Computer Graphics & Geometry
Scheduling of purposeful motions of keying units in the organized mediums on the basis of the analysis of instantaneous positions
F. N. Pritikin, S. A. Kuznetsov, V. N. Yarovoj
Omsk state engineering university
prit@omgtu.omskelecom.ru
Abstract: The diversities of points in space of instantaneous velocities of change of generalized coordinates obeying of bypass by points of an operating mechanism of the keying unit of given hindrances are explored. The procedure is grounded on usage of instantaneous reduction ratios for points of the opened kinematic chains of robots on given directions.
Key words: Intellectual robots, build-up of motions of keying units, restricted work envelopes.
1. Introduction. At projection of technological processes, bound with handling of surfaces or drawing of coverage’s and robots, executed by handle systems, one of problems is the build-up of driving of an output link on given trajectories. The functions of change of generalized coordinates j
n=fn(t) in some cases are necessary for determining in view of an arrangement of enclosing inventory and job surfaces, which appear in a role of hindrances [1,2]. If on any of sites of a trajectory there are deadlock situations the given operating mechanism is not capable to fulfil given motive problems. For clearing up of motive opportunities of the keying unit it is necessary to determine on each step of calculations a vector of instantaneous velocities of change of generalized coordinates obeying to the given requirements. In a article the results of researches under the analysis of instantaneous positions of the keying unit obeying of driving of points of an operating mechanism, with the purposes of bypass of given hindrances and security of migration of an output link on a given trajectory are reduced.
2. Definition of instantaneous reduction ratioes between linear velocities of points of links of the mechanism and velocities of a modification of generalized coordinates. Position six-linked of a flat kinematic chain of the keying unit, restricted work envelope Ð and trajectory f of driving of an output link (fig. 1) let is preset. At calculation of a vector QM of instantaneous velocities of change of generalized coordinates relevant to measure of minimization of volume of driving [3], the instantaneous trajectory of migration of a point C intersected a restricted work envelope Ð. In this case on driving of a point C is necessary to overlap a requirement, such, that a component V ac of a vector of terrain clearance velocity of a point C should be or is more than zero, either is equal to zero, or should reduce a negative value (where, the component of velocity V ac is put a side on direct a, perpendicular to a hindrance Ð). The terrain clearance velocity of a point C here is determined only by instantaneous velocities of change of generalized coordinates j
1, j
2 and j
3. For calculation of the contribution of each velocity j
i in a component of velocity V ac we shall consider a frame O xa ya (the axes O xa is parallel direct a). The transition from a frame O xaya to a system Oxy is characterized by a corner a
, and
cosa
=
Ux·
Uxà , (1)
Fig. 1. To calculation of instantaneous reduction ratios for a point C on a direction of direct a.
where Ux and Uxà - guiding unit vectors of an axes Ox and direct a. For definition of instantaneous reduction ratios of velocities j
1, j
2, j
3 and component Vñ à it is necessary to calculate matrixes M 01 a, M 02 a and M 03 a in a system O xa ya. Thus the devices of a matrix M 01 a are calculated on a corner j
1+a
. The matrixes will be defined by expression M 02 a = M 01 a ·
M 1,2, and M 03 a by expression M 03 a= M 01 a ·
M 1,2·
M 2,3. The transformation matrixes M 1,2 and M 2,3 are calculated on corners j
2 and j
3. Using known relations [4-5], we shall define instantaneous reduction ratios Jj
1 a, Jj
2 a and Jj
3 a, which geometrical meaning is clear from a fig. 1. Using coefficients Jj
1 a , …, we shall note in an analytical aspect a requirement superimposed on driving of a point C, when Vca ³
0:
Jj
1 a j
·
1 + Jj
2 a j
·
2 + Jj
3 a j
·
3 ³
0 (2)
The inequality (2) in space of instantaneous velocities of change of generalized coordinates determines two areas Q a and Q a 1 assigned by a hyperplane, which equation is obtained at equality of expression (2) zero. Let’s mark, that the points of space of instantaneous velocities of change of generalized coordinates which are not obeying to expression (2), determine instantaneous positions of an operating mechanism, at which point C is gone in a direction of a restricted work envelope. Therefore, at calculation of a vector QN of instantaneous velocities of change of generalized coordinates, it is necessary to consider, that the point N belongs to a ð-plane G and is in area Qà, defined inequality (2). The ð-plane G is set by a linear system of the equations, the defining dependence of velocities of capture on velocities of change of generalized coordinates [5] (as dimension of a ð-plane for a viewed case will be equal n - r = 3, in a further ð-plane G we shall term a 3-plane):
QN = J*Vr, (3)
where n - number of generalized coordinates of the mechanism, r - dimension of a vector Vr (the migration is carried out on a two-component vector Vr (Vx ,Vy)), J - matrix of private reduction ratios [5].
3. Superposition of geometrical requirements on driving of points of links flat
six-linked of a kinematic chain and analysis of instantaneous states of the keying unit. Let’s consider (fig. 2) links BC of the keying unit and area of possible positions of its points D
ÂÑ at an instantaneous position characterized by an instantaneous center of rotation (i. c. r.) by a point ÎÂÑ and instantaneous angular velocity w
ÂÑ. The area D
ÂÑ is constructed on the basis of map of points of a link by bound circles with center by incident i. c. r. These circles determine possible trajectories of driving of points of a link of the mechanism ÂÑ in view of gauge of map h, depending from the module of terrain clearance linear velocities of points of a link, at a given instantaneous position QM. At intersection of area D
ÂÑ with a restricted work envelope on the basis of analytical calculations we shall define parameter m
, assigning a point K on a link ÂÑ (fig. 3), the possible trajectory of which driving first intersected a hindrance Ð, if the analysis of possible trajectories of points to spend in the order from a point B to a point Ñ. If at an instantaneous positions QM of area D
ÂÑ and Ð are intersecting, it is necessary in a 3-plane G of space Q to define instead of a point Ì a point N (given vector QN), at which the given intersection will miss. As at change of a position of a point N in a 3-plane G any instantaneous states of the mechanism, a point K can be shaded slide necessarily on direct t 1 k and t 11 k permitting to bypass a restricted work envelope can be considered. It is necessary only to fulfil a requirement, that a vector of terrain clearance velocity of a point K has not appeared in area q
Ê p. In this case point K will be shaded slide outside of area q
Ê p, defined direct t 1 k and t 11 k. The vector of terrain clearance velocity of a point K also will be determined only by instantaneous velocities of change of generalized coordinates j
1, j
2 and j
3. If to consider a point K center “of spurious capture”, for the given point we can define (on, procedure circumscribed above) of the instantaneous reduction ratio Jk between the velocities j
i and components V b k, V d k, where V b k ^
t 11 k, V d k ^
t 1 k .
For definition of instantaneous reduction ratios between velocities of change of generalized coordinates and component V b k, it is necessary for length of the third relative frame link to accept not length of a cut ÂÑ, and length of a cut ÂÊ, that is to consider, that a point K becomes “by temporal center of an output link”, for which the matching components V b k and V d k of a vector of terrain clearance velocity VK of a point Ê.
Fig. 2. Relative positions of a restricted work envelope Ð and area D
ÂÑ relevant to a vector QM.
Then by instantaneous reduction ratios describing the contribution of each instantaneous velocity of change of the generalized coordinates j
1, j
2 and j
3 in a component V b k there will be distances Jj
1 b, Jj
1 b , Jj
1 b (fig. 3). Similarly we can define instantaneous reduction ratios for a vector V d k. The requirement not inhering of a vector VK of area q
KP can be defined with the help of inequalities reflecting dependence component V b k, V d k from velocities of change of generalized coordinates j
1,j
2 and j
3 as follows. If component V b k and V d k of a vector of terrain clearance velocity VK of a point K will more than zero or be equal to zero, then
Jj
1 b j
·
1 + Jj
2 b j
·
2 + Jj
3 b j
·
3 ³
0 (4)
Jj
1 d j
·
1 + Jj
2 d j
·
2 + Jj
3 d j
·
3 ³
0
where Jj
1 b, Jj
2 b, Jj
3 b - distance from centers of kinematic pairs or points O, A and C up to direct b, which is perpendicular by tangential to a hindrance Ð and transits through a point Ê. Jj
1 d, Jj
2 d, Jj
3 d - accordingly distances from point’s O, A and C up to direct d.
Fig. 3. To definition of instantaneous reduction ratios for a point K a link ÂÑ of the mechanism on a direction direct b ^
t11 k.
These coefficients of the equations is determined on dependencies:
Jj
i = | U b´
( ri - rK )| / | U b | , (5)
where U b - guiding unit vector by tangential direct b or d; ri - position vector of center i-th of a kinematic pair, rK - position vector of a point K, through which transit direct b and d. It is necessary to mark, that the sign of coefficients Jj
i will be determined by an arrangement of center of kinematic pairs O, A and C rather direct b and d. Each expression (4), equal to zero, also sets a hyperplane, which disjoints all space of instantaneous velocities of change of generalized coordinates Q on two parts. If even one inequality (4) is executed, the point K will be shaded slide outside of area q
ÊÐ. As the point N obeying to a requirement (4), should have the least removal from a point M, it is necessary in the beginning to find 2-planes q and q / dimensions equal two, as intersections of a 3-plane G (3) with hyperplanes (4). For definition of 2-planes q and q / it is necessary in turn to solve jointly linear system (3) in the beginning with the first expression of a system (4), equal to zero, and then - with second. The obtained 2-planes q and q / also set points, which ensure a prescribed motion of an output link and migration of a point K on direct t 1 k and t 11 k. Then in 2-planes q and q / it is necessary to find points N / and N / /, which will be determined by intersection of hyperplanes S
and 2-planes q and q /. The hyperplanes S
transit through a beginning of coordinates of a system Îj
1j
2j
3j
4j
5 and are perpendicular to hyperplanes (3). The procedure of definition of these planes was explained in an article [1]. From points N / and N / / it is necessary to select the point obeying (4) and having least removal from points M. Let given point is the point N /. Then if the area of possible positions of points of a link D
ÀÂ, relevant to a point N / intersected also hindrance Ð, it is necessary in an environ of a point N / for a 3-plane (3) and in areas (4) iterative fashion to find such point N at which the area D
ÀÂ does not intersect a restricted work envelope. If the point D of a link DÑ (see fig. 3) contacts to a restricted work envelope, the number of unknowns j
·
i of and coefficients Jj
i in the equations (4) will be equal to four:
Jj
1 å j
·
1+ Jj
2 å j
·
2+ Jj
3 å j
·
3+ Jj
4 å j
·
4 
0 (6)
Thus coefficients Jj
i å is defined similarly on the basis of definition of instantaneous reduction ratios between velocities j
i and component Vd å (see fig. 3) (e ^
l). Let at an instantaneous position, defined by a vector QÌ area D
ÂÑ of a link BÑ and area D
CD of a link of a CD simultaneously contact to a restricted work envelope. Calculation of areas, defined by the equations (4) (6) in space of instantaneous velocities of change of generalized coordinates in this case is necessary. The point N now should obey to inequalities (4) and (6). For definition of the indicated point N it is necessary to find the equations of a 2-plane q 1, q 2 and q 3 dimensions equal 2, on which the 3-plane G (3) will be intersecting with hyperplanes (4) and (6). In 2-planes q 1, q 2 and q 3 it is necessary to find points N 1, N 2, and N 3 having the least removal from a beginning of coordinates of a system Oj
1,j
2….j
n. For this purpose it is necessary to discover crosspoints of 2-planes q 1, q 2 and q 3 with hyperplanes S
(see. above). From these points it is necessary to select such, which will obey to inequalities (4), (6). The vector magnitude assigning the given point in the space Q, should be least that is the given point should have the least removal from a point Ì.
4. Algorithm of build-up of motions of the keying unit on the basis
of the analysis of instantaneous positions in view of a position of a restricted
work envelope. The common algorithm of build-up of motions of an operating mechanism of the keying unit on the basis of the analysis of instantaneous positions of the mechanism is shown in a fig. 4. The offered algorithm of the analysis of the information about a possible position of points of links of an operating mechanism of keying units can at adjusting driving simultaneously take into account some plants of hindrances. Thus usage of a variable value of a vector magnitude Vr allows considerably to reduce an aberration of an output link from a given trajectory and to increase exactitude of positioning. The algorithm enables to accept the solutions on adjusting driving beforehand, how the operating mechanism will achieve plants of hindrances. In a fig. 5 the results of calculation of test examples are reduced.
Thus, the searching of instantaneous positions obeying to a requirement of driving of capture on a given trajectory and a requirement of bypass with points of links of the mechanism of restricted work envelopes, can be by an essential fashion simplified. It is explained to that there is no necessity of exhaustive search of major number of points N, inhering p-plane, for the analysis of a position of areas of possible positions of points of links of the keying unit and plants of hindrances in working space. The offered method of searching of instantaneous positions of an operating mechanism can be used in automated systems of build-up of motions of the intellectual robots executing the motive jobs in organized mediums.
Fig. 4. Algorithms of build-up of motions on a given local trajectory of an output link on the basis of the analysis of instantaneous positions: 1 - definition of a vector QÌ; 2 - definitions of areas D
k (k - number of a link of the mechanism); 3 - definition of a requirement of intersections D
i and Pj (j - number of a hindrance or restricted work envelope); 4 - definition of parameter m
, assigning the first point, contacting to a hindrance, k-th of a link; 5 - definition direct b, d and etc. (see fig . 3); 6 - definition of instantaneous reduction ratios Ji b , Ji d and etc. the equations (4)(6) on directions direct b, d and etc., incident point K and vectors V b k and V d k … and etc.; 7 - definitions of areas Qà ... in space Q; 8 - definition of crosspoints N / and N // ð-planes G (3) with hyperplanes (4)(6) and S
; 9 - analysis of a position of points N / , N // ... in relation to area Qà and definition of a point NP with minimum distance up to a point Ì (the registration of a requirement of minimization of volume of driving) is necessary; 10 - build-up of areas D
K on the basis of an instantaneous position, defined by a vector Q N; 11- all possible values of a point N ðàñ in an environ of a point Np are surveyed; 12 - calculation of a new point N ðàñ of a p-plane G in an environ of
a point Np; 13 - diminution of a vector magnitude of velocity of capture Vr and gauge h; 14 - a vector magnitude of velocity of driving of capture Vr and gauge h accept a minimum value; 15 - build-up of the following configuration; 16 - Npac = NP; 17 - kinematic chains in a deadlock situation.
a
b
c
Fig. 5. Results of simulation with the help of a computer. a - migration by measure of minimization of volume of driving without the registration of presence of a hindrance; b - migration on the basis of prime exhaustive search of points of a ð-plane G; c - migration of the keying unit in view of dependencies (4)(6).
References
[1] Kobrinsky A.A., Kobrinsky A.E. Handle systems of robots. M.: Science, 1985. p. 344
[2] Malishev V.A., Timofeev A.V. Algorithm of build-up of program motions of keying units in view of a design constraints and hindrances // Information’s AS USSR. An engineering cybernetics. 1978. No. 6. p. 64 - 72.
[3] Pritikin F.N., Tevlin A.N. A method of build-up of motions of the keying unit on a given local trajectory of capture at presence of hindrances. // Mashinovedenie. 1987. No. 4. p. 35 - 38.
[4] Pritikin F.N. Definition of instantaneous states of an operating mechanism of the keying unit providing bypass of a given hindrance. The analysis and synthesis of mechanical systems. Omsk: publishing house OmSEU. 1998. p. 94 -97.
[5] Korendyasev A.I., Salamandra B.P., Tives P.N. Definition of number of degree of freedoms of the executive organ of the industrial robot. // Mashinovedenie. 1985. No. 6. p. 44 - 53.
Computer Graphics & Geometry