Evaluation of the controller is performed by selectively enabling unsteady aerodynamics, wing inertia, flexibility and motor dynamics in the evaluation model. These additional dynamic effects, unaccounted for in the control design model, add uncertainty to the closed-loop system. Conse-quently, it is of importance to evaluate the degree of robustness provided by the controller against
Table 6. Summary of the dynamic effects included in the open-loop models for the flexible wing cases shown in Figures 10 and 11.
Case Aerodynamics Inertia Flexibility
Control Design Model Quasi-steady, Linearized No No
Enhanced Unsteady ROM Yes Nonlinear ROM
FEA+DVM Quasi-steady + Unsteady Wake Yes Nonlinear FEA
Control Input (" !/!0)
Enhanced FEA+DVM Control Design Model
Figure 11. Open-Loop results computing the cycle averaged force or moment for each DOF for the flexible L3B1 configuration.
these. Listed in Table 7 are the parameters for the vehicle. The motor and linkage parameters are the same as those in [23].
The target trajectory, shown in Figure 12, was selected as a representative case study. The vehicle is commanded to start at the origin of the earth-fixed frame and to move forward by 1 m, followed by a rotation of 90◦about the x axis. The vehicle is then commanded to simultaneously move upward and forward by 1 m ending at the point (1, −1, 1). The reference trajectory for each segment of the commanded motion is prescribed as a filtered step function. The total time of the simulation is 20 s.
Table 7. Closed-loop vehicle parameters.
Description Variable Value Units
Vehicle mass mveh 25 g
Height h 10 cm
Width w 3 cm
Depth d 1 cm
Wing area moment of inertia IA 416 cm4
Wing radius R 7.5 cm
Right wing offset ∆rRB [5 1.5 0] cm Left wing offset ∆rLB [5 -1.5 0] cm
Rigid wing pitch angle α0 25◦ deg
0 −0.5 1 0.5
1.5 −1.5
−0.5 −1 0.5 0
−0.5 0 0.5 1 1.5
y (m) z (m)
x (m)
Start End
Figure 12. Commanded trajectory in the earth fixed frame.
4.2.1. Unsteady Aerodynamics and Wing Inertia
The controller is first evaluated with the addition of unsteady aerodynamics and wing inertia with a rigid wing and no motor dynamics. Studying these effects first will establish the robustness of the controller against relatively benign model uncertainty, before more prominent nonlinear and unmodeled dynamic effects like wing flexibility and motor dynamics are taken into consideration.
The simple CEM is the time accurate version of the CDM, with only quasi-steady translation, while the enhanced CEM considers unsteady aerodynamics and wing inertia effects. For both CEMs, a flapping frequency of 46.9 Hz is assumed. The abbreviations listed in the table are provided for reference and are used throughout the remainder of this work.
The closed-loop simulation results are shown in Figure 13, including the tracked vehicle state variables and the corresponding control inputs. The simulations exhibit steady state error for the y
Table 8. Results of simple vs enhanced CEM simulations.
Case Uns Inert Flex Motor Performance
Simple CEM × × × × Good
Enhanced CEM X X × × Fair
coordinate in the earth fixed frame. Due to the vehicle rotation, this is an offset in the forward and backward direction in the vehicle reference frame. Tracking the initial forward motion command appears to be a challenging task for the controller, possibly due to the dynamic coupling between translational and rotational motion. These effects are exacerbated by the enhanced CEM, due to added uncertainty of the forces and moments generated by the wings. Tracking errors are also present in the yaw (φ) response, possibly due to the contribution of wing inertia. Wing inertia could be a large source of uncertainty, since yaw tracking depends on the use of the split-cycle mechanism; as prescribing an asymmetric waveform generates a net moment due to inertia, in addition to the aerodynamic force, which is not accounted for in the CDM.
The results of each study will be henceforth qualitatively labeled as good, fair and unstable.
For the case study under consideration, the evaluation with the simple CEM is labeled as "good,"
due to the fact that the closed-loop system remains stable, with a relatively small steady-state error exhibited by the regulated outputs. Results obtained with the enhanced CEM are labeled as "fair,"
due to the significant tracking error seen over large portions of the trajectories. The results are listed in Table 8. Note, that in Table 8 the labels for the dynamic effects have been abbreviated.
4.2.2. Wing Flexibility
The effect of wing flexibility across a range of flapping frequencies on the vehicle tracking per-formance is shown in Figure 14. The hovering frequencies are determined by varying the vehicle mass, as listed in Table 9. No motor dynamics are considered in these cases. The configurations with slower flapping frequencies exhibit the same steady state offset at the end of the simulation in the output y as the rigid wing case. This may be due to tracking error in forward translation, as significant errors are observed for the output z for the first 15 s of the simulation. The 64 Hz flapping frequency exhibits instability after performing the 90◦ yaw maneuver. Due to the build up of errors in multiple DOFs, the controller is unable to correct the errors simultaneously. This may be due to changes in the aerodynamic forces applied to the vehicle caused by flexibility. A
(a) Vehicle state in earth-fixed frame.
(b) Control inputs.
Figure 13. Closed-loop simulation of the controller and rigid wings comparing the simple and enhanced CEMs. The coordinates x, y, and z are in the earth fixed frame.
Table 9. Vehicle mass and hovering frequencies for the flexible wing simulations shown in Figure 14.
Vehicle Mass (g) Hovering Frequency (Hz)
25 44
34 51
52 64
Table 10. Summarized performance of the flexible wing configurations for the controller.
Flapping Frequency Uns Inert Flex Motor Performance
44 Hz X X X × Fair
51 Hz X X X × Fair
64 Hz X X X × Unstable
summary of the performance obtained in these cases is listed in Table 10.
Further analysis is provided by the open loop results for the 44 Hz and 64 Hz flexible config-urations, shown in Figure 15. Open-loop results help diagnose the source of instability from the analysis of the cycle averaged output corresponding to a given input trajectory. The dashed lines in Figure 15 represent the linear control design model, i.e., the predicted forces for a given control input. The solid lines are determined from the cycle averaged forces from the control evaluation model for each control input. It is important to note that the control design model changes depend-ing on the flappdepend-ing frequency, as increased flappdepend-ing frequencies leads to increased aerodynamic forces. This can be seen in Fx, as the flapping frequency increases, more thrust is produced due to the coupling of increased flapping frequency resulting in an increased pitching angle caused by the aerodynamic loading deforming the wing. The DOFs that depend on the split-cycle parame-ter δ produce less force in Fz and yaw moment Mx for the 64 Hz flapping frequency case. This is due to the aerodynamic loading deforming the wing, and reducing the forces and moments produced in these DOFs. The large errors between the CDM and CEM in the 64 Hz case produce uncertainty in relating the control inputs to cycle averaged forces and moments, destabilizing the vehicle. However, despite the reduction in forces, the vehicle still maintains controllability, as changes in the forces and moments are still produced in Fz and Mx, albeit with reduced magni-tude compared to the CDM. Thus, the controller performs poorly when the forces and moments exhibit uncertainty due to flexibility.
(a) Vehicle state in earth-fixed frame.
(b) Control inputs.
Figure 14. Closed-loop simulation with flexible wings at varying hovering frequencies.
Control Input (" !/!-0.2 0 0.20)
Figure 15. Open-loop results of the 44 Hz and 64 Hz flexible configurations. The solid lines are the control evaluation model and the dashed lines are the control design model.
4.2.3. Limited Motor Torque
A fundamental consideration for flapping wing vehicles is the ability of the wing to attain the desired motion. If the motor is undersized, the ability of the inner-loop controller to reproduce the target waveform is reduced. The motor torque is limited by a saturation to the voltage prescribed to the motor via the motor controller, which limits the torque that can be produced. However, it should be noted that this method of limiting the motor torque is not physically representative of motors of varying size and torque, as motor properties such as armature inertia, back-EMF and torque constants are not varied. Additionally, the wing bias is modeled without consideration of the mechanism necessary to rotate the motor and linkage assembly, and is directly prescribed.
This is a reasonable assumption, since this mechanism could be implemented using a simple servo motor, and has minimal coupling with the motor and aeroelastic dynamics.
Closed-loop simulations of configurations with limited motor torque are shown in Figure 16.
Table 11. Summarized performance of the limited motor torque configurations.
Motor Saturation (V) Uns Inert Flex Motor Performance
5 X X × X Unstable
7 X X × X Unstable
12 X X × X Fair
None X X × × Fair
Configurations of ±5 V, ±7 V, ±12 V and no saturation are shown. These results assume a rigid wing at a 46.9 Hz flapping frequency. In both the 5 V and 7 V cases, the closed-loop systems are unable to maintain hover, and become unstable. The 12 V case performs similarly to the unsaturated case, indicating that the motor is adequately sized. A summary of these results are listed in Table 11. These results demonstrate the importance of considering motor limitations when selecting control inputs, as parameters that include modulating the wingbeat increase the torque required to generate the wingbeat and achieve closed-loop stability.
Open-loop results are presented in Figure 17. Each motor configuration can achieve the desired thrust in Fx, illustrating that an adequate thrust-to-weight ratio of the vehicle is easily achieved by each motor configuration. However, the control input effectiveness in Fz and Mx, which depend on the split-cycle parameter, are reduced. In the 5 V case, effectively no yaw moment is produced regardless of the control input. Interestingly, as the magnitude of δ/ω0 is increased, the yaw mo-ment produced actually decreases, indicating that inner-loop controller dynamics, specifically the behavior of the controller once the input has been saturated, may be a factor. From these results, it is apparent that increased torque is required to achieve the split-cycle wingbeat, due to the quick actuation required to shorten either the fore or aft stroke. This result is intuitive, as increasing the velocity of the fore or aft stroke would increase the aerodynamic loading on the wing, increasing the required motor torque.
4.2.4. Limited Motor Torque with Wing Flexibility
Qualitative results for simulations with limited motor torque and wing flexibility are listed in Table 12. The flapping frequency of the wing is 51 Hz. The 12 V configuration is unstable, despite stability exhibited by both the 51 Hz flexible case and the 12 V saturation case. This demonstrates the coupling effects result in reduced closed-loop performance, due to uncertainty in the forces and moments generated by the wing. Additionally, increased control effort is necessary by the
(a) Vehicle state in earth-fixed frame.
(b) Control inputs.
Figure 16. Closed-loop simulations with varying motor torque. Available motor torque is determined by saturating the motor voltage.
Control Input (" !/!-0.2 0 0.20)
Figure 17. Open-loop results of the configurations with limited motor torque.
motor controller to achieve the target kinematics due to the coupled interactions. The controller does not achieve "fair" performance until the saturation is increased to 25 V, approximately double the required voltage compared to a rigid wing. Thus, dynamic coupling within the vehicle is a challenge, and requires additional actuator capacity to account for unmodeled dynamics in the control evaluation model and maintain closed-loop stability.
4.2.5. Summary of Results
Ultimately, the added uncertainty of the enhanced control evaluation model compared to the con-trol design model yields poor performance and instability for the ensuing system. This is evident in the reduced tracking performance of the controller with the enhanced CEM compared to the simple CEM. In the flexible cases, increased flapping frequencies lead to increased wing deforma-tions, producing large errors between the CDM and CEM. The controller is unable to stabilize the
Table 12. Summarized performance of the limited motor torque with wing flexibility configurations.
Motor Saturation (V) Uns Inert Flex Motor Performance
12 X X X X Unstable
20 X X X X Unstable
25 X X X X Fair
vehicle in the presence of large uncertainty in the forces and moments. When the motor torque is limited, further sources of instability are introduced due to the inability of the motor to repro-duce the split-cycle waveform. In these cases, some degree of controllability of the cycle averaged forces is maintained, and it is reasonable to assume that a controller with improved robustness could achieve improved closed-loop performance, without modifying the control design model.
However, in the combined limited motor torque and flexible wing cases, the additional uncer-tainty yields a significant reduction in performance, requiring a considerable increase in actuator authority to stabilize the vehicle.
5. Conclusions
The study investigates the performance of a comprehensive vehicle dynamics model in a con-troller based on a split-cycle approach. Geometrically nonlinear wing deformation, which arises due to a combination of the flapping motion and the unsteady aerodynamic loading, is modeled using an implicit condensation approach. Important conclusions from this study are as follows:
1. Investigations reveal that moderate-to-large wing deformation increases the overall force generated by the wings but reduces the component of the aerodynamic force in the stroke plane. This leads to vehicle instability due to reduced effectiveness in the yaw moment.
2. By design, the split-cycle requires greater torque from the motor compared to a symmetric waveform. Thus undersized motors are found to produce vehicle instability due to reduced control effectiveness.
3. In some cases controllability is maintained despite a noticeable reduction in control effec-tiveness. This highlights the necessity of control techniques that maintain control while accounting for uncertainties.
The completion of this study represents an important step in the control of flapping wing flight, and highlights the importance of considering wing flexibility and motor dynamics in vehicle trol and design. The inclusion of dynamics in the control evaluation model unmodeled in the con-trol design model exacerbates the inherent vehicle instability. In particular, the work presented here motivates the development of robust control techniques that are suitable for vehicle control in the presence of unmodeled dynamics.
Acknowledgment
This research is funded by the AFRL/DAGSI Ohio Student-Faculty Fellowship along with the AFRL summer internship program. The authors would like to thank Mr. Gregory Knapik and the Biodynamics Laboratories at the Ohio State University for access to their computational resources. This work is also supported in part by an allocation of computing time from the Ohio Supercomputer Center.
References
1Pines, D. and Bohorquez, F., “Challenges facing future micro-air-vehicle development,” Journal of Aircraft, Vol. 43, No. 2, 2006, pp. 290–305.
2Platzer, M. F., Jones, K. D., Young, J., and Lai, J. C. S., “Flapping-wing aerodynamics: progress and challenges,”
AIAA Journal, Vol. 46, No. 9, 2008, pp. 2136–2149.
3Galinski, C. and Zbikowski, R., “Some problems of micro air vehicles development,” Bulletin Of The Polish Academy Of Sciences Technical Sciences, Vol. 55, No. 1, 2007.
4Anderson, M. L. and Cobb, R. G., “Toward flapping wing control of micro air vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 1, 2012, pp. 296–308.
5Deng, X., Schenato, L., Wu, W. C., and Sastry, S. S., “Flapping flight for biomimetic robotic insects: Part I-system modeling,” IEEE Transactions on Robotics, Vol. 22, No. 4, 2006, pp. 776–788.
6Deng, X., Schenato, L., and Sastry, S. S., “Flapping flight for biomimetic robotic insects: Part II-flight control design,” IEEE Transactions on Robotics, Vol. 22, No. 4, 2006, pp. 789–803.
7Orlowski, C. T. and Girard, A. R., “Dynamics, stability, and control analyses of flapping wing micro-air vehicles,”
Progress in Aerospace Sciences, Vol. 51, 2012, pp. 18–30.
8Taha, H. E., Hajj, M. R., and Nayfeh, A. H., “Flight dynamics and control of flapping-wing MAVs: a review,”
Nonlinear Dynamics, Vol. 70, No. 2, 2012, pp. 907–939.
9Shyy, W., Aono, H., Chimakurthi, S., Trizila, P., Kang, C.-K., Cesnik, C. E. S., and Liu, H., “Recent progress in flapping wing aerodynamics and aeroelasticity,” Progress in Aerospace Sciences, Vol. 46, No. 7, 2010, pp. 284–327.
10Wu, P., Stanford, B., Sällström, E., Ukeiley, L., and Ifju, P., “Structural dynamics and aerodynamics measurements of biologically inspired flexible flapping wings,” Bioinspiration & Biomimetics, Vol. 6, No. 1, 2011.
11Oppenheimer, M. W., Doman, D. B., and Sigthorsson, D. O., “Dynamics and control of a biomimetic vehicle using biased wingbeat forcing functions,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 1, 2011, pp. 204–217.
12Geder, J., Ramamurti, R., Sandberg, W., and Flynn, A., “Modeling and control design for a flapping-wing nano air vehicle,” AIAA-Paper-2010-7556, 2010.
13Rifai, H., Marchand, N., and Poulin, G., “Bounded control of a flapping wing micro drone in three dimensions,”
International Conference on Robotics and Automation, IEEE, 2008, pp. 164–169.
14Khan, Z. a. and Agrawal, S. K., “Control of longitudinal flight dynamics of a flapping-wing micro air vehicle using time-averaged model and differential flatness based controller,” American Control Conference, 2007, pp. 5284–5289.
15Doman, D. B., Oppenheimer, M. W., and Sigthorsson, D. O., “Wingbeat shape modulation for flapping-wing micro-air-vehicle control during hover,” Journal of Guidance, Control, and Dynamics, Vol. 33, No. 3, 2010, pp. 724–739.
16Chimakurthi, S. K., Stanford, B. K., Cesnik, C. E. S., and Shyy, W., “Flapping wing CFD/CSD aeroelastic formu-lation based on a co-rotational shell finite element,” AIAA-Paper-2009-2412, 2009.
17Aono, H., Chimakurthi, S. K., Cesnik, C. E. S., Liu, H., and Shyy, W., “Computational modeling of spanwise flexibility effects on flapping wing aerodynamics,” AIAA-Paper-2009-1270, 2009.
18Young, J., Lai, J. C. S., and Germain, C., “Simulation and parameter variation of flapping-wing motion based on dragonfly hovering,” AIAA Journal, Vol. 46, No. 4, 2008, pp. 918–924.
19Zhu, Q., “Numerical simulation of a flapping foil with chordwise or spanwise flexibility,” AIAA journal, Vol. 45, No. 10, 2007, pp. 2448–2457.
20Kang, C.-K., Aono, H., Cesnik, C. E. S., and Shyy, W., “Effects of flexibility on the aerodynamic performance of flapping wings,” Journal of Fluid Mechanics, Vol. 689, 2011, pp. 32–74.
21Chimakurthi, S. K., Tang, J., Palacios, R., S. Cesnik, C. E. S., and Shyy, W., “Computational aeroelasticity frame-work for analyzing flapping wing micro air vehicles,” AIAA Journal, Vol. 47, No. 8, 2009, pp. 1865–1878.
22Palacios, R. and Cesnik, C. E. S., “Geometrically nonlinear theory of composite beams with deformable cross sections,” AIAA Journal, Vol. 46, No. 2, 2008, pp. 439–450.
23Nogar, S. M., Gogulapati, A., McNamara, J. J., Serrani, A., Oppenheimer, M. W., and Doman, D. B., “Control Oriented Modeling of Actuator and Aeroelastic Interactions for Flapping Wing Micro Air Vehicles,” Submitted to Journal of Dynamics and Control, 2015.
24Doman, D. B., Tang, C., and Regisford, S., “Modeling interactions between flexible flapping-wing spars, mecha-nisms, and drive motors,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 5, 2011, pp. 1457–1473.
25Finio, B. M., Shang, J. K., and Wood, R. J., “Body torque modulation for a microrobotic fly,” International Confer-ence on Robotics and Automation, IEEE, 2009, pp. 3449–3456.
26Dickson, W. B., Straw, A. D., and Dickinson, M. H., “Integrative model of Drosophila flight,” AIAA Journal, Vol. 46, No. 9, 2008, pp. 2150–2164.
27Sakhaei, A. and Liu, G., “Model-based predictive control of insect-like flapping wing aerial micro-robot,” AIAA-Paper-2010-7706.
28Keennon, M., Klingebiel, K., Won, H., and Andriukov, A., “Development of the nano hummingbird: a tailless flapping wing micro air vehicle,” AIAA-Paper-2012-0588, 2012.
29Kang, C. and Shyy, W., “Modeling of instantaneous passive pitch of flexible flapping wings,” AIAA-Paper-2013-2469, 2013.
30Zhao, L., Huang, Q., Deng, X., and Sane, S. P., “Aerodynamic effects of flexibility in flapping wings,” Journal of The Royal Society Interface, Vol. 7, No. 44, 2010, pp. 485–497.
31Gao, N., Aono, H., and Liu, H., “A numerical analysis of dynamic flight stability of hawkmoth hovering,” Journal of Biomechanical Science and Engineering, Vol. 4, No. 1, 2009, pp. 105–116.
32Faruque, I. and Sean Humbert, J., “Dipteran insect flight dynamics. Part 1 Longitudinal motion about hover.”
Journal of Theoretical Biology, Vol. 264, No. 2, 2010, pp. 538–52.
33Faruque, I. and Sean Humbert, J., “Dipteran insect flight dynamics. Part 2: Lateral-directional motion about hover.” Journal of Theoretical Biology, Vol. 265, No. 3, 2010, pp. 306–13.
34Taylor, G. K. and Thomas, A. L., “Dynamic flight stability in the desert locust Schistocerca gregaria,” Journal of Experimental Biology, Vol. 206, No. 16, 2003, pp. 2803–2829.
35Orlowski, C. T. and Girard, A. R., “Modeling and Simulation of Nonlinear Dynamics of Flapping Wing Micro
35Orlowski, C. T. and Girard, A. R., “Modeling and Simulation of Nonlinear Dynamics of Flapping Wing Micro