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An imaging Fabry–Perot spectrometer (FPS) utilizes multiple piezoelectric actuators (PEAs) to tune the gap spacing between two optical lenses at nanometer level in order to scan the desired optical band. Due to the nonlinear hysteresis in PEAs and the mechanical coupling, the output responses of the PEAs are quite different, which cause unparallel movement of the FPS positioning stage. For precise positioning control, a model for describing the coupled hysteresis effect is proposed in this paper and a decoupling hysteresis feedforward compensator is constructed based on this model. Experimental results show that the proposed model predicts the coupled hysteresis effect very well and the compensator mitigates the coupled hysteresis efficiently.

In this paper, output feedback integral control of piezoelectric actuators is considered with respect to the hysteresis effect. The linear dynamics of the piezoelectric actuator is modeled as a linear state space system with an input nonlinearity that considers the hysteresis effect. A proof of the Lyapunov stability of the system with integral control is presented, and a method for deriving the upper bound for the regulating gain is shown. A simple example is used to illustrate the approach, and then the approach is applied for tracking a step signal with an experimental single-axis piezoelectric actuator to verify that the system is stable.

Large deployable hoop-truss antennas are widely used in satellite communication systems for their high antenna gain and small packed volume. Knowledge of the deployment dynamics of the large deployable hoop-truss antenna and its coupling with spacecraft attitude motion are essential for the design of deployment mechanisms and attitude control systems. In this paper, a spacecraft with two flexible solar panels, which are reduced by Craig–Bampton component modal synthesis, is modeled by natural coordinate formulation. A large deployable hoop-truss antenna is modeled with the absolute nodal coordinate formulation. Both lead to a constant-mass matrix without the centrifugal and Coriolis forces. The whole dynamics model is used to solve the attitude and the reaction moment during the deployment process. Nonsynchronous deployment phenomena due to the structural flexibility and the decay of the driving force are investigated, and results show that these phenomena can increase the stresses suffered by the hoop tr...

A distributed leader–follower formation tracking controller is presented in this paper. The dynamics of each agent are modeled by Euler–Lagrange equations, and all agents are guaranteed to track a desired time-varying trajectory in the workspace. The system uncertainties and external disturbances, which are equivalently described by a bounded additive noise, are considered in the controller design, and the proposed controller is robust to noise. Fault diagnosis of the nonlinear multi-agent system is also discussed with the help of differential geometry tools and an active fault recovery strategy is incorporated into the proposed control scheme. The effectiveness of the proposed controller is verified through simulations.

This paper presents a vibration control strategy for a flexible manipulator with a collocated piezoelectric sensor/actuator pair. A hybrid vibration controller is proposed by combining the input shaping technique with auto disturbance rejection controller. The parameters of the closed-loop system can be adjusted to the known values by disturbance compensation and linear feedback using the auto disturbance rejection controller. This way, input shaper can be designed without accurate parameters of the flexible manipulator. Both simulation and experiments are conducted to validate the proposed control algorithm. The results verified the effectiveness of the proposed controller in vibration suppression of flexible manipulator.

In this paper, three piezoelectric actuators (PEAs) in parallel installation are utilized in the Fabry-Perot spectrometer for adjusting the optical lens at the nanometer level. Due to the mechanical coupling and uneven preloads to the actuators, the responses of each PEA is rather different, which causes unparallel movement of the positioning stage in the Fabry-Perot spectrometer leading to poor measurement results. Therefore, the controller is highly demanded to enable the synchronized motion of the three PEAs. To this end, the linear-quadratic-Gaussian (LQG) controller combining a synchronization control strategy is developed. The effectiveness of the proposed controller is validated via the experimental tests.

This paper presents an experimental study on wrinkle reduction in a rectangular membrane structure with elliptical boundary cuts. The structure is subjected to two different heat loads: 145 °C and 205 °C. Wrinkling induced by the thermal expansion of the membrane is measured using a photogrammetry system and the root mean square of the membrane's surface above a plane of best fit. A fuzzy logic controller is developed to track a desired flatness level when the membrane is subject to either application or removal of a thermal load. Experimental verification is conducted using a membrane structure test facility and the results show that wrinkles have been reduced significantly and quickly using the proposed fuzzy logic controller.

DOI: 10.2514/1.52287 This paper presents a vibration control strategy for a flexible manipulator with a collocated piezoelectric sensor/ actuator pair. Dynamic modeling of the flexible manipulator is first shown, and then a control law is developed. The proposed vibration controller combines the input shaping technique with multimode adaptive positive position feedback. An adaptive parameter estimator based on the recursive least-square method is developed to update the system’s natural frequencies, which are used by the adaptive positive position feedback. A proportional-derivative controller is combined with the proposed vibration controller to suppress vibration while slewing the manipulator. Simulation results are presented to illustrate the efficacy of the proposed controller.

PurposeThe purpose of this paper is to develop a tri‐axis spacecraft simulator to simulate the three‐axis attitude motion of a satellite and for ground‐based hardware‐in‐the‐loop simulation.Design/methodology/approachThe structure of tri‐axis satellite attitude simulator is designed first. Full dynamic model is then derived. Based on the dynamic model, a simple proportional‐integral‐derivative controller is developed and applied to control the motion of simulator.FindingsThe effectiveness of the proposed simulator configuration has been verified through numerical simulations. The tri‐axis simulator can follow the satellite attitude motion precisely.Originality/valueThis paper is valuable for researchers working on the development of tri‐axis spacecraft attitude simulator. This work is original. The simulator configuration has been applied to a satellite mission that was launched successfully in 2006.

An adaptive learning synchronisation control strategy for six-degree-of-freedom spacecraft formation flying based on the cross-coupling concept is presented. The proposed strategy allows the spacecraft track the desired attitude and position trajectories in a synchronised fashion, in the presence of periodic disturbances and uncertainties in constant parameters. Illustrative simulations of spacecraft formation flying are conducted to verify the effectiveness of the proposed approach.

A synchronised trajectory-tracking control strategy is proposed for multiple experimental three-degrees-of-freedom helicopters. The model-based controller includes a feedforward compensation term and a PD feedback control term. Asymptotic convergence is achieved for both trajectory tracking and motion synchronisation. A proposed generalised synchronisation error concept allows the design of different synchronisation strategies so that different synchronisation performance can be reached. Experimental results verify the effectiveness of the proposed controller.

Input shaping is a feedforward approach and can be applied to suppress flexible vibration. In this paper, modified input shaping (MIS) is proposed for robustness and multimode vibration suppression. The theoretic analysis shows that the MIS poses better performance in several aspects than the traditional input-shaping technique. The controller that combines MIS with PD feedback law is designed for an experimental single-link flexible manipulator. Hardware experiments are conducted to verify the proposed method.

Positive position feedback (PPF) is an effective vibration control technique for smart actuators and has been studied in recent years. In this paper, PPF controller is analyzed and applied to the PZT actuators for suppressing multi-mode vibrations while slewing the single-link flexible manipulator. Both set-point and trajectory tracking control are considered. Moreover, PPF is compared with L-type velocity feedback for PZT actuators to verify the characteristic of anti-spillover of PPF. The hardware experiments verified the proposed method.

In this paper, a hybrid control algorithm is proposed to control the rotation of a flexible beam while suppressing the beam's vibration. The control law combines an enhanced PD feedback with a nonlinear differentiator to derive a high-quality velocity signal to control the gross motion of the beam, and a vibration control by PZT actuators bonded on the surface of the beam. There are three advantages of the proposed method: (i) the enhanced PD control is model free, and appears to be more robust against the noise; (ii) the linear velocity rather than the angular velocity is used in the PZT actuator control, a signal which is easily available; (iii) a unique solution is provided for examination of the actuator placement, based on the analysis of mode shape functions. Experimental results validate these theoretical analyses.

This paper presents a development of component synthesis vibration suppression (CSVS) method for control of flexible structures. The proposed method eliminates unwanted flexible modes of vibration while achieving the desired rigid body motion. The robustness to uncertainties of dynamic modeling parameters is analyzed. Unlike traditional input shaping, in which a numerical optimization is used, design for CSVS commands is based on analytic methodology and is relatively easy to implement. A case study is performed on a time-fuel optimal control strategy using constant amplitude reaction jet thrusters. Both simulation and experimental results validate the effectiveness of the proposed approach.

This paper describes an approach for the use of smart materials, specifically, piezoelectric materials (PZT), in control of a single-link flexible manipulator. It is investigated by a Lyapunov approach that a combined scheme of PD feedback and command voltages applied to segmented PZT actuators, which are bonded to the surface of the flexible link, can effectively control the rigid body motion and at the same time, damp link vibrations. The unique features of the proposed PZT actuator control are twofolds: First, it utilizes linear in contrast to angular velocities of particular points on the link, signals which are readily available. Second, the actuator placement is examined based on the analysis of mode shape functions. Stability of the system with the proposed PZT actuator control is analyzed using a virtual joint model. Simulation and experimental results confirm these theoretical predictions.

In recent years comfort has become an important factor when evaluating the performance of modern automobiles. One important aspect that has negative ramifications on the perception of the quality is the generated noise, which mainly contributes to the disturbing noise level of urban regions. Therefore, an important goal in current research is the attenuation of the noise level of car engines. The paper at hand presents two main approaches, an active and a passive one, to reduce the noise radiation of combustion engines, which is the main noise source of automobiles. In the active approach thin piezoelectric wafers are attached to the structure as sensors and actuators. With an appropriate controller the structural vibrations are reduced, which result in an attenuation of the sound pressure in the environment. The passive approach utilizes a full engine encapsulation, which is also designed to reduce the sound radiation of the engine. By encapsulating the motor the oil temperature can be increased such resulting in decreased fuel consumptions.

This paper presents a control scheme for piezoelectric actuators (PEAs) in high-speed triangular trajectory scanning applications. Nonlinear hysteresis is compensated using a Maxwell resistive capacitor model. Based on the frequency response of the identified linear model, the feedforward compensation is proposed. This feedforward strategy is then combined with a proportional plus integral (PI) feedback controller for high-speed triangular trajectory tracking using piezoelectric actuator. Fast and precise tracking results are demonstrated through experiments. Moreover, it is found that the feedback loop does not always improve the tracking accuracy. When the input frequency exceeds a certain value, feedforward control only will result in better control performance.

To improve the performance of piezo nano-positioning system, feedback controller is designed based on the linearized hysteresis-compensated model of piezoelectric actuator (PEA). Nonlinear hysteresis is compensated for using Maxwell resistive capacitor (MRC) model. Then, a linear system is identified with this hysteresis-compensated dynamics. Based on the identified linear model, a trajectory preshaping method is employed for open-loop scanning application and a proportional plus integral (PI) controller is designed for closed-loop tracking application. Experimental results demonstrate the effectiveness of the proposed method.

Estimation and guidance scheme is presented, which enables two missiles tracking single target cooperatively. Each missile is equipped with an infra-red sensor and measures the LOS angle only. The two missiles form a measuring baseline relative to the target in space and share measured information with each other, enabling that each identies the target state based triangulation method. However, the observability of this double line-ofsight measuring relative navigation method is weak if the separation angle between both line-of-sight vectors is small. This paper proposes a guidance law that modulates the lineof-sight angle through a performance index. Using this guidance law, the missile with large initial ling-of-sight angle maximizes this index. Whereas the other one minimizes it. Simulation results show that the proposed estimation and guidance scheme improves the homing accuracy and can yield hit-to-kill performance.

This paper presents a systematic control system design for nano-positioning of a piezoelectric actuator (PEA). PEAs exhibit hysteresis nonlinearity, which can dramatically limit the application and performance of linear feedback control theory. Thus the hysteresis is compensated for based on the Maxwell resistive capacitor (MRC) model first. Then a proportional plus integral (PI) controller and a proportional double integral plus lead compensation (PII&L) controller are designed for the hysteresis-compensated PEA to account for model uncertainty, disturbance, and noise. The robust stability of both controllers is proved. The effectiveness of the proposed control scheme is demonstrated experimentally. Both controllers achieve fast precise positioning. The 2% settling times for the PI controller and the PII&L controller are 1.5 ms and 4.7 ms, respectively. The positioning resolution is upto 1 nm for both controllers.

This paper presents the development of photogrammetry software to measure the flatness of a membrane structure for active shape control. The principle of photogrammetry technique and detailed implementation are presented in this paper. In order to verify the effectiveness of this software, experiments are conducted on a membrane structure test facility. Comparisons between this developed software and the commercial Automation Manager vision software from Creative Lifestyles Inc. are conducted. The results show that this photogrammetry software can measure the membrane flatness accurately and quickly.

This paper presents an experimental study on wrinkle reduction in a rectangular membrane structure with elliptical boundary cuts. The structure is subjected to a varying thermal load and wrinkling induced by the thermal expansion of the membrane is measured using a photogrammetry system and the root mean square of the membrane’s surface above a plane of best t. An incremental PI controller is implemented to track a desired atness level when the membrane is subject to either application or removal of a thermal load. Experimental verication is conducted on a membrane structure test facility and the results show that the PI controller is able to track a desired atness level quite well.

This paper presents an experimental study on wrinkle reduction in a rectangular membrane structure with elliptical boundary cuts. The structure is subjected to two different heat loads: 145 ◦ Ca nd205 ◦ C. Wrinkling induced by the thermal expansion of the membrane is measured using a photogrammetry system and the root mean square of the membrane’s surface above a plane of best fit. A fuzzy Llogic controller is developed in order to reduce wrinkles due to the thermal disturbance. Experimental study is conducted using our membrane structure test facility and the results show that wrinkles have been reduced significantly and quickly using the proposed fuzzy logic controller.

A vibration suppression strategy is developed for a flexible manipulator with a collocated piezoelectric sensor/actuator pair. A control law is developed based upon positive position feedback and is augmented with an adaptive parameter estimator based on the recursive least squaresmethod to update the first two natural frequencies of the structure online. For the positive position feedback control law, accurate targeting of the modes is critical for vibration control. Experiments are then conducted to show that the controller can be used to suppress the vibrations of a structure with unknown natural frequencies.

A vibration suppression strategy is developed for an unconstrained flexible manipulator with a collocated piezoelectric sensor/actuator pair. Dynamic modeling of the flexible manipulator is first shown and then a control law is developed. A proportional derivative (PD) controller is combined with various modified input shapers (MIS) to control the rigid mode and attempt to suppress the vibrations of the flexible mode. In order to suppress vibrations of a strucuture with frequency uncertainties, a positive position feedback (PPF) control law is augmented with an adaptive parameter estimator based on the recursive least squares method which updates the natural frequencies of the structure online.

This paper presents a hardware-in-the-loop (HITL) simulation approach for multiple spacecraft formation flying. Considering a leader-follower formation flying configuration, a Fuzzy Logic controller is developed first to maintain the desired formation shape under external perturbations and the initial position offsets. Cold-gas on/off thrusters are developed to be introduced to the simulation loop, and the HITL simulations are conducted to validate the effectiveness of the proposed simulation configuration and Fuzzy Logic control.

This paper presents dynamic modeling and vision-based navigation strategy for an indoor airship. A detailed six-degree-of-freedom (6-DOF) dynamic model is developed for an 7.5 ft indoor airship. Vision-based Fuzzy logic controller is designed for line following task. Flight tests validate the effectiveness of the proposed Fuzzy logic control.

An active vibration control strategy is developed for a flexible manipulator with a collocated piezoelectric sensor/actuator pair. Dynamic modeling of the flexible manipulator is first shown and then a control law is developed. The positive position feedback (PPF) control law is augmented with an adaptive parameter estimator based on the recursive least squares method which updates the natural frequencies of the structure online. Accurate targeting of the vibrational modes is critical for damping of the vibrations using PPF and simulations are conducted to demonstrate its eectiveness. I. Introduction Active vibration control of a single-link flexible manipulator is a relatively complex problem and has had numerous papers published on it. From a robotics standpoint, a flexible manipulator is desirable due to the fact that existing manipulators are large and heavy due to the need for the manipulator to be sti in order for the precise location of the tip to be known. The flexible manipulator on the other hand, would have the advantages of lighter weight, lower power consumption, and better maneuverability due to smaller size. The drawback to the flexible manipulator is that it faces large vibrations, leading to positional inaccuracy of the tip during and after movement. The same problems occur in large flexible structures for use in space, such as solar panel arrays. Since the size and mass of an object that can be put into space is severely limited, most structures are flexible due to their smaller mass. This can lead to a variety of problems in areas such as high precision attitude control and stability of motions. A significant number of papers have been written on PPF 1‐3 demonstrating its eectiveness in active vibration control. However, the eectiveness of PPF deteriorates when the natural frequencies of the structure are poorly known or changing due to, for example, the presence of a tip mass. There has been significantly less work done on adaptive PPF than there has been on just PPF. Kwak et al. 4 present a method for adaptively tuning the controller frequency of the PPF controller for a single mode of a cantilever beam, and also illustrate their implementation of a real-time controller. Rew et al. 5 propose an adaptive PPF controller for a plate in a fixed-free configuration in which estimated natural frequencies are adjusted at every time step. Baz and Hong 6 present an adaptive modal PPF controller where the AMPPF controller parameters are adjusted in an adaptive manner in order to follow the performance of an optimal reference model (for a cantilever beam). In the first section, development of a finite element model for an Euler-Bernoulli beam to take into account the eect of the sensor and actuator dynamics is presented. This model is created for use in the simulations and is based upon the work of Bandyopadhyay et al. 7 with some modifications. In the second section, a positive position feedback controller is designed for damping of the beam’s vibrations, and then an adaptive online parameter estimator based upon the recursive least squares method with forgetting factor is designed for the first two modes of vibration. Next, the results of using the adaptive estimator to update the estimates of the beam’s first two natural frequencies are presented and their eectiveness demonstrated. Finally, conclusions and possible further areas of investigation are discussed.

This paper involves a thruster hardware setup and Hardware-in-the-Loop (HITL) simulation. The hardware that was used in the simulation is a simple cold gas thruster, which was constructed out of a solenoid valve and a nozzle with N2 gas being fed through air tubing from a compressed tank. The thruster was tested to characterize its performance. The thruster characteristic like time to reach on state, consistency of maximum thrust in steady state, the time it takes to reach an off state, and the input pressure vs. thrust profile were examined. The simulation is comprised of a non-linear dynamic model of two spacecraft formation flying, a Fuzzy Logic controller used to control the relative motion of spacecraft in formation, and an actuator block to fire the thruster and read the thrust measurement. The whole process and results are further explained in the paper.

This paper presents a passivity control for snake-like robot systems. The bionic robot and underactuated robot have been put into practice. The paper mainly studies the snake robot that is close to a real snake by using biological modeling. The dynamic model in this paper has been built with the passivity controller technique. Lyapunovpsilas stability theory is applied to prove the stability of the proposed controller. Simulations are conducted to verify the effectiveness of the controller.

This paper presents a flight control system development course project using an innovative, interactive simulation platform. The project integrates design, testing, and flight simulation to give students a close-to-reality engineering development practice. Valuable lessons are learned through “flight testing” experience. It also provides feedback for design fine-tuning and improvement.

This paper proposes a robust output feedback controller for a piezoelectrically actuated system with only position sensor. This considered piezoelectric actuator (PEA) system is subjected to model imperfection, creep nonlinearity, hysteresis nonlinearity and other external effects. The designed controller employs a second-order auxiliary system and a discontinuous uncertainty and disturbance estimation term to generate filtered error signals and to compensate for the model uncertainties and system disturbance, respectively. The global stability of the proposed controller is proved through Lyapunov-based stability analysis. The feasibility and effectiveness of the proposed control approach are verified experimentally using a PEA stage. Results demonstrate that both set-point and tracking control without/with external loads are realized with good performance and the PEA system with high-accuracy can be achieved. Moreover, the robustness of the controller is verified and analyzed through the sinusoidal tracking with external disturbance.

This paper proposes a modified Maxwell-slip (MS) model to capture convex and/or concave asymmetric hysteresis in smart actuators. The original MS model can only capture antisymmetric hysteresis. To overcome this limitation, generalized elasto-slide operators (ESOs) are constructed to replace MS elements. Each ESO is characterized by its spring extension stiffness constant, maximum extension, compression stiffness constant, and maximum compression. A procedure is also proposed to identify the model parameters. Simulation and experimental results show that the proposed model is capable of reproducing convex and/or concave asymmetric hysteresis and improving the fit between the modeling and experimental results.

In this paper, the synchronized tracking controller is developed for multiple flying wings. With this controller, the synchronized tracking motion among multiple flying wings can be realized, i.e. all flying wings can track the desired trajectory at the same rate. Simulations are conducted on the nonlinear model of multiple flying wings to verify the proposed controller.

This paper presents the control system design for a piezoelectric actuator (PEA) for a high-speed trajectory scanning application. First nonlinear hysteresis is compensated for by using the Maxwell resistive capacitor model. Then the linear dynamics of the hysteresis-compensated piezoelectric actuator are identified. A proportional plus integral (PI) controller is designed based on the linear system, enhanced by feedforward hysteresis compensation. It is found that the feedback controller does not always improve tracking accuracy. When the input frequency exceeds a certain value, feedforward control only may result in better control performance. Experiments are conducted, and the results demonstrate the effectiveness of the proposed control approach.

This paper presents the development of an experimental testbed for multiple vehicles formation control. This four 3-DOF helicopter-testbed in the Flight Systems and Control (FSC) Laboratory of University of Toronto can be applied to validate the effectiveness of formation flight and cooperative (coordination) control strategy. Each helicopter has three degrees of freedom, in elevation, pitch and travel motion respectively. Two experimental schemes and results of the classical leader-follower and cooperative configurations are presented and discussed

An adaptive synchronization strategy is proposed and applied to the attitude angular velocity tracking control of multiple unmanned aerial vehicles (UAVs). It achieves global asymptotic convergence of both the attitude angular velocity tracking and the angular velocity synchronization, even in the presence of system parameter uncertainties. Simulation results of multiple UAVs in coordination verify the effectiveness of the proposed approach.

In this paper, a synchronized trajectory tracking control strategy is proposed for multiple experimental three-degrees-of-freedom (3-DOF) helicopters. This model-based controller includes a feedforward dynamic term and uses a PD control law as feedback. The convergence is achieved for both trajectory tracking and the motion synchronization. A proposed generalized synchronization concept further improves the transient performance. Numerical simulation results of four helicopters in coordination are presented.

This paper focuses on the dynamics and control of piezoactuators (PEAs) for high-speed large-range scanning applications. Firstly, the nonlinear hysteresis is modeled by using a modified Maxwell resistive capacitor (MRC) model. Secondly, an inverse-based feedforward controller is proposed for this application with hysteresis compensation. Then, the scanning trajectories are preshaped by treating the hysteresis-compensated PEA as a linear system. Finally, experiments are conducted to verify the effectiveness of the proposed approaches.

A physics-based fractional-order Maxwell resistive capacitor (FOMRC) model is proposed to characterize nonlinear hysteresis and creep behaviors of a piezoelectric actuator (PEA). The Maxwell resistive capacitor (MRC) model is interpreted physically in the electric domain for PEAs. Based on this interpretation, the MRC model is modified to directly describe the relationship between the input voltage and the output displacement of a PEA. Then a procedure is developed to identify the parameters of the MRC model. This procedure is capable of being carried out using the measured input and output of a PEA only. A fractional-order dynamics is integrated into the MRC model to describe the effect of creep, as well as the detachment of hysteresis loops caused by creep. Moreover, the inverse FOMRC model is constructed to compensate for hysteresis and creep in an open-loop positioning application of PEAs. Simulation and experiments are carried out to validate the proposed model. The PEA compensated by the inverse FOMRC model shows an excellent linear behavior. (Some figures may appear in colour only in the online journal)

A genetic algorithm is implemented to identify the transfer function of an experimental system consisting of a flexible manipulator with a collocated piezoelectric sensor/actuator pair. A multi-mode positive position feedback controller is then designed based upon the identified transfer function. To this end, the same iteratively implemented genetic algorithm is used to optimize all controller parameters by minimization of the closed loop H?-norm. The designed controller is then applied for vibration suppression on the experimental system.

In this paper, an enhanced hybrid control algorithm is proposed to control the rotation of a flexible beam while suppressing the beam's vibration. The control law combines an enhanced PD feedback with nonlinear differentiator to derive high-quality velocity signal to control gross motion of the beam, and a vibration control by PZT actuators bonded on the surface of the beam. The significance of the proposed method are threefold: i) The enhanced PD control is a non-model based control, and appears to be more robust against the noise; ii) The linear velocity in contrast the angular velocity is used in the PZT actuator control, a signal which is easily available; iii) A unique solution is provided for examination of actuator placement, based on the analysis of mode shape functions. Experimental results validate these theoretical analyses.

This paper presents a development of component synthesis vibration suppression (CSVS) method for control of spacecrafts with large flexible appendages. The proposed method eliminates unwanted flexible modes of vibrations while achieving the desired rigid body motion. Unlike traditional input shaping in which a numerical optimization is utilized, design for CSVS commands is based on analytic methodology and is relatively easy to implement. The robustness to uncertainties of dynamic modeling parameters is analyzed. A case study is performed on a time-fuel optimal control strategy using constant amplitude reaction jet thrusters. Both simulation and experimental results validate the effectiveness of the CSVS approach.