This paper focuses on modeling and control of in-ground-effect (IGE) on multirotor unmanned aerial vehicles (UAVs). As the vehicle flies and hovers over, around, or underneath obstacles, such as the ground, ceiling, and other features, the IGE induces a change in thrust that drastically affects flight behavior. This effect on each rotor can be vastly different as the vehicle's attitude varies, and this phenomenon limits the ability for precision flight control, navigation, and landing in tight and confined spaces. An exponential model describing this effect is proposed, analyzed, and validated through experiments. The model accurately predicts the quasi-steady IGE for an experimental quadcopter UAV. To compensate for the IGE, a model-based feed-forward controller and a nonlinear-disturbance observer (NDO) are designed for closed-loop control. Both controllers are validated through physical experiments, where results show approximately 23% reduction in the tracking error using the NDO compared to the case when IGE is not compensated for.

## Introduction

Multirotor aerial robots, such as quadrotor helicopters (quadcopters), are becoming smaller and more cost-effective to manufacture and deploy. In particular, small form-factor quadcopter unmanned aerial vehicles (UAVs) are used to explore, survey, and even perform high-speed maneuvers in constrained environments [13]. However, a key challenge when operating in confined and compact spaces is the proximity effect, such as in-ground-effect (IGE), which affects flight performance and, in some cases, can lead to system instabilities when the vehicle operates close to the ground, ceiling, and other features [4]. The IGE is caused by airflow interactions between the UAV rotor blades and the surrounding solid boundaries [5]. Thus, flow patterns around the vehicle are altered because of the presence of obstacles, which can lead to: (a) reduced induced velocity, (b) reduced profile drag, and (c) higher rotor efficiency. Most research work on controlling UAVs ignore or avoid IGE by keeping the aerial vehicle far away from obstacles [69].To better understand and compensate for this effect, a new quasi-steady IGE model for multirotor aerial vehicles is proposed and used for motion control. In this work: (1) the ground effect on vehicle dynamics is studied using a new model and experimental validation of the model is performed, (2) a nonlinear-disturbance observer (NDO) is designed and implemented for motion control, and finally, and (3) an IGE model-based feed-forward controller is developed for motion control.

In recent years, the main focus of IGE research is on height control and stability of aerial vehicles when subjected to IGE. The objective is to compensate for the extra force provided by IGE that mainly changes the height dynamics [4,1013]. For example, vertical acceleration data from a quadcopter flying over four different types of obstacles were recorded and used for predicting the throttle adjustment, resulting in a relatively-smooth flight path over an obstacle [4]. A robust height controller and a model-based ground-effect compensator are described in Ref. [10], where a quadcopter tracked a trajectory for landing. Recent work on multirotor platforms has only dealt with height control without considering different IGE on each rotor [13,14]. The ground effect produces an additional disturbance torque when the multirotor aerial vehicle is very close to the ground [15]. This effect on the aerial vehicle's attitude is usually ignored, and its influence on aircraft dynamics has not been fully studied. Although robust control, e.g., integral sliding-mode control, can be used to guarantee the stability of the aerial vehicle without an explicit in-ground-effect model [16,17], the performance and response of the aircraft IGE and out-of-ground-effect (OGE) may differ. Attempts to employ an empirical IGE model into the controller design process have helped to improve stability and tracking performance of the aerial vehicle [10,1719]. For example, an integral sliding-mode controller together with a ground-effect compensator was created to suppress the modeling error in the dynamics and to minimize steady-state error [17]. The ground-effect compensator in this structure acted to correct ground-effect kinematics. Similar ground-effect compensator designs are described in Refs. [20,21]. However, the controller relies on the explicit quasi-steady IGE model, which is typically valid for heights $z>0.5R$, where R is the rotor radius, and does not consider the advance ratio.

Compared to the state-of-the-art methods, the contribution of this work is an empirical quasi-steady IGE model that captures a wider range of operation with respect to the height of the vehicle from the ground. Finite maximum IGE-to-OGE ratio predicted by the model is validated with Blade element momentum theory (BEMT). The model is experimentally verified and used to simulate and evaluate the performance of a NDO and dynamics of the vehicle. Additionally, the IGE model is independently applied to each rotor so that the IGE dynamics are factored into the attitude of the vehicle. This approach has not been explored. Finally, the proposed NDO is shown to effectively deal with the IGE by estimating the extra force caused by IGE for improved closed-loop flight control.

The paper is organized as follows: Sec. 2 reviews the state-of-the-art research on in-ground-effect modeling. Section 3 describes the exponential IGE model and the finite maximum IGE ratio, and provides experimental results validating the model. Dynamics of the UAV in and out of the ground-effect regime are presented in Sec. 4. Design of the nonlinear disturbance observer is presented in Sec. 5. Section 6 presents the simulation and experimental results. Finally, concluding remarks are found in Sec. 7.

## Prior Related Work

There are three main approaches to develop, predict, and analyze rotor IGE: (1) analytical, (2) computational, and (3) empirical approaches. Analytical models started with Knight and Hefner, who proposed the use of an imaginary source below the ground plane [22]. They used a point-source model in which the flow near the propeller is parallel to the rotor disk. Later, the point source was replaced with a ring source, where the strength increases from the axis of rotation to the rotor tip [23]. The derived ring-source model was not used in the control loop directly, due to its computational complexity. Instead, it was used off-line to estimate the height above ground through an airspeed sensor attached below the propeller. BEMT used in IGE analysis incorporates the empirical IGE ratio in the calculation of thrust and power coefficient [2426]. It is assumed that the reduction of inflow follows the empirical model.

Computational approaches use computational fluid dynamics and numerical analysis to provide ground–wake interactions and tip-vortex flow-field predictions [27]. The free-vortex wake model, a relatively new computational approach, was combined with the image-source method, and the resulting IGE model provides rotor thrust, power, and induced velocity for analysis [28]. The model was later extended to analyze the fountain effect [24,25,28], in which two rotors close to each other induce stronger IGE. This effect is also observed experimentally in this work on the quadcopter platform, as described in Sec. 3.3.

Research that study in-ground-effects using computational methods or flow visualization [2934] focus on flow-field analysis to help understand the reason and cause of the behavior (e.g., the brownout and whiteout phenomenon [3236]). Empirical approaches, on the other hand, are concerned more with the output, i.e., thrust and power from the rotor. The empirical IGE models are mostly quasi-steady models that describe the relationship between IGE thrust and rotor height over the ground plane, assuming constant power for a hovering rotor. These models can be used in flight controllers and motion planners directly. A fundamental empirical in-ground-effect model was developed by Cheeseman and Bennett for helicopters [4]. This model defines a quadratic function that relate the output-rotor thrust to the rotor height from the ground, given by
$T(z)T∞=11−(R4z)2, with constant power$
(1)
where T denotes the IGE rotor thrust, and $T∞$ is the OGE hovering thrust. In Eq. (1), R and z represent the rotor radius and height above ground, respectively. The model assumes a constant rotor power and has been implemented in numerous experiments [10,18,3739]. However, this empirical model was developed for single-rotor helicopters that usually have their fuselage underneath the rotor, thus the rotor height is at least $0.5R$. However, the model's assumptions for $z>0.5R$ may not always be true for multirotor aerial vehicles [40], especially when the propellers are installed in the pusher configuration [41]. It was discovered that quadcopters have a stronger ground effect, up to $z=5R$, compared to the prediction given by Eq. (1), where the ground effect is negligible once the rotor is one diameter above the ground [20]. Li et al. [10] also indicated that the single-rotor IGE model might not be suitable for a quadcopter, and thus, a correction coefficient, ρ, was proposed, for example
$TinTout=T∞T(z)=1−ρ(R4z)2$
(2)

where Tin denotes the controller output command thrust from the quadcopter, Tout is the actual output from the four rotors IGE, ρ is the correction coefficient, which varies with platform, and $Tin≈Tout$ when the quadcopter is far from ground.

An alternative quasi-steady model was derived by Hayden [42], given by
$T(z)T∞=0.9926+0.03794×(2Rz)2$
(3)

Bernard et al. [43] compared the behaviors between a single propeller, a quadcopter, and both Cheeseman's and Hayden's models. They concluded that none of these models precisely capture the quadcopter platform IGE. Nobahari and Sharifi [21] suggested that Hayden's model can be implemented on quadcopters by replacing the radius of the propeller with an equivalent larger radius, $Req=5R/2$. Additional extensions to the Cheeseman and Bennett's model are given in Ref. [15], where the partial-ground effect is also considered.

## Multirotor Unmanned Aerial Vehicle In-Ground-Effect Modeling

The reason why a new in-ground-effect model is needed for multirotor UAVs is that most of the existing empirical IGE models share a common singularity problem. Cheeseman's and Bennett's model, and likewise Danjun's model, have a singularity at $z=R/4$. Hayden's model predicted infinite thrust when the rotor-to-ground distance is zero. These models are still applicable in certain situations, e.g., when the dimensions of the fuselage are considered that prevent the rotors from physically touching the ground. In fact, helicopters do not generally hover lower than $z/R<0.5$ [24]. Thus, these models, unfortunately, will fail to adequately predict the IGE for many multirotor UAV platforms because many designs have a low profile and some designs have inverted rotors (rotors located underneath the body). The rotors of these UAVs are usually closer to ground ($z<0.5R$) compared to traditional helicopters. In an attempt to overcome this challenge, an empirical exponential IGE model is proposed that predicts a finite maximum IGE thrust when the rotor-to-ground distance is zero.

### Finite In-Ground-Effect Thrust.

Most of the empirical models presented in Sec. 2 have singularities, but whether the theoretical maximum ground effect ratio is finite or infinite has not been studied before. Thus, BEMT is used to calculate the range of the IGE ratio. According to the literature [22,44], the induced velocity decreases as the rotor approaches the ground plane, therefore it is reasonable to make the following assumption: for a single-rotor IGE in hover, the minimum induced velocity ($minvi$) appears when the rotor is infinitesimally close to the ground plane, and the value is 0 m/s.

Given the minimum-induced velocity, the maximum angle of attack is determined solely by the rotor blade geometry. Thus, the maximum IGE thrust can be calculated using BEMT, where the result is: for a single rotor IGE in hover, the maximum IGE thrust at constant power is finite, and the value is determined by the rotor blade geometry.

The finite-maximum IGE thrust is found as follows: without loss of generality, the blade can be assumed to have a rectangular shape (i.e., pitch angle θ and chord length c are constant across the blade). As defined in Ref. [45], the BEMT thrust-coefficient CT can be expressed as
$CT=12σ∫01Clr2dr$
(4)
where σ is the rotor solidity determined by the chord length, r is the blade-element length along the propeller, and $Cl=Clα(θ−ϕ−α0)$ is the lift coefficient composed of a constant pitch angle θ, zero-lift angle of attack α0, constant two-dimensional lift-curve-slope $Clα$, and inflow angle $ϕ$. For a rectangular blade with constant pitch, the zero-lift angle of attack α0 can be combined with θ into $θ0=θ−α0$, according to Ref. [45]. The inflow angle $ϕ=λ/r$ is a function of both blade element length r and inflow ratio λ. The thrust coefficient of a rectangular-shaped blade with constant pitch becomes
$CT=12σClα(θ03−λ2)$
(5)
According to momentum theory, $λ=CT∞/2$ in the hovering state for a single-rotor OGE. Equation (5) now becomes
$CT∞=12σClα(θ03−12CT∞2)$
(6)
which can be used for solving the OGE thrust coefficient $CT∞$.
From the assumption and letting the inflow ratio $λ=vi/ΩR=0$ when z =0, the maximum IGE thrust-coefficient $CTIGE$ is given by
$maxCTIGE=12σClαθ03$
(7)
Given that $maximumCTIGE$ is finite and $CT∞$ is nonzero for rotors that provide lift, the maximum IGE ratio is
$maxT(z)T∞=maxCTIGECT∞$
(8)
which is finite. For example, given an untwisted blade, uniform inflow, $σ=0.2, θ0=25 deg$, and $Clα=0.2$, the maximum IGE over OGE ratio is $max(T(z)/T∞)≈1.52$. Taking the rotor-tip-loss into consideration [45], Eq. (5) becomes
$CT∞=12σClα(θ03−λ2B)$
(9)

where B is the Prandtl tip-loss factor that represents the increase in average induced velocity for a given thrust. Assuming B =0.9 for a quadcopter's rotor blade, the maximum IGE ratio becomes $max(T(z)/T∞)≈1.7$. This result indicates that the maximum IGE ratio depends on the geometry of the blade, e.g., the twist, solidity, and the rotor-tip-loss.

### Exponential Empirical In-Ground-Effect Model.

Based on the result that for a single-rotor IGE in hover, the maximum IGE thrust at a constant power is finite, the proposed ground effect model that predicts finite maximum IGE ratio is
$T(z)T∞=Cae−(Cb/R)z+1$
(10)

where Ca and Cb are coefficients that depend on the geometry of the blade. More specifically, $Ca=max(T(z)/T∞)−1$. According to the example shown above, for a single rotor, Ca is a function of the rotor twist, the rotor-tip-loss factor, rotor solidity, and the zero-lift angle. Since the analytical solution of inflow ratio λ IGE as a function of rotor height does not exist, the coefficient Cb is determined experimentally. The above example shows a method of predicting the maximum IGE thrust ratio for a single rotor based on BEMT. For multirotor aerial vehicles, both Ca and Cb, can be functions of the airframe geometry (e.g., number of rotors, rotor hub-to-hub distance, etc.). It is found that the model can be fitted to experimental data and accurately predicts IGE for multirotors.

The main difference between the proposed exponential ground effect model and previous models is that the proposed model does not have singularities over the range of possible heights, i.e., $z/R∈(0,∞)$. Thus, this model has broader application and useful for designing model-based flight controllers and motion planners that operate closer to obstacles.

### Experimental Validation of the In-Ground-Effect Model.

A custom-designed ground-effect test stand, shown in Fig. 1, is used to validate the in-ground-effect model given Eq. (10). The test stand is constructed of aluminum alloy and weighs approximately 80 kg. This weight also means that the test stand is less affected by external vibrations and other mechanical disturbances during routine testing. As shown, the quadcopter is mounted to an arm, where at the other end of the arm is attached to a load cell for measuring the thrust as a function of the height of the rotor above ground. The quadcopter is mounted at a distance of 4R away from the test stand to avoid any interference. Propellers are installed in an inverted configuration to allow the rotor-to-ground height to be varied between 0 and 1 m. The height of the vehicle is computer controlled using a VICON motion-capture system (Oxford, UK). The maximum-thrust OGE for the quadcopter used in the experiment with a $2R=8$-in (0.20-m) propeller is 17.6 N. The Crazyflie platform is used as the flight controller for the quadcopter [46]. An Odroid single-board computer running the robot operating system is used to log the load-cell data, control the stepper motor to position the quadcopter up and down relative to ground, with feedback from the VICON system, and publish data to the ground station (see Fig. 1). A three-cell lithium polymer (Li–Po) battery is used to power the quadcopter in order to model the voltage drop effect because of the current draw [47]. The relevant propeller and quadcopter parameters are reported in Table 1.

Fig. 1
Fig. 1
Close modal
Table 1

Parameters of the propeller and quadcopter

VariablesValue
Rotor twistθ45 deg
Average chord length$c¯$0.0150 m
Rotor hub-to-hub distancel0.2336 m
Rotor tip-to-tip distancelt0.0304 m
Quadcopter inertia in xb and ybJx, Jy$1×10−3 kg/m2$
Quadcopter inertia in zbJz$2×10−2 kg/m2$
VariablesValue
Rotor twistθ45 deg
Average chord length$c¯$0.0150 m
Rotor hub-to-hub distancel0.2336 m
Rotor tip-to-tip distancelt0.0304 m
Quadcopter inertia in xb and ybJx, Jy$1×10−3 kg/m2$
Quadcopter inertia in zbJz$2×10−2 kg/m2$

Thrust is recorded as a constant pulse width modulation (PWM) signal, sent to the electronic speed controller (ESC) at different heights. One side effect of using the Li–Po battery is that the voltage will gradually decrease as the power is consumed by the motor. To compensate this side effect, each trial (moving up or down being defined as a trail) with constant PWM is recorded first up, then down, and averaged at the end. Thrust at different PWM values and heights is shown in Fig. 2. The thrust increases as the height decreases to 0 m and, as expected, the PWM-level increases. The results also show that the higher the PWM value (or rotor power), the higher the IGE thrust (see Fig. 2 with z/R between 0 and 2). The normalized ground effect is shown in Fig. 3. Normalized GE is calculated from $T(z)/T∞$, which converts into an IGE ratio. The $T∞$ is the vehicle's thrust OGE, determined by averaging thrust from the aerial vehicle at $z>8R$. As can be seen in Fig. 3, the IGE ratio $T/T∞$ follows approximately the same curve at different PWM levels, indicating that the ratio is independent of PWM level or power. The maximum IGE ratio measured in the experiment is about 1.35 at $z/R=$ 0.015. This provides evidence that the IGE ratio has a finite maximum value. Furthermore, because the ratio is not a function of the PWM level, it can be combined and better represented in a two-dimensional plot as shown in Fig. 4.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

The curve-fitting toolbox in matlab is used to fit the proposed in-ground-effect model given by Eq. (10) to the experimental data. Given no prior knowledge of the blade geometry, both coefficients Ca and Cb in the proposed model are fitted to the experimental result, yielding $Ca=0.44$ and $Cb=1.8$. The box and whisker plots at different heights represent the distribution of the normalized IGE ratio given different PWM input levels. The previously described models from Cheeseman and Bennett [48], Hayden [42], Li et al. [10], and Nobahari and Sharifi [21] are also plotted for comparison. The coefficients in Danjun's and Nobahari and Sharifi's model are fitted to the experimental data using the same process. It should be noted that data below the heights that predict the singularities in previous models are not shown in Fig. 4. It can be seen that these four models follow the experimental data when $z>1.0R$. The modeling errors in the region $z∈(0,0.5R)$ are large, however, compared to the proposed model. The root-mean-square modeling errors (RMSE) with respect to different in-ground-effect models are given in Table 2. The small difference in modeling RMSE between the proposed model and Nobahari and Sharifi's model is because the coefficients in their models helped reduce the error in extreme ground effect compare to other models. However, the lack of accuracy can be seen in Fig. 4.

Table 2

Root-mean-square modeling error for various IGE models

ModelRMS error (%)
Hayden142.8
Cheeseman and Bennett40.4
Danjun37.9
Nobahari and Sharifi7.2
Proposed0.6
ModelRMS error (%)
Hayden142.8
Cheeseman and Bennett40.4
Danjun37.9
Nobahari and Sharifi7.2
Proposed0.6

Additional experiments were performed with a single rotor (see dashed curve in upper-right plot in Fig. 4). It can be seen that the single-propeller IGE ratio is always below the full quadcopter IGE ratio at different heights, which experimentally supports the fountain effect for quadcopters [13]. Further analysis of the fountain effect is needed to fully understand this behavior, including quantifying the ratio with respect to the distance between the rotors. The IGE model for the complete quadcopter system obtained experimentally is used in the remainder of the paper.

## Multirotor Unmanned Aerial Vehicle In-Ground-Effect Dynamics

This section presents the dynamics of the UAV derived from Newton–Euler's rigid body dynamics equations [49]. The dynamics model is then combined with the IGE model presented above.

A multirotor UAV is usually modeled as a six degrees-of-freedom (6DOF) rigid body with its body frame $Fb$ origin located at the center of gravity, the x-axis pointing out the front, the y-axis pointing to the right, and the z-axis pointing down [50] as shown in the free body diagram in Fig. 5.

Fig. 5
Fig. 5
Close modal
The dynamics of a rigid body expressed in the body frame $Fb$ are
$mv˙b+ωb×mvb=Fb$
(11)
$Jω˙b+ωb×Jωb=Mb$
(12)
where m is the mass of the aerial vehicle, $Fb≜[fx,fy,fz]T$ and $Mb≜[τϕ,τθ,τψ]T$ are the total applied forces and moments, respectively, and $vb≜[u,v,w]T, ωb≜[p,q,r]T$ are linear and angular velocities of the vehicle in frame $Fb$, respectively. Since the quadcopter is assumed symmetric about the three axes in its body frame, the inertia matrix is diagonal $J=diag(Jx,Jy,Jz)$. Therefore, the dynamics of the aerial vehicle can be written as follows:
$[u˙v˙w˙]=[rv−qwpw−ruqu−pv]+Rib[00g]+1m[00fz]$
(13)
$[p˙q˙r˙]=[Jy−JzJxqrJz−JxJyprJx−JyJzpq]+[1Jxτϕ1Jyτθ1Jzτψ]$
(14)
The kinematics of the linear and angular velocity based on the relationship between Euler angles $Φ=[ϕ,θ,ψ]T$ and angular rates $[p,q,r]T$ in body frame $Fb$ are given by
$[x˙y˙z˙]=Rbi[uvw], [ϕ˙θ˙ψ˙]=V[pqr]$
(15)
$whereV=[1sϕtθcϕtθ0cϕ−sϕ0sϕcθcϕcθ]$
(16)
and $Rbi$ is the rotation matrix from velocity in the body frame $Fb$ to position $x=[x,y,z]T$ in the inertial frame $Fi$, expressed as
$Rbi=[cθcψsϕsθcψ−cϕsψcϕsθcψ+sϕsψcθsψsϕsθsψ+cϕcψcϕsθsψ−sϕcψ−sθsϕcθcϕcθ]$
(17)

Here, $cϕ, sϕ$, and $tϕ$ denote $cos ϕ, sin ϕ$, and $tan ϕ$, respectively. The dynamics of the 6DOFs aerial vehicle is modeled by Eqs. (13)(16).

The ground effect changes the dynamics of the height response of the vehicle, and it generally provides the system with extra lift [13]. For a multirotor aerial vehicle, due to changes and variations in attitude near ground, each rotor may suffer from different levels of ground effect. Thus, the attitude loop response is highly affected by the IGE. Based on the empirical model described in Sec. 3, the IGE thrust is implemented on each of the rotors, hence
$Ti(zi)Ti∞=Cae−Cbzi/R+1$
(18)
where zi denotes the height of rotor i. The IGE contributes a nonlinear effect on the behavior of the multirotor aircraft. For a small quadcopter with rotor radius of R =0.1016 m, at velocity of $||V∞||=2 m/s$, rotor angular velocity of Ω = 480 rad/s, the advance ratio [51] is
$μ=V∞ΩR≈0.04≪0.15$
(19)
which is relatively small compared to a standard full-size helicopter. So, the advance ratio is neglected in the model.
Based on the empirical model described in Sec. 3, the IGE thrust is implemented on each of the rotors, hence for attitude-loop control with angular velocity feedback [52,53], and proportional-integral-derivative (PID) controller, for height control, are implemented to stabilize the quadcopter. The stability analysis is based on the closed-loop system with constant PID gains. The attitude and height controllers are designed as follows:
$ct=kPzez+kDze˙z+kIz∫ez$
(20)
$cr=kPϕ˙(ϕ˙d−ϕ˙)=kPϕ˙(−kPϕϕ−kDϕϕ˙−ϕ˙)$
(21)
$cp=kPθ˙(θ˙d−θ˙)=kPθ˙(−kPθθ−kDθθ˙−θ˙)$
(22)
$cy=kPψ˙(ψ˙d−ψ˙)=kPψ˙(−kPψψ−kDψψ˙−ψ˙)$
(23)
where ez is the height tracking error and ct, cr, cp, and cy are throttle, roll, pitch, and yaw commands, respectively, that are sent to the mixer given by
$[W1W2W3W4]=[1−1111−1−1−111−11111−1][ctcrcpcy]$
(24)

In the above equation, W1W4 corresponds to the PWM command shown in the experimental test described earlier. These PWM commands are converted to duty cycles and sent to ESCs 1–4, respectively.

The quadcopter system above has eight states, i.e., $[ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,z,z˙]$. It is difficult to evaluate and visualize the region of attraction of the above closed-loop system because of its high dimensionality. The stability or settling time, however, can be visualized and treated as a regulation problem by setting up the system with different initial or desired conditions for angle and height. Angular and linear velocities are set to zero at the start. The time response plot with different initial conditions and desired $ϕ¯,θ¯,ψ¯=0$ is shown in Fig. 6(a1). The dots in Fig. 6 represent the different initial conditions for roll, pitch, and height. The color of the dots denotes the settling time. It can be seen that the nonlinear dynamics result in different settling times for different initial attitudes (the color of the dots at the same height is slightly different, see Fig. 6(a3)). Settling time at lower heights is smaller (see Figs. 6(a2) and 6(a3)). The reason is that, with the help of the ground effect on each rotor, the quadcopter settles back to the hovering state faster at a lower height. The configuration of the multirotor tends to help it return to a level condition ($ϕ,θ,ψ=0$).

Fig. 6
Fig. 6
Close modal

On the contrary, by varying the desired attitude, while having the quadcopter start with zero initial conditions, the result is opposite (see Figs. 6(b2) and 6(b3)). The quadcopter has a hard time tracking and maintaining the desired attitude. This can be coupled with the previously discussed effect and causes difficulties in position control IGE. This explains one of the reasons why most commercial multirotor controllers have to be carefully configured for take-off heights above $z/R>3$ to ensure stability. The simulation results shown in Fig. 6 also show that there does exist attitude disturbance, and it is considerable when the quadcopter is close to ground, i.e., $0 [15]. The disturbance on attitude is not always a stabilizing moment if the quadcopter has a nonzero desired attitude. The ground effect could possibly affect the inner-loop controller and creates oscillations and even increases the settling time.

## Nonlinear-Disturbance Observer for Closed-Loop Control of In-Ground-Effect

From the previous analysis, the ground effect changes the response of multirotor aerial vehicles at different heights. The controller of interest should be able to deal with the ground effect and maintain the system response at different heights. Let the generic form of the dynamics of the system be
$ẍ(t)=f(x,t)+g(x,u,t)$
(25)
where x and u denote the state and the input, respectively. The ground effect is usually modeled as an additional force to the system and can be canceled directly by the control input [17,54]. For multirotor aerial vehicles, the ground effect contributes both force and torque. Thus, the input consists of IGE force and torque
$g(x,u,t)=gn(x,u,t)+gd(x,u,t)$
(26)

where $gn(x,u,t)$ is the nominal input or OGE input, and $gd(x,u,t)$ is the extra input due to IGE loads which can be treated as a disturbance. The nonlinear-disturbance observer from Ref. [55] is applied to the quadcopter system to observe and estimate the IGE. The nonlinear-disturbance observer requires the Euler angles, the angular velocity, and the vertical velocity of the quadcopter. These are all available from an on-board inertial measurement unit (IMU) sensor.

### Nonlinear-Disturbance Observer Design.

The input to the dynamic system can be separated into an OGE input and IGE disturbance. Let the OGE input to the system be torque $Mb=[τϕ,τθ,τψ]T$ and force $Fb=Rib[0,0,mg]T+[0,0,T∞]T$ from Eqs. (13) and (14), respectively. Let the IGE disturbance force and torque be df and $dτ$, respectively. Then, the dynamics of the aerial vehicle can be expressed as
$ẍb=Tf+1/m(Fb+df)$
(27)
$ω˙b=Tτ+J−1(Mb+dτ)$
(28)
where Tf and $Tτ$ are nonlinear Coriolis functions of states x and $Φ$. Differentiating Eq. (15), substituting in Eqs. (13) and (14), and moving the disturbance terms to the left gives
$dτ(t)=JV−1Φ̈(t)−JV−1V˙V−1Φ˙(t)−JTτ−Mb$
(29)
$df(t)=mRibẍ(t)−mRibRbi˙Ribx˙(t)−mTf−Fb$
(30)

Since the ground effect acting on a propeller is along zb, the disturbances on the vehicle are df on along the zb axis and $dτ$ along all three axes in $Fb$. Therefore, the disturbance force along xb and yb are ignored in the observer design. Additionally, the quadcopter does not have actuators to compensate for the disturbance force on xb and yb directly.

Define the state of the observer as $Θ=[ΦT,z]T$; then the disturbance $d=[dτT,df]T$ from Eqs. (29) and (30) can be represented by
$d(t)=J(Θ)Θ̈(t)+G(Θ,Θ˙)−W$
(31)
where $G(Θ,Θ˙)$ is the combination of the Coriolis effect and gravity, $W=[MbT,fz]T$ is the wrench input, and the equivalent inertia matrix for the nonlinear system is given by
$J(Θ)=[Jx0−Jxsθ00JycϕJycθsϕ00−JzsϕJzcϕcθ0000m/(cϕcθ)]$
(32)
The nonlinear-disturbance observer is designed as
$d̂˙(t)=−L(Θ,Θ˙)d̂(t)+L(Θ,Θ˙)[J(Θ)Θ̈+G(Θ,Θ˙)−W]$
(33)

where $d̂$ is the estimated disturbance.

Since no prior information about the derivative of the disturbance is known beforehand to the observer, an assumption is made that $d˙=0$. However, it has been illustrated in Ref. [55] that the designed observer can track time-varying disturbances. Because the angular acceleration is usually not available directly from the onboard IMU, an auxiliary variable vector $ζ(t)$ and a function vector $p(Θ˙)$ are designed such that
$L(Θ,Θ˙)J(Θ)Θ̈(t)=∂p(Θ˙)∂Θ˙Θ̈(t)$
(34)
$ζ(t)=d̂(t)−p(Θ˙)$
(35)
which enables the observer to track the disturbance without calculating the acceleration information. Combining Eqs. (34) and (35) with Eq. (33) yields
$ζ˙(t)=d̂˙(t)−dpdt=−L(Θ,Θ˙)[ζ(t)+p(Θ˙)]+L(Θ,Θ˙)[G(Θ,Θ˙)−W]$
(36)
Equations (36) and (35) are the nonlinear-disturbance observer proposed to estimate the ground effect. Let the auxiliary function vector $p(Θ˙)=cJ*Θ˙$, where $J*=diag(Jx,Jy,Jz,m)$ is a constant matrix and c is a constant observer gain. The function $L(Θ)$ is then
$L(Θ)=cJ*J(Θ)−1$
(37)

where $detJ(Θ)=(JxJyJzm)/ cos ϕ$. The matrix is invertible when the roll angle is not $±π/2$. The reason for adding $J*$ is to keep both L and p small and to prevent them from being extremely large when the observer initializes with error. The disturbance observer can be proved to be globally asymptotically stable [55]. The outputs of the observer are estimated disturbance torques and forces, which are converted to PWM on each rotor and subtracted from the controller outputs W1W4 in Eq. (24) (see Fig. 7(b)).

Fig. 7
Fig. 7
Close modal

### Calculating the Motor Parameters.

The nonlinear-disturbance observer also requires information about the input force and torque, which cannot be measured easily on the vehicle. Converting from force and torque to PWM also relies on the motor parameters; therefore, the motor parameters are needed to estimate the input and to convert the output. Motor parameters kw and kf are found through
$Ωi=kwWi$
(38)
$Ti∞=kfΩi2$
(39)
where Wi denotes the PWM command (same as in Figs. 2 and 3) for the ith motor, and it is assumed that there are no motor dynamics and the desired Ωi can be achieved instantaneously. So, the motor parameters can be combined into one expression
$Ti∞=kfkw2Wi2=kmWi2$
(40)
where km is the motor parameter that requires experiments on the propeller. Since the process of getting the parameter is tedious (usually by curve fitting using the least square method), and the parameter would change with different propellers, an empirical approach is taken. One can estimate the motor parameter easily by solving
$k̂m=mg4W¯2$
(41)

where m is the quadcopter mass and $W¯$ is the averaged PWM command in the hovering state OGE.

## Simulation and Experimental Results

Simulations are based on the full quadcopter system dynamics and the in-ground-effect model presented above. As mentioned in Sec. 5, the quadcopter's stability with angular PID control is affected by the ground effect. Thus, a position command (different heights above ground) would result in different responses.

### Simulation Results

To visualize how the ground effect would affect the dynamics and the response of the quadcopter IGE, a quadcopter with regular PID control is created and analyzed based on the structure shown in Fig. 7.

The outer loop contains a nominal PID controller on four states $[x,y,z,ψ]$. To simulate the dynamics of the motor from PWM signal to thrust, a first-order dynamics model with time constant $τm=0.01$ s is implemented. The proposed empirical exponential in-ground-effect model is augmented to the rigid body dynamics as seen in Fig. 7(c). Figure 8 shows the different tracking results for the same quadcopter, control structure, as well as trajectory with and without in-ground-effect included. The desired trajectory first takes off to a height of 0.5 m and travels to x =1 m, y =2 m along a slope. Note that the disturbance observer and ground-effect compensation are switched off for this test. The desired helical trajectories (at t >30 s) are defined as a spiral in three-dimensional space with a maximum height up to 0.9 m and a minimum height at 0.1 m
$xd(t)=2+0.25 sin(0.3πt)$
(42)
$yd(t)=1+0.25 cos(0.3πt)$
(43)
$zd(t)=0.5−0.4 sin(0.1πt)$
(44)
Fig. 8
Fig. 8
Close modal

The cascade PID controller is tuned for the OGE dynamics and the tracking error of the spiral trajectory in three-dimensional OGE is shown. The response of the height controller with the IGE model has larger tracking error in height. There also exists an interesting coupling effect between the ground effect and the integrator. The IGE height control has less tracking error in the first cycle (see Fig. 8(b)). The quadcopter simulated in this section uses the same model as the one in IGE modeling.

#### Disturbance Observer Simulation.

The disturbance observer and ground-effect compensator are implemented in the inner-loop controller, where the Euler angles, the angular velocity, and the linear velocity along zb can be estimated from onboard sensors (i.e., IMU and barometer). The benefit of designing the controller in the inner-loop system is that the NDO is independent of the reference trajectory. Therefore, the designed controller can enhance the vehicle performance even in “manual control” mode (controlling the aerial vehicle attitude and thrust using a joystick). By feeding back $d̂$ from the NDO and converting to PWM signal, the ground-effect force and torque are estimated online and are subtracted from the command PWM sent to the ESC. This subtraction, however, assumes a small disturbance force and torque so that it can be linearly subtracted. As shown in Fig. 9, maximum tracking error on height with NDO is only 0.05 m, an approximately 60% reduction in tracking error compared to PID control only from Fig. 9(b). The residual tracking error comes from the noise, the error in subtracting the additional PWM from the command signal, and the time-varying disturbances. However, it has been empirically determined that the designed disturbance observer can handle higher frequency time-varying disturbances before the linear PID controller becomes unstable and excites higher order dynamics of the vehicle at 2.5 Hz in the vertical direction. The robust ground-effect compensation controller from Ref. [10] is also simulated, and the tracking error can be seen in Fig. 9(b). Because the IGE model in Ref. [10] has a singularity at $0.25R$, the GE compensation is only applied when the vehicle's height is above $0.5R$, which is based on the range of the original model from Cheeseman and Bennett [48].

Fig. 9
Fig. 9
Close modal

The tracking error along the x and y directions is barely affected by the NDO. The first reason is that the horizontal reference trajectory is a small circle with the diameter of 0.4 m at 0.15 Hz. The average angle of tracking is 1.5 $deg$. The height differences on different rotors are not significant enough to exert the disturbance torque that affects tracking control in x and y directions. Most tracking error comes from the nominal PID controller. Second, the quadcopter used in the simulation is compact and has short arms (see Table 1). Thus, the disturbance torques are not significant.

Fig. 10
Fig. 10
Close modal

### Experimental Results.

The proposed NDO controller is implemented onboard on a quadcopter platform (see Fig. 10) with the Crazyflie flight controller [46]. The platform is the same as the one used in the IGE modeling process. The experiments are carried out in an indoor flight volume with dimensions of 5 m by 5 m by 5 m. Robot operating system is used for communication with the VICON motion capture system for position feedback and motion tracking. The desired trajectory IGE is kept the same as the one used in simulation (see Eqs. (42)(44)). The OGE trajectory in Fig. 11(a) is achieved by shifting the IGE trajectory up by 1.0 m. Comparison of tracking results with regular PID IGE and OGE can be seen in Figs. 11(a) and 11(b). Due to the additional force brought on by the IGE, the regular PID controller cannot bring the quadcopter to the desired height when $0.1 m. The NDO controller compensates for the additional force on the quadcopter and helps the quadcopter reach the desired minimum height (see Fig. 11(c)). The results do show some oscillatory behavior IGE. The main reason for the oscillation is that the NDO uses only the onboard sensor, i.e., the IMU and barometer, which inherently pick up higher noise and have a lower resolution compared to the VICON. The height velocity is estimated by fusing the pressure measurement from the barometer and the vertical acceleration from the IMU. Tracking a time-varying disturbance and the coupling effect between the NDO and the integrator in the PID controller could contribute to the oscillation as well. One of the drawbacks of the NDO is that the modeling error of the motor parameter, vehicle weight, and inertia introduce an initial estimation error on the observed disturbance, resulting in a drop in height when the NDO is switched on (see Fig. 11(c)). Tracking error of height with NDO versus PID control IGE (see Fig. 12) are evaluated at $zd<0.3$ m, where the ground effect is significant on the aerial vehicle. The RMSE of height tracking with NDO and PID are 0.024 m and 0.032 m, respectively. By implementing the NDO, the tracking error reduces by approximately 23%.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
To further test and compare the IGE model, an IGE model-based feed-forward compensator is implemented with the same helical trajectory. The compensation is only implemented for height control, which takes the thrust control output ct from Eq. (20) and divides it by the IGE ratio as follows:
$c′t=ctT(zd)/T∞$
(45)

where $c′t$ is the new input and zd is the desired height of the quadcopter. The square-root operator is used here to factor in the parabolic relationship from the PWM command to the thrust in Eq. (40). Experimental results are shown in Fig. 13. The quadcopter has an overshoot at minimum height, which indicates that the compensator overly reduces the thrust beyond what was needed. The possible reason is that, compared to the quasi-steady IGE (in hovering state with the advance ratio μ = 0), the thrust coefficient IGE could be reduced with a nonzero advance ratio [56,57]. The proposed model is a quasi-steady IGE model and it does not consider the advance ratio. In the experiment, the magnitude of the velocity of the quadcopter is maintained at a constant 0.5 m/s, which corresponds to an advance ratio of $μ≈0.01$. This indicates that further modeling of the IGE should consider the advance ratio even at low velocity magnitudes before it can be used for model-based IGE compensation.

Fig. 13
Fig. 13
Close modal

## Conclusions and Future Work

This paper presented an empirical quasi-steady in-ground-effect model which was subsequently used to design a nonliear-disturbance observer to handle IGE. Demonstration of the finite maximum IGE ratio is given with BEMT. A nonlinear-disturbance observer is designed to estimate the ground effect, and the estimated results are used to compensate for the disturbance on both force and torque. Simulation results show that the nonlinear disturbance observer can effectively compensate for the ground effect and maintain the system response at different heights. The proposed exponential IGE model with finite maximum IGE ratio and the NDO are validated experimentally, together with an IGE model-based feed-forward compensator. NDO experiment results show the effectiveness of the disturbance rejection of the quadcopter IGE. The feed-forward compensation results provide direction for future modeling of the ground effect.

The further extension includes aerodynamic analysis of the ground effect on multirotor systems (e.g., the partial ground effect where only part of the vehicle is in the IGE region) and the effect due to changes in the angle of attack.

## Funding Data

• National Science Foundation, Partnership for Innovation Program (Grant No. 1430328; Funder ID: 10.13039/501100008982).

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