Aerial Object Following Using Visual Fuzzy Servoing

This article presents a visual servoing system to follow a 3D moving object by a Micro Unmanned Aerial Vehicle (MUAV). The presented control strategy is based only on the visual information given by an adaptive tracking method based on the colour information. A visual fuzzy system has been developed for servoing the camera situated on a rotary wing MAUV, that also considers its own dynamics. This system is focused on continuously following of an aerial moving target object, maintaining it with a fixed safe distance and centred on the image plane. The algorithm is validated on real flights on outdoors scenarios, showing the robustness of the proposed systems against winds perturbations, illumination and weather changes among others. The obtained results indicate that the proposed algorithms is suitable for complex controls task, such object following and pursuit, flying in formation, as well as their use for indoor navigation

Use of calculus of variations to determine the shape of hovering rotors of minimum power and its application to micro air vehicles

In this paper, calculus of variations and combined blade element and momentum theory (BEMT) are used to demonstrate that, in hover, when neither root nor tip losses are considered; the rotor, which minimizes the total power (MPR), generates an induced velocity that varies linearly along the blade span. The angle of attack of every blade element is constant and equal to its optimum value. The traditional ideal twist (ITR) and optimum (OR) rotors are revisited in the context of this variational framework. Two more optimum rotors are obtained considering root and tip losses, the ORL, and the MPRL. A comparison between these five rotors is presented and discussed. The MPR and MPRL present a remarkable saving of power for low values of both thrust coefficient and maximum aerodynamic efficiency. The result obtained can be exploited to improve the aerodynamic behaviour of rotary wing micro air vehicles (MAV). A comparison with experimental results obtained from the literature is presented.

A General Purpose Configurable Navigation Controller for Micro Aerial Multirotor Vehicles

In this paper, we consider the problem of autonomous navigation of multirotor platforms in GPS-denied environments. The focus of this work is on safe navigation based on unperfect odometry measurements, such as on-board optical flow measurements. The multirotor platform is modeled as a flying object with specific kinematic constraints that must be taken into account in order to obtain successful results. A navigation controller is proposed featuring a set of configurable parameters that allow, for instance, to have a configuration setup for fast trajectory following, and another to soften the control laws and make the vehicle navigation more precise and slow whenever necessary. The proposed controller has been successfully implemented in two different multirotor platforms with similar sensoring capabilities showing the openness and tolerance of the approach. This research is focused around the Computer Vision Group's objective of applying multirotor vehicles to civilian service applications. The presented work was implemented to compete in the International Micro Air Vehicle Conference and Flight Competition IMAV 2012, gaining two awards: the Special Award on "Best Automatic Performance - IMAV 2012" and the second overall prize in the participating category "Indoor Flight Dynamics - Rotary Wing MAV". Most of the code related to the present work is available as two open-source projects hosted in GitHub.

Pararotor dynamics: center of mass displacement from the blade plane—analytical approach

The pararotor is a biology-inspired decelerator device based on the autorotation of a rotary wing whose main purpose is to guide a load descent into a certain atmosphere. This paper focuses on a practical approach to the general dynamic stability of a pararotor whose center of mass is displaced from the blade plane. The analytical study departs from the motion equations of pararotor flight, considering the center of mass displacement from the blade plane, studied over a number of simplifying hypotheses that allows determining the most important influences to flight behavior near equilibrium. Two practical indexes are developed to characterize the stability of a pararotor in terms of geometry, inertia, and the aerodynamic characteristics of the device. Based on these two parameters, a stability diagram can be defined upon which stability regions can be identified. It was concluded that the ability to reach stability conditions depends mainly on a limited number of parameters associated with the pararotor configuration: the relationship between moments of inertia, the position of the blades, the planform shape (associated with the blade aerodynamic coefficients and blade area), and the vertical distance between the center of mass and the blade plane. These parameters can be evaluated by computing practical indexes to determine stability behavior.

Applications of Formal Languages to Management of Manned and Unmanned Aircraft

En el futuro, la gestión del tráfico aéreo (ATM, del inglés air traffic management) requerirá un cambio de paradigma, de la gestión principalmente táctica de hoy, a las denominadas operaciones basadas en trayectoria. Un incremento en el nivel de automatización liberará al personal de ATM —controladores, tripulación, etc.— de muchas de las tareas que realizan hoy. Las personas seguirán siendo el elemento central en la gestión del tráfico aéreo del futuro, pero lo serán mediante la gestión y toma de decisiones. Se espera que estas dos mejoras traigan un incremento en la eficiencia de la gestión del tráfico aéreo que permita hacer frente al incremento previsto en la demanda de transporte aéreo. Para aplicar el concepto de operaciones basadas en trayectoria, el usuario del espacio aéreo (la aerolínea, piloto, u operador) y el proveedor del servicio de navegación aérea deben negociar las trayectorias mediante un proceso de toma de decisiones colaborativo. En esta negociación, es necesaria una forma adecuada de compartir dichas trayectorias. Compartir la trayectoria completa requeriría un gran ancho de banda, y la trayectoria compartida podría invalidarse si cambiase la predicción meteorológica. En su lugar, podría compartirse una descripción de la trayectoria independiente de las condiciones meteorológicas, de manera que la trayectoria real se pudiese calcular a partir de dicha descripción. Esta descripción de la trayectoria debería ser fácil de procesar usando un programa de ordenador —ya que parte del proceso de toma de decisiones estará automatizado—, pero también fácil de entender para un operador humano —que será el que supervise el proceso y tome las decisiones oportunas—. Esta tesis presenta una serie de lenguajes formales que pueden usarse para este propósito. Estos lenguajes proporcionan los medios para describir trayectorias de aviones durante todas las fases de vuelo, desde la maniobra de push-back (remolcado hasta la calle de rodaje), hasta la llegada a la terminal del aeropuerto de destino. También permiten describir trayectorias tanto de aeronaves tripuladas como no tripuladas, incluyendo aviones de ala fija y cuadricópteros. Algunos de estos lenguajes están estrechamente relacionados entre sí, y organizados en una jerarquía. Uno de los lenguajes fundamentales de esta jerarquía, llamado aircraft intent description language (AIDL), ya había sido desarrollado con anterioridad a esta tesis. Este lenguaje fue derivado de las ecuaciones del movimiento de los aviones de ala fija, y puede utilizarse para describir sin ambigüedad trayectorias de este tipo de aeronaves. Una variante de este lenguaje, denominada quadrotor AIDL (QR-AIDL), ha sido desarrollada en esta tesis para permitir describir trayectorias de cuadricópteros con el mismo nivel de detalle. Seguidamente, otro lenguaje, denominado intent composite description language (ICDL), se apoya en los dos lenguajes anteriores, ofreciendo más flexibilidad para describir algunas partes de la trayectoria y dejar otras sin especificar. El ICDL se usa para proporcionar descripciones genéricas de maniobras comunes, que después se particularizan y combinan para formar descripciones complejas de un vuelo. Otro lenguaje puede construirse a partir del ICDL, denominado flight intent description language (FIDL). El FIDL especifica requisitos de alto nivel sobre las trayectorias —incluyendo restricciones y objetivos—, pero puede utilizar características del ICDL para proporcionar niveles de detalle arbitrarios en las distintas partes de un vuelo. Tanto el ICDL como el FIDL han sido desarrollados en colaboración con Boeing Research & Technology Europe (BR&TE). También se ha desarrollado un lenguaje para definir misiones en las que interactúan varias aeronaves, el mission intent description language (MIDL). Este lenguaje se basa en el FIDL y mantiene todo su poder expresivo, a la vez que proporciona nuevas semánticas para describir tareas, restricciones y objetivos relacionados con la misión. En ATM, los movimientos de un avión en la superficie de aeropuerto también tienen que ser monitorizados y gestionados. Otro lenguaje formal ha sido diseñado con este propósito, llamado surface movement description language (SMDL). Este lenguaje no pertenece a la jerarquía de lenguajes descrita en el párrafo anterior, y se basa en las clearances (autorizaciones del controlador) utilizadas durante las operaciones en superficie de aeropuerto. También proporciona medios para expresar incertidumbre y posibilidad de cambios en las distintas partes de la trayectoria. Finalmente, esta tesis explora las aplicaciones de estos lenguajes a la predicción de trayectorias y a la planificación de misiones. El concepto de trajectory language processing engine (TLPE) se usa en ambas aplicaciones. Un TLPE es una función de ATM cuya principal entrada y salida se expresan en cualquiera de los lenguajes incluidos en la jerarquía descrita en esta tesis. El proceso de predicción de trayectorias puede definirse como una combinación de TLPEs, cada uno de los cuales realiza una pequeña sub-tarea. Se le ha dado especial importancia a uno de estos TLPEs, que se encarga de generar el perfil horizontal, vertical y de configuración de la trayectoria. En particular, esta tesis presenta un método novedoso para la generación del perfil vertical. El proceso de planificar una misión también se puede ver como un TLPE donde la entrada se expresa en MIDL y la salida consiste en cierto número de trayectorias —una por cada aeronave disponible— descritas utilizando FIDL. Se ha formulado este problema utilizando programación entera mixta. Además, dado que encontrar caminos óptimos entre distintos puntos es un problema fundamental en la planificación de misiones, también se propone un algoritmo de búsqueda de caminos. Este algoritmo permite calcular rápidamente caminos cuasi-óptimos que esquivan todos los obstáculos en un entorno urbano. Los diferentes lenguajes formales definidos en esta tesis pueden utilizarse como una especificación estándar para la difusión de información entre distintos actores de la gestión del tráfico aéreo. En conjunto, estos lenguajes permiten describir trayectorias con el nivel de detalle necesario en cada aplicación, y se pueden utilizar para aumentar el nivel de automatización explotando esta información utilizando sistemas de soporte a la toma de decisiones. La aplicación de estos lenguajes a algunas funciones básicas de estos sistemas, como la predicción de trayectorias, han sido analizadas. ABSTRACT Future air traffic management (ATM) will require a paradigm shift from today’s mainly tactical ATM to trajectory-based operations (TBOs). An increase in the level of automation will also relieve humans —air traffic control officers (ATCOs), flight crew, etc.— from many of the tasks they perform today. Humans will still be central in this future ATM, as decision-makers and managers. These two improvements (TBOs and increased automation) are expected to provide the increase in ATM performance that will allow coping with the expected increase in air transport demand. Under TBOs, trajectories are negotiated between the airspace user (an airline, pilot, or operator) and the air navigation service provider (ANSP) using a collaborative decision making (CDM) process. A suitable method for sharing aircraft trajectories is necessary for this negotiation. Sharing a whole trajectory would require a high amount of bandwidth, and the shared trajectory might become invalid if the weather forecast changed. Instead, a description of the trajectory, decoupled from the weather conditions, could be shared, so that the actual trajectory could be computed from this trajectory description. This trajectory description should be easy to process using a computing program —as some of the CDM processes will be automated— but also easy to understand for a human operator —who will be supervising the process and making decisions. This thesis presents a series of formal languages that can be used for this purpose. These languages provide the means to describe aircraft trajectories during all phases of flight, from push back to arrival at the gate. They can also describe trajectories of both manned and unmanned aircraft, including fixedwing and some rotary-wing aircraft (quadrotors). Some of these languages are tightly interrelated and organized in a language hierarchy. One of the key languages in this hierarchy, the aircraft intent description language (AIDL), had already been developed prior to this thesis. This language was derived from the equations of motion of fixed-wing aircraft, and can provide an unambiguous description of fixed-wing aircraft trajectories. A variant of this language, the quadrotor AIDL (QR-AIDL), is developed in this thesis to allow describing a quadrotor aircraft trajectory with the same level of detail. Then, the intent composite description language (ICDL) is built on top of these two languages, providing more flexibility to describe some parts of the trajectory while leaving others unspecified. The ICDL is used to provide generic descriptions of common aircraft manoeuvres, which can be particularized and combined to form complex descriptions of flight. Another language is built on top of the ICDL, the flight intent description language (FIDL). The FIDL specifies high-level requirements on trajectories —including constraints and objectives—, but can use features of the ICDL to provide arbitrary levels of detail in different parts of the flight. The ICDL and FIDL have been developed in collaboration with Boeing Research & Technology Europe (BR&TE). Also, the mission intent description language (MIDL) has been developed to allow describing missions involving multiple aircraft. This language is based on the FIDL and keeps all its expressive power, while it also provides new semantics for describing mission tasks, mission objectives, and constraints involving several aircraft. In ATM, the movement of aircraft while on the airport surface also has to be monitored and managed. Another formal language has been designed for this purpose, denoted surface movement description language (SMDL). This language does not belong to the language hierarchy described above, and it is based on the clearances used in airport surface operations. Means to express uncertainty and mutability of different parts of the trajectory are also provided. Finally, the applications of these languages to trajectory prediction and mission planning are explored in this thesis. The concept of trajectory language processing engine (TLPE) is used in these two applications. A TLPE is an ATM function whose main input and output are expressed in any of the languages in the hierarchy described in this thesis. A modular trajectory predictor is defined as a combination of multiple TLPEs, each of them performing a small subtask. Special attention is given to the TLPE that builds the horizontal, vertical, and configuration profiles of the trajectory. In particular, a novel method for the generation of the vertical profile is presented. The process of planning a mission can also be seen as a TLPE, where the main input is expressed in the MIDL and the output consists of a number of trajectory descriptions —one for each aircraft available in the mission— expressed in the FIDL. A mixed integer linear programming (MILP) formulation for the problem of assigning mission tasks to the available aircraft is provided. In addition, since finding optimal paths between locations is a key problem to mission planning, a novel path finding algorithm is presented. This algorithm can compute near-shortest paths avoiding all obstacles in an urban environment in very short times. The several formal languages described in this thesis can serve as a standard specification to share trajectory information among different actors in ATM. In combination, these languages can describe trajectories with the necessary level of detail for any application, and can be used to increase automation by exploiting this information using decision support tools (DSTs). Their applications to some basic functions of DSTs, such as trajectory prediction, have been analized.

Desarrollo de un datalink con el chip CC1110 de Texas Instruments para UAVs

Se trata de estudiar el comportamiento de un sistema basado en el chip CC1110 de Texas Instruments, para aplicaciones inalámbricas. Los dispositivos basados en este tipo de chips tienen actualmente gran profusión, dada la demanda cada vez mayor de aplicaciones de gestión y control inalámbrico. Por ello, en la primera parte del proyecto se presenta el estado del arte referente a este aspecto, haciendo mención a los sistemas operativos embebidos, FPGAs, etc. También se realiza una introducción sobre la historia de los aviones no tripulados, que son el vehículo elegido para el uso del enlace de datos. En una segunda parte se realiza el estudio del dispositivo mediante una placa de desarrollo, verificando y comprobando mediante el software suministrado, el alcance del mismo. Cabe resaltar en este punto que el control con la placa mencionada se debe hacer mediante programación de bajo nivel (lenguaje C), lo que aporta gran versatilidad a las aplicaciones que se pueden desarrollar. Por ello, en una tercera parte se realiza un programa funcional, basado en necesidades aportadas por la empresa con la que se colabora en el proyecto (INDRA). Este programa es realizado sobre el entorno de Matlab, muy útil para este tipo de aplicaciones, dada su versatilidad y gran capacidad de cálculo con variables. Para terminar, con la realización de dichos programas, se realizan pruebas específicas para cada uno de ellos, realizando pruebas de campo en algunas ocasiones, con vehículos los más similares a los del entorno real en el que se prevé utilizar. Como implementación al programa realizado, se incluye un manual de usuario con un formato muy gráfico, para que la toma de contacto se realice de una manera rápida y sencilla. Para terminar, se plantean líneas futuras de aplicación del sistema, conclusiones, presupuesto y un anexo con los códigos de programación más importantes. Abstract In this document studied the system behavior based on chip CC1110 of Texas Instruments, for wireless applications. These devices currently have profusion. Right the increasing demand for control and management wireless applications. In the first part of project presents the state of art of this aspect, with reference to the embedded systems, FPGAs, etc. It also makes a history introduction of UAVs, which are the vehicle for use data link. In the second part is studied the device through development board, verifying and checking with provided software the scope. The board programming is C language; this gives a good versatility to develop applications. Thus, in third part performing a functionally program, it based on requirements provided by company with which it collaborates, INDRA Company. This program is developed with Matlab, very useful for such applications because of its versatility and ability to use variables. Finally, with the implementation of such programs, specific tests are performed for each of them, field tests are performed in several cases, and vehicles used for this are the most similar to the actual environment plain to use. Like implementing with the program made, includes a graphical user manual, so your understanding is conducted quickly and easily. Ultimately, present future targets for system applications, conclusions, budget and annex of the most important programming codes.

Aeroelastic characteristics of slender wing/bodies with freeplay non-linearities

This article presents a time domain approach to the flutter analysis of a missile-type wing/body configuration with concentrated structural non-linearities. The missile wing is considered fully movable and its rotation angle contains the structural freeplay-type non-linearity. Although a general formulation for flexible configurations is developed, only two rigid degrees of freedom are taken into account for the results: pitching of the whole wing/body configuration and wing rotation angle around its hinge. An unsteady aerodynamic model based on the slender-body approach is used to calculate aerodynamic generalized forces. Limit-cycle oscillations and chaotic motion below the flutter speed are observed in this study.

Experimental investigation on box-wing configuration for UAS

The range for airframe configurations available for UAS is as diverse as those used for manned aircraft and more since the commercial risk in trying unorthodox solutions is less for the UAS manufacturer. This is principally because the UAS airframes are usually much smaller than the manned aircraft and operators are less likely to have a bias against unconventional configurations. One of these unconventional configurations is the box-wing, which is an unconventional solution for the design of the new UAS generation. The existence of two wings separated in different planes that are, however, significantly close together, means that the aerodynamic analysis by theoretical or computational methods is a difficult task, due to the considerable interference existing. Considering the fact that the flight of most UAS takes place at low Reynolds numbers, it is necessary to study the aerodynamics of the box wing configuration by testing different models in a wind tunnel to be able to obtain reasonable results. In the present work, the study is enhanced by varying not only the sweepback angles of the two wings, but also their position along the models’ fuselage. Certain models have shown being more efficient than others, pointing out that certain relative positions of wing exists that can improve the aerodynamics efficiency of the box wing configuration.

A hierarchical strategy for real-time tracking on-board UAVs

In this paper, we present a real-time tracking strategy based on direct methods for tracking tasks on-board UAVs, that is able to overcome problems posed by the challenging conditions of the task: e.g. constant vibrations, fast 3D changes, and limited capacity on-board. The vast majority of approaches make use of feature-based methods to track objects. Nonetheless, in this paper we show that although some of these feature-based solutions are faster, direct methods can be more robust under fast 3D motions (fast changes in position), some changes in appearance, constant vibrations (without requiring any specific hardware or software for video stabilization), and situations where part of the object to track is out the field of view of the camera. The performance of the proposed strategy is evaluated with images from real-flight tests using different evaluation mechanisms (e.g. accurate position estimation using a Vicon sytem). Results show that our tracking strategy performs better than well known feature-based algorithms and well known configurations of direct methods, and that the recovered data is robust enough for vision-in-the-loop tasks.

Arquitectura cognitiva híbrida para la navegación autónoma de UAVs mediante mapas topológicos visuales

El objetivo fundamental de la presente tesis doctoral es el diseño de una arquitectura cognitiva, que pueda ser empleada para la navegación autónoma de vehículos aéreos no tripulados conocidos como UAV (Unmanned Aerial Vehicle). Dicha arquitectura cognitiva se apoya en la definición de una librería de comportamientos, que aportarán la inteligencia necesaria al UAV para alcanzar los objetivos establecidos, en base a la información sensorial recopilada del entorno de operación. La navegación autónoma del UAV se apoyará en la utilización de un mapa topológico visual, consistente en la definición de un grafo que engloba mediante nodos los diferentes landmarks ubicados en el entorno, y que le servirán al UAV de guía para alcanzar su objetivo. Los arcos establecidos entre los nodos del mapa topológico, le proporcionarán de la información necesaria para establecer el rumbo más adecuado para alcanzar el siguiente landmark a visitar, siguiendo siempre una secuencia lógica de navegación, basada en la distancia entre un determinado landmark con respecto al objetivo final ó landmark destino. La arquitectura define un mecanismo híbrido de control, el cual puede conmutar entre dos diferentes modos de navegación. El primero es el denominado como Search Mode, el cual se activará cuando el UAV se encuentre en un estado desconocido dentro del entorno, para lo cual hará uso de cálculos basado en la entropía para la búsqueda de posibles landmarks. Se empleará como estrategia novedosa la idea de que la entropía de una imagen tiene una correlación directa con respecto a la probabilidad de que dicha imagen contenga uno ó varios landmarks. De esta forma, la estrategia para la búsqueda de nuevos landmarks en el entorno, se basará en un proceso continuo de maximización de la entropía. Si por el contrario el UAV identifica la existencia de un posible landmark entre los definidos en su mapa topológico, se considerará que está sobre un estado conocido, por lo que se conmutará al segundo modo de navegación denominado como Homing Mode, el cual se encargará de calcular señales de control para la aproximación del UAV al landmark localizado. Éste último modo implementa un control dual basado en dos tipos de controladores (FeedForward/FeedBack) que mediante su combinación, aportarán al UAV señales de control cada vez más óptimas, además de llevar a cabo un entrenamiento continuo y en tiempo real. Para cumplir con los requisitos de ejecución y aprendizaje en tiempo real de la arquitectura, se han tomado como principales referencias dos paradigmas empleados en diferentes estudios dentro del área de la robótica, como son el paradigma de robots de desarrollo (developmental robots) basado en un aprendizaje del robot en tiempo real y de forma adaptativa con su entorno, así como del paradigma de modelos internos (internal models) basado en los resultados obtenidos a partir de estudios neurocientíficos del cerebelo humano; dicho modelo interno sirve de base para la construcción del control dual de la arquitectura. Se presentarán los detalles de diseño e implementación de los diferentes módulos que componen la arquitectura cognitiva híbrida, y posteriormente, los diferentes resultados obtenidos a partir de las pruebas experimentales ejecutadas, empleando como UAV la plataforma robótica aérea de AR.Drone. Como resultado final se ha obtenido una validación completa de la arquitectura cognitiva híbrida objetivo de la tesis, cumplimento con la totalidad de requisitos especificados y garantizando su viabilidad como aplicación operativa en el mundo real. Finalmente, se muestran las distintas conclusiones a las cuales se ha llegado a partir de los resultados experimentales, y se presentan las diferentes líneas de investigación futuras que podrán ser ejecutadas.

A Hierarchical Tracking Strategy for Vision-Based Applications On-Board UAVs

In this paper, we apply a hierarchical tracking strategy of planar objects (or that can be assumed to be planar) that is based on direct methods for vision-based applications on-board UAVs. The use of this tracking strategy allows to achieve the tasks at real-time frame rates and to overcome problems posed by the challenging conditions of the tasks: e.g. constant vibrations, fast 3D changes, or limited capacity on-board. The vast majority of approaches make use of feature-based methods to track objects. Nonetheless, in this paper we show that although some of these feature-based solutions are faster, direct methods can be more robust under fast 3D motions (fast changes in position), some changes in appearance, constant vibrations (without requiring any specific hardware or software for video stabilization), and situations in which part of the object to track is outside of the field of view of the camera. The performance of the proposed tracking strategy on-board UAVs is evaluated with images from realflight tests using manually-generated ground truth information, accurate position estimation using a Vicon system, and also with simulated data from a simulation environment. Results show that the hierarchical tracking strategy performs better than wellknown feature-based algorithms and well-known configurations of direct methods, and that its performance is robust enough for vision-in-the-loop tasks, e.g. for vision-based landing tasks.

Aerodynamic study of a blended wing body, comparison with a conventional transport airplane.

Blended-wing-body (BWB) aircraft are being studied with interest and effort to improve economic efficiency and to overcome operational and infrastructure related problems associated to the increasing size of conventional transport airplanes. The objective of the research reported here is to assess the aerodynamic feasibility and operational efficiency of a great size, blended wing body layout, a configuration which has many advantages. To this end, the conceptual aerodynamic design process of an 800 seat BWB has been done completed with a comparison of performance and operational issues with last generation of conventional very large aircraft. The results are greatly encouraging and predict about 20 percent increase in transport productivity efficiency, without the burden of new or aggravated safety or operational problems.

Cancellation zone in supersonic lifting wing theory

BASING their work on a linear theory, Evvard1 and Krasilshchikova2'3 independently developed an expression that yields the perturbation generated by a thiri lifting wing of arbitrary planform flying at supersonic speed on a point placed on the wing plane inside its planform,1 or both on and above the wing plane.2 This point must be influenced by two leading edges, one supersonic and the other partially subsonic. Although these authors followed different approaches, their methods concur in showing the existence of a perfectly defined cancellation zone. In this Note, the Evvard approach is generalized to the case solved by Krasilshchikova. Circumventing the latter's lengthy and somewhat complex approach, Evvard's simple method seems to be useful at least for educational purposes.

Aerodynamic parametric analysis of an unconventional joined-wing aircraft configuration

In this paper, the experimental results of an unconventional joined-wing aircraft configuration are presented. The test model uses two different wings, forward and rear, both joined in tandem and forming diamond shapes both in plant and front views. The wings are joined in such a way that it is possible to change the rear wing dihedral angle values and the rear wing sweep angle values in 25 different positions that modify the relative distance and the relative height between the wings. To measure the system aerodynamic coefficients itis necessary to perform wind tunnel tests. The datapresented corresponds to the lift, drag and induced drag aerodynamic coefficients, as well as the aerodynamic efficiency and the parameter for minimum required power, from the calculated values of the lift and drag time series measured by a 6-axis force and torque sensor. The results show the influence on the aerodynamic coefficients of the rear wing sweep and dihedral angles parameters. As a main result, it can be concluded that, in general terms, the lift and induced drag aerodynamic coefficients values decrease as both the distance and height between the wings increase, on the other hand, the total drag aerodynamic coefficient decreases if the distance between the wings increases, but nevertheless shows a slight tendency to increase if the height of the rear wing increases, whereas the aerodynamic efficiency and the parameter for minimum required power increase if the distance between the wings increases