158

Chapter 6

Screw theory for instantaneous kinematics

6.1 Introduction

Screw theory was developed by Sir Robert Stawell Ball [111] in 1876, for

application in kinematics and statics of mechanisms (rigid body mechanics). It is a

way to express displacements, velocities, forces and torques in three dimensional

space, combining both rotational and translational parts. Rotational motion of a rigid

body is represented by an element of the special orthogonal group.

𝑆𝑂 3 = 𝑅 ∈ 𝑅3×3| 𝑅𝑇𝑅 = 𝐼, det 𝑅 = 1 (6.1)

Other parameterizations of 𝑆𝑂(3) include fixed and Euler angle sets and unit

quaternions. The change in position as well as orientation of a rigid body is equivalent

to a screw motion, a simultaneous translation along and rotation about the some axis

in space.

A pure screw is simply a geometric concept which describes a helix. A screw

with zero pitch looks like a circle. A screw with infinite pitch looks like a straight

line. A zero pitch screw is a pure rotation (as a screw) or a pure force (as a wrench).

Conversely, an infinite pitch screw is a pure translation (as a screw) or a pure torque

(as a wrench) for statics [112,113].

6.2 Exponential coordinates for rotation

Rotation (𝑅) is function of unit direction vector of rotation (𝜔) and angular

rotation 𝜃 in radians. Rotation of a body with constant unit velocity about the axis

𝜔, the velocity of point can be expressed as,

𝑞 𝑡 = 𝜔 × 𝑞 𝑡 = 𝜔 𝑞 𝑡 (6.2)

Integrating time invariant linear differential equation,

𝑞 𝑡 = 𝑒𝜔 𝑡𝑞 0 (6.3)

Where, 𝑞 0 is initial position of point and 𝑒𝜔 𝑡 is matrix exponential

𝑒𝜔 𝑡 = 𝐼 + 𝜔 𝑡 + 𝜔 𝑡 2

2!+

𝜔 𝑡 3

3!+ ⋯ (6.4𝑎)

The net rotation about axis 𝜔 at unit velocity for 𝜃 unit times,

𝑅 𝜔, 𝜃 = 𝑒𝜔 𝜃 = 𝐼 + 𝜔 𝜃 + 𝜔 𝜃 2

2!+

𝜔 𝜃 3

3!+ ⋯ 6.4𝑏

159

Infinite series is not useful for computation purpose and its closed form

formulation is referred as Rodrigues’ formula [114],

𝑅 𝜔, 𝜃 = 𝑒𝜔 𝜃 = 𝐼 + 𝜔 𝑠𝑖𝑛𝜃 + 𝜔 2 1 − 𝑐𝑜𝑠𝜃 , 𝜔 = 1 (6.5)

The exponential coordinates are called the canonical coordinates of the

rotation group and 𝜔 is skew symmetric matrix. The angular velocity vector of a rigid

body can be expressed with respect to a certain basis as,

𝜔 =

𝜔1

𝜔2

𝜔3

≡ 𝜔 = 0 −𝜔3 𝜔2

𝜔3 0 −𝜔1

−𝜔2 𝜔1 0 (6.6)

Let us consider a rotation about x-axis,

The x-axis is 1, 0, 0 , which is in a skew symmetric matrix form as,

𝜔 = 0 0 00 0 −10 1 0

(6.6𝑎)

𝑒𝜔 𝜃 = 𝐼 + 𝜔 𝑠𝑖𝑛𝜃 + 𝜔 2 1 − 𝑐𝑜𝑠𝜃

=

1 − 𝑣𝜃 𝜔22 + 𝜔3

2 𝜔1𝜔2𝑣𝜃 − 𝜔3𝑠𝜃 𝜔1𝜔3𝑣𝜃 + 𝜔2𝑠𝜃

𝜔1𝜔2𝑣𝜃 + 𝜔3𝑠𝜃 1 − 𝑣𝜃 𝜔12 + 𝜔3

2 𝜔2𝜔3𝑣𝜃 − 𝜔1𝑠𝜃

𝜔1𝜔3𝑣𝜃 − 𝜔2𝑠𝜃 𝜔2𝜔3𝑣𝜃 + 𝜔1𝑠𝜃 1 − 𝑣𝜃 𝜔12 + 𝜔2

2

=

𝜔12𝑣𝜃 + 𝑐𝜃 𝜔1𝜔2𝑣𝜃 − 𝜔3𝑠𝜃 𝜔1𝜔3𝑣𝜃 + 𝜔2𝑠𝜃

𝜔1𝜔2𝑣𝜃 + 𝜔3𝑠𝜃 𝜔22𝑣𝜃 + 𝑐𝜃 𝜔2𝜔3𝑣𝜃 − 𝜔1𝑠𝜃

𝜔1𝜔3𝑣𝜃 − 𝜔2𝑠𝜃 𝜔2𝜔3𝑣𝜃 + 𝜔1𝑠𝜃 𝜔32𝑣𝜃 + 𝑐𝜃

(6.7)

Where, 𝑣𝜃 = 1 − 𝑐𝑜𝑠𝜃 , 𝑠𝜃 = 𝑠𝑖𝑛𝜃 and 𝑐𝜃 = 𝑐𝑜𝑠𝜃

The general rotation matrix,

𝑅 =

𝑟11 𝑟12 𝑟13

𝑟21 𝑟22 𝑟23

𝑟31 𝑟32 𝑟33

(6.8)

𝑅𝑥 = 𝑒𝑥 𝜃 = 1 0 00 𝑐𝜃 −𝑠𝜃0 𝑠𝜃 𝑐𝜃

(6.8𝑎)

Similarly, other representations are,

𝑅𝑦 = 𝑒𝑦 𝜃 = 𝑐𝜃 0 𝑠𝜃0 1 0

−𝑠𝜃 0 𝑐𝜃 (6.8𝑏)

𝑅𝑧 = 𝑒𝑥 𝜃 = 𝑐𝜃 −𝑠𝜃 0𝑠𝜃 𝑐𝜃 00 0 1

(6.8𝑐)

160

6.3 Finite rotations

The purpose of this chapter to make aware the possible ways to tackle

kinematics and dynamics with large rotations by rigid and flexible bodies and finally

computing angular velocities and accelerations associated them.

Figure 6.1 Various approaches for finite rotations

6.3.1 Body fixed to reference frame transformation

Let 𝐴(𝑥1, 𝑦1, 𝑧1) is any point on the rigid body B (in-deformable body

hypothesis). The coordinates (𝑥1, 𝑦1, 𝑧1) are denoted with an absolute reference frame

𝑂; 𝐿, 𝑀, 𝑁 and (𝑋1, 𝑌1, 𝑍1) are denoted with respect to relative reference frame

𝑂′ ; 𝑃, 𝑄, 𝑅 rigidly attached to body B as shown in figure 6.2.

Figure 6.2 Body fixed to reference frame transformation

𝑥

𝑦

𝑧

𝐿 𝑀

𝑁

𝑂

𝑃

𝑄

𝑅

𝑂′

𝑦′

𝑥′

𝑧′

𝐴

𝐵 𝑎

𝑎 0

𝐴

Geometrical Approach

- Euler Angles

- Bryan Angles

- Euler-Chasles

representation

- Euler and Rodrigues

parameters

-

Matrix Approach

- Based on orthonormal

property of rotation

operator

- Rotation operator is

expressed by either

Euler or Rodrigues

parameters

Algebraic Approach

- Quaternion

algebra

- Matrix algebra

(Based on

differential

geometry)

Ways of finite rotations

161

Decomposition of position vector 𝐴 should be in form of,

𝑎 = 𝑎0 + 𝐴 (6.9)

Where,

𝑎 = 𝑂𝐴 , 𝑎0 = 𝑂𝑂′ and 𝐴 = 𝑂′𝐴

In matrix form,

𝑎 𝑇 = 𝑥1 𝑦1 𝑧1

𝑎0 𝑇 = 𝑥01 𝑦01 𝑧01

[𝐴]𝑇 = 𝑋1 𝑌1 𝑍1

The vectors 𝑎 , 𝑎0 𝑎𝑛𝑑 𝐴 represent unicolumn matrices as above. Here,

components of vector 𝐴 are expressed in form of body fixed reference frame

𝑂′ ; 𝑃, 𝑄, 𝑅 . The corresponding components 𝑃′ in the inertial frame can be obtained

by,

𝐴′ = 𝑅 𝐴 (6.10)

Where R represents transformations from relative reference frame to fixed reference

frame (i.e. frame 𝑂′ ; 𝑃, 𝑄, 𝑅 → 𝑂; 𝐿, 𝑀, 𝑁 )

The stated transformation preserves the length of vector 𝐴

𝐴′ 𝑇𝐴′ = 𝐴𝑇𝑅𝑇𝑅 𝐴 = 𝐴𝑇𝐴 (6.11)

The 𝑅𝑇𝑅 = 1 is an algebraic condition of this transformation matrix. These are

known as unitary or orthogonal matrices with 𝑅 = ∓1. If 𝑅 = +1 then called

proper orthogonal matrices used for rigid body rotations but 𝑅 = −1 are improper

orthogonal matrices employed for reflections. Thus, 𝑃′ = 𝑅 𝑃 with 𝑅−1 = 𝑅𝑇

represents the body reference frame coincides with fixed reference frame. So,

𝑎 = 𝑎0 + 𝑅𝐴 (6.12)

The above expression depicts the position and orientation transformation

resulting from a rigid body rotation and translation. The orthonormality property of R

can be expressed in terms of 3 independent parameters like normal, orient and

approach. Actually, R is 3×3 matrix with 9 components as expressed in

equation (6.8).

The orthonormality property of the vectors 𝑟𝑗 imposed six constraints to each

other.

𝑟𝑇𝑖𝑟𝑗 = 𝛿𝑖𝑗 , (𝑖 = 1,2. . 𝑗, 𝑗 = 1,2,3 and 𝛿𝑖𝑗 = 0 𝑜𝑟 1) (6.13)

162

The vectors are linked together by six constraints are,

𝑛𝑥2 + 𝑛𝑦

2 + 𝑛𝑧2 = 1

𝑜𝑥2 + 𝑜𝑦

2 + 𝑜𝑧2 = 1

𝑎𝑥2 + 𝑎𝑦

2 + 𝑎𝑧2 = 1

𝑛𝑥𝑜𝑥 + 𝑛𝑦𝑜𝑦 + 𝑛𝑧𝑜𝑧 = 0

𝑜𝑥𝑎𝑥 + 𝑜𝑦𝑎𝑦 + 𝑜𝑧𝑎𝑧 = 0

𝑎𝑥𝑛𝑥 + 𝑎𝑦𝑛𝑦 + 𝑎𝑧𝑛𝑧 = 0 (6.14)

6.3.2 Translational and rotational velocities

The matrix form of the any position vector A on the rigid body B,

𝑎 = 𝑎0 + 𝑅 𝐴 (6.15)

The time differentiation of the above expression gives the velocity vector,

𝑎 = 𝑎 0 + 𝑅 𝐴 + 𝑅 𝐴 (6.16)

Where,

𝑎 0 = Velocity vector at reference point 𝑂′ . 𝐴 = 0 (As rigid body)

Hence, velocity of any point on the rigid body,

𝑎 = 𝑎 0 + 𝑅 𝐴 (6.17)

Using the equation (6.15),

𝐴 = 𝑅𝑇 𝑎 − 𝑎0 (6.18)

The velocity expression becomes,

𝑎 = 𝑎 0 + 𝑅 𝑅𝑇 𝑎 − 𝑎0 (6.19)

The absolute velocity of point P can be expressed in matrix form using extrinsic

expression of vector in Cartesian coordinates,

𝑑𝑎

𝑑𝑡=

𝑑𝑎0

𝑑𝑡+ 𝜔 × 𝑎 − 𝑎0 (6.20)

Where, 𝜔 = Angular velocity vector from frame 𝑂′ ; 𝑃, 𝑄, 𝑅 to frame 𝑂; 𝐿, 𝑀, 𝑁

Owing to orthonormal property of R,

𝑅 𝑅𝑇 = 1 → 𝑅 𝑅𝑇 + 𝑅 𝑅 𝑇 = 0 (6.21)

∴ 𝑅 𝑅𝑇 = − 𝑅 𝑅 𝑇 , which is a skew symmetric

The matrix of the angular velocities can be expressed using equation 6.6 as,

𝜔 = 𝑅 𝑅𝑇 = 0 −𝜔3 𝜔2

𝜔3 0 −𝜔1

−𝜔2 𝜔1 0

163

Where, 𝜔1, 𝜔2 & 𝜔3 are the components of the vector 𝜔 with respect to a

frame 𝑂; 𝐿, 𝑀, 𝑁 .

The matrix analog of the above equation is,

𝑑𝑎

𝑑𝑡=

𝑑𝑎0

𝑑𝑡+ 𝜔 𝑎 − 𝑎0 (6.22)

6.3.3 Translation and rotational accelerations

The second time derivatives of a general spatial motion equation is expressed

in the form of

𝑎 = 𝑎 0 + 𝑅 𝐴

𝑎 = 𝑎 0 + 𝑅 𝑅𝑇 𝑎 − 𝑎0 (6.23)

The matrix 𝑅 𝑅𝑇 can be obtained using time differentiation of the angular velocity

matrix 𝜔 .

Rigid body differential motion of equation 6.22 may also be interpreted as

screw motion. The screw motion can be described as combination of translation and

rotation motion about the screw axis 𝑠 in a space. Hence, the concept of screw, screw

axis, pitch of screw, twist and wrench of screw are explained in next section.

6.4 The concept of screw, screw axis and pitch of screw

6.4.1 The screw

The screw S is nothing but a line l together with a scalar pitch h. A screw is a

six dimensional vector composed by a primal part 𝑠 and dual part 𝑠𝑂 and can be

represented in form of

$ = 𝑠𝑠𝑂

(6.24)

Where, 𝑠 is a unit vector along the screw axis, where as 𝑠𝑂 is the moment produced by

𝑠 about a point 𝑂 fixed to the reference frame , which is calculated according to the

pitch 𝑕 of the screw and vector 𝑟𝑂, denoting the position of the point 𝑂 with respect to

any point on the screw axis.

𝑠𝑂 = 𝑟𝑂 × 𝑠 + 𝑕𝑠 (6.24𝑎)

A screw is a line in space with an associated pitch, which is a ratio of linear to

angular quantities. Zero pitch is a rotation about the line, then the screw represents a

revolute joint and using Plu cker coordinates,

$ = 𝑠

𝑟𝑂 × 𝑠 (6.24𝑏)

164

Infinite pitch is a translation, then the screw pair reduces to prismatic joint and

represented by

$ = 0𝑠 (6.24𝑐)

Where, 0 = 0 0 0 𝑇 is a three dimensional null vector.

If a screw passes through the origin of the coordinate system, the screw coordinates

can be denoted as:

$ = 𝑠𝑕𝑠

(6.24𝑑)

6.4.2 Screw axis

The line l is called the (screw) axis of the motion. For a pure translation, this

axis is at infinity. Screw is a line (𝑞, 𝑞0) + a pitch 𝑕.

Define the screw coordinates to be (𝑠, 𝑠0), where

𝑠 = 𝑞

𝑠𝑂 = 𝑞0 + 𝑕𝑞 (6.25)

The pitch 𝑝 can be determined back using 𝑞 ∙ 𝑞𝑂 = 0,

𝑠 ∙ 𝑠𝑂 = 𝑞 ∙ 𝑞𝑂 + 𝑕𝑞. 𝑞 (6.26)

6.4.3 Pitch of Screw

The ratio of the translation parallel to the axis (𝑑) to a rotation about the axis

(𝜃) is known as a pitch of the screw (𝑕) with a finite motion. It is a coordinate

independent property of motion.

𝑕 =𝑑

𝜃 (6.27)

Where, 𝑑 = translational displacement

𝜃 = rotation around the axis

Using equation (6.26),

𝑕 =𝑠 ∙ 𝑠𝑂

𝑠 ∙ 𝑠 (6.28)

[A norm is invariant to the choice of a coordinate system although it can depend on

the choice of length scale. A norm is similar to the concept of a distance and distance

does not depend on coordinate system used. The distance measured can be in m, cm,

mm etc. and hence the number signifying distance will be different. Hence the pitch is

invariant to the choice of the coordinate system]

The pitch measures linear advancement along the axis per unit rotation about

the axis. The magnitude of pitch is zero, the point on nut move in circular path that

165

provides pure rotation similar to hinge or revolute joint motion. Pitch magnitude is

very large, the point on nut move in straight line path that provides pure translation

similar to sliding or prismatic joint. For finite displacements the pitch of the twist is

not unique since the rotation angle may differ by multiples of 2π.

6.5 Twist and wrench of screw

It is well known that screw theory underlies the foundation of both

instantaneous kinematics and statics. The concept of twist and wrench are also

explained by Sir Robert Stawell Ball [111] for kinematic and statics of mechanism in

1876.

6.5.1 Twist

The infinitesimal version of a screw motion is called a twist and it provides a

description of the instantaneous velocity of a rigid body in terms of its linear and

angular components.

(a) Revolute joint (b) Prismatic joint

Figure 6.3 Revolute joint and prismatic joint

Twists represent velocity of a body. For example, climbing up a spiral

staircase at a constant speed can also be represented in form of twist. In general, the

velocities of each particle within a rigid body define a helical field called the velocity

twist. A twist contains 6 quantities (three linear and three angular). Another way of

decomposing a twist is by 4 line coordinates (see Plücker coordinates), 1 scalar pitch

value and 1 twist magnitude.

The velocity of a point 𝐴(𝑡) is as shown in figure 6.3(𝑎),

𝐴 𝑡 = 𝜔 × 𝐴 𝑡 − 𝐵 = 𝜔 × 𝐴 𝑡 − 𝜔 × 𝐵 = 𝜔 𝐴 − 𝜔 × 𝐵 (6.29𝑎)

𝐴

𝜔

𝐵

v

A

A(t)

A(t)

166

Using homogeneous coordinates,

𝐴

0 =

𝜔 −𝜔 × 𝐵0 0

𝐴1 = 𝜉

𝐴1 (6.29𝑏)

Rigid body transformations can be represented as the exponentials of twists:

Where, 𝜉 = 𝜔 𝑣0 0

, 𝜔 ∈ 𝑆𝑂 3 , 𝑣 ∈ 𝑅3, 𝜃 ∈ 𝑅 , 𝜉 ∈ 𝑅4×4

The twist coordinates are 𝜉 = (𝑣, 𝜔) ∈ 𝑅6.

Conversely, given a screw one can write the associated twist. Two special

cases are pure rotation about an axis 𝑙 = 𝑏 + 𝑕𝜔 by an amount 𝜃 and pure

translation along an axis= 0 + 𝑕𝑣 .

𝜉 = −𝜔 × 𝑞

𝜔 𝜃 pure rotation (6.29𝑐)

𝜉 = 𝑣0 𝜃 pure translation (6.29𝑑)

The velocity of a point attached to a prismatic joint moving with unit velocity

as shown in figure 6.3(𝑏),

𝐴 𝑡 = 𝑣 and 𝜉 = 0 𝑣0 0

(6.30)

Figure 6.4 Generalized screw motions [114]

A twist 𝜉 = (𝑣, 𝜔) is associated with screw motion having attributes,

Pitch:

𝑕 =𝜔𝑇𝑣

𝜔 2 (6.31)

Axis:

𝑙 = 𝜔 × 𝑣

𝜔 2+ 𝑕𝜔 ∶ 𝑕 ∈ 𝑅 , 𝑖𝑓 𝜔 ≠ 0

0 + 𝑕𝑣 ∶ 𝑕 ∈ 𝑅 , 𝑖𝑓 𝜔 = 0

(6.32)

Magnitude:

𝑀 = 𝜔 , 𝑖𝑓 𝜔 ≠ 0 𝑣 , 𝑖𝑓 𝜔 = 0

(6.33)

𝑏

𝑎 𝜔

𝑑

𝜃

𝑎 − 𝑏

𝑏 + 𝑒𝜔 𝜃(𝑎 − 𝑏)

𝑏 + 𝑒𝜔 𝜃 𝑎 − 𝑏 + 𝑕𝜃𝜔

167

6.5.2 Wrench

Wrenches represent forces and torques. One way to conceptualize this is to

consider someone who is fastening two wooden boards together with a metal screw.

The person turns the screw (applies a torque), which then experiences a net force

along its axis of rotation. In screw notation force wrenches transform with a 6 × 6

transformation matrix.

A finite or infinite motion (displacement) of a rigid body in 3D space is

defined by using the concept of twist. It is possible to reduce a system of forces and

couples to a resultant force and couple about any point of interest. In general to

resultant force and couple are not collinear, however, one can show that it is possible

to find a unique axis of which both the resulting force 𝑓 and the resulting couple 𝑐

could be acted along. This force-couple combination is called a wrench. A generalized

force acting on a rigid body consists of a linear component (pure force) and an

angular component (pure moment) acting at a point. Wrenches are represented as a

force, moment pair as a vector in 𝑅6.

$ 𝑤 = 𝑓, 𝑀 ∈ 𝑅6 (6.34)

The unique axis is called the wrench axis or screw axis of the system of forces

and couples. Unit wrench is defined as ,

$ 𝑤 = 𝑠𝑟

𝑟𝑜 × 𝑠𝑟 + 𝑃𝑟𝑠𝑟

(6.35𝑎)

Where, 𝑠𝑟 is a unit vector in direction of the screw axis and 𝑟𝑜 is a position vector of

any point on the axis. The vector 𝑟𝑜 × 𝑠𝑟 defines moment of screw axis about the

origin. For a pure Force, the 𝑃𝑟 = 0

$ 𝑤 = 𝑠𝑟

𝑟𝑜 × 𝑠𝑟

(6.35𝑏)

For a pure couple, the 𝑃𝑟 = ∞

$ 𝑤 = 0

𝑠𝑟

(6.35𝑐)

A wrench of intensity 𝜌 can be expressed as,

$ 𝑤 = 𝜌$ 𝑤 (6.36)

The pitch of the wrench is defined as the ratio of the couple to the force.

𝑃𝑟 =𝑐

𝑓 (6.37)

168

First three components of the wrench represent the force f, and the last three

represent the resulting moment due to the combined effect of the force f and the

couple c about the origin of the reference frame.

Wrenches are dual to twists, so that many of the theorems which apply to

twists can be extended to wrenches. A wrench 𝐹 = 𝑓, 𝜏 is associated with a screw

having attributes,

Pitch:

𝑕 =𝑓𝑇𝑣

𝑓 2 (6.38)

Axis:

𝑙 = 𝑓 × 𝜏

𝑓 2+ 𝑕𝑓 ∶ 𝑕 ∈ 𝑅 , 𝑖𝑓 𝑓 ≠ 0

0 + 𝑕𝜏 ∶ 𝑕 ∈ 𝑅 , 𝑖𝑓 𝑓 = 0

(6.39)

Magnitude:

𝑀 = 𝑓 , 𝑖𝑓 𝑓 ≠ 0 𝜏 , 𝑖𝑓 𝑓 = 0

(6.40)

Table 6.1 Instantaneous kinematics and statics description

Instantaneous Kinematics Pitch Statics

Twist 𝜔, 𝑟 × 𝜔 + 𝑕𝜔 𝑕 Wrench 𝑓, 𝑟 × 𝑓 + 𝑕𝑓

Screw 𝑆, 𝑟 × 𝑆 + 𝑕𝑆 𝑕 Screw 𝑆, 𝑟 × 𝑆 + 𝑕𝑆

Rotation 𝜔, 𝑟 × 𝜔 𝑕 = 0 Force 𝑓, 𝑟 × 𝑓

Translation 0, 𝑣 𝑕 = ∞ Couple 0, 𝑐

Screw theory is suitable for kinetostatic analysis of rigid body. It is based on

two well known fundamental theorems: (a) Chasles’ theorem and (b) Poinshot’s

theorem.

6.5.3 Chasles theorem

Rigid body motion is equivalent to twist on a screw, rotation about a unique

axis and translation parallel to this axis. (Lipkin and Duffy, [115])

Any motion of a rigid body can be reproduced as a rotation of the body about

a unique line in space and a translation along that same line.

169

Figure 6.5 Representation of motion at different points of body

𝜔 is an angular velocity about an axis passing through point A and translational

velocity due to rotation for point B is 𝜔 × 𝐴𝐵 .

$𝑡 = 𝜔𝑥 𝜔𝑦 𝜔𝑧 𝑣𝑥 𝑣𝑦 𝑣𝑧 (6.41)

6.5.4 Poinshot Theorem

Rigid body action is equivalent to a wrench on a screw, force along a unique

line and a couple parallel to the line. (Lipkin and Duffy, [115])

Any set of forces and couples applied to a body can be reduced to a single

force acting along a specific line in space, and a pure couple acting in a plane

perpendicular to that line.

Figure 6.6 Poinshot theorem

A wrench is a screw describing the resultant force and moment of a force

system acting on a rigid body.

$𝑤 = 𝑓𝑥 𝑓𝑦 𝑓𝑧 𝑀𝑥 𝑀𝑦 𝑀𝑧 (6.42)

Constraints analysis:

Each twist has a reciprocal called a wrench expressed in form of screw

coordinates as [𝑓𝑥 𝑓𝑦 𝑓𝑧 𝑀𝑥 𝑀𝑦 𝑀𝑧]. It represents all the forces and torques that the

feature can transmit to a mating part. Where motion is allowed, no force or torque can

be transmitted, and vice versa. The intersection of all wrenches acting on a part shows

the amount of constraint on the part provided by those features. If a part is constrained

𝑓

𝐴 𝐵 ≡ ⇒

𝑓

𝐴 𝐵

𝑟

𝑓 𝐴

𝐵

𝑃𝑄

𝑚 𝐴 𝑚 𝐴

𝑓

𝑚 𝐵

𝐴

𝐵

𝜔

𝐴

𝐵

𝜔

𝑟 𝑟𝜔 ≡ ⇒

𝐴 𝐵

𝜔 𝜔

𝑣 𝑎 𝑣 𝑎

𝑣 𝑏

𝜔 × 𝐴𝐵

𝑓 × 𝐴𝐵

170

in some direction, then every feature can provide that constraint. Constraint analysis is

depicted for revolute and prismatic joints in next section of reciprocal screws.

Point lying on the screw axis:

Assumption: Point lying on the axis of the screw and nut is fixed.

Using Chasle’s theorem, Let 𝜔 be the angular velocity of the screw, any point

A on the axis of the screw at any instance moves with a linear velocity 𝑣𝑎 = 𝑕𝜔,

where h is the pitch of the screw. If any point A is known, then the vector pair 𝜔, 𝑣𝑎

is used to specify the twist.

Using poinsot’s theorem, Let f be the force on the body along the screw axis of

wrench, any point on the axis have net parallel moment 𝑀𝑎 = 𝑕𝑓. If any point A is

known, then the vector pair 𝑓, 𝑀𝑎 is used to specify the wrench.

Point lying on the body:

Let 𝜔 be the angular velocity and 𝑓 be the force on the body, then for any

point on the body the vector pairs specify the twist and wrench in that case are:

𝜔, 𝑣𝑏 = 𝜔, 𝑟𝑏 × 𝜔 + 𝑕𝜔 (6.43𝑎)

𝑓, 𝑀𝑏 = 𝑓, 𝑟𝑏 × 𝑓 + 𝑕𝑓 (6.43𝑏)

Where, 𝑟𝑏 is a vector with magnitude of perpendicular distance between point on body

and an axis.

6.6 Representation of screws in different forms

Nowadays, Screws can be described in several different forms:

Vector Representations

Dual Numbers

Plu cker Coordinates

6.6.1 Vector Representation

A screw is presented as two classical vectors:

a) angular quantity vector

b) total effect vector (total velocity + total force) as expressed in equation 6.43,

𝜔, 𝑣𝑝 = 𝜔, 𝑟𝑝 × 𝜔 + 𝑕𝜔

𝑓, 𝑀𝑝 = 𝑓, 𝑟𝑝 × 𝑓 + 𝑕𝑓

Where, 𝑟 = The vector directed from point P to the screw

𝑓 𝑎𝑛𝑑 𝑀 = The resultant and moment vector

𝜔 𝑎𝑛𝑑 𝑣 = The resultant angular and linear velocity vector

𝑕 = Pitch

171

6.6.2 Dual Number

The dual number algebra was originally developed by W K Clifford [116]. A

screw is represented by an operator called as 휀 (Clifford’s operator). It is analogous to

imaginary operator 𝑖, where 𝑖2 = −1. Existence of the complex numbers is to cater

the need of roots of polynomial equations. Imaginary unit 𝑖 (i.e. 𝑖2 = −1) was

invented and the complex field C can be defined as:

𝐶 = 𝑎 + 𝑖𝑏 ∶ 𝑎, 𝑏 ∈ 𝑅, 𝑖2 = −1 (6.44)

Dual numbers are other types of complex numbers of the form: 𝑥 + 휀𝑦. 휀 is

Clifford operator (휀2 = 0) (i.e. whose square is zero and yet is not itself zero, nor is it

‘small’). First term is scalar and second term is accompanied by 휀 , is called dual

number.

Figure 6.7 Dual angle between two screw axes is 𝜃 + 휀𝑑, where d is the directed distance

along common normal and 𝜃 is the angle measured along common normal [115]

The function of dual number has an extraordinarily simple Taylor’s series expansion

about 𝑥.

𝑓 𝑥 + 휀𝑦 = 𝑓 𝑥 + 휀𝑦 𝑑𝑓(𝑥)

𝑑𝑥 (6.45)

Since, higher order terms vanish as 휀2 = 휀3 = 휀4 = ⋯ = 0. For example,

𝑒𝑥+휀𝑦 = 𝑒𝑥 1 + 휀𝑦 (6.46)

Table 6.2 Comparison between complex number and dual number

Cartesian Form Polar Form Exponential Form

Complex Number 𝑎 + 𝑖𝑏 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃) 𝑟𝑒𝑖𝜃

Dual Number 𝑥 + 휀𝑦 𝑅(1 + 휀𝑇) 𝑅𝑒휀𝑇

𝜃 = Angle between x-axis and the radial line from origin to point 𝑎 + 𝑖𝑏.

𝑇 = Distance along the line x=1 from axis to the radial line from the origin to point

𝑥 + 휀𝑦.

𝜃

𝑑

172

Modulus: 𝑅 = 𝑥 Argument: 𝑇 =𝑦

𝑥 𝑓𝑜𝑟 𝑥 ≠ 0

Figure 6.8 Interpretation of complex numbers as points in a plane: (a) ordinary complex

number, and (b) dual number [117]

6.6.2.1 Dual number algebra

Dual number algebra is a powerful mathematical tool for the kinematic and

dynamic analysis of spatial mechanisms. The algebraic interpretation of dual numbers

can be same as two dimensional geometric senses of planar operators.

Addition: 𝑥 + 휀𝑦 + 𝑥′ + 휀𝑦′ = 𝑥 + 𝑥′ + 휀 𝑦 + 𝑦′ (6.47𝑎)

Subtraction: 𝑥 + 휀𝑦 − 𝑥′ + 휀𝑦′ = 𝑥 − 𝑥′ + 휀 𝑦 − 𝑦′ (6.47𝑏)

Multiplication: 𝑥 + 휀𝑦 ∙ 𝑥′ + 휀𝑦′ = 𝑥𝑥′ + 휀 𝑦𝑥′ + 𝑥𝑦′ (6.47𝑐)

0휀 = 휀0 = 0 𝑎𝑛𝑑 𝑦휀 = 휀𝑦

𝐼𝑓 𝑥1 + 휀𝑦1 = 𝑥2 + 휀𝑦2 𝑡𝑕𝑒𝑛 𝑥1 = 𝑥2 , 𝑦1 = 𝑦2

Thus addition, subtraction and multiplication exist for any pair and they are

commutative, associative and distributive.

Multiplication of complex number 𝑟𝑒𝑖𝜃 with a unit complex number 𝑒𝑖𝛼 rotates the

point 𝑎 + 𝑖𝑏 by an angle α in counter clockwise direction about an origin.

𝑒𝑖𝛼 𝑟𝑒𝑖𝜃 = 𝑟𝑒𝑖 𝜃+𝛼 (6.48)

Hence, the complex number 𝑒𝑖𝛼 is planar rotational operator (Rotating the

points in the plane without affecting their distance from the origin.

Multiplication of dual number 𝑅𝑒휀𝑇 with a unit dual number 𝑒휀𝜏 translates the

point (x, y) along the y axis through distance 𝑅𝜏

The net effect of multiplication operator is to shear the entire plane parallel to

y axis through the shear angle of arctan 𝜏. Multiplications of dual numbers

correspond to shear mapping.

173

Figure 6.9 Plane transformations under multiplications by (a) an ordinary complex number

𝑒𝑖𝛼 (b) dual number 𝑒휀𝜏 [117]

If the dual numbers are considered as functional arguments, using Taylors

series expansions for the dual argument 𝑥 + 휀𝑦

𝑓 𝑥 + 휀𝑦 = 𝑓 𝑥 + 휀𝑦𝑓 ′ 𝑥 (6.49)

Therefore,

𝑠𝑖𝑛 𝑥 + 휀𝑦 = 𝑠𝑖𝑛𝑥 + 휀𝑦𝑐𝑜𝑠𝑥

𝑐𝑜𝑠 𝑥 + 휀𝑦 = 𝑐𝑜𝑠𝑥 − 휀𝑦𝑠𝑖𝑛𝑥

𝑐𝑜𝑡 𝑥 + 휀𝑦 = 𝑐𝑜𝑡𝑥 −휀𝑦

𝑠𝑖𝑛2𝑥, 𝑠𝑖𝑛𝑥 ≠ 0

𝑥 + 휀𝑦 12 = 𝑥

12 +

휀𝑦

2𝑥12

, 𝑎 > 0 (6.50)

If 𝑝 = 𝑝𝑥 , 𝑝𝑦 , 𝑝𝑧 and 𝑞 = 𝑞𝑥 , 𝑞𝑦 , 𝑞𝑧 are two vectors in Euclidean space then three

components of the dual vector 𝑝 + 휀𝑞 are also dual numbers.

𝑝 + 휀𝑞 = 𝑝𝑥 + 휀𝑞𝑥 , 𝑝𝑦 + 휀𝑞𝑦 , 𝑝𝑧 + 휀𝑞𝑧 (6.51)

A dual vector multiplication with a dual scalar

𝑎 + 휀𝑏 𝑝 + 휀𝑞 = 𝑎𝑝 + 휀 𝑎𝑞 + 𝑝𝑏 (6.52𝑎)

Scalar product:

𝑝 + 휀𝑞 ∙ 𝑟 + 휀𝑠 = 𝑝 ∙ 𝑟 + 휀 𝑝 ∙ 𝑠 + 𝑟 ∙ 𝑞 (6.52𝑏)

Cross product:

𝑝 + 휀𝑞 × 𝑟 + 휀𝑠 = 𝑝 × 𝑟 + 휀 𝑝 × 𝑠 + 𝑞 × 𝑟 (6.52𝑐)

6.6.2.2 Dual Unit Quaternion

A unit quaternion of the form q = cosθ

2+ nsin

θ

2 , when applied to an

arbitrary vector r in the form of transformation, effects a fixed point rotation through

an angle of θ about the spatial axis n .

174

Dual quaternions are defined as 4-tuples of dual numbers,

𝑞 = 𝑞1 + 휀𝑞01 + 𝑖 𝑞2 + 휀𝑞02 + 𝑗 𝑞3 + 휀𝑞03 + 𝑘 𝑞4 + 휀𝑞04

= 𝑞1 + 휀𝑞01 , 𝑞2 + 휀𝑞02 , 𝑞3 + 휀𝑞03 , 𝑞4 + 휀𝑞04

= 𝑞1 + 𝑞2 + 𝑞3 + 𝑞4 + 휀 𝑞01 + 𝑞02 + 𝑞03 + 𝑞04

= 𝑞 + 휀𝑞0 (6.53)

Where, q and q0 are real quaternions.

Applying the principle of transference, the following expression for a general

screw displacement of a dual vector 𝑙 through the dual angle 𝜃 = 𝜃 + 휀𝑑 along the

dual spatial axis 𝑛 :

𝑙′ = 𝑐𝑜𝑠𝜃

2+ 𝑛 𝑠𝑖𝑛

𝜃

2 ∗ 𝑙 ∗ 𝑐𝑜𝑠

𝜃

2− 𝑛 𝑠𝑖𝑛

𝜃

2 (6.54)

Moreover, 𝑛 in the above equation must be a unit line vector to ensure that the

screw operator 𝑄 is of unit magnitude, i.e.

𝑄 = 𝑞1 + 휀𝑞01 2 + 𝑞2 + 휀𝑞02 2 + 𝑞3 + 휀𝑞03 2 + 𝑞4 + 휀𝑞04 2 = 1 (6.55)

This requirement translates into two distinct conditions

𝑞12 + 𝑞2

2 + 𝑞32 + 𝑞4

2 = 1 (6.56𝑎)

𝑞1𝑞01 + 𝑞2𝑞02 + 𝑞3𝑞03 + 𝑞4𝑞04 = 0 (6.56𝑏)

Which, imposed on the eight parameters of a general dual quaternion,

effectively reduce the number of degrees of freedom of its parameter space to six.

Hamilton [118] has introduces the algebra of quaternion.

Dual Angles and screws:

A screw is described as dual vector as shown in figure 6.9. The relationship of

two directed skew straight lines a and b in space can be expressed uniquely as:

Figure 6.10 Parameters describing relative location of two skew lines in a space [117]

175

the common normal vector 𝑛 = 𝑎 × 𝑏 between the two lines

the distance d between the lines along the common normal, and

the twist angle α, measured positive from a to b

Above figure shows parameters describing relative location of two skew lines

in space. The set of parameters defined above now allows us to specify the position

and orientation of an arbitrary line l2 in space with respect to a given line l1 by a single

rotation (through the twist angle α ) about a unique spatial axis (the common normal

n), combined with a unique translation (through the distance d ) along that same axis.

In summary, a spatial screw displacement can be thought of as a dual angular

displacement 𝛼 = 𝛼 + 휀𝑑 about a spatial axis. The six plu cker coordinates of the

screw axis along with the rotation angle α and the translation parameter d completely

describe a general rigid body displacement.

6.6.3 𝐏𝐥𝐮 𝐜𝐤𝐞𝐫 coordinates

Plu cker Coordinates are very concise and efficient for numerous chores in

field of robot mechanics. They are also directly describing lines at infinity, which

describe pure translations.

Point: A point in a space has three degrees of freedom and consequently three

dimensional space is commonly treated as an aggregate of ∞3 points.

Line: At least 4 parameters are required to completely specify a line in space. Slope

and intercept of the projection of the line onto two orthogonal planes.

It is considered that the space should be composed of an aggregate of ∞4

lines. Line has one more degree of freedom than a point, but more convenient to use

for spatial mechanisms as rotation axis, link orientation axis etc. Only 4 parameters

are needed to completely and uniquely define a line in a space. A line is represented

by a 6 parameters as a reason of mathematical convenience called Plu cker

Coordinates. The relation between Plu cker Coordinates with Cartesian space will be

described using dual vector in space as shown in figure 6.11 ,

First term is scalar and second term is accompanied by 휀 , is called dual

number.

𝐸 = 𝑒 + 휀𝑒° (6.57)

= 𝑒 + 휀(𝜌 × 𝑒)

Above equation defines a unit screw in space. Screw algebra is vector algebra of this

dual vector.

176

Figure 6.11 A unit screw with a radius vector a space

A screw can be defined using three dual coordinates in the space as

𝐸 = 𝐿 , 𝑀 , 𝑁 (6.58)

Where,

E = Unit line vector

e = Unit vector of screw axis. (Resultant Vector)

휀 = Dual number with properties 휀2 = 0, 휀 ≠ 0

e˚ = Moment of e with respect to the origin of fixed coordinate system (moment

vector). It is vector product of the radius vector and unit vector as:

𝑒° = 𝜌 × 𝑒 (6.59)

𝜌 = 𝜌(𝑥, 𝑦, 𝑧) is the radius vector to an arbitrary point on the screw axis 𝐸.

Using equation (6.88),

𝜌 × 𝑒 = 𝑑𝑒𝑡 𝑖 𝑗 𝑘𝑥 𝑦 𝑧𝐿 𝑀 𝑁

𝑃𝑖 + 𝑄𝑗 + 𝑅𝑘 = 𝑁𝑦 − 𝑀𝑧 𝑖 + 𝐿𝑧 − 𝑁𝑥 𝑗 + 𝑀𝑥 − 𝐿𝑦 𝑘

or

𝑃 = 𝑁𝑦 − 𝑀𝑧 𝑄 = 𝐿𝑧 − 𝑁𝑥 𝑅 = 𝑀𝑥 − 𝐿𝑦 (6.60)

Equations (6.57) are called as the equations of the screw axis, which are the

equations of line in space. The equations of the axis of the screw in Plu cker

coordinates are homogeneous with respect to their coordinates. One can write the

equations in form of:

(𝜆𝑁)𝑦 − (𝜆𝑀)𝑧 − (𝜆𝑃) = 0

(𝜆𝐿)𝑧 − (𝜆𝑁)𝑥 − (𝜆𝑄) = 0

(𝜆𝑀)𝑥 − (𝜆𝐿)𝑦 − (𝜆𝑅) = 0

If (𝐿, 𝑀, 𝑁, 𝑃, 𝑄, 𝑅) satisfies the line equations then 𝜆𝐿, 𝜆𝑀, 𝜆𝑁, 𝜆𝑃, 𝜆𝑄, 𝜆𝑅 also

satisfies the line equations. Since screw 𝐸 is unit, 𝐸2 = 1 + 휀 ∙ 0

177

Using equation (6.57) and definition (6.58), each dual coordinate of the screw can be

divided in two parts,

𝐿 = 𝐿 + 휀𝑃 𝑀 = 𝑀 + 휀𝑄 𝑁 = 𝑁 + 휀𝑅 (6.61)

The six real coordinates of screw of equation (6.61) are called as Plu cker

Coordinates of a unit screw 𝐸 (𝐿, 𝑀, 𝑁, 𝑃, 𝑄, 𝑅).

𝐿, 𝑀, 𝑁 = Components of the unit vector 𝑒

𝑃, 𝑄, 𝑅 = Components of the moment vector 𝑒˚

Velocity and acceleration of a unit screw in the space can be obtained from the first

and the second differentials of (6.61) with respect to time as:

𝐿 = 𝐿 + 휀𝑃 𝑀 = 𝑀 + 휀𝑄 𝑁 = 𝑁 + 휀𝑅 (6.62)

𝐿 = 𝐿 + 휀𝑃 𝑀 = 𝑀 + 휀𝑄 𝑁 = 𝑁 + 휀𝑅 (6.63)

Hence, the time derivatives of Plu cker coordinates of a unit screw as:

𝐸 = (𝐿 , 𝑀 , 𝑁 , 𝑃 , 𝑄 , 𝑅 ) 𝐸 = (𝐿 , 𝑀 , 𝑁 , 𝑃 , 𝑄 , 𝑅 )

In coordinate form,

𝐿2 + 𝑀2 + 𝑁2 = 1 + 휀 ∙ 0 (6.64)

𝐿2 = 𝐿2 + 2휀𝑃𝐿 𝑀2 = 𝑀2 + 2휀𝑄𝑀 𝑁2 = 𝑁2 + 2휀𝑅𝑁 (6.65)

Substituting (6.65) into (6.64), a resulting equation is in form of,

𝐿2 + 𝑀2 + 𝑁2 + 2휀 𝑃𝐿 + 𝑄𝑀 + 𝑅𝑁 = 1 + 휀 ∙ 0 (6.66)

Using the components of 𝑒0𝑎𝑛𝑑 𝑒 as:

𝑒. 𝑒0 = 0 𝐿𝑃 + 𝑀𝑄 + 𝑁𝑅 = 0 (6.67𝑎)

It means orientation vector 𝑒 and the moment vector 𝑒0are mutually perpendicular.

Also, since the vectors 𝑒 and 𝑘𝑒 (for 𝑘 ≠ 0) define the same spatial

orientation, one can render the representation of a line unique by restricting the

orientation vector to be of unit magnitude, i.e.,

𝑒. 𝑒 = 1 𝐿2 + 𝑀2 + 𝑁2 = 1 (6.67𝑏)

The above two equations provide necessary two constraints regarding Plu cker

coordinates. So, out of six Plu cker coordinates only four are independent. The

remaining two can be found out using equations (6.65).

• (𝑒, 𝑒0): 𝑒 ≠ 0, 𝑒0 ≠ 0: general line

• 𝑒, 𝑒0 : 𝑒 ≠ 0, 𝑒0 = 0: line through origin

• 𝑒, 𝑒0 : 𝑒 = 0, 𝑒0 ≠ 0: [not allowed]

178

Note:

Plu cker Coordinates are dependent.

The axis of screw is defined synonymously with 𝑒 (𝐿, 𝑀, 𝑁) and 𝑒˚ (𝑃, 𝑄, 𝑅).

Time derivatives of e˚ as:

- 𝑃 = 𝑁 𝑦 + 𝑁𝑦 − 𝑀 𝑧 − 𝑀𝑧

- 𝑄 = 𝐿 𝑧 + 𝐿𝑧 − 𝑁 𝑥 − 𝑁𝑥

- 𝑅 = 𝑀 𝑥 + 𝑀𝑥 − 𝐿 𝑦 − 𝐿𝑦

- 𝑃 = 𝑁 𝑦 + 𝑁𝑦 − 𝑀 𝑧 − 𝑀𝑧 + 2(𝑁 𝑦 − 𝑀 𝑧 )

- 𝑄 = 𝐿 𝑧 + 𝐿𝑧 − 𝑁 𝑥 − 𝑁𝑥 + 2(𝐿 𝑧 − 𝑁 𝑥 )

- 𝑅 = 𝑀 𝑥 + 𝑀𝑥 − 𝐿 𝑦 − 𝐿𝑦 + 2(𝑀 𝑥 − 𝐿 𝑦 ) (6.68)

Using (6.61) and (6.68), one can determine the magnitude of components of the

moment vector 𝑒˚ (𝑃, 𝑄, 𝑅) and its time derivatives from known vector 𝜌(𝑥, 𝑦, 𝑧) and

𝑒 𝐿, 𝑀, 𝑁 .

6.6.3.1 𝐏𝐥𝐮 𝐜𝐤𝐞𝐫 Coordinates of a line, twist and wrench:

The line is defined in a plane with 𝑞1 and 𝑞2 with direction ratios L and M as

shown in figure 6.12,

A line vector 𝐹12 from 𝑃1to 𝑃2whose direction ratios are L and M,

𝐿 = 𝑥2 − 𝑥1 = 𝑃12 cos 𝜃

𝑀 = 𝑦2 − 𝑦1 = 𝑃12 𝑠𝑖𝑛 𝜃 6.69

The moment of 𝐹12 about z-axis as R and using right hand rule,

𝑅 = 𝐹12 × 𝑂𝐴 = 𝑥1𝑀 − 𝑦1𝐿 = 𝑥1 𝑦1

𝐿 𝑀 (6.70)

Using equation 6.69 and 6.70,

𝑅 = 𝑥1 𝑦1

𝑥2 − 𝑥1 𝑦2 − 𝑦1

= 𝑥1 𝑦1

𝑥2 𝑦2

(6.71)

Three determinants of single matrix can be defined as,

𝑅 𝐿 𝑀 = 1 𝑥1 𝑦1

1 𝑥2 𝑦2 (6.72)

The gradient of line is given by,

𝜃 = tan−1𝑀

𝐿 (6.73)

The shortest distance from line to origin is given by,

𝑂𝐴 =𝑅

𝐿2 + 𝑀2 (6.74)

179

Figure 6.12 A line in a plane with 𝑞1 and 𝑞2 with direction ratios L and M

In homogeneous form, equation (6.71) is represented as,

𝑅 𝐿 𝑀 = 𝑤1 𝑥1 𝑦1

𝑤2 𝑥2 𝑦2 (6.75)

Hence,

𝐿 = 𝑤1 𝑥1

𝑤2 𝑥2

, 𝑀 = 𝑤1 𝑦1

𝑤2 𝑦2

, 𝑅 = 𝑥1 𝑦1

𝑥2 𝑦2

(6.76)

Similarly, Plu cker coordinates representations of line in ℜ3. The directed line

containing 𝑞2 and 𝑞1 in affine 3-Dimensional space has a third direction ratio N.

𝑁 = 𝑤1 𝑧1

𝑤2 𝑧2 (6.77)

For a orthogonal XYZ system,

𝐿 = 𝑥2 − 𝑥1 = 𝑠1

𝑀 = 𝑦2 − 𝑦1 = 𝑠2

𝑁 = 𝑧2 − 𝑧1 = 𝑠3

(6.78)

The moment about X, Y and Z axis are P, Q and R respectively,

𝑃 = 𝑦1𝑧2 − 𝑦2𝑧1 = 𝑦1𝑁 − 𝑧1𝑀 = 𝑦2𝑁 − 𝑧2𝑀

𝑄 = 𝑧1𝑥2 − 𝑧2𝑥1 = 𝑧1𝐿 − 𝑥1𝑁 = 𝑧2𝐿 − 𝑥2𝑁

𝑅 = 𝑥1𝑦2 − 𝑥2𝑦1 = 𝑥1𝑀 − 𝑦1𝐿 = 𝑥2𝑀 − 𝑦2𝐿

(6.79)

The Plu cker coordinates of 3D line will be,

𝑠1 = 𝑤1𝑥2 − 𝑤2𝑥1

𝑠2 = 𝑤1𝑦2 − 𝑤2𝑦1

𝑠3 = 𝑤1𝑧2 − 𝑤2𝑧1

= 𝜔 = 𝑞2 − 𝑞1 = Direction of line

𝑠4 = 𝑦1𝑧2 − 𝑦2𝑧1

𝑠5 = 𝑥2𝑧1 − 𝑥1𝑧2

𝑠6 = 𝑥1𝑦2 − 𝑥2𝑦1

= 𝑣 = 𝑞2 × 𝑞1 = Moment of line

L

M

𝐹12

𝜃 O

A

X

Y

180

The six coordinates of line can be written as 𝐿 𝑀 𝑁 𝑃 𝑄 𝑅

or 𝑠1 𝑠2 𝑠3 𝑠4𝑠5 𝑠6 . The first three elements of such a sextuple represent a

direction, and the last three a moment.

Thus, the Plu cker line coordinates are given by 𝐿𝑖 , where

𝐿𝑖 = 𝑠𝑖 , 𝑖 = 1,2,3

𝐿𝑖 = 𝑠𝑖 − 𝑕𝑠𝑖−3, 𝑖 = 4,5,6

The notations can be reduced further by combing each term from two sets of numbers

𝑠1 𝑠2 𝑠3 and 𝑠4 𝑠5 𝑠6 into three pairs.

𝑠 = 𝑠1, 𝑠4 , 𝑠2, 𝑠5 , 𝑠3, 𝑠6

In short, six-dimensional real vector which contains the 𝑃𝑙𝑢 𝑐𝑘𝑒𝑟 coordinates for the

line joining 𝑞1 and 𝑞2 with respect to frame i:

𝑙𝑖 = 𝑥1 𝑥2

𝑤1 𝑤2 ,

𝑦1 𝑦2

𝑤1 𝑤2 ,

𝑧1 𝑧2

𝑤1 𝑤2 ,

𝑦1 𝑦2

𝑧1 𝑧2 ,

𝑧1 𝑧2

𝑥1 𝑥2 ,

𝑥1 𝑥2

𝑦1 𝑦2

𝑇

(6.80)

can be defined using 2-by-2 determinants of the coordinates of the points.

It is easily possible to see that above equation can be written as:

𝑙𝑖 = 𝑤2𝑞1 − 𝑤1𝑞2

𝑞1 ∧ 𝑞2 =

𝑤2𝑞1 − 𝑤1𝑞2

𝑞1 ∧ 𝑞2 − 𝑞1 (6.81)

From this last expression, it is possible to consider two cases: the first one when the

defining points are finite (w1, w2 ≠ 0) and the second when the points are at infinity

(w1, w2 = 0) and therefore describe a line at infinity.

6.6.3.2 𝑷𝒍𝒖 𝒄𝒌𝒆𝒓 screw coordinates

A screw $ is defined by straight line with an associated pitch h and is conveniently

defined by six homogeneous six Plu cker coordinates,

$ = 𝑠

𝑠𝑜 + 𝑕𝑠 =

𝐿𝑀𝑁𝑠1𝑠2

𝑠3

(6.82)

Where 𝑠 denotes direction ratios pointing along screw axis, 𝑠𝑜 = 𝑟 × 𝑠 defines the

moment of screw axis about origin of the coordinate system, r is position vector of

any point on screw axis with respect to coordinate system.

Screw axis is denoted by Plu cker homogeneous coordinates,

$𝑎𝑥𝑖𝑠 = 𝑠𝑠𝑜

181

Figure 6.13 The screw

Assume

𝑠 = 𝐿 𝑀 𝑁 𝑇

𝑠𝑜 + 𝑕𝑠 = 𝑃 𝑄 𝑅 𝑇 (6.83)

Considering 𝑠. 𝑠𝑜 + 𝑕𝑠 = 𝑠. 𝑠𝑜 + 𝑠 2𝑕 = 𝑕 𝑠 2 assuming 𝑠 ≠ 0,

Pitch of screw:

𝑕 =𝑠. 𝑠𝑜 + 𝑕𝑠

𝑠 2=

𝐿𝑃 + 𝑀𝑄 + 𝑁𝑅

𝐿2 + 𝑀2 + 𝑁2 (6.84)

The axis of screw can be denoted as

$𝑎𝑥𝑖𝑠 = 𝐿 𝑀 𝑁 𝑃 − 𝐿𝑕 𝑄 − 𝑀𝑕 𝑅 − 𝑁𝑕 (6.85)

Assume that vector of projective point of the origin on the screw axis is represented

by 𝑟𝑂𝑝. The screw axis and vector 𝑟𝑂𝑝

are perpendicular to each other.

Using concept of vector triple product,

𝑠 × 𝑟𝑂𝑝× 𝑠 = 𝑠 ∙ 𝑠 𝑟𝑂𝑝

+ 𝑠. 𝑟𝑂𝑝 𝑠 = 𝑠 2𝑟𝑂𝑝

(6.86)

Using equation (6.82) and (6.83),

𝑠 = 𝐿 𝑀 𝑁 𝑇

𝑟𝑂𝑝× 𝑠 = 𝑃 − 𝐿𝑕 𝑄 − 𝑀𝑕 𝑅 − 𝑁𝑕

(6.87)

Using above equations,

𝑟𝑂𝑝=

𝑠 × 𝑟𝑂𝑝× 𝑠

𝑠 2=

1

𝐿2 + 𝑀2 + 𝑁2 𝑀 𝑅 − 𝑁𝑕 − 𝑁 𝑄 − 𝑀𝑕

𝑁 𝑃 − 𝐿𝑕 − 𝐿 𝑅 − 𝑁𝑕

𝐿 𝑄 − 𝑀𝑕 − 𝑀 𝑃 − 𝐿𝑕 6.88

If the 𝑃𝑙𝑢 𝑐𝑘𝑒𝑟 coordinates of screw are given, then

1) Unit direction vector (s)

2) The pitch (h)

3) The screw axis ($𝑎𝑥𝑖𝑠 )

4) The vector of the projective point of origin on the axis (𝑟𝑂𝑝) are easily obtained.

182

If the pitch of screw equals to zero, then $ = 𝑠𝑠𝑜

is just 𝑃𝑙𝑢 𝑐𝑘𝑒𝑟 homogeneous

coordinates of the screw axis. Assume that point 𝑂𝑃 is the projective point of the

origin on a line 𝑙 and point 𝐴 is any other point on the line. Then,

𝑟𝐴 = 𝑟𝑂𝑃+ 𝑟𝑂𝑃𝐴

= 𝑟𝑂𝑃+

𝑎𝑠

𝑠 (6.89)

Where s is direction vector of line 𝑙 and 𝑎 is length of line segment 𝑂𝑃𝐴.

The moment of line 𝑙 about the origin at point 𝐴 will be,

𝑠0 = 𝑟𝐴 × 𝑠 = 𝑟𝑂𝑃+

𝑎𝑠

𝑠 × 𝑠 (6.90𝑎)

One obtains that the moment of a line about the origin is irrelevant to the point’s

selection on the line. A screw associated with a revolute pair is a twist of zero pitch

(𝑕 = 0) pointing along the pair axis,

$ = 𝑠

𝑟 × 𝑠 (6.90𝑏)

Hence, the revolute joint is represented as a line in plucker coordinates.

A screw associated with a prismatic pair is a twist of infinite pitch (𝑕 = ∞) pointing

in the direction of the translational guide line of the pair.

$ = 0𝑠 (6.90𝑐)

Hence, a prismatic Joint (𝑝 = ∞) is represented as a line at infinity contained in a

plane perpendicular to the line represented at (6.88). Of course the magnitude of the

vector along such a line in the plane at infinity must be taken to be zero, r at the best

infinitesimal By virtue of the first three of its line coordinates being zero but its

moment about the origin can be regarded as meaningful and finite [Hunt, 1978].

The kinematic screw is often denoted in the form of Plücker homogeneous

coordinates:

$ = 𝐿 𝑀 𝑁 𝑃 𝑄 𝑅 𝑇 (6.91)

Where, the first three components denote the angular velocity, the last three

components denote the linear velocity of a point in the rigid body that is

instantaneously coincident with the origin of the coordinate system.

The displacement of a rigid body cannot be completely determined without the

specification of the amplitude or intensity of the screw axis is specified. Let 𝑞 be the

intensity of a twist, then the twist can be expressed as:

$ = 𝑞 $ (6.92)

183

Where, 𝑞 = 𝜃 for revolute joint (angular velocity) and 𝑞 = 𝑑 for prismatic joint

(linear velocity)

6.6.4 Kinematics of two unit screw in a space

A rigid body in space can be described by two unit vectors 𝐸1 and 𝐸2. Figure

6.14 shows two unit screws placed arbitrarily in space. The angle and the distance

between these two screws is defined with the dual angle, 𝐴 = 𝛼 + 휀. 𝑎

Where,

α = Twist angle

𝑎 = Shortest distance between two axes

Figure 6.14 Two arbitrary unit screws in space

The following equations describe the physical situation of the above two unit screws,

𝐸1

2 = 1 + 휀. 0

𝐸22 = 1 + 휀. 0

𝐸1 ∙ 𝐸2 = 𝑐𝑜𝑠𝐴 ⇒

𝐿12 + 𝑀1

2 + 𝑁12 = 1 + 휀 ∙ 0

𝐿22 + 𝑀2

2 + 𝑁22 = 1 + 휀 ∙ 0

𝐿1 𝐿2

+ 𝑀1 𝑀2

+ 𝑁1 𝑁2

= 𝑐𝑜𝑠𝐴

(6.93)

From screw algebra,

cos 𝐴 = 𝑐𝑜𝑠𝛼 − 휀 ∙ 𝑎𝑠𝑖𝑛𝛼 (6.94)

From six dual coordinates 𝐿1 , 𝑀1

, 𝑁1 , 𝐿2

, 𝑀2 , 𝑁2

that describe the position of rigid

body in space, just three of them are independent out of six constraints.

From equations 6.93 and 6.94 , using the real Pluckers coordinates of the screws

𝐸1 and 𝐸2 ,

𝐿12 + 𝑀1

2 + 𝑁12 + 2휀 𝐿1𝑃1 + 𝑀1𝑄1 + 𝑁1𝑅1 = 1 + 휀 ∙ 0

𝐿22 + 𝑀2

2 + 𝑁22 + 2휀 𝐿2𝑃2 + 𝑀2𝑄2 + 𝑁2𝑅2 = 1 + 휀 ∙ 0

𝐿1𝐿2 + 𝑀1𝑀2 + 𝑁1𝑁2 + 𝜔 𝑃1𝐿2 + 𝐿1𝑃2 + 𝑄1𝑀2 + 𝑄2𝑀1 + 𝑅1𝑁2 + 𝑅2𝑁1

= 𝑐𝑜𝑠𝛼12 − 휀 ∙ 𝑎12𝑠𝑖𝑛𝛼12

It is clear that unit vectors e1 and e2, moment vectors 𝑒1° and 𝑒2

° can be written using

their components as,

𝛼

𝐸1 𝐿 1 , 𝑀 1 , 𝑁 1

𝐸2 𝐿 2 , 𝑀 2 , 𝑁 2

𝑎

184

𝑒1𝑒1° = 0 ⟺ 𝐿1𝑃1 + 𝑀1𝑄1 + 𝑁1𝑅1 = 0

𝑒1𝑒1 = 1 ⟺ 𝐿12 + 𝑀1

2 + 𝑁12 = 1

𝑒2𝑒2° = 0 ⟺ 𝐿2𝑃2 + 𝑀2𝑄2 + 𝑁2𝑅2 = 0

𝑒2𝑒2 = 1 ⟺ 𝐿22 + 𝑀2

2 + 𝑁22 = 1

𝑒1𝑒2 = 𝑐𝑜𝑠𝛼12 ⟺ 𝐿1𝐿2 + 𝑀1𝑀2 + 𝑁1𝑁2 = 𝑐𝑜𝑠𝛼12 (6.95)

𝑒10𝑒2 + 𝑒1𝑒2

0 = −𝑎12𝑠𝑖𝑛𝛼12

⟺ 𝑃1𝐿2 + 𝐿1𝑃2 + 𝑄1𝑀2 + 𝑀1𝑄2 + 𝑅1𝑁2 + 𝑁1𝑅2 − 𝑎12𝑠𝑖𝑛𝛼12

The equation 6.95 describes the relative moment of two screws E1 and E2 in screw

theory.

6.7 Reciprocal screws

The concept of reciprocal screw was first introduced by Ball [111]. Mohamed

and Duffy [119] applied the theory of reciprocal screws for the Jacobian analysis of

parallel manipulators. The main idea of this concept is that if a wrench acts on a rigid

body in such a way that that it produced no work while the body undergoes an

infinitesimal twist, then both screws representing the twist and the wrench are said to

be reciprocal to each other as shown in figure 6.15.

Figure 6.15 Twist and wrench in 3D-Space [120]

One of them represents the motion of the rigid body, the other is constraint

force exerted on the body. The reciprocal product of screws can be defined as the

instantaneous virtual power developed by a force screw along the kinematic screw.

Therefore, a reciprocal screw $𝒓, denotes a wrench which produces no power for a

kinematic screw $ (or a twist).

Lipking and Duffy [115] have mentioned that reciprocity is a geometric

quantity relating two screws, $1 = 𝐿1 𝑀1 𝑁1 𝑃1∗ 𝑄1

∗ 𝑅1∗ and $2 =

𝐿2 𝑀2 𝑁2 𝑃2∗ 𝑄2

∗ 𝑅2∗ and can be identified using following condition,

𝑥

𝑧

𝑦 𝑂

$

$𝑟

Wrench: $𝑤

Twist: $𝑡

a

𝛼

𝑠𝑂

Body B

𝑠𝑟𝑂

185

$1 ∘ $2 = $1𝑇 ∆ $2 = 0 (6.96)

Where,

∆ = 03×3 𝐼3×3

𝐼3×3 03×3

In other form, two screws, $1 and $2 are called to be reciprocal if they satisfy

the equation,

𝐿1𝑃2∗ + 𝑀1𝑄2

∗ + 𝑁1𝑅2∗ + 𝐿2𝑃1

∗ + 𝑀2𝑄1∗ + 𝑁2𝑅1

∗ = 0 (6.97)

If the screws $1 and $2 represent revolute joints, then these screws are reciprocal when

their primal parts intersect or are parallel. Reciprocal screws emerges if the primal

part of $1 representing a revolute joint is perpendicular to the dual part of the screw $2,

representing a prismatic joint as shown in figure 6.17. For a screw system, twist

system represents mobility and wrench system represents the constraints. The screw is

a twist if it represents an instantaneous motion of a rigid body, and a wrench if it

represents a system of forces and couples acting on a rigid body. First three

components of a wrench represent the resultant force and the last three components

represent the resultant moment about the origin of the reference frame as discussed

earlier. Twist matrix and its corresponding wrench matrix are reciprocal to each other.

Each kinematic chain consists of (𝑚) twist screws. Normally, each screw

forms a row vector of the screw matrix. The rank of this screw matrix is known as

dimension of screw space.

In short, dimension of screw matrix and dimension of reciprocal screw

matrices are always six. Moreover, two screw systems $1 and $2 are reciprocal if all

the screws in $1 are reciprocal to all the screws in $2.

Particular Instances [111]:

Parallel or intersecting screws are reciprocal when the sum of their pitches is

zero.

Screws at right angles are reciprocal either when they intersect or when one of

the pitches is infinite.

Two screws of infinite pitch ($ ∞ and $ ′∞) are reciprocal, because a couple

could not move a body which was only susceptible of translation.

A screw whose pitch is zero or infinite is reciprocal to itself.

A zero pitch screw $ 0 is reciprocal to an infinite pitch screw $ ∞ if only if

their directions are orthogonal to each other.

186

Geometrical method to determine the reciprocal screws was explained by

Jianguo Zhao, Bing Li, Xiaojun Yang and Hongjian Yu [121]. The approach is free of

algebraic manipulation and completely based on three observations for determination

of reciprocal screw system for any kinematic pairs and chain.

6.8 Calculating twist and wrench for common kinematic pairs:

Revolute pair:

Associated screw of revolute pair is line vector. It provides a rotary movement

between two bodies about an axis in a space. Suppose, vector 𝑟 = 𝑥𝑟 , 𝑦𝑟 , 𝑧𝑟

represents the point of rotational axis and unit vector along the direction of rotation is

represented by 𝜔 = 𝑎 𝑏 𝑐 . Hence, $𝑡 = 𝜔 | 𝑣 , where, 𝑣 = 𝑟 × 𝜔 = 𝑑 𝑒 𝑓 .

The twist for the joint can be calculated with the following formula:

$𝑡 = 𝑑 𝑒 𝑓 𝑎 𝑏 𝑐

The corresponding wrench matrix will be,

$𝑤 =

1 0 0 0 0 −𝑑0 1 0 0 0 −𝑒0 0 1 0 0 −𝑓0 0 0 𝑔 0 𝑕0 0 0 0 𝑖 𝑓

(6.98)

Where, vectors 𝑔 0 𝑕 , 0 𝑖 𝑓 and 𝑎 𝑏 𝑐 are mutually perpendicular.

The plucker coordinates for the joint for a coordinate frame as shown in figure

6.16 is represented as,

$𝑡 = 0 0 1 0 0 0

Using reciprocal conditions stated in equation (6.97), following are the different

conditions for reciprocal screw system for a revolute joint axis in direction of

coordinate frame z-axis direction as shown in figure 6.16,

Screws at right angles are reciprocal either when they intersect,

$1𝑟 = 1 0 0 0 0 0

$2𝑟 = 0 1 0 0 0 0

A screw whose pitch is zero or infinite is reciprocal to itself.

$3𝑟 = 0 0 1 0 0 0

A zero pitch screw $ 0 is reciprocal to an infinite pitch screw $ ∞ if only if their

directions are orthogonal to each other.

$4𝑟 = 0 0 0 1 0 0

$5𝑟 = 0 0 0 0 1 0

187

(a) Revolute pair (b) Prismatic pair

Figure 6.16 Revolute and prismatic pair axis aligned with z-axis of coordinate frame

Prismatic pair:

The associated screw of a prismatic pair is a couple vector. The first triplet is

always null as no angular velocity with this joint. The second triplet describes vector

direction ratios of linear movement of joint in a space. Hence, $𝑡 = 0| 𝑢 .

$𝑡 = 0 0 0 𝑎 𝑏 𝑐

The wrench matrix can be derived using the reciprocity property and it

consists of five rows. Each row of the wrench matrix represents vector of the base of

the reciprocal space of the twist. It can be represented in a simplest form as,

$𝑤 =

𝑑 𝑒 𝑓 0 0 0𝑔 𝑕 𝑖 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

(6.99)

Where, vectors 𝑎 𝑏 𝑐 , 𝑑 𝑒 𝑓 and 𝑔 𝑕 𝑖 are mutually perpendicular.

One can choose any five linearly independent ones to form the reciprocal 5

system in same way as explained for revolute joint pair. Twist screw can be

determined for prismatic pair as shown in figure 6.16 as,

$𝑡 = 0 0 0 0 0 1

Using reciprocal condition,

$1𝑟 = 0 0 0 1 0 0

$2𝑟 = 0 0 0 0 1 0

$3𝑟 = 0 0 0 0 0 1

$4𝑟 = 1 0 0 0 0 0

$5𝑟 = 0 1 0 0 0 0 (6.100)

𝑥

𝑦

𝑧

$ $

𝑥

𝑦

𝑧

188

The zero-pitch reciprocal screws lie on all planes perpendicular to the axis of

prismatic joint.

Figure 6.17 Reciprocal screws

Cylindrical Pair:

Kinematically, any cylindrical joint can be replaced by a revolute joint plus a

coaxial prismatic joint as shown in figure 6.18.

Figure 6.18 Cylindrical pair

Two screws are associated with cylindrical pair,

$𝑡 = 0 0 1 0 0 0

$𝑡 = 0 0 0 0 0 1

The constraint in form of wrench form four screw system,

$1𝑟 = 1 0 0 0 0 0

$2𝑟 = 0 1 0 0 0 0

$3𝑟 = 0 0 0 1 0 0

$4𝑟 = 1 0 0 0 1 0 (6.101)

Spherical Pair:

The associated unit screws to spherical joint form a three system of zero pitch screws

(three intersecting line vectors) passing through the center of the joint.

$𝑡 = 0 0 1 0 0 0

$𝑡 = 0 1 0 0 0 0

$𝑡 = 1 0 0 0 0 0 (6.102)

𝑥

𝑦 𝑧

𝜃

189

By observation, there are no reciprocal screws of couple vectors. Hence,

constraints in form of wrench form three screws system of line vectors. In short,

reciprocal screws form a three-system of zero pitch passing through the center of the

sphere.

In short, number of twist screws represents degrees of freedom of kinematic

pair and number of reciprocal screws of wrenches indicates constraint imposed on it.

Application of reciprocal screw system:

To carry out mobility analysis of serial or parallel manipulators

It is used to identify number of degrees of freedom and associated types of

motion. Mobility analysis of 3-PRS configuration is presented in next section.

To investigate constraint and motions in assembly

A constraint condition of parts into assembly is possible to distinguish by forming

twist and wrench matrix for an exact kinematic state of the assembly.

To analyze Jacobian of serial or parallel manipulator [122]

Generalized Jacobian of parallel manipulators with 𝑛 limbs can be formulated

using reciprocal screw concept. Instantaneous twist of moving platform can be

expressed as linear combination of 𝑚 instantaneous twists of 𝑖𝑡𝑕 limb.

$𝑃 = 𝑞 𝑗 ,𝑖$ 𝑗 ,𝑖

𝑚

𝑗 =1

, 𝑖 ∈ 1,2, … 𝑛 (6.103)

Where, 𝑞 𝑗 ,𝑖 and $ 𝑗 ,𝑖 indicates intensity and unit screw associated with 𝑗𝑡𝑕 joint of a

𝑖𝑡𝑕 limb. Equation (6.103) contains many un-actuated joint rates, which should be

eliminated by applying the theory of reciprocal screws. Actuated joints are

identified as 𝑔 screws, $ 𝑟 ,𝑗 ,𝑖 for 𝑗 = 1, 2, …𝑔 of each limb. Each of them is

reciprocal to the screw system of all un-actuated joints of 𝑖𝑡𝑕 limb. Then, the

orthogonal product of both sides of equation 6.103 with each reciprocal screw

results in g equations, which can be written in a matrix form as follows:

𝐽𝑥 ,𝑖$𝑃 = 𝐽𝑞 ,𝑖𝑞 𝑖 (6.104)

Where, 𝑞 𝑖 = 𝑞 1,𝑖 𝑞 2,𝑖 … 𝑞 𝑔 ,𝑖

𝐽𝑥 ,𝑖 =

$ 𝑇

𝑟 ,1,𝑖

$ 𝑇𝑟 ,2,𝑖

⋮

$ 𝑇𝑟 ,𝑔 ,𝑖

and 𝐽𝑞 ,𝑖 =

$ 𝑇

𝑟 ,1,𝑖 ∙ $ 1,𝑖 $ 𝑇𝑟 ,1,𝑖 ∙ $ 2,𝑖 … $ 𝑇

𝑟 ,1,𝑖 ∙ $ 𝑔,𝑖

$ 𝑇𝑟 ,2,𝑖 ∙ $ 1,𝑖 $ 𝑇

𝑟 ,2,𝑖 ∙ $ 2,𝑖 … $ 𝑇𝑟 ,2,𝑖 ∙ $ 𝑔,𝑖

⋮ ⋮ … ⋮

$ 𝑇𝑟 ,𝑔 ,𝑖 ∙ $ 1,𝑖 $ 𝑇

𝑟 ,𝑔 ,𝑖 ∙ $ 2,𝑖 … $ 𝑇𝑟 ,𝑔,𝑖 ∙ $ 𝑔 ,𝑖

Equation (6.104) is written for all n limbs of the manipulator.

190

To understand singularities of parallel manipulator

Singularity analysis of a parallel manipulator is often very complicated. Screw

theory helps to analyze the relationship between different criterions of singular

configurations. Singularity analysis of 3-PRS is also carried out in next section.

In this way, screw and reciprocal screw system provides a new approach in

analysis and synthesis of mechanisms. Recently, many researchers have used this tool

for various applications as stated above.

6.9 Mobility analysis of 3-PRS using screw theory

A spatial parallel manipulator configuration consists of three limbs. Each limb

as shown in figure 6.19 has prismatic, revolute and spherical joint. Thus, each

kinematic chain has two single DOF pairs and one three DOF pair. Screw axis method

is quite different and each joint axis is identified as screw. This section will

investigate the mobility of a PRS kinematic chain using screw theory. So, the

kinematic screws for each kinematic chain can be expressed as:

$𝑗𝑖 or $𝑗 ,𝑖 is used to represent the unit twist associated with the 𝑗𝑡𝑕 kinematic pair in

the 𝑖𝑡𝑕 limb.

Figure 6.19 Screw coordinate system for 3-PRS configuration

Infinite pitch is a translation, then the screw pair reduces to prismatic joint and

represented by

$𝑡1,𝑖 =

0𝑠1,𝑖

(6.105)

S22

S

R

P

A1

A2

A3

S13

S11

S12

S21

S23

S31 S51

S41

𝑆1

𝑆3 𝑆2

$ 𝑟 ,1

$ 𝑟 ,2

$ 𝑟 ,3

𝑥 𝑦

𝑧

𝑈

𝑟 𝑟𝑖

191

Zero pitch is a rotation about the line, then the screw represents a revolute joint and

twist of revolute joint is represented using Plu cker coordinates as,

$𝑡2,𝑖 =

𝑠2,𝑖

𝑟𝑟𝑖× 𝑠2,𝑖

(6.106)

Cartesian coordinates 𝑟𝑟𝑖= 𝑥𝑟1

𝑦𝑟1𝑧𝑟1 𝑇 are denoted for 𝑖𝑡𝑕 limb plane revolute

joint as shown in figure 6.19.

For the spherical kinematic pair 𝑆1 , its centre point cartesian coordinates are

denoted by 𝑟𝑠𝑖= 𝑥𝑠1

𝑦𝑠1𝑧𝑠1 𝑇 , spherical joint with three DOF is equivalent to

three single-DOF revolute pairs and the twists for the three orthogonal axes can be

expressed as:

$𝑡𝑗 ,𝑖 =

𝑠𝑗 ,𝑖

𝑟𝑠𝑖× 𝑠𝑗 ,𝑖

𝑗 = 3, 4, 5 (6.107)

Where, 𝑠𝑗 ,𝑖 , 𝑗 = 3,4,5 indicates three orthogonal directions of spherical joints 𝑆𝑖 and 𝑖

indicates the 𝑖th limb under consideration. As far as 3-PRS configuration as shown in

figure 6.19 has 𝑟𝑟𝑖= 𝑅𝑖𝑥 𝑅𝑖𝑦 𝑅𝑖𝑧 and 𝑟𝑠𝑖

= 𝑆𝑖𝑥 𝑆𝑖𝑦 𝑆𝑖𝑧 . In fact, the Cartesian

coordinates of spherical joints can also be represented in form of,

𝑟𝑠𝑖= 𝑟𝑟𝑖

+ 𝑈𝑠5,𝑖 (6.108)

Direction of 𝑠5,𝑖 is in same direction of axis of connecting link in limb plane -1. Thus,

the kinematic screw system of the first limb consists of k=5 system.

The reciprocal screw system for 𝑖th limb motion screw system is a 6 − k system

in which any one screw is reciprocal to all the screws in the given system. It is found

from the literature that a reciprocal screw for a kinematic chain is reciprocal to the

screws associated with all pairs in the chain. Hence, there is only one reciprocal screw

in existence for 𝑖𝑡𝑕 limb of 3-PRS configuration.

Passive revolute joint axes are reciprocal to the zero pitch wrench (pure force)

applied via the actuated prismatic joint. Hence, all the passive joint axes are reciprocal

to the zero pitch wrench (pure force) applied via the actuated prismatic joint. Using

reciprocal screw theory, one can obtain the limb constraint system of limb 𝑖 as,

$ 𝑟𝑤𝑐,𝑖 =

𝑠2,𝑖

𝑟𝑠𝑖× 𝑠2,𝑖

(6.109)

Zero pitch screw is passing through the centre of spherical joint and its

direction is parallel to revolute joint axis of 𝑖𝑡𝑕 limb. It is orthogonal to all unit twist

screws in 𝑖𝑡𝑕 limb. In fact, the intersection of moving platform plane and another

192

plane passing from centre of spherical joint and revolute joint axis define the

reciprocal screw axis at any instance.

Case Study:

Assumed structural parameters of 3-PRS configuration are:

- Initially position of prismatic joint 𝑃𝑖 from fixed base =160 mm

- Connecting link length 𝑈 =482 mm

- The prismatic joint axis and revolute joint axis offset (𝑏) = 41 mm

- Distance between two prismatic joint axis on fixed base (𝑝) = 750 mm

- Distance between two spherical joint centre points 𝑞 = 300𝑚𝑚

- Fixed Base Coordinates:

𝐵1𝑥 𝐵1𝑦 𝐵1𝑧

𝐵2𝑥 𝐵2𝑦 𝐵2𝑧

𝐵3𝑥 𝐵3𝑦 𝐵3𝑧

=

−

𝑃

2−

𝑃

2 30

𝑃

2−

𝑃

2 30

0𝑃

30

= −375 −216.506 0375 −216.506 0

0 433.013 0

- Prismatic Joint Coordinates:

𝑃1𝑥 𝑃1𝑦 𝑃1𝑧

𝑃2𝑥 𝑃2𝑦 𝑃2𝑧

𝑃3𝑥 𝑃3𝑦 𝑃3𝑧

=

𝐵1𝑥 𝐵1𝑦 𝐵1𝑧 + 𝑇1

𝐵2𝑥 𝐵2𝑦 𝐵2𝑧 + 𝑇2

𝐵3𝑥 𝐵3𝑦 𝐵3𝑧 + 𝑇3

= −375 −216.506 160375 −216.506 160

0 433.013 160

- Revolute Joint Coordinates:

𝑅1𝑥 𝑅1𝑦 𝑅1𝑧

𝑅2𝑥 𝑅2𝑦 𝑅2𝑧

𝑅3𝑥 𝑅3𝑦 𝑅3𝑧

=

𝑃1𝑥 +

3

2𝑏 𝑃1𝑦 +

𝑏

2𝑃1𝑧

𝑃2𝑥 − 3

2𝑏 𝑃2𝑦 +

𝑏

2𝑃2𝑧

𝑃3𝑥 𝑃3𝑦 − 𝑏 𝑃3𝑧

= −339.4930 −196.0064 160339.4930 −196.0064 160

0 392.0127 160

- Spherical joint coordinates:

𝑆1𝑥 𝑆1𝑦 𝑆1𝑧

𝑆2𝑥 𝑆2𝑦 𝑆2𝑧

𝑆3𝑥 𝑆3𝑦 𝑆3𝑧

= −150 −86.6025 589.473150 86.6025 589.473

0 173.2051 589.473

The five twist screws form a limb-1 motion screw system can be represented using

Plucker’s coordinates,

$𝑡1,1 = 0 0 0 0 0 1

$𝑡2,1 = −0.866 0.5 0 −80 −138.56 −339.4825

$𝑡3,1 = −0.866 0.5 0 −294.7365 −510.4836 −149.9978

$𝑡4,1 = −0.4455 −0.7716 0.3931 420.7939 −203.6452 77.1586

193

$𝑡5,1 = 0.3931 0.227 0.891 −210.9732 365.3718 −0.0066

The one wrench screw as a reciprocal screw of a limb-1 as constraint screw of moving

platform

$𝑟𝑤1,1 = −0.866 0.5 0 −294.7365 −510.4836 −149.9978

5-twist screws system of limb -2 motion screw system using Plucker’s coordinates is,

$𝑡1,2 = 0 0 0 0 0 1

$𝑡2,2 = −0.866 −0.5 0 80 −138.56 −339.4825

$𝑡3,2 = −0.866 −0.5 0 294.7365 −510.4836 −150

$𝑡4,2 = 0.4455 −0.7716 0.3931 420.7939 203. 6452 −77.1586

$𝑡5,2 = −0.3931 0.227 0.891 −210.9732 −365.3718 0.0066

The one wrench screw as a reciprocal screw of a limb-1 as constraint screw of moving

platform

$𝑟𝑤1,2 = −0.866 −0.5 0 294.7365 −510.4836 −150

The five twist screws form a limb-3 motion screw system can be represented using

Plu cker’s coordinates,

$𝑡1,3 = 0 0 0 0 0 1

$𝑡2,3 = 1 0 0 0 160 −392.0127

$𝑡3,3 = 1 0 0 0 589.4730 −173.2051

$𝑡4,3 = 0 0.6504 −0.5591 −480.2322 0 0

$𝑡5,3 = 0.5141 0.5591 0.6504 −216.9218 303.0481 −89.0447

The one wrench screw as a reciprocal screw of a limb-3 as constraint screw of moving

platform,

$𝑟𝑤1,3 = 1 0 0 0 589.473 −173.2051

Reciprocity condition of two screw systems using equation (6.101) is also verified for

all limbs 𝑖 ∈ 1,2,3 as,

$𝑡𝑗 ,𝑖 ∘ $𝑟𝑤

1,𝑖 = 0 , 𝑗 = 1,2 … 5 (6.110)

Each reciprocal screw of each limb imposes constraint force (zero pitch screw)

on the moving platform.

The wrench screw system as a platform constraint system can be represented as,

$𝑤1 = −0.866 0.5 0 0 0 −0.866

$𝑤2 = −0.866 −0.5 0 0 0 −0.866

$𝑤3 = 1 0 0 0 0 1

194

Figure 6.20 Moving platform constraint screw system

The corresponding platform motion screw system which is reciprocal to

platform constraint system is obtained using reciprocity condition as expressed in

equation (6.101),

$𝑟𝑡1 = 0 0 0 0 0 1

$𝑟𝑡2 = −0.866 −0.5 0 0 0 −0.866

$𝑟𝑡3 = −0.866 0.5 0 0 0 −0.866

Translation along 𝑥, translation along 𝑦 and rotation along 𝑧 are three

constraints imposed by three reciprocal screws of platform motion screw system as

observed from platform motion screw system. Hence, 3-PRS manipulator of proposed

configuration allows three degrees of freedom. It verifies the same results as

mentioned in mobility analysis of chapter 2. The 3-PRS manipulator has two rotary

degrees of freedom along 𝑥 and 𝑦 - directions and one translational DOF along 𝑧 -

direction.

6.10 Singularity analysis of a manipulator using screw theory

The following steps to be followed for a singularity analysis of 3-DOF parallel

manipulator using screw theory.

Step1: Each PRS kinematic chain contains 3 kinematic pairs of prismatic, revolute

and spherical pair and each pair has 𝜅𝑖 DOF. The twist screw system of 𝑖𝑡𝑕

chain is represented as,

𝑇𝑖 = $𝑡1,𝑖 $𝑡

2,𝑖 $𝑡3,𝑖 $𝑡

4,𝑖 $𝑡5,𝑖

𝑇 (111)

Here, twist screw system for 𝑖𝑡𝑕 chain has 5 × 6 matrix dimension. The

number of reciprocal screws in 𝑖𝑡𝑕 chain has 6 − 𝑛 linearly independent

screws. It is observed that there is only one constraint wrench screw for an

individual limb. Moreover, they are satisfying the condition of reciprocity as

per equation (6.107),

$𝑡𝑗 ,𝑖 ∘ $𝑟𝑤

1,𝑖 = 0 , 𝑗 = 1,2 … 5 and 𝑖 ∈ 1,2,3 (112)

𝑣1

𝑥 𝑦

30°

𝑆2

𝑆3

𝑆1 𝑣2

$𝑟𝑤1,3

$𝑟𝑤1,1

$𝑟𝑤1,2

𝑣3

195

There are three wrench screws to constraint a motion of moving platform.

Hence, platform constraint screw system [123] of 3-PRS configuration is

represented by,

𝑊𝑝𝑐 = $𝑟𝑤1,1 $𝑟𝑤

1,2 $𝑟𝑤1,3

𝑇 (113)

The actuation twist screw $𝐴𝑡1,𝑖 of a prismatic actuator is part of twist screw

system 𝑇𝑖 . When the prismatic actuator is locked then same twist screw of an

actuator is not part of twist screw system and form a system 𝑇𝑖∗. Hence, matrix

dimension of twist screw system 𝑇𝑖 becomes 4 × 6. There will be a screw $𝑇𝑖

reciprocal to a screw system 𝑇𝑖∗, 𝑖 ∈ 1,2,3 .

$𝑡𝑗 ,𝑖 ∘ $𝑇 = 0 , 𝑗 = 1,2 … 5 and 𝑗 ≠ 1

$𝐴𝑤 represents a wrench from the prismatic actuator and it is transmitted to

moving platform. It is called as actuation wrench screw (AWS) system.

𝐴𝑤 = $𝐴𝑤1,1 $𝐴𝑤

1,2 $𝐴𝑤1,3

𝑇 (114)

Step 2: A spatial 3-PRS manipulator with three degrees of freedom has 3 limbs with

one actuator in each limb. Platform constraint screw system and actuation

wrench screw together form a wrench screw system of the manipulator.

𝑊 = 𝑊𝑝𝑐 𝐴𝑤 = $𝑟𝑤1,1 $𝑟𝑤

1,2 $𝑟𝑤1,3 $𝐴𝑤

1,1 $𝐴𝑤1,2 $𝐴𝑤

1,3 (115)

Step 3:Constraint wrench screw system in the manipulator has three constraint

wrenches to the moving platform as shown in figure 6.20. There might be an

existence of constraint singularity.

Step 4:Three constraint wrenches restrict 3-DOF of moving platform as mentioned in

mobility analysis.

If constraint wrench screws are less than 3DOF results in gain in one degree of

freedom and manipulator possess constraint singularity in that situation. It

should be avoided as far as possible within workspace.

Step 5:Singularity identification is possible by determining 𝑑𝑒𝑡(𝑊) = 0.

It can be conceived that for a single actuation in limb-1 the moment axis

passes through the both the centers points of spherical joints in limb plane 2 and 3

respectively and vice versa. Connecting links axes are in same plane of moving

platform axis and constraint singular configuration is observed. In such configurations

relationship between moments axes can be described by 𝑣1 × 𝑣2 ∙ 𝑣3 = 0 as shown

in figure 6.20. The manipulator is said to have a constraint singularity.

196

6.11 Summary

A recent trend of screw theory application for parallel manipulator is applied

further for its instantaneous kinematic investigation. The first part of this chapter

presented exponential coordinates of rotations. That may be further assist in

understanding Euler angle rotation matrices (See Appendix-II). The definition of

screw, screw axis, pitch based on screw theory is reviewed. Reciprocal screw systems

for R (Revolute) and P (Prismatic) joints associated with the configuration are

presented. Various applications of reciprocal screw system are presented. Mobility

analysis of 3-PRS configuration is carried out using screw axis method. Total three

reciprocal screws of all limbs constraint three degrees of freedom of 3-PRS parallel

manipulator system. It is found that unconstrained degrees of freedom are obtained as

rotation about 𝑥, rotation about 𝑦 and translation about 𝑧 for 3-PRS configuration.

Kutzbach–Gr¨ubler formula computes only number of degrees of freedom. Types of

motions associated about specific axis cannot be determined using this formula.

Platform motion screw system and platform constraint screw system completely

specify number degrees of freedom and motion types. It is observed that manipulator

under constraint configuration only when three moments axes are coplanar with a

plane of moving platform.