SU1583726A1 - L-coordinate spacial mechanism - Google Patents

Abstract

The invention relates to mechanical engineering and can be used, for example, in robotics. The aim of the invention is to increase the speed of the mechanism. The coordinate spatial mechanism contains a base 1, having from three to six spherical supports 2, a movable link 3, having five spherical supports 2 and six tons of 4, made with the possibility of changing their length and connecting the base 1 with the movable link 3 by means of spherical hinges 5, as well as the distance meter 5 between the centers of the spherical supports 2, the base 1 and the centers of the spherical supports 2 of the movable link 3. The number of points on the base 1 in the mechanism varies from three to six. 13 il.

Description

FIG. one

The invention relates to mechanical engineering and can be used, for example, in robotics.

The purpose of the invention is to increase the speed of the mechanism.

FIG. 1-12 various schemes of the proposed coordinate mechanisms are shown; in fig. 13 is a structure of a 1-coordinate mechanism, for which the determination of the coordinates of points of a moving link is carried out in an explicit form.

All schemes of 1-coordinate spatial mechanisms contain a base 1, having from three to six spherical supports 2, a mobile link 3, having five spherical supports 2 and six tons of 4, made with the possibility of changing their lengths and connecting the base 1 with the mobile link 3 by means of spherical hinges 5, as well as a measuring instrument 6 distances between the centers of the spherical supports 2 of the base 1 and the centers of the spherical supports 2 of the moving link 3.

The mechanisms shown in FIG. 1-12, differ in the number of base points 1 in which the centers of the spherical supports 2 are located. The number of points on the base 1 in these mechanisms varies from three to six. These mechanisms realize their own, different from others, 1-coordinate structure.

When controlling the motion of 1-coordinate spatial mechanisms, it is necessary to solve the direct and inverse position problem. The direct problem is reduced to determining the position of the moving link 3 by given values of the generalized coordinates. In the 1-coordinate mechanisms, the generalized coordinates are the lengths AND, 12, ..., b, six t, g 4.

The inverse problem involves the definition of circumscribed coordinates And, the mechanism for a given spatial position of the moving link 3,

A general method for solving a direct problem is based on the analysis of a system of equations, each of which establishes the dependence of the distances between two points on their coordinates. Moreover, in the general case the number of equations of the system is equal to the number of points of the moving link 3, in which the spherical supports 2 are located, multiplied by three, since each point on the moving link 3 is characterized by three coordinate values - x, y, 2; If the number of points on the moving link 3 is equal to five, the number of communication equations is equal to fifteen.

For the mechanism shown, for example, in FIG. 12, such a system has the form:

(Ha - XA) 2 + (YA - YA) 2 + (za - 2e) 2 H2,

(ha (x (Xc (Xd

(Heh (ha (ha (Hs (Hs

0 (heh (ha (xb (xs

(x 5,

-XB) 2 + (Va - HC) 2 + (Za - ZB) 2 I22.

- xsH + (y - usg + (zb - zcr 3 ,.

-XD), + (Us - PP) 2 + (2s - ZD) 2 -.

-XE) 2 + (yd - YE), + (Zd -ZE) 2 I 52 XF) 2 + (Ye VFj 2-4ze - 2F / I 6 -Hj12 + (Yah-HG „+ (Za-Zb) 2 ( ab) 2. -HeG + (Ya-hG + (us-xdf + (Us-Xd), + (Us-Xdf + (Wa

-He) 2 + (UH-Xe) 2 + (Us

-Xd) 2 + (yb

ye) 2+ (za-ze) 2 (ae) 2

/ „/„ Uv

yb) + (zc-zbf- (cb /, yd) 2 + ((Cd) 2 yd) 2 + (ze-zd) 2 (ed) 2 -yd) 2 + (za-zd) 2 (ad) 2

-E) 2 + (Zb-Ze) 2 (be) 2, EE + (Zc -ze) 2- (Ce) 2

(one)

- УЕГ + -Vdf +

() 2 (

The solution of such a system of equations is possible only by numerical methods, which takes a long time.

But there are a number of structures for which the direct problem is solved in an explicit form and 0 is reduced to the successive solution of three systems of equations, each of which contains three equations each. FIG. Figure 13 shows the structure for which the direct problem is solved explicitly and the coordinates of the 5 points a, b, and c are determined by the successive solution of the following systems:

1. To determine the coordinates of a point, the system of equations has the form:

(ha - xd) 2 + (Wa - beat) 2 + (za - 2d) 2 I 12, (Xa - XB) 2 + (We-HC) 2 + (Za- ZB) 2 I 22, (2) ( ha - XC) 2 + (YA) 2 + (z a - ZC) 2 I З2. This system, consisting of three equations with three unknowns, is solved as follows: subtract the first from the second and third equations, we get:

(XB2 + Y B2 + ZB2) - (xd2 + YD2 + ZA2) - 2xA (XB-HD) - 2ua (YB - YA) -2Za (ZB - ZA) V -I G, (3)

(XC + CS + ZC) - (HD + UA + ZA) - 2xa (xc-XA) - 2ua (us - beats) - 2za (zc - ZA) h - We introduce the following notation:

XAV XA - xv, udv ud - uv, ZAB ZA - ZB, (4)

HSA CG - XA, VCA CS - UA, ZCA ZC - ZA

The XR + VR + ZFT: C2 xg + vr +

In KB + HC + ZB; WITH

A2 XA2 + UA2 + ZA2.

Substitute (4) into (3)

xc + us + zc:

B2-A2 + 2xA XAV + 2ua yAB + 2Za ZAB 2

-I2, (5)

C-A - 2x XA - 2ua usd - 2za ZCA h - -If

Express ha in za. To do this, both equations (5) are solved for ya and let's equate the right sides

B2 -A2 + 2ha-xAH -f 2za-ZAB-l22 + li2 XCA-ZAB - xdv-ZCA to YA -2 udv XA-yyr - XAV-usd

G2 A2 9x. Hgd 97, 7g. ,, 2,, 2 Substitute (9} in (7) and (8), we get XCA 2Za ZCA1з + Iha "1-Кзга, (10)

USA5ya- K2-K4Za

We substitute (10) into one of the equations (2), half B2-A2-l22 + li2 + 2x-Xdv + 2za ZAB:

2UVA (Kl-K3Za-XA) 2 + (K2-K4Za-yA) 2 + (Za-ZA) ll

(eleven)

 C - A2 - 2x -Chod - 2za-.ZcA - 1z 2 + li 2 W. Transform (11), we obtain a square equation with respect to Za

Z aW + K42 - U - 2Z (K1 Kg + K2 K4 - KZ XA - K4UA

whence + ZA) .t (Kg + K27 + XA + UA + ZA2-2Ki XA - 2fe beats (B2-A2) usA- (122-112) usA + 2xHAUS + l) °, ..

+ 2Za ZAB UCA - CHS2 - A2) SAW + (32 -) SAW +15 (, g ",", ","

2x CSA UAV - 2 ZCA UAV, (6) W | DK24 G G $ Z

2Xa (xSA - SAW - XAV UASA) (B2 - A2) UCA -2Za + A 1 2XA) + K2 ((2

022-112) UIA + (C2-A2) UAV-032-112) UAV-20

-2zA (zcA UAV - ZAB UCA), we denote:

Кз2 + К42 + 1 mi Увс (А2 - И 2) + usd (В2 - I2 2) + УАВ (С2 - 1з 2).

ZSKSA-UAV-XAV-UBA) jfKi-XA) + K (K2 - beats) + ZA t2, (13)

25 A2-112 + K1 (K1-2xA) + K2 (K2-2uA) tz

, 12АААВ - АВ АСА.

CSA-SAW -hdv-usA Then the solution of equation (12) with regard to (13)

has the form:

where is uv uvus. (7)

Similarly, we express ya in Za. For 30 tz ± Vm2 2 - 4pttsts / 14

of this, both equations (5) are solved as a-2mi

ha and equate the right parts

 B2 -A2 + 2ua-uAV + 2za-2AB -l22 + li 2 By substituting (14) into (10), we find the value

ha-2khdv ha and ya- Thus, the coordinates of a

35 identified. 2 D2 9 - about. | 2 | 22. To determine the coordinates of point b

Xg; r: .S 22Уа УСА- Za ZcA- | з-Zlll, it is necessary to solve the system of equations

2xA (xb - ha) 2 + (xb - ya) 2 + (zb - za) 2 (ab) 2,

.2 A2, /, 2. 2ch (Xb-XD) 2 + (yb-yD) 2 + (zb-ZD) (15)

(B2 - A2) xd - (122 - YUHSA + 2U | udv XSA +40 (xb - xE) 2 + (yb - ye) 2 + (zb - zE) 2 Is2 2za ZAB XCA - ChS4 - A) xdv + ( h - I) ХАВ + 2ua similar to the system (2). ХАВ УСА + 2Г хдв ZCA.3. To determine the coordinates of the point

whence 2 22 .., j. 2 need to solve the system of equations

Wa HVS (ACh12 (E (Xc-xa) 2 + (yc-ya (zc-yjf (caf.

хсД.гдВ-хДв (У 45 (хс-хь) + (yc-yb (zc-zb) 2 (cbf, (16)

I XCA-UAV-XAV-UCA w (xc - xyy + (mustache - ugg + (zc - zpf le,

where xvs xv - xs. also similar to system (2). Decision

The designation of these systems is carried out explicitly, and the UVS (A2 - AND V usdSV-UUAvSSU-b2) takes much less time than the re-2 (xcA-uAB-XAV-usd fe ° shenG system (1}

 The increase in speed of the mechanisms shown in FIG. 1-12, it is possible to bake 2 2, 2 hours, 2 hours by equipping them with two xs (A) + XCA (P 2 J t hdv (. 1h) with 6 distance measurements from the centers of spheres- 2 (xcA UDV XAW UHA55 of supports 2 bases 1 to centers

 2-spherical bearings 2 moving link 3.

6 distance meters in each of

ZCA UDV ZAB UCA. these mechanisms are located as follows. what's the axis

xsA Udv - xdv Usdf. The gauges 6 together with the axes of the grades 4 form a l-coordinate structure for which the solution of the direct problem is carried out in an explicit form. In these mechanisms, three pyramids can be distinguished and, consequently, using equations (2), (15) and (16), to find in real time the position of the moving link 3 in space, since it is known that in space the position of a solid body is determined coordinates of its three points not lying on the same line.

For the mechanism shown in FIG. 1 — pyramids aABC, LVAa, dCab, in FIG. 2 — pyramids aABC, LVAa, dBab, in FIG. 3 — pyramids of aABC, LVAa, dAab, in FIG. 4 — pyramids of aABC, eBC, dBae, in FIG. 5 - pyramids aABC, LVAa, dCab, in FIG. 6 CBA, dAab, in FIG.

ecaa,

7

dAae, in FIG. 8 ÁBAA, dBab, in FIG. 9 LVAa, dcab, in FIG. 10 - bBAa.-dDab, BBAa, dCab,

in fig. 11 — FIG. 12 pyramids aBCD, pyramids aACD, pyramids aACD, pyramids aABD, pyramids aABD, pyramids aEBD, pyramids aABF, cCDa, dEac.

To find the coordinates of the other two points, it is necessary to solve two similar systems of equations (2-16).

The mechanism shown, for example, in FIG. 2, works as follows.

When using the mechanism as a manipulator, it is necessary to determine the position of the moving link 3 in space according to the sensor readings of length m 4 and meter 6 readings. According to the algorithm specified above, the coordinates are calculated in the manipulator control system

points a, b, c, characterizing the spatial position of the moving link 3. In this case, the direct problem of the position of the mechanism is solved.

The proposed 1-coordinate mechanism has a higher speed, which increases the efficiency of the control process and expands the scope of application of such mechanisms.

Claims (1)

The invention is an L-coordinate spatial mechanism comprising a base having from three to six spherical supports, a movable link having five spherical supports and six tons of gram made with the possibility of changing their length and connecting the base with the movable link by means of spherical hinges, with this axis of all The plugs pass through the centers of the respective hinges of the movable link, characterized in that, in order to increase the speed of the mechanism, it is equipped with two measures of the distance between center of one of the base hinges to the centers of two corresponding hinges of the movable link, or between two centers of the hinges of the base to one or two corresponding hinges of the movable link, with distance gauges placed between the hinges that are not connected by that each hinge of the moving link passes one, two or three axis tg and / or gauges. FIG. g PhieZ FIG L Fig.8 FIG. 9 Phases. AND ABOUT  / 77777-U XZ Editor N. Tupitsa Compiled by V.Zhilev Tehred M. Morgental Phys.Sh.Fig. 12 FIG. /five Proofreader S. Shekmar