Control of Closed Kinematic Chains Using A Singularly Perturbed Dynamics Model

2005 ◽  
Vol 128 (1) ◽  
pp. 142-151 ◽  
Author(s):  
Zhiyong Wang ◽  
Fathi H. Ghorbel
Keyword(s):  
Kinematic Chain ◽  
Perturbation Parameter ◽  
Parallel Robot ◽  
Differential Algebraic Equations ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
Kinematic Chains ◽  
Novel Approach ◽  
Lyapunov Function Method ◽  
Perturbed Model

In this paper, we propose a novel approach to the control of closed kinematic chains (CKCs). This method is based on a recently developed singularly perturbed model for CKCs. Conventionally, the dynamics of CKCs are described by differential-algebraic equations (DAEs). Our approach transfers the control of the original DAE system to the control of an artificially created singularly perturbed system in which the slow dynamics corresponds to the original DAE when the perturbation parameter tends to zero. Compared to control schemes that rely on solving nonlinear algebraic constraint equations, the proposed method uses an ordinary differential equation (ODE) solver to obtain the dependent coordinates, hence, eliminates the need for Newton-type iterations and is amenable to real-time implementation. The composite Lyapunov function method is used to show that the closed-loop system, when controlled by typical open kinematic chain schemes, achieves asymptotic trajectory tracking. Simulations and experimental results on a parallel robot, the Rice planar Delta robot, are also presented to illustrate the efficacy of our method.

scholarly journals Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity

2019 ◽  
Vol 24 (1) ◽  
pp. 17
Author(s):  
Clément Olivier ◽  
David Ryckelynck ◽  
Julien Cortial
Keyword(s):  
Constitutive Equations ◽  
Surrogate Model ◽  
Reference Model ◽  
Data Representation ◽  
Differential Algebraic Equations ◽  
Constitutive Law ◽  
Algebraic Equations ◽  
Tensor Representation ◽  
Novel Approach ◽  
Parametric Studies

This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients.

Non-smooth dynamical analysis and experimental validation of the cable-suspended parallel manipulator

2012 ◽  
Vol 226 (10) ◽  
pp. 2456-2466 ◽  
Author(s):  
Xing-Guo Shao ◽  
Zhen-Cai Zhu ◽  
Qing-Guo Wang ◽  
Peter CY Chen ◽  
Bin Zi ◽  
...  
Keyword(s):  
Dynamic Analysis ◽  
Parallel Manipulator ◽  
Inequality Constraints ◽  
Parallel Robot ◽  
Differential Algebraic Equations ◽  
Dynamical Analysis ◽  
Algebraic Equations ◽  
Dynamics Model ◽  
Midpoint Method ◽  
Smooth Dynamics

The cable-suspended parallel manipulator replaces the rigid links of traditional parallel robot. The unilateral property of the cable complicates the dynamic analysis of such manipulator and further induces difficulty in control problem. The set-valued tension law is proposed to model the unilateral constraint of the cable, and the dynamics of cable-suspended parallel manipulator is analyzed in the framework of non-smooth dynamics. The resulting non-smooth dynamics model consists of a set of differential–algebraic equations with inequality constraints. Its solution is found by the Moreau midpoint method. An experimental setup was established to verify and validate the effectiveness and accuracy of non-smooth dynamics. And the simulation results generally agree with the experimental results, which demonstrate that the non-smooth dynamics is effective and reasonable for the dynamic analysis of the cable-suspended parallel manipulator. The results of this article deeply reveal the dynamics of the cable-suspended parallel manipulator, and may be used to design more accurate controller for its trajectory control.

Evolutionary Design of Planar Kinematic Chains

1998 ◽  
Author(s):  
David R. Nielsen ◽  
Kazem Kazerounian
Keyword(s):  
Genetic Algorithm ◽  
Kinematic Chain ◽  
Type Synthesis ◽  
Kinematic Chains ◽  
Design Flexibility ◽  
Novel Approach ◽  
Chain Link ◽  
Synthesis Of Mechanisms ◽  
New Ideas ◽  
Crossover And Mutation

Abstract A procedure is developed to optimize planar mechanism type. A Genetic Algorithm is used to cycle populations of kinematic chain link adjacency matrices, through selection, crossover, and mutation. During this optimization, fit kinematic chains survive while unfit kinematic chains do not. Upon convergence, synthesized kinematic chains of high fitness remain. This technique was lead to be called the Genetic Algorithm for Type Synthesis (GATS). GATS introduces four new ideas for the type synthesis of mechanisms. First, it does not permute all possible kinematic chains. It searches for the best kinematic chains depending on a designer’s specifications. Second, larger size mechanisms can be generated because of the genetic algorithm’s evolutionary naturalness. Third, a novel approach was applied to genetic algorithms to allow the encodings to mutate in size. This allowed for addition or elimination of links in kinematic chains during evolution. Forth, a new property was deduced from mechanism topography that describes the mechanism design flexibility.

Kinematic Modelling and VR Simulation of a 3DOF Medical Parallel Robot with One Decoupled Motion

2013 ◽  
Vol 837 ◽  
pp. 567-572
Author(s):  
Nadia Cretescu ◽  
Mircea Neagoe ◽  
Radu Saulescu
Keyword(s):  
Kinematic Model ◽  
Kinematic Chain ◽  
Parallel Robot ◽  
Parallel Structure ◽  
Level Control ◽  
Kinematic Chains ◽  
Fixed Base ◽  
Decoupled Motion ◽  
Kinematic Modelling ◽  
High Level

The robot studied in the paper has a 3DOF parallel structure of type 1PRRR+2PRPaR, with two coupled motions and one decoupled motion, composed by a mobile platform connected to the fixed base by three kinematic chains (one open kinematic chain of Prismatic Revolute Revolute Revolute type and two kinematic chains of Prismatic Revolute Parallelogram Revolute type). An analytical kinematic modelling of the parallel robot of type 1PRRR+2PRPaR is firstly presented in this paper, followed by a numerical simulation of the closed-form kinematic model and by a Virtual Reality (VR) application with control aspects. An innovative user interface for high-level control of the parallel 1PRRR+2PRPaR type robot is developed in MATLAB - Simulink and SimMechanics environment.

scholarly journals Bifurcation in Singularity Parameterized ODEs

2016 ◽  
Vol 12 (1) ◽  
pp. 5808-5816
Author(s):  
Kamal H Yasir ◽  
Zahraa A Mutar
Keyword(s):  
Sufficient Conditions ◽  
Differential Algebraic Equations ◽  
Transcritical Bifurcation ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
Parameter Approach ◽  
New System

In this paper we will study differential algebraic equations (DAEs) through studying singularly perturbed ODEs. That's the ODEs will be transformed to an DAEs when the perturbed parameter approach to 0. This will permit us to apply the classicalbifurcation theory of ODEs for the new system (DAEs). So we will show by giving theorems, sufficient conditions for fold, pitchfork and transcritical bifurcation to be occurred in (DAEs). An illustrative example is given.

scholarly journals A singularly perturbed vector-bias malaria model incorporating bed-net control

2021 ◽  
Author(s):  
Sanaa Salman
Keyword(s):  
Time Scale ◽  
Disease Model ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Bed Nets ◽  
Bed Net ◽  
Model Dynamics ◽  
Insecticide Treated Bed Nets ◽  
Perturbed Model ◽  
Malaria Model

A malaria transmission disease model with host selectivity and Insecticide treated bed nets (ITNs), as an intervention for controlling the disease, is formulated. Since the vector is an insect, the vector time scale is much more expeditious than the host time scale. This leads to a singularly perturbed model with two distinctive intrinsic time scales, two-slow for the host and one-fast for the vector. The basic reproduction number R0 is calculated and the local stability analysis is performed at equilibria of the model when the perturbation parameter ɛ > 0. The model is analyzed when ɛ → 0 using asymptotic expansions technique. Merging bed-net control, vector-bias, and singular perturbation have a notable effect on the model dynamics. It is shown that if over %30 of humans use ITNs, malaria disease burden can be reduced. The dynamics on the slow surface indicate that the infected vectors decays very fast when ɛ = 0.001 according to the numerical simulations.

scholarly journals Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Muslima Kedir Siraj ◽  
Gemechis File Duressa ◽  
Tesfaye Aga Bullo
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
Central Difference ◽  
Convergence Of The Scheme

AbstractThis study introduces a stable central difference method for solving second-order self-adjoint singularly perturbed boundary value problems. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is developed. The obtained system of algebraic equations is solved by Thomas algorithm. The consistency and stability that guarantee the convergence of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields sixth order convergent. To validate the applicability of the method, two model examples are solved for different values of perturbation parameter ε and different mesh size h. The proposed method approximates the exact solution very well. Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.

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