Hi,
I have problems with understanding the problem of underctuation and nonholonomy in Quadcopter system. I cannot find a clear and precise definition of what nonholonomin and underactuated systems are. Could anyone explain to me exactly if a classical 4-rotor Quadcopter is an underactuated and nonholonomic system?
A quadcopter has six degrees of freedom but it has four actuators. That is why it is underactuated. Also, the relation between inputs and outputs is not holonomic and the dynamic extension algorithm must be applied. See Sira-Ramirez, "On the linear control of the quad-rotor system", American Control Conference 2011 for further details and the book of A. Bloch et al for a comprehensive analisys of the non holonomic system.
Mateusz, a non-holonomic system is a system with at least one non-holonomic constraint, i.e. a constraint imposed on its velocities (f(dx/dt, dy/dt...) = 0) that cannot be integrated and thus transformed into a constraint imposed on coordinates (f(x, y...) = 0). Holonomic constraints are treated easily: one can just integrate the constraint equation and then express one of the coordinates as a function of the others, thus reducing the system order and getting rid of the constraint. Such a procedure is impossible for non-holonomic constraints.
The simplest non-holonomic system is probably a wheel rolling on a horizontal table. If we characterize its state by the center's coordinates, x and y, and by azimuth A, then dx/dt - dy/dt tan A = 0, since the wheel center's velocity direction is completely determined by its orientation in azimuth. On the other hand, we cannot write x - y tan A = 0, so the constraint is non-holonomic.
From a purely mechanical standpoint, a quadcopter is not a non-holonomic system. It is just a free flying body with 6 degrees of freedom, which is subject to some forces. Neither holonomic, nor non-holonomic constraints, like a table, are imposed on it.
Nevertheless, some authors try to rewrite the quadcopter's dynamical equations as a direct relation between the coordinates and the control force: f(d2x/dt2, d2y/dt2... F) = 0. When they realize that this equation cannot be explicitly integrated, they call it a non-holonomic constraint. In this, somewhat improper, sense, a quadcopter is a non-holonomic system.
Thank You for both of Your answers.
You gave me things to think about.
A simple intuitive notion of an underactuated system is the following. Think about a 2-degrees of freedom planar robotic arm. Since you have 2 coordinates (angular positions) you will need two motors to move each link independently. However, if you have only one motor (at the bottom of the robot, i.e., theta 1, say) you certainly will be able of moving theta 1 (angular position 1) almost as you wish but theta 2 will not be modifiable with this scheme (you are facing an underactuated system). Theta 2 will move as a result of the action exerted on theta 1 but you can not move it correctly with only one motor. Think about the contrary, an overactuated system (this planar robot but with 4, 5 ... joints and motors). This opens other problems to deal with.I hope this helps
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