IET Control Theory & Applications
Research Article
On-line aerodynamic identification of
quadrotor and its application to tracking
control
ISSN 1751-8644
Received on 3rd July 2017
Accepted on 21st August 2017
E-First on 26th September 2017
doi: 10.1049/iet-cta.2017.0664
www.ietdl.org
Wenchao Lei1,2, Chanying Li1,2
1Key
Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,
People's Republic of China
2School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China
E-mail: cyli@amss.ac.cn
Abstract: This study investigates the aerodynamic effects and the tracking control problem of quadrotor-type unmanned aerial
vehicles. The authors first present the on-line identification of the aerodynamic parameters by using the recursive least squares
algorithm based on the measurement outputs of the accelerometer. Then, the non-linear discrete-time trajectory tracking
controllers with aerodynamic compensation have been designed. Through identifying and compensating the external
aerodynamics on line, the simulation results show that the tracking performance has been enhanced, especially when the
vehicle is in some flight envelopes where the aerodynamics have significant effects on the quadrotor dynamics, such as the
large-acceleration flight regime.
1
Introduction
An autonomous unmanned aerial vehicle (UAV) has considerable
potential for varieties of applications such as surveillance, search
and rescue, remote inspection and mobile sensor network. With the
rapid improvements in micro-electro-mechanical system, miniature
inertial measurement unit, new material and so on, small-scale
UAVs are growing in popularity. In earlier times, fixed-wing
unmanned aircrafts have been used mainly for military purpose.
Compared with these conventional fixed-wing aerial vehicles,
rotary-wing aerial vehicles are capable of vertical take-off and
landing and hovering above the target. Therefore, they are more
suitable for assignments in limited spaces and for civilian purposes,
such as indoor environments, disaster scenes and farming tasks.
In recent years, the quadrotor UAV, one type of such rotarywing UAV with four rotors, simple configuration, low-cost sensors
and high manoeuvrability, becomes a standard experimental
platform for UAV research, such as flight control method,
simultaneous localisation and mapping, navigation and formation
control [1–3]. The model of quadrotor has been studied by using
rigid body dynamics. The corresponding position and attitude
control methods are mainly presented for a simplified model
derived through the small-angle approximation or the linear
approximation [4–6]. The existing investigations on control
methods mainly include proportional–integral–derivative, feedback
linearisation, sliding mode, backstepping and so on (e.g. [7–12]).
However, as a quadrotor is dynamically underactuated and its
states are highly coupled, most of the existing simplified models
only suit for a quasi-steady flight envelope. For example, a large
number of current works model the quadrotor dynamics without
considering the complex aerodynamics generated by rotors [9–12].
As a result, the coupled relationship between the dynamics and the
aerodynamics has been neglected. Actually, the aerodynamic
effects, such as blade flapping and induced drag, have important
influence on the vehicle flight performance, as discussed in [4, 13].
On the other hand, these effects are amplified on quadrotor due to
its small scale, high sensitivity to aerodynamic perturbation and the
complexity of rotor aerodynamics. Therefore, high-performance
non-linear control methods are needed to achieve stability, fast
response, and robustness to aerodynamic perturbation. In fact,
aerodynamic effects have been thoroughly investigated for
conventional large-scale helicopters [14, 15]. For the small-scale
quadrotor, only a small part of the literature has considered these
aerodynamic effects when studying the model and analysing the
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
control performance [5, 13, 16–19]. More specifically, Bouabdallah
and Siegwart [5] present the aerodynamic model of rotor but only
use it in the simulation model without combining it with the control
problem of the dynamics. Hoffmann et al. in [16] have studied
three separate aerodynamic effects and validated their influence
through flight tests using the Stanford testbed but keep the
aerodynamic identification and controller design as the future
work. After that, they have further investigated these effects in
[17], designed the aerodynamic compensation with a simple
flapping model and validated the result in experiments. Besides,
Omari et al. [13] have also considered the flapping effect and
provided an off-line identification method. Recently, Kaya and
Kutay [18] have modelled the aerodynamics of quadrotor with a
lumped parameter model and identified the unknown parameters
off line through the wind tunnel tests. By using the momentum and
blade element theories established for helicopter [15], Bangura et
al. [19] have studied the aerodynamic theory of quadrotor in detail,
developed several models for different rotor geometries, and
discussed the aerodynamic effects. These models are suitable for
establishing the interaction between rotor aerodynamics and
quadrotor dynamics.
Motivated by the above works, we study the influence of
aerodynamics on quadrotor dynamics and aim to improve the
tracking control performance in this paper. The main contribution
is that we propose a method to identify the aerodynamic
parameters from the aerodynamic model on line by using the
recursive estimation algorithm based on the measurement outputs
of the accelerometer, and then design the non-linear discrete-time
controllers with aerodynamic compensation for the trajectory
tracking problem of quadrotor according to the ‘certainty
equivalence principle’ [20]. The designed control laws are easy to
implement on digital computers. By simulation, we show that,
when the vehicle is in some flight envelopes with large
acceleration, the designed trajectory tracking controllers with
aerodynamic compensation can improve the tracking performance
effectively.
Our results not only are consistent with the previous works on
analysing aerodynamic effects of quadrotors (e.g. [13, 16–18]), but
also provide a more useful on-line implementation method
compared with the off-line (static) test approach [18]. Another
advantage of this work is that the rotor aerodynamics are based on
the blade element theory, so that the proposed models could cover
more flight regimes than the commonly used models which are
3097
Fig. 1 ‘ +’ configuration of the quadrotor
only based on the momentum theory (e.g. [16, 17]), see Remark
1(ii) in Section 2.3. In addition, unlike the additive-affine
disturbance models used in [6], the blade element theory provides
us a more detailed aerodynamic disturbance model, which is
beneficial to the aerodynamic identification and compensation. In
the most works on the trajectory tracking control, the secondary
aerodynamic forces are usually regarded as unknown disturbances
and overcome by the robustness of the designed controllers, see [9–
12]. From this point, the modelling method under the blade
element theory can give us a better understanding of the coupling
relationship between the rotor aerodynamics and the frame
dynamics. Moreover, compared with the quadrotor dynamic
models studied in the majority of the existing literature, our system
models remove the small-angle assumption (e.g. [5, 6, 8, 12]) and
do not use the linear approximation (e.g. [6]). By this way, we
preserve the non-linear characteristics of the original system as
much as possible in our model, so that the true dynamics are better
approximated and the proposed model serves as a favourable
simulation model for the future studies on various control methods.
The remaining of this paper is organised as follows: Section 2
introduces the system modelling of quadrotor dynamics and rotor
aerodynamics. Section 3 presents the aerodynamic identification
method. The non-linear discrete-time trajectory tracking controllers
for the quadrotor and the related performance analysis have been
developed in Section 4. Section 5 presents the simulation results of
the designed tracking controllers with aerodynamic compensation.
Concluding remarks are provided in Section 6.
2
ϖi ∈ ℝ, i = 1, 2, 3, 4 denote the angular speeds of rotor i and
T i, Hi, Qi denote the magnitudes of its thrust, hub-force and drag
torque, respectively. The distance between rotor shaft and CoM is
d. The air density and acceleration of gravity are ρ and g,
respectively. For each rotor with the same geometric properties, we
use r, c, θtw, Nb, Cl and Cd denote its radius, average chord, twist
angle, number of blades, lift and drag coefficients, respectively. For
calculation, let [ ⋅ ]× denote the skew-symmetric matrix related to
the cross product, i.e. [a]×b = a × b, ∀a, b ∈ ℝ3. The notation
^
A, ∀A ∈ ℝ p × q, p, q > 0 denotes the estimation of A. The canonical
basis of ℝ3 is {e1, e2, e3}, where e1 = (1, 0, 0)T, e2 = (0, 1, 0)T and
e3 = (0, 0, 1)T, and the Euclidean norm in ℝn is denoted by ∥ ⋅ ∥.
Finally, for a matrix P ∈ ℝn × n, we use λ(P), λmin(P) and λmax(P)
denote its eigenvalues, minimum eigenvalue and maximum
eigenvalue, respectively.
2.2 Model of quadrotor dynamics
To improve the tracking control performance, we have considered
the horizontal force, usually viewed as unknown disturbance in the
previous works, in the dynamics. The equations of motion is [4]
2.1 Notations
The centre of mass (CoM) of the vehicle is denoted by G. Its mass
and inertia matrix are m and J, respectively. Let
→ → →
ℐ ≜ {O; i o, j o, k o} with a fixed point O and ℬ ≜ {G; i , j , k }
with k pointing downward be the fixed North-East-Down inertial
frame and the right-hand body-fixed frame, respectively. For the
simplicity, a ∈ ℝ3[ ↪ ℐ] (or [ ↪ ℬ]) denotes that the vector
a ∈ ℝ3 is expressed in frame ℐ (or ℬ). Let ξ ∈ ℝ3[ ↪ ℐ] denote
the position of G with respect to (w.r.t.) frame ℐ. The orientation
of frame ℬ w.r.t. frame ℐ is given by a rotation matrix
R ∈ SO(3) ≜ {A ∈ ℝ3 × 3 | AT A = I3, | A | = 1}, where In denotes
the identity matrix in ℝn × n and |A|, ∀A ∈ ℝn × n is the determinant
of A. The velocity is v = ξ̇ ∈ ℝ3[ ↪ ℐ], or denote it by
V ≜ (V x, V y, V z)T = RTv ∈ ℝ3[ ↪ ℬ]. Moreover, Ω ∈ ℝ3[ ↪ ℬ] is
the angular velocity of frame ℬ, w.r.t. frame ℐ. The Euler angle is
denoted by η ≜ (ϕ, θ, ψ)T, where roll angle: −π/2 < ϕ < π/2, pitch
angle: −π/2 < θ < π/2 and yaw angle: −π < ψ < π. Then, let
3098
Ṙ = R[Ω]×,
(2)
JΩ̇ = − [Ω]× JΩ + Γ,
(3)
−H x
F ≜ RT f = −Hy − D,
(4)
−T
d(T 4 − T 2)
d(T 1 − T 3)
Γ=
In this section, we present a modified quadrotor dynamic model,
which reflects the coupling relationship between aerodynamics and
the vehicle states and hence can be used in the aerodynamic
identification and designing the improved tracking controllers, as
will be shown in Sections 3 and 4. We focus on a class of simple
and typical quadrotor with ‘ +’ configuration (see Fig. 1) and
assume that the structure of quadrotor is rigid and symmetrical. At
first, we introduce the following basic notations.
→
(1)
where f ∈ ℝ3[ ↪ ℐ] and Γ ∈ ℝ3[ ↪ ℬ] are the nonconservative
force and torque applied to the vehicle, respectively. In our work,
the force and torque are further expressed as
System modelling
→ → →
mξ̈ = mge3 + f ,
4
∑ ( − 1)iQi
.
(5)
i=1
The body drag D is modelled by
CDxV x
Dx
1
D ≜ Dy = ρS V x2 + V y2 + V z2 CDyV y ,
2
Dz
CDzV z
(6)
where S is a chosen reference area and {CDx, CDy, CDz} ≪ Cd are
the drag coefficients of the whole vehicle [15, 21]. The hub force
H x, Hy generated by rotor are
Hx =
Vx
2
2
Vx + Vy
4
H,
Hy =
Vy
2
V x + V y2
H,
(7)
4
and T = ∑i = 1 T i, H = ∑i = 1 Hi. The rotation matrix is chosen as
[9]
cθcψ
sϕsθcψ − cϕsψ
cϕsθcψ + sϕsψ
R = cθsψ
−sθ
sϕsθsψ + cϕcψ
cϕsθsψ − sϕcψ
sϕcθ
cϕcθ
(8)
with s( ⋅ ) denotes sin( ⋅ ) and c( ⋅ ) for cos( ⋅ ).
Note that we have ignored the un-modelled forces, torques and
the modelling errors so that we can focus on analysing the
aerodynamic effects.
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
2.3 Model of rotor aerodynamics
Ti =
Now, we establish the model of aerodynamic forces and torques in
frame ℬ for the rotors. All the notations with subscript i are
defined for rotor i without declaration.
Let
Cl = Clαα,
Hi =
Qi =
(9)
Cd0
1
1
N ρcr3Clαϖi2
μ − θ λ μ + λi2 μi ,
4 b
Clα i 3 0 i i
(10)
1
N ρcr4Cd0ϖi2 1 + 3μi2 + (T iλi + Hi μi)r,
8 b
(11)
where the symbols ρ, r, c, θtw, Nb are explained in Section 2.1. The
rotor hub advance ratio μi and vertical inflow ratio λi are [15, 19]
λi =
1
ρcr2Cd0 V x2 + V y2ϖi
2
2
− ρcrθtwClα V x2 + V y2 V iI − V z
3
2 1
1
+ ρcClα V x2 + V y2 V iI − V z
,
2
ϖi
(15)
1
3
ρcr4Cd0ϖi2 + ρcr2Cd0 V x2 + V y2
4
4
1
+ T i V iI − V z + Hi V x2 + V y2
,
ϖi
(16)
Cd = Cd0,
1
2
3
N ρcr3Clαϖi2 θtw 1 + μi2 − λi ,
4 b
3
2
μi =
(14)
Hi =
where Clα > 0 is the 2D lift-curve-slope of the blade element and
assumed to be a constant without serious loss of accuracy, α is the
angle of attack and Cd0 > 0 is a constant. By the blade element
theory and [19, Equations (67)–(69) in Section 4], the
aerodynamics of rotor i including the induced drag and torque
caused by the blade flapping effect can be modelled as
Ti =
1
1
ρcr3θtwClαϖi2 − ρcr2Clα V iI − V z ϖi
3
2
+2ρcrθtwClα V x2 + V y2 ,
∥ V xe1 + V ye2 ∥
=
ϖir
V x2 + V y2
,
ϖir
∥ V iI e3 − V ze3 ∥ V iI − V z
=
.
ϖir
ϖir
(12)
(13)
The notation V iI > V z is the vertical induced velocity of the air flow
→
and its direction always coincides with that of k .
Remark 1:
i.
The blade flapping effect enters this aerodynamic model
mainly through the hub force Hi. Its first component with term
(Cd0 /Clα)μi corresponds to the profile drag of the rotor blades
when it pass through the air. The remaining part including
−(1/3)θ0λi μi + λi2 μi is the induced in-plane force induced by the
tilting of the tip path plane, i.e. the blade flapping effect, see
[19, Equations (53)–(59)]. As a result, the last term
T iλi + Hi μi r in (11) has considered both the tilt of the thrust
force and the secondary hub force included in Hi (see [19,
Equations (60)–(63)]), which are the results of blades flapping
phenomenon. Note that, as the influence of blade flapping is so
small compared with the thrust force, it has been neglected
when modelling T i.
ii. Compared with the models based on momentum theory (e.g.
[16, 17], models (9)–(13) can cover more general flight
regimes, such as vortex ring state, turbulent wake state and so
on, because the blade element theory is based on elemental
force and torque which does not require the streamtube model
and ideal fluid assumptions [15].
iii. Bangura et al. [19] have presented a more complex model of
aerodynamics for the ideal rotor blades used in the majority of
quadrotors (see [19, Sections 5 and 6]). In fact, the
identification method proposed later can easily be extended to
this complex model because its expression is similar to the
model considered here, i.e. (9)–(13).
Qi =
where the induced velocity V iI satisfies (see [15])
V iI =
T i /2ρπr2
V x2 + V y2 + (V iI − V z)2
.
(17)
Remark 2:
i.
Expressions (14)–(16) coincide with the models proposed in
[18], where the lumped parameters are calculated through offline wind-tunnel tests and do not used in the control designs.
We will show an on-line identification method in Section 3.
The identified aerodynamic parameters will help us to
compensate the aerodynamic disturbances in real time and lead
to the improved tracking controller design in Section 4.
ii. By combining (14) and (17), V iI can be solved through some
numerical iteration algorithms [15, pp. 97–98], such as the
fixed point iteration and Newton–Raphson iteration by setting
I
V hover
= (mg/4)/(2ρπr2), the induced velocity in hovering
state, as the initial value.
Then, we propose the following rearranged aerodynamic model:
T i = T im + δTi,
(18)
Hi = Him + δHi,
(19)
Qi = Qim + δQi,
(20)
where the main terms are
T im = kTmϖi2,
kTm
Him
= kHm V x2 + V y2ϖi,
kHm
Qim
= kQmϖi2,
kQm
1
ρcr3θtwClα,
3
1
≜ ρcr2Cd0,
2
1
≜ ρcr4Cd0,
4
≜
and the secondary terms δTi, δHi and δQi are the remaining parts of
T i, Hi and Qi, see (14)–(16).
With (18)–(20), the thrust T and torque Γ in (4) and (5) can be
decomposed as
T = T m + δT ,
4
Γ = Γm + δΓ,
(21)
4
where T m = ∑i = 1 T im, δT = ∑i = 1 δTi and
Choosing Nb = 2 (this means that each rotor owns two blades)
and substituting (12)–(13) into (9)–(11), we have
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
3099
Γm =
d(T 4m − T 2m)
d(δT4 − δT2)
d(T 1m − T 3m)
d(δT1 − δT3)
4
∑ ( − 1) Qim
i
,
δΓ =
i=1
4
Φk =
∑ ( − 1) δQ
i
i=1
.
−V x, kφ1, k
On-line aerodynamic identification
The aerodynamic identification is mainly accomplished on line by
virtue of the outputs of the accelerometer and using the recursive
least squares (RLS) algorithm. We assume that S is given, the basic
propeller characteristics c, r, θtw and air density ρ are already
measured through the material properties of rotors and other onboard sensors. Denote the unknown aerodynamic parameters Clα,
Cd0 and CDx, CDy, CDz as
T
ϑC = Cd0, Clα, CDx, CDy, CDz .
A three-axis accelerometer measures the non-gravitational
acceleration acting on the vehicle and its measurement is expressed
in frame ℬ, see [22] for a more detailed discussion. The measured
data can be modelled by [4, 22]
Aaccel =
1
F + Baccel + Waccel,
m
(22)
where Aaccel ∈ ℝ3[ ↪ ℬ] denotes its output, Baccel ∈ ℝ3 is a
constant bias vector and vector Waccel ∈ ℝ3 represents the
measurement noise. We assume that each component of Waccel is
the white Gaussian noise.
Remark 3: In most of the existing works, an accelerometer is used
to obtain the attitude information [4, 23]. The reason is that the
acceleration of a vehicle is very small under most circumstances,
such as indoor environments or the quasi-steady flight modes. As a
result, combining (1) and (4) results (1/m)F ≃ − RTge3, and hence
T
Aaccel ≃ − R ge3 + Baccel + Waccel .
(23)
Then, this approximated model combined with the output of the
three-axis gyroscope can be used to estimate the attitude
information. However, when the vehicle has large acceleration, the
accelerometer cannot provide correct attitude information through
(23). Therefore, as discussed in [13, 22], we use (22) to estimate
the aerodynamic parameters and hence could avoid the smallacceleration assumption. We also emphasise that the effects of
aerodynamic forces and torques are especially important when the
vehicle suffers large acceleration because the aerodynamics change
rapidly in this case, see (14)–(16) and the simulation results shown
in Section 5.
Without loss of generality, we ignore the device bias Baccel as it
can be calibrated in advance, i.e. Baccel = 0 in (22). Let the
sampling period be h and the subscript k means the variables take
values at time t = kh. Then, by (4), (7) and (14) and (15), model
(22) is simplified as the following discrete-time model:
Yk = ΦTk ϑC + Wk,
where
for
T
k ≥ 0,
Yk ≜ yx, k, yy, k, yz, k ∈ ℝ3
k ≥ 0,
3
and
Wk ≜ wx, k, wy, k, wz, k ∈ ℝ denote the values of sensor output
Aaccel and measurement noise Waccel at time t = kh, respectively.
The regression matrix Φk satisfies
3100
Φbody, k
T
∈ ℝ5 × 3,
where
i
Similarly, the hub forces H x, Hy can be calculated directly through
(7) and (19).
Now, these aerodynamic models establish the connection
between rotor aerodynamics and the vehicle states. They will be
used in the following identification process and controller design.
3
1
Φ
m rotor, k
−V x, kφ2, k
Φrotor, k ≜ −V y, kφ1, k
−V y, kφ2, k ,
0
−φ3
and Φbody, k ≜ φ4, kdiag{ − V x, k, − V y, k, − V z, k}. The regressors
φ j, j = 1, 2, 3, 4 are defined as follows:
φ1 =
4
1
ρcr2 ∑ ϖi,
2
i=1
4
4
2
1
1
φ2 = − ρcrθtw ∑ (V iI − V z) + ρc ∑ (V iI − V z)2 ,
3
2 i=1
ϖi
i=1
φ3 =
4
4
1
1
ρcr3θtw ∑ ϖi2 − ρcr2 ∑ (V iI − V z)ϖi
3
2
i=1
i=1
+8ρcrθtw(V x2 + V y2),
1
φ4 = ρS V x2 + V y2 + V z2 .
2
For any given symbol s ∈ {x, y}, denote
T
ϑs ≜ Cd0, Clα, CDs ,
Φs, k ≜
1
T
−V s, kφ1, k, − V s, kφ2, k, − V s, kφ4, k ,
m
and, for s = z,
T
ϑz ≜ Clα, CDz ,
Φ z, k ≜
1
T
−φ3, k, − V z, kφ4, k .
m
Then, we get three linear regression models
ys, k = ΦTs, kϑs + ws, k,
k ≥ 0,
s ∈ {x, y, z} .
For each s ∈ {x, y, z}, the following well-known RLS algorithm
can be used to estimate parameter ϑs [20]:
^
^
ϑs, k + 1 = ϑs, k + Ks, kεs, k,
^
εs, k = ys, k − ΦsT, kϑs, k,
Ks, k = Ps, kΦs, k ΦsT, k Ps, kΦs, k + 1
(24)
−1
,
Ps, k + 1 = Ins − Ks, kΦTs, k Ps, k,
^
where ns is the dimension of ϑs. The initial value ϑs, 0 and
Ps, 0 = δPs, 0 Ins with δPs, 0 > 0 are taken arbitrarily. However, some
better initial value can be chosen if we have some prior information
as discussed in the following Remark 4. Note that (24) can be
easily applied on line in a digital computer.
As the rotor aerodynamic coefficients Cd0, Clα are estimated
^
multiple times at each sampling time, i.e. Cd0, s, k for s ∈ {x, y} and
^
Clα, s, k for s ∈ {x, y, z}, we simply combine them with some
weights βd0, s, k ≥ 0, s ∈ {x, y}, βlα, s, k ≥ 0, s ∈ {x, y, z} in this work,
i.e. the final values of the estimated rotor aerodynamic coefficients
are given by
^
^
^
Cd0, k = βd0, x, kCd0, x, k + βd0, y, kCd0, y, k
(25)
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
^
^
^
Clα, k = βlα, x, kClα, x, k + βlα, y, kClα, y, k
(26)
^
+βlα, z, kClα, z, k,
where ∑s ∈ {x, y} βd0, s, k = 1 and ∑s ∈ {x, y, z} βlα, s, k = 1. According to
the structure of Φs, k, it is reasonable to choose
βd0, x, k
βlα, x, k |V x, k|
=
=
,
βd0, y, k
βlα, y, k |V y, k|
βlα, z, k = 1 − ε,
and ε ≥ 0 is small enough.
^
^
Next, we obtain the estimated rotor aerodynamics T i(t), H i(t),
^
^
Qi(t) and body drag D(t) at each time interval [kh, (k + 1)h] by
replacing the values of Clα, Cd0 and CDx, CDy, CDz in (6) and (14)–
(16) with the estimated values given by (24)–(26) at each sampling
^
^
^
time t = kh. Furthermore, the estimated aerodynamics D, H x, H y,
^
^
^
^
^
^
^
T m, Γm, δT , δΓ and the estimated coefficients kTm, k Hm, kQm could
also be calculated and used in the tracking controllers.
4 Tracking controller design with aerodynamic
compensation
In this section, we show how to design discrete-time control laws
so that the position of the vehicle ξ ∈ ℝ3 could track a given
desired trajectory ξd ∈ ℝ3. The secondary terms of forces and
torques (caused by D, δTi, Hi and δQi) will be compensated through
the aerodynamic identification results simultaneously.
We assume that the states of the quadrotor have been measured
from the onboard sensors or estimated from the embedded state
estimators in the flight controllers with high accuracy [8, 23].
Hence, states ξ, v, η, Ω and ϖi, i = 1, 2, 3, 4 can be used in the
parameter estimators and the feedback controllers directly. Also let
the mass m and the inertial matrix J of the quadrotor be known
(e.g. [24]). In the following controller design, the main thrust T m
and main torque Γm are chosen as the control inputs. In fact, they
are completely determined by four rotor speeds ϖi, i = 1, 2, 3, 4
through the following relationship:
T m, Γm
T
T
= Πm ϖ12, ϖ22, ϖ32, ϖ42 ,
(27)
where the allocation matrix
Πm ≜
kTm
kTm
kTm
kTm
0
−dkTm
0
dkTm
dkTm
0
−dkTm
0
−kQm
kQm
−kQm
kQm
.
The coefficients kTm, kQm are defined in Section 2.3 and their initial
estimated values can be measured through static thrust test [18] as
discussed in the following Remark 4(ii).
Remark 4:
i.
After designing T m, Γm, the rotor speed ϖi, i = 1, 2, 3, 4 are
solved from (27) by replacing the coefficients kTm, kQm in Πm
^
^
with the estimated values kTm, kQm in our simulation.
ii. The static test results correspond to (14)–(16) with V = 0. By
combining (14) and (17), it is easy to calculate that
1
T i, static ≜ kT , staticϖi2 = kTmϖi2 − ρcr2ClαV iI, staticϖi,
2
where
V iI, static ≜
1 1 2 2
2
1
c C + crθ C − cC ϖ .
2 16π 2 lα 3π tw lα 8π lα i
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
According to the parameters given in Section 5, the last term
(1/2)ρcr2ClαV iI, staticϖi is relatively small and hence kT , static can
be chosen as the initial estimated value of kTm. The same
analysis is applied to static torque Qi, static and leads to
Qi, static ≜ kQ, staticϖi2 = kQmϖi2 + T i, staticV iI, static
1
.
ϖi
The considered system is underactuated, since there are six
degrees of freedom and only four independent control inputs.
Moreover, we should note that these inputs are idealised. In
practise, the motor dynamics must overcome the load drag torques
to achieve the desired speeds. Hence, a hierarchical control
architecture is usually adopted for controlling the translational
dynamics [9, 10, 25]. The lowest-level controls the motor speeds
ϖi, i = 1, 2, 3, 4. The middle-level is designed to control the
attitude. Finally, the top-level is in control of position to track the
given desired trajectory. In our work, we do not consider the motor
control loop and hence the control inputs are given by (27) directly.
4.1 Position control
For any given position ξd with velocity vξd ≜ ξ̇d and acceleration
aξd ≜ ξ̈d to be tracked, let the tracking errors be eξ ≜ ξ − ξd,
ev ≜ v − vξd and the virtual position tracking control input
uξd = ge3 +
1
f − aξd .
m
(28)
Under the sampling period h, the translational dynamics (1)
becomes the following discrete-time system:
eξ, k + 1
ev, k + 1
I3
=
0
h2
hI3 eξ, k
I3
+ 2 uξd, k .
I3 ev, k
hI3
(29)
The classical proportional–integral form controller gives us the
following discrete-time feedback control law:
uξd, k = − Kξd, peξ, k + Kξd, dev, k,
(30)
where the diagonal matrix 0 < Kξd, p, Kξd, d ∈ ℝ3 × 3. As we mainly
show how to compensate the aerodynamics in the feedback
controller and its influence on tracking performance, the integral of
the tracking error has not been considered here.
Now, the designed virtual control law uξd, k is used to design
T m, k and give the desired attitude ηd, k (or Rd, k). From the
expression of F in (4), it yields that
f = −R(H x + Dx)e1 − R(Hy + Dy)e2
−R(T m + δT + Dz)e3 .
(31)
Substituting the estimated aerodynamics (see Section 3) into (31)
according to the certainty equivalence principle and combining it
with (28) and (30), the desired attitude should satisfy
^
^
uT , k ≜ Rd, k T m, k + δT , k − 1 + Dz, k e3
^
^
^
^
≃ −Rd, k − 1 H x, k − 1 + Dx, k e1
−Rd, k − 1 H y, k − 1 + Dy, k e2
(32)
−m uξd, k − ge3 + aξd, k .
The notation uT , k ≜ (uT x, k, uT y, k, uTz, k)T[ ↪ ℐ] denotes the desired
thrust.
Remark 5: We have introduced a sampling delay for the desired
attitude Rd in the right hand side of (32) to overcome its
unsolvability, which means that the compensation of hub force has
3101
a sampling delay. The initial value of Rd is set to be Rd, − 1 = I3. At
^
^
^
the same time, the estimated secondary aerodynamics δT , H x, H y
caused by rotors are taking values at t = (k − 1)h, because we have
the rotor speeds ϖi, k − 1, i = 1, 2, 3, 4 and will design ϖi, k,
i = 1, 2, 3, 4 by combining (27), the following (33) and (40) where
the same sampling delay strategy has been used.
Then, by taking the Euclidean norm on both sides of (32) and
note that ∥ Rd, ke3 ∥ = 1, the main thrust is given by
^
^
T m, k = ∥ uT , k ∥ − δT , k − 1 − Dz, k .
(33)
Moreover, with the predesigned yaw angle ψ d, (32) gives the
desired roll angle and pitch angle through
ϕd, k = arcsin
uT x, ksin ψ d, k − uT y, kcos ψ d, k
,
∥ uT , k ∥
(34)
θd, k = arctan
uT x, kcos ψ d, k + uT y, ksin ψ d, k
.
uT z, k
(35)
Hence, we obtain the desired attitude ηd, k = ϕd, k, θd, k, ψ d, k
sampling time t = kh.
T
at
Remark 6: To generate the desired attitude, we require that the
vertical thrust satisfies uTz, k ≠ 0 (subscript z denotes the third
component of a vector). According to (8) and (32), we need
uξd, z, k ≠
^
1 ^
H x, k − 1 + Dx, k sθd, k − 1
m
^
which gives a singular point of uξd, z, k when designing uξd, k by (30).
This can easily be modified as uξd, z, k = ūξd, z, k + ϵ with a small
enough ϵ > 0 in the digital computer.
To estimate the first and second derivatives of ϕd, we view its
third derivative ϕ⃛ d as disturbance and then the following high-gain
observer [26]:
0
e AosdsBo ϕd, k,
(36)
where
−l1 1 0
Ao = −l2 0 1 ,
−l3 0 0
l1
Bo = l2
l3
with constants l1, l2, l3 > 0, gives the estimated values of ϕ̇d and ϕ̈d
^
^
at t = kh, i.e. ϕ̇d, k = xϕ, 2, k, ϕ̈d, k = xϕ, 3, k. The initial value can be set
as
xϕ, 1, − 1, xϕ, 2, − 1, xϕ, 3, − 1
^
^
T
= 0, 0, 0 T. The same observer is
designed to get θ̇d, k and θ̈d, k. Finally, we assume ψ̇ d, ψ̈ d are given.
By combining (34)–(36), the desired value of η̇d, η̈d at t = kh are
^
^
T
^
^
T
−sϕ
(37)
the attitude dynamics can be transformed as
η̈ =
−1
d W −1(η)
dη̇ d W (η)Ω
=
=
Ω + W −1(η)Ω̇
dt
dt
dt
= −W −1(η) Ẇ(η)W −1(η)Ω − Ω̇
(38)
= −W −1(η) Ẇ(η)W −1(η)Ω
+ J−1 [Ω]× JΩ − Γm − δΓ
≜ uηd + η̈d .
Γm, δΓ are defined before and uηd is the virtual tracking control
input for attitude dynamics.
Then, the same discrete-time PD controller design as shown in
(29) and (30) yields
uηd, k = Kηd, peη, k + Kηd, deη̇, k,
(39)
where eη ≜ η − ηd, eη̇ ≜ η̇ − η̇d and the diagonal matrix
0 < Kηd, p, Kηd, d ∈ ℝ3 × 3. Note that the value of η̇ is calculated
through (37) and it is also used to compute matrix Ẇ(η).
Combining the definition of uηd in (38) with (39) results the
following main torque input:
^
+g − aξd, z, k ≜ ūξd, z, k,
h
−sθ
sϕcθ η̇ ≜ W(η)η̇,
cϕcθ
+[Ωk]× JΩk − δΓ, k − 1 .
− H y, k − 1 + Dy, k sϕd, k − 1cθd, k − 1
∫
0
cϕ
Γm, k = J W(ηk) uηd, k + η̈d, k + Ẇ(ηk)W −1(ηk)Ωk
^
xϕ, 1, k
xϕ, 1, k − 1
xϕ, 2, k = e Aoh xϕ, 2, k − 1 +
xϕ, 3, k − 1
xϕ, 3, k
1
Ω= 0
0
η̇d, k = ϕ̇d, k, θ̇d, k, ψ̇ d, k ,
η̈d, k = ϕ̈d, k, θ̈d, k, ψ̈ d, k .
4.2 Attitude control
According to (3) and the following standard kinematic relationship:
(40)
The secondary torque δΓ is replaced by its estimated value
according to the certainty equivalence principle.
4
Remark 7: Let T static ≜ ∑i = 1 T i, static and
d(T 4, static − T 2, static)
Γstatic ≜
d(T 1, static − T 3, static)
4
∑ ( − 1)iQi static
i=1
.
,
Note that, this static thrust and torque allocation structure has been
adopted in a large amount of literature on quadrotor control, such
as [5, 6, 8–12]. Then, compared with the following classical thrust
and torque controllers without aerodynamic compensation
T static, k = ∥ − m uξd, k − ge3 + aξd, k ∥ ,
Γstatic, k = J W(ηk) uηd, k + η̈d, k + Ẇ(ηk)W −1(ηk)Ωk
+[Ωk]× JΩk,
the tracking controllers we proposed in (33) and (40) can not only
identify the allocation matrix Πm on line but also compensate the
estimation of the secondary aerodynamics (such as hub forces and
induced drags) simultaneously. As a result, the final control design
process has utilised the real-time information contained in the
thrust and torque models.
4.3 Performance analysis
To verify the effectiveness of the controllers designed in Sections
4.1 and 4.2, we give a performance analysis of the corresponding
closed-loop systems here.
For the translational dynamics, decompose the non-conservative
force in (4) as F = Fm + δF with Fm ≜ − T me3 and
δF ≜ − (H x + Dx)e1 − (Hy + Dy)e2 − (δT + Dz)e3.
Assume
that
~
^
∥ δF ∥ ≤ M1 and the aerodynamic estimated error δ F ≜ δF − δF
3102
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
~
satisfies ∥ δ F ∥ < ϵ1 for some small non-negative ϵ1 ≪ M1, where
^
^
^
^
^
^
^
δF ≜ − (H x + Dx)e1 − (H y + Dy)e2 − (δT + Dz)e3. For the attitude
~
^
dynamics, define δ Γ ≜ δΓ − δΓ. We also assume that
~
∥ δ Γ ∥ ≤ ϵ2 ≪ M2 with M2 being the bound of the norm of δΓ.
Finally, we ignore the sampling delay introduced in (32), (33) and
(40).
It is easy to see that, under the control laws given by (28)–(33)
and (39) and (40), the error dynamics of both translational
dynamics and attitude dynamics can reduce to the following
cascade form with a proper dimension
ẋ1 = x2,
(41)
ẋ2 = − k p x1 − kd x2 + dx,
k p, kd > 0.
(42)
~
Since ϵ1 ≪ M1, then ∥ m−1 Rδ F ∥ ≪ ∥ m−1 RδF ∥. As a result, by
(43), it is easy to see the ultimate bound of (eξT, evT)T under the
controllers with aerodynamic compensation is significantly smaller
than that in the case without compensation. The similar conclusion
holds for the attitude dynamics.
5
Consider a quadrotor with the constant parameters given in Table
1, where the mass and distance parameters are measured from our
actual vehicle, the rotor aerodynamic coefficients are calculated in
the curve fitting toolbox of MATLAB by using the off-line test
data, and the body drag coefficients are chosen such that
{CDx, CDy, CDz} ≪ Cd0. Finally, the following inertia matrix is
computed according to [21, Equations (1a)–(1e)]:
For the position, by ignoring the underlying attitude tracking error
in (30), we have x1 = eξ, x2 = ev, k p = Kξd, p, kd = Kξd, d and
J=
~
dx = − m−1 Rδ F for the controller with aerodynamic compensation
or dx = − m−1 RδF for the controller without aerodynamic
compensation. Furthermore, the attitude dynamics correspond to
~
x1 = eη, x2 = eη̇, k p = Kηd, p, kd = Kηd, d and dx = W −1 J−1δ Γ or
dx = W −1 J−1δΓ.
As a results, we just need to consider the one-dimensional case
of cascade system (41) and (42). Now, assume |dx | ≤ Mdx for some
constant Mdx > 0 (corresponds to ϵ1, ϵ2 and M1, M2) and let
x ≜ (x1, x2)T, Δ x ≜ (0, dx)T,
A≜
0
−k p
1
.
−kd
k p p22 + kd
1
1
p22
,
Q=2
kp
0
0
kd p22 − 1
0.9560
0
0.1019
0
0.1019
0.9417 0.1019 × 10−2 kg ⋅ m2 .
0.1019 1.8762
In our simulation, the basic non-linear models (1)–(13) and (17)
are viewed as the true system with Euler difference time
Δt = 0.0001 s. The superiority of the proposed controller with
aerodynamic compensation are exhibited in the following two
examples for this quadrotor.
Example 1: For illustrating the tracking performance of the
designed non-linear discrete-time controllers (33) and (40), we
choose the following desired rectangle trajectory motivated by
Huang et al. [17]. For a given end time T end > 0 and constant
v0 > 0, we set
The system (41) and (42) can be rewritten as ẋ = Ax + Δ x. By
k p, kd > 0, we see A is Hurwitz. Then, according to [26, Theorem
4.6], for any given positive definite symmetric matrix Q, there
exists a positive definite symmetric matrix P that satisfies the
Lyapunov equation PA + AT P = − Q. For simplicity, in our
analysis, set
P=
Simulation results
ξd =
0≤t<
T
v0T end
v0T end
, v0t −
, −5 ,
4
4
T end
T end
≤t<
,
4
2
T
3v0T end
v0T end
− v0t,
, −5 ,
4
4
T end
3T end
≤t<
,
2
4
3T end
≤ t ≤ T end,
4
0, v0T end − v0t, − 5 T,
,
T end
,
4
v0t, 0, − 5 T,
which owns large acceleration in the four corners.
The desired yaw angle is ψ d ≡ 0, i.e. always points the north. Note
that, ξd is expressed in frame ℐ and hence the third element of ξd is
where constant p22 > 1/kd > 0.
Choose the Lypunov function as V(x) = xT Px. Then
V̇(x) = xT PA + AT P x + 2xT PΔ x
= − xTQx + 2xT PΔ x
T
Table 1 Parameters of the quadrotor used in simulation
Parameters
values
2
22
≤ − x Qx + 2 1 + p Mdx ∥ x ∥
≤ −(1 − γ)λmin(Q) ∥ x ∥2
2
− γλmin(Q) ∥ x ∥ − 2 1 + p22
Md x ∥ x ∥
≤ −(1 − γ)λmin(Q) ∥ x ∥2 ,
∀∥x∥ ≥
2
2 1 + p22
Md x
,
γλmin(Q)
where γ ∈ (0, 1) and λmin(Q) = 2 min {k p, kd p22 − 1}. Finally,
according to [26, Theorem 4.18], we conclude that, there is a T ≥ 0
such that
∥ x(t) ∥ ≤
2
λmax(P) 1 + p22
Md2 x
2
2
λmin(P)γ min {k p, kd p22 − 1}
,
∀t ≥ T,
where the eigenvalues of P are
λ(P) =
(k p + 1)p22 + kd ±
2
(k p − 1)p22 + kd + 4
.
2
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
(43)
m
d
r
c
θtw
0.985 kg
0.225 m
0.120 m
0.025 m
S
Clα
πd 2
3.320
15∘
Cd0
0.220
CDx
0.010
CDy
0.010
CDz
0.020
kTm
1.508 × 10−5 N/(rad/s)2
kHm
4.772 × 10−5 N/(rad/s)2
kQm
3.436 × 10−7 N ⋅ m/(rad/s)2
g
ρ
9.810 m/s2
1.205 kg/m3
3103
Fig. 3 On-line aerodynamic parameters identification results of Example
1. Note that, the time starts at t = 10 s in this figure
→
can be further seen in Fig. 2, where −k o has been used to denote
the direction of the altitude.
Then, we set T end = 60 s, v0 = 5 m/s, and the sampling period
h = 0.01 s. The initial states of quadrotor are
ξ0 = 0.2, − 0.2, − 0.1 T,
T
ξ̇0 = ξ̈0 = 0, 0, 0 T,
η0 = 1 , − 1 , 5 , η̇0 = η̈0 = 0, 0, 0 T .
The
parameters
of
high-gain
observer
satisfy
l1 = 3ωo, l2 = 3ωo2 , l3 = ωo3 , ωo = 6. The controller parameters are
given by
∘
Fig. 2 Simulation results of Example 1
→
(a) Trajectory tracking results, (b) Tracking results in the direction of − k o, (c)
Induced velocities of four rotors, (d) Time averages of the tracking error norms ∥ eξ ∥
and ∥ eη ∥
→
in the direction of k o, i.e. pointing downward. As a result, we set
the third element of ξd as −5 to get a non-negative altitude. This
3104
∘
∘
Kξd, p = diag{0.3, 0.3, 0.3},
Kξd, d = diag{0.9, 0.9, 0.9},
Kηd, p = diag{0.8, 0.8, 0.8},
Kηd, d = diag{2.4, 2.4, 2.4} .
We need to adjust these parameters so that the attitude control loop
obtains a faster convergence speed than that of the position control
loop.
The trajectory tracking results are shown in Fig. 2a, where the
tracking trajectory without aerodynamic compensation corresponds
to that the estimated secondary aerodynamics in controllers (33)
and (40) are view as zero. In particular, Fig. 2b shows the tracking
→
results in the direction of −k 0, which is consistent with the results
in [17]. More specifically, considering the thrust variation resulting
from aerodynamics could reduce the flight height effectively when
the vehicle encounters braking at the vertices of the trajectory.
Furthermore, we illustrate the change of the induced velocities of
the four rotors in Fig. 2c. It can be seen that there exists rapid
change when the vehicle encounters braking and hence it
influences the rotor thrusts. Finally, the time averages of the
tracking error norms ∥ eξ ∥ and ∥ eη ∥ are illustrated in Fig. 2d,
which coincide with the analysis results given in Section 4.3.
As illustrated in Figs. 2a–d, when the vehicle is in the rising
level, the difference between the conventional controllers without
aerodynamic compensation and the designed controllers with
compensation is small. However, when the vehicle encounters
rapid braking and cornering at the corners (the vertices of the given
desired trajectory), the controllers with aerodynamic compensation
can reduce the tracking error effectively. At the same time, the
difference of the two tracking errors becomes obvious. This
phenomenon is caused by the drastic change of the acceleration
which further leads to the sudden change of aerodynamic forces
and torques. From a theoretical point of view, the sudden change of
the aerodynamic forces and torques at the corners could increase
the bounds M1 and M2 (see Section 4.3), which can be explained by
(14)–(21) and Fig. 2c. However, as long as the RLS estimators are
sufficiently excited by the regression vectors φ j, j = 1, 2, 3, 4 (see
Section 3), the estimation error bounds ϵ1, ϵ2 in Section 4.3 could
keep small enough. Therefore, the ultimate bound of the trajectory
tracking error characterised by (43) under the controllers with
aerodynamic compensation could be smaller than that caused by
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
Finally, Fig. 3 shows the on-line aerodynamic parameter
^
T
identification results with initial values ϑs, 0 = 0.5 Cd0, Clα, CDs ,
T
^
s ∈ {x, y}, ϑz, 0 = 0.5 Clα, CDz , Ps, 0 = 50Ins, s ∈ {x, y, z} and
βlα, z ≡ 1.
To simulate the true accelerometer sensor, a bias vector
Baccel = 0.001, − 0.001, 0.001 T has been added and the
measurement noise vector Waccel satisfies ws, k ∼ N(0, 0.052),
∀s ∈ {x, y, z}. It shows that the RLS algorithm can identify the
^
aerodynamic parameters quickly. Note that, C Dz in Fig. 3 has been
influenced by the accelerometer bias.
Example 2: Here, we choose a circular orbit to show the
performance of the tracking controllers and estimators for the
concerned quadrotor. Let the end time T end = 60 s, if t ∈ [0, 20], we
set
ξd = −28cos
T
π
π
t , 28sin
t , −5 ,
20
20
else if t ∈ (20, 30], we set
ξd = 28cos
T
π
π
t − 20 , − 28sin
t − 20 , − 5 .
10
10
For t ∈ (30, 60], the vehicle will try to repeat this circular route.
We set all the initial states of vehicle as zeros, i.e. the vehicle is
locating below the centre of the circle. The desired yaw angle is
still set to zero.
The parameters of the high-gain observer, controllers and
estimators are chosen as the same as those in Example 1. Figs. 4a–
d show the trajectory tracking results, the altitude tracking results,
the time averages of ∥ eξ ∥ and ∥ eη ∥ and the identification results
of the aerodynamic parameters, respectively. All of the above show
that the tracking performance has been improved by identifying
and compensating the external aerodynamics.
6
Conclusion
In this paper, we have studied a modified quadrotor model and
compensated the identified external aerodynamics in the designed
controllers to improve the tracking performance. The estimation
algorithm of aerodynamics can also be used in other control design
methods. Future works include designing estimators for the more
comprehensive model of aerodynamics proposed in [19] (as
explained in Remark 1(iii)), analysing the closed-loop stability of
the whole system, and applying the estimators and controllers in
the actual vehicle to validate the results through experimentation.
7
[1]
[2]
[3]
[4]
[5]
[6]
Fig. 4 Simulation results of Example 2
→
(a) Trajectory tracking results, (b) Tracking results in the direction of − k o, (c) Time
averages of the tracking error norms ∥ eξ ∥ and ∥ eη ∥, (d) On-line aerodynamic
[7]
parameters identification results. The time also starts at t = 10 s
[8]
the controllers without compensation, as shown in Figs. 2d and 4c
in Example 2.
[9]
IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017
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IET Control Theory Appl., 2017, Vol. 11 Iss. 17, pp. 3097-3106
© The Institution of Engineering and Technology 2017