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The energy-optimal motion of a vibration-driven robot in a resistive medium
Journal of Applied Mathematics and Mechanics 74 (2010) 443–451
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The energy-optimal motion of a vibration-driven robot in a resistive medium夽 A.G. Yegorov ∗ , O.S. Zakharova Kazan, Russia
a r t i c l e
i n f o
Article history: Received 1 September 2009
a b s t r a c t The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections – one with a positive velocity and the other with a negative velocity of motion of the shell – is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. ©Elsevier Ltd. All rights reserved. © 2010 Elsevier Ltd. All rights reserved.
The mechanical system investigated models a vibration-driven robot – a mobile device capable of moving in a resistive medium without moving external parts (wheels, legs, caterpillar tracks, etc.). The possibility of using vibration-driven robots in medicine has been discussed earlier.1,2 A similar principle was used to describe the motion of limbless living creatures (worms and snakes).3,4 The question of the optimal motion of the system by the motion of the internal body was first posed by Chernous’ko,5,6 who examined the rectilinear motion, along a horizontal plane, of a rigid body with a cavity containing a different moving body (the “internal mass”) when there is Coulomb friction between the plane and the body. Recently, these problems have been widely discussed in the literature both for other resistance laws and for non-one-dimensional motion the internal masses.7,8 Normally, the problem of optimizing the motion of the internal mass is posed as the problem of maximizing the average velocity of the body with a constrained maximum velocity or minimum acceleration of the internal mass. In the present paper, the constraint has an energy form and is imposed on the internal motive power (the moving internal mass). The problem formulated in this way allows of a complete analytical investigation.
1. Formulation of the problem Consider a system consisting of two rigid bodies. The main body (body M – the shell) of mass M is located in a resistive medium, while the second body of mass (referred to below as the “internal mass”) is moving within it. The longitudinal periodic motions of the internal mass in relative to body M with which the entire system moves as a whole are investigated. The velocity of body M will be denoted by u, and the displacement and velocity of the internal mass relative to body M will be denoted by s and = s˙ respectively. The equations of motion of the internal mass and of body M in a fixed system of coordinates have the form (1.1)
夽 Prikl. Mat. Mekh. Vol. 74, No. 4, pp. 620–632, 2010. ∗ Corresponding author. E-mail address: [email protected] (A.G. Yegorov). 0021-8928/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2010.09.010
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A.G. Yegorov, O.S. Zakharova / Journal of Applied Mathematics and Mechanics 74 (2010) 443–451
where F is the force of interaction of the internal mass and body M, and R is the resistance of the medium to the motion of the body. Below we will assume that the resistance force is determined solely by the velocity of motion of the body and can be written in the form
Depending on the choice of the exponent ␣, we have different (anisotropic) resistance laws: when ␣ = 0, we have dry friction; when ␣ = 1, we have linear resistance; with ␣ = 2, we have quadratic resistance. For slow motion of body M in a liquid (i.e., ignoring the effects of inertia), pseudoplastic, Newtonian and dilatant liquids have corresponding ␣ values of 0 < ␣ < 1, ␣ = 1 and 1 < ␣ < ∞. A difference in the positive coefficients of resistance k+ and k− when of the body moves in the positive and negative directions can be ensured, for example, by an asymmetric body shape (a sharpened nose and a blunt tail). Omitting the force F from Eq. (1.1), we will obtain the basic equation (1.2) which describes the motion u(t) of the body with a specified law motion s(t) of the internal mass. From physical considerations it is clear that, for any specified periodic law s(t), with period T, Eq. (1.2) uniquely defines the periodic function u(t) with the same period. Using angular brackets to denote the average for a period, we will define the following functionals: U(u) = u is the average velocity of motion of the body; N(u)= u · R(u) is the average power expended in overcoming the resistance forces; L(s) = max(s)–min(s) is the amplitude of oscillations of the internal mass (the length of the body). Of course, the power N expended in overcoming the resistance forces is precisely equal to the power of the forces F · u imparted by the internal mass to the body. This can be ascertained by multiplying the second of Eqs. (1.1) by u and integrating it over to the period. To describe the energy consumption due to motion of the body by means of an internal motive power (the moving internal mass), we will introduce the energy coefficient (1.3) as the ratio of the power UR(U) necessary for the body to move at constant velocity U to the power N(u) actually expended on periodic motion. The double inequality in (1.3) follows from the convexity and non-negativity of the function uR(u). We will formulate the problem of optimizing the motion of the internal mass as follows. Problem 1. It is required to find the period T and the law s(t) of oscillations of the internal mass such that the amplitude of oscillations and the average velocity of motion of the body are fixed:
while the power of the internal motive force is a minimum:
Along with Problem 1, the following problem is considered. Problem 2. It is required to find the law s(t) of oscillations of the internal mass such that the period T of oscillations and the average velocity of motion of the body are fixed while the power of the internal motive force is a minimum. Note that the amplitude L of oscillations of the internal mass in Problem 2 is not fixed in advance, but is determined after the function s(t) has been found. Below it will be shown that the above problems are equivalent. The convenience of the formulation of Problem 2 is the fact that the initial problem is split into two simple successively solved problems: the first determines the optimal law u(t) of motion of the body, and the second restores, according to the optimal law u(t), the time dependence of the law s(t) of motion of the internal mass. The possibility of this splitting is due to the fact that no additional conditions, besides periodicity, are imposed on the function s(t). This condition is easily expressed in terms of u. In fact, relation (1.2) can be considered as the problem of finding the periodic function s(t) with a specified periodic left-hand side. The necessary condition for this problem to be solvable is obtained by taking the average of the two sides of Eq. (1.2). Taking the periodicity of the function s and u into account, we have the condition R(u) = 0, which will also be sufficient. Therefore, for any law
u(t) satisfying the constraint R(u) = 0, the periodic function s(t) is found from relation (1.2) by simple integration. The initial problem in this case reduces to finding the periodic function u(t), with period T, that minimizes the functional N(u), taking into account the constraints u = U0 and R(u) = 0. 2. The optimal motion of the body Normalizing the time t to T, the velocity u to U0 and the power N to k+ U0˛+1 , we will present the constraints imposed on u(t) as
and the problem of the optimal motion of the body in the form (2.3)
(2.4)
A.G. Yegorov, O.S. Zakharova / Journal of Applied Mathematics and Mechanics 74 (2010) 443–451
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A formulation equivalent to Eq. (2.3), written in terms of the coefficient of energy consumption , has the form
(2.5) Beneath the arrows in relations (2.3) and (2.5) we give the constraints under which minimization is carried out. Note that, in problem (2.5), the function u(t) does not come under constraint (2.1), and consequently u can have any positive value. The equivalence of problems (2.3) and (2.5) follows directly from the invariance of the functional (u) with respect to the normalization of u. From problem (2.5), in particular, it follows that the initial problem (2.3) could also be stated as the problem of maximizing the average velocity of the body for a specified motive power. Introducing the Lagrange multipliers 1 and 1 corresponding to constraints (2.1) and (2.2), we will rewrite problem (2.3) as the problem of finding the stationary point of the functional
By varying I(u) we obtain for u the algebraic equation
(2.6) from which it follows that the function u(t), which is the solution to problem (2.3), is a piecewise-constant function taking values corresponding to the roots of Eq. (2.6). We will denote these values by u1 , . . ., uk , and the size of the intervals in which they are taken by a1 , . . ., ak , in which case a1 + · · · + ak = 1. By virtue of constraint (2.2), among the values u1 , . . ., uk there should be both positive and negative values. Note that the Lagrange multiplier has a simple physical meaning. It can be established by multiplying each of the k equations (2.7) by ai ui and adding. As a result we obtain
and by virtue of conditions (2.1) and (2.2) (2.8) from which, in particular, it follows that is positive. Similarly, by multiplying Eq. (2.7) by ai and adding, the positiveness of is proved. Before carrying out a general analysis, we will begin by considering the special case ␣ = 1 of anisotropic linear friction. When ␣ = 1, Eq. (2.7) takes the form
For it to have a positive and a negative root, the inequality < < , which constrains the range of variation to the condition > 1, must be satisfied. In the opposite case ( ≤ 1), forward motion of the body in the positive direction is impossible in principle. This was noted earlier.7 When > 1, in parameters and vary in the ranges 0 < < < < ∞, the roots u+ > 0 and −u− < 0 of the equation take all possible values from the range (0, + ∞ ) and ( − ∞ , 0) respectively. Constraints (2.1) and (2.2) are written in the form (2.9) and the minimized functional as (2.10) Expressing, from Eqs (2.9), u+ and u− in terms of a, and substituting the expressions obtained into relation (2.10), we arrive at the problem of minimizing the function
in the interval (0, 1). The solution of this problem has the form
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Fig. 1.
The maximum value of the energy coefficient
obviously increases from zero to unity as increases from unity to infinity. We will return to the general case, examining Eq. (2.6) separately when 0 < ␣ < 1 and when 1 < ␣ < ∞. The qualitative form of the left-hand side of this equation is presented in Fig. 1. The stationary points f are defined as
Since, among the roots of Eq. (2.6), there should be positive and negative roots, the range of variation in the Lagrange multipliers and is constrained by the inequality f* () ≤ < ∞ when 0 < ␣ < 1 and by the inequality 0 < ≤ f* () when 1 < ␣ < ∞. The corresponding part of the (, ) plane will be denoted by S0 . When (, ) ∈ S0 , Eq. (2.6) has three roots: two positive roots and one negative root when 1 < ␣ < 1, or two negative roots and one positive root when 1 < ␣ < ∞. In the first case we will number the roots u1 (, ), u2 (, ) and u3 (, ) in increasing order, and in the second case in decreasing order. We will denote the size of the intervals in which u takes the values u1 , u2 and u3 by a, (1–a)b and (1–a)(1–b), where
and represent the constraints (2.1) and (2.2) in the form
(2.11) We can ascertain that relations (2.11) uniquely define the parameters a and b in terms of u1 , u2 and u3 , and consequently in terms of the parameters and . Here, problem (2.3) reduces to minimizing the function N(, ) under the constraints (2.12) The first of these is unimportant in the sense that, when the second constraint is satisfied, so must it be. In fact, under the second constraint of system (2.12), from the second equation of (2.11) it follows that (2.13) When a > 1 (when a < 1), from the first inequality of system (2.11) it follows that au1 > 0 (respectively 1 − au1 > 0), and hence a > 0 (a < 1). Either of the conditions a > 0 or a < 1, along with inequality (2.13), gives the first constraint of (2.12). It is clear that the admissible set S = {(, ) : 0 ≤ b(, ) ≤ 1} lies within S0 . Therefore, problem (2.5) reduces to finding the minimum of the function N(, ) ≡ in set S. However, by virtue of its linearity, the function N should take a minimum value at the boundary of S, i.e., when b = 0 or when b = 1. In both cases, the solution of problem (2.5) is a piecewise-constant function taking two values: a positive value (u+ ) in a set of size a and a negative value (−u− ) in a set of size 1 − a. Any pair u+ > 0, u− > 0 is admissible, and Eq. (2.6) acts as a linear system for finding Lagrange multipliers and from the specified u+ , u− . Constraints (2.1) and (2.2) for the piecewise-constant function mentioned take the form (2.14)
A.G. Yegorov, O.S. Zakharova / Journal of Applied Mathematics and Mechanics 74 (2010) 443–451
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Fig. 2.
Table 1 ˛
– ∞
∞ –
0 –
a
1 1
1/2 1
1
u−
0
1
0
u+
1
1
1+
1+
1 –
√ 2 ( −1) √ √ +1 √1 −1 √ √ −1
2 1
10
100
0.079 0.965
0.306 0.933
0.587 0.945
3.52
1.306
0.608
1.16
1.164
1.093
Expressing, from these, u+ and u− in terms of a, we have
(2.15) and we arrive at the problem of finding the minimum of the function of one variable
(2.16) It should be noted that the initial interval 0 < a < 1 in which the minimum of the function N is sought should be additionally constrained by the requirement u± (a) > 0. This requirement (see Eqs (2.15)) is expressed as
and reduces to the condition a > a0 when ␣ > 1 and to the condition a < a0 when ␣ < 1. Here
It can be shown by direct calculation that, in the interval a0 < a < 1 when ␣ > 1 and 0 < a < a0 when ␣ < 1, the function N(a) defined by Eq. (2.16) is strictly convex. At the ends of these intervals it reverts to +∞. Therefore, the problem of minimizing the function N(a) has a unique solution. The dependence of the maximum energy coefficient on the exponent ␣ for different values of the anisotropy parameter is presented in Fig. 2. The main characteristics of the optimal motion of the body for the limiting values of the parameters ␣ and are given in Table 1. The table also gives certain numerical values for the important practical case of a quadratic law of resistance. 3. The optimal motion of the internal mass. The basic solution It has been shown that motion of a body with the maximum energy coefficient occurs realized when the velocity of the body takes two values: u+ U0 > 0 and −u− U0 < 0. The interchanging of the intervals of forward and backward motion in this case may be arbitrary, but the total time of each of the forms of motion in the interval of periodicity should be equal to aT and (1 − a)T respectively. The dimensionless constants u+ , u− and a are entirely determined by the parameters ␣ and of the resistance law. For each of the infinite set of optimal laws of motion of the body there is a unique periodic law s(t), with period T, of motion of the internal mass. Among the entire set of optimal laws of motion of the body, we will single out the simplest (“basic”) law for which forward and backward motion of the body occur in the intervals (0, aT) and (aT, T). We will define the basic law s(t) of motion of the internal mass and calculate
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Fig. 3.
the corresponding amplitude L and maximum velocity V = max s˙ of its oscillations. Introducing the notation
we rewrite Eq. (1.2) in the regions of continuous motion as follows:
Taking into account the standard conditions at the discontinuities
and the conditions for the motion to be periodic
the solution of this equation, apart from an unimportant constant, has the form
(3.1) For short oscillation periods, namely when T < T+ = 2u0 /w− , the function s+ (t) decreases monotonically in the interval (0, aT), whereas when T > T+ , as it increases it initially increases and then, having reached a maximum value, decreases. Similarly, when T < T− = 2u0 /w+ , the function s− (t) increases monotonically in the interval (aT, T), whereas when T > T− it reaches a unique minimum in this interval. On account of this, as the oscillation period increases, in the intervals (0, T+ ), (T+ , T− ) and (T− , ∞) when a > 1/2 and in the intervals (0, T− ), (T− , T+ ) and (T+ , ∞) when a < 1/2, three optimal modes of motion of the internal mass occur successively. Taking the ratio between the inertial and viscous forces into account, they will be referred to as inertial, intermediate and viscous modes. For the case a > 1/2, these types of motion are represented schematically in Fig. 3a, b and c respectively. With obvious modification of the intermediate mode, they have a similar form when a < 1/2. The amplitude of the oscillation of the internal mass when a > 1/2 is defined as
(3.2) A similar formula with the changes T+ ↔ T− and a ↔ 1 − a occurs when a < 1/2. The function L(T) increases monotonically as in T increases from zero when T = 0 to infinity when T = ∞. For short oscillations periods, the dependence of the amplitude of the oscillations on the period has a linear form, and for long oscillations periods it is quadratic:
(3.3) As might have been expected, for short T the parameters k± and ␣, responsible for the viscous properties of the medium, do not occur in the relation L(T), and for long T the parameter M defining the inertial properties of the body, does not occur in L(T)
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The determination of the maximum dimensionless velocity of the body in modulus leads to the relation
(3.4) As can be seen, when the oscillation period increases, to ensure the same average velocity of motion of the body, the velocity of the internal mass must increase linearly. Formulae (3.1) and (3.2) represent the simplest of the optimal solutions – the basic solution of Problem 1. Taking into account the unique relation (3.2) in the basic solution between the oscillation period and the body length, they define the solution of Problem 2. 4. The extremal property of the basic solution The basic solution, besides being the simplest of the optimal solutions, possesses a remarkable extremal property. The region in the (L, T) plane corresponding to the entire set of optimal motions is defined as
Here and below, the zero superscript denotes the basic solution. Thus, for a specified oscillation period, of the body length does not exceed that determined by the basic solution, and for a specified body length the oscillations period cannot be shorter than the basic one. To prove the extremal nature of the basic solution, we will establish two facts: (a) for each constant L* ≤ L(0) (T) an optimal law of the body motion exists with L(T) = L* ; (b) for any optimal law the body motion, the inequality L(T) ≤ L(0) (T) is satisfied. By virtue of the fact that the basic solution with period T/2 occurs in the set of optimal motions with period T, it is sufficient to prove statement (a) for L* ∈ [L(0) (T/2), L(0) (T)]. We will define the one-parameter set of optimal laws of the body motion
with parameter T1 : T/2 < T1 < T. For given u, we will define the function s(t) and calculate for it the functional L = max s − min s. With a fixed value of T, the quantity L is a continuous function of T1 , taking values L(0) (T/2) and L(0) (T) at the ends of the interval [T/2, T] of variation in parameter T1 . As a consequence, L(T1 ) may take any intermediate value between L(0) (T/2) and L(0) (T). The proof of statement (b) is fairly lengthy (as far as checking inequality (4.1) is concerned), so we will therefore only present its principal idea. For the optimal non-basic motion of the body, the dimensionless velocity u(n) (t) takes the values u+ U0 and −u+ U0 in the pair of interchanging intervals T2k
We will introduce, along with u(n) (t), the sequence of other optimal motions of the body u(n−1) (t), . . ., u(0) (t), in which u(i−1) (t) is obtained from u(i) (t) by transposing the pair of interchanging time intervals and the corresponding velocity values; here, at each step i, the number of pairs of intervals decreases by unity. It is obvious that, at the final step of this procedure, a unique pair of intervals will be left, such that u(0) (t) defines the basic motion of the body. The laws of motion of the internal mass that correspond to u(i) (t) will be denoted by s(i) (t). By introducing, in the standard way, the variation of the functions s in the interval of periodicity as
it is possible to show that (4.1) It remains to point out that, for any periodic function s, the inequality 2L(s) ≤ var s is holds, strict equality being achieved when, and only when, the function s has, in the interval of periodicity, a single point of the local minimum and a single point of the local maximum. For the basic solution, this condition is satisifed, and therefore, when condition (4.1) is satisfied statement (b) turns out to be a consequence of the chain of inequalities
5. Discussion of the results We have formulated and solved the problem of energy optimizing the periodic motion of the internal mass ensuring motion of the body containing this mass with a specified average velocity U0 . The same solution obviously determines the maximum average velocity of the body motion with a fixed energy consumption.
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Fig. 4.
The oscillation period of the internal mass specified in the problem when designing of specific devices can in fact vary over a fairly wide range. The question arises as to the best choice of the period T. An analysis of formulae (3.3) and (3.4) indicates that an inertial mode is optimal, realized for high-frequency oscillations of the internal mass, when
First, as T→0, the amplitude of the oscillations of the internal mass tends to zero (see formula (3.3)), and hence the size of the body containing it ceases to be a limiting factor. Secondly, for the specified average velocity of the body, the maximum velocity of motion of the internal mass takes the smallest of the possible values as T→0 (see formula (3.4)). Finally, the impact loads, at the instant when the direction of the body motion changes, remain the same for optimal motions, irrespective of T. This conclusion, of course, is reached within the framework of the initial formulation. The most constraining assumption in the formulation is that the resistance forces are uniquely defined by the velocity u of the body motion. In the case of motion in a liquid, this is valid only for quasi-stationary processes with a slow change in the velocity u. Violation of the condition of quasi-stationarity as oscillation the frequency increases requires a more thorough analysis of the problem of the optimal choice of the period T. This analysis must be based on the combined solution of mechanical and hydrodynamic problems. In the first optimization formulations5,6 of the problem of the motion of a two-mass system at a fixed average velocity of the body, it was not the energy consumption but the maximum velocity of motion of the internal mass that was minimized. It is interesting to compare the results obtained for these two formulations. For comparison, we will use the results obtained by Chernous’ko,7 and we will limit ourselves to the most important case when T→0. For a linear anisotropic medium (␣ = 1) with anisotropy factor , for optimal motion the following was obtained7 (in the notation used here)
(5.1) For the principle of optimality employed in the present paper, the ratio of the velocities and the energy coefficient are in agreement with expressions (5.1) if, instead of the relations f1 () and f2 (), we use the relations
(5.2) As expected, solution (5.1) gives, by comparison with the case (5.2), a higher ratio of the velocities U0 /V but a lower energy coefficient (see Fig. 4). A similar situation occurs in the case of a medium with anisotropic dry friction. Acknowledgement This research was supported financially by the Russian Foundation for Basic Research (08-01-00548-a). References 1. Li H, Furuta K, Chernousko FL. Motion generation of the capsubot using inner force and static friction. In: Proc 45th IEEE Conf Decision and Control. 2006. p. 6575–80.
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2. Vartholomeos P, Papadopoulos E. Dynamics, design and simulation of a novel microrobotic platform employing oscillation microactuators. Trans ASME, J Dynam Syst Measurement, and Control 2006;128(1):122–33. 3. Zimmerman K, Zeidis I, Steigenberger J. Mathematical model of worm-like motion systems with finite and infinite degree of freedom. In: Theory and Practice of Robots Manipulators: Proc 14th CISM-IFToMM Symp. Berlin: Springer; 2002, 507–16. 4. Miller G. The motion dynamics of snakes and worms. Computer Graphics 1988;22(4):169–73. 5. Chernous’ko FL. On the motion of a body containing a mobile internal mass. Dokl Ross Akad Nauk 2005;405(1):56–60. 6. Chernous’ko FL. Analysis and optimization of the motion of a body controlled by a moving internal mass. Prikl Mat Mekh 2006;70(6):915–41. 7. Chernous’ko FL. Optimal periodic motions of a two-mass system in a resistive medium. Prikl Mat Mekh 2008;72(2):202–15. 8. Bolotnik NN, Figurina TYu. Optimal control of the rectilinear motion of a rigid body on a rough plane by means of the motion of two internal masses. Prikl Mat Mekh 2008;72(2):216–29.
Translated by P.S.C.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The energy-optimal motion of a vibration-driven robot in a resistive medium夽 A.G. Yegorov ∗ , O.S. Zakharova Kazan, Russia
a r t i c l e
i n f o
Article history: Received 1 September 2009
a b s t r a c t The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections – one with a positive velocity and the other with a negative velocity of motion of the shell – is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. ©Elsevier Ltd. All rights reserved. © 2010 Elsevier Ltd. All rights reserved.
The mechanical system investigated models a vibration-driven robot – a mobile device capable of moving in a resistive medium without moving external parts (wheels, legs, caterpillar tracks, etc.). The possibility of using vibration-driven robots in medicine has been discussed earlier.1,2 A similar principle was used to describe the motion of limbless living creatures (worms and snakes).3,4 The question of the optimal motion of the system by the motion of the internal body was first posed by Chernous’ko,5,6 who examined the rectilinear motion, along a horizontal plane, of a rigid body with a cavity containing a different moving body (the “internal mass”) when there is Coulomb friction between the plane and the body. Recently, these problems have been widely discussed in the literature both for other resistance laws and for non-one-dimensional motion the internal masses.7,8 Normally, the problem of optimizing the motion of the internal mass is posed as the problem of maximizing the average velocity of the body with a constrained maximum velocity or minimum acceleration of the internal mass. In the present paper, the constraint has an energy form and is imposed on the internal motive power (the moving internal mass). The problem formulated in this way allows of a complete analytical investigation.
1. Formulation of the problem Consider a system consisting of two rigid bodies. The main body (body M – the shell) of mass M is located in a resistive medium, while the second body of mass (referred to below as the “internal mass”) is moving within it. The longitudinal periodic motions of the internal mass in relative to body M with which the entire system moves as a whole are investigated. The velocity of body M will be denoted by u, and the displacement and velocity of the internal mass relative to body M will be denoted by s and = s˙ respectively. The equations of motion of the internal mass and of body M in a fixed system of coordinates have the form (1.1)
夽 Prikl. Mat. Mekh. Vol. 74, No. 4, pp. 620–632, 2010. ∗ Corresponding author. E-mail address: [email protected] (A.G. Yegorov). 0021-8928/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2010.09.010
444
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where F is the force of interaction of the internal mass and body M, and R is the resistance of the medium to the motion of the body. Below we will assume that the resistance force is determined solely by the velocity of motion of the body and can be written in the form
Depending on the choice of the exponent ␣, we have different (anisotropic) resistance laws: when ␣ = 0, we have dry friction; when ␣ = 1, we have linear resistance; with ␣ = 2, we have quadratic resistance. For slow motion of body M in a liquid (i.e., ignoring the effects of inertia), pseudoplastic, Newtonian and dilatant liquids have corresponding ␣ values of 0 < ␣ < 1, ␣ = 1 and 1 < ␣ < ∞. A difference in the positive coefficients of resistance k+ and k− when of the body moves in the positive and negative directions can be ensured, for example, by an asymmetric body shape (a sharpened nose and a blunt tail). Omitting the force F from Eq. (1.1), we will obtain the basic equation (1.2) which describes the motion u(t) of the body with a specified law motion s(t) of the internal mass. From physical considerations it is clear that, for any specified periodic law s(t), with period T, Eq. (1.2) uniquely defines the periodic function u(t) with the same period. Using angular brackets to denote the average for a period, we will define the following functionals: U(u) = u is the average velocity of motion of the body; N(u)= u · R(u) is the average power expended in overcoming the resistance forces; L(s) = max(s)–min(s) is the amplitude of oscillations of the internal mass (the length of the body). Of course, the power N expended in overcoming the resistance forces is precisely equal to the power of the forces F · u imparted by the internal mass to the body. This can be ascertained by multiplying the second of Eqs. (1.1) by u and integrating it over to the period. To describe the energy consumption due to motion of the body by means of an internal motive power (the moving internal mass), we will introduce the energy coefficient (1.3) as the ratio of the power UR(U) necessary for the body to move at constant velocity U to the power N(u) actually expended on periodic motion. The double inequality in (1.3) follows from the convexity and non-negativity of the function uR(u). We will formulate the problem of optimizing the motion of the internal mass as follows. Problem 1. It is required to find the period T and the law s(t) of oscillations of the internal mass such that the amplitude of oscillations and the average velocity of motion of the body are fixed:
while the power of the internal motive force is a minimum:
Along with Problem 1, the following problem is considered. Problem 2. It is required to find the law s(t) of oscillations of the internal mass such that the period T of oscillations and the average velocity of motion of the body are fixed while the power of the internal motive force is a minimum. Note that the amplitude L of oscillations of the internal mass in Problem 2 is not fixed in advance, but is determined after the function s(t) has been found. Below it will be shown that the above problems are equivalent. The convenience of the formulation of Problem 2 is the fact that the initial problem is split into two simple successively solved problems: the first determines the optimal law u(t) of motion of the body, and the second restores, according to the optimal law u(t), the time dependence of the law s(t) of motion of the internal mass. The possibility of this splitting is due to the fact that no additional conditions, besides periodicity, are imposed on the function s(t). This condition is easily expressed in terms of u. In fact, relation (1.2) can be considered as the problem of finding the periodic function s(t) with a specified periodic left-hand side. The necessary condition for this problem to be solvable is obtained by taking the average of the two sides of Eq. (1.2). Taking the periodicity of the function s and u into account, we have the condition R(u) = 0, which will also be sufficient. Therefore, for any law
u(t) satisfying the constraint R(u) = 0, the periodic function s(t) is found from relation (1.2) by simple integration. The initial problem in this case reduces to finding the periodic function u(t), with period T, that minimizes the functional N(u), taking into account the constraints u = U0 and R(u) = 0. 2. The optimal motion of the body Normalizing the time t to T, the velocity u to U0 and the power N to k+ U0˛+1 , we will present the constraints imposed on u(t) as
and the problem of the optimal motion of the body in the form (2.3)
(2.4)
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A formulation equivalent to Eq. (2.3), written in terms of the coefficient of energy consumption , has the form
(2.5) Beneath the arrows in relations (2.3) and (2.5) we give the constraints under which minimization is carried out. Note that, in problem (2.5), the function u(t) does not come under constraint (2.1), and consequently u can have any positive value. The equivalence of problems (2.3) and (2.5) follows directly from the invariance of the functional (u) with respect to the normalization of u. From problem (2.5), in particular, it follows that the initial problem (2.3) could also be stated as the problem of maximizing the average velocity of the body for a specified motive power. Introducing the Lagrange multipliers 1 and 1 corresponding to constraints (2.1) and (2.2), we will rewrite problem (2.3) as the problem of finding the stationary point of the functional
By varying I(u) we obtain for u the algebraic equation
(2.6) from which it follows that the function u(t), which is the solution to problem (2.3), is a piecewise-constant function taking values corresponding to the roots of Eq. (2.6). We will denote these values by u1 , . . ., uk , and the size of the intervals in which they are taken by a1 , . . ., ak , in which case a1 + · · · + ak = 1. By virtue of constraint (2.2), among the values u1 , . . ., uk there should be both positive and negative values. Note that the Lagrange multiplier has a simple physical meaning. It can be established by multiplying each of the k equations (2.7) by ai ui and adding. As a result we obtain
and by virtue of conditions (2.1) and (2.2) (2.8) from which, in particular, it follows that is positive. Similarly, by multiplying Eq. (2.7) by ai and adding, the positiveness of is proved. Before carrying out a general analysis, we will begin by considering the special case ␣ = 1 of anisotropic linear friction. When ␣ = 1, Eq. (2.7) takes the form
For it to have a positive and a negative root, the inequality < < , which constrains the range of variation to the condition > 1, must be satisfied. In the opposite case ( ≤ 1), forward motion of the body in the positive direction is impossible in principle. This was noted earlier.7 When > 1, in parameters and vary in the ranges 0 < < < < ∞, the roots u+ > 0 and −u− < 0 of the equation take all possible values from the range (0, + ∞ ) and ( − ∞ , 0) respectively. Constraints (2.1) and (2.2) are written in the form (2.9) and the minimized functional as (2.10) Expressing, from Eqs (2.9), u+ and u− in terms of a, and substituting the expressions obtained into relation (2.10), we arrive at the problem of minimizing the function
in the interval (0, 1). The solution of this problem has the form
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Fig. 1.
The maximum value of the energy coefficient
obviously increases from zero to unity as increases from unity to infinity. We will return to the general case, examining Eq. (2.6) separately when 0 < ␣ < 1 and when 1 < ␣ < ∞. The qualitative form of the left-hand side of this equation is presented in Fig. 1. The stationary points f are defined as
Since, among the roots of Eq. (2.6), there should be positive and negative roots, the range of variation in the Lagrange multipliers and is constrained by the inequality f* () ≤ < ∞ when 0 < ␣ < 1 and by the inequality 0 < ≤ f* () when 1 < ␣ < ∞. The corresponding part of the (, ) plane will be denoted by S0 . When (, ) ∈ S0 , Eq. (2.6) has three roots: two positive roots and one negative root when 1 < ␣ < 1, or two negative roots and one positive root when 1 < ␣ < ∞. In the first case we will number the roots u1 (, ), u2 (, ) and u3 (, ) in increasing order, and in the second case in decreasing order. We will denote the size of the intervals in which u takes the values u1 , u2 and u3 by a, (1–a)b and (1–a)(1–b), where
and represent the constraints (2.1) and (2.2) in the form
(2.11) We can ascertain that relations (2.11) uniquely define the parameters a and b in terms of u1 , u2 and u3 , and consequently in terms of the parameters and . Here, problem (2.3) reduces to minimizing the function N(, ) under the constraints (2.12) The first of these is unimportant in the sense that, when the second constraint is satisfied, so must it be. In fact, under the second constraint of system (2.12), from the second equation of (2.11) it follows that (2.13) When a > 1 (when a < 1), from the first inequality of system (2.11) it follows that au1 > 0 (respectively 1 − au1 > 0), and hence a > 0 (a < 1). Either of the conditions a > 0 or a < 1, along with inequality (2.13), gives the first constraint of (2.12). It is clear that the admissible set S = {(, ) : 0 ≤ b(, ) ≤ 1} lies within S0 . Therefore, problem (2.5) reduces to finding the minimum of the function N(, ) ≡ in set S. However, by virtue of its linearity, the function N should take a minimum value at the boundary of S, i.e., when b = 0 or when b = 1. In both cases, the solution of problem (2.5) is a piecewise-constant function taking two values: a positive value (u+ ) in a set of size a and a negative value (−u− ) in a set of size 1 − a. Any pair u+ > 0, u− > 0 is admissible, and Eq. (2.6) acts as a linear system for finding Lagrange multipliers and from the specified u+ , u− . Constraints (2.1) and (2.2) for the piecewise-constant function mentioned take the form (2.14)
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Fig. 2.
Table 1 ˛
– ∞
∞ –
0 –
a
1 1
1/2 1
1
u−
0
1
0
u+
1
1
1+
1+
1 –
√ 2 ( −1) √ √ +1 √1 −1 √ √ −1
2 1
10
100
0.079 0.965
0.306 0.933
0.587 0.945
3.52
1.306
0.608
1.16
1.164
1.093
Expressing, from these, u+ and u− in terms of a, we have
(2.15) and we arrive at the problem of finding the minimum of the function of one variable
(2.16) It should be noted that the initial interval 0 < a < 1 in which the minimum of the function N is sought should be additionally constrained by the requirement u± (a) > 0. This requirement (see Eqs (2.15)) is expressed as
and reduces to the condition a > a0 when ␣ > 1 and to the condition a < a0 when ␣ < 1. Here
It can be shown by direct calculation that, in the interval a0 < a < 1 when ␣ > 1 and 0 < a < a0 when ␣ < 1, the function N(a) defined by Eq. (2.16) is strictly convex. At the ends of these intervals it reverts to +∞. Therefore, the problem of minimizing the function N(a) has a unique solution. The dependence of the maximum energy coefficient on the exponent ␣ for different values of the anisotropy parameter is presented in Fig. 2. The main characteristics of the optimal motion of the body for the limiting values of the parameters ␣ and are given in Table 1. The table also gives certain numerical values for the important practical case of a quadratic law of resistance. 3. The optimal motion of the internal mass. The basic solution It has been shown that motion of a body with the maximum energy coefficient occurs realized when the velocity of the body takes two values: u+ U0 > 0 and −u− U0 < 0. The interchanging of the intervals of forward and backward motion in this case may be arbitrary, but the total time of each of the forms of motion in the interval of periodicity should be equal to aT and (1 − a)T respectively. The dimensionless constants u+ , u− and a are entirely determined by the parameters ␣ and of the resistance law. For each of the infinite set of optimal laws of motion of the body there is a unique periodic law s(t), with period T, of motion of the internal mass. Among the entire set of optimal laws of motion of the body, we will single out the simplest (“basic”) law for which forward and backward motion of the body occur in the intervals (0, aT) and (aT, T). We will define the basic law s(t) of motion of the internal mass and calculate
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Fig. 3.
the corresponding amplitude L and maximum velocity V = max s˙ of its oscillations. Introducing the notation
we rewrite Eq. (1.2) in the regions of continuous motion as follows:
Taking into account the standard conditions at the discontinuities
and the conditions for the motion to be periodic
the solution of this equation, apart from an unimportant constant, has the form
(3.1) For short oscillation periods, namely when T < T+ = 2u0 /w− , the function s+ (t) decreases monotonically in the interval (0, aT), whereas when T > T+ , as it increases it initially increases and then, having reached a maximum value, decreases. Similarly, when T < T− = 2u0 /w+ , the function s− (t) increases monotonically in the interval (aT, T), whereas when T > T− it reaches a unique minimum in this interval. On account of this, as the oscillation period increases, in the intervals (0, T+ ), (T+ , T− ) and (T− , ∞) when a > 1/2 and in the intervals (0, T− ), (T− , T+ ) and (T+ , ∞) when a < 1/2, three optimal modes of motion of the internal mass occur successively. Taking the ratio between the inertial and viscous forces into account, they will be referred to as inertial, intermediate and viscous modes. For the case a > 1/2, these types of motion are represented schematically in Fig. 3a, b and c respectively. With obvious modification of the intermediate mode, they have a similar form when a < 1/2. The amplitude of the oscillation of the internal mass when a > 1/2 is defined as
(3.2) A similar formula with the changes T+ ↔ T− and a ↔ 1 − a occurs when a < 1/2. The function L(T) increases monotonically as in T increases from zero when T = 0 to infinity when T = ∞. For short oscillations periods, the dependence of the amplitude of the oscillations on the period has a linear form, and for long oscillations periods it is quadratic:
(3.3) As might have been expected, for short T the parameters k± and ␣, responsible for the viscous properties of the medium, do not occur in the relation L(T), and for long T the parameter M defining the inertial properties of the body, does not occur in L(T)
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The determination of the maximum dimensionless velocity of the body in modulus leads to the relation
(3.4) As can be seen, when the oscillation period increases, to ensure the same average velocity of motion of the body, the velocity of the internal mass must increase linearly. Formulae (3.1) and (3.2) represent the simplest of the optimal solutions – the basic solution of Problem 1. Taking into account the unique relation (3.2) in the basic solution between the oscillation period and the body length, they define the solution of Problem 2. 4. The extremal property of the basic solution The basic solution, besides being the simplest of the optimal solutions, possesses a remarkable extremal property. The region in the (L, T) plane corresponding to the entire set of optimal motions is defined as
Here and below, the zero superscript denotes the basic solution. Thus, for a specified oscillation period, of the body length does not exceed that determined by the basic solution, and for a specified body length the oscillations period cannot be shorter than the basic one. To prove the extremal nature of the basic solution, we will establish two facts: (a) for each constant L* ≤ L(0) (T) an optimal law of the body motion exists with L(T) = L* ; (b) for any optimal law the body motion, the inequality L(T) ≤ L(0) (T) is satisfied. By virtue of the fact that the basic solution with period T/2 occurs in the set of optimal motions with period T, it is sufficient to prove statement (a) for L* ∈ [L(0) (T/2), L(0) (T)]. We will define the one-parameter set of optimal laws of the body motion
with parameter T1 : T/2 < T1 < T. For given u, we will define the function s(t) and calculate for it the functional L = max s − min s. With a fixed value of T, the quantity L is a continuous function of T1 , taking values L(0) (T/2) and L(0) (T) at the ends of the interval [T/2, T] of variation in parameter T1 . As a consequence, L(T1 ) may take any intermediate value between L(0) (T/2) and L(0) (T). The proof of statement (b) is fairly lengthy (as far as checking inequality (4.1) is concerned), so we will therefore only present its principal idea. For the optimal non-basic motion of the body, the dimensionless velocity u(n) (t) takes the values u+ U0 and −u+ U0 in the pair of interchanging intervals T2k
We will introduce, along with u(n) (t), the sequence of other optimal motions of the body u(n−1) (t), . . ., u(0) (t), in which u(i−1) (t) is obtained from u(i) (t) by transposing the pair of interchanging time intervals and the corresponding velocity values; here, at each step i, the number of pairs of intervals decreases by unity. It is obvious that, at the final step of this procedure, a unique pair of intervals will be left, such that u(0) (t) defines the basic motion of the body. The laws of motion of the internal mass that correspond to u(i) (t) will be denoted by s(i) (t). By introducing, in the standard way, the variation of the functions s in the interval of periodicity as
it is possible to show that (4.1) It remains to point out that, for any periodic function s, the inequality 2L(s) ≤ var s is holds, strict equality being achieved when, and only when, the function s has, in the interval of periodicity, a single point of the local minimum and a single point of the local maximum. For the basic solution, this condition is satisifed, and therefore, when condition (4.1) is satisfied statement (b) turns out to be a consequence of the chain of inequalities
5. Discussion of the results We have formulated and solved the problem of energy optimizing the periodic motion of the internal mass ensuring motion of the body containing this mass with a specified average velocity U0 . The same solution obviously determines the maximum average velocity of the body motion with a fixed energy consumption.
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Fig. 4.
The oscillation period of the internal mass specified in the problem when designing of specific devices can in fact vary over a fairly wide range. The question arises as to the best choice of the period T. An analysis of formulae (3.3) and (3.4) indicates that an inertial mode is optimal, realized for high-frequency oscillations of the internal mass, when
First, as T→0, the amplitude of the oscillations of the internal mass tends to zero (see formula (3.3)), and hence the size of the body containing it ceases to be a limiting factor. Secondly, for the specified average velocity of the body, the maximum velocity of motion of the internal mass takes the smallest of the possible values as T→0 (see formula (3.4)). Finally, the impact loads, at the instant when the direction of the body motion changes, remain the same for optimal motions, irrespective of T. This conclusion, of course, is reached within the framework of the initial formulation. The most constraining assumption in the formulation is that the resistance forces are uniquely defined by the velocity u of the body motion. In the case of motion in a liquid, this is valid only for quasi-stationary processes with a slow change in the velocity u. Violation of the condition of quasi-stationarity as oscillation the frequency increases requires a more thorough analysis of the problem of the optimal choice of the period T. This analysis must be based on the combined solution of mechanical and hydrodynamic problems. In the first optimization formulations5,6 of the problem of the motion of a two-mass system at a fixed average velocity of the body, it was not the energy consumption but the maximum velocity of motion of the internal mass that was minimized. It is interesting to compare the results obtained for these two formulations. For comparison, we will use the results obtained by Chernous’ko,7 and we will limit ourselves to the most important case when T→0. For a linear anisotropic medium (␣ = 1) with anisotropy factor , for optimal motion the following was obtained7 (in the notation used here)
(5.1) For the principle of optimality employed in the present paper, the ratio of the velocities and the energy coefficient are in agreement with expressions (5.1) if, instead of the relations f1 () and f2 (), we use the relations
(5.2) As expected, solution (5.1) gives, by comparison with the case (5.2), a higher ratio of the velocities U0 /V but a lower energy coefficient (see Fig. 4). A similar situation occurs in the case of a medium with anisotropic dry friction. Acknowledgement This research was supported financially by the Russian Foundation for Basic Research (08-01-00548-a). References 1. Li H, Furuta K, Chernousko FL. Motion generation of the capsubot using inner force and static friction. In: Proc 45th IEEE Conf Decision and Control. 2006. p. 6575–80.
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2. Vartholomeos P, Papadopoulos E. Dynamics, design and simulation of a novel microrobotic platform employing oscillation microactuators. Trans ASME, J Dynam Syst Measurement, and Control 2006;128(1):122–33. 3. Zimmerman K, Zeidis I, Steigenberger J. Mathematical model of worm-like motion systems with finite and infinite degree of freedom. In: Theory and Practice of Robots Manipulators: Proc 14th CISM-IFToMM Symp. Berlin: Springer; 2002, 507–16. 4. Miller G. The motion dynamics of snakes and worms. Computer Graphics 1988;22(4):169–73. 5. Chernous’ko FL. On the motion of a body containing a mobile internal mass. Dokl Ross Akad Nauk 2005;405(1):56–60. 6. Chernous’ko FL. Analysis and optimization of the motion of a body controlled by a moving internal mass. Prikl Mat Mekh 2006;70(6):915–41. 7. Chernous’ko FL. Optimal periodic motions of a two-mass system in a resistive medium. Prikl Mat Mekh 2008;72(2):202–15. 8. Bolotnik NN, Figurina TYu. Optimal control of the rectilinear motion of a rigid body on a rough plane by means of the motion of two internal masses. Prikl Mat Mekh 2008;72(2):216–29.
Translated by P.S.C.