Thrust computation

The variables in Equations (5.7a)–(5.7c) are CPam, CT, CH, µi,µs, λi,λs, κ and v. As

outlined in Section 5.2, the aerodynamic mechanical power Pam and therefore CPam

can be estimated by measuring the electrical power into the motor and compensating for electrical losses at the local electronic speed controller (ESC) level. Thus, there are seven unknowns CT,CH, µi,µs, λi,λs, κand four constraint equations ((5.7a), (5.7b),

§5.3 Application of Aerodynamic Mechanical Power to Thrust Computation and Control113

(5.7c) and (4.64)).

In a similar manner to drag force and IMU measurement models presented in Section 4.4.7, a number of authors have noted that the horizontal velocities vs_h,vi_h, forceH and thereforeµs,µi,CH are related to the horizontal acceleration of the vehi-

cle Abeywardena et al. [2013]; Allibert et al. [2014]; Martin and Salaun [2010] and can be measured using the accelerometers in an inertial measurement unit (IMU). Using such measurements, it should be possible to resolve the remaining four unknown variables from the algebraic constraints ((5.7a), (5.7c) and (4.64)). Although this ap- pears to be a promising approach, the ESC on a typical quadrotor is not equipped with an accelerometer and the communication link to the central IMU is far lower bandwidth (50Hz for the quadrotors developed in Chapter 3) than the ESC control loop operational frequency (1−2kHz for ESC32v2), making corrections for horizon- tal aerodynamics difficult.

Rather than take such an approach, it can be argued that the contribution of the horizontal variables to the aerodynamics of the rotor is effectively negligible for most aerial robotics applications and can be ignored for low speeds under 10m/s. This claim is supported by computational fluid dynamics results obtained in Luo et al. [2015]. Considering (5.7a) and (5.7c), noting that the µ variable appears as a quadratic µ2. For quadrotors in near hover conditions typical of many robotic applications, the advance ratio µ is naturally small and consequently its square is negligible. Furthermore, withvs

h small, consequentlyvih is also small, such that the

term CH(µs+κµi) ∝ µ2 and can be ignored. Formally, the following assumption is made.

Assumption 5.1. The advance ratioµis small such thatµ2 ≈0within the accuracy of the

aerodynamic model.

This assumption decouples the dependence of (5.7b) with (5.7a) and (5.7c). The horizontal components of forceCH and velocity ratiosµiandµsno longer contribute to the vertical aerodynamics in (5.7a) and (5.7c). This leaves four aerodynamic vari- ables CT, λs, λi and κ along with three constraints (5.7a), (5.7c) and (4.64). It is first required to simplify the notation in the sequel by using the following lumped aerodynamic coefficients c0= R, c1 = 1 2NbρctipR 3_C lα, c2= θtip, c3 = 1 2ρctipNbCd0R 4_.

The final constraint equation is obtained by equating the expression forCT derived

from momentum theory (5.6a) to the CT from blade element theory (5.7a). From

Assumption 5.1, (5.7a) and (5.7c) can be rewritten respectively as

CT =c1[c2−λ], (5.8) CPam =c3+CT κλi+λs c0. (5.9)

Recalling from Assumption 5.1 thatµ2 ≈0, then (5.6a) becomes

CT =c4λiλ, (5.10)

wherec4 =2ρAR2=2ρAc2₀.

The relationship (5.10) along with (5.8) provide the final constraint which relates λi and λs. It is convenient to make this relationship explicit rather than work with the two separate constraint equations. Thus equating (5.10) to (5.8) and collecting terms, yields, c4 λi 2 +λi(c4λs+c1) +c1(λs−c2) =0. (5.11)

In summary, one has aerodynamic variables CT, λs, λi and κ, constraint equations (4.64), (5.8), (5.9) and (5.10), depending on aerodynamic constant coefficients d0, d1

andc0,c1,c2,c3 andc4. These aerodynamic coefficients are determined offline using

linear regression described in Section 5.3.3.

The proposed iterative scheme for solving for thrust using (4.64), (5.8), (5.9) and (5.11) (implemented as described in Section 5.4) is outlined in Algorithm 1. The approach taken is tailored to exploit the fact that once the stream inflow ratio λs is known, it is straightforward to compute CT, κ andCPam sequentially, but difficult to

compute a single function of all variables. Two initial estimates are generated λs₁ and λs₂ based on the previous estimate of λs and a small offset ∆ of the previous estimate. Then for each estimate λs_k, the aerodynamic variables are computed one- by-one. Consider the implicit function

f(λs) =CPam(t)−CPam,

where CPam is the computed value based on the guess of λ

s _{and C}

Pam(t)is the mea-

sured value at the current timet. The goal is to find λs that makes f(λs) →0. The two initial guesses λs₁ andλs₂ form the first two elements of a Newton-Secant itera- tion that converges to the optimumλs. A simple stopping criteria based on decrease in f(λs_k)with a precisioneis used to exit the loop. In practice, the loop usually con- verges in 3-4 iterations (that includes the two initialisation values) and rarely runs more than 5 iterations. The arbitrary limit N > 3 on total iterations of the for loop ensures that the code meets run time requirements — although N = 20 is used in practice and have never seen the for loop run to completion.

For the proposed implicit thrust modelling scheme to work, the coefficientsc0,c1,

c2,c3,c4,d0andd1 need to be identified using methods described in Section 5.3.3