Rational Parametrization of Linear Pentapod’s Singularity Variety and the Distance to it
Abstract
A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (endeffector), which are connected via legs to the base. This manipulator has five controllable degreesoffreedom and the remaining one is a free rotation around the motion platform axis (which in fact is an axial spindle). In this paper we present a rational parametrization of the singularity variety of the linear pentapod. Moreover we compute the shortest distance to this rational variety with respect to a suitable metric. Kinematically this distance can be interpreted as the radius of the maximal singularity freesphere. Moreover we compare the result with the radius of the maximal singularity freesphere in the position workspace and the orientation workspace, respectively.
Keywords:
Pentapod, Kinematic Singularity, Rational Variety, Singularityfree zone.1 Introduction
The StewartGough platform (sometimes called simply Stewart platform) can be defined as six degreeoffreedom (DOF) parallel manipulator (PM) with six identical sphericalprismaticspherical () legs, where only the prismatic joints are active. This parallel robot is merely used in flight simulation where a replica cockpit plays the role of the moving platform.
Although the Stewart platform is the most celebrated PM, some of its subassemblies with a lower number of legs are of interest from theoretical and practical points of view. Sometimes these subassemblies are referred to as components kg2000 (). In this paper we study the socalled linebody component, which is a rigid subassembly of a Stewart PM consisting of a linear motion platform (endeffector) named and five legs, where the base anchor points can have position in . Here this component is referred to as linear pentapod, which is an alternative to serial robots for handling axissymmetric tools (see Fig. 1). Moreover we use the following notations:

The position of is given by the vector and the orientation of is defined by a unitvector .

The coordinate vector of the platform anchor point of the th leg is described by the equation for .

The base anchor point of the leg has coordinates .
Note that all vectors are given with respect to a fixed reference frame, which can always be chosen and scaled in a way that the following conditions hold:
(1) 
According to (naccepted, , Theorem 12) one possible pointmodel for the configuration space of the linear pentapod reads as follows: There exists a bijection between and all real points located on the singular quadric . Based on this notation we study the singularity loci of linear pentapods and the distance to it in the paper at hand, which is structured as follows:
We close Section 1 by a review on the singularity analysis of linear pentapods and recall the implicit equation of the singularity variety. In Section 2 we give a brief introduction to rational varieties and present a rational parametrization of the singularity loci of linear pentapods. In Section 3 we compute the minimal distance to the singularity variety with respect to a novel metric in the ambient space of the configuration space . We also compute the closest singular configuration under the constraint of a fixed orientation and a fixed position, respectively. Finally a conclusion and a plan for future research is given.
1.1 Singularity Variety of the Pentapod
Singularity analysis plays an important role in motion planning of PMs. For linear pentapods the singularities as well as the singularinvariant legrearrangements have been studied in btt2011 () for a planar base and in borras2010singularity () for a nonplanar one. A complete list of architectural singular designs of linear pentapods is given in ns (), where also nonarchitecturally singular designs with selfmotions are classified (see also n2016 ()).
Kinematical singularities occur whenever the Jacobian matrix becomes rank deficient, where can be written as follows (cf. borras2010singularity ()):
This Jacobian matrix has a rank less than five whenever the determinants of all its submatrices vanish. So by naming the determinant of the submatrix, which results from excluding the th column, with the singularity loci equals ; i.e. the variety of the ideal spanned by the polynomials . It can easily be checked by direct computations that this variety equals the zeroset of the greatest common divisor of . This singularity polynomial has the following structure:
(2) 
where the coefficients belong to the ring which evidently makes a polynomial with the totaldegree of 3 belonging to . Note that for a specified orientation the equation determines only a quadric surface in the space of positions. This property is of great importance later on.
Remark 1
It can easily be checked that the polynomial is identical with the determinant of a matrix, given in (borras2010singularity, , Eq. (4)).
2 Rational Parametrization of the Singularity Variety
In this section we rationally parametrize the singularity variety, which is given by the implicit equation . But before stepping into the computations, the presentation of a formal definition of this parametrization seems necessary.
Definition 1
Let be a field and and be irreducible affine varieties. A rational mapping from to is a function represented by
(3) 
where and satisfies the following properties:

is defined at some point of .

For every where is defined, .
Definition 2
Two irreducible varieties and are said to be birationally equivalent if there exist rational mappings and such that and be equal to and respectively.
Definition 3
A rational variety is a variety that is birationally equivalent to .
One can find the extensive discussion of above definitions in (shafarevich1977basic, , Chapters 1 and 2).
Having a rational parametrization of a variety has numerous advantages: If the coefficients of the polynomials and of Eq. (3) belong to and if is an element of , then one obtains points with rational coordinates on the singularity variety (shafarevich1977basic, , page 3). This is a matter, which is of high importance to computer aided designs, as computers can calculate rational coordinates at a much faster rate.
Moreover the rationality of the singularity variety implies that it is path connected, which means that every singular pose can be connected to any other singular pose by a continuous singular motion husty2008singularity (). This property can be used for a computationally efficient approximation of the singularityfree workspace by hierarchical structured hyperboxes, where only their boundaries have to be checked to be free of singularities. Beside the rationally parametrized singularity loci of the planar 3RPR PM husty2008singularity (), only the one of Stewart PMs with planar platform and planar base cm2015 () (see also an2016 (); bg2006 ()) are known to the authors (in the context of PMs of StewartGough type).
For the computation of the rational parametrization of the linear pentapod, we exploit the idea used in cm2015 (): By homogenizing the singularity polynomial of Eq. (2) by the extra variable with respect to the position variables , and , we obtain a homogeneous polynomial in the projective 3space with homogeneous coordinates . It turns out that the point with homogeneous coordinates is a point of the singularity variety; i.e. . Note that is the ideal point of the linear platform with orientation vector .
The side condition on the vector to be of unitlength, can be avoided by using the stereographic parametrization of the unitsphere :
(4) 
Based on this we can parametrize the lines of the bundle with vertex in the finite space of positions with coordinates as follows:
(5) 
Note that the bituple fixes the line of the bundle and the parameter determines the point on this line. By varying and setting one obtains the plane through the origin, which is orthogonal to .
Plugging into shows that the resulting expression is only linear in , as the ideal point is always one of the two intersection points of a line belonging to with the quadric . By solving this linear condition we get . Now the singular configurations of the linear pentapod can be rationally parametrized by and
(6) 
This parametrization covers the singular variety with exception of two lowdimensional subvariety: A missing 3dimensional subvariety is defined by the denominator of . In this case the residual intersection point of the line belonging to with is not determined uniquely; i.e. the complete line belongs to . As the orientation cannot be obtained by the stereographic parametrization, also the 2dimensional subvariety is missing.
3 Distance to the Singularity Variety
In singularities the number of DOFs of the mechanism changes instantaneously and becomes uncontrollable. Additionally the actuator forces can become very large and cause the break down of the platform li2007determination (). Henceforth knowing the distance of a given pose from the singularity variety is of great importance.
Fixed Orientation:
We ask for the closest singular configuration having the same orientation as the given pose . As and only differ by a translation, we can define the distance between these two poses by the length of the translation vector. Therefore has to be a pedalpoint on with respect to the point . The set of all these pedalpoints equals the variety where is the Lagrange multiplier of the Lagrange equation
(7) 
It is well known (see Appendix A) that in general consists of six points over , where the closest one to implies (see Figs. 2 and 3).
Fixed Position:
Now we ask for the closest singular configuration , which has the same position as the given pose . As and only differ in orientation, the angle enclosed by these two directions can be used as distance function. Note that this angle is the spherical distance function on .
By intersecting the singularity surface for the given position with we obtain a spherical curve of degree 4. Then has to be a spherical pedalpoint on with respect to the point (see Fig. 2). By replacing the underlying spherical distance by the Euclidean metric of the ambient space , one will not change the set of pedalpoints on with respect to . Therefore can be computed as the variety where and are the Lagrange multipliers of the Lagrange equation
(8) 
with . It can easily be checked (see Appendix B) that in general consists of 8 points over , where the one with the shortest spherical distance to implies (see Fig. 3).
Remark 2
For the practical application of this spherical distance to the singularity, we recommend to locate the position vector in the toolcenterpoint of .
General Case:
In contrast to the two special cases discussed above, the general case deals with mixed (translational and rotational) DOFs, thus the question of a suitable distance function arises. As the configuration space equals the space of oriented lineelements, we can adopt the object dependent metrics discussed in naccepted () for our mechanical device as follows:
(9) 
where and are two configurations and and denote the coordinate vectors of the corresponding platform anchor points. Note that the ambient space (of ) equipped with the metric of Eq. (9) is a Euclidean space (cf. naccepted ()).
With respect to this metric we can compute the closest singular configuration to in the following way: We determine the set of pedalpoints on the singularity variety with respect to as the variety where and are the Lagrange multipliers of the Lagrange equation
(10) 
Random examples (see Appendix C) indicate that consists of eighty points over , where the one with the shortest distance to equals (see Fig. 3).
Remark 3
Note that these minimal distances can be seen as the radii of maximal singularityfree hyperspheres li2007determination () in the position workspace (see also nag2016 ()), the orientation workspace (see also jg2009 ()) and the complete configuration space. Moreover the distance to the singularity variety can also be interpreted as quality index thus it is an alternative to the value of proposed in btoc2009 ().
4 Conclusions and future research
We presented a rational parametrization of the singularity variety of linear pentapods in Section 2 and computed the distance to it in Section 3 with respect to the novel metric given in Eq. (9), which can easily be adopted for e.g. Stewart PMs as well. As this distance is of interest for many tasks (e.g. quality index for path planning, radius of the maximal singularityfree hypersphere, …) a detailed study of it (e.g. efficient computation of , proof of , …) is dedicated to future research.
Acknowledgements.
The first author is funded by the Doctoral College Computational Design of the Vienna University of Technology. The second author is supported by Grant No. P 24927N25 of the Austrian Science Fund FWF within the project Stewart Gough platforms with selfmotions.Bibliography
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Appendix A
The set equals the variety of the ideal , which can be computed as follows:
As we are dealing with a fixed orientation we can assume without loss of generality that holds beside the conditions given in Eq. (1). It turns out that the equations are linear with respect to . By solving these equations for the variables and by plugging the obtained expressions into it is shown that the numerator , which has 354513 terms, is of degree in .
Solutions:
It turns out that for the random example (i.e. architectural parameters and pose given in the captions of Figs. 1 and 2) the equation has real solutions and complex ones. The corresponding values of are obtained by backsubstitution (cf. Table 1).
1  2.551763090  2.637467970  0.1144666998  0.0002811301  3.941223289 

2  0.4205946500  10.11287492  3.678294530  0.0045791513  13.21156707 
3  6.106365796  8.333480392  0.3367715809  0.7825158446  14.30080937 
4  39.77559922  14.40064789  6.535304462  0.6930082534  46.46478104 
Appendix B
The set equals the variety of the ideal , which can be computed as follows:
Under consideration of our assumptions given in Eq. (1), we start computing from the two equations , which are linear in and . By plugging the obtained expressions into , and we get three rational polynomials in the variables , and . We name their numerators , and , respectively. It turns out that these equations are quadratic. Since the solution set of these quadratic equations is , the number of solutions is according to Bezout’s Theorem. Additionally the number of terms of the polynomials , and are , and respectively ( is in fact the implicit equation of the sphere). Now in order to obtain these 8 solutions we use the resultant method in the following form:
(11) 
where and are dependent on the variables and . By using the resultant method again to eliminate the variable we obtain
(12) 
The greatest common divisor of yields the degree 8 polynomial in .
Solutions:
It turns out that for the random example (i.e. architectural parameters and pose given in the captions of Figs. 1 and 2) only solutions are real. The corresponding values of are obtained by backsubstitution (cf. Table 2).
1  0.3701933149  0.5523718708  0.7468883632  4.381351180  
2  0.3265950579  0.5850572044  0.7423232012  0.0000131667  6.434527722 
Appendix C
Gröbner Base:
It is possible to compute the Gröbner basis of the ideal
(13) 
by Maple using the FGb package of Faugère faugere () for a random example (e.g. architectural parameters and pose given in the captions of Figs. 1 and 2). By means of this package we can also compute the univariate polynomial in . The corresponding Maple pseudocode reads as follows:
It can easily be checked that is of degree in .
Resultant method:
We are also able to compute this polynomial by a stepwise elimination of unknowns based on resultant method executed by Maple. Details of this approach read as follows^{1}^{1}1Degrees and lengths of the given polynomials and factors are given with respect to the architectural parameters and pose given in the captions of Figs. 1 and 2, respectively.: We start by computing from the three equations , which are linear in . Plugging the obtained expressions into shows that its numerator only depends linearly on . From this condition we compute and insert it into the equations , which only depend on . The remaining equation equals with . Then we compute the following resultants:
where with denotes the numerator of . Note that is of degree in and that and are both of degree in . Moreover we have
where the number in the brackets gives the number of terms. It should also be mentioned that and are polynomials of degree with respect to and that is of degree in . Then we proceed by computing
have two common factors, which do not cause solutions as they imply zeros in the denominators of above arisen expressions. Beside these factors , , split up into
respectively, where the long factors (for ) are caused by the elimination process and do not contribute to the final solution. The factors and are of degree in and is of degree in . The greatest common divisor of and yields the univariate polynomial in .
Solutions:
The polynomial (either obtained by Gröbner basis elimination techniques or by the resultant method) has to be solved numerically. It turns out that for the random example under consideration only solutions are real and solutions are complex.^{2}^{2}2It is unknown if examples with real solutions can exist. By backsubstitution into the equations obtained during the stepwise elimination based on resultant method, we get the values for (cf. Table 3).
1  0.5559273038  0.7274604486  0.4021767380  0.0000977412  1.479192394 

2  0.7100848787  0.6097073464  0.3521880419  0.0112120286  6.370089783 
3  0.6707364219  0.6608309577  0.3367715809  0.0263760716  6.396348687 
4  0.9520812787  0.2971145357  0.0725547483  0.0518935724  6.494930694 
5  0.4198912232  0.7478308408  0.5142376826  0.0001295064  6.522840484 
6  0.6426323048  0.5670451826  0.5152508920  0.6390457179  7.901089998 
7  0.9141441020  0.2145188032  0.3439800050  0.0175335168  8.153560918 
8  0.6633066166  0.6523793948  0.3666407746  0.0198075803  9.072642063 
9  0.4968498376  0.8534603513  0.1573075574  0.0025204733  9.244102979 
10  0.4561177696  0.7759229223  0.4357753997  0.1291505456  9.308167139 
11  0.6449198390  0.5507646090  0.5298459651  0.0516473389  9.970322913 
12  0.9794782799  0.1982747821  0.0361857697  0.0967973416  10.05488078 
13  0.2161351180  0.9213140919  0.3232119349  0.0362890328  13.78049458 
14  0.1003162322  0.5648716701  0.8190583922  0.0430362461  37.60374403 
15  0.8001243699  0.1449203823  0.5820644943  0.1991801120  52.29308488 
16  0.0428010579  0.5580832041  0.8286804008  0.0557997298  65.26242524 
Addendum
We can simplify the problem by considering equiform transformations of the linear platform . This means that we can cancel the side condition . The computation can be done in a similar fashion to Appendic C with the sole difference that we set .
Random examples show that this reduced problem has only solutions over in the general case. For the architectural parameters and pose given in the captions of Figs. 1 and 2 it turns out that only solutions are real, which are given in Table 4. Moreover the global minimizer is displayed in Fig. 4. Important for application is that and therefore the value of gives us the radius of hypersphere, which is guaranteed singularity free.
1  0.5055836745  0.6656442614  0.3718172932  0.0000990198  1.4517670618  0.9148471097 

2  0.6486166479  0.5542384068  0.3254994305  0.0118010768  6.3636100364  0.9131449210 
3  0.6200661016  0.6110910452  0.3126505854  0.0283247861  6.3914193483  0.9250214183 
4  0.8789381698  0.2541301769  0.0569840426  0.0548756462  6.4897306508  0.9167124054 
5  0.5220226650  0.4591892110  0.4197574551  9.9482986641  7.8756112220  0.8121322031 
6  0.6513903749  0.9060852905  0.5534569690  0.1501405543  9.2038614723  1.2456382263 