Radhakant Padhi

Impact-Angle-Constrained Suboptimal Model Predictive Static Programming Guidance of Air-to-Ground Missiles

A nonlinear suboptimal guidance law is presented in this paper for successful interception of ground targets by air-launched missiles and guided munitions. The main feature of this guidance law is that it accurately satisfies terminal impact angle constraints in both azimuth as well as elevation simultaneously. In addition, it is capable of hitting the target with high accuracy as well as minimizing the lateral acceleration demand. The guidance law is synthesized using recently developed model predictive static programming (MPSP). Performance of the proposed MPSP guidance is demonstrated using three-dimensional (3-D) nonlinear engagement dynamics by considering stationary, moving, and maneuvering targets. Effectiveness of the proposed guidance has also been verified by considering first. order autopilot lag as well as assuming inaccurate information about target maneuvers. Multiple munitions engagement results are presented as well. Moreover, comparison studies with respect to an augmented proportional navigation guidance (which does not impose impact angle constraints) as well as an explicit linear optimal guidance (which imposes the same impact angle constraints in 3-D) lead to the conclusion that the proposed MPSP guidance is superior to both. A large number of randomized simulation studies show that it also has a larger capture region.

An account of chronological developments in control of distributed parameter systems

Control systems arising in many engineering fields are often of distributed parameter type, which are modeled by partial differential equations. Decades of research have lead to a great deal of literature on distributed parameter systems scattered in a wide spectrum. Extensions of popular finite-dimensional techniques to infinite-dimensional systems as well as innovative infinite-dimensional specific control design approaches have been proposed. A comprehensive account of all the developments would probably require several volumes and is perhaps a very difficult task. In this paper, however, an attempt has been made to give a brief yet reasonably representative account of many of these developments in a chronological order. To make it accessible to a wide audience, mathematical descriptions have been completely avoided with the assumption that an interested reader can always find the mathematical details in the relevant references.

Adaptive-critic based optimal neuro control synthesis for distributed parameter systems

A neural network based optimal control synthesis approach is presented for systems modeled by partial differential equations. The problem is formulated via discrete dynamic programming and the necessary conditions of optimality are derived. For synthesis of the controller, we propose two sets of neural networks: the set of action networks captures the mapping between the state and control, while the set of critic networks captures the mapping between the state and costate. We illustrate the solution process with a parabolic equation involving a nonlinear term. For comparison, we consider the linear quadratic regulator problem for the diffusion equation, for which the Ricatti-operator based solution is known. Results show that this adaptive-critic based systematic approach holds promise for obtaining the optimal control design of both linear and nonlinear distributed parameter systems.

Suboptimal Midcourse Guidance of Interceptors for High-Speed Targets with Alignment Angle Constraint

Using the recently developed computationally efficient model predictive static programming and a closely related model predictive spread control concept, two nonlinear suboptimal midcourse guidance laws are presented in this paper for interceptors engaging against incoming high-speed ballistic missiles. The guidance laws are primarily based on nonlinear optimal control theory, and hence imbed effective trajectory optimization concepts into the guidance laws. Apart from being energy efficient by minimizing the control usage throughout the trajectory (minimum control usage leads to minimum turning, and hence leads to minimum induced drag), both of these laws enforce desired alignment constraints in both elevation and azimuth in a hard-constraint sense. This good alignment during midcourse is expected to enhance the effectiveness of the terminal guidance substantially. Both point mass as well as six-degree-of-freedom simulation results (with a realistic inner-loop autopilot based on dynamic inversion) are presented in this paper, which clearly shows the effectiveness of the proposed guidance laws. It has also been observed that, even with different perturbations of missile parameters, the performance of guidance is satisfactory. A comparison study, with the vector explicit guidance scheme proposed earlier in the literature, also shows that the newly proposed model-predictive-static-programming-based and model-predictive-spread-control-based guidance schemes lead to lesser lateral acceleration demand and lesser velocity loss during engagement.

Automatic path planning and control design for autonomous landing of UAVs using dynamic inversion

In this paper a nonlinear control has been designed using the dynamic inversion approach for automatic landing of unmanned aerial vehicles (UAVs), along with associated path planning. This is a difficult problem because of light weight of UAVs and strong coupling between longitudinal and lateral modes. The landing maneuver of the UAV is divided into approach, glideslope and flare. In the approach UAV aligns with the centerline of the runway by heading angle correction. In glideslope and flare the UAV follows straight line and exponential curves respectively in the pitch plane with no lateral deviations. The glideslope and flare path are scheduled as a function of approach distance from runway. The trajectory parameters are calculated such that the sink rate at touchdown remains within specified bounds. It is also ensured that the transition from the glideslope to flare path is smooth by ensuring C1 continuity at the transition. In the outer loop, the roll rate command is generated by assuring a coordinated turn in the alignment segment and by assuring zero bank angle in the glideslope and flare segments. The pitch rate command is generated from the error in altitude to control the deviations from the landing trajectory. The yaw rate command is generated from the required heading correction. In the inner loop, the aileron, elevator and rudder deflections are computed together to track the required body rate commands. Moreover, it is also ensured that the forward velocity of the UAV at the touch down remains close to a desired value by manipulating the thrust of the vehicle. A nonlinear six-DOF model, which has been developed from extensive wind-tunnel testing, is used both for control design as well as to validate it.

Partial Integrated Guidance and Control of Interceptors for High-Speed Ballistic Targets

A new partial integrated guidance and control design approach is proposed in this paper, which combines the benefits of both integrated guidance and control as well as the conventional guidance and control design philosophies. The proposed technique essentially operates in a two-loop structure. In the outer loop, an optimal guidance problem is formulated considering the nonlinear six degrees-of-freedom equation of motion of the interceptor. From this loop, the required pitch and yaw rates are generated by solving a nonlinear suboptimal guidance formulation in a computationally efficient manner while simultaneously assuring roll stabilization. Next, the inner loop tracks these outer loop body rate commands. This manipulation of the six degrees-of-freedom dynamics in both loops preserves the inherent time scale separation property between the translational and rotational dynamics, while retaining the philosophy of integrated guidance and control design as well. Because of this, the tuning process is quite straightforward and nontedious as well. Extensive six degrees-of-freedom simulations studies have been carried out, considering three-dimensional engagement geometry, to demonstrate the effectiveness of the proposed new design approach engaging high-speed ballistic targets. A variety of comparison studies have also been carried out to demonstrate the effectiveness of the proposed approach.

Reactive Collision Avoidance Using Nonlinear Geometric and Differential Geometric Guidance

U NMANNED Aerial Vehicles (UAVs) hold good promise for autonomously carrying out complex civilian and military operations. However, many of these missions require them to fly at low altitudes, making them vulnerable to collision with both stationary as well as moving obstacles. Hence, it is vital that UAVs are equipped with autonomous capability to sense and avoid collisions, especially for the pop-up threats. When such a threat is sensed and a collision is predictedwithin a short time ahead, theUAV should be able to react and maneuver away quickly so that the collision is avoided. An algorithmwhich can assure such amaneuver is called a “reactive collision avoidance algorithm.” Since the available reaction time in such a scenario is usually is small and UAVs are usually limited by computational resources, such an algorithm should also be computationally very efficient (it should preferably be noniterative). It is also required that whilemaneuvering away, it should notmaneuver toomuch away from the obstacle either. This is both to avoid collision from other nearby obstacles and not to compromise on the overall mission objective. There are various attempts in the literature to develop algorithms for collision avoidance purpose, many of which are inspired from global path planning algorithms. The artificial potential field method is such an approach where the motion of the vehicle is guided under the influence of a potential field. The potential field (which is essentially a cost function) is designed in such a way that obstacles have repulsive fields while the destination has an attractive field. The safe path of the UAV is then found by optimizing the carefully selected cost function. To tune this basic philosophy for reactive collision avoidance, a model predictive control-based algorithm has been proposed in the literature. This algorithm essentially assures path following under safe conditions (i.e. if no collision is predicted in the near future) and invokes the potential field function when new collisions are sensed. However, in potential field based techniques the associated optimization process is typically done in an iterative manner. Because of this they are usually computationally intensive and hence are not suitable for reactive collision avoidance of airborne UAVs in general. A promising algorithm in collision avoidance and global path planning is the philosophy of rapidly-exploring random tree (RRT) [1], which has also been used for reactive collision avoidance. However, there are many concerns about the RRT approach, which can largely be attributed to the random nature of the algorithm. For example, the path predicted by RRT is usually a sting of connected straight lines that does not reflect the path followed by a vehicle with nonholonomic constraints. More important, it is a probabilistic approach and hence there is no guarantee of finding a feasible path within a limited finite time. Other graph search algorithms such as best-first search are also implemented for reactive collision avoidance [2]. However, this is not systematic approach and could result in the algorithm searching far too many nodes under some conditions. Moreover, precomputing motion primitives and saving them in a lookup table is infeasible for UAVs, which are usually resource-limited. An interesting perspective to collision avoidance problem is the minimum effort guidance [3], where an optimal control-based approach has been proposed after applying the collision cone philosophy to detect collisions. This method is computationally nonintensive as a closed form solution has been proposed. Even though this is an interesting idea, by minimizing the lateral acceleration, perhaps it imposes unwanted extra constraint on the problem formulation as reactive collision avoidance problems do not necessarily have to be carried out with minimum lateral acceleration. More important, one can observe that this formulation only assures position guarantee and no constraint is imposed on the velocity vector. Hence, even though it guides the vehicle to a carefully selected target point on the safety boundary (we call it the “aiming point”), it causes the vehicle to maneuver until this point. This can be risky as the vehicle may enter the safety ball before reaching the aiming point. Even though the collision cone based aiming point philosophy is a very good idea, the authors of this Note strongly believe that instead of only position guarantee, rather the velocity vector should be aligned towards the aiming point as soon as a collision is detected (which will automatically lead to position guarantee as well). Towards this objective, two new nonlinear guidance laws are proposed in this Note, which are named as nonlinear geometric guidance (NGG) and differential geometric guidance (DGG). These guidance laws are inspired by the philosophy of “aiming point guidance” (APG) [4], which has been proposed in missile guidance literature. It turns out that the APG is a simplified case of the NGG where the associated since function is replaced by its linear approximation (hence, for a systematic discussion, it is renamed as linear geometric guidance (LGG) in this Note). Both of the guidance algorithms proposed in this Note quickly align the velocity vector of the UAValong the aiming point within a part of the available time-togo, which ensures quick reaction and hence safety of the vehicle. The main feature of this philosophy is that they effect high maneuvering at the beginning, causing the velocity vector of the UAV to align with the aiming point direction quickly and then settling along it. Therefore there is no need to maneuver all the way until the aiming point is reached and hence the chance of the UAV entering into the safety ball is minimized. Using the point of closest approach (PCA) [5], the proposed NGG andDGGalgorithms have also been extended for collision avoidance with moving obstacles in both cooperative as well as ignorant scenarios. Mathematical correlations between the guidance laws have also been established, which show that the NGG and DGG are exactly correlated to each other with appropriate gain selections, while the LGG is an approximation of DGG. A “sphere-tracking algorithm” is also proposed in this Note where the UAV is guided to track the surface of the safety spherewhenever a brief violation of the safety boundary occurs after reaching the aiming point because of the location of the next aiming point (which may include the target in Presented as Paper 2010-8315 at the AIAA Guidance, Navigation and Control, Toronto, 2–5August 2010; received 26May 2010; revision received 4 October 2010; accepted for publication 6 October 2010. Copyright

Robust Reentry Guidance of a Reusable Launch Vehicle Using Model Predictive Static Programming

DOI: 10.2514/1.61615 A robust suboptimal reentry guidance scheme is presented for a reusable launch vehicle using the recently developed, computationally efficient model predictive static programming. The formulation uses the nonlinear vehicle dynamics with a spherical and rotating Earth, hard constraints for desired terminal conditions, and an innovative cost function having several components with associated weighting factors that can account for path and control constraints in a soft constraint manner, thereby leading to smooth solutions of the guidance parameters. The proposed guidance essentially shapes the trajectory of the vehicle by computing the necessary angle of attack and bank angle that the vehicle should execute. The path constraints are the structural load constraint, thermal load constraint,boundsontheangleofattack,andboundsonthebankangle.Inaddition,theterminalconstraintsinclude thethree-dimensional position andvelocity vectorcomponentsat theend of thereentry.Whereasthe angle-of-attack commandis generated directly, the bank angle commandis generated by first generating the requiredheading angle history and then using it in a dynamic inversion loop considering the heading angle dynamics. Such a two-loop synthesis of bank angle leads to better management of the vehicle trajectory and avoids mathematical complexity as well. Moreover, all bank angle maneuvers have been confined to the middle of the trajectory and the vehicle ends the reentrysegmentwith near-zerobankangle,whichis quitedesirable.Ithasalsobeendemonstratedthattheproposed guidance has sufficient robustness for state perturbations as well as parametric uncertainties in the model.

Generalized Model Predictive Static Programming and Angle-Constrained Guidance of Air-to-Ground Missiles

A new generalized model predictive static programming technique is presented for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. Two key features for its high computational efficiency include one-time backward integration of a small-dimensional weighting matrix dynamics, followed bya static optimization formulation that requires only a static Lagrange multiplier to update the control history. It turns out that under Euler integration and rectangular approximation of finite integrals it is equivalent to the existing model predictive static programming technique. In addition to the benchmark double integrator problem, usefulness of the proposed technique is demonstrated by solving a three-dimensional angle-constrained guidance problem for an air-to-ground missile, which demands that the missile must meet constraints on both azimuth and elevation angles at the impact point in addition to achieving near-zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Simulation studies include maneuvering ground targets along with a first-order autopilot lag. Comparison studies with classical augmented proportional navigation guidance and modern general explicit guidance lead to the conclusion that the proposed guidance is superior to both and has a larger capture region as well.

A Nonlinear Suboptimal Guidance Law with 3D Impact Angle Constraints for Ground Targets

Using the recently developed model predictive static programming (MPSP) technique, a suboptimal guidance law is presented in this paper considering the three-dimensional nonlinear engagement dynamics. The main feature of the guidance law is that it accurately satisfies terminal impact angle constraints in both azimuth as well as elevation, in addition to being capable of hitting the target with high accuracy. Moreover, it minimizes the control eort (i.e. the latax demand) throughout the engagement and hence leads to an optimal trajectory as well. The guidance law is primarily based on nonlinear optimal control theory and hence imbeds eective trajectory optimization concept into the guidance law. The performance of the proposed scheme is investigated using nonlinear simulation studies for stationary, moving and maneuvering ground targets, by considering both thrusted as well as unthrusted vehicles. Multiple munition engagement results are also presented to show the eectiveness of the proposed guidance scheme in such a scenario. A comparison plot for the Zero Eort Miss (ZEM) is also included, which reconfirms the superiority of the proposed optimal guidance over an augmented proportional navigation guidance available in the literature to engage maneuvering targets.