Dual Numbers Approach in Multiaxis Machines Error Modeling

Abstract

Multiaxis machines error modeling 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.

1. 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 is obtained. The basic setting takes place in the affine extension of vector space . This approach appears quite often in modern literature with minor modifications; see [5–8]; 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 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 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 and . 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 , , and for all 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 , and for all , 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.

2. 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 satisfying the following property: if and , then there are and such that and either or .

Let us recall that for any field we shall define 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.

Consider Rings , , and can be obtained as factor rings of , , and . Only is irreducible over ; thereby is a field. Neither nor are integral domains: we have for and for (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 ( 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 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 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 .

Theorem 1. An element lies in if and satisfies the identity

Proof. and thus .

Definition 2. Let . Then the set 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 .

Lemma 3. The error class is a Lie algebra with respect to the commutator operation if and only if is symmetric, that is, .

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 we obtain and thus

Furthermore, the set will be called the class of the special orthogonal matrices over .

Lemma 4. Let . Then is of the form where , and .

Proof. Because , it has to be of the form Furthermore, let us consider a matrix in the general form . As , the identity (10) has to be fulfilled:
This leads to the following equations: From the first and the last one we obtain identities and . If we substitute these into the rest of the equalities, we obtain the following calculations: Thus for the matrix we conclude that which completes the proof.

Lemma 5. Let then

Proof. and it is easy to see that : On the other hand, if then one can find and such that ; for any we can define , such that :

Theorem 6. Let . Then is of the form where , , and are the matrices of rotations around the axes , , and , respectively, and

Proof. It is known that any element can be represented as a composition of three rotation matrices , , (around axes , , and , resp.). Thus if the extension to is considered, we obtain and , according to Lemma 5.
Let us now consider the rotation around the axis in particular. The matrix will be of the form , where . If we consider a general error matrix of such type in the form , where and are the dimension two row and column vectors, respectively, from the identity (10) we obtain Thus 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 and can be computed similarly.

Let us note that the matrices 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 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.

3. 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 . It is crucial that 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 two-axis 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 , where vectors are represented as the elements . The matrix is then represented by the matrix . For we write .

The transformation matrices are the elements of . When the errors are added, we obtain

Now the transformation matrices became the elements of (note that it would make sense to consider the calculations in the algebra instead, but this is not the topic considered in this paper). The resulting matrix is then of the form

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 . If, in addition, rotation machine elements are considered, it is necessary to employ the rotation error matrices .

4. Notes on Weil Algebras

Let and let us denote the -algebra where is the -algebra of real polynomials in indeterminates and is the th power of its maximal ideals.

Definition 7. Weil algebra is an arbitrary nontrivial quotient -algebra of . For example, for we obtain the dual numbers and for we have the set of polynomials in the form .
Furthermore, the Weil algebra can be defined and the set of polynomials is obtained.
For the sake of the error analysis, the matrix class can be represented by the element of (where determines the level of accuracy of the error analysis): The choice neglects the interference of any two errors and the calculations will be similar to those over the dual numbers. In case , 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, for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the project NETME CENTRE PLUS (LO1202). The results of the project NETME CENTRE PLUS (LO1202) were cofunded by the Ministry of Education, Youth and Sports within the support programme “National Sustainability Programme I.” Publication of the results was financially supported by the project Popularization of BUT R&D results and support systematic collaboration with Czech students CZ.1.07/2.3.00/35.0004.