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Transcript of Ch.05 Forward Kinematics
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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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.02 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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.03 Forward Kinematics
I. DENAVIT-HARTENBERG NOTATION
• 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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.04 Forward Kinematics
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I. DENAVIT-HARTENBERG NOTATION
•
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.05 Forward Kinematics
I. DENAVIT-HARTENBERG NOTATION
• A DH coordinate frame is identified by four parameters: , , and
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.06 Forward Kinematics
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I. DENAVIT-HARTENBERG NOTATION
•
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.07 Forward Kinematics
I. DENAVIT-HARTENBERG NOTATION
• 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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.08 Forward Kinematics
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I. DENAVIT-HARTENBERG NOTATION
•
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
Robotics 5.09 Forward Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
I. DENAVIT-HARTENBERG NOTATION
- 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
Robotics 5.10 Forward Kinematics
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I. DENAVIT-HARTENBERG NOTATION
- 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
Robotics 5.11 Forward Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
I. DENAVIT-HARTENBERG NOTATION
• 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
Robotics 5.12 Forward Kinematics
FrameNo.
1 0 0
2 0 0
3 0 0
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I. DENAVIT-HARTENBERG NOTATION
•
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
Robotics 5.13 Forward Kinematics
FrameNo.
1 0 90 0
2 0
3 0 90 0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
I. DENAVIT-HARTENBERG NOTATION
• Example 137 (Stanford arm)
A schematic illustration of the Stanford arm is a spherical robot
⊢ ⊢ attached to a spherical wrist ⊢ ⊢
Robotics 5.14 Forward Kinematics
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.15 Forward Kinematics
II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.16 Forward Kinematics
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II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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 −
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.17 Forward Kinematics
II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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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Robotics 5.18 Forward Kinematics
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II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
FRAMES
• Example 141 ( DH transformation matrices for a 2R planar manipulator )
Figure 5.9 illustrates an ∥ planar manipulator and its DH link
coordinate frame
Robotics 5.19 Forward Kinematics
Frame
No.
1 0 0
2 0 0
II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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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Robotics 5.20 Forward Kinematics
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II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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
Robotics 5.21 Forward Kinematics
II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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
−
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.22 Forward Kinematics
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II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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
=
=
=
Robotics 5.23 Forward Kinematics
II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.24 Forward Kinematics
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II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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 ⊥ ⊢ ∥ ∥ )
Robotics 5.25 Forward Kinematics
FrameNo.
1 90 180
2 0 0
3 0 0
4 0 90 0
II. TRANSFORMATION BETWEEN TWO ADJACENT COORDINATE
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.26 Forward Kinematics
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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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.27 Forward Kinematics
III. FORWARD POSITION KINEMATICS OF ROBOTS
• 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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.28 Forward Kinematics
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III. FORWARD POSITION KINEMATICS OF ROBOTS
•
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
=
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Robotics 5.29 Forward Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
III. FORWARD POSITION KINEMATICS OF ROBOTS
• 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
Robotics 5.30 Forward Kinematics
Fram
e No.
1 0 0
2 0 0
3 0 0
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III. FORWARD POSITION KINEMATICS OF ROBOTS
•
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
= + + + + +
= + + + + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.31 Forward Kinematics
III. FORWARD POSITION KINEMATICS OF ROBOTS
• 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= + + + + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.32 Forward Kinematics
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III. FORWARD POSITION KINEMATICS OF ROBOTS
•
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
Robotics 5.33 Forward Kinematics
Frame
No.
1 0 90
2 0
3 0 90
Link No. Type
1 ⊢ (90)
2 ∥ 0
3 ⊢ 90
III. FORWARD POSITION KINEMATICS OF ROBOTS
• 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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Robotics 5.34 Forward Kinematics
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III. FORWARD POSITION KINEMATICS OF ROBOTS
•
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
Robotics 5.35 Forward Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
III. FORWARD POSITION KINEMATICS OF ROBOTS
• 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
Robotics 5.36 Forward Kinematics
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III. FORWARD POSITION KINEMATICS OF ROBOTS
=
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
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Robotics 5.37 Forward Kinematics
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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Robotics 5.38 Forward Kinematics
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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
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Robotics 5.39 Forward Kinematics
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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
Robotics 5.40 Forward Kinematics
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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
Robotics 5.41 Forward Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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
=
=
+
Robotics 5.42 Forward Kinematics
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V. ASSEMBLING KINEMATICS
•
Most modern industrial robots have a main manipulator and a series of interchangeable wrists
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Robotics 5.43 Forward Kinematics
An articulated manipulator
with three DOFA spherical wrist
V. ASSEMBLING KINEMATICS
• The articulated robot that is made by assembling the spherical wrist and
articulated manipulator is shown as follows
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V. ASSEMBLING KINEMATICS
•
Example 169 (Spherical robot forward kinematics )A spherical manipulator attached with a spherical wrist to make an
⊢ ⊢ robot
Robotics 5.45 Forward Kinematics
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
V. ASSEMBLING KINEMATICS
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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Robotics 5.46 Forward Kinematics
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V. ASSEMBLING KINEMATICS
= 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
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Robotics 5.47 Forward Kinematics
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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Robotics 5.48 Forward Kinematics
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VI. COORDINATE TRANSFORMATION USING SCREWS
•
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
Robotics 5.49 Forward Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
VI. COORDINATE TRANSFORMATION USING 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,
−
Robotics 5.50 Forward Kinematics
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
8/3/2019 Ch.05 Forward Kinematics
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Hi-Tech Mechatronics Lab 10/4/2011
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.
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.51 Forward Kinematics
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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Robotics 5.52 Forward Kinematics