Thrust force model and control

The classical model for thrust is based on a static free air model that relates thrust

T to rotor speed v. It is given by Mahony et al. [2012]; Martin and Salaun [2010]; Mellinger et al. [2012]; Pounds [2007]

T =CTv2,

whereCT is an aerodynamic thrust constant obtained from static free air tests.

Static thrust models have proven effective in a wide range of applications, how- ever, they display significant errors in the presence of gusts or when the rotor is moving. The strongest effects are due to vertical motion of the rotor or updrafts and downdrafts, although lateral motion or sideways gusts also cause small variations in rotor thrust. Using computational fluid dynamics tools, Luo et al. [2015] showed

§2.5 Quadrotor Modelling 29

that for lateral flight with velocities up to 10m/s, the maximum power saved associ- ated with the additional translational lift is only 6%. Consequently, a constant power flight will result in less that 6% gain in translational lift. The question of providing effective control for multirotors in the presence of wind gusts, or during fast and ag- gressive manoeuvres Kumar and Michael [2012b]; Shen et al. [2013]; Waslander and Wang [2009], or during large displacement of air Martin and Salaun [2010] has been raised by a number of recent papers. One approach that has proven to be highly suc- cessful for repetitive high performance aggressive manoeuvres is to use time varying parameter adaptation and iterative learning Lupashin et al. [2010]; Mellinger et al. [2012]. The resulting controller provides a learnt feedforward compensation for the highly non-linear aerodynamic conditions encountered by each rotor during a given known manoeuvre for which training has been undertaken. In the absence of learn- ing, then it is necessary to either estimate or measure the actual thrust generated by the rotor. Possible approaches include using strain gauges, or directly measuring the airspeed using pitot tubes Yeo et al. [2015]; Arain and Kendoul [2014] or estimation of the aerodynamic state using inertial measurement units (IMUs) Allibert et al. [2014]; Omari et al. [2013]. Direct force measurements using strain gauges suffer from high frequency and high noise to signal ratio JR3 Team [2014]. Direct airspeed measure- ments suffer from accuracy and slow response of pitot tubes. Arainet. al. Arain and Kendoul [2014] used a single pitot tube to measure the forward velocity of the vehi- cle while Yeoet. al. Yeo et al. [2015] used four pitot tubes mounted underneath each rotor to measure the axial velocities through the rotors. From results in Arain and Kendoul [2014], low airspeed wind estimation for quadrotors is a challenge and the airspeed measurements are unreliable for velocities under 1m/s. The authors also obtained errors of up to 2m/s for ground truth forward velocity of 6m/s. Yeoet. al.

Yeo et al. [2015] used his wind estimates in designing safe trajectories, though errors of 0.4m/s were observed for velocity measurements of 1.5m/s. More importantly, typical pitot tubes display response times of 100ms which makes them unsuitable for high performance control of quadrotors.

Shen et. al. Shen et al. [2013] used a Kalman filter to estimate thrust force in the presence of external disturbances arising from their indoor flight experiments. The 50Hz limitation in the communication between the flight control board and electronic speed controllers typical in quadrotor systems limits the performance of such an ap- proach. Another approach that has been proposed is to use analytic implicit models developed from computational fluid dynamics (CFD) Hwang et al. [2014]; Luo et al. [2015] to estimate thrust. These methods also consider the effect of wake interference during translational motion of air. The computational load of such an approach is infeasible for small scale aerial robotic systems.

To obtain a good fit for thrust to rotor speed, the static free air thrust to rotor speed model (T = CTv2) has been modified. This modification was necessary to obtain a good R2 goodness of fit between rotor speed and thrust. The modified state-of-the-art thrust to RPM model is used in the v controller on all closed-loop control electronic speed controllers. For example, ESC32v2 Autoquad Team [2015].

The model is given by

T=CT0v+CTv2, (2.11)

where CTO is an aerodynamic constant at free air conditions. With this model, the

desired rotor speedv_d is determined and regulated with a simple feedforward and proportional integral derivative (PID) feedback controller

va =vf f(vd)−Kp(v−v_d)−K_i Z t

0 (

v−v_d)dt, (2.12)

whereKp,Ki >0 are feedback gains,vf f(vd)is the feedforward voltage andva is the

motor voltage specified as a fraction of the measured bus voltage.

In Orsag and Bogdan [2009, 2012]; Bristeau et al. [2009], the authors applied mo- mentum theory and blade element theory to incorporate translational velocities in the determination of thrust. The major drawback of these works is that they require aerodynamic parameters (for example inflow ratios, induced velocity and angle of at- tack) which are difficult to measure or estimate during flight for the low-cost blades that are used on multirotor aerial vehicles.

A modified version of the static thrust model that relates rotor speed to battery voltage was proposed in 2013 Tang and Li [2013]. In the same year, Podhradsky

et. al. Podhradsky et al. [2013] proposed a similar battery voltage based model for altitude control without current and RPM feedback but included a model for the battery’s state of charge. In all these voltage based thrust models, the models failed to incorporate translational and axial velocities and therefore they algebraically reduce to the static free air thrust model (2.11). Where in this case the coefficientsCTO and CT are now expressed as electrical constants. Furthermore, these models are only

applicable to PWM open-loop ESC control which suffers from battery voltage decay. In addition to using battery voltage, Staub and Franchi [2015] considered the use of an IMU. The state of charge and the rate of discharge vary from one battery to another. This is illustrated in Figure 2.11 where the voltages of four different 4S LiPo batteries awere measured during hovering flights in static free air. It should be noted that the voltage drop of battery 4 was more significant than the other three batteries due to its age.

The problem of producing an accurate thrust model has also been considered in marine thrusters. In the case, where only axial flow is present, accurate thrust computation/estimation and control is a well studied problem for such propulsion systems Bachmayer et al. [2000]; Healey et al. [1995]; Pivano et al. [2009]. In Healey

et. al. Healey et al. [1995], a model that uses the electro-mechanical dynamics of the motor along with propeller hydrodynamics and thin-foil theory were used to produce a two-state propulsion model for marine thrusters. Bachmayeret. al. Bach- mayer et al. [2000] further developed this model to account for positive and negative flow velocities and proposed a method for generating lift and drag curves to achieve more accurate control. Sørensenet. al. Sørensen and Smogeli [2009] proposed a con- troller that does friction compensation and torque limiting through a minimisation algorithm that is unfortunately computationally infeasible in real-time on existing

§2.6 State Estimation for Quadrotors 31