The quartic singularity surfaces of planar platforms in the Clifford algebra of the projective plane

PII:

Mech. Mach. Theory Vol. 33, No. 7, pp. 931±944, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00098-0

THE QUARTIC SINGULARITY SURFACES OF PLANAR PLATFORMS IN THE CLIFFORD ALGEBRA OF THE PROJECTIVE PLANE C. L. COLLINS and J. M. McCARTHY{ Department of Mechanical and Aerospace Engineering, University of California at Irvine, Irvine, CA 92697, U.S.A. (Received 25 November 1996) AbstractÐIn this paper, we study the workspace and singular con®gurations of a planar platform supported by three linearly actuated legs, the 3-RPR parallel manipulator. The constraint equations of the platform are formulated in the Cli€ord algebra of the projective plane, C‡ …P2 †, which yields a manifold de®ning its set of reachable positions and orientations. We compute the Jacobian of these equations and derive the algebraic equation of the surface of points in C‡ …P2 † for which this Jacobian is singular, called the singularity surface of the manipulator. For the general planar platform manipulator this surface is a quartic surface with a double line. For the special case of the ``proportional'' planar platform, the surface factors into two planes and a circular hyperboloid. For the special case of the ``in-line'' planar platform, this surface reduces to a quartic ruled surface. Further special cases of this surface are examined and found to consist of pairs of hyperbolic paraboloids. # 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION

This paper studies the set of singular positions and orientations of a planar platform supported by three RPR chainsÐR denotes a revolute, or hinge, joint and P a prismatic or sliding joint. These singular positions have the property that the actuators of the platform do not completely de®ne its con®guration. Often, this is described as an instantaneous loss of constraint, or increase in degree of freedom. Because the linear actuation scheme generates pure forces at the contacts with the platform, line geometry has been a useful tool for geometrically determining singular con®gurations. Merlet [1] studied the linear dependencies among forces in a special Stewart platform and obtained a systematic description of its singular con®gurations. Collins and Long [2] used this approach to determine the singularities of a pantograph linkage-based hand controller; as did Notash and Podohorodeski [3] in their study of three-branch manipulator architectures. These same methods were applied to determine the singular con®gurations of mechanical hands by Hunt et al. [4] and McAree et al. [5]. The ®rst analysis of parallel manipulator singularities using the Jacobian of its constraint equations was presented by Gosselin and Angeles [6]. They identi®ed singular con®gurations of the platform as those that lead to a singular Jacobian matrix. The goal is to determine the location of all singular con®gurations in the workspace of the manipulator. These singularity loci have been studied in detail for planar platforms by Sefrioui and Gosselin [7, 8]. The singularity loci for parallel manipulators with four degrees of freedom are studied by Wang and Gosselin [9]. This paper studies the general planar platform following the lead of Sefrioui and Gosselin. However, rather than study the singularity loci using Cartesian coordinates and a ®xed axis rotation angle, we use the Cli€ord algebra of the projective plane to provide an algebraic representation for combined rotations and translations. The result is a three-dimensional algebraic surface representing the set of singular con®gurations, which we call the singularity surface. We derive the general form of this surface, and then study in detail its properties for two special planar platform architectures. In the ®rst case, we study the ``proportional'' planar plat{To whom all correspondence should be addressed. 931

932

C. L. Collins and J. M. McCarthy

form manipulator. These manipulators have ®xed and moving pivots arranged to form similar base and platform triangles. In the second case, we study ``in-line'' parallel platforms in detail. These platforms are constructed such that their ®xed and moving pivots lie on straight lines in the base and platform, respectively.

THE CLIFFORD ALGEBRA OF P2

The two-dimensional Euclidean plane, E2, is assigned coordinates by choosing a point O to be the origin of a coordinate frame F; the result is that each point is assigned a coordinate vector in R2 of the form x = xe1+ye2 in R2, where x and y are real numbers and ei are the natural basis vectors of R2. These coordinates can be augmented to de®ne the projective plane, P2, by embedding the coordinate space R2 in R3 as the Z = 1 plane. Coordinates of the ``ane'' part of P2, which is identical to E2, have the form x = xe1+ye2+1e3. Since each point x in the Z = 1 plane de®nes a unique line kx through the origin of R3, the projective plane is often described as the set of lines through a point in three-dimensional space, and kx are termed the homogeneous coordinates of points in P2. This point of view provides an explicit representation of points ``at in®nity'' as those lines through the origin that are parallel to the plane Z = 1. These points at in®nity are part of the P2, but not E2. We construct the Cli€ord algebra for the projective plane, denoted C…P2 †, as follows (McCarthy [10]). De®ne the product of two vectors, x = x1e1+x2e2+x3e3 and y = y1e1+y2e2+y3e2+y3e3, in R3, to be the multiplication: xy ˆ …Sxi ei †…Syj ej † ˆ Sxi yj ei ej :

…1†

Next we introduce a degenerate scalar product on these coordinates so distances and angles are measured parallel to the Z = 1 plane: hx; yi ˆ x1 y1 ‡ x2 y2 :

…2†

xy ‡ yx ˆ ÿ2hx; yi;

…3†

Finally, we introduce Cli€ord's rule: which links the product (1) to the geometry represented by the scalar product. This rule provides a mechanism to reduce repeated basis vectors to scalars (Cli€ord [11]); in particular e1e1=e2e2= ÿ 1 and e3e3=0. The product of k basis vectors of R3 is called a rank-k multivector, or k-vector. Cli€ord's rule Equation (3) limits the maximum rank multivector to be the dimension of the vector space, in this case three. Thus, the general element of C…P2 † consists of one scalar component, the three basis 1-vectors, three bi-vectors, and one 3-vector; eight components in all. Our focus will be on the even subalgebra, C‡ …P2 †, which is formed from the four even rank basis multivectors; that is the scalar term and the three bi-vector terms given by: P ˆ q 1 e 2 e 3 ‡ q2 e 3 e 1 ‡ q3 e 1 e 2 ‡ q4 :

…4†

This general element of C‡ …P2 † is equivalent to the planar subalgebra of the set of dual quaternions; that is the elements of the form: P ˆ q1 iE ‡ q2 jE ‡ q3 k ‡ q4 ;

…5†

where, i, j, and k are the well-known quaternion units [12] and E is the ``dual unit'', with the property that E2=0. See Blaschke and Muller [13], Yang and Freudenstein [14], and Bottema and Roth [15]. Equations (4) and (5) are equivalent forms of what we term a planar quaternion. The computations with the basis multivectors e2e3, e3e1, and e1e2 as elements of the Cli€ord algebra C‡ …P2 † are completely equivalent to computations using iE, jE and k as dual quaternion elements. See DeSa and Roth [16] and Ravani and Roth [17] for further study of the geometry of this space.

The quartic singularity surfaces of planar platforms

933

PLANAR DISPLACEMENTS AS ELEMENTS OF C‡ …P2 †

A transformation of coordinates in R3 can be de®ned such that it is a rigid displacement in the plane Z = 1. This is the well-known 3  3 homogeneous transform: 8 9 2 38 9 cos y ÿsin y dx < x =
…7†

where z = (x, y)T and Z = (X, Y)T are the coordinate vectors of points in the moving and ®xed frames, M and F, respectively, in the plane Z = 1. The upper left 2  2 submatrix of [T] is the planar rotation matrix, A(y), of M relative to F; and the vector d = (dx, dy)T in [T] de®nes the translation of M relative to F. The planar quaternion corresponding to a displacement [T] = (A(y), d) is given by: 9 8 9 8 1 q1 > …dx cos…y=2† ‡ dy sin…y=2†† > > > > > > > 2 = < = <1 q2 …dy cos…y=2† ÿ dx sin…y=2†† 2 : …8† ˆ Pˆ q > > > sin…y=2† > > > ; : ; > : 3> q4 cos…y=2† Notice that in order to represent a displacement the third and fourth components of the planar quaternion P = (q1, q2, q3, q4)T must satisfy the relation: q23 ‡ q24 ˆ 1:

…9†

When Equation (9) is satis®ed, the four planar quaternion parameters can be used to de®ne the homogeneous matrix in Equation (6) as 2 2 3 q4 ÿ q23 ÿ2q3 q4 q1 q4 ÿ q2 q3 …10† ‰TŠ ˆ 4 2q3 q4 q24 ÿ q23 q1 q3 ‡ q2 q4 5 0 0 1 The Equations (8) and (10) provide the means to transform from planar displacements to planar quaternion coordinates and back again. Let Y = (y1, y2, y3, y4)T be a general planar quaternion. The point YÄ on the line joining Y to the origin {0, 0, 0, 0}T which represents a planar displacement can be obtained by normalizing the last two components of Y, n o ~ ˆ 1 Y ˆ y1 ; y2 ; y3 ; y4 ; …11† Y k k k k k where k2 ˆ y23 ‡ y24 : The positive and negative roots of k, however, both de®ne the same planar displacement. Thus, general planar quaternions can be viewed as homogeneous coordinates for planar displacements; and C‡ …P2 † can be considered to be a three-dimensional projective space in which each point de®nes a planar displacement. CONSTRAINT MANIFOLD OF A PLANAR PLATFORM

Consider the planar platform manipulator supported by three RPR chains (Fig. 1). Let the location of the three ®xed pivots be speci®ed by ui=(ui, ni)T, i = 1,2,3 measured in F and let the three moving pivots be zi=(li, mi)T measured in M. Coordinates in the platform M are related to coordinate in F by the 3  3 homogeneous transformation (6), that is Zi=[T]zi. Let the distance between the ith ®xed and moving pivots be denoted ri so we have the relations:

934

C. L. Collins and J. M. McCarthy

Fig. 1. A general planar platform formed by three RPR chains.

Qi : h…ui ÿ Zi †; …ui ÿ Zi †i ˆ h…ui ÿ ‰TŠzi †; …ui ÿ ‰TŠzi †i ˆ r2i ;

…13†

where i = 1,2,3. Substituting Equation (10) into Equation (13), we obtain three quadratic equations in planar quaternion coordinates which de®ne the positions of the platform M that satisfy the constraints of the three RPR chains. This is the set of positions and orientations reachable by the planar platform manipulator represented as a manifold in C‡ …P2 †. We call this the constraint manifold of the platform (Ge and McCarthy [18]). The constraint manifold of a single RPR chain is given by the general quadric equation in four homogeneous coordinates q = (q1, q2, q3, q4)T, Q : aq21 ‡ bq22 ‡ cq23 ‡ dq24 ‡ 2fq2 q3 ‡ 2gq1 q3 ‡ 2hq1 q2 ‡ 2lq1 q4 ‡ 2mq2 q4 ‡ 2nq3 q4 ˆ where

9 8 1 > > > > > > > > 1 > > > > 2 2 > > 1 > > ‰…l ‡ u† ‡ …m ‡ v† Š > > 4 > > > > > > > > 2 2 1 > > ‰…l ÿ u† ‡ …m ÿ v† Š > > 4 > > > > > > > > l‡u > > = < 2

8 9 a> > > > > > > b> > > > > > > > > c > > > > > > > > d > > > = < > f ˆ g> > > > > > > > > > > > h > > > > > > > > > > > > > > > l > > > > > > > > > > > > m > ; > : > > > > n > > > :

:

m‡v 2

> > > > > > > > > > > > > > > > > > > > ;

0

lÿu 2 mÿv 2 muÿlv 2

r2 ; 4

…14†

…15†

The three Equations (13), when expanded as above, are equivalent to those used by Sefrioui and Gosselin [8] to study general planar platforms. The di€erence lies only in our use of planar quaternion coordinates. Following Sefrioui and Gosselin, we determine the Jacobian of the planar platform by computing the time derivative of these equations, 2qT ‰Ci Šq_ ˆ

ri r_ i ; 2

where the three matrices [Ci] have the form, 2 ai 6 hi ‰Ci Š ˆ 6 4 gi li

hi bi fi mi

i ˆ 1; 2; 3;

gi fi ci ni

3 li mi 7 7: ni 5 di

…16†

…17†

We add to this set of equations the time derivative of the constraint (9), ‰0; 0; q3 ; q4 Šq_ ˆ 0:

…18†

The quartic singularity surfaces of planar platforms

The three Equations (16) together with (18) form the Jacobian 2 38 9 2 r1 =4 0 0 qT ‰C1 Š > > > q_ 1 > 6 qT ‰C2 Š 7< q_ 2 = 6 0 r =4 0 2 6 6 7 4 qT ‰C3 Š 5> q_ 3 > ÿ 4 0 0 r 3 =4 > ; : > q_ 4 0 0 0 0 0 q3 q4

935

system of the planar platform: 38 9 0 > r_ > > = < 1> r 07 7 _ 2 ˆ 0: …19† 5 0 > r_ 3 > > > : ; 0 1

or ‰AŠq_ ÿ ‰BŠ_r ˆ 0;

…20†

where Çr=(rÇ 1, rÇ 2, rÇ 3, 0)T. This set of equations is equivalent to the Jacobian system studied by Sefrioui and Gosselin [7, 8]. The elements of the matrix [A] in Equation (20) depend linearly on the planar quaternion components q = (q1, q2, q3, q4)T which de®ne the position of the platform of this manipulator. The positions q for which [A] is singular are de®ned by the quartic equation: det‰AŠ ˆ 0:

…21†

Gosselin and Angeles [6] call these positions ``type II singularities'' of the planar platform. In our formulation, Equation (21) de®nes an algebraic surface in the three-dimensional projective space, C…P2 †, which we call the singularity surface of the planar platform manipulator.

THE GENERAL PLANAR PLATFORM

Let the ®xed pivot u1 be the origin of the ®xed frame F with its x-axis directed along the line containing the pivot u2. Similarly, let the moving pivot z1 be the origin of M with its x-axis directed along the line containing z2. Then, the pivots can be assigned the coordinates:       g2x 0 g1 ; u3 ˆ ; u2 ˆ ; …22† u1 ˆ g2y 0 0 and z1 ˆ

  0 ; 0

 z2 ˆ

These coordinates yield the matrix [A] as: 2 q2 q1 6 6 q1 ‡ …h1 ÿg1 †q4 q2 ‡ …h1 ‡g1 †q3 6 2 2 4 C1 C2 0 0

h1 0



 ; z3 ˆ

 h2x : h2y

0 …h1 ‡g1 †q2 2

‡ …h1 ‡g41 † C3 q3

…23† 3

0 2

q3

…h1 ÿg1 †q1 2

‡ …h1 ÿg41 † C4 q4

2

q4

7 7 7: 5

…24†

where C 1 ˆ q1 ‡

ÿ…h2y ‡ g2y † …h2x ÿ g2x † q4 ; q3 ‡ 2 2

C 2 ˆ q2 ‡

…h2y ÿ g2y † …h2x ‡ g2x † q4 ; q3 ‡ 2 2

C3 ˆ

ÿ…h2y ‡ g2y † …h2x ‡ g2x †2 ‡ …h2y ‡ g2y †2 h2y g2x ÿ h2x g2y …h2x ‡ g2x † q3 ‡ q4 ; q1 ‡ q2 ‡ 4 2 2 2

C4 ˆ

…h2y ÿ g2y † h2y g2x ÿ h2x g2y …h2x ÿ g2x †2 ‡ …h2y ÿ g2y †2 …h2x ÿ g2x † q1 ‡ q3 ‡ q4 : q2 ‡ 2 2 4 2

…25†

Setting the determinant of the resulting matrix [A] to zero, we obtain the algebraic equation of the singularity surface as:

936

C. L. Collins and J. M. McCarthy

S : A1 q21 q23 ‡ A2 q21 q3 q4 ‡ A3 q22 q24 ‡ A4 q22 q3 q4 ‡ A5 q1 q33 ‡ A6 q2 q34 ‡ A7 q1 q2 q23 ‡ A8 q1 q2 q24 ‡ A9 q1 q23 q4 ‡ A10 q1 q3 q24 ‡ A11 q2 q23 q4 ‡ A12 q2 q3 q24 ˆ 0;

…26†

where A1 ˆ h1 g2x ÿ g1 h2x ; A2 ˆ ÿ…h1 g2y ‡ g1 h2y †; A3 ˆ ÿ…h1 g2x ÿ g1 h2x †; A4 ˆ ÿ…h1 g2y ‡ g1 h2y †; 1 A5 ˆ ÿ …h2y g2x ÿ h2x g2y †…g1 ‡ h1 †; 2 1 A6 ˆ …h2y g2x ÿ h2x g2y †…g1 ÿ h1 †; 2 A7 ˆ h1 g2y ÿ g1 h2y ; A8 ˆ h1 g2y ÿ g1 h2y ; A9 ˆ g1 g2x …h2x ÿ h1 † ‡ h1 h2x …g2x ÿ g1 † ‡ g1 g2y h2y ‡ h1 h2y g2y ; 1 1 1 1 A10 ˆ ÿg1 g2y … h2x ÿ h1 † ‡ h1 h2y … g2x ÿ g1 † ‡ g1 g2x h2y ÿ h1 h2x g2y ; 2 2 2 2 1 1 1 1 A11 ˆ g1 g2y … h2x ÿ h1 † ‡ h1 h2y … g2x ÿ g1 † ÿ g1 g2x h2y ÿ h1 h2x g2y ; 2 2 2 2 A12 ˆ g1 g2x …h2x ÿ h1 † ÿ h1 h2x …g2x ÿ g1 † ‡ g1 g2y h2y ÿ h1 h2y g2y :

…27†

It is useful to note that A3= ÿ A1, A4=A2, and A8=A7. The singularity surface S is a quartic surface in the homogeneous coordinates q1, q2, q3 and q4. Its geometric properties are a function of the parameters de®ning the kinematic architecture of the planar parallel manipulator. We can ``dehomogenize'' these coordinates with respect to q4 to visualize this surface in three-dimensional Euclidean space. This is done by making the substitution x = q1/q4, y = q2/q4 and z = q3/q4. Further, of particular interest are the planar cross sections of S orthogonal to the z-axis, de®ned by z = s. After making these substitutions in Equation (26) we obtain: Cz …s† : A1 s2 x2 ‡ A2 sx2 ‡ A3 y2 ‡ A3 y2 ‡ A4 sy2 ‡ A5 s3 x ‡ A6 y ‡ A7 s2 xy ‡ A8 xy ‡ A9 s2 x ‡ A10 sx ‡ A11 s2 y ‡ A12 sy ˆ 0:

…28†

This cross-section yields a quadratic function of x and y which is the equation of a general conic in the z = s plane. See Sefrioui and Gosselin [8] for comparison. In projective geometry, the intersection of a plane with a quartic surface must yield a planer quartic curve. The missing components can be shown to be the ``double line'' at in®nity given by q3=0, q4=0. Quartic surfaces with a double line have been studied in detail by mathematicians since the late 1800 s and have many interesting geometric properties (see Salmon [19] for example). SPECIAL CASES

The geometric properties of the singularity surface are characterized by the locations of the platform ®xed and moving pivots. Two interesting planar platform architectures are the proportional planar platform, and the in-line planar platform. The proportional planar platform A proportional planar platform has ®xed and moving pivots that form corresponding similar or congruent (equal) base and platform triangles (see Fig. 2). For similar triangles, the pivot

The quartic singularity surfaces of planar platforms

937

Fig. 2. The proportional planar platform architecture.

dimensions must satisfy: h1 h2x h2y ˆ ˆ ˆ k: g1 g2x g2y

…29†

If the triangles are congruent, k = 1. For architectures with similar triangles, six of the coecients in Equation (26) become zero, leaving a set of terms that can be factored as: S : q3 q4 …A2 q21 ‡ A2 q22 ‡ A9 q1 q3 ‡ A10 q1 q4 ‡ A11 q2 q3 ‡ A12 q2 q4 †:

…30†

The resulting surface consists of the two planes q3=0 and q4=0, and a quadric surface. We can study the quadratic surface in three-dimensional space using the same substitution as before. Cross-sections orthogonal to the z-axis, de®ned by z = s, yield the conics: Cz …s† : x2 ‡ y2 ‡

A9 s ‡ A10 A11 s ‡ A12 x‡ y ˆ 0: A2 A2

…31†

To put this conic in a standard form, we complete the squares in x and y separately (Shilov [20]) to obtain: Cz …s† : …x ÿ kx †2 ‡ …y ÿ ky †2 ˆ k2x ‡ k2y ;

…32†

where kx= ÿ A9s + A10/2A2 and ky= ÿ A11s + A12/2A2. This is the equation of a circle with center c = (kx, ky) in the plane z = s, and radius r2=k2x+k2y. It can be shown that the x and y cross-sections are hyperbolic, and the surface is a circular hyperboloid. Figure 3 shows the projected singularity surface for a platform manipulator whose pivot locations form similar equilateral triangles. If the triangles are congruent, A10=A12=0 in Equation (30). Cross-sections of the resulting quadratic term of S orthogonal to the z-axis, de®ned by z = s, can now be written as: Cz …s† : …x ÿ skx †2 ‡ …y ÿ sky †2 ˆ s2 …k2x ‡ k2y †;

…33†

where kx= ÿ A9/2A2 and ky= ÿ A11/2A2, and s has been factored out of these parameters. This is the equation of a circle with center c = (skx, sky) in the plane z = s, and radius r2=s2(k2x+k2y). It can be shown that this surface is a circular cone. Figure 4 shows the projected singularity surface for a platform manipulator whose pivot locations form congruent equilateral triangles. The in-line planar platform We now specialize the equations derived above to the case of an ``in-line'' planar platform which is designed such that the three moving pivots lie on a line in M, and the three ®xed pivots lie on a line in F (Fig. 5). Let the ®xed pivot u1 be the origin of the ®xed frame F with its x-axis

938

C. L. Collins and J. M. McCarthy

Fig. 3. Projected singularity surface for a proportional planar platform with k$1, consisting of the z = 0 plane and a circular hyperboloid.

direct along the line containing the other two pivots u2 and u3. Similarly, let the moving pivot z1 be the origin of M with its x-axis directed along the line containing z2 and z3. Then these pivots can be assigned the coordinates:       0 g1 g2 ; u3 ˆ ; …34† ; u2 ˆ u1 ˆ 0 0 0 and

  0 ; z1 ˆ 0

 z2 ˆ

 h1 ; 0

 z3 ˆ

 h2 : 0

Fig. 4. Projected singularity surface for a proportional planar platform with k = 1, consisting of the z = 0 plane and a circular cone.

…35†

The quartic singularity surfaces of planar platforms

939

Fig. 5. An in-line planar platform with ®xed pivots u1, u2, u3 and moving pivots z1, z2, z3.

These coordinates yield the matrix [A] as: 2 q2 q1 6 6 q1 ‡ …h1 ÿg2 1 †q2 q2 ‡ …h1 ‡g2 1 †q3 6 6 4 q ‡ …h2 ÿg2 †q2 q ‡ …h2 ‡g2 †q3 1

0

2

2

0

2

0

3

0

…h1 ‡g1 †q2 2

‡ …h1 ‡g41 †

2

q3

…h2 ‡g2 †q2 2

‡ …h2 ‡g42 † q3

2

q3

7 7 7: 7 2 …h2 ÿg2 †q1 …h2 ÿg2 † q4 5 ‡ …h1 ÿg1 †q1 2 2

‡ …h1 ÿg41 † q4

2

q4

…36†

4

Setting the determinant of this matrix to zero, we obtain the algebraic equation of the singularity surface as: S : Aq21 q23 ÿ Aq22 q24 ‡ …B ÿ C†q1 q23 q4 ‡ …B ‡ C†q2 q3 q24 ˆ 0;

…37†

where A ˆ g 2 h1 ÿ g1 h2 ; B ˆ g1 g2 …h2 ÿ h1 †; C ˆ ÿh1 h2 …g2 ÿ g1 †:

…38†

Equation (37) is a quartic surface in the homogeneous coordinates q1, q2, q3 and q4, but is highly specialized compared to the general platform manipulator singularity surface Equation (26). Geometric properties of the singularity surface. We can study the properties of this surface in three-dimensional Euclidean space by dehomogenizing with respect to q4. This is done as before by making the substitution x = q1/q4, y = q2/q4 and z = q3/q4 to obtain: S : Ax2 z2 ‡ …B ÿ C†sz2 ‡ y…ÿAy ‡ …B ‡ C†z† ˆ 0:

…39†

The point (x, y, z) = (0, 0, 0) lies on S and is what Salmon [19] calls a ``bi-planar double point'' because it has the two tangent planes T 1 : y ˆ 0 and T 2 : ÿAy ‡ …B ‡ C†z ˆ 0. The tangent plane T 1 intersects this surface in four lines L1;2 : y ˆ 0, z = 0 counted twice, L3 : y ˆ 0, x = 0, and L4 : y ˆ 0, Ax + B ÿ C = 0. Thus, we ®nd that the x-axis is a double line of the surface. Planar sections of S orthogonal to the x-axis, de®ned by x = t, can be computed to be: Cx …t† : Ay2 ÿ …B ‡ C†yz ÿ t…At ‡ B ÿ C†z2 ˆ 0:

…40†

This equation de®nes set of conics, each of which is, in fact, the product of two lines: M1 : x ˆ t; y ˆ m1 …t†z

and

M2 : x ˆ t; y ˆ m2 …t†z;

…41†

940

C. L. Collins and J. M. McCarthy

where m1;2 …t† ˆ

q …B ‡ C†2 …B ‡ C†2 ‡ 4At…At ‡ B ÿ C† 2A

:

…42†

Thus, the surface S is a quartic ruled surface which has the elementary parameterization (x, y, z) = (t, m1,2(t)z, z). Cross sections of S orthogonal to the z-axis, de®ned by z = s, yield the conics: Cz …s† : s2 x2 ÿ y2 ‡

BÿC 2 B‡C s x‡ sy ˆ 0: A A

…43†

To put this conic in a standard form, we complete the squares in x and y separately (Shilov [20]). Let kx=C ÿ B/2A and ky=C + B/2A, then we obtain: Cz …s† : s2 …x ÿ kx †2 ÿ …y ÿ sky †2 ˆ k2x ÿ s2 k2y : This is the equation of a hyperbola with center c = (cx, cy) in the plane z = s, where   CÿB C‡B ;s c ˆ …kx ; sky † ˆ : 2A 2A The parameters ax and ay de®ned by, 1 q k2x ÿ s2 k2y ax ˆ s

and

ay ˆ

q k2x ÿ s2 k2y ;

…44†

…45†

…46†

are the semi-axes of the hyperbola; and we obtain the standard form: Cz …s† :

…x ÿ cx †2 …y ÿ cy †2 ÿ ˆ 1: a2x a2y

…47†

Fig. 6 shows the singularity surface for the in-line planar platform with h1=1, h2=4, g1=3 and g2=5 studied by Sefrioui and Gosselin [7]. Note that the surface appears to be composed of two surfaces having the x axis in common. Indeed, if we eliminate t and z from the parameterization in Equation (41) we ®nd that Equation (39) can be rewritten in terms of two factors as: …y ÿ m1 …x†z†…y ÿ m2 …x†z† ˆ 0;

Fig. 6. The singularity surface of the ``in-line'' planar platform with h1=1, h2=4, g1=3 and g2=5.

…48†

The quartic singularity surfaces of planar platforms

941

where m1(x) and m2(x) are de®ned by Equation (42). This factorization is complicated by the appearance of x under the square root. Special cases of this surface yield cleaner factorizations. Special cases of the singularity surface. In this section, we consider the properties of special cases of the singularity surface (37). These are obtained from the dimensions g1, g2, h1, and h2, that cause the constants A, B, and C to become zero. A = 0. The coecient A is zero when the dimensions of the platform satisfy the relation: g1 h1 ˆ : g2 h2

…49†

This states that the ratio of the distances from the ®rst to the second and third pivots of the RPR chains are the same in both the base and the platform. From Equation (38), we ®nd that B = ÿ g1/h1C, thus Equation (39) reduces to the form: S 1 : z……h1 ‡ g1 †xz ÿ …h1 ÿ g1 †y† ˆ 0:

…50†

This surface consists of the plane z = 0 and the quadric given by: H : xz ˆ Ky;

Kˆ

h1 ÿ g1 : h1 ‡ g1

…51†

The standard form for this surface is obtained by rotating the x±z frame by p/4 to obtain: H : X 2 ÿ Z 2 ˆ 2Ky;

…52†

which is the equation of a hyperbolic paraboloid. Sections normal to the y-axis de®ne hyperbolas, while sections normal to the X or Z axes de®ne parabolas. Figure 7 shows an example of this singularity surface. B = 0. The coecient B of the singularity surface becomes zero either: (a) when h1=h2=h, which occurs when the second and third moving pivots, z2 and z3, coincide; or (b) when g1=0 or g2=0, which occurs when either the second or third ®xed pivot, u2 or u3, coincides with the ®rst, u1. Case a. From Equation (38), we ®nd that h1=h2=h yields the relation C = ÿ hA, and the equation of the singularity surface becomes: S : x2 z2 ‡ hxz2 ÿ y2 ÿ hyz ˆ …xz ‡ y ‡ hz†…xz ÿ y† ˆ 0:

Fig. 7. A singularity surface with the constant A = 0.

…53†

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C. L. Collins and J. M. McCarthy

The second equality shows that S consists of the two quadric surfaces: H1 : …x ‡ h†z ‡ y ˆ 0;

and

H2 : xz ÿ y ˆ 0:

…54†

The surface H2 is a hyperbolic paraboloid similar to Equation (52). The quadric H1 is also a hyperbolic paraboloid, which can be put into standard form by shifting the x-axis by the amount ÿh, then rotating the x±z coordinates by p/4. Figure 8 is an example of this singularity surface. Case b. In this case, we set g1=0 and obtain C = ÿ h2A which yields the same singularity surface as Equation (53) with h = h2. Setting g2=0 yields C = ÿ h1A and we obtain Equation (53) with h = h1. C = 0. The properties of the singularity surface with dimensions such that the coecient C = 0 are similar to the results obtained for the case B = 0. In order for C = 0, we must have: (c) g1=g2, which occurs when the second and third ®xed pivots, u2 and u3, coincide; or (d) h1=0 or h2=0 which occurs when either the second or third moving pivot, z2 or z3, coincides with the ®rst moving pivot, z1. Case c. Substituting the condition g1=g2=g into Equation (38), we obtain B = ÿ gA. The result is the equation of the singularity surface becomes: S : x2 z2 ÿ gxz2 ÿ y2 ÿ gyz ˆ …xz ÿ y ÿ gz†…xz ‡ y† ˆ 0:

…55†

Thus, the singularity surfaces consists of the two quadrics: H1 : …x ÿ g†z ÿ y ˆ 0;

and

H2 : xz ‡ y ˆ 0:

…56†

These are both hyperbolic paraboloids which are easily transformed to the standard form (52). Case d. The condition h1=0 yields B = ÿ g2A and we obtain the same surfaces as in Equation (55) with g = g2. The condition h2=0 results in the relation B = ÿ g1A which is identical to case (c) with g = g1. B = C = 0. This case occurs when the ®xed pivots u1 and u2 are concurrent and the moving pivots z1 and z3 are concurrent. This is equivalent to the dimensions g1=0 and h2=0. The singularity surface for this platform is: S : …xz ‡ y†…xz ÿ y† ˆ 0:

…57†

This surface factors into the two hyperbolic paraboloids H1 : xz ‡ y ˆ 0 and H2 : xz ÿ y ˆ 0 is independent of the dimensions of the platform.

Fig. 8. A singularity surface with the constant B = 0.

The quartic singularity surfaces of planar platforms

943

SINGULAR POSITIONS IN THE WORKSPACE

The singularity surface de®nes the set of con®gurations of the planar platform that cause the Jacobian to be singular independent of whether or not these positions are in the workspace of the platform. The workspace is bounded by the constraint manifolds associated with the minimum and maximum extension of each of the linear actuators. The intersection of the singularity surface with this workspace identi®es the con®gurations of the system that presently control problems. Let ri,min and ri,max be the minimum and maximum extension of the ith RPR-chain, then the set of positions reachable by this chain is the semi-algebraic set in C ‡ …P2 † consisting of points q, such that W i : ri;min  qT ‰Ci Šq  ri;max :

…58†

The workspace of the general platform is the intersection of the three semi-algebraic sets W i , i = 1, 2, 3. This is a region in projective three-space bounded by six quadric surfaces. Figure 9 shows the singularity surface of a proportional planar platform manipulator along with the constraint manifolds associated with three RPR chains at their minimum prismatic link extensions. The workspace of the manipulator lies outside of these minimum constraint surfaces, but inside the surfaces associated with the maximum extensions of the prismatic joints (not shown). Figure 10 shows three constant z = q3/q4 cross-sections of the workspace and singularity surface for values equivalent to the platform angles f = 08, f = 308 and f = ÿ 308. Notice that in these cross-sections, the workspaces W i yield three circular annuli. The mutual intersection of these annuli is the set of reachable positions for a constant platform angle f. The z cross-sections of the singularity surface yield circles which are represented by the heavier line. CONCLUSION

This paper derives the singularity surface of a planar platform manipulator in a three-dimensional projective space in which each point represents a planar displacement. This projective space is constructed as the Cli€ord algebra of the projective plane, and the homogeneous coordinates of each point is known as a planar quaternion. In this space, the set of singular con®gurations of the manipulator is a quartic algebraic surface. For special parallel platform

Fig. 9. The intersection of a singularity surface and the constraint manifolds of three RPR chains with minimum actuator lengths.

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C. L. Collins and J. M. McCarthy

Fig. 10. Cross-sections of the constraint manifold and singularity surface, where ri,min=1 and ri,max=3.

architectures, this surface contains combinations of linear and quadratic factors. Examples show that the location of this surface in the workspace of the manipulator is easily determined. AcknowledgementsÐThe support of the National Science Foundation through grant DMII-9321936 is gratefully acknowledged.

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