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Abstract

This paper addresses a novel geometric analysis method of the singularity and kinestatic characteristics of robots. For non-redundant robots, there exist two uniquely determined Jacobians – the screw-based Jacobian and the reciprocal Jacobian. Here, it is shown that if some of the reciprocal products between the column screws of the two Jacobians are close to zero, the robot is in the vicinity of a singular configuration and the corresponding columns of the Jacobian are involved in the singularity. From this observation, an invariant measure of closeness to a singularity is presented using the reciprocal products. Furthermore, by considering the reciprocal products between the actuated joint screws and column screws of the Jacobian, this measure is extended so that the concept of the kinestatic characterization index is presented for parallel robots. Since it is defined as the ratio of reciprocal products from the uniqueness of the two Jacobians, it represents a unique invariant characteristic of robots. From the singularity analyses of the planar 4-bar linkage and the 3-DOF parallel manipulator with PRS-serial chains, the validity of the proposed index is examined and the concept of a kinestatically balanced configuration is introduced as well.

Keywords

Invariant measure of singularity kinestatic characterization index singularity of the robot parallel manipulator theory of screws

1. Introduction

An understanding of the kinestatic (velocity and static force) characteristics of a manipulator is important for the optimal design, the exact and efficient control and the evaluation of the velocity and force capabilities of the manipulator. The characteristics depend on the geometric configurations of a robot and the physical performances of the actuators. Thus, previous research works have focused mainly on the Jacobian matrix of the robot mechanism.

One of many efforts to measure the characteristics was the measure of manipulability, which was defined as $\sqrt{\det (J)}$ for a square Jacobian matrix ( $\sqrt{\det (J J^{T})}$ ) or $\sqrt{\det (J^{T} J)}$ for a non-square Jacobian) in [1]. It had been used extensively for the analysis and optimal design of manipulators [2 –12]. Some properties of the manipulability index, such as the invariance with respect to the coordinate transformation, were investigated in [13]. However, it was shown that by expanding the determinant of the Jacobian and observing each of its terms, the manipulability indices cannot be considered as the measures of the closeness to a singularity [14]. In [15], the physical inconsistency of the manipulability indices was discussed and a modified concept of manipulability was proposed by considering the kinetic energy of the end-effector. This concept was further generalized as a constrained optimization problem in [16]. Another index called the ‘quality index’ was defined as det(J) / det(J_m), where det(J_m) denotes the maximum values of det(J) and which was used for the optimal design of parallel manipulators [17,18]. The condition number of the Jacobian matrix has been used as the measure of the kinestatic performance [19 –21]. In [22,23], the condition number was described as the error amplification factor of the relative errors of actuated joints and, furthermore, it was pointed out that, for a robot with at least one translational and one rotational degree of freedom, the condition number has no clear physical meaning. If the 2-norm is used, the condition number is expressed as the square root of the ratio of the largest to the smallest eigenvalues (or singular values) of the Jacobian. However, it was shown that the use of the eigenvalues of the Jacobian was not meaningful due to the different physical units of their components [24]. To overcome this problem, in [25,26] the use of the characteristic link length L was introduced to form a 6×6 diagonal matrix diag(1/L, 1/L, 1/L, 1, 1, 1). In addition, a dimensionally homogeneous Jacobian J_h was suggested, such that J _h ≡ J · diag (1/L, 1/L, 1/L, 1, 1, 1). However, since L can be chosen arbitrarily, J _h depends on the choice of L. In order to obtain a dimensionally homogeneous Jacobian, one method was presented where the Jacobian relates the actuator velocities to the linear velocities of two (or three) points on the end-effector of a planar (or spatial) manipulator [27]. From the work given in ref. [28], a 6 × 9 homogeneous Jacobian matrix for a 6-DOF parallel manipulator (or a 3 × 6 Jacobian for a 3-DOF planar parallel manipulator) was derived using three points on the moving platform. For n -DOF lower mobility parallel manipulators, in [29], a method to form the overall 6 × 6 Jacobian matrix was presented in terms of the 6 × n Jacobian of actuation and the 6 × (6 – n) Jacobian of constraints, while the constraint singularities of the parallel manipulators were analysed using the determinant of the overall Jacobian. In [30], instead of finding a single invariant measure from the Jacobian matrix, the rotational and translational motions are separately considered so that they define the maximum rotation sensitivity and the maximum point-displacement sensitivity. Furthermore, in [31], the manipulability was defined as the closeness of the forward kinematic map to an isometry in view of the Riemannian geometry.

Along with the manipulability, the velocity or force transmission capabilities of manipulators were studied, mostly by considering the velocity or force ellipsoid (or polytope) which is formed by the mapping of bounded actuator inputs. In [32], the methods for the force transmission capability analyses were proposed using the force and moment transmission ellipsoids. The velocity performance of parallel manipulators with actuation redundancy was analysed by making use of the largest ellipsoid included in the velocity polytope [33], while the concept of wrench polytopes for planar parallel manipulators was investigated in view of the mapping of the parallelepiped region of the actuator input force vector [34].

The singularities of serial manipulators arise from linear dependences between the column screws of the Jacobian. Thus, the closeness to a singularity may be measured if the closeness to a linear dependence of column screws of the Jacobian can be defined properly. From the work given in [35], a method which evaluates the closeness to a linear dependence of six screws was introduced using the reciprocal product between one screw out of six screws and the screw reciprocal to the other five screws except for the screw under consideration. This concept was extended for the determination of the closeness to a linear dependence of any system of screws [36]. In that work, for n screws the maximum available reciprocal product between a screw out of the n screws and any unit combination of screws reciprocal to the rest (n-1) screws except for the test screw was used to check the closeness to a singularity of a lower mobility manipulator. For some manipulators, there exists a situation where the maximum available reciprocal product becomes an infinite, unbounded value. In this case, the use of the maximum available reciprocal product may not be adequate for the measure of a singularity.

In this paper, a new geometric method for the analyses of the singularity and kinestatic characteristics of robot mechanisms is presented. This method is based on the reciprocal Jacobian method for robot kinestatic analyses which is given in [37]. Thus, in Section 2, the fundamental concept of the reciprocal Jacobian as well as the reciprocal Jacobian forms of kinestatic relations are briefly introduced. From the reciprocal relation of the two Jacobians, in Section 3, we firstly show the necessary and sufficient condition in order to find the linear dependence on the column screws of the screw-based Jacobian. Then, the velocity and force characteristics of a serial robot mechanism in the vicinityof a singular configuration are investigated. For a parallel manipulator, by considering another reciprocal relation between the column screws of the screw-based Jacobian and the actuated joint screws, a concept of a kinestatic characterization index is introduced. With this proposed index, the velocity and static force characteristics of a parallel manipulator in the vicinity of a singularity are discussed. Finally, in Section 4, the validity of the proposed index is examined via the singularity analyses of a planar 4-bar linkage and a 3-DOF parallel manipulator with three PRS-serial chains.

2. Background: Reciprocal Jacobian Forms of the Kinestatic Relations of Robot Mechanisms

The kinestatic (velocity and static force) relations of a robot manipulator can be simply represented by use of the Jacobian of the robot. The unique form of the screw-based Jacobian can be formulated for serial and parallel robots [38]. Besides the Jacobian, in [37], it is shown that there exists another form of Jacobian which is called the ‘reciprocal Jacobian’. This is uniquely determined for a given manipulator and it performs the inverse mapping of the kinestatic relation expressed with the Jacobian.

Since the proposed method of analysis of a robot singularity is based on the kinestatic relations written with the reciprocal Jacobian, we briefly introduce, here, the reciprocal Jacobian forms of robot kinestatics.

2.1 Kinestatics of a Serial Robot

For ann-DOF (1 ≤ n ≤ 6) serial robot, the forward velocity relation can be written as: $\hat{T} = J \dot{q} = {\dot{q}}_{1} {\hat{S}}_{1} + {\dot{q}}_{2} {\hat{S}}_{2} + \dots + {\dot{q}}_{n} {\hat{S}}_{n}$ (1)

where q̇ =[ q̇ … q̇_n]^T ∈ Rⁿ^×1 is the actuator input velocity vector and T̂ (=[v^T_o; ω^T]^T ∈ R^6×1) means the end-effector twist expressed in Plücker's axis coordinates. v ₀ (∈ R^3×1) is the velocity of the end-effector coincident with the origin at the instant and ω (∈ R^3×1) is the angular velocity of the end-effector. In addition, J(≡[ Ŝ ₁, Ŝ ₂, …, Ŝ _n]∈R^6×n) denotes the screw-based Jacobian of the robot and the ith column of the Jacobian Ŝ _i is the screw with a unit magnitude expressed in Plücker's axis coordinates. It can be written as Ŝ _i =[(r_i × s_i + h_is_i)^T,, s ^T_i]^T where s _i and r _i are, respectively, the unit direction and position vectors and h_i is its pitch. If Ŝ _i is a screw of zero pitch (or a free vector), then Ŝ _i =[(r_i × s_i)^T, s ^T_i]^T (or Ŝ_i=[ s _i ^T ,0^T_3×1]^T).

From Eq. (1), the end-effector twist which is controllable by the actuators can be represented by the linear combination of column screws of the Jacobian. Thus, they span the n-dimensional space of controllable end-effector twists. The other twists that cannot be generated by actuators belong to the (6-n)-dimensional space of completely uncontrollable twists.

When the screw-based Jacobian is known for a serial robot, the inverse static force relation can easily be found in the following manner: $τ = J^{T} \hat{w}$ (2)

where τ(≡[τ₁…τ_n]^T ∈ R^n×¹) and ŵ (=[ f ^T , m _o ^T ]^T ∈ R^6×1) are, respectively, the n-dimensional vector of actuator forces (or torques) and the end-effector wrench. f (∈ R^3×1) is the force vector and m ₀ (∈ R^3×1) denotes the moment with respect to the origin.

Now, we consider the reciprocal Jacobian J_r(≡[ ŝ _r _,1' … ŝ_r,n] ∈ R^6×n) of an n-DOF serial robot. The ith column, of it ŝ _r,i is determined as the unit screw that is reciprocal to (n-1) column screws of J except the ith one Ŝ _i and which is also reciprocal to (6-n) basis screws of the completely uncontrollable twist space. ŝ _r,i s are expressed in Plücker's ray coordinates. From these 5(=n-1+6-n) reciprocity conditions, ŝ _r,i is uniquely determined even for the lower mobility robot. Furthermore, ŝ _r,i spans the n-dimensional space of unconstrained wrenches of the serial robot (see Fig. 1(a)).

Figure 1.

Twist and wrench spaces of an n-DOF robot. (a) Foraserial robot, the column screws of the Jacobian span the controllable twist space and the non-constraint wrench space is spanned by the column screws of the reciprocal Jacobian. (b) For a parallel manipulator, while column screws of the Jacobian span the non-constraint wrench space while the controllable twist space of the moving platform is spanned by the column screws of the reciprocal Jacobian.

Utilizing this reciprocal Jacobian, the inverse velocity and forward static force relations can be formulated as follows: $\dot{q} = {[r]}_{}^{- 1} J_{r}^{T} \hat{T}$ (3)

and: $\hat{w} = J_{r} {[r]}_{}^{- 1} τ$ (4)

where: ${[r]}_{} \equiv J_{r}^{T} J = J_{}^{T} J_{r}^{} = d i a g ({\hat{s}}_{r, 1}^{T} {\hat{S}}_{1}, \dots, {\hat{s}}_{r, n}^{T} {\hat{S}}_{n}) \in R^{n \times n}$ (5)

Ŝ _i ands ŝ_r,i are expressed in different forms of Plücker's coordinates and thereby the scalar product of them ŝ _r,i ^T Ŝ _i , which is the ith diagonal element of the matrix [r] denotes the reciprocal product of the screws.

2.2 Kinestatics of a Parallel Manipulator

The forward static force relation of an n-DOF (1 ≤ n ≤ 6) parallel manipulator may be written as [38 –40]: $\hat{w} = J {[J_{q}^{}]}^{-1} τ = (τ_{1} / J_{q, 1}) {\hat{s}}_{1} + (τ_{2} / J_{q, 2}) {\hat{s}}_{2} + \dots + (τ_{n} / J_{q, n}) {\hat{s}}_{n}$ (6)

where: $[J_{q}^{}] \equiv d i a g (J_{q, 1}, J_{q, 2}, \dots, J_{q, n}) \in R^{n \times n}$

and: $J \equiv [{\hat{s}}_{1}^{}, {\hat{s}}_{2}^{}, \dots, {\hat{s}}_{n}^{}] \in R^{6 \times n}$

ŝ _i is the unit screw expressed in Plücker's ray coordinates and J_q,i denotes the reciprocal product between the ith actuated joint screw with a unit magnitude and ŝ _i . Furthermore, the inverse velocity relation of the manipulator can be expressed as: ${[J_{q}]}_{} \dot{q} = J^{T} \hat{T} or J_{q, i} {\dot{q}}_{i} = {\hat{s}}_{i}^{T} \hat{T} (for i = 1, \dots, n)$ (7)

Clearly, from Eq. (6), the column screws ŝ _i of the screw-based Jacobian J span the n-dimensional space of the non-constraint wrenches. It is seen in [38] that, from the reciprocal Jacobian analysis of each serial chain, the screw-based Jacobian J can be uniquely determined even for the lower mobility parallel manipulator.

Similar to the serial robot, the ith column screw ŝ _r,i of the reciprocal Jacobian J _r (≡[ ŝ _r ,₁,…, Ŝ _r,n ] ∈ R^6×n) is found as the unit screw that is reciprocal to (n-1) column screws of the Jacobian except for the ith one ŝ _i and which is also reciprocal to (6-n) basis screws of the (6-n)-dimensional space of the constraint wrenches. It is also uniquely determined from these 5 reciprocity conditions and Š_r,i spans the n-dimensional space of the controllable end-effector twists (see Fig. 1(b)). ŝ _r,i is expressed in Plücker's axis coordinates.

Now, the inverse forms of Eqs. (6) and (7) can be obtained using the reciprocal Jacobian of the manipulator: $τ = {[r]}_{}^{- 1} [J_{q}^{}] J_{r}^{T} \hat{w}$ (8)

and: $\hat{T} = J_{r} {[r]}_{}^{- 1} {[J_{q}]}_{} \dot{q}$ (9)

The diagonal matrix [r] is defined as the same as Eq. (5).

3. Singularity Analysis

3.1 Singularity Analysis of a Serial Robot

Now, we consider the kinematic singularity of robot manipulators. When both the screw-based Jacobian and the reciprocal Jacobian are known for a robot manipulator, the following theorem provides a simple way to check whether the robot under consideration is in a singular configuration.

Theorem 1) For an n-DOF (1 < n ≤ 6) serial robot, when the two (controllable and completely uncontrollable) end-effector twist spaces are linearly independent, k(1 < k ≤ n) diagonal elements of the matrix [r] are zero if and only if the corresponding k column screws of the Jacobian are linearly dependent.

Proof) For the proof of the theorem, and without loss of generality, we consider a 3-DOF serial robot having the Jacobian J = [ Ŝ ₁, Ŝ ₂, Ŝ ₃] and the reciprocal Jacobian J_r = [ ŝ _r ,₁, ŝ _r ,₂, ŝ _r _,3], and we also assume that only two diagonal elements of [r], ŝ ^T _r ,₁ Ŝ ₁ and ŝ ^T _r,2 Ŝ ₂ are zero. If we let B̂ ₁, B̂ ₂ and B̂ ₃ be three basis screws of the 3-dimensional space of the completely uncontrollable end-effector twists, by definition ŝ _r _,1 should be reciprocal to the 5 screws { Ŝ ₂, Ŝ ₃, B̂ ₁, B̂ ₂, B̂ ₃}. All of the screws reciprocal to ŝ _r _,1 belong to the 5-system of screws spanned by these 5 screws. Thus, Ŝ ₁ has to be contained in this 5-systembecause ŝ ^T _r ,₁ Ŝ ₁ = 0. However, we assume that the two twist spaces, span{ Ŝ ₁, Ŝ ₂, Ŝ ₃} and span{ B̂ ₁, B̂ ₂, B̂ ₃} are linearly independent and, thereby, Ŝ ₁ is represented by the linear combination of only screws Ŝ ₂ and Ŝ ₃. Likewise, ŝ ^T _r _,2 Ŝ ₂=0, Ŝ ₂ can also be represented by the linear combination of Ŝ ₁ and Ŝ ₃. However, ŝ ^T _r ,₃ Ŝ ₃ ≠ 0. Thus, Ŝ ₃ does not belong to the 5-system spanned by { Ŝ ₁, Ŝ ₂, B̂ ₁, B̂ ₂, B̂ ₃}. This means that Ŝ ₃ is linearly independent on screws Ŝ ₁ and Ŝ ₂. Consequently, linear dependence arises only among Ŝ ₁ and Ŝ ₂.

Conversely, we assume that the two column screws of the Jacobian Ŝ ₁ and Ŝ ₂ are linearly dependent. Then, from the definition of the reciprocal screws ŝ _r _,1 and ŝ _r _,2, it is easy to show that s ^T _r ,₁ Ŝ ₁ = ŝ ^T _r _,2 Ŝ ₂ = 0, which completes the proof.

From the above theorem, it can be seen that if some of the reciprocal products ŝ ^T _r,i Ŝ _i are zero, only the corresponding column screws Ŝ _i of the Jacobian are involved in the singular configuration of the robot. Now, in order to examine the velocity characteristics of the serial robot in the vicinity of a singular configuration, we consider Eq. (3). This can be written as $({\hat{s}}_{r, i}^{T} {\hat{S}}_{i}) {\dot{q}}_{i} = {\hat{s}}_{r, i}^{T} \hat{T}, (for i = 1, 2, \dots, n)$ (10)

If we assume that the magnitude of the reciprocal product | ŝ ^T _r,i Ŝ _i | approaches zero then, even for the generation of an end-effector twist with a small magnitude, quite a large amount of velocity input is required for the ith actuator. This means that the controllability of the end-effector twist by the ith actuator becomes worse, as the value |ŝ ^T_r,i Ŝ _i| is close to zero. Finally, when |ŝ ^T_r,i Ŝ _i| = 0, the end-effector twist becomes completely uncontrollable for the actuator velocity input.

From this consistent behaviour of the velocity characteristics, it is expected that the magnitudes of the reciprocal products |ŝ ^T_r,i Ŝ _i| may be used for the measure of closeness to a singularity of the robot. Therefore, we define the following index γ for the measure of closeness to a singular configuration of an n-DOF robot mechanism: $γ \equiv \sqrt[n]{| \det ([r]) |} = {(_{} | {\hat{s}}_{r, 1}^{T} {\hat{S}}_{1} | \cdot | {\hat{s}}_{r, 2}^{T} {\hat{S}}_{2} | \cdot \dots \cdot | {\hat{s}}_{r, n}^{T} {\hat{S}}_{n} |_{})}_{}^{1 / n}$ (11)

If γ is close to zero, the robot is in the vicinity of a singular configuration. In that instant, and by observing each value of |ŝ^T_r,iŜ_i|, it can be known which joint screws are associated with the singularity. It is noted here that γ is composed of only the reciprocal products between the column screws of the screw-based Jacobian and the reciprocal Jacobian. Since the two Jacobians are uniquely determined for a given robot configuration, the proposed measure of singularity γ is also uniquely determined. Furthermore, since the reciprocal product of two screws is an invariant scalar product, γ is invariant with respect to the coordinate transformation and, by normalizing in terms of an appropriate value, the changes of the scale and physical units.

The force characteristics of the serial robot which is close to a singular configuration may be examined from Eq. (4). It can be expressed as: $\hat{w} = {\hat{s}}_{r, 1} (\frac{τ_{1}}{{\hat{s}}_{r, 1}^{T} {\hat{S}}_{1}}) + {\hat{s}}_{r, 2} (\frac{τ_{2}}{{\hat{s}}_{r, 2}^{T} {\hat{S}}_{2}}) + \dots + {\hat{s}}_{r, n} (\frac{τ_{n}}{{\hat{s}}_{r, n}^{T} {\hat{S}}_{n}})$ (12)

From the above equation, it is expected that if |ŝ^T_r,iŜ_i| is close to zero, most of the wrenches acting at the end-effector along the screw ŝ _r,i can be supported by the ith actuator, even for a small actuator force (or torque) input. That means that if the ith actuated joint screw is involved in a singular configuration of the serial robot, the robot mechanism may immediately become structurally constrained for the wrench input along screw ŝ_r,i.

3.2 Singularity Analysis of a Parallel Manipulator

In a similar manner as was used for the serial robot, the following theorem can be obtained for an n-DOF parallel manipulator:

Theorem 2) For an n-DOF (1 < n ≤ 6) parallel manipulator, when the two (constraint and unconstraint) end-effector wrench spaces are linearly independent, k(1 < k ≤ n) diagonal elements of the matrix [r] are zero if and only if the corresponding k column screws of the screw-based Jacobian are linearly dependent.

For the parallel manipulator, there exists a situation where the constraint and unconstraint wrench spaces are linearly dependent. In this case, a subset of the unconstraint wrenches belongs to the constraint wrench space, by which the unconstraint wrench space is degenerated. Thus, the manipulator configuration becomes singular even if the column screws of the screw-based Jacobian are linearly independent. Considering this kind of singularity, the above theorem can be generalized as the following theorem:

Corollary 2.1) For an n-DOF (1 ≤ n ≤ 6) parallel mechanism, if k(1 ≤ k ≤ n) diagonal elements of the matrix [r] are zero, the manipulator configuration becomes singular and the corresponding k column screws of the screw-based Jacobian are involved in the singular configuration of the robot.

Proof) Without loss of generality, we consider a 3-DOF parallel manipulator having the Jacobian J = [ ŝ ₁, ŝ ₂, ŝ ₃] and the reciprocal Jacobian J_r = [ ŝ _r _,1, Ŝ _r _,2, ŝ _r _,3], and we also assume that two diagonal elements of [r], ŝ ^T _r ,₁ ŝ ₁ and ŝ ^T _r ₂ ŝ ₂ are zero. If we take b̂ ₁, b̂ ₂ and b̂ ₃ as three basis screws of the 3-dimensional space of the constraint wrenches, since ŝ ^T _r _,1 ŝ ₁ = Ŝ ^T _r _,2 ŝ ₂= 0 and ŝ ^T _r _,3 ŝ ₃ ≠ 0, there exists a linear, dependence among the screws { ŝ ₁, ŝ ₂, ŝ ₁, ŝ ₂, ŝ ₃}.

If the two screw systems span{ ŝ ₁, ŝ ₂, ŝ ₃} and span{ b̂ ₁, b̂ ₂, b̂ ₃} are linearly independent, then from Theorem 2) a linear dependence arises only among ŝ ₁ and ŝ ₂ and thereby they are involved in the singular configuration. Now, we assume that span { ŝ ₁, ŝ ₂, ŝ ₃} and span{ b̂ ₁, b̂ ₂, b̂ ₃} are not linearly independent. Since there exists a linear dependence among the screws { ŝ ₁, ŝ ₂, b̂ ₁, b̂ ₂, b̂ ₃}, it becomes obvious that a subset of the screw system spanned by ŝ ₁ and ŝ ₂ belongs to the constraint wrench space spanned by b̂ ₁, b̂ ₂ and b̂ ₃. It means that the unconstraint wrench space becomes degenerated and that ŝ ₁ and b̂ ₂ are involved in the singularity, which completes the proof.

One remark may be made which is that when n=1, it is the 1-DOF parallel mechanism and it has just one column screw ŝ ₁ in the screw-based Jacobian. ŝ ₁ spans the 1-dimensional space of the unconstraint wrenches. For the mechanism, if ŝ ^T _r _,1 ŝ ₁ = 0, since there exists only one column screw in the Jacobian, the only possible singularity is the singularity caused by the degenerated unconstraint wrench space.

The velocity characteristics of the parallel manipulator in the vicinity of the singular configuration may be examined from Eq. (9). They can be rewritten as: $\hat{T} = {\hat{S}}_{r, 1}^{} {(\frac{J_{q, 1}}{{\hat{S}}_{r, 1}^{T} {\hat{s}}_{1}})}_{} {\dot{q}}_{1} + {\hat{S}}_{r, 2}^{} {(\frac{J_{q, 2}}{{\hat{S}}_{r, 2}^{T} {\hat{s}}_{2}})}_{} {\dot{q}}_{2} + \dots {\hat{S}}_{r, n}^{} {(\frac{J_{q, n}}{{\hat{S}}_{r, n}^{T} {\hat{s}}_{n}})}_{} {\dot{q}}_{n}$ (13)

Here, it is assumed that the matrix [J_q] is not singular at the instant under consideration. If | ŝ ^T _r,i ŝ _i | is close to zero for the ith actuator, it is obvious from, the above equation that the twist of the moving platform T̂ becomes very sensitive to the velocity input of the actuator. Thus, it can be inferred that, in this configuration of the manipulator, small clearance such as the backlash of gears that is associated with the ith actuator may be greatly amplified along the screw ŝ _r_,i. That is, even though the actuator is locked, a finite motion of the moving platform may be generated along screw ŝ _r,i.

Furthermore, from Eq. (8) we have: $(\frac{{\hat{S}}_{r, i}^{T} {\hat{s}}_{i}}{J_{q, i}}) τ_{i} = {\hat{S}}_{r, i}^{T} \hat{w}$ (14)

From this relation, it becomes clear that as the reciprocal product |Ŝ ^T_r,i ŝ _i| which is involved in the ith actuator is close to zero, any wrench ŵ acting at the moving platform cannot be supported by the force (or torque) input of the ith actuator.

From corollary 2.1 and the above considerations, the index γ which is defined in Eq. (11) may also be used for the measure of closeness to a singular configuration of a parallel manipulator. The index γ is also uniquely defined for the parallel manipulator from the uniqueness of the screw-based Jacobian and reciprocal Jacobian.

Parallel manipulators have another kind of singularity due to the matrix [J_q] [29,40]. This matrix represents the reciprocal relation between the actuated joint screws and the column screws of the screw-based Jacobian [38,40]. If J_q_,i is close to zero, it is obvious from Eq. (6) that the manipulator is nearly structurally constrained for the wrench input along screw ŝ _i .

Considering these singularities together, we define the kinestatic characterization index γ_q of the parallel manipulator as follows: $γ_{q} \equiv \sqrt[n]{| \det ([r] {[J_{q}]}^{- 1}) |} = {(_{} γ_{q, 1} \cdot γ_{q, 2} \cdot \dots \cdot γ_{q, n})}_{}^{1 / n}$ (15)

where: $γ_{q, i} \equiv_{} | ({\hat{s}}_{r, i}^{T} {\hat{S}}_{i}) / J_{q, i} |$ (16)

As the index γ_q approaches zero, it is expected that the manipulator becomes very sensitive to joint clearances and that some of actuators may not support the wrenches acting at the moving platform. On the contrary, if γ_q tends to infinity, the manipulator becomes structurally constrained for the wrench inputs along some screws.

Since each diagonal element of [J_q] is the reciprocal product of the actuated joint screw and the column screw of the screw-based Jacobian, and since it is also uniquely determined [38], the index γ_q also denotes a unique invariant measure of the kinestatic characteristics of the given parallel manipulator.

4. Examples

4.1 Singularity Analysis of a Planar 4-Bar Linkage

Figure 2 shows the planar 4-bar linkage lying on the XY-plane. It is composed of four revolute joints. The joint screw Û _i can be represented as the unit line (the unit screw of zero pitch)having a direction parallel to the Z-axis and passing through the centre of the revolute joint J_i. It is expressed in Plücker's axis coordinates. This 4-bar linkage may be thought of as the 1-DOF parallel mechanism. That is, the output link L₃ is considered here as the moving platform which is connected to the ground by one RRR-serial chain with the joints {J₁,J₂,J₃} and one passive revolute joint J₄. Joint J₁ is the actuated joint of this mechanism.

Figure 2.

Planar 4-bar linkage. When the link L₃ is considered as the moving platform of the 1-DOF parallel mechanism, the lines ŝ and ŝ _r consist of the column screws of the Jacobian and the reciprocal Jacobian, respectively.

For the planar mechanism, the overall twist space belongs to Hunt's 5th special three-system of screws [41]. It is spanned by two independent free vectors parallel to the XY-plane and one line perpendicular to the plane. Furthermore, the overall wrench space of the planar mechanism is contained in Hunt's 4th special three-system of screws [41]. Two independent lines lying on the plane and one free vector perpendicular to the plane span the 3-dimensional wrench space. The RRR-serial chain is a planar mechanism with 3 degrees of freedom and therefore it has no constraint wrenches. The other serial chain – the passive revolute joint J₄ – has, however, the 2-dimensional space of constraint wrenches. If we take b̂ ₁ and b̂ ₂ as the basis screws of the constraint wrench space, they become the two independent lines lying on the XY-plane and passing through J₄ (see Fig. 2). Utilizing the method for the formulation of the screw-based Jacobian given in [38], J and [J_q] can be obtained as follows: $J = [\hat{s}] \in R^{6 \times 1} and [J_{q}] = {\hat{U}}_{1}^{T} \hat{s}$

where ŝ is the unit line lying on the XY-plane and passing through the centres of the revolute joints J₂ and J₃. It is expressed in Plücker's ray coordinates. Furthermore, [J_q] is a scalar with the value of the reciprocal product of the actuated joint screw Û ₁ and the column screw ŝ of the screw-based Jacobian. Geometrically, |Û ₁^Tŝ| becomes the perpendicular distance from J₁ to the line ŝ . For the coordinate frames given in Fig. 2, two unit lines ŝ and Û ₁ can be expressed as: ${\hat{U}}_{1}^{} = [{(r_{1} \times u_{1})}^{T}, u_{1}^{T}] = [0, 0, 0, 0, 0, 1]^{T}$

and: $\hat{s} = [s_{}^{T}, {(r_{} \times s_{})}^{T}] = \frac{1}{| | P_{3} - P_{2} | |} [\begin{matrix} P_{3} - P_{2} \\ P_{2} \times (P_{3} - P_{2}) \end{matrix}]$

where u ₁ and s denote, respectively, the unit directions of Û ₁ and ŝ . Moreover, r ₁ and r mean the position vectors to a point on the lines Û ₁ and ŝ respectively. P _i is the position vector to the revolute joint J_i. Now, from the reciprocal Jacobian analysis given in [37], J_r and [r] are found as: $J_{r} = [{\hat{S}}_{r}^{}] = [{\hat{U}}_{4}^{}] \in R^{6 \times 1} and [r] = {\hat{S}}_{r}^{T} \hat{s}$

The column screw ŝ _r is determined as the joint screw Û ₄ and |Ŝ^T_rŝ| denotes, geometrically, the perpendicular distance from J₄ to the line ŝ . If we let a, b, c and d be the lengths of the links L₁, L₂, L₃ and L₄, respectively, Û ₄ can be written as Û ₄ =[0,–d,0,0,0,1]^T. The column screw ŝ of the screw-based Jacobian spans the 1-dimensional space of the unconstraint wrenches and the 1-dimensional controllable twist space is spanned by the column screw ŝ _r of the reciprocal Jacobian.

For the 4-bar mechanism, the kinestatic characterization index γ_q of Eq. (15) can be denoted as: $γ_{q} \equiv | \det ([r] {[J_{q}]}^{- 1}) | = |_{} {\frac{{\hat{S}}_{r}^{T} \hat{s}}{{\hat{U}}_{1}^{T} \hat{s}}}_{} |$

Now, we consider the 4-bar linkage which has the following geometric parameters: $a = 45 m m, b = 20 m m, c = 70 m m, d = 90 m m$

For the 4-bar linkage with these parameters, as the crank link L₁ is rotated from θ = 15° to 80°, the variation of the index γ_q is plotted as illustrated in Fig. 3(a).

Figure 3.

Kinestatic characterization curve and singular configurations of the planar 4-bar linkage. (a) Kinestatic characterization curve (variation of the kinestatic characterization index, γ_q). (b) When the crank angle θ is θ = 50.608°, two lines ŝ and Û ₁ meet each other at J₁ and, thereby, γ_q ∞ because the reciprocal product Û ^T₁ ŝ becomes zero. (c) When θ = 19.72° and 75.52°, the column screw ŝ of the Jacobian intersects the column screw ŝ _r of the reciprocal Jacobian at J₄. Since Û _r^T ŝ = 0, γ_q 0. In both cases, ŝ is linearly dependent on the basis screws b̂ ₁ and b̂ ₂ of the 2-dimensional constraint wrench space.

When θ ≈ 50.609°, γ_q nearly goes to infinity. In that instant, as shown in Fig. 3(b), the column screw ŝ of the Jacobian and the actuated joint screw Û ₁ meet each other and, thereby, Û ^T₁ ŝ ≈ 0. The 4-bar linkage becomes structurally constrained for a wrench input on the rocker L₃. On the contrary, when θ = 19.72° and 75.52°, γ_q is close to zero. At these configurations, the column screw s of the Jacobian intersects to the column screw ŝ _r of the reciprocal Jacobian at J₄ (see Fig. 3(c)). Thus, ŝ ^T _r ŝ = 0. Since, in this case, the line ŝ belongs to the pencil of lines spanned by b̂ ₁ and b̂ ₂, it becomes clear that this singularity arises due to the linear dependence among the unconstraint and constraint wrench spaces. At these instants, an arbitrary wrench acting at the rocker L₃ cannot be supported by the actuator located at the joint J₁. Clearly, from Eq. (14), the only possible wrench that can be imposed on the rocker is the wrench acting along the screw (such as the line ŝ ) that is reciprocal ŝ _r .

It is interesting to note that when γ_q ≈ 1, the kinestatic characteristics of the mechanism seem to be balanced. When γ_q > 1, it can be inferred from Eq. (14) that the kinematic structure of the mechanism plays the role of reducing the actuator torque that is required to support the wrench acting at the moving platform. Thus, this configuration may be effective for supporting external wrenches. Conversely, if 0 < γ_q <1, from Eq. (13), the kinematic structure seems to act as a mechanical amplifier of the actuator velocity input and, therefore, the end-effector twist may be realized more efficiently at this configuration.

From this observation, it is expected that this contribution or restriction imposed on the actuator force and velocity transmissions due to the kinematic structure of the mechanism may be balanced at the manipulator configuration around γ_q = 1. Figure 4 shows the two possible configurations of the 4-bar linkage when γ_q ≈ 1.

Figure 4.

Two possible configurations of the 4-bar linkage when γ_q ≈ 1.

4.2 Singularity Analysis of a 3-DOF Parallel Manipulator

Now, we consider a 3-DOF parallel manipulator with three identical PRS (Prismatic-Revolute- Spherical)-serial chains (see Fig. 5). The moving platform of the manipulator can be translated along the Z-axis and can also be rotated independently about the X- and Y- axes. The kinematic parameters and the initial configuration of the manipulator are given in Table 1. Three serial chains are located symmetrically around the base and the moving platforms. With these kinematic parameters, the inverse kinematics can be solved using the method given in [42].

Table 1.

Kinematic parameters and the initial configuration of the 3-DOF parallel manipulator.

Parameters	Values
Radius of the moving platform (r)	1 m
Length of each leg (l_i)	1 m
α	120°
β	240°
Initial position of the moving coordinate frames	[0,0,0.3] (m)
Initial orientation of the moving platform	[Ψ, θ, 0]^T = [0,0,0]^T (deg)

Figure 5.

3-DOF parallel manipulator with three PRS-serial chains.

The Jacobian matrices J and [J_q] of the manipulator are given in [38]. If we take A_i and B_i as, respectively, the centres of the spherical and revolute joints of the ith serial chain, the ith column screw ŝ _i of J is found geometrically as the unit line passing through B_i and A_i. ŝ _i spans the 3-dimensional space of the unconstraint wrenches. Furthermore, the constraint wrench of the ith serial chain b̂ _i is obtained as the unit line which is parallel to the axis of the revolute joint of the serial chain and passes through the centre of the spherical joint A_i. Then, the 3-dimensional constraint wrench space of the manipulator is spanned by the three lines b̂ _i.

Since we know the screw-based Jacobian J and the basis screws of the constraint wrench space, the column screws ŝ _r,i of the reciprocal Jacobian J_r can be easily determined. For instance, ŝ _r _,1 is defined as the unit screw reciprocal to the 5 screws { ŝ ₂, ŝ ₃, b̂ ₁, b̂ ₂, b̂ ₃}. Thus, at the given initial configuration, it is determined as the unit line passing through A₂ and A₃. It is reciprocal to { ŝ ₂, ŝ ₃, b̂ ₂, b̂ ₃} because ŝ _r _,1 intersects with the 4 lines. In addition, since ŝ _r _,1 is parallel to line b̂ ₁, they are reciprocal to each other.

Now, the moving platform is rotated only around the Y-axis from θ = −15° to 15° while fixing the angle of rotation about the X-axis, Ψ, and the Z-axis translation of the moving platform to their initial values. For the given range of motions of the moving platform, γ_q is plotted as shown in Fig. 6(a). In the figure, it can be seen that there exist two singular configurations at θ = −11.615° and θ = 8.627°. In order to examine which column screws of the screw-based Jacobian are involved in each singular configuration, the changes of the values γ_q,i(i = 1,2,3) that are defined in Eq. (16) are examined, as shown in Figs. 6(b)–(d). It can be seen from the figures and from the corollary 2.1 that when θ = −11.615°, only the two column screws ŝ ₂ and ŝ ₃ of J are involved in the singularity. On the contrary, all of the column screws are associated with the singular configuration at θ = 8.627°. These two singular configurations are illustrated in Figs. 7(a)–(b). It is obvious from the figures that, in both cases, there is no linear dependence on the column screws ŝ _i of the screw-based Jacobian and, therefore, these are not the Jacobian singularities. In addition, three basis screws b̂ _i of the constraint wrench space are also linearly independent. This means that there exist no constraint singularities [43] in these two singular configurations. Consequently, the only possible case is the degenerated unconstraint wrench space.

Figure 6.

Kinestatic characterization curves of the 3-DOF parallel manipulator.

Figure 7.

Two singular configurations of the 3-DOF parallel manipulator. a) When the moving platform is rotated about the Y-axis to θ = −11.615°, the two column screws ŝ ₂ and ŝ ₃ of the screw-based Jacobian are involved in the singularity. (b) When θ = 8.627°, all the column screws ŝ _i of the screw-based Jacobian are associated with the singularity.

We consider the first singular configuration at θ = −11.615° (see Fig. 7(a)). If we let C be the intersection of the lines b̂ ₂ and b̂ ₃, they span the pencil of lines lying on the plane parallel to the XY-plane and having the centre point at C₁. Among the lines in the pencil, we take the line b̂' that is parallel to the Y-axis. Since b̂ ₁ is also parallel to the Y-axis, any lines parallel to the Y-axis and lying on the plane formed by b̂' and b̂ ₁ can be expressed as the linear combination of the two lines. Then, among these lines, the line which passes through the intersection C₂ of the two lines ŝ ₂ and ŝ ₃ can be chosen because, at this configuration, the three points C₁, C₂ and A₁ lie on the plane formed by the two lines b̂' and b̂ ₁. Clearly, the selected line that is parallel to the Y-axis and passes through point C₂ is linearly dependent on lines ŝ ₂ and ŝ ₃. Consequently, when θ = −11.615°, there exists a linear dependence among the lines { ŝ ₂, ŝ ₃, b̂ ₁, b̂ ₂, b̂ ₃}.

In this configuration, the column screw of the reciprocal Jacobian ŝ _r ,₁ is not affected by this singularity, which means that since γ_q,1 ≠ 0, from Eq. (13), the end-effector twist along the line ŝ _r _,1 that passes through the centres of the spherical joints A₂ and A₃ is still properly controllable by the 1st actuator. However, in this instant, ŝ _r ,₂ and ŝ _r ,₃ become nearly congruent to the line which passes through the three points C₁, C₂ and A₁. Because y_q_,2 = r_q,₃ ≈ 0, it can be inferred from Eq. (13) that even though all the actuators are locked, the unintentional rotation of the moving platform about the axis ŝ _r ,₂ may be generated.

Now, we consider the second singular configuration (θ = 8.627°) that is illustrated in Fig. 7(b). b̂ ₂ and b̂ ₃ also span the pencil of lines lying on the plane parallel to the XY-plane and having the centre point at C₁. Among the lines, if we take the line b̂″ that is parallel to the X-axis, b̂″ and ŝ ₁ intersect with each other at point C₃, which is the middle of the line segment $\bar{A_{2} A_{3}}$ . Then, the two lines b̂″; and ŝ ₁ span another pencil of lines lying on the XZ-plane and centred at C₃. Obviously, the line passing through C₂ (the intersection of ŝ ₂ and ŝ ₃) and C₃ belongs to the pencil of lines and, moreover, it is contained in the space spanned by ŝ ₂ and ŝ ₃. Therefore, when θ = 8.627°, there is a linear dependence among the 5 lines { ŝ ₁, ŝ ₂, ŝ ₃, b̂ ₂, b̂ ₃}.

In this second singular configuration, all the column screws of the reciprocal Jacobian are affected by the singularity. All the ŝ _r,i s are congruent to the single line passing through A₂ and A₃. Since γ_q,1 = γ_q,2 = γ_q,3 ≈ 0, the uncontrollable rotation of the moving platform is expected about the axis ŝ _r _,i, even if all the actuators are locked. Furthermore, from Eq. (14), any wrenches acting at the moving platform may not be supported by the actuators.

In both cases, the singularities arise due to the linear dependencies on the unconstraint and constraint wrench spaces. One remark is that, among the manipulator configurations inside the interval −11.615° < θ ≤ 8.627°, it can be seen from Fig. 6(a) that the maximum value of γ_q is generated at θ = 0°. Consequently, it is expected that, within the range of motions, the manipulator may have the most kinestatically balanced configuration around its initial state. However, since γ_q < 1 for the given range of motions, the manipulator configurations considered in this example may not have good performance as a force generating machine.

5. Conclusion

In this paper, a new geometric method for the analyses of the singularity and kinestatic characteristics of robot mechanisms is presented on the basis of the reciprocal products of the screws. For a non-redundant robot mechanism, there always exists two uniquely determined Jacobians – the screw-based Jacobian and the reciprocal Jacobian. From the reciprocity of the two Jacobians, a necessary and sufficient condition is suggested in order to identify which column screws of the screw-based Jacobian are linearly dependent at a singular configuration. Using this condition, the linearly dependent column screws of the screw-based Jacobian can be simply identified by checking the values of the reciprocal products between the column screws of the screw-based Jacobian and the reciprocal Jacobian, all of which are close to zero. From this observation, the measure of closeness to a singular configuration of robot mechanisms is defined by using the norms of these reciprocal products. Since the reciprocal product of screws is an invariant scalar product and the two Jacobians are uniquely determined, the proposed index provides a unique invariant measure of closeness to a singular configuration for non-redundant serial and parallel robots.

For the parallel manipulator, the proposed measure of singularity is extended by integrating the reciprocal relations between the column screws of the screw-based Jacobian and the actuated joint screws so that a concept of the kinestatic characterization index γ_q is proposed. If γ_q tends to infinity, the manipulator becomes structurally constrained for the wrench input along some column screws of the screw-based Jacobian. However, if γ_q is close to zero, even though all the actuators are locked, finite motions of the moving platform may be generated along some column screws of the reciprocal Jacobian.

With the configuration where γ_q > 1, the kinematic structure of the mechanism plays the role of reducing the actuator forces (or torques) that are required to support the external wrench. Conversely, if 0 < γ_q < 1, the kinematic structure acts as a mechanical amplifier of the actuator velocity inputs and, accordingly, the end-effector twist may be realized more efficiently at this configuration. From this observation, the manipulator configuration at γ_q ≈ 1 is defined as the kinestatically-balanced configuration of the robot.

From the singularity analysis of a planar 4-bar linkage, it is shown that all the singular configurations of the mechanism can be successfully identified with the proposed index. Furthermore, throughout the analysis of a 3-DOF parallel manipulator with three PRS-serial chains, the column screws of the screw-based Jacobian which are involved in a singular configuration of the manipulator are clearly identified.

For the investigation into the robot singularity and manipulability, most of the works in the literature have focused on just the Jacobian of robot mechanisms. In this work, however, by considering the two uniquely determined Jacobians, it is demonstrated that a unique invariant measure of closeness to a singularity can be formulated. Therefore, the author believes that more in-depth analyses on the kinestatic characteristics of robot mechanisms may be possible by examining the screw-based Jacobian, the reciprocal Jacobian and their geometrical interrelations.

Footnotes

6. Acknowledgments

This work was supported by the R&D Programme MKE/KEIT (10035201),the ADL Support System for The Elderly and Disabled and the Global Frontier R&D Programme on Human-centred Interaction for Coexistence funded by the National Research Foundation of Korea,grant funded by the Korean Government (MEST) (NRF-M1AXA003-2010-0029748).

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