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Manipulator kinematics, joint space method, world space method, forward and backward transformation

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- 1. IPE-409 CAD/CAM Dr. Nafis Ahmad Professor Department of IPE, BUET Email:nafis@ipe.buet.ac.bd
- 3. 3 Introduction Manipulator Kinematics Accuracy and repeatability with which the robot can position its end effector
- 5. 5 Manipulator Kinematics Manipulator kinematics is concerned with the position and orientation of the robot's end of-arm or the end effector attached to it as a function of time but without regard for the effects of force or mass. Kinematic analysis neglects the effect of mass of the manipulator's links and joints, the end effector and load being carried on position and orientation. Here discussion on manipulator kinematics will be limited to the mathematical representation of the position and orientation of the robot's end-of-arm
- 6. 6 Manipulator Kinematics • Figure 1: Two manipulators with two degrees-of-freedom: (a) an OO robot and (b) an RR Robot. The robot manipulator consists of a sequence of joints (J1, J2 ) and links (L1, L2)
- 7. 7 Joint space method vs. world space method • The values of the positions of the joints relative to their respective inputs links as shown in Figure 1[a] are λ1 and λ2 . In Figure 1[b] the values are θ1 and θ2. • Figure 1[a]: Pj = ( λ1 , λ2 ) …...(5.1) • Figure 1[b]: Pj = ( θ1 , θ2 ) …...(5.2) • This is known as joint space method of representation, because it defines position and orientation (symbolized as Pj) in term, of the joint values. • Position can also be represented by Cartesian or World Coordinate known as world space method of representation.
- 8. 8 World space method • Position by Cartesian or World Coordinate known as world space method of representation. The end-of-arm position Pw is defined in Cartesian or World Coordinate as : • Pw = ( x , z ) • For a robot with six joints operating in 3-D space, the end- of-arm position and orientation Pw, can be defined as • Pw = (x, y, z, α, β, γ) • where x, y, and z specify the Cartesian coordinates in world space α, β and γ specify the angles of rotation of the three wrist joints (orientation).
- 9. 9 World space method • Orientation cannot be independently established for our two robots in Figure 1. • For the OO manipulator, the end-of-arm orientation is always vertical; and • For the RR manipulator, the orientation is determined by the joint angles θ1 and θ2. • RR robot has two possible ways of reaching a given set of x and z coordinates, and so there are two alternative orientations of the end-of-arm that are possible for all x-z values within the manipulator's reach except for those coordinate positions making up the outer circle of the work volume when θ2 is zero. Figure 2.
- 10. 10 Manipulator Kinematics Figure 2: The two alternative pairs of joint values for RR robot
- 11. 11 Forward and Backward Transformation • Mapping from joint space to world space is called forward transformation and converting from world space to joint space is called backward transformation. • The forward and backward transformations are readily accomplished for the Cartesian coordinate robot. • Forward transformation : x = λ2 and z = λ1 • Backward transformation : λ1= z and λ2 = x • Where x and z are the coordinate values in world space and λ1 and λ2 are the values in joint space.
- 12. 12 Forward and Backward Transformation • For the RR robot of the forward transformation is calculated by noting that the lengths and directions of the two links might be viewed as vectors in space : • r1 = {L1 cos θ1 , L1 sin θ2} ; r2 = {L2 cos (θ1 + θ2), L2 sin (θ1 + θ2)} • Vector addition of r1 and r2 (and taking account of link L0 ) yields the coordinate values of x and y at the end-of-arm: • x = L1 cos θ1 + L2 cos (θ1 + θ2 ) ; z = L0 + L1 sin θ1 + L2 sin (θ1 + θ2 ) • Given the link values L1 and L2 , the following equations can be derived for the two angles θ1 and θ2 :
- 13. 13 Forward and Backward Transformation • Forward and Backward Transformation for a Robot with Three Joints: • For the forward transformation, we can compute the x and z coordinates in a way similar to that used for the previous RR robot. The values of x and z can be computed as follows: • x = L1 cos θ1 + L2 cos (θ1 + θ2 ) + L3 cos (θ1 + θ2 + θ3 ) • z = L1 sin θ1 + L2 sin (θ1 + θ2 ) + L3 sin (θ1 + θ2 + θ3 ) • (0,0)--? • The angle made by the wrist with the horizontal: • α = θ1 + θ2 + θ3
- 14. 14 Forward and Backward Transformation Forward and Backward Transformation for a Robot with Three Joints. Coordinates of joint 3 X3 = x – L3 cos α Z3 = z – L3 sin α θ3 = α – (θ1 + θ2 )
- 15. 15 Examples Example 7.1: Given the world coordinates for a RR:R robot (similar to earlier one) as x = 300 mm, Z = 400 mm, and α = 30°;and given that the links have values L1 = 350 mm, L2 = 250 mm and L3 = 50 mm, determine the joint angles θ1, θ2 and θ3
- 16. 16 Manipulator Kinematics The first step is to find X3 and Z3 using given coordinates x = 300 and z = 400. X3 ~ 300 - 50 cos30 = 256.7, Z3 = 400 - 50 sin30 = 375 θ3 can be determined as follows:
- 17. 17 Robot with 4 DOF
- 18. 18 Robot with 4 DOF
- 19. 19 Accuracy and repeatability The capacity of the robot to position and orient the end of its wrist with accuracy and repeatability is an Important control attribute in nearly all industrial applications. There are several terms that must be defined in the context of this discussion: (1) control resolution, (2) accuracy, and (3) repeatability.
- 20. 20 Accuracy and repeatability Control resolution refers to the capability of the robot's controller and positioning system to divide the range of the joint into closely spaced points that can be identified by the controller. These are called addressable points because they represent locations to which the robot can be commanded to move. The capability to divide the range into addressable points depends on two factors: (1) limitations of the electromechanical components that make up each joint-link combination and (2) the controller's bit storage capacity for that joint.
- 21. 21 Accuracy and repeatability (2) Accuracy: Accuracy is a measure of the robot's ability to position the end of its wrist at a desired location in the work volume. (3) Repeatability: Repeatability is a measure of the robot's ability to position its end-of-wrist at a previously taught point in the work volume.
- 22. Thanks! Any questions? You can find me at: @ahmadn nafis@ipe.buet.ac.bd 22