Dual Numbers Approach in Multiaxis Machines Error Modeling

Multiaxis machines errormodeling is set in the context of modern differential geometry and linear algebra.We apply special classes of matrices over dual numbers and propose a generalization of such concept by means of general Weil algebras. We show that the classification of the geometric errors follows directly from the algebraic properties of the matrices over dual numbers and thus the calculus over the dual numbers is the proper tool for the methodology of multiaxis machines error modeling.


Introduction
The concepts of multiaxis machines error modeling can be found in classical literature [1,2], including the appropriate methodology.For the process description, the homogeneous transformation matrices (HTM) are used as the crucial mathematical tool for the error models; see [3,4].
The complex multiaxis machine positioning is represented by a kinematic chain.Thus, by means of the product of the transformations between successive coordinate systems associated to the mechanisms elements, from the absolute reference system to the tool reference system, the global transformation matrix  ∈ Mat 4 (R) is obtained.The basic setting takes place in the affine extension of vector space R 3 .This approach appears quite often in modern literature with minor modifications; see [5][6][7][8]; for a rare attempt to use modern advanced mathematical structures such as the algebra of quaternions, see [9].
The main uncertainty sources in the design and construction of machine tools are geometric and kinematic errors, thermal errors, stiffness error, and errors addressed to the deflection of cutting tools.Those mentioned above are the known sources.Their consequences are complex, but techniques to evaluate them or compensate their effects have being developed; see [10,11].In our paper we work with geometric and kinematic errors in any machine component which can be considered in the kinematic model as a new parameter.
For example, the coordinate transformation matrices and the corresponding error matrices for three-axis machines are  1 = ( 1 0 0  0 1 0 0 0 0 1 0 0 0 0 1 ) , where   denotes the linear errors along the th axis,   ,  ̸ =  is the straightness errors in the th axis direction when moving along the th axis,   is the angular errors around the th axis when moving along the th axis, and   is the squareness errors between the corresponding axes.
Our goal is to set the methodology of multiaxis machines geometric error in the context of modern theory of Weil algebras [12].In this paper, we essentially use Weil algebra D 1 1 and in the final section we present the directions of further research and discuss the advantages of employing more general Weil algebras into the error analysis.When assembling the kinematic chain containing geometric errors, we embed the error matrix corresponding to any kinematic joint, that is, the errors of joint translation or rotation.In particular, for the translation in the vector (, , ) direction or for the rotation around the  axis by the angle , the following error matrices apply, respectively: ) . ( The parameters , , and  represent the error rotations around the axes , , and , respectively, and  gives the proper rotation around axis .The error matrices were derived from the rotation matrices around particular axes by approximation.More precisely, for the translation of the rotation around axis , error matrix is approximated as follows: ( Thus cos   1 and sin   .In case two approximations are multiplied, the whole term vanishes.This is caused by the assumption that the errors are by order smaller than the proper rotation parameters.The above mentioned representation is a standard description of the error matrices to be embedded into the kinematic chain.For example, in case of two-axis machines with two translation joints, we obtain the following kinematic chain: If the above mentioned identities     = 0,     = 0, and     = 0 for all ,  ∈ {1, 2} are applied, we obtain the matix and the corresponding kinematic equations which are to be solved within the error analysis.Generally, in case of the system of linear equations, we proceed by Gauss elimination; for nonlinear systems we use Gröbner bases.In our setting, we compute with the matrices using the identities     = 0,     = 0 and     = 0 for all ,  ∈ {1, 2}, which resemble the identities for the imaginary parts of the dual numbers.Thus it makes sense for the whole theory to use the homogeneous transformation matrices over the dual numbers.Our further approach to error calculations will thus be based on the dual numbers calculus; more generally we use a Weil algebra.This gives us a formal setting for the geometric errors modeling.

Matrices over Dual Numbers
As we are going to calculate with matrices over a structure different from real or complex numbers, moreover a structure which is not a field but a ring only, we have to guarantee that the calculations within the inverse kinematics make sense.In mathematical language, we need the dual numbers to form the so-called Euclidean domain.
By an Euclidean domain  we understand an integral domain which is endowed with at least one Euclidean function, that is, function of the form  : −{0  } → Z + 0 satisfying the following property: if ,  ∈  and  ̸ = 0  , then there are  and  ∈  such that  =  +  and either  = 0  or () < ().
Let us recall that for any field we shall define () = 1 for any nonzero  and thus any field is Euclidean.The most important property of the Euclidean domain is that the Euclidean algorithm can be used to find the greatest common divisor of its two elements (i.e., one can easily see that any Euclidean domain is a principal ideal domain-PID).This leads to the fact that in Euclidean domains the Gauss elimination method for solving systems of linear equations can be applied.
As an example of Euclidean domain, let us recall three well-known extensions of real numbers.Only  2 +1 is irreducible over R; thereby C is a field.Neither D nor P are integral domains: we have  ×  = 0 for D and ( + ) × ( − ) = 0 for P (by equations also presented all zero divisors of these rings).
In particular, the dual numbers extend the real numbers by adjoining one new element  with the property  2 = 0 ( is nilpotent).The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers.Every dual number has the form  =  +  (7) with  and  uniquely determined real numbers.Division of dual numbers is defined when the real part of the denominator is nonzero.The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the nonreal parts.We define the following class of matrices over dual numbers: and the group of orthogonal matrices over dual numbers: Indeed, the set (, D) with standard matrix multiplication is a group as it is closed under the operation, more precisely ()   =      =    = ; the inverse to the matrix  is the matrix   and the unitary matrix is of the form  + 0.
is called the dual extension of the matrix  and the set of the imaginary parts of the matrices within the class   is called the -admissible error class or shortly the error class.
Let us note that in the special case  = , the error class   is a Lie algebra s().

Lemma 3. The error class 𝐼 𝐴 is a Lie algebra with respect to the commutator operation [𝐹, 𝐺] = 𝐹𝐺 − 𝐺𝐹 if and only if
Proof.For  ∈ (), ,  ∈   holds the following: Example.Another possible choice of the matrix  such that   is an algebra is  = , where where  is an identity × matrix.Indeed, for another matrix (     ) ∈   with , , ,  ∈ Mat  R we obtain and thus Furthermore, the set will be called the class of the special orthogonal matrices over D.
Proof.Because  ∈  (2), it has to be of the form Furthermore, let us consider a matrix  in the general form (     ).As  +  ∈ (2, D), the identity (10) has to be fulfilled:   (21) Thus for the matrix  we conclude that which completes the proof.
Let us now consider the rotation around the axis  in particular.The matrix  will be of the form ( 1 0 0  ), where  ∈  (2).If we consider a general error matrix of such type in the form (     ), where  ∈ R and ,  are the dimension two row and column vectors, respectively, from the identity (10) we obtain Thus  = 0 and  = −   which is equal to    =   and the identity    = −   for the dimension two matrices holds from Lemma 4. If, in addition, then and we obtain that ) . ( The remaining rotation represented by  2 and  3 can be computed similarly.
Let us note that the matrices    ,  ∈ {1, 2, 3} contain the errors ,  and if  is understood as the sum  +  of the rotation angle  and the rotation error angle , then all classical rotation errors are involved and the appropriate error matrix with  = 0 in addition corresponds to the classical error matrix.The geometric role of the parameter  within the matrix is unknown as it does not appear in the geometric error modeling.

Example
The following elementary example of two-axis machine will show the methodology of geometric errors modeling.The kinematic chain is described by means of the moving frame method, where the rotation matrices   ∈ () are replaced by the matrices   +     ∈ (, D).It is crucial that D is the Euclidean domain and thus the methods of Gauss elimination and Gröbner bases can be used when the inverse kinematics is solved.We will demonstrate the process in the case of twoaxis machine with one rotation axis  and one translation in the direction of axis .Thus, in the following, we shall work in the affine extension of the vector space R 3 , where vectors X = (, , )  ∈ R 3 are represented as the elements (, , , 1)  ∈ R 4 .The matrix  ∈ Mat  (3, D) is then represented by the matrix (  X 0 1 ) ∈ Mat  (3, D) ⋊ R 3 ⊂ Mat  (4, D).For  = (, X) ∈ Mat  (3, D) ⋊ R 3 we write   := (  , X).
The transformation matrices are the elements of (3) ⋊ R 3 .When the errors are added, we obtain Now the transformation matrices became the elements of (3, D) ⋊ R 3 (note that it would make sense to consider the calculations in the algebra (3, D) ⊗ D instead, but this is not the topic considered in this paper).The resulting matrix is then of the form ) . (36) This describes the case of two-axis machine completely.To apply this approach on the three-axis machine, it is enough to extend the kinematic chain by the term ( 3 +   ).If, in addition, rotation machine elements are considered, it is necessary to employ the rotation error matrices (  +    ).
Definition 7. Weil algebra is an arbitrary nontrivial quotient R-algebra of D   .For example, for D 1 1 we obtain the dual numbers D and for D 2 2 we have the set of polynomials in the form  +  +  2 +  2 + .
For the sake of the error analysis, the matrix class  1 +   1 can be represented by the element of D  4 (where  determines the level of accuracy of the error analysis): (39) The choice  = 1 neglects the interference of any two errors and the calculations will be similar to those over the dual numbers.In case  = 3, the actual interference of three errors for the term to be neglected is needed, but with the additional choice of the Weil algebra one can determine those error combinations which can be neglected or eventually replaced.For instance, the choice works similarly to the classical calculations over the dual numbers, but the interferences of two different rotation errors are not neglected, that is,     ̸ = 0 for  ̸ = .