ENME800 Assessment Brief

In this report, you are expected to present in detail your work on the following tasks:

Task 1:  Do the kinematic analysis for an ABB robot: IRB-2600-12/1.85

[60marks]

In this task, you are required to do the kinematic analysis for an ABB robot ABB-260012/1.85 (Figure 1).

Figure 1 IRB-2600-12/1.85 Robot and Axes Descriptions

The specifications of the robot and its dimensions are respectively listed in Appendix A and Appendix B.  Note the units of length, mass and time are millimetre, kilogram and second respectively.

A tool whose initial geometric dimensions (to be finalised) shown in Appendix C is attached to the robot’s end effector through a flange shown in Appendix D. Its tip moves from Point A to Point B separated by 200 mm on the surface of a table shown in Figure 2. The table surface is 300 mm above the ground.

Figure 2 Table

Figure 3 Base and Wrist Frames

(a) Check if the tool (designed by a sophomore) shown in Appendix C can be reliably assembled with the flange shown in Appendix D, adjust its dimensions if needed, and sketch it and its assembly with the flange. Provide your answer for (a) in a single A4 page.

[10]

(b) Choose the base of the robot as Link 0 and set up its frame as shown in Figure 3, then set up all the remaining link frames including the tool frame {T}, with drawings and explanations. [10]

(c) Determine and list link (DH) parameters in a table. [10]

(d) Derive the homogeneous transformation matrix linking the base to the tool frames: ^𝐵𝑇𝑇,  express the position of the tool tip, [𝑝𝑥 𝑝𝑦 𝑝𝑧]^𝑇, relative to the base frame, as functions of the joint angles. [10]

In all  the remaining tasks (e, f), assume  Joint 1 and Joint 2 angles are zero. 

(e) Sketch the configuration of the robot when the joint angles are zero. [5]

(f) Express the position of the tool tip, [𝑝𝑥 𝑝𝑦 𝑝𝑧]^𝑇 as a function of the joint angles, describe the position when Joint 5 angle is zero and solve the remaining joint angles. [15]

Task 2: Solve the following questions from the textbook 

[40 marks] 

2.20 [20] Imagine rotating a vector Q about a vector K̂ by an amount θ to form a new vector, Q′ — that is,

Q′ = R_K(θ) Q.

Use (2.80) to derive Rodrigues’s formula,

Q′ = Q cos θ + sin θ (K̂ × Q) + (1 − cos θ) (K̂ · Q) K̂.

2.31 [15] Referring to Fig. 2.26, give the value of ^A_B^T T.
2.32 [15] Referring to Fig. 2.26, give the value of ^A_C^T T.
2.33 [15] Referring to Fig. 2.26, give the value of ^B_C^T T.

FIGURE 2.26: Frames at the corners of a wedge.

2.37 [15] Given

^A_BT = [ 0.87 -0.43 0.25 7.0 ;
0.5 0.75 -0.43 -2.0 ;
0 0.5 0.87 8.0 ;
0 0 0 1 ]

what is the (1,4) element of ^A_BT?

3.9 [11] For the two-link manipulator shown in Fig. 3.32(a), the link-transformation matrices, ⁰₁T and ¹₂T, were constructed. Their product is

T₂ =[ cθ₁cθ₂ -cθ₁sθ₂ sθ₁ l₁cθ₁ ;
sθ₁cθ₂ -sθ₁sθ₂ -cθ₁ l₁sθ₁ ;
sθ₂ cθ₂ 0 0 ;
0 0 0 1 ]

The link-frame assignments used are indicated in Fig. 3.32(b). Note that frame {0} is coincident with frame {1} when θ₁ = 0. The length of the second link is l₂. Find an expression for the vector P_tip, which locates the tip of the arm relative to the {0} frame.

FIGURE 3.32: Two-link arm with frame assignments (Exercise 3.9).

3.22 [18] Show the attachment of link frames on the P3R robot shown in Fig. 3.42.
Given your frame assignments, what are the signs of d₂, d₃, and a₂?

FIGURE 3.42: Schematic of a P3R manipulator (Exercise 3.22).

4.11 [24] A 2-DOF positioning table is used to orient parts for arc-welding. The forward kinematics that locate the bed of the table (link 2) with respect to the base (link 0) are

T₂⁰ = [ c₁c₂ -c₁s₂ s₁ l₂s₁ + l₁ ;
s₂ c₂ 0 0 ;
-s₁c₂ s₁s₂ c₁ l₂c₁ + h₁ ;
0 0 0 1 ]

Given any unit direction fixed in the frame of the bed (link 2), V̂, give the inverse-kinematic solution for θ₁, θ₂ such that this vector is aligned with V̂ (i.e., upward). Are there multiple solutions? Is there a singular condition for which a unique solution cannot be obtained?

4.16 [25] A 4R manipulator is shown schematically in Fig. 4.15. The nonzero link parameters are a₁ = 1, a₂ = 45°, d₃ = √2, and a₃ = √2, and the mechanism is pictured in the configuration corresponding to Θ = [0, 90°, −90°, 0]^T. Each joint has ±180° as limits. Find all the values of θ₃ such that

P_4ORG = [1.12, 1.35, 1.2589]^T.

FIGURE 4.15: A 4R manipulator shown in the position Θ = [0, 90°, −90°, 0]^T (Exercise 4.16 and 4.33).

Full question descriptions can be found in Appendix E.

Timelines

The report with a detailed descriptions of the steps you have taken to complete Task 1 and Task 2 must be submitted online through CANVAS by 11:59PM of SUNDAY of the last respective teaching weeks (Week 6, Week 12).

Note:

• Simply including programming code, programme output printouts, or screenshots (which should only be put in appendices as supplementary materials) without sufficient explanations in the report is NOT acceptable and will result in at least 30% mark deduction.
• Zero marks will be given for any report containing plagiarism.