# Monospinner

The Monospinner are Mechanically simplest flying vehicle in existence, which uses only one moving part (the rotating propeller) to hover and control the position of system. The vehicle features no additional actuators or aerodynamic surfaces. This vehicle is controllable in three translational degrees of freedom and two attitude degrees of freedom. The monospinner cannot hover like a standard multicopper. However, an unconventional equilibrium is found by analyzing the vehicle’s dynamics. For a certain constant angular speed and propeller force, the torques are cancelled out by the cross-coupling terms in the altitude dynamics. And the monospinner is able to remain substantially in one position. Feedback control keeps the vehicle near this equilibrium.

In regular flying modes, equilibrium in flight is defined as a hovering state, where all velocities and accelerations of the vehicle are at zero. This is only possible if all forces and moments in the system are balanced. In a flying vehicle with only one rotor, forces and moments are not balanced and as a result, one cannot maintain zero velocities and accelerations in all directions. However, a new equilibrium state can be defined such that, for instance, all the angular velocities are constant and linear velocities are bounded, but periodic. In hover, the monospinner center of mass has a uniform circular motion at a constant height, while the vehicle body is rotating at a constant angular velocity ω¯ B BE in the parallel direction of gravity. Although the vehicle’s attitude is constantly changing due to the non-zero angular velocity, there exists a body-fixed unit vector n, which does not change when expressed in the inertial (ground-fixed) coordinate system. This vector may be thought as an averaged thrust direction of the vehicle. In this equilibrium state, by controlling the direction of this axis(n), the position of the vehicle can be controlled. The average thrust force of the propeller and deviations from this average thrust is used to move the drone in the air, which is used as control inputs. This allows it to determine its position.

If the position error of the vehicle (i.e. the vector to a desired point in space) is defined as P, and its velocity error as P’, where both are expressed in the inertial coordinate system. A desired acceleration commands P”des des is calculated as follows:

### P"des = -2.k.w.P' - w.2.P

Where “ k ” is a damping ratio and “ w ” is a natural frequency. If P” des can be tracked perfectly, the translational deviation P will behave like a damped second-order system. This acceleration command is used as input by the attitude controller.Even with only one motor it can both control the position and the altitude of the vehicle. This is tricky for the Monospinner, as it has only a single input (the thrust force) to control its states (compared to a conventional quadcopter that has four inputs). a cascaded controller is designed in which the faster inner loop controls the thrust direction, while the slower outer loop controls the vehicle’s acceleration and thereby position. Roughly speaking, the single control input (the thrust magnitude) is decomposed into two parts, the average part of the thrust (calculated by the outer loop) which determines the acceleration of the Monospinner and the deviation from the average thrust (calculated from the inner loop) which controls its orientation.

The control strategy only works near hover, where the vehicle spins at a particular angular velocity. To get the vehicle near the operating angular velocity, a passive platform is built or the vehicle is thrown like a frisbee. The mechanism consists of a platform, on which the monospinner rests, connected by a bearing to the ground, so that the monospinner can freely rotate about its vector n (a constant vector along which the vehicle rotates). The rotation is achieved solely through the reaction torque T of the propeller, and the thrust is slowly ramped up from zero to the equilibrium solution. Once sufficiently close to equilibrium, the full control is switched on and the vehicle takes off.

The vehicle’s approximate size and shape are based on the existing trispinner, with a Y-shape. The three major components are always taken to be coplanar and the positions of the battery and the motor are fixed to be two vertices of an equilateral triangle, while the position of the electronics is to be determined. A 2-dimensional grid search of the position of the electronics is then conducted, where two different quality metrics are considered. The first is the probability of input saturation, and is based on the linear, time-invariant model of the attitude system. The second metric uses Monte Carlo simulations of the nonlinear system, including parameter perturbations and noise, to approximate the probability that the resulting vehicle is able to maintain a hover.

Quadcopters can remain in the air even when losing one, two or three of their propellers, the Monospinner almost seemed like a logical step to take things one step further and build a flying machine with no moving parts at all.

Paper by Weixuan Zhang, Mark W. Mueller, and Raffaello D’Andrea

Written by Raghavendra K and Krishna Prajwal.