1. CHAPTER 2
FORWARD KINEMATIC 1

2. What is FORWARD Kinematic?
The study of the position and orientation of a robot hand with respect to a reference coordinate system, given the joint variables and the arm parameters, OR
The analytical study of the geometry of motion of a robot arm with respect to a reference coordinate system.
Without regard the forces of moments that cause the robot motion.
It is the first step towards robotic control.

3. What is FORWARD Kinematic?
What you are given:
The length of each link
The angle of each joint
What you can find:
The position of any point (i.e. it’s (x, y, z) coordinates

4. Kinematic Relationship
Between two frames, the is a kinematic relationship either a translation, rotation or both. The relationship can be describe by a transformation matrix. z0 x0 y0
Translation and rotation z1 x1 y1 z2 x2 y2 z3 x3 y3
Rotation
Translation
Rereference frame
{D}
{C}
{A}
{B}
Note: {D} = Frame D

5. Rotation Matrix (2D) describes the rotations of {B} w.r.t. {A} Note: yv u
Puv x y
Pxy
{A}
{B}
Note:
describes the rotations of {B} w.r.t. {A}

6. Rotation Matrix (3D) z x y w u v
Puvw
Pxyz

7. Rotation Matrix (3D) Rotation transformation matrices
Rotation about x-axis by degrees - Yaw
Roll
Yaw
Pitch z x y
Rotation about y-axis by degrees - Pitch
Rotation about z-axis by degrees - Roll

8. Rotation Matrix (3D) Ruvw is mobile with respect to the Rxyz
Roll-pitch-yaw angles (Z-Y-X Euler angle-Relative axis)
It provides a method to decompose a complex rotation into three consecutive fundamental rotations; roll, pitch, and yaw.
Use post multiplication rule.
Ruvw is mobile with respect to the Rxyz

9. Rotation Matrix (3D) Yaw-pitch-roll angles (X-Y-Z fixed angle)
Representation in yaw-pitch-roll angles allows complex rotation to be decomposed into a sequence of yaw, pitch and roll about the x, y and z axis.
Use pre-multiplication rule.
Conclusion:

10. Rotation Matrix (3D) Z-Y-Z Euler angle
Read the Z-Y-Z Euler angles on page 30 (M. Zhihong)

11. Exercises
Find the position of point P=[10 10] with respect to the global axis after it is transformed/rotated by [pi/3]
Find the position of point P=[ ] with respect to the global axis after it is transformed by [pi/4; pi/3; pi/6]

12. Transformation Matrix (3D)
Homogeneous transformations
Transforms and translates.
The homogenous transformation matrix below is used to transform and translate. R is a 3x3 rotation matrix and P is a 3x1translation/position vector.
Three fundamental rotation matrices of roll, pitch and yaw in the homogeneous coordinate system:

13. Transformation Matrix (3D)
Homogeneous transformations
Three fundamental rotation matrices of roll, pitch and yaw Hrpy in the homogeneous coordinate system: R
A point B’ can be found from the following relationship:

14. Transformation Matrix (3D)
Homogeneous transformations P Y X Z O N A
Translation without rotation A Y X Z O N
Rotation without translation

15. Transformation Matrix (3D)
Example 1: Find a point B’ in {B} w.r.t to the reference frame {A} if the origin of {B} is (5,5,5) . Given B=(1,2,3). Given
(5,5,5)
B(1,2,3) B’

16. Transformation Matrix (3D)
Solution

17. Transformation Matrix (3D)
Example 2: Find a point P’ in {N} w.r.t to the reference frame {M} if the origin of {N} is (3,5,4) . Given B=(3,2,1). {N} is rotated by
(3,5,4)
B(1,2,3) B’
{N}
{M}

18. Transformation Matrix (3D)
Solution:

19. Transformation Matrix (3D)
Example 3: Find a point P’ in {N} w.r.t to the reference frame {M} if the origin of {N} is (3,5,4) . Given B=(3,2,1). {N} is rotated by
(3,5,4)
B (1,2,3) B’
{N}
{M}

20. Quiz 1
Find a point P’ in {N} w.r.t to the reference frame {M} if the origin of {N} is (3,5,4) . Given B=(3,2,1). {N} is rotated by
(3,5,4)
B (1,2,3) B’
{N}
{M}

21. Quiz 1
A point p(7,3,1) is attached to {A} and is subjected to the following transformation. Find the coordinates of the point relative to the reference frame at the conclusion of the transformations.
Rotation of 90˚ about the z-axis.
Followed by a rotation of 90 ˚ about the y-axis.
Followed by a translation of [4,-3,7].
(Ex2.8, ItR)