SiSi SiSi SjSj SjSj Figure 3.1: Two Views of a Spatial Link a ij  ij

Figure 3.2: Kinematic Link physical link kinematic equivalent link SiSi SjSj a ij  ij

Figure 3.3: Spherical Link SiSi SjSj a ij  ij a ij = 0

Figure 3.4: Planar Link SiSi SjSj a ij  ij = 0

Figure 3.5: Revolute Joint, R a ij a jk SjSj jj SjSj

a ij a jk SjSj jj SjSj Figure 3.6: Prismatic Joint, P

Figure 3.7: Cylindric Joint, C a ij a jk SjSj jj SjSj

a ij a jk SjSj jj SjSj Figure 3.8: Screw Joint, H

x1x1 y1y1 z1z1 x2x2 y2y2 z2z2 Figure 3.9: Plane Joint, E

Figure 3.10: Simulation of Plane Pair (PRP) Figure 3.11: Simulation of Plane Pair (PPR) Figure 3.12: Simulation of Plane Pair (RPR) Figure 3.13: Simulation of Plane Pair (RRP)

Figure 3.14: Simulation of Plane Pair (RRR)

Figure 3.15: Hooke Joint, T a kl SjSj SkSk a ij a jk

x1x1 y1y1 z1z1 x2x2 y2y2 z2z2    link ij link jk Figure 3.16: Spherical Joint, S SjSj SkSk SiSi SjSj SkSk SiSi

Figure 3.19: Kinematic Chain R R CP R R

Figure 3.20: Joint Vectors Labeled S1S1 S2S2 S3S3 S4S4 S5S5 S6S6

Figure 3.21: Link Vectors Labeled S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 23 a 34 a 45 a 56 a 67

S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 23 a 34 a 45 a 56 a 67 Figure 3.22: Joint Angles Labeled 22 33 44 55 66

S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 23 a 34 a 45 a 56 a 67  12  23  34  45  56 Figure 3.23: Twist Angles Labeled

S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 23 a 34 a 45 a 56 a 67 Figure 3.24: Offset Lengths Labeled S2S2 S3S3 S4S4 S5S5 S6S6

S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 34 a 45 a 56 a 67 a 12 a 23 a 34 a 45 a 56 Figure 3.25: Link Lengths Labeled a 23

S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 34 a 45 a 56 a 67 a 23 xFxF zFzF yFyF 11 Figure 3.26: Definition of Fixed Coordinate System

Link Length, in. Twist Angle, deg. Joint Offset, in. Joint Angle, deg. a 12 = 3.25  12 = 30  1 = variable a 23 = 2.25  23 = 30 S 2 = 2.75  2 = variable a 34 =  34 = 270 S 3 = variable  3 = variable a 45 = 3.5  45 = 210 S 4 = variable  4 = 270 a 56 = 3.25  56 = 40 S 5 = 3.75  5 = variable S 6 = 4.75  6 = variable Table 3.1: Mechanism Parameters S 6 is actually a free choice

S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 Figure 3.29: Joint Vectors Labeled

Figure 3.30: Link Vectors Labeled S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 a 12 a 34 a 45 a 56 a 67 a 23

Link Length, in. Twist Angle, deg. Joint Offset, in. Joint Angle, deg. a 12 = 0  12 = 90  1 = variable a 23 = 17  23 = 0 S 2 = 5.9  2 = variable a 34 = 0.8  34 = 270 S 3 = 0  3 = variable a 45 = 0  45 = 90 S 4 = 17  4 = variable a 56 = 0  56 = 90 S 5 = 0  5 = variable  6 = variable Table 3.2: Puma Mechanism Parameters S 6 is actually a free choice

Figure 3.31: Standard Link Coordinate System SiSi SjSj a ij X Y Z

S1S1 S2S2 S3S3 a 12 a 23 Figure 3.32: First and Second Coordinate Systems X1X1 Y1Y1 Z1Z1 X2X2 Y2Y2 Z2Z2

S1S1 S2S2 S3S3 a 12 a 23 X1X1 Y1Y1 Z1Z1 X2X2 Y2Y2 Z2Z2 x y z Figure 3.33: First Coordinate System Translated Along a 12 {A}

Figure 3.34: Coordinate System Rotated  12 About a 12 S1S1 S2S2 S3S3 a 12 a 23 X1X1 Y1Y1 Z1Z1 X2X2 Y2Y2 Z2Z2 x y z {B}

Coordinate System Translated S 2 Along S 2 S1S1 S2S2 S3S3 a 12 a 23 X1X1 Y1Y1 Z1Z1 X2X2 Y2Y2 Z2Z2 x y z {C}

Coordinate System Rotated θ 2 About S 2 S1S1 S2S2 S3S3 a 23 X1X1 Y1Y1 Z1Z1 X2X2 Y2Y2 Z2Z2 a 12

Coordinate system 2 is initially aligned with coordinate system 1. It is translated a distance a 12 along the x axis. It is then rotated an angle  12 about the x axis. It is then translated a distance S 2 along the z axis. It is then rotated an angle  2 about the z axis.

In general, It can be shown that,