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Forward Kinematics University of Bridgeport 1 Introduction to ROBOTICS.

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Presentation on theme: "Forward Kinematics University of Bridgeport 1 Introduction to ROBOTICS."— Presentation transcript:

1 Forward Kinematics University of Bridgeport 1 Introduction to ROBOTICS

2 Kinematic Forward (direct) Kinematics Given: The values of the joint variables. Required: The position and the orientation of the end effector. Inverse Kinematics Given : The position and the orientation of the end effector. Required : The values of the joint variables. 2

3 Why DH notation Find the homogeneous transformation H relating the tool frame to the fixed base frame 3

4 Why DH notation A very simple way of modeling robot links and joints that can be used for any kind of robot configuration. This technique has became the standard way of representing robots and modeling their motions. 4

5 DH Techniques 1.Assign a reference frame to each joint (x-axis and z-axis). The D-H representation does not use the y-axis at all. 2.Each homogeneous transformation A i is represented as a product of four basic transformations 5

6 DH Techniques 6 Matrix A i representing the four movements is found by: four movements

7 7

8 DH Techniques The link and joint parameters : Link length a i : the offset distance between the Z i-1 and Z i axes along the X i axis. Link offset d i the distance from the origin of frame i−1 to the X i axis along the Z i-1 axis. 8

9 DH Techniques 9 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.

10 DH Techniques The four parameters: a i : link length, α i : Link twist, d i : Link offset and θ i : joint angle. The matrix A i is a function of only a single variable q i, it turns out that three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable. 10

11 DH Techniques With the i th joint, a joint variable is q i associated where All joints are represented by the z-axis. If the joint is revolute, the z-axis is in the direction of rotation as followed by the right hand rule. If the joint is prismatic, the z-axis for the joint is along the direction of the liner movement. 11

12 DH Techniques 3. Combine all transformations, from the first joint (base) to the next until we get to the last joint, to get the robot’s total transformation matrix. 4. From, the position and orientation of the tool frame are calculated. 12

13 DH Techniques 13

14 DH Techniques 14

15 DH Techniques 15

16 DH Techniques 16

17 Example I The two links arm 17 Base frame O 0 All Z ‘s are normal to the page

18 Example I The two links arm 18 Where (θ 1 + θ 2 ) denoted by θ 12 and

19 Example 2 19

20 Example 2 20

21 Example 3 The three links cylindrical 21

22 Example 3 The three links cylindrical 22

23 Example 3 The three links cylindrical 23

24 Example 3 The three links cylindrical 24

25 Example 4 Spherical wrist 25

26 Example 4 Spherical wrist 26

27 Example 4 Spherical wrist 27

28 Example 4 Spherical wrist 28

29 Example 5 The three links cylindrical with Spherical wrist 29

30 Example 5 The three links cylindrical with Spherical wrist 30 given by example 2, and given by example 3.

31 Example 5 The three links cylindrical with Spherical wrist 31

32 Example 5 The three links cylindrical with Spherical wrist 32

33 Example 5 The three links cylindrical with Spherical wrist 33 Forward kinematics: 1. The position of the end-effector: (d x,d y,d z ) 2. The orientation {Roll, Pitch, Yaw }

34 Roll Pitch Yaw The rotation matrix for the following operations: X Y Z 34

35 Example 4 The three links cylindrical with Spherical wrist 35 How to calculate Compare the matrix R with Of the matrix

36 Module 1 RRR:RRR 36 Linksαaθd 1900*10 20 * *0 4900* *0 600*0

37 HW From Spong book: page , 3.3, 3.4, 3.6, 3.7, 3.8, 3.9, 3.11 No Class on next Tuesday 37

38 Module 1 38 Where are Roll, Pitch, and Yaw

39 Representing forward kinematics Forward kinematics 39 Transformation Matrix

40 Remember For n joints: we have n+1 links. Link 0 is the base Joints are numbered from 1 to n Joint i connects link i − 1 to link i. Frame i {X i Y i Z i } is attached to joint i+1. So, frame {O 0 X 0 Y 0 Z 0 }, which is attached to the robot base (inertial frame) “joint 1”. 40


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