Screw Theory and Reciprocity

Screw Theory and Reciprocity

Latifah Nurahmi

Definition of Screw

A spatial displacement of a rigid body can be expressed as a

combination of a rotation about a line and a translation along

the same line. This combined motion is called

screw displacement

.

A unit screw $ is defined as:

                               

S

S

S

S

S

S

5 4 3 2 1 s s r s $



Definition of Screw

A unit screw $ is defined as:

s = a unit vector along the axis of the screw $

r = a position vector of any point on the screw axis of $ λ = pitch                                

S

S

S

S

S

S

6 5 4 3 2 1 s s r s $



Definition of Screw

A unit screw :

(λ = 0),          s s r s $



        s r s

$

0 (λ = ∞), _        _s 0

$

Zero-pitch screw

Infinite-pitch screw

(λ ≠ ∞, λ ≠ 0), _

_{Finite-pitch screw}

        s s r s

$

h

_

Screw Systems

A

screw system

of order n (0 ≤ n ≤ 6) comprises all the screws

that are linearly dependent on n given linearly independent screw.

Definition of Twist and Wrench

Twist

A twist represents an instantaneous motion of a rigid body.

Zero-pitch twist  pure rotation.

Infinite-pitch twist  pure translation.

0







Wrench

A wrench represents a system of forces and moments acting on

a rigid body.

Zero-pitch wrench F  pure force

Infinite-pitch wrench M  pure moment

Relations to Kinematic Joints

system



0

1



system







1

system



0

3



system







_



2

1

₀

system







_



1

1

₀

Relations to Serial Kinematic Chain

A rigid body constrained to move in the plane

The end effector is constrained to rotate and translate in the plane.

 The screw system associated with the available twists is of order 3.

Available twists

Relations to Serial Kinematic Chain

Two rigid bodies connected by three prismatic joints

The screw system describing all the available twists consists of all screws with infinite pitch.

 All translatory motions are instantaneously (in this particular example, also finite translatory motions) possible.

 The set of all twists is a third order screw system.

Body 2 Body 1

Definition of Reciprocity

Work done by a wrench on a twist

 The rate of work done by a wrench w = [fT_{, m}T _]T _{on a twist t = [}wT_{, v}T _]T is given by

 Alternatively,

Reciprocity

Two screws are said to be reciprocal if a wrench applied about one does no work on a twist about the other.

P   f v m w





P T     _  v f m t w w        _   0 I I 0 3 3 3 3 3 3 3 3

Reciprocal Screws

Two screws S

₁

(pitch h

₁

) S

₂

(pitch

h

₂

) are reciprocal if and only if

(h

₁

+ h

₂

) cos

f

- d sin

f

= 0

Remarks

 A wrench applied about S1does no

work on a twist about S₂.

 A wrench applied about S2does no

work on a twist about S1.

 The condition for reciprocity is a purely geometric relationship

f

d

f

m

v

w

S

₁

S

₂

Reciprocal Screws (continued)

Derivation

f

d

f

m

v

w

S

₁

S

₂

x

y

z

S₁ S 1 2 2 2 1 0 0 0 0 0 0                                         h h d h d , cos sin cos sin sin cos f f f f f f w f m t v    _                       _  _                    f h h d h d 1 0 0 0 0 0 0 1 2 2 , cos sin cos sin sin cos w w f f f f f f





f v  m w 0  h1h2 cosfdsinf0

Examples

Screws reciprocal to a zero pitch screw

A wrench acting on a rigid body free to rotate about a revolute joint does no work on the rigid body if one of the following is true

 The wrench is of zero pitch and the axis intersects the axis of rotation

 The wrench of infinite pitch is perpendicular to the axis of rotation

 The pitch is non zero but equal to

d tan f Note: If m cosf  f d sin f,

the wrench does no work.

q S₁

d

f

m

f

S₂

(h

₁

+ h

₂

) cos

f

- d sin

f

= 0

Examples (continued)

Screws reciprocal to an infinite pitch screw

A wrench acting on a rigid body free to translate along a prismatic joint does no work on the rigid body if one of the following is true

 The wrench is of infinite pitch.

 The pitch is zero or finite, but the axis is perpendicular to the axis of the prismatic joint. S₁

d

f

m

f

S₂ r

f v

  w 

m

0

Reciprocal Screw Systems

Definition of a screw system

The vector space of all screws generated by taking all possible

linear combinations of a finite number of screws.

Reciprocal screw systems

Two screw systems are reciprocal if every screw in the first

screw system is reciprocal to all screws in the second screw

system.

Important Property

Given a screw system, the set of all screws reciprocal to every

screw in the given screw system is another screw system.

Example

The screw system consisting of all screws

reciprocal to a given zero pitch screw

Given screw system is a first order screw

system defined by a zero pitch screw.

The reciprocal screw system consists of

 All screws of zero pitch such that the screw axes intersect the axis of the given zero pitch screw  Screws of pitch equal to d tan f

f

d

f

m

v

w

S

₁

S

₂

Properties

 Given a screw system, the set of all screws reciprocal to every screw in the given screw system is another screw system.

 The screw system reciprocal to a nth order screw system is of order (6-n)  A rigid body subject to constraints has the following property:

Available twists

dim = n

Constraint wrenches

dim = 6-n

Reciprocity

Another view point

A pure force Fperpendicular to t, cannot produce any translation along t

Twist and wrench systems for

a kinematic joint

The set of all twists allowed by a joint is reciprocal to the set of all wrenches that can be resisted (passively) by the joint.

A n degree-of-freedom joint has a screw system of order n associated with the available twists and a screw system of order (6-n) associated with the wrenches that can be transmitted by the joint.

Available twists

dim = n

Constraint wrenches

dim = 6-n

Reciprocity

Reciprocal Twist and Wrench in

Kinematic Joints

Example 1

Two rigid bodies connected by a spherical joint

The set of available twists for the moving rigid body is described by a screw system consisting of all screws with zero pitch passing through the center of the spherical joint.

 The set of all twists is a three-dimensional vector space. The screw system is order 3.

The set of all wrenches that do no work is described by the reciprocal screw system  The reciprocal screw system is of order (6-3=) 3

 It consists of all pure forces passing through the center of the spherical joint.

Fixed rigid body

Example 2

A serial kinematic chain, composed of P and R joints connecting link a and b.

If link a is fixed and link b is moving, the twist of link b is:

 The screw system associated with the available twists is of order 2.

 The screw system associated with the constraint wrenches (the wrenches that can do no work on the end effector) is of order 4.

Example 3

Two rigid bodies connected by three prismatic joints

The screw system describing all the available twists consists of all screws with infinite pitch.

 All translatory motions are instantaneously (in this particular example, also finite translatory motions) possible.

 The set of all twists is a third order screw system.

The reciprocal screw system describing all the wrenches that do no work on the constrained rigid body (Body 2) consists of all infinite pitch screws.

 No couple can do work on Body 2.

 The set of all constraint wrenches is a third order screw system.

Body 2 Body 1

Example 4

A rigid body constrained to move in the plane

The end effector is constrained to rotate and translate in the plane.

 The screw system associated with the available twists is of order 3.

 The screw system associated with the constraint wrenches (the wrenches that can do no work on the end effector) is also of order 3.

Available twists

Constraint wrenches

Type Synthesis

Type synthesis adalah salah satu proses desain untuk mensintesa jenis-jenis

atau tipe-tipe kaki robot.

Type synthesis dapat dilakukan berdasarkan Screw theory dan Virtual chain

Virtual Chain adalah sebuah serial kinematic chain yang mempunyai tipe

gerakan tertentu untuk menggambarkan jenis gerakan pada parallel manipulator.

Virtual chain diusulkan berdasarkan analisa wrench system dan virtual chain yang paling sederhana harus dipilih.

Contoh 1 virtual chain pada

parallel manipulator

Sebuah Parallel Manipulator (PM) mempunyai 3-DOF dengan

Virtual chain yang paling sederhana untuk menggambar PM tersebut adalah tiga P-joints yang terhubung secara seri.

PPP-Virtual Chain system  



3

Contoh 2 virtual chain pada

parallel manipulator

Sebuah Parallel Manipulator (PM) mempunyai 3-DOF dengan

Virtual chain yang paling sederhana untuk menggambar PM tersebut adalah tiga R-joints yang terhubung secara seri atau satu S-joint.

S-Virtual Chain

system



0 3



Contoh 3 virtual chain pada

parallel manipulator

Sebuah Parallel Manipulator (PM) mempunyai 4-DOF dengan

Virtual chain yang paling sederhana untuk menggambar PM tersebut adalah tiga P-joints dan satu R-joint yang terhubung secara seri.

PPPR-Virtual Chain system  



2

Prosedur Type Synthesis

Parallel Manipulator

Prosedur Type Synthesis

Parallel Manipulator

Langkah 3 – Merangkai kaki robot menjadi parallel manipulator

Syarat-syarat yang harus dipenuhi dalam merangkai kaki robot:

1. Setiap kaki harus mempunyai dof yang sama dengan virtual chain

2. Wrench system dari PM harus sama dengan virtual chain

Langkah 4 – Seleksi joint yang akan diaktuasi

Dalam memilih joint yang akan diaktuasi (diberi motor), beberapa kriteri berikut harus diikuti:

1. Harus terdistribusi diantara semua kaki

2. Lebih disukai yang berada di bas

Type Synthesis Schönflies Motion

Parallel Manipulator

Type Synthesis Schönflies Motion Parallel

Manipulator

Langkah 1: Dekomposisi wrench system

system







2

system







1

m c Δ c1 c2 c3 c4 c5 2 2 2 2 2 1 2 1 0 1 1 3 2 4 2 2 2 3 2 2 1 2 2 1 1 1 1 1 1 4 2 6 2 2 2 2 5 2 2 2 1 4 2 2 1 1 3 2 1 1 1 2 1 1 1 1 5 2 8 2 2 2 2 2 7 2 2 2 2 1 6 2 2 2 1 1 5 2 2 1 1 1 4 2 1 1 1 1

Kombinasi wrench system setiap kaki (2-5 kaki)

Type Synthesis Schönflies Motion Parallel

Manipulator

Langkah 2: Type synthesis dari setiap kaki Kasus (1.)

Menghitung jumlah joint di setiap kaki

8 joints termasuk PPPR Virtual chain

system







2

joints 8 ) 6 (    F c f

Type Synthesis Schönflies Motion Parallel

Manipulator

Type Synthesis Schönflies Motion Parallel

Manipulator

Langkah 2: Type synthesis dari setiap kaki Kasus (2.)

Menghitung jumlah joint di setiap kaki

9 joints termasuk PPPR Virtual chain

system







1

joints 9 ) 6 (    F c f

Type Synthesis Schönflies Motion Parallel

Manipulator

Tabel 2. Tipe-tipe kaki

ci _{Dof Class Type}

2 4 3R-1P Permutation PŔŔŔ 2R-2P Permutation PPŔŔ 1R-3P Permutation PPPŔ 1 5 5R Permutation ŔŔŔȐȐ Permutation ŔŔȐȐȐ 4R-1P Permutation PŔŔŔȐ Permutation PŔŔȐȐ Permutation PŔȐȐȐ 3R-2P Permutation PPŔŔȐ Permutation PPŔȐȐ

Type Synthesis Schönflies Motion Parallel

Manipulator

Langkah 3: Merangkai kaki

m c Δ c1 _c2 _c3 _c4 _c5 2 2 2 2 2 1 2 1 0 1 1 3 2 4 2 2 2 3 2 2 1 2 2 1 1 1 1 1 1 4 2 6 2 2 2 2 5 2 2 2 1 4 2 2 1 1 3 2 1 1 1 2 1 1 1 1 5 2 8 2 2 2 2 2 7 2 2 2 2 1 6 2 2 2 1 1 5 2 2 1 1 1 4 2 1 1 1 1 3 1 1 1 1 1

ci Dof Class Type

2 4 3R-1P Permutation PŔŔŔ 2R-2P Permutation PPŔŔ 1R-3P Permutation PPPŔ 1 5 5R Permutation ŔŔŔȐȐ Permutation ŔŔȐȐȐ 4R-1P Permutation PŔŔŔȐ Permutation PŔŔȐȐ Permutation PŔȐȐȐ 3R-2P Permutation PPŔŔȐ Permutation PPŔȐȐ 2R-3P Permutation PPPŔȐ Table 1 Table 2

Type Synthesis Schönflies Motion Parallel

Manipulator

Langkah 4: Selection of the actuated joint Kriteria:

1. Harus terdistribusi diantara semua kaki

2. Lebih disukai yang berada di bas 3. Sebaiknya tidak ada P-joint yang

tidak diaktuasi/pasif. Quadrupteron PM L_i= Pi_Ȑi 1Ȑi2Ȑi3Ŕi i = 1, 2, 3 L₄= PŔ₁Ŕ₂Ŕ₃,

system

6





c a

W

W