A Coordinate-Free Approach to the Design of Generalized Griffis-Duffy Platforms

Architectural singularity belongs to the Type II singularity, in which a parallel manipulator (PM) gains one or more degrees of freedom and becomes uncontrollable. PMs remaining permanently in a singularity are beneficial for linear-to-rotary motion conversion. Griffis-Duffy (GD) platform is a mobile structure admitting a Bricard motion. In this paper, we present a coordinate-free approach to the design of generalized GD platforms, which consists in determining the shape and attachment of both the moving platform and the fixed base. The generalized GD platform is treated as a combination of six coaxial single-loop mechanisms under the same constraints. Owing to the inversion, hidden in the geometric structure of these single-loop mechanisms, the mapping from a line to a circle establishes the geometric transformation between the fixed base and the moving platform based on the center of inversion, and describes the shape and attachment of the generalized GD platform. Moreover, the center of inversion not only identifies the location of rotation axis, but also affects the shape of the platform mechanism. A graphical construction of generalized GD platforms using inversion, proposed in the paper, provides geometrically feasible solutions of the manipulator design for the requirement of the location of rotation axis.


Introduction
When both the moving platform and the fixed base fulfill particular conditions, the Stewart-Gough (SG) platform can still admit a continuous motion, although all six linear actuators are locked at equal length, which is called self-motion.Indeed, the manipulator that undergoes the self-motion is singular in every possible configuration inside the whole workspace.These continuous singular configurations rather than discrete ones, merely caused by design parameters, were defined as architectural singularity.The notion was first introduced by Ma and Angeles in 1991 [1], and later widely used within the robotics community.On the one hand, architectural singularity belongs to the Type II singularity [2], in which a parallel manipulator (PM) gains one or more degrees of freedom and becomes uncontrollable.Thus, architectural singularity should be avoided for general applications.On the other hand, PMs remaining permanently in a singularity can be useful due to the capability of generating complex motions with only one actuator [3].An energy regenerative suspension based on the architecturally singular platform was presented in Ref. [4], with the aim of converting vibratory linear motion into rotary motion.
Over the past decades, most investigations into architectural singularity concentrated on SG platforms [5], consisting in the characterization of self-motions and design conditions.Husty and Karger [6] made a classification of all self-motions of the original SG platform, which contains translations, rotations, generalized screw motions, and other complex spatial motions.Refs.[7,8] gave the projective characterization of architecturally singular planar and non-planar SG platforms and indicated that they are invariant.Borras et al. [9,10] provided a novel geometric interpretation for a class of architecturally singular line-plane 5-SPU platforms (S stands for a spherical joint, P stands for an prismatic joint, U stands for an universal joint), which in fact are the degenerated 6-SPS PMs.Nawratil [11] classified one-parametric selfmotions of a general planar SG platform into two types.These publications followed a roadmap of analysis, synthesizing architecturally singular mechanisms may be an interesting topic.Kong [12] proposed a component method for the generation of singular SG platforms.Based on the location of limb attachments on the platforms, Wohlhart [13] constructed four types of mobile 6-SPS PMs with locked actuators.Lee and Herve [14] focused on the mechanical generation of the one degree of freedom Bricard motion and described a general structure with six SS kinematic chains, which can produce such motion.It should be emphasized that the manipulators possessing self-motions are not equivalent to architecturally singular manipulators.Ref. [15] presented a complete list of architecturally singular SG platforms with multidimensional self-motions.The phenomenon of self-motions also appears in a few other PMs.Briot et al. showed that the 3-RPR PM (R stands for a revolute joint) with similar platforms and zero offsets [16] and the Pantopteron robot [17] can admit the Cardanic self-motion.The movement also exists in the 3-PPPS PM [18].Refs.[19,20] discussed the self-motions of different types of 3-RPS PMs, which can be the butterfly motion and the spherical four-bar motion.From a theoretical point of view, the determination of these self-motions is closely related to the singularity analysis of the 6-3 type SG platform.Nurahmi et al. [21] detected the conditions for selfmotions of a 4-CRU PM (C stands for a cylindrical joint).Wu and Bai [22] proposed an analytic approach to the determination of architectural singularity, the presence of self-motions were validated in the 3-PPR planar PM and the 3-PPS spatial PM.
The aim of this paper is to present a coordinate-free approach to the design of generalized Griffis-Duffy (GD) platforms, which consists in determining the shape and attachment of both the moving platform and the fixed base.The GD platform [23] is a special 6-6 type parallel manipulator with planar triangular platforms.Ref. [24] investigated the self-motions of a class of GD platforms and indicated that the manipulator can admit a one degree of freedom Bricard motion.We focus on the mechanical generation of such motion based on the geometric construction of generalized GD platforms.Our study cannot contribute to the design of any novel manipulators, but it provides an insight of synthesis for architecturally singular GD platforms.With the introduction of inversion, it will be shown that the mapping from a line to a circle reveals the geometric transformation between the moving platform and the fixed base of the GD platform.The center of inversion plays such an important role that affects the position of rotation axis.The approach, proposed in this paper, addresses the feasible solutions to the topology design of generalized GD platforms concerned with the location of rotation axis.
The rest of this paper includes five sections, which are organized as follows.Section 2 provides the concept of inversion and the subject of our investigation.Section 3 gives the relation between an inversion and a series of single-loop SSC mechanisms under the same algebraic constraints.Section 4 deals with the coordinate-free determination of a generalized GD platform using inversion.The effect of the center of inversion on the graphical construction is discussed in Section 5. Finally, Section 6 summarizes the main points of our study.

Geometric Inversion
Inversion, a particular geometric transformation, will be applied as a powerful tool to the determination of the GD platform in our work.The concept of inversion is introduced as follows.More about the topic can be referred to Ref. [25].Given an inversion I(O, μ) and a point P other than the point O, the point Q on the ray OP is the inverse of the point P, if and only if: It is noted that the symbol | • | denotes the Euclidean distance between two points in this paper.The relationship between the points P and Q, expressed as Eq. ( 1) is called an inversion, while point O is called the center of inversion, and constant μ is called the power of inversion.
The inversion indeed describes a constraint of a pair of points with respect to a fixed point, i.e., the center of inversion.These three points are collinear, and the product of the distance from one to the center with the distance from the other to the center is constant.Such a constraint is hidden in the geometric structure of the GD platform.It follows that generalized GD platforms can be characterized and synthesized by inversion, which will be discussed in this paper.

Generalized Griffis-Duffy Platforms
A midpoint-to-vertex Griffis-Duffy platform is presented schematically in Figure 1.The mobile structure consists of a fixed base and a moving platform connected to each other through six limbs, after locking all the extensible actuators of the SPS kinematic chains.Both the fixed base (1) and the moving platform are equilateral triangular plates.Each vertex of the fixed base is linked to the midpoint of the side of the moving platform, and each midpoint of the side of the fixed base is linked to the vertex of the moving platform.The moving platform can undergo a continuous Bricard motion [14,24], which is a rotation about a certain axis, perpendicular to the fixed base, with parasitic translation along the axis, as depicted in Figure 1.
In this paper, we focus on the generalized GD platform, in which the shape and attachment of the fixed base and the moving platform is different from the midline-to-vertex GD platform, but the manipulator still admit a continuous Bricard motion.

Equivalent Mechanism with a Bricard Motion
Since there exists a virtual rotation axis in the GD platform, we start our study from the equivalent mechanism [5], which can admit a Bricard motion.A single-loop SSC mechanism appearing in Figure 2(a) is therefore taken into consideration.Let point P be the anchor point of the S joint attached to the fixed base, and point Q be the anchor point of the S joint attached to the moving platform.Point O is the foot of the perpendicular obtained from P to the axis of the C joint, and point O' is the foot of the perpendicular obtained from Q to the C joint axis.The dimension of the fixed base is defined as the distance from the anchor point P of the fixed S joint to the C joint axis, while the dimension of the moving platform is defined as the distance from the anchor point Q of the moving S joint to the C joint axis.For the equivalent mechanism, the moving platform O'Q can translate along the C joint axis and rotate about the axis.
A reference frame {Oxyz} is attached to the fixed base, in which the x-axis coincides with OP and the z-axis coincides with OO'.Therefore, the pose of the moving platform with respect to the fixed base is determined by the height s (s ≥ 0), which is measured from O to O', and the angle θ (θ ≥ 0), which is measured from Ox to O'Q.Note that the kinematics of the mechanism has been studied in Refs.[14,26].Here, the constraint equation of the mechanism is given by: As depicted in Figure 2(b), the moving platform is located at the home configuration of the single-loop SSC mechanism, in which the four points O, O', P and Q are coplanar.
Let θ be zero, then Eq. ( 2) is rewritten as: where s max denotes the maximal height, which the moving platform can reach.Taking the system of Eqs. ( 2) and (3) into consideration, we have: which indicates that the pose (s, θ) of the single-loop SSC mechanism is constrained by the dimensions of links, (2) (3) including the maximal height s max of the moving platform located at the home configuration and the product of the dimension of the moving platform with the dimension of the fixed base.Let the product be: then Eq. ( 4) can be reformulated as: Eq. ( 6) constitutes the algebraic constraint for the single-loop SSC mechanism, any of these mechanism can be described by the formulation.Once the maximal height s max is confirmed, the functional relation of two pose variables s and θ, expressed by Eq. ( 6), only depends on the unique factor μ. In this way, all the single-loop SSC mechanisms with the same constraints but different dimensions can be characterized by the factor μ. In other words, all the mechanisms with the same factor, which is the value of the product of the dimension of the moving platform with the dimension of the fixed base, can admit the same motion output.
The projection of the mechanism, when the moving platform is located at the home configuration, along the C joint axis onto the plane defined by the axes Ox and Oy is depicted in Figure 2(b).The point O determined by the location of the C joint, the anchor point P of the fixed S joint, and the projection of the point Q of the moving S joint are collinear.Then the factor μ in Eq. ( 5) can be rewritten as: which corresponds to Eq. (1).For a common assignment of the maximal height, a series of single-loop SSC mechanisms under the same constraints is characterized by the only factor μ, independent of the dimensions (5) of links.Any of them can be described as three collinear points satisfying Eq. ( 7), while the point O stands for the location of the C joint, and a pair of inverse points P and Q with respect to the inversion I(O, μ) represents the anchor points of the S joints.
Based on the projection of the single-loop SSC mechanism along the C joint axis, the description of a series of single-loop SSC mechanisms under the same constraints is linked to the concept of inversion.All of these mechanisms can admit the same motion output, although the dimensions of them are different.The link is free of dimensions and will be utilized for the graphical construction of generalized GD platforms.

Coordinate-Free Determination of Generalized GD Platforms
As a result of the existence of certain rotation axis, the selfmotion of a generalized GD platform is equivalent to the motion output of a single-loop SSC mechanism.For a generalized GD platform, the moving platform is constrained by six SS limbs.Thus, the generalized GD platform can be regarded as a combination of six coaxial single-loop SSC mechanisms admitting the same Bricard motion.These mechanisms are under the same constraints and share the same factor μ, which has been analyzed in Section 3.
In this section, two key points to the determination of generalized GD platforms are addressed.The one is the construction of a series of coaxial single-loop SSC mechanisms under the same constraints.The other is the location of anchor points of S joints for the rational attachment of the fixed base and the moving platform, which will be performed by the arrangement of limbs from the view of vertical projection.
Go back to the concept of inversion, the mathematical notion not only describes a constraint of a pair of points with respect to a fixed point, but also establishes a geometric transformation on account of the fixed point, which maps a line to a circle under the constraint.The mapping under an inversion provides a graphical way directly for the construction of generalized GD platforms from an arbitrary triangle.A coordinate-free approach to the design of generalized GD platforms is presented thereupon.

Geometric Inversion
The inversion, in nature, maps one point to another on a ray from the center of the inversion.By taking the inverse of each point of a figure with respect to a center, the inverse of the figure can be obtained, which appears as a geometric transformation between the two figures.There are anchor points being located on the sides of platforms in the generalized GD platform, the inverse of points lying on a line is first taken into consideration.As shown in Figure 3, let point P 0 be the foot of the perpendicular from point O to line l, and point Q 0 be the inverse of P 0 under the inversion I(O, μ).Suppose that any point P on l other than P 0 has an inverse point Q with respect to the point O, then they obey: There exists an identity for triangles QOQ 0 and P 0 OP as follows: In accordance with the Side-Angle-Side similarity criterion, it can be concluded that: (8 (9) ∠QOQ 0 ≡ ∠P 0 OP.
(10) QOQ 0 ∼ P 0 OP.Thus, we have: which implies that point Q lies on the circle σ with diameter OQ 0 .
The above derivation indicates that a line l can be mapped to a circle σ under an inversion with the center O.The circle passes through the center of inversion and the tangent to the circle at the center is parallel to the line.As introduced in Section 2, an inversion is defined by a certain center and a constant power.However, it is concluded that a line can be inverted into a circle that passes through the center of inversion, while the power of inversion keeps unknown but constant.Thus, this geometric transformation from the line to the circle is independent of the power of inversion.
Thanks to the mapping based on the inversion, there is a one-to-one correspondence between points on l and points on σ.Consider a series of coaxial single-loop SSC mechanisms with the same constraints, all the mechanisms are characterized by the factor μ, which corresponds to the power of inversion.Any of them, denoted by subscript i, can be described by three collinear points O, P i and Q i .Among them, point O stands for the location of the C joint axis, points P i represent the anchor points of the fixed S joints, and points Q i represent the anchor points of the moving S joints.Then all the points Q i , except for O, on the circle σ are the inverse points of P i on the line l.Owing to the property of inversion, the points O, P i and Q i obey: All the coaxial single-loop SSC mechanisms can realize the same motion output, in which anchor points of the fixed base lie on the line drawn in blue, and anchor points of the moving platform lie the circle drawn in red, as shown in Figure 3.Note that if the moving anchor point Q i lies on l, its inverse, i.e., the fixed anchor point P i lies on σ.
Contributed by the independence of the power of inversion in the geometric transformation from a line to a circle, a series of coaxial single-loop SSC mechanisms with the same motion output are constructed by a line and an inverted circle, based on a point defined as the center of inversion.The center indeed performs as the C joint axis of these coaxial mechanisms.

Construction of a Generalized GD Platform
Based on the mapping from a line to a circle, a series of coaxial single-loop SSC mechanisms with the same factor μ are constructed.The generalized GD platform will be (11)  Denote P i and Q i as the anchor points of the S joints on the fixed base and the moving platform, respectively.Given an arbitrary triangle, defined by P 1 , P 2 and P 3 , as the fixed base, any point O inside it is selected as the center of inversion.As illustrated in Figure 4, the inverted circle σ 1 from side P 1 P 2 is obtained by any diameter OD 1 perpendicular to P 1 P 2 .The circle σ 1 intersects the rays OP 1 and OP 2 at Q 1 and Q 2 , respectively.Inverted from side P 2 P 3 , the second circle σ 2 through Q 2 , with diameter OD 2 perpendicular to P 2 P 3 , intersects at the ray OP 3 at Q 3 .Due to the power keeps constant, we have: which indicates that the circle σ 3 determined by the three points Q 1 , O and Q 3 is the inverse of side P 1 P 3 .Because of the finite of segments, three sides are inverted respectively into three circular arcs under the inversion, as shown in Figure 4.For the sake of simplicity, both the circle and circular arc are denoted by σ i .
According to the one-to-one correspondence between the points on the line and the points on the inverted circle, points P i lying on the sides and Q i lying on the inverted circular arcs, shown as Figure 5, satisfy: There exists a point, denoted by Q 4 , lying on the circle σ 1 .Suppose that ray Q 4 Q 2 intersects the circle σ 2 at Q 5 and ray Q 4 Q 1 intersects the circle σ 3 at Q 6 , as depicted in Figure 5. Considering that opposite (13) angles are supplementary for the cyclic quadrilaterals

we have:
Since points Q 4 , Q 1 and Q 6 are collinear, so are points Q 4 , Q 2 and Q 5 , then we obtain: Thus, it can be concluded that: Finally, the shape of the moving platform and the arrangement of SS limbs, with respect to a triangle selected as the fixed base, has been determined graphically.
With the center of inversion O being selected inside the fixed base, a circle with arbitrary diameter through the point O is inverted from any one of three sides, then the other two inverted circles are obtained from the remaining sides, which determines the arrangement of three SS limbs connected between the vertex of the fixed base and the side of the moving platform.Two rays from a point lying on an inverted circular arcs constitute two sides of the moving platform.The existence of collinear points lying on the remaining two inverted circular arcs ensures the arrangement of the remaining three SS limbs connected between the side of the fixed base and the vertex of the moving platform.Finally, six coaxial single-loop SSC mechanisms are combined as a generalized GD platform, whose fixed base and moving platform are general ( 15) It should be noted that if a triangle is given as the moving platform, the other is then generated as the fixed base under the inversion, which can also achieve a generalized GD platform.

Discussion on the Construction
In the graphical construction of a generalized GD platform through the geometric transformation of inversion, which is free of the power of inversion, the center of inversion plays such a critical role that affects the shape of the generated triangle with respect to the given triangle.
Two different triangles are generated from a given ∆P 1 P 2 P 3 based on three inverted circles, as depicted in Figure 6.The center of inversion O inside ∆P 1 P 2 P 3 is selected, and three angles, P 1 OP 2 , P 2 OP 3 , and P 3 OP 1 , can be determined uniquely.It can be obtained that the four convex quadrilaterals OQ 1 Q 4 Q 2 , OQ 2 Q 5 Q 3 and OQ 3 Q 6 Q 1 are cyclic, the opposite angles of each of them are supplementary, thus, Since points O, P i and Q i are collinear, we get the following identities: which indicates that the shape of the generated triangles is affected by the location of the center of inversion.For any point O inside ∆P 1 P 2 P 3 , we can obtain: In accordance with the supplementation of opposite angles in cyclic quadrilateral, the generated ∆Q 4 Q 5 Q 6 obeys: Thus, we have: From Eq. ( 22), it shows that the shape of the generated moving platform, ∆Q 4 Q 5 Q 6 , is constrained by the shape of the given fixed base, ∆P 1 P 2 P 3 , besides the effect of the location of the center of inversion.In other word, it can be concluded that not any two triangles can be constructed as a generalized GD platform.Furthermore, Eq. ( 22) indicates the identification condition of a pair of triangles for the construction of generalized GD platforms.
Finally, the regions of locations of the center of inversion that lead to different shapes of generated triangles are depicted in Figure 7.The boundary between the regions that generate acute and obtuse triangles is identified by three circles with diameters obtained by ( 18) Taking an obtuse triangle given as the fixed base for instance, we generate the acute, right and obtuse triangles respectively, as depicted in Figure 8. Recall that if one triangle is given as the moving platform, the other is then generated as the fixed base under the inversion.
As mentioned before, the center of inversion stands for the C joint axis, which indeed indicates the location of the Bricard motion of the manipulator in the view of vertical projection.For the topology design of generalized GD platforms, we can obtain the geometrically feasible solutions, including the sharp and attachment of the fixed base and the moving platform, to meet the requirement of the location of rotation axis of the moving platform with respect to the fixed base through the map depicted as Figure 7.In addition, a schematic procedure for constructing a generalized GD platform with respect to the location of rotation axis, which contains nine steps from (a) to (i), is presented in Figure 9.

Conclusions
(1) This paper presents a coordinate-free approach to the design of generalized GD platforms.The generalized GD platform is regarded as a combination of six coaxial single-loop SSC mechanisms with different dimensions.These mechanisms are under the same constraints and admit the same Bricard motion, which are linked to the concept of inversion.
(2) Any of these mechanisms can be described by three collinear points, one stands for the location of the C joint axis, the remaining two represent the anchor points of the fixed base and the moving platform.
Based on an arbitrary triangle as the fixed base, the shape of the moving platform and the arrangement of SS limbs are determined through the geometric transformation from a line to circle.(3) On the one hand, the inversion from a line to a circle free of the power, provides a geometric approach for the non-dimensional construction of generalized GD platforms in the plane.The mapping from a line to a circle under the inversion performs as a geometric transformation and describes the shape and attachment of the fixed base and the moving platform of the generalized GD platform.(4) On the other hand, the center of inversion plays a critical role in the generation.The point not only identifies the location of rotation axis of the generalized GD platform, but also affects the shape of the generated platform with respect to the given one.Thus, the proposed construction is a synthesis (5) An interesting conclusion, according to our investigation, is drawn that not any two triangles can be constructed as a generalized GD platform.

Figure 3 Figure 4
Figure 3 Transformation from a line to a circle Figure 4 Circles are inverted from the sides of a given triangle

Figure 5
Figure 5 Moving platform generated from a fixed base

Figure 6 Figure 7
Figure 6 Two different triangles generated from a given triangle

Figure 8
Figure 8 Moving platform generated from a fixed base

Figure 9
Figure 9 Schematic procedure for constructing a generalized GD platform with respect to the location of rotation axis