Differential Form Integral (Work?) on C-Space of Robot Arm?

Question

So, I would like to do an Extra Credit problem for my Vector Analysis course (the last half of a traditional Calc III course) that involves robotics (my new toy). However, it's Week 05 already, I want the Extra Credit released in Week 08, and I haven't gotten as far in my robotics studies as I had hoped.

So, I'm told most robot arms have a 6-dim'l c-space (configuration space). Assuming this is the case and assuming one can do a work integral (e.g.) in the c-space and not in the workspace, would someone who knows more about robotics than me be able to suggest a robot arm with a 6-dim'l c-space $Q^6$ with $H_{1,dR}(Q)\ (= H_1(Q;\mathbb{R})) = 0$, a differential 1-form $\omega$ with $d\omega = 0$ on $Q$ (so that $\omega = df$ for a potential function $f$) representing a force field, and an oriented curve $[C]$ in $Q$ with $\displaystyle \int_{[C]} \omega = f(q)-f(p)$ representing the work done moving the robot arm "through" the c-space? I can change the form and the curve, of course; the robot arm, the c-space, and a coordinate patch for the c-space are the things I don't know yet. More than one robot arm, etc., would be nice.

See this post for doing something similar with a torque field on $SO(3)$.

Answer 1

(So, this isn't a real 6 degrees of freedom answer, but it's a start.)

1. The robot arm consists of two revolute joints and one prismatic joint. The first revolute link is 2 ft long. The second revolute link is 1 ft long. The prismatic link is 0.5 ft long.

2. We model the end effector as weighing 2 lbm $\displaystyle \left( =\frac{1}{16} \text{ slug}\right)$. We model each link as being weightless. We model the air resistance as being 0 and the frictional forces at the joints as being 0. The only force acting is gravity acting on the end effector in the workspace, $\omega = -2\ dz$. Have students show $\omega$ is closed, $d\omega = 0$.

3. The c-space $Q^3$ is a solid torus with another solid torus drilled out of the middle (core). A coordinate patch for $Q$ is $(x,y,z) = \psi(\theta, \phi, s) = \left\langle s[2+\cos(\theta)]\cos(\phi), s[2+\sin(\theta)]\cos(\phi), s\sin(\phi)\right\rangle$ for $(\theta, \phi, s) \in [0,2\pi] \times [0,2\pi] \times [0,0.5]$. Need DH tables, transformation matrices, map from domain of coordinate patch for c-space to workspace, and map from c-space to workspace. $Q$ has de Rham homology $H_{1,dR}(Q) \cong \mathbb{R}^2$. Make up parametrizations of generators of $H_1$. Even though 1-form in workspace has "natural domain" $\mathbb{R}^3$, which is simply-connected, therefore has an antiderivative because it is closed, and therefore the pullback has an antiderivative, have students do periods of $\psi^*\omega$ over generators of $H_1$ to show $\psi^*\omega$ is exact and find antiderivative $\psi^*f(\theta, \phi, s) = -2s\sin(\phi)$, pullback along $\psi$ of $f(x,y,z) = -2z$.

4. Path in coordinate patch domain is $\displaystyle \psi^*\vec{r}(t) = \left\langle t, t, \frac{1}{4\pi}t\right\rangle$ for $\displaystyle t \in \left[0, \frac{\pi}{4}\right]$. Students should do work integral two ways, one with a parametrization of the oriented curve and one with FTC for LI.