1. Time to Derive Kinematics Model of the Robotic Arm
Amirkabir University of Technology Computer Engineering & Information Technology Department

2. Direct Kinematics
Where is my hand?
Direct Kinematics:
HERE!

3. 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.

4. Degrees of Freedom
The degrees of freedom of a rigid body is defined as the number of independent movements it has.
The number of :
Independent position variables needed to locate all parts of the mechanism,
Different ways in which a robot arm can move,
Joints
Quiz: Humans have 4 DOF for each fingers. The human finger has 3 joints,

5. DOF of a Rigid Body
In a plane
In space

6. Degrees of Freedom 3 position 3D Space = 6 DOF 3 orientation
In robotics:
DOF = number of independently driven joints
positioning accuracy
As DOF
computational complexity
cost
flexibility
power transmission is more difficult

7. Robot Links and Joints {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.
In open kinematics chains (i.e. Industrial Manipulators):
{No of D.O.F. = No of Joints}

8. Lower Pair
The connection between a pair of bodies when the relative motion is characterized by two surfaces sliding over one another

9. The Six Possible Lower Pair Joints

10. Higher Pair
A higher pair joint is one which contact occurs only at isolated points or along a line segments

11. Robot Joints Revolute Joint 1 DOF ( Variable - ) Spherical Joint
Due to mechanical design considerations manipulators are generally constructed from joints which exhibit just one degree of freedom.
Most manipulators have revolute joints or have sliding joints.
In the rare case that a mechanism is built with a joint having n degrees of freedom it can be modeled as n joints of one degree of freedom connected with n-1 links of zero length.
Spherical Joint
3 DOF ( Variables - 1, 2, 3)
Prismatic Joint
1 DOF (linear) (Variables - d)

12. 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

13. The PUMA 560 has SIX revolute joints.
An Example - The PUMA 560 2 3 1 4
The PUMA 560 has SIX revolute joints.
A revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle.
There are two more joints on the end-effector (the gripper)

14. 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.

15. Link
Link n
q n+1
a n
q n
Joint n+1
Joint n
z n
x n
x n+1
z n+1
A link is considered as a rigid body which defines the relationship between two neighboring joint axes of a manipulator.

16. 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 a
In order to position an end-effecter generally in 3-space minimum of six joints is required.

17. 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.

18. 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

19. Link Length and Twist Axis i Axis i-1 ai-1 i-1
The distance between two axes in 3-space is measured along a line which is mutually perpendicular to both axes.
Link twist: If we imagine a plane whose normal is the mutually perpendicular line just constructed, we can project both axes I-1, I onto this plane and measure the angle between them. The angle is measured from i-1 to I by Right-Hand-Rule about the common normal.
Note: If the axes intersect, then a is zero, and  is still measured from axis i-1 to axis i.

20. 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 qn between the normals is measured in a plane normal to the joint axis.

21. Link and Joint Parameters
Axis i-1
Axis i
ai-1 di i
For kinematical studies we only need two more quantities to completely define the relative position of two neighboring links.
The Distance between the two common normals “di” at joint-i. This distance is called the “Link-Offset” (.
Angle of rotation about their common axis-i, between one link and its neighbor, “i”. This angleis called the “Joint-Angle” (“i” is the angle between ai-1 and ai about axis-i).
ai-1
i-1

22. Link and Joint Parameters
4 parameters are associated with each link. You can align the two axis using these parameters.
Link parameters:
a0 the length of the link.
an the twist angle between the joint axes.
Joint parameters:
qn the angle between the links.
dn the distance between the links
4 parameters are associated with each link: These parameters come in pairs: The link parameters which determines the structure of the link and the joint parameters which determine the relative position of the neighboring links.

23. 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.
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.
For a six joint robot 18 numbers would be required to completely describe the fixed portion of kinematics.

24. Links Numbering Convention
Base of the arm: Link-0
1st moving link: Link-1
Last moving link Link-n
A 3-DOF Manipulator Arm 1 2 3
Link 2
Link 3
Link 1
Link 0

25. First and Last Links in the Chain
a0= an=0.0
If joint 1 is revolute: d0= 0 and q1 is arbitrary
If joint 1 is prismatic: d0= arbitrary and q1 = 0

26. Affixing Frames to Links
In 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.
The joint axes is the axes of motion
In the case of ai=0 Xi is normal to plane of Zi and Zi+1
This convention does not result in a unique attachment of frames to links…

27. Affixing Frames to Links
First 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 qn = 0
Origin {n} such that dn=0
Prismatic joint n:
Xn such that qn = 0
Origin {n} at intersection of joint axis n and Xn when dn=0
End-Frames:
The base frame, Fo, can always be located on joint axis zo at the intersection point with the common perpendicular to axis z1. Therefore, parameter d1 can always be chosen as zero.
The end-effector frame, Fn (for an n-DOF robot), is the only frame that does not have to be located on a joint axis. It is attached to the end-effector and can always be chosen such that parameters dn, an, and an are zero if joint n is revolute or parameters qn, an, and an are zero if joint n is translational

28. Affixing Frames to Links
Link n-1
Link n
zn-1
yn-1
xn-1 zn xn yn
zn+1
xn+1
yn+1 dn an
Joint n+1
Joint n-1
Joint n
an-1
In the case of ai=0 Xi is normal to plane of Zi and Zi+1
This convention does not result in a unique attachment of frames to links…

29. Affixing Frames to Links
Note: 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.

30. The Kinematics Model
The robot can now be kinematically modeled by using the link transforms ie:
Where
0nT 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.

31. 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.

32. 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.

33. Denavit-Hartenberg Parameters
Axis i
Axis i-1
ai-1
i-1 i
Link i di

34. The Link Parameters ai = the distance from zi to zi+1.
measured along xi.
ai = the angle between zi and zi+1.
measured about xi.
di = the distance from xi-1 to xi.
measured along z i.
qi = the angle between xi-1 to xi.
measured about z i

35. 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 q,
A translation along the new x axis by a, and.
A rotation about the new x axis by a.

36. A General Transformation in D-h Convention
D-H transformation for adjacent coordinate frames:

37. 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-1
D3. 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.

38. Denavit-Hartenberg Convention
D7. Establish the hand coordinate system
D8. Find the link and joint parameters : d,a,a,q
D-H transformation for adjacent coordinate frames:

39. Example a0 a1 d2 Joint 3 Joint 1 Joint 2 Z0 X0 Y0 Z3 X2 Y1 X1 Y2 Z1 X3

40. Example

41. Example

42. Link Frame Assignments
Example (3.3):
Link Frame Assignments
Reference frame 0 aligns with frame 1 when first joint variable is zero.
All Z are parallel.
There is no link offset.

43. Example (3.3):

44. Example (3.3):

45. Example:SCARA Robot

46. 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 a2 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.

47. Example: Puma 560

48. Example: Puma 560

49. Forearm of a PUMA a3 x5 y5 x6 z6 x3 y3 x4 z4 d4
Spherical joint

50. Example: Puma 560
Different Configuration

51. Link Coordinate Parameters
PUMA 560 robot arm link coordinate parameters

52. Example: Puma 560

53. Example: Puma 560

54. 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.

55. 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.

56. Exercise
To be posted on CEAUT site:
exercise1.pdf
Due: 83/7/27

57. Next Course
Inverse Kinematics