Kinematic analysis of scissor-like elements using dual quaternions

This paper presents a standard approach, based on dual quaternions (DQ), to deal with the kinematic analysis of scissor-like elements (SLE). Kinematic relations of homogeneous SLE are simpliﬁed by using DQ exponentiation. The steps to identify kinematic singularities are detailed by an algorithm. A second algorithm is introduced to deal with the computation and plotting of reachable and dexterous workspace. In this approach, not just points and vectors, but lines and planes are represented by DQ. The motion of points, lines and planes can be appreciated in practical examples based on domotics.


Introduction
The need of making folding equipment ranges from the Mongolian yurts, to more sophisticated devices such as the Da Vinci's umbrella or the velum of the Roman Coliseum [1]. In the last decades, the idea of joining various scissors through a pivot and hinges leaded into more complex assemblies; for example, in 1985 Escrig [2] introduced an approach of translational homogeneous SLEs structures, the next year a novel configuration through polar SLE was presented [3], fourteen years later Hoberman patented a closed structure of symmetrical angulated SLEs [4], leading into the famous toy today known as the Hoberman sphere, the next year Hoberman patented a radial expansion truss structure [5] by using non-symmetrical angulated SLEs. Nowadays, SLEs have been used in aeronautical, architecture, robotics, among other applications [6,7,8,9,10,11,12].
The kinematic analysis of extensive SLE becomes complex because of the large amount of joints and links. Farrugia introduced an approach based on matrices [13] to deal with the kinematics of SLE. The main drawback of this approach is the number of rows and columns in the matrices which depends on the amount of SLEs in the mechanism. The kinematic analysis of SLEs can be developed by using Screw Theory [14]. However, the possibility to deal with the orientation of lines, planes and volumes are not allowed. In [9] it was introduced the use of DQ in the kinematic analysis of translational SLE with constant bar length; nevertheless, the kinematics of connectors was omitted, the application was limited to homogeneous assemblies, and the motion of lines and planes was not explored.
Usually, deployable structures exhibit kinematic singularities along the motion paths [15]. These critical points could lead into the change of mobility [16]. A robust strategy to control SLE based systems was shown in [17]. This control technique did not consider singularities being possible to lead into collisions that reduce or damage the utility of the mechanism. In this paper, we introduce an algorithm to identify singularities in SLE. If the singularities are detected, then the reachable and dexterous workspace can be computed and plotted by using a second algorithm. Previously, it is necessary to write the kinematic relations of the points of interest from the origin frame. The rotations and translations caused by connectors are considered in this kinematic relations. Extensive homogeneous assemblies of SLE can be simplified by DQ exponentiation being possible to reduce the computational effort [9]. One of the main advantages of DQ is the possibility to orientate lines, planes and volumes [18,19,20], we take advantage of this property giving the possibility to include lines and planes, considered as end-effectors, in the kinematic relations of SLE structures.
This document is organized as follows: in the first Section; concepts and terms used in the paper are defined, starting from Clifford algebra, as well propositions to deal with the motion of points, lines and planes in SLE. In the second Section, translational and polar SLE are represented by DQ, connectors employed to assemble three dimensional structures are represented by DQ too. Also, two algorithms to identify singularities and compute the workspace are included in this section. In the third Section; the theory introduced in the first and second Sections is applied in numerical examples based on domotics. Finally, pertinent conclusions and perspectives for future works can be found.

Rigid displacements by Clifford algebra
Quaternions have proven to be efficient at representing rotations with clear geometric meaning [21,22,23,24,25,26,27]; unfortunately, they do not handle translations. Dual quaternions were introduced by William Clifford [28] like an effort to combine rotations and translations while retaining the benefits of the quaternion representation of rotations. In this section, we introduce concepts, notations and definitions about Clifford algebra, as well, its application in the displacement of rigid bodies by dual quaternions. Well known algebraic systems like complex numbers, matrix, vector, quaternion are included in Clifford algebra and unified in a coherent mathematical language [29]. Rotation and translation of geometric entities such points, lines, planes, areas and volumes are basic operations in Clifford algebra, this characteristic will be exploited in the next sections of this paper. We start defining the Clifford algebra.
represents the signature of any generator ς i .
Aiming for simplicity, the Clifford space is depicted as Cl λ p ,λ r ,λ q (n). The dimension n of the Clifford space can be computed as the sum of the signatures λ p,r,q .
Definition 2 The elements in Cl (n) are called according to the amount of generators, i.e, an element with ς i is called vector, with ς i ς j is called bivector, with ς i ς j ς k is called trivector, and so on. The elements with zero generators are scalars.
Based on Definition 2, a multivector is a general element of Clifford algebra and the number of generators defines the degree of any element.
Definition 3 A j-dimensional subspace of Cl (n), defined by homogeneous multivectors of j degree, is denoted by M j . By using Definition 3, it is possible to decompose a Clifford space as follows In Eq.(1) " ⊕ " represents the direct sum of subspaces.
Definition 4 The subspaces of Cl (n) with even degree, i.e, M 2 , M 4 , and so on, are defined as spinors and the direct sum of the spinors of Cl (n) is denoted by Cl + (n).
The spinors are employed in rotation and translation of rigid bodies [30, 31, 32], it can be appreciated in the next subsection.
Isomorphism between Clifford algebra and dual quaternions By using the definitions established above, we introduce the concept of dual quaternion and its property to rotate and translate rigid bodies. The primal Clifford space employed in rigid displacements is Cl 0,1,3 (4), which belongs to the hyperspace P R 3 with orthonormal basis {ς r 1 , ς q 1 , ς q 2 , ς q 3 }. Note that by Definition 1 we have Also, by Definition 1 and 3 note that there exist 2 4 elements in Cl 0,1,3 (4) as follows By Definition 4, the spinors from Cl 0,1,3 (4) are extracted as follows Any element h in Cl + (4) can be written as Note that the expression above is formed by one scalar, six bivectors and one tetravector giving a total of eight elements. By Definition 1 the expression above can be rewritten as In Eq.(2) υ i , ω i ∈ R. If it is used the next relations then by replacing Eq.(3) in Eq.
(2) we have Definition 5 A dual quaternion, denoted as H 2 , is a dual number composed by eight elements H 2 ∈ R 8 ; namely where υ i , ω i are real coefficients, ε represents the dual unit of the number and i, j, k are the unit vector along the x−, y− and z−axes respectively. The elements of H 2 satisfy the properties i 2 = j 2 = k 2 = ijk = −1 and ε 2 = 0.
From Eq.(4) and Definition 5 we conclude the isomorphism between Clifford algebra and DQ.

Rotation and translation of rigid bodies by dual quaternions
In the Subsection above, starting from the isomorphism with Clifford Algebra, it was detailed the origin of DQ. Here, we use DQ to rotate and translate rigid bodies. Aiming for simplicity, Eq.(5) is represented by Eq.(6) is used in the next definition.
Definition 6 For two given DQ, h a and h b , the homogeneous product is computed as It is important to highlight that, in general, the product of DQ does not satisfy the commutative property.
Definition 7 Two conjugation relations are defined for DQ as follows Both DQ conjugations will be used in the motion of points, lines and planes in the propositions introduced in the next subsection.
Definition 8 A DQ is called unit DQ if hh * = 1. Then, it satisfies the following pair of conditions The conditions detailed in Definition 8 let visualize a unit hypersphere of three dimensions, and a three-dimensional hyperplane perpendicular to the normal at the point υ 0 , υ 1 , υ 2 , υ 3 on the hypersphere. Note that the hyperspace P R 3 ⊂ R 4 , then a DQ representing a rotation and a translation is defined in R 8 . This manifold interpretation is used in the next definition.
Definition 9 Let Υ = cos ϑ 2 +û sin ϑ 2 be a spinor from Cl 0,0,3 (3) that represents a rotation about a unit vectorû through ϑ. Its conjugate is Its corresponding conjugate is t * = −t. A geometric body ξ under the rotation Υ followed by the translation t becomes the point Υξ + t. The transformation sequence Υ, t can be compacted in the next DQ as follows The DQ presented in Definition 9 will be employed in the displacement of points, lines and planes in the next subsection.

Motion of points , lines and planes
The main objective of this paper is to use DQ to analyze the kinematics of SLE assemblies which have a point, a line or a plane such an end-effector. Therefore, in this subsection, by using the definitions depicted in the last subsections, we present the motion of points, lines and planes using DQ.
Proposition 1 The DQ presented in Definition 9 is a unit DQ.
Proof. By Definition 8 and 9 we have

Proposition 2 For a given point µ
Proof. Proof. The rotation in this case affects to the vector displacement ΥlΥ * as well the moment ΥmΥ * . On the other hand the translation only affects to the moment through t × (ΥlΥ * ). So, we have

Proposition 3 A line in DQ coordinates can be represented by
Note that the line representation in DQ coordinates is based on P lücker coordinates [29,33].
Proposition 4 The rotation and translation of a plane, represented in DQ coordinates as ξ N = n − d, where n = (0, n x i, n y j, n z k) is the normal vector represented in the hyperspace P R 3 and d = (d , 0, 0, 0) where d is the distance of the plane from the origin, is given by Proof. The rotation of a plane affects the orientation of the normal vector ΥnΥ * , while the translation affects the distance of the plane from the origin; however, it is necessary to consider the projection of the translation over the normal vector d + t · (ΥnΥ * ). Thus, we have Note that in Propositions 2, 3 ,and 4 the conjugate of DQ is used in motion of lines and the dual conjugate of DQ is used in points and planes. The definitions and propositions presented in this section are used in the kinematic analysis of SLE structures in the next section.

Kinematic analysis of SLE using DQ
There are translational SLEs with the possibility to translate along a straight trajectory in one ore more axes. Also there are polar SLEs that in addition to translation they have the possibility to rotate through a fixed axis being possible to describe curvilinear trajectories. Moreover, three dimensional folding structures can be assembled by using connectors. In practical application it is possible to find, besides of a point, a line or a plane as end-effector in SLE structures. Thus, by taking advantage of the DQ properties, in this section we introduce an approach to deal with kinematic analysis of complex homogeneous and heterogeneous assemblies of SLEs.
Definition 10 A SLE unit, also known as pantograph is a composition of two bars connected to each other at an intermediate point through a revolute joint called pivot which allows them to rotate freely about an axis perpendicular to their common plane but constrains all other degrees of freedom. The end points of these bars can be hinged to the next SLE unit, these joints are called hinges depicted on Figure 1. Also, the geometrical parameters of a standard SLE are shown in Figure 1.

Representation of translational SLE by DQ
There are two types of translational SLEs [34]: the first type with constant bar lengths and the second type with different bar lengths.
Translational SLEs with constant bar lengths: type 1 This type of folding structures can only translate without any rotation. When all bars have the same lengths a 1 = a 2 = b 2 = b 2 = a I , and the pivots are located in the middle of the bars, these assemblies describe an straight translation. Definitions 9 and 10 are employed to describe the kinematic relation of a point in the pivot or the hinges from the origin frame C through a translation spinor t I = (0, t x i, t y j, 0k) as follows.
• From the origin C to the pivot A: and t y = δ 4 . • From the origin C to the upper hinge B P : and t y = δ 2 . • From the origin C to the lower hinge C P : and t y = 0.
Note that the input parameters are the SLE aperture δ and the length of the bar a I . In extensive assemblies of SLEs the most used DQ, among the DQ defined above, is the h IC P . It happens because the hinge C P of the i SLE is linked with the hinge C of the i + 1 SLE, leading into uniform displacement of the origin frame. As seen in the following example.
Example 1 In the SLE structure in Figure 2 the kinematic relation of a point ξ P in A 3 , A 4 , B P 3 , B P 4 , C P 4 , from the origin C is given by The relations above come from Proposition 2. Also, it is possible to appreciate that the most used DQ is h IC P and its dual conjugate. Therefore we consider important to introduce DQ exponentiation for homogeneous DQ in the following point.
DQ exponentiation of translational SLEs: type1 By using Definition 6, if a point ξ P = 1 + ε0 is affected by homogeneous motions represented by h IC P then we can use where n is the number of SLEs. In the Example 1 the relation C P 4 C ξ P can be rewritten as [ C P 4 C ξ P I ] 4 , and thus reducing the computational effort [9]. DQ exponentiation can be proven here and in the sequel by using Definition 6 and Proposition 2 as follows Translational SLEs with different bar lengths: type 2 Similar to the previous case this kind of SLE can only translate without any rotation. Bar lengths are a 1 = b 2 = a y and b 1 = a 2 = b y . These assemblies describe an inclined translation. Definitions 9 and 10 are used to describe the kinematic relation of a point in the pivot or the hinges from the origin frame C through a translation spinor t Y = (0, t x i, t y j, 0k) as follows.
• From the origin C to the pivot A: Note that the input parameters are the SLE aperture δ and the lengths of the bars a Y , b Y . The DQ presented in this subsection is employed in the next example. Figure 3 the kinematic relation of a point ξ P inA 3 , B P 2 , C P 2 , from the origin C is given by

Example 2 In the SLE assembly in
Similar to the last case, DQ exponentiation for this type of SLE is presented in the following point.
DQ exponentiation of translational SLE: type 2 Homogeneous displacements represented by h Y C P in a point ξ P = 1 + ε0 can be defined through DQ exponentiation as where n is the number of SLEs. In the Example 2 the relation C P 2 C ξ P can be rewritten as

Representation of polar SLEs by DQ
The main characteristic of these structures is to deploy and contract as a part of a single arc. Besides of translation, there exists a rotation by this type of SLE. The lengths of the bars are a 1 = a 2 = a S and b 1 = b 2 = b S . Definitions 9 and 10 are used to describe the kinematic relation of a point in the pivot or the hinges from the origin frame C through a rotation spinor Υ S = cos ϑ 2 +û sin ϑ 2 and a translation spinor t S = (0, t x i, t y j, 0k) as follows.
• From the origin to the pivot A: • From the origin C to the upper hinge B P : h SB P = 1 + εt S , with t x = (a S +b S ) cos τ S 2 and t y = (a S +b S ) sin τ S 2 ; τ S , α S defined above. • From the origin C to the lower hinge C P : ; α S defined above.
Note that the input parameters are the SLE aperture δ and the lengths of the bars a S , b S . In the next example is applied the DQ presented in this subsection.

Example 3
The kinematic relation of a point ξ P in the positions A 2 , B P 2 , C P 3 , in Figure 4 from the origin C is expressed as It is important to highlight that the expression above can be employed to displace lines and planes too, being just necessary to replace ξ L , ξ N instead of ξ P . In homogeneous displacements such as C P 3 C ξ P in the Example 3, based on Definition 6, DQ exponentiation can be used as follows.
DQ exponentiation of polar SLE Homogeneous displacements represented by h SC P in a point ξ P = 1 + ε0 can be defined through DQ exponentiation as The parameters t x and t y correspond to the DQ h SCP , which relates the lower hinge from the origin. Note that in the Example 3 the expression C P 3 C ξ P can be replaced by C P 3 C ξ P S ) 3 .

Representation of connectors by DQ
The use of connectors in three dimensional SLE structures is quite common. However, the kinematic impact of these connectors generally is small being sometimes ignored or it is considered ideal, it means that the translation that produce these connectors is considered null, i.e, the folding roof presented in [9]. Thereby, aiming for accuracy, in this subsection we introduce a DQ representation of a connector taking into account the rotation and translation that it causes in the assembly. In Figure 5 is depicted a connector with L-shape. The parameters are ∆ X , ∆ Y , ∆ Z that represent translations in x−, y−, z−axes respectively, and Ω, 0 < Ω < π is the angle between the beams of the connector. The above mentioned parameters can be embedded in a DQ through translation and rotation spinors t L = (0, ∆ X , ∆ Y , −∆ Z ), Υ L = cos π−Ω 2 +û sin π−Ω 2 withû = (0, ±1, 0). The symbol ± inû is going to depend of the desired rotation according to right-hand rule. Thus, we have h L = Υ L + ε 2 t L Υ L . In Figure 5 is shown a structure composed by two translational SLEs with constant bar (in red), and one polar SLE (in green), both assemblies linked by a connector. Then, the kinematic relation of a point ξ P in B P from the origin C is given by

Analysis of singularities in SLE
Deployable structures, including SLEs, usually deal with singularities. These singularities can cause the change of mobility. Previously to compute the reachable workspace of a folding structure, it is indispensable to identify the singularity points in the path of the mechanism. Thus, in this subsection we introduce Algorithm 1, based on the aforementioned theory, to identify singularities in SLE assemblies. for i = 1 to n do 11: Calculate CP C ξ for each interaction 12:

Analysis of workspace in SLE
Analogously to serial robots, SLE structures usually have a end-effector, then it is important to know the dexterous or reachable workspace of SLE mechanisms. The singularity analysis is going to be essential in this point because we can know the limit of the input δ. Therefore, the first step of this approach is to apply Algorithm 1 to compute δ BIF . Some SLE assemblies are free of singularities, if this is the case, then Algorithm 1 must be omitted. In Algorithm 2 are detailed the steps to calculate and plot the reachable workspace of a SLE structure. If it is desired to compute the dexterous workspace, then in the line 7 of the Algorithm 2 is necessary to replace by = 100 × δ arb /δ M AX where δ arb is an arbitrary value of δ arb , δ M AX > δ arb > δ BIF > 0.
Algorithm 2 Algorithm for computation and plotting of workspace in SLE 1: procedure Parameters description(Geometry-dimensions) 2: Define the dimensions of the SLE bars 3: Compute δ M AX = a 1 + b 1 Based on Figure 1

4:
Determine the kinematic relation CP C ξ P,L,N 5: Choose the increment (S bty ) Plotting accuracy 6: procedure Workspace computation Loop 7: aux 1 = 100 : −S bty : Vector from 100 to = 100 × δ BIF /δ M AX with −S bty increment 8: n = size(aux 1 ) Number of elements in vector aux 1 9: Vector with n possible values of δ 10: for i = 1 to n do 11: Calculate CP C ξ for each interaction 12: Extract the translation spinor 13: Extract the rotation spinor 14: Plot the orientation of the point, line or plane ξ P

Numerical and practical examples
In this section is applied all the theory presented above in applications based on domotics. The first example is a retractable lamp, the chosen end-effector is a line which emulates the central beam of a light bulb. The second example is a folding furniture for displays like televisions or digital photo frames, which will be represented like a plane. Thus, we can appreciate the DQ advantage of orientate points, lines and planes.

Retractable lamp
The folding structure shown in Figure 6 can be employed to assemble a lamp that could be used over a table, in the corner of a lounge or on the streets to illuminate a public place. The lamp presented below will be used like a hall lamp, and the central beam of light is represented as a line ξ L . The proposed assembly is presented in Figure 6. The structure is composed by two translational SLEs with constant bar length (in red), two translational SLEs with different bar length (in blue) and three polar SLE (in green). Aiming to plot the reachable workspace, we define the kinematic relations of the points A 1 , B P 1 , C P 1 , C P 2 from the origin C.
The geometric dimensions of the SLE structure in Figure 6 are detailed in Table  1. In Figure 7, by using Algorithm 2, is depicted the reachable workspace of the aforementioned points. In Figure 7 the axes are orientated according to Figure 6.b. In Figure 6.c is highlighted the collision between the bars of the structure, when δ = δ BIF , leading into the lost of mobility. By using the numerical values shown in Table 1 and the Algorithm 1 we compute δ BIF = 0.19 m, it means that the reachable workspace is computed for 0.5 > δ > 0.19 m. The kinematic relation for the line ξ L depicted in Figure 6.b. is defined as follows In the expression above, the line ξ L has a vector displacement l = (1.5i, 0j, 0k) and a moment m = 0. Because of Proposition 2, note that the kinematic relation above employs conjugation of DQ instead of dual conjugation of DQ. By using A1 C ξ L , in Figure 8 is plotted the lines orientation for δ = 0.202, 0.23, 0.29, 0.36, 0.45, 0.5. Now we interpret a numerical example, for δ = 0.45 m the value of A1 C ξ L is From Eq. (7), we extract the displacement vector l = (1.37i − 0.62j + 0k) and the moment m = (0i + 0j − 0.4461k) by using l 0 = l × m , we compute l 0 = (0.2765i + 0.6094j + 0k) being any arbitrary point over the line. If it is desired to orientate the line segment as depicted in Figure 8, then we can use l = P F − P S where P F is the final point and P S is the starting point of any segment line. Thus, by using A1 C ξ P , the starting point can be extracted, then the value of l can be extracted from A1 C ξ L .

Folding furniture for displays
Each year the size of televisions increases while the area of the rooms decreases.
One possible solution to this contradiction is to fix the display on the wall. The main drawback is that the display can not be inclined. The furniture presented in this subsection was designed to be fixed on the roof of a room (see figure 9.a), moreover offers the possibility to regulate the inclination. Then, in this example we intend to apply the orientation of planes emulating flat displays, as well the use of connector to create a three dimensional structure. The structure is composed by one translational SLE with constant bar length (in red), one connector (see Figure  9.b), one translational SLE with different bar length (in blue), and two polar SLEs (in green). The kinematic relations of the points A 1 , C P 1 , C P 2 from the origin frame C can be written as follows The connector h L in Eq. (8) has Ω = π/2 and ∆ X = 0.5, ∆ Z = −1 cm according to Figure 5. By using Algorithm 1 the value of δ at singularity time was computed as δ BIF = 0.2 m. This time the singularity was caused by a jam as depicted in Figure  9.b. The dimensions of the bars are the same depicted in Table 1. In Figure 10, by using Algorithm 2, it is plotted the reachable workspace of the arbitrary chosen points A 1 , C P 1 , C P 2 in Figure 9.b. Because of the connector Figure 10, becomes three dimensional. Now, we assume that there is a television display fixed in the point A 1 . Based on Proposition 4 the plane of the display can be represented by ξ N = n − εd, where n = (0, i, 0j, 0k) and d = 0 at the origin. The kinematic relation of the plane from the origin is given by Note that in Eq.(9) is employed the dual conjugation according to Proposition 4. In Figure 11, the orientation of planes are plotted according to various values given to δ. Now, we interpret one of the plotted result, in the case of δ = 0.43 m the orientation of the plane was which means a plane with normal vector n = (0i−0.26j−0.97k) at distance d = 0.48 m from the origin. However, if it is desired to compute a point in the plane, then we can use A1 C ξ P from Eq. (8). Through the development of the numerical examples shown in this section, we applied all the theory introduced above, based on these results and other experiences acquired throughout this research in the next section we detail the main conclusions and possible future work.

Conclusions and future work
This paper addresses the kinematic analysis of SLE structures based on DQ. It starts from the representation of SLE, connectors, and end-effectors by DQ. Kinematic relations of homogeneous assemblies of SLEs are simplified by using DQ exponentiation. The analysis of singularities is developed by Algorithm 1. If there exist singularities in the SLE structure, then , by using Algorithm 2, they are considered in the computation and plotting of reachable or dexterous workspace. In the practical examples, the theory developed along the paper was applied. Also, it was possible to appreciate the dual quaternions characteristics to orientate points, lines and planes. In future works, we intend to present a complete analysis and interpretation of singularities in SLE; thereby, control strategies to deal with these singularities. Also, we perceived that this approach can be applied to serial and parallel mechanisms or robots, by using the same concept, it means to represent mechanical elements by DQ, then the kinematic relation can be written in the same way presented in this paper. Differential kinematics based on DQ is expected to be explored being possible to define conditions and theorems to deal with the kinematic control of SLEs.

Declaration
Availability of data and materials Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Competing interests
The authors declare that they have no competing interests.

Funding
This paper had the financial support of the Brazilian funding agency CAPES (Grant n.40410) during its study and writing.
Author's contributions ERP described the mathematical approach related to Clifford algebra as well its relation with dual quaternions, and the use of them in the positioning of points, lines and planes as end-effectors in SLEs. JGGG programmed and simulated the use of dual quaternions in SLEs, and was a major contributor in writing the manuscript. Both authors read and approved the final manuscript.          Tables Figure 11 Folding furniture for TV. Orientation for planes emulating a flat display.