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Kinematics
Time to Derive Kinematics Model of the Robotic Arm
Amirkabir University of TechnologyComputer Engineering & Information Technology Department
Direct Kinematics
Where is my hand?
Direct Kinematics:HERE!
Kinematics of Manipulators
Objective:
To drive a method to compute the position and orientation of the manipulator’s end-effector relative to the base of the manipulator as a function of the joint variables.
Degrees of Freedom
The number of :
• Independent position variables needed to locate all parts of the mechanism,
• Different ways in which a robot arm can move,
• Joints
The The degrees of freedomdegrees of freedom of a rigid body is defined as of a rigid body is defined as the number of independent movements it has.the number of independent movements it has.
DOF of a Rigid Body
In a planeIn a plane
In spaceIn space
Degrees of Freedom
As DOF
3 position3 orientation
3D Space = 6 DOF
In robotics:DOF = number of independently driven joints
computational complexitycostflexibilitypower transmission is more difficult
positioning accuracy
Robot Links and Joints
In open kinematics chains (i.e. Industrial Manipulators):
{No of D.O.F. = No of Joints}
A manipulator may be thought of as a set of bodies (links) connected in a chain by joints.
Lower Pair
The connection between a pair of bodies when the relative motion is characterized by two surfaces sliding over one another
The Six Possible Lower Pair Joints
Higher PairA higher pair joint is one which contact occurs only at isolated points or along a line segments
Robot Joints
Spherical Joint3 DOF ( Variables - 1, 2, 3)
Revolute Joint1 DOF ( Variable - )
Prismatic Joint1 DOF (linear) (Variables - d)
Robot Specifications
Number of axes
Major axes, (1-3) => position the wrist
Minor axes, (4-6) => orient the tool
Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration
An Example - The PUMA 560
The PUMA 560 has SIX revolute joints.A revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle.
1
23
4
There are two more joints on the end-effector (the gripper)
Note on Joints
Without loss of generality, we will consider only manipulators which have joints with a single degree of freedom. A joint having n degrees of freedom can be modeled as n joints of one degree of freedom connected with n-1 links of zero length.
Link
A link is considered as a rigid body which defines the relationship between two neighboring joint axes of a manipulator.
Link n
n+1a n
n
Joint n+1
Joint n
z n
x n
x n+1
z n+1
x n
z n
The Kinematics Function of a Link
The kinematics function of a link is to maintain a fixed relationship between the two joint axes it supports.This relationship can be described with two parameters: the link length a, the link twist
Link Length
Is measured along a line which is mutually perpendicular to both axes.
The mutually perpendicular always exists and is unique except when both axes are parallel.
Link twist
Project both axes i-1 and i onto the plane whose normal is the mutually perpendicular line, and measure the angle between them
Right-hand sense
Link Length and Twist Axis i
Axis i-1
a i-1
i-1
Joint Parameters
A joint axis is established at the connection of two links. This joint will have two normals connected to it one for each of the links.
The relative position of two links is called link offset dn whish is the distance between the links (the displacement, along the joint axes between the links).
The joint angle n between the normals is measured in a plane normal to the joint axis.
Link and Joint Parameters
a i-1
Axis i-1
Axis i
dii
i-1
ai-1
Link and Joint Parameters4 parameters are associated with each link. You
can align the two axis using these parameters.
Link parameters:
a0 the length of the link.
n the twist angle between the joint axes.
Joint parameters:
n the angle between the links.
dn the distance between the links
Link Connection Description:
For Revolute Joints: a, , and d. are all fixed, then “i” is the. Joint Variable.
For Prismatic Joints: a, , and . are all fixed, then “di” is the. Joint Variable.
These four parameters: (Link-Length ai-1), (Link-Twist i-1(, (Link-Offset di), (Joint-Angle i) are known as the Denavit-Hartenberg Link Parameters.
A 3-DOF Manipulator Arm
0
1
23
Links Numbering Convention
Base of the arm:Link-01st moving link:Link-1
. .
. .
. .
Last moving linkLink-n
Link 0
Link 1
Link 2Link 3
First and Last Links in the Chain
a0= n=0.0
0= n=0.0
If joint 1 is revolute: d0= and 1 is arbitrary
If joint 1 is prismatic: d0= arbitraryand 1 =
Affixing Frames to LinksIn order to describe the location of each link
relative to its neighbors we define a frame attached to each link.
The Z axis is coincident with the joint axis i.
The origin of frame is located where ai perpendicular intersects the joint i axis.
The X axis points along ai( from i to i+1).If ai = 0 (i.E. The axes intersect) then Xi is perpendicular to axes i and i+1.
The Y axis is formed by right hand rule.
Affixing Frames to LinksFirst and last links
Base frame (0) is arbitrary Make life easy Coincides with frame {1} when joint parameter is 0
Frame {n} (last link) Revolute joint n:
Xn = Xn-1 when n = 0 Origin {n} such that dn=0
Prismatic joint n: Xn such that n = 0 Origin {n} at intersection of joint axis n and Xn when dn=0
Link n-1
Link n
zn-1 yn-1
xn-1
zn
xn
yn
zn+1
xn+1
yn+1
dnn
Joint n+1
an
Joint n-1Joint n
an-1
Affixing Frames to Links
Affixing Frames to LinksNote: assign link frames so as to cause as many
link parameters as possible to become zero!
The reference vector z of a link-frame is always on a joint axis.
The parameter di is algebraic and may be negative. It is constant if joint i is revolute and variable when joint i is prismatic.
The parameter ai is always constant and positive.
a i is always chosen positive with the smallest possible magnitude.
The robot can now be kinematically modeled by using the link transforms ie:
Where 0
nT is the pose of the end-effector relative to base; Ti is the link transform for the ith joint; and n is the number of links.
The Kinematics Model
nin TTTTTT 3210
The Denavit-Hartenberg (D-H) Representation
In the robotics literature, the Denavit-Hartenberg (D-H) representation has been used, almost universally, to derive the kinematic description of robotic manipulators.
The Denavit-Hartenberg (D-H) Representation
The appeal of the D-H representation lies in its algorithmic approach. The method begins with a systematic approach to assigning and labeling an orthonormal (x,y,z) coordinate system to each robot joint. It is then possible to relate one joint to the next and ultimately to assemble a complete representation of a robot's geometry.
Denavit-Hartenberg Parameters
Axis i-1
a i-1
i-1
Axis i
Link i
di
i
The Link Parametersai = the distance from zi to zi+1.
measured along xi.
i = the angle between zi and zi+1.
measured about xi.
di = the distance from xi-1 to xi.
measured along z i.
i = the angle between xi-1 to xi.
measured about z i
General Transformation Between Two Bodies
In D-H convention, a general transformation between two bodies is defined as the product of four basic transformations:
A translation along the initial z axis by d,
A rotation about the initial z axis by ,
A translation along the new x axis by a, and.
A rotation about the new x axis by .
A General Transformation in D-h Convention
D-H transformation for adjacent coordinate frames:
44,,,,
1
ITTTT xaxdzz
i
iT
Denavit-Hartenberg Convention
D1. Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1.D2. Initialize and loop Steps D3 to D6 for I=1,2,….n-1D3. Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1.
D4. Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis.
D5. Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel.
D6. Establish Yi axis. Assign to complete the right-handed coordinate system.
),,( 000 ZYX
iiiii ZZZZX 11 /)(
iiiii XZXZY /)(
0Z
Denavit-Hartenberg Convention
D7. Establish the hand coordinate systemD8. Find the link and joint parameters : d,a,,
D-H transformation for adjacent coordinate frames:
44,,,,
1
ITTTT xaxdzz
i
iT
Example
Joint i i ai di
i
1 0 a0 0 0
2 -90 a1 0 1
3 0 0 d2 2
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X33O
2O1O0O
Z2
Joint 1
Joint 2
Joint 3
Example
))()(( 2103213
0 TTTT
1000
01iii
iiiiiii
iiiiiii
ii dCS
SaCSCCS
CaSSSCC
T
Example
Joint i i ai di
i
1 0 a0 0 0
2 -90 a1 0 1
3 0 0 d2 2
1000
0100
0cosθsinθ
0sinθcosθ
00
00
000
00
1 sin
cos
a
a
T
1000
000
sinθ
cosθ
1
1
1 1
sincos0
cossin0
111
111
2
a
a
T
1000
0sinθcosθ 22
22
223
100
00cossin
0
dT
Example (3.3):
Link Frame Assignments
Example (3.3):
1000
0100
00
0
1000
0100
00
0
1000
0100
00
00
1000
0
33
233
23
22
122
12
11
11
1000101
1000101
011
01
cs
Lsc
Tcs
Lsc
T
cs
sc
dccscss
dsscccs
asc
T
Example (3.3):
.
1000
0.00.10.00.0
0.0
0.0
12211123123
12211123123
03
slslcs
clclsc
TTBW
Example:SCARA Robot
The location of the sliding axis Z2 is arbitrary, since it is a free vector. For simplicity, we make it coincident with Z3 . thus 2 and d2 are arbitrarily set.
The placement of O3 and X3 along Z3 is arbitrary, since Z2 and Z3 are coincident. Once we choose O3, however, then the joint displacement d3 is defined.
We have also placed the end link frame in a convenient manner, with the Z4 axis coincident with the Z3 axis and the origin O4 displaced down into the gripper by d4.
Example:SCARA Robot
Example: Puma 560
Joint i i
i ai(mm) di(mm)
1 1 0 0 0
2 2 -90 0 d2
3 3 0 a2 d3
4 4 90 a3 d4
5 5 -90 0 0
6 6 0 0 0
Example: Puma 560
Forearm of a PUMA
a3
x5
y5
x6
z6
x3
y3
x4
z4
d4
Spherical joint
Example: Puma 560Different Configuration
Link Coordinate Parameters
Joint i i i ai(mm) di(mm)
1 1 -90 0 0
2 2 0 431.8 149.09
3 3 90 -20.32 0
4 4 -90 0 433.07
5 5 90 0 0
6 6 0 0 56.25
PUMA 560 robot arm link coordinate parameters
Example: Puma 560
Example: Puma 560
The Tool Transform
A robot will be frequently picking up objects or tools.
Standard practice is to to add an extra homogeneous transformation that relates the frame of the object or tool to a fixed frame in the end-effector.
Kinematic Calibration
How one knows the DH parameters? Certainly when robots are built, there are design specifications. Yet due to manufacturing tolerances, these nominal parameters will not be exact. The process of kinematic calibration determines these nominal parameters experimentally. Kinematic calibration is typically accomplished with an external metrology system, although alternatives that do not require a metrology system exist.
Exercise
To be posted on CEAUT site:exercise1.pdf
Due: 83/7/27
Next Course
Inverse Kinematics