Mikhail Yu. Shavin1
1Moscow Institute of Physics and Technology, Dolgoprudny, Russia
We propose a control framework for a tilt-motor quadrotor. By letting the rotors with propellers tilt with respect to the quadrotor main body, we increase the number of the system’s control inputs and decouple the UAVs translational and rotational motion. The schematic of the quadrotor prototype is shown in Fig. 1. There is a hub in the center with four beams supporting the four rotors.
The prototype implements the so-called x-configuration, and the beams’ ends are vertices of a square with the center in the center of mass of the main body.
The adjacent propellers have different positive directions of rotation. Each rotor can be tilted around the beam it is mounted on.
1 3 2
4 X
Y Z
xi yi
zi
θi
Fig. 1. Quadrotor prototype and reference frames
The UAV’s dynamics is described with respect to an inertial reference frame, which is designated by the upper indexI. Upper indexB denotes the body-frame of the quadrotor, whose axes coincide with the main body’s princi- pal axes of inertia. We also use an auxiliary body-frameXYZ, whose axes XandYare aligned with the beams andZcoincides withZ (XYZ can be arrived at by rotating theXY Zframe aroundZ by−π/4). Upper indices Ridenote the four tilting rotors’ reference frames.
Position and attitude of the quadrotor are described by three coordinates of its center of mass comprising vectorrI, and the attitude quaternionqIB, which relates the quadrotor body frame and inertial frame. The translational dynamics
of the quadrotor of massM is given by
M¨rI =FIg+FIa+FIt, (1) whereFIgis the gravity force,FIais the aerodynamic drag force, andFItis the thrust force with absolute valuekω˜i|ω˜i|,k being the empirical aerodynamic coefficient andω˜ithe propeller rates.
The rotational motion equations take the form JBΩ˙B+ΩB×JBΩB=
τBi+
rBi×FBt,i, (2) τBi =τBa,i−qBRi◦(JRiω˙Rii+ωRii×JRiωRii)◦q˜BRi,
whereωi are the rotors’ angular velocities,τa,iare the aerodynamic torques with absolute valuesbω˜i|ω˜i|,bbeing the empirical aerodynamic coefficient.
Equations (1) and (2) along with the quaternion kinematic equations com- prise the closed system, which is used to simulate the dynamics of a tilt-motor quadrotor.
Given the required trajectoryr0(t)and attitudeq0(t)of the quadrotor, we denote byΔr(t) =r(t)−r0(t)andΔq(t) = ˜q0(t)◦q(t)the difference between the required and actual position and attitude. The following combinations of the propellers’ rotation ratesω˜iand the propellers’ tilt anglesθiare chosen as control inputsu:
u= ωu
θu
, ωu = (˜ω1|ω˜1|,ω˜2|ω˜2|,ω˜3|ω˜3|,ω˜4|ω˜4|)T, θu = (θ1, θ2, θ3, θ4)T.
The PD controllers for decoupled position and attitude tracking systems is given by:
¨r(t) = ¨r0(t) +Kr1
r˙(t)−r˙0(t)
+Kr2
r(t)−r0(t) , Ω˙ (t) = ˙Ω0(t) +KΩ1
Ω(t)−Ω0(t)
+KΩ2Δq, (3) whereΔqis the vector part of the attitude error quaternion,KriandKΩiare diagonal controller gain positive-definite matrices. The exponential conver- gence of both position and attitude errors to zero is guaranteed by the proper choice of gain coefficients in (3), which is determined by Routh-Hurwitz crite- rion.
To implement a control loop according to (3) we invert the system’s dy- namics. This is achieved by rearranging the model equations as follows. We rewrite the equations (1) and (2) in body-frameXYZ, whose axesXandY
are aligned with the beams. We then leave in the right-hand sides of the equa- tions only those terms that explicitly depend on the chosen control parameters u, moving all other terms to the left-hand side. Thus we obtain:
F(¨r,r˙, qIB) =kFthr(θu)ωu, T
Ω˙,Ω
= (kLTthr(θu)−bTaero(θu))ωu, (4) whereLis the length of each beam and the matricesFthr,Tthr, andTaeroin the right-hand side are expressed in terms of the rotor tilt angles. Calculation of the left-hand sides of (4) requires the estimates of the state vector variables rI,qIB,r˙I,ΩBand the controller (3) output – ¨r(t)andΩ˙ (t). Then for given F(¨r,r˙, qIB) = (FX,FY,FZ)T andT( ˙Ω,Ω) = (TX,TY,TZ)T the system (4) consists of six equations, and there are eight control parameters to be de- termined from it. We may also point out that in theXYplane these relations are decoupled as theXcomponent equations contain only control parameters with indices 2 and 4, whereas theYcomponents ofFandT are only related by (4) to control parameters with indices 1 and 3. At the same time all the control parameters are present in theZcomponent equations.
It is shown that we can completely decouple the equations (4) with re- spect to the control parameters pertaining to the pairs of nonajacent rotors, by adding certain additional relations responsible for the force and torque balance between the pairs of rotors attached to the parallel beams. Substituting the balance equations into (4), we obtain a system of eight equations that can be split up into two subsystems of four equations, which both admit an analytical solution with respect to the control parameters.
ωui= 1
2kL ε2T+Ai(F,T)εT+Bi(F,T), θi= arctan Piθ(F,T) Qθi(F,T)+εT
, (5) whereεT is the balance equations tuning parameter,Biis a quadratic form in the components ofF andT. Ai,Pi andQi are first degree polynomials in the components ofFandT. It is very straightforward, then to find from the first set of equations (5) the range ofεT to satisfy the rotor rates constraint
|ω˜i| ≤ ω˜max and check from the second set of (5) whether there exist such values ofεTin this range that satisfy the tilt angles constraint|θi| ≤θmax. If the solution inεTis not found we can go back to the controller (3) and scale its output. If the solution is not unique, one can pick the value forεT to minimize the power consumption (expressed as|ωu|).