A Novel Three-loop Parallel Robot with Full Mobility: Kinematics, Singularity, Workspace and Dexterity Analysis

A novel parallel robot, dubbed the SDelta, is the subject of this paper. SDelta is a simpler alternative to both the well-known Stewart-Gough platform (SGP) and current three-limb, full-mobility parallel robots, as it contains fewer components and all its motors are located on the base. This reduces the inertial load on the system, making it a good candidate for high-speed operations. SDelta features a symmetric structure; its forward-displacement analysis leads to a system of three quadratic equations in three unknowns, which admits up to eight solutions, or half the number of those admitted by the SGP. The kinematic analysis, undertaken with a geometrical method based on screw theory, leads to two Jacobian matrices, whose singularity conditions are investigated. Instead of using the determinant of a 6 × 6 matrix, we derive one simple expression that characterizes the singularity condition. This approach is also applicable to a large number of parallel robots whose six actuation wrench axes intersect pairwise, such as the SGP and three-limb parallel robots whose limbs include, each, a passive spherical joint. The workspace is analyzed via a geometric method, while the dexterity analysis is conducted via discretization. Both show that the given robot has the potential to offer both large workspace and good dexterity with a proper choice of design variables.


Introduction
A parallel robot, a.k.a. a parallel-kinematics machine (PKM), is defined as a multi-degree-of-freedom (multi-dof) articulated mechanical system composed of one moving platform (MP) and one base platform (BP), connected by at least two serial limbs [1].Compared with their serial counterparts, PKMs offer many attractive features, as per the pertinent literature [1][2][3].On the other hand, PKMs suffer of limited workspace, multiple singularities and coupled motion 1 , which makes their control especially challenging.These issues have motivated extensive research on PKMs.A special 3-CPS robot, dubbed SDelta 6 , together with its kinematic and singularity analyses, is discussed.The novel architecture is described in Section 2. Sections 3 and 4 pertain to the forward-displacement and kinematic analyses, respectively.In Section 5, the singularity analysis is provided, which leads to a compact formulation.Sections 6 and 7 pertain to the workspace and dexterity analyses; Section 8 includes a discussion on the performance, and Section 9 summarizes the paper and provides concluding remarks.

The SDelta Architecture
First and foremost, what distinguishes the SDelta from other three-limb (or three-loop) parallel robots is its topology, as it is based on three parallel actuation mechanisms, each with a RHHR topology [43], as displayed in Figs. 1 and 2. In these figures, 0 denotes the BP, while the closed subchain 0-1-2-3-0 denotes one of the three actuation mechanisms, one on each side of the equilateral triangle of Fig. 1.In these subchains, 2 is the driving link of one limb, 5 and 8 those of the two other limbs.Moreover, 1 and 3 denote the "left" and the "right" screws driven by corresponding rotary motors.Moreover, 10, 11 and 12 denote the links of the passive C-joints making up each limb, which are coupled to the MP (13) by means of spherical joints.
PKMs with all actuators located on the base are preferred in some applications, such as those calling for high-speed operations, and in haptic devices.For this type of PKMs, motor weight will not add to the load on the actuation system, thus resulting in lower inertial loads, higher load-carrying capacity and better dynamic characteristics.Designers have come up with different solutions: Sorli et al. [10] used three double-parallelogram mechanisms to realize this goal, but this approach resorted to many extra components, which led to extra interference.Chen et al. [11] proposed a six-dof haptic device using two-dof planetary-belt systems, but these systems unavoidably introduce belt slip and flexibility, thus making it unsuitable for high-speed or high-torque applications.One major problem with these designs is that most of the actuation systems either are extremely complex or introduce many extra moving parts, thereby exacerbating the link interference, while limiting the workspace.Many other parallel robots carrying three limbs have been proposed [7,[12][13][14][15][16], either failing to have all actuators mounted on the base or achieving this at the expense of a complex actuation system.
Despite the vast number of designs, several common issues persist: many of the existing designs involve either too many components or complex actuation systems, which affects their performance in terms of workspace, accuracy, singularity and stability.Moreover, most of the designs have their motors mounted on moving links, which leads to a waste of installed power; higher inertia also influences robot performance in high-speed applications and the quality of the system highfrequency response.Our design, as shown in the graphs of Figs. 1 and 2, departs from current practice.
The SDelta Robot bears a special 3-CPS topology, each C joint being realized with a RHHR mechanism [43], as shown in Figs. 3 and 4. Most existing three-limb parallel robots employ three serial limbs; moreover, since each limb has to carry two motors, it is not possible to use a simple limb serial chain to locate all the motors on the base.Hence, we introduce a parallel substructure in each serial limb, i.e., the two-dof cylindrical actuator.More specifically, we use a two-dof actuator, dubbed the C-drive [43], as shown in Fig. 4. The C-drive carries a single-loop closed kinematic chain of the RHHR type; 6 Abbreviation of Six-dof Delta.
JMR-16-1286, Li, 3 the two H joints, of identical pitches but of opposite hands, lead to a 2 × 2 isotropic, constant Jacobian matrix of the drive mechanism, which is the simplest possible.The C-drive operates as a differential: when the two motors turn in the same sense 7 at the same rate, the collar undergoes a pure rotation; when the two motors turn in opposite senses at the same rate, the collar undergoes a pure translation.Each limb is thus driven in both translation along a given direction and rotation about an axis parallel to the same direction.
The SDelta Robot has the advantage that its architecture is simple, with fewer components than other three-limb designs, which reduces the complexity of its architecture and hence, simplifies its dynamics model and its control.This simplicity also reduces the potential interference among the limbs, thus resulting in a larger workspace.Moreover, mounting all motors on the base reduces the inertial load on the system, thereby making it suitable for high-speed operations.A similar topology (a 3-PRPS structure) was proposed by Behi [8], whose first two joints can be regarded as a C joint, but in this design, the two prismatic joints in each limb are actuated, and hence, each limb has one floating motor, significantly increasing the inertia of the system.

Geometry: the Forward-displacement Analysis
The architecture of the SDelta Robot is shown in Figs. 3 and 5, where the MP and the BP are represented by equilateral triangles, of sides a and b, respectively.The elimination method suggested by Nanua et al. [44] is adopted for its forwarddisplacement analysis.In the foregoing paper, the authors analyzed a class of Stewart-Gough platform with three pairs of concurrent limbs, ending up with a 16th-degree resolvent polynomial; in our case, the forward-displacement problem leads to a simpler model, namely, an octic polynomial.In this vein, the actuators, i.e., the cylindrical joints, are locked, while the lengths of the limbs, l j for j = 1, 2, 3, are unknown.Let s j denote the position vector of point S j , the center of the spherical joint of the jth limb, for j = 1, 2, 3, the relations below readily following: In the forward-displacement analysis, the sliding r j of the jth C joint and its rotation φ j are prescribed, the pose of the MP being unknown.The MP pose is found upon locating points S j , which is possible when the limb lengths l j are known, their computation being outlined below.Proceeding exactly as Nanua et al. did [44], three quadratic equations in the three limb lengths l j are derived: 7 As viewed from the same side of the drive layout.

BP
Fig. 5: Notation for the kinematic chain of the jth limb of the SDelta Robot The three foregoing equations thus admit up to 2 3 = 8 roots.This is half the number of the solutions of the forwarddisplacement problem admitted by the SGP with triangular BP and MP.

Kinematics 4.1 The Jacobian Matrix of the SDelta Robot
The architecture of the SDelta Robot, whose three cylindrical drives are actuated, is shown in Figs. 3 and 5. Let c be the position vector of the operation point C on the moving platform 8 , t = [ω ω ω T ċT ] T the twist of the MP, with ω ω ω denoting the angular velocity of the MP, and ċ the velocity of C. Furthermore, ψ ψ ψ = ψL1 ψR1 ψL2 ψR2 ψL3 ψR3 T represents the array of six motor rates, three to the right (R), three to the left (L) of point O j of the C-Drive collar in Fig. 5, for j = 1, 2, 3. Then we need to find the mapping between t and ψ ψ ψ.It is known [32] that the array of actuated-joint rates ψ ψ ψ and the MP twist are related by two Jacobian matrices, J and K, namely, However, since the C-drive of the SDelta Robot carries, each, two screw pairs, the derivation of its Jacobian matrices is not as straightforward.We thus introduce a new array: where φ j , ṙ j represent the turning and the sliding rates of the collar of the jth C-drive, for j = 1, 2, 3. Now we can express Eqn.(3) in the form 8 Depicted in Fig. 5 as the centroid C of the equilateral triangle S 1 S 2 S 3 .
JMR-16-1286, Li, 5 with matrices D, J m and K as yet to be displayed.Therefore, the Jacobian matrix J can be expressed as

The K and D Matrices
Firstly we introduce our notation: Vector p j is defined as that joining the jth spherical joint center S j with the operation point C, while e j and f j , for j = 1, 2, 3, all shown in Fig. 5, represent the unit vectors associated with the jth C-drive axis and the jth-limb axis; g j is defined as the direction vector along e j × f j .It is noteworthy that {e j , f j , g j } is an orthonormal, right-handed triad, as illustrated in Fig. 5.
Matrices K and D relate the end-effector twist with the array of the turning and sliding rates of the collars of the three C-drives.These Jacobian matrices can be conveniently derived based on screw theory [23,45].It is known that every joint in the robot bears an axis (i.e., a finite line or a line at infinity) associated with a corresponding array of Plücker coordinates; in this vein, let us assume that the ith joint variable of φ φ φ is associated with the jth joint of the kth limb; then, the ith row of K must be a screw 9 that is reciprocal [46] to the Plücker coordinates of all the axes in the kth limb, but the one associated with the ith component of φ φ φ.Moreover, it is recalled that a finite line can be regarded as a zero-pitch screw, while a line at infinity as an infinite-pitch screw.Furthermore, (a) Two zero-pitch screws are reciprocal when they are coplanar.(b) Two infinite-pitch screws are always reciprocal.(c) A zero-pitch screw is reciprocal to an infinite-pitch screw when their directions are orthogonal.Now consider line K j1 , associated with the wrench corresponding to the rotational degree of freedom of the jth C-drive.Since there exists a spherical joint at S j , K j1 must pass through S j ; moreover, its direction vector must be normal to the direction vector of J j2 and J j3 .Thus, K j1 passes through S j , with its direction vector normal to e j and f j , i.e., along g j , as shown in Fig. 5.
Next, consider line K j2 , associated with the wrench corresponding to the translational degree of freedom of the jth Cdrive.Since K j2 passes through S j , it is coplanar with J j1 , and normal to J j3 ; we can then conclude that K j2 passes through S j and is parallel to the axis of the jth C-drive, as shown, again, in Fig. 5. Based on the above analysis, we can obtain the K matrix as Under this notation, g j and e j represent the unit vectors parallel to lines K j1 and K j2 , respectively, for j = 1, 2, 3. We will henceforth denote K j1 and K j2 as G j and E j , respectively, for simplicity, as shown in Fig. 5, their Plücker arrays being k j1 = g T j , (g j × p j ) T T and k j2 = e T j , (e j × p j ) T T .Once K is available, D is straightforward to derive as a diagonal matrix, whose ith component is the reciprocal product of the line corresponding to the ith row of K and that associated with the joint of the ith component of φ φ φ.D is found to bear the form with l i representing the length of the ith limb, thereby deriving the two Jacobian matrices, K in Eqn.(7) and D in Eqn.(8).

The J m Matrix
Matrix J m relates the array of six motor rates with the array of the turning and sliding rates of the collars of the three C-drives.The Jacobian matrix J C of the C-drive [43] relates the speeds of the two screws with the turning and sliding rates 9 In its ray coordinates JMR-16-1286, Li, 6 of the collar, namely: where γ γ γ C and ψ ψ ψ C represent the speeds of the collar and of the screws, respectively, ψL , ψR the angular speeds of the left and right screws of the C-drive, respectively, and hence, of the C-drive motors.
It is noteworthy that the three C-drives are identical, the above relation thus applying to all of them.Apparently, then J m is a block-diagonal matrix for the three C-drives: thereby obtaining relation (5).

Redefinition of J m
If the rates of the C-drive are redefined as γ γ γ C = [ φT , 2πṙ/p T ] T , then, we can rewrite Eqn. ( 9) as In this vein, we redefine J m as which is a constant isotropic matrix because J C in Eqn. ( 12) is isotropic.Correspondingly, D is redefined as in order to preserve the equality in Eqn.(6).Finally, the J matrix becomes where D, as displayed in Eqn.(14), is a diagonal matrix whose entries have all units of length, while J m is a 6 × 6 dimensionless block-diaongal isotropic matrix.

Singularity Analysis
The singularities of parallel robots pertain to 1) those occurring in the serial Jacobian matrices J j of any of the limbs and 2) those occurring in matrices K and D. The latter are what are known as type-I (for K) and type-II for (D) singularities, respectively [32].

Singularities of the Serial Jacobian Matrices
Since each limb of the SDelta Robot has a decoupled architecture-i.e., a spherical joint coupling the MP with each limb-the singularity analysis of their Jacobian matrices is straightforward.The pertinent singularities can be classified into JMR-16-1286, Li, 7 wrist and shoulder singularities: Wrist singularities occur when the three wrist axes become coplanar.This means that the axes of the three concurrent revolute joints are coplanar; the shoulder singularity occurs when point C lies in the hyperboloid defined by the first three limb axes [47].This condition means that there exists one line L j passing through S j that is coplanar with J j1 , normal to J j2 and J j3 .This can only happen when the length of the jth limb is zero, which is physically impossible for the SDelta architecture.

Type-I Singularity
Since D is a diagonal matrix, it becomes singular when any of its diagonal entries vanishes.From Eqn. ( 14) it can be readily seen that this happens when the length of one of the limbs is zero, which is, again, physically impossible in general.This condition is the same as that for the shoulder singularity of serial Jacobian matrices, and hence, excluded.

Type-II Singularity
This occurs when K becomes singular, and the robot gains extra mobility 10 .The singularity of three-limb PKMs with one passive S joint at each limb end has been investigated extensively [31,[48][49][50][51], based on: instability analysis [31]; screw theory [48]; passive joint velocities [49]; the pure condition [50]; and the characteristic tetrahedron [51].Moreover, the singularity of a more general class of PKMs, namely, three-limb PKMs whose limbs, each, includes a passive S joint somewhere, has also been investigated based on passive joint velocities [28] and Grassmann-Cayley algebra [29,30].It has been shown that the singularity of this class of robots yields a straightforward geometrical interpretation, namely, the four planes-three planes composed by the three pairs of intersecting wrench axes plus the plane of the MP triangle-share at least one common point [28][29][30].Starting from this geometric condition, we propose a simple formulation for the singularity condition without involving any determinant calculation [28,31,[49][50][51] or passive joint velocities [28,49].
We have obtained K as shown in Eqn.(7), whose six rows can be regarded as the Plücker coordinates of six actuated wrenches, intersecting pairwise at the center of the three spherical joints.Since these six wrench lines of K intersect in three pairs in our case, it has been found that K becomes singular when the four planes-three planes composed by the three pairs of intersecting wrench lines together with the plane of the MP triangle-share at least one common point [28][29][30][31]50,51].We denote the plane defined by the intersecting lines E j , G j as Π j , its normal as n j , for j = 1, 2, 3.It is noteworthy that n j = f j for the SDelta Robot, but we use n j for generality.Moreover, we denote the MP plane as Π 4 , its normal as n 4 .Next we conduct the singularity analysis based on this geometrical interpretation.
x y a b z ′ Fig. 6: The reference posture of the MP First, denote the intersecting line between Π j and Π 4 as L j , for j = 1, 2, 3; it is noteworthy that L j passes through S j , and lies in the common plane Π 4 ; the foregoing geometrical condition is then equivalent to requiring that the three lines L j , for j = 1, 2, 3, share common points.In this way, we can reduce the analysis to the plane Π 4 .Furthermore, let us denote the intersecting point of L 1 with L 2 and with L 3 as R 2 and R 3 , respectively; then, the condition leads to requiring that the position vectors of R 2 and R 3 be identical.
Let us denote the direction vector of L j as l j ; then, l j must be normal to both n j and n 4 .We do not require l j to be of unit norm here, and hence, we can assign ) 10 In the ensuing derivation, p j denotes the position vector pointing from S j , the jth spherical joint center to the operation point C on the MP, these centers being the vertices of the MP when the three spherical joints are connected directly to the MP, as in our case.
JMR-16-1286, Li, 8 Furthermore, upon denoting the position vector of the common point as r, we have where k j , j = 1, 2, 3, are as yet to be determined.From the above relations, we have Next we cross-multiply both sides of Eqn.(18a) with l 2 , of Eqn.(18b) with l 3 , which leads to It can be seen that both sides of Eqns.(19a) and (19b) are parallel to the z-axis of the MP frame.Next, upon dot-multiplying the LHS of Eqn.(19a) with the RHS of Eqn.(19b), then equating this product with that of the corresponding sides of Eqns.(19a) and (19b), we obtain where the common factor k 1 has been eliminated.Moreover, plugging Eqn. ( 16) into Eqn.(20), after some manipulations, leads to Furthermore, it is noted that n 4 is normal to p i − p j , for i, j = 1, 2, 3, i = j; we can thus simplify Eqn.(21) to which is the singularity condition sought.It is observed that, when represented in the MP frame, p j and n 4 in Eqn.(22) become constant, which reduces the computational cost greatly.As a result, we choose to express all the vectors in the MP frame.Then, we only need to find n j , for j = 1, 2, 3, in the MP frame.It is noteworthy that, for the SDelta robot, n j is nothing but f j , and hence, parallel to the axis of the jth limb; moreover, its norm does not affect the relation in Eqn.(22).Assuming that the perpendicular foot of S j on the axis of the jth C-drive is O j , as shown in Figs. 5 and 6, we can use −−→ O j S j to substitute n j in Eqn.(22), where which yields in which only Q and c are variable.Then we can use −−→ O j S j to substitute n j in Eqn.(22), for j = 1, 2, 3.
It can be readily verified that this equality also holds even if some of the four planes Π j are coincident, or when some of the three intersecting lines L j coincide.

Case Study: the Fixed-Orientation Singularity Locus
Since the singularity locus of a six-dof PKM is impossible to visualize, its fixed-orientation subset has been mostly investigated in the literature, which means that the orientation, i.e., the Q matrix, is fixed.Further, −−→ O j S j is linear in c, while (n j × n 1 ) • n 4 is quadratic in c, for j = 2, 3; then follows that Eqn. ( 22) yields a cubic surface in c.A numerical example is given below for illustration.
It is found that the design parameters impact on singularity distribution, workspace and dexterity.Hence, we plot these three items for two typical sets of design variables, namely, the sides of the MP and the BP are assumed to obey the relation a/b = 0.3 for Design I and a/b = 1 for Design II, respectively.Furthermore, we define the reference pose of the MP as that under which the operation point C coincides with O in the BP, while the MP orientation is as shown in Fig. 6, with the BP and MP planes coincident.
Several typical orientations are selected, under which the singularity loci are plotted as the surfaces shown in Figs.7 to 9, with the open and closed surfaces representing the singularity locus and the corresponding workspace boundary, respectively.In these figures, the orientation is given by axis e and angle φ of the rotation matrix that carries the BP from its reference to its current attitude.It is observed that, when the MP rotates about the Z-axis, the singularity is characterized by three vertical planes, which can be analyzed by means of Grassmann geometry: at the reference orientation, the three wrench axes E i corresponding to the three C-Drive axes are coplanar, in a horizontal plane; then, when the middle link of the ith limb is vertical, the wrench line corresponding to g i lies also in the same horizontal plane, these four lines becoming a linear variety [24] of rank 3, which leads to a singular configuration.Hence, the three planes can be found when the MP translates to a configuration in which S i lies in the vertical plane that passes through the ith C-Drive axis.Furthermore, when the MP undergoes a rotation about the Z-axis, the three lines E i are still in the same horizontal plane, which again, leads to three vertical singularity planes; this is simple to characterize.However, when the MP rotates about some other axes, the singularity surface has, generally, a complex shape.

Workspace Analysis
In this section we investigate the fixed-orientation position workspace to have a general idea of the workspace volume.We consider only the limits of the active and passive joints first, then verify whether singularities exist within the workspace thus obtained.
We propose a geometrical method capable of obtaining the workspace efficiently, and thus, can be used for workspace JMR-16-1286, Li, 10 regard the robot as a serial chain), denoted as W i ; then, the workspace of interest will be the intersection of {W i } 3 1 .Next, we explain the procedure for finding W i .
Since the orientation of the MP is fixed for the fixed-orientation workspace, the center of the spherical joint S i , undergoes the same motion as the operation point C. Hence, we firstly find the "position workspace" of S i -denoted as W Si .
Considering the stroke of the C-drive, denoted r s , the upper and lower bounds of the telescopic arm, denoted l min and l max , the position of S i is found to lie within the region between two co-axial cylinders of radii l min and l max , respectively, whose height is the stroke r s , as shown in Fig. 10(a).Furthermore, we consider the constraint of the S joint, which we assume to be realized by a ball-and-socket joint, whose working angle range can be modelled as a cone [2], its maximum denoted δ max .Apparently, the translation of the MP along the direction e i of the axis of the C-drive, or along the direction f i of the ith limb, will not change the relative orientation of the two links connected by the spherical joint (i.e. the upper link of the telescopic boom and the MP); only the translation of the MP in the direction of g i will change the foregoing relative orientation, which corresponds to the rotation of the C-drive.The motion of the S joint has a limit only on the feasible range of the angle of rotation of the C-drive; this limit remains constant when S i translates in the direction of e i or f i .This means that, when we further consider the limits of the S joint, we will no longer have a cylindrical shape, but a pie slice, as shown, again, in Fig. 10(a), whose angular range can be derived from the projection of the cone onto the plane normal to e i .Until now, we have found W si ; it is a simple matter to translate this region by p i to find W i , the feasible region of the operation point under the constraint of the ith limb, as shown in Fig. 10(b).Once all the three regions {W i } 3  1 are available, their intersection yields the fixed-orientation workspace sought.Researchers usually discretize one of the three coordinates, e.g., the Z-coordinate, then find the workspace shape on different layers.With the aid of computer-algebra, this kind of intersection operation of geometrical objects can be handled directly; the software in use also provides the workspace volume and the workspace boundary.

Case Study
The fixed-orientation worksapce is provided for a robot, again, with two sets of design parameters: a = 0.2b, l min = 0.45b, l max = 0.85b and a = b, l min = 0.6b, l max = 1.13b JMR-16-1286, Li, 11  where l min and l max represent the minimum and maximum lengths of each limb; It is noteworthy that we choose different l min and l max values in order to keep the height of the MP similar at the reference pose for the two set of different design parameters.Moreover, we assume the maximum angle attained by the S joint to be δ max = 45 • ; then, the workspace under the reference orientation is plotted in Fig. 11 for each case, yielding volume values of 0.049d 3 and 0.067b 3 , respectively.The workspace is also evaluated under several other orientations, as shown in Figs. 12 and 13, with the volumes 0.045b 3 , 0.007b 3 , 0.047b 3 , 0.042b 3 , respectively.It is, however, noteworthy that the singularity surface sometimes crosses the workspace for the given set of design parameters and selected orientation, as shown in Figs.7 to 9, similar to the case of the Stewart-Gough platform [52].Finally, in order to reveal the effect of the ratio a/b on the workspace volume (V ), we plot V vs. a/b, in the range of 0.2 to 1.5, which range we believe to be sufficient in general applications.Moreover, even though we used different l min and l max for Designs I and II in order to make the height of the MP similar at the reference pose, we fix them here in order to reveal the effect of solely the ratio a/b; more specifically, l min = 0.55b, l max = b.Then, the workspace volumes are, again, plotted under the three different orientations, i.e., the reference orientation, the orientation e = [0, 0, 1] T with φ = 15 • JMR-16-1286, Li, 12 and e = [0, 1, 0] T with φ = 15 • , respectively, as shown in Fig. 14.Apparently, the workspace volume V under the reference orientation remains largely unchanged for different values of a/b; however, V decreases significantly when the MP is rotated about the Z-axis; when the MP is rotated about other directions, the workspace volume can either decrease or increase.It is noteworthy that the above workspace volume is different from those of Designs I and II, indicating that the range of the passive limb length affects the workspace as well.
The above result indicates that, when the range of the passive limb length is fixed, the workspace volume generally decreases as the ratio a/b increases; however, for a larger MP, (e.g., a = b), the robot allows for a larger range of the passive limb length 11 , which can end up with a bigger workspace volume.The reciprocal of the condition number of the Jacobian matrix, is adopted here [53] to quantify the dexterity; more specifically, we use the condition number based on the Frobenious norm because it is analytical.Since D is diagonal, its condition number can be readily monitored as the ratio of the largest absolute value to the smallest abosolute value of its nonzero entries, which is well bounded since the limb lengths are designed to be always positive.Moreover, as J m is orthogonal, it is isotropic.Hence, we only look at matrix K here, its Frobenius-norm condition number [46], represented as κ F , being defined as Given that the entries of the Jacobian matrices bear different units, we introduce a pertinent characteristic length L [54] to resolve this dimensional inhomogeneity.To this end, we redefine the twist and K in their dimensionally homogeneous form: The characteristic length is defined as the value L that minimizes the condition number of K h ; as reported in a forthcoming paper, the characteristic length of the SDelta robot can be obtained as the radius of the circle circumscribing the equilateraltriangular MP, i.e., as L = a/ √ 3.Then, the dexterity is analyzed via a numerical algorithm based on discretization.Four different Z-values are chosen, while the dimensions in the other directions are discretized with an interval of b/100; the dexterity distribution is plotted in Figs. 15 and 16 for different orientations 12 .The value log 2 (1/κ F (K h )), denoted λ for brevity, is plotted vs. X-Y , those surfaces with higher altitudes representing a layer with smaller Z, i.e., the condition 06.This means that when the MP moves near the plane of the BP, the condition number is low 13 .It is found that the dexterity is generally small as long as the posture is not close to a singularity: For the worst case in each orientation (i.e., on the surfaces at the bottom of Figs. 15 and 16, corresponding to Z = 1.2), at least 75% of the area has a condition number smaller than 8.0, at least 93% has a condition number smaller than 32.0, and at least 99% has a codition number smaller than 256.0.

Discussion
From the above figures for which two sets of arbitrarily chosen design parameters are used, it is apparent that the singularity locus, workspace and dexterity are greatly affected by the ratio a/b and the range of the limb length.
It was found that, when the MP is under the reference orientaion, the singularity locus is the union of three vertical planes, each corresponding to a set of postures under which one of the three points S j lies in the vertical plane passing through the axis of the jth C-Drive, for j = 1, 2, 3.As a result, |a − b/2| must be as large as possible, so that the distance from S j to the vertical plane passing through the axis of the jth C-Drive is as large as possible, the singularity surface becoming farther from the desired workspace region.Furthermore, when the MP rotates about the Z axis, the singularity surface is still the union of three vertical planes.As the MP rotates about an axis other than the Z-axis, the singularity surfaces generally show the tendency to both rotate about this axis, and deform in such a way that their shapes become more complex.
Besides the ratio a/b, the workspace volume is greatly affected by the range of the limb length: the workspace volume increases as the stroke of each limb increases; moreover, as the average of l min and l max increases, the "center" of the workspace region lies higher and the workspace volume generally becomes larger.On the other hand, the dexterity worsens as the "center" of the workspace region lies higher; however, the condition number is generally small enough as long as the operation point is not too close to the singularity surface.As these indices do not obey a monotonic relation, a compromise

Fig. 7 :Fig. 8 :
Fig. 7: The workspace and singularity of the SDelta under the reference orientation (a) Design I (b) Design II

Fig. 11 :
Fig. 11: The workspace of the Sdelta under the reference orientation (a) Design I (b) Design II

Fig. 14 :
Fig. 14: The workspace volume w.r.t. the ratio of a/b, under different orientations