Design and Testing of a Rotary Wing MAV with an Active Structure for Stability and Control Paul Samuel Asst. Research Scientist pdsamuel@umd.edu
Jayant Sirohi Asst. Research Scientist sirohij@glue.umd.edu
Felipe Bohorquez Grad. Research Asst. fbohorq@glue.umd.edu
Ronald Couch Grad. Research Asst. rncouch@glue.umd.edu
Alfred Gessow Rotorcraft Center Department of Aerospace Engineering University of Maryland College Park, MD 20742-3015
Abstract The design and testing of a rotary wing Micro Aerial Vehicle (MAV) using an active, flexible structure for stability and control is discussed in this paper. The vehicle configuration consists of a set of coaxial, counter-rotating rotors independently driven by two coaxial electric motors. Multi-functional structures are used as the primary structural components with the ultimate goal of vehicle weight minimization. Light-weight carbon fiber composite beams comprise the MAV structure. The inherent flexibility of the structure enables the realization of a rotor system with marginal passive stability and minimal complexity. Thin, lightweight shape memory alloy (SMA) wires are incorporated in the structure. The wires serve as lateral control actuators as well as a portion of the structural support. Structural vibration is mitigated through the integration of constrained viscoelastic damping material into the composite structural elements. In addition, power is provided to the upper electric motor through conduits integrated into the composite structural elements. A lithium polymer battery pack is used to power the vehicle and serves as the structural support for the electronics and sensors. Preliminary tests show that the SMA actuator has a bandwidth of approximately 1 Hz and can provide sufficient authority for lateral control.
Introduction For many years, there has been much interest in the development of very small flying vehicles, referred to as Micro Aerial Vehicles (MAVs), for sensory applications such as covert imaging, biological and chemical agent detection, and urban intelligence gathering. One of the earliest examples was a mechanical dragonfly with an integrated listening device developed by the CIA in the 1970s.1 A common requirement of many of the missions envisioned for MAVs is the ability both to loiter for long periods of time and to effectively maneuver in confined spaces. Since 1999, the Alfred Gessow Rotorcraft Center at the University of Maryland has focused on the development of rotary-wing MAVs. The first vehicle resulting from this effort was MICOR (MIcro Presented at the AHS 61st Annual Forum, Grapevine, TX, c 2005 by AHS International, 1-3 June 2005. Copyright 째 Inc. All rights reserved.
COaxial Rotorcraft).2, 3 MICOR has two counterrotating rotors arranged in a coaxial configuration, with each rotor driven by an independent electric motor. Since the development of the first prototype, MICOR has been significantly refined.4 Most recently, an innovative swashplate was added to the vehicle for lateral control. This research has demonstrated the effectiveness of the coaxial configuration for rotary-wing MAVs. The original MICOR prototype was powered by three Tadiran LIMnO2 3V batteries. This vehicle could hover for approximately four minutes and had a negligible payload capacity. At the time, the large current flow (about 3A) required by the electronics and motors exceeded the batteries capabilities, which were designed to provide a maximum continuous current of 1 A. When forced to discharge at higher current levels, their effective energy density was reduced, shortening the flying time. Since then, battery technology
Electric Motors with Integrated Gearboxes
Flexible Structure
Coaxial Counter-Rotating Rotor System
Flexible Central Support
Electronics Supported by Li-Poly Battery Pack Figure 1. Flexible MAV.
has improved significantly, in particular with the advent of lithium polymer (Li-Poly) batteries. These batteries have a high energy density and can handle the large current flow required to fly MICOR. However, even the most advanced LiPoly batteries do not provide the power required to achieve a sufficient mission duration. Thus, the operating efficiency of the vehicle must be improved. Great strides have been made in the efficiency of lift generation of the rotor. The figure of merit (FOM) of the original MICOR rotor was approximately 0.42.2 A comprehensive aerodynamic investigation was subsequently performed which suggested a number of changes to the design of the rotor blades. These advancements have enabled the current MICOR rotor to reach a FOM of approximately 0.64.5 These rotor and battery improvements have lead to an estimated hover endurance of 20 minutes with an estimated payload capacity of 20 grams,
which is approaching the level necessary for practical application. However, improvements are still necessary in order to further increase the endurance and payload capacity of the vehicle. One potential area of improvement is the vehicle weight. A reduction in vehicle weight leads to an increase in payload capacity or an increase in endurance due to a reduction in the power required to hover. The incorporation of multifunctional structures is an effective technique for minimizing the weight of an MAV by reducing the structural weigh fraction of the vehicle. In addition, minimizing the power drawn by other vehicle systems such as control actuators leads to an increase in endurance. The focus of the current research is the development of a rotary-wing MAV that uses an active, flexible structure to achieve stability and control. The inherent flexibility of the support structure is expected to enable the vehicle to demon-
strate marginal passive stability, minimizing the work done and thus the power drawn by the control system actuators. Weight minimization is achieved by making the flexible structure active through the use of integrated Shape Memory Alloy (SMA) wires. The SMA wires provide both actuation and a portion of the structural support for the vehicle. In addition, a Li-Poly battery pack powers the vehicle and is used as the support structure for the MAV electronics and sensors, further decreasing the structural weight fraction of the vehicle.
Configuration
Central Support Shaft
Integrated Bearings
The design of a hovering MAV is driven by the desire for efficient operation resulting in a maximum flight duration. Efficient operation is achieved through efficient rotor design, weight minimization, and actuator workload reduction. Concurrent research at the University of Maryland has lead to the development of efficient rotors for a coaxial MAV.4, 5 The rotor system used for this vehicle is based on this work. Weight reduction is achieved through the incorporation of multi-functional structures. Multi-functional structures are necessary vehicle systems that are designed to serve as structural components. For the current research, the battery and the actuators are multi-functional. In order for the structural supports to double as actuators, a vehicle architecture had to be developed that would allow small changes in the shape of the structure to be used as control inputs. The resulting MAV developed in this research is shown in figure 1. The rotors and motors are mounted coaxially. The previous work on MICOR2, 3 has shown that the coaxial rotor configuration is effective for rotary-wing MAVs from the perspective of both stability and controllability. The motors are supported by four flexible members, each with integrated SMA wires (the SMAs are not shown in figure 1). Thus, each flexible member serves as a control actuator. The flexible central support structure, shown in figure 2, is kept in tension by the 4 members and serves to keep the motors appropriately aligned. The central support consists of a section of cable protected by silicon tube and is attached at each end by a bearing mounted in each rotor hub. The bearings allow the rotors to counter-rotate without twisting the central support.
Figure 2. Central Support Structure.
Stabilizer Bar Teeter Stops Integrated Bearing
Blade Root Clamps
Gearbox
Carbon Blades
Teeter Hinge
Figure 3. Rotor Hub. Each rotor consists of two blades and a stabilizer bar. The rotational inertia of the stabilizer bar minimizes any change in the tip path plane of the rotor that may be caused by small perturbations in the orientation of the vehicle. The weights at the tip of the stabilizer bar increase the rotational inertia of the bar and are shaped to minimize drag. Each rotor is free to teeter about the axis of the stabilizer bars. However, teeter-stops have been incorporated to ensure that the rotor does not collide with the structure. The rotor hub is shown in figure 3. As stated above, an efficient rotor blade design resulting from the work on
Thrust
Tensile Force Motor
Rotor Planes
Actuator
Central Shaft
Actuator Gravity
Figure 4. Vehicle Schematic. MICOR4, 5 has been incorporated. The Li-Poly battery pack powers the vehicle and serves as a structural support for the vehicle electronics and payload. Currently, in addition to the battery pack, the vehicle electronics consist of a six-channel receiver, a yaw gyro, two motor speed controllers and four small SMA control boards that interpret the receiver signals and regulate the voltage sent to the SMA actuators accordingly. Currently, the complete vehicle weight without payload is estimated to be 118 grams. For comparison, MICOR currently weighs 135 grams.5 However, certain components have not currently been optimized for weight and thus further weight reduction is possible.
Structural Flexibility In order to develop a flexible structure that enables passive stability and to determine the appropriate location of the SMA wires for control, the appropriate deformation for a desired rotor response must first be determined. Consider the schematic of the vehicle with two in-plane actuators, given in figure 4.
Motor Separation Distance
Motor Tensile Force Center Line
Figure 5. Basic Kinematic Diagram. included. The central support is assumed to be rigid and pinned at each end. Both motors have a sliding pin at the base and a pinned joint at the connection to the central shaft. The composite beams are bent and attached to the base of each motor thus applying the tensile force indicated in the figure. The tensile force serves to keep the structure aligned acting against any rotation at the pinned joints as this rotation, herein referred to as motor tilt, decreases the motor separation distance. Hence, the tensile force can be replaced by relatively soft torsional springs located at the pinned joints between the central shaft and the motors, as indicated in figure 6. The composite structural support beams are rigidly attached to the base of each motor, perpendicular to the motor axis of rotation. In addition to reducing the motor separation distance, motor tilt causes a deformation, effectively a local bending, of the composite beams. The beams resist this local bending more strongly than than they resist a decrease in motor separation distance for a given degree of motor tilt. This effect is represented by a relatively stiff torsional spring placed at the base of each motor, as indicated in figure 7.
Kinematic Representation A kinematic diagram of the vehicle can be used to more clearly understand the vehicle deformation. Consider the simple kinematic diagram presented in figure 5. Only internal forces have been
Control Actuation Directional control is achieved by tilting the plane of both rotors toward the desired direction of motion. Thus, deformation as shown in figure 8
Actuation Moment
Motor
Rotor Planes
Soft Torsional Springs
Motor Separation Distance
Motor
Rotor Planes
Actuation Moment
Center Line
Center Line
Figure 6. Modified Kinematic Diagram.
Stiff Torsional Spring Actuator
Rotor Planes
Soft Torsional Springs
Actuator Stiff Torsional Spring
Figure 8. Kinematic Diagram with Actuation Moments Imposed. rotor plane to tilt in the desired direction. Based on this kinematic analysis, an analytical model is under development that should aid in further refinement of the vehicle structure. The properties of the current support structure are being determined for inclusion in the model. It is anticipated that actuation of the SMA wires will lead to an increase in the stiffness of the structural supports, effectively increasing the stiffness of the torsional springs during actuation. However, it has not yet been determined whether the change in stiffness is non-negligible. Passive Stability
Center Line
Figure 7. Improved Kinematic Diagram. must be imposed by the actuators. Note that the moments are applied directly to the stiff torsional springs and thus only the soft torsional springs resist the deformation. In addition, it should be mentioned that since the rotor blades are free to teeter, only the planes of the stabilizer bars are directly affected. The deformation changes the collective pitch of the rotors in a cyclic manner such that the collective pitch of each advancing blade is at a minimum and the collective pitch of each retreating blade is at a maximum perpendicular to the plane of actuation. This causes each
Passive stability is achieved by allowing the rotor to react to small perturbations in the orientation of the vehicle such that any change in the tip path plane of the rotor is minimized. The flexibility of the support structure acts as a virtual teeter hinge for the stabilizer bar and is represented in the model by pinned joints at the bottom and top of each motor. This virtual hinge allows the rotational inertia of the stabilizer bar to keep the tip-path-plane of the rotor steady when small external disturbances change the orientation of the vehicle. Often, the requirements for hover stability and maneuverability conflict, particularly when the vehicle is designed to be passively stable. Passive stability is particulary desirable for MAVs as
power is at a premium and thus the power used by an active stability system could significantly diminish the flight duration of the vehicle. The motor/rotor system is designed to take advantage of the inherent flexibility of the structure in order to achieve marginal stability. However, the stability of the vehicle is dependent upon the flexibility of the structure. As mentioned above, actuation of the SMAs is expected to stiffen the structure, potentially changing (reducing) the passive stability of the vehicle and improving controllability and maneuverability. Vibration Mitigation One problem inherent to flexible structures is vibration. Initial vehicle testing demonstrated that the structure was extremely sensitive to rotor vibration. For a real system, the rotor will always cause some degree of vibration since it can never have perfect track and balance. Hence, the structure must be designed to tolerate this vibration. Two modifications were made in order to mitigate the effect of vibration. First, the stiffness of the structure was changed such that the natural frequency of the structure would not coalesce with the excitation frequency of the rotor during hover. However, changes in the rotational velocity of the rotors are used to control altitude and yaw. Thus, ensuring that the rotor excitation frequency will not coalesce with the natural frequency of the structure over the entire operational envelope of the vehicle is nearly impossible. To further mitigate vibration, a second modification was made. A constrained layer of viscoelastic polymer damping material was integrated into each composite beam. Each support structure is comprised of a 10 mil layer of viscoelastic polymer sandwiched between two layers of carbon fiber. The sandwich structure is shown in figure 9. After the modifications were implemented on the vehicle, additional tests were performed. These tests demonstrated that the modifications had the desired effect and the structure sufficiently damped over the operational envelope.
Lateral Control As stated above, four actuators are incorporated into the vehicle, spaced 90 degrees
Carbon Fiber Beams 10 mil Viscoelastic Polymer Figure 9. Viscoelastic Sandwich Structure. apart about the rotational axis of the rotors. Actuation is achieved through the use of thin, light-weight SMA wires mounted on the beam surface. These wires offer a relatively large force and displacement while providing an operational bandwidth sufficient for stability and control. SMA Properties Shape Memory Alloys (SMAs) belong to a class of materials that have the ability to ‘memorize’ their shape at a low temperature.6 Strains imparted to the material at a low temperature can be recovered when the material is exposed to a sufficiently elevated temperature. On subsequent cooling, the material does not undergo any further change in geometry. This phenomenon is known as the Shape Memory Effect (SME).7, 8 A widely known material that exhibits the SME is Nitinol, which is an alloy of Nickel and Titanium. A large number of actuators based on Nitinol are commercially available.9 Nitinol is most commonly available in the form of thin wires. It is typically capable of recovering strains on the order of 6-8%. The thermal activation of the wires can be conveniently achieved by passing a current through the wire and making use of the self-heating of the wire. These properties make it attractive for use as a simple, lightweight, high stroke force generator in MAV applications. Typical Nitinol properties are summarized in Table 1. The basic phenomenon responsible for the shape memory effect is a thermally activated phase transformation.10 At high temperatures, the material exists in the austenite phase. On cooling, the austenite phase transforms to martensite phase, with a large change in Young’s modulus. At a molecular level, the phase transformation proceeds at the local speed of sound of the
Maximum recoverable strain Youngs modulus (martensite), Gpa Youngs modulus (austenite), Gpa Activation temperature, F Density, g/cc Wire diameter, in Linear resistance, ohms/in
6% 23.7 53.5 94 6.45 0.005 1.7
Table 1. Typical Properties of SMA Wire
material. However, as the material is thermally activated, the overall speed of the transformation is limited by the heat transfer rate in the material, which is typically much slower than the local speed of sound. Consequently, actuation based on the shape memory effect is usually very slow, and typical actuators operate at frequencies of less than 1 Hz. The major factors affecting the bandwidth of SMA wire are the heating and cooling rates. While the heating rate can be increased by increasing the current passed through the wire, the cooling rate depends primarily on the convective heat transfer from the wire to the surrounding medium. The cooling rate can be increased by decreasing the diameter of the wires. Therefore, the wires must be thin enough to achieve the required bandwidth, but thick enough to sustain the required actuation force. Figure 10 shows a schematic of the stress-strain behavior of a SMA wire acting against springs of different stiffness. Also shown are the stress-strain curves of the SMA at low and high temperatures. The curves labeled (1) and (2) depict the strain recovery of the SMA wire acting against a nonlinear spring, while the curve labeled (3) depicts the strain recovery of the SMA wire acting against a linear spring. Curves (1) and (2) are of interest for the present application because an SMA wire behaves as a non-linear spring. SMA Actuation Figure 11 shows a schematic of an antagonistic SMA actuator with feedback. SMA wire 1 is heated by passing current through it from the DC voltage source. SMA wire 2 is kept at room temperature and small current is passed through it to measure its resistance. The wires are initially pre-strained by the same amount (approximately 3%). Heating of wire 1 causes it to recover the pre-strain and increase the pre-strain in wire 2.
Figure 10. SMA Wire Strain Recovery with Different Loading Conditions. motion SMA 1
SMA 2
(actuator)
(sensor)
current
low current source DC voltage
Figure 11. Antagonistic SMA Wire Actuator. The resistance in wire 2 changes as a result of its deformation and is measured to provide a position feedback for the actuator. Each actuator implemented on this vehicle is an antagonistic SMA actuator, arranged as a bimorph. In this configuration, an SMA wire is attached to either side of a stiff central structure. Passing current through one of the wires causes it to contract, bending the structure in the direction of the actuated wire. Passing current through the opposing wire causes the structure to bend in the opposite direction. As in the actuator shown in figure 11, the non-actuated wire can be used for position, or bending, feedback. Alternately, if a more accurate measure of deformation is required,
Composite Strip
Composite Strips
(a)
(a)
(b)
(b) Direction of Motion
Direction of Motion
SMA Mounting Brackets
SMAs
Integrated SMAs
Actuated SMA
Figure 12. SMA Bi-Morph Actuator with Integrated Wires. strain gauges can be integrated into the structure. Two SMA bi-morph architectures have been considered. The first is an SMA sandwich structure. This actuator consists of two SMA wires and three carbon strips arranged as seen in figure 12(a). When one SMA wire is actuated, the curvature of the actuator is changed. This is demonstrated in figure 12(b). Integrating the SMA wires into the structure has the advantage of protecting the wires. In the second bi-morph architecture, the wires are placed external to the carbon structure and are held in place using small brackets, as shown in figure 13(a). Once again, when one SMA wire is actuated, the curvature of the actuator is changed and demonstrated in figure 13(b). Although this architecture does not offer the same degree of protection for the wires as the first, it offers a larger actuation moment since the wires are placed further from the axis of symmetry. In addition, the external wires are more easily cooled, allowing faster actuation response times. From a practical standpoint, this configuration offers the additional advantage that the wires are much easier to replace if broken. Both SMA bi-morph architectures were evaluated. It was determined that for the first prototype of
Actuated SMA
Figure 13. SMA Bi-Morph Actuator with External Wires. the vehicle, the external wire architecture was preferable as it offered larger actuation moments and ease of implementation. Actuator Implementation and Testing Recall that for a desired lateral motion, both rotor planes must be tilted in the direction of the desired motion and thus the vehicle must be deformed as shown in figure 8. Once again, consider the schematic of the vehicle given in figure 4. Assume that a leftward motion is desired. For the actuators to impose the appropriate deformation, the curvature of the upper half of the left actuator and the lower half of the right actuator must decrease while the curvature of the the lower half of the left actuator and the upper half of the right actuator must increase. This is demonstrated in figure 14 In order for a single SMA wire to cause opposing changes in curvature in the upper and lower halves of the actuator, the wire must switch from one side of the actuator to the other between the halves. Thus, both wires in a given SMA bimorph must cross at the middle of the actuator as shown in figure 15. A single implemented actuator is shown in figure 16. The mounting brackets for this actuator
Increase Curvature
Thrust Decrease Curvature
SMA Actuator
Direction of Motion
Increase Curvature
Decrease Curvature Gravity
Figure 14. Actuated Vehicle Schematic.
Figure 16. SMA Actuator Mounted on Vehicle.
Composite Strip
(a)
(a)
(b)
(b)
SMA Wire
SMA Cross-Over Point
Opposing Curvature
Cross-Over Point
SMA Mounting Brackets
Figure 17. SMA Actuator Close-Up. SMAs
Actuated SMA
Figure 15. SMA Actuator with Cross-Over. are small slotted disks which slide onto the structure. Holes in the disks support the SMA wire. The actuator was attached to the vehicle electronics and tested. The resulting deformation was observed, and the displacement of the central shaft from the rotational axis of the motors was found to be 1.4 mm at the center point of the shaft. The bandwidth of the actuator was found to be on the order of 1 Hz. Note that this deformation was achieved using a single SMA wire actuator. Improved performance is anticipated when all actuators are implemented as two SMA wires will be used to impose any desired moment.
A new set of mounting brackets has been manufactured that are integral to the composite structure. These brackets consist of small segments of teflon tube attached to the structure using epoxy. For comparison, an enlarged view of the cross-over point for each type of bracket is shown in figure 17. The original brackets in figure 17(a) are shown with SMA wires mounted, however, the wires have not yet been mounted in the new brackets shown in figure 17(b). Actuator fabrication is progressing and a comprehensive assessment of their control effectiveness will follow. However, preliminary tests indicate that the rotor planes deflect as expected when the SMA wire is actuated. Further testing is required to determine whether the deflection provides sufficient control authority for the vehicle.
Yaw Control (a)
Yaw control is achieved through differential rotor RPM. A yaw gyro is placed in series with the lower rotor. The gyro makes small adjustments to the RPM of the rotor so that the orientation of the vehicle is held relatively steady when no yaw is commanded. When a yaw is commanded, the RPM of both rotors change so that the vehicle can yaw while holding a relatively constant altitude. In addition, thrust is controlled by varying the RPM of the rotors in tandem. Hence, the yaw gyro is required to ensure that yaw is kept to a minimum when a change in thrust is commanded. This approach has been show to be effective in past research.2–5
(b)
Power Conduit in Viscoelastic
Solder Point for Conduit
Electronics Integration As stated above, the vehicle electronics consist of a Li-Poly battery pack, a six-channel receiver, a yaw gyro, two motor speed controllers and four small SMA control boards. The Li-Poly battery pack powers the vehicle and serves as the structural support the vehicle electronics and payload.
Figure 18. Integrated Power Conduit.
The primary challenge involved with integrating the electronics is providing power to the upper motor since, as seen in figure 1, the batteries are located at the base of the vehicle. To solve this problem, a single power conduit has been incorporated into each structural support. The conduit and solder point are shown in figure 18. The conduit is integrated in the sandwich structure between two 5 mil layers of viscoelastic polymer, as shown in figure 18(a). Currently, the vehicle uses coreless motors and thus two of the conduits are used, one for the positive motor lead and one for the negative motor lead. However, since four conduits are available, the vehicle can easily be upgraded to use three phase brushless motors. A schematic of the resulting final structure with all current electronics integrated is shown in figure 19. If it is determined that strain sensors are necessary for actuator feedback, the can be easily integrated as shown in figure 20.
Summary and Conclusion The focus of this paper is the development of a rotary wing MAV that uses an active,
Carbon Fiber Beams 5 mil Viscoelastic Integrated SMA Mounting Brackets
SMA Wire
Power Conduit
Figure 19. Structure Cross Section.
Integrated Strain Sensors
Carbon Fiber Beam 5 mil Viscoelastic
Figure 20. Structure Cross Section with Integrated Strain Guages.
flexible structure to achieve stability and control. The inherent flexibility of the support structure is designed to enable the vehicle to demonstrate marginal passive stability, minimizing the work and thus the power drawn by an active stability system. Weight minimization is achieved through the incorporation of multi-functional structures. The flexible structure is made active by integrating SMA wires. The SMA wires provide both actuation and a portion of the structural support for the vehicle. A Li-Poly battery pack is used as the support structure for the MAV electronics and sensors, further reducing the vehicle weight. An actuator was fabricated and tested. The actuator behaved as expected, producing the deformation required for lateral motion. The actuator was found to have a bandwidth on the order of 1 Hz. An assessment of the control effectiveness of the actuator in in progress.
Acknowledgements This research is supported by the Army Research Office through the MAV MURI Program (Grant No. ARMY-W911NF0410176) with Dr. Gary Anderson serving as Technical Monitor.
References 1. “CIA Exhibits Spy Gadgets with Bond Edge,” Reuters, Retrieved from http://www.cnn.com/2003/ TECH/10/28/tech.cia.reut/index.html, October 28, 2003. 2. Samuel, P., Sirohi, J., Rudd, L., Pines, D. and Perel, R., “Design and Analysis of a Micro Coaxial Rotorcraft,” Proceedings of the AHS Vertical Lift Aircraft Design Conference, San Francisco, CA, January 1921, 2000. 3. Bohorquez, F., Samuel, P., Sirohi, J., Rudd, L., Pines, D. and Perel, R., “Design, Analysis and Performance of a Rotary Wing MAV, Journal of the American Helicopter Society, Vol. 48, No. 2, pp. 80-90, April, 2003. 4. Bohorquez, F. and Pines, D., “Hover Performance and Swashplate Design of a Coaxial Rotary Wing Micro Air Vehicle,” Proceedings of the AHS 60th Forum, Baltimore, MD, June 7-10, 2004. 5. Bohorquez, F. and Pines, D., “Rotor Design for Efficient Rotary Wing Micro Air Vehicles,” Proceedings of the AHS International Specialists’ Meeting - Unmanned Rotorcraft: Design, Control and Testing, Chandler, AZ, January 18-20, 2005. 6. Buehler, W., Wiley, R. and Wang, F., “Nickel-Based Alloys,” US Patent 3,174,851,23, March, 1965.
7. Chopra, I., “Review of State of Art of Smart Structures and Integrated Systems,” AIAA Journal, Vol. 40, No. 11, pp. 2145-2187, November, 2002. 8. Prahlad, H., and Chopra, I., “Experimental Characteristics of Ni-Ti Shape Memory Alloys under Uniaxial Loading Conditions,” Journal of Intelligent Material Systems and Structures, Vol. 11, No. 4, pp. 272-282, 2001. 9. Dynalloy Inc., “Website: http://www.dynalloy.com, Technical Characteristics of Flexinol TM Actuator Wires,” Dynalloy, Inc., 3194-A Airport Loop Dr., Costa Mesa, CA 92626-3405, 2002. 10. Wayman, C., and Duerig, T., “Engineering Aspects of Shape Memory Alloys,” in An introduction to martensite and shape memory, ButterworthHeinemann Ltd., 1990. 11. Fay, J., The helicopter: history, piloting and how it flies, Newton Abbot: David & Charles, 4th ed. 1987, c1954.