The Experts below are selected from a list of 282 Experts worldwide ranked by ideXlab platform
Kouichi Taji - One of the best experts on this subject based on the ideXlab platform.
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Parametric Excitation Walking for Four-linked Bipedal Robot
IFAC Proceedings Volumes, 2020Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:Abstract In the bipedal walking, mechanical energy needs to be restored because of energy lost by heel strike collisions. Harata et al. have applied Parametric Excitation method to a kneed bipedal robot, and have shown that sustainable gait is generated with only knee torque. In this paper, we propose the method that combines the Parametric Excitation method for swing-leg with that for support-leg to improve the gait efficiency. To do this, we first extend the three-link bipedal robot to four-link bipedal robot by adding a support-leg knee, and then apply the Parametric Excitation method to the support-leg. Consequently, Parametric Excitation method for swing-leg and support-leg restores energy twice a step, and gait of the proposed method grows in efficiency. In the method proposed by Harata et al., the robot has large shin masses to restore much energy and has large semicircular feet to decrease the energy lost by heel strike. These features are unfavorable because common bipedal robots do not have such features. By simulation, we show that the bipedal robot with small shin masses or small feet can walk sustainably because the proposed method increases the quantity of restored energy. For example, the ratio of shin mass to thigh mass is one and the foot radius is reduced to one fifth of the previous method.
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Ornithoid Gait Generation Based on Parametric Excitation
Journal of the Robotics Society of Japan, 2020Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:We have applied a Parametric Excitation method to a kneed biped robot with semicircular feet and have shown that the robot can walk sustainably with only knee torque. A swing-leg of the kneed biped robot has similar mechanism to an acrobot, and many acrobots are controlled in inverse direction like ornithoid walking. These suggest that inverse bending of a knee restores more mechanical energy than forward bending, and hence, the ornithoid walking can be more efficient. In this paper, we first compare the forward bending with the inverse bending for a double pendulum, and show by numerical simulation that the mechanical energy of the inverse bending increases more than that of the forward bending like human walking. We then propose a Parametric Excitation based ornithoid gait for a kneed biped robot, and show sustainably walking by numerical simulation. Finally, we compare Parametric Excitation based ornithoid gait with Parametric Excitation based human gait, and we show that ornithoid gait is more efficeint.
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An Application of Delayed Feedback Control to Parametric Excitation Walking
Journal of Control and Systems Engineering, 2014Co-Authors: Hajime Asaka, Yuji Harata, Kouichi TajiAbstract:This paper proposes a new applying method of delayed feedback control (DFC) which can suppress the period-doubling bifurcation in Parametric Excitation walking. Parametric Excitation walking for a kneed biped robot was realized by controlling knee angle of the robot with knee joint torque. When the energy input increased, Parametric Excitation walking exhibited period-doubling bifurcation and period-two walking emerged. In this study, based on DFC we redesigned the knee joint torque, which suppresses the period-doubling bifurcation, and makes period-two walking converge to period-one walking. Numerical simulations showed that the resulting period-one walking with DFC was more efficient than the period-two walking. Regions of attraction for period-one Parametric Excitation walking were calculated and it was shown that the proposed method can improve the stability of Parametric Excitation walking.
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efficient Parametric Excitation walking with delayed feedback control
Nonlinear Dynamics, 2012Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:In passive dynamic walking proposed by McGeer, mechanical energy lost by heel strike is restored by transporting potential energy to kinetic energy as walking down a slope. When energy input is larger as a slope is steeper, the bifurcation of a walking cycle occurs. In the Parametric Excitation walking, which is to realize passive dynamic-like walking on the level ground, the bifurcation of a walking cycle has also been observed when walking speed is fast. Recently, Asano et al. have shown that bifurcation exerts an adverse influence upon walking performance by using a rimless wheel model. In this paper, we apply the delayed feedback control (DFC), originally used in chaos control, to Parametric Excitation walking to suppress bifurcation. We show in numerical simulation that the proposed method makes period-two walking to period-one walking, and improves energy efficiency. In addition, the proposed method can generate a sustainable gait in the region where a biped robot cannot walk without DFC. The analyses using a Poincare map reveal that period-one walking with DFC corresponds to an unstable periodic orbit and reveal that a robot model in this paper satisfies the sufficient condition of applicability of DFC.
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IROS - Development and experiment of a kneed biped walking robot based on Parametric Excitation principle
2011 IEEE RSJ International Conference on Intelligent Robots and Systems, 2011Co-Authors: Yoshihisa Banno, Yuji Harata, Kouichi TajiAbstract:Parametric Excitation walking is one of methods to realize dynamic walking on a level ground. This method has first applied to a biped robot with telescopic legs and later to a robot with actuated knee joints. In Parametric Excitation walking, mechanical energy is increased by periodic up-and-down motion of the center of mass. While Parametric Excitation walking with telescopic legs has verified by an experimental robot, that with actuated knees has not yet as far as we know. The purpose of this paper is to present demonstration experiment of Parametric Excitation walking with a kneed biped robot. To do this, we develop an experimental kneed biped robot having four parallel legs with semicircular feet. In the experiment, the robot achieves walking on a level ground more than 15 steps. We also measure the movements of the robot during walking by a 3D motion capture and compare with simulation results.
Yuji Harata - One of the best experts on this subject based on the ideXlab platform.
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Parametric Excitation Walking for Four-linked Bipedal Robot
IFAC Proceedings Volumes, 2020Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:Abstract In the bipedal walking, mechanical energy needs to be restored because of energy lost by heel strike collisions. Harata et al. have applied Parametric Excitation method to a kneed bipedal robot, and have shown that sustainable gait is generated with only knee torque. In this paper, we propose the method that combines the Parametric Excitation method for swing-leg with that for support-leg to improve the gait efficiency. To do this, we first extend the three-link bipedal robot to four-link bipedal robot by adding a support-leg knee, and then apply the Parametric Excitation method to the support-leg. Consequently, Parametric Excitation method for swing-leg and support-leg restores energy twice a step, and gait of the proposed method grows in efficiency. In the method proposed by Harata et al., the robot has large shin masses to restore much energy and has large semicircular feet to decrease the energy lost by heel strike. These features are unfavorable because common bipedal robots do not have such features. By simulation, we show that the bipedal robot with small shin masses or small feet can walk sustainably because the proposed method increases the quantity of restored energy. For example, the ratio of shin mass to thigh mass is one and the foot radius is reduced to one fifth of the previous method.
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Ornithoid Gait Generation Based on Parametric Excitation
Journal of the Robotics Society of Japan, 2020Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:We have applied a Parametric Excitation method to a kneed biped robot with semicircular feet and have shown that the robot can walk sustainably with only knee torque. A swing-leg of the kneed biped robot has similar mechanism to an acrobot, and many acrobots are controlled in inverse direction like ornithoid walking. These suggest that inverse bending of a knee restores more mechanical energy than forward bending, and hence, the ornithoid walking can be more efficient. In this paper, we first compare the forward bending with the inverse bending for a double pendulum, and show by numerical simulation that the mechanical energy of the inverse bending increases more than that of the forward bending like human walking. We then propose a Parametric Excitation based ornithoid gait for a kneed biped robot, and show sustainably walking by numerical simulation. Finally, we compare Parametric Excitation based ornithoid gait with Parametric Excitation based human gait, and we show that ornithoid gait is more efficeint.
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An Application of Delayed Feedback Control to Parametric Excitation Walking
Journal of Control and Systems Engineering, 2014Co-Authors: Hajime Asaka, Yuji Harata, Kouichi TajiAbstract:This paper proposes a new applying method of delayed feedback control (DFC) which can suppress the period-doubling bifurcation in Parametric Excitation walking. Parametric Excitation walking for a kneed biped robot was realized by controlling knee angle of the robot with knee joint torque. When the energy input increased, Parametric Excitation walking exhibited period-doubling bifurcation and period-two walking emerged. In this study, based on DFC we redesigned the knee joint torque, which suppresses the period-doubling bifurcation, and makes period-two walking converge to period-one walking. Numerical simulations showed that the resulting period-one walking with DFC was more efficient than the period-two walking. Regions of attraction for period-one Parametric Excitation walking were calculated and it was shown that the proposed method can improve the stability of Parametric Excitation walking.
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efficient Parametric Excitation walking with delayed feedback control
Nonlinear Dynamics, 2012Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:In passive dynamic walking proposed by McGeer, mechanical energy lost by heel strike is restored by transporting potential energy to kinetic energy as walking down a slope. When energy input is larger as a slope is steeper, the bifurcation of a walking cycle occurs. In the Parametric Excitation walking, which is to realize passive dynamic-like walking on the level ground, the bifurcation of a walking cycle has also been observed when walking speed is fast. Recently, Asano et al. have shown that bifurcation exerts an adverse influence upon walking performance by using a rimless wheel model. In this paper, we apply the delayed feedback control (DFC), originally used in chaos control, to Parametric Excitation walking to suppress bifurcation. We show in numerical simulation that the proposed method makes period-two walking to period-one walking, and improves energy efficiency. In addition, the proposed method can generate a sustainable gait in the region where a biped robot cannot walk without DFC. The analyses using a Poincare map reveal that period-one walking with DFC corresponds to an unstable periodic orbit and reveal that a robot model in this paper satisfies the sufficient condition of applicability of DFC.
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IROS - Development and experiment of a kneed biped walking robot based on Parametric Excitation principle
2011 IEEE RSJ International Conference on Intelligent Robots and Systems, 2011Co-Authors: Yoshihisa Banno, Yuji Harata, Kouichi TajiAbstract:Parametric Excitation walking is one of methods to realize dynamic walking on a level ground. This method has first applied to a biped robot with telescopic legs and later to a robot with actuated knee joints. In Parametric Excitation walking, mechanical energy is increased by periodic up-and-down motion of the center of mass. While Parametric Excitation walking with telescopic legs has verified by an experimental robot, that with actuated knees has not yet as far as we know. The purpose of this paper is to present demonstration experiment of Parametric Excitation walking with a kneed biped robot. To do this, we develop an experimental kneed biped robot having four parallel legs with semicircular feet. In the experiment, the robot achieves walking on a level ground more than 15 steps. We also measure the movements of the robot during walking by a 3D motion capture and compare with simulation results.
T.x. Wu - One of the best experts on this subject based on the ideXlab platform.
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Parametric Excitation of wheel/track system and its effects on rail corrugation
Wear, 2008Co-Authors: T.x. WuAbstract:In this work the effects on rail corrugation due to the wheel/track Parametric Excitation is studied by combining wheel/track dynamics, contact mechanics and wear. Firstly, the wheel/track interaction due to the Parametric Excitation by the varying dynamic stiffness of a discrete track is studied using a low-order time-varying track model in the base of the frequency domain analysis, coupled with a mass wheel. Secondly, the wheel/rail rolling contact mechanics and wear are briefly analyzed according to Hertz theory. Then a quasi-static method is used to simulate rail corrugation formation due to the wheel/track Parametric Excitation by combining wheel/track dynamics, rolling contact mechanics and wear to form a closed positive feedback loop of corrugation growth. It is found that rail corrugation due to the uneven wear is closely related to the variation pattern of the wheel/rail interaction due to the Parametric Excitation. Although in practice both initial roughness and Parametric Excitation take effect, rail corrugation can be developed by the wheel/track Parametric Excitation alone without initial roughness on the railhead. ?? 2008 T.X. Wu.
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on the rolling noise generation due to wheel track Parametric Excitation
Journal of Sound and Vibration, 2006Co-Authors: T.x. Wu, D.j. ThompsonAbstract:As a discretely supported railway track is essentially periodic, when a wheel rolls over the rail, it experiences the varying dynamic stiffness in a sleeper bay of the track, and thus the wheel and rail is periodically excited at the sleeper-passing frequency. The Parametric Excitation due to the varying track stiffness, in addition to the roughness or discontinuities on the wheel and rail rolling surfaces, also causes vibration and noise emission. A frequency–time domain methodology is applied for simulation of the wheel/rail interaction due to the Parametric Excitation. The wheel/rail interaction forces are calculated and Track–Wheel Interaction Noise Software (TWINS) is used to predict the noise radiation due to the Parametric Excitation at various train speeds. The results are compared with those from a moving irregularity model where no Parametric Excitation is generated. It is found that the components due to the Parametric Excitation are not significant at lower speeds compared with those due to the roughness Excitation. Use of a moving irregularity model without considering the wheel/track Parametric Excitation may under-estimate the noise emission level at high speeds.
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On the rolling noise generation due to wheel/track Parametric Excitation
Journal of Sound and Vibration, 2006Co-Authors: T.x. Wu, David ThompsonAbstract:As a discretely supported railway track is essentially periodic, when a wheel rolls over the rail, it experiences the varying dynamic stiffness in a sleeper bay of the track, and thus the wheel and rail is periodically excited at the sleeper-passing frequency. The Parametric Excitation due to the varying track stiffness, in addition to the roughness or discontinuities on the wheel and rail rolling surfaces, also causes vibration and noise emission. A frequency–time domain methodology is applied for simulation of the wheel/rail interaction due to the Parametric Excitation. The wheel/rail interaction forces are calculated and Track–Wheel Interaction Noise Software (TWINS) is used to predict the noise radiation due to the Parametric Excitation at various train speeds. The results are compared with those from a moving irregularity model where no Parametric Excitation is generated. It is found that the components due to the Parametric Excitation are not significant at lower speeds compared with those due to the roughness Excitation. Use of a moving irregularity model without considering the wheel/track Parametric Excitation may under-estimate the noise emission level at high speeds.
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On the rolling noise generation due to wheel/track Parametric Excitation
Journal of Sound and Vibration, 2006Co-Authors: T.x. Wu, D.j. ThompsonAbstract:As a discretely supported railway track is essentially periodic, when a wheel rolls over the rail, it experiences the varying dynamic stiffness in a sleeper bay of the track, and thus the wheel and rail is periodically excited at the sleeper-passing frequency. The Parametric Excitation due to the varying track stiffness, in addition to the roughness or discontinuities on the wheel and rail rolling surfaces, also causes vibration and noise emission. A frequency-time domain methodology is applied for simulation of the wheel/rail interaction due to the Parametric Excitation. The wheel/rail interaction forces are calculated and Track-Wheel Interaction Noise Software (TWINS) is used to predict the noise radiation due to the Parametric Excitation at various train speeds. The results are compared with those from a moving irregularity model where no Parametric Excitation is generated. It is found that the components due to the Parametric Excitation are not significant at lower speeds compared with those due to the roughness Excitation. Use of a moving irregularity model without considering the wheel/track Parametric Excitation may under-estimate the noise emission level at high speeds. ?? 2006 Elsevier Ltd. All rights reserved.
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on the Parametric Excitation of the wheel track system
Journal of Sound and Vibration, 2004Co-Authors: T.x. Wu, D.j. ThompsonAbstract:As a wheel moves along a discretely supported track, it can experience Parametric Excitation due to the varying dynamic stiffness of the track. In order to study this, an equivalent time-varying model is developed for the track, according to the space-varying receptance in a sleeper bay. Using this track model combined with a mass representing the wheel, the wheel/rail interaction and response to the Parametric Excitation are simulated. The results are compared with those from a moving irregularity model and the differences between the moving wheel and moving irregularity models are examined from various aspects of wheel/rail dynamics. The wheel/rail interaction force due to the Parametric Excitation may be significant compared with that due to the roughness Excitation especially at low frequencies and increases in magnitude with the running speed of a train. Because of the Parametric Excitation the wheel/rail contact force spectra contain many harmonics with a basic component at the sleeper-passing frequency, and the components around the pinned–pinned resonance frequency show a higher level.
Fumihiko Asano - One of the best experts on this subject based on the ideXlab platform.
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Ornithoid Gait Generation Based on Parametric Excitation
Journal of the Robotics Society of Japan, 2020Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:We have applied a Parametric Excitation method to a kneed biped robot with semicircular feet and have shown that the robot can walk sustainably with only knee torque. A swing-leg of the kneed biped robot has similar mechanism to an acrobot, and many acrobots are controlled in inverse direction like ornithoid walking. These suggest that inverse bending of a knee restores more mechanical energy than forward bending, and hence, the ornithoid walking can be more efficient. In this paper, we first compare the forward bending with the inverse bending for a double pendulum, and show by numerical simulation that the mechanical energy of the inverse bending increases more than that of the forward bending like human walking. We then propose a Parametric Excitation based ornithoid gait for a kneed biped robot, and show sustainably walking by numerical simulation. Finally, we compare Parametric Excitation based ornithoid gait with Parametric Excitation based human gait, and we show that ornithoid gait is more efficeint.
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Parametric Excitation Walking for Four-linked Bipedal Robot
IFAC Proceedings Volumes, 2020Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:Abstract In the bipedal walking, mechanical energy needs to be restored because of energy lost by heel strike collisions. Harata et al. have applied Parametric Excitation method to a kneed bipedal robot, and have shown that sustainable gait is generated with only knee torque. In this paper, we propose the method that combines the Parametric Excitation method for swing-leg with that for support-leg to improve the gait efficiency. To do this, we first extend the three-link bipedal robot to four-link bipedal robot by adding a support-leg knee, and then apply the Parametric Excitation method to the support-leg. Consequently, Parametric Excitation method for swing-leg and support-leg restores energy twice a step, and gait of the proposed method grows in efficiency. In the method proposed by Harata et al., the robot has large shin masses to restore much energy and has large semicircular feet to decrease the energy lost by heel strike. These features are unfavorable because common bipedal robots do not have such features. By simulation, we show that the bipedal robot with small shin masses or small feet can walk sustainably because the proposed method increases the quantity of restored energy. For example, the ratio of shin mass to thigh mass is one and the foot radius is reduced to one fifth of the previous method.
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efficient Parametric Excitation walking with delayed feedback control
Nonlinear Dynamics, 2012Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:In passive dynamic walking proposed by McGeer, mechanical energy lost by heel strike is restored by transporting potential energy to kinetic energy as walking down a slope. When energy input is larger as a slope is steeper, the bifurcation of a walking cycle occurs. In the Parametric Excitation walking, which is to realize passive dynamic-like walking on the level ground, the bifurcation of a walking cycle has also been observed when walking speed is fast. Recently, Asano et al. have shown that bifurcation exerts an adverse influence upon walking performance by using a rimless wheel model. In this paper, we apply the delayed feedback control (DFC), originally used in chaos control, to Parametric Excitation walking to suppress bifurcation. We show in numerical simulation that the proposed method makes period-two walking to period-one walking, and improves energy efficiency. In addition, the proposed method can generate a sustainable gait in the region where a biped robot cannot walk without DFC. The analyses using a Poincare map reveal that period-one walking with DFC corresponds to an unstable periodic orbit and reveal that a robot model in this paper satisfies the sufficient condition of applicability of DFC.
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Parametric Excitation-based inverse bending gait generation
Robotica, 2011Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:In a gait generation method based on the Parametric Excitation principle, appropriate motion of the center of mass restores kinetic energy lost by heel strike. The motion is realized by bending and stretching a swing-leg regardless of bending direction. In this paper, we first show that inverse bending restores more mechanical energy than forward bending, and then propose a Parametric Excitation-based inverse bending gait for a kneed biped robot, which improves gait efficiency of Parametric Excitation walking.
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biped gait generation based on Parametric Excitation by knee joint actuation
Robotica, 2009Co-Authors: Yuji Harata, Fumihiko Asano, Kouichi TajiAbstract:Restoration of mechanical energy dissipating on impact at the ground is necessary for sustainable gait generation. Parametric Excitation is one approach to restore the mechanical energy. Asano et al. (“Parametric Excitation mechanisms for dynamic bipedal walking,” IEEE International Conference on Robotics and Automation (2005) pp. 611–617.) applied Parametric Excitation to a biped robot with telescopic-legs, in which up-and-down motion restores total mechanical energy like playing on the swing. In this paper, Parametric Excitation principle is applied to a kneed biped robot with only knee actuation and it is shown that the robot walks successively without hip actuation. We also examine influences of several parameters and reference trajectory on walking performance.
Yoshihisa Banno - One of the best experts on this subject based on the ideXlab platform.
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IROS - Development and experiment of a kneed biped walking robot based on Parametric Excitation principle
2011 IEEE RSJ International Conference on Intelligent Robots and Systems, 2011Co-Authors: Yoshihisa Banno, Yuji Harata, Kouichi TajiAbstract:Parametric Excitation walking is one of methods to realize dynamic walking on a level ground. This method has first applied to a biped robot with telescopic legs and later to a robot with actuated knee joints. In Parametric Excitation walking, mechanical energy is increased by periodic up-and-down motion of the center of mass. While Parametric Excitation walking with telescopic legs has verified by an experimental robot, that with actuated knees has not yet as far as we know. The purpose of this paper is to present demonstration experiment of Parametric Excitation walking with a kneed biped robot. To do this, we develop an experimental kneed biped robot having four parallel legs with semicircular feet. In the experiment, the robot achieves walking on a level ground more than 15 steps. We also measure the movements of the robot during walking by a 3D motion capture and compare with simulation results.
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Parametric Excitation based bipedal walking: Control method and optimization
Numerical Algebra Control and Optimization, 2011Co-Authors: Yuji Harata, Yoshihisa Banno, Kouichi TajiAbstract:In Parametric Excitation walking, energy lost by a heel strike is restored by bending and stretching a swing leg, and then a sustainable gait is generated with only knee torque. In this paper, we first propose the method that combines the Parametric Excitation method for a swing leg with that for a support leg to improve gait efficiency. Next, we improve gait efficiency of the combined Parametric Excitation walking by the optimization method for reference trajectories. Numerical results show that the specific resistance of the combined method is reduced to about one tenth of those of the previous results. In addition, the results of multi-objective optimization method are presented by reformulating a single-objective optimization problem.
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Development and experiment of a kneed biped walking robot based on Parametric Excitation principle
2011 IEEE RSJ International Conference on Intelligent Robots and Systems, 2011Co-Authors: Yoshihisa Banno, Yuji Harata, Kouichi TajiAbstract:Parametric Excitation walking is one of methods to realize dynamic walking on a level ground. This method has first applied to a biped robot with telescopic legs and later to a robot with actuated knee joints. In Parametric Excitation walking, mechanical energy is increased by periodic up-and-down motion of the center of mass. While Parametric Excitation walking with telescopic legs has verified by an experimental robot, that with actuated knees has not yet as far as we know. The purpose of this paper is to present demonstration experiment of Parametric Excitation walking with a kneed biped robot. To do this, we develop an experimental kneed biped robot having four parallel legs with semicircular feet. In the experiment, the robot achieves walking on a level ground more than 15 steps. We also measure the movements of the robot during walking by a 3D motion capture and compare with simulation results.
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IROS - Optimal trajectory design for Parametric Excitation walking
2009 IEEE RSJ International Conference on Intelligent Robots and Systems, 2009Co-Authors: Yoshihisa Banno, Yuji Harata, Kouichi TajiAbstract:Parametric Excitation walking is one of methods that realize a passive dynamic like walking on the level ground. In Parametric Excitation walking, up-and-down motion of the center of mass restores mechanical energy and sustainable gait is generated. Walking ability and walking performance strongly depend on the reference trajectory of the center of mass. In this paper, we propose an optimization method for the reference trajectory of Parametric Excitation walking. There are two problems for optimization. One is that search space of a reference trajectory is inherently infinite dimensional. Another is that it takes long simulation time to generate steady gait for a given reference trajectory. Therefore, the proposing optimization method adopts the following strategy. For the former, we confine the reference trajectory to the quartic spline curve and take the parameter of spline curve as decision variables. For the latter, we discretize the search space and adopt a local search method usually used in combinational optimization problems. We apply the proposed method to a kneed biped robot, and optimize the reference trajectory of its swing leg.
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Optimal trajectory design for Parametric Excitation walking
2009 IEEE RSJ International Conference on Intelligent Robots and Systems, 2009Co-Authors: Yoshihisa Banno, Yuji Harata, Kouichi TajiAbstract:Parametric Excitation walking is one of methods that realize a passive dynamic like walking on the level ground. In Parametric Excitation walking, up-and-down motion of the center of mass restores mechanical energy and sustainable gait is generated. Walking ability and walking performance strongly depend on the reference trajectory of the center of mass. In this paper, we propose an optimization method for the reference trajectory of Parametric Excitation walking. There are two problems for optimization. One is that search space of a reference trajectory is inherently infinite dimensional. Another is that it takes long simulation time to generate steady gait for a given reference trajectory. Therefore, the proposing optimization method adopts the following strategy. For the former, we confine the reference trajectory to the quartic spline curve and take the parameter of spline curve as decision variables. For the latter, we discretize the search space and adopt a local search method usually used in combinational optimization problems. We apply the proposed method to a kneed biped robot, and optimize the reference trajectory of its swing leg.