Assignment I Solution

Part I: Rotation




Rotations and different ways of representation are subject of this part of the assignment. Note that a question may be composed of several sub-questions.




1. Are the matrices R1 and R2 rotation matrices? Give mathematical proof!

R1="
0:6314
0:6301
0:4520
#
and R2 = "
7
5
3
#
:
0:7033
0:2197
0:6761
1
3
4


0:3267
0:7448
0:5818




6
9
2







[6 points]

p p p




Compute the rotation matrix R3 to rotate a point by rad about an unit axis with !^ = [1= 3; 1= 3; 1= 3]T . Show that rotating a point twice by R3 returns it to its original position. Afterwards, compute the corresponding quaternion (R3) and exponential coordinates.



[4 points]




3. From the explicit formula (B.12) in Lynch & Park [1], it can be seen that both quaternions and produce the same rotation matrix and, therefore, lead to same rotated point. Show that both quaternions and produce the same rotated point by using the sandwich product 1, where = 0 + x{ + y| + zk is a pure quaternion embedding the position vector p = [x; y; z]T 2 R3.

[2 points]




Use 1 to rotate the vector p = [0; 1; 0]T about the x-axis by 90 . Check your results by converting the angle and axis to an elemental rotation matrix and perform a matrix multiplication.



[8 points]




Convert the matrices R4 and R5 into the corresponding quaternion by using the method proposed by Shep-perd [2]. Then convert both matrices directly to angle-axis representation by considering one of the three elemental rotation matrices. Afterwards, construct the corresponding quaternion by using the angle-axis-representation and check your previous results.
R4
=
2p




p








0
3
and R5
=
"
0
1
0
#
:
2=2
2=2




p
2=2
p2=2
0








1
0
0








4














5






0
0
1








0


0


1










[10 points]




6. Compute Rz ( ) Ry ( ) Rx ( ) for the angles = = = =2. Can you describe

Rz ( =2) Ry ( =2) Rx ( =2) with an elemental rotation? Determine the angle and rotation axis of this

elemental rotation. Give an alternative set of Euler angles for the ZYX sequence based on your finding. Now, compute the Euler angles for the ZYX sequence based only on the computed matrix. According to [1], the angles can be determined with




= atan2 (r21
; r11) ;


(1)










= atan2
r31; q
r112 + r212
; and
(2)
= atan2
(r32
; r33) ;


(3)



respectively. What is the name of this phenomenon which arises through an unfortunate rotation sequence and angles?




[10 points]




























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Part II: Transformations between Frames




The scene illustrated in Fig. 1 depicts a common test bed for a peg-in-hole task. Six frames are shown in the robot workspace: the base frame fbg, the tool-centre-point frame fgg of the gripper, an intermediate hover frame fig, the peg frame fpg of the peg, the operational frame fhg for the peg-and-hole task, and an auxiliary frame fag. It is assumed that the bar with three holes has the dimension K, 4B, and 2L, where 0 < K < 2L < 4B holds. Further, it is assumed




that the frame of the peg fpg is hp units above the base frame fbg and it is rotated around the z-axis of the base frame fbg by 135 .




Write down the homogeneous transformation matrices Tbg, Tbp, Tba, Tbi and Tbh in terms of the dimension given in Fig. 1.



[5 points]




Compute the transformation matrix Tgp using the results of II-1 and determine the Euclidean distance between the tool-center-point (origin of frame fgg) and the handle of the peg (origin of frame fpg).
[8 points]




3. Compute the transformation matrices Tai and Tih in terms of the dimensions given in Fig. 1.




[12 points]




Find the Euclidean distance between the origin of frame fhg and the origin of frame fag. Now suppose that someone moves the bar with the three holes within the workspace of the robot such that Tba computed in II-1 does not hold any more. Is the previous determined Euclidean distance changing? Show mathematically that for any homogeneous transformation matrix Tba(new) the Euclidean distance between the origin of frame fhg and the origin of frame fag is not affected by Tba(new).
[15 points]




Remember that the bar has been moved, therefore, the transformations Tba, Tbi, and Tbh are no longer valid. However, all other transformations are still correct. How can Tba(new), Tbi(new), and Tbh(new) be determined without moving the bar again? It is assumed that the robot’s joints can be freely moved by the user and that the robot’s software interface can provide Tbg. Further, suppose that the peg can be rigidly attached to the gripper and that the orientation between the peg and hole can be determined, if the peg is in the hole. Outline your approach and provide the transformation sequences leading to Tba(new), Tbi(new), and Tbh(new).



[20 points]




References




Lynch, K.M. & Park, F.C. , Modern Robotics: Mechanics, Planning, and Control, Cambridge University Press, 2017.



Shepperd , S.W. , "Quaternion form Rotation Matrix,"Journal of Guidance and Control, Vol. 41, No. 3, pp. 223–224, 1978.































































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Figure 1: Peg-in-hole testbed. Six frames are shown in the robot workspace: the base frame fbg, the tool-centre-point frame fgg of the gripper, an intermediate hover frame fig, the peg frame fpg of the peg, the operational frame fhg for the peg-and-hole task, and an auxiliary frame fag. All shown dashed lines are aligned with

one of the axes of the base frame f
b
g. The
x
,
y
, and
z
axis of
each right-handed frames are coloured with red, green, and blue, respectively. For the sake of clarity,








T
w.r.t. the auxiliary frame fag. Note that D 6= 2L.
the farthest corner of the bar has the coordinate [2L; 4B; K]