Periodic Non-Reverse Rectilinear Motion of a Two-Body System on a Rough Plane⁎

Proceedings of the 9th Vienna International Conference on Mathematical Proceedings ofModelling the 9th Vienna International Conference on Mathematical Proceedings ofModelling the 9th Vienna International Conference on Vienna, Austria, February 21-23, 2018 Mathematical Modelling Vienna, Austria, February 21-23, 2018 Available online at www.sciencedirect.com Mathematical Modelling Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018

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IFAC PapersOnLine 51-2 (2018) 226–231 Periodic Non-Reverse Rectilinear Motion Periodic Non-Reverse Rectilinear Motion Periodic Non-Reverse Rectilinear Motion  Periodic Non-Reverse Rectilinear Motion of a Two-Body System on a Rough Plane of a Two-Body System on a Rough Plane  of on a Rough Plane of a a Two-Body Two-Body System System on a Rough Plane ∗ ∗∗

Nikolay Nikolay Nikolay Nikolay

N. N. N. N.

Bolotnik Yu. Bolotnik ∗∗ Tatiana Tatiana∗∗∗ Yu. Figurina Figurina ∗∗ ∗∗ Bolotnik Tatiana Yu. Pavel A. Gubko ∗∗∗ ∗ Pavel A. Tatiana Gubko Yu. Figurina Bolotnik Figurina ∗∗ Pavel A. Gubko ∗∗∗ Pavel A. Gubko ∗∗∗ ∗ ∗ Ishlinsky Institute for Problems in Mechanics RAS, 101-1 Vernadsky ∗ Ishlinsky Institute for Problems in Mechanics RAS, 101-1 Vernadsky Problems in Mechanics RAS, 101-1 Vernadsky Ave,Institute Moscowfor 119526 Russia, e-mail:[email protected] ∗ Ishlinsky Ave, Moscow 119526 Russia, e-mail:[email protected] Institute for Problems in Mechanics RAS, 101-1 Vernadsky ∗∗Ishlinsky Ave, Moscow 119526 Russia, e-mail:[email protected] Institute for Problems in Mechanics ∗∗ Ishlinsky Problems in e-mail:[email protected] Mechanics RAS, RAS, 101-1 101-1 Vernadsky Vernadsky Ave,Institute Moscow for 119526 Russia, ∗∗ Ishlinsky Institute for Problems in Mechanics RAS, 101-1 Vernadsky Ave, Moscow 119526 Russia, e-mail:t [email protected] ∗∗ Ishlinsky Ave, Moscow 119526 Russia, e-mail:t [email protected] Ishlinsky Institute for Problems in Mechanics RAS, 101-1 Vernadsky ∗∗∗ Ave, Moscow 119526 Russia, e-mail:t [email protected] Institute of and Technology, 9 ∗∗∗ Moscow Institute119526 of Physics Physics ande-mail:t Technology, 9 Institutskiy Institutskiy per., per., Ave, Moscow Russia, [email protected] ∗∗∗ Moscow Institute Moscow of Physics and Technology, 9 Institutskiy Dolgoprudny, Region, 141700, e-mail: ∗∗∗ Moscow Dolgoprudny, Region, 141700, Russia, Russia, e-mail: per., Moscow Institute Moscow of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia, e-mail: gubkopaul.yandex.ru gubkopaul.yandex.ru Dolgoprudny, Moscow Region, 141700, Russia, e-mail: gubkopaul.yandex.ru gubkopaul.yandex.ru Abstract: A periodic rectilinear motion Abstract: A periodic rectilinear motion of of a a two-body two-body system system along along a a rough rough plane plane is is considered. considered. Abstract: A periodic rectilinear motion of a two-body system along a rough motion plane isisconsidered. The system is controlled by the force of interaction of the bodies. A periodic defined The system A is periodic controlled by the force of interaction of the bodies. A periodic defined as as Abstract: rectilinear motion of a two-body system along a rough motion plane isisconsidered. The system is controlled by the force of interaction ofand thetheir bodies. A periodic motion is defined as a motion in which the distance between the bodies velocities relative to the plane a motion in which the distance between the bodies and their velocities relative to the plane are are The system is controlled by the force of interaction of the bodies. A periodic motion is defined as arepresented motion in by which the distance betweenwith the the bodies and their The velocities relative to the plane the are time-periodic functions same period. friction that between time-periodic functions same period. friction that acts acts between arepresented motion in by which the distance betweenwith the the bodies and their The velocities relative to the plane the are represented by time-periodic functions with the same period. The friction that acts between the bodies and the plane is Coulomb’s dry friction. Necessary and sufficient conditions for possibility bodies and the plane is Coulomb’s dry friction. Necessary and The sufficient conditions for possibility represented by time-periodic functions with the same period. friction that acts between the bodies and thenon-reverse plane is Coulomb’s dry friction. Necessary andneither sufficient conditions for possibility of motion of system, in of the changes the of aa periodic periodic motion dry of the the system, in which which of conditions the bodies bodiesfor changes the bodies and thenon-reverse plane is Coulomb’s friction. Necessary andneither sufficient possibility of a periodic non-reverse motion of the system, in which neither of the bodies changes the direction of its motion, are proved. These conditions are expressed by inequalities that involve direction of its motion, are proved. These conditions are expressed by inequalities that involve of a periodic non-reverse motion of the system, in which neither of the bodies changes the direction of of its the motion, are proved. These areof expressed bythese inequalities involve the system’s bodies the coefficients friction bodies against the the masses masses system’s bodies and and theconditions coefficients friction of of bodies that against the direction of of its the motion, are proved. These conditions areofexpressed bythese inequalities that involve the masses of the system’s bodies and the coefficients of friction of these bodies against the underlying plane. Non-reverse motions provide a minimum for friction-induced energy losses underlying motions provide a minimum for friction-induced losses the masses plane. of the Non-reverse system’s bodies and the coefficients of friction of these bodiesenergy against the underlying plane. Non-reverse motions provide a minimum for friction-induced energy losses per per unit unit path. path. underlying plane. Non-reverse motions provide a minimum for friction-induced energy losses per unit path. per2018, unitIFAC path.(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © Keywords: Keywords: limbless limbless locomotion, locomotion, Coulomb’s Coulomb’s friction, friction, non-reverse non-reverse motion motion Keywords: limbless locomotion, Coulomb’s friction, non-reverse motion Keywords: limbless locomotion, Coulomb’s friction, non-reverse motion 1. is 1. INTRODUCTION INTRODUCTION is defined defined by by a a piecewise piecewise continuous continuous function function of of time, time, 1. INTRODUCTION is proposed defined by a piecewise continuous function time, by Chernousko (2001). One of the bodies (2001). One of theof 1. INTRODUCTION is proposed defined bybya Chernousko piecewise continuous function ofbodies time, is proposed by Chernousko (2001). One of the bodies does not change the direction of its motion, whereas the Two interacting bodies that are moving progressively does not change the direction (2001). of its motion, the proposed by Chernousko One ofwhereas the bodies Two interacting bodies that are moving progressively is does not change the direction of itsdirection motion, during whereassome the other body moves in the opposite Two interacting bodies that are moving progressively along a straight line on a plane with dry friction provide other body moves in the opposite direction during some not change the direction of its motion, whereas the along interacting a straight line on a that planeare withmoving dry friction provide does Two bodies progressively other body moves in the opposite direction during some time intervals. The friction between the bodies and the along a straight line on a plane with dry friction provide aa simple model for locomotion by means of crawling. time intervals. Theinfriction between the bodies andsome the body moves the opposite direction during simple model line for on locomotion by means of crawling. along a straight a plane with dry friction provide other time The friction the bodies the plane obeys law, the of friction aThe simple model for locomotion of crawling. crawling systems move to the in plane intervals. obeys Coulomb’s Coulomb’s law,between the coefficients coefficients of and friction intervals. The friction between the bodies and the crawling systems move due due by to means the change change in the the time aThe simple model for locomotion by means of crawling. plane obeys Coulomb’s law, the coefficients of friction being different for different bodies in the general case. The The crawling systems move due to the change in the relative positions of their parts while contacting with being different for different bodies the generalofcase. The plane obeys Coulomb’s law, the incoefficients friction relative positions of their partsdue while contacting with The crawling systems move to the change in the being different for different bodies in the the general case. The optimal parameters that mean of relative positions ofby their parts while contacting with the underlying surface all their segments. Such way of parameters that maximize maximize mean velocity velocity of being different for different bodies in the the general case. The underlying surfaceof by all parts their while segments. Such aawith waythe of optimal relative positions their contacting optimal parameters that maximize the mean velocity of the system are found. underlying surface by all their segments. Such a way of motion is inherent in some terrestrial limbless animals, the system are found. optimal parameters that maximize the mean velocity of motion is inherent in some terrestrial limbless animals, underlying surface by all their segments. Such a way of the system are found. motion is inherent in some terrestrial limbless animals, e.g., snakes and worms. This principle of motion can system are found. e.g., snakes and worms. This principlelimbless of motion can the A motion in which the bodies are moving apart from one motion is inherent in some terrestrial animals, motion in which the bodies are moving apart from one e.g., snakes in and worms.locomotion This principle of motion can A be artificial mechanisms (mobile motionduring in which theof bodies are moving apart one be utilized utilized artificial mechanisms (mobile another a the and are moving e.g., snakes inand worms.locomotion This principle of motion can A another a part part the period period and then then arefrom moving A motionduring in which theof are moving apart one be utilized in artificial locomotion mechanisms (mobile robots). Such mechanisms have of during a part ofbodies the period and isthen arefrom moving robots). Such have a a number number of advantages advantages toward each other at constant velocities constructed by be utilized in mechanisms artificial locomotion mechanisms (mobile another toward each other at constant velocities constructed by another during a part of the period and isthen are moving robots). Such mechanisms have a number of advantages over mobile systems of other types. In particular, they are each other at the constant velocities isapart constructed by over mobile systems of otherhave types. In particular, they are toward Chernousko (2011), rates of moving from and robots). Such mechanisms a number of advantages Chernousko (2011), the rates of moving apart from and toward each other at constant velocities is constructed by over mobile systems of other types. In particular, they are simple in design, do not require complex gear trains to (2011), the rates of moving apart from and simple in design, doofnot require gear trains to Chernousko toward each other being different in the general case. In over mobile systems other types.complex In particular, they are toward each other being different in the general case.and In (2011), the rates of moving apart from simple design,from do not requireto complex gear trains to Chernousko transmit motion the the devices, each other being different in the general case. In transmitin the motors motors the propelling propelling devices, this motion, one of the bodies (the main body) moves persimple inmotion design,from do not requiretocomplex gear trains to toward this motion, one of the bodies (the main body) moves pertoward each other being different in the general case. In transmit motion from the motors to the propelling devices, and to The major drawback of motion, one direction, of the bodies (the main body) movesbody perand are are easy easy to miniaturize. miniaturize. The to major drawback devices, of these these this manently in one while the other (auxiliary) transmit motion from the motors the propelling manently in one while themain otherbody) (auxiliary) this motion, one direction, of the bodies (the movesbody perand are easy to miniaturize. The major drawback of these mechanisms are irreversible energy losses for compensation direction, while (auxiliary) stage body mechanisms irreversible energy lossesdrawback for compensation moves in the same direction at the moving-toward and are easyare to miniaturize. The major of these manently in in theone same direction at the theother moving-toward manently in one direction, while the other (auxiliary) stage body mechanisms areforce irreversible losses for compensation moves of work. in the same direction at the moving-toward stage of the the friction friction work. energy and in the opposite direction at the moving-apart stage. mechanisms areforce irreversible energy losses for compensation moves and in in thethe opposite directionatatthe themoving-toward moving-apart stage. moves same direction stage of the friction force work. and the opposite direction at thesurface moving-apart stage. The friction against underlying is to of friction force Of most interest for crawling The in friction against the the underlying is assumed assumed to in the opposite direction at thesurface moving-apart stage. Of the most interest for work. crawling systems, systems, is is a a periodic periodic mode mode and The friction against the underlying surface is assumed to be Coulomb’s friction for the main body and anisotropic Of most interest for crawling systems, is a periodic mode of motion, in which the distance between the bodies be Coulomb’s frictionthe forunderlying the main body anisotropic friction against surfaceand is assumed to of motion, in which the distance the bodies Of most interest for crawling systems,between is a periodic mode The be Coulomb’s friction for the main body and anisotropic Coulomb’s friction (with the friction coefficient depending of motion, in whichand the between thethe bodies changes periodically the entire moves same friction (with coefficient depending be Coulomb’s friction forthe thefriction main body and anisotropic changes periodically thedistance entire system system moves same Coulomb’s of motion, in whichand the distance between thethebodies Coulomb’s frictionof (with the friction depending on motion) for auxiliary The changes the entire system moves the same distance during period. For systems on on the the direction direction motion) for the the coefficient auxiliary body. body. The frictionof(with the friction coefficient depending distance periodically during each eachand period. For two-body two-body systems on a a Coulomb’s changes periodically and the entire system moves the same on the direction of motion) for the auxiliary body. The existence conditions for the desired motion are obtained distance during each period. For two-body systems on a plane with dry friction, such motions were constructed, existence conditions for the desired motion are obtained on the direction of motion) for the auxiliary body. The plane with dry friction, such motions were constructed, distance during each period. For two-body systems on a existence conditions for the desired motion are obtained in the form of inequalities that are imposed on the paraplane with dry friction, such motions were constructed, investigated, and optimized by a number of researchers in in the form of inequalities are imposed on the paraconditions for thethat desired motion are obtained investigated, andfriction, optimized by motions a numberwere of researchers in existence plane with dry such constructed, in the form inequalities areof imposed on The the paraof system and mode its rates investigated, and optimized by a number of researchers in meters the fields mechanics, engineering mechanics, of the theof and the thethat mode its motion. motion. rates in the form ofsystem inequalities that areofimposed on The the parathe fields of of theoretical theoretical mechanics, engineering mechanics, investigated, and optimized by a number of researchers in meters meters of the system and the mode of its motion. The rates of apart and toward motions at which the average velocity the fields of theoretical mechanics, engineering mechanics, and biomechanics. A motion mode, in which the distance of apart and toward motions at which the average velocity meters of the system and the mode of its motion. The rates and biomechanics. A motion mode, in which the distance the fields of theoretical mechanics, engineering mechanics, of apart and toward motions at which the average velocity the system is a maximum are found, and an estimate and biomechanics. A motion mode, in which the distance between the bodies changes within prescribed limits and the system is a maximum found, an estimate apart and toward motions atare which theand average velocity between the bodiesAchanges prescribed limits and of and biomechanics. motion within mode, in which the distance a maximum are found, and an estimate for the energy losses per is between the bodies within prescribed limitsforce and of the force that the of forthe the system energy is losses per unit unit path path is given. given. the system is a maximum are found, and an estimate the interaction interaction forcechanges that plays plays the role role of a a control control between the bodies within prescribed limitsforce and of for the energy losses per unit path is given. the interaction forcechanges that plays the role of a control force for the energy losses per unit path is given. The conditions, subject to which a two-body the interaction force that plays the role of a control force  This The conditions, subject to which a two-body system system can can was partly supported by the Russian Foundation for  This study study was partly supported by the Russian Foundation for The conditions, subject to which a two-body system can move periodically on a horizontal rough plane at a nonzero  move periodically on a horizontal rough plane at a nonzero Basic Research (projects 17-01-00652, 17-51-12025). This study was partly supported by the Russian Foundation for The conditions, subject to which a two-body system can  Basic (projects move periodically on a horizontal rough plane at a nonzero ThisResearch study was partly 17-01-00652, supported by17-51-12025). the Russian Foundation for Basic Research (projects 17-01-00652, 17-51-12025). move periodically on a horizontal rough plane at a nonzero Basic Research (projects 17-01-00652, 17-51-12025).

Copyright © 2018 IFAC 1 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2018 responsibility IFAC 1 Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.03.039

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227

2. NON-REVERSE MOTION: STATEMENT OF THE PROBLEM

average velocity are sought and investigated by Wagner and Lauga (2013). The friction of the bodies against the plane is assumed to be Coulomb’s friction. In addition, it is assumed that each period of the motion involves one phase in which the distance between the bodies increases and one phase in which it decreases. It is stated that the motion with nonzero average velocity is impossible, if the magnitudes of sliding friction forces are the same for both bodies or if the change in the distance l between the bodies satisfies the relation l(t) = l(T − t), where t is the current time and T is the period of the motion. Otherwise, the motion with the desired properties is possible. Examples of such motions are given. In these examples, at least one of the bodies move in the direction opposite to that of the average velocity in some time intervals.

Consider a system of two interacting bodies of masses M and m, respectively, on a horizontal plane (Fig. 1).

Fig. 1. Model of the locomotion system We assume Coulomb’s dry friction force to act between the bodies and the underlying plane. No constraints are imposed on the force by means of which the bodies are interacting with each other. This internal force causes the change in the velocities of the bodies, which leads to the change in the friction forces that are external forces for the system under consideration. Therefore, the control of the force of interaction between the bodies allows the control of the motion of the system’s center of mass. We assume that the bodies move along a fixed straight line l on the plane. We will treat the bodies as mass points. Let x and y denote the coordinates along the line l, and v and V the velocities of bodies m and M , respectively. Let km and kM be the coefficients of friction against the plane for bodies m and M , and let g denote the acceleration due to gravity.

The motion of a two-body system in an environment with the resistance (friction) characterized by a powerlaw function of the velocities of the motion of the bodies is considered by Bolotnik et al. (2016). The exponents of the resistance law are the same for both bodies, whereas the coefficients of friction may be different. This class of resistance laws involves dry friction as a limiting case corresponding to zero value of the exponent. As was the case for Wagner and Lauga (2013), the control of the system is defined by a time-periodic function that characterizes the change in the distance between the bodies. The friction force is assumed to be small as compared with the force of interaction of the bodies. A condition necessary and sufficient for the average velocity of the system’s center of mass in the steady-state motion to be nonzero is obtained. For the case of Coulomb’s friction, this condition requires that the sliding friction forces be different for different bodies and, in addition, that the total time during which the distance between the bodies increases be different from the total time during which this distance decreases. Solutions of a number of problems of dynamics and control of periodic motions for a two-body locomotion system in different environments with small friction, including dry friction (isotropic and anisotropic), are presented in the monograph by Zimmermann et al. (2009).

We assume that the sliding friction forces are different for different bodies and, hence, km m = kM M . Without loss of generality we assume km m > kM M. (1) The motion of the system along the line is governed by the equations x˙ = v, y˙ = V, mv˙ = F + F1 , M V˙ = −F + F2 , (2) where F denotes the force applied to body m by body M , and F1 and F2 denote Coulomb’s friction forces applied to the bodies by the plane. The friction forces are defined by the relations F1 = −km mg sgn v, v = 0, |F1 | ≤ km mg, v = 0, (3) F2 = −kM M g sgn V, V = 0, |F2 | ≤ kM M g, V = 0. We will consider the motions of the system for which the distance between the bodies and the velocities of both bodies are expressed by time-periodic functions with a fixed period T : y(t + T ) − x(t + T ) ≡ y(t) − x(t), (4) v(t + T ) ≡ v(t), V (t + T ) ≡ V (t). In other words, we will consider the motions characterized by the same displacement of both bodies for the period: y(t + T ) − y(t) ≡ x(t + T ) − x(t) ≡ const. (5) We will call such motions periodic motions. We will study the possibility for the non-reverse motion in which the velocity of each of the bodies is nonnegative during the entire interval of motion: v(t) ≥ 0, V (t) ≥ 0. (6) We will consider the system’s motion during one period, t ∈ [0, T ]. Without loss of generality, let the coordinates of both bodies be equal to zero at the initial time instant: x(0) = y(0) = 0. (7)

For all examples of periodic motions of a two-body system known to the authors from the literature, at least one of the bodies moves in the direction opposite to that of the displacement of the system for the period in some time intervals. Of great interest and importance is the issue of whether it is possible (and, if possible, under which conditions) to provide a periodic motion along a straight line on a horizontal plane for a two-body system such that neither of the bodies move in the direction opposite to that of the average velocity of the system; friction between the bodies and the underlying plane is assumed to obey Coulomb’s law. We will call such motions non-reverse motions. The non-reverse motions minimize the energy losses for compensation of the work of the friction forces per unit path, increasing thereby the energy efficiency of the crawling locomotion system. For chains of bodies that consist of no less than three bodies, the possibility of nonreverse motions is established by Figurina (2015). For a two-body system, an answer to this question is given below.

2

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We can make the coordinates of the bodies equal to each other at the initial time instant by choosing appropriate origins for coordinates of bodies m and M on the line l. The periodicity conditions (4) imply that the velocities of each of the bodies are the same at the beginning and at the end of the interval [0, T ]: v(0) = v(T ), V (0) = V (T ) (8) and, with reference to (7), that the coordinates of the bodies are the same at the end of this interval: x(T ) = y(T ). (9)

Lemma. For motions that provide a maximum (minimum) for ∆, there do not exist time intervals in which the velocities of both bodies are positive. Proof. Assume the contrary, i.e., let the velocities of both bodies be positive in some time interval of an optimal motion: v(t) > 0, V (t) > 0, t ∈ (t1 , t2 ). (19) Denote by vi and Vi , i = 1, 2, the velocities of bodies m and M , respectively, at the ends of this interval: v(ti ) = vi , V (ti ) = Vi , i = 1, 2. (20) The motion of the bodies in the interval [t1 , t2 ] is governed by the equations v˙ = −u − 1, V˙ = µu − k. (21)

Introduce the dimensionless variables and parameters x = km gT 2 x ˜, , y = km gT 2 y˜, t = T t˜, kM m , k= , (10) M km F Fi , i = 1, 2, u = − . fi = km mg km mg In the new variables, equations (2) and (3) and inequality (1) become x˙ = v, y˙ = V, (11) v˙ = −u + f1 , V˙ = µu + kf2 , t ∈ [0, 1], µ=

f1 = −sgn v,

µ > k, v = 0, |f1 | ≤ 1,

Add the second equation to the first equation multiplied by µ to obtain µv˙ + V˙ = −µ − k, which implies that the quantity µv + V monotonically decreases. Therefore, µv1 + V1 > µv2 + V2 . (22) For the motions subject to the conditions of (19) and (20), we will seek a maximum and a minimum for the quantity ∆xy defined as the difference of the distances moved by the bodies during the interval [t1 , t2 ]:   ∆xy = y(t2 ) − y(t1 ) − (x(t2 ) − x(t1 )) → max min . (23)

(12)

u

v = 0,

Denote

(13) |f2 | ≤ 1, V = 0. f2 = −sgn V, V = 0, Here and in what follows, the dot stands for differentiating with respect to the new time and the tildes are omitted.

T ∗ = t 2 − t1 ,

f (t) =

t

u(s)ds.

u

(24)

t1

Integrate equations (21) to obtain v(t) = v1 − f (t) − (t − t1 ),

The boundary conditions (7), (8), and (9) become x(0) = y(0) = 0, (14) v(0) = v(1), V (0) = V (1), (15) x(1) = y(1). (16) The non-reverseness condition (6) is expressed by the inequalities v(t) ≥ 0, V (t) ≥ 0, t ∈ [0, 1]. (17) We assume that no constraints are imposed on the dimensionless force u(t); in particular, an impulsive force represented by Dirac’s delta function is allowed. In addition, we assume that the velocities of the bodies may undergo at most a finite number of jump discontinuities in the interval [0, 1].

(25) V (t) = V1 + µf (t) − k(t − t1 ). Substitute t = t2 into these relations to find V2 − V1 + k(v1 − v2 ) , f (t2 ) = µ+k (26) V1 − V2 + µ(v1 − v2 ) T∗ = . µ+k Integrate equations (25) to calculate the distances moved by bodies m and M : t2 T∗2 x(t2 ) − x(t1 ) = v1 T∗ − − f (t)dt, 2 t1 (27) t2 2 T∗ + µ f (t)dt. y(t2 ) − y(t1 ) = V1 T∗ − k 2

Problem. Find conditions that allow a motion of the system that satisfies relations (11) – (13), boundary conditions (14) – (16), and non-reverseness conditions (17).

t1

3. AUXILIARY PROBLEM

Accordingly, T2 ∆xy = (V1 − v1 )T∗ − (k − 1) ∗ 2 t2 +(µ + 1) f (t)dt.

Introduce the quantity ∆, defined as the difference of the distances moved by the bodies for the period: ∆ = y(1) − x(1). (18) We will seek a minimum and a maximum for ∆ on the set of motions x(t), y(t) that satisfy relations (11) – (13), boundary conditions (14) and (15), and non-reverseness condition (17). Based on truth or falsehood of the inclusion 0 ∈ [min ∆, max ∆], we will conclude about the possibility of a non-reverse motion desired in Problem stated in Section 1.

(28)

t1

We will find a minimum and a maximum for the quantity ∆xy , provided that the values vi and Vi , i = 1, 2, are fixed. The first two terms in expression (28) do dot depend on the motion, since, according to (26), the duration of the motion T∗ is uniquely defined by the values of the 3

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velocities at the beginning and at the end of the motion. Hence, with reference to the inequality µ > 0, the quantity ∆xy becomes a minimum simultaneously with the quantity t2 f (t)dt. Therefore, problem (23) is equivalent to the

with a value that is less than the values provided by the motions with positive velocities of both bodies. Subject to the control u+ (t), body m instantaneously comes to a stop at the time instant t1 , transmitting its momentum completely to body M , and then remains at rest until the time instant t2 . At the instant t2 , body m takes part of the momentum from body M and acquires the velocity v2 . Body M at the beginning and at the end of the motion instantaneously changes its velocity and moves according to equation (33) in the interval (t1 , t2 ). Subject to the control u− (t), body M instantaneously comes to a stop at the time instant t1 , transmitting its momentum completely to body m and then remains at rest until the time instant t2 . At the instant t2 , body M takes part of the momentum of body m and acquires the velocity V2 . Body m at the beginning and at the end of the motion instantaneously changes its velocity and moves according to equation (34) in the interval (t1 , t2 ).

t1

following problem: t2 t1

f (t)dt → max (min). u

u

(29)

The second relation of (24) implies the boundary condition f (t1 ) = 0. (30) From the positivity of the velocities v(t) and V (t) and relations (25) we arrive at the inequalities k(t − t1 ) − V1 < f (t) < v1 −(t−t1 ), t ∈ (t1 , t2 ). (31) µ If these inequalities had been non-strict, a minimum and a maximum for the functional (29) would have been provided, respectively, by the functions k(t − t1 ) − V1 f+ (t) = v1 − (t − t1 ), . (32) f− (t) = µ Extend the set of the examined motions, i.e., the motions with positive velocities in the entire interval of the motion (inequalities (19)) and specified velocities at the end of this interval (relations (20)), by adding two elements. Specifically, we will consider the motions for which the velocities are specified by relations (20) at the ends of the interval of motion, while in the interior of this interval they satisfy the relations v(t) ≡ 0, V˙ (t) = −µ − k, t ∈ (t1 , t2 ) (33)

Thus, any motion with positive velocities of both bodies provides the functional ∆xy with a value that is less than the value provided by the motion defined by equations (33), for which only body M moves and body m remains at rest, and is greater than the value provided by the motion defined by equations (34), for which body m moves, while body M remains at rest. By the hypothesis, the optimal motion involves the interval (t1 , t2 ) in which the velocities of both bodies are positive. However, by changing the motion in the interval (t1 , t2 ) to the motion with only one moving body we obtain greater or less value for the functional ∆, depending on which of the bodies does not move. This contradicts the hypothesis of optimality for the original motion. Therefore, the optimal motion does not involve a time interval during which both bodies are moving, which proves the lemma.

for the first motion and the relation k t ∈ (t1 , t2 ) (34) V (t) ≡ 0, v(t) ˙ = − − 1, µ for the second motion. The controls u+ and u− that generate the first and the second motions, respectively, are defined by u+ (t) = v1 δ(t − t1 ) − v2 δ(t − t2 ) − 1,

(35)

V2 V1 k u− (t) = − δ(t − t1 ) + + δ(t − t2 ). µ µ µ

(36)

229

Corollary. For the optimal motions that correspond to min ∆ or max ∆, the bodies move alternately, instantaneously exchanging their momenta, and, hence, v(t)V (t) ≡ 0 for t ∈ [0, 1]. 4. SOLUTION OF THE NON-REVERSE MOTION PROBLEM Proposition 1. If µ > 1 (m > M ), then the non-reverse motion is impossible. Proof. Consider a motion that satisfies relations (11) – (13), boundary conditions (14) and (15), and the nonreverse condition (17). We will show that for µ > 1, the inequality min ∆ > 0 holds. Relying on the corollary from Lemma, we will seek min ∆ on the set of the system’s motions for which the bodies move alternately. Let the velocities v(t) and V (t) characterize such a motion. We will assume that body m moves first in the interval [0, 1], and body M moves last. (Due to periodicity, this assumption does not restrict generality.) Let ti , i = 1, . . . , 2n − 1, be the instants of exchange of momentum between the bodies, body m moving in the intervals (t2j , t2j+1 ) and body M in the intervals (t2j+1 , t2j+2 ). As a result, we have the relations t0 < t1 < ...t2n−1 < t2n , (37) t0 = 0, (38) t2n = 1, (39)

These two motions, as was the case for the motions in which both bodies move forward, are governed by equations (21). Therefore, all reasonings that prove the equivalence of problems (23) and (29) remain valid for the extended set of motions that involves the original set of motions with positive velocities and two added motions. Unlike the case for the original set, strict inequalities (31) are replaced by non-strict inequalities and become equalities for the functions f+ (t) and f− (t) defined by expressions (32). One can readily verify that the functions f+ (t) and f− (t) correspond to the functions u+ (t) and u− (t) defined by expressions (35) and (36). Therefore, the motion generated by the control u+ (t) provides the functional ∆xy with a value that is greater than the value provided by any motion with positive velocities of both bodies, and the motion generated by the control u− (t), 4

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Nikolay N. Bolotnik et al. / IFAC PapersOnLine 51-2 (2018) 226–231

Let us find now a minimum for the displacement of body M during the time interval [t2j+1 , t2j+2 ] for given values of V2j+1 and V2j+2 . Since body m is not moving, equations of motion (11) – (13) imply that |u| ≤ 1; hence V˙ ∈ [−µ − k, µ − k]. (50)

v(t) > 0,

V (t) = 0, t ∈ (t2j , t2j+1 ), (40) j = 0, . . . , n − 1, V (t) > 0, v(t) = 0, t ∈ (t2j+1 , t2j+2 ), (41) j = 0, . . . , n − 1. Denote by vi and Vi the velocity of the respective bodies at the instants of exchange of momenta: v2j = v(t2j + 0), V2j = V (t2j − 0), (42) j = 0, . . . , n,

Since k < µ, body M may either decelerate or accelerate. The minimum of the path traveled by body M occurs for the time-minimal acceleration, if V2j+2 > V2j+1 , or for deceleration with maximal intensity, if V2j+2 < V2j+1 ; for these cases, V˙ = µ − k or V˙ = −µ − k, respectively. Indeed, let V2j+2 < V2j+1 . Then the displacement of body M during the time τ for the initial velocity V2j+1 is no less than the displacement of this body in the case of deceleration with maximal intensity, V˙ = −µ − k. The displacement of body M moving at the acceleration V˙ = −µ − k increases as τ increases. The minimal value of the time τ during which body M can change its velocity from V2j+1 to V2j+2 also occurs for deceleration with maximum intensity. Therefore, the displacement of body M is a minimum for this body decelerating according to the equation V˙ = −µ − k. For V2j+2 > V2j+1 , the proof completely repeats the proof given above, with the change of time t˜ = −t. As a result, we obtain  2 2  V2j+2 − V2j+1 t 2j+2  , V2j+2 ≥ V2j+1 ,  2(µ − k) min V dt = (51) 2 2 − V2j+2 V2j+1    , V < V . t2j+1 2j+2 2j+1 2(µ + k) Denote (v 2 − vi2 )µ , i = 0, . . . , 2n − 1. (52) δi = i+1 2(µ − k) The definition for the quantities δi and relation (45) imply that 2n−1  δi = 0. (53)

v2j−1 = v(t2j−1 − 0), V2j−1 = V (t2j−1 + 0), (43) j = 1, . . . , n. The conservation of momentum of the system at the instants ti implies the relations i = 0, . . . , 2n. (44) µvi = Vi , From the periodicity of the motion, it follows that (45) v0 = v2n . We will call the motion that satisfies relations (37) – (45) motion A. We will prove the inequality ∆ > 0 for motion A. Consider a set of motions that are defined by relations (37) and (40) – (45) for the velocities vi and Vi , i = 0, . . . , 2n, coinciding with the respective velocities for motion A, for arbitrary switching instants ti . This set of motions contains motion A. Introduce the quantity t2j+1 t2j+2 n−1 n−1     ∆∗ = V (t)dt − v(t)dt. (46) j=0 t

2j+1

j=0 t 2j

The values of ∆ and ∆∗ coincide for motion A. To prove the inequality∆ > 0 it suffices to prove the inequality min ∆∗ > 0. (47)

To calculate min ∆∗ one should minimize each term in the first sum and maximize each term in the second sum in expression (46), i.e., one should minimize and maximize the displacements of bodies M and m, respectively, for each interval of their motion. First, we will find a maximum for the displacement of body m during the interval [t2j , t2j+1 ] for given values of v2j and v2j+1 . Since body M is not moving, the equations of motion (11) – (13) imply that |u| ≤ k/µ; hence   k k v˙ ∈ −1 − , −1 + . (48) µ µ Since k < µ, body m is necessarily decelerating, and, hence, v2j > v2j+1 . The path moved by this body for given initial and terminal velocities is a maximum when the intensity of the deceleration is a minimum, i.e., v˙ = −1 + k/µ. In fact, the path moved by the body during a fixed time τ for given initial velocity v2j and the conditions of (48) does not exceed the path moved during the same time when decelerating according the the law v˙ = −1+k/µ, the maximal path monotonically increasing as τ increases. The maximum value of the time τ , during which the body can change its velocity from the value v2j to the value v2j+1 , provided that inclusion (48) is valid, also occurs for the case of minimum deceleration intensity, v˙ = −1 + k/µ. Therefore, the path is a maximum for the deceleration according to the equation v˙ = −1 + k/µ, which implies max u

t 2j+1

t2j

2 2 − v2j+1 v2j µ. v(t)dt = 2(µ − k)

i=0

With reference to (44) and (52), relations (49) and (51) can be represented as follows: t 2j+1 max v(t)dt = −δ2j , (54) t2j

min

t 2j+2

V (t)dt =

t2j+1

 

µδ2j+1 , δ2j+1 ≥ 0, µ−k δ2j+1 , δ2j+1 < 0.  −µ µ+k

(55)

Relations (46), (54), and (55) imply the equality min ∆∗ = −





δ2j +

δ2j+1 <0



δ2j+1 >0

µ

µδ2j+1

µ−k δ2j+1 . µ+k

(56)

The inequality min ∆∗ >

2n−1  i=0

δi

(57)

occurs, since the coefficient µ > 1 multiplies the terms δ2j+1 > 0 in relation (56) and all terms δ2j+1 < 0 in the sum of (57) are replaced by positive terms in the sum of (56). (At least one of the terms δ2j+1 must be nonzero

(49)

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Nikolay N. Bolotnik et al. / IFAC PapersOnLine 51-2 (2018) 226–231

and, hence, the speeding up and slowing down of the respective bodies occur at the accelerations equal in magnitude.

because of relation (53) and negativity of all δ2j ). Using relations (57) and (53) we obtain min ∆∗ > 0. (58) Since ∆ ≥ min ∆∗ , we have ∆ > 0 and, hence, x(1) < y(1), which proves Proposition 1.

Thus, for the case of µ ≤ 1, we have proposed a motion, defined by expressions (60) – (63), (67), and (69), that satisfies relations (11) – (17) and represents thereby a solution of the non-reverse motion problem. This completes the proof of Proposition 2.

Proposition 2. If µ ≤ 1 (m ≤ M ), then the non-reverse motion is possible. Proof. We will construct a control that leads to a nonreverse motion for µ ≤ 1 in an explicit form. Let the system be in a state of rest at the initial time instant; hence the initial conditions are given by V (0) = v(0) = 0. (59) We will seek a non-reverse motion that consists of two stages in the time interval [0, 1]. At the first stage, body M speeds up at a constant acceleration a, with body m being at rest. Then body M transmits its momentum entirely to body m, and at the second stage, body m moves slowing down at a constant acceleration −b, body M remaining at rest. For this case, the motion of the system is governed by the following equations with boundary conditions: x˙ = v, y˙ = V, x(1) − x(0) = y(1) − y(0), (60) V˙ = a, v˙ = −b,

V (0) = 0, v(1) = 0,

v ≡ 0,

Being represented in terms of the original dimensional variables, this solution takes the form m M V˙ = (km mg − kM M g) , M   (71) M V (0) = 0, v ≡ 0, t ∈ 0, T , m+M mv˙ = kM M g − km mg,   (72) M v(T ) = 0, V ≡ 0, t ∈ T, T . m+M 5. CONCLUSIONS The ability to perform a periodic non-reverse motion along a straight line on a rough horizontal plane is studied for a system of two bodies that interact with one another and with the underlining plane. Coulomb’s dry friction is assumed to act between the plane and the system’s bodies. The system is controlled by the force of interaction of the bodies. Non-reverse motion is defined as the motion in which neither of the bodies changes the direction of its velocity. Within the framework of this model, it is proved that the non-reverse motion may occur if and only if the forces of sliding friction against the plane are different for different bodies, and, in addition, the mass of the body characterized by the less sliding friction force is not less than the mass of the other body. This implies, in particular, that non-reverse motion is impossible for a system the bodies of which have the same coefficient of friction against the underlying plane.

t ∈ [0, t∗ ), (61)

V (t∗ − 0) = µv(t∗ + 0),

231

(62)

V ≡ 0, t ∈ (t∗ , 1], (63)   k k a ∈ [0, µ − k], b ∈ 1 − ,1 + . (64) µ µ Constraints (64) stem from Coulomb’s law for dry friction and the fact that one body is at rest at each stage. Solve equations (61) and (63), subject to the boundary conditions and relation (62), to find the functions v(t) and V (t) and the interstage instant t∗ : µb . (65) t∗ = a + µb Solve then the equations with boundary condition (60) to obtain the relation at2∗ = b(1 − t∗ )2 , which implies that both bodies travel the same distance during the interval [0, 1]. Substitute expression (65) for the interstage instant into this relation to obtain a/b = µ2 . (66) Taking into account this expression, we obtain the final expression for t∗ : 1 . (67) t∗ = 1+µ Thus, the interstage instant is independent of the parameters a and b. From inclusions (64), it follows that a/b ∈ [0, µ]. (68) 2 Since µ ∈ [0, µ] for µ ≤ 1, any a and b that satisfy relation (66) satisfy inequality (68); in particular, one may let a = (µ − k)µ, b = (µ − k)/µ. (69) Straightforward verification shows that the parameters a and b defined by relation (69) satisfy the inclusions of (64). For µ < 1, the solution defined by (66) and (64) is nonunique, while for µ = 1, relation (69) defines the unique solution for the non-reverse motion problem. For this case, a = 1 − k, b = 1 − k (70)

REFERENCES F.L. Chernousko. The optimum rectilinear motion of a two-mass system. Journal of Applied Mathematics and Mechanics, 66:1–7, 2002. F.L. Chernousko. Analysis and optimization of the rectilinear motion of a two-body system. Journal of Applied Mathematics and Mechanics, 75:1–7, 2011. G. Wagner G. and E. Lauga. Scallop crawling: Frictionbased locomotion with one degree of freedom. J. Theor. Biol., 32:42–51, 2013. N. Bolotnik, M. Pivovarov, I. Zeidis, and K. Zimmermann. The motion of a two-body limbless locomotor along a straight line in a resistive medium. ZAMM, 96:429–452, 2016. K. Zimmermann, I. Zeidis, and C. Behn. Mechanics of Terrestrial Locomotion with a Focus on Nonpedal Motion Systems. Springer, Heidelberg, 2009. T.Yu. Figurina. Optimal control of system of material points in a straight line with dry friction. Journal of Computer and Systems Sciences International, 55:671– 677, 2015.

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