FORWARD KINEMATICS: THE DENAVIT-HARTENBERG ...

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86CHAPTER 3. FORWARD KINEMATICS: THE DENAVIT-HARTENBERG CONVENTION and T matrices are ⋄ A1 = A2 = A3 = T 0 3 = A1A2A3 = ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ Example 3.3 Spherical Wrist x4 z4 z3, x5 z5 θ4 θ5 c1 −s1 0 0 s1 c1 0 0 0 0 1 d1 0 0 0 1 1 0 0 0 0 0 1 0 0 −1 0 d2 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 d3 0 0 0 1 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ (3.27) ⎡ c1 ⎢ s1 ⎢ ⎣ 0 0 0 −1 −s1 c1 0 ⎤ −s1d3 ⎥ c1d3 ⎥ ⎥. d1 + d2 ⎦ (3.28) 0 0 0 1 θ6 To gripper Figure 3.8: The spherical wrist frame assignment. The spherical wrist configuration is shown in Figure 3.8, in which the joint axes z3, z4, z5 intersect at O. The Denavit-Hartenberg parameters are shown in Table 3.3. The Stanford manipulator is an example of a manipulator that possesses a wrist of this type. In fact, the following analysis applies to virtually all spherical wrists.
3.3. EXAMPLES 87 Table 3.3: DH parameters for spherical wrist. Link ai αi di θi 4 0 −90 0 θ ∗ 4 5 0 90 0 θ ∗ 5 6 0 0 d6 θ ∗ 6 ∗ variable We show now that the final three joint variables, θ4, θ5, θ6 are the Euler angles φ, θ, ψ, respectively, with respect to the coordinate frame o3x3y3z3. To see this we need only compute the matrices A4, A5, and A6 using Table 3.3 and the expression (3.10). This gives T 3 6 = A4A5A6 = A4 = A5 = A6 = ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ Multiplying these together yields � � = R 3 6 O 3 6 0 1 c4 0 −s4 0 s4 0 c4 0 0 −1 0 0 0 0 0 1 c5 0 s5 0 s5 0 −c5 0 0 −1 0 0 0 0 0 1 c6 −s6 0 0 s6 c6 0 0 0 0 1 d6 0 0 0 1 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ ⎤ (3.29) (3.30) ⎥ ⎥. (3.31) ⎦ ⎡ c4c5c6 − s4s6 ⎢ s4c5c6 + c4s6 ⎢ ⎣ −s5c6 −c4c5s6 − s4c6 −s4c5s6 + c4c6 s5s6 c4s5 s4s5 c5 c4s5d6 s4s5d6 c5d6 0 0 0 1 (3.32) Comparing the rotational part R 3 6 of T 3 6 with the Euler angle transformation (2.51) shows that θ4,θ5,θ6 can indeed be identified as the Euler angles φ, θ and ψ with respect to the coordinate frame o3x3y3z3. ⋄ ⎤ ⎥ ⎦ .
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: 3.3. EXAMPLES 85 x2 d2 x1 x0 d3 O2
Page 19 and 20: 3.3. EXAMPLES 89 where r11 = c1c4c5
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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