Computer-Aided Design 39 (2007) 1075–1080www.elsevier.com/locate/cad

On the application of CAD technology for the synthesis of spatialrevolute–revolute dyads

Kevin Russella,∗, Wen-Tzong Leeb, Raj S. Sodhic

a Armaments Engineering and Technology Center, US Army Research, Development and Engineering Center, Picatinny Arsenal, NJ 07806-5000, USAb Department of Information Management, Leader University, Taiana, 70970, Taiwan

c Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA

Received 15 February 2006; accepted 15 July 2007

Abstract

This work presents a method based on Computer Aided Design or CAD for facilitating the synthesis of Revolute–Revolute (R–R) dyads withadjustable moving pivots. The CAD-based method presented in this work ensures that all prescribed rigid-body parameters used to synthesize theR–R dyad satisfy particular kinematic requirements of an R–R dyad. Through the application of this CAD method, five of the six general R–R dyadconstraint equations are satisfied and therefore not essential for the synthesis of the R–R dyad. By reducing the number of dyad design constraintsfrom six to one, the user can synthesize R–R links with adjustable moving pivots for multi-phase motion and path generation applications. Theexample included demonstrates the use of the CAD method in the synthesis of an RRSS path generator with adjustable moving pivots.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: RRSS mechanism; Path generation; Adjustable mechanisms; Spatial mechanisms

1. Introduction

1.1. The revolute–revolute dyad

Fig. 1 illustrates a spatial R–R dyad. Fig. 2 illustrates theRRSS (Revolute–Revolute–Spherical–Spherical) and RRSC(Revolute–Revolute–Spherical–Cylindrical) mechanisms—twofour-link, single degree of freedom spatial mechanisms thatinclude the R–R dyad. Being closed-loop spatial kinematicchains, the RRSS and RRSC offer both spatial motion and agreater degree structural soundness than comparable open-loopkinematic chains.

Having a link with both a grounded revolute joint anda revolute joint as its moving pivot restricts any “passive”degree of freedom of the coupler in the RRSS and RRSCmechanisms. The elimination of a passive degree of freedom(passive meaning the link can rotate about the imaginary axisbetween its two joints) of a link is essential if that link isaffixed to a rigid body whose motion is to be prescribed. Thispassive degree of freedom is prevalent in the coupler of the

∗ Corresponding author. Tel.: +1 973 724 6073; fax: +1 973 724 6027.E-mail address: kevin.russell1@us.army.mil (K. Russell).

0010-4485/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2007.07.004

RSSR mechanism [1,2], and as a result, the RSSR mechanismmakes a poor candidate for coupler motion and path generationapplications.

1.2. Literature review and scope of work

Several authors have made significant contributions in theareas of RRSS and RRSC mechanism synthesis and analysis.Su, Collins and McCarthy [3,4] classified the movement ofthe RRSS spatial linkage in terms of its link dimensions. Leeand Yoon [5] proposed a method of using Euler parametersand quaternion algebra for the kinematic analysis of an RRSSspatial motion generator. Lee, Youm and Chung [6] developedthe quadratic input–output relation and the associated mobilitycondition for the RRSS mechanism. Russell and Sodhi [7–10]presented an instant screw axis-based synthesis method for theRRSS mechanism [7], multi-phase motion generation methodsfor the RRSS mechanism using precise prescribed rigid-body positions and rigid-body positions with tolerances [8,9] and a method to synthesize RRSS mechanisms to achievemultiple phase of prescribed finite and multiply separatedpositions [10].

This work presents a CAD-based method to facilitate thesynthesis of Revolute–Revolute dyads with adjustable moving

1076 K. Russell et al. / Computer-Aided Design 39 (2007) 1075–1080

Fig. 1. The spatial R–R dyad.

pivots. The CAD-based method presented in this work ensuresthat all prescribed rigid-body parameters used to synthesizethe R–R dyad satisfy specific kinematic requirements of anR–R dyad—thus reducing the number of required constraintequations for the R–R dyad from six to one. With sucha substantial reduction in R–R dyad design constraints, theuser can synthesize such a dyad with adjustable movingpivots for multi-phase motion and path generation applications.The example in this work demonstrates the application ofCAD technology in the synthesis of an RRSS mechanismwith adjustable moving pivots to approximate two groups ofprescribed rigid-body parameters.

2. R–R dyad spatial rigid-body guidance and CADtechnology

2.1. R–R dyad spatial rigid-body guidance

Given a fixed pivot a0, a moving pivot a1 and fixed andmoving pivot joint axes ua0 and ua1 respectively, Eq. (1)through (4) [1,2] represent the constraints that are consideredfor the displacement of a spatial R–R dyad (see Fig. 3). Thecondition associated with Eq. (1) is that both revolute jointunit vectors (ua0 and ua1) remain orthogonal to the link vectora0 − a1 for an arbitrary displacement of the R–R dyad. Thecondition associated with Eq. (3) is that the scalar length oflink a0 − a1 remains constant for an arbitrary displacement ofthe R–R dyad. The condition associated with Eq. (4) is that thescalar distance between the endpoints of the unit vectors ua0and ua1 remains constant for an arbitrary displacement of theR–R dyad. Eq. (4) is a secondary constant length condition thatconstrains the position of ua1 with respect to ua0.

In some kinematic motion models for mechanisms includingR–R dyads (e.g., RRSS and RRSC mechanisms) the R–Rdyad joint axes are not required to be orthogonal to the linkitself [1]. As a result of Eq. (1), the R–R dyad calculated is aparticular subset of R–R dyad. Since the CAD-based methodto be presented in this work includes Eq. (1) through (4) as thefoundation for its methodology, the conditions in Eq. (1) willbe preserved, and subsequently, only R–R dyads with joint axesthat are orthogonal to the link can be produced.

Eq. (1) through (4) form a set of 12 design equations with 12unknown scalar components of ua0, ua1, a0 and a1 [1,2]. Giventhese equations, the maximum number of rigid-body positionsthat can be specified for an R–R dyad for rigid-body guidanceis three, with no arbitrary choice of parameter [1,2].

(ua0)T(a j − a0) = 0, (ua j )

T(a j − a0) = 0,

j = 1, 2, 3, (1)

(ua0)T(ua0) = 1, (ua1)

T(ua1) = 1, (2)

(a j − a0)T(a j − a0) = (a1 − a0)

T(a1 − a0), j = 2, 3, (3)

[(a j + ua j ) − (a0 + ua0)]T[(a j + ua j ) − (a0 + ua0)]

= [(a1 + ua1) − (a0 + ua0)]T[(a1 + ua1) − (a0 + ua0)],

j = 2, 3. (4)

2.2. R–R dyad spatial rigid-body guidance with CADtechnology

The absolute position of a rigid body in an R–R dyad isthe result of two angular displacements [1,2]. One angulardisplacement is the rotation of the rigid body about the joint axisvector ua1. The other angular displacement is the rotation of therigid body about the joint axis vector ua0. This motion can beaccurately replicated in CAD software with relative ease sincemodern 3D CAD tools enable the user to create and manipulategeometry in three-dimensional space.

Fig. 4 illustrates the CAD approach to manipulating spatialcoordinates so that they follow rotations about an arbitraryua1 and a ua0 that is parallel to the global z-axis of a spatialx–y–z coordinate frame. The spatial coordinates are then usedto characterize the motion of the R–R dyad rigid body. Furtherdiscussion on rigid-body motion characterization is included inSection 3.

To prescribe joint axis ua1, a plane is defined that isorthogonal to ua1. This plane includes a prescribed rigid-bodypoint. In Fig. 4, two parallel planes are defined that includea rigid-body point p and a rigid-body point q. Both planesare orthogonal to ua1. To produce additional positions for theprescribed rigid-body points, the points can be translated withintheir respective planes and the planes themselves can be rotatedabout ua0 (the x-axis in Fig. 4). This CAD-based approachnot only ensures that the prescribed rigid-body points rotateabout joint axes ua1 and ua0, but it also reduces the requiredconstraint equations to the constant length condition onlyEq. (3) because Eqs. (1), (2) and (4) are satisfied as a resultof the CAD approach.

K. Russell et al. / Computer-Aided Design 39 (2007) 1075–1080 1077

Fig. 2. The RRSS and RRSC mechanisms.

Fig. 3. Spatial R–R dyad in arbitrary rigid-body positions (1 and “ j”).

3. RRSS multi-phase path generation and synthesisequations

3.1. RRSS multi-phase path generation

For the remainder of this work, the mechanism offocus for synthesis is the RRSS mechanism to demonstratethe application of the presented CAD-based method. Inconventional path generation, the objective is to calculate themechanism parameters required to approximate or achievea single group or phase of prescribed rigid-body pathpoints [1,2]. In multi-phase path generation the objective is tocalculate the mechanism parameters required to approximate orachieve multiple groups of prescribed rigid-body path points(e.g., rigid-body path points p and q in Fig. 5). For thelatter, the calculated mechanism includes fixed pivots, movingpivots and/or link lengths that can be adjusted to achieve eachphase of prescribed rigid-body path points. Fig. 5 illustrates

Fig. 4. CAD approach for prescribing R–R dyad rigid-body positions.

an RRSS mechanism with adjustable moving pivots. Whenmoving pivots a1 and b1 are used, one phase of prescribed rigid-body path points are achieved and another phase of rigid-bodypath points are achieved when moving pivots a1n and b1n areincorporated.

3.2. RRSS multi-phase path generation equations

As mentioned in Section 2.2, by incorporating the CAD-based rigid-body point selection method, the sole parameterrequired to synthesize an R–R dyad is the constant lengthconstraint Eq. (3). An R–R link with adjustable moving pivotsand a constant link length has a fixed pivot a0, moving pivotsa1 and a1n and a scalar link length R1. Since the R–R link isconstrained to rotate in the x–y plane when joint axis ua0 isaligned with the global z-axis (as shown in Fig. 4), the fixedand moving pivot variables for this link are the following:

a0 = (a0x , a0y) a1 = (a1x , a1y) a1n = (a1nx , a1ny).

1078 K. Russell et al. / Computer-Aided Design 39 (2007) 1075–1080

Fig. 5. RRSS mechanism with adjustable moving pivots and rigid-body points.

Eq. (5) through (10) are used to calculate the six unknownsin a0, a1 and a1n . The scalar link length R1 is specified.Eq. (5) through (10) are used to calculate the phase 1 and phase2 adjustments of the R–R link (a0 − a1 and a0 − a1n).

Eq. (11) is the rigid-body displacement matrix for Eq. (5)through (10) [1,2]. This displacement matrix incorporates thex and y-coordinates of a rigid-body point p and the z-axisrotational displacement of the rigid body.

([D12]a1 − a0)T([D12]a1 − a0) − R2

1 = 0 (5)

([D13]a1 − a0)T([D13]a1 − a0) − R2

1 = 0 (6)

([D14]a1 − a0)T([D14]a1 − a0) − R2

1 = 0 (7)

([D15]a1n − a0)T([D15]a1n − a0) − R2

1 = 0 (8)

([D16]a1n − a0)T([D16]a1n − a0) − R2

1 = 0 (9)

([D17]a1n − a0)T([D17]a1n − a0) − R2

1 = 0 (10)

[D1 j ] =

cos α1 j − sin α1 j p j x − p1x cos α1 j + p1y sin α1 j

sin α1 j cos α1 j p j y − p1x sin α1 j − p1y cos α1 j

0 0 1

.

(11)

The sole parameter required to synthesize an S–S dyad is theconstant length constraint [1,2]. An S–S link with adjustablemoving pivots and an adjustable link length has a fixed pivotb0, moving pivots b1 and b1n and scalar link lengths R2 andR3. Since the S–S link is free to travel in 3D space, the fixedand moving pivot variables for this link are the following:

b0 = (b0x , b0y, b0z) b1 = (b1x , b1y, b1z)

b1n = (b1nx , b1ny, b1nz).

Eq. (12) through (17) are used to calculate the six of the nineunknowns in b0, b1 and b1n . The scalar link lengths R2 andR3 are specified. Eq. (11) through (16) are used to calculate thephase 1 and phase 2 adjustments of the S–S link (b0 − b1 andb0 − b1n).

Eq. (18) is the rigid-body displacement matrix for Eq. (12)through (17) [1,2]. This displacement matrix incorporates thex, y and z-coordinates of a rigid-body point p and the z, and y-axis rotational displacements of the rigid body. In the rotationaldisplacement matrix Eq. (19) rotational displacements of α andγ about the z and y-axes respectively align the rigid body inposition “ j” to position 1 and a rotational displacement of β

about the z-axis align both rigid-body positions to the x-axis.The authors elected to employ Eq. (19) over the commonly usedspatial z–y–x rotation (by α, β and γ respectively), which isalso employable in Eq. (18).

Rigid-body point p is approximated using the pathgeneration method presented in this section. To measure theprescribed rigid-body rotation angles in Eqs. (11) and (18) (α,β and γ ) an additional “dummy” rigid-body point is requiredto produce a line (from which the angles are measured). Rigid-body point q in Fig. 5 serves this purpose.

([D12]b1 − b0)T([D12]b1 − b0) − R2

2 = 0 (12)

([D13]b1 − b0)T([D13]b1 − b0) − R2

2 = 0 (13)

([D14]b1 − b0)T([D14]b1 − b0) − R2

2 = 0 (14)

([D15]b1n − b0)T([D15]b1n − b0) − R2

3 = 0 (15)

([D16]b1n − b0)T([D16]b1n − b0) − R2

3 = 0 (16)

([D17]b1n − b0)T([D17]b1n − b0) − R2

3 = 0 (17)

[D1 j ] =

[[Rα1 j β1 j γ1 j ] p j − [Rα1 j β1 j γ1 j ]p1

0 0 0 1

](18)

[Rαβγ ] = [Rβ,z]−1

[Rγ,y][Rβ,z][Rα,z]. (19)

4. RRSS moving pivot adjustment example

Table 1 includes two phases of prescribed rigid-bodypositions and their respective displacement angles. The rigid-body positions in this table were determined using the CAD-based rigid-body selection approach described in Section 2.2.The axis orthogonal to the prescribed planes (where the rigid-body points) is the moving pivot joint axis ua1 and the axisthese planes rotate about is the joint axis ua0. By rotating theprescribed planes about ua0 and rotating and translating therigid-body points in their respective planes, new rigid-bodypoints were obtained.

Eqs. (5) and (10) were used to calculate six of the eightunknowns in a0, a1 and a1n . The variable R1 has specified valueof 1.

Using the following initial guesses:

a0 = (0.01, 0.01), a1 = (0.01, 1), a1n = (−0.1, 0.9)

the adjustable R–R link solutions converge to

a0 = (−0.0877, −0.0445, 1), a1 = (−0.1356, 0.9537, 1),

K. Russell et al. / Computer-Aided Design 39 (2007) 1075–1080 1079

Table 1Prescribed rigid-body positions and orientation angles for the adjustable RRSSmechanism

PHASE 1p α (◦) β (◦) γ (◦)

Position 1 0.5000, 1.2500, 0.9000Position 2 0.3205, 1.2314, 0.8948 2.4104 −36.1004 0.6954Position 3 0.1503, 1.1833, 0.8919 4.8188 −38.5088 1.2251Position 4 −0.0065, 1.1091, 0.8919 7.3064 −40.9965 1.5776

PHASE 2

Position 1 0.5000, 1.2500, 0.9000Position 5 0.3275, 1.2298, 0.8964 0.2224 −33.9124 0.8652Position 6 0.1611, 1.1791, 0.8903 0.3979 −34.0880 1.7044Position 7 0.0062, 1.0990, 0.8903 0.4693 −34.1594 1.7433

Note: ua0 = [0, 0, 1], ua1 = [sin 10◦, 0, cos 10◦] and q1 =

[0.0444, 1.0700, 0.9562].

Fig. 6. The synthesized adjustable RRSS path generator.

a1n = (−0.3070, 0.9308, 1).

Eq. (12) through (17) were used to calculate six of the elevenunknowns in b0, b1 and b1n . The variables R2 and R3 werespecified to 1 and 0.9110 respectively and b0 = (1, 0, 1).

Using the following initial guesses:

b1 = (1.1, 0.9, 0.9), b1n = (0.9, −0.9, 0.9)

the adjustable S–S link solutions converge to

b1 = (1.1396, 0.8472, 0.6937),

b1n = (1.0134, 0.9081, 0.5779).

The synthesized adjustable RRSS path generator isillustrated in Fig. 6. Table 2 includes the rigid-body pointsachieved by the adjustable RRSS path generator. In Table 2,rigid-body positions 2, 3 and 4 were achieved with crankdisplacement angles of 9.9◦, 19.8◦ and 30◦. Rigid-bodypositions 6, 7 and 8 were achieved with crank displacementangles 9.7◦, 19.7◦ and 29.8◦.

5. Discussion

Although the mechanism of focus for the example in thiswork is the RRSS mechanism (due to the amount of recent

Table 2Rigid-body points achieved by the synthesized adjustable RRSS path generator

PHASE 1p

Position 1 0.5000, 1.2500, 0.9000Position 2 0.3191, 1.2349, 0.8942Position 3 0.1483, 1.1913, 0.8906Position 4 −0.0131, 1.1208, 0.8895

PHASE 2

Position 5 0.5000, 1.2500, 0.9000Position 6 0.3269, 1.2372, 0.8935Position 7 0.1550, 1.1951, 0.8896Position 8 −0.0071, 1.1251, 0.8888

Note: ua0 = [0, 0, 1] and ua1 = [sin 10◦, 0, cos 10◦] and q1 =

[0.0444, 1.0700, 0.9562].

work done on the RRSS mechanism), the CAD-based methodpresented can be applied to synthesize any R–R dyad whereone revolute joint is grounded and the other revolute joint isthe moving pivot (as illustrated in Fig. 3). Other mechanismsthat include such a configurations are the RRSC mechanism, thespatial RRRR mechanism [1,2] and any three-link (ground linkincluded) open-loop kinematic chain that includes a link witha grounded revolute joint and a moving pivot revolute joint.This work considers moving pivot adjustments of the RRSSmechanism only. Although the RRSS mechanism can undergofixed pivot adjustments, such an adjustment would require aCAD method that is distinct from the method presented in thiswork. The rotational displacement matrix presented as Eq. (18)uses z–y–z rotations. The user however can incorporate anyrotation matrix sequence that he/she prefers (e.g., commonlyused z–y–x rotations). The particular CAD software used toaccomplish the procedure in Section 2.2 is AutoCAD. Thesoftware Mathematica was used to used to calculate solutionsfor the equation sets in Section 3.2.

6. Conclusion

The presented CAD-based rigid-body point selectionmethod for the design of R–R dyads was demonstrated to beeffective for the synthesis of a moving pivot adjustable R–Rdyad of an RRSS path generator. Through the application ofthis CAD method, only the constant length R–R dyad constraintequation is required—reducing the number of constraintequations from six to one. By reducing the number of dyaddesign constraints to a single constraint type, the user cansynthesize R–R links with adjustable moving pivots for multi-phase motion and path generation applications.

References

[1] Suh CH, Radcliffe CW. Kinematics and mechanism design. New York:John Wiley and Sons; 1978.

[2] Sandor GN, Erdman AG. Advanced mechanism design: Analysis andsynthesis. Englewood Cliffs: Prentice-Hall; 1984.

[3] Haijun S, Collins CL, McCarthy MJ. Classification of RRSS linkages.Mechanism and Machine Theory 2002;37(11):1413–33.

[4] Haijun S, McCarthy MJ. Classification of designs for RRSS linkages. In:2001 ASME design engineering technical conference and computers andinformation in engineering conference. 2001.

1080 K. Russell et al. / Computer-Aided Design 39 (2007) 1075–1080

[5] Lee K, Yoon Y. Kinematic synthesis of RRSS spatial motion generatorsusing Euler parameters and quaternion algebra. Proceedings of theInstitution of Mechanical Engineers, Part C: Journal of MechanicalEngineering Science 1993;207(5):355–9.

[6] Lee D, Youm CY, Chung W. Mobility analysis of spatial 4 and 5-linkmechanisms of the RS class. Mechanism and Machine Theory 1996;31(5):673–90.

[7] Russell K, Sodhi RS. Instant screw axis point synthesis of the RRSSmechanism. Mechanism and Machine Theory 2002;37(10):1117–26.

[8] Russell K, Sodhi RS. Kinematic synthesis of adjustable RRSSmechanisms for multi-phase motion generation. Mechanism and MachineTheory 2001;36(8):939–52.

[9] Russell K, Sodhi RS. Kinematic synthesis of adjustable RRSSmechanisms for multi-phase motion generation with tolerances.Mechanism and Machine Theory 2002;37(3):279–94.

[10] Russell K, Sodhi RS. Kinematic synthesis of RRSC mechanisms formulti-phase finite and multiply-separated positions. Transactions of theCSME/de la SCGM 2002;26(3):263–80.