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Research Papers

Energetic and Dynamic Analysis of Multifrequency Legged Robot Locomotion With an Elastically Suspended Load

[+] Author and Article Information
Xingye Da

e-mail: xda@purdue.edu

Jeffrey Ackerman

e-mail: ackermaj@purdue.edu

Justin Seipel

e-mail: jseipel@purdue.edu
School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907

Contributed by Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 24, 2013; final manuscript received May 24, 2013; published online September 17, 2013. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 9(2), 021006 (Sep 17, 2013) (10 pages) Paper No: CND-13-1022; doi: 10.1115/1.4024778 History: Received January 24, 2013; Revised May 24, 2013

Elastically suspended loads can reduce the energetic cost and peak forces of legged robot locomotion. However, legged locomotion frequently exhibits multiple frequency modes due to variable leg contact times, body pitch and roll, and transient locomotion dynamics. We used a simple hexapod robot to investigate the effect of multiple frequency components on the energetic cost, dynamics, and peak forces of legged robot locomotion using a high-speed motion tracking system and the fast Fourier transform (FFT). The trajectories of the robot body and the suspended load revealed that the robot was excited by both a body pitching frequency and the primary locomotion frequency. Both frequency modes affected the dynamics of the legged robot as the natural frequency of the elastic load suspension was varied. When the natural frequency of the load suspension was reduced below the primary locomotion and body pitching frequencies, the robot consumed less average power with an elastically suspended load versus a rigidly attached load. To generalize the experimental results more broadly, a modified double-mass coupled-oscillator model with experimental parameters was shown to qualitatively predict the energetic cost and dynamics of legged robot locomotion with an elastically suspended load. The experimental results and the theoretical model could help researchers better understand locomotion with elastically suspended loads and design load suspension systems that are optimized to reduce the energetic cost and peak forces of legged locomotion.

Copyright © 2014 by ASME
Topics: Robots , Stress
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Figures

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Fig. 1

The double-mass coupled-oscillator model of locomotion with an elastically suspended load [2]

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Fig. 2

The block diagram of the input and output in the double-mass coupled-oscillator model

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Fig. 3

Diagram of the legged robot with an elastically suspended load. The hollow circles represent the infrared Vicon tracking markers used to measure the position of the load mass, the body mass, and the front of the robot in 3D space.

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Fig. 4

The rotation of the bell crank suspension mechanism with small oscillation angles can be approximated by an effective vertical oscillation. The effective stiffness and damping of the load suspension were determined from the free oscillation response of the load mass.

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Fig. 5

The free oscillation of the load mass can be observed from the displacement of the load's infrared marker using the Vicon tracking system. The figure shows a configuration with a low suspension stiffness and low damping ratio.

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Fig. 6

Video from a high-speed camera showed the robot pitching with respect to its center of mass (Θ is pitching angle). This diagram is representative of the dynamics observed in the video.

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Fig. 7

The amplitude of the rigidly attached load trajectory does not match the amplitude of the robot body trajectory. Since the load is rigidly attached, one would expect these trajectories to overlap.

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Fig. 8

The position of the load without body pitching is Load’, where the marker of the front aligns horizontally with the body

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Fig 9

After subtracting the body pitch, the amplitude, frequency, and phase of the robot body and the load trajectories are approximately the same. The slight differences between the body and the load may originate from the assumption that the center of rotation is always about the body infrared marker.

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Fig. 10

The amplitudes of the three main frequency modes of the robot body trajectory data vary with the natural frequency of the load mass. The amplitude of each mode tended to increase slightly when the natural frequency of the load mass was tuned near the frequency of the mode.

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Fig. 11

The amplitudes of each of the three main frequency modes of the load mass trajectory data increase when the natural frequency of the load suspension is tuned near each mode. Since the damping ratio of the load suspension was low, the load mass achieved resonance near the 3.5 Hz pitching frequency and oscillated with high amplitude when the natural frequency of the load mass was tuned near 3.27 and 4.01 Hz. A time-variant plot is given on Fig. 14(c).

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Fig. 12

The dynamic peak forces of each mode are maximized when the natural frequency of the load suspension is tuned near each mode. The peak forces of each mode are reduced when the natural frequency of the load suspension is tuned below the given mode.

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Fig. 13

The absolute amplitude of the dynamic peak force of the load acting on the body increases as the natural frequency of the load suspension is reduced near the resonant pitching frequency of 3.5 Hz. The peak forces will tend to decrease when the natural frequency of the load suspension is reduced below the pitching frequency. This figure illustrates that both the absolute peak force and the peak forces of each frequency mode should be considered.

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Fig. 14

The experimental average power shows an S-shaped-curve with an extra peak at 4.17 Hz (a). The theoretical model with a single frequency input predicts that the average power will only increase when the natural frequency of the load suspension is near 7 Hz (b). The experimental data show complicated trajectories such as a beating pattern (c), or multiple frequency modes, 3.5 Hz, 5 Hz, and 7 Hz (e). The body and load trajectories in the theoretical model show a single frequency mode (d), (f).

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Fig. 15

The average power of the modified model with pitching frequency in steady state and transient state shows an extra peak with a load natural frequency near 3.5 Hz (a), (b). The graphs demonstrate that the pitching frequency does increase the average power when the natural frequency is near the pitching frequency. A similar phenomenon was observed in the experiment (Fig. 3(a)). The body and load trajectories have multiple modes (c), (d) and the trajectories in the transient state can exhibit the beating pattern (d) and multiple frequency modes (f). The trajectories in the theoretical mode qualitatively match the experimental results (Fig. 3(c), 14(e)).

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