1. Robotics Chapter 3 – Forward Kinematics
Dr. Amit Goradia

2. Topics Introduction – 2 hrs Coordinate transformations – 6 hrs
Forward Kinematics hrs
Inverse Kinematics hrs
Velocity Kinematics hrs
Trajectory Planning hrs
Robot Dynamics (Introduction) hrs
Force Control (Introduction) - 1 hrs
Task Planning hrs

3. Robot – Mechanical Structure

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

5. Forward Kinematics
Given the various joint and link parameters, compute the end-effector (tool) location in task space.
Given joint variables
End-effector position and orientation, -Formula? or
Joint
Frame
World
Forward
Inverse

6. Kinematic Chains A robot is a set of rigid links connected at joints
All joints have a single degree of freedom
Joints can be
Revolute – rotation about one axis
Variable: angle of rotation
Prismatic – linear motion along an axis
Variable: amount of linear displacement.
More complex joints which are a combination of one or more revolute or prismatic joints can be thought of as a succession of 1DOF revolute or prismatic joints with zero link length between them.

7. Kinematic Chains Rigid bodies Kinematic Pairs of Rigid Bodies
Robot Manipulators
Kinematic Chains : Link Joint
Rigid bodies Kinematic Pairs of Rigid Bodies
Basic Topologies of Kinematic Chain
Necklace
Open Chain
Tree

8. Kinematic Chain Representation
n+1 link manipulator
Base of robot is link 0
Joints are numbered 1 to n
joint variable denoted by
Attach a coordinate frame to each link
(total n+1 coordinate frame)
is the homogenous transformation from coordinate frame to

9. Kinematic Chain Representation
is a variable matrix with 1 variable
Transformation Matrix
Forward Kinematics: Represent a point on end effector w.r.t. base as a function of the joint variables
Attach a coordinate frame to each link and represent the end effector using the transformation between various coordinate frames

10. Link Parameters Link Length Link twist
Is the shortest distance measured along the common normal between joint axes
The common normal always exists and is unique except when both axes are parallel.
Link twist
The angle between the projection of axis of joint i and axis of joint i+1 on the plane perpendicular to the common normal between the two joints

11. 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 which is the distance between the links (the displacement, along the joint axes between the links).
The joint angle between the normals is measured in a plane normal to the joint axis.

12. Link and Joint Parameters

13. Link and Joint Parameters
ai = link length = the distance from zi to zi+1.measured along xi.
ai = link twist = the angle between zi and zi+1.measured about xi.
di = link offset = the distance from xi-1 to xi. measured along z i.
qi = joint angle = the angle between xi-1 to xi.measured about z i
Note: Since the joint has only 1 DOF, 3 out of the 4 parameters are constant and 1 represents the state of the link

14. Links and Joints 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

15. Denavitt-Hartenberg Convention
Systematic method to choose coordinate axes attached to links and represent the homogenous transformation between successive coordinate frames
Analysis is possible without D-H, but it helps to be systematic
Transformation is a composition of 4 basic transformations:

16. D-H Transformation
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.

17. Important Note
Homogenous transformation between coordinate systems have 6 parameters
How come Ai has only 4 parameters???
This is done by eliminating 2 or more parameters by choice of location of coordinate frame
Note that Ai is only comprised of a rotation and translation about only the X and Z axes.
We place the coordinate frames such that there is no movement about the Y axes.
This is codified by using DH conventions.

18. Existence and Uniqueness
There exists a unique transform between adjacent frames
Conditions to follow:
DH1: axis xn perpendicular to zn-1
DH2: axis xn intersects zn-1
Show that given we can, in all cases, uniquely derive the link parameters and vice versa

19. Existence and Uniquness
Satisfying DH1:
Column 1 of is an expression of in frame 0
Now show that there exist unique angles and such that
Row and column vectors of R are orthonormal
So and have a unique solution
Given and we can easily show that remaining elements of R have the said form

20. Existence and Uniqueness
Satisfying DH2: Displacement between o0 and o1 can be expressed as a linear combination of vectors and
Combining results from DH1 and DH2, we can show that the link parameters can be uniquely determined from Homogenous transformation matrix

21. The Arm Matrix
The arm matrix is the composite transformation from the base frame to tool frame
n, s, a, p are respectively called the Normal, Sliding, Approach and Position Vectors

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

23. Assigning Coordinate Frames
Assign Zi axes
Identify all revolute joints and assign Zi-1 axis as axis of rotation
Identify all prismatic joints and assign Zi-1 axis as axis of translation
Xi is chosen perpendicular to both Zi and Zi-1
If zi and zi-1 donot intersect, xi is the common normal of Zi and Zi-1. Its positive direction is from zi-1 to zi
If zi and zi-1 intersect, assignment of Xi can have 2 solutions
If zi-1 and zi are parallel, then there are infinitely many solutions
Yi is choosen to complete the right handed set

24. Assignment of X axis

25. DH Algorithm – Pass 1 Assign Coordinate Frames

26. DH Algorithm – Pass 2 Make Kinematic Parameter Table

27. Link Coordinate Diagram Example 1
4 DOF SCARA type robot

28. Kinematic Parameter Table Example 1

29. Kinematic Parameter Table Example 1
Location of sliding axis Z2 is arbitrary. For simplicity, make it coincident with Z3 . Thus a2 and d2 (d2 = 0) are arbitrarily set.
Placement of O3 and X3 along Z3 is arbitrary, since Z2 and Z3 are coincident. Once we choose O3, 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.

30. Link Coordinate Diagram Example 2
5 DOF Articulated

31. Kinematic Parameter Table Example 2

32. Example 3 – 3DOF Anthropomorphic Arm

33. Example 4 – Alpha II

34. Example 4 – Micro Alpha IIz0 x0 x1 z1 x2 z2 x3 z3 x4 z4 x5 z5
shoulder
elbow
Tool pitch
Tool roll
base
Tool 1 5 2 3 4

35. Example 4 – Micro Alpha II
Link
Var  d  a 1 1
215
-/2 2 2
177.8 3 3 4 4
4-/2 5 5
129.5

36. Example 4 – Micro Alpha II

37. Example 5 a0 a1 d2 Joint 3 Joint 1 Joint 2 Z0 X0 Y0 Z3 X2 Y1 X1 Y2 Z1

38. Example 5

39. Tool Configuration Vector
Another method to express position and orientation
Approach vector (a) represents the tool pitch and yaw angles.
Also, a is a unit vector.
So, append tool roll info into approach vector and we have another representation for position
Note: existence and uniqueness of inverse transformation needs to be established.

40. TCV
Scaling function for roll must be invertible
TCV