Inertia forces of robots and manipulators

klectTanism rune M41chine Theory, 1977, Voi 12, pp. 387-401, Pergamon Press. Printed in Groat Britain

Inertia Forces of Robots and Manipulators M. S. Konstantinovt Abstract Design of remote manipulators and industrial robot/manipulators requires the derivation of the inertia forces. They determine the forces produced on the arm, and consequently, the actuator torque and arm structure. A technique is developed for the determination of the dynamical forces with the help of point mass models. To this end, a generalized arm structure is introduced. This structure is decomposable in typical units, which represent the elements of a modular design and assembly system for robots and manipulators. Suggestions are made by two examples to assist the designer in obtaining good results. Introduction

P~UOTE manipulator units with suitable viewing schemes and specialized end-effectors can significantly extend the operator at the control station in performing a wide variety of activities. Existing manipulators are sometimes classified as being either "mass-handling" manipulators, or "man equivalent" manipulators. Today, the requirements for the manipulator system have been stated in both broad and simple terms. The broadest terms fall into three categories of functions when remotely manned aerospace systems are considered: payload handling, docking assistance and servicing. The payload handling problem is in associating specific performance characteristics of the manipulator in the face of incompletely understood payload characteristics. The static precision characteristics depend on the ability of the operator to compensate for dynamic effects, whereas passivation of inertia forces is required to position and align free-flying payloads for attachment in the cargo bay. Obviously, the requirements for accelerating or decelerating a load attached to the arm is one of the most significant design parameters of the remote manipulator units. It determines the forces produced on the arm, and consequently, the actuator torque and arm structure. Thus, according to Smith and DeRochcr[1] the factors to be considered for stiffness in manipulator design arc: dynamic interaction with orbiter, arm weight, maximum allowable payload velocity and handling clearance. Industrial robots are used to replace human operators engaged in repetitive tasks, while industrial robot/manipulators are employed for heavy materials handling. At a work (loading) station an industrial robot (robot/manipulator) is usually programmed to repeatedly perform the task of picking up an object from a location (receptacle, pile, conveyor etc.) and placing it to another location (machine fixture, pile, lorry etc.). Robot handling time is defined as the time elapsed between the completion of grabbing the object to the time at which it is released[2]. The problem of reducing robot handling time is an important economic consideration. As a matter of fact, the problem could be attacked from many aspects and a number of factors should be considered, such as the configuration of the manipulator's degrees of freedom, simultaneous joint movements, transient responses associated with the servomechanism, dynamic forces. Therefore, the design of a manipulator which satisfies various combinations of positions, velocities, accelerations, force and torque transmissions, stresses, strains etc. at a predetermined operating speed, requires the derivation of the inertia forces. The manipulator members are assumed to be rigid and are considered as having distributed masses. The reference variable is time. tCentral Laboratoryfor Manipulators and Robots,Higher Institute of Mechanicaland Electrical Engineerin$, Sofia, Bulgaria. 387

388

Manipulators are in fact spatial mechanisms. Such a mechanism, in essence, consists of a system of rigid bodies interconnected through a series of externally imposed geometric constraints. The description of a "mass-handling" manipulator in motion with a load attached to the arm inevitably involves complicated and lengthly terms, due to the complexity of the Newtonian dynamic equations. These terms are hardly susceptible for geometric interpretation and are the cause of time-consuming operations in digital computation. An analytical approach to the study of kinetics of spatial mechanisms with closed kinematic chains was reported recently by Yang[3]. The formulation of the method is broad based and general in scope so that it is useful to the study of dynamics of rigid members in any complex mechanical system, inclusive manipulator arm/hand systems. The bearing reactions and inertia torque are obtained in closed form dual-number expressions. Such concise analytic expressions offer better geometric insight and are well adapted for digital computation. Our search for a simple formulation of the kinematic specifications referred to the mass of a rigid body (solid) led us to the study of Reye[4]. Indeed, he proves in an original way that a solid performing a spatial motion (with 6 degrees-of-freedom) is equivalent with respect to its centrifugal force (the force due to its angular velocity) to a discrete model having a minimum of four concentrated masses. The refined topological investigations of Reye were partially supplemented by his followers. It was demonstrated in [5, 6] that such a discrete model is equivalent to the solid also with respect to the inertia forces due to the angular acceleration. It was proved in a recent paper [7] that it is convenient with regard to the dynamical investigation of spatial mechanisms to replace the mechanisms (manipulator) links by point mass models. This paper, which is a continuation of two previous papers [8, 9], deals with the simulation of the dynamical behavior of a loaded manipulator arm system in motion, It attempts through the use of dynamical point mass models and the determination of the inertia forces a comprehensive study of the dynamics of manipulator arms in any complex mechanical robot system.

The "Matador" Moduhir System Let us consider the basic structure of a manipulator arm system as shown in Fig. l(a). It is notable that the links of this structure are not limited in number. The joints having axes parallel to the coordinate axis x are cylindrical (C) joints, and these with axes parallel to the coordinate axis : are revolute (R) joints. Therefore, relative rotation ~0.... , = ~0, is possible about the axes specified by uneven indices (s = 1.3, 5...) and two elementary relative motions, (a) rotation

\

.~

~ . . ~ 5 \

Zs'Zs-1

)'s

Rgures l(a, b).

~" ,

1

,..

389 @w j_, = @j and (b) straight line translation Sjj-, = h~-,, are possible about the axes denoted by even indices (/" = 2, 4, 6...). Additional constraints in the joints lead to the partial structures (configurations) inherent to the mono-arm and single open kinematic chain systems of all known robots and manipulators of this type. The graph of this structure is plotted in Fig. l(b). Let us remember that we can define every kinematic chain in terms of its interchange graph, with each link represented by a vertex, and each joint by an edge. Every such interchange graph is finite, and is connected, having no loops, nor multiple edges. For further discussion of modern graph theory and terminology one may refer to the references item [10]. The graph of Fig. l(b) could be decomposed in partial graphs, which are duals of the kinematic chains I, 2 (see Fig. l(a)) a n d s - I, s and s, s + l (see Figs. l(a) and 2). Thus, we have shown respectively in Figs. 3(a--c) the decomposed structure graphs and in Figs. 3(d-f) the interchange graphs of the former, as they were named by Crossley[10]. Wherever the former has an edge, its interchange has a vertex, and vice versa: the vertices of the interchange graph are joined by an edge when, but only when, the edges of the original are incident in a common vertex. In our further discussion we will use only the interchange graphs as defined hereinbefore. Reverting to Figs. 3(d-f), note that these interchange graphs are described by their edges as CCR, CRC and RCR interchange graphs, respectively.

Z$ :Z$. I

s+1

\

", ,

/~A--)

~'

/

Figure 2.

(o) I

(b) 2

S

(c) S-I

S

S-I-I

Figures ~a-~. It should be recognized, however, that complex open kinematic chains consisting of many four-bar linkages connected by quaternary links with revolute (R) joints, such as these discussed in Ref. [11], cannot be generated by additional constraints in the joints of the basic structure shown in Fig. 1(a). Examples of such complex chains are illustrated in Figs. 4(a, b). We may observe that the plane four-bar linkages I and 3 of these manipulator arm systems are degenerate forms of a RCCC linkage (Fig. 5a). In other words that is the inverted form of a kinematic chain obeying the Kutzbach single degree-of-freedom criterion[I2] for a tridimensional helical motion. The interchange graph of this RCCC linkage (Fig. 5b) could be decomposed into two interchange subgraphs CCR (Fig. 5c) of identical form. Note that the linkage interchange of these subgraphs is a partial kinematic chain of the Assur's type [13], and that this

39o

Figure 4(a).

~1 .......I

,,,',

(b)

Figure 4(b).

C!I i

¢

(b)

,

¢

co)

Figures 5(I-.C). partial kinematic chain is identical with the partial kinematic chain of the basic structure (Figs. l(a) and 2), covered by the interchange graph CCR (Fig. 3d). Clearly, the basic structure, shown in Fig. l(a), consists of partial kinematic chains which cover these of all hitherto known structures of manipulator (robot) arm systems. The most systematic method of designing a mechanical movement for any desired machine is by type synthesis. We have found that the simplest types of kinematic chains, covered by the interchange graphs CCR, CRC and RCR, are capable of producing the desired motion of any desired manipulator arm system. The synthesis is then to procure as complete a set as possible of the above partial kinematic chains, and select from this set those that fulfill further desirable characteristics of the manipulator arm system sought. Obviously the partial interchange graphs CCR, CRC and RCR in Figs. 3(d-f) are very appropriate, as basic units, to the composition of an arbitrary complex interchange graph, which is dual to an arbitrary complex manipulator arm structure. For instance, a set of CCR interchange graphs and the interchange graph of a RCCC linkage (see Fig. 6a) could be joined

391

(a)

(b)

R

C.

C

R

C

Figures 6(a.-¢). together in a complex interchange graph (Fig. 6b). This connected graph is dual to the structures shown in Figs. 4(a, b) under the condition that all C and C. joints are deprived from sliding relative motion, i.e. become revolute (R) joints. Alternatively, when the edges C. of the above connected graph are inhibited, it can be traversed unicursally. Then, the unicursal form of this complex interchange graph (Fig. 6b) is dual to the structure shown in Fig. l(a), which interchange diagram is shown in Fig. 6(c). Note that the structure graph of Fig. ICo) is the dual to a mechanism, i.e. of a linkage attached to the frame by cylindric (C) joint, while the interchange graph of Fig. 6(c) is dual only to the kinematic chain without the frame C joint. Thus far, we have considered manipulators with single arm systems. An extension of the one arm system is to add a second similar or identical arm to the manipulator (robot). This concept permits assembly of two objects in space with neither stabilized except by the manipulator arms. The concept could be applied also to industry for assembly purposes. Moreover, for fastest possible handling of workpieces between three machining operations and an auxiliary device, industrial robots are fitted with three arms. Thus, several workpieces are being machined simultaneously which means a high degree of productivity. The concept of a multi-gripper hot forge manipulator was discussed in Refs. [8 and 14]. Obviously, the interchange graph of Fig. (,Co) should be extended to cover these concepts. By an approach through graph theory, we have found that the most complex multi-arms (multi-grippers) manipulator structure may be described in terms of its "Family Tree" inR

R

R

C

C

C

R

C

C

- -

~

C

A

(o1

Rgures 7(a, b)

R

R

R

R

C C

R

C

R

C

C

C

I

R

(b)

R

R

R

R

392

terchange graph, as plotted in Fig. 7(a) with reference to Fig. 6(b). Another form of this interchange graph, we call it the "'Poseidon Trident" interchange diagram (graph), is plotted in Fig. 7(b). The trident interchange graph takes the form plotted in Fig. 8 when referred to Fig. 6(c). However, the "Poseidon Trident" interchange graphs of Figs. 7(b) and 8 are duals to manipulator (robot) arm(s) structures. These duals are the interchange diagrams for the chains, such that for every link of the chain there is a vertex in the graph, and vertices are joined by an edge whenever the corresponding links are connected by a simple joint (R or C). It may at once be asked: might it be possible to use the partial kinematic chains CCR, CRC and RCR (see Fig. 6) as modular units of an arbitrary complex arm kinematic chain? One first needs assurance that the duals of the partial kinematic chains are the modules of the interchange graph, the "Family Tree" graph of Fig. 7(a). Thus, we arrived at the "Matador" concept[9], as a modular design and assembly system for all kinds of industrial robots and all types of manipulators.

R

R

R

R

R

R

R

R

R

o, 9 ,o o, 0 ,o o, 0 ,o C

C

R

C

C

C

R

c

C

C

c

R

C

R

0 c 1

Figure 8.

The Potnt t k t u Models The theory of point mass models was thoroughly discussed in Ref. [7]. Briefly, with regard to the dynamical investigation of spatial mechanisms it is convenient to replace the links of the mechanism by point mass models. For the determination of the inertia forces with the help of point mass models it is sufficient to know the paths and the accelerations of single points of the solid. Robots and manipulators mechanical arm systems represent spatial mechanisms with an open kinematic structure. For inertia force analysis, after the structure type synthesis, it is convenient to attach concentrated masses on the elements of the structure, i.e. to replace the rigid links by point mass models. Thus, we construct the physical model or the skeleton of the arm system. According to Reye[4], three postulates are necessary and sufficient for dynamical equivalence; (a) the mass of the solid and the total of the concentrated masses should be equal, (b) the center of gravity of the solid and of the discrete system comprising the point mass model should coincide and (c) the inertial ellipsoid of the solid and this of the discrete system should be equal. The above postulates are expressed by

• k=l

mk

=

m

(1)

393

(3)

k-I

where (4)

(5) (~£ZkXk

ff~Z,: J

f'12Zky,

The components

T~, = f(,)xy dm

T,,~ = f , , x: d m ;

(6)

of the inertia tensor {TT} are integrals of the inertia and centrifugal moments, respectively, of the link s (see Figs. 1, 2 and 9) in reference to the coordinate system x , B , y , z , which is rigidly attached to the link. Whereas, x~y, and z, of the tensor {tS~&}are the components of point K in the mentioned coordinate system, and G is the center of gravity of the link. The postulates (I). (2), (3) can be reduced to the following scalar equations ~mk = m

•

(7)

xkrnk = mxG;

k~l

y, m k = m y c ; kw!

(8)

2,mk = rnzc kml

xky, rn.k = T,,,.;

x,z

, = T,,~ ;

y~z~m, =

:

(9) ~.~ k~l

x,:mk =

.2

2

,2

2

T,~ = m,=: ~ , ~,'k m k = T , = m r , : ~ , zk m k = T,~ = mi:-,. k~]

k~l

Note that eqns (7) and (8) express the forces equivalence, and that eqns (9) express the moments equivalence.

Y ..,cs

Figure g.

rs

,",

394

Considering an infinite spatial motion of a solid having an arbitrary configuration, i.e. disposition of the masses, eqns (7)-(9) permit four point mass concentrations as a minimum dynamical model. Let B, = G and let x,, y,, z, be main inertia axes of the solid, then eqns (8) and (9) are simplified to

~x~mk=O;~ ykmk---O;~ zkmk=O

k~l

k-I

(Sa)

kit

~.~ xky~mk = 0; ~

x , z k m , = 0; ~ y , z , ra. = 0

(9a)

±

±

x~2mk = mi:~;

kl|

y~2m~ = mz,,,

kul

zk:m~

= miZ..~.

kul

We take four points (Fig. 4) in the coordinate system planes x , z , and y,z, and we locate any pair of these points in a way to be symmetrically spaced with regard to the coordinate axes x, and y,, so that all four points are equally spaced from the coordinate plane x,y,, therefore o

m,

; m2

0 ' m~ y

L--Z

m,

tZ

y" l

(lo)

Z

Following four point'model is obtained by introducing eqn (I0) in scalar eqns (7),(Sa) and (9a) m , = m2 = m3 = m , = O,25m

t

(il)

x = V'2i,=; y = V : 2 i , , ; z = i==.

W e may observe that: (a) the coordinates of all four points arc described by three scalar quantities x, y, z and (b) the point masses of all four points are equal. Evidently, only three accelerations are sufficientto determine the inertiaforces of the solid (Fig. 10).

m 4

m 3

A ~

Figure 10.

The Acoeie¢~uons Referring to F;g. 2, the relative disposition of the three adjacent links s - l, s, s + l inherent to the structure shown in Fig. l(a), can be described by the recurrence operator algorithm

395

(,{:::): (;-:)o,..;

(12) (A,)=~, \&.,:(£'+"}e",-'; £, = ,~,+. where the conditional term

expresses a complex vector. The vector coordinate of the reference point (pole) B,, which is a point of the solid s, is to be found from s-2

r, = ho;o÷ ~E h,£.

(t4)

$-I

Here, the unit vector of axis x, is ~s

(15)

Os -Os = a,,Ao+ a Os : ~-o + a3,vo

where ~o,/20 and ~o arethe unit vectors of the fixed coordinate system axes X. Y, Z. The matrix transformation between two arbitrary coordinate systems p and q attached to the corresponding links of the arm system structure, could be expressed by

[tj=,o,:

~qi Fall a21 a3l~

F*"p]

:,

(16)

where l>p>q~s

and coefficients a~, are calculated by the help of algorithm (12) for a complete variation of the indices p and q in the range zero through six, and are listed in Table 1. Turning to the positioning eqn 04) one could obtain the acceleration of pole Bs by double differentiation s-2

~. =/~o~o + ~ (/~,;[, + 2h,,~, + h,L).

(17)

Furthermore, the acceleration of an arbitrary point of the link (solid) s could be calculated by the help of eqns (16) and (17) as

F~k] F..o, -o, a,,z,~F 0"] ,_:~a,,h,+2a,,h,+a,,h/~ ajlXk

+ QI2Yk +

-os

"?',i .=la,,J"°*x~+ a,,y~"°" + a:,z,|+|O|+~|a:,h.+2m,h.+a=,h.|'°" L Z "", J

La,,x, *.os

+a,,.y,+a,z,J LhoJ "°"

..o,

"

,-,

os "

-os "

o. -

.o,"

-os

-o,

La,,h, 2a,,h,+,,,,h,_J. o,"+

.o,"

(;8)

..o,

Evidently, the eqn (18) could be applied to any of the points of the minimum dynamical model.

The Dynamic Forces As stated, the time required to perform a pick and place operation is defined as the time elapsed between the completion of grabbing an object to the time at which it is released. At this time the grabbed object and the output link of the arm system represent a solid, which is

396

Table 1. Matrix coefficients aenn p°

Pq

0..-1

t-2

0-2

1-3

Ct

l

C~

C~

0-3

I.-4

0-4

\ dll

C~¢3 -- S~C2S3

C~

- C~S 3 -- SaC:C~

--S3C,

S , $ : S ~ -- C~(C~S3 + StCzC3)

C~S~

StS 2 5tC~ * CiCz$3

S~$~ CzS~

S1$:C~ + S,{CtS ~ + SiC2C3) SiC~ + ClC~S3

CzC3

~51S3 + ClCaC 3

a12 t~13 d'2l

0

0

s,s~

t)

St

0

S~

ax2

¢t

Cz

CtC~

"/23 a3t

0

--Sx

-Ct$~

a]3

0

0

szs~

0

S2

Se

S~C~

l

C:

Cz

C2

°\

g/l| I;112

--$25 ~ * C2C~C 4

S~Sa 51C3 C~

SzS~ CzSa + SzC;C~ C~C, - 5:C3$~

1-5

52S3 Cz$~ + $2C3C~ ClC¢ -- S2C3S~

0-5

c~(ctc3 - stc2s3) + s s / s , s ~ s , - c , ( c , s 3 + stc2c3)l - s s ( c , c 3 - s,czs~) + c~/s,s~s, - c ~ c , s 3 + stCzCO/

C3Cs - $~C~$s -¢35s

--C13254 -- C~{SlS ~ -- CLC2C3)

-S2

0

\ oq

CtC~ -- 5,C2S3

- - S3C.¢~

S~S~C, + S,(CtS3 + S~CzC3)

53$~

Cz$3C~ - S s ( S x 5 . - ¢~C3C,) -¢zS3S~ - C~($~$, - ¢zC3¢~) -SxC~ - CzC3S.

c,(s,c3 + c , c ~ s 3 ) - s , / c , s ~ s * * c , ( s , s 3 - c~c~c3)l -Ss($~C3 + GCzS~) - C3]C~Sz$~ ÷ C,(S~S3 - ¢~C~C3)1 - C ~ S : C . + S,(S~$3 - C~C=C~)

$z8~C~ + $s(C2$, + $zC3C,)

$~S~Cs ÷ S~(CzS~ + SzC3C,)

C2C~ - 5~C35~

CzC~ -- S~C3S~

a12

1--6

d~2

C3Cs - $~C.~5s S~S.d, - Cn(C3Ss + S~C~.C~)

ll21

S~S.,C.. + S*.(C3$s + S~C..Cs) CzS~Cs - Ss(S:~$, - C2C3C,~)

- s , ( s 2 c , + czc~s,) - cdc,.s3ss + cdszs,, - czc3c.,)/ -c,(srtc4 + c~c35,) + s,[c2s3ss + cs(s.zs, - C2C3C.,)/ Sr~S~Cs + Ss(C:S4+ S2¢3C~,)

s d c z c , - s:c3s,) - cds2s3s~ - c3(czs, + s..c~c,)l c,,(c2c, - s~c3s,) + s,Js2s3ss - c~(c._s, + szc3c,)l

d13

i 0--6

a12 a2!

Ct(C;Cs - s ~ c 4 $ s ) - s J c z s ~ c s - S s ( s , s , - czc~c,)/ cds3s,.~, - c d c 3 s s + s~c,cs)/ + S~CdC2S~Ss + cs(s2S, - c 2 c 3 c , ) / + s ~ s d s 2 c , * c,.c3s,)

cds~s,c,~ + sdc3s, ÷ s~c,c,)l - s,sdc=s3s~ + c,(s~,s,, - c=c3c,)l + s,c,,(s:c,. + c:,c~s.,) s,('c3c~- s~c,s,) + c,/c~s3c~ s~ls~s,,- c~c3c,)t sds3s,*,,- c,,(c3s, + s,c,c~)l - c , c d c ~ s , s , + c~(s2s,, - c~c3c,)l - c,s,,(s2c, * c~c3s,) s,/s3s,c, + s,(c3s, + s , c , c , ) / + c,sdc,.s3s, + c , ( s 2 s , - czc3c,)l - c,c,(s2c, + c2c3s,) -

S2S3C5 4- Ss(C284 + S2C3C4)

a31 a3~

s6(c2c, - szc~s,,) - cds2s3ss - c d c : s , + s2c3c,)/ c~(c2c, - s:c3s,) + sdszs3s~ - c,(c,.s, + s2c3c,)I

l~33

~g~chemes=~ - -

=~5 =6-6

the outer solid of the manipulator arm system, connected through a joint to the adjacent link (rigid body). It is to note that the magnitude of the velocity and this of the acceleration of the outer solid are both very high. Therefore, the inertia forces generated by the solid are dominant for the handling equipment as a controlled mechanical system.

397 We may write, based on eqns (11) and (18), the equations from which the inertia forces of the four point mass model can be computed

= -

?,

m,

k = 1, 2, 3, 4.

(19)

The four inertia forces, acting in the discrete points of the model, need not to be reduced, for practical purposes, to a dynama of the inertia forces (resulting inertia force and resulting moment in a point). In practice, the superposition of the reactions due to the inertia forces applied in sequence, is an useful approach which helps the robot designer select the optimum dimensions of his project.

Applications Because the level of detail on payloads, in space operations, ranges from minimal to nonexistent, the Payload Accomodation System (PAS) designer must conceive and design a system for which there are few "hard" requirements but with the knowledge that the system will be called to do tasks that are not even imagined at this time. A significant progress has been made by Martin Marietta Corporation[l] with the one-arm manipulator system shown in Fig. I l(a). This is the conventional manipulator with forearm, upper arm, and 6 degrees-of-freedom plus a terminal device. We have chosen this manipulator, as a first example, for the purposes of our inertia force analysis. Let us assume that (a) the joints of this manipulator are frictionless, (b) the manipulator carries a payload and (c) the forearm, the wrist and the terminal device with the grabbed payload represent the outer solid. Our problem is, given prescribed state of motion of the outer solid during manipulator handling time, to determine the inertia forces generated by the solid. It is assumed that only regional movements (arm motions) are performed, so that the outer solid is

Terminal Device Wrist Roll Drive

Lig~$ T V Ca'nero T V Control

Unit

Wrist Yaw Drive

335.0

Wrist Pitch Drive

v - B ~ I0" D~o

.

Couplers

Tube Wrist Aisem Suppoet Cra~le and Latch

IL"

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Tube

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Forearm ~Stow~a Po=mon 7,','------------

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Elbow PItch Drive

/

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398

• X5

nTZ~ ~'~ O'L

Z$

..2G.', /713

"•

(b)

(c)

i ~ i ~ u 111b, ©l. in rotational motion: shoulder yaw (~), shoulder pitch (~3) and elbow pitch (,ps), as denoted in Fig. ll(b). Referring to this figure, we note that it represents the structural arrangement of the manipulator in Fig. ll(a) with the dynamical point mass model of the outer solid. Thus, Fig. ll(b) represents also the sceleton of the manipulator, and Fig. ll(c) is the interchange graph which is dual to its kinematic structure. The inertia forces, reduced to a unit mass, have been calculated by the help of eqns (I8) and (19) and Table 1, The results are listed in Table 2. where the additional constraints (¢, = 0;

,

±,

i, °

li

, 1'2

I

Ia)

X ~

Ih5"

h, 2~

R~

(b)

y Figures 12(a.-c).

K¸

(c)

399

Table 2. "03

-o~ --05 rxkall+Yka|2+ge,

rail7

-05 a,21

K = he V°: / ÷ / .a:r + r.a: + z.a2°g/ 2, d ? = a,,

a,, -

U-I

B:Y'f~

~,l,

a,2)

a'"

a `4,

a '5)

~ll

0

at2

0

-a,j

0

~t2

0

0

-a,2

0

0

13.,

0

0

0

0

0

O2t

C253.*~

~2~

--a3l

~2.*

--021

--02]

~0t32

O~

C2C3.s

~22

--tlr32

--(121

--~22

l~

a~

-s2

~2~

-a32

0

-a2.,

0

a~l

$=$~,,_~

~3~

--a'21

0'22

--031

--n'3~

"2.=

a32

$2c2÷5

~22

~¢v3]

--032

--0'32

--/1'21

-a3~

0

0

12i05

cIii

011

C2÷5

012

--$~'5

a.

-a.

111'12 a22

0

~ll

0

f,

a;2 ¢,+¢, -03 a,, = - ~3s2- ~2c~

6;

_-02

26~(~b,+ @,)

'

"

a=, = ~,,c~c2 - #=s2s, - ( * / + ~/)c2s2 - 2¢,2¢3s=c2 -03__ " " a,, - ~3s2c2+ O=c2s, - (~:: + (D/)s2s~ + 2~=(~,c,c~

--ZJ

I

2

3

4

-ll,

x~ + i,~

x~ + i~,

x~ - i,~

x~-in

o

o

v i,,

v%

-v%

o

Y~ Z,

The inertia radiuses/,.,, i,, and i,, are calculatedin the system xGyz, where G ~ B,; x~x,; z~z,.

o

~, = 0; ho = const; h, = const; h~ = const) are taken into consideration with regard to Fig. l(a). Industrial robot/manipulators today are programmable, memory-controlled, open-loop several degree-of-freedom machines with grippers which can perform materials handling. We have chosen such a robot/manipulator, as a second example, to verify our dynamic analysis procedure. The industrial robot for heavy materials handling shown in Fig. (12a), compsises a manipulator with a boom system consisting of an inner boom (3 m 4) and an outer boom (5, 6) which is equipped with a hydraulically controlled extension, a loader body (2), and a slewing unit (1). It has 4 degrees-of-freedom and the speed is continuously controlled. Note that the simple hinges between links 6, 7 and links 8, 9, respectively, are not actuated in order to deprive the sceleton (Fig. 12b) from inertia torque. The interchange graph of Fig, 12(c) is dual to the kinematic structure (Fig. 12b) of the manipulator. For inertia forces calculation, we use the same procedure as given hereinbefore. Conclusions

We may observe that there are a number of notable features to the determination of the inertia forces with the help of point mass models. The procedure can be applied to any one of the minipulator links. Numerical computations can be easily performed via digital computer. We may observe also that we are able to solve some problems concerned with the control strategies in "mass-handling" manipulators: i.e. the force feedback, the type of the hand controller, and the effects of realistic structural flexibility on system controlability, operation and performance.

References I, G. W. Smith and W. L. DeRocher Jr.,Shuttlepayloadaccommodationsystemteleoperator.Proc. Ist ,Vat. Conf. o n Remotely Manned Systems. Pasadena. California (1972), Published by Califorma lnstttute of Technology 1, 85-102 ( 1973~. 2. J. R. Birk and D. E. Franklin, Pitching workpieces to minimize the cycle time of industrial robots. The lad, Robot 1. 217-222 (1974). 3. A. T. Yang, Inertia force analysis of spatial mechanisms. Trans, ASME, J. gagng Incl. 9315, 27-33 (1971). 4. Th. Reye, Beitrag zu der Lehre yon den Tr~eitsmomenten, Zeit. ,Math. u. Phys. tO, 433 (1865). 5. M. S, Konstantinov, P. Genova and V. Topentcharov. Syst~me de masses concentrees, equivalentes ~ un corps solide. Annuaire de I'l~titut de M~chanique Appliqu[e et d'F,lectrotechnique 6, 260.-266. Sofia (1959). 6. M. S. Konstantinov, P. Genova and V. Topentcharov, Theorie de I'equivalence dynamique. Collection of Papers to the Int. Con[. Mechanisms and Machines !. 245-270. Varna, Bulgaria (1965). 7. M. S. Konstantinov and P. J. Genova. Dynamical point mass models of spatial mechanisms. ASME Paper No. 72~Mech-57 (1972). 8. M. S. Konstantinov and Z, I.Zankov,A kinematicalalgorithmand dynamicalpointmass simulationappliedin robots and manipulators. Proc. [st CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators 1. 1, 69-72, Springer Verlag, Berlin (1974). 9. M. S. Konstantinov, Structural and kinematical analysis of robots and manipulators. Proc. 4th Int, Syrup. Ind. Robots. 1,311-327. lIRA, Tokyo (1974). 10. F. R. E. Crossley, The permutations of kinematic chains of eight members or less from the graph.theoretic viewpoint. Developments in Theoretical and Appt. Mech. 2, 467-.486 (Edited by W. A, Shaw). Pergamon Press, Oxford (1965). 11. M.S. Konstantinov, P. I. Genova, W. Galabov, G. Berov and St. Parlor, Positioning accuracy of parallelogram chains for manipulators. Proc. 4th World Congr. Theory of ,Machines and Mechanisms 5. 1163-1168. I. Mech. E. (1975). 12. M. Savage, Gross motion classifications of the RCC¢ chain. ASMF, Paper No. 70-Mech-56 (1970), 13. L. V. Assur, lnvestilation of planar linkage mechanisms with the simplest joints--from the viewpoint of structure and classification (in Russian). [zdat. Akad. Nauk. Moscow (1952). 14. M, S. Konstantinov and Z. I. Zankov, Multigrippers hot forge manipulator. Proc. 4th Int. Syrup. Ind. Robots, 1.405-428. lIRA, Tokyo (1974). T R ~ G H E I T S E ~ F T E VO N ,ROBOT~N UND r,~NIPULATOR~ M.S. Konstantinov Kurzfass~ng -Oer Entwurf yon Fernmanipuiatoren und industrierobotern fordert die Ableitung der Tri~heitskrifte. Zum Beispiel, sind die Bedingungen fur die Beschleunigung oder diese fSr die Verz~gerung einer mit der Hand des Manipulators festgehaltenen Last, die der bedeutendsten und bei dem ~ntwurf yon Fernmanipulatoren fur die Kosmonautia an der ersten Stelle zu berGc:~sichti~en Kenngr55en. Diese Kenngr~Sen werden zur Bestimmung der an den Manipulatorarm wir~enden Zrifte eingesetzt, so da5 man die Berechnung des Stellantriebdrehmomentes und die eigentliche Konstruktion des Armes durchf~hren kann. Anderseits, wird die Bestimmung der Dynamikkr~fte auch fur die Minimisierung der Handhabungszeit der Industrieroboter eingesetzt. Die Aufgabe der Ableitung d~r Trighelts~r~fte wird in dieser Arbeit durch die BenGtzung yon ZassenpunKtmodellen [4,7] gel~st. Um aber diese Aufgabe gewlssermaSen zu erlelchtern, wird zuerst der Begriff des "Matador"-Modulsystems ~r~rtert. Bild la zeigt die Grundstruktur sines Manipulatorarms mit beliebiger Zahl yon Gliedern und Bild Ib ist das Graph dieses Armsystems. Das Graph im Bild Ib wird in Austauschgraphen ("interchange graphs" [10]) CCR, CRC und RCR (siehe Bilder 5d,e,f) zerlegt. Man merkt auf den ersten Blick, da5 diese Austauschgraphen als Grundeinhelten fur die Zusammensetzung sines belieblg komplexen Austauschgraphs benUtzt werden kSnnsn, ias der Zwitter (dual) einer beliebig ~omplexen Armstruktur ist. Die ~truktur sines Doppelgliederi~ettenManipulatorarms (Bilder 4a,b) ist mit CCR Austauschgraphen aufzubauen, wenn man die Bedingungen in Bilder 5b,c berGc~sichtigt. Bilder 7a,b und Bild 8 zeigen die verschiedenen Formen der komplexen Austauschgraphen f~r Doppelglieder~etten bzw. ~ingliederketten des "Matador"-Modulsystems. Fdr die Bestimmung der Trigheitskr~fte mit der Hills yon ~assenpun~tmodellen ist es hinreichend senug die Bahnen und die Beschleunigungen yon einzelnen ~ n k t e n des in B~tracht ko:~menden festen KSrpers zu kennen [7J. FUr die Analyse der T r ~ g h e i t s ~ f t e des Armsystezs sines Roborers oder '[anipulators werden ~onzentrierte '~assen an den Tlementen der Armstru~tur verhaftet, d.n. die stelfen Glieder des Armsystems wetden durch MassenpunKtmodellen ersetzt. Folglich, werden die Forderungen (I),(~),(3) zu den zehn Skalargleichungen (7),(~),(~) reduziert und G1. (8),(9) werden zu G1. (~a),(ga) vereinfacht. ~s werien vier Punate ml,m2,m3,m4 mit den entsprechenden Koordina~en (siehe G1. (10)) angenommen (Bild ~0) und diese Koordinaten werden jeweils in Gi. (7),(Sa),(ga) eingesetzt um ein Vierpunktmodell zu erhalten. ~s wird festgestellt, da~ nur drei Skalarbeschleunigungen ausreichen, um die Tr~gheits~r~fte sines festen K~rpers Oestimmen zu k~nnen.

401

Um die Beschleuni@un@en ~er 31ieder s-!, s, s+! (~ild 2), die zu =er im ~±Zi la ~ar@estellten Struktur leh~ren, zu berechnen, wir~ die relative Anordnung !ieser ~!ieder dutch g3. {12) beschrieben. Die Vektorkocrdinate des Pols Ba (Biii 10) wird lurch gi. (]4) @efun~en und die BeschZeuniEun E des Po!s ~urch doppelte D~fferenzierung ~ni Tabelie 1 berechnet. Di~ ~esultate sind in ?abeZie 2 zu linden. Die 5e~echnungsgrDze~ur ist aucn auf den zwei=en BeispieZ (~i!der ~2a-c) anwenibar.

MMT VOl 12 No