The adjoint trigonometric representation of displacements and a closed‐form solution to the IKP of general 3C chains

Based on the representation of rigid body displacements as adjoint matrices, the article introduces the adjoint trigonometric representation of displacements (ATRD) as a further generalization of the trigonometric representation of rotations. In comparison to the dual Rodrigues–Euler–Gauß–Gelman equation, recently reported for affine screw displacements with arbitrary, fixed pitches, the ATRD is built upon a product of a unit line and a dual angle, instead of upon a product of a unit screw and a real angle. Due to this conceptual difference, the ATRD requires four independent parameters of a unit line instead of five when parametrizing a displacement along a unit screw. As a consequence for computational kinematics, the ATRD permits transferring the analytic solution to the inverse kinematics problem (IKP) of 3‐DOF, general, spherical 3R‐chains into a closed‐form solution to the IKP of 6‐DOF, general, affine 3C‐chains.

'geometric screw parameters' simplify if (6 × 6)-adjoint matrices are employed in place of (4 × 4)-homogeneous matrices to represent rigid body displacements [22]. The present article extends this previous work by expressing the adjoint representation of displacements via 'geometric line parameters', as a product of an invariant line with a dual angle, in coherence to the analysis reported in [5].
The main contributions are two-fold: the adjoint trigonometric representation of displacements (ATRD) is reported as a novel, principal formula together with a proof. Based on this fundamental advancement, a closed-form solution to a problem of computational kinematics is derived: Applying the principle of transference [23][24][25][26] to the solution approach for the inverse kinematics problem (IKP) of general, spherical three-revolute (3R) chains with three degrees-of-freedom (DOF) and employing the ATRD, a novel analytic method for solving the IKP of 6-DOF general, affine three-cylindric (3C) chains is achieved. The method complements former approaches in which spatial 3C chains are analyzed as subchains in spatial closed RCCC loops [7,[27][28][29][30] and closed 3CCC parallel platforms [31]. In the field of kinematic analysis, spatial kinematic chains equipped with cylindric joints are of significant importance since they appear as subchains in other mechanisms [10] but also since can be regarded as 'relaxed' -in the sense of 'sub-constrained' -versions of all chains featuring 'simple' (revolute, prismatic, and/or helical) joints. In order to achieve these main contributions, the article further develops technical tools to analyze the geometry of lines in space: these include the concept of a dual directed angle between two oriented lines relative to a third, the definition of novel dual number functions, as well as a dual inner product for the (6 × 6) cross-matrix representation of oriented lines.
The article is arranged according to the following structure. In Section 2, technical preparations are prepared for the remainder of the document. In Section 3, a comprehensive overview of the trigonometric representation of rotations (TRR) is provided: the section introduces an inner product for the space of skew-symmetric (3 × 3)-matrices, 'cross-matrices' for short, that matches the usual inner product of the corresponding vectors. Finally, the trigonometric formula by Rodrigues, Euler, Gauß, and Gelman (REGG) is stated together with a proof based on an analysis of the periodicity of the cross-matrix powers. The main theoretical contributions of the article are presented in Section 4. The principle of transference is applied to the TRR, yielding the adjoint trigonometric representation of displacements (ATRD), a certain generalization of the REGG-rotation formula. In the presented shape, the generalization employs the matrix algebra of real (6 × 6)-adjoint matrices and avoids the usage of matrix algebra over dual numebrs. Coarsely, the proceeding follows the structure of the preceding section. In Section 5, two auxiliary functions for dual numbers are introduced: the dual bivariate inverse tangent function, for computing the two dual solutions of a dual trigonometric equation, and the dual trivariate inverse cosine function, for computing the dual relative directed angle between two lines in space with respect to a third. The main practical contribution of the article is presented in Section 6. The inverse kinematics problem of general affine 3C chains is formulated in the adjoint representation. Based on the ATRD and the two dual trigonometric functions, a novel closed-form solution is developed. The IK computations are illustrated for an example instance of a general spatial 3C chain. In Section 7, the structural geometric and algebraic properties of the TRR formula and of the ARTD formula are briefly reflected. Finally, the article is concluded in Section 8 with an outline of its contributions and potential continuations. A notation overview, further technical details, and an alternate parametrization via Cayley maps are compiled in an supplementary appendix.

PREREQUISITES
In this section, a set of definitions and conventions is compiled in preparation of the main parts of the document. The orientation of a set of three vectors , , ∈ ℝ 3 is defined as the trivariate function A normalized vector is emphasized by using a hat. In particular, a unit vector, a vector with length one, is indicated aŝ= 1 ‖ ‖ ⋅ . The operator * expresses the combination of transposition and multiplication. In particular * ≔ ⋅ denotes the inner product of to vectors and * ≔ ⋅ denotes its matrix generalization. 1 The operator ⊗ expresses the outer product, for example, as ⊗ ≔ ⋅ , determining the dyadic tensor. A Plücker vector 2 is a six-dimensional vector ( ), consisting firstly of a direction , and secondly of a moment , in the so-called ray-coordinate order [2]. An oriented line (spear) in 3D-space 1 The scalar = * and the matrix = * represent cosine similarities for (matrices with columns of) normalized vectors. 2 A 'Plücker vector' is also called a 'motor' [32]: 'Dem Kundigen mag der Hinweis genügen, daß der Motor in gewissem Sinn als der Sechservektor der reellen vierdimensionalen Liniengeometrie erscheint.' in form of a Plücker vector is briefly denoted here as = ( ); its moment is defined by the cross product of an arbitrary, fixed point of the line, an anchor , with the direction of the line, as = × . 3 A unit Plücker vector is a Plücker vector that features either ‖ ‖ = 1, or ‖ ‖ = 1 and ‖ ‖ = 0; here indicated by a hat, aŝ= (̂̂). The corresponding unit Plücker vector (̂̂) associated to a Plücker vector ( ) is obtained by means of the 'normalization function' , with the Plücker vector length In particular, the two parts of a unit screw are the unit direction̂= 1 ⋅ and the normalized moment̂= 1 ⋅ . A screw in 3Dspace in form of a unit Plücker vector is briefly denoted as$ = (̂̂); its moment is determined bŷ= ×̂+ ℎ ⋅̂with a pitch ℎ. A finite twist, representing a spatial displacement, is obtained as product of an angle with a unit screw, as $ = ⋅$. A Plücker vector is expressed in form of a dual 3-vector$ = + ⋅ (Appendix B), instead of the form as a real 6-vector. The (4 × 4)cross-matrix form of a unit screw$ ≅ (̂,̂) reads$ ⊗ = (̂⊗̂0). The homogeneous matrix displacement = ( 1 ) ∈ is computed as the matrix exponential of the product of an angle and the (4 × 4)-cross-matrix form$ ⊗ of a unit screw as Finite-sum expressions, that involve cubical and fractional expressions for general displacements, are, for instance, stated in [11][12][13]. For a given homogeneous displacement matrix = ( 1 ), the components and are retrieved, by means of the projection operators = ( ) ∈ ℝ 4×3 and = ( 1 ) ∈ ℝ 4×1 , via the matrix products The (6 × 6)-left adjoint matrix of a displacement [13,22] is denoted in the sequel briefly as̀= Ad( ). In correspondence to Equation (3), the adjoint matrix is given as = adsp where ⊗ expresses the skew-symmetric matrix (Appendix C) associated to , the translation vector. A simplified finite-sum expression has recently been reported in [22] for the adjoint representation. 4 The dual inner product of two screws, given as Plücker vectors $ and $ , is defined as and is typically deduced from the interpretation of a Plücker vector as dual vector$ = + ⋅ and from the definition of dual multiplication (compare Equation (46) in Appendix B). For unit lineŝand̂, the dual inner product of Equation (5) matches the 'dual cosine similarity'̃=̂⊛̂= (̂) ⊛ 3 Given a line in form of a Plücker vector , the anchor can generally not be retrieved. The point ⋆ = ( × )∕( * ) is the point of that features the minimal distance to the origin/the least norm. 4 In the present article, the adjoint representation is not expressed via the 'arguments' of an angle and a unit twist$ but via the 'arguments' of a dual anglẽ and a unit linê. This reparametrization is the origin for the formulation of the adjoint trigonometric representation of displacements in Section 4 and its consequences. the dual cosine (Appendix B) of the dual anglẽ= + ⋅ which describes the rotational offset, , and translational offset, , of the two lineŝand̂in space. 5 Applying the dual inverse cosine function (Appendix B) to the dual cosine similaritỹ, as |̃| =ãcos(̃) =ãcos( + ⋅̊) = acos( ) + ⋅ −s in(acos( )) , the dual absolute value (Appendix B) of the dual angle |∡ | =ãcos(̂⊛̂) between the two unit lines is obtained. The operation extends the primal inverse cosine function that determines the absolute value of the angle between two unit vectors, | ∡ | = acos(̂ * ̂). 6

TRIGONOMETRIC REPRESENTATION OF ROTATIONS
This section introduces the rotation formula by Rodrigues-Euler-Gauß in the trigonometric form, documented by Gelman [21], via three mutually reciprocal base matrices. All three matrix tensors are associated to a 'rotation 3-vector' and are equipped with dedicated names and operator symbols due to their geometric and algebraic significance. A supplementary overview is provided with Table 5 in the appendix.

Orthogonal base matrices
In accordance with [16,21,33], three (3 × 3)-matrices, the cross-matrix ⊗ , the unit-matrix ⊚ , and the square-matrix ⊘ , associated to a vector ∈ ℝ 3 are defined as The unit-matrix and the square-matrix can be characterized as 'product matrices' of 'inner type' and of 'outer type', respectively, as it can be observed by the identities The product-character of the two matrices is also observed by their sum: the two matrices add up to the 'inner product in diagonal form': Geometrically, the three matrices serve as particular operators if applied for unit vectors in matrix-vector products [22]: the projection vector, the rejection vector, and the orthogonal vector of a vector with respect to a unit vector̂are obtained as ( An example for the application as geometric operators is given in Figure 1. Algebraically, the three matrices are rank-deficient for ≠ with rank( ⊗ ) = 2 rank( ⊚ ) = 2 rank( ⊘ ) = 1 .

Matrix inner product
For the scope of this article, the Lie algebra of (3 × 3)-matrices is equipped with the particular inner product 5 A method for extracting the corresponding unit line (four parameters) from a given finite screw (six parameters) is described in Equation (25) and in Equation (31). 6 The directed angle between vectors and between lines is determined in Section 5 (Equation (39) and Equation (42)). The product ⟨ , ⟩ 33 connects the matrix algebra consistently to the typical inner product of vector algebra. In particular, the matrix inner product of two cross-matrices, ⊗ and ⊗ , matches (based on Equation 53) the usual vector inner product: 7 ⟨ ⊗ , ⊗ ⟩ 33 = * .
The three matrices of Equation (6) are pairwise reciprocal, as it can be read-off the equations The inner product of Equation (10) relates the squared norm of a cross-matrix consistently to the squared norm of a vector (via Equation 11) The squared norms of the three base matrices (Equation 6) for unit vectors correspond to

Periodic powers
For the positive powers of the unit vector's cross matrix̂⊗, the recursive relations hold. Based on this observation, the generalized powers (for integer exponents) for a unit vector's cross-matrix are defined as the augmented definition reflects the fact that the cross-matrix geometrically corresponds to a rotation of 2 within the planê ⟂ . The cyclic relation is illustrated in Figure 2. As the unit matrix̂⊚ and the square-matrix̂⊘ represent projections, they are idempotent, i.e., the powers equal the matrices themselves 7 For the unit-matrix and the square matrix, the inner products are ⟨ ⊚ , ⊚ ⟩ 33 = 1 2 ⋅ (3 ⋅ ( * ) − 2 ⋅ ( * ) 2 + ( * ) ⋅ ( * )) and ⟨ ⊘ , F I G U R E 2 Illustration for the periodicity of the generalized powers of cross-matriceŝ⊗ and̂ : the expression (×) is a placeholder for̂⊗ , the (3 × 3)-cross-matrix of a unit vector̂, and for̂ , the (6 × 6)-cross-matrix of a unit linê. Multiplying with (the inverse of) a cross-matrix corresponds to a ∕2-rotation, counter-clockwise (clockwise) with respect to the orientation of the rotation axis

Trigonometric rotation formula
The proper orthogonal matrix for a rotation by an angle about an axiŝ, is determined as the trigonometric representation [16,21,33] of rotations as In terms of the introduced (3 × 3)-base matrices, the rotation matrix is obtained as an affine-trigonometric combination. As an alternative to the exponential map, the rotation matrix is obtained via the Cayley map for the argument 'tan( ∕2) ⋅̂⊗' (Rodrigues' vector) in Appendix D. A formal proof for the trigonometric representation of rotation in Equation (15) is obtained by applying Taylor series expansion and employing the periodicity of the powers of the cross-matrix (Equation 13). By means of the trigonometric representation of rotations, the invariance ⋅̂=̂follows immediately ⋅̂= cos( ) ⋅̂⊗ * ̂⊗ ⋅̂+ sin( ) ⋅̂⊗ ⋅̂+̂⋅̂ * ̂=̂, using the cross product identitŷ⊗ ⋅̂=̂×̂= and the inner product identitŷ * ̂= 1.

ADJOINT TRIGONOMETRIC REPRESENTATION OF DISPLACEMENTS
The rotation formula by Rodrigues, Euler, Gauß, and Gelman is generalized in this section for spatial displacements in a particular manner. As previously, the formula is stated by means of three mutually reciprocal base matrices. Each of the three matrix tensors is associated to a 'screw Plücker 6-vector' and here equipped -due to its geometric and algebraic significance -with a dedicated name and symbol. Supplementary overviews are provided with Figure 12 and Table 5 in the appendix.

Left adjoint representation
The left adjoint representation of a displacement, with homogeneous matrix =  . The (6 × 6)-left adjoint matrix is defined as that operator which transforms the Plücker vector (of a screw) in correspondence to the left adjoint action of a homogeneous displacement on the homogeneous representation (of the screw), formally as The main purpose of the adjoint representation in this article is not to serve as 'a matrix operator for the adjoint action on Plücker vectors', but as 'a matrix representation of rotations and translations for rigid bodies'. In this context, the adjoint matrix representation can be regarded as an alternative to the homogeneous matrix representation (see Figure 4 at the end of this section). The adjoint matrix representation can also be further seen as the 'real manifestation' of the displacement representation using dual (3 × 3)-matrices which has been employed in spatial kinematic analysis in the past [6,8,34,35].

Primal and dual unit matrices
For emphasizing the close relationship between the adjoint matrix representation and the dual matrix representation, the 'primalunit-matrix' and the 'dual-unit-matrix' are defined as in analogy to the definition of dual numbers (Section B). The matrices and serve as 'primal-dual unit matrices' to express the left adjoint matrix representation conveniently. Via and , and by means of the lifting map Γ( ) = 1 * ⋅ 1 + 2 * ⋅ 2 , with auxiliary matrices 1 = ( ) ∈ ℝ 3×6 and 2 = ( ) ∈ ℝ 3×6 , the adjoint representation reads as in terms of its 'constituting' components, the rotation matrix and the translation vector . The product-shaped decomposition based on and in Equation (18) is comparable to the expression of Study parameters in terms of the rotational and the translational parameters [4]. Similarly to Equation (4), for a given adjoint displacement matrix̀, the components and are formally retrieved, by means of the projection operators̀= ( ) ∈ ℝ 6×3 and̀= ( ) ∈ ℝ 6×3 , via the matrix products The operations for obtaining and from a homogeneous and from an adjoint displacement matrix in Equation (4) and Equation (19) are subsumed in a schematic overview within the upper part of Figure 4 in Section 4.

Dual number matrix form
A dual number̃= + ⋅̊is lifted into the form of a compatible (6 × 6)-matrix by substituting the real unit by the primal unit matrix and the dual unit by the dual unit matrix . The embedding is formalized by the operator defined as For the remaining parts of this section, the matrix expressions corresponding to a dual angle and to its dual trigonometric functions (Equation 50) are stated explicitly.

Adjoint orthogonal base matrices
As affine line generalizations to the three (3 × 3)-matrices defined in Equation (6), three (6 × 6)-matrices are stated in accordance with [22]. The cross-matrix $ , the unit-matrix $ , and the square-matrix $ , associated to a six-dimensional Plücker vector $ = ( ), are defined as For sake of a compact description, the matrix product ⊗ + is used as an abbreviation for the sum of two 'twisted' matrix products as By means of this symmetric product (further properties are stated in Section C), two equalities are stated as line-generalization of the two vector-identities in Equation (7). In terms of the primal-dual unit matrices and from Equation (17), the line cross-matrix reads The line unit-matrix and the line square-matrix are expressed in terms of the primal-dual unit matrices and of the symmetric matrix product from Equation (22) as Geometrically, the three matrices, $ , $ , and $ , serve as geometric operators for Plücker vectors if applied in matrix-vector multiplications [2]: the line projection, the line rejection, and the line orthogonal 8 of a line with respect to a unit linêare obtained as The obtained results from of the three line operations, generally involve non-zero pitches, indicated in compact notation as . For deriving the corresponding line vectors, satisfying the orthogonality of moment and direction, × = 0, the 'alignment function' ∶ $  → , is defined in accordance with [22] as . All lines pass the closest point to on̂('point projection' in Equation 26) and is applicable to obtain the lines from the computed screws, as = ( $ ), = ( $ ), and = ( $ ). In Figure 3, an example for the projection applications is given. In that figure, the point projection of a line , for ∈ { , } , onto the unit linêare computed via

Adjoint matrix inner product
For the scope of this article, the Lie algebra of (6 × 6)-matrices is equipped with the particular, dual inner product ⟨̀,̀⟩ generalizing the inner product for (3 × 3)-matrices introduced in Equation (10). The adjoint matrix inner product ⟨̃,̃⟩ 66 is consistent to the usual inner product of two screws: for two cross-matrices, $ and $ of Plücker vectors $ and $ , the adjoint matrix inner product evaluates via Equation (53) to identical to Equation (5). In compact form, the identity holds as a generalization of Equation (11). For unit lines, the expression resembles the dual-cosine similarity ⟨̂,̂⟩ 66 from Equation (5), cos − ⋅ ⋅ sin , of the rotational offset and the translational offset between two lines in space,̂and̂. By means of the adjoint matrix inner product, the pairwise reciprocity of the three base matrices is observed with The squared norm of a cross-matrix is determined with Equation (28) as In consistency to the norms of Plücker vectors, the squared norm for a cross-matrix of a unit linêis evaluated to ‖̂ ‖ 2 = 1. For all three base matrices of a unit line, the squared norms are determined with in consistency to the results for unit vectors in Section 3. By means of the dual square root in Appendix B, the norm of a cross-matrix is computed as simplifying to ‖ ‖ = ‖ ‖ and to ‖̂ ‖ = 1 for lines and unit lines. Based on the derived norm, a normalization for the 'adjoint representation of Plücker vectors' is given via the fraction 1 ‖$ ‖ , expressed as an adjoint matrix with from Equation (20). In detail, the cross-matrix normalization is computed by the equation chain The normalization (for the cross-matrix) of a screw yields the (cross-matrix for the) corresponding unit line, indicating the screw axis in normalized form. In the chain of Equations (31), the normalization function , from Equation (2), and the alignment function , from Equation (25), are recovered 'by definition'. The steps of the normalization process are illustrated in the lower part of Figure 4.

Periodic powers
For the positive powers of the unit line's cross matrix̂ , the recursive relations hold. Based on this observation, the generalized powers (for integer exponents) for a unit line's cross-matrix are defined, as a generalization of Equation (13), as The augmented definition reflects the fact that the (6 × 6) cross-matrix geometrically corresponds to a rotation of 2 about the affine linêin the planê⟂. The scheme in Figure 2 illustrates the definition of Equation (32). The matriceŝ and̂ are idempotent projections, their powers equal generalizing the corresponding Equation (14) for unit vectors.
The adjoint trigonometric representation of displacements (ATRD) expresses the left adjoint matrix as an affine-trigonometric combination of the three (6 × 6)-base matrices introduced in advance. As an alternative to the exponential map, the displacement matrix̀is obtained via the Cayley map for the argument '(tan(̃∕2)) ⋅̂ ' (generalized Rodrigues vector) in Appendix D.
The structure of the formal proof for the ATRD corresponds to the spherical counterpart in Section 3 with employing the periodic nature of the powers in Equation (32). The invariancè⋅̂=̂, as an affine generalization to Equation (16), follows immediately by means of the adjoint trigonometric representation of displacements ⋅̂= −(cos̃) ⋅̂ ⋅̂ ⋅̂+ (siñ) ⋅̂ ⋅̂+̂ ⋅̂=̂ (34) using the reciprocity condition̂ ⋅̂= and the idempotence condition̂ ⋅̂=̂. Two matrix representations of spatial displacemcents are compared in Figure 4. The homogeneous matrix representation, indicated on the left hand side, is opposed to the adjoint matrix representation, on the right hand side. The structural similarity between the two matrix representations and their conceptual differences can be read-off the graphics. This concerns the 'composition' (products) from 'geometric parameters' (angles, lines, screws) and the decomposition (projections) into rotational (matrix), translational (vector), and invariant (screws, lines) subcomponents.

NOVEL CONCEPTS FOR DUAL QUANTITIES
In this section, two novel concepts for dual quantities are introduced. First, the dual trigonometric equatioñ⋅ cos̃+̃⋅ siñ= is stated together with its closed-form solution via the novel dual bivariate inverse tangent function 'ãtan2'. Second, the concept of the dual relative directed angle between two lines with respect to a third is presented and computed via the novel dual trivariate inverse cosine functionãcos3.

Trigonometric equations
The two angles solving the primal trigonometric equation are computed, with = The two solutions to Equation (35) can be interpreted geometrically as the intersection points of a line and a circle [37]. An example is provided in Figure 6 together with the numerical values. The number of distinct solutions to Equation (35) can be determined, by means of the normalization term ≔ √ 2 + 2 and the normalized parameter̂≔ , via as one of three possible cases. 9 Applying the principle of transference to Equation (35), the dual trigonometric equatioñ ⋅cos̃+̃⋅siñ=̃, is obtained. Applying the same dualization principle to the solution approach Equation (35), the two dual angles solving the dualized equation are determined, with̃= √̃2 +̃2 −̃2, via the expressioñ The definitions for the square and for the square root of a dual number are provided in Appendix B. The dual bivariate inverse tangent functionãtan2 required in Equation (37) is defined as atan2(̃,̃) =ãtan2( + ⋅̊, + ⋅̊) = atan2( , ) + ⋅̊−2 9 The degenerate setup with = 0, ≠ 0, and ≠ 0 corresponds to the inversion of sine function, the degenerate setup with ≠ 0, = 0, and ≠ 0 corresponds to the inversion of cosine function. In both setups, the line intersecting the circle is parallel to one of the two coordinate axes.
F I G U R E 5 Concepts of angles and interrelations identical to the first Taylor approximation of 'atan2' (in correspondence with the principle of transference, applied to real function [24], Appendix B) and generalizing the dual inverse tangent function (Equation 50). A direct geometric interpretation of the dual constraint in Equation (37) is cumbersome due to the dimension of the stated problem; the two solutions can be interpreted as the (dual) intersection points of a (dual) line with a (dual) circle, as in the primal case. 10

Dual relative directed angles
The dual directed angle between two lines relative to a third line is introduced as a generalization to the dual angle between two lines (Section 2) as well as to the directed angle between two vectors relative to a third vector [33]. See Figure 5 for a graphical illustration.
As the absolute value of the directed angle between two vectors, and , relative to a third vector̂, matches the 'absolute angle' between and if the third vector̂is orthogonal to the vectors and , the absolute value of the dual directed angle between two lines, and , relative to a third linê, matches the 'dual absolute angle' between and if the third linê is reciprocal to the lines and . 11 The relative directed angle from to , measured with respect to the (directed) axiŝ, is computed, in accordance with [33], using the trivariate inverse trigonometric function compare Figure 7 for an illustration. The concept is extended from the spherical case, of intersecting lines, to the affine case, of non-intersecting lines. The dual relative directed angle∡ from to measured with respect to the (directed) unit linêis formally determined via the dual trivariate inverse trigonometric function 'ãcos3' as The trivariate inverse dual trigonometric function 'ãcos3' is developed within four steps. In the first step, the unit lines of the line rejections of̂and̂with respect tôare computed ⟂ = ( ( (̂;̂)))̂⟂ = ( ( (̂;̂))) .
The required functions are introduced in the previous sections. See Equation (21) for the line rejection ' ', Equation (25) for the alignment function ' ', and Equation (2) for the normalization function ' '. Alternately, the matrix normalization 'nrml' in Equation (31) could be applied. In the second step, the dual inner product of the lineŝ⟂ and̂⟂ , see Equation (5) and Equation (28), is determined:̃⟨ The dual number̃represents the dual cosine similarity of the two lines measured within the planê⟂. In the third step, the dual cosine similarity is transformed into an dual absolute relative angle |̃⟨ ⟩ | =ãcos (̃⟨ ⟩ ) , 10 As an alternative to using the dual bivariate inverse tangent function, the two solutions to Equation (36)  In particular, the condition det( ) = 0 covers the 'constellations' (the geometric posture of all joint axes of a mechanism, given by design) of parallel, antiparallel, or coincident lines and : In Figure 8, two illustrations of geometry of a generic and of a degenerate line constellation are provided. By means of the introduced tools, the dual inverse cosine functionãcos3, for computing the dual directed angle from linet ôrelative to linê, is defined by the compact expressioñ In contrast to the dual absolute directed angle between two lines, given in Equation (5) in terms of dual vector algebra and in Equation (11) in terms of real matrix algebra, the dual relative directed angle, computed in Equation (42), respects a 'third' line. Geometrically, a dual relative directed anglẽ⟨ ⟩ =̃has the following interpretation: the real part of the dual anglẽrepresents that angle which 'rotates' , by computing ′ = exp(( + ⋅ 0) ⋅̂ ) ⋅ , so 'far' such that the transformed direction of ′ projects to the same vector in the planê⟂ as the direction of does. The dual part of the dual anglẽrepresents that shifts which 'translates' , by computing ′ = exp((0 + ⋅ ) ⋅̂ ) ⋅ , so 'far' such that the transformed location of ′ projects to the same point on the linêas the line does. In conclusion, the rotational part of̃⟨ ⟩ corresponds to the directed arc length between the projections onto the unit circle within the planêand the translational part of̃⟨ ⟩ corresponds to the directed vector length between the projections onto the linêitself.

ANALYTIC SOLUTION TO THE IKP OF GENERAL 3C CHAINS
This section provides the novel analytic solution to the inverse kinematics problem of general affine 3C chains in the adjoint representation. The solution is based on the ATRD, the generalized REGG formula for spatial displacements, introduced in Section 4, and on the novel dual inverse trigonometric functions 'ãtan2' and 'ãcos3', introduced in Section 5. The method is illustrated with two examples.

Problem statements
The finite inverse kinematics problem (IKP) of a generic spatial 3C chain can be stated as the task to find joint configuration vectors = ( , , , , , ) that satisfy  (3) in form of a homogeneous transformation matrix. In the formulation, each joint displacement acts 'with respect to the origin' and the equation is a product-of-exponentials [39]. 12 Alternately to the homogeneous matrix form in Equation (43), the inverse kinematics problem for a general spatial 3C chain is restated as a product-of-exponentials in the adjoint matrix form in the equatioǹ Each of the three cylindric joints is represented by a single (6 × 6) left-adjoint matrix, determined as the matrix exponential adsp and computed via the adjoint trigonometric formula. The provided closed-form solution to the IK problem of 3C chains is based on the adjoint problem formulation in Equation (44), on the adjoint trigonometric representation of displacements in Equation (33), and on the novel dual concepts of Section 5.

Solution method
The analytic solution to the inverse kinematics problem of general 3C chains form is achieved in four steps. In the first step, the problem is stated based on the ATRD from Section 4. In the second step, the second dual anglẽis computed by means of the dual bivariate inverse tangent function from Section 5. In the third and fourth step, the first and the third dual angles,ã nd̃, are determined based on the concept of the dual relative directed angle from Section 5.

=̀⋅̀⋅̀.
In the first step, the ansatz [22,41] for computing the second dual angle is given by exploiting the invariance of the first and third axis with respect to those displacements, explicitly stated in Equation (34), aŝ As a second step, the trigonometric representation of displacement Equation (33)  Similarly, the two dual angles for the third joint are obtained with Overall, two solutions to the inverse kinematics problem of 3C chains in Equation (44)

Example
The application of the closed-form solution method for the IKP of spatial 3C chains is illustrated by a kinematic chain which features joint axes that resemble the line constellation depicted in Figure 3 Figure 9), the effector is positioned at = (−4, +4, 10) and oriented as the global origin. The solution method is once applied for zero reference pose (in Figure 9) and once for the pose depicted in Figure 10. In the second case, the displacement 14 of the effector with respect to the origin corresponds to In both figures, the links are indicated as three dimensional splines, connecting to the adjacent joints asymptotically to the directions of the respective joint axes. The cylindric joint are indicated by tuples of blue and gray cylinders. Markings on the cylinder surfaces indicate the rotational joint offsets. The yellow 'rod' connecting the cylinders highlight the translational joint offsets. The illustrations exemplify the consistency of the solutions that are obtained by the analytic method from Section 6. Table 2 provides an overview of the (approximate) numerical values for the inverse kinematics solutions.

DISCUSSION
The adjoint trigonometric formula ATRD in Equation (33) In order to interpret these trigonometric formulas in an 'intuitive' manner, Figure 11 provides two drawings that outline their common 'geometric' structure. The figure indicates the reciprocity of the three base matrics and comprehends the period-four cyclicity of the cross-matrix ( Figure 2). It further motivates the two trigonometric formulas as particular extensions 13 of Euler's formula [42]̂= exp( ⋅ ) = cos( ) + ⋅ sin( ). For comparability with the pattern in Equation (45), Euler's formula is also circumscribed by exp(phi ⋅ imag) = cos(phi) ⋅ one + sin(phi) ⋅ imag + zero.
T A B L E 3 Comparison of four algebraic representations of oriented unit lines. In the first three rows, the four representation are characterized by their names, their used symbols, and their ambient space. In the fourth row it is emphasized that a (3 × 1) dual vector generally contains non-real elements. Next, its is highlighted that the set of Plücker vector does not feature a Lie algebra structure (there is no exponential map the lifts (6 × 1) Plücker vectors to a Lie group of rigid body displacements). Further, only the matrix representations permit using techniques of linear algebra (for example, the computation of a determinant or trace). The subsequent row highlights the property of the (6 × 6)-representation to allow the definition of three orthgonal, geometrically meaningful base matrices ( Figure 11). The last row indicates that only the representations of (6 × 6) matrices and dual vectors preserve structure when generalizing from a linear (intersecting lines) to an affine-linear (skew lines) geometric setting

CONCLUSIONS
The concept of the adjoint trigonometric representation of displacements has been introduced as the major theoretical contribution of the article. The concept is obtained as a novel 'dual-angle/affine-line generalization in shape of adjoint matrices' of the REGG-rotation formula, the trigonometric representation of rotations. In the future, the adjoint trigonometric formula could find applications in various contexts of computing displacements in spatial rigid body systems. As the mayor practical contribution, the adjoint trigonometric representation has been employed in the adjoint formulation of the inverse kinematics problem of general affine 3C chains to obtain a novel, closed-form solution to this particular problem. The analytic solution could be used as a subroutine in solution schemes for solving kinematic problems of related mechanical designs. In context to these major results, the article features additional contributions. The set of left adjoint (6 × 6) matrices is identified as an alternate tool for dealing with rigid body motions in a comparison to the set of homogeneous (4 × 4) matrices. It has been indicated that the left adjoint matrices represent a real-number version of the (3 × 3) matrices with dual entries. The findings are summarized in a brief discussion. With regard to line geometry and screw theory, the cross-matrix representation of an oriented line is presented as an alternative to (6 × 1) Plücker vectors and to dual (3 × 1) vectors. The algebra of cross-matrices is equipped with a dual inner product that induces those properties that are established for the Plücker vector representation of lines. For example, the inner product provides the means for obtaining the unit line of the axis associated to a screw via the induced normalization of the cross-matrix representation. With respect to dual angles, the article extends the concept of a dual angle lines to the concept of a dual directed angle between two oriented lines relative to a third. Its computation is conducted via a novel, trivariate dual inverse cosine function respecting the orientation of three lines in space accordingly. Further, a bivariate dual inverse tangent function is introduced via the transference principle that permits solving a dual trigonometric equation in closed form. The obtained tools for studying spatial lines and screws can be employed in a broad context of computational geometry and kinematics in the future.

APPENDIX A: NOMENCLATURE
The scheme in Figure A1 indicates the interrelations of the representation of a screw as a 6-vector, as a (4 × 4)-cross matrix, and as a (6 × 6)-cross matrix. Table A2 gives an overview of the matrices that serve as the reciprocal base for the REGG equations in (3 × 3)-shape and (6 × 6)-shape. An overview of the symbols used in this article is provided in Table A1.

Cross-matrices and products
The matrix entries of a cross matrix ⊗ is and its transpose equals it negative. As an expansion of ⊚ = ( * ) ⋅ − ⊘ (Equation 8), the identity is observed. For the product of two-cross matrices, the entries are given as For the cross-matrix products ⊗ * ⊗ = −( ⊗ ⋅ ⊗ ), the identity relating to the inner product and the outer product, holds as a 'dyadic/bivariate' generalization to Equation (8) and Equation C2 .