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Research Papers

# Kinematic Analysis of a Novel Kinematically Redundant Spherical Parallel ManipulatorOPEN ACCESS

[+] Author and Article Information
Jérôme Landuré

Départment de génie mécanique,
Université Laval,
e-mail: jerome.landure.1@ulaval.ca

Clément Gosselin

Départment de génie mécanique,
Université Laval,
e-mail: gosselin@gmc.ulaval.ca

Manuscript received September 20, 2017; final manuscript received December 6, 2017; published online February 12, 2018. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 10(2), 021007 (Feb 12, 2018) (10 pages) Paper No: JMR-17-1303; doi: 10.1115/1.4038971 History: Received September 20, 2017; Revised December 06, 2017

## Abstract

This paper introduces a new architecture of spherical parallel robot which significantly extends the workspace when compared to existing architectures. To this end, the singularity locus is studied and the design parameters are chosen so as to confine the singularities to areas already limited by other constraints such as mechanical interferences. First, the architecture of the spherical redundant robot is presented and the Jacobian matrices are derived. Afterwards, the analysis of the singularities is addressed from a geometric point of view, which yields a description of the singularity locus expressed as a function of the architectural parameters. Then, the results are applied to an example set of architectural parameters, which are chosen in order to illustrate the advantages of the redundant architecture over current designs in terms of workspace.

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## Introduction

Parallel spherical three-degrees-of-freedom (3DOF) devices have been studied in the literature for several decades. In Ref. [1], for example, a mechanism that simulates the movements of a shoulder is presented. Architectures of spherical robots with the required minimum number of actuators to ensure the control of the three main rotations have been proposed and studied in the literature. Designs with prismatic actuators have been proposed, for instance, in Refs. [2] and [3] while architectures based on revolute actuators were introduced in Refs. [4] and [5] where a spherical manipulator referred to as the “agile eye” was presented. Numerous studies have highlighted the singularity loci: Gosselin et al. [6] provides numerical algorithms to determine the loci, and Sefrioui and Gosselin [2] finds an analytical description of the loci in the Cartesian space for a specific case.

Other studies deal with spherical robots but study different approaches. For example, Cammarata and Sinatra [7] proposes a mathematical model to determine the stiffness of a 3-CRU spherical robot in order to choose the best materials and dimensions. A vibration analysis is also presented. Li and Guo [8] studies the torques needed from the actuators to ensure a correct trajectory and speed for a 3-RRP spherical robot. In Ref. [9], the precision of a 3-URU spherical robot is discussed through several performance indices. In Ref. [10], the architecture of a 3-RUU spherical robot, where the internal unnecessary mobilities are suppressed, is presented. Other studies deal with less general subjects, but the architecture chosen is often quite different from a classical one and tailored to the needs of a specific area of application. For instance, Yu et al. [11] deals with a 2DOF spherical manipulator which orients camera lenses, i.e., in which torsion is ignored. In Ref. [12], a shoulder rehabilitation device is shown, Congzhe et al. [13] deals with a knee and ankle rehabilitation device and Hayes et al. [14] presents a 6DOF robot whose rotations are decoupled from translations. Moreover, the rotations are produced by friction driving rollers. Recently, Wu et al. [15] presents a spherical robot designed to ensure unlimited torsional movement in any configuration.

Many mechanisms with wrists like motions have been studied over the years, Stanisic and Duta [16] describes a 4DOF spherical serial manipulator using kinematic redundancy for workspace improvement purposes, Leguay-Durand and Reboulet [17] presents a spherical hybrid (serial and parallel) mechanism with actuation redundancy to avoid singularities and locally improve dexterity. Other studies, such as Ref. [18] or Ref. [19], present new families of spherical parallel manipulators whose designs are proven to be optimized for certain motions. Mathematical indices are presented in Ref. [20] to evaluate the dexterity and stiffness of 3DOF spherical parallel manipulators over their workspace. Robots are closely related to biomedical applications, and some spherical architectures are studied for that purpose. For instance, Lum et al. [21] presents architectures for medical assistive devices to help surgeons in the case of minimally invasive surgery.

In all these studies, the rotational range of motion is an important challenge mainly limited by the singularity loci. It is proposed here to enlarge the workspace of a spherical robot by using kinematic redundancy. Indeed, a singular configuration is a configuration which leads to the loss of control over Cartesian coordinates, joint coordinates or both due to a specific geometric configuration. Such configurations lead to limitations in the workspace. In addition to the mechanical restrictions, the designer must deal with these unintuitive constraints to reach the desired performances. This problem is one of the main reasons for which spherical parallel manipulators are limited to applications that require rather small ranges of motion. Hence, the architecture introduced in this paper has the potential to enlarge the range of applications of spherical robots.

The use of redundancy to reduce the singularity loci has already been studied in the literature. For example, Kurtz and Hayward [22] introduces a spherical robot with four prismatic actuators. However, the redundant architectures proposed in the literature are based on actuation redundancy, i.e., they include more actuators than degrees-of-freedom. The drawback of actuation redundancy is that internal forces can be generated by the actuators, which requires advanced control and sensing. On the other hand, in the architecture introduced here, two of the legs are connected to a common link that is in turn connected to the platform. Using this approach, the resulting redundancy is purely kinematic. The robot has four degrees-of-freedom and the actuators can be position controlled, just like in nonredundant robots. The architecture proposed here can be thought of as an extension to the spherical case of the planar redundant architecture presented in Ref. [23]. Indeed, the same subarchitecture composed of the two legs connected to a common link which is in turn connected to the platform is used for the singularity avoidance.

This paper is structured as follows: First, a description of the novel parallel redundant architecture is given. Then, the Jacobian matrices are derived. Afterwards, the inverse kinematic problem is presented, and finally, the singularity loci are studied to show the advantages of the redundant architecture. An example design is also introduced in order to illustrate the advantages of the redundant device over nonredundant ones. The performances of the manipulators are compared for two trajectories.

## Architecture of the Manipulator

The architecture of the proposed robot can be described as $S/[(2−UP¯)RS]/2−UP¯S$. This notation means that the architecture is composed of four kinematic chains connecting the fixed frame and the effector, namely: one S chain, two $UP¯S$ chains and one $(2−UP¯)RS$ chain, where R stands for a revolute joint, S stands for a spherical joint, U stands for a Hooke joint, and P stands for a prismatic joint and where underlined joints are actuated. In other words, the robot is composed of a platform mounted on a passive spherical joint and connected to the base by two $UP¯S$ legs as well as one double leg which includes two $UP¯$ legs connected to a common revolute joint that is connected to a link that is in turn connected to the platform via a spherical joint, as shown in Fig. 1.

## Kinematic Model

Referring to Fig. 2, point O is defined as the center of the passive spherical joint on which the platform is mounted.

The platform undergoes pure rotations around point O, which is also chosen as the origin of the base frame. Point Ai is defined as the center of the Hooke joint of the ith leg (two indices are used for the double leg) while point Bi is defined as the center of the spherical joint attaching leg i to the platform. Point C is defined as the intersection of the axes of the prismatic joints of the double leg, i.e., the center of the revolute joint connecting the two sublegs of the double leg. The axis of this revolute joint is chosen orthogonal to the plane described by the three points A11, A12, and B1. The position vector of point Ai with respect to the origin of the fixed frame is noted $ai$ and the position vector of point Bi is noted $bi$. Similarly, the position vector of point C is noted c.

Joint coordinates are given by the extension of the prismatic joints, namely, l11 and l12, respectively, corresponding to the distance between A11 and C, and between A12 and C which compose the double leg of the architecture. The other two joint coordinates are referred to as l2 and l3 and correspond, respectively, to the distance between A2 and B2, and between A3 and B3. The kinematic model is represented schematically in Fig. 2.

The orientation of the platform (Cartesian coordinates) is represented by a matrix using the tilt and torsion angles as described in Ref. [24]. The tilt is an angle (referred to as θ in the mathematical model) around an axis in the horizontal plane located by an angle $ϕ$, and the torsion is an angle (referred to as σ in the mathematical model) around an axis orthogonal to the plane defined by the three attachment points on the effector (B1, B2, and B3).

The matrix defining the orientation of the platform is then written as Display Formula

(1)$R(ϕ,θ,σ)=[CϕCθCσ−ϕ−SϕSσ−ϕ−CϕCθSσ−ϕ−SϕCσ−ϕCϕSθSϕCθCσ−ϕ+CϕSσ−ϕ−SϕCθSσ−ϕ+CϕCσ−ϕSϕSθ−SθCσ−ϕSθSσ−ϕCθ]$

where C stands for “cosine” and S stands for “sine.”

Any point P on the platform has the resulting position vector p given as Display Formula

(2)$p=Rp0$
with $p0$ the position vector of P in a configuration chosen as the reference configuration, which is defined by $ϕ=θ=σ=0$.

Moreover, the velocity of point P can be written as Display Formula

(3)$p˙=R˙p0=ω×Rp0=ω×p$
with $ω$ the angular velocity vector of the platform.

The architecture of the robot introduces constraint equations on which the derivation of the Jacobian matrices is based.

For legs 2 and 3, one has Display Formula

(4)$∀i∈{2,3}:(bi−ai)T(bi−ai)=li2$
with li the distance between Ai and Bi which is the joint variable associated with the ith leg.

For the double leg composed of A11, A12, C, and B1, similarly to the above equations, one obtains Display Formula

(5)$∀i∈{11,12}:(c−ai)T(c−ai)=li2$

The revolute joint at point C also introduces a constraint. In order to express this constraint, the following vectors are introduced: Display Formula

(6)$e=a12−a11||a12−a11|| and n=(b1−a11)×e$

where n is a vector orthogonal to the plane formed by points A11, A12, and B1, which leads to the constraint Display Formula

(7)$nT(c−a11)=0$

## Solution of the Inverse Kinematic Problem

It is rather straightforward to write the joint coordinates in terms of the Cartesian coordinates. Indeed for the first two legs, one has Display Formula

(8)$∀i{2,3}:li2=(bi−ai)T(bi−ai)$

where $bi=Rbi0$ and R is the matrix giving the orientation of the platform as described above while $bi0$ is the position vector of point Bi in the reference orientation.

For the double leg, one has Display Formula

(9)$∀i{11,12}:li2=(c−ai)T(c−ai)$

where Display Formula

(10)$c=b1+(c−b1)=Rb10+Qu$

where $b10$ is an architectural constant that characterizes the geometry of the platform and $u=[l cos βl sin β0]T$ describes the coordinates of C in the plane defined by points A11, A12, and B1. In this plane, the motion of point C describes a circle of radius l, where l is the length of the connecting rod $B1C$, centered in B1. This rotation is described by matrix Q, which corresponds to a rotation of angle β around an axis orthogonal to the plane defined by points A12, A11, and B1. The motion of C in this plane is independent from the Cartesian coordinates $ϕ$, θ, and σ. Indeed, angle β is the control parameter of the redundant degree-of-freedom while $ai$ is an architectural constant.

## Jacobian Matrices

Based on the above derivation, the Jacobian matrices of the robot can now be established. The constraint equation (8) is differentiated with respect to time, yielding Display Formula

(11)$∀i∈{2,3}:[bi×(bi−ai)]Tω=lil˙i$
with $ω$ the angular velocity vector of the platform.

Then, Eqs. (7) and (9) are also differentiated, which leads to $∀i∈{11,12}:$Display Formula

(12)$[(b1−a11)×e]Tc˙+[b1˙×e]T(c−a11)=0$
Display Formula
(13)$(c−ai)Tc˙=lil˙i$

The circular permutation property of the triple product leads to $[b1˙×e]T(c−a11)=[e×(c−a11)]Tb1˙$. Equations (12) and (13) can then be written in matrix form as Display Formula

(14)$Hc˙=w$

where Display Formula

(15)$H=[(c−a11)T(c−a12)T[(b1−a11)×e]T]$

(16)$w=[l11l˙11l12l˙12[(c−a11)×e]Tb˙1]$

Vector $c˙$ can be obtained as a function of the joint coordinates using $c˙=H−1w$ provided that matrix H is not singular. One can write Display Formula

(17)$H−1=1d[h1h2h3]$

where Display Formula

(18)$d=[(c−a11)×(c−a12)]T[(b1−a11)×e]$
Display Formula
(19)$h1=(c−a12)×[(b1−a11)×e]$
Display Formula
(20)$h2=[(b1−a11)×e]×(c−a11)$
Display Formula
(21)$h3=(c−a11)×(c−a12)$

By inspection of the expression for d given in Eq. (18), it can be observed that H is singular only when C is aligned with A11 and A12. This condition is always easily avoided if the distance OB1 is chosen smaller than the distance between O and the line $A11A12$.

Under this assumption, one can then write Display Formula

(22)$c˙=H−1w$
Display Formula
(23)$=1d(h1l11l˙11+h2l12l˙12+h3[(c−a11)×e]b˙1)$

Moreover, the constraint introduced by the link connecting the rod whose length is l connecting points B1 and C can be written as Display Formula

(24)$(c−b1)T(c−b1)=l2$

The differentiation of this expression then yields Display Formula

(25)$(c−b1)Tc˙=(c−b1)Tb˙1$

Substituting Eq. (23) into Eq. (25) and knowing that $b˙1=ω×b1$, one then obtains Display Formula

(26)$(c−b1)T(ω×b1)=(c−b1)T1d(h1l11l˙11+h2l12l˙12+h3[(c−a11)×e]Tb1)$

Moreover, the left-hand side of Eq. (26) can be rearranged as Display Formula

(27)$(c−b1)T(ω×b1)=[b1×(c−b1)]Tω$

Also, noting that vector $h3$ is orthogonal to vector $(c−b1)$, one obtains Display Formula

(28)$[b1×(c−b1)]Tω=(c−b1)Tk11l˙11+(c−b1)Tk12l˙12$

where Display Formula

(29)$k11=l11dh1$
Display Formula
(30)$k12=l12dh2$

Finally, from Eq. (28) and Eq. (11) the velocity equations can be written in matrix form as Display Formula

(31)$Jω=Kl˙$

where $l˙=[l˙11l˙12l˙2l˙3]T$ is the joint velocity vector and the Jacobian matrices are, respectively, written as Display Formula

(32)$J=[[b1×(b1−c)]T[b2×(b2−a2)]T[b3×(b3−a3)]T]=[[c×b1]T[a2×b2]T[a3×b3]T]$

(33)$K=[(b1−c)Tk11(b1−c)Tk120000l20000l3]$

## Singularity Analysis

The singularities referred to as type II singularities in Ref. [25] are studied here because they correspond to the loss of control of the moving platform.

Type II singularities occur when matrix J is not of full rank. The transpose of J is used here to simplify the expression and because its rank is equal to that of J.

The singularity loci of spherical 3DOF parallel mechanisms have been studied in the literature (for instance, see Ref. [2,26,27], or [28]). The approach taken in these references consists in expanding the determinant of the Jacobian matrix and deriving mathematical conditions for which this determinant vanishes. It is proposed here to adopt a geometric approach. Based on vector analysis, this method gives similar results but it is easier to perform and it provides direct architectural constraints to visualize the singularity loci.

A 3 × 3 matrix is not of full rank if and only if one of the following conditions is verified, namely (i) one of its columns is zero, (ii) two of its columns are collinear or (iii) the three columns are coplanar. Mathematically, these conditions can be written as Display Formula

(34)c×b1=0ora2×b2=0ora3×b3=0}alignment
Display Formula
(35)$(a2×b2)×(a3×b3)=0or(c×b1)×(a2×b2)=0or(c×b1)×(a3×b3)=0}collinearity$
Display Formula
(36)$[(a2×b2)×(a3×b3)]T(c×b1)=0}coplanarity$

Point C of the robot is now considered. The redundant actuator of the robot enables to position point C of the robot in different locations without moving the platform. The fact the joint at C is a revolute joint constrains C to lie on a circle around B1 in the plane formed by points A11, A12 and B1. This redundant degree-of-freedom makes the following cases of singularity avoidable: Display Formula

(37)$c×b1=0$
Display Formula
(38)$(c×b1)×(a2×b2)=0ifa2×b2≠0$
Display Formula
(39)$(c×b1)×(a3×b3)=0ifa3×b3≠0$
Display Formula
(40)$[(a2×b2)×(a3×b3)]T(c×b1)=0if{(a2×b2)×(a3×b3)≠0a2×b2≠0a3×b3≠0$

One important question can then be formulated as follows: are there configurations or architectures of the robot for which the singularities described above cannot be avoided despite the redundancy?

The cases mentioned above are treated separately: For the case described by Eq. (37), $c×b1=0$ occurs if either of the vectors is zero or if they are aligned. Furthermore, vector $b1$ is never equal to $0$ because it is located on a sphere centered in O with radius $||b1||=OB1=constant$. Vector c is located on a circle centered on point B1. Therefore, it is easy to prevent vector c from vanishing by designing the length of the rod connecting B1 and C as $B1C=||b1−c||≠||b1||$. Finally, regarding a possible alignment of points C, B1 and O (which would make vectors c and $b1$ aligned), all the configurations in which C, B1 and O are aligned represent a line in space (the line intersecting B1 and O). On the other hand, all the positions that point C can reach are on a circle. Yet, a line and a circle either do not intersect, or intersect in one point or intersect in two points which is the case here because the line goes through the center of the circle and is located on the same plane. However, there always exists a number of positions for point C on the circle for which $c×b1≠0$. Therefore, this singularity can always be avoided.

For the case described by Eqs. (38) and (39), for the same reasons as given above, vectors c and $b1$ cannot be aligned and so it will always be possible to avoid $(c×b1)×(a2×b2)=0$ or $(c×b1)×(a3×b3)=0$ by exploiting the redundancy.

For the case described by Eq. (40), it is clear that vector $c×b1$ can be oriented arbitrarily in a plane orthogonal to $b1$ using the redundant degree-of-freedom.

Therefore, the configurations described above can always be avoided except if n and $b1$ are collinear, where Display Formula

(41)$n=(a2×b2)×(a3×b3)$

This case is the only one that must absolutely be avoided because in this situation the redundancy cannot be used to avoid the singular configuration.

## Redundancy Resolution for Singularity Avoidance

For control purposes, one must choose the appropriate coordinate for the redundant degree-of-freedom to avoid effectively the singular configurations. To this end, a method is presented here based on an analytical solution for the determinant of the Jacobian matrix. Combining the solution of the inverse kinematic problem with the scalar triple product form of the determinant one can obtain Display Formula

(42)$det(J)=det(JT)$
Display Formula
(43)$=det([[c×b1][a2×b2][a3×b3]])$
Display Formula
(44)$=[(a2×b2)×(a3×b3)]T(c×b1)$
Display Formula
(45)$=[(a2×b2)×(a3×b3)]T(Q[l cos βl sin β0]×b1)$

Matrix Q is the rotation matrix from the base frame to the frame in which the movement of the leg of length l is constrained in the ($qx, qy$) plane, namely Display Formula

(46)$Q=[qxqyqz]$

where Display Formula

(47)$qx=a11−b1||a11−b1||$
Display Formula
(48)$qz=(a11−b1)×(a11−a12)||(a11−b1)×(a11−a12)||$
Display Formula
(49)$qy=qz×qx$

The following vector is then defined: Display Formula

(50)$v=(a2×b2)×(a3×b3)$
which is independent from the motion associated with the redundancy. The expression of the determinant of J becomes Display Formula
(51)$det(J)=lvT(qx×b1)cos β+lvT(qy×b1)sin β$
with β the redundant parameter. We define Display Formula
(52)$k1=lvT(qx×b1)$
Display Formula
(53)$k2=lvT(qy×b1)$
as new parameters. The singularity locus of the platform in a specific orientation can be obtained with the following equation: Display Formula
(54)$det(J)=k1 cos β+k2 sin β=0$

The solution of this equation is Display Formula

(55)$β=π2−γ+nπ$

where n$∈Z$ and γ is defined as Display Formula

(56)$sin γ=−k2k12+k22$
Display Formula
(57)$cos γ=k1k12+k22$

Therefore, the two solutions of Eq. (51) in the range $[−π,π]$ can be written as Display Formula

(58)$η1=π2−γ$
Display Formula
(59)$η2=−π2−γ$

Hence, it is clear that the redundant mobility parameter β can be chosen arbitrarily over a range of amplitude equal to π while avoiding the coplanarity singularity case.

Based on the above derivation, two main observations can be made. First, Eq. (51) can be easily differentiated with respect to the parameter β, which indicates that the local maximum for the determinant (the theoretically farthest configuration from a singularity) is located in the middle of the admissible range. Therefore this value is the best if no other constraints are considered. Second, k1 and k2 sum up the results of the vector analysis for the case in which the redundancy cannot avoid singularities. Indeed, these configurations are the ones where $k1=k2=0$, and so whatever the value of β is, the robot is in an unavoidable singularity state.

This method presented above for the resolution of the redundancy is simple, computationally efficient and robust. It also gives flexibility in the control trajectory planning because of the range of admissible positions for the redundant mobility parameter.

Figure 3 illustrates the admissible values for parameter β which satisfy the considered constraints. This figure represents the orientation of the link between C and B1 in its mobility plane defined by its normal $qz$. Angles η1 and η2 are the angles that correspond to the singular state described above and ηconstraint represents an additional independent constraint such as a mechanical interference.

The workspace of the link connecting C and B1 is the shaded area, which is limited by two nearest constraints, according to the initial orientation of the link. Since the constraints are configuration dependent, it would be difficult to show, for a particular design, that this workspace can never be reduced to zero in any of its configurations.

## Particular Architecture

This section presents a particular case of the architecture described above. In this case, the robot is designed such that points B1, B2, B3, and O are coplanar. Moreover A2 and A3 are placed on an axis intersecting point O, which means that A2, A3, and O are collinear. Vector z is defined as a unit vector along this axis. An example of such a robot is presented in Fig. 4.

Vector $n=(a2×b2)×(a3×b3)$ can then be easily located. Indeed $a2×b2$ is orthogonal to $a2$ which is collinear with z and hence $a2×b2$ is orthogonal to z. Similarly $a3×b3$ is orthogonal to $a3$ which is collinear with z and hence $a3×b3$ is orthogonal to z too. Therefore, n is collinear with z. In this situation, the condition identified above, namely, that vectors n and $b1$ are aligned, can be rewritten by transitivity Display Formula

(60)$b1//n⇔b1//z$

However if $b1$ and z are collinear it means that A2, A3, B2, B3, and O are coplanar and so $(a2×b2)//(a3×b3)$, which yields Display Formula

(61)$b1//z⇒(a2×b2)//(a3×b3)$
which is equivalent to Display Formula
(62)$b1//n⇒(a2×b2)//(a3×b3)$

It is recalled that $b1//n$ corresponds to the only situation in which the redundant degree-of-freedom cannot be used to avoid coplanar singularities. Also, it can be observed, from Eq. (32) that the case in which $(a2×b2)//(a3×b3)$ corresponds to a situation where two lines of the Jacobian matrix are collinear which is a singularity that cannot be avoided using the redundant degree-of-freedom.

Hence, Eq. (62) means that with the particular architecture defined in this section, the coplanar singularity to be avoided has been “pushed” to a case that cannot be avoided anyway, which makes this architecture particularly attractive. This is possible thanks to the alignment of points O, A2, and A3.

One of the most significant advantages of the special architecture described above is an improved orientational workspace compared to the workspace of nonredundant architectures. Indeed, as mentioned in the introduction, singularities are the main limitation for the workspace of the robot. A more detailed analysis of the workspace based on tilt and torsion angles is given in Appendices A and B.

The result of this workspace analysis is that the singularity locus of the proposed architecture reduces to $θ=±(π/2)$ in the space of tilt and torsion angles. Hence, the theoretical workspace for this robot is much larger than the one of a nonredundant spherical robot.

## Example

Example trajectories are presented in this section in order to demonstrate the ability of the proposed robot architecture to avoid singular configurations. The joint coordinates are calculated, for a given trajectory, using the method described previously. Two trajectories have been chosen for this example. The trajectories can be defined with the T and T angles as follows:

$trajectory 1: {ϕ=−0.9θ=0.50≤σ≤2π$
$trajectory 2: {0≤ϕ≤2πθ=1.8σ=0$

The architectural parameters chosen are given in Table 1.

For each trajectory the joint coordinates and the determinant of J during the movement are shown in Figs. 58.

Figures 5 and 6 present the joint coordinates for the two trajectories for the redundant architecture. Figures 7 and 8 present the determinant of the Jacobian matrices for the redundant architecture and a nonredundant one for spherical robots. The nonredundant architecture is designed by replacing the double leg by a simple leg connecting A11 and B1.

Two main observations can be made from this example. The first is the ability of the redundant robot to follow the trajectories with no deviation between the desired and the obtained trajectory with smooth joint coordinates. Also, in both trajectories the nonredundant architecture goes through several singular configurations during the movement while the redundant architecture does not. Indeed the determinant of the Jacobian matrix for the nonredundant robot crosses 0 several times.

## Prototyping Using Revolute Actuators

A prototype was designed and built in order to validate the architecture and redundancy resolution algorithm proposed in this paper. For ease of prototyping, revolute actuators are used instead of prismatic actuators. Therefore, the $RP¯R$ legs (or sublegs) are replaced with $RR¯R$ legs. It is pointed out that the second R joint of each leg must be actuated (and not the first one) for the kinematic analysis presented in the paper to remain valid. If the base revolute joints were actuated, the kinematic and singularity analyses would have to be revised. In a real application though, prismatic joints should be favored, which is why an architecture equivalent to the one using actuated prismatic joints is prototyped here.

The $RR¯R$ leg architecture used in the prototype is illustrated in Fig. 9, where is it compared with the original $RP¯R$ architecture. The motor controls the angle between the two leg links in order to vary the distance between points Bi and Ai. A driving belt enables to place the motor closer to the base, where its amplitude of motion is significantly reduced and where its weight does not induce large actuation forces. The parameters used to build the prototype are given in Table 2. The parameters were chosen according to the analyses presented in the paper regarding singularity locus and computer-aided design models for mechanical interferences, complemented with a simplified collision detection analysis. The lengths of the links to be used in the $RR¯R$ legs can be determined based on the required range of the original $RP¯R$ legs, namely ρmin and ρmax and the limits chosen at the actuated R joint, namely the maximum and minimum angle (to avoid near singular legs). The geometric modeling of the $RR¯R$ leg yields, based on the law of cosines Display Formula

(63)$l2=l12+l22−2l1l2 cos γ$

where l is the distance between Bi and Ai, l1 and l2 are the lengths of the links of the leg and γ is the angle between the links (actuated joint angle). This relation can be written for the minimum and maximum extensions noted ρmin and ρmax, with the corresponding angles γmin and γmax, namely, Display Formula

(64)$ρmax2=l12+l22−2l1l2 cos γmax$
Display Formula
(65)$ρmin2=l12+l22−2l1l2 cos γmin$

If ρmin, ρmax, γmin and γmax are prescribed, Eqs. (64) and (65) can be solved for l1 and l2 as follows: Subtracting Eq. (65) from Eq. (64), one obtains Display Formula

(66)$ρmax2−ρmin2=−2l1l2(cos γmax−cos γmin)$
which can be rewritten as Display Formula
(67)$l2=−ρmax2−ρmin22l1(cos γmax−cos γmin)=Al1$

where Display Formula

(68)$A=−ρmax2−ρmin22(cos γmax−cos γmin)$

Substituting Eq. (67) into Eq. (64) then yields Display Formula

(69)$ρmax2=l12+A2l12−2A cos γmax$
which can be rewritten as Display Formula
(70)$l14+Bl12+A2=0$

where Display Formula

(71)$B=−(ρmax2+2A cos γmax)$

Equation (70) can then be solved for $l12$ as Display Formula

(72)$l12=−B±B2−4A22$

and using Eq. (67) one finally gets Display Formula

(73)$l1=±−B±B2−4A22$
Display Formula
(74)$l2=Al1˙$

The above equations can be used to determine the link lengths of the $RR¯R$ leg from given values of ρmin and ρmax, γmin and γmax. Up to four solutions are obtained and the best solution can then be determined based on other practical considerations. The values of γmin and γmax are selected to ensure that the legs never approach serial singularities (fully aligned or fully folded links).

The prototype built using 3D-printed links is shown in Fig. 10. It uses four small electric motors and pulleys to drive the intermediate joint of each of the $RR¯R$ sublegs. Although the prototype validates the concept, its performance is rather limited due to the inaccuracies of the fabrication process. A simulation of a trajectory performed with the computer-aided design model of the prototype can be visualized at the website.1 In this trajectory, the robot first performs a torsional motion of the platform around the vertical axis with a range of motion of 155 deg (110 deg to −45 deg). Then, the robot performs a tilting motion to 80 deg and to −70 deg. Finally, the tilting angle of −70 deg is maintained while the platform undergoes a full rotation around the vertical axis. This trajectory demonstrates the large workspace that is made possible by the use of kinematic redundancy.

## Conclusion

This paper presents a kinematically redundant architecture for a spherical parallel robot. The orientable connecting link of the redundant architecture allows one to avoid almost all singularities. The new architecture is similar to that of the conventional nonredundant spherical parallel robot and is therefore easy to understand and use. Also, the supporting forces remain mostly concentrated in the spherical joint between the platform and the fixed frame. Due to the redundancy, the proposed manipulator has a theoretical workspace much larger than that of the nonredundant spherical robot. This result could significantly extend the range of applications of spherical parallel robots, for example in machine tools.

## Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

## Appendices

###### Appendix A: Determination of the Alignment and Collinear Singularity Loci

Singularity by alignment: The orientation matrix which corresponds to the singularity case when A, B and O are aligned can be obtained by the product of two rotation matrices: the first which configures the platform to get the alignment and the second which corresponds to the rotation around the axis of alignment and includes all the movements allowed that keep the robot in the singular configuration.

For all $i={2,3}$ corresponding to legs 2 or 3 of the robot, the first rotation places vector $bi0$ (initial position for this point) in the position where $bi$ is aligned with $ai$ and can be designed as follows: the rotation axis is $e=(bi0×ai)/(||bi0×ai||)$ and the angle of the rotation α is defined by $cos α=bi0Tai/(||bi0||||ai||)$. The alignment of the two vectors is π periodic so the cosine of this angle is sufficient. One has then Display Formula

(A1)$Q1=eeT+cos α(13−eeT)+sin α(13×e)$

The second rotation is taking place around the axis describing the alignment and corresponds to all the rotations possible which maintain the alignment state; in other words, this second matrix enables to describe all the sets of angles $(ϕ,θ,σ)$ for which the alignment singularity occurs. One can write Display Formula

(A2)$Q2=uiuiT+cos η(13−uiuiT)+sin η(13×ui)$

where η is an arbitrary angle and $ui$ is a unit vector defined as Display Formula

(A3)$ui=ai||ai||$

The global matrix which defines this case of singularity is $Q=Q1Q2$. Then, this matrix is equated with the T and T matrix to determine the triplet $(ϕ,θ,σ)$ which corresponds to these singular configurations Display Formula

(A4)$R(ϕ,θ,σ)=Q$

The advantage of this method compared to expanding the determinant of J and looking for its roots is that the equations to solve here are much simpler. Indeed, the equation obtained by expanding the determinant requires to solve all the singularity cases at the same time by solving a trigonometric nonlinear equation whereas this method decouples the solutions with nine equations in three variables including at least five very simple equations.

The reference configuration of the platform is chosen as one in which all the z coordinates of vectors $bi0$ are equal to 0. Moreover the architecture is such that the x and y coordinates for a2 and a3 are equal to 0. This yields Display Formula

(A5)$q33=cos θ=0 i.e. θ=±π2 independent from ϕ and σ$

Moreover, it is assumed that $bi0/||bi0||=[ cos γ sin γ0]$ yielding Display Formula

(A6)${q13=−cos(α+γ)=±cos(ϕ)q23=sin(α+γ)=±sin ϕ$

and so $ϕ=((1±1)/2)˙π−(α+γ)=−α+constant$ which means $ϕ$ is free to take any value Display Formula

(A7)${q31=−sin θ cos(σ−ϕ)=sign(ai3)cos γq32=sin θ sin(σ−ϕ)=−sign(ai3)sin γ$

Therefore, $σ−ϕ=((1±sign(ai3)1)/2)π+γ$ which can be simplified as $σ−ϕ=constant$.

Finally, all the alignment singularity loci in the Cartesian space can be expressed as Display Formula

(A8)${ϕ=ϕθ=±π2σ=ϕ+constant$
which is a line in the Cartesian space of the robot.

Collinear singularities: another singularity case occurs when $a2×b2$ and $a3×b3$ are collinear. These two vectors are defined as orthogonal to two other vectors; hence, we can assume that these two vectors are collinear when the vectors they are defined with are coplanar. Moreover, vectors $b2$ and $b3$ are in the plane of the platform. Therefore, the coplanar condition can be reduced to seeking the conditions for which $a2$ and $a3$ are in the plane of the platform. To this end, vectors $b20×b30$ and $a2×a3$ are calculated and the rotation matrix which aligns these two vectors is calculated. Then, the rotation matrix of the degree-of-freedom allowed to remain in the singular configuration is computed. This matrix is the matrix of the rotation around vector $a2×a3$ with a free angle. However, in this particular state, $a2×a3$ is equal to $0$ because $a2$ and $a3$ are aligned. It is sufficient to calculate the rotation matrix defined by the cosine between $a2$ and its projection in the reference plane of the platform $a2p=(13−(((b20×b30)⋅(b20×b30)T)/(||b20×b30||2)))a2$ around the axis $a2p×a2$. So, the platform is free to rotate around $(b2×b3)$ and $a2$ and the singularity state is maintained. The method used to locate the alignment singularity loci can be used here too but two matrices rather than one must be introduced to represent the freedom allowed for the robot to remain in the singularity state. As before, $Q1$ positions the platform in the collinear singularity state and $Q2$ is the free rotation around $(b2×b3)$ and finally $Q3$ is the free rotation around $a2$. We get $Q=Q1Q2Q3$ which is equated to $R(ϕ,θ,σ)$. Similar to the previous case, five equations among nine are simple to solve and yield Display Formula

(A9)${ϕ=ϕθ=±π2σ=σ$

Other singularity cases: the other singularity cases can be avoided with the redundancy as shown above.

###### Appendix B: Example of the Method Used to Determine c as a Function of the Joint Coordinates

Vector c is subjected to the following constraint equations: Display Formula

(B1)$l112=(c−a11)T(c−a11)=cTc+a11Ta11−2a11Tc$
Display Formula
(B2)$l122=(c−a12)T(c−a12)=cTc+a12Ta12−2a12Tc$
Display Formula
(B3)$l2=(c−b1)T(c−b1)=cTc+b1Tb1−2b1Tc$

Subtracting the equations above from one another to eliminate $cTc$, one obtains Display Formula

(B4)$l112−l122=a11Ta11−a12Ta12−2(a11−a12)Tc$
Display Formula
(B5)$l112−l2=a11Ta11−b1Tb1−2(a11−b1)Tc$

Moreover, the coplanar condition between A11, A12, B1, and C can be written as Display Formula

(B6)$[(a11−a12)×(b1−a12)]T(c−a11)=0$

Expressed in matrix form, Eqs. (B4)(B6) become Display Formula

(B7)$[(l112−a11Ta11)−(l122−a12Ta12)(l112−a11Ta11)−(l2−b1Tb1)[(a11−a12)×(b1−a12)]Ta11]=[−2(a11−a12)T−2(a11−b1)T[(a11−a12)×(b1−a12)]T]c$
which is readily solved for c for all the range of motion required.

## References

Cox, D. J. , and Tesar, D. , 1989, “ The Dynamic Model of a Three Degree of Freedom Parallel Robotic Shoulder Module,” Advanced Robotics, Springer, Berlin, pp. 475–487.
Sefrioui, J. , and Gosselin, C. M. , 1994, “ Étude et représentation des lieux de singularité des manipulateurs parallelles sphériques a trois degrés de liberté avec actionneurs prismatiques,” Mech. Mach. Theory, 29(4), pp. 559–579.
Kong, X. , Gosselin, C. M. , and Ritchie, J. M. , 2011, “ Forward Displacement Analysis of a Linearly Actuated Quadratic Spherical Parallel Manipulator,” ASME J. Mech. Rob., 3(12), p. 011007.
Gosselin, C. M. , and Lavoie, E. , 1993, “ On the Kinematic Design of Spherical Three-Degree-of-Freedom Parallel Manipulators,” Int. J. Rob. Res., 12(4), pp. 393–402.
Gosselin, C. M. , St-Pierre, E. , and Gagné, M. , 1996, “ On the Development of the Agile Eye: Mechanical Design, Control Issues and Experimentation,” IEEE Rob. Autom. Soc. Mag., 3(4), pp. 29–37.
Gosselin, C. M. , Perreault, L. , and Vallancourt, C. , 1995, “ Simulation and Computer-Aided Kinematic Design of Three-Degree-of-Freedom Spherical Parallel Manipulators,” J. Rob. Syst., 12(12), pp. 857–869.
Cammarata, A. , and Sinatra, R. , 2008, “ The Elastodynamics of the 3-CRU Spherical Robot,” Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, Sept. 21–22, pp. 159–165.
Li, R. , and Guo, Y. , 2014, “ Research on Dynamics and Simulation of 3-RRP Spherical Parallel Mechanism,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–8.
Huda, S. , Takeda, Y. , and Hanagasaki, S. , 2008, “ Kinematic Design of 3-URU Pure Rotational Parallel Mechanism to Perform Precise Motion Within a Large Workspace,” Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, Sept. 21–22, pp. 49–56.
Hervé, J. M. , and Karouia, M. , 2002, “ The Novel 3-RUU Wrist With No Idle Pair,” First International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Quebec City, QC, Canada, Oct. 3–4, pp. 284–286.
Yu, J. , Lu, D. , and Hao, G. , 2014, “ Design and Analysis of a 2-DOF Compliant Parallel Pan-Tilt Platform,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–8.
Shen, H. , Yang, L. , Deng, J. , Li, J. , and Zhang, X. , 2014, “ A Parallel Shoulder Rehabilitation Training Mechanism and Its Kinematics Design,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–8.
Congzhe, W. , Yuefa, F. , Sheng, G. , and Zhihong, C. , 2014, “ Design and Kinematics of a Reconfigurable Robot for Ankle and Knee Rehabilitation,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–10.
Hayes, M. J. D. , Weiss, A. , and Langlois, R. G. , 2008, “ Atlas Motion Platform Generalized Kinematic Model,” Second International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Montpellier, France, Sept. 21–22, pp. 227–234.
Wu, G. , Caro, S. , and Wang, J. , 2015, “ Design and Transmission Analysis of an Asymmetrical Spherical Parallel Manipulator,” Mech. Mach. Theory, 94, pp. 119–131.
Stanisic, M. M. , and Duta, O. , 1990, “ Symmetrically Actuated Double Pointing Systems: The Basis of Singularity-Free Robot Wrists,” IEEE Trans. Rob. Autom., 6(5), pp. 562–569.
Leguay-Durand, S. , and Reboulet, C. , 1997, “ Optimal Design of a Redundant Spherical Parallel Manipulator,” Robotica, 15(4), pp. 399–405.
Di Gregorio, R. , 2002, “ A New Family of Spherical Parallel Manipulators,” Robotica, 20(4), pp. 353–358.
Bai, S. , 2010, “ Optimum Design of Spherical Parallel Manipulators for a Prescribed Workspace,” Mech. Mach. Theory, 45(2), pp. 200–211.
Liu, X.-J. , Jin, Z.-L. , and Gao, F. , 2000, “ Optimum Design of 3-DOF Spherical Parallel Manipulators With Respect to the Conditioning and Stiffness Indices,” Mech. Mach. Theory, 35(9), pp. 1257–1267.
Lum, M. J. , Rosen, J. , Sinanan, M. N. , and Hannaford, B. , 2006, “ Optimization of a Spherical Mechanism for a Minimally Invasive Surgical Robot: Theoretical and Experimental Approaches,” IEEE Trans. Biomed. Eng., 53(7), pp. 1440–1445. [PubMed]
Kurtz, R. , and Hayward, V. , 1992, “ Multiple-Goal Kinematic Optimization of a Parallel Spherical Mechanism With Actuator Redundancy,” IEEE Trans. Rob. Autom., 8(5), pp. 644–651.
Gosselin, C. M. , Laliberté, T. , and Veillette, A. , 2015, “ Singularity-Free Kinematically Redundant Planar Parallel Mechanisms With Unlimited Rotational Capability,” IEEE Trans. Rob., 31(2), pp. 457–467.
Bonev, I. A. , Zlatanov, D. , and Gosselin, C. M. , 2002, “ Advantages of the Modified Euler Angles in the Design and Control of PKMs,” Third Chemnitz Parallel Kinematics Seminar/2002 Parallel Kinematic Machines International Conference, Chemnitz, Germany, Apr. 23–25, pp. 171–188.
Gosselin, C. M. , and Angeles, J. , 1990, “ Singularity Analysis of Closed-Loop Kinematic Chains,” IEEE Trans. Rob. Autom., 6(3), pp. 281–290.
Wang, J. , and Gosselin, C. M. , 2004, “ Singularity Loci of a Special Class of Spherical 3-DOF Parallel Mechanisms With Prismatic Actuators,” ASME J. Mech. Des., 126(2), pp. 319–326.
Bonev, I. A. , and Gosselin, C. M. , 2005, “ Singularity Loci of Spherical Parallel Mechanisms,” IEEE International Conference on Robotics and Automation (ICRA), Barcelona, Spain, Apr. 18–22, pp. 2968–2973.
Bonev, I. A. , and Gosselin, C. M. , 2006, “ Analytical Determination of the Workspace of Symmetrical Spherical Parallel Mechanisms,” IEEE Trans. Rob., 22(5), pp. 2968–2973.
View article in PDF format.

## References

Cox, D. J. , and Tesar, D. , 1989, “ The Dynamic Model of a Three Degree of Freedom Parallel Robotic Shoulder Module,” Advanced Robotics, Springer, Berlin, pp. 475–487.
Sefrioui, J. , and Gosselin, C. M. , 1994, “ Étude et représentation des lieux de singularité des manipulateurs parallelles sphériques a trois degrés de liberté avec actionneurs prismatiques,” Mech. Mach. Theory, 29(4), pp. 559–579.
Kong, X. , Gosselin, C. M. , and Ritchie, J. M. , 2011, “ Forward Displacement Analysis of a Linearly Actuated Quadratic Spherical Parallel Manipulator,” ASME J. Mech. Rob., 3(12), p. 011007.
Gosselin, C. M. , and Lavoie, E. , 1993, “ On the Kinematic Design of Spherical Three-Degree-of-Freedom Parallel Manipulators,” Int. J. Rob. Res., 12(4), pp. 393–402.
Gosselin, C. M. , St-Pierre, E. , and Gagné, M. , 1996, “ On the Development of the Agile Eye: Mechanical Design, Control Issues and Experimentation,” IEEE Rob. Autom. Soc. Mag., 3(4), pp. 29–37.
Gosselin, C. M. , Perreault, L. , and Vallancourt, C. , 1995, “ Simulation and Computer-Aided Kinematic Design of Three-Degree-of-Freedom Spherical Parallel Manipulators,” J. Rob. Syst., 12(12), pp. 857–869.
Cammarata, A. , and Sinatra, R. , 2008, “ The Elastodynamics of the 3-CRU Spherical Robot,” Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, Sept. 21–22, pp. 159–165.
Li, R. , and Guo, Y. , 2014, “ Research on Dynamics and Simulation of 3-RRP Spherical Parallel Mechanism,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–8.
Huda, S. , Takeda, Y. , and Hanagasaki, S. , 2008, “ Kinematic Design of 3-URU Pure Rotational Parallel Mechanism to Perform Precise Motion Within a Large Workspace,” Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, France, Sept. 21–22, pp. 49–56.
Hervé, J. M. , and Karouia, M. , 2002, “ The Novel 3-RUU Wrist With No Idle Pair,” First International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Quebec City, QC, Canada, Oct. 3–4, pp. 284–286.
Yu, J. , Lu, D. , and Hao, G. , 2014, “ Design and Analysis of a 2-DOF Compliant Parallel Pan-Tilt Platform,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–8.
Shen, H. , Yang, L. , Deng, J. , Li, J. , and Zhang, X. , 2014, “ A Parallel Shoulder Rehabilitation Training Mechanism and Its Kinematics Design,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–8.
Congzhe, W. , Yuefa, F. , Sheng, G. , and Zhihong, C. , 2014, “ Design and Kinematics of a Reconfigurable Robot for Ankle and Knee Rehabilitation,” Third International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Tianjin, China, July 7–8, pp. 1–10.
Hayes, M. J. D. , Weiss, A. , and Langlois, R. G. , 2008, “ Atlas Motion Platform Generalized Kinematic Model,” Second International Workshop on Fundamental Issues and Future Directions for Parallel Mechanisms and Manipulators, Montpellier, France, Sept. 21–22, pp. 227–234.
Wu, G. , Caro, S. , and Wang, J. , 2015, “ Design and Transmission Analysis of an Asymmetrical Spherical Parallel Manipulator,” Mech. Mach. Theory, 94, pp. 119–131.
Stanisic, M. M. , and Duta, O. , 1990, “ Symmetrically Actuated Double Pointing Systems: The Basis of Singularity-Free Robot Wrists,” IEEE Trans. Rob. Autom., 6(5), pp. 562–569.
Leguay-Durand, S. , and Reboulet, C. , 1997, “ Optimal Design of a Redundant Spherical Parallel Manipulator,” Robotica, 15(4), pp. 399–405.
Di Gregorio, R. , 2002, “ A New Family of Spherical Parallel Manipulators,” Robotica, 20(4), pp. 353–358.
Bai, S. , 2010, “ Optimum Design of Spherical Parallel Manipulators for a Prescribed Workspace,” Mech. Mach. Theory, 45(2), pp. 200–211.
Liu, X.-J. , Jin, Z.-L. , and Gao, F. , 2000, “ Optimum Design of 3-DOF Spherical Parallel Manipulators With Respect to the Conditioning and Stiffness Indices,” Mech. Mach. Theory, 35(9), pp. 1257–1267.
Lum, M. J. , Rosen, J. , Sinanan, M. N. , and Hannaford, B. , 2006, “ Optimization of a Spherical Mechanism for a Minimally Invasive Surgical Robot: Theoretical and Experimental Approaches,” IEEE Trans. Biomed. Eng., 53(7), pp. 1440–1445. [PubMed]
Kurtz, R. , and Hayward, V. , 1992, “ Multiple-Goal Kinematic Optimization of a Parallel Spherical Mechanism With Actuator Redundancy,” IEEE Trans. Rob. Autom., 8(5), pp. 644–651.
Gosselin, C. M. , Laliberté, T. , and Veillette, A. , 2015, “ Singularity-Free Kinematically Redundant Planar Parallel Mechanisms With Unlimited Rotational Capability,” IEEE Trans. Rob., 31(2), pp. 457–467.
Bonev, I. A. , Zlatanov, D. , and Gosselin, C. M. , 2002, “ Advantages of the Modified Euler Angles in the Design and Control of PKMs,” Third Chemnitz Parallel Kinematics Seminar/2002 Parallel Kinematic Machines International Conference, Chemnitz, Germany, Apr. 23–25, pp. 171–188.
Gosselin, C. M. , and Angeles, J. , 1990, “ Singularity Analysis of Closed-Loop Kinematic Chains,” IEEE Trans. Rob. Autom., 6(3), pp. 281–290.
Wang, J. , and Gosselin, C. M. , 2004, “ Singularity Loci of a Special Class of Spherical 3-DOF Parallel Mechanisms With Prismatic Actuators,” ASME J. Mech. Des., 126(2), pp. 319–326.
Bonev, I. A. , and Gosselin, C. M. , 2005, “ Singularity Loci of Spherical Parallel Mechanisms,” IEEE International Conference on Robotics and Automation (ICRA), Barcelona, Spain, Apr. 18–22, pp. 2968–2973.
Bonev, I. A. , and Gosselin, C. M. , 2006, “ Analytical Determination of the Workspace of Symmetrical Spherical Parallel Mechanisms,” IEEE Trans. Rob., 22(5), pp. 2968–2973.

## Figures

Fig. 1

Architecture of the proposed 4DOF kinematically redundant spherical robot

Fig. 2

Kinematic modeling of the 4DOF redundant spherical parallel robot

Fig. 3

Illustration of the admissible values for parameter β

Fig. 4

Particular architecture of a spherical 4DOF redundant parallel robot. Vector z (not represented) is collinear with vectorn.

Fig. 5

Joint coordinates of the robot during trajectory 1 with optimized redundancy

Fig. 6

Joint coordinates of the robot during trajectory 2 with optimized redundancy

Fig. 7

Determinant of matrix j during trajectory 1 for the redundant architecture and the nonredundant architecture. The determinant for the redundant architecture is multiplied by 10 for scale considerations.

Fig. 8

Determinant of matrix j during trajectory 2 for the redundant architecture and the nonredundant architecture. The determinant for the redundant architecture is multiplied by 10 for scale considerations.

Fig. 9

Architecture of the RPR leg and of the equivalent RRR leg

Fig. 10

Prototype of the spherical parallel manipulator

## Tables

Table 1 Architectural parameters chosen for the spherical robot in the example section
Table 2 Architectural parameters chosen for the prototype of spherical robot. Lengths are given in millimeters.

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