In simpler terms, a holonomic system is when the number of controllable degrees of freedom is equal to the total degrees of freedom.
Consider a mobile robot such as the one depicted to the right, moving in the two-dimensional plane. Imagine that three omnidirectional wheels are mounted on the frame of the robot. Each wheel is described by its coordinates , so that a configuration of the robot can be given by the six scalars . Also, each wheel can impulse a velocity to the robot. However, because all three wheels are connected by the rigid robot frame, their relative velocities are zero (unless the frame breaks):
These velocity constraints integrate into positional constraints
where are scalar constants. The system is thus holonomic.
Let us finally look at the degree of freedom of the robot. We initially used six coordinates to describe a configuration of the robot. Yet, each of the positional constraints "consumes" a degree of freedom. For instance, implies that , i.e., . The coordinate can then be replaced by the appropriate root of this quadratic polynomial. Repeating the process thrice leaves us with three irreducible coordinates, corresponding to the three degrees of freedom of the system.
Note that the simplest generalized coordinates for this system are , where and denote translation along the plane axes, and is the orientation of the robot.
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