Parallel manipulators feature relatively high payload and accuracy capabilities compared to their serial counterparts. However, they suffer from small workspace and low maneuverability. Kinematic redundancy for parallel manipulators can improve both of these characteristics. This paper presents a family of new kinematically redundant planar parallel manipulators with six actuated-joint degrees of freedom based on a $3-PṞRR$ architecture obtained by adding an active prismatic joint at the base of each limb of the $3-ṞRR$ manipulator. First, the inverse displacement of the manipulators is explained, then their reachable and dexterous workspaces are obtained. Comparing the proposed redundant manipulators to the original $3-ṞRR$ nonredundant manipulator, both reachable and dexterous workspaces are substantially larger. Next, the Jacobian matrices of the manipulators are derived, and different types of singularities are analyzed and demonstrated. It is shown that the vast majority of singularities can be avoided by using kinematic redundancy.

## Introduction

2Higher accuracy, speed, and payload-to-weight ratio are the major advantages of parallel manipulators compared to serial manipulators. A smaller workspace, reduced dexterity, as well as complex kinematic and dynamic models are their major drawbacks. Much research has been conducted on kinematics and dynamics of parallel manipulators (1,2,3,4,5,6,7,8), as well as on singularity and workspace analyses (9,10,11,12,13).

The vast majority of the studies on parallel manipulators have focused on nonredundant parallel manipulators. Redundant parallel manipulators have been introduced to alleviate some of the aforementioned shortcomings of parallel manipulators. Redundancy in parallel manipulators was introduced in Refs. 14,15,16. Redundancy can be divided into two main types: actuation redundancy and kinematic redundancy. Actuation redundancy is defined as replacing existing passive joints of a manipulator by active ones. Actuation redundancy does not change the mobility or reachable workspace of a manipulator but entails the manipulator having more actuators than are needed for a given task and may be used to reduce the singularities within the manipulator’s workspace (17). For example, the planar $3-ṞRR$ parallel manipulator, shown in Fig. 1, becomes redundantly actuated when a normally passive revolute joint such as $Bi$ or $Di$ is replaced with an active joint.

Figure 1
Figure 1
Close modal

Kinematic redundancy increases the mobility and actuated-joint degrees of freedom (ADOFs) of parallel manipulators. Kinematic redundancy takes place when extra active joints and links (if needed) are added to manipulators. For instance, by adding one extra active prismatic joint to one limb of the $3-ṞRR$, it is converted into a kinematically redundant parallel manipulator (see Fig. 2). In this example, the resulting redundant parallel manipulator has 4-ADOF, one more than the planar task space. In general, kinematic redundant parallel manipulators need more controlling parameters than required for a set of given tasks (17). A manipulator with sufficient kinematic redundancy can avoid all interior singularities, has a larger workspace, and improved maneuverability (18). Using kinematic redundancy can also allow manipulators to cope with joint-jam failure (e.g., Ref. 19).

Figure 2
Figure 2
Close modal

Compared to other research that has been pursued for parallel manipulators, redundancy in such manipulators has not adequately been studied. In serial manipulators, the concept of kinematic redundancy has been broadly studied (see, for instance, Refs. 20,21,22).

In the present work, a family of 6-ADOF kinematically redundant planar parallel manipulators is introduced. Note that the task space of all the manipulators that are considered in this work is a three degrees of freedom planar motion. First, the proposed $3-PṞRR$ manipulator family obtained by adding an active prismatic joint at the base of each limb of a $3-ṞRR$ manipulator is presented. Then, their inverse displacement problem (IDP) is explained. Thereafter, the reachable and dexterous workspaces of the original nonredundant $3-ṞRR$ manipulator are obtained and compared to those of the redundant manipulators. Finally, the direct, inverse, and combined singularities of the aforementioned manipulators are analyzed and illustrated.

## Proposed Architectures

Adding one degree of kinematic redundancy (1-DOKR) can considerably reduce the number of direct kinematic singularities of a $3-ṞRR$ planar manipulator but not all of them (17). By adding 2-DOKR, it is possible to avoid all direct kinematic singularities of the $3-ṞRR$ manipulators. A symmetrical architecture is usually desirable. Also, as is shown later, kinematic redundancy increases the workspace; hence, 1-DOKR is added here to each limb of the $3-ṞRR$ manipulator producing manipulators with a total of 3-DOKR. Therefore, the family of redundant manipulators proposed here has 6-ADOF for a planar task, three of which are redundant. The added kinematic redundancies enable the manipulators to avoid kinematic singularities, improve their maneuverability, and enlarge their reachable and dexterous workspaces.

Figures 3,4,5 illustrate a new family of three 6-ADOF redundant planar parallel manipulators. The notation used for each manipulator is based on the shape of the guides on which the redundant prismatic actuators slide. All $3-PṞRR$ manipulators are based on the nonredundant $3-ṞRR$ planar parallel manipulator proposed in Ref. 23 (Fig. 1). Each limb of the $3-PṞRR$ manipulators has one prismatic actuator at its base. Adding prismatic redundant actuators close to the base causes less dynamic effects due to the weight of the actuators.

Figure 3
Figure 3
Close modal
Figure 4
Figure 4
Close modal
Figure 5
Figure 5
Close modal

The redundant prismatic actuators slide on their respective guides that can take the shape of a triangle, a star, and a circle. The notation is thus $3-PṞRR▵$, $3-PṞRR★$, and $3-PṞRR○$ for guides that have a triangle, star, or circle shape, respectively. Note that the $3-PṞRR○$ manipulator is, in fact, a $3-RṞRR$ manipulator, since a circular prismatic joint with a finite radius is, in fact, a revolute joint. The $3-PṞRR○$ notation is maintained to more readily identify the similarities of each manipulator in the proposed family. An actuated revolute joint is mounted on the prismatic actuator at Point $Ai$ (throughout the present work, $i=1,2,3$). Note that the solid circles in all the figures represent active revolute joints, whereas the empty circles represent passive ones. Finally, two passive revolute joints are at $Di$ and $Bi$, where Point $Bi$ is attached to the end effector.

## Inverse Displacement of the Proposed Architectures

Kinematic redundancy in manipulators results in having an infinite number of solutions to the IDP. That is, rather than having a finite number of solutions, there may be one or more loci of joint variable values (angles or lengths) for a given position and orientation of the end effector. Figure 2 illustrates a situation where, by adding an extra prismatic actuator to the first limb, 1-DOKR is added to that limb of a $3-ṞRR$ planar parallel manipulator. The resulting kinematically redundant manipulator has 4-ADOF for a planar task. That is, as long as two circles centered at $B1$ and $A1$ and radii of $l1$ and $d1$, respectively, intersect each other, there are infinite solutions for the IDP of the manipulator.

The hatched region in Fig. 2 shows the area that Point $D1$ of Link $A1D1$ can cover when the prismatic Actuator $A1$ slides within a certain range and Link $A1D1$ rotates around Point $A1$. As it is evident from Fig. 2, Point $D1$ as a part of Link $B1D1$ (pinned at Point $B1$) can rotate a full circle around Point $B1$. Therefore, all possible solutions of the IDP for that limb lie on the intersection of the aforesaid regions. Assuming that the IDP exists for Limb 1, the redundant manipulator has a locus of solutions shown as arc $RD1Ŝ$. Note that the IDP for each limb of the original nonredundant $3-ṞRR$ has at the most two solutions.

Finding the best inverse displacement solution for redundant manipulators is normally formulated as a numerical optimization problem (21). To find an optimal solution of the IDP, different cost functions may be defined for different purposes such as minimum energy, distance, time, or condition number (21,24). Note that once a position for one of the active Joints $Ai$ or $θi$ is chosen, there are only two solutions for the IDP of each limb for a given position and orientation of the end effector.

Figure 6 shows the details of the end effector and its associated dimensions, which are the same for all the manipulators. Considering that the end-effector location is measured from Point $P$ and the geometry, of the $3-PṞRR$ manipulator family as shown in Figs. 3,4,5,6, the IDP of the manipulator can be written as
$DiBi¯=DiAi¯+AiO¯+OP¯+PBi¯$
1
Equation 1 can be expressed as
$xDiBi¯2+yDiBi¯2=x(DiAi¯+AiO¯+OP¯+PBi¯)2+y(DiAi¯+AiO¯+OP¯+PBi¯)2$
2
$xDiBi¯2+yDiBi¯2=li2$
3
Figure 6
Figure 6
Close modal
For the $3-PṞRR▵$ and the $3-PṞRR★$ manipulators, the inverse displacement equations can be written as
$li2=(xp+ric(ϕ+ψi)−Rcβi−ρicτiΔ,⋆−dicθi)2+(yp+ris(ϕ+ψi)−Rsβi−ρisτiΔ,⋆−disθi)2$
4
where $c∡$ and $s∡$ represent $cos(∡)$ and $sin(∡)$, respectively. All angles except $ψi$ (which is measured relative to $B1B2$, see Fig. 6) are measured with respect to the $X$-axis. Also, a double architecture identification symbol indicates the angle to be used depending on the considered kinematic architecture. For example $cτ3▵,★$ refers to the cosine of $τ3▵$ or $τ3★$ depending on the architecture. Angles $τi$ are defined as5
$τ1Δ=0τ1⋆=β1−β2+π2$
$τ2Δ=β3−β1+2π2τ2⋆=β2−β1+π2$
$τ3Δ=β3−β2+4π2τ3⋆=2β3−β2−β1+π2$
Similarly, for the $3-PṞRR○$ manipulator, the inverse displacement equations can be written as
$li2=(xp+ric(ϕ+ψi)−Rcβi−dicθi)2+(yp+ris(ϕ+ψi)−Rsβi−disθi)2$
6

For the $3-PṞRR▵$ and $3-PṞRR★$ manipulators, the end effector pose and the actuator vectors can be expressed as $x=[xp,yp,ϕ]T$ and $q=[ρ1,θ1,ρ2,θ2,ρ3,θ3]T$, respectively. For the $3-PṞRR○$ manipulator, they can be defined as $x=[xp,yp,ϕ]T$ and $q=[β1,θ1,β2,θ2,β3,θ3]T$, respectively.

Regardless of the optimization method that is chosen, the selected inverse displacement solution depends on the initial pose and configuration of each limb as well as the cost function that is used (18,24). This work mainly focuses on introducing a family of kinematically redundant planar manipulators and analyzing the manipulators’ workspace and kinematic singularities. Optimal path selection is an extensive topic on its own and is left for subsequent research.

## Workspace Comparison of the Manipulators

In this work, two types of workspaces are studied: reachable and dexterous. The reachable (maximal) workspace is considered here as the region that contains every point that can be reached by the end effector in at least one orientation (25). The condition for being accepted as a point in the reachable workspace is the existence of a solution to the IDP while considering only the position of the end effector. The dexterous workspace, on the other hand, is considered as the region of the reachable workspace that can be reached by the end effector with all possible orientations (from zero to $2π$) (25). The dexterous workspace of each of the four manipulators analyzed here is geometrically obtained, as shown in Figs. 7b,8b,9b,10b, for the $3-ṞRR$, $3-PṞRR▵$, $3-PṞRR★$, and $3-PṞRR○$, respectively. For every branch of any of the $3-PṞRR$ manipulators, each leg can slide at Point $Ai$. The dexterous workspace of each branch can be represented by a circle whose radius is $li+di−ri$ and is swept along slider $i$. The intersection of the three regions made by the reachable workspace of the three branches will be the dexterous workspaces of the manipulator (26). For the $3-ṞRR$, the same processs is followed except that the circle representing the reachable workspace of each branch is kept with its center at a constant location $Ci$.

Figure 7
Figure 7
Close modal
Figure 8
Figure 8
Close modal
Figure 9
Figure 9
Close modal
Figure 10
Figure 10
Close modal

The reachable workspaces of the manipulators can be geometrically obtained too but, due to the complexity and lengthiness of the procedure, it is not explained here. For details, one can refer to the work that was done in Ref. 26 for the nonredundant $3-RP̱R$. Here, the reachable workspace of each of the manipulators is numerically obtained and will be used for comparing the manipulators’ capabilities.

The reachable and dexterous workspaces of the $3-ṞRR$ nonredundant manipulator are shown in Fig. 7. The workspaces of the redundant $3-PṞRR$ manipulators are compared in Figs. 8,9,10. In Figs. 8,9,10, the base platforms (the sliders’ guides) are identified as the triangle $C1C2C3$, the star $C1C2C3$, and the circle, respectively. The geometric parameters for the $3-ṞRR$ and all the $3-PṞRR$ manipulators analyzed here are $R=0.6m$, $li=0.6m$, $di=0.6m$, $ri=0.1155m$, $ψ1=7π∕6$, $ψ2=11π∕6$, $ψ3=π∕2$, and $∡C1OC2=∡C1OC3=2π∕3$.

Table 1 shows the absolute and relative sizes (written as Abs. and Rel., respectively) of the reachable and dexterous workspaces of the $3-PṞRR$ manipulators relative to $3-ṞRR$. As expected, both the reachable and dexterous workspaces of the $3-PṞRR$ manipulators are considerably larger than those of the $3-ṞRR$ manipulator.

Table 1

Workspace comparison for the nonredundant and redundant manipulators. All absolute quantities are in $m2$.

 Reachable Dexterous Ratio $DexterousReachable$ Abs. Rel. Abs. Rel. $3-ṞRR$ 1.89 1 0.92 1 0.49 $3-PṞRRΔ$ 4.54 2.40 2.90 3.15 0.64 $3-PṞRR★$ 5.00 2.64 3.70 4.02 0.74 $3-PṞRR○$ 9.37 4.96 8.93 9.71 0.95
 Reachable Dexterous Ratio $DexterousReachable$ Abs. Rel. Abs. Rel. $3-ṞRR$ 1.89 1 0.92 1 0.49 $3-PṞRRΔ$ 4.54 2.40 2.90 3.15 0.64 $3-PṞRR★$ 5.00 2.64 3.70 4.02 0.74 $3-PṞRR○$ 9.37 4.96 8.93 9.71 0.95

Considering the last column of Table 1, it can also be inferred that kinematic redundancy results in considerable increase in the ratio of the dexterous over the reachable workspace area compared to the nonredundant $3-ṞRR$ manipulator. This ratio varies for the different arrangements of the prismatic redundant actuators, with the maximum value being for the circular prismatic redundant actuators and the minimum for the triangle-shape path.

## Kinematic Singularity Analysis

### Types of Kinematic Singularities

Jacobian matrices transform the velocity vector of the end effector into a velocity vector of the active joints. That is,
$Jxẋ=Jqq̇$
7
where $ẋ$ is the velocity vector of the end effector $ẋ=[ẋp,ẏp,ϕ̇]T$, and $q̇$ is the velocity vector of the associated active joints, $q̇=[ρ̇1,θ̇1,ρ̇2,θ̇2,ρ̇3,θ̇3]T$ for the $3-PṞRR▵$ and $3-PṞRR★$ manipulators and $q̇=[β̇1,θ̇1,β̇2,θ̇2,β̇3,θ̇3]T$ for the $3-PṞRR○$ manipulator. Considering Eq. 7, three types of kinematic singularities can be defined for parallel manipulators (27).
• 1

Direct kinematic singularities when $Jx$ is singular.

• 2

Inverse kinematic singularities when $Jq$ is singular.

• 3

Combined (complex) singularities when $Jx$ and $Jq$ are singular.

Direct kinematic singularities (i.e., $∣Jx∣=0$) take place when there are some nonzero velocities of the end effector that are possible with zero velocities at the actuators. Also, direct kinematic singularities are referred to as force uncertainties or as force unconstrained poses (28). On the other hand, inverse kinematic singularities (i.e., when $∣Jq∣=0$) happen when there exist some nonzero actuator velocities that cause zero velocities for the end effector. In order to perform a kinematic singularity analysis on the introduced $3-PṞRR$ manipulator family, their Jacobian matrices are obtained in the following section.

### Jacobian Matrices of the 3-PṞRR Manipulators

By differentiating Eq. 4 with respect to time, the Jacobian matrices are given by
$Jx∀=[t11∀t12∀t13∀t21∀t22∀t23∀t31∀t31∀t33∀]3×3$
8
$Jq∀=[u1∀v1∀000000u2∀v2∀000000u3∀v3∀]3×6$
9
where $∀=▵,★,○$, depending on the manipulator. For the $3-PṞRR▵$ and $3-PṞRR★$ manipulators, the elements are
$ti1Δ,⋆=xp+ric(ϕ+ψi)−Rcβi−ρicτiΔ,⋆−dicθi$
10
$ti2Δ,⋆=yp+ris(ϕ+ψi)−Rsβi−ρ2sτ2Δ,⋆−disθi$
11
$ti3Δ,⋆=−ris(ϕ+ψi)ti1Δ,⋆+ric(ϕ+ψi)ti2Δ,⋆$
12
$uiΔ,⋆=cτiΔ,⋆ti1Δ,⋆+sτiΔ,⋆ti2Δ,⋆$
13
$viΔ,⋆=−disθiti1Δ,⋆+dicθ1ti2Δ,⋆$
14
whereas for the $3-PṞRR○$, the elements are
$ti1○=xp+ric(ϕ+ψi)−Rcβi−dicθi$
15
$ti2○=yp+ris(ϕ+ψi)−Rsβi−disθi$
16
$ti3○=−ris(ϕ+ψi)ti1○+ric(ϕ+ψi)ti2○$
17
$ui○=−Rsβiti1○+Rcβiti2○$
18
$vi○=−disθiti1○+dicθiti2○$
19

### Direct Kinematic Singularities of the 3-PṞRR Manipulators

The direct kinematic singularities of planar parallel manipulators with architectures whose distal links have passive revolute joints at both ends are the same. They all take place when the lines defining the distal links meet at a common point. The reason for this is that forces can only be transmitted in the distal link directions, and when all of them meet at a common point, the end effector can infinitesimally rotate around that point while all the actuators are locked. As shown in Figs. 3,4,5, all manipulators have passive revolute joints at both ends of the distal links (i.e., $Bi$ and $Di$).

The direct Jacobian matrix of all manipulators can be rewritten as
$Jx∀=[xl1yl1−r1s(ϕ+ψ1)xl1+r1c(ϕ+ψ1)yl1xl2yl2−r2s(ϕ+ψ2)xl2+r2c(ϕ+ψ2)yl2xl3yl3−r3s(ϕ+ψ3)xl3+r3c(ϕ+ψ3)yl3]3×3=[l1cα1l1sα1r1l1s(α1−ϕ−ψ1)l2cα2l2sα2r2l2s(α2−ϕ−ψ2)l3cα3l3sα3r3l3s(α3−ϕ−ψ3)]3×3$
20
where $xli$ and $yli$ are the projections of Links $BiDi$ onto the $X$ and $Y$ axes, respectively. The determinant of $Jx∀$ is zero, when there are linear dependencies between any two or more rows or columns. That is,
$λ1W1+λ2W2+λ3W3=0$
21
where $λi$ are scalar coefficients of linear dependency of which only up to two can be simultaneously zero and vector $Wi$ represents the $ith$ row (or column) of $Jx∀$. Having only one $λi$ nonzero represents configurations where all the elements of a row or a column are zero. Considering Eq. 20, the elements of the first or second column are zero when $xli=0$ or $yli=0$, respectively. These two conditions occur when all the distal links are parallel to the $Y$-axis or $X$-axis, respectively. In both cases, the distal links meet at a common point at infinity. The third column becomes zero, when all the distal links meet at Point $P$. Also, linear dependency between the rows have the same meaning as columns. Linear dependency between any two rows happens when two distal links are aligned with the side of the end effector that is between them.

Note that kinematic singularities are independent of the coordinate system. For the discussed manipulators, a singularity configuration only depends on relative positions and orientations of the distal links. Therefore, the aforementioned direct kinematic singularities can take place when all the distal links are parallel regardless of the common direction or where they all meet at a common point.

### Inverse Kinematic Singularities of the 3-PṞRR Manipulators

For redundant parallel manipulators, the matrix $Jq∀$ is not square, and therefore, the inverse kinematic singularities can be said to occur when the rank of $Jq∀$ is lower than the degrees of freedom of the end effector (16), that is, the number of rows of $Jq∀$. Therefore, a kinematically redundant parallel manipulator is in an inverse kinematic singular configuration when any minor square matrix extracted from $Jq∀$ is singular. This degeneracy can also be identified as the condition that sets the determinant of $Jq∀Jq∀T$ to zero. For the family of manipulators being presented,
$Jq∀Jq∀T=[u12+v12000u22+v22000u32+u32]3×3$
22
The elements of the matrix in Eq. 22 for the $3-PṞRR▵$ and the $3-PṞRR★$ can be expressed as
$uiΔ,⋆=lic(αi−τi),viΔ,⋆=dilis(αi−θi)$
23
whereas those for the $3-PṞRR○$ can be expressed as
$ui○=Rlis(αi−βi),vi○=dilis(αi−θi)$
24
Since matrix $Jq∀Jq∀T$ is diagonal, its determinant becomes zero when any one or more of the diagonal elements become zero, that is, $ui∀2$ and $vi∀2$ would have to be simultaneously zero. For the $3-PṞRR▵$ and the $3-PṞRR★$ manipulators, each diagonal element would be zero when both of the following conditions are satisfied:25
$αiΔ,⋆=2n1+12π+τiΔ,⋆,n1=0,1,2,…and$
$αiΔ,⋆=n2π+θiΔ,⋆,n2=0,1,2,…$
The above conditions show that inverse kinematic singularities for the $3-PṞRR▵$ and $3-PṞRR★$ manipulators occur when the sliding paths of the prismatic actuators (i.e., $CiCi+1$ for the $3-PṞRR▵$ and $CiO$ for the $3-PṞRR★$ manipulators) are perpendicular to $BiDi$ and also limb $AiDiBi$ is fully stretched or fully folded (i.e., $AiDi∥DiBi$), as shown in Figs. 11a,11b.
Figure 11
Figure 11
Close modal

For the $3-PṞRR○$ manipulator, each diagonal element of $Jq∀Jq∀T$ becomes zero when both of the following conditions are satisfied.26

$αi○=n1π+βi○,n1=0,1,2,…and$
$αi○=n2π+θi○,n2=0,1,2,…$
The above conditions for the inverse kinematic singularities of the $3-PṞRR○$ imply that the inverse kinematic singularities take place when at least one limb is fully stretched or fully folded and also passes through the center of the circle (i.e., Point $O$). Figure 11c illustrates an inverse kinematic singular configuration when the first and third limbs are fully stretched and fully folded, respectively, and also pass through the center of the guide’s circle for the prismatic actuators. It should be noted that for the entire family of $3-PṞRR$ manipulators, as long as the manipulator is not in its workspace boundary, it is possible to choose one or more sets of joint displacements that avoid the inverse kinematic singularity conditions outlined here.

### Combined (Complex) Kinematic Singularities of the 3-PṞRR Manipulators

The combined kinematic singularities happen when both direct and inverse kinematic singularities take place; in other words, when both $Jx∀$ and $Jq∀Jq∀T$ are singular. Figures 12a,12b,12c show sample configurations in which both the direct and inverse singularities occur at the same time for the three manipulators of the $3-PṞRR$ family introduced here.

Figure 12
Figure 12
Close modal

## Conclusions

A family of kinematically redundant planar parallel manipulators was proposed. Their IDPs were explained. Then, their reachable and dexterous workspaces were determined and compared to those of a similar-sized nonredundant manipulator. It was shown and compared that the novel manipulators have substantially larger reachable and dexterous workspaces. Thereafter, the kinematic singularity analyses were presented and their geometrical interpretations were explained for the $3-PṞRR$ manipulator family. It can be generally concluded that inverse kinematic singularities occur for the whole family of $3-PṞRR$ manipulators when at least one limb is fully stretched or fully folded while the limb is perpendicular to the sliding direction of the prismatic actuator, i.e., when the entire limb is collinear with the line that passes through the center of curvature of the slider’s path at that point.

Generally, the proposed manipulators have loci of solutions for every pose inside their workspaces. Therefore, they have the capability to be programed to optimize different cost functions such as minimum time and energy, while constrained to avoid kinematic singularities. The path shape of the prismatic redundant actuators (e.g., triangle, star, and circle) has a significant effect on the workspace shape. It was shown that kinematic redundancy not only increases the workspace of the manipulators but also significantly adds to the ratio of the dexterous workspace over the reachable workspace.

## Acknowledgment

The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for its support in funding this research.

The terminology used is the following. A $3-ṞRR$ mechanism indicates that the end effector is connected to the base by three serial kinematic chains (limbs), each consisting of an active (and therefore underlined) revolute joint (R) connected to the base, followed by two passive revolute joints, the second of which connects the limb to the end effector. In the adopted notation, a prismatic joint would be shown by a P and would be underlined if it were active.

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