Robotics Forwards Kinematics Example SCARA TEMPUS IV Project

Robotics Forwards Kinematics Example: SCARA TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS

SCARA – Forward Kinematics Use the DH Algorithm to assign the frames and kinematic parameters TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 2

2 3 4 -Tool Roll 1 Number the joints 1 to n starting with the base and ending with the tool yaw, pitch and roll in that order. Note: There is no tool pitch or yaw in this case TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 3

k=0 z 0 y 0 x 0 Assign a right-handed orthonormal frame L 0 to the robot base, making sure that z 0 aligns with the axis of joint. Set k=1 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 4

k=1 Common Normal 2 z 1 z 0 y 0 x 0 Align zk with the axis of joint k+1. Locate the origin of Lk at the intersection of the zk and zk-1 axes If they do not intersect use the intersection of zk with a common normal between zk and zk-1. (can point up or down in this case) TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 5

k=1 x 1 y 1 z 0 y 0 x 0 Select xk to be orthogonal to both zk and zk-1. If zk and zk-1 are parallel, point xk away from zk-1. Select yk to form a right handed orthonormal co-ordinate frame Lk TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 6

k=2 Vertical Extension Common Normal x 1 y 1 z 0 z 2 y 0 x 0 Align zk with the axis of joint k+1. Again zk and zk-1 are parallel the so we use the intersection of zk with a common normal. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 7

k=2 x 1 y 1 z 0 x 2 y 2 z 2 y 0 x 0 Select xk to be orthogonal to both zk and zk-1. Once again zk and zk-1 are parallel, point xk away from zk-1. Select yk to complete the right handed orthonormal co-ordinate frame TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 8

k=3 x 1 y 1 z 1 x 2 y 2 z 2 4 z 3 z 0 y 0 x 0 Align zk with the axis of joint k+1. Locate the origin of Lk at the intersection of the zk and zk-1 axes TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 9

k=3 x 2 x 1 y 2 z 1 x 3 y 3 z 0 z 2 z 3 y 0 x 0 Select xk to be orthogonal to both zk and zk-1. Again xk can point in either direction. It is chosen to point in the same direction as xk-1 Select yk to complete the right handed orthonormal co-ordinate frame TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 10

k=4 x 2 x 1 y 2 z 1 z 2 x 3 y 3 z 0 y 4 y 0 x 4 z 4 x 0 Set the origin of Ln at the tool tip. Align zn with the approach vector of the tool. Align yn with the sliding vector of the tool. Align zn with the normal vector of the tool. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 11

x 2 x 1 y 2 z 1 z 2 x 3 y 3 z 3 y 4 z 0 y 0 x 4 z 4 x 0 With the frames assigned the kinematic parameters can be determined. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 12

k=4 x 2 x 1 y 2 z 1 z 2 x 3 y 4 z 0 y 0 x 4 z 3 b 4 z 4 x 0 Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 13

k=4 x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 x 3 z 3 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is 90 degrees (clockwise +ve) i. e. 4 = 90º But this is only for the soft home position, 4 is the joint variable. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 14

k=1 x 2 x 1 y 1 z 1 y 2 4 y 3 y 4 z 0 y 0 z 2 x 4 x 3 z 3 d 4 b 4 z 4 x 0 Compute dk as the distance from the origin of frame Lk-1 to point bk along zk-1 Compute ak as the distance from point bk to the origin of frame Lk along xk In this case these are the same point therefore a 4=0 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 15

k=4 x 2 x 1 y 1 z 1 y 2 4 y 3 y 4 z 0 y 0 z 2 x 4 x 3 z 3 d 4 b 4 z 4 x 0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z 3 to z 4 about x 4 is zero i. e. 4 = 0º TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 16

k=3 x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 b 3 z 3 x 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Locate point bk at the intersection of the xk and zk-1 axes. If they do not intersect, use the intersection of xk with a common normal between xk and zk-1 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 17

k=3 x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 b 3 z 3 x 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from xk-1 to xk about zk-1 is zero i. e. 3 = 0º TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 18

k=3 x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 b 3 z 3 d 3 x 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute dk as the distance from the origin of frame Lk-1 to point bk along zk-1 Since joint 3 is prismatic d 3 is the joint variable Compute ak as the distance from point bk to the origin of frame Lk along xk In this case these are the same point, therefore ak=0 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 19

k=3 x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 b 3 z 3 d 3 x 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z 2 to z 3 about x 3 is zero i. e. 3 = 0º TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 20

k=2 b 2 x 1 y 1 z 1 y 2 z 2 4 y 3 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Once again locate point bk at the intersection of the xk and zk-1 axes If they did not intersect we would use the intersection of xk with a common normal between xk and zk-1 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 21

2 k=2 b 2 x 1 y 1 z 1 y 2 z 2 4 y 3 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from x 1 to x 2 about z 1 is zero i. e. 2 = 0º But this is only for the soft home position, 4 is the joint variable. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 22

a 2 2 k=2 b 2 x 1 y 1 z 1 y 2 z 2 4 y 3 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute dk as the distance from the origin of frame Lk-1 to point bk along zk-1 In this case these are the same point therefore d 2=0 Compute ak as the distance from point bk to the origin of frame Lk along xk TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 23

a 2 2 k=2 b 2 x 1 y 1 z 1 y 2 z 2 4 y 3 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from zk-1 to zk measured about xk It can be seen here that the angle of rotation from z 1 to z 2 about x 2 is zero i. e. 2 = 0º TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 24

k=1 a 2 2 b 1 x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 For the final time locate point bk at the intersection of the xk and zk-1 axes TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 25

a 2 k=1 2 bk x 2 x 1 y 1 z 1 y 2 z 2 4 y 3 1 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from xk-1 to xk measured about zk-1 It can be seen here that the angle of rotation from x 0 to x 1 about z 0 is zero i. e. 1 = 0º But this is only for the soft home position, 1 is the joint variable. TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 26

a 1 k=1 a 2 2 b 1 x 2 x 1 y 1 z 1 d 1 y 2 z 2 4 y 3 1 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute dk as the distance from the origin of frame Lk-1 to point bk along zk-1 Compute ak as the distance from point bk to the origin of frame Lk along xk TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 27

a 1 k=1 a 2 2 b 1 x 2 x 1 y 1 z 1 d 1 y 2 z 2 4 y 3 1 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Compute k as the angle of rotation from zk-1 to zk measured about xk-1 It can be seen here that the angle of rotation from z 0 to z 1 about x 1 is 180 degrees TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 28

a 1 a 2 2 x 1 y 1 z 1 d 1 y 2 z 2 4 y 3 1 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 From this drawing of D-H parameters can be compiled TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 29

a 1 a 2 2 x 1 y 2 z 1 d 1 z 2 4 y 3 1 d 3 x 3 z 3 d 4 y 4 z 0 y 0 x 4 z 4 x 0 Joint d a 1 1 d 1 180º 2 2 0 a 1 a 2 0º 0º 3 0º 0 0º dmax 4 4 d 3 d 4 0 0º 90º Home q 0º 30

Joint d a Home q 1 2 3 4 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 31

References: http: //www. zimtok 5. com/eee/EE 4009. Fwd. Kinematic s. SCARA. ppt#277, 23, Slide 23 TEMPUS IV Project: 158644 – JPCR Development of Regional Interdisciplinary Mechatronic Studies - DRIMS ROBOTICS 32