Fundamental Actuation Properties of Multirotors: Force–Moment Decoupling and Fail–Safe Robustness

Abstract

In this paper we shed light on two fundamental actuation capabilities of multi-rotors. The first is the amount of coupling between the total force and total moment applied by the propellers to the whole body. The second is the ability to robustly fly completely still in place after the loss of one or more propellers, when the used propellers can only spin in one direction. These two actuation properties are formalized through the definition of some algebraic conditions on the control allocation matrices. The theory is valid for any multi-rotor, with arbitrary number, position and orientation of the propellers, including the more classic ones. As a show case for the general theory we show and explain why standard star-shaped hexarotors with collinear propellers are not able to robustly fly completely still at a constant spot using only five of their six propellers. To deeply understand this counterintuitive result, it is enough to apply our theory, which clarifies the role of the tilt angles and locations of the propellers in the vehicle. The theory is also able to explain why, on the contrary, both the tilted star-shaped hexarotor and the Y-shaped hexarotor can fly with only five out of six propellers. The analysis is validated with both simulations and experimental results testing the control of multi-rotor vehicles subject to rotor loss.

Hierarchical nonlinear control for multi-rotor asymptotic stabilization based on zero-moment direction

“…The arbitrariness of the orientation is fundamental for the feasibility of Problem 1, which is in general solvable only if certain steady-state attitudes are realized by the platform (static hoverability realizability [25]). Nevertheless, a solution can always be found whether matrices F and M satisfy some suitable properties.…”

“…♦ Remark 1 Assumption 1 essentially enables a sufficient level of decoupling between f c and τ τ τ c ensuring the possibility to identify (at least) a direction along which the control force can be freely assigned guaranteeing zero control moment. Referring to the nomenclature introduce in [25], Assumption 1 are fulfilled for platforms having at least a decoupled force direction (D1).…”

“…where ω ω ω d ∈ R 3 is an additional virtual input that should be selected so that the actual input to the translational dynamics (14) eventually converges to the state feedback (25). In other words, ω ω ω d should be set to drive to zero the following mismatch, motivated by (14) and (23),…”

“…For such generic platforms, we propose a non-linear hovering control law that rests upon the identification of a socalled zero-moment direction. This concept, introduced in [24,25], refers to a virtual direction along which the intensity of the control force can be freely assigned being the control moment equal to zero. The designed controller exploits a sort of dynamic feedback linearization around this preferential direction which is assumed to be generically oriented (contrarily to the state-of-the-art multi-rotor controllers).…”

“…The designed controller exploits a sort of dynamic feedback linearization around this preferential direction which is assumed to be generically oriented (contrarily to the state-of-the-art multi-rotor controllers). Its implementation asymptotically stabilizes the platform to a given constant reference position, constraining its linear and angular velocities to be zero (static hover condition [25]). The proposed control strategy requires some algebraic prerequisites on the control matrices that map the motors input to the vehicle control force and torque.…”