An Alternative Formulation of Motion Equations in Redundant Coordinates for the Inverse Dynamics of Constrained Mechanical Systems

2011 ◽  
Author(s):  
Andreas Mu¨ller
Keyword(s):  
Inverse Dynamics ◽  
Mass Matrix ◽  
Alternative Formulation ◽  
Constrained System ◽  
Motion Equations ◽  
Constrained Mechanical Systems ◽  
Implementation Effort ◽  
Regular Configuration ◽  
Minimal Coordinates ◽  
Coordinate Partitioning

The basis for any model-based control of dynamical systems is an efficient formulation of the motion equations. These are preferably expressed in terms of independent coordinates. In other words the coordinates of a constrained system are split into a set of dependent and independent ones. It is well-known that such coordinate partitioning is not globally valid. A remedy is to switch between different possible sets of minimal coordinates. This drastically increases the numerical complexity and implementation effort, however. In this paper a formulation of the motion equations in redundant coordinates is presented for general non-holonomic systems. This gives rise to a redundant system of differential equations. The formulation is valid in any regular configuration. Because of the singular mass matrix it is not directly applicable for solving the forward dynamics but is tailored for solving the inverse dynamics. An inverse dynamics solution is presented for general full-actuated systems.

Differential-Geometric Methods in Multibody Dynamics and Control

2005 ◽  
Author(s):  
P. Maißer
Keyword(s):  
Configuration Space ◽  
High Performance ◽  
Multibody System ◽  
Equations Of Motion ◽  
Geometric Approach ◽  
Motion Equations ◽  
Constrained Mechanical Systems ◽  
Dynamics And Control ◽  
Set Up ◽  
Lagrangian Motion

This paper presents a differential-geometric approach to the multibody system dynamics regarded as a point dynamics in a n-dimensional configuration space Rn. This configuration space becomes a Riemannian space Vn the metric of which is defined by the kinetic energy of the multibody system (MBS). Hence, all concepts and statements of the Riemannian geometry can be used to study the dynamics of MBS. One of the key points is to set up the non-linear Lagrangian motion equations of tree-like MBS as well as of constrained mechanical systems, the perturbed equations of motion, and the motion equations of hybrid MBS in a derivative-free manner. Based on this approach transformation properties can be investigated for application in real-time simulation, control theory, Hamilton mechanics, the construction of first integrals, stability etc. Finally, a general Lyapunov-stable force control law for underactuated systems is given that demonstrates the power of the approach in high-performance sports applications.

Forward dynamics of variable topology mechanisms—The case of constraint activation

2016 ◽  
Vol 230 (4) ◽  
pp. 442-454
Author(s):  
Andreas Müller
Keyword(s):  
Human Machine Interaction ◽  
Motion Equations ◽  
Industrial Manipulator ◽  
The Core ◽  
Topology Changes ◽  
Ideal Contact ◽  
Variable Topology ◽  
Minimal Coordinates ◽  
Machine Interaction ◽  
Computational Properties

Many mechanical systems exhibit changes in their kinematic topology altering the mobility. Ideal contact is the best known cause, but also stiction and controlled locking of parts of a mechanism lead to topology changes. The latter is becoming an important issue in human–machine interaction. Anticipating the dynamic behavior of variable topology mechanisms requires solving a nonsmooth dynamic problem. The core challenge is a physically meaningful transition condition at the topology switching events. Such a condition is presented in this article. Two versions are reported, one using projected motion equations in terms of redundant coordinates, and another one using the Voronets equations in terms of minimal coordinates. Their computational properties are discussed. Results are shown for joint locking of a planar 3R mechanisms and a 6DOF industrial manipulator.

Adaptive and Singularity-Free Inverse Dynamics Models for Control of Parallel Manipulators With Actuation Redundancy

2011 ◽  
Author(s):  
Andreas Mu¨ller ◽  
Timo Hufnagel
Keyword(s):  
Range Of Motion ◽  
Inverse Dynamics ◽  
Null Space ◽  
Parallel Manipulators ◽  
Point Of View ◽  
Torque Control ◽  
Alternative Formulation ◽  
Dynamics Model ◽  
Model Based ◽  
Control Forces

Redundant actuation of parallel kinematics machines (PKM) is a way to eliminate input-singularities and so to enlarge the usable workspace. From a kinematic point of view the number m of actuator coordinates exceeds the DOF δ of a redundantly actuated PKM (RA-PKM). The dynamics model, being the basis for model-based control, is usually expressed in terms of δ independent actuator coordinates. This implies that the model exhibits the same singularities as the non-redundant PKM, even though the RA-PKM is not singular. Consequently the admissible range of motion of the RA-PKM model is limited to that of the non-redundant PKM. In this paper an alternative formulation of the dynamics model in terms of the full set of m actuator coordinates is presented. It leads to a redundant system of m motion equations that is valid in the entire range of motion. This formulation gives rise to an inverse dynamics formulation tailored for real-time implementation. In contrast to the standard formulation in independent coordinates, the proposed inverse dynamics formulation does not involve control forces in the null space of the control matrix, i.e. it does not allow for the generation of internal prestresses, however. This is not problematic as the latter is usually not exploited. The proposed method is compared to the recently proposed adaptive coordinate switching method. Experimental results are reported if the inverse dynamics solution is introduced in model-based computed torque control scheme of a planar 2DOF RA-PKM.

Toward a New Modeling Paradigm for Constrained Mechanical Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 80-87
Author(s):  
T. M. Hodges ◽  
P. N. Sheth
Keyword(s):  
Nonholonomic Constraints ◽  
Matrix Analysis ◽  
Joint Models ◽  
Modeling Framework ◽  
Constrained Mechanical Systems ◽  
Modeling Paradigm ◽  
The Matrix ◽  
Comprehensive Modeling ◽  
Coordinate Partitioning ◽  
Special Case

The motivational background and a comprehensive modeling framework for incorporating localized design details of joints in the overall multibody models are described in this paper. A direct contact model of two bodies in contact is utilized as a basic building block to develop spatial kinematic analysis procedures based on matrix analysis. A substructure partitioning method for large, detailed models and the concept of local Jacobians for localized joint models is introduced. The nonholonomic constraints are directly incorporated in the matrix method, with the special case of rolling without slipping described in this paper. The coordinate partitioning process is shown to play a fundamental role in the simulation algorithms. The developments reported in this paper represent initial steps toward a comprehensive modeling paradigm for multibody systems with a focus on the localized details of contacts between the bodies.

scholarly journals O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives

Robotica ◽  
2007 ◽  
Vol 25 (6) ◽  
pp. 739-750 ◽  
Author(s):  
Kiju Lee ◽  
Yunfeng Wang ◽  
Gregory S. Chirikjian
Keyword(s):  
Inverse Dynamics ◽  
Mass Matrix ◽  
Rotation Group ◽  
Rigid Bodies ◽  
Euclidean Group ◽  
Serial Manipulators ◽  
The Past ◽  
Polypeptide Chains ◽  
Serial Chains ◽  
And Robotics

SUMMARYOver the past several decades, a number of O(n) methods for forward and inverse dynamics computations have been developed in the multibody dynamics and robotics literature. A method was developed by Fixman in 1974 for O(n) computation of the mass-matrix determinant for a serial polymer chain consisting of point masses. In other of our recent papers, we extended this method in order to compute the inverse of the mass matrix for serial chains consisting of point masses. In the present paper, we extend these ideas further and address the case of serial chains composed of rigid-bodies. This requires the use of relatively deep mathematics associated with the rotation group, SO(3), and the special Euclidean group, SE(3), and specifically, it requires that one differentiates real-valued functions of Lie-group-valued argument.

scholarly journals Lagrangian Dynamics Analysis of a XY-Theta Parallel Robotic Machine Tool

2017 ◽  
Vol 61 (2) ◽  
pp. 107 ◽  
Author(s):  
Javad Enferadi ◽  
Mohammad Tavakolian
Keyword(s):  
Machine Tool ◽  
Inverse Kinematics ◽  
Good Accuracy ◽  
Inverse Dynamics ◽  
Constraint Forces ◽  
Lagrangian Multipliers ◽  
Motion Equations ◽  
The Subject ◽  
Parallel Machine Tool ◽  
Dynamics Problems

Dynamics of a highly stiff parallel machine tool is the subject of this paper. High stiffness, good accuracy, relatively large workspace and free of singularities on the whole workspace makes the manipulator suitable for machining applications as an XY-Theta precision table. First, obtaining kinematics constraints, inverse kinematics analysis and velocity analysis are performed. Next, using six redundant generalized coordinates, we obtain Lagrangian of the manipulator. Also, a Lagrangian approach is proposed to obtain dynamics equations of the machine tool using three Lagrangian multipliers. This method allows elimination of constraint forces and moments at the joints from the motion equations. Dynamic equations of the manipulator are formed as inverse dynamics and direct dynamics problems. Finally, two examples are presented that confirms the obtained dynamics equations.

A Novel Approach for the Dynamic Analysis and Simulation of Constrained Mechanical Systems

2003 ◽  
Author(s):  
Jo´zsef Ko¨vecses ◽  
Jean-Claude Piedboeuf
Keyword(s):  
Dynamic Analysis ◽  
Potential Application ◽  
Nonholonomic Constraints ◽  
Constrained Systems ◽  
Space Robotics ◽  
Lagrange Principle ◽  
Constrained System ◽  
Constrained Mechanical Systems ◽  
Novel Approach ◽  
Accuracy And Stability

In this paper we will outline a formulation for the dynamics of constrained systems. This formulation relies on the D’Alembert-Lagrange principle and the physically meaningful decomposition of the virtual displacements and the generalized velocities for the system where a redundant, non-minimum set of variables is used. The approach is valid for general constrained system with holonomic and/or nonholonomic constraints. We will also discuss a potential application of this formulation in improving the accuracy and stability of the simulation of constrained systems. This application will be demonstrated by an example drawn from space robotics.

Extending the Modified Inertia Representation to Constrained Rigid Multibody Systems

2019 ◽  
Vol 87 (1) ◽  
Author(s):  
X. M. Xu ◽  
J. H. Luo ◽  
Z. G. Wu
Keyword(s):  
Kinetic Energy ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constrained System ◽  
Energy Error ◽  
Numerical Performance ◽  
Rigid Multibody Systems ◽  
Rigid Multibody

Abstract The inertia representation of a constrained system includes the formulation of the kinetic energy and its corresponding mass matrix, given the coordinates of the system. The way to find a proper inertia representation achieving better numerical performance is largely unexplored. This paper extends the modified inertia representation (MIR) to the constrained rigid multibody systems. By using the orthogonal projection, we show the possibility to derive the MIR for many types of non-minimal coordinates. We present examples of the derivation of the MIR for both planar and spatial rigid body systems. Error estimation shows that the MIR is different from the traditional inertia representation (TIR) in that its parameter γ can be used to reduce the kinetic energy error. With preconditioned γ, numerical results show that the MIR consistently presents significantly higher numerical accuracy and faster convergence speed than the TIR for the given variational integrator. The idea of using different inertia representations to improve the numerical performance may go beyond constrained rigid multibody systems to other systems governed by differential algebraic equations.

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