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Forward kinematics of planar parallel manipulators in the Clifford algebra of P2
Mechanism and Machine Theory 37 (2002) 799–813 www.elsevier.com/locate/mechmt
Forward kinematics of planar parallel manipulators in the Clifford algebra of P2 Curtis L. Collins Bourns College of Engineering, University of California, Riverside, CA 92521, USA Received 2 October 2000; accepted 6 August 2001
Abstract This paper presents the forward kinematics of planar platform manipulators composed of three RPR chains. The Clifford Algebra of projective two space is used to represent planar displacements. Also known as planar quaternions, the representation allows us to formulate the kinematic constraints in an algebraic form with rich geometric content. The forward kinematics problem reduces to the intersection of three circular hyperboloids. An algorithm for parameterizing the forward kinematics solution is presented. The general kinematics function is shown to yield six solutions. Furthermore, it is shown that real solutions only exist if the roots of a certain polynomial are on the interval from 1 to 1. A series of platform architectures is then studied to show how the kinematics algorithm specializes in these cases. Degenerate architectures are also identified. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Parallel manipulator; Forward kinematics; Planar quaternions
1. Introduction A planar parallel manipulator is formed when two or more planar kinematic chains act together on a common rigid platform. The most common planar platform architecture is composed of three RPR chains, where the notation RPR denotes the planar chain made up of a revolute joint, a prismatic joint, and a second revolute joint in series. The spatial analog to the planar platform, known as the Stewart–Gough platform, is composed of six SPS spatial chains, where the notation SPS denotes the spatial chain made up of a spherical joint, a prismatic joint, and a spherical joint in series.
E-mail address: [email protected] (C.L. Collins). 0094-114X/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 2 3 - X
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The forward kinematics of general parallel manipulators is a well-established problem. The general spatial parallel manipulator has been shown to have up to 40 solutions using elimination methods in both Cartesian coordinates [14] and soma coordinates [10,17]. The planar parallel manipulator has been studied using Cartesian coordinates and has at most six solutions [5]. A number of special architectures have been identified [6]. General planar parallel manipulators have also been studied using algebraic coordinates [9]. In this paper, we seek to elucidate the special architectures of planar parallel manipulators and the operational use of planar quaternion coordinates using a blend of algebra and geometry. We are motivated by our recent work studying the singularity surfaces of planar parallel manipulators [2] which allowed us to organize special manipulator architectures based on the geometric features of their singularities, and to further demonstrate the use of kinematic mappings in the study of manipulator systems. A number of references are available that discuss the mathematical and geometric construction of these mappings [1,7,11]. A solid understanding of the forward kinematics problem is a necessary first step in developing algorithms for parallel manipulator design. Algebraic coordinates provide a powerful mathematical machinery for these algorithms [13]. In addition, the parameter variation of the actuator lengths define the workspace of the manipulator through the forward kinematics mapping. Recent studies have focused on algorithms to define the boundaries of different types of workspace projections [12]. Dasgupta and Mruthyunjaya [3] have identified the need for more focused efforts in these areas. The paper is organized as follows. First, we review the construction and manipulation of planar quaternions and define the constraint manifold imposed by an RPR chain. Next we introduce our forward kinematics algorithm as a blend of geometric and algebraic arguments and discuss its features. Then, we discuss a variety of special parallel manipulator architectures and the effect of their geometric properties on the forward kinematics function. We also identify a number of degenerate cases where the general algorithm becomes singular.
2. Planar quaternions and constraint manifolds A transformation of coordinates in R3 can be defined such that it is a rigid displacement in the plane Z ¼ 1. This is the well-known 3 3 homogeneous transform: 8 9 2 38 9 cos h sin h dx < x =
ð2Þ T
T
where z ¼ ðx; yÞ and Z ¼ ðX ; Y Þ are the coordinate vectors of points in the fixed and moving frame, M and F respectively, in the plane Z ¼ 1. The upper left 2 2 matrix AðhÞ defines the T planar rotation, and the vector d ¼ ðdx ; dy Þ is the translation vector which defines the location of M relative to F.
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The pole of the displacement is the eigenvector associated with the eigenvalue k ¼ 1 of [T], and is used to identify a planar displacement with an element of the even Clifford algebra of P 2 . The resulting element can be viewed as a four-vector and is also known as a planar quaternion (see [11] for details of this construction). The planar quaternion corresponding to a displacement ½T ¼ ðAðhÞ; dÞ is given by: 8 9 8 9 1 q1 > > ðdx cos h2 þ dy sin h2Þ > > > > 2 > > > > >1 > = > > < < q2 ðdy cos h2 dx sin h2Þ = 2 P¼ : ð3Þ ¼ q3 > > > > sin h2 > > > > > > > > > ; : q4 > ; : cos h2 Notice that in order to represent a displacement, the components of the planar quaternion P ¼ ðq1 ; q2 ; q3 ; q4 ÞT , the third and fourth components must satisfy the relation: q23 þ q24 ¼ 1:
ð4Þ
When (4) is satisfied, the four planar quaternion parameters can be used to define the homogeneous matrix in Eq. (1) as 2 2 3 q4 q23 2q3 q4 2ðq1 q4 q2 q3 Þ ½T ¼ 4 2q3 q4 q24 q23 2ðq1 q3 þ q2 q4 Þ 5: ð5Þ 0 0 1 Eqs. (3) and (5) provide the means to transform from planar displacements to planar quaternion coordinates and back again. The composition rules of the Clifford algebra elements naturally allow us to concatenate displacements using polynomial multiplication. The vector interpretation of planar quaternions allows us to view planar displacements as points in a four-dimensional projective space, called the image space (see [1] for an alternate derivation of the image space coordinates and a discussion of its geometric properties). In this space, planar displacements correspond to lines through the origin. Associated with the constraint equation (4) are two points on each of these lines. From the quadratic nature of [T] in Eq. (5) it can be seen that the vectors q and q define the same planar displacement. Because of this association, the set of planar quaternions is a double covering of the set of planar displacements. Now consider the planar platform manipulator supported by three RPR chains as shown in Fig. 1. Let the location of the three fixed pivots be specified by ui ¼ ðui ; vi ÞT , i ¼ 1; 2; 3 measured in F and let the three moving pivots be zi ¼ ðki ; li ÞT measured in M. Coordinates in the platform M are related to coordinate in F by the 3 3 homogeneous transformation (1), that is Zi ¼ ½T zi . Let the distance between the ith fixed and moving pivots be denoted qi so we have the relations: Qi : hðui ½T zi Þ; ðui ½T zi Þi ¼ q2i ;
i ¼ 1; 2; 3:
ð6Þ
Substituting (5) into (6), we obtain three quadratic equations in planar quaternion coordinates which define 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 the image space of planar quaternions. This is called the constraint manifold of the manipulator.
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Fig. 1. General planar parallel manipulator with 3-RPR chains.
The constraint manifold of a single RPR chain is given by the quadric equation in four homogeneous coordinates q ¼ ðq1 ; q2 ; q3 ; q4 ÞT , aq21 þ bq22 þ cq23 þ dq24 þ 2fq2 q3 þ 2gq1 q3 þ 2hq1 q2 þ 2lq1 q4 þ 2mq2 q4 þ 2nq3 q4 ¼ where the coefficients are given by 9 8 9 8 1 a > > > > > > > > > > > > > > 1 > > > > b > > > > > > > 2 2 > 1 > > > > > > > > ½ðk þ uÞ þ ðl þ vÞ c > > > > 4 > > > > > > > 2 2 > 1 > > > > > > > > ½ðk uÞ þ ðl vÞ d > > > > 4 > > < = > = kþu 2 ¼ : lþv g> > > > > > > > 2 > > > > > > > > > > > 0 h> > > > > > > > > > > > > ku > > > > l > > > > 2 > > > > > > > > lv > > > > > > > > m > > 2 : > > ; > > : ; lukv n 2
q2 ; 4
ð7Þ
ð8Þ
The three equations (6) when expanded as above are equivalent to those used by Sefrioui and Gosselin [15] to study general planar platforms. The difference lies only in our use of planar quaternion coordinates. It can be shown [1,11] that the above quadratic equation defines a circular hyperboloid. As the locations of the fixed and moving pivots changes, the hyperboloids remain circular, but become skewed. It is important to note that the points J1 ¼ ð1; i; 0; 0Þ and J2 ¼ ð1; i; 0; 0Þ satisfy Eq. (7). These points are imaginary and are invariants in the geometry of the image space. They belong to the quadrics, but do not represent a planar displacement since they do not satisfy the constraint equation (4). Using planar quaternions, or the Clifford Algebra of P 2 , the positions and orientations of an RPR linkage is represented as a circular hyperboloid written in the coordinates q1 , q2 , q3 and q4 . When three RPR linkages come together to form a planar platform manipulator, the resulting kinematics is defined by the intersections of three circular hyperboloids.
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3. Forward kinematics solutions 3.1. Geometric arguments The foundation of our forward kinematics algorithm comes from the geometry of quadrics [16], although it can be argued from a purely algebraic standpoint as well. A well known result from algebraic geometry is that three quadric surfaces intersect in eight points. This is a direct result of the application of Bezout’s theorem. We seek to further clarify the nature of these solutions and relate them to assembly configurations of planar platform manipulators. Geometrically, any linear combination of two quadrics defines the curve of intersection of the quadrics. That is, any quadric of the form Q ¼ QA þ kQB passes through the intersection of QA and QB . Q is known as a pencil of quadrics. Consider the quadric QAB ¼ QB QA . QAB passes through the curve of intersection of QA and QB . Next, consider the quadric QAC ¼ QC QA which passes through the curve of intersection of of QA and QC . QAB and QAC intersect in their own curve, but geometrically, this curve must intersect QA in exactly the eight points of intersection of QA , QB , and QC . The quadrics defining our constraint manifolds are circular hyperboloids, and thus two of the points of intersection of three of these special quadrics are the imaginary points at infinity (J1 and J2 ), thus leaving only six other possible real solutions. We now present the analytic formulation of our geometric argument. 3.2. Analytic solution The fixed and moving reference frames for any planar platform can always be assigned such that the fixed frame origin is located at one of the fixed revolute joints while its x-axis is along the line between two of the fixed pivots. Similarly, the moving frame origin can be located at one of the moving revolute joints while its x-axis is along the line between two of the moving pivots corresponding to the fixed pivots used to assign the fixed frame (see Fig. 1 for example). Let the lengths of the three prismatic actuators be a, b and c. Let the coordinates of the fixed and moving pivots of actuator 2 be defined by u2 ¼ ðg; 0ÞT and z2 ¼ ðh; 0ÞT ; and actuator 3 by u3 ¼ ðxf ; yf ÞT T and z3 ¼ ðxm ; ym Þ . These parameters will now be used to define the three quadrics QA , QB , and QC . Substituting into Eq. (6) we find that quadric QA has the form q21 þ q22 14a2 q23 14a2 q24 ;
ð9Þ
quadric QB has the form 2
2
q21 þ q22 þ ðg þ hÞq2 q3 þ ðg þ hÞq1 q4 þ 14ððg þ hÞ b2 Þq23 þ 14ððg hÞ b2 Þq24 ;
ð10Þ
and quadric QC has the form q21 þ q22 þ ðxf þ xm Þq2 q3 ðxf þ xm Þq1 q4 ðyf þ ym Þq1 q3 þ ðyf þ ym Þq2 q4 þ ðxm yf þ xf ym Þq3 q4 þ 14ððxf þ xm Þ2 þ ðyf þ ym Þ2 c2 Þq23 þ 14ððxf þ xm Þ2 þ ðyf þ ym Þ2 c2 Þq24 : Note that we have homogenized the equations by applying the constraint equation (4).
ð11Þ
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Taking QAB ¼ QB QA and QAC ¼ QC QA we find that QAB has the form ðg þ hÞq2 q3 þ ðg þ hÞq1 q4 þ 14ðða2 b2 Þ þ ðg þ hÞ2 Þq23 þ 14ðða2 b2 Þ þ ðg hÞ2 Þq24 ;
ð12Þ
and QAC has the form q1 ðq4 ðxf þ xm Þ þ q3 ðyf ym ÞÞ þ q2 ðq3 ðxf þ xm Þ þ q4 ðyf þ ym ÞÞ þ q3 q4 ððxm yf Þ þ xf ym Þ þ 14ða2 c2 þ ðxf þ xm Þ2 þ ðyf þ ym Þ2 Þq23 þ 14ða2 c2 þ ðxm xf Þ2 þ ðym yf Þ2 Þq24 :
ð13Þ
Note that the quadrics QAB and QAC are linear in the variables q1 and q2 . Solving for q1 and q2 gives two equations in q3 and q4 which we will denote f1 ðq3 ; q4 Þ and f2 ðq3 ; q4 Þ respectively. The denominators, d, of f1 ðq3 ; q4 Þ and f2 ðq3 ; q4 Þ are equal and can be written as d ¼ 2q4 q3 ðhxf gxm Þ þ ðg þ hÞðyf þ ym Þ 2q24 ðhyf þ gym Þ:
ð14Þ
By applying the constraint q23 þ q24 1 ¼ 0 we find a parameterization of the curve of intersection of QAB and QAC of the form q1 ¼ f1 ðq3 ; q4 Þ; q2 ¼ f2 ðq3 ; q4 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffi q3 ¼ 1 q24 ;
ð15Þ
q4 ¼ q4 : Substituting this parametrization into QA gives an irrational function of q4 . The denominator of this function is simply d 2 . The roots of the numerator of this function lead to the forward kinematics solutions for the planar platform manipulator by substituting the values for q4 back into the original parametrization Eq. (15). The numerator, which is referred to here as the forward kinematics function, has the form qffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ k0 þ k2 q24 þ k4 q44 þ k6 q64 þ ðk1 q4 þ k3 q34 þ k5 q54 Þ 1 q24 ¼ 0; where ki are functions of the structural parameters and the link lengths. Simplified expressions for ki can be found in Appendix A. Eq. (16) defines the consistency condition that holds when the curve represented in (15) intersects QA . When expanded out, Eq. (16) contains 795 individual terms. We can clear the square root terms by bring the terms to the right hand side, squaring them, and then bringing them back. The result is the 12th order even polynomial, k02 ðk12 2k0 k2 Þq24 þ ðk12 þ k22 2k1 k3 þ 2k0 k4 Þq44 ðk32 2k1 ðk3 k5 Þ 2ðk2 k4 þ k0 k6 ÞÞq64 þ ðk32 þ k42 þ 2k1 k5 2k3 k5 þ 2k2 k6 Þq84 2 2 12 þ ð2k3 k5 k52 þ 2k4 k6 Þq10 4 þ ðk5 þ k6 Þq4 ;
ð17Þ
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whose roots occur in pairs. This is equivalent to solving a 6th order polynomial. The 12 roots of this polynomial will lead to 12 forward kinematics solutions. Closer examination of these solutions reveals that only six of them are unique, the other six being the negatives of the first set; a consequence of the double covering of planar quaternions. It is for this reason that only the positive solution for q3 appears in the parameterization (15). Another interesting and useful fact about our solution method, is that only roots of q4 in the range 1 6 q4 6 1 will result in real forward kinematics solutions. Indeed, you can only plot the function on this interval, otherwise the function is complex. Thus, when we solve for the roots numerically, we only have to solve for roots in the interval ½1; 1 to find all the real solutions. 3.3. Numerical example As an example and verification of the method, we choose to reproduce a result reported in [6]. While their notation is slightly different, we find the following system of parameters for comparison: a ¼ 14:89;
b ¼ 15:38;
g ¼ 15:91;
h ¼ 17:04;
xf ¼ 0;
yf ¼ 10;
c ¼ 12;
xm ¼ 13:2364;
ym ¼ 16:0967:
Substituting these parameters into our forward kinematics expression in q4 given by Eq. (16) gives the six degree irrational function of q4 : 3:9852 108 1:0039 109 q24 þ 7:2338 108 q44 1:17775 108 q64 qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 3 9 5 þ ð8:8575 10 q4 þ 2:0091 10 q4 1:1177 10 q4 Þ 1 q24 :
ð18Þ
Solving for the roots of this equation and substituting the values into the parameterization given in Eq. (15) gives the results shown in Table 1. Finally applying Eq. (5) allows us to solve for the position and orientation of the moving platform in each position. The results are provided in Table 2. These are exactly the values reported in [6]. A rendering of the planar platform in each of these configurations is shown in Fig. 2.
Table 1 Forward kinematics solutions in planar quaternion form Solution
q4
q3
q2
q1
1 2 3 4 5 6
0.8807 0.9997 0.9925 0.9574 0.8771 0.4832
0.4737 0.0237 0.1223 0.2887 0.4803 0.8755
3.2946 7.0308 1.6968 1.2495 4.1670 7.1513
6.7265 2.5822 7.2953 7.3850 6.2219 2.2267
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Table 2 Forward kinematics solutions in Cartesian form Solution
h
dx
dy
1 2 3 4 5 6
0.9870 0.0473 0.2453 0.5857 1.0020 2.1329
8.7266 5.4957 14.8961 13.4199 14.9201 14.6739
12.1757 13.9355 1.5830 6.6563 1.3379 3.0126
Fig. 2. Example forward kinematics solutions.
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4. Special platform architectures Motivated by the results of Collins and McCarthy [2] where singularity surfaces degenerated when the kinematic architecture assumed special forms, we apply these forms to the forward kinematics problem. This work expands on the results of Gosselin and Merlet [6] and demonstrates the advantages of working with the algebraic representation afforded by using planar quaternions. The notion of special architectures of spatial platform manipulators has been studied by Faugere and Lazard [4] where their primary focus was on classes of manipulators with different combinations of coincident spherical joints. As we will show, there are other geometric features that are useful to identify and consider. We have studied the following cases: Proportional case. The planar platform architecture is characterized by having proportional fixed and moving pivot locations. That is, the base and top platforms are similar triangles (see Fig. 3). In-Line case. The platform architecture is characterized by having colinear fixed and moving pivots (see Fig. 3). Coincident pivots case. The platform architecture is characterized by having either a pair of fixed or moving pivots coincident (see Fig. 4).
Fig. 3. Proportional and in-line platforms.
Fig. 4. Platforms with coincident fixed or moving pivots.
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Fig. 5. In-line platforms with coincident fixed or moving pivots.
Combination cases. These architectures are characterized by combinations of the previous cases, such as the in-line platforms with proportional pivot locations (see Fig. 3), and in-line platforms with different combinations of coincident fixed and moving pivots as shown in Fig. 5. These will be discussed in the context of the other cases. Degenerate cases. When the denominator, d, of the parameterization in Eq. (15) becomes zero, the algorithm fails to compute the forward kinematics solutions. Four classes of architectures have been identified that lead to this degenerate condition. In what follows, the coefficients ki are simply the specializations of the ki found in Eq. (16) and listed in general form in Appendix A. Coefficients li are new coefficients found from factoring the simplified polynomials in ki . 4.1. Proportional architectures We can express the condition of proportionality by requiring that the ratio of the coordinates of the fixed and moving pivots be equal for each actuator pair (Fig. 3). This can be written as: h xm ym ¼ ¼ t: ¼ yf g xf
ð19Þ
The forward kinematics function given by Eq. (16) reduces to a 6th order even polynomial, k0 þ k2 q24 þ k4 q44 þ k6 q64 ¼ 0;
ð20Þ
that factors into a quadratic and an even quartic polynomial ð1 þ 2t 4q24 t þ t2 Þðl0 þ l2 q24 þ l44 Þ ¼ 0:
ð21Þ
The quadratic term only depends on the constant of proportionality. It turns out that the values for q4 found from factoring this term are imaginary and the solutions degenerate since the denominator also contains this factor. The quartic term is an even polynomial and can be solved making the substitution p ¼ q24 . The resulting quadratic can be solved in closed form.
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4.2. In-line architectures By requiring that the revolute joints in the fixed base and moving platform to be colinear we can define the in-line platforms (Fig. 3) using the condition yf ¼ ym ¼ 0:
ð22Þ
In this case, the forward kinematics Eq. (16) reduces to a 6th order even polynomial, k0 þ k2 q24 þ k4 q44 þ k6 q64 ¼ 0:
ð23Þ
It does not factor in general. As in the proportional case, we can apply the substitution, p ¼ q24 , and solve the resulting cubic equation in closed form. 4.3. Coincident pivot architectures The conditions for having coincident fixed pivots is fyf ¼ 0; g ¼ 0g
or fg ¼ xf g:
ð24Þ
The conditions for having coincident moving pivots is fym ¼ 0; h ¼ 0g
or fh ¼ xm g:
In both cases (Fig. 4) the forward kinematics equation (16) takes on the form qffiffiffiffiffiffiffiffiffiffiffiffiffi k0 þ k2 q24 þ k4 q44 þ ðk1 q4 þ k3 q34 Þ 1 q24 ¼ 0;
ð25Þ
ð26Þ
which is a 4th order function of q4 . Similar to the general case, we can convert this equation to an 8th order even polynomial. Then by applying the substitution p ¼ q24 , the problem reduces to the solution of a quartic equation. By further constraining the pivots to be colinear (Fig. 5), we have the special cases of in-line platforms with various combinations of coincident fixed or moving pivots. In any of these cases, the forward kinematics equation has the form k0 þ k2 q24 þ k4 q44 ¼ 0;
ð27Þ
which is a 4th order even polynomial. It is equivalent to solving a quadratic equation in p. 4.4. Degenerate architectures The geometric elimination algorithm and parameterization degenerates when the denominator, d, in Eq. (14) becomes zero for all ðq3 ; q4 Þ pairs. For this to be true, each coefficient must become zero simultaneously. This occurs in four distinct sets of architectures, two of which are singular.
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The singular architectures satisfy the conditions fg ¼ 0; h ¼ 0g
or fxf ¼ 0; xm ¼ 0; yf ¼ 0; ym ¼ 0g;
ð28Þ
and fg ¼ 0; xf ¼ 0; yf ¼ 0; ym ¼ 0g or fh ¼ 0; xm ¼ 0; yf ¼ 0; ym ¼ 0g: ð29Þ Geometrically, the architectures in Eq. (28) have two RPR joints with identical fixed and moving pivots. The architectures in Eq. (29) are characterized by having three coincident fixed or moving pivots. These architectures are singular since the lines defining the prismatic joints always intersect in a common point. Alternatively, it can be shown that the determinant of the Jacobian of the forward kinematics mapping is identically zero for all qi (see [2] or [15]). The non-singular architectures satisfy the conditions g ¼ h;
xf ¼ xm ;
yf ¼ ym ;
ð30Þ
yf ¼ 0;
ym ¼ 0;
h xm ¼ : g xf
ð31Þ
and
Geometrically, Eq. (30) is satisfied when the fixed and moving platforms are mirror images of each other. The forward kinematics function represented by Eq. (16) does not loose any terms in this case. Eq. (31) represents architectures that have colinear fixed and moving pivots whose locations are in proportion. We call these in-line proportional platforms. In this case. the forward kinematics function reduces to a 6th order even polynomial and factors into a quadric and a double quadric. ð1 þ 2t 4q24 t þ t2 Þðl0 þ l2 q24 Þ2 ¼ 0:
ð32Þ
This gives at most two real double solutions. It is important to note that while our geometric parameterization of the forward kinematics solutions degenerates in these two nonsingular cases, we can still solve for the assembly configurations using standard elimination and substitution methods. 5. Conclusions The forward kinematics of the general planar parallel manipulator has been presented along with a numerical example for verification purposes. The methodology utilizes planar quaternions which allow us to develop kinematics equations in a four-dimensional space of algebraic coordinates. The geometry of the constraint imposed by an RPR chain appears in this space as a circular hyperboloid. The three RPR chains that make up a planar parallel manipulator correspond to three circular hyperboloids whose intersections define the forward kinematics solutions. The geometry of quadrics allows us to construct a geometric and analytic solution that results in a 6th order irrational function of a single coordinate q4 . Real solutions to the problem lie on the interval 1 6 q4 6 1. The geometry of the platform architecture dramatically effects the characteristics of the kinematics function. As the revolute joints attached to the base and platform become proportional, linear, or coincident, the forward kinematics equation reduces in degree, or
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degenerates. In all cases, the forward kinematics solution reduces to finding the roots of an even polynomial in q4 .
Appendix A k0 ¼ fða cÞ2 ða þ cÞ2 ðg þ hÞ2 þ ðg þ hÞ2 x4f 2ðg þ hÞða2 b2 þ g2 þ 2gh þ h2 Þx3m þ ðg þ hÞ2 x4m 2ðg þ hÞx3f ða2 b2 þ g2 þ 2gh þ h2 2ðg þ hÞxm Þ 2
2ðg þ hÞða2 b2 þ g2 þ 2gh þ h2 Þxm ðða cÞða þ cÞ þ ðyf þ ym Þ Þ þ ðyf þ ym Þ2 ða4 2a2 b2 þ b4 2b2 ðg þ hÞ2 þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ þ ðg þ hÞ2 ðyf þ ym Þ2 Þ þ x2m ða4 þ b4 2b2 ðg þ hÞ2 þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ þ a2 ð2b2 þ 4ðg þ hÞ2 Þ þ 2ðg þ hÞ2 ðyf þ ym Þ2 Þ þ 2xf ða2 b2 þ g2 þ 2gh þ h2 2ðg þ hÞxm Þðða2 b2 þ g2 þ 2gh þ h2 Þxm ðg þ hÞx2m ðg þ hÞðða cÞða þ cÞ þ ðyf þ ym Þ2 ÞÞ þ x2f ða4 þ b4 2b2 ðg þ hÞ2 þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ 2
2
þ a2 ð2b2 þ 4ðg þ hÞ Þ þ 2ðg þ hÞð3ða2 b2 þ ðg þ hÞ Þxm þ 3ðg þ hÞx2m þ ðg þ hÞðyf þ ym Þ2 ÞÞg;
ðA:1Þ
k1 ¼ f4ððhða2 b2 þ ðg þ hÞ2 Þ 2ðg þ hÞ2 xm Þyf3 þ ððg þ 2hÞða2 b2 þ ðg þ hÞ2 Þ þ 2ðg þ hÞ2 xf 4ðg þ hÞ2 xm Þyf2 ym þ ym ð2ðg þ hÞ2 x3f þ x2f ðð3g 2hÞ ða2 b2 þ ðg þ hÞ2 Þ þ 4ðg þ hÞ2 xm Þ þ xf ða4 þ b4 2b2 ðg þ hÞ2 þ a2 ð2b2 þ 4gðg þ hÞÞ þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ 2ð2g þ hÞða2 b2 þ g2 þ 2gh þ h2 Þ xm þ 2ðg þ hÞ2 x2m þ 2ðg þ hÞ2 ym2 Þ gð4a2 ðg þ hÞxm þ ða2 b2 þ ðg þ hÞ2 Þ x2m þ ða2 b2 þ ðg þ hÞ2 Þða2 c2 þ ym2 ÞÞÞ þ yf ðð2g þ 3hÞða2 b2 þ ðg þ hÞ2 Þ x2m 2ðg þ hÞ2 x3m þ x2f ðhða2 b2 þ ðg þ hÞ2 Þ 2ðg þ hÞ2 xm Þ þ ða2 b2 þ ðg þ hÞ2 Þðða cÞða þ cÞh þ ð2g þ hÞym2 Þ xm ða4 þ b4 2b2 ðg þ hÞ2 þ a2 ð2b2 þ 4hðg þ hÞÞ þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ þ 2ðg þ hÞ2 ym2 Þ 2xf ððg 2hÞða2 b2 þ ðg þ hÞ2 Þxm þ 2ðg þ hÞ2 x2m 2ðg þ hÞðða2 hÞ þ ðg þ hÞym2 ÞÞÞÞg;
ðA:2Þ
k2 ¼ f4ðða2 c2 Þ2 gh þ ghx4f gða2 b2 þ ðg þ hÞðg þ 3hÞÞx3m þ ghx4m þ x3f ððhða2 b2 þ ðg þ hÞð3g þ hÞÞÞ þ 2ðg2 þ 4gh þ h2 Þxm Þ þ hyf2 ð2ððb2 gÞ 2a2 h þ gðc þ g þ hÞðc þ g þ hÞÞ þ gyf2 Þ þ yf ða4 2a2 b2 þ b4 2b2 g2 2c2 g2 þ g4 þ 8gða2 b2 c2 þ g2 Þh 2ðb2 þ c2 7g2 Þh2 þ 8gh3 þ h4 þ 2ðg2 þ 4gh þ h2 Þyf2 Þym þ 2ðgð2a2 g ðb2 þ c2 Þh þ hðg þ hÞ2 Þ
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þð2g2 þ 7gh þ 2h2 Þyf2 Þym2 þ 2ðg2 þ 4gh þ h2 Þyf ym3 þ ghym4 þ x2f ððððg þ hÞð3g þ hÞðg þ 4hÞÞ a2 ð3g þ 4hÞ þ b2 ð3g þ 4hÞÞxm þ 2ð2g2 þ 7gh þ 2h2 Þx2m þ 2hð2a2 ðg þ hÞ þgðb2 c2 þ g2 þ 2gh þ h2 Þ þ gyf2 Þ þ 2ðg2 þ 4gh þ h2 Þyf ym 2ð2g2 þ 3gh þ 2h2 Þym2 Þ þxm ððg3 a2 ðg 4hÞ þ b2 ðg 4hÞ þ 5gh2 þ 4h3 Þyf2 2ða2 ð4g þ hÞ b2 ð4g þ hÞ þðg þ hÞð4g2 þ 7gh þ h2 ÞÞyf ym gða2 b2 þ ðg þ hÞðg þ 3hÞÞða2 c2 þ ym2 ÞÞ þ2x2m ðð2g2 3gh 2h2 Þyf2 þ ðg2 þ 4gh þ h2 Þyf ym þ gð2a2 ðg þ hÞ þhðb2 c2 þ g2 þ 2gh þ h2 Þ þ hym2 ÞÞ þ xf ððððg þ hÞð4g þ hÞðg þ 3hÞÞ a2 ð4g þ 3hÞ þb2 ð4g þ 3hÞÞx2m þ 2ðg2 þ 4gh þ h2 Þx3m hða2 b2 þ ðg þ hÞð3g þ hÞÞða2 c2 þ yf2 Þ 2ða2 ðg þ 4hÞ b2 ðg þ 4hÞ þ ðg þ hÞðg2 þ 7gh þ 4h2 ÞÞyf ym þ ð4g3 þ a2 ð4g hÞ þ5g2 h h3 þ b2 ð4g þ hÞÞym2 þ xm ða4 þ b4 2b2 g2 2c2 g2 þ g4 8gðb2 þ c2 g2 Þh 2ðb2 þ c2 7g2 Þh2 þ 8gh3 þ h4 þ a2 ð2b2 þ 4ðg þ hÞ2 Þ þ 2ðg2 þ 4gh þ h2 Þyf2 þ8ð2g þ hÞðg þ 2hÞyf ym þ 2ðg2 þ 4gh þ h2 Þym2 ÞÞÞg;
ðA:3Þ
k3 ¼ f16ððghðh 2xm Þyf3 Þ ðhða2 b2 þ 4gh þ h2 þ 2gxf Þ 2ðg2 þ 4gh þ h2 Þxm Þyf2 ym þ ym ð2ghx3f þ x2f ðhða2 b2 þ ð2g þ hÞ2 Þ 2ðg2 þ 4gh þ h2 Þxm Þ þ g2 ð2a2 xm þ hx2m þ hða2 c2 þ ym2 ÞÞ 2gxf ðða2 þ b2 g2 4gh 2h2 Þxm þ hx2m þ hða2 b2 c2 þ g2 þ 2gh þ h2 þ ym2 ÞÞÞ þ yf ððghx2f ðh 2xm ÞÞ þ 2xf ða2 h2 hða2 b2 þ 2g2 þ 4gh þ h2 Þxm þ ðg2 þ 4gh þ h2 Þx2m ðg2 þ 4gh þ h2 Þym2 Þ 2
þ gðða2 þ c2 Þh2 ða2 b2 þ ðg þ 2hÞ Þx2m þ 2hx3m þ ða2 b2 þ gðg þ 4hÞÞym2 þ 2hxm ða2 b2 c2 þ g2 þ 2gh þ h2 þ ym2 ÞÞÞÞg;
ðA:4Þ
k4 ¼ f16ðghx3f ðh 2xm Þ þ g2 hx3m þ ða gÞða þ gÞh2 yf2 2ghyf ða2 b2 c2 þ g2 þ 3gh þ h2 þ yf2 Þym ðg2 ða2 þ h2 Þ þ ðg2 þ 6gh þ h2 Þyf2 Þym2 2ghyf ym3 þ x2m ððg2 ða2 þ h2 ÞÞ þ ðg2 þ 6gh þ h2 Þyf2 2ghyf ym Þ þ x2f ðða2 g2 Þh2 þ hða2 b2 þ 4g2 þ 6gh þ h2 Þxm ðg2 þ 6gh þ h2 Þx2m 2ghyf ym þ ðg2 þ 6gh þ h2 Þym2 Þ þ xf ðgðða2 b2 þ g2 þ 6gh þ 4h2 Þx2m 2hx3m þ h2 ða2 c2 þ yf2 Þ 2hxm ða2 b2 c2 þ g2 þ 3gh þ h2 þ yf2 ÞÞ þ 2ðhða2 b2 þ 2g2 þ 6gh þ h2 Þ 2ðg2 þ 6gh þ h2 Þxm Þyf ym gða2 b2 þ g2 þ 6gh þ 2hxm Þym2 Þ þ xm ððhða2 b2 þ 6gh þ h2 Þyf2 Þ þ 2gða2 b2 þ g2 þ 6gh þ 2h2 Þyf ym þ g2 hða2 c2 þ ym2 ÞÞÞg; k5 ¼ f64ghððg 2xf Þðh xm Þxm yf ðh 2xm Þððg xf Þxf þ yf2 Þym þ ðg 2xf Þyf ym2 Þg;
ðA:5Þ ðA:6Þ
k6 ¼ f64ghðxm ðh þ xm Þððg xf Þxf þ yf2 Þ ðg 2xf Þðh 2xm Þyf ym ððg xf Þxf þ yf2 Þym2 Þg: ðA:7Þ
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813
References [1] O. Bottema, B. Roth, Theoretical Kinematics, Dover Publications Inc., New York, 1990. [2] C.L. Collins, J.M. McCarthy, The quartic singularity surface of planar platforms in the Clifford algebra of the projective plane, Mech. Mach. Theory 33 (1998) 931–944. [3] B. Dasgupta, T.S. Mruthyunjaya, The Stewart platform manipulator: a review, Mech. Mach. Theory 35 (2000) 15– 40. [4] J.C. Faugere, D. Lazard, Combinatorial classes of parallel manipulators, Mech. Mach. Theory 30 (1995) 765–776. [5] C.M. Gosselin, J. Sefrioui, M.J. Richard, Polynomial solutions to the direct kinematic problem of planar three degree-of-freedom parallel manipulators, Mech. Mach. Theory 27 (1992) 107–119. [6] C.M. Gosselin, J.P. Merlet, The direct kinematics of planar parallel manipulators: special architectures and number of solutions, Mech. Mach. Theory 29 (1994) 1083–1097. [7] M.J.D. Hayes, M.L. Husty, P.J. Zsombor-Murray, Solving the forward kinematics of a planar three-legged platform with holonomic higher pairs, ASME J. Mech. Des. 121 (1999) 212–219. [9] M.L. Husty, Kinematic Mapping of planar three-legged platforms, Proceedings of 15th Canadian Congress of Applied Mechanics, CANCAM 1995, vol. 2, 1995. pp. 876–877. [10] M.L. Husty, An algorithm for solving the direct kinematics of general Stewart–Gough platforms, Mech. Mach. Theory 31 (1996) 365–380. [11] J.M. McCarthy, Introduction to Theoretical Kinematics, The MIT Press, Cambridge, 1990. [12] J.P. Merlet, C.M. Gosselin, N. Mouly, Workspaces of planar parallel manipulators, Mech. Mach. Theory 33 (1998) 7–20. [13] A.P. Murray, F. Pierrot, P. Duachez, J.M. McCarthy, On the design of parallel manipulators for a prescribed workspace, in: 5th International Symposium on Advances in Robotics, Portoroz-Bernardin, Slovenia, June 1996, pp. 22–26. [14] M. Raghavan, The Stewart platform of general geometry has 40 configurations, ASME J. Mech. Des. 115 (1993) 277–282. [15] J. Sefrioui, C.M. Gosselin, On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators, Mech. Mach. Theory 30 (1995) 533–551. [16] V. Snyder, C. Sisam, Analytic Geometry of Space, Holt and Co., New York, 1941. [17] C.M. Wampler, Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates, Mech. Mach. Theory 31 (1996) 331–337.
Forward kinematics of planar parallel manipulators in the Clifford algebra of P2 Curtis L. Collins Bourns College of Engineering, University of California, Riverside, CA 92521, USA Received 2 October 2000; accepted 6 August 2001
Abstract This paper presents the forward kinematics of planar platform manipulators composed of three RPR chains. The Clifford Algebra of projective two space is used to represent planar displacements. Also known as planar quaternions, the representation allows us to formulate the kinematic constraints in an algebraic form with rich geometric content. The forward kinematics problem reduces to the intersection of three circular hyperboloids. An algorithm for parameterizing the forward kinematics solution is presented. The general kinematics function is shown to yield six solutions. Furthermore, it is shown that real solutions only exist if the roots of a certain polynomial are on the interval from 1 to 1. A series of platform architectures is then studied to show how the kinematics algorithm specializes in these cases. Degenerate architectures are also identified. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Parallel manipulator; Forward kinematics; Planar quaternions
1. Introduction A planar parallel manipulator is formed when two or more planar kinematic chains act together on a common rigid platform. The most common planar platform architecture is composed of three RPR chains, where the notation RPR denotes the planar chain made up of a revolute joint, a prismatic joint, and a second revolute joint in series. The spatial analog to the planar platform, known as the Stewart–Gough platform, is composed of six SPS spatial chains, where the notation SPS denotes the spatial chain made up of a spherical joint, a prismatic joint, and a spherical joint in series.
E-mail address: [email protected] (C.L. Collins). 0094-114X/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 2 3 - X
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The forward kinematics of general parallel manipulators is a well-established problem. The general spatial parallel manipulator has been shown to have up to 40 solutions using elimination methods in both Cartesian coordinates [14] and soma coordinates [10,17]. The planar parallel manipulator has been studied using Cartesian coordinates and has at most six solutions [5]. A number of special architectures have been identified [6]. General planar parallel manipulators have also been studied using algebraic coordinates [9]. In this paper, we seek to elucidate the special architectures of planar parallel manipulators and the operational use of planar quaternion coordinates using a blend of algebra and geometry. We are motivated by our recent work studying the singularity surfaces of planar parallel manipulators [2] which allowed us to organize special manipulator architectures based on the geometric features of their singularities, and to further demonstrate the use of kinematic mappings in the study of manipulator systems. A number of references are available that discuss the mathematical and geometric construction of these mappings [1,7,11]. A solid understanding of the forward kinematics problem is a necessary first step in developing algorithms for parallel manipulator design. Algebraic coordinates provide a powerful mathematical machinery for these algorithms [13]. In addition, the parameter variation of the actuator lengths define the workspace of the manipulator through the forward kinematics mapping. Recent studies have focused on algorithms to define the boundaries of different types of workspace projections [12]. Dasgupta and Mruthyunjaya [3] have identified the need for more focused efforts in these areas. The paper is organized as follows. First, we review the construction and manipulation of planar quaternions and define the constraint manifold imposed by an RPR chain. Next we introduce our forward kinematics algorithm as a blend of geometric and algebraic arguments and discuss its features. Then, we discuss a variety of special parallel manipulator architectures and the effect of their geometric properties on the forward kinematics function. We also identify a number of degenerate cases where the general algorithm becomes singular.
2. Planar quaternions and constraint manifolds A transformation of coordinates in R3 can be defined such that it is a rigid displacement in the plane Z ¼ 1. This is the well-known 3 3 homogeneous transform: 8 9 2 38 9 cos h sin h dx < x =
ð2Þ T
T
where z ¼ ðx; yÞ and Z ¼ ðX ; Y Þ are the coordinate vectors of points in the fixed and moving frame, M and F respectively, in the plane Z ¼ 1. The upper left 2 2 matrix AðhÞ defines the T planar rotation, and the vector d ¼ ðdx ; dy Þ is the translation vector which defines the location of M relative to F.
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The pole of the displacement is the eigenvector associated with the eigenvalue k ¼ 1 of [T], and is used to identify a planar displacement with an element of the even Clifford algebra of P 2 . The resulting element can be viewed as a four-vector and is also known as a planar quaternion (see [11] for details of this construction). The planar quaternion corresponding to a displacement ½T ¼ ðAðhÞ; dÞ is given by: 8 9 8 9 1 q1 > > ðdx cos h2 þ dy sin h2Þ > > > > 2 > > > > >1 > = > > < < q2 ðdy cos h2 dx sin h2Þ = 2 P¼ : ð3Þ ¼ q3 > > > > sin h2 > > > > > > > > > ; : q4 > ; : cos h2 Notice that in order to represent a displacement, the components of the planar quaternion P ¼ ðq1 ; q2 ; q3 ; q4 ÞT , the third and fourth components must satisfy the relation: q23 þ q24 ¼ 1:
ð4Þ
When (4) is satisfied, the four planar quaternion parameters can be used to define the homogeneous matrix in Eq. (1) as 2 2 3 q4 q23 2q3 q4 2ðq1 q4 q2 q3 Þ ½T ¼ 4 2q3 q4 q24 q23 2ðq1 q3 þ q2 q4 Þ 5: ð5Þ 0 0 1 Eqs. (3) and (5) provide the means to transform from planar displacements to planar quaternion coordinates and back again. The composition rules of the Clifford algebra elements naturally allow us to concatenate displacements using polynomial multiplication. The vector interpretation of planar quaternions allows us to view planar displacements as points in a four-dimensional projective space, called the image space (see [1] for an alternate derivation of the image space coordinates and a discussion of its geometric properties). In this space, planar displacements correspond to lines through the origin. Associated with the constraint equation (4) are two points on each of these lines. From the quadratic nature of [T] in Eq. (5) it can be seen that the vectors q and q define the same planar displacement. Because of this association, the set of planar quaternions is a double covering of the set of planar displacements. Now consider the planar platform manipulator supported by three RPR chains as shown in Fig. 1. Let the location of the three fixed pivots be specified by ui ¼ ðui ; vi ÞT , i ¼ 1; 2; 3 measured in F and let the three moving pivots be zi ¼ ðki ; li ÞT measured in M. Coordinates in the platform M are related to coordinate in F by the 3 3 homogeneous transformation (1), that is Zi ¼ ½T zi . Let the distance between the ith fixed and moving pivots be denoted qi so we have the relations: Qi : hðui ½T zi Þ; ðui ½T zi Þi ¼ q2i ;
i ¼ 1; 2; 3:
ð6Þ
Substituting (5) into (6), we obtain three quadratic equations in planar quaternion coordinates which define 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 the image space of planar quaternions. This is called the constraint manifold of the manipulator.
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Fig. 1. General planar parallel manipulator with 3-RPR chains.
The constraint manifold of a single RPR chain is given by the quadric equation in four homogeneous coordinates q ¼ ðq1 ; q2 ; q3 ; q4 ÞT , aq21 þ bq22 þ cq23 þ dq24 þ 2fq2 q3 þ 2gq1 q3 þ 2hq1 q2 þ 2lq1 q4 þ 2mq2 q4 þ 2nq3 q4 ¼ where the coefficients are given by 9 8 9 8 1 a > > > > > > > > > > > > > > 1 > > > > b > > > > > > > 2 2 > 1 > > > > > > > > ½ðk þ uÞ þ ðl þ vÞ c > > > > 4 > > > > > > > 2 2 > 1 > > > > > > > > ½ðk uÞ þ ðl vÞ d > > > > 4 > >
q2 ; 4
ð7Þ
ð8Þ
The three equations (6) when expanded as above are equivalent to those used by Sefrioui and Gosselin [15] to study general planar platforms. The difference lies only in our use of planar quaternion coordinates. It can be shown [1,11] that the above quadratic equation defines a circular hyperboloid. As the locations of the fixed and moving pivots changes, the hyperboloids remain circular, but become skewed. It is important to note that the points J1 ¼ ð1; i; 0; 0Þ and J2 ¼ ð1; i; 0; 0Þ satisfy Eq. (7). These points are imaginary and are invariants in the geometry of the image space. They belong to the quadrics, but do not represent a planar displacement since they do not satisfy the constraint equation (4). Using planar quaternions, or the Clifford Algebra of P 2 , the positions and orientations of an RPR linkage is represented as a circular hyperboloid written in the coordinates q1 , q2 , q3 and q4 . When three RPR linkages come together to form a planar platform manipulator, the resulting kinematics is defined by the intersections of three circular hyperboloids.
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3. Forward kinematics solutions 3.1. Geometric arguments The foundation of our forward kinematics algorithm comes from the geometry of quadrics [16], although it can be argued from a purely algebraic standpoint as well. A well known result from algebraic geometry is that three quadric surfaces intersect in eight points. This is a direct result of the application of Bezout’s theorem. We seek to further clarify the nature of these solutions and relate them to assembly configurations of planar platform manipulators. Geometrically, any linear combination of two quadrics defines the curve of intersection of the quadrics. That is, any quadric of the form Q ¼ QA þ kQB passes through the intersection of QA and QB . Q is known as a pencil of quadrics. Consider the quadric QAB ¼ QB QA . QAB passes through the curve of intersection of QA and QB . Next, consider the quadric QAC ¼ QC QA which passes through the curve of intersection of of QA and QC . QAB and QAC intersect in their own curve, but geometrically, this curve must intersect QA in exactly the eight points of intersection of QA , QB , and QC . The quadrics defining our constraint manifolds are circular hyperboloids, and thus two of the points of intersection of three of these special quadrics are the imaginary points at infinity (J1 and J2 ), thus leaving only six other possible real solutions. We now present the analytic formulation of our geometric argument. 3.2. Analytic solution The fixed and moving reference frames for any planar platform can always be assigned such that the fixed frame origin is located at one of the fixed revolute joints while its x-axis is along the line between two of the fixed pivots. Similarly, the moving frame origin can be located at one of the moving revolute joints while its x-axis is along the line between two of the moving pivots corresponding to the fixed pivots used to assign the fixed frame (see Fig. 1 for example). Let the lengths of the three prismatic actuators be a, b and c. Let the coordinates of the fixed and moving pivots of actuator 2 be defined by u2 ¼ ðg; 0ÞT and z2 ¼ ðh; 0ÞT ; and actuator 3 by u3 ¼ ðxf ; yf ÞT T and z3 ¼ ðxm ; ym Þ . These parameters will now be used to define the three quadrics QA , QB , and QC . Substituting into Eq. (6) we find that quadric QA has the form q21 þ q22 14a2 q23 14a2 q24 ;
ð9Þ
quadric QB has the form 2
2
q21 þ q22 þ ðg þ hÞq2 q3 þ ðg þ hÞq1 q4 þ 14ððg þ hÞ b2 Þq23 þ 14ððg hÞ b2 Þq24 ;
ð10Þ
and quadric QC has the form q21 þ q22 þ ðxf þ xm Þq2 q3 ðxf þ xm Þq1 q4 ðyf þ ym Þq1 q3 þ ðyf þ ym Þq2 q4 þ ðxm yf þ xf ym Þq3 q4 þ 14ððxf þ xm Þ2 þ ðyf þ ym Þ2 c2 Þq23 þ 14ððxf þ xm Þ2 þ ðyf þ ym Þ2 c2 Þq24 : Note that we have homogenized the equations by applying the constraint equation (4).
ð11Þ
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Taking QAB ¼ QB QA and QAC ¼ QC QA we find that QAB has the form ðg þ hÞq2 q3 þ ðg þ hÞq1 q4 þ 14ðða2 b2 Þ þ ðg þ hÞ2 Þq23 þ 14ðða2 b2 Þ þ ðg hÞ2 Þq24 ;
ð12Þ
and QAC has the form q1 ðq4 ðxf þ xm Þ þ q3 ðyf ym ÞÞ þ q2 ðq3 ðxf þ xm Þ þ q4 ðyf þ ym ÞÞ þ q3 q4 ððxm yf Þ þ xf ym Þ þ 14ða2 c2 þ ðxf þ xm Þ2 þ ðyf þ ym Þ2 Þq23 þ 14ða2 c2 þ ðxm xf Þ2 þ ðym yf Þ2 Þq24 :
ð13Þ
Note that the quadrics QAB and QAC are linear in the variables q1 and q2 . Solving for q1 and q2 gives two equations in q3 and q4 which we will denote f1 ðq3 ; q4 Þ and f2 ðq3 ; q4 Þ respectively. The denominators, d, of f1 ðq3 ; q4 Þ and f2 ðq3 ; q4 Þ are equal and can be written as d ¼ 2q4 q3 ðhxf gxm Þ þ ðg þ hÞðyf þ ym Þ 2q24 ðhyf þ gym Þ:
ð14Þ
By applying the constraint q23 þ q24 1 ¼ 0 we find a parameterization of the curve of intersection of QAB and QAC of the form q1 ¼ f1 ðq3 ; q4 Þ; q2 ¼ f2 ðq3 ; q4 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffi q3 ¼ 1 q24 ;
ð15Þ
q4 ¼ q4 : Substituting this parametrization into QA gives an irrational function of q4 . The denominator of this function is simply d 2 . The roots of the numerator of this function lead to the forward kinematics solutions for the planar platform manipulator by substituting the values for q4 back into the original parametrization Eq. (15). The numerator, which is referred to here as the forward kinematics function, has the form qffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ k0 þ k2 q24 þ k4 q44 þ k6 q64 þ ðk1 q4 þ k3 q34 þ k5 q54 Þ 1 q24 ¼ 0; where ki are functions of the structural parameters and the link lengths. Simplified expressions for ki can be found in Appendix A. Eq. (16) defines the consistency condition that holds when the curve represented in (15) intersects QA . When expanded out, Eq. (16) contains 795 individual terms. We can clear the square root terms by bring the terms to the right hand side, squaring them, and then bringing them back. The result is the 12th order even polynomial, k02 ðk12 2k0 k2 Þq24 þ ðk12 þ k22 2k1 k3 þ 2k0 k4 Þq44 ðk32 2k1 ðk3 k5 Þ 2ðk2 k4 þ k0 k6 ÞÞq64 þ ðk32 þ k42 þ 2k1 k5 2k3 k5 þ 2k2 k6 Þq84 2 2 12 þ ð2k3 k5 k52 þ 2k4 k6 Þq10 4 þ ðk5 þ k6 Þq4 ;
ð17Þ
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whose roots occur in pairs. This is equivalent to solving a 6th order polynomial. The 12 roots of this polynomial will lead to 12 forward kinematics solutions. Closer examination of these solutions reveals that only six of them are unique, the other six being the negatives of the first set; a consequence of the double covering of planar quaternions. It is for this reason that only the positive solution for q3 appears in the parameterization (15). Another interesting and useful fact about our solution method, is that only roots of q4 in the range 1 6 q4 6 1 will result in real forward kinematics solutions. Indeed, you can only plot the function on this interval, otherwise the function is complex. Thus, when we solve for the roots numerically, we only have to solve for roots in the interval ½1; 1 to find all the real solutions. 3.3. Numerical example As an example and verification of the method, we choose to reproduce a result reported in [6]. While their notation is slightly different, we find the following system of parameters for comparison: a ¼ 14:89;
b ¼ 15:38;
g ¼ 15:91;
h ¼ 17:04;
xf ¼ 0;
yf ¼ 10;
c ¼ 12;
xm ¼ 13:2364;
ym ¼ 16:0967:
Substituting these parameters into our forward kinematics expression in q4 given by Eq. (16) gives the six degree irrational function of q4 : 3:9852 108 1:0039 109 q24 þ 7:2338 108 q44 1:17775 108 q64 qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 3 9 5 þ ð8:8575 10 q4 þ 2:0091 10 q4 1:1177 10 q4 Þ 1 q24 :
ð18Þ
Solving for the roots of this equation and substituting the values into the parameterization given in Eq. (15) gives the results shown in Table 1. Finally applying Eq. (5) allows us to solve for the position and orientation of the moving platform in each position. The results are provided in Table 2. These are exactly the values reported in [6]. A rendering of the planar platform in each of these configurations is shown in Fig. 2.
Table 1 Forward kinematics solutions in planar quaternion form Solution
q4
q3
q2
q1
1 2 3 4 5 6
0.8807 0.9997 0.9925 0.9574 0.8771 0.4832
0.4737 0.0237 0.1223 0.2887 0.4803 0.8755
3.2946 7.0308 1.6968 1.2495 4.1670 7.1513
6.7265 2.5822 7.2953 7.3850 6.2219 2.2267
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Table 2 Forward kinematics solutions in Cartesian form Solution
h
dx
dy
1 2 3 4 5 6
0.9870 0.0473 0.2453 0.5857 1.0020 2.1329
8.7266 5.4957 14.8961 13.4199 14.9201 14.6739
12.1757 13.9355 1.5830 6.6563 1.3379 3.0126
Fig. 2. Example forward kinematics solutions.
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4. Special platform architectures Motivated by the results of Collins and McCarthy [2] where singularity surfaces degenerated when the kinematic architecture assumed special forms, we apply these forms to the forward kinematics problem. This work expands on the results of Gosselin and Merlet [6] and demonstrates the advantages of working with the algebraic representation afforded by using planar quaternions. The notion of special architectures of spatial platform manipulators has been studied by Faugere and Lazard [4] where their primary focus was on classes of manipulators with different combinations of coincident spherical joints. As we will show, there are other geometric features that are useful to identify and consider. We have studied the following cases: Proportional case. The planar platform architecture is characterized by having proportional fixed and moving pivot locations. That is, the base and top platforms are similar triangles (see Fig. 3). In-Line case. The platform architecture is characterized by having colinear fixed and moving pivots (see Fig. 3). Coincident pivots case. The platform architecture is characterized by having either a pair of fixed or moving pivots coincident (see Fig. 4).
Fig. 3. Proportional and in-line platforms.
Fig. 4. Platforms with coincident fixed or moving pivots.
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Fig. 5. In-line platforms with coincident fixed or moving pivots.
Combination cases. These architectures are characterized by combinations of the previous cases, such as the in-line platforms with proportional pivot locations (see Fig. 3), and in-line platforms with different combinations of coincident fixed and moving pivots as shown in Fig. 5. These will be discussed in the context of the other cases. Degenerate cases. When the denominator, d, of the parameterization in Eq. (15) becomes zero, the algorithm fails to compute the forward kinematics solutions. Four classes of architectures have been identified that lead to this degenerate condition. In what follows, the coefficients ki are simply the specializations of the ki found in Eq. (16) and listed in general form in Appendix A. Coefficients li are new coefficients found from factoring the simplified polynomials in ki . 4.1. Proportional architectures We can express the condition of proportionality by requiring that the ratio of the coordinates of the fixed and moving pivots be equal for each actuator pair (Fig. 3). This can be written as: h xm ym ¼ ¼ t: ¼ yf g xf
ð19Þ
The forward kinematics function given by Eq. (16) reduces to a 6th order even polynomial, k0 þ k2 q24 þ k4 q44 þ k6 q64 ¼ 0;
ð20Þ
that factors into a quadratic and an even quartic polynomial ð1 þ 2t 4q24 t þ t2 Þðl0 þ l2 q24 þ l44 Þ ¼ 0:
ð21Þ
The quadratic term only depends on the constant of proportionality. It turns out that the values for q4 found from factoring this term are imaginary and the solutions degenerate since the denominator also contains this factor. The quartic term is an even polynomial and can be solved making the substitution p ¼ q24 . The resulting quadratic can be solved in closed form.
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4.2. In-line architectures By requiring that the revolute joints in the fixed base and moving platform to be colinear we can define the in-line platforms (Fig. 3) using the condition yf ¼ ym ¼ 0:
ð22Þ
In this case, the forward kinematics Eq. (16) reduces to a 6th order even polynomial, k0 þ k2 q24 þ k4 q44 þ k6 q64 ¼ 0:
ð23Þ
It does not factor in general. As in the proportional case, we can apply the substitution, p ¼ q24 , and solve the resulting cubic equation in closed form. 4.3. Coincident pivot architectures The conditions for having coincident fixed pivots is fyf ¼ 0; g ¼ 0g
or fg ¼ xf g:
ð24Þ
The conditions for having coincident moving pivots is fym ¼ 0; h ¼ 0g
or fh ¼ xm g:
In both cases (Fig. 4) the forward kinematics equation (16) takes on the form qffiffiffiffiffiffiffiffiffiffiffiffiffi k0 þ k2 q24 þ k4 q44 þ ðk1 q4 þ k3 q34 Þ 1 q24 ¼ 0;
ð25Þ
ð26Þ
which is a 4th order function of q4 . Similar to the general case, we can convert this equation to an 8th order even polynomial. Then by applying the substitution p ¼ q24 , the problem reduces to the solution of a quartic equation. By further constraining the pivots to be colinear (Fig. 5), we have the special cases of in-line platforms with various combinations of coincident fixed or moving pivots. In any of these cases, the forward kinematics equation has the form k0 þ k2 q24 þ k4 q44 ¼ 0;
ð27Þ
which is a 4th order even polynomial. It is equivalent to solving a quadratic equation in p. 4.4. Degenerate architectures The geometric elimination algorithm and parameterization degenerates when the denominator, d, in Eq. (14) becomes zero for all ðq3 ; q4 Þ pairs. For this to be true, each coefficient must become zero simultaneously. This occurs in four distinct sets of architectures, two of which are singular.
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The singular architectures satisfy the conditions fg ¼ 0; h ¼ 0g
or fxf ¼ 0; xm ¼ 0; yf ¼ 0; ym ¼ 0g;
ð28Þ
and fg ¼ 0; xf ¼ 0; yf ¼ 0; ym ¼ 0g or fh ¼ 0; xm ¼ 0; yf ¼ 0; ym ¼ 0g: ð29Þ Geometrically, the architectures in Eq. (28) have two RPR joints with identical fixed and moving pivots. The architectures in Eq. (29) are characterized by having three coincident fixed or moving pivots. These architectures are singular since the lines defining the prismatic joints always intersect in a common point. Alternatively, it can be shown that the determinant of the Jacobian of the forward kinematics mapping is identically zero for all qi (see [2] or [15]). The non-singular architectures satisfy the conditions g ¼ h;
xf ¼ xm ;
yf ¼ ym ;
ð30Þ
yf ¼ 0;
ym ¼ 0;
h xm ¼ : g xf
ð31Þ
and
Geometrically, Eq. (30) is satisfied when the fixed and moving platforms are mirror images of each other. The forward kinematics function represented by Eq. (16) does not loose any terms in this case. Eq. (31) represents architectures that have colinear fixed and moving pivots whose locations are in proportion. We call these in-line proportional platforms. In this case. the forward kinematics function reduces to a 6th order even polynomial and factors into a quadric and a double quadric. ð1 þ 2t 4q24 t þ t2 Þðl0 þ l2 q24 Þ2 ¼ 0:
ð32Þ
This gives at most two real double solutions. It is important to note that while our geometric parameterization of the forward kinematics solutions degenerates in these two nonsingular cases, we can still solve for the assembly configurations using standard elimination and substitution methods. 5. Conclusions The forward kinematics of the general planar parallel manipulator has been presented along with a numerical example for verification purposes. The methodology utilizes planar quaternions which allow us to develop kinematics equations in a four-dimensional space of algebraic coordinates. The geometry of the constraint imposed by an RPR chain appears in this space as a circular hyperboloid. The three RPR chains that make up a planar parallel manipulator correspond to three circular hyperboloids whose intersections define the forward kinematics solutions. The geometry of quadrics allows us to construct a geometric and analytic solution that results in a 6th order irrational function of a single coordinate q4 . Real solutions to the problem lie on the interval 1 6 q4 6 1. The geometry of the platform architecture dramatically effects the characteristics of the kinematics function. As the revolute joints attached to the base and platform become proportional, linear, or coincident, the forward kinematics equation reduces in degree, or
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degenerates. In all cases, the forward kinematics solution reduces to finding the roots of an even polynomial in q4 .
Appendix A k0 ¼ fða cÞ2 ða þ cÞ2 ðg þ hÞ2 þ ðg þ hÞ2 x4f 2ðg þ hÞða2 b2 þ g2 þ 2gh þ h2 Þx3m þ ðg þ hÞ2 x4m 2ðg þ hÞx3f ða2 b2 þ g2 þ 2gh þ h2 2ðg þ hÞxm Þ 2
2ðg þ hÞða2 b2 þ g2 þ 2gh þ h2 Þxm ðða cÞða þ cÞ þ ðyf þ ym Þ Þ þ ðyf þ ym Þ2 ða4 2a2 b2 þ b4 2b2 ðg þ hÞ2 þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ þ ðg þ hÞ2 ðyf þ ym Þ2 Þ þ x2m ða4 þ b4 2b2 ðg þ hÞ2 þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ þ a2 ð2b2 þ 4ðg þ hÞ2 Þ þ 2ðg þ hÞ2 ðyf þ ym Þ2 Þ þ 2xf ða2 b2 þ g2 þ 2gh þ h2 2ðg þ hÞxm Þðða2 b2 þ g2 þ 2gh þ h2 Þxm ðg þ hÞx2m ðg þ hÞðða cÞða þ cÞ þ ðyf þ ym Þ2 ÞÞ þ x2f ða4 þ b4 2b2 ðg þ hÞ2 þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ 2
2
þ a2 ð2b2 þ 4ðg þ hÞ Þ þ 2ðg þ hÞð3ða2 b2 þ ðg þ hÞ Þxm þ 3ðg þ hÞx2m þ ðg þ hÞðyf þ ym Þ2 ÞÞg;
ðA:1Þ
k1 ¼ f4ððhða2 b2 þ ðg þ hÞ2 Þ 2ðg þ hÞ2 xm Þyf3 þ ððg þ 2hÞða2 b2 þ ðg þ hÞ2 Þ þ 2ðg þ hÞ2 xf 4ðg þ hÞ2 xm Þyf2 ym þ ym ð2ðg þ hÞ2 x3f þ x2f ðð3g 2hÞ ða2 b2 þ ðg þ hÞ2 Þ þ 4ðg þ hÞ2 xm Þ þ xf ða4 þ b4 2b2 ðg þ hÞ2 þ a2 ð2b2 þ 4gðg þ hÞÞ þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ 2ð2g þ hÞða2 b2 þ g2 þ 2gh þ h2 Þ xm þ 2ðg þ hÞ2 x2m þ 2ðg þ hÞ2 ym2 Þ gð4a2 ðg þ hÞxm þ ða2 b2 þ ðg þ hÞ2 Þ x2m þ ða2 b2 þ ðg þ hÞ2 Þða2 c2 þ ym2 ÞÞÞ þ yf ðð2g þ 3hÞða2 b2 þ ðg þ hÞ2 Þ x2m 2ðg þ hÞ2 x3m þ x2f ðhða2 b2 þ ðg þ hÞ2 Þ 2ðg þ hÞ2 xm Þ þ ða2 b2 þ ðg þ hÞ2 Þðða cÞða þ cÞh þ ð2g þ hÞym2 Þ xm ða4 þ b4 2b2 ðg þ hÞ2 þ a2 ð2b2 þ 4hðg þ hÞÞ þ ðg þ hÞ2 ð2c2 þ ðg þ hÞ2 Þ þ 2ðg þ hÞ2 ym2 Þ 2xf ððg 2hÞða2 b2 þ ðg þ hÞ2 Þxm þ 2ðg þ hÞ2 x2m 2ðg þ hÞðða2 hÞ þ ðg þ hÞym2 ÞÞÞÞg;
ðA:2Þ
k2 ¼ f4ðða2 c2 Þ2 gh þ ghx4f gða2 b2 þ ðg þ hÞðg þ 3hÞÞx3m þ ghx4m þ x3f ððhða2 b2 þ ðg þ hÞð3g þ hÞÞÞ þ 2ðg2 þ 4gh þ h2 Þxm Þ þ hyf2 ð2ððb2 gÞ 2a2 h þ gðc þ g þ hÞðc þ g þ hÞÞ þ gyf2 Þ þ yf ða4 2a2 b2 þ b4 2b2 g2 2c2 g2 þ g4 þ 8gða2 b2 c2 þ g2 Þh 2ðb2 þ c2 7g2 Þh2 þ 8gh3 þ h4 þ 2ðg2 þ 4gh þ h2 Þyf2 Þym þ 2ðgð2a2 g ðb2 þ c2 Þh þ hðg þ hÞ2 Þ
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þð2g2 þ 7gh þ 2h2 Þyf2 Þym2 þ 2ðg2 þ 4gh þ h2 Þyf ym3 þ ghym4 þ x2f ððððg þ hÞð3g þ hÞðg þ 4hÞÞ a2 ð3g þ 4hÞ þ b2 ð3g þ 4hÞÞxm þ 2ð2g2 þ 7gh þ 2h2 Þx2m þ 2hð2a2 ðg þ hÞ þgðb2 c2 þ g2 þ 2gh þ h2 Þ þ gyf2 Þ þ 2ðg2 þ 4gh þ h2 Þyf ym 2ð2g2 þ 3gh þ 2h2 Þym2 Þ þxm ððg3 a2 ðg 4hÞ þ b2 ðg 4hÞ þ 5gh2 þ 4h3 Þyf2 2ða2 ð4g þ hÞ b2 ð4g þ hÞ þðg þ hÞð4g2 þ 7gh þ h2 ÞÞyf ym gða2 b2 þ ðg þ hÞðg þ 3hÞÞða2 c2 þ ym2 ÞÞ þ2x2m ðð2g2 3gh 2h2 Þyf2 þ ðg2 þ 4gh þ h2 Þyf ym þ gð2a2 ðg þ hÞ þhðb2 c2 þ g2 þ 2gh þ h2 Þ þ hym2 ÞÞ þ xf ððððg þ hÞð4g þ hÞðg þ 3hÞÞ a2 ð4g þ 3hÞ þb2 ð4g þ 3hÞÞx2m þ 2ðg2 þ 4gh þ h2 Þx3m hða2 b2 þ ðg þ hÞð3g þ hÞÞða2 c2 þ yf2 Þ 2ða2 ðg þ 4hÞ b2 ðg þ 4hÞ þ ðg þ hÞðg2 þ 7gh þ 4h2 ÞÞyf ym þ ð4g3 þ a2 ð4g hÞ þ5g2 h h3 þ b2 ð4g þ hÞÞym2 þ xm ða4 þ b4 2b2 g2 2c2 g2 þ g4 8gðb2 þ c2 g2 Þh 2ðb2 þ c2 7g2 Þh2 þ 8gh3 þ h4 þ a2 ð2b2 þ 4ðg þ hÞ2 Þ þ 2ðg2 þ 4gh þ h2 Þyf2 þ8ð2g þ hÞðg þ 2hÞyf ym þ 2ðg2 þ 4gh þ h2 Þym2 ÞÞÞg;
ðA:3Þ
k3 ¼ f16ððghðh 2xm Þyf3 Þ ðhða2 b2 þ 4gh þ h2 þ 2gxf Þ 2ðg2 þ 4gh þ h2 Þxm Þyf2 ym þ ym ð2ghx3f þ x2f ðhða2 b2 þ ð2g þ hÞ2 Þ 2ðg2 þ 4gh þ h2 Þxm Þ þ g2 ð2a2 xm þ hx2m þ hða2 c2 þ ym2 ÞÞ 2gxf ðða2 þ b2 g2 4gh 2h2 Þxm þ hx2m þ hða2 b2 c2 þ g2 þ 2gh þ h2 þ ym2 ÞÞÞ þ yf ððghx2f ðh 2xm ÞÞ þ 2xf ða2 h2 hða2 b2 þ 2g2 þ 4gh þ h2 Þxm þ ðg2 þ 4gh þ h2 Þx2m ðg2 þ 4gh þ h2 Þym2 Þ 2
þ gðða2 þ c2 Þh2 ða2 b2 þ ðg þ 2hÞ Þx2m þ 2hx3m þ ða2 b2 þ gðg þ 4hÞÞym2 þ 2hxm ða2 b2 c2 þ g2 þ 2gh þ h2 þ ym2 ÞÞÞÞg;
ðA:4Þ
k4 ¼ f16ðghx3f ðh 2xm Þ þ g2 hx3m þ ða gÞða þ gÞh2 yf2 2ghyf ða2 b2 c2 þ g2 þ 3gh þ h2 þ yf2 Þym ðg2 ða2 þ h2 Þ þ ðg2 þ 6gh þ h2 Þyf2 Þym2 2ghyf ym3 þ x2m ððg2 ða2 þ h2 ÞÞ þ ðg2 þ 6gh þ h2 Þyf2 2ghyf ym Þ þ x2f ðða2 g2 Þh2 þ hða2 b2 þ 4g2 þ 6gh þ h2 Þxm ðg2 þ 6gh þ h2 Þx2m 2ghyf ym þ ðg2 þ 6gh þ h2 Þym2 Þ þ xf ðgðða2 b2 þ g2 þ 6gh þ 4h2 Þx2m 2hx3m þ h2 ða2 c2 þ yf2 Þ 2hxm ða2 b2 c2 þ g2 þ 3gh þ h2 þ yf2 ÞÞ þ 2ðhða2 b2 þ 2g2 þ 6gh þ h2 Þ 2ðg2 þ 6gh þ h2 Þxm Þyf ym gða2 b2 þ g2 þ 6gh þ 2hxm Þym2 Þ þ xm ððhða2 b2 þ 6gh þ h2 Þyf2 Þ þ 2gða2 b2 þ g2 þ 6gh þ 2h2 Þyf ym þ g2 hða2 c2 þ ym2 ÞÞÞg; k5 ¼ f64ghððg 2xf Þðh xm Þxm yf ðh 2xm Þððg xf Þxf þ yf2 Þym þ ðg 2xf Þyf ym2 Þg;
ðA:5Þ ðA:6Þ
k6 ¼ f64ghðxm ðh þ xm Þððg xf Þxf þ yf2 Þ ðg 2xf Þðh 2xm Þyf ym ððg xf Þxf þ yf2 Þym2 Þg: ðA:7Þ
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