STUDY OF THE INFLUENCE OF A NONLINEAR CONTROL ALLOCATION FREQUENCY VARIATION ON A QUADROTOR TILT-ROTOR AIRCRAFT STABILITY

This paper presents a study on the influence of the frequency variation of a nonlinear 1 control allocation technique execution, developed by the author [1], named by Fast Control 2 Allocation (FCA) for the Quadrotor Tilt-Rotor (QTR) aircraft. Then, through Software In The 3 Loop (SITL) simulation, the proposed work considers the use of Gazebo, QGroundControl, and 4 Matlab applications, where different frequencies of the FCA can be implemented separated in 5 Matlab, always analyzing the QTR stability conditions from the virtual environment performed in 6 Gazebo. The results showed that the FCA needs at least 200 Hz of frequency for the QTR safe flight 7 conditions, i. e., 2 times smaller than the main control loop frequency, 400 Hz. Lower frequencies 8 than this one would case instability or crashes during the QTR operation. 9


Introduction
The applications of Unmanned Aerial Vehicles (UAVs) to perform tasks previously 13 developed by humans are increasing every day, performing from medical to military 14 tasks [2][3][4][5]. Among all possible applications, their topologies became a very important 15 part of the whole project and execution, which are classified as fixed-wings (planes 16 for example) and rotary wings (helicopters for example). A quick search on internet 17 shows some reliable and stable UAVs, such as multicopters [6,7] and fixed-wings [8,9], 18 for example. 19 Aiming to include the fixed-wing and rotary-wing classifications at the same vehicle, 20 the hybrid topologies have emerged. A classic example is the Tilt-Rotors, which have 21 tilting mechanisms for their propellers, making their maneuvers a partial combination 22 of the two mentioned topologies. These tilt-rotors may or may not have fixed wings, just 23 as many propellers may be required. These combinations let them being considered as 24 overactuated vehicles, which are defined as UAV with more actuators than the respective 25 Degrees of Freedom (DoF). Also, the overactuation directly interfere the system control 26 allocation technique choice, which by definition is responsible for generating signals to 27 the actuators from the control actions resulted from the controllers. 28 It is important to highlight that the vast majority of robotic systems do not require 29 a complex method of control allocation, both for under and overactuated types. In 30 addition, there are still cases where an overactuated UAV can be simplified as underac-31 tuated [10,11]. However, depending on the aircraft physical characteristics and design 32 requirements, there is a strict necessity to use a non-linear and complex method [12]. 33 an appropriate configuration to reach the requirements [16]. 48 In contrast to this methodology, Linear Programming (LP) minimizes the weighted 49 error between the desired and estimated VCAs. Thus, an optimization problem with 50 geometric / polyhedral constraints is represented. Using defined cost functions, the 51 resulting problem is linearly programmable and can be solved using iterative numerical 52 algorithms such as the simplex method [17].

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Taking the Non Linear Programming (NLP) into consideration, there will always 54 be a single optimal solution if all the weights in the cost function are necessarily positive 55 using slack variables [18,19]. Referring to the numerical methods of solving, 3 algorithms 56 deserve to be highlighted: active-set, interior point and fixed point methods.

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However, an important point needs to be highlighted: as the solution complexity 58 increases, the computational effort becomes a critical point when real-time execution is 59 needed, due to the possibility of finding different local minima and numerical sensitivity 60 in the evaluation and validation procedures. Also, the control allocation frequency 61 evaluation becomes a crucial step in the whole UAV design, which means that if the 62 RCAs are not obtained correctly, the system can become unstable and uncontrollable 63 [12]. Aiming this aspect, it is possible to observe a large gap in the studies reporting 64 these analysis, since incompatible control allocation technique frequencies can lead to 65 instability on the vehicle, therefore generating accidents.

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It is in this context that this work is proposed, presenting a study on the influence of 67 the control allocation execution frequency variation for some simulation flight scenarios, 68 being analyzed on the vehicle stability. Regarding the control allocation considered, it is 69 used the nonlinear technique applied to the QTR developed in [1] and [12], illustrating a 70 SITL representation.

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The main contribution of this work is focused on establishing the minimum and 72 safe frequency of the control allocation task, developed in work [1] and [12], where it was 73 considered to be run in real-time and embedded execution at 400 Hz (same frequency as 74 the QTR attitude control loop). Then, this work will study the minimum frequency able 75 to still keep the QTR safely flying with the same control requirements previously set.

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This paper is divided as follows: Section 2 presents the UAV kinemactics and 77 dynamics modelling, strictly necessary for understanding the control allocation design 78 and the simulation results; Section 3 depicts the considered controller topology, much as 79 a brief overview of the Fast Control Allocation Technique proposed by [1,12]; Section 4 80 illustrates the SITL scheme to perform the results; Section 5 shows the simulation results,

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Here is presented and depicted the aircraft kinematics and dynamics modelling, 86 also considering the servomotor tilting angles. For better illustration, Fig. 1 shows the 87 UAV and its axis rotations: where k 1 is the constant of propulsion, characteristic to each propulsion system (set of

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Then, the vehicle position is defined by the vector η 1 ∈ R 3 in F I (Inertial Frame), 103 while its angles are defined by η 2 ∈ R 3 in F v (Vehicle Frame). Moreover, ν 1 ∈ R 3 and 104 ν 2 ∈ R 3 are the linear and the angular velocities, measured in F b (Body-Fixed Frame).

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According to [21], (7) presents the nomenclature: where is the linear velocity vector, and ν 2 ∈ R 3 is 107 the angular velocity vector, both in F b .

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The 6 DoFs rigid body kinematics is expressed in (10): whereη ∈ R 6 is the velocity vector in F I , ν ∈ R 6 is the general velocity vector in F b and 110 J ∈ R 6×6 is the Jacobian matrix, where the position vector η = η is in F I .

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The QTR dynamics can be described by differential equations from Newton-Euler 112 method, such as shown in (11).

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It is known that UAV control systems are multiloops, that is, Multiple-Input- To illustrate all the loops in a simplified form, Figure 2 shows the QTR overall 136 control structure, where the control allocation task is marked with the circle number 3.  (14): where η d ∈ R 5 is the setpoint desired vector, and K b ∈ R 5 are the actions from controller More details about the QTR controllers and their tuning can be found in work [1]. established in the literature [24].

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Taking this into account, the QTR CEM was broken into two different problems 167 such as: Also, u a ∪ u b = u and M a ⊂ M b ⊂ M [12].

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By consequence, these subsystems are shown in (17) and (18): To obtain u a , it is necessary to 173 take the first 4 elements of u ′ a .

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It is important to remark that the chosen subsystem combination allows the QTR 175 to perform maneuvers using the motors differential rotation speeds and/or tilting its

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More details about the control allocation technique are depicted in [1,12].  To illustrate the procedure, Figure 3 shows the SITL block diagram.  This index was chosen because a system based on it would have reasonable damp-212 ing and a satisfactory transient response, punishing errors in the quadratic weights 213 during the experiment, regardless of the time they occur [25,26].  in [27]. However, several other control methodologies could provide similar results 224 [28,29]. 225

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This section will present some simulation results that aim to analyze the QTR 227 dynamics responses considering 3 different delay times that the FCA technique spent to 228 process the VCAs, demanded by the 5 controllers.

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In order to illustrate the controlled responses, Figure 5

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This section shows simulation results in three-dimensional graphics for different 245 SPs, close to the SP range considered in the controller tuning, aiming to obtain the 246 minimum frequency for each case. This range was based on the parameters presented by 247 the author in the work [1,12]. In this way, the pitching SP was kept at 0 degrees flying at 248 10 m altitude. The rolling SP ranged from −10 to 10 degrees and the forward/backward 249 velocity were from −3 to 3 m/s, both requested in 10 seconds of flight after take off.   requirements considered in [1,12].

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Discontinuities between some operating points in their respective neighborhoods