Dynamics and Control of a Biomimetic Single-Wing Nano Air Vehicle

Transcription

1 g 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeA01.6 Dynamics and Control of a Biomimetic Single-Wing Nano Air Vehicle Kingsley Fregene, Senior Member, IEEE and Cortney L. Bolden Abstract This paper describes the Samarai a rotating mono-wing nano air vehicle (NAV) designed at Lockheed Martin Advanced Technology Laboratories. Challenges uniquely associated with developing this type of vehicle are identified and a dynamic modeling and control synthesis procedure is described for tackling them. It is shown that, by modeling the rotating wing as a lower-complexity spinning disk, and using the disk states for feedback control, a very simple control design results. This approach is validated in simulation of flight maneuvers that include vertical, hovering and translational motion segments. N I. INTRODUCTION ano UAVs (defined by the U.S. Defense Advanced Research Projects Agency [1]) as no larger than 7.5cm weighing no more than 10g) offer niche operational advantages over larger UAVs. These smaller, more compact vehicles open up many more mission avenues including close range surveillance with low probability of detection, indoor and outdoor operations, more flexibility in delivering small payloads and safer operation. However, the design and operation of these vehicles present several significant challenges, including propulsion, aero/dynamics modeling, flight control and navigation. While each of these is significant, the focus of this paper will be on vehicle dynamics and control. Understanding vehicle dynamics and control at this scale is particularly challenging because the associated size/ weight constraints results in airfoils that operate in ultra-low Reynolds number (<<15000) aerodynamic flow regimes. Indeed the difficulty in generating sufficient lift at this scale has caused many nano UAV designs to be biologically inspired. That is, the flight design is (in some sense) inspired by nature s flyers and gliders including flapping wing designs motivated by insect or bird flight and those based on single-winged fruits and seeds (or samaras). Other more conventional designs based on various rotorcraft forms including standard helicopters and dual coaxial rotor-copters have also been proposed [2, 3]. Flight control and control actuation are also considerable Manuscript received September 15, This work was supported in part by the U.S. Defense Advanced Research Projects Agency (DARPA) under contract number W31P4Q-06-C Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA. Distribution Statement A (Approved for Public Release, Distribution Unlimited). Kingsley Fregene and Cortney Bolden are with the Lockheed Martin Advanced Technology Laboratories, 3 Executive Campus, Ste 600, Cherry Hill, New Jersey, 08002, USA (corresponding author phone: ; kocfrege@ieee.org). challenges because the constraints identified above and size, weight and power (SWaP) considerations dictate that conventional flight control surfaces, even if they fit, need to be actuated in novel ways. Control authority is also quite limited, making stable and robust flight difficult. Another challenge is the development of effective flight control algorithms that are computationally inexpensive and fully respect limitations on control surface type, size and actuation modality. In this paper, we describe the Samarai, a single-wing UAV inspired by nature s maple seeds and an approach for vehicle dynamic modeling and control that attempts to overcome some of these problems. Unlike the flapping wing and helicopter-based designs, a rotating single-wing vehicle like the Samarai is inherently stable in hover, mechanically simple and robust, has clean aerodynamics and very few moving parts. It is biomimetic, yet does not depend on frangible feathers, delicate wings or precision moving parts to operate. Furthermore, the fact that the vehicle is fuselageless coupled with its dynamic stability simplifies the control problem to that of controlling a virtual disk created by the spinning blade. This disk has equations of motion similar to but simpler than a helicopter or flapping wing vehicle. Moreover, it is well known that, for a given dimensional constraint, the greatest Reynolds number for a flight vehicle is achievable with a rotating wing. A samara configuration can have a wing loading one tenth that of a conventional helicopter or flapping wing design of the same size. The lower loading greatly reduces the power requirement for a given endurance. Indeed, it was shown recently that a single rotating wing requires half as much power as a flapping wing to achieve a comparable level of flight performance [4]. Examples of other works that have described single-wing or mono-copter designs are [5-7] but none has reported detailed modeling and control design at the vehicle scale described in this paper. Also, these references focused on larger mono-copters in which many of the challenges peculiar to nano UAVs are less of a concern. The rest of this paper is organized as follows: Section II provides a more detailed description of the Samarai, including its distinguishing features and aerodynamics as well as factors necessary to achieve controlled flight. Dynamic models of the vehicle suited for simulation and controls development are described in Section III, followed by a discussion of the control strategy in Section IV. Simulation results are presented and discussed in Section V followed by concluding remarks in Section VI /10/$ AACC 51

2 II. SAMARAI DESCRIPTION The Samarai is a single-bladed rotary wing UAV that mimics a samara or winged seed, such as a maple seed shown in Fig. 1. parameters that ensure the vehicle in rotation is stable about the feather axis (y-axis in Fig. 3). Without this stability, active control of the wing deflection will be required during each rotation cycle, which poses prohibitive requirements for sensing, processing, and actuation [10]. Fig. 1. The maple seed (samara). The Samarai configuration (depicted in Fig. 2) consists of a single wing, payload, trailing-edge flap, guidance and control system that yields a mechanically simple, globally stable platform. It is propelled by a fuel-powered tip jet and controlled by a single cyclic/collective flap. This design differs in aerodynamics from nature s samaras because it is powered and exhibits several controlled flight modes including hover, transition, and forward flight, while nature s samaras exhibit only uncontrolled autorotation [8]. Fig. 2a. Major components of the Samarai Nano UAV [10]. Fig. 3. Aerodynamic behavior of a stylized Samarai vehicle. Observe tennis racket effect or tendency to want to flatten the disk about the feather axis. The equilibrium is inherently stable in the flap or z-axis, but needs to have payload and vehicle mass distributed in a spatulate (blade-like) shape for feather axis stability. This is in fact the form found in nature s samara as shown in Fig. 1. The x-axis is the primary axis of vehicle rotation [10]. (For ease of exposition, we will use the more traditional coordinate system with z as the spin axis in Section III and forward). For the purposes of control synthesis, the Samarai is treated as if it were a fully articulated helicopter. Thus, it achieves forward flight via the application of cyclic lift to produce rolling and pitching moments from advancing and retreating blades as shown in Fig. 4. Cyclic lift modulation is required to set forward flight equilibrium. One way to achieve cyclic lift, as is done in standard helicopter flight, is by varying the pitch on the rotor blade through the rotational cycle. Because of the feather axis stability, forward flight equilibrium is extremely difficult to do in the Samarai. Instead, the design uses direct modulation of the lift of the rotor blade without changing the pitch of the blade itself. This is accomplished by modulating the trailing edge flap shown in Fig. 2a. Fig. 2b. Samarai Nano UAV geometry and dimensions [10]. The vehicle has a 5.8cm rotor radius, 1 cm chord, 8.2cm overall length and 10g gross takeoff weight with a rotation rate of 8000 RPM and 15% chord flap for lift control. The unique fuselage-less design along with its high speed of rotation and limitations in sensing and control actuation, present a critical challenge of ensuring an aerodynamic design that provides as much inherent stability as possible. The fact that maple seeds exhibit passive stability during auto-rotational hover suggests that the Samarai vehicle be stable by design. This is done by selecting vehicle Fig. 4. Illustration of the application of cyclic lift modulation to achieve forward flight [9]. 52

3 The same mechanism used for cyclic lift control to achieve translational flight is also used as collective pitch for vertical flight. In this case, thrust changes are necessary to match required power. In Fig. 4, Ω is the vehicle rotation rate, R is the wing radius (i.e., ΩR is the tip speed in hover) and V fwd is the forward speed. III. DYNAMIC MODEL The dynamics and control aspects of the Samarai vehicle are shown in Fig. 5. The controller (shown outside the dashed line on Fig. 5) computes flap deflections δ flap and engine throttle setting δ throt. While the model (1) is suitable for high fidelity simulations of the vehicle s hovering and forward flight dynamics, it is ill-suited for controller design that satisfies the constraint of a nano-class UAS. To do this, a simpler disk model that averages the translational and spin states of the vehicle is developed. This model is described in detail in [10] and leads to a simplified set of equations of motion defined by motion of the center of gravity and disk orientation. The disk frame can be thought of as the typical helicopter frame with its equations derived by integrating the blade equation around one pure vertical rotation and performing rotations from the World frame (R W ) to Body frame (R B ); then from Body frame to Disk frame (R D ). Equation (2) describes the simplified disk equations of motion while Fig. 6 is an example of a simulated rotor blade displaying angular rates in several tilted directions. Fig. 5. Overview of Samarai s dynamics and controls. The vehicle dynamics model (shown in dashed box) is based on the samara (maple seed) auto-rotation flight described in [8]. It was extended for Samarai use by including a rigid body representation of the Samarai vehicle, with blade geometry and inertias based on a CAD model of the vehicle. This model receives flap deflections and engine throttle setting as input. These are then mapped to forces (F) and moments (M) through tables of aerodynamic and propulsion coefficients developed from wind tunnel testing or as a hybrid of empirical data and derived data using standard simplifying assumptions. The response of the vehicle to these inputs is encapsulated in the state vector X= [P V ω q], where P= [P x P y P z ] is the position vector of the center of gravity, V= [V x V y V z ] is the translational velocity, ω= [ω x ω y ω z ] is the angular velocity vector and q= [q 0 q 1 q 2 q 3 ] is the attitude quaternion. Vehicle state evolution is governed by the 6DOF rigid body equations of motion (EOM) given by (1). where V b is the body frame velocity, m is the mass, is the skew symmetric matrix of angular velocities, is the Inertia matrix, the rotation matrix from body to inertial frame and Q ˆ is the quaternion wedge matrix given by: Q ˆ = 0 ω ω T ω ˆ Fig. 6. Single-bladed Rotor Disk Simulation [9] The variables in (2) are as follows: translational velocity; is the disk frame the disk frame rotational velocity; is the collective control gain;, are cyclic control gain and the cyclic angle respectively. The disk model is possible based on the quasi-steady assumption (established in [11]) that the blade acts much faster in flap, lag and feather than the disk. Under this concept, the rotor flap, feather and lag degrees of freedom occur at a higher rate than that for the translational degrees of freedom. The assumption alleviates the need to deal with flap, lag, and feather axes directly. This basically means the rotor disk reacts instantly to produce angular rates or forces from control inputs. The disk state space X meas then consists only of the states needed for control. IV. CONTROLLER SYNTHESIS The controller generates appropriate flap and throttle commands to achieve desired position for the Samarai. A 53

4 top-level block diagram of the controller is shown in Fig. 5. Commanded positions P cmd are converted into desired velocities V desired which are subsequently mapped to specific collective and/or cyclic flap deflections, depending on whether commanded motion is mostly in the vertical (z-) or translational (xy-) plane. Collective flap inputs are generated in response to ascent/descent/hover commands while cyclic flap inputs are generated in response to translational motion commands and hover commands (to counter the effect of side forces). Engine throttle or desired thrust setting is computed indirectly by using a vehicle rotation rate controller. Control design is greatly simplified by using the reducedcomplexity disk model (rather than the full six DOF) for controls. In this case, it turns out that simple combinations of proportional, proportional+integral or proportional + derivative type controllers suffice to compute the required control signals. By keeping this design simple, it is able to operate under the constraints on computing power associated with nano-class UAVs. The disk model is a key enabler for this design approach to work. The control design is outlined next. The engine throttle is controlled indirectly by controlling the vehicle spin rate. While issues like aerodynamic loading would tend to make the relationship between these two variables inexact, the compactness of the vehicle concentrates the mass distribution in ways that reduce the need for more fine-grained representation of aero effects. This approximation enables the use of a P controller with feed-forward term given by: All the gains in (5) are typically positive constants while the error signals are between desired and disk velocities in the x-, y- and z-directions. For an example 6 m/s lateral translation along a 45-degree heading, the controller (5) produces the flap commands depicted in Fig. 7. Observe that, when the vehicle s spin angle equals the desired direction of flight (45 o ), the cyclic flap deflection (solid line) is almost +6 deg, tilting the disk down. When the spin angle is 180 deg out of phase of the desired direction, the cyclic flap is almost -6 deg, pushing the disk up. The overall effect is that the disk tilts into the direction of flight, with the angle of tilt determining how fast the vehicle flies. This control signal is shown (dot-dash line) with a collective offset included to account for ascent and altitude hold flights. for vehicle rotation rate control. Rot des is the desired spin rate and Rot meas is the disk spin rate. K rot, K FF > 0 are respectively proportional and feed-forward gains. Desired spin rate may be set a priori or dynamically computed in response to motion in the vertical plane. For vehicle position control, define the position tracking error as e P = P cmd - P meas, the desired velocity is then computed by using the following PI-type structure: where K 1pos, K 2pos >0 are proportional and derivative gains. Define the velocity tracking error as e V =[e Vx e Vy e Vz ] = V desired - V meas (where V meas is the disk velocity), the cyclic component of the overall flap command is computed using a P controller while the collective component uses a PD-type structure as shown in (5). A harmonic function of vehicle spin angle Ψ together with some phase offset angle ψ o is used in the cyclic term to ensure that differential forces applied to the vehicle do tilt the disk in the desired translational direction. Fig. 7. Example control signals from the flap control system V. SIMULATION RESULTS The control system obtained by feeding back the reducedcomplexity disk states was validated on the higher fidelity 6DOF model simulator based on (1). The values for the relevant parameters are: I = diag[6.122e E E-6]kg.m 2, m=10g, K 1pos = [ ], K 2pos = [ ]; K Vx, K Vy = [ ] ; K rot = 0.1, K FF = 1 In the simulation, the vehicle was commanded to climb to 6m and hold station till t=5s, at which time it would then hover momentarily and translate to the point (12,-12) meters relative to the start point. The results are shown in Figs. 8a and 8b. The following observations and annotations apply to these figures. a. Vehicle attains and holds commanded height, with slight overshoot and negligible lateral motion. b. Vehicle exhibits delayed response to lateral motion commands, but tilts to execute forward motion resulting in a momentary altitude perturbation. The controller recovers nicely to keep the altitude close to desired. c. Vehicle executes commanded lateral displacement with slight overshoot and almost zero steady state error. 54

5 Fig. 9a. Engine control signals (N) for a climb-hover-translate flight maneuver. Fig. 8a. Vehicle height response for the example climb-hover-translate maneuver Fig. 9b. Flap control signals (in radians) for the climb-hover-flight translate maneuver. Fig. 8b. Vehicle lateral position response for the example climb-hovertranslate maneuver d. Speed maxima are reached only during brief periods of the maneuver. In Fig. 9a, the engine thrust response shows full throttle only at points of maximum speed maneuver. In Fig. 9b, for 0< t <5s, the flap deflection is essentially all collective flap required for climb and altitude hold. Given limitations on the flap authority, it is evident that the flap deflections required for climbing and holding altitude leaves a lot of authority for lateral maneuvers. These begin at the t=5s mark and continue for the rest of the maneuver. Taken together, the control signals provide some evidence that this simple control scheme is quite effective at combining flap deflection and engine thrust for adequate authority during general motion and maneuvers. VI. CONCLUSION This paper described the Samarai Nano Air Vehicle design, identified challenges unique to it and described several beneficial characteristics of this type of rotating mono-wing airframe. An approach for carrying out dynamic modeling and control was presented with accompanying simulation results that showed effective performance for vertical and translational motion. In addition to its effectiveness for its intended use, the simplicity of the control approach meets one of the key challenges of nano- UAVs algorithms that work within the computational constraints of this vehicle class. There are three main areas in which we wish to extend this work. First, it would be important to investigate the lag between the initiation of cyclic flap commands and the onset of translational motion as seen in Figs. 8 and 9. Second, we would also want to explore the stability margins of the control design described in this paper by simulating scenarios that involve more aggressive maneuvers and analyses of same. Finally, a full analysis of the performance and stability properties of the overall closed loop system needs to be done in order to better characterize the control actions observed. These activities and efforts to validate variants of the scheme in flight experiments remain an area of ongoing research. ACKNOWLEDGMENT The authors would like to thank Mr. Steve Jameson of Lockheed Martin Advanced Technology Laboratories for fostering an enabling environment for this work to be done 55

6 in the Intelligent Robotics Laboratory. Numerous discussions and ongoing consultation with Mr. Harold Youngren of AeroCraft Consulting, Portland, ME, have been invaluable in shaping key aspects of this work. REFERENCES [1] DARPA Defense Sciences Office Nano Air Vehicle, Online documentation available at: (last accessed September 5th 2009). [2] The Delfly Micro, Online documentation available at: (last accessed September 5th 2009). [3] The Black Hornet. Prox Dynamics Inc, Online documentation available at: ype_is_flown_outdoors/ (last accessed September 5th 2009). [4] D. Lentink and M.H. Dickinson, Biofluidynamic scaling of flapping, spinning and translating fins and wings, Journal of Experimental Biology, vol. 16, July 31, [5] W. Hoburg, Fly-by-wire Control of a Monocopter, Technical Report, MIT, Online document available at: (last accessed Sept 10th 2009). [6] F. Graham, Monocopters, East Liverpool, OH, Perigree Press, [7] A. Kellas, The Guided Samara: Design and Development of a Controllable Single-Bladed Autorotating Vehicle, MIT Master s Thesis, Cambridge, MA, September [8] A. Rosen and D. Seter, Autorotation of a Single-Winged Samara, ASME Joint Applied Mechanic/Bioengineering Conference, Ohio State University, Columbus, OH, June 16-19, [9] S. Jameson, B. Satterfield, C. Bolden, H. Youngren, and N. Allen, SAMARAI Nano Air Vehicle A Revolution in Flight, Association for Unmanned Vehicle Systems International Unmanned Systems North America 2007, Washington, DC, August 6-9, [10] S. Jameson, H. Youngren, and B. Satterfield, Design of the SAMARAI Monowing Rotorcraft Nano Air Vehicle. American Helicopter Society 65 th Annual Forum, Grapevine, Texas, May 27-29, [11] R. Prouty, Helicopter Performance, Stability and Control, Krieger Publishing Company, Florida,