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Håkan Wettergren

Helicopter Landing on Ship: Model and Simulation

December 16, 2015 — Håkan Wettergren, Applications Engineer, SystemModeler (MathCore)

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Today, many helicopters launch from and land on ships at sea. Some are conventional helicopters, both commercial and military, and some are drones. In Wolfram SystemModeler, we now have a system for simulating helicopter landings and launches that includes waves and ships. The models have been used for the design of mechanical parts, autopilots, landing criteria, and operational limits.

Major components of the system

The aim has been to develop a model with an accurate depiction of the waves, ship motion, and helicopters in such a way that the results can be used not only qualitatively but also quantitatively in real industrial applications.

The first task is to calculate the motion of the landing platform mounted on the ship’s deck. There is commercially available historical wave data for different seas and oceans. Since access to this data is expensive, we will instead describe the waves mathematically. A model of the forces on the ship’s hull was developed with classical analytical theory. With the waves and ship hull forces, the motion of the ship’s landing platform can be calculated. If we assume that the helicopter landing does not influence the landing platform motion, the system is simplified. We speed up the simulation by storing the motion in a database for the different wave heights, lengths, and directions, and the ship’s speed. Typically the database will include wave heights of 1, 2, 3, and 4 m; wave directions 0, 30, 60, 90, 120, 150, and 180 degrees; wave lengths 100, 150, and 200 m; and ship speeds of 5 and 10 knots. The helicopter was modeled with the MultiBody library. It includes mechanical parts such as rotors with gyroscopic effects and landing gear with hydraulic dampers. Friction models for wheel-deck interface and flexible beams for the rotor blades have been developed. We have also developed a simple autopilot where the landing algorithm is implemented and tested. For one application, the model has been run with the actual autopilot as hardware in the loop.

Sea waves

A sea state is the condition of the free surface of water with respect to wind, waves, and swell. Statistics, including the wave height, period, and power spectrum, characterize a sea state. The sea state varies with time, as the wind conditions or swell conditions change. The sea state can either be assessed by an experienced observer, such as a trained mariner, or through instruments like weather buoys, wave radars, or remote sensing satellites.

A commonly used scale for the sea state is called the Douglas Sea Scale, where 0 is calm (glassy), 5 is rough (approximately 2.5- to 4-meter waves), and 9 is phenomenal (over 14-meter waves). A safe helicopter landing can be performed up to, say, sea state 4.

The equation for the linear oceanographic wave model can be written as

Equation for linear oceanographic wave model

where the coefficients are generated from a frequency spectrum—for instance, a modified Pierson–Moskowitz spectrum for a fully developed sea. The Pierson–Moskowitz spectrum is an empirical relationship that defines the distribution of energy with frequency within the ocean. The Pierson–Moskowitz spectrum is one of the simplest descriptions for the energy distribution. It assumes that if the wind blows steadily for a long time over a large area, then the waves will eventually reach a point of equilibrium with the wind, known as a fully developed sea. Pierson and Moskowitz developed their spectrum from measurements in the North Atlantic during 1964 and found this relationship between energy distribution and wind.

Ship motion

The ship is continually disturbed by waves, wind, ocean currents, and propulsion loads, leading to heave, pitch, sway, and yaw motions of the ship, and accordingly, motions on the landing platform mounted on the ship’s deck. In this work the motions due to waves will be taken into consideration. The ship is assumed to cruise at a steady speed. For a helicopter landing, the ship typically cruises at 10 knots with headwind and, as much as possible, head waves during the landing. The ship’s hull geometry is modeled in Mathematica below:

Ship's hull geometry modeled in Mathematica

To determine the motion, you must first describe the geometrical and mass properties of the ship and its hull. Then derive the corresponding hydrodynamic forces during operation. Generally, you can solve for these forces by integrating Bernoulli’s equation over the wetted surface. (It is almost impossible to solve for a ship moving in wind-generated waves, especially when included in a dynamic analysis of the helicopter.) Instead, a well-known and widely used linear solution has been applied. If we assume that the squared particle velocities are small, the problem can be subdivided into individually solvable subproblems:

F = F radiation+Frestoring+FFroude-Krylov+Fdiffraction

The forces are then calculated for a finite number of ship cross-sections and integrated across the hull as described below.

Radiation force

The radiation forces are the forces acting on an oscillating ship on the surface of a fluid. Cross-section shapes are mapped to cylinders using Lewis’ conformal mapping on the discretized segments along the ship.

Since it is not possible to solve the hydrodynamic equations of an arbitrary ship hull, the shape must be transformed into something in which analytical solutions exist. Lewis did this already in 1929, transforming a ship’s hull into convenient circular sections:

Transforming a ship's hull into convenient circular sections

In about 1949, Ursell calculated the forces for an oscillating cylinder in free water, which we then use to calculate frequency-dependent hydrodynamic coefficients. In the example below, there is one damping and one added mass. B33 is the damping coefficient [Ns/m] and A33 the mass added [kg] in the heave direction expressed as a function of frequency [rad/s]:

Calculating frequency-dependent hydrodynamic coefficients

The added mass and damping approach is used in most cases when the water is in motion. A simple way of thinking about this is that if you move an object in the water, you also need to move some of the water.

We use Cummins’ theory to transform the damping coefficients to the time domain:

Transforming the damping coefficients to the time domain

So FRadiation is calculated by doing a conformal mapping of the current ship segment, calculating an Ursell number and a damping coefficient, and then transforming it back with the Cummins method.

Archimedes’ principle and the Froude–Krylov force

According to Archimedes’ principle, the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces. If only these forces are included, the system behaves as a floating pontoon. In fact, for drone helicopters, floating pontoons were used for early sea landing tests.

The Froude–Krylov force is the force introduced by the unsteady pressure field generated by undisturbed waves. Together these two forces generate a pressure field as

Written expression of pressure field

where p is pressure, ρ is density, z is the height coordinate, and ϕ is the potential.

The boundary surface of each ship segment is discretized and the pressure and resulting force and torque are calculated at each point.

Diffraction force

The diffraction force is the effect of the floating body disturbing the waves. The forces act on a fixed surface in an oscillating fluid:

Using Cummins' theory to transform the damping coefficients to the time domain

Modelica library implementation

The ship used here has been divided into 16 segments. For each segment, the forces described here have been calculated. Finally, the ship is assembled in SystemModeler as shown below:

Ship assembled in SystemModeler


Several different helicopter model principles are used here. One of these helicopter models, modeled with the MultiBody library, includes such mechanical parts as rotors with gyroscopic effects and landing gears with hydraulic dampers. A friction model for the wheel-deck interaction and stiff beams for the rotor blades is also used.

A simple autopilot gives the lifting force without needing to model the aerodynamics around the blades. This model is used when studying, for instance, touchdown. It describes the actual behavior during this landing well.

For example, the delay from touchdown to “lifting force” cutoff is included. When either the left and right skid or all wheels are in contact with the deck, the lifting force should be eliminated. Otherwise, the helicopter has no down-force and may slide or tilt. There is a delay both in the sensors indicating contact and in the sensor detecting the lifting force.


This is the standard model. When studying landing approach and touchdown algorithms for a helicopter in flight, there is no need to have flexible landing gear or rotating blades. Therefore, a simpler model is used during these calculations. As when studying the aerodynamic behavior caused by flexible rotating blades, the computational time would be much longer if, for instance, the landing gear was included.

Landing aid system

The helicopter can either land manually with some aid or automatically. One of the challenges is to calculate the landing platform motion at the moment of touchdown. It is nearly impossible to predict in advance the exact motion. When a pilot lands the helicopter, the pilot must judge, based on experience, whether a safe landing is possible. In the case of a drone landing automatically without ship-based support or a pilot, a statistical approach needs to be used. Different systems have been tested, but so far none of them are used commercially. The goal is to touch down without exceeding the maximum allowable forces and without the helicopter slipping or toppling before it is secured.

The system developed by Wolfram MathCore uses an energy index called iTol to predict if it is safe to land. It is likely that a low index at a certain time gives low enough motions 5–10 seconds later, which is the time it takes from go-ahead until touchdown.

Several thousand simulated touchdowns were made to optimize the procedure.

Landing when the waves’ amplitude is approximately 2 meters, i.e. in sea state 4, may take up to 20 minutes because of the wait to find a moment when the touchdown spot is calm enough.

The iTOL energy index is described below. The coefficients ai are estimated dynamically based on the current sea state, ship orientation, and ship speed during the touchdown:

Written expression of iTOL energy index

where y, z, ϕ, and θ are sway, heave, roll, and pitch, respectively.

Where y, z, ϕ, and θ are sway, heave, roll, and pitch

Examples of limits

Limits may vary between applications. But in reality, they are very similar for helicopters of all sizes.

Red (iTOL >1)

Max sway velocity > 0.2 m/s

Max heave velocity > 2.5 m/s

Max pitch angle > 10 deg

Max roll angle > 10 deg

Green (iTOL ≲ 0.2)

Max sway acceleration < 0.75 m/s^2

Max heave acceleration < 0.75 m/s^2

Orange (iTOL ≳ 0.5)

Max heave velocity > 1.5 m/s

Yellow (0.2 ≲ iTOL ≲ 0.5)

Landing and abort criteria

Traffic signal

When the helicopter is hovering a few meters above the landing platform, the iTol shows green. If the sea state subsequently changes to yellow, it should be safe for the helicopter to continue landing because, most likely, the sea state will not change fast enough to cause a problem. However, if it changes to orange, the situation is evaluated to see if the helicopter can rise fast enough to avoid a hard landing. If it changes to red, the landing is aborted. Most likely, it will not be successful and the helicopter will still land, but the force impact on the landing gear will be reduced.

Green ⇒ Start landing.
Yellow ⇒ Continue landing if already started.
Orange ⇒ Abort landing if possible.
Red ⇒ Abort landing.

Landing procedure

Helicopter landing pad

The procedure for landing a helicopter at sea is complex, because of both safety reasons and the need for abort and backup systems in case of component failures. Typical examples of failures are for a drone to lose the GPS signal or the development of communication problems between ground and helicopter. If we do not include abort points in the procedure, a landing may look like this: after the mission is complete, the helicopter flies to landing standby point, say 200–300 meters behind the ship. The ship is made ready for landing and ground personnel take position. The ship’s captain gives the go-ahead for landing. The helicopter flies to the touchdown standby point, approximately 10–20 meters behind and beside the ship. The flight officer gives the go-ahead for touchdown. A piloted helicopter will wait for a moment of low landing platform motion and attempt to touch down, most likely from beside the ship, while an unmanned drone helicopter will hover several meters above the touchdown point and wait for a prediction of low platform motion. The last part, the touchdown, is schematically described below with examples of positions to make it clearer. This will be shown in one of the videos below:

1.Standby position: {Δx,Δy,\Δz} = {-15, -15, 15} m

2.OK to land from flight officer ⇒ {0, 0, 6} m

3.OK from iTOL ⇒ {0, 0, 0} m

    3.1. Abort ⇒ {0, 0, 6} m

    3.2. Continue landing

Landing diagram


The model is used to understand the behavior during the approach, hover, and landing and touchdown but also during the on-deck securing, handling, and lashing of the helicopter in rough weather. For example, the model allows for checking landing requirements prior to sea-trail. Such requirements can be landing forces and accelerations. The model also allows one to check wheel slippage and the risk of the helicopter toppling during bad weather.

The model allows for the calculation of peak securing forces, operational envelopes, deck clearances, and fatigue spectra for secured aircraft. It also allows for calculation of lashing forces and it establishes under what weather conditions the helicopter can be lashed or needs to be stowed away.

A major task after the sea-trail is to establish what is called the Ships Helicopter Operating Limit (SHOL). With an accurate simulation model, this work can be sped up to reduce the required testing and shorten the test period. The test period can be up to one year before the complete envelope has been tested. During this period, the flight envelope is restricted.

The model can also be used in design to optimize the landing gear and procedures. The behavior and transmitted forces from the touchdown are highly correlated to the landing gear design.

Work is ongoing to include the real dynamic blade forces. So far, hovering can be included. In contrast to many other applications, flexible beams can be used for the blades, which yields a more accurate modeling of the lifting force.

Operational limits and overhaul requirements can be calculated. For instance, methods have been developed for including the response after depth charge shocks.

Many helicopters, both land- and sea-based, have a camera mounted underneath. A method for reducing vibrations has been developed for use by the film industry.

In the simulation below, one can see a landing without lifting force (a drop test), a shock wave, or a tilting of the deck leading to sliding, and finally contact with an obstacle.

Example 1

Video 1 shows the landing algorithm for a drone. The helicopter is flying to the touchdown standby point, and waits there till approximately 30 seconds. Then it proceeds to a hovering position and waits there until the iTol is green before landing. To shorten the video, the in flight from the standby point is not shown. Touchdown is after 72 seconds.

Example 2

The second example is an Agusta 109 helicopter. In video 2, it is dropped from 0.5 m, a standard design requirement. After 5 seconds, a depth bomb is exploded, putting the ship in motion; finally, the ship starts to roll until sliding occurs. An obstacle is present on the landing platform. A side force of a helicopter during landing is bad, especially when the lifting force is still present. The system may easily become unstable and a lot is done to avoid this. Note how the gyroscopic forces tilt the helicopter unexpectedly.

Video 3 is the same as video 2 except that it is zoomed in on the helicopter.

Download this post as a Computable Document Format (CDF) file.

Posted in: SystemModeler
Leave a Comment


Bill Schacht

Under the heading “Archimedes’ principle and the Froude–Krylov force” it seems that the generated pressure field “equals” the radiation force. Is this correct? This is definitely an excellent paper!

Posted by Bill Schacht    December 27, 2015 at 11:26 am
Håkan Wettergren

Bill, thanks for the positive feedback and also thanks for finding the “copy and paste error”. The equations are now updated!

Posted by Håkan Wettergren    January 5, 2016 at 9:52 am
Jerry Vargas

Awesome article, the videos were super cool to watch.

Thanks and keep up the good work!


Posted by Jerry Vargas    January 23, 2016 at 5:53 pm


I couldn’t get this modelling at my end. May I request if you could share your modelica file?


Posted by vijay    May 12, 2016 at 3:55 pm
    Håkan Wettergren

    “Hi Vijay, unfortunately we have not made this model publicly available yet, but if you contact us at mathcore@wolfram.com we can see what we can do to help you.”

    Posted by Håkan Wettergren    June 3, 2016 at 2:30 pm

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