Gust Disturbance Alleviation with Incremental Nonlinear Dynamic Inversion

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1 Delft University of Technology Gust Disturbance Alleviation with Increental Nonlinear Dynaic Inversion Seur, Ewoud; de Croon, Guido; Chu, Qiping DOI.9/IROS Publication date 26 Docuent Version Peer reviewed version Published in Proceedings of the 26 IEEE/RSJ International Conference on Intelligent Robots and Systes (IROS) Citation (APA) Seur, E., de Croon, G., & Chu, Q. (26). Gust Disturbance Alleviation with Increental Nonlinear Dynaic Inversion. In Proceedings of the 26 IEEE/RSJ International Conference on Intelligent Robots and Systes (IROS): Daejeon, Korea DOI:.9/IROS Iportant note To cite this publication, please use the final published version (if applicable). Please check the docuent version above. Copyright Other than for strictly personal use, it is not peritted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Coons. Takedown policy Please contact us and provide details if you believe this docuent breaches copyrights. We will reove access to the work iediately and investigate your clai. This work is downloaded fro Delft University of Technology. For technical reasons the nuber of authors shown on this cover page is liited to a axiu of.

2 Gust Disturbance Alleviation with Increental Nonlinear Dynaic Inversion Ewoud J.J. Seur and Guido C.H.E. de Croon2 and Qiping Chu3 Abstract Micro Aerial Vehicles (MAVs) are liited in their operation outdoors near obstacles by their ability to withstand wind gusts. Currently widespread position control ethods such as Proportional Integral Derivative control do not perfor well under the influence of gusts. Increental Nonlinear Dynaic Inversion (INDI) is a sensor-based control technique that can control nonlinear systes subject to disturbances. This ethod was developed for the attitude control of MAVs, but in this paper we generalie this ethod to the outer loop control of MAVs under gust loads. Significant iproveents over a traditional Proportional Integral Derivative (PID) controller are deonstrated in an experient where the drone flies in and out of a fan s wake. The control ethod does not rely on frequent position updates, so it is ready to be applied outside with standard GPS odules. I. INTRODUCTION Micro Aerial Vehicles (MAV) have the potential to perfor any useful tasks, such as search and rescue [], package delivery, aerial iaging [2], etc. For applications where the MAV needs to operate close to obstacles or close to the ground, accurate position control is of paraount iportance. However, currently widespread position control ethods such as Proportional Integral Derivative control (PID) [3] do not perfor well under the influence of gusts. Iagine a search and rescue scenario where a drone needs to fly into a house through an open window to look for survivors of a disaster. If the conditions are windy outdoors, the drone will need to provide a certain force to counteract this wind. The oent the drone flies into the house, the drag fro the wind disappears and the drone will start to accelerate. With traditional control ethods, the drone is likely to hit soething in a confined indoor space. Outdoor UAV issions can encounter significant gusts due to atospheric turbulence [4]. Shen et al. even observed these wind disturbances indoor [5], and they needed an augentation of the controller that could cope with slowly varying wind disturbances. To cope with fast changing wind gusts, a solution could be to use onboard wind sensors to estiate the wind field [6]. However, this increases the syste coplexity and cost. *This work was supported by the Delphi Consortiu Ewoud J.J. Seur is with Faculty of Aerospace Engineering, Delft University of Technology, 2629HS Delft, The Netherlands e.j.j.seur@tudelft.nl 2 Guido C.H.E. de Croon is with Faculty of Aerospace Engineering, Delft University of Technology, 2629HS Delft, The Netherlands G.C.H.E.deCroon@tudelft.nl 3 Qiping Chu is with Faculty of Aerospace Engineering, Delft University of Technology, 2629HS Delft, The Netherlands q.p.chu@tudelft.nl Fig.. The quadcopter in front of the fan during one of the experients. Alternatively, the wind velocity could be estiated onboard through odeling [7], [8]. The downside of this approach is that it is very dependent on the odel. If the odel does not represent reality well enough due to odeling errors or airfrae changes, the gust alleviation perforance will degrade. Gardner et al. focused on creating a fraework to assess the ability of different physical platfors to withstand gusts [9], but they did not discuss how the controller can take advantage of this ability. In other research, the use of the acceleroeter for the control in the vertical axis is discussed []. A siilar research has been worked out for the vertical control of a helicopter using the ain collective []. In the latter, the ethod of Increental Nonlinear Dynaic Inversion (INDI) is applied, but only in one axis. It is shown that the controller is able to track a reference in siulation, but no disturbance rejection properties are discussed. The approach of using the acceleroeter sees proising, since disturbances are easured, as is pointed out by Wang et al. [2]. However, to fully take advantage of this fact, all axes should be taken into account. In this paper we introduce a gust resistant controller through generaliation of INDI to the outer loop control. This controller does not require any inforation on the aerodynaic drag of the quadrotor. It is ipleented on a Parrot Bebop quadrotor running the Paparai open source autopilot software [3]. Experients are perfored that show only five c position changes while entering and leaving an industrial fan s wake. A benchark PID controller gives position errors of half a eter for the sae test.

3 II. INCREMENTAL NONLINEAR DYNAMIC INVERSION APPLIED TO LINEAR ACCELERATIONS Consider the quadrotor shown in Figure 2. The distance fro the center of gravity to each of the rotors along the body X axis is given by l and along the Y axis by b. Two reference fraes will be used throughout this paper, the body frae, with subscript B and the North East Down (NED) frae, with subscript N. The subscripts will only be used to avoid confusion, the position ξ and velocity v of the MAV will always be in the NED frae. M Y X M 4 Z Fig. 2. The Bebop Quadcopter used in the experients with body axis definitions. We start with a description of the syste, in this case the position dynaics. These follow fro Newtons second law of otion: b l M 2 M 3 ξ = g + F (v) + T N (η, T ) () Where ξ = [x, y, ] T is the position and F is the aerodynaic force working on the airfrae as a function of the velocity v of the MAV. T N is the thrust vector in the NED frae as a function of the attitude η = [φ, θ, ψ] T and the total thrust produced by the four rotors T. Finally, g is the gravity vector and is the ass of the drone. The thrust vector in the NED frae can be obtained by taking the thrust vector in the body frae, defined as T B = [,, T ] T, and rotating it using the rotation atrix M NB (η). Since the thrust vector in the body frae only has a Z coponent, only the last colun of the rotation atrix is relevant. The thrust vector in the NED frae is therefore given by the following: T N (η, T ) = M NB (η)t B = (sφsψ + cφcψsθ)t (cφsψsθ cψsφ)t (cφcθ)t (2) where the sine and cosine functions are abbreviated by the letters s and c respectively. Now we can apply a first order Taylor expansion to equation, resulting in equation 3. ξ = g + F (v ) + T N (η, T ) v F (v) v=v (v v ) φ T N(η, T ) φ=φ (φ φ ) θ T N(η, T ) θ=θ (θ θ ) T T N(η, T ) T =T (T T ) (3) The first ter can be siplified to the acceleration at the previous tiestep: g + F (v ) + T N (η, T ) = ξ. This acceleration can be obtained by rotating the accelerations easured in the body axes to the NED frae and adding the gravity vector. Furtherore, we assue that the second ter, the partial derivative of F with respect to v, is sall copared to the other three partial derivatives. This is coonly referred to as the principle of tie scale separation. Cobining this with Eq. 2 and 3 we end up with: ξ = ξ + G(η, T /)(u u ) (4) where u = [φ θ T/] T and G(η, T/) = (cφsψ sφcψsθ)t/ (cφcψcθ)t/ sφsψ + cφcψsθ ( sφsψsθ cψcφ)t/ (cφsψcθ)t/ cφsψsθ cψsφ cθsφt/ sθcφt/ cφcθ (5) The easured accelerations, necessary to obtain ξ, are typically noisy due to vibrations in the airfrae introduced by the spinning propellers. Therefore, the accelerations need to be filtered. Fro the literature, we adopted the use of a second order filter [4], given by: ω 2 n H(s) = s 2 + 2ζω n s + ωn 2 This filter also introduces delays in the signal, resulting in delayed acceleration easureents. In previous research [5], we showed that by applying this sae filter on the input as well, the input is not increented further before the result of the previous increent is known. If we denote filtered signals with subscript f and invert Eq. 4, we get the INDI control law for linear accelerations: (6) u c = u f + G (η, T /)(ν ξ ξ f ) (7) We have replaced ξ with the virtual control ν ξ to indicate that this is now an input to the equation (the desired acceleration), and we added the subscript c to u to indicate that this is the coand that will be sent to the inner loop controller. We also define the increent ũ = u c u f, so clearly Eq. 7 is an increental control law. III. IMPLEMENTATION The ipleentation of the control law given by Eq. 7 is shown in Figure 3. Note how the increent in specific thrust coand T is an output of this diagra. This is because the specific thrust is not a control variable in itself, instead the rotors are used to provide a certain thrust. Therefore, the specific thrust will have to go through a second inversion step, to find the rotor angular rate increents that will result in the coanded specific force increent. The rotors are also used by the inner loop INDI controller to control the angular acceleration of the MAV. In order to find rotor increents that satisfy both the increent in angular acceleration as well as the increent in specific thrust, we will expand the inner loop inversion step to include the relation of specific force and rotor angular rates. This way, increents for the

4 ν ξ ξerr + G(η, T ) [ ] + [ φ φc θ θ c [ φf T ] Inner loop ] H() [ φ θ MAV ] ξ θ f ξ f H() ξ Fig. 3. The outer INDI control structure. angular rates of the rotors can be found that satisfy both the desired increent in angular accelerations as well as the desired increent in specific force. The inner INDI loop is shown in Figure 4. It was derived in our previous work [5] using siilar ethods as used in this paper for the outer loop controller. For a coplete derivation, including stability analysis, we refer to that paper. The angular rate of rotor one through four is denoted by the vector ω and the angular rates of the vehicle by Ω. Note that this diagra contains two different control effectiveness atrices, G and G 2. G is a 4x4 atrix defined as the control effectiveness of the four rotors on the angular acceleration vector and the acceleration in the Z B axis. G 2 is a 4x4 atrix introduced as an extension to G to account for changes in the angular oentu of the propellers. Changing the rotational speed of the rotors changes their angular oentu, which produces a torque in the yaw axis. G 2 therefore has one row of nonero values, corresponding to yaw axis. A. Estiation of the Specific Thrust Throughout the derivation of the outer loop INDI controller, we ade use of the specific thrust T, for instance in the atrix G(η, T/). One way to obtain the specific thrust would be to odel the thrust/rotational rate curve of the propellers and easure the ass of the drone. In this paper, we chose to do soething else: we assued that the aerodynaic forces in the B direction are sall copared to the thrust. Then the specific thrust can be approxiated by the specific force easured by the onboard acceleroeter in the B direction. Furtherore, since the propellers have a quadratic thrust curve, their control effectiveness changes depending on their current rotational rate. In this paper, we assue that the control effectiveness of the rotors with respect to the specific force can be approxiated by a static one. In future research, we will investigate the benefits of using the thrust/rotational rate curve of the propellers in the controller. B. Position Control The acceleration of the vehicle is accurately controlled by the syste shown in Figure 3. To control the position of the MAV, an acceleration reference needs to be passed to the outer INDI controller that will steer the drone towards its target position. This can be done by a Proportional Derivative (PD) controller. The gains of this PD controller were anually tuned. They depend ainly on two things: the update rate of the position estiate and the speed of the inner loop controller, which is only dependent on the actuator dynaics. This is the case because all other coponents are inverted in the inversion step of the inner and outer loop. C. Filtering The easured accelerations are filtered to reove noise. This filtering also introduces a delay. To ake sure we only increent the control signal when we are able to easure the result of the previous increent, the control signal and the easureent need to be synchronied by applying the sae filter, and hence the sae delay. For the roll and pitch controls this is straightforward, as is shown in figure 3. The specific thrust increent is added to the rotor angular rates after a second inversion step in Figure 4. Therefore, the rotor angular rates should also be filtered with the sae filter. However, the angular rates of the rotors are also used to control the angular acceleration of the vehicle. This is done with the inner INDI control loop shown in Figure 4. Here the increent of the rotor angular rates is calculated fro the easured angular acceleration, which is obtained fro the gyroscopes. This easureent is also noisy, and needs to be filtered. Because of this, the angular rates of the rotors should be filtered with the sae filter. Since both the inner and the outer loop ake use of the sae actuators, the rotational rate of the rotors, their filters need to be the sae. For the experient, we chose a filter with a ω n = 5 rad/s and ζ =.55. Choosing a lower cutoff frequency will result in less noise, but ore delay. This eans that it will take longer for disturbances to be easured and counteracted. Choosing a higher cutoff frequency will have the reverse effect, ore noise will end up in the control signals but disturbances are counteracted faster. D. Lineariation The control of the acceleration is nonlinear in ters of the inputs, especially roll and pitch, as can be seen fro

5 T ν + + (G + G 2 ) ω ω c Ω Ωerr + A() MAV Ω G 2 ω f H() ω T s Ω f T s Ω f H() Ω Fig. 4. The inner INDI control structure. Equation 2. In Equation 5 it can even be seen that soe of the control derivatives can change sign, for instance φ for different values of φ. What this eans in practice is that if the increents in the input are large, because suddenly a large lateral acceleration is required, they will result in a different acceleration than intended. A solution ay be to ipleent a nonlinear ethod of finding increents in the input that give the desired increent in the acceleration. In this paper, we solved this issue by bounding the acceleration increent such that the resulting change in inputs can still be approxiated linearly. IV. EXPERIMENTAL SETUP The goal of the experient is to test how well the controller can handle gust disturbances. The experient will be perfored indoors, such that there is a controlled environent in which repeatable experients can be perfored. The drone will be coanded to fly back and forth between two waypoints at the sae altitude, which are about one eter apart in the east direction. The source of the disturbance is a Master DF3P 465 W fan placed in front of one of the waypoints, blowing towards north. The fan produces a non-unifor wind with airspeeds ranging fro.3 /s in the center to 4. /s towards the edge of the fan, which was easured downstrea. When the drone reaches the waypoint with the fan, it will suddenly experience the wind. When the drone leaves the waypoint, it will fly out of the wind again. The drone will spend eight seconds at each waypoint and repeat this three ties. The perforance of the INDI controller will be copared to a PID controller which is anually tuned to give the fastest response possible. This PID controller also akes use of the inner loop INDI controller for attitude control, but it does not use the outer loop INDI controller. The P, I and D gains work directly on the position and velocity to produce a reference roll, pitch and thrust. For the PID controller, there is a trade-off to be ade. By increasing the integral gain, faster offset copensation can be obtained. This way the quadrotor can adjust to the disturbance of the fan faster. However, by increasing the integral gain, the quadrotor will have ore overshoot in reference tracking tasks such as sudden position changes. This trade-off is non-existent for the INDI controller. The MAV used for the experients is the Bebop quadrotor fro Parrot. Instead of the stock firware, it is running the Paparai open source autopilot syste. The control algorith, as well as the onboard acceleroeter and gyroscope, were running at 52 H. An infrared otion tracking syste called Optitrack was used to obtain position inforation. This syste can easure the drone s position with illieter accuracy at a frequency up to 2 H. But because we want the experient to be realistic for outside scenarios and since ost Global Positioning Syste (GPS) odules can only provide position updates at four H, the data was only sent to the drone at a frequency of four H. The Optitrack syste can deliver better accuracy than a GPS odule, so it ight see that the experient is still not realistic, regardless of the low update rate. However, the ain contribution of this paper is a controller that can cope with gust disturbances. This is achieved through effective use of the acceleroeter and an increental control schee, and does not depend on the position accuracy. V. RESULTS For the outer loop INDI experient, Figures 5 and 6 show the acceleration of the MAV in the East and North directions respectively. The acceleration signal shown is filtered on the drone with the second order filter given by Eq. 6 with ζ =.55 and ω n = 5 rad/s. Figure 5 shows the acceleration in the direction orthogonal to the disturbance. Here we see accurate tracking of the acceleration reference. In Figure 6, we can see the effect of the wind on the acceleration at 35 s and 43 s when the MAV enters and leaves the wake of the fan. Figures 7 and 8 show the position of the quadrotor during the experient for the outer loop INDI controller. Figure 7 shows that the INDI controller is able to track step responses with inial overshoot. Fro Figure 8 we can see that entering and leaving the fan s wake, for instance at 35 s and 45 s, typically results in a position change of about five c. This error is rejected in two seconds after its occurrence. Note the offset that can be observed in Figure 8. In our previous work on the inner loop attitude control we did not see such offsets in the attitude angles. The reason for this is

6 ÿ [/s 2 ] x [] ÿ ref ÿ Fig. 5. Acceleration in the East direction for the INDI controller. x ref x Fig. 8. Position in the North direction for the INDI controller with fan disturbance..5 ẍ [/s 2 ].5.5 ẍ ref ẍ Fig. 6. Acceleration in the North direction for the INDI controller with fan disturbance. y [] y ref y Fig. 7. Position in the East direction for the INDI controller. that the angular acceleration easureent was bias-free, as it was derived fro the angular rate easureent. For the linear acceleration, we rely on a direct easureent. Sensor drift or an error in the attitude estiation can lead to errors in the estiate of the acceleration. In the coplete interval of [3,8] s in the x axis, the average easured acceleration was -.8 /s 2. However, because of estiation errors there is no real acceleration and a bias in the position is the result. This proble can be solved by estiating the acceleroeter bias through the derivative of the speed estiate, which will be considered in future work. Copare this with the position of the quadrotor during the experient for the PID controller in Figures 9 and. Fro Figure 9 it can be seen that the relatively large integral gain resulted in soe overshoot of the step reference, which did not happen without the integral gain. However, fro Figure it can be observed that even with this large integral gain, the quadrotor is blown away ore than half a eter and it takes about five seconds for it to get back to the reference. The oent it flies back to the first waypoint it overshoots in the other direction, because now it suddenly flies out of the fan s wake. Because the fan was blowing toward the north, the average position error in the North direction is a easure of the perforance. Fro Figures 8 and the average error in x is 6.4 c for the INDI controller and 9.3 c for the PID controller. As was entioned before, the position error for the INDI controller is ostly due to the acceleration bias. Estiation of this bias can result in even better perforance. Finally a top view of the experient is depicted in figure. It shows the track of the quadrotor for the experients with the INDI and PID controller for the interval of [3,5] s and [2,4] s respectively. The figure shows how well the INDI controller can reject the disturbance of the fan copared to a traditional PID controller.

7 y [] x [] y ref y Fig. 9. Position in the East direction for the PID controller. x ref x Fig.. Position in the North direction for the PID controller with fan disturbance. x [] PID INDI Ref Fan y [] Fig.. Top view of the trajectories of the INDI and PID controller (best viewed in color). VI. CONCLUSIONS We have generalied Increental Nonlinear Dynaic Inversion (INDI) for the control of linear accelerations of a quadrotor subject to disturbances. The experients show that the perforance of the INDI controller is three ties better than that of a traditional Proportional Integral Derivative (PID) controller in ters of average position error. In the experients, the quadrotor received four H position updates, which eans that the technique can readily be applied outdoors with standard GPS odules. This outer loop INDI controller enables Micro Aerial Vehicles to perfor tasks that require accurate position control under gusty conditions, such as flying near obstacles and entering a building through a window. REFERENCES [] A. Ryan and J. Hedrick, A ode-switching path planner for UAVassisted search and rescue, in 44th IEEE Conference on Decision and Control, 25, p. pp [2] J. Ki and S. Sukkarieh, Airborne siultaneous localisation and ap building, in IEEE International Conference on Robotics and Autoation, 23. [3] D. Mellinger, N. Michael, and V. Kuar, Trajectory generation and control for precise aggressive aneuvers with quadrotors, The International Journal of Robotics Research, vol. 3, no. 5, pp , 22. [4] K. Alexis, G. Nikolakopoulos, and A. Tes, Constrained-Control of a Quadrotor Helicopter for Trajectory Tracking under Wind-Gust Disturbances, in IEEE Mediterranean Electrotechnical Conference, 2. [5] S. Shen, N. Michael, and V. Kuar, Autonoous Multi-Floor Indoor Navigation with a Coputationally Constrained MAV, in International Conference on Robotics and Autoation, May 2. [6] N. Sydney, B. Syth, and D. A. Paley, Dynaic control of autonoous quadrotor flight in an estiated wind field, in IEEE Conference on Decision and Control (CDC), Deceber 23. [7] S. L. Waslander and C. Wang, Wind Disturbance Estiation and Rejection for Quadrotor Position Control, in AIAA Infotech@Aerospace Conference and AIAA Unanned...Unliited Conference, April 29. [8] F. Schiano, J. Alonso-Mora, K. Rudin, P. Beardsley, R. Siegwart, and B. Siciliano, Towards Estiation and Correction of Wind Effects on a Quadrotor UAV, in International Micro Air Vehicle Conference and Copetition (IMAV), August 24. [9] R. C. Gardner and J. S. Hubert, Coparative Fraework for Maneuverability and Gust Tolerance of Microhelicopters, Journal of Aircraft, vol. 5, no. 5, pp , 24. [] G. M. Hoffann, H. Huang, S. L. Waslander, and C. J. T. c, Precision flight control for a ulti-vehicle quadrotor helicopter testbed, Control Engineering Practice, vol. 9, no. 9, pp , 2. [] P. Siplicio, M. Pavel, E. van Kapen, and Q. Chu, An acceleration easureents-based approach for helicopter nonlinear flight control using Increental Nonlinear Dynaic Inversion, Control Engineering Practice, vol. 2, no. 8, pp , aug 23. [2] J. Wang, T. Raffler, and F. Holapfel, Nonlinear Position Control Approaches for Quadcopters Using a Novel State Representation, in Guidance, Navigation and Control Conference. AIAA Paper , 22. [3] G. Hattenberger, M. Bron, and M. Gorra, Using the Paparai UAV Syste for Scientific Research, in International Micro Air Vehicle Conference and Copetition (IMAV), 24. [4] B. J. Bacon, A. J. Ostroff, and S. M. Joshi, Reconfigurable NDI Controller Using Inertial Sensor Failure Detection & Isolation, IEEE Transactions On Aerospace And Electronic Systes, vol. 37, no. 4, pp , Oct 2. [5] E. J. J. Seur, Q. P. Chu, and G. C. H. E. de Croon, Adaptive Increental Nonlinear Dynaic Inversion for Attitude Control of Micro Aerial Vehicles, Journal of Guidance, Control, and Dynaics, vol. 39, no. 3, pp , March 26.