Rational Parametrization of Linear Pentapod's Singularity Variety and   the Distance to it

Arvin Rasoulzadeh; Georg Nawratil

arXiv:1701.09107·cs.RO·December 1, 2017

Rational Parametrization of Linear Pentapod's Singularity Variety and the Distance to it

Arvin Rasoulzadeh, Georg Nawratil

PDF

Open Access

TL;DR

This paper provides a rational parametrization of the singularity variety for linear pentapods and calculates the shortest distance to this variety, aiding in understanding and avoiding singular configurations in robotic manipulators.

Contribution

It introduces a novel rational parametrization of the linear pentapod's singularity variety and computes the minimal distance to it, enhancing singularity analysis.

Findings

01

Rational parametrization of the singularity variety achieved

02

Shortest distance to the singularity variety computed

03

Maximal singularity-free sphere radius determined

Abstract

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via extendible legs to the base. This manipulator has five controllable degrees-of-freedom 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 free-sphere.

Tables4

Table 1. Table 1: The 4 real solutions in ascending order with respect to the length l 𝑙 l of the translation vector towards the given position.

	$p_{x}$	$p_{y}$	$p_{z}$	$λ$	$l$
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

Table 2. Table 2: The 2 real solutions in ascending order with respect to the spherical distance s 𝑠 s to the given orientation.

	$u$	$v$	$w$	$λ_{1}$	$λ_{2}$	$s$
1	0.3701933149	0.5523718708	0.7468883632	$0.0000002748$	4.381351180	${48.41786560}^{\circ}$
2	-0.3265950579	-0.5850572044	-0.7423232012	0.0000131667	-6.434527722	${131.6726142}^{\circ}$

Table 3. Table 3: The 16 real solutions in ascending order with respect to the distance d 𝑑 d (given in Eq. ( 9 )) from 𝔊 𝔊 \mathfrak{G} . The corresponding values of missing variables p x , p y , p z , λ 1 subscript 𝑝 𝑥 subscript 𝑝 𝑦 subscript 𝑝 𝑧 subscript 𝜆 1 p_{x},p_{y},p_{z},\lambda_{1} are obtained by substituting u , v , w , λ 2 𝑢 𝑣 𝑤 subscript 𝜆 2 u,v,w,\lambda_{2} into the expressions for p x , p y , p z , λ 1 subscript 𝑝 𝑥 subscript 𝑝 𝑦 subscript 𝑝 𝑧 subscript 𝜆 1 p_{x},p_{y},p_{z},\lambda_{1} . For the global minimizer (solution 1) these values are p x = 2.296688437 subscript 𝑝 𝑥 2.296688437 p_{x}=2.296688437 , p y = 3.479406728 subscript 𝑝 𝑦 3.479406728 p_{y}=3.479406728 , p z = 1.835729103 subscript 𝑝 𝑧 1.835729103 p_{z}=1.835729103 and λ 1 = − 4.720444174 subscript 𝜆 1 4.720444174 \lambda_{1}=-4.720444174 .

	$u$	$v$	$w$	$λ_{2}$	$d$
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

Table 4. Table 4: The 6 real solutions in ascending order with respect to the distance d 𝑑 d (given in Eq. ( 9 )) from 𝔊 𝔊 \mathfrak{G} . The scaling factor of the corresponding equiform displacement of the platform is given by μ 𝜇 \mu . The corresponding values of missing variables p x , p y , p z subscript 𝑝 𝑥 subscript 𝑝 𝑦 subscript 𝑝 𝑧 p_{x},p_{y},p_{z} are obtained by substituting u , v , w , λ 2 𝑢 𝑣 𝑤 subscript 𝜆 2 u,v,w,\lambda_{2} into the expressions for p x , p y , p z subscript 𝑝 𝑥 subscript 𝑝 𝑦 subscript 𝑝 𝑧 p_{x},p_{y},p_{z} . For the global minimizer 𝔑 𝔑 \mathfrak{N} (solution 1) these values are p x = 2.5031164070 subscript 𝑝 𝑥 2.5031164070 p_{x}=2.5031164070 , p y = 3.7266491740 subscript 𝑝 𝑦 3.7266491740 p_{y}=3.7266491740 and p z = 1.9989579769 subscript 𝑝 𝑧 1.9989579769 p_{z}=1.9989579769 .

	$u$	$v$	$w$	$λ_{2}$	$d$	$μ$
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

Equations38

x_{1} = y_{1} = z_{1} = y_{2} = z_{2} = z_{3} = 0 and x_{2} = 1.

x_{1} = y_{1} = z_{1} = y_{2} = z_{2} = z_{3} = 0 and x_{2} = 1.

J = (l_{1} \hat{l}_{1} \dots \dots l_{5} \hat{l}_{5})^{T} with l_{j} = p_{x} + r_{j} u - x_{j} p_{y} + r_{j} v - y_{j} p_{z} + r_{j} w - z_{j}, \hat{l}_{j} = z_{j} (p_{y} + r_{j} v) - y_{j} (p_{z} + r_{j} w) x_{j} (p_{z} + r_{j} w) - z_{j} (p_{x} + r_{j} u) y_{j} (p_{x} + r_{j} u) - x_{j} (p_{y} + r_{j} v) .

J = (l_{1} \hat{l}_{1} \dots \dots l_{5} \hat{l}_{5})^{T} with l_{j} = p_{x} + r_{j} u - x_{j} p_{y} + r_{j} v - y_{j} p_{z} + r_{j} w - z_{j}, \hat{l}_{j} = z_{j} (p_{y} + r_{j} v) - y_{j} (p_{z} + r_{j} w) x_{j} (p_{z} + r_{j} w) - z_{j} (p_{x} + r_{j} u) y_{j} (p_{x} + r_{j} u) - x_{j} (p_{y} + r_{j} v) .

F := (A_{1} p_{y} + A_{2} p_{z}) u^{2} + [(A_{3} p_{x} + A_{4} p_{y} + A_{5} p_{z} + A_{6}) v + (A_{7} p_{x} + A_{8} p_{y} + A_{9} p_{z} + A_{10}) w + (A_{11} p_{y} + A_{12} p_{z}) p_{x} + A_{13} p_{y}^{2} + (A_{14} p_{z} + A_{15}) p_{y} + A_{16} p_{z}^{2} + A_{17} p_{z}] u + (A_{18} p_{x} + A_{19} p_{z} + A_{20}) v^{2} + [(A_{21} p_{x} + A_{22} p_{y} + A_{23} p_{z} + A_{24}) w + A_{25} p_{x}^{2} + (A_{26} p_{y} + A_{27} p_{z} + A_{28}) p_{x} + (A_{29} p_{z} + A_{30}) p_{y} + A_{31} p_{z}^{2} + A_{32} p_{z}] v + (A_{33} p_{x} + A_{34} p_{y} + A_{35}) w^{2} + [A_{36} p_{x}^{2} + (A_{37} p_{y} + A_{38} p_{z} + A_{39}) p_{x} + A_{40} p_{y}^{2} + (A_{41} p_{z} + A_{42}) p_{y} + A_{43} p_{z}] w

F := (A_{1} p_{y} + A_{2} p_{z}) u^{2} + [(A_{3} p_{x} + A_{4} p_{y} + A_{5} p_{z} + A_{6}) v + (A_{7} p_{x} + A_{8} p_{y} + A_{9} p_{z} + A_{10}) w + (A_{11} p_{y} + A_{12} p_{z}) p_{x} + A_{13} p_{y}^{2} + (A_{14} p_{z} + A_{15}) p_{y} + A_{16} p_{z}^{2} + A_{17} p_{z}] u + (A_{18} p_{x} + A_{19} p_{z} + A_{20}) v^{2} + [(A_{21} p_{x} + A_{22} p_{y} + A_{23} p_{z} + A_{24}) w + A_{25} p_{x}^{2} + (A_{26} p_{y} + A_{27} p_{z} + A_{28}) p_{x} + (A_{29} p_{z} + A_{30}) p_{y} + A_{31} p_{z}^{2} + A_{32} p_{z}] v + (A_{33} p_{x} + A_{34} p_{y} + A_{35}) w^{2} + [A_{36} p_{x}^{2} + (A_{37} p_{y} + A_{38} p_{z} + A_{39}) p_{x} + A_{40} p_{y}^{2} + (A_{41} p_{z} + A_{42}) p_{y} + A_{43} p_{z}] w

ϕ : V ⇢ W with ϕ (x_{1}, \dots, x_{m}) = (\frac{f _{1} ( x _{1} , \dots , x _{m} )}{g _{1} ( x _{1} , \dots , x _{m} )}, \dots, \frac{f _{n} ( x _{1} , \dots , x _{m} )}{g _{n} ( x _{1} , \dots , x _{m} )})

ϕ : V ⇢ W with ϕ (x_{1}, \dots, x_{m}) = (\frac{f _{1} ( x _{1} , \dots , x _{m} )}{g _{1} ( x _{1} , \dots , x _{m} )}, \dots, \frac{f _{n} ( x _{1} , \dots , x _{m} )}{g _{n} ( x _{1} , \dots , x _{m} )})

x : (t_{3}, t_{4}) \mapsto (\frac{2 t _{3}}{t _{3} ^{2} + t _{4} ^{2} + 1}, \frac{2 t _{4}}{t _{3} ^{2} + t _{4} ^{2} + 1}, \frac{t _{3} ^{2} + t _{4} ^{2} - 1}{t _{3} ^{2} + t _{4} ^{2} + 1}) .

x : (t_{3}, t_{4}) \mapsto (\frac{2 t _{3}}{t _{3} ^{2} + t _{4} ^{2} + 1}, \frac{2 t _{4}}{t _{3} ^{2} + t _{4} ^{2} + 1}, \frac{t _{3} ^{2} + t _{4} ^{2} - 1}{t _{3} ^{2} + t _{4} ^{2} + 1}) .

\mathcal{B}:\left(\begin{array}[]{c}p_{x}\\ p_{y}\\ p_{z}\end{array}\right)=a\mathbf{x}(t_{3},t_{4})+t_{1}\frac{\partial\mathbf{x}(t_{3},t_{4})}{\partial t_{3}}+t_{2}\frac{\partial\mathbf{x}(t_{3},t_{4})}{\partial t_{4}}.

\mathcal{B}:\left(\begin{array}[]{c}p_{x}\\ p_{y}\\ p_{z}\end{array}\right)=a\mathbf{x}(t_{3},t_{4})+t_{1}\frac{\partial\mathbf{x}(t_{3},t_{4})}{\partial t_{3}}+t_{2}\frac{\partial\mathbf{x}(t_{3},t_{4})}{\partial t_{4}}.

ξ_{4} ξ_{5} ξ_{6} = 2 \frac{a ( t _{1} , t _{2} , t _{3} , t _{4} ) t _{3}}{t _{3} ^{2} + t _{4} ^{2} + 1} - 2 \frac{t _{1} ( t _{3} ^{2} - t _{4} ^{2} - 1 )}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} - 4 \frac{t _{2} t _{3} t _{4}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}}, = 2 \frac{a ( t _{1} , t _{2} , t _{3} , t _{4} ) t _{4}}{t _{3} ^{2} + t _{4} ^{2} + 1} - 4 \frac{t _{1} t _{3} t _{4}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} + 2 \frac{t _{2} ( t _{3} ^{2} - t _{4} ^{2} + 1 )}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}}, = \frac{a ( t _{1} , t _{2} , t _{3} , t _{4} ) ( t _{3} ^{2} + t _{4} ^{2} - 1 )}{t _{3} ^{2} + t _{4} ^{2} + 1} + 4 \frac{t _{1} t _{3}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} + 4 \frac{t _{2} t _{4}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} .

ξ_{4} ξ_{5} ξ_{6} = 2 \frac{a ( t _{1} , t _{2} , t _{3} , t _{4} ) t _{3}}{t _{3} ^{2} + t _{4} ^{2} + 1} - 2 \frac{t _{1} ( t _{3} ^{2} - t _{4} ^{2} - 1 )}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} - 4 \frac{t _{2} t _{3} t _{4}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}}, = 2 \frac{a ( t _{1} , t _{2} , t _{3} , t _{4} ) t _{4}}{t _{3} ^{2} + t _{4} ^{2} + 1} - 4 \frac{t _{1} t _{3} t _{4}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} + 2 \frac{t _{2} ( t _{3} ^{2} - t _{4} ^{2} + 1 )}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}}, = \frac{a ( t _{1} , t _{2} , t _{3} , t _{4} ) ( t _{3} ^{2} + t _{4} ^{2} - 1 )}{t _{3} ^{2} + t _{4} ^{2} + 1} + 4 \frac{t _{1} t _{3}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} + 4 \frac{t _{2} t _{4}}{( t _{3} ^{2} + t _{4} ^{2} + 1 ) ^{2}} .

L (p_{x}, p_{y}, p_{z}, λ) = (p_{x} - g_{4})^{2} + (p_{y} - g_{5})^{2} + (p_{z} - g_{6})^{2} + λ F .

L (p_{x}, p_{y}, p_{z}, λ) = (p_{x} - g_{4})^{2} + (p_{y} - g_{5})^{2} + (p_{z} - g_{6})^{2} + λ F .

L (u, v, w, λ_{1}, λ_{2}) = (u - g_{1})^{2} + (v - g_{2})^{2} + (w - g_{3})^{2} + λ_{1} F + λ_{2} G

L (u, v, w, λ_{1}, λ_{2}) = (u - g_{1})^{2} + (v - g_{2})^{2} + (w - g_{3})^{2} + λ_{1} F + λ_{2} G

d (L, L^{^{'}})^{2} := \frac{1}{5} j = 1 \sum 5 ∥ b_{j} - b_{j}^{^{'}} ∥^{2}

d (L, L^{^{'}})^{2} := \frac{1}{5} j = 1 \sum 5 ∥ b_{j} - b_{j}^{^{'}} ∥^{2}

L (u, v, w, p_{x}, p_{y}, p_{z}, λ_{1}, λ_{2}) := d (M, G)^{2} + λ_{1} G + λ_{2} F .

L (u, v, w, p_{x}, p_{y}, p_{z}, λ_{1}, λ_{2}) := d (M, G)^{2} + λ_{1} G + λ_{2} F .

R_{1} := R es (F_{2}, F_{3}, u), R_{2} := R es (F_{1}, F_{3}, u), R_{3} := R es (F_{1}, F_{2}, u),

R_{1} := R es (F_{2}, F_{3}, u), R_{2} := R es (F_{1}, F_{3}, u), R_{3} := R es (F_{1}, F_{2}, u),

G_{1} := R es (R_{2}, R_{3}, v), G_{2} := R es (R_{1}, R_{3}, v), G_{3} := R es (R_{1}, R_{2}, v) .

G_{1} := R es (R_{2}, R_{3}, v), G_{2} := R es (R_{1}, R_{3}, v), G_{3} := R es (R_{1}, R_{2}, v) .

⟨ \frac{\partial L}{\partial u}, \frac{\partial L}{\partial v}, \frac{\partial L}{\partial w}, \frac{\partial L}{\partial p _{x}}, \frac{\partial L}{\partial p _{y}}, \frac{\partial L}{\partial p _{z}}, \frac{\partial L}{\partial λ _{1}}, \frac{\partial L}{\partial λ _{2}} ⟩

⟨ \frac{\partial L}{\partial u}, \frac{\partial L}{\partial v}, \frac{\partial L}{\partial w}, \frac{\partial L}{\partial p _{x}}, \frac{\partial L}{\partial p _{y}}, \frac{\partial L}{\partial p _{z}}, \frac{\partial L}{\partial λ _{1}}, \frac{\partial L}{\partial λ _{2}} ⟩

w i t h (F G b); GB := fgb_gbasis ([\frac{\partial L}{\partial u}, \frac{\partial L}{\partial v}, \frac{\partial L}{\partial w}, \frac{\partial L}{\partial p _{x}}, \frac{\partial L}{\partial p _{y}}, \frac{\partial L}{\partial p _{z}}, \frac{\partial L}{\partial λ _{1}}, \frac{\partial L}{\partial λ _{2}}], 0, [], [u, v, w, λ_{2}, λ_{1}, p_{x}, p_{y}, p_{z}]) : P := fgb_gbasis_elim (GB, 0, [v, w, λ_{2}, λ_{1}, p_{x}, p_{y}, p_{z}], [u, v, w, λ_{2}, λ_{1}, p_{x}, p_{y}, p_{z}]) :

w i t h (F G b); GB := fgb_gbasis ([\frac{\partial L}{\partial u}, \frac{\partial L}{\partial v}, \frac{\partial L}{\partial w}, \frac{\partial L}{\partial p _{x}}, \frac{\partial L}{\partial p _{y}}, \frac{\partial L}{\partial p _{z}}, \frac{\partial L}{\partial λ _{1}}, \frac{\partial L}{\partial λ _{2}}], 0, [], [u, v, w, λ_{2}, λ_{1}, p_{x}, p_{y}, p_{z}]) : P := fgb_gbasis_elim (GB, 0, [v, w, λ_{2}, λ_{1}, p_{x}, p_{y}, p_{z}], [u, v, w, λ_{2}, λ_{1}, p_{x}, p_{y}, p_{z}]) :

H_{1} := R es (G_{v}, G, w), H_{2} := R es (G_{w}, G, w), H_{3} := R es (G_{λ_{2}}, G, w),

H_{1} := R es (G_{v}, G, w), H_{2} := R es (G_{w}, G, w), H_{3} := R es (G_{λ_{2}}, G, w),

H_{1} [1230], H_{2} [1271], H_{3} [1252],

H_{1} [1230], H_{2} [1271], H_{3} [1252],

K_{1} := R es (H_{2}, H_{3}, λ_{2}), K_{2} := R es (H_{1}, H_{3}, λ_{2}), K_{3} := R es (H_{1}, H_{2}, λ_{2}) .

K_{1} := R es (H_{2}, H_{3}, λ_{2}), K_{2} := R es (H_{1}, H_{3}, λ_{2}), K_{3} := R es (H_{1}, H_{2}, λ_{2}) .

K_{1, 1} [2016] K_{1, 2} [11175], K_{2, 1} [1938] K_{2, 2} [11097] and K_{3, 1} [1653] K_{3, 2} [11371],

K_{1, 1} [2016] K_{1, 2} [11175], K_{2, 1} [1938] K_{2, 2} [11097] and K_{3, 1} [1653] K_{3, 2} [11371],

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsControl and Dynamics of Mobile Robots · Robotic Locomotion and Control · Robotic Mechanisms and Dynamics

Full text

11institutetext: Center for Geometry and Computational Design, Vienna University of Technology, Austria, 11email: {rasoulzadeh, nawratil}@geometrie.tuwien.ac.at

Rational Parametrization of Linear Pentapod’s Singularity Variety and the Distance to it

Arvin Rasoulzadeh and Georg Nawratil

Abstract

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via $\mathrm{S\underline{P}S}$ legs to the base. This manipulator has five controllable degrees-of-freedom 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 free-sphere. Moreover we compare the result with the radius of the maximal singularity free-sphere in the position workspace and the orientation workspace, respectively.

Keywords:

Pentapod, Kinematic Singularity, Rational Variety, Singularity-free zone.

1 Introduction

The Stewart-Gough platform (sometimes called simply Stewart platform) can be defined as six degree-of-freedom (DOF) parallel manipulator (PM) with six identical spherical-prismatic-spherical ( $\mathrm{S}\underline{\mathrm{P}}\mathrm{S}$ ) 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 sub-assemblies with a lower number of legs are of interest from theoretical and practical points of view. Sometimes these sub-assemblies are referred to as components kg2000 . In this paper we study the so-called line-body component, which is a rigid sub-assembly of a Stewart PM consisting of a linear motion platform (end-effector) named $\ell$ and five $\mathrm{S}\underline{\mathrm{P}}\mathrm{S}$ legs, where the base anchor points can have position in $\mathbb{R}^{3}$ . Here this component is referred to as linear pentapod, which is an alternative to serial robots for handling axis-symmetric tools (see Fig. 1). Moreover we use the following notations:

The position of $\ell$ is given by the vector $\mathbf{p}=(p_{x},p_{y},p_{z})^{T}$ and the orientation of $\ell$ is defined by a unit-vector $\mathbf{i}=(u,v,w)^{T}$ .

The coordinate vector $\mathbf{b}_{j}$ of the platform anchor point of the $j$ th leg is described by the equation $\mathbf{b}_{j}=\mathbf{p}+r_{j}\mathbf{i}$ for $j=1,\ldots,5$ . 2. 3.

The base anchor point of the leg $j$ has coordinates $\mathbf{a}_{j}=(x_{j},y_{j},z_{j})^{T}$ .

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:

[TABLE]

According to (naccepted, , Theorem 12) one possible point-model for the configuration space $\mathcal{C}$ of the linear pentapod reads as follows: There exists a bijection between $\mathcal{C}$ and all real points $\mathfrak{C}=(u,v,w,p_{x},p_{y},p_{z})\in\mathbb{R}^{6}$ located on the singular quadric $\Gamma:\,\,u^{2}+v^{2}+w^{2}=1$ . 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 $\mathbb{R}^{6}$ of the configuration space $\mathcal{C}$ . 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 singular-invariant leg-rearrangements have been studied in btt2011 for a planar base and in borras2010singularity for a non-planar one. A complete list of architectural singular designs of linear pentapods is given in ns , where also non-architecturally singular designs with self-motions are classified (see also n2016 ).

Kinematical singularities occur whenever the Jacobian matrix $\mathbf{J}$ becomes rank deficient, where $\mathbf{J}$ can be written as follows (cf. borras2010singularity ):

[TABLE]

This $5\times 6$ Jacobian matrix $\mathbf{J}$ has a rank less than five whenever the determinants of all its $5\times 5$ sub-matrices vanish. So by naming the determinant of the $5\times 5$ sub-matrix, which results from excluding the $j$ th column, with $F_{j}$ the singularity loci equals $V(F_{1},\dots,F_{6})$ ; i.e. the variety of the ideal spanned by the polynomials $F_{1},\dots,F_{6}$ . It can easily be checked by direct computations that this variety equals the zero-set of the greatest common divisor $F$ of $F_{1},\dots,F_{6}$ . This singularity polynomial $F$ has the following structure:

[TABLE]

where the coefficients $A_{i}$ belong to the ring $\mathrm{R}=\mathbb{R}[x_{3},x_{4},x_{5},y_{3},y_{4},y_{5},z_{4},z_{5},r_{1},\ldots,r_{5}]$ which evidently makes $F$ a polynomial with the total-degree of 3 belonging to $\mathrm{R}[u,v,w,p_{x},p_{y},p_{z}]$ . Note that for a specified orientation $(u,v,w)$ the equation $F=0$ determines only a quadric surface $\Omega(u,v,w)$ in the space of positions. This property is of great importance later on.

Remark 1

It can easily be checked that the polynomial $F$ is identical with the determinant of a $7\times 7$ matrix, given in (borras2010singularity, , Eq. (4)). $\diamond$

2 Rational Parametrization of the Singularity Variety

In this section we rationally parametrize the singularity variety, which is given by the implicit equation $F=0$ . But before stepping into the computations, the presentation of a formal definition of this parametrization seems necessary.

Definition 1

Let $\mathbb{K}$ be a field and $V\subset\mathbb{K}^{m}$ and $W\subset\mathbb{K}^{n}$ be irreducible affine varieties. A rational mapping from $V$ to $W$ is a function $\phi$ represented by

[TABLE]

where $\frac{f_{i}}{g_{i}}\in\mathbb{K}(x_{1},\dots,x_{m})$ and satisfies the following properties:

$\phi$ is defined at some point of $V$ . 2. 2.

For every $(a_{1},\dots a_{m})\in V$ where $\phi$ is defined, $\phi(a_{1},\dots a_{m})\in W$ .

Definition 2

Two irreducible varieties $V$ and $W$ are said to be birationally equivalent if there exist rational mappings $\phi:V\dashrightarrow W$ and $\psi:W\dashrightarrow V$ such that $\phi\circ\psi$ and $\psi\circ\phi$ be equal to $\mathrm{id}_{W}$ and $\mathrm{id}_{V}$ respectively.

Definition 3

A rational variety is a variety that is birationally equivalent to $\mathbb{K}^{n}$ .

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 $f_{i}$ and $g_{i}$ of Eq. (3) belong to $\mathbb{Q}$ and if $(x_{1},\ldots,x_{m})$ is an element of $\mathbb{Q}^{m}$ , 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 singularity-free 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 3-RPR 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 Stewart-Gough type).

For the computation of the rational parametrization of the linear pentapod, we exploit the idea used in cm2015 : By homogenizing the singularity polynomial $F$ of Eq. (2) by the extra variable $p_{0}$ with respect to the position variables $p_{x}$ , $p_{y}$ and $p_{z}$ , we obtain a homogeneous polynomial $F_{h}\in\mathrm{R}(u,v,w)[p_{x},p_{y},p_{z},p_{0}]$ in the projective 3-space $\mathbb{P}^{3}$ with homogeneous coordinates $(p_{x}:p_{y}:p_{z}:p_{0})$ . It turns out that the point $\sf B$ with homogeneous coordinates $(u:v:w:0)$ is a point of the singularity variety; i.e. $\sf B\in V(F_{h})\subset\mathbb{P}^{3}$ . Note that $\sf B$ is the ideal point of the linear platform $\ell$ with orientation vector $\mathbf{i}$ .

The side condition on the vector $\mathbf{i}=(u,v,w)^{T}$ to be of unit-length, can be avoided by using the stereographic parametrization of the unit-sphere $S^{2}$ :

[TABLE]

Based on this we can parametrize the lines of the bundle $\mathcal{B}$ with vertex $\sf B$ in the finite space $\mathbb{R}^{3}$ of positions with coordinates $(p_{x},p_{y},p_{z})$ as follows:

[TABLE]

Note that the bituple $(t_{1},t_{2})$ fixes the line of the bundle $\mathcal{B}$ and the parameter $a$ determines the point on this line. By varying $(t_{1},t_{2})\in\mathbb{R}^{2}$ and setting $a=0$ one obtains the plane through the origin, which is orthogonal to $\mathbf{i}$ .

Plugging $\mathcal{B}(a,t_{1},t_{2},t_{3},t_{4})$ into $F=0$ shows that the resulting expression is only linear in $a$ , as the ideal point ${\sf B}$ is always one of the two intersection points of a line belonging to $\mathcal{B}$ with the quadric $\Omega(\mathbf{x}(t_{3},t_{4}))$ . By solving this linear condition we get $a(t_{1},t_{2},t_{3},t_{4})$ . Now the singular configurations $\mathfrak{X}=(\xi_{1},\ldots,\xi_{6})\in\mathbb{R}^{6}$ of the linear pentapod can be rationally parametrized by $(\xi_{1},\xi_{2},\xi_{3}):=\mathbf{x}(t_{3},t_{4})$ and

[TABLE]

This parametrization covers the singular variety with exception of two low-dimensional sub-variety: A missing 3-dimensional sub-variety is defined by the denominator of $a(t_{1},t_{2},t_{3},t_{4})$ . In this case the residual intersection point $\in\mathbb{R}^{3}$ of the line belonging to $\mathcal{B}$ with $\Omega(\mathbf{x}(t_{3},t_{4}))$ is not determined uniquely; i.e. the complete line belongs to $\Omega(\mathbf{x}(t_{3},t_{4}))$ . As the orientation $(0,0,1)$ cannot be obtained by the stereographic parametrization, also the 2-dimensional sub-variety $\Omega(0,0,1)$ is missing.

Moreover for a given singular pose $\mathfrak{X}=(\xi_{1},\ldots,\xi_{6})\in\mathbb{R}^{6}$ we can trivially compute $t_{1},\ldots,t_{4}$ in a rational way from $\xi_{1},\ldots,\xi_{6}$ , thus the singularity variety is a rational one (according to the Definitions 1, 2 and 3).

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 $\mathfrak{G}=(g_{1},\ldots,g_{6})\in\mathbb{R}^{6}$ from the singularity variety is of great importance.

Fixed Orientation:

We ask for the closest singular configuration $\mathfrak{O}$ having the same orientation $(g_{1},g_{2},g_{3})$ as the given pose $\mathfrak{G}$ . As $\mathfrak{G}$ and $\mathfrak{O}$ only differ by a translation, we can define the distance between these two poses by the length of the translation vector. Therefore $\mathfrak{O}$ has to be a pedal-point on $\Omega(g_{1},g_{2},g_{3})$ with respect to the point $(g_{4},g_{5},g_{6})$ . The set $\mathcal{O}$ of all these pedal-points equals the variety $V(\tfrac{\partial L}{\partial p_{x}},\tfrac{\partial L}{\partial p_{y}},\tfrac{\partial L}{\partial p_{z}},\tfrac{\partial L}{\partial\lambda})$ where $\lambda$ is the Lagrange multiplier of the Lagrange equation

[TABLE]

It is well known (see Appendix A) that in general $\mathcal{O}$ consists of six points over $\mathbb{C}$ , where the closest one to $(g_{4},g_{5},g_{6})$ implies $\mathfrak{O}$ (see Figs. 2 and 3).

Fixed Position:

Now we ask for the closest singular configuration $\mathfrak{P}$ , which has the same position $(g_{4},g_{5},g_{6})$ as the given pose $\mathfrak{G}$ . As $\mathfrak{G}$ and $\mathfrak{P}$ only differ in orientation, the angle $\in[0,\pi]$ enclosed by these two directions can be used as distance function. Note that this angle is the spherical distance function on $S^{2}$ .

By intersecting the singularity surface for the given position $(g_{4},g_{5},g_{6})$ with $S^{2}$ we obtain a spherical curve $\omega(g_{4},g_{5},g_{6})$ of degree 4. Then $\mathfrak{P}$ has to be a spherical pedal-point on $\omega(g_{4},g_{5},g_{6})$ with respect to the point $(g_{1},g_{2},g_{3})\in S^{2}$ (see Fig. 2). By replacing the underlying spherical distance by the Euclidean metric of the ambient space $\mathbb{R}^{3}$ , one will not change the set $\mathcal{P}$ of pedal-points on $\omega(g_{4},g_{5},g_{6})$ with respect to $(g_{1},g_{2},g_{3})$ . Therefore $\mathcal{P}$ can be computed as the variety $V(\tfrac{\partial L}{\partial u},\tfrac{\partial L}{\partial v},\tfrac{\partial L}{\partial w},\tfrac{\partial L}{\partial\lambda_{1}},\tfrac{\partial L}{\partial\lambda_{2}})$ where $\lambda_{1}$ and $\lambda_{2}$ are the Lagrange multipliers of the Lagrange equation

[TABLE]

with $G=u^{2}+v^{2}+w^{2}-1$ . It can easily be checked (see Appendix B) that in general $\mathcal{P}$ consists of 8 points over $\mathbb{C}$ , where the one with the shortest spherical distance to $(g_{4},g_{5},g_{6})$ implies $\mathfrak{P}$ (see Fig. 3).

Remark 2

For the practical application of this spherical distance to the singularity, we recommend to locate the position vector $\mathbf{p}$ in the tool-center-point of $\ell$ . $\diamond$

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 $\mathcal{C}$ equals the space of oriented line-elements, we can adopt the object dependent metrics discussed in naccepted for our mechanical device as follows:

[TABLE]

where $\mathfrak{L}$ and $\mathfrak{L^{{}^{\prime}}}$ are two configurations and $\mathbf{b}_{j}$ and $\mathbf{b}^{{}^{\prime}}_{j}$ denote the coordinate vectors of the corresponding platform anchor points. Note that the ambient space $\mathbb{R}^{6}$ (of $\mathcal{C}$ ) equipped with the metric $d$ of Eq. (9) is a Euclidean space (cf. naccepted ).

With respect to this metric $d$ we can compute the closest singular configuration $\mathfrak{M}$ to $\mathfrak{G}$ in the following way: We determine the set $\mathcal{M}$ of pedal-points on the singularity variety with respect to $\mathfrak{G}$ as the variety $V(\tfrac{\partial L}{\partial u},\tfrac{\partial L}{\partial v},\tfrac{\partial L}{\partial w},\tfrac{\partial L}{\partial p_{x}},\tfrac{\partial L}{\partial p_{y}},\tfrac{\partial L}{\partial p_{z}},\tfrac{\partial L}{\partial\lambda_{1}},\tfrac{\partial L}{\partial\lambda_{2}})$ where $\lambda_{1}$ and $\lambda_{2}$ are the Lagrange multipliers of the Lagrange equation

[TABLE]

Random examples (see Appendix C) indicate that $\mathcal{M}$ consists of eighty points over $\mathbb{C}$ , where the one with the shortest distance $d$ to $\mathfrak{G}$ equals $\mathfrak{M}$ (see Fig. 3).

Remark 3

Note that these minimal distances can be seen as the radii of maximal singularity-free hyperspheres li2007determination in the position workspace (see also nag2016 ), the orientation workspace (see also jg2009 ) and the complete configuration space. Moreover the distance $d(\mathfrak{M},\mathfrak{G})$ to the singularity variety can also be interpreted as quality index thus it is an alternative to the value of $F$ proposed in btoc2009 . $\diamond$

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 singularity-free hypersphere, …) a detailed study of it (e.g. efficient computation of $\mathfrak{M}$ , proof of $\#\mathcal{M}=80$ , …) 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 24927-N25 of the Austrian Science Fund FWF within the project Stewart Gough platforms with self-motions.

Appendix A

The set $\mathcal{O}$ equals the variety of the ideal $\langle\tfrac{\partial L}{\partial p_{x}},\tfrac{\partial L}{\partial p_{y}},\tfrac{\partial L}{\partial p_{z}},\tfrac{\partial L}{\partial\lambda}\rangle$ , which can be computed as follows:

As we are dealing with a fixed orientation we can assume without loss of generality that $r_{1}=0$ holds beside the conditions given in Eq. (1). It turns out that the equations $\tfrac{\partial L}{\partial p_{x}}=\tfrac{\partial L}{\partial p_{y}}=\tfrac{\partial L}{\partial p_{z}}=0$ are linear with respect to $p_{x},p_{y},p_{z}$ . By solving these equations for the variables $p_{x},p_{y},p_{z}$ and by plugging the obtained expressions into $\tfrac{\partial L}{\partial\lambda}$ it is shown that the numerator $K$ , which has 354513 terms, is of degree $6$ in $\lambda$ .

Solutions:

It turns out that for the random example (i.e. architectural parameters and pose $\mathfrak{G}$ given in the captions of Figs. 1 and 2) the equation $K=0$ has $4$ real solutions and $2$ complex ones. The corresponding values of $p_{x},p_{y},p_{z}$ are obtained by back-substitution (cf. Table 1).

Appendix B

The set $\mathcal{P}$ equals the variety of the ideal $\langle\tfrac{\partial L}{\partial u},\tfrac{\partial L}{\partial v},\tfrac{\partial L}{\partial w},\tfrac{\partial L}{\partial\lambda_{1}},\tfrac{\partial L}{\partial\lambda_{2}}\rangle$ , which can be computed as follows:

Under consideration of our assumptions given in Eq. (1), we start computing $\lambda_{1},\lambda_{2}$ from the two equations $\tfrac{\partial L}{\partial u}=\tfrac{\partial L}{\partial v}=0$ , which are linear in $u,v$ and $w$ . By plugging the obtained expressions into $\tfrac{\partial L}{\partial w}$ , $\tfrac{\partial L}{\partial\lambda_{1}}$ and $\tfrac{\partial L}{\partial\lambda_{2}}$ we get three rational polynomials in the variables $u$ , $v$ and $w$ . We name their numerators $F_{1}$ , $F_{2}$ and $F_{3}$ , respectively. It turns out that these equations are quadratic. Since the solution set of these quadratic equations is $V(F_{1},F_{2},F_{3})=V(F_{1})\cap V(F_{2})\cap V(F_{3})$ , the number of solutions is $8$ according to Bezout’s Theorem. Additionally the number of terms of the polynomials $F_{1}$ , $F_{2}$ and $F_{3}$ are $896$ , $348$ and $4$ respectively ( $F_{3}$ 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:

[TABLE]

where $R_{1},R_{2}$ and $R_{3}$ are dependent on the variables $v$ and $w$ . By using the resultant method again to eliminate the variable $v$ we obtain

[TABLE]

The greatest common divisor of $G_{1},G_{2},G_{3}$ yields the degree 8 polynomial in $w$ .

Solutions:

It turns out that for the random example (i.e. architectural parameters and pose $\mathfrak{G}$ given in the captions of Figs. 1 and 2) only $2$ solutions are real. The corresponding values of $u,v,w,\lambda_{2}$ are obtained by back-substitution (cf. Table 2).

Appendix C

Gröbner Base:

It is possible to compute the Gröbner basis of the ideal

[TABLE]

by Maple using the FGb package of Faugère faugere for a random example (e.g. architectural parameters and pose $\mathfrak{G}$ given in the captions of Figs. 1 and 2). By means of this package we can also compute the univariate polynomial $P$ in $u$ . The corresponding Maple pseudo-code reads as follows:

[TABLE]

It can easily be checked that $P$ is of degree $80$ in $u$ .

Resultant method:

We are also able to compute this polynomial $P$ by a stepwise elimination of unknowns based on resultant method executed by Maple. Details of this approach read as follows111Degrees and lengths of the given polynomials and factors are given with respect to the architectural parameters and pose $\mathfrak{G}$ given in the captions of Figs. 1 and 2, respectively.: We start by computing $p_{x},p_{y},p_{z}$ from the three equations $\tfrac{\partial L}{\partial p_{x}}=\tfrac{\partial L}{\partial p_{y}}=\tfrac{\partial L}{\partial p_{z}}=0$ , which are linear in $p_{x},p_{y},p_{z}$ . Plugging the obtained expressions into $\tfrac{\partial L}{\partial u}$ shows that its numerator only depends linearly on $\lambda_{1}$ . From this condition we compute $\lambda_{1}$ and insert it into the equations $\tfrac{\partial L}{\partial v}=\tfrac{\partial L}{\partial w}=\tfrac{\partial L}{\partial\lambda_{2}}=0$ , which only depend on $u,v,w,\lambda_{2}$ . The remaining equation $\tfrac{\partial L}{\partial\lambda_{1}}=0$ equals $G=0$ with $G=u^{2}+v^{2}+w^{2}-1$ . Then we compute the following resultants:

[TABLE]

where $G_{i}$ with $i\in\left\{v,w,\lambda_{2}\right\}$ denotes the numerator of $\tfrac{\partial L}{\partial i}(u,v,w,\lambda_{2})$ . Note that $G_{v}$ is of degree $8$ in $w$ and that $G_{w}$ and $G_{\lambda_{2}}$ are both of degree $9$ in $w$ . Moreover we have

[TABLE]

where the number in the brackets gives the number of terms. It should also be mentioned that $H_{1}$ and $H_{2}$ are polynomials of degree $14$ with respect to $\lambda_{2}$ and that $H_{3}$ is of degree $12$ in $\lambda_{2}$ . Then we proceed by computing

[TABLE]

$K_{1},K_{2},K_{3}$ have two common factors, which do not cause solutions as they imply zeros in the denominators of above arisen expressions. Beside these factors $K_{1}$ , $K_{2}$ , $K_{3}$ split up into

[TABLE]

respectively, where the long factors $K_{j,2}$ (for $j=1,2,3$ ) are caused by the elimination process and do not contribute to the final solution. The factors $K_{1,1}$ and $K_{2,1}$ are of degree $62$ in $v$ and $K_{3,1}$ is of degree $56$ in $v$ . The greatest common divisor of $Res(K_{1,1},K_{3,1},v)$ and $Res(K_{2,1},K_{3,1},v)$ yields the univariate polynomial $P$ in $u$ .

Solutions:

The polynomial $P$ (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 $16$ solutions are real and $64$ solutions are complex.222It is unknown if examples with $80$ real solutions can exist. By back-substitution into the equations obtained during the stepwise elimination based on resultant method, we get the values for $u,v,w,\lambda_{2}$ (cf. Table 3).

Addendum

We can simplify the problem by considering equiform transformations of the linear platform $\ell$ . This means that we can cancel the side condition $G=1$ . The computation can be done in a similar fashion to Appendic C with the sole difference that we set $\lambda_{1}=0$ .

Random examples show that this reduced problem has only $28$ solutions over $\mathbb{C}$ in the general case. For the architectural parameters and pose $\mathfrak{G}$ given in the captions of Figs. 1 and 2 it turns out that only $6$ solutions are real, which are given in Table 4. Moreover the global minimizer $\mathfrak{N}$ is displayed in Fig. 4. Important for application is that $d(\mathfrak{N},\mathfrak{G})\leq d(\mathfrak{M},\mathfrak{G})$ and therefore the value of $d(\mathfrak{N},\mathfrak{G})$ gives us the radius of hyper-sphere, which is guaranteed singularity free.

Bibliography16

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1(1) Aigner, B., Nawratil, G.: Planar Stewart Gough platforms with quadratic singularity surface. New Trends in Mechanisms Science - Theory and Industrial Applications (P. Wenger, P. Flores eds.), pages 93-102, Springer (2016)
2(2) Bandyopadhyay, S., Ghosal, A.: Geometric characterization and parametric representation of the singularity manifold of a 6-6 Stewart platform manipulator. Mech. Mach. Theory 41 (11) 1377–1400 (2006)
3(3) Borràs, J., Thomas, F.: Singularity-Invariant Leg Substitutions in Pentapods. IEEE/RSJ Int. Conference on Intelligent Robots and Systems, Taipei, Taiwan, October 18–22 (2010)
4(4) Borràs, J., Thomas, F., Ottaviano, E., Ceccarelli, M.: A Reconfigurable 5-DOF 5-SPU Parallel Platform. ASME/IF To MM Int. Conference on Reconfigurable Mechanisms and Robots, King’s College of London, London, UK, June 22-24 (2009)
5(5) Borràs, J., Thomas, F., Torras, C.: Singularity-Invariant Families of Line-Plane 5-S P U Platforms. IEEE Transactions on Robotics 27 (5) 837–848 (2011)
6(6) Coste, M., Moussa, S.: On the rationality of the singularity locus of a Gough-Stewart platform – Biplanar case. Mech. Mach. Theory 87 82–92 (2015)
7(7) Faugère, J.-C.: F Gb: A Library for Computing Gröbner Bases. Mathematical Software ICMS 2010 (K. Fukuda et al, eds.), pages 84–87, Springer (2010)
8(8) Husty, M., Gosselin, C.: On the Singularity Surface of Planar 3-RPR Parallel Mechanisms. Mechanics Based Design of Structures and Machines 36 411–425 (2008)