Robotics Tutorial Edited

8/12/2019 Robotics Tutorial Edited

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Robotics and Automation MEM665/T/LCYRev.01-2011

UNIVERSITI TEKNOLOGI MARAFACULTY OF MECHANICAL ENGINEERING

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Program : Bachelor of Engineering (Hons) (Mechanical) EM220Bachelor of Mechanical Engineering (Manufacturing) (Hons) EM221

Course : Robotics and AutomationCode : MEM665 / KJP 626

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Tutorial 1 Object Location

Q1 Frame zero, F 0. is the fixed global frame. For each of the cases below find T 01:

(a) F 1 is rotated by an angle about z o.

(b) F 1 is rotated by about x o.

(c) F 1 is rotated by about y o.

Q2 Referring to Figure 1, find T 12, T 13 and T 23.

Figure 1


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Q3 In Q2, let P 1, P2 and P 3 be the co-ordinates of P in Frame 1, Frame 2 and Frame 3 respectively:-

(a) Given P 2 = (0, 0, 0,1), find P 1

(b) Given P 2 = (1, 0, 0, 1), find P 1

(c) Given P 3 = (0,0,0,1) find P 1 and P 2

Q4 A vector P (4,6,0) is rotated anticlockwise through 30 around Z-axis, what are the new co-ordinates of P.

Q5 A vector P (1,0,0) is rotated

(a) ROT (Z,90)

(b) ROT (Y,90)

(c) ROT (Z, -90)

For each case find the new co-ordinates of P.

Q6 Find the composite transformation matrix for each of the following cases

(a) F 1 is rotated by 90 about y 0 and then F 2 is rotated by 90 about z 0.

(b) F 1 is rotated by 90

about y 0 and then F 2 is rotated by 90

about z 1.

(c) F 1 is rotated by 45 about y 0 and then F 2 is rotated by 90 about z 1 and then F 3 is rotated by 90 about x 2.

Q7 A vector P (1,0,0) is subjected to the following rotations.

(a) ROT (Z,90) and then ROT (Y,90)

(b) ROT (Y,90) and then ROT (Y,90)

(c) ROT (X,90) and then ROT (Y,90)

Find the new co-ordinates of P after each of the above rotation.


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Q8 Given the two-dimensional transformation of A=Trans (2,0), B= Trans (0,2) and C=Rot (45), draw the following transformations: AB, AC, BC, CA, CB, ABC, CBA, ACBA, ABCA, CCA,ACCAB

Q9 Find the transformation matrix which solves the simple assembly problem in Figure 2. Thewedge-shaped block is to be placed on the assembly so that p and r are touching, and s and q aretouching and the final assembly is a rectangular block. Use the following method:

Figure 2

(a) Assign co-ordinate frames to each object.

(b) Develop transformation matrixes for each frame with respect to word co-ordinates.

(c) Calculate the transformation matrix.

(d) Use the matrix to transform point s .

Q10 Consider the diagram of figure 3. A robot is set up 1 meter from a table, two of whose legs are onthe y 0 axis as shown. The table top is 1 meter high square. A frame o 1x1y1z1 is fixed to the edgeof the table as shown. A cube measuring 20cm on a side is placed in the center of the table withframe o 2x2y2z2 established at the center of the cube as shown. A camera is situated directly abovethe center of the block 2 m above the table top with frame o 3x3y3z3 attached as shown. Find thehomogeneous transformations relating each of these frames to the base frame o

0xo

0y

0z

0. Find the

homogeneous transformation relating the frame o 2xo2y2z2 to the camera frame o 3xo3y3z3.


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Figure 3


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Tutorial 2 Kinematics

Q1 For each of the planar robots shown in Figure 1, find the forward kinematic equations using thevector-loop method.

For each of the planar robots shown in Figure 1, find the forward kinematic equations using theDenavit-Hartenbert method.

Find the tool position and orientation corresponding to each of the following joint configurations.

(a) Figure (a) : 1 = 45 ; 2 = 60

; 3 = 30

(b) Figure (b) : 1 = 30 ; h = 5; 2 = 45

All angles are measured from the positive sense of the x-axis. The link lengths are given by R 1 = R 2 = R 3 = 10 units.

Figure 1


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Q2 A three link cylindrical robot is symbolically represented in Figure 2. Derive the forwardkinematic equations.

Figure 2

Q3 A Standford manipulator is symbolically represendted in Figure 3. Derive the forward kinematicequations.


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Figure 3

Q4 Derive the forward kinematic equations for the three-link articulated robot shown in Figure 4.

Figure 4


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Q5 Derive the forward kinematic equations for the cylindrical manipulator with a spherical wrist,shown in Figure 5.

Figure 5

Q6 Derive the forward kinematic equations for the PUMA manipulator shown in Figure 6.

Find the tool position and orientation for the following joint configurations:

(a) 1 = 2 = 3 = 4 = 5 = 6 = 0

(b) 1 = 3 = 4 = 5 = 6 = 90 and 2 = 0


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Figure 6


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Q7 The tool of the robot in Figure 7 is at the (x,y) = (40,50) position. Find the corresponding jointco-ordinates. It is required to move the tool to the (x,y) = (45, 55) position. Compute thecorresponding joint co-ordinates.

Figure 7

The link lengths of the robots in Figure 8 are (R1, R2) = (80, 60). Find the joint co-ordinateswhen the tool is at the (x, y) = (60, 50) position. The tool is to be moved to the (x, y) = (55, 45)

position. Compute the corresponding joint co-ordinates.

Figure 8

The planar robot with 2 rotary joints ( 1, 2) and a translator joint (h) is shown in Figure 1(b). Thelink length are R 1 = 4.2 and R 2 = 2.0. It is required to move the tool tip to point P (6,7) with thetool tip pointing at an angle 120 with the positive x-axis. Determine the corresponding joint co-ordinates.

The link lengths of the robot in Figure 1(a) are (R1, R2, R3) = (30, 40, 20). Determine the jointangles when the end effector is at the point (x, y) = 60, 50) with an orientation of 60 .


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Q8 Derive the inverse kinematic equations for the three link cylindrical robot given in Figure 2.

Let d 1 = 10.

Find ( , d2, d3) when the end effector is at point (5, 10, 15).

What is the end effector orientation at this position.

Q9 Derive inverse kinematic equations for the three-link articulated robot shown in Figure 4.

Let R 1 = R 2 = R 3 = 10.

Find 1, 2, 3 when the end effector is at point (7,8,9). What is the corresponding tool orientation.

Q10 Derive the inverse kinematic equation for the cylindrical manipulator with a spherical wrist,shown in Figure 5.

Q11 Derive the invese kinematic equation for the PUMA manipulator shown in Figure 6.


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Tutorial 3 Jacobian Matrices

Q1 Derive the Jacobian matrix for each of the robot arms in Figures 1 to 8 in the Kinematics Tutorial.

Q2 For each of the robot arms in Figures 1 to 8 in the Kinematics Tutorial, derive the singularconfigurations.

Q3 For each of the robot arms in Figures 1 to 8 in the Kinematics Tutorial, it is required to increaseeach of the tool coordinates by 5 unit from the current position. Determine the correspondinginverse kinematic solutions using the Jacobian matrices.

Q4 For each of the robot arms in Figures 1 to 8 in the Kinematics Tutorial, the tool is required toexert a force vector against the environment. All the elements of the force vector is one unit.Find the corresponding joint actuator forces.

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Robotics Tutorial Edited