IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARYFEBRUARY 1993 237

0018-9472/93$03.00 0 1993 IEEE

Determining Optimally Ordered Pairs Using Edge-Coloring of Graphs

Ian Cloete and Wilma G. Cloete

Abstmct-Tables to find optimally ordered pairs for the assessment of judgemental information were published. The optimality of such orders according to spacing and balance requirements is proved using edge- colorings of graphs. In addition, the constructive proof leads to an elegant

1) Maintain the greatest possible spacing between any pairs involving the same altemative. For n odd the bounds for optimal spacing for all pairs is the following: Pairs involving the same alternative are separated by at most (n - 1)/2 pairs, and by at least (n - 3)/2 pairs. Both the upper and the lower bound have to be attained for optimal spacing.

2) Present an alternative equally often as the first member and as the second member of a pair. Pairs are said to be balanced.

For example, for n = 5 the 10 ordered pairs given by Ross are algorithm for constructing orders by hand without consulting tables.

I. INTRODUCTION

Determining relative preferences among different criteria (“stim- uli”) plays an important role in many areas of research. The assess- ment of judgemental information belongs traditionally to a branch of psychology known as psychometrics. The assumption is that relative preferences can be established if the criteria are lined up in pairs and presented for comparison. The order of pairs in the stimulus series is important since it influences judgement. In such an order the idea is to minimize judgemental bias by suitably arranging the order of comparisons because regular repetitions may influence judgement undesirably. In the order the greatest possible spacing should be achieved between pairs involving identical members, and the pairs should be balanced so that a given member of a pair does not occur more often on the left than on the right.

One of the scientists who made an important contribution to the method of presenting pairs in an optimum order was Ross [l], [2]. This paper gives a detailed account that the Ross method for determining pairs of alternatives for comparison can be explained by edge-colorings of graphs. In search of a general algorithm, Ross gave a method that makes use of two tables to construct an order [l]. We present a simpler algorithm that chooses elements to compare based on their position on an imaginary circle. The advantage is that the order can easily be constructed by hand without consulting tables.

The paper is organized as follows: Firstly, the Ross method is explained and an example is given. In Section 111 relevant terms of graph theory are given, and it is shown when and why perfect balance of pairs can be achieved. In Section IV the Ross method is developed in terms of edge-colorings of complete graphs. It is proved that the order is optimal with respect to the given criteria. Lastly, in Section V an algorithm for generating a Ross order for n elements is presented based ,on the properties obtained from graph theory. The algorithm is also suitable for computer implementa- tion.

11. THE ROSS METHOD OF PAIRED COMPARISONS

The method of paired comparisons, and in particular the Ross method, is explained as follows: Assume n alternatives exist, each of which must be compared with every other alternative exactly once. This gives ( ) comparisons. The Ross method is concerned with the order in which these pairs should be presented for comparison. His order satisfies the following restrictions:

Manuscript received December 22, 1990; revised April 20, 1992. The authors are with the Department of Computer Science, University of

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He also suggests for n even that the order for n + 1 be determined first, and that all pairs involving the additional alternative then be eliminated. In the following therefore, consider only odd n, n 2 5 .

111. DEFINITIONS AND GRAPH REPRESENTATION

Before Ross’ method is developed by edge-colorings of graphs a few definitions [3], [4] are in order. Complete graphs on n vertices, denoted by li,, are simple graphs in which each pair of distinct vertices is joined by an edge. Edges are adjacent if they have a common vertex. A subset E, of the edge-set E of a graph G is called a matching if no two of its elements are adjacent. A matching E, of G is called a maximum matching if G has no matching E3 with IEjI > IEtl. An m-edge coloring of G is constructed by assigning 7n colors 1 ,2 , . . . , m to the edges of G. This coloring is proper if no two adjacent edges have the same color. An m - edge coloring is also a partition ( E l , E2, . . . , E,) of E where E, is the set of edges assigned color i . Each E, is a matching if and only if they represent a proper m-edge coloring. The edge chromatic number X‘( G ) is the minimum m for which G has a proper m-edge coloring.

Pairs can be represented by a graph in which the vertices represent the alternatives and an edge represents the comparison between two alternatives. Because every alternative, say 2, is compared exactly once with each other alternative, say y, the comparison (I, y) can be represented as edges of a complete graph Kn with n alternatives or vertices. If an order satisfying restriction 1) has been constructed, the balance restriction 2) can be met in the following way: Because all the vertices of I<, (n odd) are of even degree IC, is Eulerian [3], i.e., I<, contains an Euler tour 0 0 e 1 v 1 e 2 . . ‘emuO. Using this tour perfect balance can always be achieved.

Iv. CONSTRUCHON OF AN ORDER

In this section we determine the upper and lower bounds for optimal spacing and show how such an order is to be constructed. Using graph theoretic reasoning properties of the order to satisfy the spacing requirement are derived. The graph representation now allows the construction of an order satisfying two restrictions:

a) An edge appears exactly once. b) No two consecutive edges in the order are adjacent. Restriction a) requires that alternatives be compared once only,

while b) requires spacing between alternatives, i.e., vertices are not repeated in any two consecutive edges. The following argument derives the upper and lower bounds for maximum spacing.

238 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANIJARYFEBRUARY 1993

From a) and b) it is clear that consecutive edges in this order can be grouped in, say m, consecutive (ordered) matchings E l , E2, . . , E, where E, n E, = 0,; # j , and UE, = E . Thus ( E E , . . - , E m ) . is a partition of E. Note that the E, are ordered sets and the partition represents the order E, , . , E,. This represents a proper m-edge coloring of E where color i is assigned to matching E,. To obtain maximum spacing (Ross’ restriction 1) every !Ell must be as large as possible, thus m is required to be as small as possible. Because X‘(K.) = n for n odd and n 2 3 [4], m = n gives the smallest attainable value for m. Then, because n is odd all E, are maximum matchings in K , where 1E.I = (n - 1)/2. Thus it is established from the graph representation and the properties of edge-colorings that no vertex repeats in a matching, that all matchings have the same number of edges, and that these matchings have the largest size possible.

Now consider the minimum number of edges S,,, and the maximum S,,, that may occur in the order between two edges that have a vertex in common. Note that these two edges belong to different matchings, and that in any maximum matching E, exactly one vertex of h’, does not occur. This result will establish the order of edges in consecutive maximum matchings E, and E,+, .

To obtain optimal spacing between any two pairs (Ross’ restriction 1) it is clear that the lower bound S,,, must be as large as possible for every pair in the order having a vertex in common. Let V, denote the vertex-set of E,. Note the following: Every maximum matching for m minimum (i.e., m = n) contains the same number of edges, Le., (V.1 = n - 1 and (E, ( = (n - 1) /2 for all 2 . In such a proper m-edge coloring there exists for all sets a unique vertex V k $! V,, which occurs in all other V , , j # 2 , so that any vertex-sets V. and V, , j # i , have n - 2 vertices in common.

Consider the first maximum matching E1 and its first edge (v,., v,). The lower bound for maximum spacing between E1 and E2 for vertices v, and v, satisfies S,,, 2 (n - 3)/2 because (E,I = (n - 1)/2, for all i. Now suppose that S,,, > (n - 3)/2 can be achieved. Then v, or v3 cannot be a vertex of the first edge in Ea, otherwise this edge contains v, or v,. Because exactly one vertex V k V I , it then follows for one of the vertices of the first edge of E2 that S,,, < (n - 3)/2. This contradicts the attempt S,,, > (n - 3)/2 because a smaller lower bound is then obtained for some of the n - 2 common vertices in % and V2. Therefore, S,,, = (n - 3)/2, and the candidate vertices that satisfy this minimum for the first edge of E2 are up, v3 E VI and V k $! V I . Vertex V k occurs in all remaining V,, i = 2, . . . , n. Using V k in similar reasoning to the aforementioned followed for the first edge of E1 shows that if V k is not placed in the first edge of E2 it causes S,,, < (n - 3)/2 for some vertices common to V2 and Vs. One or both of v, and v, must occur in V2. So, assume (v, ,vk) to be the first edge of E2. From restriction (a) S,,, # S,,,, therefore S,,, 2 (n - 1)/2. For the second edge of E2 only the two vertices of the second edge of El and v, satisfy S,,,. If v, $! Vz then again it causes S,,, < (n - 3)/2 for some vertices common to 1‘1 and V2. Thus v, must be a vertex of the second edge of E2. Therefore,

In general, similar reasoning for the successive remaining edges of E1 and E2, and then for E, and E,+1, for all z = 2,e.s ,n - 1, verifies that S,,, = (n - 3 ) / 2 and S,, = (n - 1)/2. The previous reasoning also shows how an order with the Ross spacing requirements are to be constructed.

Edges of the matchings E, therefore have the following properties: The pth edge of E, is (TI,., v,) if up or v3 (say v,) is one of the vertices in the pth edge of E,+1 and the other vertex (Le., v,) is one of the vertices in the p + l th edge of E,+1. Whenever p = (n - 1)/2 (i.e.,

Smax # (n - 1)/2.

the last edge in a maximum matching), the v, is one of the vertices in the first edge of E,+2, and because every E, is a maximum matching this vertex v, is exactly that vertex not occurring in the matching E , + I . These properties are used to construct an algorithm that produces an optimal order of comparisons.

V. AN ALGORITHM An algorithm that constructs an order achieving the optimal spacing

bounds is as follows: Construct E1 by choosing a maximum matching for El,. Construct every subsequent E, to be a maximum matching satisfying the properties given previously. The resulting order is a proper n-edge coloring with the spacing bounds as proved.

However, an algorithm to construct exactly the Ross order has additional constraints when choosing edges of the matchings E, because the previous explanation shows that different orders ex- ist with identical spacing bounds. This algorithm is given in the following.

Construct K , by placing n vertices on an imaginary circle. (Recall that n is odd.) Label these vertices from 1 to n clockwise. Any vertex has a neighbor on either side on the circle, e.g., the left (anti- . clockwise) neighbor of 1 is n, and the right (clockwise) neighbor is 2. Let N record the set of vertices of K, from which neighboring vertices on the circle are chosen. Initially N = { 1, . , n} . Whenever an edge is chosen, its vertices are removed from N. A variable x is used to guide choices to obtain the Ross order. Edges are chosen using three rules Rl-R3. R1 is used when IN1 2 2, R2 when N contains one vertex, and R3 when N is empty. R2 selects two edges. Always choose (1, 2) as the fmt edge of E1 to obtain the Ross order.

The complete algorithm for Ross’ order is: 1) Choose (1, 2) as the first edge of E l . 2) Construct maximum matchings using the applicable rule R1,

R2, or R3 until all edges of K, have been chosen. Then go to step 3.

When IN1 2 2 : If the previous edge chosen has the number 1 as one of its vertices, e.g., (1, v3), then the next edge chosen uses the neighboring vertices in the set N on either side of the vertex v, on the circle. (The left neighbor becomes the first coordinate of the edge.) Set x = v3. Remove the chosen vertices from N. Otherwise, when the previous edge, say (v,,v3), does not contain the vertex numbered 1, choose the left neighbor of vt (as first coordinate of the edge) and the right neighbor of v3 in the N as the next edge. Remove v, and v3 from N. When exactly one element v, remains in the set N, choose this vertex and the vertex numbered 1 to ob- tain the edge (1, v3). Also choose the next edge as (z + 1, z). Set N = (2, . - , n}-{z , z + 1) and there- after x = z + 1. Comment: The edge (1, v,) is followed by an edge that is obtained using the property relating edges of E, and E,+1. The spacing requirement of S,,, = (n - 3)/2 and S,,, = (n - 1)/2 determines a choice between two edges (vr, v3) and (vr, vh), which is made by taking the edge (vr,vr + 1) to obtain the Ross order. (Due to the way neighbors are chosen up and v, + 1 always exist in N.) When there are no remaining elements in N choose the edge (1, z). Set N = { 2 , . . - , n } - {x}. Comment: A choice between two edges (v,, v,) and (v,, vk), which is determined by the spacing requirement as in R2, is made by taking the numerically largest of v 3 and V k to obtain the Ross order.

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARYFEBRUARY 1993 239

3) Balance the vertices. Note, in order to easily know when to terminate the algorithm, a

counter could count the number of edges chosen until ( ) is reached, or simply stop when the edge containing vertices 2 and n is reached.

If the algorithm is applied to I i 7 , the following Ross order is obtained (after balancing) in which the matchings E, are ordered edge-sets:

The algorithm is easily followed on the circle by noting the regularity of the edges chosen with the aid of the variable s. Successive edges chosen construct triangles using the vertex s in a clockwise rotational sequence until the edge ( ( n + 1) /2 , ( n + 3) /2) is reached (start of the last maximum matching). For I i 7 the edges obtained using the vertex T are (1,2), (2,3), (1,3), (3,4), (1,4), and (43).

VI. CONCLUSION We have given a constructive proof in graph-theoretic terms of the

Ross bounds on spacing for n odd, and have shown how to construct orders that have identical bounds on spacing. The order given by Ross follows as a special case. An algorithm was also given that allows the construction of the Ross order without consulting tables, and which i s easily done by hand.

ACKNOWLEDGMENT

The authors would like to thank Prof. G Geldenhuys for helpful comments on an earlier draft of this paper.

REFERENCES

R. T. Ross, “Optimum orders for the presentation of pairs in the method of paired comparisons,” J. Educ. Psychol., vol. 25, pp. 375-382, 1934. -, “Optimal orders in the method of paired comparison,” J . Exp. Psychol., vol. 25, pp. 414-424, 1939. J. A. Bondy and U.S.R. Murty, Graph Theory with Applications. New York MacMillan, 1978, p. 51. S. Fiorini and R. J. Wilson, Edge-Colourings of Graphs. London: Pitman, 1977, p. 23.

Linearization of Manipulator Dynamics Using Spatial Operators

A. Jain and G. Rodriguez

Abstract-Linearized dynamics models for manipulators are useful in robot analysis, motion planning, and control applications. Techniques from the spatial operator algebra are used to obtain closed form operator expressions for two types of linearized dynamics models, the linearized inverse and forward dynamics models. Spatially recursive algorithms of O ( n ) and O ( n 2 ) complexity for the computation of the perturbation vector and coefficient matrices for the linearized inverse dynamics model (LIDM) are developed first. Subsequently, operator factorization and inversion identities are used to develop corresponding closed-form expres- sions for the linearized forward dynamics model (LFDM). Once again, these are used to develop algorithms of O(n) and O ( n 2 ) complexity for the computation of the perturbation vector and the coefficient matrices. The algorithms for the LFDM do not require the explicit computation of the mass matrix nor its numerical inversion and are also of lower complexity than the conventional O( n3 ) algorithms.

I. INTRODUCTION Linearized dynamics models for manipulators are useful in robot

analysis, motion planning, and control applications [ 11. Optimal trajectory design methods for manipulators require the minimization of a trajectory cost function [2]. The trajectory cost gradient needed during the optimization process is obtained using linearized dynam- ics models. Trajectory sensitivity models are obtained by driving linearized dynamics models with gradients of the nonlinear forces with respect to the manipulator parameters [3] , [4]. These sensitivity models relate the errorshncertainties in the kinematical and inertial manipulator model parameters to errors in the trajectory. In the face of uncertainties, such as from payload variations and friction, ‘‘linear’’ feedback control based on linearized dynamics models in combination with nonlinear feedforward control can be used to enhance robot performance. Linearized dynamics models are also used for adaptive control of manipulators [5], nonlinear decoupling and control in the presence of actuator dynamics [6], and for system identification.

In this paper, techniques from the spatial operator algebra [7] are used to derive closed form operator cxpressions and new recursive al- gorithms for linearized dynamics models for robot manipulators. The spatial operator algebra is a robot modeling and analysis framework that makes use of spatial operators to provide a compact description of robot dynamics, and to derive efficient recursive algorithms for robotics computations.

Linearized models consist of a linearization of the nonlinear dynamics of a manipulator about a nominal trajectory. The nonlinear dynamics of an 11 degree of freedom robot manipulator can be expressed in the form:

T(6.i .k’) = . M ( 6 ) 8 + C ( 8 . i ) (1)

where 0 and T are the vectors of hinge angles and hinge forces respectively, .M E E” is the mass matrix and C E ‘R” is the vector of Coriolis and centrifugal hinge torques. A model relating \lariations in hinge torques 6T to hinge angle perturbations {60,68, b e } , is obtained by a linearization of the nonlinear “inverse dynamics”

Manuscript received May 4, 1991; revised April 24, 1992. The authors are with the Jet Propulsion Laboratory, California Institute of

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