kinematic synthesis of a four-link mechanism with rolling...

Mathematical and Computer Modelling 48 (2008) 805–817www.elsevier.com/locate/mcm

Kinematic synthesis of a four-link mechanism with rolling contactsfor motion and function generation

Jinn-Biau Sheu, Sheng-Lun Hu, Jyh-Jone Lee∗

Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC

Received 9 January 2007; received in revised form 14 October 2007; accepted 23 October 2007

Abstract

In this paper, the kinematic synthesis of a four-link mechanism with rolling contacts is investigated. This mechanism comprisesa two-fingered gripper and a grasped object. The synthesis equations used for motion generation and function generation areestablished. The number of free choices in design variables for the kinematic synthesis is also discussed. Furthermore, theoptimization-based numerical technique is applied to solve the design equations. The optimized solutions are illustrated to discussthe kinematic states of the mechanism. It is also shown that the optimization-based method is effective in finding the admissiblesynthesis solution of the mechanism.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Rolling contacts; Motion generation; Function generation; Optimization

1. Introduction

The kinematic synthesis of hinged mechanisms has been studied extensively in connection with four-barmechanism. Graphical and analytical methods have been established for solving the problem that is related to motion,path, and function generation [1–4]. However, to our knowledge the kinematic synthesis of mechanism with rollingcontacts received little attention in the previous literature. As shown in Fig. 1a, the mechanism is commonly usedas gripping mechanism in industry. Usually, it is composed of two crank links, each may be pin-jointed to theground link. While gripping action takes place, the two links, connected to one actuator via certain linkage, movesimultaneously and hence are able to grip the object. The kinematic and static analyses of such mechanism have beenwell investigated by many researchers [5]. Chen [6,7] examined the kinematic structures and the force analysis ofsuch grippers. Vassura and Nerozzi [8] presented a study of object handling where a multi-finger gripping mechanismmoved an object through various positions and orientations. They showed the possibility of manipulating the objectfrom one position to another by coordinating the gripper’s fingers in stages. We now consider the following situation.If the object is gripped by the two links and the contacts between the links are assumed to be pure rolling, the gripperand object form a four-link mechanism with one degree of freedom (Fig. 1b). Therefore, a limited manipulation ofthe object can be achieved through the motion of the two cranks. Moreover, in certain application, the object may be

∗ Corresponding author.E-mail address: [email protected] (J.-J. Lee).

0895-7177/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2007.10.019

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806 J.-B. Sheu et al. / Mathematical and Computer Modelling 48 (2008) 805–817

Nomenclature

r radius of the circular object (link 3)R1 first position vector of the coupler point on link 3R j j th position vector of the coupler pointZi vectors representing the mechanism in its first reference position as shown in Fig. 2Z′

i j th position of the vector Ziα j angular displacement of link 3β j angular displacement that the object rolls with respect to link 2 when link 2 rotates an angle φ jδ j δ j = R j − R1θ j angular displacement of link 4φ j angular displacement of link 2

required to pass through several prescribed positions and orientations, or the motions of the cranks are related to eachother in a specific manner to complete a given task. In the latter condition, for example, when a robot hand grips anobject, the manipulation of the object may be subject to mechanical constraint embedded in the finger such that theangular motion of the finger segments must be controlled according to certain function. Under such circumstances,the mechanism designer is then required to determine the design parameters of the mechanism to meet with a givenset of requirements. The kinematic analysis of the four-link mechanism with pure rolling pairs has been well studiedby [9–11] or in the related field of multi-jointed dexterous hands design [12–14]. Nonetheless, not much literaturecould be found for the kinematic synthesis of such mechanism. Lee and Tsai [15] studied the synthesis of four-linkmechanism for path generation. This work extends Lee and Tsai’s work to investigate the syntheses of the mechanismfor motion generation and function generation. First, the synthesis equations for the mechanism will be derived usingthe complex number expression. Then, the optimization method will be applied to solve the synthesis equations. Itwill be shown that the maximum number of positions that can be synthesized for motion generation is four and forfunction generation is five. Finally, four numerical examples will be used to show the results of synthesis.

2. Synthesis equations

To adapt to the scope of this work, the following assumptions of the mechanism are made.

1. The contact between the object and the link is pure rolling.2. The object has a circular shape of known radius r .

The gripper–object mechanism with its link represented by vector segment can be now schematically described asfollows. Shown in Fig. 2, a vector Z2 is defined from the pivot of link 2, Oa, to the contact point A between link 2and link 3 (object) and a vector Z3 is defined from point A to the center of link 3. On the other side of the mechanism,a vector Z5 is defined from the pivot of link 4, Ob, to the contact point B between link 4 and link 3 and a vector Z6 isdefined from point B to the center of link 3. Vectors Z1 and Z4 are two vectors defined from the origin to the pivot ofthe revolute joints, respectively. Since we are interested in the motion synthesis of the mechanism, a reference vectorZ7 resting on the coupler link, is defined from the center of link 3 to an arbitrary coupler point P1 on link. Then,vector-loop equations containing the vector on coupler link can be defined by the two independent loop equations as

Z1 + Z2 + Z3 + Z7 = R1 (1a)

Fig. 1a. A two-fingered gripping mechanism.

J.-B. Sheu et al. / Mathematical and Computer Modelling 48 (2008) 805–817 807

Fig. 1b. A four-link mechanism after gripping.

Fig. 2. The vector representation of the mechanism in its two positions.

Z4 + Z5 + Z6 + Z7 = R1 (1b)

where Zi ’s are vectors representing the mechanism in its first reference position and R1 is the first position vector ofthe coupler point on link 3. After some movement, the j th position vector-loop equations can then be written as

Z1 + Z′

2 + Z′

3 + Z′

7 = R j (2a)

Z4 + Z′

5 + Z′

6 + Z′

7 = R j (2b)

where vector Z′

i is the j th position of the vector Zi and R j is the j th position vector of the coupler point. By subtractingEq. (1) from (2), the motion-increment equations can be written as

(Z′

2 − Z2) + (Z′

3 − Z3) + (Z′

7 − Z7) = δ j (3a)

(Z′

5 − Z5) + (Z′

6 − Z6) + (Z′

7 − Z7) = δ j (3b)

where the vector δ j is defined as δ j = R j −R1, and where φ j , α j , and θ j are the angles of link 2, 3, and 4, respectively,measured with respect to their own reference position. As a result, when the object is manipulated vectors Z′

2 andZ′

5 will change both direction and magnitude, while vectors Z′

3, Z′

6, and Z′

7 change only direction. Therefore, theirkinematic relations during the motion must be realized to find the independent design parameters. A free body diagramshowing the kinematic relation between Z2 and Z′

2 is depicted in Fig. 3. Let β j be the angular displacement that theobject rolls with respect to link 2 when link 2 rotates an angle φ j . Then the magnitude of vector Z′

2 can be obtainedfrom the original magnitude of Z2 subtracted by the distance that link 3 has rolled on the link 2. This yields

Z′

2 = Z2eiφ j (1 − β j |Z3|/|Z2|). (4)

Since vector Z3 is defined such that it is always perpendicular to Z2, the following relations can be obtained:

Z3 = Z2ei(−π/2)(|Z3|/|Z2|) (5a)

Z′

3 = Z3eiφ j = −iZ2(|Z3|/|Z2|)eiφ j . (5b)


Fig. 3. Relations of links rotation of left-hand side part of the mechanism.

Similarly, Z′

7 can be written as

Z′

7 = Z7eiα j . (6)

Note that we have introduced the variable β j in Eq. (4). This variable can be further related to the combined motionof link 2 and 3 from the superposition viewpoint. As shown in Fig. 3, the initial position of link 3 is given by a linemarked with an arrow m directing from the center to a designated direction. After some rotation, link 3 is moved tothe new position while the mark is moved to m′. The angular motion of link 3 as given by the angle between m and m′

can be obtained as follows. First fix link 3 to link 2 as a whole link and rotate an angle φ j and then roll link 3 an angleβi with respect to link 2. The sum of the above angular displacements will give the resultant angular displacement oflink 3. This yields

β j = α j − φ j . (7)

Substituting Eq. (7) into Eq. (4) to eliminate β j and again back substituting Eqs. (4), (5a) and (5b) into (3a), resultsin

Z2(eiφ j − 1) + Z2 A[i − ieiφ j − (φ j − α j )eiφ j ] + Z7(eiα j − 1) = δ j (8a)

where A = |Z3|/|Z2|. Eq. (8a) is the motion-increment vector equation for the left-hand part of the mechanism.Similarly, the motion-increment vector equation for the right-hand part of the mechanism can be derived as

Z5(eiθ j − 1) + Z5 B[ieiθ j − i + (θ j − α j )eiθ j ] + Z7(eiα j − 1) = δ j (8b)

where B = |Z6|/|Z5|. Letting Z2 = Z2x + iZ2y , Z5 = Z5x + iZ5y , Z7 = Z7x + iZ7y and δ j = δ j x + iδ j y anddeploying the two complex number equations into scalar form, yielding

[Z2x + AZ2y − AZ2x (φ j − α j )] cos φ j + [−Z2y + AZ2x + AZ2y(φ j − α j )] sin φ j

−Z2x − AZ2y + Z7x cos α j − Z7y sin α j − Z7x = δ j x (9a)

[Z2y − AZ2x − AZ2y(φ j − α j )] cos φ j + [Z2x + AZ2y − AZ2x (φ j − α j )] sin φ j

−Z2y + AZ2x + Z7x sin α j + Z7y cos α j − Z7y = δ j y (9b)

[Z5x − B Z5y + B Z5x (θ j − α j )] cos θ j + [−Z5y − B Z5x − B Z5y(θ j − α j )] sin θ j

−Z5x + B Z5y + Z7x cos α j − Z7y sin α j − Z7x = δ j x (9c)

[Z5y + B Z5x + B Z5y(θ j − α j )] cos θ j + [Z5x − B Z2y + B Z5x (θ j − α j )] sin θ j

−Z5y − B Z5x + Z7x sin α j + Z7y cos α j − Z7y = δ j y . (9d)

We summarize that in Eqs. (9a)–(9d), the design parameters are Z2, Z5, Z7, r(|Z3|), δ j , φ j , θ j , and α j . By countingthe number of design equations versus the number of unknown design variables (Z2, Z5, Z7, φ j and θ j ), we can obtain


Table 1The number of precision positions vs. the number of free design variables for motion generation

No. of positions No. of scalar equations No. of unknowns No. of free choices

2 4 8 4(Z2x , Z2y , Z5x , Z5y , Z7x , Z7y , φ2, θ2)

3 8 10 (above +φ3, θ3) 24 12 12 (above +φ4, θ4) 0

Table 2The number of admissible positions vs. the number of free design variables for function generation

No. of positions No. of scalar equations No. of unknowns No. of free choices

2 2 5 3(Z2x , Z2y , Z5x , Z5y , α2)

3 4 6 (above +α3) 24 6 7 (above +α4) 15 8 8 (above +α5) 0

the set of free choices of design variables of the two-fingered gripping mechanism for motion generation as listed inTable 1. It can be concluded that the maximum number of positions with free choice of design variables is four.

For the mechanism used as function generator, the function-increment equation of the mechanism can be writtenas

(Z′

2 − Z2) + (Z′

3 − Z3) − (Z′

5 − Z5) − (Z′

6 − Z6) = 0. (10a)

Substituting Eqs. (4), (5a) and (5b) into (10a) and rearranging, yields

Z2(eiφ j − 1) + Z2 A[i − ieiφ j − (φ j − α j )eiφ j ] − Z5(eiθ j − 1) − Z5 B[ieiθ j − i + (θ j − α j )eiθ j ] = 0. (10b)

Eq. (10b) can be further deployed into two scalar equations as

[Z2x + AZ2y − AZ2x (φ j − α j )] cos φ j + [−Z2y + AZ2x + AZ2y(φ j − α j )] sin φ j − Z2x − AZ2y

−[Z5x − B Z5y + B Z5x (θ j − α j )] cos θ j − [−Z5y − B Z5x − B Z5y(θ j − α j )] sin θ j + Z5x − B Z5y = 0 (11a)

[Z2y − AZ2x − AZ2y(φ j − α j )] cos φ j + [Z2x + AZ2y − AZ2x (φ j − α j )] sin φ j − Z2y + AZ2x

−[Z5y + B Z5x + B Z5y(θ j − α j )] cos θ j − [Z5x − B Z2y + B Z5x (θ j − α j )] sin θ j + Z5y + B Z5x = 0. (11b)

Table 2 also illustrates the number of admissible positions versus the number of free design variables for functiongeneration. It can be observed that the maximum number of positions with free choice of design variables for thefunction generation is five. It can be also noted that Eqs. (9) and (11) contain the terms φ cos φ and φ sin φ; therefore,the system of equations is transcendental and no closed form solutions can be found. Thus, a numerical method needsto be developed to solve this system of nonlinear equations.

3. Solution procedure

Many different numerical methods for solving the nonlinear kinematic equations were proposed in the past. Acommon method is to use the Newton–Raphson method (NRM) to solve nonlinear equations. However, the majordisadvantage of this method is that it is sensitive to the starting estimate of the solution, i.e., if the starting estimateis not sufficiently accurate, this method usually does not converge. Further, information about the starting mechanismthat can be provided for the kinematic synthesis problem is always little. Another method is to use the optimization-based technique. The advantage of using the optimization method lies in that the starting vectors need not be veryclose to the final solution. In addition, the design constraints are easier to be dealt with provided that the problem isformulated in a constrained optimization form [16–18]. In this work, we shall use the optimization-based algorithmto obtain an optimized mechanism with sufficient accuracy. The methodology is described as follows.


In order to solve Eqs. (9a)–(9d), an initial estimate of the unknown parameters is made. The approximate scalarequations of the motion increment of the mechanism are calculated as

dL j x = [(Z2x )∗

+ A∗(Z2y)∗

− A∗(Z2x )∗(φ∗

j − α j )] cos φ∗

j + [−(Z2y)∗

+ A∗(Z2x )∗

+ A∗(Z2y)∗(φ∗

j − α j )] sin φ∗

j

−(Z2x )∗

− A∗(Z2y)∗

+ (Z7x )∗ cos α j − (Z7y)

∗ sin α j − (Z7x )∗ (12a)

dL j y = [(Z2y)∗

− A∗(Z2x )∗

− A∗(Z2y)∗(φ∗

j − α j )] cos φ∗

j + [(Z2x )∗

+ A∗(Z2y)∗

− A∗(Z2x )∗(φ∗

j − α j )] sin φ∗

j

−(Z2y)∗

+ A∗(Z2x )∗

+ (Z7x )∗ sin α j + (Z7y)

∗ cos α j − (Z7y)∗ (12b)

dR j x = [(Z5x )∗

− B∗(Z5y)∗

+ B∗(Z5x )∗(θ∗

j − α j )] cos θ∗

j + [−(Z5y)∗

− B∗(Z5x )∗

− B∗(Z5y)∗(θ∗

j − α j )] sin θ∗

j

−(Z5x )∗

+ B∗(Z5y)∗

+ (Z7x )∗ cos α j − (Z7y)

∗ sin α j − (Z7x )∗ (12c)

dR j y = [(Z5y)∗

+ B∗(Z5x )∗

+ B∗(Z5y)∗(θ∗

j − α j )] cos θ∗

j + [(Z5x )∗

− B∗(Z5y)∗

+ B∗(Z5x )∗(θ∗

j − α j )] sin θ∗

j

−(Z5y)∗

− B∗(Z5x )∗

+ (Z7x )∗ sin α j + (Z7y)

∗ cos α j − (Z7y)∗ (12d)

j = 2, 3, . . . , n,

where (dL j x , dL j y) represents the x and y components of the left-hand motion increment, (dR j x , dR j y) the x and ycomponents of the right-hand motion increment of the mechanism, and (#)∗ the approximate value of (#). To obtain theoptimal values of the design variables, one can define the objective function as the sum of the squares of the differencebetween each component of the approximate motion-increment vector (dL j x , dL j y) and the desired motion-incrementvector δ j (δ j x , δ j y) and the difference between the approximate rotation increment dα j and the desired α j as

f =

n∑j=2

[(dL j x − δ j x )2+ (dL j y − δ j y)

2+ w1(dα j − α j )

2] (13)

where w1 is a weighting factor for the angular difference.In the meantime, both the approximate left-hand and right-hand motion increments must be compatible, i.e., they

are subjected to the equality constraints

dL j x − dR j x = 0 (14a)

dL j y − dR j y = 0 j = 2, 3, . . . , n. (14b)

The above constraints also imply that the loop closure formed by the vectors in the new position and originalposition remains closed. The procedure can now be summarized as

(i) State the problem as

Minimize f =

n∑j=2

[(dL j x − δ j x )2+ (dL j y − δ j y)

2+ w1(dα j − α j )

2]

Subject to

dL j x − dR j x = 0

dL j y − dR j y = 0

where dL j x , dL j y , dR j x , and dR j y are given in Eq. (12) and dα j is calculated from the kinematic analysis.(ii) If necessary, add estimated ranges of the unknown variables to the optimization problem as inequality constraint.

These steps can be expressed as

c1 ≤ Z2x ≤ d1

c2 ≤ Z2y ≤ d2

c3 ≤ Z3x ≤ d3

...

where ci ’s and di ’s are the lower- and upper-bound values. Adding the inequality constraints may allow theoptimization algorithm to search in a smaller region, thus resulting in more efficiency than an unconstrainedproblem. Besides, with specified regions, it is also helpful in identifying multiple solutions.


(iii) Solve the constrained optimization problem by using the optimization algorithm. In this work, the sequentialquadratic programming (SQP) algorithm provided by MATLAB optimization toolbox [19] is utilized to solve theproblem since it allows us to closely mimic Newton’s method for constrained optimization.

3.1. Function generation problems

A similar procedure for developing the objective function for function generation can also be obtained. Two typicalfunction generation problems are discussed in this work. One is given the initial angles of input link (φ1) and outputlink (θ1) while the other is given the fixed distance (D) between the two points Oa and Ob shown in Fig. 1b. For thefirst case, two more constraint equations need to be considered, and are defined as

φ1 = tan−1(Z2y/Z2x ) (15a)

θ1 = tan−1(Z5y/Z5x ). (15b)

The objective function can be defined as a least-square error:

Minimize f =

n∑j=2

[(dL j x − dR j x )2+ (dL j y − dR j y)

2+ w1(dφ1 − φ1)

2+ w2(dθ1 − θ1)

2] (16a)

where (dφ1, dθ1) represents the approximate values of φ1 and θ1, and (w1, w2) are the weighting factors for the angulardifference. Note that the maximum number of positions with free choice of design variables is reduced to three. Forthe second case, the constraint equation can be obtained in terms of Z2x , Z2y , Z5x , and Z5y :

D = {[|Z2| cos φ1 + r(sin φ1 + sin θ1) − |Z5| cos(θ1)]2+ [|Z2| sin φ1 − r(cos φ1 + cos θ1) − |Z5| sin(θ1)]

2}1/2

where φ1 and θ1 are given in Eq. (15). The objective function is then defined as:

Minimize f =

n∑j=2

[(dL j x − dR j x )2+ (dL j y − dR j y)

2+ (dD − D)2

] (16b)

where dD represents the approximate value of distance D. Note that the maximum number of positions with freechoice of design variables is four. These problems are unconstrained minimization problems; however, they becomeconstrained minimization problem after added the inequality constraints of the unknown variables.

4. Numerical examples

In this section, four examples have been selected to illustrate the design procedure. To show the effectiveness ofnumerical algorithm, three-position and four-position synthesis are executed for both motion and function generations.In these examples, all weighting factors are set to be 1 for simplicity.

Example 1. Synthesis of three-position motion generation.

Fig. 4a depicts the coupler link, a disk with radius r = 28 mm, whose center is to pass three precision pointswith certain prescribed orientation. Hence, vector Z7 = 0. The three positions with respect to the reference origin areP1 = (40, 65), P2 = (24.5, 57), P3 = (13.5, 45) while the rotation-increments of the disk are α2 = 12◦ (or 0.2094rad), α3 = 28◦ (or 0.4887 rad). Thus, the desired motion-increment vectors δ j ( j = 2, 3) are

δ2 = (24.5 − 40) + i(57 − 65) = −15.5 − i8, and δ3 = (13.5 − 40) + i(45 − 65) = −26.5 − i20.

We assume that the design variables are bounded as −100 ≤ Z2x , Z2y , Z5x , Z5y ≤ 100, and 0 ≤ all angles ≤ 3.The initial estimates are given somewhat in a trial and error manner. Table 3 lists the initial estimate and optimizedsolution. Fig. 4b shows the synthesized mechanism associated with the configuration at each precision position.We have also run the optimization problem using different initial estimate and the results are listed in Table 4 forcomparison. The synthesized configuration is shown in Fig. 4c. It can be seen that the mechanism will need to changeits assembly condition when transiting from position 2 to position 3 as is the branch problem occurred in the kinematicsynthesis of hinged four-bar linkage.


Fig. 4a. Three-position to be guided.

Fig. 4b. Example 1 — synthesized mechanism.

Fig. 4c. Example 1 — a different synthesized mechanism.


Table 3Results of numerical analysis for three-position motion-generation synthesis

Parameters Initial Optimized

Z2x 40 −20.9743Z2y 10 31.7886Z5x −15 −2.4537Z5y 60 79.5863Z7x 0 0Z7y 0 0φ2 0.3 0.4834φ3 0.9 1.0210θ2 0.15 0.2068θ3 0.42 0.3898Objective value – 9.9563e–009

Table 4A different result for three-position motion-generation synthesis


Z2x 30 −17.5918Z2y 15 29.0679Z5x −35 −2.4565Z5y 70 79.6001Z7x 0 0Z7y 0 0φ2 0.3 0.5745φ3 0.5 2.1771θ2 0.2 0.2068θ3 0.4 0.3897Objective value – 3.3215e–008

Example 2. Synthesis of four-position motion generation.

In this example, four-position synthesis is studied. An extra point is added between points P2 and P3 inExample 1. The coordinates for the four points are now P1 (40, 65), P2 (24.5, 57), P3 (19, 53), and P4 (13.5,45) while the rotation increment are α2 = 12◦ (or 0.2094 rad), α3 = 19◦ (or 0.3316 rad), and α4 = 28◦

(or 0.4887 rad). In such a condition, no free variables can be assigned in the optimization problem. The designvariables are left unconstrained. The initial values are given as (Z2x , Z2y, Z5x , Z5y, Z7x , Z7y, φ2, φ3, φ4, θ2, θ3, θ4) =

(−21, 32, −3, 79, 0.05, 0.05, 0.5, 0.8, 0.9, 0.2, 0.3, 0.4) and the optimized solution set is found as (−33.5748,41.9312, −38.4299, 140.3626, 20.4728, −13.8462, 0.4180, 0.6053, 0.8909, 0.1460, 0.2074, 0.2879) with the objectivevalue minimized to 3.0852e–007. The synthesized mechanism is shown in Fig. 5a. Similarly, by assigning a differentset of initial values (−25, 27, −10, 70, 0.01, 0.02, 0.2, 0.6, 1.1, 0.3, 0.4, 0.5), a different optimized result can beobtained as (−23.4587, 25.6439, −43.6646, 152.8923, 14.7850, 6.7267, 0.5608, 0.8253, 2.1674, 0.1053, 0.1444,0.1938) with the objective value minimized to 1.0260e–007. The configuration is shown in Fig. 5b, where a branchproblem also occurred when the mechanism transited from position 3 to position 4.

Example 3. Synthesis of three-position function generation.

In this function generation problem, the disk radius r is set as 1.5 inches, and the initial positions of link 2 andlink 4 are specified as φ1 = 98◦ (or 1.7104 rad), θ1 = 82.5◦ (or 1.4399 rad). While link 2 rotates 10◦ (φ2 = 10◦,or 0.1745 rad, ccw) and 30◦ (φ3 = 30◦, or 0.5236 rad, ccw), link 4 rotates 9◦ (θ2 = 9◦, or 0.1571 rad, ccw) and27◦ (θ3 = 27◦, or 0.4712 rad, ccw), respectively. In this example, the objective function Eq. (16a) is used and theQuasi-Newton algorithm provided by the MATLAB optimization toolbox is applied to find a more accurate initialvalue. Then, with the found initial value the Gauss–Newton algorithm provided by the same toolbox is used to find


Fig. 5a. Example 2 — synthesized mechanism.

Fig. 5b. Example 2 — a different synthesized mechanism.

the optimized solution. Table 5 illustrates the initial estimate and optimized solution. Fig. 6 shows the synthesizedmechanism associated with the configuration at each precision position.

Example 4. Synthesis of four-position function generation.

In this example, four-position function generation synthesis is studied where the objective function Eq. (16b) isapplied. The distance (D) between two fixed points, disk radius (r ), and three angular displacements of input andoutput links are specified as D = 6.4, r = 2.0 (inches), φ2 = 10◦ (or 0.1745 rad), φ3 = 20◦ (or 0.3491 rad), φ4 = 30◦

(or 0.5236 rad), θ2 = 5◦ (or 0.0873 rad), θ3 = 10◦ (or 0.1745 rad), and θ4 = 15◦ (or 0.2618 rad). We assume thatthe design variables are subjected to the constraints 2r ≤ |Z2|, 2r ≤ |Z5|, and 0.5r ≤ (Obx − Oax ). The sameoptimization algorithm used in Example 1 can be applied to solve the problem. Table 6 shows the initial estimates andoptimized solution. The synthesized mechanism associated with the configuration at each precision position is shownin Fig. 7.


Table 5Results for three-position function generation synthesis


Z2x −1 −0.5214Z2y 3.5 3.7104Z5x 0.1 0.5212Z5y 2.0 3.9592α2 0.02 0.0596α3 0.04 0.1871Objective value – <1e–14

Table 6Results for four-position function generation synthesis


Z2x −1 −0.9582Z2y 3 6.5553Z5x 1 1.9971Z5y 5 12.8590α2 0.2 0.0984α3 0.5 0.2212α4 1.0 0.3370Objective value – 6.4e–5

Fig. 6. Example 3 — synthesized mechanism.

5. Discussion

There exist many other numerical methods for solving a system of nonlinear equations. As with most numericaliteration methods, the optimization methodology does not guarantee in finding admissible solutions given a setof initial estimates. This effect implies that many local solutions may exist in the system of equations such thatthe algorithm is still sensitive in finding the global minima of the optimization in our problem. Nonetheless, theoptimization methodology does diminish the burden of guessing the ‘right’ initial values. In this work, we have alsotried two searching algorithms to find proper initial values: general genetic algorithm [20] and cyclic coordinatedescent (CCD) method [21,22]. Our experience shows that the CCD method does not converge as well as the


Fig. 7. Example 4 — synthesized mechanism.

method used in this paper while the general genetic searching algorithm yields no solution most of the time. Despitethese, advanced searching algorithms may be worth further exploring. Another advantage of using the optimization-based method is that adding upper/lower bounds for design variables as inequality constraints is easily treated in theformulation. This characteristic may be helpful if more design constraints, which are derived from engineering viewpoints, are to be imposed in the problem.

6. Conclusion

In this paper, a methodology for the kinematic synthesis of gripping mechanism with rolling pairs has beendeveloped. Design equations for the mechanism as rigid-body guidance and function generator are derived. It isshown that up to four precision positions of the object can be prescribed for the rigid-body guidance synthesis, andfive positions for function generation synthesis. Further, the optimization method for the synthesis is also applied tosolve the design equations. Through the above-mentioned four examples, we have successfully shown how to computethe numerical solutions based on the optimization method. It is hopeful that the result of this paper may be useful inthe object manipulation planning of the robotic field in the future.

Acknowledgment

The authors wish to gratefully acknowledge the financial support of the National Science Council of Taiwan viaGrant NSC-93-2212-E-002-056.

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