Solution Region Synthesis Methodology of RCCC Linkages for Four Poses

This paper presents a synthesis methodology of RCCC linkages based on the solution region methodology, R denoting a revolute joint and C denoting a cylindrical joint. The RCCC linkage is usually synthesized via its two defining dyads, RC and CC. For the four poses problem, there are double infinite solutions of the CC dyad, but there is no solution for the RC dyad. However, if a condition is imposed that leads to a coupling of the two dyads, a maximum of four poses can be visited with the RCCC linkage. Unfortunately, until now, there is no methodology to synthesize the RCCC linkage for four given poses besides optimization method. According to the coupling condition above, infinite exact solutions of RCCC linkages can be obtained. For displaying these RCCC linkages, we first build a spherical 4R linkage solution region. Then solutions with circuit and branch defects can be eliminated on this solution region, so that the feasible solution region is obtained. An RCCC linkage can be obtained by using the prescribed spatial positions and selected a value on the feasible solution region. We take values on the feasible solution region by a certain step length and many exact solutions for RCCC linkages can be obtained. Finally we display these solutions on a map, this map is the solution region for RCCC linkages.


Introduction
The synthesis of spatial RCCC linkages has been received much more attentions recently.A formulation based on dual algebra is proposed for the approximate synthesis of RCCC linkage motion generation (Angeles, 2014).An approach is introduced for the synthesis of the axode of an RCCC linkage (Figliolini et al., 2016).An RCCC linkage is synthesized by a robust optimization method (Al-Widyan and Angeles, 2012).Fourier series theory is used for path generation of RCCC linkage (Sun et al., 2012(Sun et al., , 2017)).A semi-graphical approach is proposed for the synthesis of 4C linkage for five given poses to obtain a robust solution (Bai and Angeles, 2012).A complete classification scheme is developed for planar, spherical 4R and spatial RCCC mechanisms (Murray and Larochelle, 1998).Larochelle addressed the issue of branch and circuit analysis of spatial 4C mechanisms for rigid body guidance (Larochelle, 2000).
There are no exact solutions for the synthesis of RCCC linkage for four prescribed coupler poses (Bai and Angeles, 2012).In some literatures (Bai and Angeles, 2015), RC dyad is synthesized by optimization method and CC dyad is syn-thesized by line congruence, only one approximate solution for RCCC linkage can be obtained for four poses.In this paper, we add constraints between RC dyad and CC dyad, thus the number of unknowns more than the number of equations and infinite exact solutions for RCCC linkage are obtained.For synthesizing more RCCC linkages, we propose a solution region methodology by which infinite exact solutions can be obtained.The synthesis process of this methodology is separated into two parts.The first part is to build a spherical 4R linkage solution region based on Burmester curves.The Burmester curves are determined by four orientations from the four poses.Then we eliminate those solutions having circuit and branch defects and obtain the feasible solution region.In second part, we couple the constraint conditions for an RC dyad and a CC dyad, and obtain 23 equations with 24 unknowns.Therefore, there is one parameter to be chosen freely.Finally we select different values on the feasible solution region, and solve these equations using Bertini software, and build an RCCC linkage solution region for displaying these solutions.The solution region methodology introduced in this paper has been applied in the synthesis of planar and Published by Copernicus Publications.spherical linkages (Han and Qian, 2009;Yang et al., 2011Yang et al., , 2012;;Yin et al., 2012;Cui and Han, 2016).We also synthesized the 5-SS platform linkage by solution region methodology (Han and Cui, 2017).We proposed the primary idea of the synthesis of RCCC linkages for four poses by solution region map (Cao and Han, 2016;Han and Cao, 2017), but the method of building solution region isn't given.

Problem formulation
A spatial CC dyad is shown in Fig. 1.The CC dyad is meant to carry the rigid body through a set of given poses, specified by coordinates p i (p ix , p iy , p iz ) and orientations (θ i , α i , β i ) respect to a reference coordinate frame (xyz) as shown in Fig. 1, where orientation angles θ i , α i , β i are three Euler angles.The unit vectors u 1 and v are along the CC dyad cylindrical pairs moving and fixed axes, respectively.Vector u i denotes the unit vector of u 1 at the ith pose.The CC dyad can be geometrically regarded as a link composed of two skew lines, jointed each other by means of a third line, their common perpendicular (Fig. 1).M 1 is the intersection point of the common perpendicular and moving axis.F 1 is the intersection point of the common perpendicular and fixed axis.h donates the distance between the moving and the fixed axes of the CC dyad.S denotes the sliding displacement of the cylindrical pair from the first to the ith pose, S is defined as where S i denotes the sliding displacement of the cylindrical pair from the pose i-1 to the pose i.When the sliding direction of the cylindrical pair (while the coupler moves from one pose to the next) is in the same direction with vector v, S i is positive, whereas negative.Note that the CC dyad must satisfy two constraints.The angle of twist between the fixed axis and the moving axis must remain constant from 1st pose to ith pose.The moment of vector u i about axis v must remain constant from 1st pose to ith pose.Hence, the CC dyad  constraint equations can be written as follows.
In Eq. ( 2), m 1 , m i , f 1 and f i are the position vectors of M 1 , M i , F 1 and F i respectively.Vectors u 1 , u i and v are unit vectors, Therefore For ensuring the vector m 1 − f 1 is orthogonal to their unit vectors, we have In Eq. ( 1), u i can be expressed as In Eq. ( 5), R i is a rotation matrix which rotates rigid body from 1st to ith orientation.For rotating rigid body from 1st to ith orientation, first, we rotate the rigid body to make its orientation coincide with the reference coordinate by three Euler angles (-β 1 , -α 1 and -θ 1 around the z-, xand z-axes, respectively), then rotate it to the ith by three Euler angles (θ i , α i and β i around the z-, xand z-axes, respectively).Therefore R i can be expressed as where X, Z are pure rotation matrixes around x-, z-axes, respectively.All rotation matrixes follow the right-hand rule.m i and f i can be expressed as matrix forms where U 1x , U ix , V x and P ix are cross-product matrixes of vectors u 1 , u i , v, and p i .
Equations ( 1) to (4) constitute a set of 2n+2 (2(n−1)+4 = 2n + 2) constraint equations with twelve unknowns, which are scalar components of u 1 , v, m 1 and f 1 .Therefore, the maximum number of poses, which can be specified for a CC dyad to be used for rigid body guidance, is five.
The RC dyad shown in Fig. 2 must satisfy all of the constraints imposed by a CC dyad plus an additional displacement constraint, the sliding S i along the fixed axis is zero Equations ( 1), ( 2), (3), ( 4) and ( 9) constitute a set of 3n + 1 (3(n − 1) + 4 = 3n + 1) constraint equations with twelve unknowns, which are scalar components of u 1 , v, m 1 and f 1 .Therefore, the maximum number of poses, which can be specified for a CC dyad to be used for rigid body guidance, is three with arbitrary choice of any two scalars.
An RCCC linkage is shown in Fig. 3.We use u 1 , v, m 1 and f 1 to represent the RC dyad and use u * 1 , v * , m * 1 and f * 1 to represent the CC dyad.If we synthesize RC dyad and CC dyad separately, then simply couple the RC dyad and CC dyad with a coupler link for a RCCC linkage to be used www.mech-sci.net/9/297/2018/Mech.Sci., 9, 297-305, 2018 for rigid body guidance, the maximum number of specified poses is three.When we add a constraint between two dyads instead of Eq. ( 9), we have where g 1 is the position vector of G 1 as shown in Fig. 3. g 1 can be expressed by vectors v, f 1 , v * and f * 1 .Then the RCCC linkage synthesis problem involves 5n+3 (5(n−1)+ 8 = 5n + 3) equations, Eqs. ( 1), ( 2), ( 3), ( 4) and ( 10).These equations are to be solved for 24 unknowns, which are components of the eight vectors The maximum number of poses, which can be specified for an RCCC linkage to be used for rigid body guidance, is four with arbitrary choice of one scalar.

Process of synthesizing RCCC linkages
We separate the synthesis process into two parts.For determining the range of α c (α c is arbitrary choice from 24 unknowns, here α c is defined as the latitude of a point on spherical circle curve which will be mentioned below), we build a spherical 4R linkage feasible solution region.Then we set up the synthesis formulation for synthesizing RCCC linkages.

Determination of spherical Burmester curves and
Building of the spherical 4R linkage solution regions In this section, we determine the Burmester curves and build spherical 4R linkage solution region by these curves.On this solution region, we can classify the mechanism types and eliminate circuit and branch defects.

Spherical Burmester curves
Unit vector v is along the fixed axis, whereas unit vector u 1 is along the moving axis.For the four given poses problem, n = 4. Let then Eq. ( 1) can be rewritten as Since unit vector v cannot be zero, B must be non-full rank.Therefore, its determinant is zero, i.e., |B| = 0.The spherical Burmester circle point curve can be expressed as Likewise, the spherical Burmester center point curve can be expressed as where (Bai and Angeles, 2012).We use spherical coordinates on the unit sphere, i.e., latitude and longitude, to describe the unit vectors of all four cylindrical directions.Let α c and β c be the longitude and latitude of a point on spherical circle curve, and α o and β o be the latitude and longitude of a point on spherical center curve. Therefore Each point on the Burmester circle curve corresponds to the point on the Burmester center curve.Connecting the corresponding points, we obtain an RR dyad.A spherical 4R linkage can be synthesized by selecting two different points on the Burmester circle (or center curve).

Building of the spherical solution regions
Let α c (the latitude of Burmester circle point curve) as the xand y-axes, we build a solution region for displaying infinite spherical 4R linkage solutions, as shown in Fig. 6a.Building process in detail is shown in Sect. 4. The circuit and branch defects can be eliminated on this solution region.
The feasible solution region is built after eliminating the defects (Larochelle, 2000) from spherical 4R solution region, as shown in Fig. 6b.

The synthesis formulation of RCCC linkages
We have discussed the number of the unknowns and the equations for synthesizing RCCC linkages for four poses above.For synthesizing driving link, substituting Eq. ( 5) into Eq.( 1), we have www.mech-sci.net/9/297/2018/Mech.Sci., 9, 297-305, 2018 Substituting Eqs. ( 7) and (8) into Eq.( 2), we have Because the Eqs.( 16) and ( 17) can apply for both CC and RC dyad, we use u 1 , v, m 1 , f 1 to represent the direction and position vectors for RC dyad (driving link); we use u * 1 , v * , m * 1 , f * 1 to represent the direction and position vectors for CC dyad (driven link).The two sets of vectors can be applied for Eqs. ( 16) and ( 17).Therefore, substituting the unit vectors and position vectors of driven link into the Eqs.( 16) and ( 17), we have The twelve equations are list below Equations ( 3) and ( 4) are unit vectors equations and orthogonality equations (Suh and Radcliffe, 1978) for driving link.We rewrite the Eqs.( 3) and ( 4) with unknowns u * 1 , v * , m *  nality equations for driven link Equation ( 10) constrains the sliding displacement between the driving link and the fixed link.So, there are 23 equations (Eqs. 3,4,10,20 and 21) to solve 24 unknowns, i.e., u 1 , v, m 1 , f 1 , u * 1 , v * , m * 1 and f * 1 .We have free choice of one from 24 unknowns.
In this paper, we select α c on x-axis of spherical 4R linkage feasible solution region as the arbitrary choice scalar.Then 23 equations can be solved for other 23 unknowns using the Bertini software (Bates et al., 2006).

Illustrative examples
In Table 1, α i , β i , θ i are the Euler angles and p i (p ix , p iy , p iz ) are the spatial coordinates.The design demands are: -Linkage type: crank rocker; -The ranges of x-axis of points M 1 and M * 1 are restricted in (−100, 100); -No circuit and branch defects.
Let α c (the latitude of Burmester circle point curve) as the xand y-axes, we build a solution region for displaying infinite spherical 4R linkage solutions, as shown in Fig. 6.We classified the solution region with nine parts with different colors.In the white region, there is no linkage.The rest eight parts are classified by eight types of mechanisms as listed in Fig. 6. Circuit and branch defects can also be eliminated in this solution region.After eliminating the defects, we obtained the feasible solution region as shown in Fig. 6b.In
Non-italic R and C represent R joint and C joint respectively

Figure 4 .
Figure 4. Spherical Burmester center and circle point curves.

Figure 5 .
Figure 5. Burmester circle and center point curves in the α − β plane.

Figure 7 .
Figure 7. RCCC solutions corresponding the parts on the feasible spherical 4R linkage solution region.

Figure 8 .
Figure 8.The solution region of RCCC linkages.

Figure 9 .
Figure 9. Four configurations of the RCCC linkage for point p 1 .

1Figure 10 .
Figure 10.Four configurations of the RCCC linkage for point p 2 .

Table 1 .
Four specified rigid body poses.