Important notes:
- This is actually a question I asked in Physics Stack Exchange and despite being more suitable there I think I would get something interesting here, once I am interested in showing mathematically the robot degrees of freedom.
- I am also very aware that this is a very specific question for a robotic system that is not very common
A Ball Balancing Robot is dynamically stable robot capable of omnidirectional motion. It possesses non-holonomic properties and is a special case of underactuated system, classified as a Shape-Accelerated Underactuated System. About this robot, more specifically, when it comes to the versions:
- Rezero, developed by ETH Zurich
- Ball Balancing Robot developed in Tohoku Gakuin University, present in this paper
Both versions implemented a driving mechanism with three motors symmetrically fixed at intervals of $120$ degrees with zenith angle of $45$ or $40$ degrees. What I am curious about is the number of degrees of freedom of this type of robot. An underactuated system such that the Euler-Lagrange equations of motion for that mechanical system:
$$\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}}} - \frac{\partial{L}}{\partial{q}}= F(q)\tau$$
where, $q\in\mathbb{R}^n$ is the configuration vector $\tau\in\mathbb{R}^n$ is the control input, $F(q)\in\mathbb{R}^{n\times m}$ is the force matrix such that $m<n$ because it is an underactuated system.
Considering the paper Dynamic Constraint-based Optimal Shape Trajectory Planner for Shape-Accelerated Underactuated Balancing Systems , we have the following property of Shape-Accelerated Underactuated Balancing Systems:
There are equal number of actuated and unactuated variables, i.e., $\dim(q_x) = \dim(q_s) = m$ and hence $n = 2m$.
For that reason, I would say the robot has $4$ degrees of freedom, which means $2$ for each orthogonal plane. On the other hand, it does make sense to me that the ballbot possess $3$ degrees of freedom for the body and more $2$ degrees of freedom for the position of the ball. It sums $5$ degrees of freedom.
The ballbot presented in Dynamic Constraint-based Optimal Shape Trajectory Planner for Shape-Accelerated Underactuated Balancing Systems could be described as two inverted pendulum in two orthogonal system and could not rotate in the vertical axis without an additional motor, while the ballbots using three motors with omnidirectional could rotate around the vertical axis.
It means that the first type (with orthogonal rollers) has $4$ degrees of freedom and the one with omniwheels have $5$ degrees of freedom? Or the claim of $4$ degrees of freedom is incorrect?