Dimensional synthesis of spatial manipulators for velocity and force transmission for operation around a speciﬁed task point

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Dimensional synthesis of spatial manipulators for

velocity and force transmission for operation around

a speciﬁed task point

Akkarapakam Suneesh Jacob1,* and Bhaskar Dasgupta1,+

1Afﬁliation: Indian Institute of Technology Kanpur, Kanpur, India

*Emails: sunjac@iitk.ac.in, suneeshjacob@gmail.com

+Email: dasgupta@iitk.ac.in

ABSTRACT

Dimensional synthesis refers to design of the dimensions of manipulators by optimising different kinds of performance indices.

The motivation of this study is to perform dimensional synthesis for a wide set of spatial manipulators by optimising the

manipulability of each manipulator around a pre-deﬁned task point in the workspace and to ﬁnally give a prescription of

manipulators along with their dimensions optimised for velocity and force transmission. A systematic method to formulate

Jacobian matrix of a manipulator is presented. Optimisation of manipulability is performed for manipulation of the end-effector

around a chosen task point for 96 1-DOF manipulators, 645 2-DOF manipulators, 8 3-DOF manipulators and 15 4-DOF

manipulators taken from the result of enumeration of manipulators that is done in its companion paper devoted to enumeration

of possible manipulators up to a number of links. Prescriptions for these sets of manipulators are presented along with their

scaled condition numbers and their ordered indices. This gives the designer a prescription of manipulators with their optimised

dimensions that reﬂects the performance of the end-effector around the given task point for velocity and force transmission.

Keywords: dimensional synthesis, manipulability, condition number, jacobian, optimisation

1 Introduction

In the context of this study, a manipulator is deﬁned as an as-

sembly of kinematic links connected through kinematic joints,

with a ﬁxed base link and a moving end-effector link, in which

the end-effector link’s motion and force-transformation are

controlled by actuating one or more kinematic joints. A kine-

matic link is a rigid link that can transfer motion and forces,

and a kinematic joint is a connection between two links that

allows relative motion between the two links. Dimensional

synthesis of a manipulator is the problem of designing an

appropriate set of dimensions for the manipulator of deﬁned

topology

to achieve a requirement. Dimensional synthesis

problems are generally solved by taking one or more objec-

tives into consideration, such as energy optimisation and well-

conditioning of the end-effector’s motion. This study focuses

on the objective of well-conditioning for velocity and force

transmission around a given task point. Well-conditioning is

a measure of being distant from singularity. Singularity is

related to the local degeneracy of input-output motion/force

transformation. A manipulator is said to have a singularity at a

point in the joint space if at that conﬁguration the end-effector

of the manipulator is unable to be controlled by the actuating

joint velocities at least in one direction, either in translation or

by rotation. The singularity of a manipulator is often analysed

using its Jacobian, the transformation matrix of the joint ve-

locities to the velocity of the end-effector. Using the principle

of virtual work, it can be shown that for a system with neg-

1A speciﬁc set of connections of links with speciﬁc types of joints

ligible frictional power losses the force transmission matrix

is exactly the transpose of the velocity transmission matrix.

This gives an advantage in designing the dimensions of the

manipulators for both inverse force transformation and direct

velocity transformation by using the same matrix. Hence, this

matrix is used to analyse the singularities of the manipulator.

The task of dimensional synthesis is to design the dimensional

parameters of various manipulators with the criterion of the

maximum extent of singularity avoidance when a represen-

tative point in the workspace about which the manipulator’s

end-effector operates is given along with the bounds of the

environment. The criterion used in this study is the manip-

ulability index. The manipulability index for a manipulator

would typically be a function of dimensional parameters as

variables. The manipulability index with ellipsoidal approach

[1] is as follows.

µ=pdet([J][J]T)if nj>nt

pdet([J]T[J]) if nj≤nt

(1)

where

is the dimension of the joint space and

is the

dimension of the task space of the manipulator.

The values of the dimensional parameters that would max-

imise this function would describe the dimensions of the ma-

nipulator that would give good performance around the given

end-effector point. This concept with ellipsoidal approach is

illustrated below for a simple case of a Jacobian matrix that

maps linear velocities alone of a 2R-serial planar manipulator.

For a two-link serial manipulator of link lengths

and

arXiv:2210.04446v1 [cs.RO] 10 Oct 2022

and relative joint angles

θ1

and

θ2

as shown in ﬁgure 1, the

Jacobian matrix for mapping the linear velocities alone, is

given by

vx

vy=−l1sinθ1−l2sin(θ1+θ2)−l2sin(θ1+θ2)

l1cosθ1+l2cos(θ1+θ2) +l2cos(θ1+θ2)˙

θ1

θ2

⇒ve= [J]{˙

θ}

Since the Jacobian matrix in this case is a square matrix,

pdet([J]T[J]) = det([J])

. The determinant of Jacobian (of

transformation of linear velocities) would be

det([J]) = −l1sin θ1−l2sin (θ1+θ2)−l2sin(θ1+θ2)

l1cosθ1+l2cos(θ1+θ2) +l2cos(θ1+θ2)

⇒det([J]) = l1l2sinθ2

Figure 1. Manipulability varying with θ2.

For the determinant to be maximum, the term

l1l2sinθ2

needs to be maximum. Since

and

are independent of

θ2

, by assuming them to be constant,

θ2=90◦

gives the

maximum contribution of manipulability independent of

and

, as shown in the ﬁgure 1. And by assuming

and

be varying,

l1=l2

gives its maximum contribution, as shown

in the ﬁgure 2. Hence, for a given location of the end-effector,

the values θ1,θ2,l1and l2can be found.

In the current study, the manipulability index for a speci-

ﬁed end-effector point is maximised for various manipulators,

and prescriptions are presented for manipulators for same

kind of task around the speciﬁed point. The same kind of

task is reﬂected by the DOF, and hence the prescriptions of

manipulators for each DOF are provided.

Figure 2. Manipulability varying with l1and l2.

2 Literature review

Manipulability of a manipulator is calculated by using its Ja-

cobian. In the context of robotics, Jacobian is the matrix that

maps the joint-space velocities to the task-space velocities.

There are several methods to formulate Jacobian available

in the literature. In serial manipulators, only two types of

joints, namely prismatic and revolute, are actuated. A stan-

dard method to formulate Jacobian for serial manipulators can

be found in many books. However, for parallel manipulators,

Jacobian formulation becomes more complicated, as the ma-

nipulator can contain other types of joints, such as prismatic

and spherical joints and can contain passive revolute and pris-

matic joints. Kevin et al. [

] formulated Jacobian for a novel

6-DOF parallel manipulator by splitting the mapping into

two parts, one being the mapping between the end-effector

forces/torques and the forces at the spherical joints, and the

other being the mapping between the forces at the spherical

joints and the torques at the actuating joints. Kim et al. [

]

presented a formulation for obtaining analytic Jacobian of

parallel manipulators using screw theory. Kim et al. [

] pre-

sented the formulation of a new dimensionally homogeneous

Jacobian matrix by taking three end-effector points. In their

paper, the formulation of Jacobian is done by taking three

points of the end-effector platform to interpolate any random

point on the platform and differentiating it with respect to

time, and ﬁnally rewriting the passive joint velocities in terms

of active joint velocities and establishing the Jacobian. Geof-

frey et al. [

] also formulated a dimensionally homogeneous

Jacobian matrix of parallel manipulators for dexterity analy-

sis by using three points on the end-effector platform with a

better physical signiﬁcance of the properties of Jacobian. Liu

2/18

et al. [

] presented a method to systematically formulate the

dimensionally homogeneous Jacobian when the manipulator

has just one type of actuator, i.e., either prismatic or revolute.

Their paper shows steps to formulate the generalised Jaco-

bian by using the notation of screw theory. Hu [

] presented

a formulation of uniﬁed Jacobian for serial-parallel manip-

ulators, i.e., a set of parallel manipulators linked one upon

the other, serially. Even though there are many systematic

formulations of Jacobian in the literature, there appears to

be no unique method of systematic formulation of Jacobian

that is applicable for both serial and parallel manipulators

(including closed-loop spatial kinematic chains). In this study,

a systematic formulation of Jacobian is presented that is ap-

plicable for both serial and parallel manipulators (including

closed-loop spatial kinematic chains) containing four types of

joints, namely revolute, prismatic, cylindrical and spherical.

Condition number of Jacobian is the ratio of maximum

and minimum singular values of the Jacobian. It gives a mea-

sure of how much the manipulability is distributed to each

singular value. Sometimes the manipulability measure alone

might be unable to capture the information about singularity at

a conﬁguration in which case the condition number could give

more information. For example, if a Jacobian has three singu-

lar values

100000

0.00002

and

, the singular value

0.00002

shows that it is close to singularity. But the manipulability

would be

, from which it is not very clear that it is close to sin-

gularity, whilst the condition number would be

5×109

which

indicates that the manipulability is not equally distributed.

Hence, condition numbers at the optimal conﬁgurations of

the manipulators are important to analyse the performances.

Jacobian is a mapping from joint velocities to end-effector

velocities. This includes both the linear and the angular veloc-

ities of the end-effector that have different units. Similarly, the

transpose of Jacobian maps the joint torques/forces to both

forces and moments of the end-effector, which again have

different units. This difference in units makes it difﬁcult to

assess the signiﬁcance of the condition number of Jacobian.

It is discussed in the literature that due to the non-uniform

dimensions of the elements of the Jacobian matrix, the condi-

tion number may have little physical signiﬁcance. Doty et al.

[

] identiﬁed the problem of difference in units of elements

of Jacobian and the difﬁculties associated with it. Angeles

[

] used the concept of natural length to present the Jacobian

matrix in dimensionless form. In his paper, the natural length

is chosen such that the condition number of the Jacobian ma-

trix is minimised. Stocco et al. [

] used a scaling matrix

with which the Jacobian matrix is to be multiplied in order

to normalise the units of the Jacobian matrix and to use it

with a performance goal. Their paper chooses the scaling

matrix based on maximum desired forces. Ma et al. [

]

discussed non-uniformity of units in the elements of Jacobian

matrix, and used characteristic length to homogenise the Jaco-

bian matrix. In their paper, a homogenised form of Jacobian

matrix is presented for a 6-DOF platform manipulator, with

characteristic length as a chosen parameter. In their paper,

they suggested a scaling of the Jacobian matrix by multiply-

ing it with the matrix

[S] =







L0 0 0 0 0

L0 0 0 0

0 0 1

L000

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1







where

is called the characteristic length. As mentioned in their

paper, there are many ways to deﬁne the characteristic length

depending on the interest of the user. Various proposals of

characteristic length were made in the literature, depending on

the orientation of the problem that is to be solved. Their paper

chose

, for platform manipulators of spherical joints, as the

average of the distances from the centroid of the moving plate

to each of the joints.

Yoshikawa [

] used the concept of singular values of Ja-

cobian matrix as the manipulability measure to describe the

performance of manipulators. Lee [

] used the concept of

manipulability polytypes as a competing measure of manip-

ulability. In his paper, the manipulability measure through

manipulability ellipsoids is compared with that of manipu-

lability polytopes. The paper concludes that the polytope

approach can represent the manipulability more accurately

than that of the ellipsoid approach, as the ellipsoid would not

be covering the entire region of the set of unit joint velocities

in the joint-space. Even though the polytope approach rep-

resents a better description of manipulability measure than

the ellipsoid approach, the ellipsoid approach is simple to

implement. To reduce the computational complexity and for

simplicity, the ellipsoid approach is implemented in this study.

Khezrian et al. [

], in their paper, designed a spherical

3-DoF parallel manipulator by maximising Global Dynamic

Conditioning Index. In their paper, the Global Dynamic Con-

ditioning Index of the manipulator is optimised, and the op-

timal dimensional parameters are presented. Hazarathaiah

[

] presented a prescription of several planar manipulators of

revolute joints with corresponding optimised manipulability

indices and condition numbers for a speciﬁed task position.

In his thesis, several planar manipulators are presented, and

the manipulability index of each manipulator is calculated

as a function of its dimensional parameters. For each of the

manipulators, the manipulability index is maximised, and the

corresponding dimensions are presented. But this is limited to

planar manipulators with revolute type of joints. The current

study presents the prescriptions for several spatial manipu-

lators with four types of joints, namely revolute, prismatic,

cylindrical and spherical joints.

3 Methodology

3.1 Methodology to formulate Jacobian

The method used to generate Jacobian matrices for the ma-

nipulators in the context of this study is based on identifying

all possible connecting paths from the base link to the end-

effector link and kinematically formulating linear and angular

velocities of the end-effector link through these paths, and

3/18

ﬁnally transforming the passive joint velocities in terms of

active joint velocities.

Identifying all possible connecting paths:

Figure 3. A mechanism with loops for illustration.

For the manipulator depicted in ﬁgure 3, the connecting

paths from the base link to the end-effector link are shown

below.

Path 1: L1−j1−L2−j2−L3−j3−L4−j5−L6

Path 2: L1−j4−L4−j5−L6

Path 3: L1−j6−L5−j7−L6

Formulating linear and angular velocities:

Each consecutive pair of links of the connecting path is

considered to cumulatively formulate linear and angular veloc-

ities from the base link to the end-effector link, as each joint

contributes linear and angular velocities to the end-effector

link. For each consecutive pair of links in the connecting path,

the contributions to the angular velocity and the linear velocity

of the end-effector are given by

ωi j

and

~vi j

respectively in

various cases, as shown in table 1.

Joint type ~

ωi j ~vi j

Revolute ˙

θi j ˆni j ˙

θi j ˆni j ×(~a−~ri j)

Prismatic 0 ˙

di j ˆni j

Cylindrical ˙

θi j ˆni j ˙

θi j ˆni j ×(~a−~ri j) + ˙

di j ˆni j

Spherical ~

ωi j ~

ωi j ×(~a−~ri j)

Table 1. Result of the optimisation problem corresponding

to 2D-M71 manipulator.

And the angular and linear velocities of the end-effector,

described through the connecting path, are given by the sum

of all the components of angular velocities and the sum of all

the components of linear velocities contributed to the end-

effector, respectively. Here,

~ri j =ri jx ri jy ri jzT

and

ˆni j =ni jx ni jy ni jzT

represent the position vector of the

instantaneous location and the axis of the instantaneous mo-

tion respectively, of the joint connected to the links

and

ωi j =ωi jx ωi jy ωi jzT

represents the relative angu-

lar velocity vector of the link

relative to the link

, of the

adjacency matrix.

~a=axayazT

is the position vector

of the end-effector, and

θi j

and

di j

represent the translational

and the rotational joint velocities respectively, of the joint

connected to the links

and

of the adjacency matrix, mea-

sured relative to the link Li.

Applying the above process to each connecting path from

base link to end-effector link gives the description of velocity

of the end-effector in terms of linear combinations of active

and passive joint velocities, the coefﬁcients being functions

of the conﬁguration parameters of the manipulator, i.e.,

ri jx

ri jy

ri jz

βi j

φi j

and

, where

~a=axayazT

the position vector of the end-effector point. For the particular

kind of manipulator shown in ﬁgure 3, there are three paths

and hence three descriptions of pairs of angular and linear

velocities can be formulated as

v1

ω1

v2

ω2

and

v3

ω3

each expressed in the form of linear combinations of active

and passive velocities as shown in equation (2) for all i.

v(i)

ω(i)=hJ(i)i˙

θa

Ω=hJ(i)

1i˙

θa+hJ(i)

2iΩ(2)

Each of

v(i)

ω(i)

would represent the velocity components

of the end-effector. Since the Jacobian matrix is a mapping

of end-effector velocities with active joint velocities, all the

passive joint velocities, i.e., all the variables

θi j

di j

ωi jx

ωi jy

and

ωi jz

other than the actuating joint velocities, are to

be written in terms of active joint velocities. The number of

independent formulations of velocities (through the identiﬁed

connecting paths from the base link to the end-effector link)

should be more than or equal to the number of passive joint

velocities that are to be eliminated. Once the passive joint

velocities are written in terms of linear combinations of ac-

tuating joint velocities, the velocities of the end-effector can

be expressed as a mapping of actuating joint velocities alone,

and the mapping matrix thus generated would be the Jacobian

matrix. The convention followed in this study in order to

formulate the Jacobian matrix is to consider

v1

ω1

as the pair

of linear and angular velocities of the end-effector as shown in

equation

(3)

and to consider

vi−v1

ωi−ω1

for all

i6=1

to form

the matrices

[A1]

and

[A2]

, as shown in equation

(4)

, where

is the number of paths from base link to end-effector link of

the manipulator.

v

ω=v1

ω1= [J1]˙

θa+ [J2]Ω(3)











v2−v1

v3−v1

···

vN−v1

ω2−ω1

ω3−ω1

···

ωN−ω1











= [A]˙

θ= [A1]˙

θa+ [A2]Ω=0 (4)

4/18

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DimensionalsynthesisofspatialmanipulatorsforvelocityandforcetransmissionforoperationaroundaspeciedtaskpointAkkarapakamSuneeshJacob1,*andBhaskarDasgupta1,+1Afliation:IndianInstituteofTechnologyKanpur,Kanpur,India*Emails:sunjac@iitk.ac.in,suneeshjacob@gmail.com+Email:dasgupta@iitk.ac.inABSTRACTDimen...

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Dimensional synthesis of spatial manipulators for velocity and force transmission for operation around a speciﬁed task point

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