Chapter 1

Basics of Screw Theory

1.1 Introduction

Screw theory is a powerful mathematical tool for the analysis of spatial mechanisms.

A screw consists of two three-dimensional vectors. A screw can be used to denote

the position and orientation of a spatial vector, the linear velocity and angular

velocity of a rigid body, or a force and a couple, respectively. Therefore, the concept

of a screw is convenient in kinematics and dynamics, while the transformation

between the screw-based method and vector and matrix methods is straightforward.

When applied in mechanism analysis, screw theory has the advantages of clear

geometrical concepts, explicit physical meaning, simple expression and convenient

algebraic calculation. It is worth noting that the preliminary requirements for screw

theory are only linear algebra and basic dynamics in undergraduate level. Thus,

screw theory has been widely applied and researchers have used screw theory to

make great contribution to many frontier problems in mechanism theory.

Screw theory was established in the nineteenth century. First, Chasles (1830)

proposed the concept of twist motion of a rigid body, which was further developed

by Poinsot (1848). Then Pl€ucker gave his research and proposed his screw expres-

sion [1]. Then, in his classic book Screw theory (1875), Ball discussed the kinematics

and dynamics of a rigid body under complex constraints using screw theory [2]. In the

last 60 years, many researchers, such as Dimentberg [3], Yang and Freudenstein [4],

Waldron [5], Roth [6], Hunt [7], Phillips [8], Duffy [9] and Angeles [10] etc, have

made important contributions to screw theory.

This chapter first addresses the expression of a straight line and its Pl€ucker linecoordinates in three-dimensional space. The line vector and screw as well as their

characteristics and the screw algebra are introduced. In the last part, this chapter

Z. Huang et al., Theory of Parallel Mechanisms, Mechanisms and Machine Science 6,

DOI 10.1007/978-94-007-4201-7_1, # Springer Science+Business Media Dordrecht 2013

1

also introduces the expression for the instantaneous motion of a rigid body and the

statics of the body using screw theory.1

1.2 Equation of a Line

Two distinct points Aðx1; y1; z1Þ and Bðx2; y2; z2Þ determine a line, as shown in

Fig. 1.1. The vector S denoting the direction of the line can be expressed as

S ¼ ðx2 � x1Þiþ ðy2 � y1Þj þ ðz2 � z1Þk; (1.1)

where i, j, and k are unit vectors corresponding to each coordinate axis.

If we let

x2 � x1 ¼ L

y2 � y1 ¼ M

z2 � z1 ¼ N; ð1:2Þ

then substituting Eq. (1.2) into Eq. (1.1), we have

S ¼ LiþMj þ Nk; (1.3)

where L, M, and N are direction ratios.

The distance between the two points is given by

jSj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þM2 þ N2

p: (1.4)

Let

l ¼ L=jSjm ¼ M=jSjn ¼ N=jSj; ð1:5Þ

where l, m and n are direction cosines of the line. Then Eq. (1.4) reduces to

l2 þ m2 þ n2 ¼ 1: (1.6)

1 The content of screw theory in this book is based on the teaching material presented by Dr. Duffy

at Florida University in 1982. At that time, the first author of this book listened attentively to the

lectures and was deeply inspired by the course content. The author wishes to express here once

again his acknowledgments to Dr. Duffy.

2 1 Basics of Screw Theory

Note that a line can be determined by its direction and a point on it. We can

write the vector equation of the line as

ðr � r1Þ � S ¼ 0: (1.7)

Equation (1.7) can also be expressed as

r � S ¼ S0; (1.8)

where

S0 ¼ r1 � S (1.9)

is the moment of the line about the origin O.

The vectors ðS; S0Þ are called the Pl€ucker coordinates of the line and satisfy the

orthogonality condition

S � S0 ¼ 0: (1.10)

Note that ðS; S0Þ is homogeneous, since multiplying both sides of Eq. (1.8) by a

scalar l yields the same line.

Expanding Eq. (1.9) leads to

S0 ¼i j kx1 y1 z1L M N

������

������: (1.11)

Equation (1.11) can be expressed in the form

S0 ¼ Piþ Qj þ Rk; (1.12)

where

P ¼ y1N � z1M

R ¼ x1M � y1L: (1.13)

Z

X

r1

Y

B

r 2

r

O

A

S

Fig. 1.1 Pl€ucker Coordinatesof a line

1.2 Equation of a Line 3

Expanding Eq. (1.8) and considering Eq. (1.12), we have

yN � zM � P ¼ 0

zL� xN � Q ¼ 0

xM � yL� R ¼ 0: ð1:14Þ

From Eqs. (1.3), (1.12) and (1.10), the orthogonality condition can be written as

LPþMQþ NR ¼ 0: (1.15)

The six Pl€ucker coordinates of the line ðL;M;N;P;Q;RÞ are illustrated in

Fig. 1.2.

ðL;M;NÞ consists of the direction ratios of the line and ðP;Q;RÞ are the x, y and zcomponents of the moment of the line about the origin. The coordinates ðS; S0Þ arerelated by Eqs. (1.6) and (1.11). Therefore only four of the six scalars ðL;M;N;P;

Q;RÞ are independent, and there are 14 lines in space.

The distance of the line from the origin is determined by the length of a vector Pfrom O and perpendicular to the line. From Eq. (1.8), we have P� S ¼ S0 , andtherefore

S� ðP� SÞ ¼ S� S0:

Expanding the left side of the above equation, we have

S� ðP� SÞ ¼ ðS � SÞP� ðS � PÞS ¼ ðS � SÞP;

and so solving equation for P gives

P ¼ S� S0S � S : (1.16)

Z

LM

S

N

R

P

Q

O

P YX

Fig. 1.2 Distance between

the line and origin

4 1 Basics of Screw Theory

This can be expressed in the form

P ¼ jSjjS0jjSjjSj e ¼

jS0jjSj e; (1.17)

where e is a unit vector perpendicular to S� S0. Therefore

jPj ¼ jS0jjSj : (1.18)

When S0 ¼ 0, the line passes through the origin and the Pl€ucker line coordinatesare ðS; 0Þ or ðl m n; 0 0 0Þ. When S ¼ 0, the line lies in a plane at infinity

and the Pl€ucker coordinates are ð0; S0Þ or ð0 0 0; l m nÞ.

1.3 Mutual Moment of Two Lines

The vector equations of two skew lines separated by a perpendicular distance of a12and a twist angle of a12 (see Fig. 1.3) are given by

r1 � S1 ¼ S01 (1.19)

r2 � S2 ¼ S02: (1.20)

The projection of the moment vector a12a12 � S2 on the line S1 is given by a12a12 � S2 � S1 and is called the moment of S2 about the line S1. This scalar quantity isusually called the mutual moment of the two lines and can also be obtained by

projecting the moment vector � a12a12 � S1 on the line S2, namely,

Mm ¼ a12a12 � S2 � S1 ¼ a12a21 � S1 � S2: (1.21)

a12

r1

r2

S2

S1

Z

A

YX

O

B

a12

a12

Fig. 1.3 Mutual moment

of two lines

1.3 Mutual Moment of Two Lines 5

Expanding the scalar triple product and considering r2 � r1 ¼ a12a12, we have

ðr2 � r1Þ � S2 � S1 ¼ r2 � S2 � S1 þ r1 � S1 � S2: (1.22)

Substituting Eqs. (1.19) and (1.20) into Eq. (1.22) yields

Mm ¼ S1 � S02 þ S2 � S01: (1.23)

Since both S1 and S2 are unit vectors, S1 � S1 ¼ S2 � S2 ¼ 1, and so

S2 � S1 ¼ �a12 sin a12: (1.24)

The mutual moment of the two lines is then given by

Mm ¼ ðr2 � r1Þ � S2 � S1 ¼ �a12 sin a12: (1.25)

From this expression, it is clear that the mutual moment of two lines is

independent of the coordinate-frame.

Alternatively, Eq. (1.23) can be written as

Mm ¼ l1p2 þ m1q2 þ n1r2 þ p1l2 þ q1m2 þ r1n2: (1.26)

If the lines are parallel or intersect at infinity, we have a12 ¼ 0 and their mutual

moment is zero. If the two lines intersect, which means the perpendicular distance

between the two lines is zero, we have a12 ¼ 0. Therefore, when two lines are

coplanar, the mutual moment of the two lines is always zero:

S1 � S02 þ S2 � S01 ¼ 0: (1.27)

1.4 Line Vectors and Screws

1.4.1 The Line Vector

This section will introduce two important concepts. One is the line vector and the

other is the screw. Recall that in Sect. 1.2 we established the equation of a line:

r � S ¼ S0: (1.28)

The dual vector ðS; S0Þ, which denotes a straight line in space, is also called a

line vector. When jSj ¼ 1 , S is a unit vector and ðS; S0Þ is a unit line vector.

However,S0 is in general not a unit vector. The two vectorsS andS0 are orthogonal,

6 1 Basics of Screw Theory

so that S � S0 ¼ 0. The unit line vectors ðS; S0Þ stand in one-to-one correspondence

with the 14 lines in space.

The vector S is not origin-dependent. The vector S0, which remains constant as Sis being moved along the line, is the moment of S about the origin O. Clearly, S0 isorigin-dependent, and if the origin is shifted from point O to point A, then the

moment of S about A can be written as (Fig. 1.4)

SA ¼ rA � S ¼ ðABþ rBÞ � S: (1.29)

Substituting SB ¼ rB � S into Eq. (1.29), we have

SA ¼ SB þ AB� S: (1.30)

When the line vectors are in different positions in space, the Pl€ucker coordinatesare different. In particular, when the line vector is located in certain special positions

with respect to the coordinate system, many of the components are zero, as shown in

Fig. 1.5. This is convenient for screw analysis.

S

A

AB

rB

rA

B

Fig. 1.4 Line moment is not origin-dependent

(abc;def )

(abc;de0 )

SkewParallel to coordinate axisParallel to coordinate plane

(ab0 ;def)(ab0 ;de0 )

(abc;000 )(100;0ab )(100;010 )

(100;000 )(ab0;00c )

Z

Y

X

Fig. 1.5 Various forms of line vector

1.4 Line Vectors and Screws 7

1.4.2 The Screw

When the two vectors of a dual vector do not satisfy the orthogonality condition

S � S0 6¼ 0,2 which is the more general case, the dual vector is called a screw and is

denoted by

$ ¼ ðS; S0Þ; S � S0 6¼ 0: (1.31)

When jSj ¼ 1, $ is a unit screw.

The vector S of a screw is also not origin-dependent. The vector S0 is

origin-dependent and if the origin is shifted from point O to point A, the moment

of S about A can be obtained as

SA ¼ S0 þ AO� S: (1.32)

Multiplying both sides of this equation by S, we have

S � SA ¼ S � S0: (1.33)

Equation (1.33) shows thatS � S0 is not origin-dependent. IfS 6¼ 0, we can obtain

the origin-independent variable

h ¼ S � S0S � S ¼ lpþ mqþ nr

l2 þ m2 þ n2; (1.34)

which is called the pitch of a screw. The line vector is a special screw with a zero

pitch. A screw with an infinite pitch is called a couple and is denoted by (0; S).The number of the unit screws in 3D space is 15, and the number of screws

in 3D space is 16.

A line vector corresponds to a straight line in space, and a screw also has its axis

line. To determine the axis line that the screw lies on, S0 is decomposed into two

parts, which are parallel and perpendicular to S, respectively, as shown in Fig. 1.6:

ðS; S0Þ ¼ ðS; S0 � hSþ hSÞ: (1.35)

Obviously, S0 � hS is normal to S, and S0 � hS ¼ S0. The equation of the axis ofthe screw is given by

r � S ¼ S0 � hS: (1.36)

2 For the convenience of readers, to distinguish between line vector and screw, the dual component

of screw is expressed as S0.

8 1 Basics of Screw Theory

The Pl€ucker coordinates of the line are ðS; S0 � hSÞ. From Eq. (1.25), a screw

can be expressed as

$ ¼ ðS; S0Þ ¼ ðS;S0 � hSÞ þ ð0; hSÞ; (1.37)

or

$ ¼ ðS; S0Þ ¼ ðS; r � Sþ hSÞ ¼ ðS; S0 þ hSÞ ¼ ðS; S0Þ þ ð0; hSÞ: (1.38)

This indicates that a line vector and a couple can combine to form a screw,

or that any screw with non-zero finite pitch can be considered the summation of a

line vector and a couple. There are four factors that determine a screw: position and

direction of the axis, and the magnitude and pitch of the screw.

Screw: ðS; S0Þ : S 6¼ 0; S � S0 6¼ 0; 1 6¼ h 6¼ 0

Line vector: ðS; S0Þ : S 6¼ 0; S � S0 ¼ 0; h ¼ 0

Couple: ð0; SÞ : S 6¼ 0; h ¼ 1

Example 1.1. l m n; hl hm hnð Þ is a screw with pitch h and passing through

the origin.

Example 1.2. 1 0 0; 1 0 0ð Þ is a screw with pitch h ¼ 1 and passing through the

origin, since

h ¼ ðS � S0Þ=ðS � SÞ ¼ 1;

r � S ¼ S0 � hS ¼ 0:

Example 1.3. 1 1 1; 1 1 1ð Þ= ffiffiffi3

pis also a unit screw with pitch h ¼ 1 that passes

through the origin and points in the direction 1 1 1ð Þ.Example 1.4. Identify $ ¼ ðS; S0Þ ¼ 1 1 0; 1 0 0ð Þ and determine its axis.

Solution. The pitch of the screw is h ¼ ðS � S0Þ=ðS � SÞ ¼ 1=2. Considering that

S0 ¼ S0 þ hS ¼ r � Sþ hS, the equation of the axis is r � S ¼ S0 � hS.

S0

S

hS

S0−hS

Fig. 1.6 Axis of a screw

1.4 Line Vectors and Screws 9

Thus we have

r � S ¼ 1=2� 1=2 0ð ÞT:

1.5 Screw Algebra

Screws obey the following algebraic operations [11], and these operations have

special meanings.

1.5.1 Screw Sum

The sum of two screws $1 ¼ S1; S01

� �, and $2 ¼ S2; S

02

� �is defined as follows

$1 þ $2 ¼ S1 þ S2; S01 þ S02

� �(1.39)

Equation (1.39) shows that the sum of two screws is still a screw.

Theorem 1.1. The sum of two line vectors is a line vector only if their axes arecoplanar and the sum of their first vectors is non-zero, namely S1 þ S2 6¼ 0.

Proof. If $1 and $2 are two line vectors, then S1 � S01 ¼ 0 and S2 � S02 ¼ 0. If the

two line vectors are coplanar, then their mutual moment is equal to zero. That is,

S01 � S2 þ S02 � S1 ¼ 0. Therefore, we obtain

ðS1 þ S2Þ � ðS01 þ S02Þ ¼ 0: (1.40)

Equation (1.40) shows that the sum of two line vectors is still a line vector and its

pitch is zero.

Theorem 1.2. If two line vectors intersect, their sum is a line vector passing theintersection point.

Proof. Since the sum of two line vectors is still a line vector, it can be expressed as

r � ðS1 þ S2Þ ¼ S01 þ S02: (1.41)

If r1 denotes the position vector of the intersection point, r1 lies on both of the

lines and satisfies the equations of both lines:

r1 � S1 ¼ S01; r1 � S2 ¼ S02: (1.42)

Thus we have

r1 � ðS1 þ S2Þ ¼ S01 þ S02: (1.43)

10 1 Basics of Screw Theory

Equation (1.43) shows that the point r1 lies on the line determined by the sum of

the two line vectors.

Note that when two lines are not coplanar, the sum of the two line vectors is a

screw with non-zero pitch instead of a line vector. Further, the sum of a line vector

and a couple is not a line vector.

Theorem 1.3. The sum of two couples, if not zero, is another couple.

1.5.2 Product of a Scalar and a Screw

The product of a scalar l and a screw $ is defined by

l$ ¼ ðlS; lS0Þ: (1.44)

1.5.3 Reciprocal Product

The reciprocal product of two screws, say $1 ¼ S1; S01

� �, $2 ¼ S2; S

02

� �, is defined by

$1 � $2 ¼ S1 � S02 þ S2 � S01 (1.45)

where the symbol ○ denotes the reciprocal product of two screws.

When the origin of the coordinate system shifts from point O to point A, the twoscrews $1 and $2 become

$A1 ¼ S1; SA1

� � ¼ S1; S01 þ AO� S1

� �;

$A2 ¼ S2; SA2

� � ¼ S2; S02 þ AO� S2

� �: (1.46)

The reciprocal product of the two new screws is

$A1 � $A2 ¼ S1 � S02 þ AO� S2� �þ S2 � S01 þ AO� S1

� � ¼ $1 � $2: (1.47)

This result indicates that the reciprocal product of two screws is not origin-

dependent.

The reciprocal product of two screws represents the work produced by a wrench

acting on a rigid body undergoing an infinitesimal twist.

1.6 Instantaneous Kinematics of a Rigid Body

The most general motion in three-dimensional space is screw motion, which means

simultaneous translational and rotational motion. Pure translation and pure rotation

are special cases of screw motion. In this section, we will discuss pure rotation,

1.6 Instantaneous Kinematics of a Rigid Body 11

translation, and screw motion, including how to determine the Pl€ucker coordinates,pitch, and the equation of axis.

1.6.1 Instantaneous Rotation

In Fig. 1.7, the rigid body labeled by 2 is rotating about another rigid body labeled

by 1. The axis of rotation is S. Such a rotation can be described by the angular

velocity line vector

o$ ¼ oðS; S0Þ ¼ ðoS;oS0Þ; (1.48)

whereo is the amplitude of the rotation andS is the unit vector in the direction of theline.

The equation of the axis of rotation is given by

r � S ¼ S0: (1.49)

The second component of Eq. (1.48) is

oS0 ¼ or � S ¼ r �v ¼ v0; (1.50)

which is the velocity of a point coincident with the origin, or the tangent velocity of

the point coincident with the origin. Equation (1.48) can also be rewritten as

follows:

o$ ¼ ðv; v0Þ: (1.51)

Therefore, the dual vectors denoting the rotation of a rigid body include the

angular velocity, v, and the linear velocity, v0 , of the point coincident with the

origin. The Pl€ucker coordinates of a rotating rigid body are oðS; S0Þ or ðv; v0Þ.When the rotational axis passes through the origin, the Pl€ucker coordinates of therotational axis are o$ ¼ ðv; 0Þ or ðv; 0Þ.

Z

X

Y

2

1

r

v0 Sw

O

$

Fig. 1.7 Instantaneous

rotation of a body

12 1 Basics of Screw Theory

1.6.2 Instantaneous Translation

An instantaneous translation of one rigid body relative to another one can be

conveniently modeled by connecting the two bodies by a prismatic or sliding pair

(as illustrated in Fig. 1.7). The translational velocity n can be expressed as a scalar

multiple of the instantaneous linear displacement n and a unit vectorS parallel to thedirection of motion. The vector S is for convenience drawn through the center line

of the joint. However, all points in the moving body have the same linear velocity

v ¼ vS, and a self-parallel displacement of the vector S does not affect the motion.

The same motion is obtained when the joint is reconnected parallel to its original

attachment and the body is given the same instantaneous translational displacement

v. For this reason,S is called a free vector and the motion can be quantified by taking

a scalar multiple of the free vector vð0; SÞ or ð0; nÞ. Some readers may prefer to

consider instantaneous translation to be an instantaneous rotation about an axis that

is orthogonal to S and that lies in the plane at infinity. The Pl€ucker coordinates ofthis axis are ð0; SÞ, and the instantaneous rotation about this axis can be expressed asthe scalar multiple vð0; SÞ.

1.6.3 Instantaneous Screw Motion

When the motion of one body 2 relative to another includes rotation about the axis

S1 and translation in the direction S1, the situation is more complex, as shown in

Fig. 1.8. The body rotates about the axis S1, with the instantaneous wrench o1ðS1; S01Þ, where ðS1; S01Þ is unit screw. The body also translates with screw v2ð0;S1Þalong the axisS1 at the same time. The absolute motion of the body is the sum of the

two parts.

Si

r i× w i

O

w i

vi = hiw iSi

w iSi0 = vi

0

vi

Fig. 1.8 Twist motion

of a body

1.6 Instantaneous Kinematics of a Rigid Body 13

That is,

oi $i ¼ ðo1S1;o1S01Þ þ ð0; v1S1Þ ¼ ðo1S1;o1S01 þ ho1S1Þ¼ o1 S1; S

01

� �; (1.52)

or

oi $i ¼ o1 S1; S01

� � ¼ ðv1; v0Þ; (1.53)

where v1 is the angular velocity of the body and v0 is the velocity of a point in the

body coincident with the origin. Note that the directions of v0 andv1 are different in

general, unless the axis of the screw passes the origin. The pitch is

h ¼ S1 � S01S1 � S1 ¼

v1 � v0v1 �v1

: (1.54)

1.7 Statics of a Rigid Body

1.7.1 A Force Acting on a Body

Analogous to instantaneous rotation, unit line vectors can be used to express the

action of a force on a body, as shown in Fig. 1.9. A force f can be expressed as a

scalar multiple fS of the unit vector S bound to the line. The moment of the force C0

about a reference point O can be expressed as a scalar multiple fS0 of the moment

vector S0 ¼ r � S . The action of the force upon the body can thus be elegantly

expressed as a scalar multiple f$ of the unit line vector

Z

YX

Or

f

Fig. 1.9 Force acting

on a body

14 1 Basics of Screw Theory

f$ ¼ ðf ; fS0Þ ¼ ðf ;C0Þ; (1.55)

where $ is unit line vector, S � S ¼ 1 and S � S0 ¼ 0. The line vector can be used to

express the magnitude, direction and the acting line in space.

The Pl€ucker coordinates of the force line vector are f ðS; S0Þ, ðfS; fS0Þ or ðf ;C0Þ,where C0 is the moment of force f about the origin, that is, C0 ¼ fS0 ¼ f r � S0.When the force f passes through the origin, the moment vanishes (C0 ¼ 0), and the

Pl€ucker coordinates of the force are ðf ; 0Þ.

1.7.2 A Couple Acting on a Body

Figure 1.10 illustrates a rigid body subjected to two equal and opposite forces f 1and f 2. These two forces constitute a couple, the moment of which is given by

C ¼ ðr2 � r1Þ � f 2 ¼ ðr1 � r2Þ � f 1 (1.56)

The vector C is clearly normal to the plane containing the forces, and C can

thus be expressed as a scalar multiple CS of any unit vector normal to the plane.

The vector S is therefore a free vector and the couple vector C can be given a self-

parallel displacement without altering the statics of the body.

The couple C can thus be expressed as a scalar multiple C(0; SÞ of the free vectorð0; SÞ . Alternatively, the couple can be considered a force acting upon the body

along a line that is orthogonal to S and that lies in the plane at infinity. The Pl€uckercoordinates of this line are ð0; SÞ, and the force acting upon the body can be

expressed as the scalar multiple C(0;SÞ.

1.7.3 A Twist Acting on a Body

A general system of forces and couples acting upon a rigid body can be reduced to a

single force f1 Si; S0i

� �and a single couple C(0;SÞ. This force couple combination

Z

O

X

Y

S

f1

f2

r 1

r2

Fig. 1.10 A couple acting

on a body

1.7 Statics of a Rigid Body 15

was called a dyname by Pl€ucker and Hunt. The above mentioned force screw f1Si;S

0i

� �can be expressed as the sum of a force f1 Si; S

0i

� �and a couple C(0; SÞ with

the same direction as the force. When the origin is located on the axis S1, this forcescrew can be expressed as

f1 S1; S01

� � ¼ ðf 1;C0Þ: (1.57)

When the origin does not lie on the axis S1, the force screw is

f1 S1; S01

� � ¼ ðf 1;C0Þ; (1.58)

where C0 is the moment about the origin.

References

1. Pl€ucker J (1865) On a new geometry of space. Philos Trans 155:725–791

2. Ball RA (1900) Treatise on the theory of screws. Cambridge University Press, Cambridge,

pp 1–30

3. Dimentberg FM (1950) Determination of the motion of spatial mechanisms. Akad Nauk,

Moscow (Russian)

4. Yang AT, Freudenstein F (1964) Application of dual-number quaternion algebra to the

analysis of spatial mechanisms. Trans ASME 86E:300–308 (J Appl Mech 31)

5. Waldron KJ (1966) The constraint analysis of mechanisms. J Mech 1:101–114

6. Roth B (1967) On the screw axes and other special lines associated of a rigid body. J Eng Ind

89:102–109

7. Hunt KH (1978) Kinematic geometry of mechanisms. Oxford University Press, Oxford

8. Phillips J (1990) Freedom in machinery. Cambridge University Press, Sydney, pp 147–168

9. Duffy J (1982). The screw theory and its application. Class note of University of Florida,

Gainesville

10. Angeles J (1994) On twist and wrench generators and annihilators. In: Seabra Pereira MFO,

Ambrosio JAC (eds) Computer-aided analysis of rigid and flexible mechanical system. Kluwer

Academic, Dordrecht

11. Brand L (1947) Vector and tensor analysis. Wiley, New York, pp 51–83

16 1 Basics of Screw Theory