# A Novel Control Allocation Method for Yaw Control of Tailless Aircraft

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Method

#### 3.1. Wing Aerodynamic Models

#### 3.2. Validation of Model Predictions Due to Control Surface Deflection

#### 3.3. Control Allocation

**R**(i.e., solutions of $\delta $ which satisfy $\mathbf{R}\delta =0$). The number of null space vectors (unique solutions) is determined by the degree to which matrix $\mathbf{R}$ is rank deficient. Each of the null space vectors can be thought of as analogous to a mode shape (i.e., Eigen vector of the system). Any linear combination of these modes will ensure that the constraints of lift, pitching moment and rolling moment are satisfied but will produce an asymmetric load distribution. As an example, the modes of an untapered straight wing with three control surfaces on each semi-span are shown in Figure 4.

## 4. Results and Discussion

#### 4.1. Wing Geometries

#### 4.2. Unswept Wing

#### 4.3. Thirty Degree Sweep

#### 4.4. Forty-Five Degree Sweep

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RCS | Radar Cross Section |

TEU | Trailing Edge Up |

TED | Trailing Edge Down |

SDR | Split Drag Rudder |

## Appendix A. Model Selection

#### Appendix A.1. Overview

#### Appendix A.2. Straight Rectangular Wing

**Figure A1.**(

**a**) Lift distribution prediction at $\alpha ={4.2}^{\circ}$ and (

**b**) lift curve slope for AR5 straight wing. All methods considered predict the shape of the lift distribution well, however both lifting surface and VLM predict a slightly lower lift curve slope.

**Figure A2.**Induced drag coefficient as a function of angle of attack (

**a**) and lift coefficient (

**b**). The results of the modified LLT match those from the classical LLT, as would be expected. Whilst the induced drag predictions for the LS and VLM appear to under predict the induced drag as a function of angle of attack, it can be seen that this is due to the lower lift prediction shown in Figure A1. As a function of lift coefficient, the induced drag coefficient prediction from the LS model agrees well with the LLT prediction, whilst VLM slightly over predicts the induced drag coefficient.

#### Appendix A.3. Forty-Five Degree Swept, Untapered Wing

**Figure A3.**(

**a**) Spanwise lift distribution at 4.2 degrees and (

**b**) lift curve slope for the low order methods considered. All methods predict the lift curve slope well enough, however the modified lifting line model skews the lift to the outboard sections of the wing. For our analysis purposes, this would lead to large errors in the predicted induced drag and aerodynamic moments.

**Figure A4.**Drag prediction comparison. Compared to the experimental data, the modified LLT predicts higher induced drag than the total drag measured in the experiment. The LS and VLM models appear to under predict the total drag compared to the experiment. This is expected as they only evaluate the induced drag component. As such, when adding the profile drag contribution of the RAE101 aerofoil used for the experiment, the predicted drag coefficient is in good agreement for the LS model, and is slightly over predicted for the VLM model.

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**Figure 1.**Two different load distributions that produce the same lift and no rolling moment but different levels of induced drag. Solid line, symmetric elliptical loading; dashed line, equivalent distribution tailored to produce a net yawing moment through laterally asymmetric induced drag.

**Figure 2.**Comparison of predicted yawing moment against experimental results for opposing deflections of dual surfaces on the starboard wing. Positive deflection is outer control surface TEU and inner control surface TED (see inset). For positive deflection, the sideslip trim closely follows the lower bound on the uncertainty from the experiment. However, for negative deflections, the yawing moment is over predicted for defection greater than six degrees ($\delta >{6}^{\circ}$). This is due to the reduction in effectiveness of the outer control surface due to the increasing influence of spanwise flow.

**Figure 4.**Visualisation of null space vectors for an untapered straight wing. Any linear combination of the three modes will ensure that the constraints on lift, pitching moment and rolling moment are satisfied whilst producing a yawing moment. Defections are exaggerated for clarity.

**Figure 5.**Iso-surfaces of constant yawing moment for an untapered straight wing. For each sideslip angle case presented. Left-hand plots show the values of

**k**to attain the given sideslip angle and the centre and right-hand plots show the results of the optimisation functions X and Y, respectively.

**Figure 6.**Deflection required to generate yawing moment to trim at a given sideslip angle, ${\beta}_{trim}$, for a straight wing with objectives of maximising aerodynamic efficiency (

**a**,

**b**) and minimising aggregate control deflection (

**c**,

**d**) for a taper ratio 0.2 (

**a**,

**c**) and 1 (

**b**,

**d**). Dashed lines represent port surfaces and solid lines are for starboard surfaces. Positive deflections are defined as trailing edge down. For visualisation purposes, insets show deflections required to trim at one, five and eight degrees of sideslip.

**Figure 7.**Relative contribution of the induced drag to the total drag (

**a**) and the yawing moment (

**b**) for unswept wing. In all cases, the induced drag is responsible for over 99% of the yawing moment produced by this method. This is despite the total drag comprising 58% profile drag.

**Figure 8.**Optimisation functions aerodynamic efficiency (

**a**) and aggregate control deflection (

**b**) objectives for an unswept wing normalised by the same metrics for a SDR. Variations are shown up to a given sideslip trim angle of 10 degrees.

**Figure 9.**Deflection required to generate yawing moment to trim at sideslip for a 30 degree swept wing with objectives of maximising aerodynamic efficiency (

**a**,

**b**) and minimising aggregate control deflection (

**c**,

**d**) for a taper ratio 0.2 (

**a**,

**c**) and 1 (

**b**,

**d**). Dotted lines indicate port surfaces and solid lines are starboard surfaces, positive deflections trailing edge down. Inset shows deflections required to trim at one, five and eight degrees of sideslip.

**Figure 10.**Relative contribution of the induced drag to the total drag (

**a**) and the yawing moment (

**b**) for the 30 degree swept wing. In all cases, the induced drag is responsible for over 99% of the yawing moment produced by this method and induced drag is responsible for approximately 42% of the total drag.

**Figure 11.**Optimisation functions aerodynamic efficiency (

**a**) and aggregate control deflection (

**b**) objectives for a 30 degree swept wing normalised by the same metrics for a SDR. Variations are shown up to a given sideslip trim angle of 10 degrees.

**Figure 12.**Deflection required to generate yawing moment to trim at sideslip for a 45 degree swept wing for objectives of maximising effectiveness (

**a**,

**b**) and minimising aggregate control deflection (

**c**,

**d**) with taper ratios 0.2 (

**a**,

**c**) and 1 (

**b**,

**d**). Dotted lines indicate port surfaces and solid lines are starboard surfaces, positive deflections trailing edge down. Inset shows deflections required to trim at one, five and eight degrees of sideslip.

**Figure 13.**Relative contribution of the induced drag to the total drag (

**a**) and the yawing moment (

**b**) for the 45 degree swept wing. In all cases, the induced drag is responsible for over 99% of the yawing moment produced by this method. This is despite the total drag comprising over 50% profile drag.

**Figure 14.**Optimisation functions aerodynamic efficiency (

**a**) and aggregate control deflection (

**b**) objectives for a 45 degree swept wing normalised by the same metrics for a SDR. Variations are shown up to a given sideslip trim angle of 10 degrees.

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**MDPI and ACS Style**

Shearwood, T.R.; Nabawy, M.R.A.; Crowther, W.J.; Warsop, C.
A Novel Control Allocation Method for Yaw Control of Tailless Aircraft. *Aerospace* **2020**, *7*, 150.
https://doi.org/10.3390/aerospace7100150

**AMA Style**

Shearwood TR, Nabawy MRA, Crowther WJ, Warsop C.
A Novel Control Allocation Method for Yaw Control of Tailless Aircraft. *Aerospace*. 2020; 7(10):150.
https://doi.org/10.3390/aerospace7100150

**Chicago/Turabian Style**

Shearwood, Thomas R., Mostafa R. A. Nabawy, William J. Crowther, and Clyde Warsop.
2020. "A Novel Control Allocation Method for Yaw Control of Tailless Aircraft" *Aerospace* 7, no. 10: 150.
https://doi.org/10.3390/aerospace7100150