FORWARD KINEMATICS: THE DENAVIT-HARTENBERG ...

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88CHAPTER 3. FORWARD KINEMATICS: THE DENAVIT-HARTENBERG CONVENTION Example 3.4 Cylindrical Manipulator with Spherical Wrist Suppose that we now attach a spherical wrist to the cylindrical manipulator of Example 3.3.2 as shown in Figure 3.9. Note that the axis of rotation of d2 d3 θ1 θ5 θ4 θ6 n s Figure 3.9: Cylindrical robot with spherical wrist. joint 4 is parallel to z2 and thus coincides with the axis z3 of Example 3.3.2. The implication of this is that we can immediately combine the two previous expression (3.28) and (3.32) to derive the forward kinematics as T 0 6 = T 0 3 T 3 6 a (3.33) with T 0 3 given by (3.28) and T 3 6 given by (3.32). Therefore the forward kinematics of this manipulator is described by T 0 6 = = ⎡ c1 ⎢ s1 ⎢ ⎣ 0 0 0 −1 −s1 c1 0 ⎤ ⎡ −s1d1 c4c5c6 − s4s6 ⎥ ⎢ c1d3 ⎥ ⎢ s4c5c6 + c4s6 ⎥ ⎢ d1 + d2 ⎦ ⎣ −s5c6 −c4c5s6 − s4c6 −s4c5s6 + c4c6 s5c6 c4s5 s4s5 c5 ⎤ c4s5d6 ⎥ s4s5d6 ⎥ (3.34) ⎥ c5d6 ⎦ 0 ⎡ 0 0 1 ⎤ 0 0 0 1 r11 ⎢ r21 ⎢ ⎣ r31 r12 r22 r32 r13 r23 r33 dx dy dz ⎥ ⎦ 0 0 0 1
3.3. EXAMPLES 89 where r11 = c1c4c5c6 − c1s4s6 + s1s5c6 r21 = s1c4c5c6 − s1s4s6 − c1s5c6 r31 = −s4c5c6 − c4s6 r12 = −c1c4c5s6 − c1s4c6 − s1s5c6 r22 = −s1c4c5s6 − s1s4s6 + c1s5c6 r32 = s4c5c6 − c4c6 r13 = c1c4s5 − s1c5 r23 = s1c4s5 + c1c5 r33 = −s4s5 dx = c1c4s5d6 − s1c5d6 − s1d3 dy = s1c4s5d6 + c1c5d6 + c1d3 dz = −s4s5d6 + d1 + d2. Notice how most of the complexity of the forward kinematics for this manipulator results from the orientation of the end-effector while the expression for the arm position from (3.28) is fairly simple. The spherical wrist assumption not only simplifies the derivation of the forward kinematics here, but will also greatly simplify the inverse kinematics problem in the next chapter. ⋄ Example 3.5 Stanford Manipulator Consider now the Stanford Manipulator shown in Figure 3.10. This manipulator is an example of a spherical (RRP) manipulator with a spherical wrist. This manipulator has an offset in the shoulder joint that slightly complicates both the forward and inverse kinematics problems. We first establish the joint coordinate frames using the D-H convention as shown. The link parameters are shown in the Table 3.4. It is straightforward to compute the matrices Ai as A1 = ⎡ ⎢ ⎣ c1 0 −s1 0 s1 0 c1 0 0 −1 0 0 0 0 0 1 ⎤ ⎥ ⎦ (3.35)
Page 1 and 2: Chapter 3 FORWARD KINEMATICS: THE D
Page 3 and 4: 3.1. KINEMATIC CHAINS 73 z1 x0 y1 z
Page 5 and 6: 3.2. DENAVIT HARTENBERG REPRESENTAT
Page 13 and 14: 3.3. EXAMPLES 83 Step 9: Form T 0 n
Page 15 and 16: 3.3. EXAMPLES 85 x2 d2 x1 x0 d3 O2
Page 17: 3.3. EXAMPLES 87 Table 3.3: DH para
Page 21 and 22: 3.3. EXAMPLES 91 T 0 6 is then give
Page 23 and 24: 3.3. EXAMPLES 93 follows. A1 = ⎡
Page 25 and 26: 3.4. PROBLEMS 95 Figure 3.14: Two-l
Page 27 and 28: 3.4. PROBLEMS 97 Figure 3.18: Elbow
Page 29 and 30: 3.4. PROBLEMS 99 Figure 3.21: Rhino
Page 31 and 32: 3.4. PROBLEMS 101 Figure 3.23: GMF

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