This paper presents a neural network assisted entry guidance law that is designed by applying Bézier approximation. It is shown that a fully constrained approximation of a reference trajectory can be made by using the Bézier curve. Applying this approximation, an inverse dynamic system for an entry flight is solved to generate guidance command. The guidance solution thus gotten ensures terminal constraints for position, flight path, and azimuth angle. In order to ensure terminal velocity constraint, a prediction of the terminal velocity is required, based on which, the approximated Bézier curve is adjusted. An artificial neural network is used for this prediction of the terminal velocity. The method enables faster implementation in achieving fully constrained entry flight. Results from simulations indicate improved performance of the neural network assisted method. The scheme is expected to have prospect for further research on automated onboard control of terminal velocity for both reentry and terminal guidance laws.
In the past half century, entry guidance law has been of particular interest for research. One of the reasons for this growing importance is the rise in missions to other planets, which is further emphasized by recent endeavor for prompt global strike capability. Although present day guidance technologies are able to attain very high precision in conventional guided munitions, the same cannot be said for planetary entry vehicles. This is partly because in entry guidance certain path constraints are considered, which are not considered for conventional munition guidance. These path constraints cannot be violated in entry flight; as such, their satisfaction is the primary concern in such guidance. Due to this unavoidable reason, some terminal constraints seem to have been compromised in many entry guidance laws. However, with the growing requirement for terminally more accurate entry mission, there is a need to address these terminal constraints with much more weightage. With this in mind, an entry guidance law is presented, which, under a wide range of uncertainty, appears to be able to satisfy terminal constraints in position, velocity, and angular states as well as comply with the hard path constraints.
Entry guidance law was first successfully applied in the Apollo program [
With future planetary entry and global payload delivery mission in consideration, neither precomputed profile tracking nor predictorcorrector guidance law seems capable enough in achieving a multiconstrained entry flight. Guidance laws that depend on a precomputed profile are assumed to be more susceptible to error in the presence of atmospheric and aerodynamic uncertainty. However, in such guidance laws path constraints can be easily satisfied. On the other hand, predictorcorrector guidance laws could be suggested to be more autonomous, yet involve highly demanding onboard computation. Under these circumstances, it is believed that a guidance law would be better suited for future requirements if it involved a simple computational method for onboard modification of an offline profile that complies with path constraints. This paper presents a guidance law that is designed to work in such manner. The proposed law uses a threedimensional Bézier curve in approximating the vehicle trajectory. It is shown that this approximation significantly simplifies both offline reference generation and onboard trajectory correction. The properties of Bézier curve are employed to enforce initial and terminal states and path constraints. This constrained approximation is then used in generating the guidance command. The solution thus attained readily satisfies terminal position and angular constraints. However, for the satisfaction of terminal velocity constraint, the method requires to predict the terminal velocity under disturbance. At a previous work as reported in [
The focus of the paper is to present a guidance law for atmospheric entry flight. An atmospheric entry flight usually starts at an altitude of 120 km and terminates at around 25 to 30 km. However, from the literature it can be seen that the majority of the guidance laws are designed to actively guide the entry vehicle from 55 to 50 km down to 30 to 25 km, which is due to the ineffectiveness of control measure over the altitude of 60 km. Some of the guidance methods adopt a quasiequilibrium glide (QEG) flight under the assumption of the quasiequilibrium glide condition (QEGC) [
For the trajectory dynamics, a point mass vehicle model over a round earth is adopted. The rotating earth effects can be assumed to be compensated by the feedback nature of the guidance law. Thus, the threedegreeoffreedom (3DOF) dynamics of a point mass entry vehicle model (in a geodetic coordinate frame) can be described through the following equations of motion:
In the QEG flight, the vertical component of acceleration and the flight path angle are assumed to be small. Under this supposition, setting
The slope of a QEG trajectory (
Although in the presented problem a QEG flight is adopted, the associated path constraints cannot be assumed to be satisfied under disturbances. Therefore, additional measure is required to ensure satisfaction of the path constraints. The associated path constraints are heating rate, normal aerodynamic load factor, and dynamic pressure. These are expressed as
Terminal constraints are set as restrictions on the vehicle’s position, angle, and velocity as per the requirements for TAEM, where terms with the superscript “
Besides the constraints of entry flight and TAEM requirement, the flight vehicle is assumed to have limitations on the control parameters. The vehicle needs to be guided within the following limitations:
The reason for using a Bézier approximation is to facilitate an inverse solution of the entry dynamics. The inverse problem approach is essentially the core of this method. Exposition of the method comprises the inverse dynamics formulation followed by the techniques for the Bézier approximation and the enforcement of constraints.
The first step in the inverse approach is to derive an expression for the control parameters (angle of attack
From (
Analysis of (
In the presented method, the entry trajectory is approximated by a Bézier curve. This is because the derivative of this approximation can be easily obtained which is shown in the formulation. Theoretically an entry trajectory can be described precisely using a Bézier curve. However, the degree of the curve depends on the entry trajectory. For some trajectories a 2ndorder or 3rdorder curve may suffice, whereas a curve with a higher degree of freedom may be required for other entry trajectories. As such, the degree of the curve depends on the trajectory profile. The method proposed in this paper is applicable only for a Bezier curve of 3rd degree. As such, for curves with higher degree of freedom, the trajectory needs to be partitioned into 3rd degree Bézier curves.
Mathematically, a Bézier curve is defined as a parametric curve
Bézier curves of third degree and their control polygons.
In the proposed work, the entry trajectory is approximated by a Bezier curve in a local 3D coordinate system. As such, two coordinate transformations are required, first from the geodetic to the earthcenteredearthfixed, and then to the local coordinate frame as shown in (
After the above transformations, the vehicle trajectory can be represented as three Bézier curves as shown in Figure
The approximated Bézier curve and its control points are shown in Figure
3D Bézier curve and its control points.
Effect of the adjustment of interior control points. Direction of the tangent of the curve at the initial control point remains unchanged if the interior control point is adjusted on the same tangent line. Moving the control point outside of the tangent line changes the tangent and its direction.
The potential of adjusting a Bézier curve, while keeping the end points and directions unchanged, is very significant. This means that if a flight trajectory can be approximated as a Bézier curve, then it is possible to adjust its flight by moving the interior control points along the end tangents, thus keeping end points and directions unchanged. In order to avail this technique, parameters
Bézier parameter representation.
Off board parameter optimization process.
Mathematically, the Bezier parameters can be defined as
Using the Bézier parameters, the following equations can be derived for the intersection point and interior control points:
For a terminally constrained flight, it is thus possible to complete the approximation provided that appropriate values of the parameters
Using the above equations, the previously unknown terms of (
In the formulation presented so far, the constraints of terminal position are addressed through (
From the above acceleration commands, the angle of attack and bank angle commands can be found:
Heat rate and dynamic pressure constraints (
The final control command may then be obtained as follows:
The control command thus obtained complies with all path and boundary constraints except that of velocity. For velocity constraint, a technique is proposed. In this approach, a relation between terminal velocity and the parameter
The guidance method is implemented in two steps. The first part is the parameter optimization process where the Bézier parameters
The parameter optimization process is aimed at obtaining the parameters
Satisfaction of the terminal velocity constraint is ensured by adjusting the parameter
Velocity control through the adjustment of Bézier parameter
The polynomial in (
Using the above relation, a desired terminal velocity can be maintained through
Artificial neural networks (ANN) are inspired by the biological neural systems. These networks are composed of artificial neurons which are designed to receive input and generate activation signal which in turn triggers an output. A network of these artificial neurons can be trained to solve complex problems. In several research on entry guidance methods [
Architecture of feed forward network.
The onboard implementation starts with inputs of initial state, terminal state, and the optimized Bézier parameters. A flow chart of the process is shown in Figure
From the current state
Using the stored values of
The acceleration commands are obtained and the path constraints are applied.
At specified intervals, the terminal velocity is predicted using the ANN, and if necessary,
Guidance law implementation.
Performance and robustness of the guidance law have been evaluated through simulations of full nonlinear dynamics for LockheedMartin’s CAVH vehicle [
A nominal trajectory was generated for the desired boundary conditions using the off board parameter optimization process. The specified boundary conditions and the obtained Bézier parameters are shown in Table
Nominal profile and Bézier parameters.
Nominal profile  

Parameter  Initial position  Terminal position 
Altitude  50 km  25 km 
Velocity  7500 m/sec  2630 m/sec 
Flight path angle  −0.001°  −0.6° 
Azimuth angle  0°  0° 


Parameters 




Bézier parameters  0.848  0.717 
The ANN for the prediction of the terminal velocity was set up using the neural network toolbox of MATLAB. A twolayer feed forward back propagation network was used with 1000 neurons and it was trained for 1000 cases. Comparison of the ANN result with actual simulation data indicated 99% accuracy as shown in Figure
Accuracy of ANN in predicting terminal velocity.
The polynomial expression of
Terminal velocity and
Simulations were carried out for 500 cases with normally distributed errors in the initial state and random ±10% error in aerodynamic and atmospheric modeling. The perturbation specifications are shown in Table
The perturbations in Monte Carlo run.
Parameter  Value 

Range  ±1.5 km 
Altitude  ±1.5 km 
Cross range  ±1.5 km 
Aerodynamic modelling  ±10% 
Velocity  ±75 m/sec 
Flight path angle  ±0.06° 
Azimuth angle  ±0.06° 
Atmospheric modelling  ±10% 
The state plots from the 500 cases are shown in Figure
Monte Carlo simulation results.
Parameter  Miss distance 




Maximum  159 m  0.002°  0.02°  19.8 m/sec 
Mean  0.22 m  0.0001°  0.002°  5.01 m/sec 
Std. deviation  4.54 m  0.001°  0.003°  7.23 m/sec 
Simulation results. (a) Altitude, (b) velocity, (c) flight path angle, and (d) azimuth angle.
Statistics of terminal errors for the 500 runs.
Graphical representation of the terminal error statistics is shown in Figure
Control history from the simulations. (a) Angle of attack profile and (b) bank angle profile.
Attack and bank angle profiles corresponding to the Monte Carlo simulation are plotted in Figure
Path constraint plot for the runs. (a) Dynamic pressure remains within 110 kPa, (b) heating rate stays within 1.5 MW/m^{2}, and (c) the normal load is within 1 g.
The result shown in Figure
Effect of terminal velocity control through
In this paper, an inverse guidance law is presented. The central aspect of the method is the representation of a constrained trajectory using a Bézier curve. In the formulation, the approximation is made by using two proposed parameters. These parameters are derived from the Bézier control points and are utilized for ensuring boundary constraints. The method also employs ANN for predicting and controlling terminal velocity. The proposed method is evaluated through a 500run simulation considering perturbations in the initial state, and error in aerodynamic and atmospheric modelling. The results indicate that the guidance scheme performs remarkably in satisfying terminal and path constraints. The presented method is however limited to trajectories with limited curvature variation. Further research on use of a higher degree Bézier curve and a faster parameter optimization process can increase the method’s applicability in global payload delivery systems as well as future planetary missions.
The authors declare that there is no conflict of interests regarding the publication of this paper.