Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 176 (2017) 387 – 393
Dynamics and Vibroacoustics of Machines (DVM2016)
Modeling of vibroimpact processes which occurs in feet changing of the walking units at viscoelastic grounds V.V. Chernyshev*, A.A. Goncharov, V.V. Arykantsev Volgograd State Technical University, Volgograd, Russia
Abstract At research of planets walking machines could be more effective than wheeled and tracked ones. They have higher ground and shape passableness. Walking machines traction properties, in execution of ground works for example, are also higher. It is especially important in traction weight deficit, characterized by weakened gravity. In the paper executed mathematical modeling of vibro-impact processes, which appears during feet changing, and investigated influence on walking unit dynamic in weakened gravity. During modeling on feet changing walking unit has been investigated as system of bodies — frame and weightless movers (feet). As design scheme was accepted the two-feet one with vertical translational motion of frame and antiphase leg motion. The frame movement is executed by the impact of kinematic impulses created by the movers, their character determined by relative contact points motion of walking mechanisms in contact phase with ground. Modes with periodical feet motion have been modelled. Time dependencies of relative vertical motions of walking mechanisms contact points was approximated by trigonometric polynomial. In mathematical describing of grounds was used the viscoelastic model with different relations of viscous and elastic properties. Elastic force, which proportional to the ground deformation and unidirectional viscous friction force, which proportional to deformation speed, were introduced. The modeling results have shown, in feet changing occurs their oscillations at the ground. The oscillations are accompanied by support reactions alteration and their maximum values could be bigger than static by few times. The effect occurs at different grounds. This property can be used for implementation of increased traction forces in individual cycle phases. At increasing of movement speed the amplitude of support reactions changing is rising. In weakened gravity conditions that could result in transformation of static stable walking to running already at low velocities. The results can be demand in walking robots development, designed to planet surface investigation. © Published by Elsevier Ltd. This ©2017 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the Dynamics and Vibroacoustics of Machines (DVM2016). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines
Keywords: mobile robots, rovers, walking mover, the interaction with the soil, the vibroimpact processes, support reactions, periodic gaits, mathematical modeling
* Corresponding author. Tel.: +7-8442-24-80-99; fax: +7-8442- 24-81-13. E-mail address:
[email protected].
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines
doi:10.1016/j.proeng.2017.02.336
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1. Introduction During the research of planets walking machines could be more effective than wheeled and tracked ones [1-3]. In extreme conditions they have higher ground and shape passableness [4-9]. Also, walkng machines have decreased tractional forces costs on movement resistance — for walking movers, in contrast of wheeled and tracked, ground is not an obstacle for movement, it only requires a necessary power costs on its pressing [10-13]. That's why traction properties of walking machines in ground work execution, for example, are also higher. This is significantly important in adhesion weight deficite, provided by weakened gravitation. Field tests of walking modular hardware system MAK-1 (Fig. 1), designed in Volgograd State Technical University (VSTU) for investigation of traction properties and optimization of control methods of walking robotic systems movement, have shown that traction properties significantly depends on foot interaction dynamic with ground. In the works some results of mathematical modeling of vibro-impact processes, occured in feet changing, and investigated their influence on walking unit dynamic in weakened gravitation conditions.
Fig. 1. Walking machine МАК-1
Nomenclature Ai Bi cz Fi h j kCz Ni (t) m M Ri T Ui z0 za(r) zi max zi (r)
coefficient of the trigonometric polynomial coefficient of the trigonometric polynomial normal rigidity of a ground force, proportional to ground deformations depth of footprint number of the harmonic wave coefficient, which characterize increasing of ground rigidity as result of his ductile deformations normal reactions under feet quantity of the function values mass of walking machine frame force of viscous friction cycle period unit function, described state of i-th leg initial state of first foot relatively to machine body the amplitude of relative movements of feet in vertical direction maximum ground deformation vertical movement law in relative motion
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zk μz α φ ω
values of approximated excosecant viscous resistance coefficient initial phase of first foot relatively to machine body turn angle of mover driving link angular speed of driving link (walk frequency)
2. Dynamic model Main difficult of theoretical analysis of interaction of walking mover with ground consist in reliable analytical describing of process of foot with ground in contact phase and in feet changing. With diversity of grounds and its different mechanical properties and design of the foot, in considered case, it is necessary to consider the speed of ground loading [14]. Moreover, in contrast of traditional wheeled and tracked vehicles, there is no opportunity to consider the dynamic of walking mover interaction with supporting surface without consideration of dynamic of a whole walking machine, which performs dimensional oscillations, caused by the walking type of movement. For this reason, models of ground, used for wheel and tracked machines, unreliable for walking machines and difficulty of theoretical determining of stressed state and deformations of the ground increases significantly. In modeling, walking machine considered as a system of bodies — frame and united as boards weightless walking movers. Walking movers means as kinematically connected walking mechanisms (legs), equipped with feet. Investigated the case of statically stable machines. As mathematical model, assumed two-legged scheme with vertical translational movement of frame (Fig. 2). Influence of straight movement on dynamic of supporting reactions alteration was ignored, that is equal to walking with low velocity.
Fig. 2. Block diagram of walking machine: 1, 2 — feet; 3 — frame; 4 — ground. Vertical state of feet determined by translational motion of machine frame and their relative motion to frame. zi z zi( r )
(1)
where z z (t ) — vertical movement of position of the center of gravity; zi( r ) zi( r ) (t ) — vertical movement law in relative motion (i=1, 2). Movement of the machine frame performing due to impacts of kinematic disturbances, created by movers. Their character determined by relative trajectory of support points of walking mechanisms. In modeling, modes with periodic leg motion were considered. Dependencies zi( r ) (t ) for cyclic mechanisms of walking in range (0, T), where T 2 — cycle period, ω — angular speed of driving link (walk frequency), were approximated by trigonometrical polynomial. n
zi( r ) (t ) A0 2 ( A j cos( jt ) B j sin( j t )) , j 1
where Aj, Bj — coefficients equal, accordingly to
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2 m 1 2 m 1 zk cos( j k ), B j zk sin( j k ), m k 0 m k 0 where m — quantity of the function values and n
