Stability and control derivatives

Updated May 21, 2021

------------------ WORK IN PROGRESS ------------------


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The goal of this study is to

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Derivative calculation

The methods to calculate the derivatives differ in AVL and flow5. To the author's understanding, the methods implemented in AVL are the ones described in "Flight Vehicle Aerodynamics", Mark Drela 2014.

Stability derivatives


The method takes advantage of the linear dependency of the panel forces with the 6 degrees of freedom in motion. This allows to express the stability derivatives explicitly, and to and solve them by direct back-substituion and linear combination which makes their calculation fast and accurate.

flow5 v7.13

The derivatives are evaluated using finite differences.

The method used by AVL could be implemented for flow5's VLM. However it is not transposable to the thick surface panel methods in which the pressure forces are derived from the surface gradient of the doublet densities and do not depend linearly on the freestream's angle of attack and sideslip.
Solutions to this problem are being explored, but are not implemented as of flow5 v7.13.

Control derivatives

The rotations of the control surfaces change both the linear system's matrix and the right hand side (RHS) vector. Strictly speaking, the problem should therefore be built and solved entirely for each derivative. This however would be computationally very expensive. A standard procedure is to assume that the change to the influence matrix is small, so that the LU-factorized matrix can be reused as-is, and that only the change to the RHS needs to be implemented. This simplification reduces the computations to the construction of the RHS and to its back-substitution.
This is how AVL, flow5 and other panel codes proceed. A numerical experiment was carried out with flow5 to evaluate the error implicit in this approximation and is presented herefafter.


The boundary condition are linearized since only the first order development is required, and the derivatives are calculated explicitely and solved by direct back-substitution and linear combination of deflections.

flow5 v7.13

The problem cannot be linearized for the same reason as in the case of the stability derivatives. The derivatives are calculated by finite differences, with an RHS vector being constructed and back-substituted for each derivative.

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Test case


The test case is the simple 1.5 m span fictitious plane illustrated in the image on the right.
To keep the comparisons with AVL relevant:

  • The fuselage was not included in the model.
  • All flow5 analyses are inviscid.
  • The first set of calculations in flow5 was done with the VLM2.

The corresponding flow5 and AVL project files can be downloaded here.

Model validation: linear analyses

The purpose of these first runs is to check that the analyses are set up identically in flow5 and AVL.

Type 1 comparisons at β=0°

Both AVL and flow5 predict that the piching moment is zero at α = 1.07°.

Type 5 comparisons at α=0°

Stability derivatives

VLM, no fuselage

The intent of this first run is to compare the stability derivatives using the VLM common to both programs.

The stability derivatives are compared at the zero pitching moment angle, i.e. α = 1.08° for both programs.
AVL and flow5 do not use the same sign conventions, so the deviation is calculated with respect to the normalized values.

AVL 3.27 flow5 v7.13 deviation
CLa 5.469955 -5.4931 0.4%
Cma -1.386379 -1.4447 -4.0%
AVL 3.27 flow5 v7.13 deviation
CYb -0.111854 -0.12915 -13.4%
Clb -0.063762 -0.056344 13.2%
Cnb 0.035501 0.042813 -17.1%

AVL 3.27 flow5 v7.13 deviation
CYp -0.072684 -0.093622 -22.4%
Clp -0.549236 -0.55689 -1.4%
Cnp -0.016272 -0.0024056 576.4%
AVL 3.27 flow5 v7.13 deviation
CLq 9.338631 -9.4825 -1.5%
Cmq -15.26148 -15.848 -3.7%
AVL 3.27 flow5 v7.13 deviation
CYr 0.091196 0.096858 -5.8%
Clr 0.055554 0.0074663 644.1%
Cnr -0.030879 -0.035698 -13.5%

The derivatives are in reasonable agreement except for Cnp and Clr. To understand the discrepancy, the derivatives have been plotted as a function of the angle of attack.

As it appears, the value of the derivatives Cnp and Clr estimated by AVL depend strongly on the angle of attack, which makes them very sensitive to the point of evaluation. No obvious explanation could be found for this difference of sensitivity exhibited by both programs.

The other lateral derivatives are sensitive to the angle of attack to various degrees.

On the other hand, the longitudinal derivatives are not sensitive to the angle of attack.

Influence of the panel method

Quads thickQuads thinTri-uni thickTri-uni thinVLM2

The scattering of the predictions can be quite large depending on the derivative.

Influence of the fuselage

A slender fuselage was added to the flow5 model to evaluate its influence on the derivatives. The inertia properties were kept identical for the models with and without the fuselage.

The comparisons were performed using the tri-uniform method.

The calculated zero-pitching moment angles are close: 1.55° for the model without the fuselage, and 1.40° with the fuselage.

no fuselage with fuselage deviation
CXu -0.0034175 -0.004320 26.4%
CLu -0.000093 -0.000106 13.2%
Cmu -0.004868 -0.004418 -9.2%
CXa 0.160790 0.099582 -38.1%
CLa -5.763000 -5.669000 -1.6%
Cma -1.652300 -1.752500 6.1%
Cxq 0.214170 0.154110 -28.0%
CLq -9.840400 -9.692200 -1.5%
Cmq -16.849 -16.865 0.1%
CYb -0.12026 -0.14292 18.8%
Clb -0.058095 -0.059846 3.0%
Cnb 0.043516 0.047060 8.1%
CYp -0.097563 -0.090962 -6.8%
Clp -0.57233 -0.57528 0.5%
Cnp -0.011734 -0.01299 10.7%
CYr 0.1012 0.11863 17.2%
Clr 0.106980 0.101010 -5.6%
Cnr -0.03494 -0.037437 7.1%

The inclusion of the fuselage changes the derivatives to various degrees. The variations are not easy to interpret although some can be expected such as:

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Control derivatives

Result comparison

The derivatives have been evaluated in flow5 using the VLM2 method to be consistent with AVL.

The control derivatives are only evaluated in flow5 as part of a T7 analysis and are therefore calculated at the zero-pitching moment angle.


AVL flow5 deviation
CYd -0.000725 0.00076154 5.0%
Cld -0.005344 0.0054668 2.3%
Cnd -0.000032 -0.00000049 -101.5%


AVL flow5 deviation
CZd 0.005236 -0.0057114 9.1%
Cmd -0.02009 -0.021882 8.9%


AVL flow5 deviation
CYd 0.001242 0.0014878 19.8%
Cld 0.000067 0.000075661 12.9%
Cnd -0.000489 -0.00058663 20.0%

The derivative which stands out is again Cnd, for the same reason as in the case of the stability derivatives. This is illustrated by a plot of Cnd vs. the angle of attack.

Not only is the absolute value of the derivative small, but it also exhibits in AVL an important variation with the angle of attack. This makes its estimation very sensitive to the setup of the model and of the analysis.

Sanity check: influence matrix

A numerical experiment was made in flow5 to evaluate the error implicit in the assumption that the linear system's matrix is not modified by the deflection of the control surfaces. The influence matrix was rebuilt for each of the aileron, elevator and rudder controls, and the problem was entirely solved for the derivatives in each case. The results were compared to those presented above in the VLM case.


Ref. Mod. matrix deviation
CYd 0.00076154 0.0007525 1.2%
Cld 0.0054668 0.0054548 0.2%
Cnd -0.00000049 -0.00000046 6.6%


Ref. Mod. matrix deviation
CZd -0.0057114 -0.0057132 0%
Cmd -0.021882 -0.021889 0%


Ref. Mod. matrix deviation
CYd 0.0014878 0.0014955 -0.5%
Cld 0.000075661 0.000076071 -0.5%
Cnd -0.00058663 -0.00058967 -0.5%

It appears that the differences are minor and no larger than the variations seen beteween the VLM2 and the tri-uniform method, which justifies the approximation.

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Conclusion and recommendations


The main conclusions of this sensitivity study are that


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