Improved State Estimation in Quadrotor MAVs: A Novel Drift-Free Velocity Estimator

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1 Iproved State Estiation in Quadrotor MAVs: A Novel Drift-Free Velocity Estiator Dinuka Abeywardena, Sarath Kodagoda, Gaini Dissanayake, and Rohan Munasinghe arxiv:9.3388v [cs.ro] Sep Abstract This paper describes the synthesis and evaluation of a novel state estiator for a Quadrotor Micro Aerial Vehicle. Dynaic equations which relate acceleration, attitude and the aero-dynaic propeller drag are encapsulated in an extended Kalan filter fraework for estiating the velocity and the attitude of the quadrotor. It is deonstrated that exploiting the relationship between the body frae accelerations and velocities, due to blade flapping, enables drift free estiation of lateral and longitudinal coponents of body frae translational velocity along with iproveents to roll and pitch coponents of body attitude estiations. Real world data sets gathered using a coercial off-the-shelf quadrotor platfor, together with ground truth data fro a Vicon syste, are used to evaluate the effectiveness of the proposed algorith. INTRODUCTION Quadrotor Micro Aerial Vehicles (MAV) are siple robotic platfors to construct. In its basic for, it is no ore than two counter rotating propeller pairs attached syetrically to a rigid cross-like frae, along with the eans to control the speed of each individual propeller. This syetric design is what has enabled the quadrotor to becoe a siple yet powerful vertical take-off and landing aerial platfor that is popular aong the robotics counity. With this siplicity coes the burden of controlling otion in 3D space with the use of just four actuators. Underactuated and coupled dynaics of the quadrotor ake it nearly ipossible for a huan pilot to gain control of it, unless a well tuned control syste is in place. Such a control syste is also vital if autonoy is a goal, as is the case with ost MAVs. Estiates of controlled states and their derivatives are essential for any control syste, and where those estiates are accurate and frequent in tie, it has been deonstrated that quadrotors have extree aneuverability and agility[]. However, MAVs are - by design - liited in their payload capacity and with those liitation, obtaining accurate and fast state estiates becoes a challenge. For exaple, MEMS inertial sensors can provide fast but coarse state estiates [], while exteroceptive sensors such as lasers and caeras[3] render ore accurate state estiates albeit at a slower rate. Attepts to erge these two sensing doains are frequent in MAV literature [4], [] and an application of siilar ideas to quadrotors was presented in [6]. One aspect coon to ost MAV state estiators is their use of inertial sensors. Typically gyroscopes, acceleroeters D. Abeywardena S. Kodagoda and G. Dissanayake are with the Centre for Autonoous Systes, University of Technology Sydney, NSW, 7 Australia. R. Munasinghe is with the Departent of Electronic and Telecounication Engineering, University of Moratuwa, Sri Lanka and agnetoeters are used for the purpose of attitude estiation [7]. Based on a long history of research in inertial navigation systes, sensor fusion algoriths usually eployed for this task ake use of the equations of otion of the sensing unit in three-diensional space. The ain advantage of this approach is that these generic estiators are specific only to the sensor package geoetry and as such can be used independently of the platfor on which the sensors are ounted. However, they fail to exploit the dynaics of the vehicle under consideration in the estiation process, leading to a potentially sub-optial result. The value of using specific dynaic characteristics of the vehicle has been reported in the case of land vehicles [8] and air vehicles[9]. Siilarly, in this paper we deonstrate that the influence of blade flapping in a quadrotor leads to a set of dynaic equations that can aid state estiation using inertial sensors. The rest of the paper is arranged as follows. First, soe background on quadrotor state estiators and a discussion on what otivated us to look beyond the state-of-the-art is presented. We will then briefly present the quadrotor dynaic equations that are of interest to the state estiation process. After highlighting the shortcoings of the generic design, a novel state estiator design is presented along with experiental results which deonstrate the accuracy and consistency of estiates. The article is conclude by exploring the iplications of the novel algorith. BACKGROUND AND MOTIVATION MAV attitude estiators that fuse gyroscope and acceleroeter easureent using generic algoriths are frequently reported in literature [], [7], []. In a nutshell, these algoriths operate by fusing easureents of a triad of body ounted gyroscopes and acceleroeters. Gyroscope easureents are a source of high frequency attitude rate inforation, but they alone are not sufficient for drift free attitude estiation due to bias and various other fors of noise present in a typical low cost sensor. Attitude estiators for MAVs overcoe this issue by assuing that acceleroeters predoinantly easure gravitational acceleration and are thus capable of providing low frequency inforation about MAV orientation with respect to gravity. Clearly, when the vehicle accelerations are significant, as in the case of quadrotor, this assuption does not hold []. Furtherore, such estiators are incapable of drift free velocity estiation, as they can only be generated by integrating noisy acceleroeter easureents. To coplicate the atters even further, acceleroeter easureents need to be copensated for gravity before this integration, and

2 such a copensation requires an accurate attitude estiate. As entioned before, one proising way to overcoe these deficiencies is to exaine the behaviour of the MAV in question, in-order to identify suitable characteristics that would assist the estiation process. Martin et. al. [] have analysed the behaviour of a quadrotor MAV in detail and also presented equations describing easureents of an acceleroeter ounted on-board a quadrotor. Their results otivated us to reforulate the state estiators for quadrotors and to redesign the considering the true sensor behaviour as opposed to conventional vehicle independent assuptions. In addition to iproving the accuracy of the attitude estiate, the design presented here provides a drift free estiate of the horizontal coponents of translational velocity of the quadrotor. Recently, a siilar idea was presented in [3] where two separate non-linear copleentary filters were utilised to estiate attitude and velocity of a quadrotor MAV. The filter forulation presented in this paper is different fro [3] and we also present experiental results validating the concept. The velocity estiates thus derived are of critical iportance to control and navigational tasks of a quadrotor, as will be discussed in concluding rearks. QUADROTORS: WHAT MAKES THEM UNIQUE? A thorough derivation and analysis of the quadrotor dynaics can be found in [4] and []. Rather than reiterating the derivation, here we ai to briefly suarise the iportant equations and to provide an intuitive description of the ost salient features of the dynaic behaviour that akes quadrotors a unique MAV. Let {E} be the earth fixed inertial frae, and a vector [ x y z ] T denote the position of the centre of ass of the quadrotor expressed in {E} (See Fig. ). Let {B} [ b b b 3 ] T be a body fixed frae positioned at the centre of ass of the quadrotor. angular velocity Ω [ ω x ω y ω z ] of {B} with respect to {E}, to Euler rates can be expressed as: φ tan θ sin φ tan θ cos φ ω x θ = cos φ sin φ ω y () ψ sin φ/ cos θ cos φ/ cos θ ω z The equation describing the evolution of translational otion of the quadrotor as derived in [] is of special interest to the estiator design that will be presented in following sections. where V = g k T ωi b 3 λ i= ω i Ṽ () i= V = Velocity of {B} as observed fro an inertial frae g = gravity vector k T = thrust coefficient of propellers λ = a positive coefficient known as rotor drag coefficient ω i = rotational velocity of i th rotor, i {,, 3, 4} Ṽ = projection of V on to the propeller plane = ass of the quadrotor Equation () sheds light on two key aspects of the quadrotor. First and the ost obvious is the fact that the thrust force is perpendicular to the propeller plane, and thus has no effect on otion along that plane. Secondly and ore iportantly, we see the presence of a force which is proportional to the translational velocity of the quadrotor. For an intuitive description of this force, we refer readers to Fig. which shows a cross section of a quadrotor in flight, and provide below a siplified explanation of the origin of this force. Fig.. Coordinate frae definitions for the quadrotor dynaic odel Fig.. Scheatic of a quadrotor iediately after tilting sideways, but before it starts oving. ΣT is the suation of propeller thrusts and corresponds to the second ter in () The orientation of {B} with respect to {E} is defined using a cuulative rotation of Euler angles ψ (Yaw), θ (Pitch) and φ (Roll) in that order, around b 3, b and b, respectively. R is defined as the rotational transforation atrix fro {B} to {E}. The kineatic equation relating the instantaneous Fig. shows a quadrotor in a hypothetical state where it has tilted sideways to initiate a translation in a horizontal direction but iediately before it gains any translational otion. At this point thrust fro propellers and gravity are the only forces acting on our siplified quadrotor odel. As is obvious, in

3 3 this particular state, thrust force generated fro the propellers is perpendicular to the propeller plane. Fig. 3. Quadrotor, after tilting, starts oving sideways. f and f are the orthogonal coponents of ΣT the aount of blade flapping is dependent on the translational velocity of the quadrotor, the coponent of thrust force along the body plane is also a function of that velocity. The last ter in () odels the ipact of this coponent of the thrust force on the translational otion of the quadrotor. If one is to place an acceleroeter on-board the quadrotor, with its sensing axis parallel to the propeller plane, that acceleroeter will easure a force that is roughly proportional to the velocity of the quadrotor along the sae axis. In fact, in the next section, it is shown that this is the only significant force that the said acceleroeter will sense. (Interestingly, () ignores the aerodynaic drag experienced by a body oving through air, which is usually a function of the square of the velocity. This can be justified for quadrotor MAVs that ove at relatively low speeds.) This is the unique characteristic of quadrotor MAVs that will later be exploited to the benefit of the state estiator. To conclude this section, we re-write () using b V (i.e. V in {B} frae) to facilitate the estiator design. After neglecting the second order ters that appear due to coordinate frae transforation, the first two coponents of b V { b v x, b v y, b v z } can be written as where b v x g sin θ k b v x b v y g cos θ sin φ k k = λ i= b v y In what follows, we assue that k is a positive constant considering the fact that the suation of propeller rotational rates are fairly constant during sooth flight. ω i (3) Fig. 4. As the propeller blades rotate, flapping is deterined by their position with respect to the direction of otion of the propeller as a whole. As stated, the state depicted in Fig. is hypothetical in the sense that even the slightest tilt of the quadrotor will induce translational otion. Fig. 3 shows a ore realistic situation in which quadrotor now oves right with a nonzero velocity. For a propeller with two blades, we can now identify a retreating and an advancing blade, as shown in blue and green respectively in Fig. 4. The velocity of the advancing blade with respect to free air is higher than that of the retreating blade, due to the translational velocity of the whole quadrotor. This creates a force ibalance between the two blades of the sae propeller and thus causes the blades to flap up and down as they rotate. Blade flapping forces the propeller to rotate out of plane and the flapping angle of a blade is at a axiu just before it transitions fro advancing state to retreating state or vice versa. As shown in Fig. 3, blade flapping causes the thrust force of the propeller to be tilted in a direction which opposes the otion of the quadrotor. As INERTIAL SENSORS IN QUADROTORS This article is concerned with the quadrotor state estiators based on inertial sensors and specifically with acceleroeters and gyroscopes. For siplicity, we assue that a triad of acceleroeters and gyroscopes are ounted at the centre of ass of the quadrotor body. For both types of sensors we adhere to standard MEMS error odels []. Gyroscopes easure the instantaneous rotational rate of the body with respect to the inertial frae, and their easureents can be odelled independently of the equations of otion of the oving platfor to which they are attached. g i = Ω i + β gi + w gi (4) β gi = β gi + w βgi τ gi () where β gi is the bias of i th gyroscope and τ gi is the tie constant of i th gyroscope bias. w gi and w βgi are zero ean White Gaussian Noise (WGN) ters. In contrast, acceleroeters easure a cobination of inertial and gravitational acceleration, and their easureents can be expressed using the equations of otion governing the body they are ounted on. Perhaps one of the best

4 4 exaple of the value of this strategy is the case of a triad of acceleroeters ounted on a quadrotor platfor. Denoting by ã i the acceleration that would be easured by an ideal acceleroeter, we cobine the acceleroeter easureent odel with () to arrive at: ã = V g = k T ωi b 3 λ i= ω i Ṽ (6) Equation (6) describing the readings obtained fro an onboard triad of acceleroeter is unique to quadrotors and is of critical iportance to a state estiator in that context. As stated in the previous section, equation (6) shows that the acceleroeters along b and b coordinate axes are only sensitive to a force which is dependant on the projection of the quadrotor translational velocity on to b, b plane. Furtherore, the coponent of the gravitational acceleration in the body frae (which is typically large copared to inertial accelerations of slow oving vehicles) no longer influences the acceleroeter easureent. In the following section, we will exploit this unique property to design a better state estiator for quadrotors. i= ESTIMATOR DESIGN The goal here is to design a state estiator for the quadrotor, giving due regard to the dynaic and kineatic equations presented in the previous sections. For this, we propose a six state, Extended Kalan Filter (EKF) based state estiator. The filter states are: φ Roll angle in current orientation estiate θ Pitch angle in current orientation estiate β gx Bias in X axis gyroscope β gy Bias in Y axis gyroscope b v x X velocity coponent of quadrotor in body frae b v y Y velocity coponent of quadrotor in body frae Process Model Equations (), (3) - () for the EKF process equation. Out of the three Euler angles we can only estiate φ and θ as the equations are expressed in a for independent of the yaw angle ψ. φ = (g x β gx + w gx ) + tan θ cos φ(g z β gz ) + tan θ sin φ(g y β gy + w gy ) (7) θ = cos φ(g y β gy + w gy ) sin φ(g z β gz ) β gx = β gx + w βgx τ gx β gy = β gy + w βgy τ gy (8) b v x = g sin θ k b v x + w αx b v y = g cos θ sin φ k b v y + w αy where, w αx and w αy are WGN ters included to account for the odel iperfections in (3). Equations (7), (8) and (9) together describe the process dynaics of the estiator. The resulting syste can be represented as a non-linear function of states, control inputs and noise ters. Measureent Model ẋ = f(x, u, w) Observations of the EKF are the easureents fro X and Y acceleroeters, which are aligned respectively with b and b. Measureent equations can be easily derived fro (6), after including acceleroeter noise ters, which are assued to be Gaussian. a x = k b v x + w ax (9) () a y = k b v y + w ay where a x and a y are respectively the easureents fro the X and Y axis acceleroeters on-board the quadrotor. Here we assue that acceleroeter biases are rando constant values which can be copensated for, offline. EKF Mechanization Equations For the echanization of the Extended Kalan Filter, the discrete state transition atrix A k should be calculated. For this we first calculate F, which is the Jacobian atrix of partial derivatives of f with respect to x. Then A k is calculated by discretization of the Jacobian atrix. F (t) = f(x, u, w) x ˆxk,u k Discretization is perfored with a truncated Taylor series approxiation and a saple tie of T s, resulting in, A k = I + F (t)t s In deriving the discrete process noise atrix Q k, we assue that noise ters in (7) and (8) are uncorrelated with each other as well as with acceleroeter noise ters. w = [ ] T w gx w gy w βgx w βgy w αx w αy W (t) = diag [ ] σgx σgy σβgx σβgy σαx σαy Q(t) = G(t)W (t)g T (t) The first four ters of the (t) are the noise variances of gyroscope sensors and their biases. These can be found by experientation with actual sensors. Last two ters, which

5 correspond to the uncertainty in (9) were approxiated first and then fine tuned for optiu perforance of the estiator. Also, G(t) = f(x, u, w) w Discretization of Q(t) results in Q k. ˆxk,u k Q k = Ts AQ(τ)A T dτ Measureent atrix H required for the EKF can be directly obtained fro () as, H = [ ] k / k / Assuing uncorrelated errors in acceleroeter easureents, easureent noise atrix R k becoes diagonal, consisting only of the noise variances of the X and Y acceleroeters. R k = diag [ σ ax ] σay For initialisation, all states of the filter are set to zero and their error covariances are set to sall positive values reflecting the uncertainty in initial estiate. With ultiple experiental runs, it was found that changes of up to % in the initial values and the noise variances have negligible effect on filter perforance. We attribute this robustness of the estiator to the linear easureent odel and not-so-strong non-linearities in the process equations. EKF state prediction was carried out with the use of a nd order Runga-Kutta integrator. Covariance projection, Kalan gain calculation, state update and covariance update equations of the estiator take their standard fors as detailed in [6]. ARDRONE QUADROTOR AND THE EXPERIMENTS The quadrotor platfor used for the experients presented in this article is the Parrot ARDrone [7] (see Fig. ) ARDrone weighs about 4g including the protective hull and has a flight tie of about inutes. Straight out of the box, ARDrone is an extreely stable quadrotor platfor and therefore is an excellent platfor for quadrotor based research. It is equipped with a wide array of sensors including triad of acceleroeters, triad of gyroscopes, two caeras -one facing front and other facing down- and downward pointing sonar sensors. All sensor data fro the ARDrone are wirelessly transitted to a ground station PC either running Windows or Linux. An open source C API is provided which can be easily extended to develop application on the ground station to process incoing sensor data and to send out control coands to the ARDrone. It is also equipped with a preprograed closed source attitude control syste, which takes care of the low-level stabilisation and control tasks, while providing users the ability to develop applications for higher level navigational tasks. It is desirable to have ground truth states trajectories for perforance evaluation of the proposed estiator. Therefore, Fig.. ARDrone Quadrotor used for experients all our ARDrone experients were perfored in a Vicon otion capture environent. The Vicon otion capture syste uses a set of reflective arkers rigidly attached to the quadrotor body, which are observed by 8 fixed IR caeras to directly copute the attitude and position of the quadrotor with respect to the Vicon coordinate frae. In a typical experient, ARDrone was anually piloted within the Vicon environent (approxiately ) using a joy stick attached to the ground station coputer. The inertial sensor data were continuously streaed to the ground station coputer at Hz and were stored for post processing. Vicon generated state estiated were also stored in a separate PC. Matlab coputing environent was used for post processing of both inertial and Vicon data. A critical paraeter that needs to be precoputed for the estiator is the rotor drag coefficient λ. Since a theoretical calculation of this paraeter is a coplex task, we resorted to an experiental estiation ethod. The basic ethodology adopted here is to obtain the acceleroeter easureents and ground truth velocity data of a few flight tests. A rough estiate of the paraeter k (which incorporates λ ) can then be obtained by forulating () as a least-squares proble. For the ARDrone, the best estiate for the paraeter k was found to be.7. This paraeter estiation task was run only once and the derived k values was used for all subsequent estiation tasks. EXPERIMENTAL RESULTS During one experient, the AR Drone was anually operated within the Vicon environent, oving freely while keeping the height approxiately constant. A three-diensional trace of the path taken by the MAV in a typical experient is shown in Fig. 6. The results presented in the following sections are based on the data gathered fro this experient. Fig. 7 shows the attitude estiates of the proposed EKF together with the ground truth obtained fro the Vicon syste. For coparison purposes, we have also plotted the attitude estiates fro a generic estiator as detailed in [] in Fig. 9. It is iportant to note the iproveent in the pitch estiate of the proposed estiator over the generic estiator. This iproveent is ore pronounced in places where the quadrotor changes its flight direction (for exaple around 4.6 and 7.8 sec). During those intervals, the quadrotor undergoes

6 6 Ground truth Inertial estiate. Angle (degrees) Z () Y () Start End.. X (). Ground truth Inertial estiate Fig. 6. Three-diensional flight path of the ARDrone experient high inertial accelerations and the assuption that the acceleroeter easureents are doinated by gravitational acceleration fails to hold. Thus generic attitude estiators based on this assuption produce erroneous results. As expected the proposed EKF attitude estiates agrees ore with the ground truth because such an assuption is not utilized in that design. However, when the quadrotor is not undergoing considerable accelerations, the two attitude estiates converge and the generic estiator can perfor just as well as the proposed ethod. Fig. and present a coparison between the errors in the roll and pitch attitude estiates of both the proposed EKF and the generic estiator. Even with the proposed EKF, unodelled dynaics (such as displaceent of acceleroeter fro the centre of ass of the quadrotor) causes an increase in estiation error when the quadrotor undergoes large accelerations. But overall, it is clear that the errors in the proposed design are considerably less than those of the generic design. Fig. 8 presents the velocity estiate fro the proposed EKF together with the ground truth. Again for coparison, Fig. shows the velocity estiates in a generic design where, velocity is estiated by integrating inertial accelerations calculated by copensating the acceleroeter easureents for gravity. A coparison between the errors in velocity estiate obtained fro the proposed estiator and the generic estiator is shown in Fig. (c), where total velocity error is the su of root square errors of both X and Y axes. What is iportant to note is that the proposed strategy produces velocity estiates in which errors do not grow with tie, while estiating velocity through direct integration of accelerations as ipleented in the conventional design leads to a significant drift. As zero velocity updates, that can be used to correct this behaviour in land vehicles, are no longer viable with an MAV without soe deliberate control strategies, this points to a significant advantage of the estiator proposed in this article. CONCLUSION In this article, we presented a novel state estiator for quadrotor MAVs, where clear iproveents in estiates steing fro the incorporation of quadrotor specific dynaical constraints were deonstrated. Our design is based on an Angle (degrees) 3 3 Fig. 7. Coparison of ground truth and inertial attitude estiates of AR Drone. Roll angle (φ), Pitch angle (θ) Velocity (/s) Velocity (/s)... Ground truth Inertial estiate Ground truth Inertial estiate 3 3 Fig. 8. Coparison of ground truth and inertial velocity estiates of AR Drone. X Velocity (V x), Y Velocity (V y) EKF and is capable of estiating both roll and pitch angles of the attitude in addition to X and Y coponents of the body frae translational velocities within a bounded error. This estiator is applied to inertial data gathered fro real world flight experients. The resulting attitude and velocity estiates obtained atch closely with the ground truth and are drift free. Before concluding the discussion on the estiator perforance, we note that our design by itself is not a perfect solution to the proble of quadrotor state estiation. We

7 7 Inertial Estiate Ground Truth 6 4 Error in generic estiator Error in proposed estiator Angle (degrees) Angle error (degrees) Inertial Estiate Ground Truth Error in generic estiator Error in proposed estiator Angle (degrees) Angle error (degrees) 3 3 Fig. 9. Coparison of ground truth and inertial attitude estiates of AR Drone, obtained fro the generic estiator. Roll angle (φ), Pitch angle (θ) Velocity (/s) Inertial Estiate Ground Truth Total velocity error (/s) Proposed estiator Generic estiator 3 3 Fig.. Estiation errors of both estiator designs. Roll angle (φ) estiation error, Pitch angle (θ) estiation error, (c) Total velocity estiation error (c) velocity (/s).... Inertial Estiate Ground Truth. 3 3 Fig.. Coparison of ground truth and inertial velocity estiates of AR Drone, obtained fro the generic estiator. X Velocity (V x), Y Velocity (V y) with exteroceptive sensors such as caeras and GPS. The two caeras in the ARDrone akes it an ideal platfor for visual Siultaneous Localisation And Mapping (SLAM). One key drawback in eploying onocular SLAM for MAVs is the unavailability of odoetry for scale recovery. Another ore obscure proble is the alignent of caera with the MAV body frae. Fro a control theoretic perspective, orientation of the body frae is what atters and isalignent of caera and body fraes can lead to poor control perforance in a SLAM only MAV state estiator. Both these probles can be solved by tightly integrating the estiation algorith presented here with a onocular SLAM algorith. We believe this to be an exciting research avenue. believe that two key iproveents need to be ade to our design. First, an online estiation of the paraeter λ and acceleroeter biases will iprove estiation accuracy and ease the filter design process. Secondly, the estiation ψ angle and velocity b v z will iprove the autonoy of the quadrotor. Our current research focuses on these iproveents. In addition, we also expect to fuse the inertial inforation ACKNOWLEDGEMENTS This work is supported by the Centre for Autonoous Systes, University of Technology Sydney. REFERENCES [] D. Mellinger, N. Michael, and V. Kuar, Trajectory generation and control for precise aggressive aneuvers with quadrotors, in Proc. International Syposiu on Experiental Robotics, Dec.

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