Trajectory Tracking in the Sagittal Plane: Decoupled Lift/Thrust Control via Tunable Impedance Approach in Flapping-Wing MAVs

Transcription

1 Trajectory Tracking in the Sagittal Plane: Decouple Lift/Thrust Control via Tunable Impeance Approach in FlappingWing MAVs Hosein Mahjoubi, an Katie Byl Abstract Flappingwing microaerial vehicles (MAVs are a relatively new fiel of research in the robotics community. Inspire by insects, their small size an unique means of force prouction present many challenges, from morphological construction an power supply to control methoology. Over the past couple of ecaes, investigating the aeromechanics of insect flight an evelopment of prototypes have been a major focus of most researchers in this fiel. Those works concentrating on force manipulation an motion control often rely on moifying the properties of wingbeat profile. However, such changes affect both lift an thrust simultaneously, making it very ifficult to control these forces inepenently. The tunable impeance (TI approach is an alternate metho of force manipulation that moifies mechanical impeance properties of the wings rather than their stroke characteristics. In this work, we show how TI can be use to achieve ecouple lift/thrust control. A motion controller evelope base on this feature enables a flysize moel to track given trajectories in the sagittal plane. Results of simulate experiments with various types of trajectories emonstrate a high egree of precision an maneuverability. Keywors Tunable Impeance; MicroAerial Robotics; Trajectory Tracking; Lift/Thrust Control; Insect Flight; Maneuverability; Simulation. O I. INTRODUCTION VER recent ecaes, progress in miniaturization, from sensor, actuator an microprocessor esign to microfabrication methos an lowpower technologies, has turne the evelopment of microaerial vehicles (MAVs into a real possibility. These vehicles have the potential to revolutionize observation an information gathering operations an may be employe in a wie range of applications incluing environmental monitoring an homelan security. As a result, a growing number of researchers are now focusing on flight analysis, esign an control of such systems. Design solutions are often inspire by research on insect flight an the involve aeroynamics [][6]. The main focus of recent research on flappingwing MAVs is evelopment of flapping mechanisms that are capable of proucing sufficient lift for levitation. Several prototypes capable of takeoff an vertical flight have been evelope over the past few years [7][9]. With such platforms H. Mahjoubi an K. Byl are with the Robotics Laboratory, Department of Electrical an Computer Engineering, University of California at Santa Barbara, Santa Barbara, CA 936 USA ( h.mahjoubi@ece.ucsb.eu, katiebyl@ece.ucsb.eu. available, force manipulation an motion control are the next challenges that must be face. Recent work focuse on vertical acceleration an altitue control has le to some impressive results [][]. Research on forwar flight control has been less evelope in comparison []. The majority of works concentrating on force control employ moifications in wingbeat profile to achieve their objective [3][]. While this approach proves to be successful in separate control of lift [5] or horizontal thrust [6], simultaneous control of these forces is more challenging. Both forces are irectly influence by velocity of air flow over the wings, a factor that is heavily etermine by the rate of wingbeat [7]. Therefore, moifying wingbeat profile will lea to a coupling between lift an thrust forces, i.e. an inconvenience for motion control an maneuverability of the vehicle. Passive pitch rotation of the wing [8] is another phenomenon that plays a significant role in prouction an manipulation of lift/thrust through ajusting the angle of attack (AoA of the wing relative to air flow [7]. The tunable impeance (TI approach is one of the few methos evelope for manipulation of wing pitch rotation [9][]. It employs a mechanical structure at the wing base which allows for manipulation of mechanical impeance properties of the joint. These moifications can influence the pitch profile of wings without requiring changes in stroke profile [][]. Through ajusting the two impeance properties of stiffness an equilibrium point of the pitch joint, it is now possible to control lift an thrust more freely. Although TI is able to provie a higher egree of controllability compare to traitional wingbeat moification approaches [3], its lift an thrust manipulation proceures are not always completely ecouple []. In this work, we first ientify the conitions that guarantee maximum ecoupling between lift an thrust manipulation when TI approach is use. Uner such conitions, sagittal motion of the vehicle is fully controllable. We then use these finings to evelop a motion controller that is capable of tracking various trajectories in the sagittal plane with a consierably high egree of precision as verifie by our simulate experiments. The rest of the paper is organize as follows. Section II provies a brief review of the tunable impeance approach an how it is use for force manipulation. Decoupling conitions are then investigate in Section III. The MAV

2 moel use in simulations an the esigne motion controller are iscusse in Section IV. Section V presents the results of several simulate trajectory tracking experiments. Finally, Section VI conclues the paper. II. TUNABLE IMPEDANCE APPROACH: A REVIEW In [], we argue that synchronous muscles may be insects primary means for manipulation of wings pitch profiles. It was shown that a joint an its clinging pair of muscles can be moele as a torsional spring, i.e.: k ( ( rot where τ is the torque prouce ue to rotation of the joint while k rot an ψ are the stiffness an equilibrium point of the spring, respectively. The wing pitch angle ψ is use as a measure of joint s rotation. The torque in ( shoul be in balance with the pitch torque prouce by stroke motion of the wing, i.e.: z F b J ( N in which, J ψ an b ψ are the moment of inertia an passive amping coefficient of the wing along its pitch rotation axis, respectively. F N is the aeroynamic force component that is perpenicular to the surface of wing at its center of pressure (. As illustrate in Fig., z is the istance of from wing s pitch rotation axis. To fin the pitch profile of a wing for any given set of impeance parameters, it is also necessary to have an approximation of aeroynamic forces in Fig.. The quasisteaystate moel escribe in [] will be use to estimate these forces. The values of wing parameters use in all future simulations are reporte in Table I. Note that for the chosen wing shape in Fig., z is equal to 6.73% of the wing s span [], i.e. R W. The tunable impeance approach suggests that by moification of impeance properties, one can manipulate pitch profile of either wing while its stroke profile remains (a F N F D Downstroke Phase F T F z z x' (b F D F X Wing's Pitch Rotation Axis F Y y y' x' x Stroke Direction Fig.. (a Wing crosssection (uring ownstroke at center of pressure, illustrating the pitch angle ψ. Normal an tangential aeroynamic forces are represente by F N an F T. F an F D represent the lift an rag components of the overall force. (b Overhea view of the wing/boy setup which emonstrates the stroke angle ϕ. F X an F Y are components of F D which represent forwar an lateral thrust, respectively. TABLE I PHYSICAL PARAMETERS USED IN SIMULATIONS Symbol Description Value ρ air ensity at sea level.8 kg/m 3 R W average span of each wing in a flysize.5 m flappingwing MAV J ψ moment of inertia of each wing (pitch N.m.s rotation b ψ passive amping coefficient of each wing (pitch rotation 5 N.m.s estimate mass of a twowinge MAV 7 5 kg with similar imensions g stanar gravity at sea level 9.8 m/s m boy unperturbe in this work ϕ = 5 sin(πtβ where the small an slowly variable bias angle β is the only applicable change to stroke profiles of both wings an has an insignificant effect on force prouction []. In fact, the value of k rot primarily etermines the range of variations in ψ (Fig..a which in turn affects the magnitue of aeroynamic forces (Fig..bc. Note that when ψ =, pitch an thrust profiles are osymmetric, i.e. the average thrust is. Introucing a nonzero value of ψ perturbs this symmetry (Fig..f by biasing the pitch profile (Fig... In short, tunable impeance allows us to control overall lift an thrust by manipulating k rot an ψ, respectively []. However, Fig..c an.e suggest that each impeance parameter may also influence the other force. III. DECOUPLED LIFT/ THRUST CONTROL: CONDITIONS To investigate the possibility of ecouple lift/thrust control via tunable impeance approach, we first nee to moel the relationships between impeance parameters an (eg 5 (a Pitch Angle of the Wing vs. k rot Lift (gforce Thrust (gforce (b Instantaneous an Average Lift: F (c Instantaneous an Average Thrust: F X k rot =k * k rot =.5k * k rot =5k * (eg 5 ( Pitch Angle of the Wing vs Lift (gforce Thrust (gforce (e Instantaneous an Average Lift: F (f Instantaneous an Average Thrust: F X = o = o = o Fig.. (a Pitch angle evolution of one wing for ψ = an ifferent values of k rot (k * =.7 6 N.m/ra. The corresponing instantaneous (blue an average (re lift an forwar thrust are plotte in (b an (c, respectively. ( Pitch angle evolution of one wing for k rot = k * an ifferent values of ψ. The corresponing instantaneous (blue an average (re lift an forwar thrust are plotte in (e an (f, respectively. All forces are normalize by the estimate weight of the vehicle.

3 (gforce Average F (gforce (a Average F an F X vs. k rot ( = o.8 Lift.6 Thrust k op = 7.e k rot ( Nm / ra these forces. We choose the average values of lift an thrust over intervals of. s i.e. full stroke cycles as our measure for influence of impeance parameters on aeroynamic forces. Fig. 3.a illustrates the steay state values of these forces over a wie range of values for k rot while ψ is kept at. It is easy to observe that for a MAV with two wings, k rot = k op = 7. 6 N.m/ra an ψ = is an ieal operation point for hovering. In aition, it seems that in the vicinity of this point, average lift has an approximately linear relationship with log (k rot (Fig. 3.c. Note that small values of ψ have little influence on overall lift. Fig. 3.b illustrates the steay state values of these forces over a wie range of values for ψ while k rot is kept at k op. When impeance values are close to k rot = k op an ψ =, a linear relationship between average thrust an ψ is also observable (Fig. 3.. Note that slight changes in k rot from k op have little influence on overall thrust. The force values in Fig. 3 are calculate for constant impeance parameters. However, the linear relationships between steaystate values suggest that a linearize moel aroun k rot = k op an ψ = will be able to provie a reasonable approximation of ynamic behavior of these forces in response to varying impeance properties. To fin this linear moel, we employ system ientification techniques computation/smoothing of Empirical Transfer Function Estimates followe by subspace ientification metho [] for transfer function matching in orer to ientify components of the nonlinear MIMO system that relate impeance parameters to aeroynamic forces: F ( s G ( s K ( s G ( s (3 K F ( s G ( s K ( s G ( s ( X (eg XK (c log (k rot (gforce Average F X (gforce X (b Average F an F X vs. (k rot = k op.6. Lift Thrust. 6 (eg 6 ΔF an ΔF X represent the changes in lift an forwar thrust ue to moifications in k rot an ψ. Note that ΔK =.... (eg ( log (k rot Fig. 3. (a Average values of lift an thrust vs. k rot when ψ =. At k rot = k op = 7. 6 N.m/ra, two wings will be able to prouce sufficient lift for levitation. (b Average values of lift an thrust vs. ψ when k rot = k op. (c Average lift for impeance values aroun k rot = k op an ψ =. ( Average thrust for impeance values aroun k rot = k op an ψ =. All force values are normalize by the estimate weight of the vehicle. Magnitue (B Phase (eg Magnitue (B Phase (eg (a G K (b G 6 6 Frequency (ra/s (c G XK 6 6 Frequency (ra/s Magnitue (B Phase (eg Magnitue (B Phase (eg 6 Frequency (ra/s log (k rot / k op. Each G represents a transfer function from an impeance parameter to a force component as inicate by its subscripts. In all system ientification experiments, limits of ΔK <.5 an ψ <. ra were applie. The Boe iagrams of ientifie subsystems are plotte in Fig.. In low frequencies, both G K an G XΨ have steaystate gains close to B, while in comparison, the same gains for G Ψ an G XK are well below unity. Therefore, we expect that respective isturbances ue to stiffness/ equilibrium point manipulation on thrust/lift control will be insignificant in the low frequency range, allowing ecouple control of these forces. To choose a suitable upper limit for this range, we use the evaluation parameters that are illustrate in Fig. 5. To calculate these values, three groups of numerical experiments have been performe. In the first group, sinusoial waves of magnitue.5 an ifferent frequencies were use for ΔK while ψ = ra. Lift an ( G X 6 Frequency (ra/s Fig.. The Boe iagrams of ientifie subsystems from impeance parameters to average aeroynamic forces: (a G K, (b G Ψ, (c G XK, ( G XΨ..5.5 i Xi f K f K (a (c f f.5.5 f K (b Fig. 5. Correlation coefficient vs. frequency of mechanical impeance manipulation for (a esire an overall lift, (b isturbance an overall lift, (c esire an overall thrust, ( isturbance an overall thrust. X f K ( f f

4 thrust outputs in this case are treate as esire an isturbance, respectively. In the secon group, sinusoial waves of magnitue. ra an ifferent frequencies were use for ψ while k rot = k op. Lift an thrust outputs in this case are treate as isturbance an esire, respectively. Finally, the thir group of experiments uses both escribe nonzero inputs at the same time. Lift an thrust outputs in this case are the overall results of impeance manipulation. Correlation values between esire (isturbance an overall lift are plotte in Fig. 5.a (5.b. Corresponingly, correlation values between esire (isturbance an overall thrust are plotte in Fig. 5.c (5.. From Fig. 5, when the frequency content of both inputs is below Hz, the correlation between esire an overall values of either force is above.97 while the correlation between isturbance an overall values remains below.3. Thus, by removing the frequency content of ΔK an ψ above Hz, ecoupling of lift/thrust control will be significantly improve. IV. MAV MODEL AND FLIGHT CONTROL Base on the observations in Section III, a simple motion controller is evelope to track reference trajectories in the sagittal plane. Simulate experiments in Section V employ this controller along with a six DoF moel of the MAV. A brief review of this moel an the esigne controller are presente next. A. Dynamic Moel of the MAV The freeboy iagram of a typical twowinge flappingwing MAV [7] is shown in Fig. 6. The six DoF ynamic moel [7] of the vehicle is evelope by applying Newton s equations of motion in this boy frame. Relate etails are available in []. Employe values for moel parameters are reporte in Table II. B. Motion Controller Fig. 7 illustrates the block iagram of the esigne motion controller. Pitch angle of the boy, i.e. θ pitch, is stabilize at ra through smooth biasing ( β.6 ra of the stroke TABLE II CHARACTERISTICS OF THE MODELED FLAPPINGWING MAV Symbol Description Value b ω passive amping coefficient of the 3 6 N.m.s boy (rotation b v J boy viscous friction coefficient of the boy when moving in the air inertia matrix of the boy relative to N.s /m 3I N.m.s CoM H(ψ= istance of from transverse plane. 3 m of the boy (xy in Fig. 6 when ψ = R(ϕ= istance of from sagittal plane of.9 m the boy (xz in Fig. 6 when ϕ = U(ϕ= istance of from coronal plane of.8 m the boy (yz in Fig. 6 when ϕ = W boy boy with at the base of wings.6 3 m angle []. A PD subcontroller,.5s/ ( 3 s, is use for this purpose. The observer block in Fig. 7 applies the force approximations in (3 an ( along with (5 an (6 to estimate velocity of the moel along an X axes: m ˆ F b ˆ ˆ (5 m (a Frontal View boy boy Xˆ F X v b v Xˆ Xˆ where Ẑ an Xˆ represent these estimations, respectively. Mass of the moel is shown by m boy (Table I an b v is the viscous friction coefficient (Table II. isplacements an approximate velocities are use by manually tune proportional lift (gain: [ 5] an thrust (gain: [ ] subcontrollers to manipulate impeance properties of the wings ΔK an ψ are limite to [..6] an [.35 ra.35 ra], respectively. Note that both wings always have the same values of β, k rot an ψ, resulting in ieally lateral thrust an roll/yaw torques ue to symmetry. Thus, we assume that position of the boy along Y axis an its roll/yaw angles remain throughout all experiments. Simulations are then effectively restricte to motion in the sagittal plane (X. ref X ref X F r (b Overhea View U r ref pitch = ra pitch H r F X r R r PD Pitch Controller Proportional Lift Controller Proportional Thrust Controller ^ X ^ z x LPF with f c = Hz LPF with f c = Hz LPF with f c = Hz Observer MAV Moel Pitch CoM K Mag. =.873 ra f = Hz Sinusoi Wave Generator Fig. 7. Block iagram of the propose motion controller in interaction with MAV moel. Cutoff frequency f c of each employe filter is set to Hz. Both wings use the same values of β, k rot an ψ at all times. R l H l F X l F l x k rot U l y y k op Roll x Yaw Fig. 6. Freeboy iagram moel of a flappingwing MAV: (a frontal an (b overhea views. H, R an U are the istance components of center of pressure ( of each wing from center of mass (CoM of the moel. For simulation purposes, Euler angles in TaitBryan XY convention are employe to upate the orientation of the boy. z (6

5 V. SIMULATED EXPERIMENTS Motion of the moel along various trajectories in the X plane has been simulate. In every experiment, the moel is initially hovering at X = m an = m, i.e. X ref = m an ref = m. Later, new reference position profiles are introuce to subcontrollers that guarantee a smooth velocity profile over esire trajectory. We observe that as long as acceleration emans of these profiles are within the capabilities of the moel primarily ue to enforce limits on ΔK an ψ tracking will be achieve with a consierably high egree of precision. The results of simulate experiments with three important types of trajectories are presente next. A. Strictly Horizontal (Vertical Maneuvers As the most trivial group of maneuvers, the vehicle must be able to maintain its position on one axis while moving along the other one. Fig. 8 emonstrates the results of tracking a square trajectory that requires all of these maneuvers, i.e. takeoff, laning an forwar/backwar motion. As it can be seen, the controller successfully hanles such maneuvers an keeps close to the reference trajectory (Fig. 8.g. B. Simultaneous Displacements: A Linear Trajectory To investigate the effectiveness of our approach to ecouple lift/thrust control, ifferent trajectories with isplacements along both X an axes were examine. A simple example of such trajectories is a straight line with a slope of. Simulation results for the corresponing experiment are illustrate in Fig. 9. A combination of (eg pitch (eg (eg.6.. (a Displacement along Xaxis. 6 8 (c Pitch Equilibrium of the Wings 6 8 (e Pitch Angle of the Boy 6 8 (f Stroke Bias Angle of the Wings k rot ( Nm / ra.6.. (b Displacement along axis x 5 ( Stiffness of the Joints (g Trajectory in X plane Fig. 8. Results for tracking a square trajectory: isplacement along (a X an (b axes, (c pitch equilibrium of the wings ψ, ( stiffness of the joints k rot, (e pitch angle of the boy θ pitch, (f stroke bias angle of the wings β an (g overall trajectory in X plane. (eg pitch (eg (eg. (a Displacement along Xaxis (c Pitch Equilibrium of the Wings 6 (e Pitch Angle of the Boy 6 (f Stroke Bias Angle of the Wings backwar flight an takeoff motion starting at t = s enables the vehicle to reach X =.5 m an =.5 m by t = s. At this point, the irection of motion is reverse. Through simultaneous escent an forwar flight, the vehicle returns to its original position at t = 3 s. All the while, the slope of its trajectory remains close to (Fig. 9.g. C. Motion along Curves k rot ( Nm / ra Curves are a more complex case of trajectories with simultaneous isplacement along both X an axes. The results of an experiment with a partially circular reference trajectory are plotte in Fig.. The controller still manages to follow this trajectory with high precision (Fig..g. Note that compare to other trajectories, correction of motion over curves requires smaller changes in impeance parameters (Fig..c an. This is partially ue to the fact that stable motion along such trajectories is rarely accompanie by high velocities an/or acceleration rates. VI. CONCLUSION. 6 Inspire by insect flight, tunable impeance has been previously propose as a semipassive metho for force manipulation in flappingwing MAVs. This approach states that pitch rotation profile of each wing can be moifie through changes in mechanical impeance properties of its joint. Consequently, magnitue of variations an average value of this profile are influential factors in regulation of aeroynamic forces (Fig.. The main avantage of TI over common methos of force control is its ability to operate with a fixe stroke profile (b Displacement along axis.5 x 5 ( Stiffness of the Joints (g Trajectory in X plane Fig. 9. Results for tracking a linear trajectory with a slope of : isplacement along (a X an (b axes, (c pitch equilibrium of the wings ψ, ( stiffness of the joints k rot, (e pitch angle of the boy θ pitch, (f stroke bias angle of the wings β an (g overall trajectory in X plane.

6 (eg pitch (eg (eg.6.. (a Displacement along Xaxis (c Pitch Equilibrium of the Wings 6. (e Pitch Angle of the Boy. 6 8 (f Stroke Bias Angle of the Wings k rot ( Nm / ra. 6 8 This effectively simplifies the control system an can reuce the amount of onboar harware in an actual prototype. Uner this approach, the system is also more controllable an has a higher egree of mobility [3]. However, lift an thrust control are not completely ecouple which at times may result in limite maneuverability of the vehicle. In this work, we showe that in a system with TI as its unerlying strategy for force control, coupling between lift an thrust effectively occurs when either impeance variable carries significant components at frequencies close to the wingbeat rate (Fig. 5. Removing high frequency content from both impeance values will consierably improve ecoupling between lift an thrust an results in better performance compare to our previous work [3]. In fact, a simple control structure such as Fig. 7 is sufficient to stabilize motion along given trajectories. Results of simulate experiments verify the tracking capabilities of this metho. It was observe that as long as a trajectory can be efine by smooth velocity profiles, the only concern for successful tracking is the corresponing acceleration requirements. While impose limits on impeance parameters guarantee linear relationships with forces, they also restrict acceleration capabilities of the moel (Fig. 3. Thus, successful tracking also requires that timeerivatives of these velocity profiles lie within acceleration limits of the moel. In a real system, esigning such profiles relies on information from both environment an vehicle itself. A high level controller must first come up with an appropriate trajectory base on surrounings of the vehicle an relative location of its target. The current velocity an acceleration limits of the MAV must then be treate as constraints of (b Displacement along axis x 6 ( Stiffness of the Joints (g Trajectory in X plane Fig.. Results for tracking a partially circular trajectory: isplacement along (a X an (b axes, (c pitch equilibrium of the wings ψ, ( stiffness of the joints k rot, (e pitch angle of the boy θ pitch, (f stroke bias angle of the wings β an (g overall trajectory in X plane. esigning a suitable velocity profile for tracking purposes. REFERENCES [] M. F. M. Osborne, Aeroynamics of flapping flight with application to insects, J. Exp. Biol., 8(, pp. 5, 95. [] C. P. Ellington, The aeroynamics of hovering insect flight I. the quasisteay analysis, Philos. Trans. R. Soc. Lon. B. Biol. Sci., 35(, pp. 5, 98. [3] M. H. Dickinson, F. Lehmann, an S. P. Sane, Wing rotation an the aeroynamic basis of insect flight, Science, 8 (5, pp , 999. [] D. Floreano, J. C. ufferey, M. V. Srinivasa, an C. P. 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