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.

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).

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.

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
For the 3-PṞRR and the 3-PṞRR manipulators, the inverse displacement equations can be written as 
li2=(xp+ric(ϕ+ψi)RcβiρicτiΔ,dicθi)2+(yp+ris(ϕ+ψi)RsβiρisτiΔ,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 as
5
 
τ1Δ=0τ1=β1β2+π2
 
τ2Δ=β3β1+2π2τ2=β2β1+π2
 
τ3Δ=β3β2+4π2τ3=2β3β2β1+π2
Similarly, for the 3-PṞRR manipulator, the inverse displacement equations can be written as 
li2=(xp+ric(ϕ+ψi)Rcβidicθi)2+(yp+ris(ϕ+ψi)Rsβidisθi)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+diri 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.

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=7π6, ψ2=11π6, ψ3=π2, and C1OC2=C1OC3=2π3.

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.

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=[t11t12t13t21t22t23t31t31t33]3×3
8
 
Jq=[u1v1000000u2v2000000u3v3]3×6
9
where =,,, depending on the manipulator. For the 3-PṞRR and 3-PṞRR manipulators, the elements are 
ti1Δ,=xp+ric(ϕ+ψi)RcβiρicτiΔ,dicθi
10
 
ti2Δ,=yp+ris(ϕ+ψi)Rsβiρ2sτ2Δ,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)Rcβidicθi
15
 
ti2=yp+ris(ϕ+ψi)Rsβidisθi
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=[xl1yl1r1s(ϕ+ψ1)xl1+r1c(ϕ+ψ1)yl1xl2yl2r2s(ϕ+ψ2)xl2+r2c(ϕ+ψ2)yl2xl3yl3r3s(ϕ+ψ3)xl3+r3c(ϕ+ψ3)yl3]3×3=[l1cα1l1sα1r1l1s(α1ϕψ1)l2cα2l2sα2r2l2s(α2ϕψ2)l3cα3l3sα3r3l3s(α3ϕψ3)]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 JqJqT to zero. For the family of manipulators being presented, 
JqJqT=[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(αiτi),viΔ,=dilis(αiθi)
23
whereas those for the 3-PṞRR can be expressed as 
ui=Rlis(αiβi),vi=dilis(αiθi)
24
Since matrix JqJqT is diagonal, its determinant becomes zero when any one or more of the diagonal elements become zero, that is, ui2 and vi2 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., AiDiDiBi), as shown in Figs. 11a,11b.

For the 3-PṞRR manipulator, each diagonal element of JqJqT 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 JqJqT 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.

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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