Application of information theory for structure based comparison of in-parallel planar manipulators.
The work so far reported on the structural aspects of the kinematic chains, pertains to generation of distinct kinematic chains. All the good works reported so far will not have much significance, if quantitative methods are not developed to have a comparison between the distinct kinematic chains with the same number of links and degree of freedom in order to perform various tasks. Hence quantitative methods, simple and less time consuming; are desirable to compare the chains quantitatively at the conceptual stage of design for characteristics like workspace, rigidity etc., This will be more significant when multi degree of freedom chains are considered for application as in-parallel robotic structures.
When we identify a kinematic chain as a distinct chain, it will differ from other chains in respect of any one of the following aspects.
Link assortment i.e., type of links in a chain i.e., Binary, Ternary or Quaternary etc., and their number, to form the chain.
Joint Assortment i.e., type of joints and their numbers
Loop Assortment i.e., type of loop and the number of links forming the loop.
It may be noted that a binary link has only one design parameter, while a ternary link has three and a quaternary link has five design parameters respectively. Since the accuracy of motion generation by a chain depends upon the number of design parameters and their effective deployment, different link assortments can be expected to lead to different performance under identical conditions.
A Joint represents motion or energy loss and hence the magnitude of losses will depend upon the type of joint Ternary-Binary (or) Ternary-Quaternary etc., Motion (or) Energy will be distributed among the loops of a chain, the manner of distribution depends on the size of the loops.
Keeping in view that all the above factors, influence the performance of a chain. By assigning quantitative measures to each of the above, the ability of a chain can be quantified.
The assigned quantitative measures may not be true values, but they serve the purpose of comparing the chains accurately.
A kinematic chain with a specified number of links and degree -of -freedom will have the same number of design parameters irrespective of the link assortment. The chains however, with different link assortments. When designed, under the identical conditions to generate a function or a specified path will not perform equally well. The difference in performance is essentially due to the difference in type of links. Best performance i.e., least deviation from the desired motion can be expected of a chain, when all its links are equally effective in motion generation. This is possible when and only when all the links have the same number of design parameters which is impossible physically in view of the relationship between the number of links and number of joints to be satisfied for a specified degree-of-freedom.
For example let us consider an eight link chain with a link assortment of Four binary and Four ternary links. The total number of design parameters for an eight link chain irrespective of link assortment is sixteen. For ideal performance all these parameters should be shared by equally by the eight links i.e., each link should consist of two design parameters, which is physically impossible. The motion transmission by a chain as a whole can be thought upon as a combination of motion transmitted by each link, for the above example. The contribution of various links, in terms of design parameters can be expressed in the form of a numerical scheme as
1/16, 1/16, 1/16, 1/16,3/16, 3/16, 3/16, 3/16.
On the other hand, if the scheme for an eight link chain with a link assortment of two quaternary and six binary links will be
The sum of all the numbers in each of the schemes is unity, and hence the sum cannot be used directly to compare the chains for performance.
It is necessary to satisfy the following requirements in assigning quantitative measure to accuracy of a chain.
1) the quantum of accuracy is maximum when all the links of a chain are identical.
2) No single link contributes to the entire accuracy.
3) The quantum of accuracy of a fixed link (s) of a chain is zero.
Here the "Information theory" used in communications will come into picture.
Wherein one mathematical expression is used to quantify the information, which is
[K.sub.i] log[K.sub.i] -- (1)
Here logarithms are taken to the base '2' just to be in line with the Information theory, used in communications.
To check the validity of the expression used in Information theory, consider an eight link chain with four binary and four ternary links. Ideally each link must consist of two design parameters, so that
K = [K.sup.1] = [K.sup.2] = .... = [K.sub.n] = 2/16 = 1/8
Hence the quantum of accuracy transmitted by the ideal chain is
- [8.summation over (i=1)] [K.sub.i] [log.sub.2] = 8 [-1/8 [log.sub.2] (1/8)] = 3.0000
Now in the actual designed case, [K.sub.i] : value for the binary link = 1/16 and for the ternary link = 3/16. Hence, the quantum of accuracy transmitted by the actual chain is:
-4[ 3/16 [log.sub.2] (3/16)] -4 [1/16 [log.sub.2](1/16)] = 2.8112
It may be seen that the actual designed chain generates motion less accurately compared to, the ideal value.
Now let us consider another eight link chain with a link assortment of two quaternary and six binary links.
Then as stated earlier,
[K.sub.i] for binary link = 1/16
and [K.sub.i] for quaternary link = 5/16
There by the quantum of accuracy transmitted by the chain
= -6 [ 1/16 [log.sub.2] (1/16)] -2 [ 5/16 [log.sub.2] (5/16)] = 2.5481
Comparison of the two eight link chains with different link assortment shows that the quantum of accuracy is higher in case of chains with ternary and binary links and hence it can generate path or function more accurately.
Thus it may be concluded that the presence of links with higher connectivity will result in poor performance.
Effect of Joints
Once the chains are identified and separated on the basis of link assortments, then the next analysis is the examine the chains with same link assortment. Such chains may differ in terms of joints (or) loops. The purpose of a joint between two links is to transmit motion and energy from one link to another link and there will be loss of motion/energy with every joint. Taking this concept of energy loss, the chains are compared based on the type of joints and their number.
Though it is impossible to estimate the amount of energy transmitted by a joint based on the structure; it is possible to quantify the motion or energy, transmitted by a joint and there by chain for the purpose of comparison.
A joint involving two binary links is assigned value '2' since it involves two design parameters and such a joint can transmit motion to only two of its adjacent joints. A joint involving binary and ternary is assigned a value of '3' since such a joint involves three design parameters and can transmit motion to only three of its adjacent joints. Like this, every joint can be assigned with a value.
The total joint value of chain i.e., sum of the values of the joints for a specified link assortment can be written as
[n.summation (i=1)] [C.sub.i][C.sub.i] - 1] -- (2)
Where [C.sub.i] is the connectivity of [i.sup.th] link.
n is the total number of links in a chain.
Connectivity of a link is the no. of other links to which it can be connected through simple joints.
Let 'J' be the joint value, which is a measure of energy quantum. The joint value refers to the no. of other joints to which it is directly joined.
The quantum of energy that can be transmitted will be higher in case of joints with greater joint values, like the quantum of discharge will be more from a tank with more outlets.
The following expression can be used to quantify the energy ( E) that is expected to be transmitted by a joint.
[E.sub.i] = [([J.sub.i] / J ) [log.sub.2] ([J.sub.i]/J)] -- (3)
Where [E.sub.i] : Energy transmitted by the [i.sup.th] joint
[J.sub.i] : Joint value of the [i.sup.th] joint
J : Total joint value of the chain
The above expression is chosen since it satisfies the following requirements
1. Energy transmitted is maximum, when all the joints are of equal value.
2. Fixed link in a joint will not contribute to the joint value.
For illustration of the above theory consider two eight link chains with the same link assortment represented in Fig 1 and Fig 2
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
For the chain in Fig 1
Joint value -- 8 joints with 3/32
2 Joints with 4/32
Hence the value of E = -8 [(3/32) [log.sub.2] (3/32)] - 2 [(4/32) [log.sub.2] (4/32)] = 3.328
For the chain in Fig 2
The numerical joint value is
2 joints with 2/32
4 joints with 3/32
& 4 joints with 4/32
The corresponding energy output of the chain is :
E = - 2 [2/32 [log.sub.2] (2/32)] - 4 [ 3/32 [log.sub.2] (3/32)] -4 (4/32 [log.sub.2](4/32)]
Comparison the results shows that the chain represented at Fig 1 has a greater capacity to transmit energy and hence is superior.
Effect of Loops
When the chains are isolated on the basis of link and joint assortment, it remains to compare the chains having same link and joint assortment.
The independent loops including the peripheral loop will satisfy the condition.
4 [L.sub.4] + 5 [L.sub.5] + 6 [L.sub.6] + ... = 2 J
Where [L.sub.4] - No. of independent four bar loops
[L.sub.5] - No. of independent five bar loops
Since the value '2J' is same for chains with the same number of links, this cannot be used as a criteria to judge the quality of energy circulation.
Ideal chain is the one in which all the independent loops are of same size. The quantum of energy shared by a loop can be expressed by a numerical scheme [n.sub.1]/2J,[n.sub.2]/2J, [n.sub.3]/2J, .... [n.sub.L]/2J
Where [n.sub.1], [n.sub.2], ... [n.sub.L] are the no. of links participating in independent loops 1,2, .... L etc.,
L is the no. of independent loops including peripheral loop.
The Quantum of energy as per the first principles of Information theory circulated by loop 'i' is expressed by the term
-([n.sub.i] / 2J) [log.sub.2] ([n.sub.i] / 2J)
Hence the total energy circulated by a chain, can be expressed as
- [L.summation over (i=1)] [n.sub.i]/2j [log.sub.2] ([n.sub.i]/2j) -- (4)
The above expression will satisfy the following requirements
The energy is circulated most uniformly when all independent loop are of same size.
Fixed links, if any will not take part in the circulation of motion or energy and hence they should not be included in deciding the size of the loop.
To illustrate the above, let us consider the chains represented at Fig.3 and Fig.4 which are having same link and joint assortment.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
The chains in Fig (3) and Fig(4) can be compared on the basis of loops.
For the chain in Fig (3), the numerical scheme is
2(4/20) and 2(6/20). Hence the Quantum of Energy = 1.9662, calculated from the above expression represented at equation .4
And for the chain in Fig (4), the numerical scheme is 4(5/20).
Hence the Quantum of energy = 2.0000
Comparison of the result shows that the chain in Fig. 4 is superior than the Fig.3.
Like this we can compare the chains based on loop assortment.
In-parallel manipulator structures with multi degree-of-freedom can be compared using the first principles of Information Theory used in communications.
Link assortment is the most important factor in deciding the superiority of the chain.
When a chain is formed with higher connectivity links the chain is inferior.
Similarly when a chain is formed with higher joint values, it will lead to inferior results.
The links with larger connectivity should be preferred as the Input link.
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Prof. A.B. Srinivasa Rao (1) and Dr. A.M.K. Prasad (2)
(1) Dean(Academics), Aditya Engineering College, Surampalem, E.G.Dt, AP, India E-mail: firstname.lastname@example.org
(2) Professor & HOD, Mechanical Engineering Dept., College of Engg., Osmania University, Hyderabad, AP, India
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|Author:||Rao, A.B. Srinivasa; Prasad, A.M.K.|
|Publication:||International Journal of Applied Engineering Research|
|Date:||Aug 1, 2009|
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