Robotics and Control : Theory and Practice - Week 2 solution

 If a point  

p(x,y,z)$�(�,�,�)$ in a coordinate frame rotates about the z−axis$�-����$  with angular velocity 0.05 rad$0.05���$. per second, then:

x˙=0.05y, y˙=−0.05x$\dot{�}=0.05�,\dot{�}=-0.05�$

x˙=−0.05y, y˙=0.05x$\dot{�}=-0.05�,\dot{�}=0.05�$

x˙=−0.05x, y˙=0.05y$\dot{�}=-0.05�,\dot{�}=0.05�$

x˙=0.05x, y˙=−0.05y$\dot{�}=0.05�,\dot{�}=-0.05�$

1 point

If the zk${�}_{�}$  and zk+1${�}_{�+1}$  axes of a robot joint coordinate frames are non intersecting then:

xk+1${�}_{�+1}$  is the common normal to zk${�}_{�}$  and zk+1${�}_{�+1}$.

xk${�}_{�}$ is the common normal to zk${�}_{�}$ and zk+1.${�}_{�+1}.$

yk+1${�}_{�+1}$ is parallel to zk.${�}_{�}.$

yk${�}_{�}$ is parallel to zk.${�}_{�}.$

1 point

If  i−1Ti;i=1,2,...6${}^{�-1}{�}_{�};�=1,2,...6$ denotes ith coordinate frame with respect to i−1th$�-1^{�ℎ}$ coordinate frame of a 6 DOF manipulator, then the 4th$4^{�ℎ}$ column of the Jacobian matrix is obtained using:

0T4${}^0{�}_4$

4T6${}^4{�}_6$

3T4${}^3{�}_4$

3T6${}^3{�}_6$

1 point

If
J=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢3100000010000−10010⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥$�=\left[\begin{array}{ccc}3& 0& 0\\ 1& 0& -1\\ 0& 1& 0\\ 0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$

is the manipulator Jacobian of a 3-link manipulator at a particular time and if

q˙=⎡⎣⎢0.010.1−0.2⎤⎦⎥$\dot{�}=\left[\begin{array}{c}0.01\\ 0.1\\ -0.2\end{array}\right]$

 is the joint velocity at that instant then the velocity of the end effector is given by:

[0.03,0.21,0,0,−0.2,0.1]$[0.03,0.21,0,0,-0.2,0.1]$

[0.03,0.21,0.1,0,−0.2,0]$[0.03,0.21,0.1,0,-0.2,0]$

[0,−0.2,0]$[0,-0.2,0]$

[0.03,−0.2,0.1]$[0.03,-0.2,0.1]$

1 point

If x$�$ denotes position and orientation of end effector, qq$�$ denotes joint angles and J denotes Jacobian Matrix then:

x˙=Jq˙$\dot{�}=�\dot{�}$

q˙=Jx˙$\dot{�}=�\dot{�}$

x=Jq$�=��$

q=Jx$�=��$

1 point

The joint co-ordinate transformations of a robot manipulator are given below:
0T1=⎡⎣⎢⎢⎢cosθ1sinθ10000−10−sinθ1cosθ10000101⎤⎦⎥⎥⎥,1T2=⎡⎣⎢⎢⎢100000−10010000q21⎤⎦⎥⎥⎥,2T3=⎡⎣⎢⎢⎢cosθ3sinθ300−sinθ3cosθ30000100051⎤⎦⎥⎥⎥,0<=q2<=6${}^0{�}_1=\left[\begin{array}{cccc}���{�}_1& 0& -���{�}_1& 0\\ ���{�}_1& 0& ���{�}_1& 0\\ 0& -1& 0& 10\\ 0& 0& 0& 1\end{array}\right]{,}^1{�}_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& {�}_2\\ 0& 0& 0& 1\end{array}\right]{,}^2{�}_3=\left[\begin{array}{cccc}���{�}_3& -���{�}_3& 0& 0\\ ���{�}_3& ���{�}_3& 0& 0\\ 0& 0& 1& 5\\ 0& 0& 0& 1\end{array}\right],0<={�}_2<=6$
The arm matrix T=0T3 at θ1=π/2,q2=5 and θ3=0 is given by:$�{=}^0{�}_3��{�}_1=�/2,{�}_2=5���{�}_3=0���������:$

⎡⎣⎢⎢⎢0100100000−10−5051⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 1& 0& -5\\ 1& 0& 0& 0\\ 0& 0& -1& 5\\ 0& 0& 0& 1\end{array}\right]$

⎡⎣⎢⎢⎢010000−10001030−51⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 0& 0& 3\\ 1& 0& 0& 0\\ 0& -1& 1& -5\\ 0& 0& 0& 1\end{array}\right]$

⎡⎣⎢⎢⎢0100100000−1003−51⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 3\\ 0& 0& -1& -5\\ 0& 0& 0& 1\end{array}\right]$

⎡⎣⎢⎢⎢0100100000−1030−51⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 1& 0& 3\\ 1& 0& 0& 0\\ 0& 0& -1& -5\\ 0& 0& 0& 1\end{array}\right]$

1 point

∂T∂q2$\frac{\mathrm{\partial }�}{\mathrm{\partial }{�}_2}$ in (6) for mentioned values is given by:

⎡⎣⎢⎢⎢0000000000000100⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

⎡⎣⎢⎢⎢000000000000−1000⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

⎡⎣⎢⎢⎢00000000000000−50⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -5\\ 0& 0& 0& 0\end{array}\right]$

⎡⎣⎢⎢⎢0000000000000001⎤⎦⎥⎥⎥$\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$

1 point

Second column of manipulator Jacobian in (6) for mentioned values is given by:

[0 −1 0 0 0 0]T$[0-10000{]}^{�}$

[−1 0 0 0 0 0]T$[-100000{]}^{�}$

[0 1 0 0 0 0]T$[010000{]}^{�}$

[1 0 0 0 0 0]T$[100000{]}^{�}$

1 point

For the manipulator shown below, OA=15, BC=10, DE=11, EF=4.

If the position and orientation of the end-effector E with respect to base O is given by:
⎡⎣⎢⎢⎢⎢1/2–√1/2–√001/2–√−1/2–√0000−104351⎤⎦⎥⎥⎥⎥$\left[\begin{array}{cccc}1/\sqrt{2}& 1/\sqrt{2}& 0& 4\\ 1/\sqrt{2}& -1/\sqrt{2}& 0& 3\\ 0& 0& -1& 5\\ 0& 0& 0& 1\end{array}\right]$

What is value of  . Here O and E are revolute joints and A and C are  

θ1+θ4${�}_1+{�}_4$

Prismatic?

π/4$�/4$

π/2$�/2$

−π/4$-�/4$

−pi/2$-��/2$

1 point

In previous question (9), distance variables d2 and d3${�}_2���{�}_3$ are:

 5 and 6 respectively

 5 and 9 respectively

 4 and 6 respectively

 4 and 9 respectively