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This paper proposed an analytical iterative guidance method with the desired instantaneous impact point constraint for solid rockets in “burn-coast-burn” trajectory mode. Solid rocket motors expect to remove the thrust termination mechanism to increase the structural strength and launch reliability, which induce new difficulties and challenges to the guidance problems. In terms of the “Hohmann transfer” principle, a pointing algorithm is deduced in depth to establish the theoretical relations among the ignition time, the required velocity vector, and the orbital element constraints and provides the analytical expression of the ignition time. Then, an analytic solution of the required velocity vector is derived based on orthogonal and nonorthogonal velocity vectors, and a complete guidance logic is used to solve the target orbit elements satisfying the desired instantaneous impact point. Finally, the application of the developed theoretical algorithm in this paper is conducted using a two-stage solid rocket. The proposed guidance method is verified by Monte Carlo simulations, and the testing results demonstrate the adaptability, strong robustness, and excellent performance for different desired impact point missions.

Solid rockets without the thrust termination mechanism can be launched rapidly with high manoeuvrability and agility, fulfilling the longed-for requirement of responsive launching missions, which brings new difficulties and challenges to its guidance technology [

Over the past few decades, numerous efforts have been devoted to improve the accuracy and robustness of ascent guidance for launch vehicles. For a liquid launch vehicle shutting off controlled by the guidance command, a strongly adaptive iterative guidance method is designed according to the optimal analytical solution under multiple constraints, which is derived by the optimal control principle with the optimal fuel as the performance criteria. Countering the main characteristics of liquid launch vehicles, reference [

For solid rocket motors, removing the thrust termination mechanism can not only increase the strength of the structure but also reduce the cost, which is the inevitable trend of development. These have posed new difficulties and challenges on the ascent guidance in vacuum, which indicate that solid rockets must depend on the advanced guidance to satisfy the terminal constraints and shut off by fuel exhaustion. To adapt to the guidance problem of exhaustion shutdown, the closed-loop guidance method that is insensitive to the cut-off time connects the flight mission with the thrust vector direction of the rocket engine by introducing the concept of the required velocity vector [

In order to satisfy the instantaneous impact point constraints of solid rocket with coast arcs, this paper presents an analytical instantaneous impact point guidance method that combines the solution of ignition time and the iterative calculation of the target orbit. The study is organized as follows: In Section II, the dynamic model for solid rockets and the instantaneous impact point guidance problem is established. In Section III, the pointing algorithm for burn arc is proposed, and the analytical solution of ignition time is obtained. In Section IV, the iterative calculation of the target orbit considering the deleted shutdown and the solution logic of the proposed guidance method are presented. In Section V, the numerical simulation results are discussed. In Section VI, conclusions are drawn.

A multistage solid rocket is investigated in this paper, in which the last stage flies in the exoatmospheric, and the typical ascent launch process is as shown in Figure

Typical launch process of the instantaneous impact point guidance for solid rockets.

The three-dimensional point-mass equations of motion ascending through the atmosphere can be expressed in an inertial frame as

The thrust magnitude

Note that the centrifugal force, Coriolis force, and the J2 gravitational term are ignored or simplified in the design of the guidance model but treated as interference to verify the accuracy and robustness of the guidance algorithm in the simulation test. In the “burn-coast-burn” trajectory mode, the thrust magnitude

Due to the loss of throttle modulation and the thrust termination mechanism, solid rockets cannot control the thrust magnitude and the engine cut-off; so, Eq. (

An instantaneous impact point of the solid rocket is the intersection point between the Earth’s surface and the Kepler trajectory of the rocket in the absence of specific acceleration. According to the properties of the Kepler orbit, the IIP unit vector

And the range angle

By considering the Earth’s rotation during the flight time, the IIP longitude in the Earth-centered Earth-fixed (ECEF) is given as

Lambert’s problem seeks for the orbit that connects the two points in a prespecified transfer time

Following the initially propelled ascending phase and already beyond the atmosphere, a pointing algorithm based on the concept of impulsive orbit transfer is established, which concentrates on how the given direction the continuous thrust vector is equivalent to the velocity vector impulse in the orbit transfer [

The principle of the pointing algorithm and the relationship between vectors.

Where

In this process, solid rockets with uncontrolled thrust intensity and direction are injected to the target orbit. So, the equations of motion in Eq. (

In accordance with the character of Kepler orbits, the moment of momentum of the vehicle is conserved. Thus, the change of the moment of momentum caused by the continuous thrust is expressed as follows.

It is assumed that the body longitudinal direction

Using Eq. (

Depending on the crossproduct of the vectors with the same direction

In the vacuum phase, when the engine ignition time satisfies the time constraint, it is always possible to find an intersection line

Slimily, on the target orbit, the following equation exists

Substituting Eq. (

The target orbit is constructed based on the requirement of the mission before launching; so, the intersection point

Note that

The relationship of the states (

The velocity vector relations of orbits at the point

where

According to the equivalent pulse theory of the PA method, the velocity vectors can be calculated by the orbital elements. It is known that in the Kepler orbit, the following equations hold

The velocities

By substituting Eq. (

For an elliptical orbit, the eccentric anomalies at the initial and the final points are calculated to determine the transfer time as follows.

Then, the transfer time between the two points

In addition, it is noted that

Then, the coast time

Because of the Earth’s rotation during the flight time, the IIP longitude

The pointing algorithm establishes an equivalent theoretical relationship between the continuous propulsion process and the instantaneous velocity increment process by solving the ignition time. To achieve the desired impact point, the velocity vector

Calculation target orbit composed of the equivalent pulse point

According to Eq. (

Then, according to the equivalent relation in the principle of PA, an instantaneous impact point

where the

And the required velocity vector

In addition, the velocity increment generated by the solid rocket motor is uncontrollable during the course of exhausting shutdown, and Eq. (

The velocity vector

where function

There are several possible iteration schemes to search for the required velocity vector

Calculation required velocity vector according to different velocity components.

The required velocity vector

Relationships among the velocity components in perpendicular axes (

It is known that for the family of orbits that connect the two position vectors

In Eq. (

Since

In order to obtain the analytic relationship between iteration variables and constrained variables, Eq. (

The maximum flightpath angle

In general, the desired instantaneous impact point of solid rockets should be within a certain range, especially the lower bound of range is one of the important mission indicators. Due to the influence of interference and uncertainty in the flight process, the rocket cannot reach the target even under the optimal flight command. Thus, to ensure the iteration variable

According to Eqs. (

An instantaneous impact point guidance method is presented in the previous sections. The pointing algorithm establishes the relationship between the ignition time and the required velocity vector at the point

To evaluate the performance of the proposed guidance method, several cases are applied in this section. Monte Carlo simulations are widely used in science, engineering, and finance to assess risks and enable decision making. Reference [

The solid rocket consists of two booster stages with a total mass of 15,000 kg and a rated maximum range of 4600 km. The offline guidance commands are implemented in the first boost stage. Then, the proposed guidance method is applied in the 2nd stage, and the ignition time is calculated by Eq. (

At present, the existing guidance methods are mainly GEM [

Profiles of the state parameters compared IIP and GEM with the SEM method. (a) The pitch angle vs. time. (b) The altitude vs. range.

The simulation results show that the IIP, GEM, and SEM methods for solid rockets can satisfy the terminal longitude and latitude constraints. However, the pitch angle of the proposed IIP method is smaller and smoother than that of GEM and SEM, which is beneficial to the stability of the control system of solid rocket. Then, the performance of the solid rocket engine is significantly affected by ambient temperature. The nominal operating time deviation reaches 10% between the highest and lowest temperature conditions, which brings difficulties and challenges to the adaptability and robustness of guidance methods. The deviation and uncertainty in the simulation are shown in Table

Parameter deviation and uncertainty conditions.

Parameter errors | Unit | 1st stage | 2nd stage |
---|---|---|---|

Total mass | Kg | ±100 | ±20 |

Engine working time | s | ±10 | ±3.0 |

Average thrust of engine | % | ±12 | ±5 |

Aerodynamic coefficients | % | ±15 | — |

Atmospheric density | % | ±15 | — |

Wind | Stationary wind and random direction |

In each Monte Carlo simulation run, random dispersion, and uncertainties, vehicle mass properties, aerodynamics, and atmosphere are unknown to the guidance algorithm. In the simulations, the guidance algorithm is called at a 10 Hz rate to provide the guidance commands. Based on the above discussion, the desired range was set at 4500 km, and 250 Monte Carlo simulations are conducted. The profiles of the states are shown in Figure

Profiles of the state parameters of 250 Monte Carlo runs for solid rockets. (a) The altitude. (b) The velocity. (c) The flight path angle. (d) The guidance commands.

Figure

To verify the accuracy and robustness of the proposed guidance approach under different desired impact point conditions, extensive Monte Carlo simulations [

Figure

Profiles of the state parameters of 750 Monte Carlo runs under different missions. (a) Altitude. (b) Flight path angle.

The guidance commands the history with deviation and uncertainty under different missions.

Monte Carlo simulation tests demonstrate the proposed guidance method has high accuracy and robustness, in which the mean deviation of the range reaches the magnitude of 15.84 (km), and the standard deviation is less than 42.31 (km); the mean deviation and standard deviation of longitude and latitude are magnitudes of 10^{-1} (°), respectively. The statistic results on the same 750 dispersed trajectories are listed in Table

Statistics results of 750 Monte Carlo runs for solid rockets under different missions.

Parameter errors | Unit | Mean | Stdv | |
---|---|---|---|---|

1500 km | Range error | Km | 15.84 | 25.08 |

Longitude error | °/10^{-1} | 1.59 | 2.58 | |

Latitude error | °/10^{-1} | -0.66 | 0.94 | |

3000 km | Range error | Km | 8.76 | 42.22 |

Longitude error | °/10^{-1} | 0.49 | 3.96 | |

Latitude error | °/10^{-1} | -0.85 | 1.73 | |

4500 km | Range error | Km | 1.17 | 42.31 |

Longitude error | °/10^{-1} | 6.06 | 3.51 | |

Latitude error | °/10^{-1} | -3.82 | 2.04 |

Drogue deployment accuracy under the proposed guidance method.

This paper presents an analytical guidance method with coast arcs for solving an instantaneous impact point constrained problem during exhausting shutdown. A pointing algorithm with the terminal orbital elements constraints is developed to establish the theoretical relationship between the ignition time and the required velocity vector in “burn-coast-burn” trajectory, which is applicable to solid motor shutting off by fuel exhaustion. Then, according to Lambert’s principle, the analytic solution of the required velocity vector is derived, and the iteration of orbit elements satisfying the desired impact point is constructed. The theoretical relationship between the burn arc and the coast arc of the rocket is established by the constraints of the orbital elements satisfying the desired impact point, which improves the adaptability of the guidance algorithm to the impact point missions.

Monte Carlo simulations using a two-stage solid rocket and different desired impact point (range) missions are performed. The guidance logic and process from the ignition time of the rocket to the desired impact point constraint are given in this paper, and the guidance commands are calculated onboard by the current state provided by the navigation. The simulation testing results demonstrate that the proposed instantaneous impact point guidance has high accuracy and strong robustness against to dispersion and uncertainty and strong adaptability to different desired impact points.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this article.