NATIONAL UNIVERSITY OF SINGAPORE. (Semester I: 1999/2000) EE4304/ME ROBOTICS. October/November Time Allowed: 2 Hours

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1 NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION FOR THE DEGREE OF B.ENG. (Semester I: 1999/000) EE4304/ME445 - ROBOTICS October/November Time Allowed: Hours INSTRUCTIONS TO CANDIDATES: 1. This paper contains FIVE (5) questions in Sections and comprises SIX (6) pages.. Answer all questions (Q. 1, and 3) in Section A, and an 1 question (Q. 4 or 5) in Section B. 3. All questions carr equal marks. 4. This is an open-book eamination.

2 SECTION A: COMPULSORY (Answer all three questions in this section.) EE4304/ME445 Robotics / Page Q.1(a) Figure 1 shows a robot with three rotational joints. The ais motion of the first joint (Joint A) is along the -ais of Frame U. The ais of motion for the nd joint (Joint B) is along the link AB. The 3 rd joint (Joint C) rotates about an ais defined b BC CD (where is the vector cross product), i.e., the ais is normal to the plane formed b BCD. The three joint variables φ 1, φ, φ 3 have the following zero positions: φ 1 is zero when AB is parallel to the -ais of Frame U, φ is zero when ABCD is in the z plane of Frame U, φ 3 is zero when CD is aligned with BC. The length of each link is 1 m. AB is alwas perpendicular to BC. (i) Assign Frames to the robot according to the Denavit Hartenberg (DH) Convention discussed in class. (4 Marks) (ii) Determine the 4 DH parameters for each of the three links, i.e., complete the table of kinematic (DH) parameters. Indicate which parameter is the joint coordinate, q i. (4 Marks) (iii) Determine the relationships between q i and φ i. ( Marks) (iv) Frame D is attached to the last link such that its origin is at D and it s z ais is along the link CD. Derive the epressions describing the position U p D and the orientation vector U z D as functions of the three joint variables φ 1, φ, φ 3. You do not need to simplif the epressions. (You can leave the epression in matrivector and/or matri-matri product terms.) (5 Marks) φ B z D z U D A φ 1 C φ 3 U Figure 1

3 EE4304/ME445 Robotics / Page 3 Q.1(b) Frames B and C are attached to the same rigid bod with B T C given. Frames A and D are fied to the universe with A T D given. The bod is initiall at A T B. The bod undergoes the following ordered sequence of motions: 1> Rotation about z-ais of Frame A b 30 degrees > Rotation about -ais of Frame C b 40 degrees 3> Rotation about -ais of Frame D b 50 degrees Determine the final position and orientation of Frame B with respect to Frame A, A T B. You do not need to simplif our answer. Epress our answer in terms of a matri product epression. Q. The orientation of a rigid bod can be described b the three angles α, β, γ where R = Rot (, α) Rot (, β) Rot (, γ) = i Cos@βD Sin@βD Sin@γD Cos@γD Sin@βD Sin@αD Sin@βD Cos@αD Cos@γD Cos@βD Sin@αD Sin@γD Cos@βD Cos@γD Sin@αD Cos@αD Sin@γD k Cos@αD Sin@βD Cos@γD Sin@αD + Cos@αD Cos@βD Sin@γD Cos@αD Cos@βD Cos@γD Sin@αD Sin@γD { (a) Derive the complete inverse kinematic equations that compute α, β, γ given n R = n n z o o o z a a a z (15 Marks) (b) Frames E and F are attached to the same rigid bod with E T F given. The rigid bod is in motion and at a certain instant of time, the bod is at given orientation (α, β, γ) such that A R E = Rot (, α) Rot (, β) Rot (, γ), and the position coordinates of E in Frame A is given as (a,b,c). At this same instant, the angles (α, β, γ) are changing at the given rates (α, β, γ ) and the translational velocit of E is given as A u E = (a, b, c ) T. (i) Determine the angular velocit of Frame F with respect to Frame A, A ω F. (ii) Determine the translational velocit of Frame F with respect to Frame A, A u F. (Epress our answers in terms of the given quantities. You do not need to simplif.)

4 EE4304/ME445 Robotics / Page 4 Q.3 Assume that the robot shown in Figure is in the vertical plane, the mass of each link is concentrated at a point in the indicated mass centers with m 1 and m being the equivalent masses of links 1 and respectivel, L 1 c is the length from the center of mass m 1 to joint 1, L 1 and L are the lengths of links 1 and, τ 1 and τ are the output torques of motors 1 and which are located at the joints. (a) Give two different sets of generalized coordinates for the robot manipulator. Draw two separate figures of the manipulator indicating the generalized coordinates that ou choose. Indicate the tpe of joint for the each of the two joints as well. (3 Marks) L 1 L L 1c m 1 m g Figure (b) Derive the Lagrange-Euler equations of the form D( q)&& q + C( q, q&) q& + G( q) = τ where, q = [q 1, q] T, τ = [τ 1, τ ] T, D(q) is the inertia matri, C( q, q&) is the matri defined b the so called Christoffel Smbols, and G(q) represent the gravitational forces. (14 marks) (c) Design a computed torque controller for this robot such that the resulting closed-loop sstem is decoupled, criticall damped, and with a natural frequenc ω = 4 rad/s. (8 Marks)

5 EE4304/ME445 Robotics / Page 5 SECTION B (Answer Onl One out of the Two Questions in this Section) Q.4 (a) The equations of motion of the two-link robot arm in Fig. 3 can be written in a compact matri-vector form as: d d 11 1 ) ) d 1 d ) 1 + c 1 ) + c1 ) 1 ) 1 c 1 g1( 1, + g 1, ) τ = ) τ 1 l 1 1 l 0 z z 0 m m 1 Figure 3 z i. Choose an appropriate state variable vector (t) and a control vector u(t) for this dnamic sstem. (5 Marks) ii. Epress the equations of motion of this robot arm eplicitl in terms of d ij s, c ij s, and g i s in a state space representation with the chosen statevariable vector and control vector. (b) Briefl discuss the basic idea of the independent joint control scheme and, the advantages and disadvantages in implementing such a scheme. (6 Marks) (c) Briefl discuss the advantages and disadvantages of the computed torque method. (4 Marks)

6 EE4304/ME445 Robotics / Page 6 Q.5 (a) A single-link rotar robot is required to move from (0) = 30 to () = 100 in s. The joint velocit and acceleration are both zero at the initial and final positions. The trajector ma be composed of one or more polnomial segments whose orders are all the same. i. What is the lowest degree polnomial that can be used to accomplish the motion? (3 Marks) ii. Determine the coefficients of a quadric polnomial that accomplishes the motion. You ma split the joint trajector in to several trajector segments. (You can leave the epression in matri-vector and/or matri-matri product terms.) (b) Figure 4 shows a planar robot with 7 rotational joints. The ais of motion of each joint is parallel to the z ais of Frame U. (3 Marks each) i. How man degrees of freedom does this robot have in terms of its positioning and orienting capabilit? ii. iii. iv. It is known that robots with redundant joints can also have singularities. At a singular configuration, what is the rank of the Jacobian matri (that relates the joint and end-effector velocities)? Draw the robot at a singular configuration and indicate the rank of the Jacobian at the drawn singular configuration. What is the minimum number of joints for this planar robot to be able to have the same positioning and orienting capabilit as the 7 joints? (The positioning and orienting capabilities are independent.) U U Figure 4 END OF PAPER