Robust Attitude Controller Design for an Uncommon Quadrotor With Big and Small Tilt Rotors

Uninhabited aerial vehicles (UAVs) have drawn a lot of interest in both academic research and commercial areas in the last decade. Among all UAVs, quadrotors have become the most common configuration due to their vertical takeoff and landing, hover in place, and maneuverability abilities. The reduction of cost and complexity has increased the use of quadrotors in various military and civil applications [1]. Apart from these favorable properties, quadrotors suffer from high energy consumption. Therefore, the capability of the power system has limited the usage of quadrotors in different applications.

Power source and efficiency of transferring that energy to the air via rotors both limit flight endurance and payload capacity. Typical small commercial quadrotors have less than 20-min flight endurances and payload sizes less than 1 kg [2,3]. This is due to the inadequate stored energy per unit mass of batteries to supply brushless DC motors. For multirotors, the weights of the vehicles are mostly determined by batteries [2]; thus, increasing battery sizes does not contribute much to flight duration since it also increases required power. Therefore, significant progress in flight endurance requires improvements to current battery technology. Since this is a slow process, other methods, such as using an alternative power source and increasing the efficiency of a lifting system, have been studied recently.

For a typical multirotor, the power requirement to maintain thrust over a rotor disc area changes exponentially with total weight and the inverse of the rotor disc area [3]. Consequently, quadrotors suffer from lower efficiency compared to helicopters because of the smaller rotor area in the given unit diameter footprint. On the other hand, the mechanically complex rotor assemblies of helicopters are fragile and require intensive maintenance. Therefore, combining simple and robust quadrotor structure and efficiency of helicopters has also been studied recently.

The triangular quadrotor (Fig. 1(a)) introduced in Ref. [3] uses large rotor in the center to provide lift and canted three small rotors for control and counter torque. Even if a 15% efficiency improvement is reported, this configuration gives degraded hover attitude control performance due to the uncompensated gyroscopic torque of the main rotor compared to a standard quadrotor. In Ref. [4], bi-copter with tilting rotors (Fig. 1(b)) is introduced, and its theoretical efficiency improvement over traditional configurations is reported. For that configuration, roll torque depends on the vertical distance between the center of gravity and the plane of the propellers. Therefore, the roll response of this vehicle is slow since limited torque can be obtained due to this small distance. In addition, roll transients cause linear movement in the y-direction [8]. Moreover, in bi-copter, thrust is obtained from rotors placed over tilt mechanisms. This may cause extra vibrations and resonance problems in the structure for larger bi-copter systems.

Using hydrocarbon fuel as an alternative power source significantly outperforms the battery powered propulsion system, and it gives relatively large flight endurance. In Ref. [5], a coaxial inverting thruster using two gasoline engines is placed at the center of a standard quadrotor (Fig. 1(c)). In this case, gyroscopic and counter torques are compensated; however, coaxial rotors lose some of their efficiencies [9]. In Ref. [6], a quadrotor with variable pitch rotors that are powered from a gasoline engine (Fig. 1(d)) is reported. This configuration requires a complex drivetrain and four variable pitch rotors. In Ref. [7], a quadrotor based on four gasoline engines (Fig. 1(e)) is reported. This vehicle can carry large payloads, but the existence of four engines increases the cost dramatically.

In this study, inspired by these works, an uncommon quadrotor configuration illustrated in Fig. 2 is introduced to address the uncompensated gyroscopic torque problem for a single large rotor, the low efficiency problem for coaxial large rotors, the limited roll torque, possible vibrations, and resonance problems of bi-copter design. Therefore, two large rotors are placed on the longitudinal axis which are counter-rotating to minimize their gyroscopic and rotor drag torques. On the lateral axis, small counter-rotating rotors are used for attitude control. In addition, the distance between the small rotor and body is kept small to avoid a large vehicle horizontal width. In this configuration, large rotors carry a major portion of the vehicle weight. Alternate rotors are not counter-rotating to minimize rotor drag and gyroscopic torques in this configuration unlike a standard case. Therefore, small rotors are equipped with a tilt mechanism to gain the control of yaw motion. In addition, sufficient roll control performance can be obtained with this layout. Herein, main thrust sources are directly attached to the main frame which reduces the possible vibration and resonance problems. Therefore, this configuration is more suitable for larger scale compared to a bi-copter. Moreover, if one aims for longer flight duration, large propellers can easily be powered from gasoline engines. In this way, some drawbacks of the gasoline engine configurations discussed above can be eliminated. In this article, battery powered propulsion system is studied. Theoretical efficiency improvements and earlier flight tests of the proposed vehicle prototype are reported previously in Ref. [10].

For multirotors, controllers are generally designed based on an ideal dynamic model derived from physical laws. Thrust and rotor drag coefficients are obtained from static thrust tests. Moreover, inertia matrices are obtained from pendulum tests [1, Ch. 6]. With measuring mass and distances between rotors and mass center, approximate linear model can easily be constructed. Based on this model, proportional-integral-derivative (PID) type controllers can be designed, and sufficient control performance around hover can be reached [1, Ch. 11]. An alternative way of using the ideal dynamic model directly is to assign uncertainties to parameters. From this perspective, robust adaptive [11] and nonlinear $H∞$ attitude controllers [12] are applied to satisfy robustness against uncertainty in mass and inertia values. Similar studies can be added, but they all mainly aim for robustness against parametric uncertainties in ideal models.

If higher control performance is demanded, one needs to obtain a more accurate model using a system identification method. From the frequency response function (FRF), dynamic model can be obtained. In this way, the dynamics of the sensors, flexible modes of the airframe, and couplings of different axes due to an inaccurate rotor mixing matrix can be included in the model. In Ref. [13], model of the aircraft is estimated from FRF obtained by injecting multisine signals. Similarly, system identification of multirotors is studied in Refs. [14,15] previously. However, in these studies, perfect decoupling of axes is assumed, and only diagonal entries of the overall transfer matrix are computed. In this article, in addition to diagonal entries, coupling dynamics are also analyzed, and the FRF of the overall three-input three-output attitude model is estimated.

From a robust control point of view, nominal model and uncertainty structure selections are very important for high performance. It is possible to minimize the robust control criterion during identification of the nominal model and uncertainty if a specific dual-Youla–Kucera structure is used [16]. With this structure, control synthesis can be performed in a non-conservative manner, and similar high performance can be obtained for the entire model set. This method gives promising results in motion control applications [17,18]. Herein, this method is successfully applied for an attitude control problem for the first time.

In the robust control framework, nominal model should be accompanied by minimum uncertainty such that uncertain model and measured outputs from true system are consistent. In this study, the validation-based uncertainty modeling procedure introduced in Ref. [19] is used for that purpose. To the author’s knowledge, this study is the first implementation of this technique to UAV flight control problem. After that, the uncertain model and required robustness are determined. It is known that standard μ synthesis optimizes robustness and performance at the same time [20, Ch. 8]. Therefore, this method is not suitable when the required robustness is already known. Hence, uncommon skewed-μ synthesis is used for robust controller design.

The main contribution of this paper is to apply the combined identification and robust control procedure for the attitude control problem. To the author’s knowledge, similar methods, which include identification of a multivariable attitude model, validation-based uncertainty quantification, and skewed-μ synthesis, are not investigated in the previous studies for UAV flight control problem. In this respect, this article contributes to the studies in this area and shows performance improvements compared to traditional methods.

The outline of this article is as follows. In Sec. 2, a nonlinear dynamic model of the proposed quadrotor configuration is obtained. This model is later used in Sec. 3 to find an approximate linear model around hover. In Sec. 4, robust-control-relevant identification and its main motivation are explained. In Sec. 5, skewed-μ synthesis and in Sec. 6, prototype of the vehicle are introduced. Then, in Sec. 7, several experimental results are illustrated, and the attitude control performances of robust and proportional-integral (PI) controllers are compared.

As in the standard case, the proposed quadrotor configuration can be viewed as a connection of five rigid bodies, quadrotor body B with four rotor groups P_i attached to it. However, the vehicle illustrated in Fig. 2 differs in that it consists of big rotors on the longitudinal axis (i = 1, 3) and small rotors equipped with tilting mechanisms on the lateral axis (i = 2, 4).

Let $FE:{OE;xE,yE,zE}$ be an earth inertial frame and $FB:{OB;xB,yB,zB}$ be a quadrotor body frame attached to its center of gravity. Moreover, let fixed rotor frames $FPi:{OPi;xPi,yPi,zPi},i=1,2,3,4$⁠, be aligned with each other and body frame. The orientation of second and forth rotors are changed by rotation around $yPi$ axis by an angle of α_i. This rotation creates a new frame for small rotors as shown in Fig. 2, and they are represented by $FP¯i:{OP¯i;xP¯i,yP¯i,zP¯i},i=2,4$⁠.

To balance gyroscopic and counter torques in hover, rotating directions of similar type motors must be opposite. Therefore, rotors 1 and 2 revolve in clockwise direction, and rotors 3 and 4 revolve in counter clockwise direction.

Let the vector $ξ=[xyz]T$ denote translational coordinates of O_B relative to $FE$⁠, $η=[ϕθψ]T$ denote the orientation of the quadrotor determined by roll angle ϕ, pitch angle θ, and yaw angle ψ (⁠$Z−Y−X$ Euler angles), $RBE$ denote the overall rotation matrix from body frame to inertial frame, and $RP¯iB$ denote the rotation matrix from rotating rotor frame $FP¯i$ to body fixed frame $FB$ [21].

The full nonlinear model derived in Eqs. (1) and (2) is useful for simulation purpose despite some neglected aerodynamic effects. However, especially rotational motion equation is fairly complex for control purpose. Therefore, suitable reduced model is needed. First, it is assumed that bandwidth of rotor speed control is sufficiently high such that transients on rotor speed w_i are neglected. Therefore, w_i’s are considered as control inputs. For a quadrotor in slow flight conditions, the second-order inertial and gyroscopic terms are smaller than the forces and torques generated by propeller actuation. Therefore, these second-order terms are considered as disturbances that need to be canceled by the controller. Therefore, simplified rotational motion equation $Ω˙=I−1τB$ is used instead of Eq. (2) for controller design purpose where τ^B is now equal to $τB=τAB+τDB$⁠.

Approximate linear model around hover is obtained using Eq. (1) and $Ω˙=I−1τB$⁠, relation between rates of Euler angles and body rates, manipulated variables $u=[wTαT]T$⁠, and states $x=[(VE)TηTΩT]T$⁠. Herein, $w=[w12w22w32w42]T$ includes square of the rotor speeds and $α=[α2α4]T$ includes tilt angles.

Herein, $kfb$⁠, $kfs$ are the thrust factors, l_s is the small arm length, l_sh is the distance between mass center and small rotor along $z$-axis, $kdb$⁠, $kds$ are the rotor drag factors, and w_eqi is the square of the hover speed of the ith rotor. In addition, $ξ¯=z$ and δ denotes the perturbation of the parameter from the equilibrium value. Then, nonlinear quadrotor model is approximated as $x¯=x¯eq+δx¯$ and u = u_eq + δu where $δx¯=[δz˙δΩT]T$ and $δu=[δwTδαT]T$⁠.

A hierarchical control strategy commonly used for quadrotors is also applicable to the proposed configuration. At the lowest level, according to commands sent from the flight controller, rotor speeds and angles of the tilt mechanisms are controlled by related electronics. Generally, these loops have very high bandwidth, and their transients can be neglected. In the second level, attitude control loops exist to track desired angles. At the top level, position controllers exist that determine required angles according to desired position commands [1, Ch. 11]. This cascaded control structure, which includes these levels, is illustrated in Fig. 3.

However, due to inexact decoupling, gyro sensor delay, and other negligible dynamics, the model differs from the ideal. In this article, a more accurate plant model will be obtained using a system identification procedure.

To control the attitude of quadrotors, the cascaded control structure in Fig. 4 is used. Therefore, the accuracy of the attitude control is mainly determined from inner rate control loop performance. Herein, designing high performance rate loop which minimizes the effects of neglected and nonlinear dynamics is the main motivation. Since model-based robust control technique will be used, the equivalent model P_o in Fig. 4 for the inner loop is needed. This model will be obtained from flight tests when the quadrotor is stabilized by manually tuned PI rate and proportional (P) angle controllers.

T(P, C) includes sensitivity, process sensitivity, control sensitivity, and complementary sensitivity functions. Compared to standard mixed sensitivity problems, this four-block problem eliminates unwanted pole/zero cancelations during controller synthesis. In addition, this problem also appears in $H∞$ loop shaping method in Ref. [23] where the shaped plant is robustly stabilized with respect to coprime factor uncertainty [24]. Since similar design method and uncertainty structure are targeted, this four-block problem is used in this article.

Right coprime factorization (RCF) of $P^$ is denoted by the pair ${N^,D^}$ if $D^$ is invertible, $N^,D^∈RH∞$⁠, $P^=N^D^−1$⁠, and $∃Xr,Yr∈RH∞$ satisfying Bezout identity $XrD^+YrN^=I$⁠. Similarly, it is assumed that the stabilizing controller C_exp has a RCF {N_c, D_c}. Then, using $PDY$ in Fig. 6 gives the LFT representation (Eq. (14)).

Even if bounded performance is reached, the minimization of Eq. (15) is not clear due to frequency dependent $M^21$ and $M^12$ terms. Further simplification is explained in Sec. 4.5 by using specific coprime factorization of C_exp.

This shows that uncertainty bound γ directly affects the worst-case performance $JWC(PRCR,Cexp)$⁠. In other words, by minimizing γ separately with nominal model identification and quantification of model uncertainty Δ_u, a robust-control-relevant model set in terms of Eq. (11) is obtained [17].

Solution technique of skewed-μ synthesis is D–K iterations as in the standard μ case. However, the criterion of the D–K iterations is slightly modified as $minK,D‖DUM^D−1‖∞$ by introducing a new matrix $U=diag(In,(1/μs)Inp)$ with appropriate dimensions that is updated at each iteration [20, Ch. 8].

Most of the electrical and mechanical parts of the first prototype are constructed using low-cost off-the-shelf devices available in the UAV laboratory of the METU EEE department. While two T-Motor KV400 motors driving 16 in. (40.6 cm) propellers are used for the large rotors, two T-Motor KV830 motors driving 9 in. (22.9 cm) propellers correspond to the small rotors. Two 3D printed tilt mechanisms together with two Savöx SA-1256 servo motors are used to tilt the small rotors as depicted in Fig. 8. A Hobbywing Quattro four-channel electronic speed controller (ESC) and a Gens ace 4S (14.8 V) Lipo battery are used to power the brushless motors. An Ublox Neo 7 GPS-Compass module, a 3DR 433 Mhz radio telemetry kit, and a Futaba 2.4 Ghz R2008SB receiver are also attached to the top frame. Moreover, a clone mRo Pixhawk flight controller board compatible with the Pixhawk/PX4 open source flight control software is mounted with a vibration damper in the middle of the top frame. This prototype given in Fig. 8 has a nominal mass of 2.7 kg. The desired motor speeds and tilt angles are regulated with pulse width modulation (PWM) signals sent to the ESC and servo motors. In this study, to reach a convenient solution, available PX4 flight control software is customized for this uncommon quadrotor configuration.

The input signal is periodic with a period of 4 s. Moreover, the phases are selected according to Schröer rule [26, Ch. 4]. The identification tests take nearly 120 s which correspond to 30-period input excitation tests, which gives sufficiently accurate FRFs and covariance estimate of the disturbances for the model validation procedure [19].

In classical control theory, robustness is often specified by considering limits on gain margin (GM) and phase margin (PM). According to common rules of thumb, 2 < GM < 5 and 30 deg < PM < 60 deg are often aimed in practice [20, Ch. 2]. Previously, the PI rate controller was tuned manually in flight, and these margins were not known since the estimate of P_o had not been available. Now, estimate $P~o$ is found using identification, and using the diagonal entries of it and the PI controller, gain and phase margins are investigated. In roll axis, 2 Hz bandwidth, 37 deg PM, and 3.9 GM are obtained. In the pitch axis, 2.65 Hz bandwidth, 34 deg PM, and 2.6 GM are achieved. For yaw axis, approximately 0.28 Hz bandwidth and 2 GM are reached. Phase loss due to the finite bandwidth of motor speed control and gyro sensor delay does not allow any gain increase in the PI controller for roll and pitch axes to maintain sufficient margins. For the yaw axis, resonance around 20 Hz limits the achievable gain and bandwidth for a standard PI controller. Adding extra lead and lag filters is necessary to improve the performance, but it is not in the scope of this study. Hence, the PI rate controller that is available during the identification test is also used in the performance evaluation part. This diagonal controller which gives stability margins above is denoted by C_exp in this section.

The main motivation is to reach approximately 3 Hz bandwidth in the roll and pitch axes and 0.5 Hz bandwidth in the yaw axis to increase the disturbance rejection performance. It is assumed that coupling effects between different axes are small in the low-frequency range. Therefore, both P and C are approximately diagonal which leads to three different bandwidths definition corresponding to each axis. In this study, three different bandwidths f_roll, f_pitch, and f_yaw refer to crossover frequencies in the corresponding axes satisfying |P^rrC^rr(2πf_roll)| = 1, |P^ppC^pp(2πf_pitch)| = 1, and |P^yyC^yy(2πf_yaw)| = 1 where P^rr, P^pp, P^yy and C^rr, C^pp, C^yy terms indicate the diagonal elements of P and C for the roll, pitch, and yaw axes, respectively. In addition, to use similar formulation, the notation f_bw is used which may represent f_roll, f_pitch, or f_yaw according to axis selection.

In this article, weighting functions are selected to shape the loop similar to the method in Ref. [23]. These weighting filters are designed such that W₂P W₁ has desired open loop shape of PC. Since $P^$ is not found yet, the estimate $P~o$ can be used directly. In this study, W₁ is selected to have an integrator for good disturbance rejection, and its cutoff is at f_bw/3 as shown in Fig. 10 where f_bw = 3 Hz for the roll and pitch axes and f_bw = 0.5 for the yaw axis. W₂ is selected to satisfy 0 slope around f_bw such that desired open loop shape has −1 slope in this region for sufficient robustness. Moreover, high frequency roll-off beyond 4f_bw is also enforced with W₂. Finally, W₂ and shaped open loop W₂P W₁ are depicted in Fig. 10.

The two-block problem considered in Ref. [23] is equivalent to four-block problem considered in this article in terms of control criterion if the weighting filters are selected as $W=diag(Wy,Wu)=diag(W2,W1−1)$ and $V=diag(V2,V1)=diag(W2−1,W1)$ in Eq. (5). In this article, this method is followed.

Herein, although the performance bound in (24) is not affected with tight dynamic overbound W_γ, possible conservatism during controller synthesis can be reduced since $Pdyn∈Psta$ is satisfied. These model sets are visualized using the method introduced in Ref. [28], and elementwise singular values are depicted in Fig. 12. It shows that $Pdyn$ is narrower in the frequency domain compared to $Psta$⁠, reducing possible conservatism. In addition, both uncertain model sets get narrower around the crossover region which is very essential for high robust performance. Conversely, model sets are uncertain in both low and high frequency regions since they are relatively large there.

The uncertain model set $Pdyn$ is used to synthesize a robust controller. Gains of the resulting robust controller C_RP and PI controller C_exp are illustrated in Fig. 13 where the input of the controller is error $ε=r2−y$⁠, and the output of the controller is control signal u when r₁ is zero. In Fig. 13, controllers are mostly single variate in roll and pitch axes since off-diagonal elements are relatively small compared to diagonal entries in large frequency interval. However, in yaw axis, C_RP shows multivariable characteristics. This may improve the performance in this axis. In addition, better disturbance rejection capability can be expected from C_RP due to larger controller gains in the low frequencies compared to C_exp. Next, nominal and worst-case performances of these controllers for $Pdyn$ are given in Table 1. It shows that C_RP gives improved performance compared to C_exp. Herein, smaller performance criterion indicates that achieved bandwidth is closer to the desired one. As reported in Ref. [23], J(P, C) < 4 is an indicator of successful loop shape. Therefore, C_RP has closer properties to desired loop shape. Table 1 also indicates that all candidate models in the uncertain model set give similar high performance for C_RP since J and J_WC are almost equal in this case. Therefore, $Pdyn$ is indeed robust control relevant. In other words, this tight model set shown in Fig. 12 enables non-conservative controller synthesis as expected.

Main duties of the designed attitude rate controllers are to follow angular rate references from the angle controller and to attenuate exogenous disturbances such as wind. Since attitude error is an important performance indicator, effects of these controllers to attitude error is investigated instead of angular rate. During hovering, small attitude reference signals are generated from the outer local velocity controller. Therefore, reference tracking problem is considered where the performance variable $ε=r2−y$ denotes the attitude error. When the vehicle is stabilized around hover, attitude errors are obtained for the two controllers. In Fig. 14, attitude errors in roll, pitch, and yaw axes are depicted. It shows that attitude errors are attenuated with C_RP due to higher controller gains in the low-frequency region for roll and pitch axes and higher achieved bandwidth for yaw axis. More importantly, C_RP handles the roll and yaw axis coupling more accurately, thanks to its multivariable structure. Therefore, as depicted in Fig. 14, C_RP gives smaller attitude errors in all channels compared to C_exp. Standard deviations of the attitude errors in hover are given in Table 2. It shows that C_RP improves the attitude tracking performances around 40% in the roll, 55% in the pitch, and 62% in the yaw axes relative to the PI controller (C_exp) baseline. Moreover, good disturbance rejection capability is also essential. Therefore, to analyze the disturbance rejection performance, a signal is injected at the plant input for each axis. This torque disturbance corresponds to 1/3 of the maximum available torque, i.e., τ_d = τ_max/3, for each axis. In Fig. 15, attitude errors and torque disturbance injection points are illustrated, and standard deviations of the attitude errors are given in Table 3. These results show that disturbance rejection performances are improved around 57% in the roll, 56% in the pitch, and 35% in the yaw axes over the baseline. In view of these findings, attitude control performance is significantly improved with the robust rate controller C_RP compared to the PI rate controller C_exp around hover.

In this article, an uncommon efficient quadrotor configuration is proposed. An uncertain model set of this vehicle which facilitates subsequent high performance robust control is constructed, and a robust controller is designed with skewed-μ synthesis. First, the proposed vehicle is stabilized using PID type controllers with sufficient attitude control performance. Second, performance improvement is studied with the robust controller. Results of several flight experiments illustrate that the proposed control method significantly outperforms the classical method. Both command following and disturbance rejection performances are improved with the proposed design. In conclusion, the proposed quadrotor configuration can be very useful over the conventional configuration, and it is worthy of further research. In addition, the robust control technique can be helpful to improve the performance achieved with common control methods. Some parts of this study can be improved with further research. Experimental efficiency of this vehicle can be calculated. Alternatively, large rotors that are powered from gasoline engines can be used to improve the flight endurance. Rotor speed and tilt angle control dynamics can be added to the ideal model. Larger deviations from hover position can also be considered as future works.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.