Is it possible to control a robotic system with two or more generalized coordinates by one actuator?

Question

I have robotic system that has one actuator as control input. In the dynamic modeling (Lagrange dynamic) of this system and if I choose one coordinate as generalized coordinate the dynamic will be very complicated so I thought that it will be easier if I choose 2 generalized coordinates for this system but I don't know if it is possible to control the dynamic with two or more generalized coordinates by one actuator? that means two degree of freedoms with one actuator input. If yes then how that will be?

Answer 1

This is not possible if the dynamics in each coordinate is the same. Though, I am not sure if in general having different dynamics is a sufficient condition for it to be possible. For linear time invariant systems it is sufficient (if each sub system is controllable). A simple example of this would be

$$ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} a_1 & 0 \\ 0 & a_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \end{bmatrix} u. $$

Namely the controllability matrix can be shown to be

$$ \mathcal{C} = \begin{bmatrix} 1 & a_1 \\ 1 & a_2 \end{bmatrix}, $$

which is full rank if $a_1 \neq a_2$ (which implies that when $u=0$ $x_1$ and $x_2$ have different dynamics). The controller could then be designed using state feedback using for example pole placement or LQR.

Answer 2

As far as I understand, if you consider a mechanical system, then you cannot choose the number of your generalized coordinates. This number is a physical property of your system. If it is not the case, then please clarify the question.

You should probably search for 'underactuated mechanical systems', there exist several works on this topic. In brief: you can force your system to track some trajectories, but not necessarily an arbitrary one. Just think about an inverted pendulum on a cart, where the cart is the only actuated degree of freedom. You can swing up the pendulum, you can stabilize it at any equilibrium, but you cannot stabilize the system in an arbitrary position.