ch.05 forward kinematics

8/3/2019 Ch.05 Forward Kinematics

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Hi-Tech Mechatronics Lab 10/4/2011

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HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

05. Forward Kinematics

Robotics 5.01 Forward Kinematics

• If we have joint variables and geometrical characteristics of a robot, we

are able to determine the position and orientation of every link of robot

• We attach a coordinate frame to every link and determine its

configuration in the neighbor frames using rigid motion method

•

An analysis is called forward kinematics






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I. DENAVIT-HARTENBERG NOTATION

•

A series robot with joints will have + 1 links• Numbering of links starts from (0) for the immobile grounded base link

and increases sequentially up to () for the end-effector link

• Numbering of joints starts from 1, for the joint connecting the first

movable link to the base link, and increases sequentially up to

• The link () is connected to its lower link ( 1)

at its proximal end by joint and is connected to its

upper link ( + 1) at its distal end by joint + 1




• Figure 5.2 shows links ( 1), () and ( + 1) of a serial robot, along

with joint 1, and + 1

• Every point is indicated by its axis, which may be translational or

rotational

Attach a local coordinate frame

to each link () at joint + 1

based on the following standard

method, known as Denavit-

Hartenberg (DH) method






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•

According Denavit-Hartenberg method, local frame is defined as- The -axis is aligned with the + 1 joint axis

- The -axis is defined along the common normal between the − and

axes, pointing from the − to the -axis

- The -axis is determined by the right-hand rule, = ×

• Generally speaking, we assign reference frames to each link so that one

of the three coordinate axes , , or (usually ) is aligned along the

axis of the distal joint

• By applying the DH method, the origin of the frame , , , is

placed at the intersection of the joint axis + 1 with the common normal

between the − and axes




• A DH coordinate frame is identified by four parameters: , , and






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•

Link length is the distance between − and axes along the -axis, is the kinematic length of link

• Link twist is the required rotation of the −-axis about the -axis to

become parallel to the -axis

• Joint distance is the distance between − and axes along the −-

axis, joint distance is also called link offset

• Joint angle is the required rotation of −-axis about the −-axis to

become parallel to the -axis




• The parameters and are called joint parameters, since they define

the relative position of two adjacent links connected at joint

• For a revolute joint at joint , the value of is fixed, while is the

unique joint variable

•

For a prismatic joint , the value of is fixed and is the only jointvariable

• The joint parameters and define a screw motion because is a

rotation about the −-axis, and is a translation along the −-axis

• The parameters and are called link parameters, because they define

relative positions of joint and + 1 at two ends of link

• The link parameters and define a screw motion because is a

rotation about the -axis, and is a translation along the -axis






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•

Example 134 (Simplification comments for the DH method )There are some comments to simplify the application of the DH frame

method

- Showing only and axes is sufficient to identify a coordinate frame.

Drawing is made clearer by not showing axes

- If the first and last joint are , then

= 0 , = 0

= 0 , = 0

In these cases, the zero position for , and can be chosen arbitrarily,and link offsets can be set to zero

= 0 , = 0




- If the first and last joint are , then

= 0 , = 0

And the zero position for , and can be chosen arbitrarily, but

generally we choose them to make as many parameters as possible to 0

- If the final joint is , we choose to align with − when = 0 and the origin of is chosen such that = 0

If the final joint is , we choose such that = 0 and the origin of

is chosen at the intersection of − and joint axis that = 0

- Each link, except the base and the last, is a binary link and is connected

to two other links

- The parameters and are determined by the geometric design of the

robot and are always constant. The distance is the offset of the frame

with respect to − along the −-axis, ≥ 0





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- The angles and are directional. Positive direction is determined bythe right-hand rule according to the direction of and −

- For industrial robots, the link twist angle, , is usually a multiple of

2 radian

- The DH coordinate frames are not unique because the direction of -

axes are arbitrary

- The base frame , , = () is the global frame for an

immobile robot




• Example 135 ( DH table and coordinate frame for 3D planar

manipulator )

An ∥ ∥ manipulator is a planar robot with 3 parallel revolute joints

The link coordinate frames can be set up as shown in the figure

The DH table can be filled as follows


FrameNo.

1 0 0

2 0 0

3 0 0




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•

Example 136 (Coordinate frames for a 3R PUMA robot )A PUMA manipulator shown in figure has ⊢ ∥ main revolute

joints, ignoring the structure of the end-effector of the robot


FrameNo.

1 0 90 0

2 0

3 0 90 0



• Example 137 (Stanford arm)

A schematic illustration of the Stanford arm is a spherical robot

⊢ ⊢ attached to a spherical wrist ⊢ ⊢


Frame

No.

1 0 90

2 0 90

3 0 0 0

4 0 90 0

5 0 90 0 5

6 0 0




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II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE

FRAMES

• The coordinate frame is fixed to the link and the coordinate frame

− is fixed to the link 1

• The following set of two rotations and two translations is a straightforward

method to move the frame to coincide with the frame −

- Rotate frame through about the −-axis

- Translate frame along the −-axis by distance

- Rotate frame through about the −-axis

- Translate frame along the −-axis by distance




FRAMES

• Based on the Denavit-Hartenberg convention, the transformation matrix

− to transform coordinate frames to − is represented as a product

of four basic transformations using the parameters of link and joint

= ,

,

,

,

−

=

0

0

0

0 1

,=

1 0 0 00 0

0

0

0

0

0 1

,=

1 0 0

0 1 0 00

0

0

0

1 0

0 1






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FRAMES

,=

0 0 0 0

00

00

1 00 1

,=

1 0 0 00 1 0 000

00

1

0 1

• Therefore the transformation equation from coordinate frame , ,

to its previous coordinate frame − −, −, − is−

−−

1

=

1

−

• Matrix − may be partitioned into two submatrices, which represent a

unique rotation combined with a unique translation to produce the same

rigid motion require to move from to −




FRAMES

• The inverse of the homogeneous transformation matrix − , or the

transformation to move from − to is

− =

−−

=

0

0

0

0 1






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FRAMES

• Example 141 ( DH transformation matrices for a 2R planar manipulator )

Figure 5.9 illustrates an ∥ planar manipulator and its DH link

coordinate frame


Frame

No.

1 0 0

2 0 0


FRAMES

Based on the DH Table, we can find the transformation matrices from frame

to frame − by

=

0

0

00

00

1 00 1

=

0

0

00

00

1 00 1

Consequently, the transformation matrix from frame to is

=

=

+ + 0 + +

+ + 0 + +

0

0

0

0

1 0

0 1






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HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan


FRAMES

• Example 142 ( Link with ∥ or ∥ joints)

When the proximal joint of link is revolute and the distal joint is either

revolute or prismatic, and the joint axes at two ends are parallel then

- = 0 or = 180

- is distance between joint axes

- is only variable parameter

- is distance between origin of and −

along , usually = 0



FRAMES

The transformation matrix − for a link with = 0 and ∥ or ∥

joints, known as ∥ 0 or ∥ 0 is

=

0

0

0

0

0

0

1

0 1

−

The transformation matrix − for a link with = 180 and ∥ or

∥ joints, known as ∥ 180 or ∥ 180 is

=

0

0

0

0

0

0

1

0 1

−






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FRAMES

• Example 150 ( DH coordinate transformation based on vector addition)

The DH transformation from a coordinate frame to the other can also be

described by a vector addition. The coordinates of a point in frame are

given by vector equation

= +

Where

=

=

=



FRAMES

However, they must be expressed in the same coordinate frame

= , + , + , +

= , + , + , +

= , + , + , + 1 = 0 + 0 + 0 + 1

The transformation can be rearranged to be described with the homogeneous

matrix transformation

1

=

, , ,

, , ,

,

0

,

0

,

0 1

1

This matrix is correspond to matrix − by some assumption






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FRAMES

• Example 152 ( DH application for a slider-crank planar linkage)

For a closed loop robot or mechanism there would also be a connection

between the first and last links, so the DH convention will not be satisfied

by this connection (planar slider-crank linkage ⊥ ⊢ ∥ ∥ )


FrameNo.

1 90 180

2 0 0

3 0 0

4 0 90 0


FRAMES

Applying a loop transformation leads to

= =

Where the transformation matrix contains elements that are functions of

, , , , , , and . The parameters , and are constant while, , , and are variable.

Assuming is input and specified, we may solve for other unknown

variables , and by equating corresponding elements of and






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III. FORWARD POSITION KINEMATICS OF ROBOTS

•

The forward kinematics (direct kinematics) is the transformation of kinematic information from the robot joint variable space to the Cartesian

coordinate space

• Finding the end-effector position and orientation for a given set of joint

variables is the main problem in forward kinematics

• This problem can be solved by determining transformation matrices

to describe the kinematic information of link in the base link frame

• The traditional way of producing forward kinematic equations for robotic

manipulators is to proceed link by link using Denavit-Hartenberg notation

• Hence, the forward kinematics is basically transformation matrixmanipulation




• For a six DOF robot, six DH transformation matrices, one for each link,

are required to transform the final coordinates to the base coordinates

• The position and orientation of the end-effector is also a unique function

of the joint variables






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•

The kinematic information includes: position, velocity, acceleration and jerk. However, forward kinematics generally refers to position analysis

• The forward position kinematics is equivalent to a determination of a

combined transformation matrix

=

⋯

−

• To find the coordinates of a point in the base coordinate frame, when its

coordinates are given in the final frame, we do as follows

=





• Example 154 (3R planar manipulator forward kinematics)

The robot is an ∥ ∥ planar manipulator

Using the DH parameters, we can find the transformation matrices −

for = 3,2,1


Fram

e No.

1 0 0

2 0 0

3 0 0




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•

We have

=

0

0

0

0

0

0

1 0

0 1

=

0

0

0

0

0

0

1 0

0 1

=

0

0

0

0

0

0

1 0

0 1

• Transformation matrix to relate end-effector frame to base frame is

= =

+ + + + 0

+ + + + 0

00

00

1 00 1

= + + + + +

= + + + + +




• The position of the origin of the frame , which is the tip point of the

robot, is at

000

1

=

+ + + + +

+ + + + +

0

1

• We can find the coordinate of the tip point in the base Cartesian

coordinate frame if we have the geometry of the robot and all joint

variables

X= + + + + +

Y= + + + + +






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•

Example 155 ( ⊢ ∥ articulated arm forward kinematics)An ⊢ ∥ arm has the DH parameter table and link classification for

set-up of the link frames as follows


Frame

No.

1 0 90

2 0

3 0 90

Link No. Type

1 ⊢ (90)

2 ∥ 0

3 ⊢ 90


• The complete transformation matrix has the following expression

=

=

+ +

+ + +

+ 0 00 +

0 1

• The tip point of the third arm is at 0 0 in . So, its position in

the base frame would be at

=

=

+ + +

+ + +

+ +

1






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•

Example 157 (Working space)Consider an arm ⊢ ∥

Assume that every point joint can turn 360. Theoretically, point must

be able to reach any point in the sphere

= +

+ 0.174 + 0.48 = 1.96

Point must be out of the sphere

=

+ 0.174 + 0.48 = 0.01

The reachable space between and is

called working space of the manipulator




• Example 158 (SCARA robot forward kinematics)

Consider the ∥ ∥ ∥ robot, we have the following transformation

matrices

=

0

0 00

00

1 00 1

=

0

0

00

00

1 00 1





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=

0 0 0 0

0

0

0

0

1 0

0 1

=1 0 0 00 1 0 000

00

1 0 1

Therefore, the configuration of the end-effector in the base coordinate

frame is

=

=

+ + + + 0 + +

+ + + + 0 + +

00 00 1 0 1



IV. SPHERICAL WRIST

• The spherical joint connects two links: the forearm and hand

• Axis of forearm and hand are assumed to be colinear at the rest position

• A spherical wrist is a combination of links and joints to simulate a

spherical joint and provide three rotational DOF for the gripper link

• It is made by three ⊢ links with zero length and zero offset






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IV. SPHERICAL WRIST

•

A Roll-Pitch-Yaw spherical wrist has following transformation matrix

=

5 5 5 0

+ 5 5 5 0

5

0

5

0

5 0

0 1




IV. SPHERICAL WRIST

• Example 160 ( DH frames of a roll-pitch-roll spherical wrist )

We consider a roll-pitch-roll spherical wrist in rest position and motion

position





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IV. SPHERICAL WRIST

•

Example 161 ( Roll-pitch-roll or Eulerian wrist )A roll-pitch-roll wrist has: indicates its dead and indicates its living

coordinate frames

The transformation matrix , is a rotation about the dead axis

followed by a rotation about the -axis

= = ,,

= ,,

=

0



IV. SPHERICAL WRIST

The transformation matrix is a rotation about the local axis

= = ,

= , =

0

0

0 0 1

Therefore, the transformation matrix between the living and dead wrist

frames is

=

=

+





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V. ASSEMBLING KINEMATICS

•

Most modern industrial robots have a main manipulator and a series of interchangeable wrists



An articulated manipulator

with three DOFA spherical wrist


• The articulated robot that is made by assembling the spherical wrist and

articulated manipulator is shown as follows






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•

Example 169 (Spherical robot forward kinematics )A spherical manipulator attached with a spherical wrist to make an

⊢ ⊢ robot


FrameNo.

1 0 90 0

2 0 90

3 0 0 0

4 0 90 0

5 0 90 0 5

6 0 0 0


The configuration of the wrist final coordinate frame in the global

coordinate frame is

= 5 =

0

0

0 1

5

= + 5 + 5 +

= + 5 + 5 +

= + 5 5

= 5 + 5 +

= 5 + 5 +

= 5 5






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= 5 + 5 + = 5 + 5 +

= 5 5

= +

= +

=

The end-effector kinematics can be solved by multiplying the position of

the tool frame with respect to the wrist point, by

=

Where =

1 0 0 00 1 0 000

00

1

0 1



VI. COORDINATE TRANSFORMATION USING SCREWS

• It is possible to use screws to describe a transformation matrix between

two adjacent coordinate frames and −

• We can move to − by a central screw , , − followed by

another central screw , , −

= , , − , , −−

=

cos sin cos sin sin cos

sin cos cos cos sin sin

0

0

sin

0

cos

0 1






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•

Example 172 (Spherical robot forward kinematics based on screws )Application of screws in forward kinematics can be done by determining

the class of each link and applying the associated screws




The class of links for the spherical robot are

Therefore, the configuration of end-effector frame in based frame is

= 0, , − 0, , − 0, , − 0, , −

,0 ,− ,0, − 0, , − 0, , −

0, ,

− 0, ,

− 0,

,

−

,0,

−


Link No. Class Screw transformation

1 ⊢ 90 = 0, , − 0, , −

2 ⊢ 90 = 0, , − 0, , −

3 ∥ 0 = , 0 , − ,0 , −

4 ⊢ 90 = 0, , − 0,, −

5 ⊢ 90 5 = 0, , − 0, , −

6 ∥ 0 = 0, , − ,0 , −5




VII. NON DENAVIT-HARTENBERG METHODS

•

The Denavit-Hartenberg (DH) method is the most common method used• However, the DH method is not the only method used, nor necessarily the

best. There are other methods with advantages and disadvantages when

compared to the DH method.

• In the Sheth method, we define a coordinate frame at each joint of a link,

so an joint robot would have 2 frames.



VII. NON DENAVIT-HARTENBERG METHODS

• This figure shows the case of a binary link where a first frame

, , is attached at the origin of the link and a second frame

, , to the end of the link



ch.05 forward kinematics

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