Topological characteristics of linkage mechanisms with particular reference to platform-type robots

~

Mech. Math. Theoo' Vol. 30, No. 1, pp. 33-42, 1995

Pergamon

0094-114X(94)E0008-8

TOPOLOGICAL

CHARACTERISTICS

Copyright ti 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0094-114X/94 $7.00 + 0.00

OF LINKAGE

MECHANISMS WITH PARTICULAR REFERENCE TO PLATFORM-TYPE ROBOTS A. C. RAO Technical Education, Government of M.P., 4th floor, Satpura Bhavan, Bhopal (M.P.), India

(Received 3 May 1993; received for publication 22 February 1994)

A b s t r a e t - - A comparative study of chains and mechanisms at the conceptual stage of design is expected to help the designer in selecting the best possible chain or mechanism for the specified task. To accomplish this the designer should be able to read the characteristics of the kinematic chains based on their topology. It is only necessary to associate logically certain characteristics, weakness and strength of a chain to perform a task, with the structure of the chain and then generalize. Based on this belief work has been initiated to assess the ability of a chain to reveal some of the characteristics like structural error performance, dynamic behavior etc. in a comparative sense. In this paper criteria and measurements to compare kinematic chains and inversions for other characteristics like static behaviour (mechanical advantage), compactness, stiffness and suitability as platform type robots, which are gaining importance, are presented.

INTRODUCTION

Thorough knowledge of kinematic chains is essential for a mechanical engineering designer and persistent efforts are made to know about kinematic chains as much as possible. Until now most of the work is directed to study isomorphism among chains and to know the type of freedom. Dealing with linkages, the designer has no guidelines regarding the selection of a particular chain out of the numerous chains consisting of the same number of links and joints. The criteria for selection can be: (i) Generation of specified motion more accurately, i.e. less structural error. (ii) Static performance. (iii) Dynamic performance etc. Unless the designer has tools at his disposal a method of comparing all possible chains from different aspects of performance the work reported so far will not have much significance. Recently the author [1-4] has attempted to compare the chains for their structural error performance. Also, selection of the best combination of frame, input and output links is reported by the author [5]. In order of complexity the analysis [6] can be considered under three headings: (i) static analysis; (ii) kineto-static analysis; and (iii) dynamic analysis. The complexity of kinematic chains increases with the number of links. Of the chains with the same number and type of links, the structure or the topology of the chain is expected to play a prominent role. Exact comparison of chains even for kinematic or static performance is impossible without knowing the actual dimensions. The author believes that the structure of a kinematic chain reveals its capabilities and limitations and accordingly an attempt was made to compare chains for structural error [1-3] and dynamic performance. Static analysis is mainly concerned with assessing the chain from the viewpoint of mechanical advantage (MA). The concept of compactness of a chain is introduced and its correlation with static performance is included in this work. Application of these concepts to compare platform-type robots, which are gaining importance, is dealt with. MECHANICAL ADVANTAGE Mechanical advantage and pin reactions describe the static behaviour of kinematic chains, One of the major criterion of which a designer must be aware is the ability of a particular chain or 33

34

A . C . RAO

mechanism to transmit force or torque. It is necessary to know the relationship between force-out and force-in. Mechanical advantage (MA) is the ratio of the magnitude of force-out (Fo) over force-in (F~). Consider Fig. 1 which shows a lever and the MA that can be attained: MA = Fo/Fi = ri/r o.

(1)

The above relation indicates that MA for a lever is essentially dimension-dependent. If the relative motion between the fulcrum and the lever is prevented no MA can be obtained, i.e. a structure cannot offer any MA. In other words, it is possible to get MA when there is relative motion along links or members. Linkages can obviously be considered for obtaining MA. For chains with the same structure but different link dimensions the MA will be different and this is attributed to difference in dimensions. It is not very clear how the structure of a chain influences MA. Structure refers to the formation of loops or relative deposition of links in a chain without consideration for dimensions. When two chains are to be compared only on the basis of structure i.e. reveals the inherent merit of one chain over the other, it is not necessary to consider dimensions. For instance, once a chain is selected for a task, its dimensions can be optimised for the best performance. A different chain with the same number and type of links may also be selected for the same task and its dimensions can also be optimised. However, the performance, when compared, of one chain will be better than the other and this can be attributed to the structural superiority. This paper deals with this aspect and a measure is proposed to compare the chains. For any ideal mechanical system, i.e. when power loss such as friction loss within the system is neglected, the work put in to the system is exactly equal to the work output. Mathematically: F,. AX = Fo. A Y,

(2)

where F~ is the input force, Fo is the output force, AX is the displacement of the input force and A Y is the displacement of the output force. Hence: MA = Fo/F~ = A X / A Y.

(3)

From relation (3) it is obvious that a higher MA can be obtained from the chains which are designed to yield smaller output displacements for larger input displacements. A system with nonlinear relationships between the input and output motions can be expected to fulfil the above requirement. For example, consider two types of motion: Y = k .x

(4)

and Y=k

(5)

. x 2.

Then, for the linear motion [equation (4)]:

(6)

AY = k • Ax and for the nonlinear motion [equation (5)]: AY = 2k . x .Ax

(7)

Y = log x

(8)

Let us consider another relationship:

i

F

i

i //

//

rO

I-

/ /I T/I// Fig. 1. Simple lever.

_1 -I

Topological characteristics of linkage mechanisms

35

A Y = 1/x • Ax.

(9)

Then

Observing relations (6), (7) and (9) and generalising Ax and Ay as just two interrelated displacements it can be interpreted to mean that a high nonlinear relationship leads to magnification of one displacement with respect to the other. In the case of the linkages one can choose the link with smaller displacement as the output link and the link with larger displacement as the input link in order to get a greater MA. Thus the linkages (chains) with a potential to generate motion of a higher nonlinearity are preferable from the viewpoint of MA. To judge the kinematic chains with the same type of links and joints for their relative ability to generate nonlinear motion the concept of compactness is introduced and explained below. COMPACTNESS It is always desirable to know the structural compactness of a chain at least in a comparative sense. Compact chains, besides having good static behaviour as explained later, can be expected to occupy less space. A chain in which the links are close to one another may be considered as a compact chain. Closeness or nearness of any two links may be indicated by the distance between the two links. The distance between the two links is equal to the least number of links that separate them. In graph theory, the distance that fall between two vertices is defined as the least number of edges that fall between the two vertices out of all possible paths between them. Adapting this concept, the distance between two links in a kinematic chain is equal to the least number of joints that separate them. A distance matrix D can be written for every chain in which: dij = least number of joints between links i and j and dii = 0. For example, distance matrices for the two distinct six-link chains will be given below. (a) (b)

3

(c) 4

(d) 4

2

1

I

Fig. 2. (a) Stephenson chain; (b) Watt chain; (c) vnd (d) two eight-link chains.

36

A . C . RAO

For Stephenson's chain Fig. 2(a):

D=

0

1

2

1

2

1

!

0

1 2

2

2

2

1

0

1

1

2

1

2

1 0

2

2

2

2

I

2

0

1

1

2

2

2

1 0

(lO)

The sum of all the elements in the D matrix may be called the distance value of the chain, which for this chain is 46. For Watt's chain, Fig. 2b:

D=

0

I

2

1

2

1

1

0

1 2

3

2

2

1

0

2

3

1

2

1 0

2

3

2

1 0

1

1

2

3

2

0

1

(ll)

1 2 1

The distance value of the chain is 50. The comparison of distance values for the six-link chains shows that the links in Stephenson's chain are close to one another and hence it is structurally more compact. Also, consider two eight-link chains (Fig. 2c, d) distance values of which are respectively 96 and 100 indicating that the chain in Fig. 2a is more compact. The eight-link chain makes it clear that the forward kinematic analysis of the chain (Fig. 2c) is more difficult compared to the other due to the absence of four link loops, a point that will help in understanding the relationship between the compactness of the chain and the nonlinearity of motion. The input and output relationship in a linkage is usually nonlinear. For example the fourbar chain Fig. 3 which is the basic closed kinematic chain of 1 d.o.f., the relationship is: = f (O, a, b, c, d) = 2 arctan{[A - x / A 2 + B 2 - - C 2 ) ] / ( B + C)},

where A = sin 0, B = ( d / a ) + cos 0, C = ( d / c ) c o s 0 + (a 2 - b 2 + c : + d Z ) / 2 a c . 3

/4 /

///// Fig. 3. Four-bar linkage.

(12)

Topological characteristics of linkage mechanisms (a)

c

cm

D

37

) Fi "3%

E

(b)

~

B

,

C

A

5 cm

D

5 cm

F

10kg

Fig. 4. (a) Stephenson chain; (b) Watt chain.

The nonlinearity of a kinematic chain increases with the number of links. For example, a six-link Watt chain (Fig. 2b) can be considered as two four-bar 4 chains in series, the first consisting of links 1, 2, 3 and 4, the second consisting of links 1, 4, 5 and 6. If link 1 is the ground link the output of link 4 can be expressed following equation (12) in terms of the input of link 2 and link dimensions. The output of link 4 in turn becomes the input to the second four bar. Thus, the output of link 6, in terms of the input of link 2 and linkage dimensions, becomes more nonlinear compared to that of a single four-bar chain. The structure of a chain also contributes to the nonlinearity e.g. with six links Stephenson's chain (Fig. 2b), the output of link 6 in terms of the input of links 2 and linkage dimensions will be more nonlinear compared to that of the Watt linkage due to the presence of a five bar loop consisting of links 1, 3, 4, 5 and 6 which needs to be considered for analysis. From the above one can say that greater MA can be obtained from Stephenson's chain. To test the statement, both the chains with links of identical dimensions are analysed for the

Table 1. Joint reactions and input force for load F0= 10 kg Watt chain Stephenson chain Joint Reaction (kg) Joint Reaction (kg) A 2.455 A 1.150 B 7.000 B 6.850 C 7.000 C 6.850 D 6.850 D 5.710 E 6.850 E 2.190 F 3.150 F 6.850 G 8.550 G 6.850 F, = 9.455 kg F, = 8.000 kg

38

A . C . RAO

same output force and position (see Fig. 4 and Table 1). The values of F~ confirm the above statement. PIN R E A C T I O N S It is desirable to have some idea regarding pin reactions at the conceptual stage of design as it helps the designer in making his choice depending upon the task. The designer can get an understanding of pin reactions in a comparative sense from the expected performance of a chain from the viewpoint of MA. Each link in a chain can be viewed as a lever supported at a fulcrum (pin). For a lever with greater MA the pin reaction will be less for the simple reason that for a given Fo the value of F~ is less and the reaction is nothing but vector sum of Fo and F~. This concept is supported by the numerical values of the example problems in which two six-link chains with identical dimensions are considered (Fig. 4 and Table 1). Correlating the results with the compactness it may be said that the compact chains in general lead to greater MA and less pin reactions. STIFFNESS The actual stiffness of a link or a member of a chain depends upon its dimensions, elasticity and support(s). Naturally, if all the links of a chain are stiffer the chain can be expected to possess greater stiffness and this in turn provides insight into the load-carrying capacities and natural frequencies. Greater structural stiffness will lead to lighter chains or links. The knowledge of frequencies will be of help in deciding the running speeds. When the chains are to be compared on the basis of their structure, dimensions do not come in to consideration. However, the modulus of elasticity of all the links and dimensions as in Fig. 4a, b can be considered to be the same so that the structure provides information regarding link supports. The designer can certainly get some idea about the stiffness of chains at least in the comparative sense. A pin jointed member can be considered to have been elastically supported by as many other members as there are joints. Also, a member becomes stiffer if the number of supports are more. For example, a ternary link can be conceived as a member with three supports while a binary link is a member supported at two places only. Thus a ternary link is stiffer. Likewise the stiffness of a link increases with the connectivity of the link. Now, as far as the interaction of links is concerned a stiffer link is considered to offer more support reaction to the link it is connected to. For example a binary link connected to a ternary at one end and to a quaternary at the other, the binary can be considered as an elastically supported member, the stiffness of the supports being proportional to the connectivity of supporting links. In other words, the end deflections of the members are inversely proportional to the connectivity of the supporting or adjacent links. In order to assess the stiffness of a chain comparatively, a numerical measure is necessary and to compute the numerical measure formulation of stiffness matrix S is proposed. The elements (s!j) in the S matrix will be as follows: Sij = C i ,

where C~ is the connectivity of the ith link. When two links i a n d j are separated by other links the elements can be found out by the relation: 1 Sij

(1/Ck +

" +

I/G)'

the denominator includes all intermediate links and not the ith link that falls in the shortest path between links i and j. These links are considered to be springs connected in series. In the case of alternative shortest paths, the path which results in least stiffness may be taken as the objective is to estimate the least stiffness of the chain. In other words, the path consisting of links with lesser connectivity should be chosen. To illustrate the above, the two six-link chains are considered.

Topological characteristics of linkage mechanisms

39

For the Stephenson's chain (Fig. 2a):

S=

3

2

6/5

1

2

1

3

0

3

6/5

6/5

6/5

6/5

2

3

2

2

1

3

6/5

3

0

6/5

6/5

6/5

6/5

3

6/5

0

2

3

6/5

6/5

6/5

2

0

(13)

The sum of the elements of a row represent the stiffness of the concerned link and the sum of all elements in the matrix represent the stiffness of the chain. Thus, the stiffness values of links 1-6 are, respectively, 65/5, 58/5, 56/5, 58/5, 53/5 and 53/5 while the stiffness of the chain is 334/5. For the Watt's chain (Fig. 2b):

S =

3

2

1

3

1

2

3

2

2

6

8

6

6/5

2

2

3

6/5

8/5

3

1

2

3

2

1

6/5

8/5

6/5

3

2

2

3

6/5

8/5

6/5

2

2

(14)

The stiffness values of links 1~6 are, respectively, 12, 322/30, 322/30, 12, 322/30 and 322/30 and the stiffness value of the chain is 2008/30. Comparison of the stiffness values shows that the links of Watt's chain are stiffer. Thus Watt's chain can be expected to have a higher natural frequency under identical conditions and the running speed can also be fixed. It may, however, be recalled that this chain was found less compact compared to Stephenson's chain. Application of the above approach to two randomly chosen eight-link chains (Fig. 2c, d) reveals that the chain (Fig. 2d) has a greater stiffness than that of Fig. 2c. It may be recalled that the chain (Fig. 2c) has a greater compactness. It may, hence be generalised and stated that more compact chains will be less stiffer and vice versa. The above statement does not cover degenerate chains. Among two linkages having identical dimensions (lengths) such as Fig. 4a, b, the Watt linkage (Fig. 4b) is stiffer and hence the cross-section of its links can be smaller which in turn results in less weight.

APPLICATION

TO P L A T F O R M - T Y P E

ROBOTS

Robot arms which received greater attention in the recent past are open-chain linkages. Each joint in these robot arms is actuated independently. While possessing many advantages such as large work space and manoeuverability they do suffer from disadvantages such as less rigidity, accumulation of mechanical errors from a shoulder to the open end effector, control problems etc. The alternative to the open chain robot arms is the in-parallel actuator arrangement often referred to as the platform-type robot as it has greater rigidity, lightness, load carrying capacity and the actuators can be attached or adjacent to a fixed base [7-10]. The disadvantage of the platform type rotor is that the forward kinematic problem is more complex than the inverse kinematics of open chain robots but recently some closed form solutions are presented for forward kinematic analysis of platform-type robots, planar and spatial [9, 10]. Successful applications of platform robots are proposed by many investigators and the same are reported in Ref. [10], making their practical application feasible. The concept of compactness helps the designer in studying certain aspects of platformtype robots and make his choice. To illustrate, let us consider two eight-link chains (Fig. 5) with 3 d.o.f. The distance matrices for both the chains are given below.

40

A . C . RAO

(a) (b) 4

5

3

7

1

I

Fig. 5. Two eight-link, 3 d.o.f, chains.

F o r the chain of Fig. 5a: -0

D =

1

2

3

2

1

2

1

1

0

1

2

3

2

3

2

2

1

0

1

2

3

2

3

3

2

1

0

1

2

1

2

2

3

2

1

0

1

2

3

1

2

3

2

1

0

3

2

2

3

2

1

2

3

0

1

1

2

3

2

3

2

1

0

0

1

2

2

1

2

2

1

1

0

!

2

2

3

3

2

2

1

0

1

2

3

4

3

2

2

1

0

1

2

3

3

1

2

2

1

0

1

2

2

2

3

3

2

1

0

1

2

2

3

4

3

2

1

0

i

1

2

3

3

2

2

1

0

(15)

The distance value of the chain is 108. F o r chain of Fig. 5b:

D=

(16)

Distance value of the chain is 110. Thus, the chain o f Fig. 5a is more c o m p a c t and its forward kinematic analysis m o r e difficult. The concept o f compactness can also be utilised to select the best frame or fixed links with connectivity equal to d.o.f, of the chain or m o r e are available so that the (a)

(b)

4

6

5

7

1

2

Fig. 6. Two seven-link, 2 d.o.f, chains.

Topological characteristics of linkage mechanisms

41

actuators can be attached to the fixed frame. To understand this, let us consider 2 d.o.f, chain (Fig. 6) with seven links. As a first case, fix link i (Fig. 6a) and count the joints between links ensuring the paths that consist of link 1 as the fixed link does not receive or transmit motion. With this assumption:

D =

0

0

0

0

0

0

0

0

0

1 2

3

3

4

0

1 0

1 2

2

3

0

2

1 0

0

3

2

1 0

1

2

3

0

3

2

1 2

0

l

0

4

3

2

1 0

3

1 2

(17)

The distance value of the inversion is 62. Let us now consider the same chain with link 2 fixed (Fig. 6b). The distance matrix:

D =

0

0

3

2

1 2

1

0

0

0

0

0

0

0 3

3

0

0

1

2

2

2

0

1 0

1

1 2

1

0

2

1

0

2

2

0

2

1

2

0

1

1

0

3

2

2

1

0

(18)

2

The distance value of the inversion is 52. Obviously the inversion obtained by fixing binary link 2 is more compact. CONCLUSIONS The concept of compactness helps us draw the following conclusions= (1) The lesser the distance value of a chain or inversion, greater is the compactness. (2) For chains with greater compactness a higher MA and lesser pin-reactions can be obtained or large static loads can be carried and such robots are suitable for slow-speed operations. (3) For more compact chains, forward kinematic analysis is difficult and complex. Thus, excessive requirements will be placed upon the computational resources. Programming becomes cumbersome and real-time operation becomes difficult. (4) It is known that resolution (control) of a robot becomes poor with the increase in the number of joints. Distance values of a chain or inversion shows the total number of joints which interact motion-wise. Thus a compact chain has a less number of joints involved, i.e. joint interaction is less and the control will be accurate. (5) Loss of motion and power increase with the number of joints, hence compact chains or inversions are better from this viewpoint. (6) Reachability or accessibility can be expected to be more in the case of compact chains and inversions. For example, if link 6 (Fig. 6) is the platform or output link, it is separated from the base by three joints in the case of Fig. 6b and two joints in the case of Fig. 6a. (7) Choosing a link with lesser connectivity as the frame is preferable if the connectivity is not less than the degree of freedom of the chain. (8) Compact chains are not usually stiff and hence tend to become too heavy to carry the same loads. This comparison is permissible among chains consisting of an equal number and type of links. (9) The natural frequencies of compact chains will be less and hence more suitable for slow speeds. (10) For the mechanisms to be stiff, the link of a chain whose stiffness is low should be grounded. The choice of link 2, for example Fig. 6b, appears to be reasonable from this point of view also.

42

A . C . RAO

REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10.

A. C. Rao, Trans. Can. Soc. Mech. Engng 10, 213 (1986). A. C. Rao, Trans. Can. Soc. Mech. Engng 12, 99-102 (1988). A. C. Rao and C. N. Rao, Mech. Mach. Theory 28, 1 [3 (1993). A. C. Rao, Indian J. Technol. 30, 381 (1992). A. C. Rao and C. N. Rao, Mech. Mach. Theory 28, 129 (1993). A. G. Erdman and G. N. Sandor, Mechanism Design--Analysis and Synthesis, Vol. I, p. 219. Prentice Hall, New Jersey (1984). K. H. Hunt, A S M E J. Mech. Transmiss. Automat. Des. 105, 705 (1983). K. L. Ting and G. H. Tsai, 9th Appl. Mechanisms Con£, Kansas City, MO, Vol. 1, Session--l, pp. III.I 111.8. G. R. Pennock and D. I. Kassner, ASME J. Mech. Des., Vol. 114, pp. 87-95 (1992). V. Parneti-Castelli and C. Innocenti, A S M E J. Mech. Des., Vol. 114, 68-72 (1992).