CHAPTER 2 FORWARD KINEMATIC 1

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.

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

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

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

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

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

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

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:

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

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=[10 10 10] with respect to the global axis after it is transformed by [pi/4; pi/3; pi/6]

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:

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:

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

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’

Transformation Matrix (3D) Solution

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}

Transformation Matrix (3D) Solution:

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}

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}

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)