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Aude Billard - One of the best experts on this subject based on the ideXlab platform.
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a Dynamical System Approach for detection and reaction to human guidance in physical human robot interaction
Autonomous Robots, 2020Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:A seamless interaction requires two robotic behaviors: the leader role where the robot rejects the external perturbations and focuses on the autonomous execution of the task, and the follower role where the robot ignores the task and complies with human intentional forces. The goal of this work is to provide (1) a unified robotic architecture to produce these two roles, and (2) a human-guidance detection algorithm to switch across the two roles. In the absence of human-guidance, the robot performs its task autonomously and upon detection of such guidances the robot passively follows the human motions. We employ Dynamical Systems to generate task-specific motion and admittance control to generate reactive motions toward the human-guidance. This structure enables the robot to reject undesirable perturbations, track the motions precisely, react to human-guidance by providing proper compliant behavior, and re-plan the motion reactively. We provide analytical investigation of our method in terms of tracking and compliant behavior. Finally, we evaluate our method experimentally using a 6-DoF manipulator.
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A Dynamical System Approach for detection and reaction to human guidance in physical human–robot interaction
Autonomous Robots, 2020Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:A seamless interaction requires two robotic behaviors: the leader role where the robot rejects the external perturbations and focuses on the autonomous execution of the task, and the follower role where the robot ignores the task and complies with human intentional forces. The goal of this work is to provide (1) a unified robotic architecture to produce these two roles, and (2) a human-guidance detection algorithm to switch across the two roles. In the absence of human-guidance, the robot performs its task autonomously and upon detection of such guidances the robot passively follows the human motions. We employ Dynamical Systems to generate task-specific motion and admittance control to generate reactive motions toward the human-guidance. This structure enables the robot to reject undesirable perturbations, track the motions precisely, react to human-guidance by providing proper compliant behavior, and re-plan the motion reactively. We provide analytical investigation of our method in terms of tracking and compliant behavior. Finally, we evaluate our method experimentally using a 6-DoF manipulator.
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a Dynamical System Approach for adaptive grasping navigation and co manipulation with humanoid robots
International Conference on Robotics and Automation, 2020Co-Authors: Nadia Figueroa, Salman Faraji, Mikhail Koptev, Aude BillardAbstract:In this paper, we present an integrated Approach that provides compliant control of an iCub humanoid robot and adaptive reaching, grasping, navigating and co-manipulating capabilities. We use state-dependent Dynamical Systems (DS) to (i) coordinate and drive the robots hands (in both position and orientation) to grasp an object using an intermediate virtual object, and (ii) drive the robot's base while walking/navigating. The use of DS as motion generators allows us to adapt smoothly as the object moves and to re-plan on-line motion of the arms and body to reach the object's new location. The desired motion generated by the DS are used in combination with a whole-body compliant control strategy that absorbs perturbations while walking and offers compliant behaviors for grasping and manipulation tasks. Further, the desired dynamics for the arm and body can be learned from demonstrations. By integrating these components, we achieve unprecedented adaptive behaviors for whole body manipulation. We showcase this in simulations and real-world experiments where iCub robots (i) walk-to-grasp objects, (ii) follow a human (or another iCub) through interaction and (iii) learn to navigate or comanipulate an object from human guided demonstrations; whilst being robust to changing targets and perturbations.
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A Dynamical System Approach to task-adaptation in physical human–robot interaction
Autonomous Robots, 2019Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:The goal of this work is to enable robots to intelligently and compliantly adapt their motions to the intention of a human during physical Human–Robot Interaction in a multi-task setting. We employ a class of parameterized Dynamical Systems that allows for smooth and adaptive transitions between encoded tasks. To comply with human intention, we propose a mechanism that adapts generated motions (i.e., the desired velocity) to those intended by the human user (i.e., the real velocity) thereby switching to the most similar task. We provide a rigorous analytical evaluation of our method in terms of stability, convergence, and optimality yielding an interaction behavior which is safe and intuitive for the human. We investigate our method through experimental evaluations ranging in different setups: a 3-DoF haptic device, a 7-DoF manipulator and a mobile platform.
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a Dynamical System Approach to task adaptation in physical human robot interaction
Autonomous Robots, 2019Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:The goal of this work is to enable robots to intelligently and compliantly adapt their motions to the intention of a human during physical Human–Robot Interaction in a multi-task setting. We employ a class of parameterized Dynamical Systems that allows for smooth and adaptive transitions between encoded tasks. To comply with human intention, we propose a mechanism that adapts generated motions (i.e., the desired velocity) to those intended by the human user (i.e., the real velocity) thereby switching to the most similar task. We provide a rigorous analytical evaluation of our method in terms of stability, convergence, and optimality yielding an interaction behavior which is safe and intuitive for the human. We investigate our method through experimental evaluations ranging in different setups: a 3-DoF haptic device, a 7-DoF manipulator and a mobile platform.
Marek Szydlowski - One of the best experts on this subject based on the ideXlab platform.
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polynomial f r palatini cosmology Dynamical System Approach
Physical Review D, 2018Co-Authors: Marek Szydlowski, Aleksander StachowskiAbstract:We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the Dynamical System methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth Dynamical System. This System is is reduced to a 2D Dynamical System of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like' FLRW spacetimes are $C^0$-extendible. We point out that importance of Dynamical System of Newtonian type with non-smooth right-hand sides in the context of Palatini cosmology. In this framework we can investigate singularities which appear in the past and future of the cosmic evolution. We consider cosmological Systems in both Einstein and Jordan frames. We show that at each frame the topological structures of phase space are different.
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Dynamical System Approach to running lambda cosmological models
European Physical Journal C, 2016Co-Authors: Aleksander Stachowski, Marek SzydlowskiAbstract:We study the dynamics of cosmological models with a time dependent cosmological term. We consider five classes of models; two with the non-covariant parametrization of the cosmological term \(\Lambda \): \(\Lambda (H)\)CDM cosmologies, \(\Lambda (a)\)CDM cosmologies, and three with the covariant parametrization of \(\Lambda \): \(\Lambda (R)\)CDM cosmologies, where R(t) is the Ricci scalar, \(\Lambda (\phi )\)-cosmologies with diffusion, \(\Lambda (X)\)-cosmologies, where \(X=\frac{1}{2}g^{\alpha \beta }\nabla _{\alpha }\nabla _{\beta }\phi \) is a kinetic part of the density of the scalar field. We also consider the case of an emergent \(\Lambda (a)\) relation obtained from the behaviour of trajectories in a neighbourhood of an invariant submanifold. In the study of the dynamics we used Dynamical System methods for investigating how an evolutionary scenario can depend on the choice of special initial conditions. We show that the methods of Dynamical Systems allow one to investigate all admissible solutions of a running \(\Lambda \) cosmology for all initial conditions. We interpret Alcaniz and Lima’s Approach as a scaling cosmology. We formulate the idea of an emergent cosmological term derived directly from an approximation of the exact dynamics. We show that some non-covariant parametrization of the cosmological term like \(\Lambda (a)\), \(\Lambda (H)\) gives rise to the non-physical behaviour of trajectories in the phase space. This behaviour disappears if the term \(\Lambda (a)\) is emergent from the covariant parametrization.
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dynamics of the diffusive dm de interaction Dynamical System Approach
arXiv: General Relativity and Quantum Cosmology, 2016Co-Authors: Z Haba, Aleksander Stachowski, Marek SzydlowskiAbstract:We discuss dynamics of a model of an energy transfer between dark energy (DE) and dark matter (DM). The energy transfer is determined by a non-conservation law resulting from a diffusion of dark matter in an environment of dark energy. The relativistic invariance defines the diffusion in a unique way. The System can contain baryonic matter and radiation which do not interact with the dark sector. We treat the Friedman equation and the conservation laws as a closed Dynamical System. The dynamics of the model is examined using the Dynamical Systems methods for demonstration how solutions depend on initial conditions. We also fit the model parameters using astronomical observation: SNIa, $H(z)$, BAO and Alcock-Paczynski test. We show that the model with diffuse DM-DE is consistent with the data.
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Dynamical System Approach to running lambda cosmological models
arXiv: Cosmology and Nongalactic Astrophysics, 2016Co-Authors: Aleksander Stachowski, Marek SzydlowskiAbstract:We discussed the dynamics of cosmological models in which the cosmological constant term is a time dependent function through the scale factor $a(t)$, Hubble function $H(t)$, Ricci scalar $R(t)$ and scalar field $\phi(t)$. We considered five classes of models; two non-covariant parametrization of $\Lambda$: 1) $\Lambda(H)$CDM cosmologies where $H(t)$ is the Hubble parameter, 2) $\Lambda(a)$CDM cosmologies where $a(t)$ is the scale factor, and three covariant parametrization of $\Lambda$: 3) $\Lambda(R)$CDM cosmologies, where $R(t)$ is the Ricci scalar, 4) $\Lambda(\phi)$-cosmologies with diffusion, 5) $\Lambda(X)$-cosmologies, where $X=\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi$ is a kinetic part of density of the scalar field. We also considered the case of an emergent $\Lambda(a)$ relation obtained from the behavior of trajectories in a neighborhood of an invariant submanifold. In study of dynamics we use Dynamical System methods for investigating how a evolutional scenario can depend on the choice of special initial conditions. We showed that methods of Dynamical Systems offer the possibility of investigation all admissible solutions of a running $\Lambda$ cosmology for all initial conditions, their stability, asymptotic states as well as a nature of the evolution in the early universe (singularity or bounce) and a long term behavior at the large times. We also formulated an idea of the emergent cosmological term derived directly from an approximation of exact dynamics. We show that some non-covariant parametrizations of Lambda term like $\Lambda(a)$, $\Lambda(H)$ give rise to pathological and nonphysical behaviour of trajectories in the phase space. This behaviour disappears if the term $\Lambda(a)$ is emergent from the covariant parametrization.
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Dynamical dark energy models Dynamical System Approach
arXiv: General Relativity and Quantum Cosmology, 2005Co-Authors: Marek Szydlowski, Orest HrycynaAbstract:We study the Friedmann-Robertson-Walker model with Dynamical dark energy modelled in terms of the equation of state $p_{x}=w_{x}(a(z)) \rho_{x}$ in which the coefficient $w_{x}$ is parameterized by the scale factor $a$ or redshift $z$. We use methods of qualitative analysis of differential equations to investigate the space of all admissible solutions for all initial conditions on the two-dimensional phase plane. We show advantages of representing this dynamics as a motion of a particle in the one-dimensional potential $V(a)$. One of the features of this reduction is the possibility of investigating how typical are big rip singularities in the future evolution of the model. The properties of potential function $V$ can serve as a tool for qualitative classification of all evolution paths. Some important features like resolution of the acceleration problem can be simply visualized as domains on the phase plane. Then one is able to see how large is the class of solutions (labelled by the inset of the initial conditions) leading to the desired property.
Isuru Dasanayake - One of the best experts on this subject based on the ideXlab platform.
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SU‐GG‐T‐16: A Dynamical System Approach for Real‐Time IMRT Optimization
Medical Physics, 2010Co-Authors: Isuru Dasanayake, I. El NaqaAbstract:Purpose: Real‐time adaptation of IMRTtreatment plans to daily changes in anatomy is a necessary task for adaptive radiotherapy. Re‐optimization of radiation beam weights is maybe needed for some treatment fraction, which is computationally expensive and practically inefficient. Therefore, we are developing a new optimization algorithm for real‐time IMRT re‐planning using a Dynamical Systems Approach. Method and Materials:IMRT planning is viewed as an adaptive control design and is solved by a discrete‐time Dynamical Systems Approach. We developed a constrained Kaiman filter (CKF) to obtain optimal estimates of the desired beam weights subject to prescribed constraints recursively according to the changes in tumor geometry. Our proposed method eliminates the necessity of performing the fluence‐map optimization at each treatment and, instead, adopts previous optimization to recursively compute optimal fluence‐maps for subsequent treatment fractions. As a validation we demonstrated our method using digital phantom data. Results: To generate benchmark data, we used an offline solver to calculate the optimal pencil beam weights for two treatments, where an organ‐at‐risk is represented by disk shape and a horse shoe shape structure was selected as the PTV in one treatment and was deformed in the second treatment, which is used as a test case. In our experiments, the CKF yielded similar performance to the offline solver on the test case in terms of mean‐squared error and dose‐volume histograms with fraction of the required time. It also showed robustness to initialization conditions. Conclusion: We presented a new efficient algorithm based on recursive filtering for IMRT planning. This algorithm is capable of developing complete treatment plan taking into account changes in tumor geometry while avoiding re‐optimization of radiation beam weights for each treatment fraction. Conflict of Interest: Part of this work was supported by NSF grant and Varian Medical Systems.
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su gg t 16 a Dynamical System Approach for real time imrt optimization
Medical Physics, 2010Co-Authors: Isuru Dasanayake, El I NaqaAbstract:Purpose: Real‐time adaptation of IMRTtreatment plans to daily changes in anatomy is a necessary task for adaptive radiotherapy. Re‐optimization of radiation beam weights is maybe needed for some treatment fraction, which is computationally expensive and practically inefficient. Therefore, we are developing a new optimization algorithm for real‐time IMRT re‐planning using a Dynamical Systems Approach. Method and Materials:IMRT planning is viewed as an adaptive control design and is solved by a discrete‐time Dynamical Systems Approach. We developed a constrained Kaiman filter (CKF) to obtain optimal estimates of the desired beam weights subject to prescribed constraints recursively according to the changes in tumor geometry. Our proposed method eliminates the necessity of performing the fluence‐map optimization at each treatment and, instead, adopts previous optimization to recursively compute optimal fluence‐maps for subsequent treatment fractions. As a validation we demonstrated our method using digital phantom data. Results: To generate benchmark data, we used an offline solver to calculate the optimal pencil beam weights for two treatments, where an organ‐at‐risk is represented by disk shape and a horse shoe shape structure was selected as the PTV in one treatment and was deformed in the second treatment, which is used as a test case. In our experiments, the CKF yielded similar performance to the offline solver on the test case in terms of mean‐squared error and dose‐volume histograms with fraction of the required time. It also showed robustness to initialization conditions. Conclusion: We presented a new efficient algorithm based on recursive filtering for IMRT planning. This algorithm is capable of developing complete treatment plan taking into account changes in tumor geometry while avoiding re‐optimization of radiation beam weights for each treatment fraction. Conflict of Interest: Part of this work was supported by NSF grant and Varian Medical Systems.
Aleksander Stachowski - One of the best experts on this subject based on the ideXlab platform.
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polynomial f r palatini cosmology Dynamical System Approach
Physical Review D, 2018Co-Authors: Marek Szydlowski, Aleksander StachowskiAbstract:We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the Dynamical System methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth Dynamical System. This System is is reduced to a 2D Dynamical System of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like' FLRW spacetimes are $C^0$-extendible. We point out that importance of Dynamical System of Newtonian type with non-smooth right-hand sides in the context of Palatini cosmology. In this framework we can investigate singularities which appear in the past and future of the cosmic evolution. We consider cosmological Systems in both Einstein and Jordan frames. We show that at each frame the topological structures of phase space are different.
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Dynamical System Approach to running lambda cosmological models
European Physical Journal C, 2016Co-Authors: Aleksander Stachowski, Marek SzydlowskiAbstract:We study the dynamics of cosmological models with a time dependent cosmological term. We consider five classes of models; two with the non-covariant parametrization of the cosmological term \(\Lambda \): \(\Lambda (H)\)CDM cosmologies, \(\Lambda (a)\)CDM cosmologies, and three with the covariant parametrization of \(\Lambda \): \(\Lambda (R)\)CDM cosmologies, where R(t) is the Ricci scalar, \(\Lambda (\phi )\)-cosmologies with diffusion, \(\Lambda (X)\)-cosmologies, where \(X=\frac{1}{2}g^{\alpha \beta }\nabla _{\alpha }\nabla _{\beta }\phi \) is a kinetic part of the density of the scalar field. We also consider the case of an emergent \(\Lambda (a)\) relation obtained from the behaviour of trajectories in a neighbourhood of an invariant submanifold. In the study of the dynamics we used Dynamical System methods for investigating how an evolutionary scenario can depend on the choice of special initial conditions. We show that the methods of Dynamical Systems allow one to investigate all admissible solutions of a running \(\Lambda \) cosmology for all initial conditions. We interpret Alcaniz and Lima’s Approach as a scaling cosmology. We formulate the idea of an emergent cosmological term derived directly from an approximation of the exact dynamics. We show that some non-covariant parametrization of the cosmological term like \(\Lambda (a)\), \(\Lambda (H)\) gives rise to the non-physical behaviour of trajectories in the phase space. This behaviour disappears if the term \(\Lambda (a)\) is emergent from the covariant parametrization.
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dynamics of the diffusive dm de interaction Dynamical System Approach
arXiv: General Relativity and Quantum Cosmology, 2016Co-Authors: Z Haba, Aleksander Stachowski, Marek SzydlowskiAbstract:We discuss dynamics of a model of an energy transfer between dark energy (DE) and dark matter (DM). The energy transfer is determined by a non-conservation law resulting from a diffusion of dark matter in an environment of dark energy. The relativistic invariance defines the diffusion in a unique way. The System can contain baryonic matter and radiation which do not interact with the dark sector. We treat the Friedman equation and the conservation laws as a closed Dynamical System. The dynamics of the model is examined using the Dynamical Systems methods for demonstration how solutions depend on initial conditions. We also fit the model parameters using astronomical observation: SNIa, $H(z)$, BAO and Alcock-Paczynski test. We show that the model with diffuse DM-DE is consistent with the data.
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Dynamical System Approach to running lambda cosmological models
arXiv: Cosmology and Nongalactic Astrophysics, 2016Co-Authors: Aleksander Stachowski, Marek SzydlowskiAbstract:We discussed the dynamics of cosmological models in which the cosmological constant term is a time dependent function through the scale factor $a(t)$, Hubble function $H(t)$, Ricci scalar $R(t)$ and scalar field $\phi(t)$. We considered five classes of models; two non-covariant parametrization of $\Lambda$: 1) $\Lambda(H)$CDM cosmologies where $H(t)$ is the Hubble parameter, 2) $\Lambda(a)$CDM cosmologies where $a(t)$ is the scale factor, and three covariant parametrization of $\Lambda$: 3) $\Lambda(R)$CDM cosmologies, where $R(t)$ is the Ricci scalar, 4) $\Lambda(\phi)$-cosmologies with diffusion, 5) $\Lambda(X)$-cosmologies, where $X=\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi$ is a kinetic part of density of the scalar field. We also considered the case of an emergent $\Lambda(a)$ relation obtained from the behavior of trajectories in a neighborhood of an invariant submanifold. In study of dynamics we use Dynamical System methods for investigating how a evolutional scenario can depend on the choice of special initial conditions. We showed that methods of Dynamical Systems offer the possibility of investigation all admissible solutions of a running $\Lambda$ cosmology for all initial conditions, their stability, asymptotic states as well as a nature of the evolution in the early universe (singularity or bounce) and a long term behavior at the large times. We also formulated an idea of the emergent cosmological term derived directly from an approximation of exact dynamics. We show that some non-covariant parametrizations of Lambda term like $\Lambda(a)$, $\Lambda(H)$ give rise to pathological and nonphysical behaviour of trajectories in the phase space. This behaviour disappears if the term $\Lambda(a)$ is emergent from the covariant parametrization.
Mahdi Khoramshahi - One of the best experts on this subject based on the ideXlab platform.
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a Dynamical System Approach for detection and reaction to human guidance in physical human robot interaction
Autonomous Robots, 2020Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:A seamless interaction requires two robotic behaviors: the leader role where the robot rejects the external perturbations and focuses on the autonomous execution of the task, and the follower role where the robot ignores the task and complies with human intentional forces. The goal of this work is to provide (1) a unified robotic architecture to produce these two roles, and (2) a human-guidance detection algorithm to switch across the two roles. In the absence of human-guidance, the robot performs its task autonomously and upon detection of such guidances the robot passively follows the human motions. We employ Dynamical Systems to generate task-specific motion and admittance control to generate reactive motions toward the human-guidance. This structure enables the robot to reject undesirable perturbations, track the motions precisely, react to human-guidance by providing proper compliant behavior, and re-plan the motion reactively. We provide analytical investigation of our method in terms of tracking and compliant behavior. Finally, we evaluate our method experimentally using a 6-DoF manipulator.
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A Dynamical System Approach for detection and reaction to human guidance in physical human–robot interaction
Autonomous Robots, 2020Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:A seamless interaction requires two robotic behaviors: the leader role where the robot rejects the external perturbations and focuses on the autonomous execution of the task, and the follower role where the robot ignores the task and complies with human intentional forces. The goal of this work is to provide (1) a unified robotic architecture to produce these two roles, and (2) a human-guidance detection algorithm to switch across the two roles. In the absence of human-guidance, the robot performs its task autonomously and upon detection of such guidances the robot passively follows the human motions. We employ Dynamical Systems to generate task-specific motion and admittance control to generate reactive motions toward the human-guidance. This structure enables the robot to reject undesirable perturbations, track the motions precisely, react to human-guidance by providing proper compliant behavior, and re-plan the motion reactively. We provide analytical investigation of our method in terms of tracking and compliant behavior. Finally, we evaluate our method experimentally using a 6-DoF manipulator.
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A Dynamical System Approach to task-adaptation in physical human–robot interaction
Autonomous Robots, 2019Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:The goal of this work is to enable robots to intelligently and compliantly adapt their motions to the intention of a human during physical Human–Robot Interaction in a multi-task setting. We employ a class of parameterized Dynamical Systems that allows for smooth and adaptive transitions between encoded tasks. To comply with human intention, we propose a mechanism that adapts generated motions (i.e., the desired velocity) to those intended by the human user (i.e., the real velocity) thereby switching to the most similar task. We provide a rigorous analytical evaluation of our method in terms of stability, convergence, and optimality yielding an interaction behavior which is safe and intuitive for the human. We investigate our method through experimental evaluations ranging in different setups: a 3-DoF haptic device, a 7-DoF manipulator and a mobile platform.
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a Dynamical System Approach to task adaptation in physical human robot interaction
Autonomous Robots, 2019Co-Authors: Mahdi Khoramshahi, Aude BillardAbstract:The goal of this work is to enable robots to intelligently and compliantly adapt their motions to the intention of a human during physical Human–Robot Interaction in a multi-task setting. We employ a class of parameterized Dynamical Systems that allows for smooth and adaptive transitions between encoded tasks. To comply with human intention, we propose a mechanism that adapts generated motions (i.e., the desired velocity) to those intended by the human user (i.e., the real velocity) thereby switching to the most similar task. We provide a rigorous analytical evaluation of our method in terms of stability, convergence, and optimality yielding an interaction behavior which is safe and intuitive for the human. We investigate our method through experimental evaluations ranging in different setups: a 3-DoF haptic device, a 7-DoF manipulator and a mobile platform.
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a Dynamical System Approach for softly catching a flying object theory and experiment
IEEE Transactions on Robotics, 2016Co-Authors: Seyed Sina Mirrazavi Salehian, Mahdi Khoramshahi, Aude BillardAbstract:Catching a fast flying object is particularly challenging as it consists of two tasks: extremely precise estimation of the object's motion and control of the robot's motion. Any small imprecision may lead the fingers to close too abruptly and let the object fly away from the hand before closing. We present a strategy to overcome for sensorimotor imprecision by introducing softness in the catching Approach. Soft catching consists of having the robot moves with the object for a short period of time, so as to leave more time for the fingers to close on the object. We use a dynamic System-based control law to generate the appropriate reach and follow motion, which is expressed as a linear parameter varying (LPV) System. We propose a method to approximate the parameters of LPV Systems using Gaussian mixture models, based on a set of kinematically feasible demonstrations generated by an offline optimal control framework. We show theoretically that the resulting DS will intercept the object at the intercept point, at the right time with the desired velocity direction . Stability and convergence of the Approach are assessed through Lyapunov stability theory. The proposed method is validated Systematically to catch three objects that generate elastic contacts and demonstrate important improvement over a hard catching Approach.