Robotics kinematics and Dynamics

Transcription

1 Robotics kinematics and Dynamics C. Sivakumar Assistant Professor Department of Mechanical Engineering BSA Crescent Institute of Science and Technology 1

2 Robot kinematics KINEMATICS the analytical study of the geometry of motion of a mechanism: with respect to a fixed reference co-ordinate system, without regard to the forces or moments that cause the motion. In order to control and programme a robot we must have knowledge of both its spatial arrangement and a means of reference to the environment. 2

3 Open Chain Kinematics Mechanics of a manipulator can be represented as a kinematic chain of rigid bodies (links) connected by revolute or prismatic joints. One end of the chain is constrained to a base, while an end effector is mounted to the other end of the chain. The resulting motion is obtained by composition of the elementary motions of each link with respect to the previous one 3

4 Robot kinematics Joint labeling: started from 1 and moving towards end effector, base being joint 1 4

5 Two Basic Joints 5

6 Position representation Kinematics of RR robot is difficult compared to LL robot Analyzing in 2-D 6

7 Position of end of the arm can be represented using: Joint space method: using joint angles World space : using cartesian coordinate system. 7

8 Transformation from one representation to other is necessary for many application. Type of transformation: Forward transformation or forward kinematics going from joint space to world space Reverse transformation or inverse kinematics going from world space to joint space. 8

9 Direct (also forward) kinematics Given are joint relations (rotations, translations) for the robot arm. Task: What is the orientation and position of the end effector? Inverse kinematics Given is desired end effector position and orientation. Task: What are the joint rotations and orientations to achieve this? 9

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13 Reverse transformation of 2 DOF arm 13

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15 Figure

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18 3 DOF arm in two dimension 18

19 3 DOF arm in two dimension 19

20 4 DOF manipulator in three dimensions 2

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23 Robot Dynamics Accurate control of manipulator depends on precise control of joints Control of joints depends on forces and intertias acting on them 23

24 a. Static analysis 24

25 Balancing the forces to know the torque 25

26 Compensating for gravity 26

27 Robot arm dynamics 27

28 Torque requirement 28

29 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. 29

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

31 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. 31

32 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 32

33 DH Techniques Matrix A i representing the four movements is found by: four movements 1. Rotation of about current Z axis 2. Translation of d along current Z axis 3. Translation of a along current X axis 4. Rotation of about current X axis A Rot Trans Trans Rot i z, z, d x, a x, i i i i 33

34 34 i i i i i i i i i i i i i i i i i c -c s s s a c s c c -s c a s s c d 1 A i C S S C x Rot R x 1 ), (, 1 ), (, C S S C z Rot R z i i i i i i i i i i i C S S C a d C S S C A

35 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. 35

36 DH Techniques 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. 36

37 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. 37

38 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. 38

39 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. T A. A... A n From T n, the position and orientation of the tool frame are calculated. n 39

40 DH Techniques 4

41 DH Techniques 41

42 DH Techniques 42

43 DH Techniques A i ci -cisi sisi aici si cici -sici ais i si ci di 1 43