ENG 4406 ROBOTICS AND MACHINE VISION KINEMATICS ANALYSIS

ENG 4406 ROBOTICS AND MACHINE VISION KINEMATICS ANALYSIS OF ROBOTS (Part 2) PART 2 LECTURE 9

Kinematics Analysis of Robots II This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. After this lecture, the student should be able to: • Put into practice the concept of inverse kinematics analysis of robots • Derive inverse transformation matrix between coupled links • Formulate the inverse kinematics of articulated robots in terms of the link transformation matrices • Solve problems of robot inverse kinematics analysis using transformation matrices

Inverse Transformation A typical transformation matrix can be partitioned as follow: to give where and

Inverse Transformation The inverse transformation For example, find the inverse of is given by: :

Inverse Transformation of

Inverse Transformation of Therefore

Inverse Transformation of Find the inverse of

Inverse Transformation of and Therefore

Inverse Transformation of Find the inverse of

Inverse Transformation of and Therefore

Inverse Kinematics Problem Robot forward kinematics involve finding the gripper position and orientation given the angles of rotation of the linkages Now consider the following problem for the planar robot: The arm moves to a new location and the position along with the orientation of the gripper w. r. t. frame {0} is known. Can you determine the total rotations of 1, 2, and 3 required for the arm to reach that new position? This problem is called the robot inverse kinematics problem.

Inverse Kinematics of the Planar Robot The inverse kinematics problem for the planar robot can be stated as follow: Given the global gripper orientation and position, i. e. given Find the joint angles 1, 2, and 3 required for the arm to reach that new position

Inverse Kinematics To solve the inverse kinematics problem, we know that: Or We can equate the elements in the above two matrices to try to determine the joint angles

Inverse Kinematics example Let = 1+ 2+ 3. Equate elements (1, 1) and (2, 1): Provided that We can find using =arctan(ny, nx):

Inverse Kinematics example Next, equate elements (1, 4) and (2, 4): Square both of these equations to get: Adding the above two equation yields:

Inverse Kinematics example To solve for 1, we again reuse the following equations: where and can be found as 2 was previously determined. Let where Substitute r and into px and py to get:

Inverse Kinematics example We can then find 1 using the known : Summary of inverse kinematics for the planar robot: 1) Find = 1+ 2+ 3 from 2) Find 2 from 3) Find 1 from 4) Find 3 from 3= -( 1+ 2)

Inverse Kinematics exercise 1 The gripper position and orientation for the planar robot is at: Where A 1=3, A 2=2. Find the joint angles and hence determine the robot configuration.

Inverse Kinematics exercise 1 Assuming an elbow down configuration for link 2: Remember, positive 2 means clockwise rotation as follow: Y 2 X 2 2

Inverse Kinematics exercise 1 After 2 has been found: Results:

Visualization of Inverse Kinematics exercise 1 To visualize the arm movement, the robot should looks like this after rotating 1 = 90° ( 1 is the angle from X 0 to X 1 measured along Z 1). X 3 Y 3 X 2 Y 2 1=90° Y 0, X 1 Y 1 X 0

Visualization of Inverse Kinematics exercise 1 To visualize the arm movement, the robot should looks like this after rotating 1 = 90° ( 1 is the angle from X 0 to X 1 measured along Z 1), and after rotating 2 = -90° ( 2 is the angle from X 1 to X 2 measured along Z 2) Y 2 Y 3 X 2 Note that 3 = 0° X 3 2=-90° Y 0, X 1 Y 1 X 0

Visualization of Inverse Kinematics exercise 1 The orientation of the gripper is as follow: X 3 is in the positive X 0 direction Y 3 is in the positive Y 0 direction Z 3 is in the positive Z 0 direction Location of gripper in frame {0} Y 2 Y 3 X 2 X 3 A 2 Y 0, X 1 Y 1 X 0 A 1

Inverse Kinematics exercise 2 The gripper position and orientation for the planar robot is at: Find the joint angles and hence determine the robot configuration.

Inverse Kinematics exercise 2 After 2 has been found: Results:

Visualization of Inverse Kinematics exercise 2 To visualize the arm movement, the robot should finally looks like this after rotating 1 = 90° ( 1 is the angle from X 0 to X 1 measured along Z 1), 2 = 0° ( 2 is the angle from X 1 to X 2 measured along Z 2) and 3 = 0°. X 3 Y 3 X 2 1=90° Y 2 Y 0, X 1 Y 1 X 0

Inverse Kinematics – the general approach For a more complex robot with n>3 links, it is generally not possible to solve for all the joint variables by just equating the elements in To solve for the other joint angles, it may be necessary to find The elements in the above equation are compared to solve for the other joint angles. It may be necessary to repeat the process with premultiplication of the LHS and RHS with more inverse transformation matrices.

Inverse Kinematics – Summary • • Robot inverse kinematics involve finding the joint variables given the robot arm global position and orientation This problem can be solved by equating the overall transformation matrix with the matrix containing the given information and comparing their elements The process may need to be repeated with premultiplication of the above equation with inverse transformation matrices Although we have use a planar robot to illustrate the concept, the approach can be applied to any robot moving in 3 -D space.

Inverse Kinematics – Summary • The approach involves the following: 1. Solve for joint variables by comparing elements in 2. Repeat step 1 with etc. until all the joint variables are obtained

Summary This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. The following were covered: • The concept of inverse kinematics analysis of robots • Inverse transformation matrix between coupled links • Inverse kinematics of articulated robots in terms of the link transformation matrices • Robot inverse kinematics analysis using transformation matrices