I NTRODUCTION TO R OBOTICS CPSC Lecture 3A – Forward Kinematics

DH T ECHNIQUES A link j can be specified by two parameters, its length aj and its twist α j Joints are also described by two parameters. The link offset dj is the distance from one link coordinate frame to the next along the axis of the joint. The joint angle θ j is the rotation of one link with respect to the next about the joint axis.

DH T ECHNIQUES Link twist α i :the angle from the Z i-1 axis to the Z i axis about the X i axis. The positive sense for α is determined from z i-1 and z i by the right-hand rule. Joint angle θ i the angle between the X i-1 and X i axes about the Z i-1 axis.

DH T ECHNIQUES 4

The four parameters for each link a i : link length α i : Link twist d i : Link offset θ i : joint angle With the i th joint, a joint variable is q i associated where 5

T RANSFORMATION M ATRIX 6 Each homogeneous transformation A i is represented as a product of four basic transformations

T RANSFORMATION M ATRIX 7

The matrix A i is a function of only a single variable, as three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable, depending on whether it is a revolute or prismatic link

DH N OTATION S TEPS 9

10

DH N OTATION S TEPS From, the position and orientation of the tool frame are calculated.

T RANSFORMATION M ATRIX

E XAMPLE I - T WO L INK P LANAR A RM 13 Base frame O 0 All Z ‘s are normal to the page

E XAMPLE I - T WO L INK P LANAR A RM 14 Where (θ 1 + θ 2 ) denoted by θ 12 and

E XAMPLE 2 15

F ORWARD K INEMATICS OF E XAMPLE 2 16

E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 17

E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 18

E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 19

E XAMPLE 3 - T HREE L INK C YLINDRICAL M ANIPULATOR 20

E XAMPLE 4 – T HE S PHERICAL W RIST 21

E XAMPLE 4 – T HE S PHERICAL W RIST 22

E XAMPLE 4 – T HE S PHERICAL W RIST 23

E XAMPLE 4 – T HE S PHERICAL W RIST 24

E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 25 derived in Example 2, and derived in Example 3.

E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 26

E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 27

E XAMPLE 5 - C YLINDRICAL M ANIPULATOR WITH S PHERICAL W RIST 28 Forward kinematics: 1. The position of the end-effector: (d x,d y,d z ) 2. The orientation {Roll, Pitch, Yaw }

R OTATION – R OLL, P ITCH, Y AW The rotation matrix for the following operations: X Y Z 29

E XAMPLE 4 T HE THREE LINKS CYLINDRICAL WITH S PHERICAL WRIST 30 How to calculate Compare the matrix R With the rotation part of