Robotics and Control : Theory and Practice - Week 1 solutions

 

1 point

Degree of freedom for a rigid body in space is:

 Three

 One

 Six

 Four

1 point

Finding position and orientation of end effector given the joint variables is known as:

 Inverse Kinematics

 Forward Dynamics

 Forward Kinematics

 Inverse Kinematics

1 point

If vector x⃗ $\overrightarrow{�}$   is given by:

x⃗ =2i^+j^+2k^$\overrightarrow{�}=2\widehat{�}+\widehat{�}+2\widehat{�}$

then direction cosine of x⃗ $\overrightarrow{�}$  is given by

 2,1,2

 2/3,1/3,2/3

cos−12/3,cos−11/3,cos−12/3$��{�}^{-1}2/3,��{�}^{-1}1/3,��{�}^{-1}2/3$

cos2,cos1,cos3$���2,���1,���3$

1 point

If aij;i,j<=4${�}_{��};�,�<=4$denotes the elements of 4×4$4\times 4$ homogeneous transformation matrix, then displacement vector is given by:

[a14 a24 a34]T$[{�}_{14}{�}_{24}{�}_{34}{]}^{�}$

[a11 a12 a13]T$[{�}_{11}{�}_{12}{�}_{13}{]}^{�}$

[a11 a21 a31]T$[{�}_{11}{�}_{21}{�}_{31}{]}^{�}$

[a41 a42 a43]T$[{�}_{41}{�}_{42}{�}_{43}{]}^{�}$

1 point

Three vectors a⃗ ,b⃗  and c⃗ $\overrightarrow{�},\overrightarrow{�}���\overrightarrow{�}$ lie on a plane if:

a.(b×c)=|a||b||c|$�.(�\times �)=|�||�||�|$

a×(b×c)=0$�\times (�\times �)=0$

(b×a).(b×c)=|(b×a)||(b×c)|$(�\times �).(�\times �)=|(�\times �)||(�\times �)|$

(b×a).(b×c)=0$(�\times �).(�\times �)=0$

1 point

If the matrix for a general rotation about a unit vector by an angle θ$�$ is given by:
⎡⎣⎢⎢⎢⎢13√13√13√−12√12√016√16√−26√⎤⎦⎥⎥⎥⎥$\left[\begin{array}{ccc}\frac{1}{\sqrt{3}}& -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{6}}\\ \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{6}}\\ \frac{1}{\sqrt{3}}& 0& -\frac{2}{\sqrt{6}}\end{array}\right]$
Then approximate value of sin2θ$��{�}^2�$ will be:

 0.467

 0.9292

 0

 0.783

1 point

A  triangle with coordinates A(0,0), B(6,0) and C(3,3) rotated about z-axis by 900${90}^0$  anticlockwise, now triangle will be described by:

 A(0,0),B(3,-3),C(0,6)

 A(0,0),B(-3,3),C(0,6)

 A(0,0),B(0,6),C(-3,3)

 A(0,0),B(0,-6),C(3,-3)

1 point

Let F be the coordinate frame with veci,vecj,k⃗ $����,����,\overrightarrow{�}$   as the coordinate axes and M be the coordinate frame with k⃗ ,j⃗ ,−i⃗ $\overrightarrow{�},\overrightarrow{�},-\overrightarrow{�}$   as its coordinate axes. Suppose the coordinates of point p with respect to frame M are M[p]=[1 2 3]T${}^{�}[�]=[123{]}^{�}$. What are the coordinates of p with respect to fixed frame F.

[−3 1 2]T$[-312{]}^{�}$

[−3 2 1]T$[-321{]}^{�}$

[2 1 3]T$[213{]}^{�}$

[−2 1 3]T$[-213{]}^{�}$

1 point

If R$�$ is a 3×3$3\times 3$ rotation matrix, R′${�}^{{}^{\prime }}$ is its transpose, P∈R3$�\in {�}^3$ is a column vector and [R0 0 0P1]$\left[\begin{array}{cc}�& �\\ 000& 1\end{array}\right]$ is a 4×4$4\times 4$ homogeneous transformation matrix, then T−1${�}^{-1}$

[R′0 0 0RP1]$\left[\begin{array}{cc}{�}^{{}^{\prime }}& ��\\ 000& 1\end{array}\right]$

[R′0 0 0−R′P1]$\left[\begin{array}{cc}{�}^{{}^{\prime }}& -{�}^{{}^{\prime }}�\\ 000& 1\end{array}\right]$

[−R′0 0 0−R′P1]$\left[\begin{array}{cc}-{�}^{{}^{\prime }}& -{�}^{{}^{\prime }}�\\ 000& 1\end{array}\right]$

[−R′0 0 0R′P1]$\left[\begin{array}{cc}-{�}^{{}^{\prime }}& {�}^{{}^{\prime }}�\\ 000& 1\end{array}\right]$

1 point

Consider following differential coordinate transformation Matrix represented by

 Vectors n, o, a and p. Which of the following is true?
T=⎡⎣⎢⎢⎢⎢nxnynz0oxoyoz0axayaz0pxpypz1⎤⎦⎥⎥⎥⎥$�=\left[\begin{array}{cccc}{�}_{�}& {�}_{�}& {�}_{�}& {�}_{�}\\ {�}_{�}& {�}_{�}& {�}_{�}& {�}_{�}\\ {�}_{�}& {�}_{�}& {�}_{�}& {�}_{�}\\ 0& 0& 0& 1\end{array}\right]$

n×o=−a$�\times �=-�$

a×n=o$�\times �=�$

o×a=−n$�\times �=-�$

a×o=n