robotic manipulators of parallel architecture. In particular, the dependency of the dynamic equations on the manipulator’s configuration within the workspace is analyzed. The proposed approach is to linearize the dynamic equations locally throughout the workspace and to plot the corresponding natural frequencies and damping ratios. While the results are only applicable for small velocities of the manipulator, they present a first step towards the classification of the nonlinear dynamics of parallel manipulators. The method is applied to a sample manipulator with two degrees-of-freedom. The corresponding numerical results demonstrate the extreme variation of its natural frequencies and damping ratios throughout the workspace.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

scholarly journals Minimization of dynamic joint reaction forces of the 2-DOF serial manipulators based on interpolating polynomials and counterweights

2015 ◽  
Vol 42 (4) ◽  
pp. 249-260 ◽  
Author(s):  
Slavisa Salinic ◽  
Marina Boskovic ◽  
Radovan Bulatovic
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Symbolic Form ◽  
Serial Manipulators ◽  
Joint Reaction Forces ◽  
Reaction Forces ◽  
Inertia Forces ◽  
Joint Reaction ◽  
Selection Of

This paper presents two ways for the minimization of joint reaction forces due to inertia forces (dynamic joint reaction forces) in a two degrees of freedom (2-DOF) planar serial manipulator. The first way is based on the optimal selection of the angular rotations laws of the manipulator links and the second one is by attaching counterweights to the manipulator links. The influence of the payload carrying by the manipulator on the dynamic joint reaction forces is also considered. The expressions for the joint reaction forces are obtained in a symbolic form by means of the Lagrange equations of motion. The inertial properties of the manipulator links are represented by dynamical equivalent systems of two point masses. The weighted sum of the root mean squares of the magnitudes of the dynamic joint reactions is used as an objective function. The effectiveness of the two ways mentioned is discussed.

Locally Linearized Dynamic Analysis of Parallel Manipulators and Application of Input Shaping to Reduce Vibrations

2004 ◽  
Vol 126 (1) ◽  
pp. 156-168 ◽  
Author(s):  
Kris Kozak ◽  
Imme Ebert-Uphoff ◽  
William Singhose
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Model Simulation ◽  
Parallel Architecture ◽  
Robotic Manipulators ◽  
Parallel Manipulators ◽  
Dynamic Equations ◽  
Input Shaping ◽  
Multiple Degrees Of Freedom

Input Shaping is a technique that seeks to reduce residual vibrations through modification of the reference command given to a system. Namely the reference command is convolved with a suitable train of impulses. Input shaping has proven to be successful in reducing the vibrations of a great variety of linear systems. This article seeks to apply input shaping to robotic manipulators of parallel architecture. Such systems have multiple degrees-of-freedom and non-linear dynamics and therefore standard input shaping techniques cannot be readily applied. In order to apply standard input shaping techniques to such systems, this article linearizes the dynamic equations of the system locally and determines the configuration-dependent natural frequencies and damping ratios throughout its workspace. Techniques are developed to derive the dynamic equations directly in linearized form. The method is demonstrated for a sample manipulator with two degrees-of-freedom. A linearized dynamic model is derived and input shaping is locally tuned according to the linearized dynamic model. Simulation results are provided and discussed.

Design and Development of H∞ and μ-Synthesis Controllers for Nanorobotic Manipulation and Grasping Applications

2007 ◽  
Author(s):  
Reza Saeidpourazar ◽  
Beshah Ayalew ◽  
Nader Jalili
Keyword(s):  
Degrees Of Freedom ◽  
Controller Design ◽  
Equations Of Motion ◽  
Dynamic Equations ◽  
Robust Controller ◽  
Accuracy And Precision ◽  
Highly Nonlinear ◽  
Linearized Model ◽  
High Level ◽  
Robust Controller Design

This paper presents the development of H∞ and μ-synthesis robust controllers for nanorobotic manipulation and grasping applications. Here a 3 DOF (Degrees Of Freedom) nanomanipulator with RRP (Revolute Revolute Prismatic) actuator arrangement is considered for nanomanipulation purposes. Due to the sophisticated complexity, and expected high level of accuracy and precision (of the order of 10−7 rad in revolute actuators and 0.25 nm in the prismatic actuator) of the nanomanipulator, there is a need to design a suitable controller to guarantee an accurate manipulation process. However, structure of the nanomanipulator employed here, namely MM3A, is such that the dynamic equations of motion of the nanomanipulator are highly nonlinear and complicated. Linearizing these dynamic equations of the nanomanipulator simplifies the controller design process significantly. However, linearization could suppress some critical information about the system dynamics. In order to achieve the precise motion of the nanomanipulator utilizing the simple linearized model, H∞ and μ-synthesis robust controller design approaches are proposed. Following the development of the controllers, numerical simulations of the proposed controllers on the nanomanipulator are used to verify the positioning performance.

scholarly journals Modular Multibody Formulation for Simulating Off-Road Tracked Vehicles

2014 ◽  
Vol 1 (2) ◽  
pp. 77 ◽  
Author(s):  
Mohamed A Omar
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Dynamic Equations ◽  
Solution Stability ◽  
Simulation Code ◽  
Tracked Vehicles ◽  
Main System ◽  
6 Degrees Of Freedom

This paper presents a formulation and procedure for incorporating the multibody dynamics analysis capability of tracked vehicles in large-scale multibody system.  The proposed self-contained modular approach could be interfaced to any exiting multibody simulation code without need to alter the existing solver architecture.  Each track is modeled as a super-component that can be treated separate from the main system.  The super-component can be efficiently used in parallel processing environment to reduce the simulation time.  In the super-component, each track-link is modeled as separate body with full 6 degrees of freedom (DoF).  To improve the solution stability and efficiency, the joints between track links are modeled as complaint connection.  The spatial algebra operator is used to express the motion quantities and develop the link’s nonlinear kinematic and dynamic equations of motion.  The super-component interacts with the main system through contact forces between the track links and the driving sprocket, the support rollers and the idlers using self-contained force modules.  Also, the super-component models the interaction with the terrain through force module that is flexible to include different track-soil models, different terrain geometries, and different soil properties.  The interaction forces are expressed in the Cartesian system, applied to the link’s equation of motion and the corresponding bodies in the main system.  For sake of completeness, this paper presents dynamic equations of motion of the links as well as the main system formulated using joint coordinates approach.

scholarly journals Application of software tools for symbolic description and modeling of mechanical systems

2020 ◽  
Author(s):  
A. Banshchikov ◽  
A. Vetrov
Keyword(s):  
Mechanical System ◽  
Software Package ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Mechanical Systems ◽  
Software Tools ◽  
Rigid Bodies ◽  
Symbolic Form ◽  
The Stability ◽  
Lagrange Formalism

The paper presents two software tools (graphical editor and software package). The editor is designed for the formation of a symbolic description of a mechanical system using the Lagrange formalism. A system of the absolutely rigid bodies connected by joints is considered as a mechanical system. The editor is a user interface by which one sets the structure of the interconnection of bodies (system configuration) as well as the geometric and kinematic characteristics for each body of the system. The created structure and the entered data are automatically presented in the form of a text file, which is used as an input file for the software package for modeling mechanical systems in a symbolic form with a computer. The use of these software tools is shown in detail in the example of the analysis of the dynamics of a satellite with a gravitational stabilizer in a circular orbit. For this system, the kinetic energy and force function of an approximate Newtonian gravitational field were obtained, nonlinear and linearized equations of motion were constructed, and the question of the stability of the relative equilibrium position was considered.

Formulation of Equations of Motion for a Chain of Flexible Links Using Hamilton’s Principle

1994 ◽  
Vol 116 (1) ◽  
pp. 81-88 ◽  
Author(s):  
M. Benati ◽  
A. Morro
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Dynamic Equations ◽  
Relative Deformation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Flexible Links ◽  
A Chain

The dynamic equations of a chain of flexible links are determined by means of Hamilton’s principle. First a continuous model is adopted and the boundary conditions are determined, along with the partial differential equations of motion. Then a model with a finite number of degrees of freedom is set up. The configuration of each link is described through the line which joins the end points and the relative deformation is described in terms of appropriate trial functions. The boundary conditions are incorporated into a set of basic trial functions. The time-dependent coefficients of the remaining shape functions play the role of Lagrangian coordinates. The dynamic equations are then derived and the procedure is contrasted with other methods for reduction of a system of links to a system with a finite number of degrees of freedom.

Automatic Formulation of Dynamic Equations of Motion of Robot Manipulators

1988 ◽  
Vol 202 (6) ◽  
pp. 397-404 ◽  
Author(s):  
D Pan ◽  
R S Sharp
Keyword(s):  
Degrees Of Freedom ◽  
Robot Manipulators ◽  
Equations Of Motion ◽  
Chain Structure ◽  
Algebraic Manipulation ◽  
Dynamic Equations ◽  
Open Chain ◽  
Scalar Form ◽  
Memory Space ◽  
Manipulation Language

Based on the use of homogeneous transformation matrices with Denavit-Hartenberg notation and the Lagrangian formulation method, a general computer program ROBDYN.RED for the symbolic derivation of dynamic equations of motion for robot manipulators has been developed and is discussed in this paper. The program is developed by using REDUCE, an algebraic manipulation language, and is versatile for open-chain structure robot manipulators with any number of degrees of freedom and with any combination of types of joint. Considerations are also given to saving computer memory space required for execution and to minimizing the runtime. Several examples are included to demonstrate the use of the program. Equations of motion in scalar form can be automatically transferred to FORTRAN format for later numerical simulations. The efficiency of the resulting equations in terms of numerical integration is also discussed and some further developments to improve the efficiency are suggested.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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