Analysis of the Inverse Kinematics Problem for 3-DOF Axis-SymmetricParallel Manipulators with Parasitic Motion

Mats Isaksson, Anders Eriksson and Saeid Nahavandi

Abstract— Determining an analytical solution to the inversekinematics problem for a parallel manipulator is typicallya straightforward problem. However, lower mobility parallelmanipulators with 2-5 degrees of freedom (DOFs) often sufferfrom an unwanted parasitic motion in one or more DOFs.For such manipulators, the inverse kinematics problem canbe significantly more difficult. This paper contains an analysisof the inverse kinematics problem for a class of 3-DOFparallel manipulators with axis-symmetric arm systems. Allmanipulators in the studied class exhibit parasitic motion inone DOF. For manipulators in the studied class, the generalsolution to the inverse kinematics problem is reduced to solvinga univariate equation, while analytical solutions are presentedfor several important special cases.

I. INTRODUCTION

The field of kinematics is the study of motion withoutconsidering the forces that cause the motion. Simply put, theinverse kinematics (IK) problem means finding the actuatedjoint angles of a manipulator, knowing the position andorientation of its end-effector (EEF). For an industrial robot,the main use of the IK is to calculate the joint angles corre-sponding to a programmed path (positions and orientations)of the EEF. In addition to that, the derived expressions areuseful for kinematic analysis of a manipulator, includingstudies of its isotropy and stiffness.

The section of the workspace where solutions to theIK exist is called the reachable workspace. The reachableworkspace is limited by the arm lengths of the manipulator,joint limitations and collisions. If a solution does exist, itmay not be unique. The IK could have multiple or eveninfinite solutions. The parallel manipulators (PMs) studiedhere are typically designed so that only one solution to theIK is valid. However, manipulators of the investigated typecould be designed with the possibility of utilising multipleIK solutions. In [1], this possibility is explored for a 6-DOFvariant of the manipulators studied here.

It is often loosely stated that the IK problem is complexfor a serial manipulator (SM) whereas straightforward for aPM. However, while this is often the case, several exceptionsexist. First of all, the IK for an SM does not have to becomplex. Consider for example a serial 6-DOF mechanismwhose base is attached to a 2-DOF joint with the samefunctionality as a universal joint, except that both joint axesare actuated. Assume this joint is connected to an actuated

M. Isaksson and S. Nahavandi are with the Centre for Intelli-gent Systems Research (CISR), Deakin University, Australia. E-mail:mats.isaksson@gmail.com, saeid.nahavandi@deakin.edu.au. A. Eriksson iswith the School of Computer Science, University of Adelaide, Australia.E-mail: anders.eriksson@adelaide.edu.au.

telescopic link, which in turn is connected to a manipu-lated platform via a 3-DOF joint with three concurrent andperpendicular joint axes that are all actuated (kinematicallyequivalent to a spherical joint). If the tool centre point (TCP)is assumed to be in the joint axes intersection point ofthe 3-DOF joint, the IK of this decoupled mechanism isstraightforward with only one possible solution. However,for a typical 6-DOF serial industrial robot, the manipulatorDOFs are not decoupled, leading to a more complex IKproblem. For such manipulators, the solution for each jointis typically dependent on the solution of other joints in thekinematic chain and can not be determined independently.Furthermore, such mechanisms typically feature multiple IKsolutions.

Similarly, the IK for a parallel robot is not always straight-forward. As most definitions of a parallel robot allows formore than one actuated joint per kinematic chain, a 6-DOFparallel robot may have as few as two kinematic chains andone of these chains may include five actuated joints. In thiscase, the complexity of the IK problem is similar to that ofa 5-DOF SM and hence more intricate. A redundant parallelrobot could be composed of two 6-DOF SMs connected atthe EEF, in which case the complexity of the IK problem ismore than double that of a 6-DOF SM.

For a fully parallel manipulator, each kinematic chain onlyfeatures one actuated joint. Hence, each joint coordinate cantypically be calculated independently of the other joints. De-riving an analytical solution to the IK for such manipulatorsis usually uncomplicated. However, as will be demonstratedin this paper, finding analytical solutions to the IK for fullyparallel manipulators may also be intricate. This is often thecase for lower mobility PMs, as mechanisms of this typecommonly suffer from an unwanted coupled motion of theEEF. Such motion was named parasitic motion in [2]. Dueto the parasitic motion, the position and orientation of theEEF is not completely known. Even though the unknownEEF coordinates are not required to be determined explic-itly, solving the IK problem for mechanisms with parasiticmotion typically involves calculating both the unknown EEFcoordinates and the coordinates of the actuated joints.

This paper provides a study of the IK problem for aclass of 3-DOF PMs with axis-symmetric arm systems. Theactuated upper arms of the investigated axis-symmetric PMshave a common axis of rotation, leading to equal manipulatorproperties in all radial half planes defined by this commonaxis. All manipulators in the investigated class suffer fromparasitic motion of the EEF in one DOF, which complicatesthe IK problem.

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The studied manipulator variants are introduced in SectionII. This class of manipulators includes all variants of theSCARA-Tau robot [3]. The IK of the original SCARA-Tauprototype was first presented in [4]. In [5], an alternativederivation of the IK for this manipulator was proposed anda solution for a SCARA-Tau variant with triangular linkconfiguration, proposed in [6], was introduced. However,both [4, 5] only consider the case when the TCP is equalto the position of a platform joint. Additional variants ofthe SCARA-Tau have been proposed in [7]. This paperpresents solutions to the IK problem for general SCARA-Tau variants with arbitrary choice of TCP. Additionally, theIK problem is solved for all variants of the SymmetricSCARA manipulator proposed in [7] and for another similarmechanism introduced in [8].

A table of abbreviations used throughout the paper isprovided in Table I.

II. 3-DOF AXIS-SYMMETRIC MANIPULATORS

Figure 1 shows six variants of the 3-DOF axis-symmetricPMs studied in this paper. All the investigated manipulatorscomprise a cylindrical base column and an axis-symmetricarm system. The arm system is composed of three actuatedupper arms with a common axis of rotation, a manipulatedplatform, and six 5-DOF or 6-DOF lower arm linkages(LALs) connecting the upper arms and the manipulatedplatform. The LALs are constructed by a fixed-length linkwith a 3-DOF spherical joint on one end and either anotherone of these joints or a 2-DOF universal joint on the otherend. All actuators are mounted on the fixed base column.As the links in such LALs are only susceptible to axialforces, they can have a light-weight construction, for exampleusing carbon fibre. The axis-symmetric arm system leads toan extensive positional workspace while the possibility forinfinite rotation of the arm system means the manipulatorscan always use the shortest path between two programmedpositions.

For the studied class of manipulators, two pairs of LALsare mounted to form parallelograms in separate planes par-allel to the common axis of rotation of the upper arms. Suchparallelograms are in the future text referred to as verticalparallelograms, even if that is only true if the common axisof rotation of the upper arms is vertical. The two verticalparallelograms keep two rotational DOFs of the manipulatedplatform constant in the entire workspace. The only possibleplatform rotation is around an axis parallel to the commonaxis of rotation of the upper arms. Ideally, this platformrotation should remain constant during vertical and radialmotions of the manipulated platform. However, this is not

TABLE ITABLE OF ABBREVIATIONS

IK Inverse kinematicsPM Parallel manipulatorSM Serial manipulatorTCP Tool centre pointEEF End-effectorLAL Lower arm linkage

00.5

11.5

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

1.5

x(m)y(m)

z(m

)

(a) Triangular 3/2/1

00.5

11.5

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

1.5

x(m)y(m)

z(m

)

(b) Quadrilateral 3/2/1

00.5

11.5 −1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1.5

y(m)x(m)

z(m

)(c) Parallel 2/2/2

00.5

11.5 −1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1.5

y(m)x(m)

z(m

)

(d) Quadrilateral 2/2/2

00.5

11.5 −1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1.5

y(m)x(m)

z(m

)

(e) Triangular 4/1/1

00.5

11.5 −1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1.5

y(m)x(m)

z(m

)

(f) Triangular 4/1/1

Fig. 1. 3-DOF axis-symmetric manipulators with different link clustering.For the manipulator in (a), all upper arm joints in the cluster of three linksare collinear while this is not the case in for the manipulator in (b). For themanipulators in (e) and (f), the upper arm joints in the cluster of four linksare collinear and the vertical distances between neighbouring joints on theupper arm are the same as the vertical distances between the correspondingjoints on the manipulated platform.

the case, meaning these manipulators suffer from a parasiticrotation of the manipulated platform. Parasitic rotation isfurther discussed in Section V-A.

The manipulators in Fig. 1(a) and (b) are variants of theSCARA-Tau manipulator [3]. For the original SCARA-Taumanipulator, the LALs in the cluster of three are parallel. TheSCARA-Tau variants in Fig. 1(a) and (b) were introduced in[6] and [7], respectively. They are identical to the originalSCARA-Tau manipulator except for a different position andorientation of the LAL in the cluster of three that is not partof the vertical parallelogram. Different arrangements of thisLAL may be used to minimise the parasitic rotation of theplatform or to maximise the manipulator isotropy [7, 8]. Thenames Triangular and Quadrilateral are based on the shapes

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formed by a projection of the cluster of three links in the x-yplane. Other variants of the SCARA-Tau manipulator wereproposed in [7, 8].

The manipulators in Fig. 1(c) and (d) are two variants ofthe Symmetric SCARA mechanism introduced in [7]. Oneadvantage of the 2/2/2 clustering compared to the 3/2/1 clus-tering of the SCARA-Tau is a more equal force distributionin the LALs. Both the Symmetric SCARA and the SCARA-Tau may also be designed with a collinear arrangement ofthe platform joints of the two vertical parallelograms [7].

The possibility of utilising a 4/1/1 clustering of the LALswere discussed in [8]. The manipulators in Fig. 1(e) and (f)are two examples of such manipulators. The 4/1/1 clusteringtypically leads to a small angle between the two planescontaining the vertical parallelograms and the stiffness forsuch variants must be carefully evaluated. Possibilities tomaximise this stiffness (e.g. by using a wider manipulatedplatform) are discussed in [8]. Even though the 4/1/1 con-figurations and the 3/2/1 configurations have very differentmanipulator properties, their IK solutions are equivalent.Hence, the IK solutions for the 4/1/1 mechanisms are notexplicitly discussed in this paper.

For all the illustrated manipulator variants, the linkagesin both parallelograms form equal angles with a horizontalplane; however, this is not a necessary condition and otherarrangements are also possible [8].

III. NOTATION

A. Bodies

The bodies and kinematic parameters of a 3/2/1 variantof the studied mechanisms are shown in Fig. 2. For a 2/2/2configuration, the notations U13, P13, L13, a13, l13, h13 arereplaced by U32, P32, L32, a32, l32, h32.

The cylindrical base column B is connected to severalupper arms Ai via 1-DOF revolving joints Ri. These jointsare actuated with a common axis of rotation in the centre ofthe base column B. The upper arms Ai are not necessarilyhorizontal, which makes it possible to minimise the height ofthe base cylinder. The index i is determined by the orderingof the upper arms, where i = 1 for the upper arm containingthe lowest upper arm joint. If the lowest upper arm joints oftwo upper arms are mounted at equal height, the upper armwith a ‘right-arm’ solution is given the lower value of i.

Each upper arm Ai is connected via a joint Ui j to a lowerarm link Li j. The index j is used to differ between joints onthe same upper arm. The index is determined by the height ofthe upper arm joints and the lowest joint on each upper armis assigned j = 1. If two joints on one upper arm are mountedat the same height, the lowest value of j is assigned to thejoint with the largest perpendicular distance to the commonaxis of rotation.

The notation U refers to the location of the joint on theupper arm and does not describe the type of joint. Theupper arm joints Ui j are typically spherical joints or jointskinematically equivalent to spherical joints, allowing rotationaround three concurrent joint axes. However, they could alsobe 2-DOF universal joints, in which case the corresponding

U22

P31

U13

U31

U21

P22

P11

A3

L31

L21 L13

A2

L22

L11

R3

R2

R1

1-DOF2-DOF3-DOF

P12

P13

P21

A1 U12

U11 L12B

P

a31

l31

l11

l13l12

x

yMz

h22 h31h12 h13

a21

a22

l22

l21

a11

a13

a12

z

Fy

x

(a) Bodies and joints (3/2/1)

U22

P31

U13

U31

U21

P22

P11

A3

L31

L21 L13

A2

L22

L11

R3

R2

R1

1-DOF2-DOF3-DOF

P12

P13

P21

A1 U12

U11 L12B

P

a31

l31

l11

l13l12

x

yMz

h22 h31h12 h13

a21

a22

l22

l21

a11

a13

a12

z

Fy

x

(b) Kinematic parameters (3/2/1)

Fig. 2. Bodies, joints, defined coordinate systems and kinematic parametersof a SCARA-Tau manipulator (3/2/1).

joints on the manipulated platform would be 3-DOF sphericaljoints.

Each lower arm link Li j is connected via a joint Pi j to amanipulated platform P. The notation P refers to the locationof the joint on the manipulated platform and not to thetype of joint. These joints are typically 2-DOF universaljoints. However, they could also be spherical joints or jointskinematically equivalent to spherical joints. In order tosimplify the kinematics, all 2-DOF and 3-DOF joints areassumed to have concurrent axes of rotation.

The studied mechanisms may utilise joints which provideinfinite rotation around the lower arm link axis and +-90◦

around both the other axes. Examples of such joints areprovided in [9], which also shows how the joint limitationsfor such joints may be implemented.

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B. Joint Positions on the Upper Arms

The kinematic parameters of the studied manipulators areindicated in Fig. 2(b). A fixed coordinate system F is definedwith its z-axis being the common rotation axis of the jointsRi. The x-axis and y-axis of the coordinate system F aredefined to be in the plane perpendicular to the z-axis passingthrough the common intersection point of the joint axes ofU11. Because the manipulator is axis-symmetric, the x-axisis defined to be an arbitrary direction in this plane while they-axis is defined according to the right hand rule.

The angles qi of the upper arms Ai are measured from thex-axis of F according to the right-hand rule. Each upper armjoint Ui j is mounted at a perpendicular distance ai j from thez-axis of F with a z-coordinate hi j. The value of h11 is bydefinition zero. The position of each joint Ui j is expressedin F as

Fui j = [ui jx,ui jy,ui jz]T = [ai j cos(qi),ai j sin(qi),hi j]

T. (1)

C. Joint Positions on the Manipulated Platform

A tool coordinate system M is defined on the manipulatedplatform P with its origin in the TCP. The position of eachjoint Pi j on the manipulated platform P is expressed in M as

Mpi j = [M pi jx,M pi jy,

M pi jz]T. (2)

The distance between a joint Ui j on an upper arm and thecorresponding joint Pi j on the manipulated platform is li j.This is the kinematic length of the link Li j.

The position of the TCP in the fixed coordinate system,F, is given by the three translations x, y, z. The platformorientation remains constant except for a position dependentyaw rotation φ around an axis parallel to the common axis ofrotation of the three upper arms. The platform joint positionsin the fixed coordinate system are

Fpi j = [pi jx, pi jy, pi jz]T = FoM + FRM

Mpi j, (3)

where

FoM = [x,y,z]T, (4)

FRM = Rz(φ) =

cos(φ) −sin(φ) 0sin(φ) cos(φ) 0

0 0 1

. (5)

IV. SUB-PROBLEM OF THE INVERSE KINEMATICS

A sub-problem of the IK problem is determining thejoint angle qi of an upper arm Ai, connected via a lowerarm link Li j to a platform joint Pi j, if the position of thejoint Pi j in the fixed coordinate system F is known. Thisproblem is relatively straightforward and has been presentedpreviously; however, for completeness it is included here.For manipulators without parasitic motion, the position andorientation of the manipulated platform is completely knownand the position of all joints Pi j are immediately found from(3), which means this sub-problem comprises the completeIK problem.

The joint angles can be determined using a geometricapproach or an algebraic approach. The algebraic approach

involves defining length equations for the constant distancesbetween the passive links in a mechanism and thereaftersolving the derived non-linear equation system withoutfurther regard to the geometry of the manipulator. Whilethis approach often seems more systematic, sometimes theequations end up unnecessarily complex and the solutionoften fails to provide physical insight into a mechanism.A geometric approach uses understanding of the geometricstructure of a manipulator to derive equations, for example,noting that certain angles or distances remain constant orutilising trigonometric functions, such as the law of cosines.This approach is often less systematic; however, it oftenleads to more insight into a mechanism. Both approachesare demonstrated here. In general, they involve the samecomputational complexity and it is not unusual to end upsolving the same equations.

A. Algebraic Approach

If the positions of the platform joints are known, thejoint angles may be determined algebraically from the lengthequations of the lower arm links Li j. Squaring those equa-tions gives

|Fpi j− Fui j|2− l2i j = 0. (6)

Using (1) and (3) to expand the equations (6) leads to

ci ja + ci jb sin(qi)+ ci jc cos(qi) = 0, (7)

where parameters ci jk have been introduced according to

ci ja = p2i jx + p2

i jy +(pi jz−hi j)2 +a2

i j− l2i j,

ci jb =−2ai j pi jy, (8)ci jc =−2ai j pi jx.

If ci ja 6= ci jc, each equation (7) has two solutions. Onesolution qiR corresponds to a right upper arm and onesolution qiL corresponds to a left upper arm:

qiR =−2arctan

ci jb +√−c2

i ja + c2i jb + c2

i jc

ci ja− ci jc

,

qiL =−2arctan

ci jb−√−c2

i ja + c2i jb + c2

i jc

ci ja− ci jc

. (9)

The positions of a right and a left upper arms can be seenin Fig. 3. For the special case of ci ja = ci jc, one of the twopossible angles is always 180◦, while the other angle is either

−arcsin(

2ci jaci jbc2

i ja+c2i jb

)or 180◦+ arcsin

(2ci jaci jbc2

i ja+c2i jb

), depending

on which of these solutions that fulfil (7).

B. Geometric Approach

The solution in (9) is compact but does not providemuch physical insight into the mechanism. Determining theactuated joint angles geometrically leads to a somewhatdifferent solution and an increased understanding of themechanism.

Figure 3 shows the two possible upper arm angles, qi,knowing the x and y coordinates of one platform joint Pi j,

5739

aij

x

y

Uij

Pijproj

F

qiR

qiL

qim

qid

qid

lijproj

Fig. 3. The two possible joint angles qiR and qiL of an upper arm Ai fora known position of the joint Pi j .

together with the upper arm length ai j and the length li jproj.The length li jproj is the length of Li j projected in the xy-plane. It is determined from Pythagoras’ theorem:

li jproj =√

l2i j− (pi jz−hi j)2. (10)

As can be seen from Fig. 3, the two possible solutions are:

qiR = qim−qid,

qiL = qim +qid, (11)

where

qim = arctan2(

pi jy, pi jx),

qid = arccos

p2i jx + p2

i jy +a2i j− l2

i jproj

2ai j

√p2

i jx + p2i jy

. (12)

The first term, qim, is calculated using only pi jx and pi jy,given by (3). The case when both pi jx and pi jy are zerolacks solutions due to collisions. The second term, qid, isobtained from pi jx, pi jy and the manipulator arm lengthsusing the law of cosines. Two solutions always exist onboth sides of qim, except when the upper arm and the lowerarm link are collinear, in which case qid becomes zero.The two solutions separate different working modes of themechanism. A configuration with only one solution signifiesa Type 1 or Type 3 singularity [10].

V. COMPLETE INVERSE KINEMATICS

A. Parasitic Motion

Calculating analytical solutions to the IK for a lowermobility axis-symmetric PM exhibiting parasitic motion canbe an intricate problem. If the six LALs of the manipulatorsin Fig. 1 were connected to arbitrary positions on the

manipulated platform, actuation of the three upper armscould lead to 6-DOF motion of the platform. For such amanipulator, the motion in three DOFs can be controlledwhereas the motion in the other three DOFs is a coupledparasitic motion. In this case, the IK must be determinedwhen only three of the six coordinates describing the plat-form position and orientation are explicitly known, which isan exceedingly difficult problem. However, a manipulatorexhibiting parasitic motion in three DOFs would also beof limited practical use. Axis-symmetric PMs including twovertical parallelograms, as in Fig. 1, exhibit parasitic motionin only one DOF. The variation of the yaw angle duringradial and vertical motion of the TCP is considered parasiticand should ideally remain constant. The yaw angle φ maybe expressed as

φ(x,y,z) = arctan2(y,x)+φm +φp(x,y,z), (13)

where φm is the average of φ(x,y,z) calculated over the planewhere y = 0. Both φm and the parasitic yaw angle φp(x,y,z)depend on the manipulator architecture and the choice ofTCP on the manipulated platform. However, the term φmcan always be eliminated by modifying the definition of thecoordinate system M relative to the manipulated platform. Ifthe orientation of M is selected depending on the platformnormal, modifying M corresponds to adding a constantrotation offset to the platform.

A manipulator has the potential to be free of parasitic yawrotation if it is possible to select a TCP on the manipulatedplatform such that the term φp(x,y,z) is zero in the entireworkspace. For the 3-DOF manipulators in Fig. 1 this is notpossible, which means finding analytical solutions to the IKfor these mechanisms is a difficult problem.

Solving the IK for the 3-DOF PMs in Fig. 1 meansdetermining three joint angles when only the position ofthe EEF (x, y, z) is explicitly known. Two tilt angles of theplatform are implicitly known, as they remain constant inthe entire workspace. Due to the parasitic motion, the yawangle (φ ) is unknown.

Preferably, the IK should be solvable for an arbitrarymounting of the EEF on the manipulated platform. However,it is sometimes possible to find an analytical solution for aparticular choice of TCP, even if an analytical solution foran arbitrary TCP cannot be determined. As will be shown,for the studied manipulators the solution is simplified if theTCP is assumed to have the same horizontal position as oneof the joints on the manipulated platform. Even if findingan analytical solution for an arbitrary TCP is preferable,finding an analytical solution for a non-arbitrary TCP maystill be useful for kinematic studies of the manipulator. Alsoin this case, the derived expressions completely describethe motion of the manipulated platform and can be used tocalculate Jacobians, singular values, condition numbers andfor studying the parasitic motion of the manipulator.

B. General Case

For both the algebraic solution (9) and the geometric so-lution (11), the upper arm angles qiL and qiR are determined

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from the position of one platform joint Pi j in the fixedcoordinate system F. This position (pi jx, pi jy, pi jz) is in turndependent on φ according to (4).

In this section we instead use the notation qi jR and qi jL forthe upper arm angles determined from one platform joint Pi j.Let qikR and qikL be different calculations of the same upperarm angles based on another platform joint Pik connected bya LAL Lik to the same upper arm, where Lik must have adifferent horizontal projection than Li j. As both calculationsof the upper arm angles should give the same result, itfollows that qi jR = qikR and qi jL = qikL.

For the Symmetric SCARA variants in Fig. 1(c) and (d),the two links must be Li j = L31 and Lik = L32. For theillustrated variants the correct solution is q3 = q3L. Hence, theyaw angle of the illustrated Symmetric SCARA manipulatorsmay be determined by solving:

q31L(φ)−q32L(φ) = 0. (14)

For the SCARA-Tau variants in Fig. 1(a) and (b), the linksused are selected to be Li j = L11 and Lik = L12 accordingto Fig. 2. The correct solution for these variants is q1 =q1R. Hence, the corresponding equation for the SCARA-Taumanipulator is

q11R(φ)−q12R(φ) = 0. (15)

The only unknown variable in (14) and (15) is the yawangle φ . These equations have two solutions, where only oneis physically possible due to collisions and joint limitations.Typically, the studied manipulators are designed to minimizeφm and φp(x,y,z) in (13), meaning arctan2(y,x) is a goodinitial value to numerically find the valid solution. Afterdetermining φ from (14) or (15), q1, q2 and q3 can becalculated using (9) or (11).

The equations (14) and (15) may be expanded using eitherthe algebraic IK solution (9) or the geometric IK solution(11).

C. Non-arbitrary TCP

If the TCP (origin of M) is selected to have the samehorizontal position as a joint Pi j, then pi jx, pi jy and pi jz in(8) are equal to x, y, and z+M pi jz, respectively, in whichcase qi is immediately determined from (9). Thereafter, alength equation for a second link Lik on the same upper armAi can be reformulated as

dika(qi)+dikb(qi)sin(φ)+dikc(qi)cos(φ) = 0, (16)

with variables

dika(qi) = x2 + y2 + z2 +a2ik− l2

ik +M p2

ikx +M p2

iky

+(M pikz−hik)2 +2(M pikz−hik)z

−2aikysin(qi)−2aikxcos(qi),

dikb(qi) =−2M pikyx+2M pikxy

−2aikM pikx sin(qi)+2aik

M piky cos(qi), (17)

dikc(qi) = 2M pikxx+2M pikyy

−2aikM piky sin(qi)−2aik

M pikx cos(qi).

This equation is of the same type as (7) and can be solved thesame way. After φ has been calculated, the two remainingupper arm angles can be determined using (9) or (11).

The presented solution for a non-arbitrary TCP is validfor all Symmetric SCARA variants if the TCP has the samehorizontal position as Pm1 or Pm2, where m is the index of theupper arm not including a vertical parallelogram. Similarly,the solution is valid for all SCARA-Tau variants if the TCPhas the same horizontal position as Pm1, Pm2 or Pm3, wherem is the index of the upper arm including three LALs.

D. Parallel and Triangular Link Configurations

For the Parallel Symmetric SCARA in Fig. 1(c) and thecorresponding SCARA-Tau variant (the original SCARA-Tau), analytical solutions are possible also for an arbitraryTCP (origin of M). This is also the case for the TriangularSCARA-Tau in Fig. 1(a) and the corresponding triangularconfiguration of a Symmetric SCARA manipulator.

Figure 4 shows projections of the parallel and triangularSCARA-Tau manipulators in a horizontal plane. Only twoof the upper arms and the LALs connected to these armsare included. For the SCARA-Tau manipulator, one of theupper arms must be the one holding the cluster of threeLALs while the other arm must be the one holding thecluster of two LALs. The notation in Fig. 4 and the solutionpresented here are for two SCARA-Tau variants; however,the solution is equivalent for the corresponding SymmetricSCARA variants. For the corresponding projections of theSymmetric SCARA manipulator, one upper arm must be theone not including a vertical parallelogram and the other upperarm must have a different arm solution (qiL or qiR) whencompared to this arm.

The length l11proj in Fig. 4(a) is the length of L11’sprojection in the xy-plane given by (10). The required heightspi jz of the platform joint positions in the fixed coordinatesystem F are immediately determined from the z-coordinate(pi jz = z+M pi jz). The angle φ0 is constant and known.

The IK is calculated by applying Pythagoras’ theorem inthe two right-angled triangles C1C2C3 and FC2M shown inFig. 4(a). The distances c1 and c2 are constant and known.The variable s2 is always positive, while the variable s1 canbe both negative and positive. Pythagoras’ theorem gives

l211proj = s2

1 + s22, (18)

x2 + y2 = (a12 + c1 + s1)2 +(c2 + s2)

2.

The condition s2 > 0 leads to a unique solution of (18). Afters1 has been determined from (18), the angle q1 is calculatedaccording to

q1 = arctan2(y,x)− arccos

(a12 + c1 + s1√

x2 + y2

). (19)

The yaw angle φ is determined from q1 and the constantrotational offset φ0 according to

φ = q1 +π

2−φ0. (20)

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Simplify

a12

s1

s2

l11proj

c1

x

y

P12proj

U12

U21

P21proj

M

C2

C1

C3

F

q1q2

c1

xy

c2

U11

P11proj

0

(a) Parallel SCARA-Tau

a11=a12

x

y

P12proj

U11,U12U21

P21proj

M

F

q1

q2

P11proj

cb

yx

caproj

0

l11proj

l12proj

Mp11proj

Mp12proj

(b) Triangular SCARA-Tau

Fig. 4. Horizontal projections of two variants of the SCARA-Taumanipulator. Only two of the three kinematic chains have been included.

After φ has been calculated, it is straightforward to determinethe two remaining upper arm angles using (9) or (11).

The IK problem for the Triangular SCARA-Tau inFig. 1(a) and the corresponding Triangular SymmetricSCARA manipulator can also be solved analytically for anarbitrary choice of TCP. Fig. 4(b) show a projection ofthe Triangular SCARA-Tau in the xy-plane. Only two ofthe upper arms and the LALs connected to these arms areincluded. The distance cb is constant and known while thedistance caproj is the projection of a constant and knownlength in the xy-plane. The angle φ0 is constant and known.

The position of U11 can be determined from the intersectionof one circle with centre point in the origin of M and radiuscaproj+cb and another circle with centre point in the origin ofthe coordinate system F and radius a11 (equivalent to solvingthe length equations for the constant distances a11 andc1aproj + c1b). This leads to two solution Fu11. For the validsolution, the z-element in the vector product Fu11×([x,y,0]T

is positive.The length equations for the known distances l11proj and

|Mp11proj| are used to determine the position of P11, whilethe length equations for l12proj and |Mp12proj| are used todetermine the position of P12. The yaw angle φ can thenbe calculated according to φ =−arctan2(p12x− p11x, p12y−p11y)− φ0 while the position of P21 can be determinedusing (3). These calculations lead to multiple solutions;however, the valid solution may be selected by the conditions|Mp11proj −Mp12proj| = |Fp11proj − Fp12proj| and |Mp11proj −Mp21proj|= |Fp11proj− Fp21proj|.

After φ has been calculated, it is straightforward to deter-mine all three upper arm angles using (9) or (11).

VI. VERIFICATION

Figure 5 provides a verification of the derived equations.The IK is calculated for three of the manipulators in Fig. 1during a radial motion of the TCP (origin of M). For allthree manipulators, the TCP is selected to be in the centreof the manipulated platform between the four platform jointsof the parallelograms. For all evaluated positions, the valuesof y and z are zero while the value of x has been varied.The minimum value of x is limited by collision between thelower arm links and the base column. The collisions havehere been identified by visual inspection in the MATLABmodels in Fig. 1; however, an automatic collision detectionfor manipulators of this type may be implemented accordingto the description in [9]. The maximum value of x is limitedby the lengths of the upper arms and lower arm links.

For each of the three studied manipulators, the values ofq1, q2, q3 and φ have been calculated by one semi-numericalmethod and one analytical method. The applied analyticalmethod differs between the three manipulators whereas thesemi-numerical method used is the same in all three cases.The semi-numerical method involves determining φ by solv-ing (14) or (15) using MATLAB’s fsolve function and thencalculating q1, q2 and q3 using (9). The equations (14) and(15) were expanded by the geometric IK solution (11).

Figure 5(a) shows the results of the IK calculations forthe Parallel Symmetric SCARA manipulator in Fig. 1(c).The analytical solution is calculated by applying the methoddescribed in relation to Fig. 4(a). The largest difference be-tween corresponding angles calculated by the semi-numericaland the analytical methods in any evaluated position is2×10−8 degrees.

Figure 5(b) shows the results of the IK calculations for theQuadrilateral Symmetric SCARA manipulator in Fig. 1(d).For this manipulator, the selected TCP has the same horizon-tal position as joint P31, which means it is possible to applythe analytical method described in Section V-C. The largest

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0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2−150

−100

−50

0

50

100

150

x (m)

Ang

les (

deg.

)

(a) Parallel Symmetric SCARA (2/2/2)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2−150

−100

−50

0

50

100

150

x (m)

Ang

les (

deg.

)

(b) Quadrilateral Symmetric SCARA (2/2/2)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2−150

−100

−50

0

50

100

150

x (m)

Ang

les (

deg.

)

(c) Triangular SCARA-Tau (3/2/1)

Fig. 5. Verification of the derived equations for the manipulators inFig. 1(c), (d) and (a). The yaw angle φ is plotted in thick solid green,q1 in thin dashed blue, q2 in thin solid red and q3 in thick dashed black.

difference between corresponding angles calculated by thesemi-numerical and the analytical methods in any evaluatedposition is 7×10−10 degrees.

Figure 5(c) shows the results of the IK calculations forthe Triangular SCARA-Tau manipulator in Fig. 1(a). An an-alytical solution to the IK problem is calculated by applyingthe method described in relation to Fig. 4(b). The largestdifference between corresponding angles calculated by the

semi-numerical and the analytical methods in any evaluatedposition is 4×10−8 degrees.

As can be seen in Fig. 5, the size of the parasitic yawangle variation is very different for the three manipulatorswith the largest variation for the Parallel Symmetric SCARAmanipulator in Fig. 1(c). While the yaw angle variation maynot be eliminated, the Quadrilateral variants minimises thisvariation [8]. The optimal quadrilateral shape depends onthe length ratio between the lower arm links and the upperarms. If this ratio is large, the optimal quadrilateral shape iscloser to a triangle, while for ratios closer to one, the optimalquadrilateral shape is closer to a parallelogram.

VII. CONCLUSION AND FUTURE WORK

This paper analysed the IK problem for a class of 3-DOF axis-symmetric PMs. The manipulators in this classare composed of three actuated upper arms and six 5-DOFLALs, four of which are mounted to form two verticalparallelograms. All manipulators in this class exhibit anunwanted parasitic platform rotation in one DOF, whichmeans the IK problem for these manipulators is intricate.It has been demonstrated that, in the general case, the IKproblem can be reduced to solving one univariate equationnumerically whereas analytical solutions have been presentedfor several important special cases.

As the IK will typically be used for real-time automaticcontrol, valuable future work includes a comparison of theefficiency between the analytical and semi-numerical algo-rithms. A fully numerical algorithm should also be includedas a benchmark.

REFERENCES

[1] M. Isaksson and M. Watson, “Workspace Analysis of a Novel 6-DOF Parallel Manipulator with Coaxial Actuated Arms,” Journal ofMechanical Design, vol. 135, no. 10, 104501, 2013.

[2] J. A. Carretero, R. P. Podhorodeski, M. A. Nahon, and C. M. Gosselin,“Kinematic Analysis and Optimization of a New Three Degree-of-Freedom Spatial Parallel Manipulator,” Journal of Mechanical Design,vol. 122, no. 1, pp. 17–24, 2000.

[3] S. Kock, R. Oesterlein, and T. Brogardh, “Industrial robot,” WO Patent03/066289 A1, Aug. 14, 2003.

[4] H. Cui, Z. Zhu, Z. Gan, and T. Brogardh, “Kinematic analysis and errormodeling of TAU parallel robot,” Robotics and Computer IntegratedManufacturing, vol. 21, no. 6, pp. 497–505, 2005.

[5] M. Isaksson, T. Brogardh, I. Lundberg, and S. Nahavandi, “Improvingthe Kinematic Performance of the SCARA-Tau PKM,” in IEEEInternational Conference on Robotics and Automation (ICRA’10),Anchorage, AK, USA, 2010, pp. 4683–4690.

[6] T. Brogardh, S. Hanssen, and G. Hovland, “Application-Oriented De-velopment of Parallel Kinematic Manipulators with Large Workspace,”in 2nd International Colloquium of the Collaborative Research Center,562: Robotic Systems for Handling and Assembly, Braunschweig,Germany, 2005, pp. 153–170.

[7] M. Isaksson, T. Brogardh, and S. Nahavandi, “Parallel Manipulatorswith a Rotation-Symmetric Arm System,” Journal of MechanicalDesign, vol. 134, no. 11, 114503, 2012.

[8] M. Isaksson, “Analysis and Synthesis of Parallel Manipulators withRotation-Symmetric Arm Systems,” Ph.D. dissertation, Deakin Uni-versity, 2013.

[9] M. Isaksson, T. Brogardh, M. Watson, S. Nahavandi, and P. Crothers,“The Octahedral Hexarot — A novel 6-DOF parallel manipulator,”Mechanism and Machine Theory, vol. 55, pp. 91–102, 2012.

[10] C. Gosselin and J. Angeles, “Singularity Analysis of Closed-LoopKinematic Chains,” IEEE Transactions on Robotics and Automation,vol. 6, no. 3, pp. 281–290, 1990.

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