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A computationally efficient method for calculating near-optimal solutions to the three-objective, linear control allocation problem is disclosed. The control allocation problem is that of distributing the effort of redundant control effectors to achieve some desired set of objectives. The problem is deemed linear if control effectiveness is affine with respect to the individual control effectors. The optimal solution is that which exploits the collective maximum capability of the effectors within their individual physical limits. Computational efficiency is measured by the number of floating-point operations required for solution. The method presented returned optimal solutions in more than 90% of the cases examined; non-optimal solutions returned by the method were typically much less than 1% different from optimal and the errors tended to become smaller than 0.01% as the number of controls was increased. The magnitude of the errors returned by the present method was much smaller than those that resulted from either pseudo inverse or cascaded generalized inverse solutions. The computational complexity of the method presented varied linearly with increasing numbers of controls; the number of required floating point operations increased from 5.5 i, to seven times faster than did the minimum-norm solution (the pseudoinverse), and at about the same rate as did the cascaded generalized inverse solution. The computational requirements of the method presented were much better than that of previously described facet-searching methods which increase in proportion to the square of the number of controls.

Several frequently used departure prediction indicators for both open- and closed-loop control of flight are developed using a unified, rigorous analytical approach applied to a linear version of the aircraft model. These criteria are for departure caused by aerodynamic disturbances only. It is shown that these indicators are limited in their accuracy because of restrictive assumptions and terms omitted. A second approach is presented that leads to the same results as the first, but is more applicable to the nonlinear problem. Some ideas concerning the application of the linear methods to the nonlinear problem are presented.

The relationships between pilot control stick inputs and control effector deflections are examined. Specifically, we address multiply redundant control effector arrangements and command-driven control laws. During high-angleof-attack, low-dynamic-pressure maneuvering, there is both a control power and control coordination problem. Control effector deflections are not one to one with pilot inputs, and the maximum capabilities of effectors to respond to pilot inputs varies dynamically with the state of the airplane. The problem is analyzed in the context of a generic control law that continuously regulates sideslip. A means is presented to relate the fixed control effector limits to the dynamically varying control response limits. This information may be used to re-establish the one-to-one correspondence of pilot inputs to control capabilities. Nomenclature F = force G = gearing ratio / = moment of inertia L = transformation matrix L, D, C = aerodynamic lift, drag, and sideforce, wind axes L,M,N = rolling, pitching, and yawing moments m = mass m = vector of moments (or moment coefficients) p,q,r = rolling, pitching, and yawing rates T = thrust u = vector of control effectors V = velocity a = angle of attack ft = sideslip angle

This paper describes the results of recent research into the problem of allocating several flight control effectors to generate moments acting on a flight vehicle. The results focus on the use of various generalized inverse solutions and a hybrid solution utilizing daisy chaining. In this analysis, the number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. The control effectors are assumed to be individually linear in their effects throughout their ranges of motion and independent of one another in their effects. A standard of comparison is developed based on the volume of moments or moment coefficients a given method can yield using admissible control deflections. Details of the calculation of the various volumes are presented. Results are presented for a sample problem involving 10 flight control effectors. The effectivenesses of the various allocation schemes are contrasted during an aggressive roll about the velocity vector at low dynamic pressure. The performance of three specially derived generalized inverses, a daisy-chaining solution, and direct control allocation are compared.

The velocity-vector roll is defined as an angular rotation of an airplane about its instantaneous velocity vector, constrained to be performed at constant angle of attack (AOA), no sideslip, and constant velocity. Consideration of the aerodynamic force equations leads to requirements for body-axis yawing and pitching rotations that must be present to satisfy these constraints. Here, the body-axis rotations and the constraints are used in the moment equations to determine the aerodynamic moments required to perform the velocity-vector roll. The total aerodynamic moments, represented in the reference body-axis coordinate system, are then analyzed to determine the conditions under which their maxima occur. It is shown, for representative tactical airplanes, that the conditions for maximum pitching moment are strongly a function of the orientation of the airplane, occurring at about 90 deg of bank in a level trajectory. Maximum required pitching moment occurs at peak roll rate and is achieved at an AOA in excess of 45 deg. The conditions for maximum rolling moment depend on the value of the roll mode time constant. For a small time constant (fast response) the maximum rolling moment occurs at maximum roll acceleration and zero AOA, largely independent of airplane orientation; for a large time constant, maximum required rolling moment occurs at maximum roll rate, at maximum AOA, and at 180 deg of bank in level flight. The maximum yawing moment occurs at maximum roll acceleration and maximum AOA and is largely independent of airplane orientation. Results are compared with those obtained using conventional assumptions of zero pitch and yaw rates and show significant improvement, especially in the prediction of maximum-pitching-moment requirements.

Nonlinear model-following control design is applied to the problem of control of the six degrees of freedom of an airplane that lacks direct control of lift and side force. The nonlinear expressions for the error dynamics of the model-following control are examined using Lyapunov stability analysis. The analysis results in nonlinear feedforward and feedback gains that are functions of the airplane and model states. As a consequence, gain scheduling requirements for the implementation of the model-following control are reduced to only those involving the estimation of stability and control derivatives of the airplane. The use of these gains is shown through an example application to the control of a nonlinear aerodynamic and engine model provided by NASA Ames-Dryden Flight Research Facility. The model being followed is based on a trajectory generation algorithm, and represents a form of dynamic inversion. HE design methodology to be used is based on the applica- tion of nonlinear model-following to the problem of the control of the six degrees of freedom of an airplane. This methodology is related to nonlinear inverse model theory. It is a more complete approach in that it provides a means for analysis of the dynamics of the errors involved in model-follow- ing. The particular approach has been successfully applied to the control of a nonlinear aerodynamic model of a high-angle- of-attack research vehicle (HARV) through large attitude and angle of attack changes. In general, model-following control attempts to make an actual airplane behave similarly to a prescribed mathematical model of an airplane with different force and moment character- istics than the actual airplane. The model behavior may be based on desirable flying qualities, and the matching of those flying qualities is taken to be the design objective. In this case the pilot controls are applied to the model (either conceptually or literally, to a simulation) and the airplane controls are deter- mined. Alternatively, the mathematical model may be a simplified representation of the actual airplane being controlled, in which case model-following control becomes a solution to the inverse problem. Here the state trajectory of the model is determined from a specification of a particular flight path or maneuver, and the airplane controls required to follow it are determined. Perfect, explicit model-following solutions to the inverse prob- lem provide more than the open-loop controls required to fly a maneuver, since this formulation allows control of the errors between the airplane and model during the maneuver. It is this application of model-following control that is used in this paper. To develop the nonlinear model-following controller, we will first review the model-following concepts used here. Initially a standard form of the airplane and model equations is presented with the conditions for perfect dynamic matching presented. Associated with the conditions for perfect dynamics matching are differential equations for the error. In many cases these error equations are linearized and standard linear control ideas applied to guarantee stability (i.e., they tend to go to zero in time). Hence one is led to a gain scheduling scheme. In the method presented, however, using an approach based on the stability theory of Lyapunov, a set of gains which insure stability of the nonlinear error dynamics can be found. These require no updates but are functions of the current state. The result of this analysis is illustrated through application to the nonlinear airplane simulation provided by NASA Ames- Dryden Flight Research Facility. In this application, the model being followed is a simplified description of the airplane being controlled. The model is not, however, directly flown by exter- nally applied (pilot) controls. Rather, it represents the states and state rates required to execute some prescribed maneuver.

This paper presents a method for the solution of the constrained control allocation problem for the case of three moments. The control allocation problem is to find the "best" combination of several flight control effectors for the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. The controls are assumed to be individually linear in their effect throughout their ranges of motion and complete in the sense that they generate moments in arbitrary combinations. The best combination of controls is taken to be an apportioning of the controls that yields the greatest total moment in a specified ratio of moments without exceeding any control constraint. The method of solving the allocation problem is presented as an algorithm and is demonstrated for a problem of seven aerodynamic controls on an F-18 airplane.

This paper addresses the problem of the allocation of several airplane flight controls to the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. They are assumed to be individually linear in their effect throughout their ranges of motion and independent of one another in their effects. The geometries of the subset of the constrained controls and of its image in moment space are examined. A direct method of allocating these several controls is presented that guarantees the maximum possible moment can be generated within the constraints of the controls. It is shown that no single generalized inverse can yield these maximum moments everywhere without violating some control constraint. A method is presented for the determination of a generalized inverse that satisfies given specifications which are arbitrary but restricted in number. We then pose and solve a minimization problem that yields the generalized inverse that best approximates the exact solutions. The results are illustrated at each step by an example problem involving three controls and two moments.

For cases in which perfect model-following is not possible for a particular desired model, a class of candidate models is defined that can be followed perfectly by the given plant. A candidate model that most closely matches the dynamics of the desired model is then determined through constrained parameter optimization. The result is perfect model-following of a model that has an eigenstructure which resembles that of the desired model. In the development of this method, a new variation on perfect model-following control law development is shown. This method explicitly displays the feed-forward and feedback gains that determine the system error dynamics, which may be artibrarily selected by conventional pole placement methods if the plant is completely controllable. The method is applied to a problem involving the linearized lateral-directional equations of motion of the B-26 airplane. The results show that a candidate model can be found that has virtually the same dynamic behavior as the desired model, and that it can be followed perfectly by the original plant with arbitrarily assigned error dynamics.

A study was conducted to determine if an adaptive controller can recover level 1 handling qualities on a piloted, simulated aircraft in a degraded state. Four different aircraft models of varying handling qualities levels were used as a test data-set to determine if all four could be augmented via the adaptive controller to level 1 handling qualities. Two experienced test pilots were used to judge the aircraft flying qualities using the Cooper Harper Rating Scale, and then an attempt to correlate those results to a handling qualities rating is made.

This paper details the development and curricular impact of Virginia Tech’s new course on “Flight Test Techniques.” By using the motion based flight simulator at Virginia Tech, students enjoy a semester-long flight testing experience that gives the realism of flight testing, without the costs, risks, and delays of actual aircraft. The first oering of this course, in the Spring of 2006, has shown high potential as a capstone course in the aerospace engineering curriculum assimilating material from dynamics and control courses with hands-on, real-world application. for the first time a course titled ‘Flight Test Techniques, AOE 4984’. The course is designed to expose senior level undergraduates to industry and government accepted methods used in flight testing aircraft by introducing real world problems into the department’s curriculum in a controlled environment. It also serves as a capstone course where concepts previously taught in several classes are integrated to give students an overarching view of aircraft operation. In order to facilitate a learning environment and mitigate safety issues associated with using real aircraft, two modern high performance aircraft models in the Virginia Tech Flight Simulation Laboratory are used instead of test aircraft. This allows for accomplishment of targeted learning objectives. It also gives students the ability and opportunity to serve in all of the various roles required to flight test an aircraft, from test pilot, to test conductor, to discipline engineer. The course is based on the principal that hands-on application of concepts learned in an academic setting is key to deep understanding. Due to safety issues, the cost of owning and operating aircraft and liability concerns, application of aerospace concepts learned in a classroom is often delayed until students arrive in industry. In industry employees are expected to perform their tasks without time to revisit concepts learned in school. This course oers students a means to apply lessons learned in a conventional classroom so that when students arrive in the workplace they already have the requisite experience in applying their academic skills to real-world problems. The goals of the course are threefold, namely to reinforce concepts taught in aircraft performance and stability and control classes, expose students to flight testing by reproducing the flight test environment in a classroom setting, and teach students flight test techniques based on currently used manuals in government evaluation of aircraft to prepare them for careers on flight test teams. The team teaching and supporting this course has unique credentials to enable this eort branching the fields of flight test engineering, test

Thii paper describes the use of manned Sight simulation for flight dynamics education. The uses of a simulator for instruction and some of the capabilities that make these uses possible are discussed. Specific examples of the use of simulation in classes in the Virginia Tech Aerospace Engineering curriculum are presented. Lessons learned from these examples included the diiculty in scheduling student use of the simulator, the importance of sufficient preparation before demonstrations, and the need for students to be involved in preand post-experiment briefings and demonstrations.

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Real-time evaluation of a particular rate-based control allocation scheme which accounts for control effector position and rate constraints is performed. This type of control allocation takes as input the moment demands of the control law and is capable of finding admissible control solutions which produce the desired control-generated moment vector so long as the aircraft has enough control authority to do so. This is in general not true of other control allocation methods. Confirmation of the performance of control allocation with rate limiting, previously tested in batch, is sought for an aircraft with a multiply redundant control suite. Examples of maneuvers which produce significant rate demands are simulated. In particular, the ramifications of the observed phenomena previously referred to as control wind-up are considered. As the frame-wise control allocation algorithms evolve the system, the control effectors sometimes tend to migrate towards the effector position limits, a condition that would seem to be intuitively undesirable. It is determined that this phenomenon, called null-space saturation (NSS), does not significantly affect the maneuvering performance of the aircraft. Allocation algorithms which make use of previously examined remedies for null-space-saturated control positions (called restoring methods) are compared with non-restoring algorithms in a real-time simulation to confirm the innocuous nature of NSS. Preliminary timing statistics are presented which address the computational intensiveness of the algorithms.

In the age of modern aircraft and fly-by-wire control systems, the inclusion of mechanical backup systems for handling instances of control failures is becoming more uncommon. As a result, pilots are forced to fully rely on the failure immunities designed into these systems and be assured that, should such failures occur, the aircraft maintain adequate flying qualities long enough for a safe ejection or an emergency landing. This paper demonstrates the control reconfiguration aspects of a control allocation with rate limiting algorithm to adapt to various control failures while still achieving desired moments, thus providing a safe environment for the pilot.

Given a probability distribution for each of the unknown aerodynamic stability and control parameters, a Monte Carlo simulation of the time response allows for a statistical evaluation of the uncertain aerodynamic parameter effects on the aircraft behavior. Prior to integrating the system equations over the desired time of flight, each fractional variation in the aerodynamic coefficient was computed via a random number generator. After each simulation, flying qualities metrics were evaluated and the binary results were recorded: the aircraft either passed or failed the metric. The process was repeated a sufficient number of times such that the resulting probability of violation was statistically significant based on confidence bounds. Experiments were performed using a dynamic inversion and a model-following control law to command angle-of-attack. The results show that the properly designed model-follower has better performance robustness than a dynamic inverter with the same performance objectives.

Continued research into the direct control allocation problem with rate limiting has led to consideration of a modified problem in which control generated drag is considered as a fourth objective (in addition to the 3 moments). This allocation scheme uses the controls' effectiveness in generating moments as well as their effectiveness in generating drag to obtain an admissible solution in moment space while, simultaneously, driving the controls toward their minimum drag configurations. The control allocation algorithm was tested in a non-linear, six degree of freedom simulation of an F-18. Also implemented in the simulation were a maneuver generator and a dynamic inversion control law to provide the moment inputs to the allocator. The simulation results show that while minimum drag allocation is not an optimization algorithm, it continuously drives the controls toward their minimum drag configurations while satisfying the control law moment demands during maneuvers. It is also shown that in non-maneuvering flight, the allocation scheme actually achieves a minimum drag configuration.

The similarities and differences of dynamic inversion control and model-following control laws are examined. For the forms of these control laws assumed in this paper it is shown that dynamic inversion may be considered a special case of model-following. For any given dynamic inversion control law there is a model-following control law that achieves exactly the same response and therefore is in every way equivalent to it. This same model-following control law may be modified in its error dynamics without changing the desired response implied by the dynamic inversion law. The modification in error dynamics may be used to improve the tracking of the desired response in the presence of modeling errors. (Author)

The effect of the choice of control allocation scheme upon individual control effector rate demands is examined. Three previously reported and one new variation of control allocation schemes are described and compared. The new allocation scheme exploits the maximum attainable moment rates of a given control effector configuration and is called moment-rate allocation. The bases of comparison are single-axis sinusoidal and triangular sawtooth, and multiaxis helical time-varying moment demands placed upon the allocation schemes. It is shown that 1) the choice of allocation scheme greatly affects the onset of effective rate limiting for a particular time-varying input but not consistently for all inputs; 2) widely varying results are obtained depending on whether the input is singleor multiaxis and on the amplitude and shape of the input; 3) none of the observed behavior (except as regards generalized inverses) was easily predictable for arbitrary time-varying inputs; and 4) from the point of view of moment and moment-rate generating capabilities, moment-rate allocation clearly yielded best results.

The relationships between pilot control stick inputs and control effector deflections are examined. Specifically, we address multiply redundant control effector arrangements and command-driven control laws. During high-angleof-attack, low-dynamic-pressure maneuvering, there is both a control power and control coordination problem. Control effector deflections are not one to one with pilot inputs, and the maximum capabilities of effectors to respond to pilot inputs varies dynamically with the state of the airplane. The problem is analyzed in the context of a generic control law that continuously regulates sideslip. A means is presented to relate the fixed control effector limits to the dynamically varying control response limits. This information may be used to re-establish the one-to-one correspondence of pilot inputs to control capabilities. Nomenclature F = force G = gearing ratio / = moment of inertia L = transformation matrix L, D, C = aerodynamic lift, drag, and sideforce, wind axes L,M,N = rolling, pitching, and yawing moments m = mass m = vector of moments (or moment coefficients) p,q,r = rolling, pitching, and yawing rates T = thrust u = vector of control effectors V = velocity a = angle of attack ft = sideslip angle

The velocity vector roll is an angular rotation of an aircraft about its instantaneous velocity vector, constrained to be performed at constant angle-of-attack (AOA), no sideslip, and constant velocity. Consideration of the aerodynamic force equations leads to requirements for body-axis yawing and pitching rotations that satisfy these constraints. Here, the body axis rotations, and the constraints, are used in the moment equations to determine the aerodynamic moments required to perform the velocity vector roll. For representative tactical aircraft, the conditions for maximum pitching moment are a function of orientation, occurring at about 90 deg of bank in a level trajectory. Maximum required pitching moment occurs at peak roll rate, and is achieved at AOA above 45 deg. The conditions for maximum rolling moment depend on the value of the roll mode time constant. For a small time constant (fast response) the maximum rolling moment occurs at maximum roll acceleration and zero AOA, largely independent of aircraft orientation; for a large time constant, maximum required rolling moment occurs at maximum roll rate, at maximum AOA, and at 180 deg of bank in level flight. Maximum yawing moment occurs at maximum roll acceleration, maximum AOA, and is largely independent of airplane orientation.

Current departure prediction indicators for both open and closed loop flight are developed using the same rigorous analytical approach applied to a linear version of the aircraft model. Emphasis on the assumptions made and terms omitted indicate why the results are somewhat limited. A second approach is presented which is shown to lead to the same results as the first, but is more applicable to the nonlinear problem. Some ideas concerning the application of the linear methods to the nonlinear problem are presented.