In this paper, grazing bifurcations on bifurcation trees in a discontinuous dynamical oscillator are discussed. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Thus, grazing bifurcations on a bifurcation tree of period-1 to period-2 motions varying spring stiffness are presented in a discontinuous oscillator with three domains divided by circular boundaries. The stability and bifurcations of period-1 and period-2 motions are discussed. From analytical predictions, periodic motions are simulated numerically. Stiffness effects on the periodic motions are discussed. Such studies will help one understand parameter effects in discontinuous dynamical systems, which can be applied for system design and control.
{"title":"Period-1 to Period-2 Motions in a Discontinuous Oscillator","authors":"Siyu Guo, A. Luo","doi":"10.1115/detc2020-22712","DOIUrl":"https://doi.org/10.1115/detc2020-22712","url":null,"abstract":"\u0000 In this paper, grazing bifurcations on bifurcation trees in a discontinuous dynamical oscillator are discussed. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Thus, grazing bifurcations on a bifurcation tree of period-1 to period-2 motions varying spring stiffness are presented in a discontinuous oscillator with three domains divided by circular boundaries. The stability and bifurcations of period-1 and period-2 motions are discussed. From analytical predictions, periodic motions are simulated numerically. Stiffness effects on the periodic motions are discussed. Such studies will help one understand parameter effects in discontinuous dynamical systems, which can be applied for system design and control.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131711295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article illustrates a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g. if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this contribution is the development of an iterative, gradient based solution strategy for solving such problems. As an example, a Moon-landing as in the Apollo program, will be considered. In detail, we discuss the ascent, descent and abort maneuvers of the Apollo Lunar Excursion Module (LEM) to and from the Moon’s surface in minimum time. The goal is to find the control of the thrust nozzle of the LEM to minimize the final time.
{"title":"The Adjoint Gradient Method for Time-Optimal Control of a Moon Landing: Ascent, Descent, and Abort","authors":"Philipp Eichmeir, K. Nachbagauer, W. Steiner","doi":"10.1115/detc2020-22034","DOIUrl":"https://doi.org/10.1115/detc2020-22034","url":null,"abstract":"\u0000 This article illustrates a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g. if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this contribution is the development of an iterative, gradient based solution strategy for solving such problems. As an example, a Moon-landing as in the Apollo program, will be considered. In detail, we discuss the ascent, descent and abort maneuvers of the Apollo Lunar Excursion Module (LEM) to and from the Moon’s surface in minimum time. The goal is to find the control of the thrust nozzle of the LEM to minimize the final time.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122608744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
DARTS is a rigid/flexible multibody dynamics toolkit for the modeling and simulation of aerospace and robotic vehicles for engineering applications. In this paper we describe an on-line, browser-based environment using Jupyter notebooks to support training needs for the DARTS software. The suite of curated tutorial notebooks is organized into different topic areas, and into multiple themes within each topic area. The notebooks within a theme use a progression of examples for users to expand their understanding of the software. The topic areas include one on the DARTS multibody dynamics software and another one on the theory underlying the multibody dynamics formulation. We also describe a number of Jupyter extensions that were used — and some developed in house — to enhance the notebook interface for use with the dynamics simulation software. One significant extension we implemented allows the embedding of live 3D visualizations within simulation notebooks.
{"title":"A Jupyter Notebook Environment for Multibody Dynamics","authors":"A. Gaut, J. Cameron, Abhinandan Jain","doi":"10.1115/detc2020-22572","DOIUrl":"https://doi.org/10.1115/detc2020-22572","url":null,"abstract":"\u0000 DARTS is a rigid/flexible multibody dynamics toolkit for the modeling and simulation of aerospace and robotic vehicles for engineering applications. In this paper we describe an on-line, browser-based environment using Jupyter notebooks to support training needs for the DARTS software. The suite of curated tutorial notebooks is organized into different topic areas, and into multiple themes within each topic area. The notebooks within a theme use a progression of examples for users to expand their understanding of the software. The topic areas include one on the DARTS multibody dynamics software and another one on the theory underlying the multibody dynamics formulation. We also describe a number of Jupyter extensions that were used — and some developed in house — to enhance the notebook interface for use with the dynamics simulation software. One significant extension we implemented allows the embedding of live 3D visualizations within simulation notebooks.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"2 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123808442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.
{"title":"Period-3 Motions in a Parametrically Exited Inverted Pendulum","authors":"A. Luo, Chuan Guo","doi":"10.1115/detc2020-22176","DOIUrl":"https://doi.org/10.1115/detc2020-22176","url":null,"abstract":"\u0000 In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"2010 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127341221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yang Lv, H. Fang, Jian Xu, Qining Wang, Xiaoxu Zhang
By considering the coupling effect between the healthy lower-limb and the passive prosthesis, this paper builds a heterogeneous dynamic model for gait analysis, where the motions of the healthy limb and the prosthesis are driven by the central pattern generator (CPG) and the hip joint swing, respectively. The foot-ground contact is modelled as the process of unilateral force reaction rather than the constraint to get a refined representation of the gait motion. The response of the heterogeneous model, solved by numerical calculation, is then analyzed by comparison with a real gait test. Preliminary results show that the heterogeneous model not only describes the amputee’s gait well but also reveals a new gait feature of period-doubling. Parameter analysis further indicates that the period-doubling gait will return to the single-period pattern by amplifying the vertical motion of the hip joint at the amputated side. This dynamic bifurcation, which mimics the process of hip swing adaption, provides new insight into the compensatory mechanism for lamely walking.
{"title":"A Heterogeneous Model for Gait Analysis of the Lower-Limb and the Prosthesis Coupled System","authors":"Yang Lv, H. Fang, Jian Xu, Qining Wang, Xiaoxu Zhang","doi":"10.1115/detc2020-22392","DOIUrl":"https://doi.org/10.1115/detc2020-22392","url":null,"abstract":"\u0000 By considering the coupling effect between the healthy lower-limb and the passive prosthesis, this paper builds a heterogeneous dynamic model for gait analysis, where the motions of the healthy limb and the prosthesis are driven by the central pattern generator (CPG) and the hip joint swing, respectively. The foot-ground contact is modelled as the process of unilateral force reaction rather than the constraint to get a refined representation of the gait motion. The response of the heterogeneous model, solved by numerical calculation, is then analyzed by comparison with a real gait test. Preliminary results show that the heterogeneous model not only describes the amputee’s gait well but also reveals a new gait feature of period-doubling. Parameter analysis further indicates that the period-doubling gait will return to the single-period pattern by amplifying the vertical motion of the hip joint at the amputated side. This dynamic bifurcation, which mimics the process of hip swing adaption, provides new insight into the compensatory mechanism for lamely walking.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"07 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127753425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although the Hertzian contact theory is widely utilized in railway vehicle simulations with new wheel and rail profiles, the Hertzian contact assumptions would lead to inaccurate contact prediction for severely worn wheel and rail profiles due to their geometric conformity, causing non-elliptical contact shapes as well as pressure distribution. For this reason, various non-Hertzian contact models have been studied for use in vehicle dynamics simulations. Among others, a method proposed by Piotrowski and Kik has gained acceptance in predicting non-elliptical wheel-rail contact for vehicle dynamics simulations. Despite the elegant formulation and its accuracy, detailed online geometric calculation for non-elliptical contact shape is required for all the contact patches at every iteration, along with iterative evaluation of the force-deflection relationship. It leads to computation burdens for use in long-distance vehicle simulations. Therefore, in this study, an off-line based numerical procedure for non-Hertzian contact model is developed and integrated in the quasi-steady railway vehicle motion solver.
{"title":"Numerical Procedure for Non-Hertzian Wheel-Rail Contact Model Integrated in Quasi-Steady Railway Vehicle Motion Solver","authors":"Takayuki Tanaka, H. Sugiyama","doi":"10.1115/detc2020-22066","DOIUrl":"https://doi.org/10.1115/detc2020-22066","url":null,"abstract":"\u0000 Although the Hertzian contact theory is widely utilized in railway vehicle simulations with new wheel and rail profiles, the Hertzian contact assumptions would lead to inaccurate contact prediction for severely worn wheel and rail profiles due to their geometric conformity, causing non-elliptical contact shapes as well as pressure distribution. For this reason, various non-Hertzian contact models have been studied for use in vehicle dynamics simulations. Among others, a method proposed by Piotrowski and Kik has gained acceptance in predicting non-elliptical wheel-rail contact for vehicle dynamics simulations. Despite the elegant formulation and its accuracy, detailed online geometric calculation for non-elliptical contact shape is required for all the contact patches at every iteration, along with iterative evaluation of the force-deflection relationship. It leads to computation burdens for use in long-distance vehicle simulations. Therefore, in this study, an off-line based numerical procedure for non-Hertzian contact model is developed and integrated in the quasi-steady railway vehicle motion solver.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130494753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper a data-driven approach for model-free control of nonlinear systems with slow dynamics is proposed. The system behavior is described using a local model respectively a neural network. The network is updated online based on a Kalman filter. By predicting the system behavior two control approaches are discussed. One is obtained by calculating a control input from the one step ahead prediction equation using least squares, the other is obtained by solving a standard linear model predictive control problem. The approaches are tested on a constrained nonlinear MIMO system with slow dynamics.
{"title":"Adaptive Neural Network Based Predictive Control of Nonlinear Systems With Slow Dynamics","authors":"Mark Spiller, F. Bakhshande, D. Söffker","doi":"10.1115/DETC2020-22358","DOIUrl":"https://doi.org/10.1115/DETC2020-22358","url":null,"abstract":"\u0000 In this paper a data-driven approach for model-free control of nonlinear systems with slow dynamics is proposed. The system behavior is described using a local model respectively a neural network. The network is updated online based on a Kalman filter. By predicting the system behavior two control approaches are discussed. One is obtained by calculating a control input from the one step ahead prediction equation using least squares, the other is obtained by solving a standard linear model predictive control problem. The approaches are tested on a constrained nonlinear MIMO system with slow dynamics.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115991912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Dynamic Modeling and Analysis Toolbox DynManto is an acedemic Matlab code which allows the modeling, simulation and sensitivity analysis of spatial multibody systems. The kinematics of rigid and flexible bodies is described by the floating frame of reference formulation and the body properties are provided by standard input data files. In this way the evaluation of the equations of motion is computationally efficient and an arbitrary parameterization of the system can be achieved. The latter is important in the automated adjoint sensitivity analysis of multibody systems, which yields gradient information for system analyses, parameter identifications or gradient-based optimizations. The capabilities of DynManto are demonstrated by the application examples of a flexible two-arm manipulator and Chebyshev’s Lambda Mechanism.
{"title":"DynManto: A Matlab Toolbox for the Simulation and Analysis of Multibody Systems","authors":"A. Held, A. Moghadasi, R. Seifried","doi":"10.1115/detc2020-22336","DOIUrl":"https://doi.org/10.1115/detc2020-22336","url":null,"abstract":"\u0000 The Dynamic Modeling and Analysis Toolbox DynManto is an acedemic Matlab code which allows the modeling, simulation and sensitivity analysis of spatial multibody systems. The kinematics of rigid and flexible bodies is described by the floating frame of reference formulation and the body properties are provided by standard input data files. In this way the evaluation of the equations of motion is computationally efficient and an arbitrary parameterization of the system can be achieved. The latter is important in the automated adjoint sensitivity analysis of multibody systems, which yields gradient information for system analyses, parameter identifications or gradient-based optimizations. The capabilities of DynManto are demonstrated by the application examples of a flexible two-arm manipulator and Chebyshev’s Lambda Mechanism.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114065606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The complex dynamics of human gait is yet to be completely understood. Researchers have quantified stability of walking gait using Floquet multipliers as well as Lyapunov exponents. In this article, we utilize the techniques and tools from dynamical system theory and invariant manifolds to map the gait data onto a time invariant representation of a dynamical system. As an example, the complex behavior of the joint angle during walking was studied using a conformal mapping approach that transformed the time periodic system into a time invariant linear system. Time-delay embedding was used to reconstruct the dynamics of the original gait system with time series kinematic data. This minimal realization of the system was used to construct a Single Degree of Freedom (SDOF) oscillator. The time evolution of the linear oscillatory system was mapped back using the conformal mapping derived using Lyapunov-Floquet Theory. This algorithm was verified for walking gait kinematics data for two healthy human subjects. A comparison was drawn between the phase space behavior of the original time periodic system and the remapped time invariant system. The two systems showed good correlation. The algorithm resulted in a well correlated phase space representation.
{"title":"Invariant Manifolds in Human Joint Angle Analysis During Walking Gait","authors":"Sandesh G. Bhat, T. Sugar, S. Redkar","doi":"10.1115/detc2020-22241","DOIUrl":"https://doi.org/10.1115/detc2020-22241","url":null,"abstract":"\u0000 The complex dynamics of human gait is yet to be completely understood. Researchers have quantified stability of walking gait using Floquet multipliers as well as Lyapunov exponents. In this article, we utilize the techniques and tools from dynamical system theory and invariant manifolds to map the gait data onto a time invariant representation of a dynamical system. As an example, the complex behavior of the joint angle during walking was studied using a conformal mapping approach that transformed the time periodic system into a time invariant linear system. Time-delay embedding was used to reconstruct the dynamics of the original gait system with time series kinematic data. This minimal realization of the system was used to construct a Single Degree of Freedom (SDOF) oscillator. The time evolution of the linear oscillatory system was mapped back using the conformal mapping derived using Lyapunov-Floquet Theory. This algorithm was verified for walking gait kinematics data for two healthy human subjects. A comparison was drawn between the phase space behavior of the original time periodic system and the remapped time invariant system. The two systems showed good correlation. The algorithm resulted in a well correlated phase space representation.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116080547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stability of the collocated position control of a mass is studied when a pendulum is attached to it. The simplest proportional-derivative (PD) controller is applied, but the relevant constant time delay is taken into account. The linearized governing equations of the system are investigated. Stability charts are constructed for different pendulum parameters. Closed form expression is derived for the critical time delay; for delay values larger than the critical one, the PD controller cannot stabilize the desired position of the mass. The frequencies of the self-excited vibrations at the stability boundaries have essential role in identifying the types of loss of stability.
{"title":"Collocated Position Control of Oscillatory System in Presence of Delay","authors":"Bence Szaksz, G. Stépán","doi":"10.1115/detc2020-22362","DOIUrl":"https://doi.org/10.1115/detc2020-22362","url":null,"abstract":"\u0000 The stability of the collocated position control of a mass is studied when a pendulum is attached to it. The simplest proportional-derivative (PD) controller is applied, but the relevant constant time delay is taken into account. The linearized governing equations of the system are investigated. Stability charts are constructed for different pendulum parameters. Closed form expression is derived for the critical time delay; for delay values larger than the critical one, the PD controller cannot stabilize the desired position of the mass. The frequencies of the self-excited vibrations at the stability boundaries have essential role in identifying the types of loss of stability.","PeriodicalId":236538,"journal":{"name":"Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC)","volume":"187 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121925133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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