CC BY
17
Управление в технических системах
DOI: 10.14529/ctcr170206
MATHEMATICAL MODELING OF WALKING MACHINES WITH ONE-AXIS BODY
I.V. Voynov, mail@miass.susu.ru,
A.I. Telegin, teleginai@susu.ru,
D.N. Timofeev, goshanoob@mail.ru
South Ural State University, Miass, Russian Federation
The models walking machine (WM) with one-axis body are considered. Their kinematic schemes are providing the maximum carrying capacity and minimum actuators' power consumption for implementing the specified body movement. The solution of the dynamics equations (DE) for one-axis WM (OWM) are obtained. These DE contain OWM-N kinematic, geometric and simulation parameters, where N is any real number more than 5. The number of mathematical operations obtained in DE are minimal. DE are presented in two forms: first, as a system of differential-algebraic equations where differential equations contain the dynamic reaction at the support points, and algebraic equations describe the relations between the support feet and supporting plane. And secondly, as a system of N second degree differential equations with the excluded relation reactions. Formula of calculating dynamic reactions at the support points is as simple as possible. Formulas for calculating dynamic reactions at pivot points of such WM are derived. The authors also describe algorithms for solving dynamics tasks arising while studying WM walking and give examples.
Keywords: walking machine, plane models, dynamics equations, first task of dynamics, dynamic reactions, moving forces and force moments.
1. Introduction
Finding kinematic schemes and propellers providing the maximum carrying capacity (deadweight) and minimum actuators' power consumption for implementing the specified body movement is an urgent task for WM [1]. Four- and six-legged WM have from 12 to 18 actuators (three actuators on each leg) and provide high kinematic capabilities, maximum smoothness of body movements [1, 2].
If the WM is designed for transporting technological equipment or manipulators, its body does not require to move smoothly. To implement discrete-continuous cycle walk - acceleration, steady motion, deceleration providing a predetermined body movement - it does not need to have four or six universal legs and several motion freedoms in the three-point state. It is enough to have, for example, two legs, each of which has two feet and one actuator, and one support leg (crutch) with one foot and low-power actuators [3, 4]. The kinematic capabilities of such WM are minimum. The WM like this has one motion freedom while making a three-point step. We should expect that the minimum number of legs and actuators of electromechanical WM, as well as the efficient walking control with power recuperation of actuators in the deceleration cycle will provide a high specific carrying capacity (WM deadweight ratio) and low specific power consumption. This paper continues the investigation described in articles [2, 4] and contains the research results for such WMs.
The paper [2] shows WMs modeled with some linkwork (L) on a plane and proposes kinematic analysis models of the walking process for a three-point WM with one, two and three motion freedoms. The paper [4] proposes walking modeling and animation algorithms for such WMs. Specific examples show techniques of WM visualization in statics and dynamics, approaches for manual and automatic WM motion animation and describe simulation environment for manual walking control system. The use of open distributed information technologies (SVG, CAB, JavaScript) for these purposes is also described. MVC architecture is applied to build a simulation environment for manual walking control.
From the WMs considered in the papers [2, 4] we select WM with one body motion freedom. We identify such WM as one-axis_and denote them OWM. OWM class can be divided into subclasses
depending on the total number of units N. We denote the specific subclass as OWM-N. For example, Fig. 1 show schemes of OWM-5, Fig. 2 - schemes OWM-6, Fig. 3 - schemes OWM-7, Fig. 4 - schemes of OWM-8. These OWM hinges axes on the supporting plane (SP) are directed against the gravitation, i.e. the actuators do not overcome these forces. If the feet mechanisms of such OWM have an irreversible gearbox and provide their reciprocating motion along the axes that are collinear to the actuators' hinges axes, then the feet actuators do not work against the gravitation forces in the reference position. These properties of the OWM schemes and the opportunity of using only one actuator (e.g., OWM-5 front axle actuator in Fig. 1a) provide low power consumption.
c) d)
Fig. 2. OWA with 6 units
Fig. 4. OWA with 8 units
О
Fig. 5. OWA with contour units
The solution of the dynamics equations (DE) for OWM is published in the present paper. These DE must contain OWM-N kinematic, geometric and simulation parameters, where N is any real number more than 5. The number of mathematical operations obtained in DE shall be minimal. DE shall be presented in two forms: first, as a system of differential-algebraic equations where differential equations contain the dynamic reaction at the support points, and algebraic equations describe the relations between the support feet and SP. And secondly, as a system of N second degree differential equations with the excluded relation reactions. Formula of calculating dynamic reactions at the support points shall be as simple as possible.
The result of these equations was done with new methods of deriving bodies system DE described in the papers [3, 5-7]. It made possible to obtain DE with explicit options in a simple form that did not allow further simplification of these equations. The subsystems masses, static and inertia moments of complemented bodies introduced in the papers [3, 5-7] are constant inertial parameters.
In fact, DE and other design formulas proposed in this paper, apply to a larger set of L than described in the paper title. There are many L of this set with contour units (elements) that form the structure shown in Fig. 5, where each contour unit can carry any tree-type L, i.e. every contour unit can be a base for L with open branches. The contour units with serial number 3 is a body which has suspended single-unit legs (Fig. 5 shows only supporting legs 4 and 5) and multi-unit legs (Fig. 5 shows only one supporting two-unit leg). The hinges at points O, A and B connecting the SP with the end units (shins)
of legs model support feet. The other legs and their units are not shown in Fig. 5. But if any, they are transferred to a new (target) support position on the SP. And portable legs can be suspended not only to the body (unit 3), but also to any of the elements with serial number 1, 2, 4, 5. While walking, in each position, the contour units must have the structure and numbers as in Fig. 5. For this purpose, the elements are renumbered in a new state of OWM.
We introduce the following notation for the moments of driving forces generated by the actuators in the hinges of the leg units: Mb is the moment of force about the pivot pin [2] connecting the body (unit 3 in Fig. 5) and the thigh (unit 2 in Fig. 5) of a two-unit support leg; Mg is the moment of force about
the pivot pin connecting the thigh and the shin (unit 1 in Fig. 5) of a two-unit support leg; M4 is the moment of force about the pivot pin connecting the body and the shin of the supporting single-unit leg numbered 4; M5 is the moment of force about the pivot pin connecting the body and the shin of the supporting single-unit leg numbered 5. If N > 5 (N is the number of OWM units), then Miis the moment of force about the pivot pin connecting the removable unit numbered i with the preceding unit (towards the body).
The common features for all OWM are the following. First, they have five contour units (elements). Secondly, two contour units (units 4 and 5), forming the hinges with the SP, have one common base (body). These hinges simulate the supporting OWM feet at points A and B. A three-point OWM can be considered as L with two branches closed at points A and B [3]. The first unit (in order) is the element forming the hinge with SP at the support point O. The other units (if any) are portable.
2. OWM DE with relations at support points A and B
We denote as j.i a formula or a statement (i) in the paper (j) from the references. Statement 1. OWM DE can be represented in the following vector-matrix form, H • a + h • a2 - G - S • M = Mr, where the upper left blocks of 5x5 matrices H, h and S are represented as
(1)
Г J H21 H31 H 41 H51 ^ Г 0 -h21 -h31 -h41 h51
H21 J2 H32 H 42 H52 h21 0 -h32 -h42 -h52
Ho - H31 H32 J3 H 43 H53 , ho - h31 h32 0 -h43 -h53
H41 H42 H43 J4 0 h41 h42 h43 0 0
l H51 H52 H53 0 J5 V 1 h51 h52 h53 0 0 V
'1 -1 0 0 0 >
0 1 -1 0 0
So - 00 1 -1 -1
00 0 1 0
V 00 4 0 0 1 V
The column vector of the moments of driving forces in the hinges is M = (0, -Mg, -Mb, M4, M5, ...)T, where the driving moments of the portable leg units (if any) are in place of dots of N-dimensional column vector M, i.e. M6, M7, M8, if N > 5. Non-zero elements of the column vector
Mr = (Mr1, Mr2, Mr3, Mr4, Mr5,...)T are calculated by the following formulas
Mr1 = R2 (ya + yb)cpi - R2 (xa + Xb )spi , Mr2 = R3 fra + yb )cp2 - R3 (xa + Xb )sp2 , Mr3 = R4 (yacP3 - xasp3 ) + R5 ^bcy3 - xbsy3 ) , Mr4 = Ra fracp4 - Xasp4) , Mr5 = Rb(ybcP5 - XbsP5).
If a three-point OWM has portable units, i.e. if N > 5, there are zeros in place of dots of the N-dimensional column vector. The following geometrical relations are imposed on the rotation angles of the contour units
|R2CP1 + R3cp2 + R4cp3 + Racp4 = A^ [R2CP1 + R3cp2 + R5CY3 + Rbcp5 = B^
R2sp1 + R3sp2 + R4sp3 + Rasp4 - Ay, R2spi + R3sp2 + R5sY3 + Rbsp5 - By,
(2)
Войнов И.В., Телегин А.И., Математическое моделирование
Тимофеев Д.Н. шагающих аппаратов с одноподвижным корпусом
where Ra is the distance from the pivot pin of the fourth unit to the reference point A; Rb is the distance from the pivot pin of the fifth unit to the reference point B; Ax, Ay are the coordinates of the reference point A in the coordinate system (CS) Oij, rigidly connected with SP; Bx, By are the coordinates of reference points B in the CS Oij; xa, ya are the projections of reaction force at the reference point A on the axis Oi, Oj; xb, yb are the projections of the reaction force at the reference point B on the axis Oi, Oj; spi = sin(Pi + ai), cpi = cos(Pi + ai), ai is the angle from the axis Oi to the axis Oiii directed to mass center of the i-th augmented WM unit [3]; pi is the angle from axis Oiii to axis Oiei+1 for i = 1, 2, 3 (ei = Oi-1Oi /Ri), p4 is the angle from axis O4i4 to axis O4A, p5 is the angle from axis O5i5 to axis O5B; sy3 = sin(y + a3), cy3 = cos(y + a3), у is the angle from axis O3i3 to axis O3e5. The other notations and
values are described in statement 3.7, in particular, the Hki elements of matrix Н = (Hki}NxN are calculated according to the formula (3.21), the elements hki of matrix h = {hki}NxN are calculated according to the formula (3.22), the elements Ski of matrix S = {Ski}NxN are calculated according to the formula
(3.23), G = (Gb G2, G3, ..., Gn)T.
Proof. We can use the following algorithm to write L DE [3]:
1. We select one of the units forming a hinge with the SP as the first unit in order. We select the shin of a two-unit leg.
2. All units are numbered 2, 3, ..., N consistently. We use the numbers as in Fig. 5.
3. We break mentally the relations (hinges) at the support points A and B and replace them with the reaction forces Fr4 = Fa and Fr5 = Fb where 4 and 5 are the numbers of units that form broken relations with SP.
According to (3.21), (3.22), (3.23) we get the blocks Ho, ho, So of matrices H, h, S for contour OWM units. Really, the only zero element located under diagonal to blocks Ho and ho is situated at the intersection of the 5th row and 4th column, as 4 i {1...5} [3]. In the matrix S the element S45 = 0 because the base of the 5th level is the third unit, not the fourth. The elements S34 = S35 = -1, as the 3rd unit is the base for the 4th and 5th units, etc. 1
The element M of N-dimensional column vector Mj=0 since the first contour level (see Fig. 5) forms a hinge with the SP. The driving forces of the actuator don't work in this hinge. The element M2 = -Mg since Mg is the moment applied to the shin of a two-unit leg about a thigh of this leg, and according to the definition [3] M2 is the moment applied to the 2nd unit of the L (to the thigh of a two-unit OWM leg) about 1-st unit of L (shin of a two-unit OWM leg). The element M3 = -Mb since Mb is the moment applied to the thigh of a two-unit leg about the OWM body, and according to the definition [3] M3 is the moment applied to the 3rd unit of L (to the OWM body) about the 2nd unit of L (OWM thigh).
By the definition [3] Mrk is the moment of force acting on the k-th unit, and is caused by the reaction forces of broken connections and Mrk = Lkk • pk x Frk + Z Rik • xZ Frj, where k is the unit vec-
i,k j>i
tor of normal to AB; Frk is the reaction forces applied to the k-th unit (if the k-th unit is not closed on SP, then Frk = 0); Lk is the distance from the pivot point of the k-th unit to the point of force application Frk; pk is an ort directed from point Ok to the point of force application Frk = xrki + yrkj according to (3.25).
According to ei = cPi_1i + sPi_1j, Pk = cpki + spk j and k • Pk x Frk = k x Pk • Frk , k x Pk = cpkj" spki,
k x ei = cpi—1j _ spi_1i , we get
Mrk = Lk (cpkj _ spki) • (xrki + yrkj) + Z Ri Z (cpi_1j _ spi_1i) • (xrji + Уг] j) =
i,k j>i
= Lk(yrkcpk _ xrkspk) + Z Ri Z (cpi_1yrj _ spi—1xrj) ' i,k j>i
For OWM Fri = 0 (i = 1, 2, 3), Fr4 = Fa, FrS = Fb, L4 = Ra, L5 = Rb, xr4 = xa, yr4 = ya, xr5 = xb, Уг5 = Уь. Hence
Mr1 = R2(cp1ya _ sp1xa + cp1yb _ sp1xb ), Mr2 = R3 С^Уя _ sp2xa + cp2yb _ sp2xb ) ,
Mr3 = R4(cp3ya _ sp3xa) + М^Уь _ Sy3xb), Mr4 = Ra(yacp4 _ xasp4) , Mr5 = Rb(ybcp5 _ xbsp5) .
Bond equations (2) are a coordinate form of demonstrable vector equations
OO2 + O2O3 + O3O4 + O4A = OA, OO2 + O2O3 + O3O5 + O5B = OB. The statement is proved.
Please pay attention to the following observations before using statement 1.
Note 1. Statement 1 allows to record DE of OWM in the form containing the relation response only in two reference points A and B. The reference point O is represented by a hinge connecting the shin of a two-unit leg with SP. In this case the number of DE and geometric relations equations are minimal. The analytical work is simplified for the relationships minimization. If there is no problem of calculating responses in the reference point O, for example, if OWM moves on a plane and a two-unit leg is the supporting one (crutch) [3], i.e. the vector of dynamic response at the point O is perpendicular to SP, so this approach is optimal as a minimization of computational work.
Note 2. Another approach to derive DE of OWM has a relation with the corollary 6.3 which presents the L DE on the free base. OWM body is considered to be free. The number of DE will increase by two and they will contain the dynamic response at three points (O, A and B). The number of geometric relations will also increase by two. The formulas to calculate the dynamic responses in all control points can be obtained from these equations. Then they can be used to exclude reactions from OWM DE. Both approaches lead to the same form of OWM DE with the excluded relations reactions. This fact can be used to check and control the proper formulation and numerical experiments.
Note 3. The number of equations in (1), (2) is N + 4. Hence, we get N rotation angles of the units ai(t) (i = 1, ..., N) and four projections xa(t), ya(t), xb(t), yb(t) of reactions at the support points A and B from the system of equations (1), (2) in a given initial state of OWM, for example, with the given values of elements of N-dimensional column vectors a(0), a(0) and with the given laws of changing the moments of driving forces in the hinges of the leg. To do this, first we can use bond equations (2) as the first given integrals of the differential equations (1). These integrals allow to exclude from (1) four of the required rotation angle of contour units and their derivatives. Then we can deduce xa, ya, xb, yb from the first four equations of the system (1) and substitute them in the fifth equation. This equation together with the others (if N > 5) will not contain dynamic reactions and can be used to calculate rotation angles of the portable units (if N > 5) and the angle of rotation of one contour unit. First we can use the system (1) to exclude bond reactions. We will do it in the next section, i.e. we will use DE (1) for deriving formulas of calculating the dynamics reactions at the support points A, B and the showing the OWM DE with excluded bond reactions.
Example 1. We denote OWM-5 in Fig. 1 as OWM-5a. We write DE (1) and relation equations (2) for OWM-5a. In this and the following examples we assume that the mass centers of single-unit legs are on the axes of their rotation, the lengths of these legs are equal (Ra = Rb = R ) for this OWM-5a. The body mass center is at the point of suspension of a two-unit leg. The axis O3i3 is directed to the point O4. R4 = R5 = L is equal a half of the body length, R2 = a is the shin length, R3 = b is the thigh length of a two-unit leg. The thigh mass center is on axis O2O3. The shin mass center is on axis OiO2. It is true that Pi = 0 for all i, y = n and d3 = d4 = d5 = 0, G3 = G4 = G5 = 0 for this OWM-5a. Non-zero elements of the matrices H and h according to (3.21), (3.22) are calculated by the formulas Hki = mkdkRi+1cos(ei+1 ,ik), hki = mkdkRi+1sin(ei+1,ik). Therefore, H3i = H4i = H5i = 0 and h3i = h4i = h5i = 0. Under the definition (e2,i2) = a2 -a1, i.e. H21 = d• cos(a2 -a1), h21 = d• sin(a2 -a1) where d = m2d2R2 = m2d2a. We substitute the values in the matrices and the column vectors of DE (1) and perform matrix operations. Then we get the desired system of five nonlinear differential equations of 2nd degree in the following form
JA + H21CX 2 - h^a 2 - G1 - Mg = a(ya + yb) cos a! - a(xa + xb) sin a!, H21CX1 + J2CX2 + h21CX12 - G2 + Mg - Mb = b(ya + yb)cos C2 - b^ + xb)sin C2, J3a3 + Mb + M4 + M5 = L(ya - yb) cos a3 - L(xa - xb) sin a3, J4a4 - M4 = R(ya cosa4 - xa sina4), J5a5 - M5 = R(yb cosa5 -xb sina5).
It is obviously that relation equations (2) in the accepted notation for OWM-5a have the form facosa1 + bcosa2 + Lcosa3 + Rcosa4 = Ax, asina1 + bsina2 + Lsina3 + Rsina4 = Ay,
acosai + bcosa2 - Lcosa3 + Rcosa5 = Bx,
asinai + bsina2 - Lsina3 + Rsina5 = By.
3. Formula of calculating dynamic reactions at the support points
For walking it is necessary for the dynamic reactions at the support points to get into the friction cones [1]. Otherwise, the feet in the support points will be out of the SP. The contact with the friction cones can be achieved due to proper distribution of driving force moments at hinges of the leg units.
Statement 2. For OWM the projection of force reactions at the support points A and B are calculated by the formulas
Xa = (cP4D + cp5D4 /Ra )/sp54 , ya = (sP4D + sp5D4/Ra)/sp54 , (3)
Xb = (CP2D1/R2 " CP1D2/R3)/sP21 " Xa , yb = (sp2D1/R2 " spiD2/R3)/sp21 " ya , (4)
where Di is the left part of the i-th equation of system (1),
D = (sp52D1/R2 " sp51D2 /R3 )/sp21 + D5 /Rb , spji = sin(pj + aj — pi — ai) .
Proof. The proof contains an algorithm for deducing formulas of calculating the desired reactions. This algorithm is based on solving a system of 4 linear algebraic equations that are the 1st, 2nd, 4th and 5th of DE (1) with quantities xa, ya, xb, yb. From the first two equations of system (1) we will get the following system of linear algebraic equations and its determinant A1 to calculate xa + xb, ya + yb
f—sp1 (xa + xb) + cp1 (ya + yb) = D1/R2, 1"sp2 (xa + xb) + cp2 (ya + yb) = D2 /R3,
By using sp21 = A1 = sin(p2 + a2 — p1 — a1) and the formula of sine of two numbers difference, we get
A =
spi cpi - sp2 cp2
= sp2cpi cp2spi .
xa + xb =
Уа + УЬ =
D1/R2 D2/R3
"pi ср2
/Ai = (cP2D1/R2 - CpiD2/R3)/sp2i ,
-spi Di/R2 -SP2 D2/R3
/Ai = (sp2Di/R2 - SpiD2/R3)/sp2i.
By using Dx = (cp2D1/R2 — Cp1D2 /R3 )/sp21, Dy = (sp2D1 /R2 — sp1D2/R3)/sp21 we get Xa + Xb = Dx ,
ya + yb = Dy.
From the fifth equation of system (1) will get ybcp5 — xbsp5 = D5/Rb. We put expressions xb = Dx — xa , yb = Dy — ya instead of xb, yb. Then we get the following equation
xasp5 — yacp5 = Dxsp5 — Dycp5 + D5 /Rb .
Taking into account Dx, Dy and spji the right part of equation (5) can be represented in the form
D = [(cp2D1/R2 — Cp1D2/R3)sp5 — (sp2D1/R2 — sp1D2 /R3 )cp5 ]/sp21 + D5/Rb = = [D1 (sp5Cp2 — Cp5sp2 )/R2 — D2(sp5Cp1 — Cp5sp1)/R3]/sp21 + D5/Rb =
(5)
= (^52^2 — D2sp51/R3)/sp21 + D5/Rfe. (6)
By using the fourth equation of system (1) we get the following system of linear algebraic equations to calculate xa, ya
J—sp4xa + cp4ya = ^^ [sp5xa — Wa = D.
The determinant of this system is A2 = sp4cp5 — cp4sp5 = —sp54 . So
D4/Ra cp4 „
F. /( " sp5'
D — Cp^ s,
(7)
Cp4D + Cp5D4/Ra
p54
Уа =
" sp4 D4/Ra sp5 D
sP4D + sp5D4/Ra
/(- w=-—-—
sp54
These expressions prove the validity of formulas (3), (4) according to notations D, Dx, Dy and expressions xb = Dx - xa , yb = Dy - ya. The statement is proved.
From the formulas (1), (3), (4) it is obvious that to calculate the dynamic reactions at the support points A and B we must know the rotation angle, angular velocity and acceleration of OWM units and the driving forces moment in the hinges of these units. This is not a limitation of the obtained formulas but a consequence of the mechanics principles. The restrictions of the practical use of statement 2 are under consideration in the following notes.
Note 4. The case sp54 = 0 is special because it doesn't let us use the formula (3) for the calculation
xa, ya. It can occur at any point of time while WM is walking by different gaits, for example, O4A||O5B. Calculations are performed by special algorithms at these times. This special case may occur while making a step discussed in sections 4 and 5 of this article. Note 5. The case sp2i
= 0 is special because it doesn't help to use the formula (4) for the computation of xb and yb. To exclude this case from the consideration we assume that the first two units are not aligned along a single line in the three-point state, i.e. the two-unit leg is always in the configuration of the knee forward or backward in the reference state.
Statement 3. For OWM a projection on the axis Ok (k is the normal to SP) of force reactions in the support points A and B are calculated by the formulas
za = (Bxgs - Bygc)/(AyBx - AxBy), zb = (Aygc - Axgs)/(AyBx - AxBy), (8)
N N N N
where gs = gcosP^midisinai +^(A;a; -B;a2 j, gc = gcosP^m;d;cosa; + B;a; + A;a2 j, i=1 i=1 i=1 i=1
A: = i • I1i • k, B; = j • I1i • k, I1i = I; - miOOi - ^Rjmj , m is the mass of the i-th augmented unit,
j,;
di is the center of mass of the i-th augmented unit, p is the angle of SP to the horizon, values I;, mi, mj are defined in article [6].
Proof. The moment of force M1 of the reference point O is calculated by the formula N _
M1 =Z(I1; • £i + ®i xI1; • ®i + ®i®i • I1i j-mj xg- Ri4 xFr4 -R15 xFr5 according to (6.26). This mo-
i=1
NN
ment is acting on the shin of a two-unit leg from the side of SP. We will get mj = ^mi = ^midiii
i=1 i=1
according to formulas (6.21), (6.2) for OWM. The contact of this OWM shin with the SP is a point, i.e. i-M1 = 0, j-M1 = 0. The rotation of all OWM units is parallel with SP , i.e. w, = a;k, s, = a;k. And by
processing step there are R14 = OA, R15 = OB, Fr4 = Fa, Fr5 = Fb. Therefore,
N
0 = i • M1 =Z(a;i • I1; • k + a;2i • k xI1; • k -m;d;i • ii x gj-i • OAxFa -i • OB xFb ,
N
0 = j • Ml =Д at j • Iii • k + a2 j • k X Iii • k - m;d; j • ii X g)-j • OA x Fa - j • OB x Fb .
i=1
N
~1; " + a,2j• k xI; • k -m;d;j• i x g |-j • OA xF -
i=1
After elementary calculations
T • OA x Fa = i x (Ax i + Ay j) • Fa = Ayk • Fa = AyZa, i • OB x Fb = ByZb , i • k x IH • k = -j • IH • k,
j • OB x Fb = j x (Bx i + By j) • Fb =-Bx k • Fb =-BxZb, j • OA x Fa =-AxZa, j • k x I1; • k = i • I1i • k , i • \. x g = i x i, • g = k • gsina;, j • i, x g = j x i, • g = k • gsin(a; - n/2) = -k • gcosa;, k • g = -gcosP we get the following system of equations relative to za, zb.
Войнов И.В., Телегин А.И., Математическое моделирование
Тимофеев Д.Н. шагающих аппаратов с одноподвижным корпусом
N
Ayza + Byzb = gS, gs = Z(Aiai " Bidi2 + gmidiSinaiC0SP) ,
i=1
N
AxZa + BxZb = g^ gc = Z(Bidi + Aid2 + gmidicosaicosP) .
Jxzb - g^ gc -Z' + ^ i=1
It is obvious that formula (8) is the solution of this system. The expression for IH is obtained from (6.24) taking into account RXi - OOi, i • E • k - j • E • k - 0. The statement is proved.
Corollary 1. In the three-point OWM or in the case of the equalities ¡Xf = IyZ = 0 for all i values za, zb are calculated according to the formulas (8). In these formulas there are gs - mgrxcos P, gc - mgry cosP, where m is the OWM mass, rx, ry are projections of the vector OC on the axis Oi, Oj;
C is the OWM mass centre.
Proof. If OWM is stopped in the three-point state, then ai - ai - 0 for all i. Therefore,
N N
gs - gmXcosP , gc - gmycosP , where mx - Zmidisinai , my - Zmid^osai .
i-1 i-1
If the elements IXZ , IyZ of the matrix IH are zero then Ai - Bi - 0 and gs - gmXcosP,
gc - gmycosP . According to (6.21) mj is the static moment of the first subsystem relative to the point
O=Oi. By definition [6] the system of bodies and its first subsystem are the same. In our case mj is
the static moment of OWM for the point O. Therefore mx - i • mj is the projection of mj on the axis Oi;
my - j • mj is the projection of mj on the axis Oj; rx - mx/m, ry - my/m. The corollary is proved.
Note 6. The equations Ayza + Byzb - m g rxcosP, Axza + Bxzb - m g rycosP can be obtained from
the conditions of static equilibrium of a rigid body. This rigid body is supported at three points on the plane located at angle P to the horizon. So in case of a three-point motionless state of OWM the co-rollar 1 can be proved using the equations of static equilibrium of a rigid body.
To calculate the dynamic response of OWM to the base point O we can use the following statement. Statement 4. For OWM the projection of the reaction force at the reference point O on the axis Ok, Oi, Oj are calculated by the formulas zo = mgcos(P) - za - zb,
NN
xo - -Zmidi (aisi + a2ci ) " m i • g " xa " xb , yo - Zmidi (dici " d2si ) " m j • g " ya " Уb, i-1 i-1
where si - sinai, ci - cosai.
Proof. According the formulas (7.1), (7.3) It is obvious that the reaction force of OWM at the sup-
N
port point O is calculated by the formula Fo - Zmidi (aiji - a2iiI - m1g " Fa " Fb . Therefore, its projec-
i-1
tions xo, yo, zo on the axis Oi, Oj, Ok are calculated by the formulas
N
x„ - i • f -Zmd I ai • j - an • is I-m,i • g -x0 -x.
'o = i • Fo =Z midi ("i1 • ji " a2i • »1 )" m1> • g"
i=1
N 1 \
Уо = j • Fo =Z midi (aij • ji " d2j • »i )" m1j • g " Уа " Уь ,
i=1
N
Zo = k • Fo =Z midi (aik • ji " a2k • ii )" m1k • g " Za " Zb.
Hence, according to the equalities k • ii - k • ji - 0, k • g - -gcosP , m1 - m, i • ii - j • ji - cosai - ci, i • ji - cos(ai + n/2) - -sinai - -si, j • ii - cos(ai - n/2) - sinai - si we will get the desired formula.
The statement is proved.
Example 2. We will write formulas of the dynamic reactions at the support points A, B, O for OWM-5A in case of s54 = sin(a5 -a4) ^ 0, s21 = sin(a2 -a^ ^ 0, Ijf=IyZ =0 for all i. According to the formula (3) we will receive
xa = (Dcosa4 + D4cosa5/R)/s54, ya = (Dsina4 + D4sina5/R)/s54, xb = (D1cosa2/a - D2cosa1/b)/s21 - xa, yb = (D1sina2/a - D2sina1/b)/s21 - ya, where D = [Djsin(a5 -a2)/a-D2sin(a5 -ax)/b]/s21 + D5/R . The left part of DE (1) has been obtained in example 1: D1 = J1a1 + H21a2 - h21a2 - G1 - Mg, D2 = H21a1 + J2a2 + h21a2 - G2 + Mg - Mb,
D4 = J4a4 - M4, D5 = J5a5 - M5. The values za, Zb are calculated by the formulas (8) where accor-
2 2 ding to the corollary 1 gs = gcosp^midisinai, gc = gcosp^midicosai . According to statement 4
i=1 i=1
22
xo = -Zmidi («isi + a2ci) -m i ' g-xa -xb, yo = Zmidi («ici -a2si) -m j' g-ya -Уb, Zo = m g -Za -Zb. i=1 i=1
The values i-g , j'g are defined by the axes orientation Oi, Oj in inclined SP. If SP is horizontal (P = 0)
then i-g = j-g = 0.
4. OWM DE with excluded relation reactions
The third equation of system (1) has not been used in the proof of statement 2. If we substitute the formulas (3) and (4) we get next statement.
Statement 5. The DE of OWM third unit with the excluded relation reactions has the form
bA + b2D2 - D3 + b4D4 + bsD5/Rb = 0, (9)
where
b = bssp52 - R5sy32 b = R5sy31 - bssp51 b = R4sp53 - R5sy53 b = R4sp43 - R5sy43
R2sp21 R3sp21 Rasp54 sp54
Proof. The third equation of system (1) has the form D3 = R4(yacp3 -xasp3) + R5(ybcy3 -xbsy3). Using (4), we will exclude xb, yb from it. Then we get
D3 = R5cy3(sp2D1/R2 - sp1D2 /R3 )/sp21 - R5sy3(cp2D1/R2 - cp1D2/R3)/sp21 + + (R5sy3 - R4sp3)(cp4D + cp5D4/Ra)/sp54 + (R4cp3 - R5cy3)(sp4D + sp5D4/Ra)/sp54 .
Using (3), we will exclude xa, ya from it. Then we get
D3 =-R5[D1(sy3cp2 - cy3sp2)/R2 - D2(sy3cp1 - cy3sp1)/R3]/sp21 +
+ (R5sy3 - R4sp3)(cp4D + cp5D4/Ra)/sp54 + (R4cp3 - R5cy3)(sp4D + sp5D4/Ra)/sp54 .
We will use the notation sy3i = sin(a3 + y- ai -pi) = sy3Opi - oy3spi (i = 1,2) and do the elementary transformations. Then we get
D3 = R5(D2sy31/R3 - D1sy32/R2)/sp21 + {[ - R5 (sp4cy3 - cP4sy3) + R4 (sp4cp3 cp4sp3 )]D + + [ - r5(sP5cy3 - cP5sy3 ) + R4 (sp5cp3 - cp5sp3 )]D4 /Ra }/sp54.
Therefore,
D3 = R5(D2sy31/R3 - D1sy32/R2)/sP21 + [(R4sp53 - R5sY53)D4/Ra + (R4sp43 - R5sy43)D]/sP54 .
We will select the Di multipliers and substitute the expression (6) instead of D. Then we get
f
D3 =
Rs
^Y32
R2sß
f
Di +
Rs
5aY31
R3sß
D2 + b4D4 + bs [(Di Sß52/R2 - D2sß51/R3 )/sß21 + D5/Rb
t2°ß21 ) ^ R3sß21 )
where b4 = (R4sp53 -R5sy53)/(Rasp54), bs = (R4sp43 -R5sy43)/sp54 . We will cast similar terms of Di, D2. Then we get the formula (9). The statement is proved.
Statement 6. OWM-5 DE in the three-point state with the excluded relation reactions can be represented as the following nonlinear ordinary differential equation of the second degree with respect to the generalized coordinate (GC) q
H(q)q + h(q)q2 + G(q) = (bi - b2)Mg + (1 + b2)Mb + (1 + b4^ + (1 + bs /Rb )M5. Where H, h, and G are given q functions and are calculated by the formulas
H = tbkHk, h = ]Tbkhk, G = -]TbkGk, Gk = gmkdkg • jksinp,
(10)
(11)
k=1 k-1
k=1
k=1
k -1
Hk =1 Hkifqi + Jkfqk +1 Hikfqi ,hk ^Hf + hkifq2 ) + Jk^ + Z(Hikfqi - hf ) , (12)
i i>k i i>k
where b3 =-1; b5 = bs/Rb; the absolute rotation angles of units are associated with GC q by the dependencies ai = fi (q) and fqi = dfi (q)/dq = dai/dq, fqi = d2fi (q)/dq2 = d2ai/dq2 . For example, the rotation angle of one contour unit may be used as q.
Proof. The left part of the k-th DE of the system (1) has the form [3]:
k-1
Dk = Z (Hk!a! + hkiaf ) + Jkak + Z (H!ka! - ^ ) - Gk - Mk + Z M! .
i i>k i,k
(13)
For contour units of relation equations (2) or as a result of kinematic analysis we will receive ai = fi (q) where q is GC of OWM-5 is the rotation angle of the dog [2], for example. After double differencing of functions ai = fi (q) of t we will get ai = fqiq, ai = fqiq + fq-q2. We will substitute these expressions in (13). Then taking into account the notation (12), we will get
k-1
Dk = Z [Hki (fqiq + fqiq2) + hkifq-q2 ] + Jk (fqkq + ) + Z [Hk (fqiQ + fqiq2) - hfiq
i >k
2
( k -1
i,k
-Gk - Mk +Z Mi = Z Hkifqi + Jkfqk +Z Hikfqi q - Gk - Mk +Z Mi
V i
ik
i >k /
+
+
Z-1 (Hkifqqi + hkifqi ) + Jkfqk + Z (Hikfqqi - hf )
i>k
i,k •2
q2 = Hkq + hkq2 - Gk - Mk +Z Mi.
i,k
We will substitute the found expression for Dk into the formula (9) that can be written as ^bkDk = 0 ,
k=1
where b3 = -1, b5 = bs /Rb . Then we will get
ZbkDk =Zbk Hkq + hkq2 -Gk -Mk +ZMi
k=1
k=1
i,k
= qZbkHk + q2 Zbkhk-Z bkGk-Zbk Mk-Z Mi
k=1
k=1
k=1
k=1
i,k
= 0.
Hence, taking into account the notations (11), we will get DE (9) in the form Hq + hq2 + G = M where
M = b1(M1 - M2) + b2 (M2 - M3) - (M3 - M4 - M5) + b4M4 + bsM5 /Rb . Taking into account that M1 = 0, M2 = -Mg, M3 = -Mb, we will get M = bjMg + b2(Mb - Mg) + Mb + M4 + M5 + b4M4 + bsM5 /Rb .
After casting similar terms of Mg, Mb, M4, M5 we will get the reqired form (10) of OWM-5 DE.
The statement is proved.
Example 3. We write DE for OWM-5A in the case s54 = sin(a5 - a4) ^ 0, s21 = sin(a2 - a1) ^ 0 . According to statement 6 for OWM-5A we will get the required DE as a form (10) where according to statement 5 we have
b1 = [Lsin(a3 - a2) + bssin(a5 -a2)]/as21, b2 = [ -Lsin(a3 -a1)-bssin(a5 - a1)]/bs21, b4 = 2Lsin(a5 - a3)/Rs54, bs = 2Lsin(a4 - a3)/s54 . Using the formulas (12) we will have
H1 = f + H21fq2 , H2 = H21fq1 + J2fq2 , h1 = f + ^2 - h21fq2 , h2 = H21fqq1 + h21fq2 + f ,
Hk = Jkfqk, hk = f k = 3, 4, 5.
Using the formulas (11) we will have
H = b1H1 + b2H2 - H3 + b4H4 + bsH5/R, h = b1h1 + b2h2 - h3 + b4h4 + bsh5 /R,
G = -b^ - b2G2, G1 = gm1d1g • jjsinp, G2 = gm2d2g • j2sinp . 5. OWM DE for the step forward
If the OWM has the equal lengths of single-unit legs (Ra=Rb), and while taking a step the absolute rotation angles of simple legs are equal (O4A||O5B), then the body (the 3rd unit) of OWM makes a trans-lational displacement, i.e. a3(t) = const. Therefore, we will name the executed step as step forward (SF). It is easy to prove that Ra = Rb is necessary for the implementation of SF.
According to note 4 for SF the DE (9) or (10) and formula (3) cannot be used. To study SF we can use statement 7.
Statement 7. If OWM has Ra = Rb = R and by taking a step we have O4A||O5B then DE of its simple-unit leg is represented as
D1sp42/R2 - D2sp41/R3 + (D4 + D5)sp21/R = 0. (14)
Proof. If there are O4A||O5B then sp4 = sp5, cp4 = cp5 and after summing the equations of system (7) we get xa (sp5 - sp4) + ya (cp4 - cp5) = D4/Ra + D =0. We will substitute the expression (6) instead of D. Then we will get (sp52D1/R2 - sp51D2/R3)/sp21 + D4/Ra + D5/Rb = 0. Taking into account R = Ra = Rb and sp42 = sp52 , sp41 = sp51 we will get the formula (14). The statement is proved.
If we consider OWM-5 then the equations (9) or (14) (for SF) are the only DE with excluded relation reactions. If N > 5 then the portable units must be added to DE (9) or (14), namely, we need to add equation 6 of the system (1) for OWM-6, we need to add the equation 6 and 7 of the system (1) for OWM-7, we need to add the equation 6, 7 and 8 of the system (1) for OWM-8. All of the added equations do not have dynamic reactions at the support points.
To study the dynamics of OWM-5 by taking a SF it can be used corollary 2.
Corollary 2. SF DE of OWM-5 can be represented as the following nonlinear differential equation of 2nd degree about the absolute angle q of the single-unit leg
H(q)q + h(q)q2 + G(q) = (b -b2)Mg + b2Mb + b4^ + M5), (15)
where b1 = sq2/R2 , b2 ^q!^ b4 = sp21/R , sq1 = sin(q - ^ sq2 = sin(q - ^ .
Proof. The left part of DE (15) is proved on the basis of the DE (14) and is similar to the proof
of statement 6. We will take the components ^Mi - Mk for Dk in (14) from (13) and transfer them to
i,k
the right side of DE (14). Then DE (14) has the following form
H(q)q + h(q)q2 + G(q) = -sp42 (M - M1 ^ + sp41 (M3 - M2 )/R3 - sp21( - M4 - M5 )/R.
If we use sp4i = sqi and take into account Mj = 0, M2 = -Mg, M3 = -Mb then we will get DE (15) from
the last equation. The corollary is proved.
Corollary 3. SF ED of OWM-5A can be represented as
Jq + + G = (fql - fq2 )Mg + fq2Mb + M4 + M5, (16)
•e J = J4 + J5 + Jlfql + J2fq22 + 2dfqlfq2c2b d = m2d2a , G =-fqiGi - fq2G2 , fqi = ^2^2^
fq2 =-Rsq1/bs21, c21 = cos (a2 - a1), s21 = sin (a2 - a1), a is the thigh length, b is the shin length of a two-unit leg, q is the absolute rotation angle of single-unit legs.
Proof. There are a3 = const, a4 = a5 = q for SF. Hence, fq3 = fq3 = 0, fq4 = fq5 = 1, fq4 = fq5 = 0 and H3 = h3 = 0, Hk = Jk, hk = 0 for k = 4, 5. Therefore, by using the data and results of examples 1, 3 according to the formulas (11), we will get G = -b1G1 - b2G2,
H = b^ + b2H2 + b4J4 + b5J5 = b1 (Jifq1 + H21fq2 ) + b2 (H21fq1 + J2fq2 ) + b4J4 + b5J5, (17)
h = b^1 + b2h2 = b1(J1fqq1 + H21fqq2 -h21fq22) + b2(H21fqq1 + h21fq21 + J2fqq2) , (18)
where bj = sq2/a, b2 = -sq1/b , b4 = b5 = s21/R , R2 = a is the shin length, R3 = b is the thigh length of
two-unit legs.
From the system (2) for OWM-5A we will get fac1 + bc2 = -Rcq, cq = cos(q), sq = sin(q), h = OD,
|as1 + bs2 = h - Rsq, ci = cos(ai), si = sin(ai). ( )
For deducing the calculation formulas fqi (i = 1, 2) we will find q derivative of equations of system (19). Then we get the system of linear equations for fq1, fq2
as1fq1 + bs2fq2 = -Rsq
, and its determinant A = ab(s1c2 - c1s2) = -abs21. (20)
ac1fq1 + bc2fq2 =-Rcq
Therefore, the solution of this system
fq1 = -RSq bs2 /А = Rb(S2Cq - C2Sq) -Rsin(q - a2) Rsq2 b1
RCq bc2 -abs2i -as21 as21 b4
fq2 = as1 ac1 - RSq - RCq /А = Ra(SqCi - CqSi) _ -abs2i Rsin(q - a1) -bs21 Rsq1 _ bs21 b2 b4 '
We will divide DE (15) and b4. Then the right side becomes the right side of equation (16), the formula for calculating G will take the required form, and the coefficient with q will take the form
J = H/b4 = fq1 (Jfq1 + H21fq2 ) + fq2 (H21fq1 + J2fq2 ) + J4 + J5 according to (17). The last form will be
equal to the required form due to H21 = d • cos(a2 - a1). We will make a derivative J in q. Then taking into account dH21 /dq = -d • sin(a2 - ax) • d(a2 - ax )/dq = -h21 (fq2 - fq1) we will get
dJ/dq = f^1 (J1fq1 + H21fq2 ) + fq1 (J1fq^ - h21(fq2 - fq1)fq2 + Hf ) + +fq2 (H21fq1 + J2fq2 ) + fq2 ( h21 (fq2 - fq1 )fq2 + H21fcql + J2fqq2 ) = = fq1(2Jf1 + 2H21fq\ - 2h21fq^) + fq2(2H2fq + 2h2f2 + 2^) .
From (18) we get h/^ = ^(f + H21fq2 - h21fq22) + fq2(H21fqq1 + h21fq21 + J2fq2), that proves the equality
— = -1— . The corollary is proved. b4 2 dq
1 dH
Note 7. We can represent the left part of DE (10) in the form Hq +---q2 + G as a consequence of
2 dq
2 1 dH 2
statement 1 and 6. For single-moved mechanical systems the equality Hq + hq = Hq +---q , i.e.
2 dq
1 dH
h =--, is well-known [8]. The corollary 3 is proved to check the correctness of the obtained
2 dq
formulas.
According to note 4 it is impossible to calculate values xa, ya by the formulas (3) for SF (a3 = const). We can use statement 8.
Statement 8. The values xa, ya for SF (a3 = const) can be calculated by the formulas
xa = (Aqcp4 - cpD4 /R)/(cpcp4 + cpsp4) , ya = (Aqsp4 + spD4/R)/(cpcp4 + cpsp4) , (21)
where Aq = D3 + [D1(sYCp2 - cYsp2)/R2 + D2(cYsp1 - SyCpl)/R3]/sp21,
sy = R5sin(y + a3), cy = R5cos(y + a3), sp = sy - R4sin(p3 + a3), cp = R4cos(p3 + a3) - cy . Proof. The third equation of system (1) was not used to derive SF DE. This equation has the form
R4(yacp3 - xasp3) + R5 frb^ - xbsy3 ) = D3 . We substitute xb = Dx - xa , yb = Dy - ya at this equation
and we will get
(R5sy3 - R4sp3 )xa + (R4cp3 - R5cy3)ya = D3 + R5(Dxsy3 - Dycy3) .
We use the notation sy = R5sy3, cy = R5cy3, sp = sy - R4sp3, cp = R4cp3 - cy . Then we will get the following system of two linear equations to calculate xa, ya together with the fourth equation of system (1)
spxa + СрУа = D3 + SyDx - CyDy = Aq,
5рЛа
-sR4xa +
= d4/r.
p4xa + cp4ya = D4
Formulas (21) are the solutions of this system. The determinant of this system is A = spcp4 + cpsp4 . For example, for OWM-5A R4 = R5 = L, 2L is the body length, a3 = 0, a4 = q, y = n. Therefore, sy = R5sinn = 0, cy = R5cosn = -L, sp = 0, cp = L - (- L) = 2L and A = 2Lsinq , i.e. there are no problems with the calculation of dynamic reactions at the support points, if SF is limited by 0 < q < n .
We substitute the known expressions that are introduced in the proof of statement 2 instead of Dx, Dy. Then we get
Aq = D3 + sy(cp2D1/R2 - Cp1D2 /R3 )/sp21 - cy(sp2D1/R2 - sp1D2/R3)cp5]/sp21 = = D3 + [D1(sYCp2 - cYsp2)/R2 + D2(cYsp1 - sTcp1)/R3]/sp21 .
The statement is proved.
6. Algorithms for computing the OWM DE coefficients
To solve the dynamics of OWM on the basis of the received DE it is recommended to use known research methods of machines dynamics, for example, that were described in the fourth part of the book [8] on p. 223-235 or Chapter 2 of reference [9]. Moreover, we should write algorithms of calculations
fqi, fqi, H(q) h(q) and G(q) and arrange all the formulas in order to use them in the numerical experiments. We must do it for each specific OWM and their steps. The dependences ai = fi(q) for the selected GC q in the mechanisms and machines theory are known as position functions (PF). and their q derivatives, i.e. the values fqi, fqi are called transfer functions of the 1st and 2nd degrees, respectively [8, 9]. We will consider examples of deducing the formulas for their computations.
Example 4. We do not have calculation formulas of values fq, fq2 so far. They are necessary for writing the algorithm for calculating H(q) and h(q) for OWM-5A. To do it we will find q derivatives of the equation system (20). Then we get the system of linear equations for fq, fq in the following form
as1fqq1 + bs2fqq2 = RCq - aC1fq21 - bs2fq22 = A,
ac1fqq1 + bc2fqq2 = Rsq + af2 + bs2fq22 = B.
We will write the solution of this system
fq -
rqi -
A bs2 B bco
/д - b(Ac2 - Bs2) - Bs2 - Ac2 f q -
-abs
21
as
21
as1 A ac1 B
/Д -
a(Bs1 - Aq) _ Ac1 - Bs1
-abs
21
bs
21
where
Bs2 - Ac2 - R(SqS2 + cqc2 ) + afq1(s1s2 + c1c2> + bfq2(§2 + ^ - Rcq2 + ac21fq1 + bfq2 -
q1
q2
q2
q1
q2
-Ac1 + BS1 - R(SqS1 + c^) + afq1(Sj + 01> + bfq2(S1S2 + c1c2) - Rcq1 + afq 1 + bfq2c21.
q2
q1 q1
q2
Войнов И.В., Телегин А.И., Математическое моделирование
Тимофеев Д.Н. шагающих аппаратов с одноподвижным корпусом
Taking into account the formulaS derived in the proof of corollary 3 we will write down the algorithm for calculating H(q) h(q) for SF of OWM-5A on the level (G = 0).
d - m2d2a - conSt, Sq - Sin(q), cq - coS(q), Si - Sin(ai), ci - coS(ai),
Sqi - Sqci - cqSi , Cqi - SqSi + cqci , (i = 1 2), S21 - S2c1 - C2S1, c21 - S1S2 + c1^ H21 - dc21, h21 - dS21,
b1 - Sq2/а, b2 - Sq1/b , b4 - S21/R , fq1 - b1/b4, fq2 - W
fqq1 - (Rcq2 + ac21fq21 + Ь^У^) , fq\ - + f + bfq22c21)/ (bS21) ,
H - b1 (J1fq1 + H21fq2 ) + b2 (H21fq1 + J2fq2 ) + b4(J4 + J5) ,
h - b1(J1fqq1 + H21fq\ - h21f£) + ^2^1 + h2f2 + J2fq2) .
Here it iS conSidered to be that the PF a1(q) and a2(q) are known. It Should be noted that the analytical formS of PF are very bulky and contain inverSe trigonometric functions. The PF can be found aS the Solution of a SyStem of nonlinear equationS (19) or directly from the drawingS of the relevant kinematic SchemeS by elementary geometric reaSoning. The methodS of deriving formulaS for L PF are well known from the mechaniSmS and machineS theory, for example in [8]. As an example, we will derive the PF a1(q) and a2(q) for the case of equal lengths of the thigh and shin of a two-unit leg. To do this, we will find the solution of system (19) when a = b.
fcosa1 + cosa2 --Racosq, Ra - R/a, ha - h/a,
1 1 2 a a a (22)
[sina1 + sina2 - ha - Rasinq .
After squaring and summing equations of the system (22) we will get the equation 2 + 2(cosa1cosa2 + sina1sina2) - R^ + hj - 2haRasinq. Hence, taking into account the symbols a1 - (R^ + h^)/2 -1, a2 - Raha we will get cos(a2 - a1) - aq - a1 - a2sinq . Therefore, if |aq| < 1, then a2 - a1 - arccos(aq).
We will subtract the square of the second equation of system (22) from the square of the first one we will get
e - cos2 a1 - sin2 a1 + 2(cos a1 cos a2 - sin a1 sin a2) + cos2 a2 - sin2 a2 --cos2a1 + cos2a2 + 2cos(a1 +a2) - R2(cos2q-sin2q)-hj + 2haRa sinq-= R2 - h2 + 2haRa sin q - 2R2 sin2 q . Taking into account cos2a1 + cos2a2 - 2cos[(2a1 + 2a2)/2] • cos[(2a1 - 2a2)/2] we will get
e = 2cos(a1 + a2) • cos(a1 - a2) + 2cos(a1 + a2) - cos(a1 + a2)[2 + 2cos(a1 - a2)]. And taking into account cos(a1 - a2) - a1 - a2sinq - aq we will get
e =2cos(a1 + a2)(1 + aq) - 2ao + 2a2sinq - 2R2sin2q, where ao - (R^ - h2)/2 and cos(a1 + a2) - bq - (ao + a2sinq - R2sin2q)/(1 + aq). Therefore, if |bq| < 1, then a1 + a2 - arccos(bq).
Thus, if |aq| < 1 and |bq| < 1, then the required SF are represented in the form a1 - [arccos(bq) - arccos(aq)]/2 , a2 - [arccos(bq) + arccos(aq )]/2, where, aq - a1 - a2sinq, bq - (ao + a2sinq - R2sin2q)/(1 + aq).
Example 5. If OWM makes a step that is different from SF then the expression for fqi, fq are derived from the system (2). If we accept a5 as GC, i.e., q = a5, then for calculating fq3, fq4, f(q3, fq from (2) we will get
i.e.
[Ay - R4sp3 - Rasp4 + R5sy3 + Rbsp5 = By, [Ax - R4cp3 - RaCp4 + R5cy3 + RbCp5 = Bx,
J-R4sp3 + R5sy3 - Rasp4 = By - Ay - Rbsp5, 1-R4CP3 + R5cy3 - Racp4 = Bx - Ax - Rbcp5.
When we make double q differencing of the last system we will get J (R4Cp3 - R5Cy3)fq3 + RaCp4fq4 = Rbcpq, [(R4sp3 - R5sy3)fq3 + Rasp4fq4 = Rbspq, (R4Cp3 - R5CY3)fqq3 + RaCp4fqq4 = A,
(R4Sp3 - R5sy3)fqq3 + RaSp4fqq4 = B,
where A = (R4sp3 - R5sy3 )fq3 + Rasp4fq4 - Rbspq, B = -(R4Cp3 - R5Cy3 )fq3 - RaCp4fq4 + Rbcpq , spq = sin(p4 + q)
, Cpq = cos(p4 + q). The solutions of these systems are elementary. To derive the formulas for values fq1 , fq2 from the system (2) we will get
JR2Cp1 + R3Cp2 = Ax - R4Cp3 - RaCp4, JR2sp1 + R3sp2 = Ay - R4sp3 - Rasp4.
Here
J R2sp1fq1 + R3sp2fq2 = -R4sp3fq3 - Rasp4fq4, [R2Cp1fq1 + R3Cp2fq2 = -R4Cp3fq3 - RaCp4fq4. Here the latest system is to CalCulate values fq1, fq2 , in whiCh we use the previously derived formulas for fq3, fq4 . When we make q differentiation of this system, we will get a system of two linear equations. The notation of the formulas and their simplifiCation for a speCifiC OWM is elementary but tedious analytiCal work.
7. Conclusion
ReCeived OWM DE and formulas for CalCulating the dynamiC reaCtions at the support points allow to organize the whole range of numeriCal experiments in the study of OWM induding speCial Cases and gaits.
The works was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.0011.
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Received 7 March 2017
УДК 531.3 DOI: 10.14529/^сг170206
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ
ШАГАЮЩИХ АППАРАТОВ С ОДНОПОДВИЖНЫМ КОРПУСОМ
И.В. Войнов, А.И. Телегин, Д.Н. Тимофеев
Южно-Уральский государственный университет, филиал в г. Миассе
Рассматриваются модели шагающих аппаратов (ША) с одноподвижным корпусом, кинематические схемы которых позволяют создавать ША с максимальной удельной грузоподъемностью и минимальным энергопотреблением приводов на реализацию заданного перемещения корпуса. Получены уравнения динамики (УД) таких ША в трехопорном состоянии. Эти УД в явном виде содержат кинематические, геометрические и инерционные параметры ОША-Ы, где N - любое натуральное число больше пяти. Количество математических операций в полученных УД минимально. УД представлены в двух видах: во-первых, в виде системы дифференциально-алгебраических уравнений, в которых дифференциальные уравнения содержат динамические реакции в опорных точках, а алгебраические - описывают геометрические связи опорных стоп с опорной поверхностью (ОП); во-вторых, в виде системы N дифференциальных уравнений второго порядка с исключенными реакциями связей. Формулы вычисления динамических реакций в опорных точках имеют максимально простой вид. Выведены формулы вычисления динамических реакций в опорных точках таких ША. Описаны алгоритмы решения задач динамики, возникающие при исследовании ходьбы рассматриваемых ША. Приведено четыре примера. В первом примере рассмотрен ША с одним силовым приводом.
Ключевые слова: шагающий аппарат, плоские модели, уравнения динамики, первая задача динамики, динамические реакции, движущие силы и моменты сил.
Статья выполнена при поддержке Правительства РФ (Постановление № 211 от 16.03.2013 г.), соглашение № 02.A03.21.0011.
Литература
1. Охоцимский, Д.Е. Механика и управление движением автоматического шагающего аппарата / Д.Е. Охоцимский, Ю.Ф. Голубев. - М. : Наука, 1984 - 312 с.
2. Телегин, А.И. Структурный синтез и кинематический анализ плоских моделей шагающих аппаратов / А.И. Телегин // Вестник ЮУрГУ. Серия «Машиностроение». - 2008. - Вып. 11, № 10 (110). - С. 3-14.
3. Телегин, А.И. Алгоритмы выписывания уравнений динамики плоских шарнирных механизмов / А.И. Телегин // Вестник ЮУрГУ. Серия «Машиностроение». - 2009. - № 29 (205). -С. 4-12.
4. Телегин, В.А. Моделирование и анимация ходьбы плоских моделей шагающих аппаратов. /
B.А. Телегин, М.И. Кайгородцев //ВестникЮУрГУ. Серия «Машиностроение». - 2008. - Вып. 11, № 10 (110). - С. 15-23.
5. Телегин А.И. Общий и частные виды уравнений динамики систем абсолютно твердых тел / А.И. Телегин //Вестник ЮУрГУ. Серия «Машиностроение». - 2007. - Вып. 9, № 11 (83). - С. 3-13.
6. Телегин, А.И. Новые уравнения для решения задач динамики и синтеза систем твердых тел / А.И. Телегин // Вестник ЮУрГУ. Серия «Машиностроение». - 2006. - Вып. 8, № 11 (66). -
C. 3-14.
7. Телегин, А.И. Новые формулы для динамического силового анализа плоских рычажных механизмов. / А.И. Телегин // Вестник ЮУрГУ. Серия «Машиностроение». - 2007. - Вып. 10, № 25 (97). - С. 3-11.
8. Озол, О.Г. Теория механизмов и машин / О.Г. Озол; под ред. С.Н. Кожевникова. - М.: Нау-ка, Гл. ред. физ.-мат. лит., 1984. - 432 с.
9. Динамика машин и управление машинами: справочник / В.К. Асташев, В.И. Бабицкий, И.И. Вульфсон и др.; под ред. Г.В. Крейнина. - М.: Машиностроение, 1988. - 240 с.
Войнов Игорь Вячеславович, д-р техн. наук, профессор, директор, Южно-Уральский государственный университет, филиал в г. Миассе; mail@miass.susu.ru.
Телегин Александр Иванович, д-р физ.-мат. наук, профессор, декан электротехнического факультета, Южно-Уральский государственный университет, филиал в г. Миассе; teleginai@susu.ru.
Тимофеев Дмитрий Николаевич, аспирант, лаборант, кафедра автоматики, Южно-Уральский государственный университет, филиал в г. Миассе; goshanoob@mail.ru.
Поступила в редакцию 7 марта 2017 г.
ОБРАЗЕЦ ЦИТИРОВАНИЯ
Voynov, I.V. Mathematical Modeling of Walking Machines with One-Axis Body / I.V. Voynov, A.I. Tele-gin, D.N. Timofeev // Вестник ЮУрГУ. Серия «Компьютерные технологии, управление, радиоэлектроника». -2017. - Т. 17, № 2. - С. 65-82. DOI: 10.14529/ctcr170206
FOR CITATION
Voynov I.V., Telegin A.I., Timofeev D.N. MathematiCal Modeling of Walking MaChines with One-Axis Body. Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2017, vol. 17, no. 2, pp. 65-82. DOI: 10.14529/CtCr170206
