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The work traces a general procedure for the design of a flight simulation tool still representative of the major flight physics of a parachute-payload system along decelerated trajectories. An example of limited complexity simulation models for a payload decelerated by one or more parachutes is given, including details and implementation features usually omitted as the focus of the research in this field is typically on the investigation of mission design issues, rather than addressing general implementation guidelines for the development of a reconfigurable simulation tool. The dynamics of the system are modeled through a simple multibody model that represents the expected behavior of an entry vehicle during the terminal deceleration phase. The simulators are designed according to a comprehensive vision that enforces the simplification of the coupling mechanism between the payload and the parachute, with an adequate level of physical insight still available. The results presented for a realistic case study define the sensitivity of the simulation outputs to the functional complexity of the mathematical model. Far from being an absolute address for the software designer, this paper tries to contribute to the area of interest with some technical considerations and clarifications.

The purpose of a parachute is to decelerate and provide stability to a payload in flight. The aerodynamic and stability characteristics of the parachute system are governed by the geometry of the parachute as such careful consideration is paid to this in the design process. The effects of deployment and opening force are critical in the safe operation of the parachute and the integrity of the payload. The opening characteristics also feature heavily in the selection of geometry and other parameters in the design process.

Parachutes for aerospace applications [

There are a number of different kinds of parachutes that have been designed for various applications. The different applications parachutes are typically used for pilot, drogue, deceleration, descent, extraction, supersonic drogue and stabilization, flight termination, and landing.

The dynamics of parachutes are complex and difficult to model accurately. During both the inflation process and the terminal descent stage, the dynamics of a parachute are governed by a coupling between the structural dynamics of the parachute system and the surrounding fluid flow. Both of these dynamic systems must be addressed as a coupled system to gain a proper representation of the dynamic system as a whole.

When the parachute is in a steady state, the air flowing around the decelerator will separate at some location on the canopy. The shedding of the vortices from the canopy can affect the stability and cause a periodic motion of both parachute and payload. The wake from a porous parachute consists of air that flowed around the canopy and air that flowed through the canopy. A payload body in the speed range of parachute usage sheds a very turbulent wake. Part of the flow that is entering the parachute is therefore of a disturbed nature and should be considered regarding the aerodynamic performance of the parachute. For many types of parachutes, this change in oncoming airflow can be quite significant during the time required for the parachute to inflate. The implication of a rapid deceleration is that second-order effects are likely to be present.

To summarize, calculation of parachute deployment, inflation, and deceleration requires the numerical solution to the equations of motion for a viscous, turbulent, separated airflow. The parachute is also a flexible body having dynamic behavior coupled with the behavior of the flow, which passes through and around it. From the above description it is obvious that a full-time dependent solution of this system is far from being easily feasible. To make a mathematical model that is feasible, simplifications must be made, as long as the model can be validated satisfactorily by experiment or by comparison with reference data.

The overall behavior of parachutes is related to various parameters: added masses, filling time, parachute shape (inflated canopy elongation), porosity, suspension line length, reefing, clustering, snatch loads at deployment, and aero-mechanical and inflation instability. In the past, most of these effects could be generally modeled in an imprecise way by simulation tools. A comprehensive computational technique is presented in [

According to the different missions, several types of payloads have been used in combination with aerodynamic decelerators: paratroops, equipment, hardware, materiel, weapons, missiles, aircraft, unmanned aerial vehicles, aerospace lifting, and nonlifting spacecraft. The present analysis is focused on aerospace applications for planetary and atmospheric entry vehicles, where the payload is typically a blunt body. The purpose of a blunt body is primarily to provide a large source of drag to facilitate a deceleration. Applications of blunt bodies can be seen with both manned reentry and planetary exploration missions. An outline of aerodynamic decelerators for robotic planetary exploration missions is given in [

Blunt bodies for which there exist examples can be classified into two categories, large angle cones and capsules [

The flow field that is associated with the capsule configuration is highly complex. For almost the entire speed range that the capsules operate over, the flow remains attached on the forward face. At the point of maximum diameter, the flow is accelerated such that the boundary layer rapidly grows to the point of separation. After the maximum diameter, the flow then remains separated and turbulent. This flow is unstable and coupled with an unsteady near wake the dynamic instability associated with capsules is produced [

As dynamic instability exists for blunt bodies over various speed ranges, to complete missions successfully, there is then a requirement for some kind of accurate stability assessment with a potential impact on both stability augmentation (if any reaction control system is implemented) and mission design (parachute deployment sequence). This issue applies for probes and capsules as thrusters and parachutes are the unique available sources of additional damping.

The measurement of stability derivatives for probes and capsules was a concern of designers since the origin of space flight [

Several studies analyze the descent and landing trajectories of parachute-payload systems. Generally, these analyses consist of performing a simulation (typically a 3–6 DOFs rigid body model is adopted) of the atmospheric entry phase to predict deceleration peaks, descent attitudes, and terminal conditions. In addition, a stochastic dispersion analysis (Monte Carlo or similar) is usually performed to assess the impact of off-nominal conditions that may arise to determine the robustness of the mission design.

A two-dimensional parachute model is presented in [

A first mathematical representation of a parachute-payload system (designated as SM1) is defined according to a very comprehensive approach that enforces the simplification of the coupling mechanism between the payload and the parachute, with an adequate level of physical insight still available in terms of sensitivity of system performance metrics to design parametric changes. In modeling a payload-parachute system, two bodies and one device have been used. They are the parachute canopy and the payload, which are then connected by a single riser. In reality the parachute canopy is connected to the payload by suspension lines, a riser section, and then a set of bridles. In the simulation only the riser and the bridles are modeled in terms of a dynamic response (suspension lines are neglected). However, in terms of connection points only the riser has been modeled. The riser is assumed to have flexible connections at both ends and provides stiffness above a certain threshold distance and zero stiffness below this distance (slack conditions).

The payload is considered as a rigid body with six degrees of freedom. The forces and the moments that act on the body are provided by its aerodynamics, by the weight, by the inertial actions, and, finally, by the force applied by the riser in the suspension point. The parachute canopy acts as a rigid body and, by the action of aerodynamic drag, strains the riser which then transmits a force to the payload. Figure

Flow chart of the simulation model.

In terms of a dynamic response, the riser is modeled as having linear stiffness and damping, where the force at any given point in time is given by

The performance of the parachute has been modeled using an approximation for the added mass

The significance of fluid inertial effects is relevant. As an example, opening times and loads increase with altitude. The reason for this can be seen when it is considered that at increasing altitude the true airspeed will increase for constant indicated airspeed, as such the inertial effects of the air are greater. After comparison with flight test data, the added mass approximation tends to overestimate the deceleration that will occur. From a design point of view, this means that results obtained from the added mass approximation can be viewed as conservative. So while the concept of added mass is only an approximation, the effects of fluid inertia in transient parachute aerodynamics are significant enough to be included in modeling. At present this method of accounting for the inertial effects of air is the most practical tool available to be used in the transient phase of parachute operation.

The parachute as previously stated is modeled as a separate body (rigid canopy). The deceleration of this body is therefore calculated by extension of (

In (

Due to the complexity involved with the filling process, filling time cannot be calculated by a purely analytical method. The alternative once again lies with an empirical approach that is validated by flight test. Knacke defined the following empirical relation for filling time:

An underlying assumption used in the modeling of the parachute canopy is that it remains aligned with the velocity vector at all times. This corresponds to neglecting the lift and moment coefficients of the parachute.

The second major assumption made used in the parachute canopy model is that the added mass remains constant throughout inflation and parachute operation.

Whilst in the model the value of added mass is kept constant at the value for the fully inflated parachute (conservative approach for the estimation of deceleration peak), the value for drag area is ramped up as the parachute inflates. It is this feature of the model that provides a simulation of the inflation process. Inflation modeling is done by ramping the value of the drag area,

The final assumption made is with the clustered parachutes. In the simulation one parachute of equivalent size has been used, with a cluster efficiency factor applied.

In Figure

Layout of the suspension system (parachute-payload configuration).

The forces and moments acting on the payload are calculated as the relevant contributions are added. The equations are

In the above-presented model, there are a number of featured that need to be discussed. The aerodynamic data for the payload—which the simulation uses as input—are given in the longitudinal plane only. In order to calculate the correct coefficient, the center of gravity must be shifted accordingly. This is done using the following equations:

The lateral-directional coefficients are obtained assuming the geometric symmetry of the payload. In taking this approach, the aerodynamic response to angle of attack and angle of sideslip have been decoupled. The implications of this decoupling are that there are no aerodynamic roll moments (

The body axis system has been adopted, and this system can be seen in Figure

The reference frame and the aerodynamic angles for the payload.

The system written in residual form can be seen below:

The magnitude of the velocity vector, that is, the airspeed, is found from the three body-axis velocities:

The conversion from the Euler angles to quaternion values is computed by the following set of equations:

The atmospheric profile (density and speed of sound) is approximated with a cubic curve fit of the reference atmosphere.

A second simulation model (designated as SM2) was defined, implemented, and validated in [

the aerodynamics of the bodies are defined in terms of total angle of attack due to the geometrical symmetry of payload and parachute;

the dynamics of the parachute are modeled with a full-state rigid body six degree-of-freedom representation;

the added masses of the parachute are defined by a tensor including rotational inertial properties [

the aerodynamics of the parachute include lift and damping coefficients;

the suspension system is represented by a more realistic layout with distributed elements and suspension links whose strain is estimated with an iterative numerical method;

the effects of atmospheric turbulence and asymmetries (either geometrical or inertial unbalance) can be included.

The simulation software is written in Fortran language. This program simulates the dynamics of the parachute-payload system (time domain integration). The user is offered predefined initial conditions from reference flight conditions or the option of trimming the payload at any desired altitude and using those generated initial conditions. The inertial properties of the payload (mass, moments of inertia, and position of the center of gravity) and the parachute characteristics (size, mass, opening, and staging sequences) are defined for a selected altitude ranges (mission table lookup).

The solver used in the simulation program is DASSL (differential algebraic system solver) originally developed by Petzold [

The initial conditions for the payload (with or without decelerator deployed) may be computed for a given initial altitude ^{−7}. Longitudinal and directional planes are considered separately. The system of equations for the longitudinal plane is defined as follows:

The reference mission described in [

The parachute deployment sequence.

Sequence | Time (s) | Altitude (m) |
---|---|---|

Mortar firing (Pilot Jettison) | 0.00 | 13994 |

Back cover separation | — | 13666 |

Drogue snatch/1st stage inflation | 3.34 | 13430 |

Drogue 2nd stage inflation/disreef | 9.34 | 12546 |

Drogue bridles cut | — | 6461 |

Main snatch/1st stage inflation | 82.30 | 6364 |

Main 2nd stage inflation | 88.30 | 6092 |

Main 3rd stage inflation/disreef | 94.55 | 5950 |

Main bridle 1 cut | — | 5360 |

The aerodynamic data for the payload—which the simulation uses as input—were obtained with wind tunnel static and forced oscillation tests performed at Politecnico di Torino [

The experimental setup [

The reentry profile is correctly reproduced by the two simulators—as presented in Figure

The simulation of the trajectory profile for the parachute-payload system.

The ability of the SM1 model to estimate the riser and parachute loading is outlined in Figure

The loads acting on the riser and the decelerator (drogue and main parachute).

The projected trajectory is compared in Figure

The simulation of the trajectory profile for the parachute-payload system.

The aerodynamic angles of the payload are plotted in Figure

The simulation of the angles of attack and sideslip for the payload.

The aerodynamic coefficients of the payload (pitching moment static stability) [

The aerodynamic coefficients of the payload (pitching moment dynamic stability) [

In order to verify the modal response, at least for the longitudinal short-period natural frequency, the pitch rate spectral response for the two simulation models is compared with the elaboration of available flight logs in Figures

The index of similarity for the pitch rate spectral response.

Flight test | Flight test | ||||

Drogue parachute | Drogue parachute | ||||

I1 | I2 | I3 | I1 | I2 | I3 |

1.392 | 1.005 | 0.989 | 0.645 | 1.145 | 1.110 |

Main parachute | Main parachute | ||||

I1 | I2 | I3 | I1 | I2 | I3 |

0.578 | 1.022 | 1.029 | 0.463 | 1.088 | 1.098 |

The effect of model complexity on pitch rate spectral response (drogue parachute).

The effect of model complexity on pitch rate spectral response (main parachute).

The present work outlines a comparative analysis of two simulation models (SM1 and SM2) of a parachute-payload system with different levels of complexity. An example of limited complexity reconfigurable simulation models for a payload decelerated by one or more parachutes is given, including details and implementation features usually omitted as the focus of the research in this field is typically on the investigation of mission design issues, rather than addressing general implementation guidelines for the development of a reconfigurable simulation tool. The dynamics of the system are modeled through a simple multibody model that represents the expected behavior of an entry vehicle during the final deceleration phase by means of a system of aerodynamic decelerators. The simulators are designed according to a comprehensive vision that enforces the simplification of the coupling mechanism between the payload and the parachute, with an adequate level of physical insight still available in terms of sensitivity of system performance metrics to design parametric changes.

A reference mission described was used to evaluate the two simulation models. The aerodynamic data for the payload—which the simulation uses as input—were obtained with wind tunnel static and forced oscillation tests performed at the Politecnico di Torino.

The trajectory is reproduced by the two simulators as the profiles, mainly shaped by the parachute opening sequence, are matched by both SM1 and SM2 models, even if the use of total angle of attack for the interpolation and reconstruction of payload aerodynamic coefficients (as in SM2 model) provides a more accurate fit. The riser and parachute loading are estimated in a way that is coherent with the deceleration profiles.

The major discrepancy between the two models is the level of dynamic stability of the modal response (short-period dynamics). This can be explained with the extreme sensitivity of the aerodynamic coefficients to center of gravity location and angle of attack. Another contributing element is the dynamics of the parachute which in the higher fidelity model (SM2) allows for attitude changes decoupled from the suspended body, providing a more realistic behavior of the parachute-payload system. Differently, the parachute trajectory in the SM1 model is aligned instantly with the velocity vector of the payload, probably a too crude approximation if the purpose of the analysis is the attitude dynamics of the suspended vehicle. Furthermore, a multilink suspension system (as modeled in SM2 implementation) induces a steering effect on the payload, neglected by the single-point suspension of SM1 model.

The spectral results show that the range for the natural frequency of the short-period response—as measured in flight—is matched by both simulation models as demonstrated by the fact that the indices of similarity

As a general conclusion, the lower fidelity simulation tool SM1 exhibits a limited advantage in terms of computational workload, mainly providing a simplified approach for preliminary trajectory estimation and parachute sizing.

Damping coefficient

Body-to-Earth axis transformation matrix

Body-to-wind axis transformation matrix

Parachute drag coefficient

Auxiliary transformation matrix

Roll moment coefficient

Roll moment coefficient due to roll rate

Pitching moment coefficient

Pitching moment coefficient due to pitch rate

Yaw moment coefficient

Yaw moment coefficient due to yaw rate

Wind-to-Body axis transformation matrix

Force coefficient in

Force coefficient in

Force coefficient in

Force coefficient in

Force coefficient in

Force coefficient in

Payload reference length

Canopy nominal diameter

Parachute drag

Quaternion parameters

Riser force (tension)

Acceleration of gravity

Canopy height

Altitude

Index of similarity

Moment of inertia

Product of inertia

Product of inertia

Moment of inertia

Product of inertia

Moment of inertia

Stiffness

Added mass coefficient

Bridle stiffness

Riser stiffness

Rolling moment

Offset length

Riser length

Payload mass

Pitching moment

Parachute total mass

Parachute added mass

Parachute mass

Yawing moment

Canopy porosity

Roll rate

Pitch rate

Yaw rate

Canopy radius

Numerical residual

Payload displacement

Parachute displacement

Parachute reference area

Time

Velocity in

Velocity in

Payload-free stream velocity

Payload velocity

Parachute velocity

Parachute velocity (at line stretch)

Velocity in

North displacement (trajectory)

Force in

East displacement (trajectory)

Force in y body direction

Down displacement (trajectory)

_{:}

Force in

Angle of attack

Angle of sideslip

Flight path angle

Strain

Strain rate

Wake penalty factor

Pitch angle

Damping ratio

Air density

Measurement error

Roll angle

Yaw angle

Frequency.

The author wishes to acknowledge the support given by Mr. Michael Gordon.