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Forward Kinematics University of Bridgeport 1 Introduction to ROBOTICS

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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

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

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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

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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

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

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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

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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.

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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

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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

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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

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DH Techniques 13

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DH Techniques 14

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DH Techniques 15

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DH Techniques 16

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

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

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Example 2 19

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Example 2 20

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Example 3 The three links cylindrical 21

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Example 3 The three links cylindrical 22

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Example 3 The three links cylindrical 23

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Example 3 The three links cylindrical 24

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Example 4 Spherical wrist 25

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Example 4 Spherical wrist 26

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Example 4 Spherical wrist 27

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Example 4 Spherical wrist 28

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Example 5 The three links cylindrical with Spherical wrist 29

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

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Example 5 The three links cylindrical with Spherical wrist 31

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Example 5 The three links cylindrical with Spherical wrist 32

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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 }

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

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

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

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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

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Module 1 38 Where are Roll, Pitch, and Yaw

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Representing forward kinematics Forward kinematics 39 Transformation Matrix

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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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