Problem:
Solution:
a)
I will skip this one, the picture is pretty obvious.
b) 3 variables.
The angle $ \theta_1 $ between the first arm and an arbitrary axis on the plane
The angle $ \theta_2 $ between the second arm and first arm
The angle $ \theta_3 $ between the third arm and the second arm.
c)
If we define the "state" as the space for all the angles, then they are all independent! They are in the space of $ [-\pi, \pi]^3 $, no equation is needed to govern them.
However, if we are thinking about the "state" as position and orientation, then it can get quite complex, we can still reason it with affine transforms, we simply stack the transforms up.
$ \left(\begin{array}{ccc} \cos \theta_1 & -\sin \theta_1 & 0 \\ \sin \theta_1 & \cos \theta_1 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} \cos \theta_2 & -\sin \theta_2 & 0 \\ \sin \theta_2 & \cos \theta_2 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} \cos \theta_3 & -\sin \theta_3 & 0 \\ \sin \theta_3 & \cos \theta_3 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right) $
The product is the affine transformation that takes a coordinate (in homogeneous coordinates) in the third arms frame (i.e. origin at third arm's tip, x coordinate axis parallel with the third arm) to the world frame.
d)
The intuitive dimension of the variety of states is 3.
Solution:
a)
I will skip this one, the picture is pretty obvious.
b) 3 variables.
The angle $ \theta_1 $ between the first arm and an arbitrary axis on the plane
The angle $ \theta_2 $ between the second arm and first arm
The angle $ \theta_3 $ between the third arm and the second arm.
c)
If we define the "state" as the space for all the angles, then they are all independent! They are in the space of $ [-\pi, \pi]^3 $, no equation is needed to govern them.
However, if we are thinking about the "state" as position and orientation, then it can get quite complex, we can still reason it with affine transforms, we simply stack the transforms up.
$ \left(\begin{array}{ccc} \cos \theta_1 & -\sin \theta_1 & 0 \\ \sin \theta_1 & \cos \theta_1 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} \cos \theta_2 & -\sin \theta_2 & 0 \\ \sin \theta_2 & \cos \theta_2 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} \cos \theta_3 & -\sin \theta_3 & 0 \\ \sin \theta_3 & \cos \theta_3 & 0 \\ 0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right) $
The product is the affine transformation that takes a coordinate (in homogeneous coordinates) in the third arms frame (i.e. origin at third arm's tip, x coordinate axis parallel with the third arm) to the world frame.
d)
The intuitive dimension of the variety of states is 3.
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