MEV442_module_2

MEV442Introduction to RoboticsModule 2

Dr. Santhakumar MohanAssistant ProfessorMechanical EngineeringNational Institute of Technology Calicut

1

Jacobians: Velocities and static forces

IntroductionNotation for time-varying position and orientationLinear and rotational velocity of rigid bodiesMore on angular velocityMotion of the links of a robotVelocity “propagation” from link to linkJacobiansSingularitiesStatic forces in manipulatorJacobians in the force domainCartesian transformation of velocities and static forces

Introduction• In this chapter, we expand our consideration of

robot manipulators beyond static position problems.

• We examine the notions of linear and angularvelocity of a rigid body and use these conceptsto analyze the motion of a manipulator.

• We also will consider force acting on a rigidbody, and then use these ideas to study theapplication of static force with manipulators.

• It turns out that the study of both velocity andstatic forces leads to a matrix entity called theJacobian of the manipulator, which will beintroduced in this chapter.

2

Central Topic -Simultaneous Linear and Rotational Velocity

• Vector Form• Matrix Form

3

Definitions - Linear Velocity

• Linear velocity - The instantaneous rate of change in linear position of a point relative to some frame.

4

Definitions - Linear Velocity• The position of point Q in frame {A} is represented

by the linear position vector

• The velocity of a point Q relative to frame {A} is represented by the linear velocity vector

5

Definitions - Angular Velocity - Vector

• Angular Velocity Vector: A vector whose direction is the instantaneous axis of rotation of one frame relative to anotherand whose magnitude is the rate of rotation about that axis.

6

Definitions - Angular Velocity• Angular Velocity: The instantaneous rate of change in the orientation of one frame relative to another.

7

Definitions - Angular Velocity• Just as there are many ways to represent orientation (Euler Angles, Roll-Pitch-Yaw Angles, Rotation Matrices, etc.) there are also many ways to represent the rate of change in orientation.

• The angular velocity vector is convenient to use because it has an easy to grasp physical meaning. However, the matrixform is useful when performing algebraic manipulations.

8

Linear & Angular Velocities - Frames• When describing the velocity (linear or angular) of anobject, there are two important frames that are being used:

– Represented Frame (Reference Frame) :This is the frame used to represent (express)the object’s velocity.

– Computed FrameThis is the frame in which the velocity is measured(differentiate the position).

9

Frame - Velocity• As with any vector, a velocity vector may be described in

terms of any frame, and this frame of reference is noted witha leading superscript.

• A velocity vector computed in frame {B} and represented inframe {A} would be written

Computed(Measured)

Represented(Reference Frame)

10

Linear Velocity - Rigid Body• Given: Consider a frame {B}

attached to a rigid body whereas frame {A} is fixed. The orientation of frame {A}with respect to frame {B} is not

changing as a function of time

• Problem: describe the motion of of the vector relative to frame {A}

• Solution: Frame {B} is locatedrelative to frame {A} by a positionvector and the rotation matrix(assume that the orientation is notchanging in time ) expressingboth components of the velocity interms of frame {A} gives

11

Linear Velocity - Rigid Body

12

Angular Velocity - Rigid Body• Given: Consider a frame {B} attached to a rigid body whereas frame {A} is fixed. The vector is constant as view from frame {A}

• Problem: describe the velocityof the vector representing the point Q relative to frame {A}

• Solution: Even though the vector is constant as view from frame {B} it is clear that point Q will have a velocity as seen from frame {A} due to the rotational velocity

13

Angular Velocity - Rigid Body -Intuitive Approach

14


15


16


17


18


19


• The figure shows to instants of time as the vector rotates around . This is what an observer in frame {A} would observe.

• The Magnitude of the differential change is

• Using a vector cross product we get

20

Angular Velocity - Rigid Body -Intuitive Approach• In the general case, the vector Q may also be changing with respect to the frame {B}. Adding this componentwe get.

• Using the rotation matrix to remove the dual-superscript,and since the description of at any instance iswe get

21

MORE ON ANGULAR VELOCITY• A property of the derivative of an orthonormal matrix

Note: S is called the skew-symmetric matrix.

SRR ��

22

Velocity of a point due to rotating reference frame

• or

tconsPB tan�

23

Skew-symmetric matrices and the vector cross product

• Vector Form• Matrix Form

angular velocity matrix angular velocity vector

24

Gaining physical insight concerning the angular velocity vector

Rotating about the fixed frame: operator

( )x x x y z x z y

K x y z y y y z x

x z y y z x z z

k k v c k k v k s k k v k s

R k k v k s k k v c k k v k s

k k v k s k k v k s k k v c

� � � � � �

� � � � � � �

� � � � � �

� ��

� � � ��

� � �� ˆsin , cos , 1 cos [ , , ]A T

x y zs c vs K k k k� � � � ��

(2.80)

25


0

0

00

lim ( )

z y

z x

y x

t

k k

k kk k

R R tt

� �

� ��

�

� � ��

� � ��

��

0

0 ( )0

z y

z x

y x

k k

R k k R tk k

� �

� ��

� ��

� ��

� �

� ��

� �

1

( ) 11

z y

K z x

y x

k k

R k kk k

� �

� � ��

� � ��

� � � � ��

30

( )lim ( )kt

R IR R tt�

�

� ��

10

00

z y

z x

y x

k k

RR k kk k

� �

� ��

�

� ��

� ��

� �

� ��

� �

26


10 0

0 00 0

z y z y

z x z x

y x y x

k k

RR k k Sk k

� �

� ��

�

� ��

� � � � ��

� �

� ��

� �

ˆx x

y y

z z

kk K

k

��

�

� � � ��

� � � � ��

�

� �

�

K̂

• The physical meaning of the angular-velocity vector � is that, atany instant, the change in orientation of a rotating frame can be viewed as a rotation about some axis .

• This instantaneous axis of rotation, taken as a unit vector and thenscaled by the speed of rotation about the axis ( ) , yields the angular-velocity vector.

��

SRR ��

angular velocity matrix

angular velocity vector

27

Angular Velocity - Matrix & Vector Forms

S=

28

Simultaneous Linear and Rotational Velocity

• The final results for the derivative of a vector in a movingframe (linear and rotation velocities) as seen from a stationary frame

• Vector Form

• Matrix Form

29

Simultaneous Linear and Rotational Velocity -Vector Versus Matrix Representation

• Vector Form

• Matrix Form

30

Position Propagation• The homogeneous transform matrix provides a complete

description of the linear and angular position relationship between adjacent links.

• These descriptions may be combined together to describe the position of a link relative to the robot base frame {0}.

• A similar description of the linear and angular velocities between adjacent links as well as the base framewould also be useful.

31

Motion of the Link of a Robot• In considering the motion of a robot link we will always use link

frame {0} as the reference frame

Where: - is the linear velocity of the origin of link frame (i)with respect to frame {0}

- is the angular velocity of the origin of link frame (i)with respect to frame {0}

32

Frame – Velocity-review• As with any vector, a velocity vector may be described in

terms of any frame, and this frame of reference is noted witha leading superscript.

• A velocity vector computed in frame {B} and represented inframe {A} would be written

Computed(Measured)

Represented(Reference Frame)

33

Velocities - Frame & Notation• Expressing the velocity of a frame {i} (associated with link i )

relative to the robot base (frame {0}) using our previous notation is defined as follows:

• The velocities differentiate (computed) relative to the base frame {0} are often represented relative to other frames {k}.The following notation is used for this conditions

34

Velocity Propagation• Given: A manipulator - A

chain of rigid bodies each one capable of moving relative to its neighbour

• Problem: Calculate the linearand angular velocities of the link of a robot

• Solution (Concept): Due to the robot structure we can compute the velocities ofeach link in order startingfrom the base.

The velocity of link i+1 will be that of link i , plus whatever new velocity components were added by joint i+135

Velocity of Adjacent Links - Angular Velocity 1/5

• From the relationship developed previously

• we can re-assign link names to calculate the velocity of anlink i relative to the base frame {0}

• By pre-multiplying both sides of the equation by ,we canconvert the frame of reference for the base {0} to frame {i+1}

36


• Using the recently defined notation, we have

-Angular velocity of frame {i+1} measured relative to the robot base, and expressed in frame {i+1}

-Angular velocity of frame {i} measured relative to the robot base, and expressed in frame {i+1}

-Angular velocity of frame {i+1} measured relative to frame {i} and expressed in frame {i+1}

37


• Angular velocity of frame {i} measured relative to the robotbase, expressed in frame {i+1}

• Angular velocity of frame {i+1} measured (differentiate) in frame {i} and represented (expressed) in frame {i+1} is shown in the second term

In the second term

38


• Assuming that a joint has only 1 DOF. The joint configuration can be either revolute joint (angular velocity) or prismatic joint (Linear velocity).• Based on the frame attachment convention in which we assign the Z axis pointing along the i+1 joint axis such that the two are coincide (rotations of a link is preformed only along its Z- axis) we can rewrite this term as follows:

39


The result is a recursive equation that shows the angular velocityof one link in terms of the angular velocity of the previous link plus the relative motion of the two links.

Since the term depends on all previous links through this recursion, the angular velocity is said to propagate from the base to subsequent links.

40

Velocity of Adjacent Links - Linear Velocity 1/5

• Simultaneous Linear and Rotational Velocity

• The derivative of a vector in a moving frame (linear and rotation velocities) as seen from a stationary frame

• Vector Form

• Matrix Form

41

Velocity of Adjacent Links - Linear Velocity 2/5• From the relationship developed previously (matrix form)

• we re-assign link frames for adjacent links (i and i +1) with the velocity computed relative to the robot base frame {0}

• By pre-multiplying both sides of the equation by , we canconvert the frame of reference for the left side to frame {i+1}

Q

42


-Linear velocity of frame {i+1} measured relative to frame {i} and expressed in frame {i+1}

• Assuming that a joint has only 1 DOF. The joint configuration can be either revolute joint (angular velocity) or prismatic joint (Linear velocity).

• Based on the frame attachment convention in which we assign the Z axis pointing along the i+1 joint axis such that the two arecoincide (translation of a link is preformed only along its Z- axis) we can rewrite this term as follows:

43

Velocity of Adjacent Links - Linear Velocity 4/5• With replacement:

Definition

44

Angular Velocity - Matrix & Vector Forms

S=

45


• The result is a recursive equation that shows the linear velocity of one link in terms of the previous link plus the relative motion of the two links.

• Since the term depends on all previous links through this recursion, the angular velocity is said to propagate from thebase to subsequent links.

46

Velocity of Adjacent Links - Summary• Angular Velocity

0 - Prismatic Joint

• Linear Velocity

0 - Revolute Joint47

Example-2R RobotFind and

48

Example-2R Robot

49

Example-2R Robot

• For i=0

= =0

50

Example-2R Robot

=0

• For

51

Example-2R Robot

• or

=0•For

52

Example-2R Robot

•or

53

Angular and Linear Velocities - 3R Robot - Example

• For the manipulator shown in the figure, compute the angularand linear velocity of the “tool” frame relative to the base frameexpressed in the “tool” frame (that is, calculate

54


• Frame attachment

55


• DH Parameters

56


• From the DH parameter table, we can specify the homogeneous transform matrix for each adjacent link pair:

R01

57


• Compute the angular velocity of the end effector frame relative to the base frame expressed at the end effector frame.

• For i=0 TRR 01

10 �

58


59


• Compute the linear velocity of the end effector frame relative to the base frame expressed at the end effector frame.

• Note that the term involving the prismatic joint has been dropped from the equation (it is equal to zero).

60


• For i=0

• For i=1

61


62


2 3 14

2 3 3 3 2

1 2 2 3 23 3

0 00 ( )

0 0

L sL c L L J

L L c L c

��

� ��

�

� �

�

63


• Note that the linear and angular velocities ( ) of the end effector where differentiate (measured) in frame {0} however represented (expressed) in frame {4}

• In the car example: Observer sitting in the “Car”

Observer sitting in the “World”

64


• Multiply both sides of the equation by the inverse transformation matrix, we finally get the linear and angular velocities expressedand measured in the stationary frame {0}

65

Kinematics Relations - Joint & Cartesian Spaces

• A robot is often used to manipulate object attached to its tip (end effector).

• The location of the robot tip may be specified using one of thefollowing descriptions:

• Joint Space

• Cartesian Space

Euler Angles66

Kinematics Relations - Forward & Inverse

• The robot kinematic equations relate the two description of therobot tip location

Tip Location inJoint Space

Tip Location inCartesian Space

67

Kinematics Relations - Forward & Inverse

Tip Velocity inJoint Space

Tip velocity inCartesian Space

68

Jacobian Matrix - Introduction• The Jacobian is a multi dimensional form of the derivative.

• Suppose that for example we have 6 functions, each of which is a function of 6 independent variables

• We may also use a vector notation to write these equations as

69

Jacobian Matrix - Introduction• If we wish to calculate the differential of as a function of the

differential we use the chain rule to get

• Which again might be written more simply using a vector notation as

70

Jacobian Matrix - Introduction• The 6x6 matrix of partial derivative is defined as the Jacobian

matrix

• By dividing both sides by the differential time element, we can think of the Jacobian as mapping velocities in X to those in Y

• Note that the Jacobian is time varying linear transformation

71

Jacobian Matrix - Introduction• In the field of robotics the Jacobian matrix describe the relationship between the joint angle rates ( ) and the translation and rotationvelocities of the end effector( ).

This relationship is given by:

72

Jacobian Matrix - Introduction• This expression can be expanded to:

• Where:

- is a 6x1 vector of the end effector linear and angular velocities– is a 6xN Jacobian matrix– is a Nx1 vector of the manipulator joint velocities– N is the number of joints

73

How to obtain the Jacobian for a given robot?

• According to definition of the Jacobian is given in the Eqs. (5.58) through (5.62), a Jacobian can be obtained by directly differentiating the kinematic equations of the mechanism of thegiven robot. i.e.,

• The kinematic equations of a robot can be obtained via the geometric approach or algebraic approach (chapter three).

• However, while this is straightforward for linear velocity, there is no 3 × 1 orientation vector whose derivative is �.

74

How to obtain the Jacobian for a given robot?• Alternatively, we can use the velocity “propagation” method

to derive the Jacobian using successive application of the recursive equation

• For revolute joint:

• For prismatic joint:

75

Jacobian Matrix - Calculation Methods

76

Jacobian Matrix by Differentiation -3R - 1/3

321

321321212

321321211

)sin()sin(sin)cos()cos(cos

��

��

LLLyLLLx

• Consider the following 3 DOF Planar manipulator

(Geometric approach)

77


• The forward kinematics gives us relationship of the end effectorto the joint angles:

• Differentiating the three expressions gives

(Geometric approach)

78


• Using a matrix form we get

• The Jacobian provides a linear transformation, giving a velocity map and a force map for a robot manipulator.

• For the simple example above, the equations are trivial, but can easily become more complicated with robots that have additional

degrees a freedom. • Before tackling these problems, consider this brief review of linear

algebra.

79

Jacobian: Velocity propagation

X�

• The recursive expressions for the adjacent joint linear andangular velocities describe a relationship between the joint angle rates ( ) and the transnational and rotational velocities of the end effector ( ):

��

80

Jacobian: Velocity propagation• Therefore the recursive expressions for the adjacent joint linearand angular velocities can be used to determine the Jacobian in the end effector frame

• This equation can be expanded to:

81



82


• Compute the angular velocity of the end effector frame relative to the base frame expressed at the end effector frame.

• For i=0 TRR 01

10 �

83


2 3 14

2 3 3 3 2

1 2 2 3 23 3

0 00 ( )

0 0

L sL c L L J

L L c L c

��

� ��

�

� �

�

84

Jacobian: Velocities and singularities

• Mapping from velocities inthe joint space (joint) –tovelocities in the Cartesian space (end effector) –

• At certain points, calledsingularities, thismapping is not invertible.

85

Singularity - The Concept

• Motivation: We would like the hand of a robot (end effecror) tomove with a certain velocity vector in Cartesian space. Using linear transformation relating the joint velocity to the Cartesian velocity we could calculate the necessary joint rates at each instance along the path.

• Given: a linear transformation relating the joint velocity to the Cartesian velocity (usually the end effector)

• Question: Is the Jacobian matrix invertible? (Or) Is it nonsingular?Is the Jacobian invertible for all values of ?If not, where is it not invertible?

86

How to find the singularities?• The roots of the following equations determine the singularities:

• where Det[J (� )] is the determinant of the Jacobian under consideration.

• Certainly, a nonsquare Jacobian is not invertible. However, even if a Jacobian is in a square matrix format, the determinant of the Jacobian may still be zero.

87

What is the geometric or physical meanings of singularities?

• Cleary, when a robot is in a singular configuration, it haslost one or more degree of freedom as viewed from Cartesian space.

• This means that there is some direction (or subspace) in Cartesian space along which it is impossible to move the hand of the robot no matter which joint rates are selected.

• It is obvious this happens at the workspace boundary of robots.

88

Where do singularities occur?• All mechanisms have singularities at the boundary of their

workspace, and most have loci of singularities inside theirworkspace. Hence singularities can be classified into two categories:

1. Workspace boundary singularities are those which occur when the manipulator is fully stretched out or folded back on itself such that the end-effector is near or at the boundary of the workspace.

2. Workspace interior singularities are those whichoccur away from the workspace boundary andgenerally are caused by two or more joint axeslining up, or due to the mechanism structure sothat Det[J (� )] = f (� ) = 0 .

Viewed from Cartesian

space89

More on singularity• Since different structures of the robot yields different link

parameters, therefore, Det[J (� )] = f (� ) = 0 occur in thedifferent places.

• By modifying the link parameters, one can eliminate the singularities inside the workspace.

• Note: Det[J (� )] = f (� ) = 0 provides only one equations.

For f (� ) = 0 , - if there is only one variable inside the equation, the singularity

is only at individual points; - if there are two variables inside the equation, the singularities

occur alone a line;- if more than three variables, the singularities form a hyper-

surface.

90

Singularity -Summary

• Losing one or more DOF means that there is a some direction(or subspace) in Cartesian space along which it is impossible to move the hand of the robot (end effector) no matter which joint rate are selected

91

Singularity -Summary• Answer (Conceptual): Most manipulator have values of where

the Jacobian becomes singular . Such locations are called singularities of the mechanism or singularities for short

�

92

Properties of the Jacobian -Velocity Mapping and Singularities• Where are the singularities?• What is the physical explanation of the singularities?

• Are they workspace boundary or interior singularities?

• From example 5.3

93

Properties of the Jacobian -Velocity Mapping and Singularities

•They are workspace boundary singularities because they exist at the edge of the manipulator’ workspace.

• At both case, the motion of the end-effector is possible only along one Cartesian direction (the one perpendicular to the arm). Therefore,the mechanism has lost one degree of freedom.

• At singularity configuration, the inverse Jacobian blows up! This results in joint rates approaching infinity as the singularity isapproached. It is very dangers in a robot control system.

94


95


• Example: Planar 3R

96


• The manipulator loses 1 DEF. The end effector can only move along the tangent direction of the arm. Motion along the radial direction is not possible.

97



Workspace Interior Singularities?

98


2 3 14

2 3 3 3 2

1 2 2 3 23 3

0 00 ( )

0 0

L sL c L L J

L L c Lc

��

� ��

�

� �

�

0))(det(4 ��J0)())(det( 332233221`

4 �� sLLcLcLLJ �00 332233221̀ �� sLLcLcLL

0))(()()()( 4044

4044

0 �� JDetRDetVRDetVDet

Draw the locus of the singular points of the robot in its workspace (or in its base frame). (Interior singularities?)99


dtdPJV

viewsideLLZ

viewtopLLLY

viewtopLLLX

Tip

Tip

Tip

Tip

/)(

)sin(sin

sin))cos(cos(

cos))cos(cos(

04

0

323220

13232210

13232210

��

��

��

��

��

��

��

��

�

How to find ?)(04

0 �JorV

Geometric approach:

100

Jacobian: Velocities and Static Forces

• Mapping from velocities inthe joint space (joint) –tovelocities in the Cartesian space (end effector) –

• Mapping from thejoint force/torques –� to forces/torque inthe Cartesian spaceapplied on the endeffector - F

101

Static Analysis Protocol - Free Body Diagram 1/

Step 1Lock all the joints - Converting themanipulator (mechanism) to a structure

Step 2Consider each link in the structure as a free body and write the force / moment equilibrium equations

Step 3Solve the equations - 6 Eq. for each link.Apply backward solution starting fromthe last link (end effector) and end up atthe first link (base)

102


• Special Symbols are defined for the force and torque exerted by the neighbor link

- Force exerted on link i by link i-1- Torque exerted on link i by link i-1

Referencecoordinatesystem {B}

Force f ortorque n

Exerted on link A by link A-1

• For easy solution superscript index(B) should the same as the subscript (A)

103


• For serial manipulator in static equilibrium (joints locked), the sum the forces and torques acting on link i in the link frame {i}are equal to zero.

104


• Procedural Note: The solution starts at the end effectorand ends at the base

• Re-writing these equations in order such that the known forces (or torques) are on the right-hand side and the unknown forces (or torques) are on the left, we find

105


• Changing the reference frame such that each force (and torque) is expressed upon their link’s frame, we find the static force(and torque) propagation from link i+1 to link i

• These equations provide the static force (and torque) propagation from link to link. They allow us to start with the force and torque applied at the end effector, and calculate the force and torque at each joint all the way back to the robot base frame.

106


• Question: What torques are needed at the joints in order to balance the reaction forces and moments acting on the link.

• Answer: All the components of the force and moment vectors are resisted by the structure of mechanism itself, except for the torque about the the joint axis.

• Therefore, to find the joint the torque required to maintain the static equilibrium, the dot product of the joint axis vector with the moment vector acting on the link is computed

Prismatic Joint

Revolute Joint

107

Example - 2R Robot - Static Analysis

Problem

Given:

- 2R Robot-A Force vector is applied bythe end effector

Compute:The required joint torque as a function of the robot configuration and the applied force

108


Solution

• Lock the revolute joints

• Apply the static equilibrium equations starting from the end effector and going toward the base

109

Example - 2R Robot - Static Analysis• For i=2

Vector cross product

110

Example - 2R Robot - Static Analysis• For i=1

111


112


• Re-writing the equations in a matrix form

113

Jacobians in the force domain

• We have found joint torques that will exactly balance forces at the end-effector in the static situation.

• When force act on a mechanism, work (in the technical sense) is done if the mechanism moves through a displacement.

• Work is defined as a force acting through a distance and is a scalar with units of energy.

• The principle of virtual work allows us to make certainstatement about the static case by allowing the amount ofthis displacement to go to an infinitesimal.

Note : The units of energy are invariant after coordinate changing.

114

The principle of virtual work• In the multidimensional case, work is the dot product ofa vector force or torque and a vector displacement:

F is a 6×1 Cartesian force-torque vector acting at the end-effctor,�X is a 6×1 infinitesimal Cartesian displacement of the end-effector,

is a 6×1 vector of torques at the joints of the robot,�� is a 6×1 vector of infinitesimal joint displacements.

• Inner product can also be expressed as:• where

•it must hold for all �� , so we have

115

Jacobians: Velocities and static forces

• For velocity:

• For force-torque:

• If the Jacobian is with respect to frame {0}, we have

• When the Jacobian loses full rank, there are certaindirections in which the end-effector cannot exert staticforces as desired.

116

Jacobians in the force domain

• If the Jacobian is singular, F could be increased ordecreased in certain directions (those defining the null-space of the Jacobian) with no effect on the value calculated for .

• This also means that near singular configurations, mechanicaladvatage tends towards infinity such that with small joint torques large forces could be generated at the end-effector.

117

Properties of the Jacobian -Force Mapping and Singularities

• The relationship between joint torque and end effector force and moments is given by:

• The rank of is equals the rank of .

• At a singular configuration there exists a non trivial forcesuch that

• In other words, a finite force can be applied to the end effectorthat produces no torque at the robot’s joints. In the singular configuration, the manipulator can “lock up.”

118

Properties of the Jacobian -Force Mapping and Singularities

• This situation is an old and famous one in mechanical engineering.

• For example, in the steam locomotive, “top dead center”refers to the following condition

• The piston force, F, cannot generate any torque around the drivewheel axis because the linkage is singular in the position shown.

119

Cartesian transformation of velocities and static forces

120

Cartesian transformation of velocities and static forcesIf the frames {A} and {B} are rigidly connected, the matrix operator form to transform general velocity vectors in frame {A} to their description in frame {B} is given as follows:

• It can be shown by substituting into equations (5.45) and (5.47) and re-expressed the two equations in matrix form.

�

��

��

��

� ��

�

��

��

��

�

�

iii

i

ii

iii

iii

ii

ii v

RPRRv

�� 11

11

1

1

0

?

121


By defining a 6 ×6 operator: a velocity transformation,which maps velocities in {A} into velcoties in {B}, we have

• The inverse transformation which maps velocities in {B} to{A} is given in the similar matrix form as

• or

11

1 ��

�� iii

iii R ��

ii

ii

iii

i

ii

ii

iii

iii

PvR

PvRv

�

�

��

��

��

�

��

�

111

1

111

1•Note that

iii

iii R �� 1

11 �

��

)( 11

11

��

�� i

ii

ii

iiii

i PvRv �

122


•where is used to denote a force-momonet transformation.

• Similarly, with the static forces relations

• We have the

�

��

��

��

�

��

�

��

�

��

��

��

�

111

1

111

1 0

iii

i

ii

iii

i

ii

iii

i

nf

RRPR

nf

123


Velocity and force transformations are similar to Jacobiansin that they related velcotities and force in differentcoordinate systems. Similarly to Jacobians we have that

• as can be verified by examining by the following equations

124

Example

where

• Given:

• Find:

-the output of the forceSensor.-the transformationrelated the tool frame tothe sensor frame.

• Solution:

125

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MEV442_module_2