scholarly journals Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Dynamical Systems ◽  
Optimal Design ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Parametric Uncertainty ◽  
Least Square ◽  
Optimization Process ◽  
Entire Family ◽  
New Framework

This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work, uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or off-set, the resulting Pareto optimal trade-off curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.

scholarly journals Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Dynamical Systems ◽  
Optimal Design ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Parametric Uncertainty ◽  
Least Square ◽  
Optimization Process ◽  
Entire Family ◽  
New Framework

This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear, have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or off-set, the resulting Pareto optimal trade-off curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.

Motion Planning of Uncertain Fully-Actuated Dynamical Systems: A Forward Dynamics Formulation

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Least Square ◽  
Forward Dynamics ◽  
Optimal Motion ◽  
New Framework

This work presents a novel nonlinear programming based motion planning framework that treats uncertain fully-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; ignoring uncertainty during design may lead to poor robustness and suboptimal performance. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through a forward dynamics formulation where deterministic actuator inputs are prescribed and uncertain state trajectories are quantified. The benefits of the ability to quantify the resulting state uncertainty are illustrated in an effort optimal motion planning case-study of a serial manipulator pick-and-place application. The resulting design determines a feasible effort optimal motion plan—subject to actuator and obstacle avoidance constraints—for all possible systems within the probability space. Variance of the system’s terminal conditions are also minimized in a Pareto optimal sense. The inverse dynamics formulation (using deterministic state trajectories and uncertain actuator inputs) is presented in a companion paper.

Motion Planning of Uncertain Under-Actuated Dynamical Systems: A Hybrid Dynamics Formulation

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Design System ◽  
Double Pendulum ◽  
Hybrid Dynamics ◽  
New Framework

This work presents a novel nonlinear programming based motion planning framework that treats uncertain under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiencies of this approach enable the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through a hybrid dynamics formulation for under-actuated systems where actuated state and unactuated input trajectories are prescribed and uncertain unactuated states and actuated inputs are quantified. The benefits of the ability to quantify the resulting uncertainties are illustrated in a power optimal motion planning case-study for an inverting double pendulum problem. The resulting design determines a motion plan that minimizes the required input power—subject to actuator and terminal condition variance constraints—for all possible systems within the probability space.

Motion Planning of Uncertain Fully-Actuated Dynamical Systems: An Inverse Dynamics Formulation

2011 ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Inverse Dynamics ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Optimal Motion ◽  
Time Optimal ◽  
New Framework

This work presents a novel nonlinear programming based motion planning framework that treats uncertain fully-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; ignoring uncertainty during design may lead to poor robustness and suboptimal performance. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through an inverse dynamics formulation where deterministic state trajectories are prescribed and uncertain actuator inputs are quantified. The benefits of the ability to quantify the resulting actuator uncertainty are illustrated in a time optimal motion planning case-study of a serial manipulator pick-and-place application. The resulting design determines a feasible time optimal motion plan—subject to actuator and obstacle avoidance constraints—for all possible systems within the probability space. The forward dynamics formulation (using deterministic actuator inputs and uncertain state trajectories) is presented in a companion paper.

Analysis of Hybrid Dynamical Systems With Uncertainty in Initial Conditions

2015 ◽  
Author(s):  
Yuhua He ◽  
Arpan Mukherjee ◽  
Rahul Rai
Keyword(s):  
Dynamical Systems ◽  
Systems Engineering ◽  
Initial Conditions ◽  
Real Life ◽  
Parametric Uncertainty ◽  
Superior Performance ◽  
Hybrid Dynamical Systems ◽  
Unscented Transform ◽  
Output Variability ◽  
Novel Method

Hybrid dynamical systems (HDS) models are backbone of modeling a myriad of systems found in systems engineering, buildings, manufacturing, auto-pilot control, and chemical processes domains among others. Uncertainty quantification (UQ) techniques to ascertain output variability in HDS with parametric uncertainty is relatively understudied topic. In this paper, we present a novel method to enable UQ of HDSs. Specifically, we outline a computational pipeline to solve different types of Stochastic Hybrid Systems (SHS) with uncertainty in initial conditions. The developed method is based on a numerical integration technique Conjugate Unscented Transform (CUT) and discrete UQ technique. The developed method has been applied to different range of problems including theoretical problems and real life mechanical systems modeled in Simulink environment. Performance of the proposed method has been compared against Monte Carlo and Unscented Transform methods. Results indicate superior performance of the proposed technique over the existing methods.

scholarly journals Motion Planning of Uncertain Ordinary Differential Equation Systems

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Joe Hays ◽  
Adrian Sandu ◽  
Corina Sandu ◽  
Dennis Hong
Keyword(s):  
Nonlinear Programming ◽  
Dynamical Systems ◽  
Motion Planning ◽  
Inverse Dynamics ◽  
Initial Conditions ◽  
Practical Importance ◽  
Real Life ◽  
Underactuated Systems ◽  
Forward Dynamics ◽  
Hybrid Dynamics

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and underactuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it is not accounted for in a given design. In this work uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and underactuated systems, prescribes deterministic actuator inputs that yield uncertain state trajectories. The inverse dynamics formulation is the dual to that of forward dynamics, and is only applicable to fully actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to underactuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories that yield uncertain unactuated states and uncertain actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space.

scholarly journals D-optimal Design in Linear Model With Different Heteroscedasticity Structures

2020 ◽  
Vol 9 (2) ◽  
pp. 7 ◽  
Author(s):  
BODUNWA, O. K. ◽  
FASORANBAKU, O. A.
Keyword(s):  
Optimal Design ◽  
Linear Model ◽  
Extreme Values ◽  
Real Life ◽  
Simulated Data ◽  
Least Square Method ◽  
Correction Method ◽  
Least Square ◽  
Sequential Method ◽  
Explanatory Variables

In this paper, we developed D-optimal design in linear model with two explanatory variables in the presence of heteroscedasticity. A sequential method of getting D-optimal design was adopted. Two different structures were used based on the literatures; it was found that the optimal design takes the extreme values of the design region. The results of simulated data was justified with real life data from the kinematic viscosity of a lubricant, in stokes, as a function of temperature and pressure which was used as discussed in Linssen (1975). The relative efficiency of other designs with respect to D-optimal designs was determined. Three correction methods was adopted from weighted least square method for heteroscedasticity problem, it was found that the correction method tagged HCW1 performed better.

scholarly journals On Kaufmann's theory of the impact of the pianoforte hammer

1920 ◽  
Vol 97 (682) ◽  
pp. 99-110 ◽  
Keyword(s):  
Initial Conditions ◽  
Practical Importance ◽  
Prima Facie ◽  
Point Of View ◽  
Infinite Length ◽  
Modes Of Vibration ◽  
Rigorous Treatment ◽  
The Subject ◽  
Set Up ◽  
The Impact

The theory of the vibrations of the pianoforte string put forward by Kaufmann in a well-known paper has figured prominently in recent discussions on the acoustics of this instrument. It proceeds on lines radically different from those adopted by Helmholtz in his classical treatment of the subject. While recognising that the elasticity of the pianoforte hammer is not a negligible factor, Kaufmann set out to simplify the mathematical analysis by ignoring its effect altogether, and treating the hammer as a particle possessing only inertia without spring. The motion of the string following the impact of the hammer is found from the initial conditions and from the functional solutions of the equation of wave-propagation on the string. On this basis he gave a rigorous treatment of two cases: (1) a particle impinging on a stretched string of infinite length, and (2) a particle impinging on the centre of a finite string, neither of which cases is of much interest from an acoustical point of view. The case of practical importance treated by him is that in which a particle impinges on the string near one end. For this case, he gave only an approximate theory from which the duration of contact, the motion of the point struck, and the form of the vibration-curves for various points of the string could be found. There can be no doubt of the importance of Kaufmann’s work, and it naturally becomes necessary to extend and revise his theory in various directions. In several respects, the theory awaits fuller development, especially as regards the harmonic analysis of the modes of vibration set up by impact, and the detailed discussion of the influence of the elasticity of the hammer and of varying velocities of impact. Apart from these points, the question arises whether the approximate method used by Kaufmann is sufficiently accurate for practical purposes, and whether it may be regarded as applicable when, as in the pianoforte, the point struck is distant one-eighth or one-ninth of the length of the string from one end. Kaufmann’s treatment is practically based on the assumption that the part of the string between the end and the point struck remains straight as long as the hammer and string remain in contact. Primâ facie , it is clear that this assumption would introduce error when the part of the string under reference is an appreciable fraction of the whole. For the effect of the impact would obviously be to excite the vibrations of this portion of the string, which continue so long as the hammer is in contact, and would also influence the mode of vibration of the string as a whole when the hammer loses contact. A mathematical theory which is not subject to this error, and which is applicable for any position of the striking point, thus seems called for.

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