[american institute of aeronautics and astronautics aiaa modeling and simulation technologies...

A Highly Robust Trim Procedure for Rotorcraft

Simulations

Chen Friedman∗ Omri Rand †

Faculty of Aerospace Engineering,

Technion - Israel Institute of Technology, Haifa, 32000, Israel

A new method for trimming generic rotorcraft models which is suitable for time domain

models is presented. The method is related only to the classical six degrees of freedom and

is independent of the number of states in the model, or its level of accuracy. The method is

shown to be highly robust, fast, accurate, and capable of obtaining trim solutions even from

rather poor initial guesses and for unstable configurations. An extension to this method is

also presented allowing an additional trim state which may be utilized for determining the

required descent velocity for autorotation.

∗Research Engineer†Professor

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American Institute of Aeronautics and Astronautics

AIAA Modeling and Simulation Technologies Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6361

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Nomenclature

U∞ Rotorcraft forward velocity.

MRChord Main rotor airfoil chord.

ψ Blade # 1 azimuth angle.

u, v, w Velocity in the x, y, and z directions respectively.

p, q, r Angular rate about the x, y, z axes respectively.

(), () First and second time derivatives, respectively.

ΩMR Main rotor angular rate.

θ0, θ1c, θ1s Collective, lateral cyclic, and longitudinal cyclic commands, respectively.

θTR0 Tail rotor collective angle command.

θF Fuselage pitch angle (down pitch - positive).

φF Fuselage rolling angle (right roll - positive).

∆t Simulation time step.

ǫ Convergence criterion.

I. Introduction

Trimming a detailed rotorcraft model is still a complex challenge. Trim is the basic condition for measur-

ing the efficiency of any new design and is a fundamental mode of any evaluation. Rotorcraft simulators also

require a trimmed rotorcraft as an initial condition for the simulation start time. For given flight conditions,

a typical trim procedure should supply the initial pilot commands and fuselage angles, as well as the all the

other rotorcraft states (flapping angles, downwash etc.).

Several methods are being used today for the formulation and solution of a rotorcraft trim problem. One

very common method is the ”harmonic-balance” technique. In this method, each degree of freedom of the

rotorcraft is considered to be a sum of harmonics of the main rotor angular frequency. The method exploits

the periodicity of all states for the solution of the trim commands.1 Note that the number of degrees of

freedom involved is not the basic six degrees of freedom of rigid body motion. It may also include flapping

angles of each blade, elastic states, and any other rotorcraft periodical property that has an influence on the

forces and moments. This results in a large coupled set of algebraic equations that is usually solved in an

iterative fashion. Some advances within this approach have lead to a significant decrease in computational

resources by exploiting some periodicity characteristics of the rotorcraft.2 However, the number of states

remains large (typically several dozens).

In some cases, it is possible to write a set of differential equations, and design an appropriate set of

controllers that can bring those equations to the required trim state.3 However, considerable thought must

be given for the design and use of the controller. In the work by Peters,3 the converged solutions are still

oscillating, the trim solution depends on the states that are measured, and the convergence with respect to

the number of iterations is relatively slow.

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Enns4 shows the possibility of solving the trim states using neural networks, however this technique is

fairly new and requires substantial knowledge in neural network design and operation. Peters, Bayly, and

Sihao5 used a periodic shooting method combined with an autopilot that searches for the trim solution. The

autopilot is expected to assist the periodic solution in cases of large systems. A similar way to obtain trim

with an ”autopilot” is designing several control loops for all six rotorcraft degrees of freedom. This option

is used also for fixed wing aircraft, but it requires the knowledge (or definition) of several transfer functions

between all inputs and outputs of the system, and it also requires the careful design of the control loop with

the appropriate gains and controllers. A more detailed review of trim methods is also available in the work

by Peters and Barwey.6 This work also introduces a well established and complete definition of rotorcraft

trim.

This work presents a new and highly robust trim procedure that is suitable for use in any time-based

rotorcraft simulation. The procedure was already implemented in two rotorcraft simulators for conventional

and non-conventional configurations. The results from this trim procedure are used to calculate the properties

and performance of a helicopter at various flight conditions, including forward flight, climb, descent, and

autorotation. In this work, results for a typical conventional tail helicopter (similar to the Bell 206) are

presented.

II. The Rotorcraft Model

To demonstrate the present trim approach, a rotorcraft model written in Matlab/Simulink (using time

integration) has been developed. The model is based on a blade-element formulation for the main rotor and

individual modeling for each of the helicopter’s components that generates forces and moments about the

center of gravity. The model is capable of simulating both identical and non-identical blades (with respect

to inertial and aerodynamic characteristics, mainly for purposes of vibration analysis). Euler angles are used

throughout the model for coordinate transformations and calculation of the velocity that each helicopter

component encounters at each time step. Each component is defined with its own cartesian coordinate

system.

For the downwash model, a simple uniform inflow calculation was implemented that takes into account the

inflow due to forward flight.7 Although this is a fairly simple downwash model, the trim method formulation

will remain the same for other, more complex models. Tabulated aerodynamic data is used for the evaluation

of the lift, drag, and pitching moment, according to the local Mach number and angle of attack at each blade

section at each time step.

All forces and moments generated by the different rotorcraft blocks are summed-up and inserted into a

block that performs the time integration of the 6 degrees of freedom equations. This block allows for the

implementation of the trim procedure (Section III) which, as will be shown later on, requires zero rotorcraft

linear/angular accelerations.

The model’s computational time depends on the time step (which is constant throughout the simulation),

the time integration method, the number of rotor blades (in cases of non-identical blades), and the time span

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of the simulation. Specifically, the time step for the simulation must be properly chosen according to the

angular velocity of the rotor. The azimuthal angle should be kept in the order of 10 between two consecutive

time steps in order to obtain smooth results of all the rotorcraft properties with respect to time.

For the example helicopter presented in this paper (a typical Bell 206), a two bladed rotor using a time

step of ∆t = 0.005 and 3rd order Runge-Kutta integration method results in a fast simulation which is

capable of real time simulation on an ordinary PC.

III. Trim Procedure

When trimming fixed wing aircraft, the reaction of the various components (wing, fuselage, tail, and

other surfaces) is constant with time. Once the aircraft attitude and velocity are known, the forces that are

generated by each component are fixed in time and can be rapidly evaluated, usually through the knowledge

of several transfer function (which are actually the aircraft modeling scheme). On the other hand, rotary

wing aircraft have an inherent time dependent forces and moments (and therefore acceleration and angular

acceleration).

Since the forces that are the result of pilot commands on a rotorcraft are oscillatory, we consider the

average over the rotor’s last several cycles as the average value of any oscillatory signal from the model. The

number of cycles that are used to calculate this average is a user input; however, it was found that for most

cases, two rotor cycles are enough for a satisfactory representative average result.

An innovative trim procedure was implemented that makes use of the model’s own output and therefore

takes into account any non-linearity of the rotorcraft model. The time histories of the rotorcraft six degrees

of freedom are brought to zero (in the average sense) linear/angular accelerations by six controllers: four

pilot commands and two fuselage angles (typically pitch and roll, assuming a constant, given side-slip angles).

This is the same as saying that the helicopter is at equilibrium in all 6 degrees of freedom.

The trim procedure begins with a set of initial guesses for the pilot commands (collective, two cyclic

commands, and pedals), and two fuselage angles (in this work: fuselage pitch and roll). Typically, all initial

guesses are chosen as zero except for some collective to produce non-zero downwash at first.

III.A. ”Hold” Mode

When discussing rotorcraft, steady state means that the average values are fixed, but the values with respect

to time can be oscillating about the average value (e.g. rotorcraft vibrations). This is a major difference

that is inherent in rotorcraft compared to fixed wing aircraft.

For each stage of the trim method the model is run until all the accelerations u, v, w and angular rates

p, q, r are either constant or oscillating around the same mean value (past the initial response of the complete

rotorcraft model, flapping motion, and especially downwash dynamics are well developed). However, in order

to achieve the pure response of the rotorcraft, without the effect of the transients that develop before the

steady state, both the linear and the angular accelerations are not integrated over time. This is one of the

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key features of the trim method. The accelerations are calculated but only to get the values at the end

simulation time span (which should be enough for the model to reach constant oscillations). Since they are

not integrated, properties like velocity, fuselage angles, and position do not evolve over time, and in fact

remain the same as their initial guess or required input (e.g. rotorcraft forward velocity).

After obtaining the results, each controller is slightly perturbed and the simulation is run again (using

the same method of ”holding” the rotorcraft in place). Using the results of each perturbed set of commands

(denoted by an uppercase ” + ” sign), a [6X6] matrix of linear derivatives of each acceleration component

with respect to each pilot command is derived, shown in Eq. (1). The resultant coupled system of equations

is solved, using the negative sign of the current accelerations as the right hand side of the system, to yield a

set of adjustment values to the pilot commands. The new pilot commands are then used as a starting point

for the next iteration.

This procedure actually linearizes the rotorcraft responses about the current set of pilot commands,

under the initial conditions set by the user. The procedure is iterated until convergence is achieved or the

maximum number of iterations is reached (both are defined by the user, typically no more than 10).

−

uF

vF

wF

pF

qF

rF

=

DuF

θ0· · · · · · · · · · · · DuF

ϕF

DvF

θ0

......

...

......

......

DrF

θ0· · · · · · · · · · · · DrF

ϕF

∆θ0

∆θ1c

∆θ1s

∆θTR0

∆θF

∆ϕF

(1)

Where, for example: DuF

θ0=

u+

F−uF

∆θ0

and ∆θ0 is the perturbation magnitude for the collective command.

Note that the current method formulation computes the entire derivative matrix in each iteration. The

authors are aware of the possibility of using the same derivative matrix for two subsequent iterations or even

more, but this is not attempted in the current work. Note that the method is currently shown using forward

differences for the sensitivity derivatives, but it can be easily formulated using central differences or second

order forward differences.

The convergence criterion is defined as:√

u2 + v2 + w2 + p2 + q2 + r2 < ǫ, where ǫ is a certain user

defined criterion. Note that the linear acceleration are measured in [m/s2] and the angular accelerations are

measured in [rad/s2], but since the method is highly robust, the convergence is defined on the sum of all

values, regardless of the different units, so that if convergence is achieved - no value of any acceleration can

be above ǫ. Typically ǫ = 0.001 would suffice for both linear and angular accelerations. This means that

u2, v2, w2 < 0.001 and p2, q2, r2 < 0.001[rad/s2]. Since a rotorcraft states are usually oscillatory by nature,

the definition of the average acceleration is discussed below.

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III.A.1. Decay of Transient Phenomena

In addition to the oscillatory nature of all the rotorcraft output, rotorcraft may have several phenomena that

develop and decay in time. For example, in the present model, we have the dynamics of the downwash and

the flapping motion. The downwash velocity develops due to the rotor accelerating the air downwards to

produce the required lift forces. Since the lift depends on the downwash, any downwash model that would

be used in the model would have to be run over a certain period of time in order to bring the lift and the

downwash velocity to equilibrium.

For the examples presented in this model, the downwash at each time step is calculated based on the

thrust value from the previous time step. In steady state, the downwash velocity that would be produced

by the rotor would be suitable to the amount of thrust that is generated by the rotor and then the thrust

output would become constant with time. The rotorcraft flapping motion must also be given sufficient time

to reach its steady state oscillations (or constant coning angle). Note that the flapping motion depends on

the downwash development, but it was found to have considerably shorter time constants compared to the

downwash. This means that the downwash development is the key for acquiring accurate model sates.

Generally it was found that about 10 rotor cycles are quite enough for the downwash and flapping

motion to be fully develop (which affects the forces and moments and subsequently all the linear/angular

accelerations). The trim procedure was used to ensure the validity of the entire model in terms of helicopter

performance by comparing general results for total power required and total thrust to known flight test

experiments (not shown here).

The use of Simulink allows for re-inserting a certain trim solution as input to a time dependent simulation

and examine the stability of the helicopter at that trim conditions (for example a certain forward light

velocity, hover, etc.). Note that it is necessary to allow about one second of simulation time in ”Hold” mode

prior to the time-dependent free-flight simulation. This allows for the proper development of both the inflow

through the rotor disc and the flapping motion of all blades. Several trim solutions have been examined

using this method and the helicopter model was found to be completely trimmed (including at maximum

forward flight conditions).

III.B. Trim States Adjustments’ Limiter

This trim procedure was found to be quite robust for various rotorcraft configurations at various flight

conditions. However since a direct solution of Eq. (1) is used by calculating the inverse matrix, the resulting

adjustments to the trim states are not bounded and can become quite large in some cases. For that reason, a

limiter is introduced that does not allow the trim state adjustments to be larger than several degrees in each

iteration. Generally it was found that limiting the adjustments to about 5 is quite sufficient to guarantee

that the solution would not diverge.

The authors state that this limiter does not harm the capabilities and the robustness of the solver since

the limiting value that is used is in the order of the final solutions of the various states and thus the

solution is, in fact, almost completely free. Moreover, it was found that after about 3 iterations (for most

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cases attempted thus far), the adjustments are smaller than the limiter value anyway, and thus the solution

becomes unlimited.

III.C. Advantages Over Current Methods

The advantages of this trim process over currently used methods such as the harmonic balance technique is

the small number of degrees of freedom: only six variables for all 6 degrees of freedom, while making use

of the model itself for the solution of an equilibrium problem. The problem formulation is relatively simple

and instead of having a large matrix where each force component has several coefficients (as the number of

harmonic frequencies that is considered), the forces are summed up and the equilibrium is formulated on the

entire helicopter model.

When using the method of harmonic balance, each component of the helicopter is considered as a sum of

harmonic signals with the maximum harmonic frequency defined by the user after some careful examination

of the problem at hand (as a function of the rotorcraft parameters, mainly its main rotor’s angular rate). If

we denote Mh as the number of harmonic frequencies to consider in the trim analysis, and N as the number

of signals that need to be balanced, then the amount of variables in the harmonic balance method adds up

to: N(2Mh + 1) since there are to coefficients (one for sine and one for cosine signals) and one constant

(mean value) for each signal.

However, in the current trim method, since the model is in time domain, one may acquire time history

for every state of the rotorcraft, but the number of trim states to be resolved remains the same. Harmonic

Furier analysis can then be used to get all the harmonic frequencies in each time signal. This technique was

successfully implemented using the current model for the purpose of track and balance analysis.9

The actual number of rotorcraft states (or in general oscillating properties) is irrelevant to this method,

but it is still available to the user after trim. Not all the properties need to be measured at all in order

to achieve a trimmed rotorcraft. In addition, the current trim method uses the rotorcraft model itself so

once the rotorcraft components are modeled, it makes use of their combined modeling for both trim and

simulation.

Note that although trim results are often used for model validation, the trim method has no relation to

the validity of the rotorcraft model. This means that even if there is a modeling error - a trim solution will

still be found by the method.

III.D. Computational Performance

As stated above, this trim procedure was found to be quite robust and trim solutions can be obtained form

any condition without the need to tune up an initial guess that suits the require flight condition. Moreover,

there is no waist of computational efforts on undesired harmonic frequencies that do not contribute to the

helicopter equilibrium.

The computational time that is required for a single simulation is, in fact seven times the computational

time it takes to simulate one condition, since one simulation is required for determining the current state

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and 6 more simulations for the six perturbations in each one of the trim states. Generally, for the purpose of

the trim procedure brought herein, about 10 rotor cycles should be simulated in order to assure the proper

calculation of the forces and moments and the accelerations in the six degrees of freedom. Therefore for the

example helicopter in this work, a trim iteration takes less than 10 seconds on an ordinary laptop, and a

complete trim solution for a certain velocity is achieved after about 30 seconds.

Naturally when using more advanced physical modeling, the computational increases since the physical

models require more computational time. In this model, an advance flapping solver was implemented, which

requires far more computational resources compared to the more common flapping equation solver. When

this solver was used, the time required to trim each velocity case was increased from half a minute to half

an hour. However the comparison between two trim methods lies in the number of iterations, not the time

it takes to run the model, so comparisons must be made on the basis of similar physical models.

III.E. Method Compatibility

It is important to separate the trim method from the Simulink model. The accuracy of the model has no

relation to the ability of the method to find the accurate trim solution. In some cases, while the model was

still under development, trim solutions were found even in inherently erroneous models. This did not prevent

the validity of the current trim method from reaching a well converged trim solution.

The trim method is therefore compatible with many other physical models of rotorcraft, provided that

the models are in time domain. The authors state that the method was successfully tested with various

models (all built in Simulink) including various tail rotor configurations, coaxial rotorcraft, and an autogyro

model (introducing the thrust as an additional trim state, instead of the tail rotor collective).

IV. Results

The method was found to be impressively robust for all flight conditions, initial guesses, and rotorcraft

settings and requires about 3-5 iterations for achieving a trimmed helicopter. However, in order to test the

robustness, the method was tested for various configurations and initial guesses and conditions. In this work

results for a conventional tail-rotor rotorcraft with parameters similar to those of a Bell-206 helicopter.

IV.A. Trim Results

The results of a trim solution are the 6 trim states. The term states is used since for different rotorcraft

configurations, the trim variables are different. For a regular tail-rotor helicopter, the 6 trim states are

defined as:[θ0, θ1c, θ1s, θTR0 , θF , φF ], however, for example when an autogyro is simulated, the main rotor

angular velocity is determined so that the autogyro would be in equilibrium (rather than the constant value

in most rotorcraft), and hence is replaced with the tail rotor command.

The results for the six trim states of the bell 206 example helicopter are presented in Figure 1. These

results were obtained using a convergence criterion of ǫ = 0.001. This means that no acceleration or angular

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0 10 20 30 40 50 60 70 80 90 100

−5

0

5

10

15C

ontr

ols

[deg

]

Trim − all velocities

θ

0θ1

cθ1

sθ

0TR θ

Fφ

F

0 10 20 30 40 50 60 70 80 90 100

−1

−0.5

0

0.5

1

1.5x 10

−3

Forward Velocity [Kts]

Acc

eler

atio

ns

U V W P Q R

Figure 1. All trim commands for a forward flight velocity sweep

acceleration in any degree of freedom was above this value. The velocity sweep was carried from the case

of hover to a maximum velocity of 50[m/s] which is considered to be the maximum forward velocity of the

helicopter.

IV.B. Solver Robustness Example #1

In this example the trim solver starts from a zero initial guess for all 6 trim states and finds the trim conditions

for a single helicopter flight velocity case. Two examples of the trim method convergence process are shown

in Figures 2 and 3, for hover and for the maximum forward flight velocity of 50[m/s] respectively (maximum

forward flight velocity, taken from Figure 1). These figures show the convergence of the accelerations and

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0 1 2 3 4

−5

0

5

10

Iterations

Con

trol

s [d

eg]

Trim for Vx= 0.00 Vy= 0.00 Vz= 0.00 [m/s]

θ

0θ1

cθ1

sθ

0TR θ

Fφ

F

0 1 2 3 4

−12

−10

−8

−6

−4

−2

0

2

Iterations

Acc

eler

atio

ns

U V W P Q R

Figure 2. Trim solution convergence for hover

the trim states development for that case, and it is easy to see that the solution converges within 4, and

5 iterations for the hover and maximum forward flight conditions respectively. This means that the trim

solution was obtained after less than one minute on a regular 1600[MHz] PC.

It is known to the authors that other simulation obtain the trim solution for relatively high forward flight

velocities by first obtaining the solution for a midrange velocity, and then using that solution as the initial

conditions for a higher velocity case. In some cases, the difference between the velocities in each case must

be quite small, (as in less than 5[m/s]), otherwise the solution does not converge although it’s initial guess

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0 1 2 3

−5

0

5

10

Iterations

Con

trol

s [d

eg]


θ

0θ1

cθ1

sθ

0TR θ

Fφ

F

0 1 2 3

−12

−10

−8

−6

−4

−2

0

Iterations

Acc

eler

atio

ns

U V W P Q R

Figure 3. Trim solution for V= 50[m/s] with zero initial guess for all commands and fuselage angles

was a converged and valid trim solution of a relatively close velocity. This method of using trim solutions

as initial guesses is used also when obtaining a trimmed solution for the case of hover since it is considered

to be a more difficult case compared to a midrange forward flight velocity.

It is easy to see from Figures 2 and 3 that there is no difficulty in obtaining a trim solution for a single

velocity case, whichever it may be. This example was examined for various velocities for many rotorcraft

configurations, including velocities in climb, descent, and combined velocities. The method was found to be

robust at the majority of the cases, although some isolated cases of extreme climb or descent velocities had

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to be given a better initial guess by using a somewhat lower vertical velocity.

0 1 2 3 4

−5

0

5

10

IterationsC

ontr

ols

[deg

]


θ

0θ1

cθ1

sθ

0TR θ

Fφ

F

0 1 2 3 4

−10

−8

−6

−4

−2

0

Iterations

Acc

eler

atio

ns

U V W P Q R

(a) Initial guess switched between the two fuselage angles

0 1 2 3 4

−5

0

5

10

Iterations

Con

trol

s [d

eg]


θ

0θ1

cθ1

sθ

0TR θ

Fφ

F

0 1 2 3 4−12

−10

−8

−6

−4

−2

0

Iterations

Acc

eler

atio

ns

U V W P Q R

(b) Initial guess switched between the two cyclic commands

Figure 4. Solver robustness examples for Vx = 50[m/s]

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IV.C. Solver Robustness Example #2

In this example same maximum forward flight velocity of 50[m/s] is solved but this time with a non-zero

initial guess. The solver is given the results of the pitch angle θ as the input for the initial roll angle φ and

vice versa. Figure 4(a) shows the evolution of the 6 trim states and the accelerations convergence, and it is

easy to see that the solver switches between the two fuselage angles and reaches the same converged solution

as in the previous example (Figure 3).

The same example is employed when switching the solutions of the lateral and longitudinal cyclic com-

mands as the initial inputs for the trim solver. The results for the trim solution evolution are given in

Figure 4(b), and it is easy to see that the solver had no problem of handling this case as well.

IV.D. Unstable Rotorcraft Trim

In the work by Peters and Barwey,6 they claim that a trim method that utilizes time integration until all

transient decay cannot be used to trim unstable systems. However, the current method was found to be

capable of trimming even unstable rotorcraft configuration.

The conventional tail rotor helicopter that was used in the current study was found to be an unstable

rotorcraft: after obtaining trim for hover conditions, the rotorcraft roll, pitch and yaw rates diverged after a

short time (less than 10 seconds) as can be easily seen in Figure 5(a). Note that the rotorcraft is in ”Hold”

mode for two seconds in Figures 5(a) and 5(b) (to allow transients to decay, see Sub-Section III.A and

Sub-Section III.A.1). Therefore the values of all the angular rates are zero during that time.

The rotorcraft model was introduced with a simple closed loop control that was applied on all three

channels p, q, r, using a simple proportional gain controllers on θ1c, θ1s, θTR0 . commands respectively. A

proportional control law is given in Eq. (2) for the roll rate.

θ1c = θcommand1c + kpp (2)

Note that since the rotorcraft is in the so called ”Hold” mode while the trim method is employed, the

angular rates p, q, r are by definition zero (angular accelerations are not integrated over time). This means

that the gain controllers have no effect on the trim solution. Figure 5(b) presents the response of the same

tail rotor configurartion, with the exact same trim solution that was applied in Figure 5(a). The control

gains kp, kq, kr are set to 0.1, 1, 1 respectively. These values were found by trial and error to yield satisfactory

results with minimum control system workload.

It is easy to see that the oscillations that exist in this figure are three orders of magnitude smaller and

are well within the definitions of rotorcraft vibrations. This means that the current trim method was able

to trim an unstable rotorcraft. This capability was reproduced several times by the authors on various

rotorcraft configurations and trim conditions.

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0 2 4 6 8 10 12−160

−140

−120

−100

−80

−60

−40

−20

0

20

t[sec]

p,q,

r [d

eg/s

ec]

(a) Unstable Rotorcraft (without closed loop control)

0 2 4 6 8 10 12−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t[sec]

p,q,

r [d

eg/s

ec]

(b) Stable Rotorcraft (with closed loop control)

Figure 5. Capability of trimming unstable configurations

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IV.E. Autorotation Trim Solver

Autorotation is the situation of a rotorcraft where it does not require any power to maintain a constant

velocity. Instead, the rotorcraft converts potential energy to mechanical energy by stabilizing itself at a

certain descent velocity that causes the main rotor blades to rotate and produce lift that maintains that

same descent velocity. The maneuver can be used to perform emergency landing in case of engine failure.

The method proposed here can also be extended to solve the trim state of a rotorcraft in autorotation

mode. For this purpose a new trim state was introduced: the descent velocity. Thus, the number of trim

states has increased to seven. The overall method is identical except for the derivative matrix which is now

7 by 7, and include the sensitivity derivatives of all the helicopter states with respect to the descent velocity.

For the case of hover, the convergence of the seven trim states requires several more iterations compared

to the more conventional trim (for example: 6 more iterations for the same convergence criterion in hover).

This was expected since the number of trim states has increased, and moreover the rotorcraft enters the

descent velocity which is know to be considerably more non-linear compared to hover or climb. Of course

the results for the trim commands are different, however, the convergence figure is quite similar in general,

hence only the convergence of the descent velocity is presented to present this capability of the trim method.

1 2 3 4 5 6 7 8 9 10

−16

−14

−12

−10

−8

−6

−4

−2

0

Iterations

Des

cent

vel

ocity

[m/s

]

Figure 6. Descent velocity convergence for hover

15 of 18


IV.F. Convergence And Model Non Linearity Evaluation

10−15

10−10

10−5

100

3

4

5

6

7

8

9

Convergence Criterion, ε

Num

ber

of It

erat

ions

HoverV

x=50[m/s]

Figure 7. Trim solution for V= 50[m/s] with the initial guess switched between the two cyclic commands

The Simulink model is non-linear by nature since no linearization assumptions were made during its

development. However, the trim method is, in fact, a linearization of the model about a certain trim velocity

condition. Therefore it is interesting to test the degree of the model’s non-linearity to asses the validity of

linear models compared to the more accurate non-linear ones.

In general, it was found that about 3 or 4 iterations are sufficient for convergence of a trim solutions for

most velocity cases with a convergence criterion of ǫ = 0.001. However, in some rare occasions, the user

would like to get the rotorcraft linear/angular accelerations within tighter limits. Figure 7 shows the number

of iterations it takes to achieve convergence with a series of increasingly stringent convergence criterions.

Note that figures like Figure 7 can be used to testify as for the linearity of the model in the examined trim

velocity. If the number of iterations does not increase significantly with a more stringent convergence criterion

it means that the assumption of linearity about the current trim state is better. A detailed examination of

the trim solution revealed that all of the six trim states exhibited negligible difference in their value with

the application of more stringent convergence criterions.

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V. Conclusions And Future Work

This work presents a new, relatively intuitive and highly robust method for solving the trim problem for

various rotorcraft configurations (although the method can also be used for fixed wing simulators). Results

were presented for a conventional tail rotor helicopter with similar parameters to the Bell 206.

The method was found to be highly robust in many aspects and introduces considerable advantages over

current trim methods such as:

• Easy to plug into existing models (time domain only)

• Makes use of the simulation model itself (no need to formulate separate equations)

• No need to update the trim solver when adding new elements to the model

• Independent of the number of harmonic frequencies considered

• Independent of the modeling accuracy, complexity, or number of states

• Insensitive to the initial guesses

• Low number of iterations

• Low computational cost (solving only for the trim states)

• Capable of trimming unstable rotorcraft configurations

The only limiter that was necessary for the convergence to be assured is a limiter on the amount of

change that is permitted to each trim state. However, the limiter is simple, straight forward, without any

special characteristics for each trim state, and the limitation value is of the same order of magnitude as the

trim state itself.

Since this method makes use of the simulation results themselves, the resulting trim solution can be

directly inserted back into the time-dependent simulation in order to obtain the initial conditions for a time-

based analysis that requires a trimmed rotorcraft. The trim solution is guaranteed to be suitable for the

simulation model.

The method of holding the rotorcraft in place and preventing the accelerations from being integrated

over time was found to be efficient in predicting the pure response of the rotorcraft to each set of trim states

that are injected into the model.

Acknowledgments

The authors would like to thank Dr. Vladimir Khromov for his kind assistance.

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References

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