Muhammad Asghar Khan

Insaf Ullah

Neeraj Kumar

Omar Sami Oubbati

I Qureshi

Fazal Noor

Fahim Ullah

An Efficient and Secure Certificate-Based Access Control and Key Agreement Scheme for Flying Ad hoc Networks

Keywords: hoc Networks (FANETs), Drones, Security, Access Control, Key-Agreement, AVISPA

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Flying Ad-hoc Networks (FANETs) provide a decentralized communication mechanism that evolves due to coordination between a group of small drones [START_REF] Bekmezci | Flying ad-hoc networks (FANETs): A survey[END_REF], [START_REF] Noor | A Review on Communications Perspective of Flying Ad-Hoc Networks: Key Enabling Wireless Technologies, Applications, Challenges and Open Research Topics[END_REF]. Although a FANET system is characterized by an ad-hoc mechanism, some of its features distinguish it from the predecessor ad-hoc networks, such as Mobile Ad-hoc Networks (MANETs) and Vehicular Ad-hoc Networks (VANETs). For instance, the nodes in a FANET system have higher mobility than those in VANETs and MANETs. Certainly, FANET nodes can either be static, particularly in the relaying network purposes or dynamic, where FANET nodes have ample freedom of movement and are agile enough to move and rotate in the three-dimensional (3D) space [START_REF] Guillen-Perez | Flying ad hoc networks: A new domain for network communications[END_REF]. Such distinctive attributes make FANETs a suitable choice for time-limit and mission-critical tasks. However, on the other hand, it inherits challenges in terms of security and privacy [START_REF] Lv | The Security of Internet of Drones[END_REF], [START_REF] Rodday | Exploring security vulnerabilities of unmanned aerial vehicles[END_REF]. Thus, the participating drones in a FANET system should, in addition to handling the requests securely, maintain the integrity, and ensure an uninterrupted availability.

Efficient and secure communication among the drones, for the network layer, has received ample attention from the researchers. The research endeavors, however, have barely explored the security facet at the Medium Access Control (MAC) layer. The pivotal role played by the MAC layer cannot be ignored as it allows easy access to the broadcast radio channel [START_REF] Khan | A Quality of Service-Aware Secured Communication Scheme for Internet of Things-Based Networks[END_REF], [START_REF] Alzahrani | UAV assistance paradigm: State-of-the-art in applications and challenges[END_REF]. There are a few solutions in the existing literature that suggest exploring new and efficient MAC communication mechanisms for drones [START_REF] Choudhary | Sustainable and secure trajectories for the military internet of drones (IoD) through an efficient medium access control (mac) protocol[END_REF]. So far, system optimization for improved efficiency and fair common channel access has remained the primary aim of the existing solutions. However, the problem of security lapse is not well investigated.

Typically, the design consideration of small drones rarely addresses the security dimensions. Therefore, small drones are susceptible to cyber intrusions and physical attacks [START_REF] Lin | Security and Privacy for the Internet of Drones: Challenges and Solutions[END_REF], [START_REF] Zhi | Security and Privacy Issues of UAV: A Survey[END_REF], [START_REF] Guo | Covert Spoofing Algorithm of UAV Based on GPS/INS-Integrated Navigation[END_REF], [START_REF] Eldosouky | Drones in Distress: A Game-Theoretic Countermeasure for Protecting UAVs Against GPS Spoofing[END_REF], [START_REF] Krishna | A review on cybersecurity vulnerabilities for unmanned aerial vehicles[END_REF], [START_REF] Arteaga | Analysis of the GPS Spoofing Vulnerability in the Drone 3DR Solo[END_REF]. For instance, merely granting the adversary access to the network can result in irreparable damages. The adversary can waste bandwidth by transmitting multiple reservation requests. Also, the attacker can listen to the control messages, resort to power-efficient jamming or falsify the information [START_REF] Khan | Multi-access Edge Computing (MEC) Enabled Flying Ad-hoc Networks with Secure Deployment using Identity Based Generalized Signcryption[END_REF]. The inherent physical constraints arising due to limited on-board energy and limited computing capability, further exacerbate the situation. Moreover, a drone can become a luring target when hovering over a hostile environment.

In a FANET environment, the participating drones interact communication standards. Such wireless communication standards not only provide connectivity, but also offer a communication link that is light-weight and cost-effective [START_REF] Khan | Hybrid Communication Scheme for Efficient and Low-Cost Deployment of Future Flying Ad-Hoc Network (FANET)[END_REF], [START_REF] Khan | Dual-Radio Dual-Band Configuration for Flexible Communication in Flying Ad-hoc Network (FANET)[END_REF]. Such standards usually rely on the Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) protocol. Therefore, the decision to transmit energy is conditional upon the scenario when a legitimate node occupies a wireless medium. In the worst-case scenario, when a jammer throngs the same channel, data packets might collide and cause significant degradation in the network performance. It is observed that the issue of CRC (cyclic redundancy check) persists during packet collision. The sender attempts to retransmit the lost packets, or the corrupted packets. Thus, the packet collision qualifies as a DoS attack. To prevent suchlike untoward incidents from happening, there is a need to design a secure, computationally efficient, and cost-effective authentication scheme for FANETs.

A. Research Motivation and Contributions

Superb efforts have been made to address the underlying challenges on the way to ensure optimum usage of FANETs. The limitations of constrained on-board vitality and restricted computational ability severely hamper such efforts. The drones are, therefore, restricted from flying for long intervals and from performing complex cryptographic tasks. This constraint requires efficient security mechanisms that can stabilize the battery lifetime and incur lower computational and communication costs. We aim to meet the objectives mentioned above and do so by coming forth with an efficient scheme that is certificate-based and aggregates access control and key agreement. The efforts, particularly, add additional security attributes to the Medium Access Control (MAC) layer. Salient features of our research work are as follows:

• We approach the concept of Hyperelliptic Curve Cryptography (HECC) to come forth with certificatebased access control and key agreement scheme. This supplements the work done by Das et al. [START_REF] Das | Provably Secure ECC-Based Device Access Control and Key Agreement Protocol for IoT Environment[END_REF], Malani et al. [START_REF] Malani | CertificateBased Anonymous Device Access Control Scheme for IoT Environment[END_REF], and Odelu et.al [START_REF] Odelu | Efficient and Secure Time-Key Based Single Sign-On Authentication for Mobile Devices[END_REF]. An evident benefit of a hyperelliptic curve is its compactness since it requires an 80-bit key, which is way smaller than that needed by elliptic curve. • The proposed scheme is analyzed using formal and informal analysis techniques. It demonstrates to be sturdy enough to thwart attacks. For carrying out the formal analysis, the methodologies of AVISPA [START_REF] Avispa | Automated Validation of Internet Security Protocols and Applications[END_REF] and ROR [START_REF] Abdalla | Password-based authenticated key exchange in the three-party setting[END_REF] are used. • To improve efficiency, we propose a dual-radio strategy that caters to the demands of low power and high data rate operations. The proposed approach harnesses the robust features of each of the two options: the high-speed data transmission rate of IEEE 802.11; and the lowpower consumption characteristics of IEEE 802.15.4. • A detailed comparative analysis has been carried out to evaluate the feasibility in terms of computational costs, security features, and functionalities. The findings manifest the superiority of the proposed scheme, as compared to the existing solutions.

B. Organization of the Paper

The manuscript is structured section-wise. The related work is portrayed in Section II. A commentary about the system models is presented in Section III. The proposed scheme is exhibited in Section VI. Security analyses, both formal and informal, are discussed in Section V. The AVISPA tool is used for formal security verification and the findings are presented in Section VII. A comparative analysis is presented in Section VII. Section VIII, the final section, offers a succinct conclusion.

II. RELATED WORK

A. Security Schemes in Flying Ad-hoc Networks

Uninterrupted communication plays a vital role in a FANET environment, since it is the primary pre-requisite for enabling the applications that require continuous connectedness. Therefore, the deficiencies that result in security lapses, reduced efficiency and lower reliability, need to be addressed [START_REF] Gupta | Survey of Important Issues in UAV Communication Networks[END_REF]. The existing security mechanisms for FANETs, primarily, deal with authenticity, confidentiality and data integrity. The issue of data protection has, so far, remained unaddressed. Incorporating efficient data protection schemes in a FANET environment can offer a strong defense against the intrusions. In [START_REF] Arnosti | Evaluation of cryptography applied to broadcast storm mitigation algorithms in FANETs[END_REF], [START_REF] Pires | The Broadcast Storm Problem in FANETs and the Dynamic Neighborhood-Based Algorithm as a Countermeasure[END_REF] the authors aim to counter the broadcast storm problem that occurs in FANETs. In [START_REF] Islam | An expedite group key establishment protocol for Flying Ad-Hoc Network (FANET)[END_REF] the authors propose a secure group key establishment protocol that assists drones to form groups within the network. The protocol, however, focuses on the main goal ignoring the issues encountered in the transmission stage, such as a hidden terminal, exposed terminal and proximity problems.

A blind signature scheme for FANETs in a certificateless setting, primarily for authentication, has been proposed by Khan et al. in [START_REF] Khan | An Efficient and Provably Secure Certificateless Blind Signature Scheme for Flying Ad-Hoc Network Based on Multi-Access Edge Computing[END_REF]. The scheme, however, does not accommodate the provision of confidentiality and authentication in one go. Therefore, the key-encapsulated certificateless signcryption scheme proposed by Khan et al. in [START_REF] Khan | Efficient and Provably Secure Certificateless Key-Encapsulated Signcryption Scheme for Flying Ad-Hoc Network[END_REF] is characterized by offering confidentiality as well as authentication in a simultaneous manner. Still, neither of the schemes addressed the issue of coping with threats, both known as well as unknown, at the MAC layer.

B. Security Schemes at Medium Access Control layer

The research efforts on the topic of wireless MAC security have, so far, remained limited to addressing the concomitant threats in isolation. For instance, from the existing literature, studies undertaken so far focus on following issues: Denial of Service (DoS) attacks to jam the system [START_REF] Liu | Exploiting jamming-caused neighbor changes for jammer localization[END_REF], [START_REF] Xu | Channel surfing: defending wireless sensor networks from interference[END_REF], [START_REF] Gummadi | Understanding and mitigating the impact of rf interference on 802.11 networks[END_REF], [START_REF] Reed | Letter Response to FirstNet Conceptual Network NOI[END_REF], [START_REF] Chang | Simple MAC: a jamming resilient MAC-layer protocol for wireless channel coordination[END_REF], [START_REF] Chang | Redundancy offset narrow spectrum: countermeasure for signal-cancellation based jamming[END_REF]; transmitting bogus requests to the reserve channels [START_REF] Bellardo | 802.11 Denial-of-Service Attacks: Real Vulnerabilities and Practical Solutions[END_REF], [START_REF] Hu | Ariadne: A Secure On-Demand Routing Protocol for Ad Hoc Networks[END_REF], [START_REF] Negi | DoS attacks on a reservation-based MAC protocol[END_REF], [START_REF] Gupta | Denial of Service Attacks at the MAC Layer in Wireless Ad Hoc Networks[END_REF]; falsifying information at the communication feedback point(s) [START_REF] Tung | Vulnerability and protection of channel state information in multiuser mimo networks[END_REF]; sleeping-based MAC [START_REF] Manweiler | Avoiding the rush hours: Wifi energy management via trafc isolation[END_REF], [START_REF] Zhang | E-mili: Energy-minimizing idle listening in wireless networks[END_REF], [START_REF] Silva | Collaborative cellular tail energy reduction: Feasibility and fairness[END_REF], [START_REF] Brownfeld | Wireless sensor network denial of sleep attack[END_REF] that is vulnerable especially in case an attacker knows the MAC layer information [START_REF] Raymond | Effects of denial of sleep attacks on wireless sensor network mac protocols[END_REF]; and This section presents a literature review that comprises of three main subsections: the first subsection explains security schemes in FANETs, the second subsection describes security schemes at MAC layer, and the third subsection presents access control and key-agreement schemes.

energy DoS attacks [START_REF] Halperin | Pacemakers and implantable cardiac defibrillators: Software radio attacks and zeropower defenses[END_REF], [START_REF] Chang | Power-positive networking: wireless-charging-based networking to protect energy against battery DoS attacks[END_REF]. Reportedly, there is no research attempt aimed at addressing the issue of multiple threats originating at the same time. Therefore, the system remains vulnerable to attackers, who can launch multiple attacks. Moreover, other security requirements of FANETs, such as integrity, confidentiality and non-repudiation etc., have also not been given due consideration. Therefore, the need to ponder over and formulate a newer security scheme has further intensified. The very scheme must be able to cope with multiple security challenges. The approaches considered as a reference in our study aim to enhance the security and privacy features of the network. By presenting a solution that addresses the security requirements in a holistic manner, our proposed approach takes such research to a new level.

C. Access Control and Key-Agreement Schemes

A novel authentication and key agreement (AKA) scheme have been proposed by Semal et al. in [42] between users and nodes. The scheme makes use of hash function and bitwise XOR operation. It does not require a gateway node and is suitable for resource-constrained nodes. The scheme, however, was prone to man-in-the-middle and node impersonation attacks. Further, it lacked the capability to trace the users and to conceal the identification. Later, a new AKA scheme proposed by Farash et al. addressed and proposed viable solutions to the very issues in [START_REF] Farash | an efficient user authentication and key agreement scheme for heterogeneous wireless sensor network tailored for the internet of things environment[END_REF] Afterwards, Amin et al. [START_REF] Amin | Design of an anonymity-preserving three-factor authenticated key exchange protocol for wireless sensor networks[END_REF] highlighted the deficiencies in the work of Farash et al. [START_REF] Farash | an efficient user authentication and key agreement scheme for heterogeneous wireless sensor network tailored for the internet of things environment[END_REF] and proposed an AKA scheme based on smart card. The significant shortcomings addressed in [START_REF] Amin | Design of an anonymity-preserving three-factor authenticated key exchange protocol for wireless sensor networks[END_REF], for instance, included disposition to temporary information attack, off-line password guessing attack and user impersonation attack etc. However, the claim was dismissed by Jiang et al. [START_REF] Jiang | Lightweight three-factor authentication and key agreement protocol for internet-integrated wireless sensor networks[END_REF] who identified plausible weaknesses, such as smart card loss attack and off-line password guessing attack, in [START_REF] Farash | an efficient user authentication and key agreement scheme for heterogeneous wireless sensor network tailored for the internet of things environment[END_REF].

Challa et al. [START_REF] Challa | Secure signature-based authenticated key establishment scheme for future iot applications[END_REF] came forth with a new signature-based AKA scheme based on elliptic curve cryptography. Compared to its predecessors, the approach is characterized by increased communication and computation overhead. Wazid et al. [START_REF] Wazid | Design and analysis of secure lightweight remote user authentication and key agreement scheme in internet of drone's deployment[END_REF] proposed a novel lightweight user AKA scheme for deploying an Internet of Drones (IoD) setup. The proposed scheme relied on one-way cryptographic hash functions and bitwise XOR operations. However, the proposed scheme fails to provide a session key agreement. To overcome such deficiency, Zhang et al. [START_REF] Zhang | A lightweight authentication and key agreement scheme for internet of drones[END_REF] proposed an improved alternate.

Li et al. [START_REF] Li | Practical access control for sensor networks in the context of the Internet of Things[END_REF] designed an access control mechanism that is based on an identity-based access control (IBAC) model and uses bilinear pairing operations. The mechanism paves ways for a sender-receiver connection whereby the sender, from a certificate-less cryptography (CLC) environment, can link up and transmit a message to a receiver, who is in the identitybased cryptography (IBC) environment. For implementation, the mechanism demands a gateway node between the two IoT smart devices, and it also incurs high costs.

The efficient access control protocol scheme proposed by Luo et al. [START_REF] Luo | Secure and efficient access control scheme for wireless sensor networks in the cross-domain context of the IoT[END_REF] relies on IBC and bilinear pairing to link up with a smart device having different system parameters. The scheme demands a separate gateway node and is not economical. Further, the two-phase approach proposed by Das et al. [START_REF] Das | Provably Secure ECC-Based Device Access Control and Key Agreement Protocol for IoT Environment[END_REF], Malani et al. [START_REF] Malani | CertificateBased Anonymous Device Access Control Scheme for IoT Environment[END_REF] and Odelu et al. [START_REF] Odelu | Efficient and Secure Time-Key Based Single Sign-On Authentication for Mobile Devices[END_REF] aims for a secure communication scenario between two sensing nodes. It involves node authentication, key agreement and the concept of elliptic curve cryptography. An apparent drawback of such scheme is the hefty costs We aim to address the aforesaid deficiencies and, as a coping mechanism, propose a certificate-based access control scheme for FANETs. The scheme is based on hyperelliptic curve and demonstrates to be far more secure and efficient. Another important feature is the compactness of its key size (80 bits), which is half as much required by the elliptic curve (160 bits).

III. SYSTEM MODELS

Two models, i.e. network model and threat model, have been utilized to demonstrate the applicability of the proposed scheme.

A. Network Model

There are "n" drones, where n ≥ 2 as shown in Fig. 1. The drones are categorized into either of the two groups: Sensor Drone (S-Drone) and Gateway Drone (G-Drone). Drones from both the groups are placed in the geographical clusters that collectively make up the mission area. Each of the drones, from both G-Drones and S-Drones, are assigned a unique ID. A cluster has fixed number of drones out of which there must be a G-Drone that is linked to the ground station. A drone has following three layers: physical layer (bottom part), data link layer (middle part) and upper layer (top port). The IEEE 802.15.4 (ZigBee) system is installed on Sensor Drones (S-Drones). Gateway Drones (G-Drones) leverage both the radio technologies i.e. IEEE 802.15.4 (ZigBee) and IEEE 802.11a (Wi-Fi). In this way, the features promised by IEEE 802.11a (high-speed data transmission) and IEEE 802.15.4 (low-power consumption) are utilized by the proposed system. The process of network formation kicks off as soon as a drone lifts off.

Here, the drones are, supposedly, fed the information about neighbor's zone ID, location, altitude and speed etc. Further, the information does include the height sensors, IMU, GPS unit and the flight controller etc. The associated drones are interlinked together using the discovery function, which makes use of the beacon signals. Transmission of data between the S-Drones and G-Drones is accomplished using IEEE 802.15.4 at the frequency of 2.4 GHz. On the other hand, the data is routed between G-Drones and the ground station using IEEE 802.11a at the frequency of 5 GHz. An immediate pay off of the scheme is lower computational cost on the ground station since it only retains the information directed to it. B. Threat Model Dolev-Yao (DY) model is employed, which entails an insecure channel communication and an untrustworthy nature between the parties [START_REF] Dolev | On the security of public key protocols[END_REF]. Thus, the malicious attacker can easily eavesdrop and tamper the exchanged messages. The worst-case scenario, for instance, might involve seizing a drone, that is hovering, and then compromising its data. The prevalent "Canetti and Krawczyk's adversary model (CKadversary model)" is, reportedly, the de facto standard for modelling the authenticated key exchange protocols. The CKadversary model entails that the adversary can also hack the secret credentials, secret keys and the session states.

Therefore, it has become an essential requirement that "the leakage of some forms of secret credentials, such as session ephemeral secrets or secret key, should have the minimum possible consequence on the secrecy of other secret credentials of the communicating participants" [START_REF] Srinivas | Cloud Centric Authentication for Wearable Healthcare Monitoring System[END_REF].

IV. PROPOSED SCHEME

This section is dedicated to present the proposed scheme, which is based on hyperelliptic curve (HEC) and utilizes the one-way cryptographic hash function. The proposed approach is composed of following four steps i.e., Setup, Drones Registration, Drones Access Control Process and New Drone Addition Phase. Each of the steps is explained as follows.

A. Setup

The setup step is executed by the certifier's authority (𝒞 𝑟𝑠 𝒜). The prime intention is to generate private and public keys. Further, a public parameter set is prepared in this step. The following sequence is followed for performing such computations.

1. 𝒞 𝑟𝑠 𝒜 first choose 𝒹 𝑟𝑠 from the set {1,2,3, . … . . , n -1} as his private key.

It computes the public key as follows:

𝓌 𝑟𝑠 = 𝒹 𝑟𝑠 . 𝐷, where, 𝐷 is the divisor on a hyper elliptic curve. 3. It selects the hash function 𝒽, which has the capacity to avoid collisions and is characterized by irreversibility.

It selects the public parameter set as:

𝜂=( 𝓌 𝑟𝑠 , 𝐷, n, 𝒽) and publishes it.

B. Drones Registration

The drone registration step takes into consideration all of the deployed drones, 𝒟ℛ𝒩 𝑢 . It is supposed that the authority, 𝒞 𝑟𝑠 𝒜, wants to register them, in the offline manner. The process proceeds in the following sequence:

1

C. Drones Access Control Process

Assume that two drones, say 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 , wish to interconnect for having a key establishment mechanism. To undertake the process, there are four phases.

Phase # 1.

If 𝒟ℛ𝒩 𝑢 wants to establish a key with 𝒟ℛ𝒩 𝑣 , the actions are performed as follows:

• It chooses ε 𝑢 from {1,2,3, . … . . , n -1}

V. SECURITY ANALYSIS

Definition 1 (Collision-Resistant Cryptographic One-Way Hash Function): A ''collision-resistant cryptographic one-way hash function'' 𝒽: {0, 1} * → {0, 1} n is a ''deterministic mathematical function that produces a fixed length output string of n bits against a variable length input string''.

Definition 2 (Hyper elliptic curve Discrete logarithm problem (HECDLP)):

The HECDLP is a hard problem in which the attacker attempts to extract 𝑓 from the relation 𝐿=𝑓. D, where 𝑓 is the uniformly selected number from{1,2,3, . … . . , 𝑛 -1}.

A. Formal Security Analysis Through ROR Model

The widely-accepted "Real-Or-Random (ROR) model" [START_REF] Abdalla | Password-based authenticated key exchange in the three-party setting[END_REF] has been used to prove the existence of semantic security (secret key security) in the proposed scheme (ACKA-FANETs). Under the ROR model, it is assumed that there exists an intruder 𝒜 aiming to target a drone 𝒟 𝒮 during communication at a stage 𝒮 𝒯𝒽 . In the proposed scheme, the drone 𝒟 𝒮 is represented by the variables 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 . The variables 𝒟ℛ𝒩 𝑢 𝒮 and 𝒟ℛ𝒩 𝑢 𝒰 are used to depict 𝒮 𝒯𝒽 and 𝒰 𝒯𝒽 occurrences of 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 respectively. The queries to be initiated by the intruder 𝒜, that can harm the operations, can be classified as follows:

1. Execute query: The execute query allows the intruder 𝒜 to capture the transmitted messages while the communication link is being established between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 .

Reveal query:

The reveal query is intended to disclose the secret key established between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 .

Test query:

In the test query the intruder 𝒜 requests the drone 𝒟 𝒮 for a secret key. The drone 𝒟 𝒮 reverts with a probabilistic result 𝒞, where 𝒞 denotes the randomly-chosen bits. We also make it clear that a hash function is used for a random oracle, which is accessible to the intruder 𝒜 as well as other connected drones. By applying Theorem 1, explained in the following para, we demonstrate that our proposed scheme has a robust and effective secret key security mechanism. Theorem 1. Suppose the intruder 𝒜, operating in a polynomial time 𝒯, intends to break the security of a secret key using the games 𝐺 𝑥 , 𝐺 𝑦 and 𝐺 𝑧 . Here, the variables |𝒽𝑎𝑠ℎ|, 𝒬 ℎ𝑎𝑠ℎ and 𝐴𝑑 𝒜 𝐻𝐸𝐶𝐷𝐿𝑃 (𝒯) represent the size of irreversible hash function 𝒽(. ) , the number of hash queries and the non-negligible winning advantages for 𝒜 respectively. The pay-off that prompts the intruder 𝒜 to jeopardize the secret key security between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 in the access control step of the designed 𝐴𝐶𝐾𝐴 scheme can be written as:

𝐴𝑑 𝒜 𝐴𝐶𝐾𝐴-𝐹𝐴𝑁𝐸𝑇𝑠 (𝒯) ⩽ 𝒬 ℎ𝑎𝑠ℎ 2 |𝒽𝑎𝑠ℎ| ⁄ +2. 𝐴𝑑 𝒜 𝐻𝐸𝐶𝐷𝐿𝑃 (𝒯).
Proof. We prove the existence of semantic security (secret key security) in our proposed ACKA scheme. The security is based on equations 1 -5. Here, it is being ascertained whether the ACKA scheme can shield against the secret key exposure attack or not. To do so, three games are considered. The winning advantage for the intruder 𝒜 is given as: 𝐴𝑑 𝒜, 𝐺 𝑥 , 𝐺 𝑦 ,and 𝐺 𝑧 𝐻𝐸𝐶𝐷𝐿𝑃 = 𝑝𝑟𝑜𝑏 [𝑤𝑖𝑛𝑠 𝒜 𝐺 𝑥 , 𝐺 𝑦 ,and 𝐺 𝑧 ].

𝑮𝒂𝒎𝒆 𝑮 𝒙 : In game 𝐺 𝑥 , the intruder 𝒜 employs the working capacity of the ROR model to attack and disturb the proposed 𝐴𝐶𝐾𝐴 scheme. While the game is being initiated, the bit 𝒞 is chosen uniformly. The following output is obtained: 𝐴𝑑 𝒜 𝐴𝐶𝐾𝐴-𝐹𝐴𝑁𝐸𝑇𝑠 (𝒯) = |2. 𝐴𝑑 𝒜, 𝐺 𝑥 , 𝐺 𝑦 ,and 𝐺 𝑧 𝐻𝐸𝐶𝐷𝐿𝑃 -1|

(1) 𝑮𝒂𝒎𝒆 𝑮 𝒚 : In game 𝐺 𝑦 , the intruder 𝒜 makes use of the "execute query" to tap the communication between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 . The activity takes place during the access control step of the proposed 𝐴𝐶𝐾𝐴 scheme. The prime intention is to break the secret key security by capturing 𝛹 1 , 𝛹 2 and 𝛹 3 . As a next step, 𝒜 checks whether the secret key 𝒦 𝑢𝑣 , obtained from the communication between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 , is original or randomly selected. It does so to the reveal and test queries. Therefore, the secret key 𝒦 𝑢𝑣 , between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 , can be produced as follows: Even if the intruder 𝒜 performs the hash query, no collision occurs. Also, it is difficult to distinguish between 𝐺𝑎𝑚𝑒 𝐺 𝑦 and 𝐺𝑎𝑚𝑒 𝐺 𝑧 . Therefore, due to 𝐻𝐸𝐶𝐷𝐿𝑃 and owing to the concept of birthday paradox, the following output is obtained:

{𝒦 𝑢𝑣 = 𝒽( 𝐼𝐷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ‖ 𝜇 𝑢𝑣 ‖ 𝐼𝐷 𝑣 )= 𝒽( 𝐼𝐷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ⦁ ‖ 𝜇 𝑢𝑣 ⦁ ‖
|𝐴𝑑 𝒜, 𝐺 𝑦 𝐻𝐸𝐶𝐷𝐿𝑃 -𝐴𝑑 𝒜, 𝐺 𝑧 𝐻𝐸𝐶𝐷𝐿𝑃 | ⩽ 𝒬 ℎ𝑎𝑠ℎ 2 2|𝒽𝑎𝑠ℎ| ⁄ +𝐴𝑑 𝒜 𝐻𝐸𝐶𝐷𝐿𝑃 (𝒯)(3)
When 𝒜 performs all of the possible queries and once it guesses the bit 𝒞, it also performs the reveal and test query. This results in the following output: 𝐴𝑑 𝒜, 𝐺 𝑦 𝐻𝐸𝐶𝐷𝐿𝑃 = 1 2 ⁄ (4) From equations ( 1) and ( 2), we obtain the following output:

1 2 ⁄ 𝐴𝑑 𝒜 𝐴𝐶𝐾𝐴-𝐹𝐴𝑁𝐸𝑇𝑠 (𝒯) = |𝐴𝑑 𝒜, 𝐺 𝑥 , 𝐻𝐸𝐶𝐷𝐿𝑃 -1 2 ⁄ | = |𝐴𝑑 𝒜, 𝐺 𝑦 , 𝐻𝐸𝐶𝐷𝐿𝑃 -1 2 ⁄ | (5)
Similarly, making use of equations ( 1), ( 2) and (3) leads to the following output:

1 2 ⁄ 𝐴𝑑 𝒜 𝐴𝐶𝐾𝐴-𝐹𝐴𝑁𝐸𝑇𝑠 (𝒯) = |𝐴𝑑 𝒜, 𝐺 𝑦 𝐻𝐸𝐶𝐷𝐿𝑃 -𝐴𝑑 𝒜, 𝐺 𝑧 𝐻𝐸𝐶𝐷𝐿𝑃 | ⩽ 𝒬 ℎ𝑎𝑠ℎ 2 2|𝒽𝑎𝑠ℎ| ⁄ +𝐴𝑑 𝒜 𝐻𝐸𝐶𝐷𝐿𝑃 (𝒯) (6)
Multiplying equation ( 6) by "2" results in the following equation: In case of the variable 𝑁𝑂𝑁 𝑒𝑢𝑣 , 𝒜 will need to have the value of 𝒶 𝑢 , which can be obtained by utilizing the equation 𝒷 𝑢 = 𝒶 𝑢 . 𝐷.

𝐴𝑑 𝒜 𝐴𝐶𝐾𝐴-𝐹𝐴𝑁𝐸𝑇𝑠 (𝒯) = |𝐴𝑑
In short, if the intruder 𝒜 attempts to pose as 𝒟ℛ𝒩 𝑣 , complex mathematical operations are required. Such mathematical maneuvers are equivalent to computing the hyper elliptic curve discrete logarithm problem for up to five times. This is infeasible. Therefore, the proposed ACKA scheme offers protection from the Drone impersonation attacks.

3) Replay Attack: We assume that the intruder 𝒜, with the aim of intercepting communication between 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 , attempts to capture and then replay the message, Ψ 1 :

Ψ 1 = ( NON 𝑒𝑢 , Ω 𝑢 , Λ 𝑢 , 𝒷 𝑢 , 𝒳 𝑢 , 𝒞𝓇 𝑢 )
To proceed ahead, 𝒜 will be required to compute the value of NON 𝑒𝑢 = 𝛦 𝒷 𝑣 ( N 𝑢 ). However, to do so, the value 𝒶 𝑣 needs to be extracted from the relation 𝒷 𝑣 = 𝒶 𝑣 . 𝐷. The same process needs to be repeated in case replay is demanded for 𝛹 2 and 𝛹 3 .

Here, again, performing such computational effort is equivalent to solving the hyper elliptic curve discrete logarithm problem, which is far too complex to be resolved by the intruder 𝒜. Therefore, the proposed scheme guarantees protection from the Replay Attack.

4) Mutual Authentication:

The proposed scheme provides mutual authentication. Suppose if 𝒟ℛ𝒩 𝑢 dispatches the Ciphertext Ψ 1 = ( NON 𝑒𝑢 , Ω 𝑢 , Λ 𝑢 , 𝒷 𝑢 , 𝒳 𝑢 , 𝒞𝓇 𝑢 ) to 𝒟ℛ𝒩 𝑣 . After receiving the key 𝛹 1 , 𝒟ℛ𝒩 v performs the following computations:

• It decrypts 𝑁 𝑢 = 𝐷 𝒶 𝑣 ( 𝑁𝑂𝑁 𝑒𝑢 ) and checks for the freshness of 𝑁 𝑢 to the memory of 𝒟ℛ𝒩 𝑓𝑎𝑘𝑒 However, in reality, to generate a genuine certificate for the fake drone 𝒟ℛ𝒩 𝑓𝑎𝑘𝑒 , the intruder must have the value of 𝒶 𝑢 , which can be obtained from the relation 𝒷 𝑢 = 𝒶 𝑢 . 𝐷. Unfortunately, manipulating the very relation is as hard as computing an HECDLP. Therefore, in addition to other features, our solution protects the system from Malicious Drone Deployment Attacks. 7) Anonymity Preservation: Anonymity is preserved in our scheme.

The Ciphertext i.e. Ψ 1 = ( NON 𝑒𝑢 , Ω 𝑢 , Λ 𝑢 , 𝒷 𝑢 , 𝒳 𝑢 , 𝒞𝓇 𝑢 ) does not contains drone identity directly. 𝒜 can only extract ID 𝑢 from Λ 𝑢 = 𝒞𝓇 𝑢 + 𝒽( ID 𝑢 ‖𝒞𝓇 𝑢 ‖ 𝒷 𝑢 ‖ 𝒳 𝑢 )( ε 𝑢 + 𝒶 𝑢 ). In order to get access to drone identity, random number ε 𝑢 and the private key 𝒶 𝑢 of 𝒟ℛ𝒩 u will be required. For that 𝒜 will need to compute hyperelliptic curve discrete logarithmic problem twice. Further, it is hard to recover 𝐼𝐷 𝑢 from 𝒽( ID 𝑢 ‖𝒞𝓇 𝑢 ‖ 𝒷 𝑢 ‖ 𝒳 𝑢 ), because of the one-way property of hash function. Thus, from the above debate it has been cleared that the proposed scheme ensures anonymity. 8) Resistance against Cloning Attack: In this particular attack, 𝒜 can physically capture the drone device and copy the secret.

However, this attack is not be possible in the proposed scheme due to the updating of secret key for each session. 9) Resistance against De-synchronization Attack: During the communication between sender and receiver, 𝒜 can preclude the synchronized updating of the secret information. To prevent such attack, the receiver drone has the capacity of storing the old messages with their nonce in the database.

Here, if 𝒜 sends the cipher text Ψ 1 = ( NON 𝑒𝑢 , Ω 𝑢 , Λ 𝑢 , 𝒷 𝑢 , 𝒳 𝑢 , 𝒞𝓇 𝑢 ) to the receiving drone, then the receiver drone decrypts the value NON 𝑒𝑢 such as 𝑁 𝑢 = 𝐷 𝒶 𝑣 ( 𝑁𝑂𝑁 𝑒𝑢 ) and check freshness of 𝑁 𝑢 . In this way, the receiver drone can avoid a de-synchronization attack.

10) Resistance against DoS Attack

The scheme is secured against the DoS attack. For example, if 𝒟ℛ𝒩 v transmits 𝛹 2 , where 𝛹 2 = ( 𝑁𝑂𝑁 𝑒𝑢𝑣 , 𝛺 𝑣 , 𝛬 𝑣 , 𝒷 𝑣 , 𝒳 𝑣 , 𝒞𝓇 𝑣 , 𝒱𝒦 𝑢𝑣 , 𝐼𝐷 𝑣 ), to 𝒟ℛ𝒩 u , then recalling phase #3 : after receiving 𝛹 2 , 𝒟ℛ𝒩 𝑢 proceeds with the series of computations as follows. It first decrypts ( 𝑁 𝑢 , 𝑁 𝑣 ) = 𝐷 𝒶 𝑢 ( 𝑁𝑂𝑁 𝑒𝑢𝑣 ) and checks for the freshness of 𝑁 𝑣 and certificate of the source drone respectively to ascertains the applicability of the following condition: (𝓌 𝑟𝑠 + 𝒳 𝑣 . 𝒽( 𝐼𝐷 𝑣 ‖ 𝒳 𝑣 )) ≟ 𝒞𝓇 𝑣 . 𝐷, and finally verifies the signature by checking for the applicability of the following condition:

(𝒞𝓇 𝑣 . 𝐷 + ( 𝛺 𝑣 + 𝒷 𝑣 ). 𝒽( 𝒳 𝑣 ‖ 𝛺 𝑣 ‖𝒞𝓇 𝑣 ‖𝒷 𝑣 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 )) ≟ 𝛬 𝑣 . 𝐷 However, the aforementioned computation is hard for 𝒜 to extract information for DoS attack.

VI. FORMAL SECURITY VERIFICATION USING AVISPA SIMULATION STUDY

The algorithms A1 and A2 succinctly portrays the manner the proposed scheme is implemented. Both Gateway Drones (G-Drones) and Ground Station (GS) use High-level protocol specification language (HLPSL). For running the simulation, we use a Hair Workstation having following specifications: Intel (R) Core TM i3-4010U @ 1.70 GHz, Windows 8.1 64bit OS, Oracle VM Virtual Box (V 5.2.0.118431) and SPAN (V SPAN-ubunti-10.10-light_1). The algorithms A3 and A4 were used to carry out the execution tests, keeping in view OFMC and CL-AtSe back-ends, in order to evaluate the system's sturdiness to attacks. The simulations excluded SATMC and TA4SP as they are not compatible with bitwise XOR operations. A strong monitoring mechanism is also required and therefore, the back-ends surveil the probability of man-in-the-middle attack(s). The SPAN (Specific Protocol Animator for AVISPA) tool is also used for simulation purposes. The results, including the outcomes obtained from OFMC and AtSe, depicted in Fig. 2 and Fig. 3, demonstrate the effectiveness of the proposed scheme.

A. Computational Cost

In this subsection, the proposed scheme is compared with the schemes proposed by Das et al. [START_REF] Das | Provably Secure ECC-Based Device Access Control and Key Agreement Protocol for IoT Environment[END_REF], Malani et al. [START_REF] Malani | CertificateBased Anonymous Device Access Control Scheme for IoT Environment[END_REF], Semal et al. [START_REF] Semal | A Certificateless Group Authenticated Key Agreement Protocol for Secure Communication in Untrusted DRONE Networks[END_REF] and Odelu et al. [START_REF] Odelu | Efficient and Secure Time-Key Based Single Sign-On Authentication for Mobile Devices[END_REF]. The key results obtained from the comparison are shown in Table II to determine the effectiveness of the proposed scheme. The existing schemes either utilize elliptic curve scalar multiplication or bilinear pairings, both of which are more costly computational options. Therefore, we apply the hyperelliptic divisor multiplication. It has been shown from the results that the time it takes for a single scalar multiplication to be processed differs significantly: elliptic curve point multiplication (ECPM), 0.97 milliseconds; bilinear pairing, 14.90 ms; pairing-based point multiplication, 4.31 ms and modular exponentiation, 1.25 ms [START_REF] Zhou | Certificateless key insulated generalized signcryption scheme without bilinear pairings[END_REF]. MIRACL, or Multi-precision Integer and Rational Arithmetic C Library, is used to assess the performance of the proposed scheme [START_REF]Miracl library[END_REF]. The library applies a large number of tests, up to 1000, on basic cryptographic operations. An Intel Core i7-4510U CPU having 2.0 GHz processor, 8 GB RAM and Windows 7 is used to run the simulations [START_REF] Zhou | Certificateless key insulated generalized signcryption scheme without bilinear pairings[END_REF]. The Hyperelliptic Curve Divisor Multiplication (HCDM) is believed to be 0.48 milliseconds in duration [START_REF] Khan | An Efficient and Provably Secure Certificateless Blind Signature Scheme for Flying Ad-Hoc Network Based on Multi-Access Edge Computing[END_REF], [START_REF] Khan | Efficient and Provably Secure Certificateless Key-Encapsulated Signcryption Scheme for Flying Ad-Hoc Network[END_REF] due to a smaller key size of 80-bits. It is clear from the results in Table II and Table III that our method is much more effective in terms of the cost of computing, as shown in Fig. 4.

B. Communication Cost

This subsection is aimed at discussing the comparison results from the perspective of communication costs. The proposed approach is compared with the existing schemes presented by Das et al. [START_REF] Das | Provably Secure ECC-Based Device Access Control and Key Agreement Protocol for IoT Environment[END_REF], Malani et al. [START_REF] Malani | CertificateBased Anonymous Device Access Control Scheme for IoT Environment[END_REF], Semal et al. [START_REF] Semal | A Certificateless Group Authenticated Key Agreement Protocol for Secure Communication in Untrusted DRONE Networks[END_REF] and Odelu et al. [START_REF] Odelu | Efficient and Secure Time-Key Based Single Sign-On Authentication for Mobile Devices[END_REF]. In comparative analysis, the variables used, along with the respective values, are depicted in Table V and

C. Storage Requirement

In this subsection, we analyze the storage requirement of the proposed scheme with the existing counterpart presented by Das et al. [START_REF] Das | Provably Secure ECC-Based Device Access Control and Key Agreement Protocol for IoT Environment[END_REF], Malani et al. [START_REF] Malani | CertificateBased Anonymous Device Access Control Scheme for IoT Environment[END_REF], Semal et al. [START_REF] Semal | A Certificateless Group Authenticated Key Agreement Protocol for Secure Communication in Untrusted DRONE Networks[END_REF] and Odelu et al. [START_REF] Odelu | Efficient and Secure Time-Key Based Single Sign-On Authentication for Mobile Devices[END_REF]. The computed values are shown in Table VI. In the proposed scheme using a hyperelliptic curve (HEC), we consider identity, certificate, public key, and private key. The NIST standard key size for hyper elliptic curve is 80-bits, 160-bits for an elliptic curve, and 1024 bits for bilinear pairing.

Schemes

Approximate size (in bits) Das et al [START_REF] Das | Provably Secure ECC-Based Device Access Control and Key Agreement Protocol for IoT Environment[END_REF] 640 Malani et al [START_REF] Malani | CertificateBased Anonymous Device Access Control Scheme for IoT Environment[END_REF] 640 Semal et al [START_REF] Semal | A Certificateless Group Authenticated Key Agreement Protocol for Secure Communication in Untrusted DRONE Networks[END_REF] 3072 Odelu et al [START_REF] Odelu | Efficient and Secure Time-Key Based Single Sign-On Authentication for Mobile Devices[END_REF] 320 Proposed 320

D. Comparison of Security Attributes

This section is dedicated to present a comprehensive comparison of the proposed scheme with the existing schemes, primarily within the context of security functionalities. A brief comparison is depicted in Table VII. It can be observed from the table that none of the existing schemes address the Cloning, DoS and De-synchronization attacks.

VIII.

In Flying Ad-hoc Networks (FANETs), multiple small drones are supposed to interconnect and accomplish the assigned tasks autonomously, in an efficient manner. However, the associated stringent constraints on part of the small drones, such as limited on-board energy, restricted computing capability and insufficient bandwidth etc., hinder their ability to perform complex cryptographic operations. Further, a slow response time and a deteriorated performance is the immediate aftermath of bidding to carry out computationally intensive tasks. Our work aimed to counter such drawback by offering an efficient key agreement scheme in the certificate-based settings. Hyperelliptic curve, a muchadvanced version of elliptic curve, is employed to come forth with the solution. The proposed scheme is characterized by a multitude of advantages. For instance, it has a smaller key size, incurs lower computational, as well as communication, costs and offers superior security. Further, the scheme is capable of hindering malicious attacks. The resistance characteristic is further endorsed by the results obtained from a detailed security analysis that includes formal analysis, using ROR model and the widely-accepted AVISPA tool, as well as an informal analysis. Therefore, the proposed scheme caters to ever increasing needs of the resource-constrained small drones. It is reckoned that incorporation of the very scheme for FANETs will pave way for further realization of a reliable FANET communication arena.

Fig. 1 .

 1 Fig.1. Network model

Fig. 2 .Fig. 3 .

 23 Fig.2. Simulation results for on-the-fly model-checker (OFMC

  illustrated in Fig 5.

Fig. 4 .Fig. 5 .

 45 Fig.4. Computational cost (in ms)

  and various other journals. Currently, he holds the position of Vice Dean of Graduate Studies and Scientific Research at FCIS. He was Program Coordinator for Master of Computer Science program. He has received best faculty award in 2007. He has been a TPC member of many conferences. He is a fellow member of IAER. He has been QA evaluator for Computer Engineering program. His research interests are in AI, FANETS, Neural Networks, Embedded Systems, Signal Processing, Network Security, IoT, Optimization Algorithms, and Parallel and Distributed computing. Fahim Ullah Khanzada holds Bachelor's degree in electronic engineering from Baluchistan University of Information Technology, Engineering and Management Sciences (BUITEMS), Quetta and Master's degree in Electrical Engineering from the University of Nottingham, Nottingham, UK. His experience encompasses academia, industry and standardization. He is presently associated with Descon Engineering Limited, Lahore, Pakistan.

  . For each 𝒟ℛ𝒩 𝑢 , 𝒞 𝑟𝑠 𝒜 selects an identity ID 𝑢 and a private key, 𝒶 𝑢 , from the set {1,2,3, . … . . , n -1} 2. It computes the public ID 𝑢 using the following relation: 𝒷 𝑢 = 𝒶 𝑢 . 𝐷 3. It selects 𝒻 𝑢 from {1,2,3, . … . . , n -1} and computes the value of 𝒳 𝑢 as: 𝒳 𝑢 = ( 𝒻 𝑢 + 𝒶 𝑢 ). 𝐷 4. It calculates the certificate for ID 𝑢 as: 𝒞𝓇 𝑢 = 𝓌 𝑟𝑠 + ( 𝒻 𝑢 + 𝒶 𝑢 )𝒽( ID 𝑢 ‖ 𝒳 𝑢 ). 5. In the final step, 𝒞 𝑟𝑠 𝒜 pre-loads the identity set ( ID 𝑢 , 𝒞𝓇 𝑢 , 𝒷 𝑢 , 𝒶 𝑢 , 𝒳 𝑢 ) to the memory of 𝒟ℛ𝒩 𝑢 .

  • It computes Ω 𝑢 = ε 𝑢 . 𝐷 • It selects a fresh nonce N 𝑢 • It encrypts the NON 𝑒𝑢 𝑎𝑛𝑑 ID 𝑢 as NON 𝑒𝑢 = 𝛦 𝒷 𝑣 ( N 𝑢 , ID 𝑢 ) 𝑢 , Λ 𝑢 , 𝒷 𝑢 , 𝒳 𝑢 , 𝒞𝓇 𝑢 ) to 𝒟ℛ𝒩 𝑣 via open network. Phase # 2. Here, after the receiving the key 𝛹 1 , 𝒟ℛ𝒩 v performs the following computations: • It decrypts 𝑁 𝑢 , ID 𝑢 = 𝐷 𝒶 𝑣 ( 𝑁𝑂𝑁 𝑒𝑢 ) and checks for the freshness of 𝑁 𝑢 • It checks for certificate of the source drone by applying the condition: (𝓌 𝑟𝑠 + 𝒳 𝑢 . 𝒽( 𝐼𝐷 𝑢 ‖ 𝒳 𝑢 )) ≟ 𝒞𝓇 𝑢 . 𝐷. • It attempts to verify the signature and, to do so, it checks for the relation: (𝒞𝓇 𝑢 . 𝐷 + ( 𝛺 𝑢 + 𝒷 𝑢 ). 𝒽( 𝒳 𝑢 ‖ 𝛺 𝑢 ‖𝒞𝓇 𝑢 ‖𝒷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢 )) ≟ 𝛬 𝑢 . 𝐷 • If the signature is valid, it computes the value of 𝛺 𝑣 using the following relation: 𝛺 𝑣 = 𝜀 𝑣 . 𝐷, where 𝜀 𝑢 belongs to {1,2,3, . … . . , 𝑛 -1} • It computes 𝛬 𝑣 using the following relation: 𝛬 𝑣 = 𝒞𝓇 𝑣 + 𝒽( 𝐼𝐷 𝑣 ‖𝒞𝓇 𝑣 ‖ 𝒷 𝑣 ‖ 𝒳 𝑣 )( 𝜀 𝑣 + 𝒶 𝑣 ) • It performs the following calculations: 𝛶 𝑢𝑣 = 𝜀 𝑣 . 𝛺 𝑢 = 𝜀 𝑣 . 𝜀 𝑢 . 𝐷 𝜇 𝑢𝑣 = 𝒶 𝑣 . 𝒷 𝑢 = 𝒶 𝑣 . 𝒶 𝑢 . 𝐷 • It computes the value of 𝑁𝑂𝑁 𝑒𝑢𝑣 using the relation 𝑁𝑂𝑁 𝑒𝑢𝑣 = 𝛦 𝒷 𝑢 ( 𝑁 𝑢 , 𝑁 𝑣 ) 𝐼𝐷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ‖ 𝜇 𝑢𝑣 ‖ 𝐼𝐷 𝑣 ) • Here, 𝒦 𝑢𝑣 can be verified since 𝒱𝒦 𝑢𝑣 = 𝒽( 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝒦 𝑢𝑣 ) • Finally, 𝒟ℛ𝒩 v transmits 𝛹 2 , where 𝛹 2 = ( 𝑁𝑂𝑁 𝑒𝑢𝑣 , 𝛺 𝑣 , 𝛬 𝑣 , 𝒷 𝑣 , 𝒳 𝑣 , 𝒞𝓇 𝑣 , 𝒱𝒦 𝑢𝑣 , 𝐼𝐷 𝑣 ), to 𝒟ℛ𝒩 u via open network. In this phase, after receiving 𝛹 2 , 𝒟ℛ𝒩 𝑢 proceeds with the computations as follows. • It first decrypts ( 𝑁 𝑢 , 𝑁 𝑣 ) = 𝐷 𝒶 𝑢 ( 𝑁𝑂𝑁 𝑒𝑢𝑣 ) and checks for the freshness of 𝑁 𝑣 • It checks for certificate of the source drone and ascertains for applicability of the following condition: (𝓌 𝑟𝑠 + 𝒳 𝑣 . 𝒽( 𝐼𝐷 𝑣 ‖ 𝒳 𝑣 )) ≟ 𝒞𝓇 𝑣 . 𝐷 • It verifies the signature by checking for the applicability of the following condition: (𝒞𝓇 𝑣 . 𝐷 + ( 𝛺 𝑣 + 𝒷 𝑣 ). 𝒽( 𝒳 𝑣 ‖ 𝛺 𝑣 ‖𝒞𝓇 𝑣 ‖𝒷 𝑣 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 )) ≟ 𝛬 𝑣 . 𝐷 • It performs following calculations: 𝛶 𝑢𝑣 ⦁ = 𝜀 𝑢 . 𝛺 𝑣 = 𝜀 𝑢 . 𝜀 𝑣 . 𝐷 and 𝜇 𝑢𝑣 ⦁ = 𝒶 𝑢 . 𝒷 𝑣 = 𝒶 𝑢 . 𝒶 𝑣 . 𝐷 • It computes the value of 𝑁𝑂𝑁 𝑒𝑢𝑣 using the relation: 𝑁𝑂𝑁 𝑒𝑢𝑣 = 𝛦 𝒷 𝑣 ( 𝑁 𝑢 , 𝑁 𝑣 ) In case if 𝑁 𝑢 is found to be anew, it calculates 𝒱𝒦 𝑢𝑣 * * using the relation: This phase involves addition of a new drone 𝒟ℛ𝒩 𝑛𝑒𝑤 after establishment of a network. 𝒟ℛ𝒩 𝑛𝑒𝑤 includes 𝒞 𝑟𝑠 𝒜 in an offline manner. The process proceeds stepwise as follows: • For 𝒟ℛ𝒩 𝑛𝑒𝑤 , 𝒞 𝑟𝑠 𝒜 choses an identity ID 𝑛𝑒𝑤 and assigns a private key 𝒶 𝑛𝑒𝑤 from {1,2,3, . … . . , n -1} • It computes public identity ID 𝑛𝑒𝑤 using the relation: 𝒷 𝑛𝑒𝑤 = 𝒶 𝑛𝑒𝑤 . 𝐷. • It selects 𝒻 𝑛𝑒𝑤 from {1,2,3, . … . . , n -1} and performs the following computation: 𝒳 𝑛𝑒𝑤 = ( 𝒻 𝑛𝑒𝑤 + 𝒶 𝑛𝑒𝑤 ). 𝐷 • It calculates certificate for the identity ID 𝑛𝑒𝑤 using the relation: 𝒞𝓇 𝑢 = 𝓌 𝑟𝑠 + ( 𝒻 𝑛𝑒𝑤 + 𝒶 𝑛𝑒𝑤 )𝒽( ID 𝑛𝑒𝑤 ‖ 𝒳 𝑛𝑒𝑤 ) ID 𝑛𝑒𝑤 , 𝒞𝓇 𝑛𝑒𝑤 , 𝒷 𝑛𝑒𝑤 , 𝒶 𝑛𝑒𝑤 , 𝒳 𝑛𝑒𝑤 ) to the memory of 𝒟ℛ𝒩 𝑛𝑒𝑤

		𝒱𝒦 𝑢𝑣 • It also checks if the relation 𝒱𝒦 𝑢𝑣 * * = 𝒽( 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝒦 𝑢𝑣 )	* * ≟ 𝒱𝒦 𝑢𝑣	⦁ holds
		true
		• The following equality can also be authenticated:
		𝒱𝒦 𝑢𝑣 ≟ 𝒱𝒦 𝑢𝑣	⦁
		D. New Drone Addition Phase
	• It 𝒽( ID 𝑢 ‖𝒞𝓇 𝑢 ‖ 𝒷 𝑢 ‖ 𝒳 𝑢 )( ε 𝑢 + 𝒶 𝑢 ) as a signature computes Λ 𝑢 = 𝒞𝓇 𝑢 + on ε 𝑢 • Finally, it dispatches the key Ψ 1 = ( NON 𝑒𝑢 , Ω • It finds out the value of session key, to be shared with 𝒟ℛ𝒩 𝑢 , from the relation 𝒦 𝑢𝑣 = relation: 𝒦 𝑢𝑣 ⦁ = 𝒽( 𝐼𝐷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ⦁ ‖ 𝜇 𝑢𝑣 ⦁ ‖ 𝐼𝐷 𝑣 ) • At this step, 𝒦 𝑢𝑣 ⦁ can be verified since 𝒱𝒦 𝑢𝑣 ⦁ = 𝒽( 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝒦 𝑢𝑣 ⦁ ) • Finally, 𝒞 𝑟𝑠 𝒜 pre-loads the set 𝒽( Phase # 3. • It finds out the value of session key using the (
	• Finally, 𝒟ℛ𝒩 𝑢 transmits 𝛹 3 = ( 𝑁𝑂𝑁 𝑒𝑢𝑣 , 𝒱𝒦 𝑢𝑣	⦁ )

to 𝒟ℛ𝒩 v via open network

Phase # 4. Here, after the receiving 𝛹 3 , 𝒟ℛ𝒩 𝑣 proceeds with the computations as follows.

• It decrypts ( 𝑁 𝑢 , 𝑁 𝑣 ) = 𝐷 𝒶 𝑣 ( 𝑁𝑂𝑁 𝑒𝑢𝑣 ) and checks if 𝑁 𝑢 is fresh •

  𝐼𝐷 𝑣 )} To carry on the activity, 𝒜 needs to know the following unknown values: 𝜀 𝑢 representing the randomly generated number 𝒟ℛ𝒩 𝑢 ; 𝜀 𝑣 representing the randomly generated number 𝒟ℛ𝒩 𝑣 ; 𝒶 𝑢 representing the private key 𝒟ℛ𝒩 𝑢 ; and 𝒶 𝑣 representing the private key 𝒟ℛ𝒩 𝑣 . When the very four values are known to the intruder 𝒜, its probability to win reduces significantly. Hence, it is difficult to distinguish between 𝐺𝑎𝑚𝑒 𝐺 𝑥 and 𝐺𝑎𝑚𝑒 𝐺 𝑦 as depicted in the following relation: In game 𝐺 𝑧 , by exploiting the hash query, the intruder 𝒜 can launch an aggressive attack against the proposed 𝐴𝐶𝐾𝐴 scheme. In 𝛹 1 , 𝛹 2 and 𝛹 3 the values such as 𝒞𝓇 𝑢 , Λ 𝑢 , 𝛬 𝑣 , 𝒞𝓇 𝑣 , 𝒱𝒦 𝑢𝑣 and 𝒱𝒦 𝑢𝑣 ⦁ are protected through a collision resistant/one way hash function. Also, it is worth mentioning that even if the intruder is successful in capturing 𝛺 𝑢 and 𝛺 𝑣 , it would be impracticable to generate the following two keys: 1) { 𝛶 𝑢𝑣 = 𝜀 𝑣 . 𝛺 𝑢 = 𝜀 𝑣 . 𝜀 𝑢 . 𝐷 = 𝛶 𝑢𝑣 ⦁ = 𝜀 𝑢 . 𝛺 𝑣 = 𝜀 𝑢 . 𝜀 𝑣 . 𝐷} 2) { 𝜇 𝑢𝑣 = 𝒶 𝑣 . 𝒷 𝑢 = 𝒶 𝑣 . 𝒶 𝑢 . 𝐷 = 𝜇 𝑢𝑣 ⦁ = 𝒶 𝑢 . 𝒷 𝑣 = 𝒶 𝑢 . 𝒶 𝑣 . 𝐷} It is difficult to perform the aforementioned computations because of the following two reasons: 1) In order to find out values of the involved variables, such as 𝜀 𝑣 , 𝜀 𝑢, , 𝒶 𝑣 and 𝒶 𝑢 , cumbersome 𝐻𝐸𝐶𝐷𝐿𝑃 calculations are required. 2) The collision resistant property of hash function obstructs an attempt to find the values of 𝜀 𝑣 , 𝜀 𝑢 , 𝒶 𝑣 , and 𝒶 𝑢 from Λ 𝑢 and Λ 𝑣

	𝐴𝑑 𝒜, 𝐺 𝑥 𝐻𝐸𝐶𝐷𝐿𝑃 = 𝐴𝑑 𝒜, 𝐺 𝑦 𝐻𝐸𝐶𝐷𝐿𝑃	(2)
	𝑮𝒂𝒎𝒆 𝑮 𝒛 :

  Ψ 1 = ( NON 𝑒𝑢 , Ω 𝑢 , Λ 𝑢 , 𝒷 𝑢 , 𝒳 𝑢 , 𝒞𝓇 𝑢 ) Thus, to be successful in its malicious attempt, the intruder 𝒜 will need to estimate the values of 𝛬 𝑢 and NON 𝑒𝑢 :𝛬 𝑢 = 𝒞𝓇 𝑢 + 𝒽( 𝐼𝐷 𝑢 ‖𝒞𝓇 𝑢 ‖ 𝒷 𝑢 ‖ 𝒳 𝑢 )( 𝜀 𝑢 + 𝒶 𝑢 ) NON 𝑒𝑢 = 𝛦 𝒷 𝑣 ( N 𝑢 )In other words, 𝒜 must have the values of 𝜀 𝑢 , 𝒶 𝑢 , and 𝒶 𝑣 , which can only be obtained from the following relations: Ω 𝑢 = ε 𝑢 . 𝐷; 𝒷 𝑢 = 𝒶 𝑢 . 𝐷 and 𝒷 𝑣 = 𝒶 𝑣 . 𝐷. Performing such mathematical manoeuvre is impracticable since it is equivalent to solving the hyper elliptic curve discrete logarithm problem three times. Therefore, our scheme proves to be sturdy enough to resist the Man-in-the-Middle Attacks.2) Drone Impersonation Attack: Suppose the intruder 𝒜 attempts to impersonate 𝒟ℛ𝒩 𝑣 in order to generate the following message: 𝛹 2 = ( 𝑁𝑂𝑁 𝑒𝑢𝑣 , 𝛺 𝑣 , 𝛬 𝑣 , 𝒷 𝑣 , 𝒳 𝑣 , 𝒞𝓇 𝑣 , 𝒱𝒦 𝑢𝑣 , 𝐼𝐷 𝑣 ) Such attempt will, in turn, demand extensive computations in order to generate the variables 𝛬 𝑣 , 𝒱𝒦 𝑢𝑣 and 𝑁𝑂𝑁 𝑒𝑢𝑣 . The mathematical relations along with the inherent calculation requirements are given as follows: 1. 𝛬 𝑣 = 𝒞𝓇 𝑣 + 𝒽( 𝐼𝐷 𝑣 ‖𝒞𝓇 𝑣 ‖ 𝒷 𝑣 ‖ 𝒳 𝑣 )( 𝜀 𝑣 + 𝒶 𝑣 ) For 𝛬 𝑣 to execute such equation, it is essential for to extract the values of 𝜀 𝑣 and 𝒶 𝑣 from the relations Ω 𝑣 = ε 𝑣 . 𝐷and 𝒷 𝑣 = 𝒶 𝑣 . 𝐷 respectively. 2. 𝒱𝒦 𝑢𝑣 = 𝒽( 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝒦 𝑢𝑣 ) Here, to approach the value of 𝒱𝒦 𝑢𝑣 , 𝒜 is required to solve the following equation: 𝒦 𝑢𝑣 = 𝒽( 𝐼𝐷 𝑢 ‖𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ‖ 𝜇 𝑢𝑣 ‖ 𝐼𝐷 𝑣 ). This, in turn, demands calculation to generate 𝛶 𝑢𝑣 and 𝜇 𝑢𝑣 using the relations 𝛶 𝑢𝑣 = 𝜀 𝑣 . 𝛺 𝑢 and 𝜇 𝑢𝑣 = 𝒶 𝑣 . 𝒷 𝑢 respectively. 3. 𝑁𝑂𝑁 𝑒𝑢𝑣 = 𝛦 𝒷 𝑢 ( 𝑁 𝑢 , 𝑁 𝑣 )

		𝒜, 𝐺 𝑦 𝐻𝐸𝐶𝐷𝐿𝑃 -𝐴𝑑 𝒜, 𝐺 𝑧 𝐻𝐸𝐶𝐷𝐿𝑃 | ⩽
	𝒬 ℎ𝑎𝑠ℎ	2 ⁄ |𝒽𝑎𝑠ℎ| +2. 𝐴𝑑 𝒜 𝐻𝐸𝐶𝐷𝐿𝑃 (𝒯)

B. Informal Security Analysis 1)

Man-in-the-Middle Attack: Suppose the intruder 𝒜 intends to alter the message Ψ 1 , being transmitted between the drones 𝒟ℛ𝒩 𝑢 and 𝒟ℛ𝒩 𝑣 . The message Ψ 1 comprises of multiple parameters and is given as:

  • It checks for certificate of the source drone by applying the condition: (𝓌 𝑟𝑠 + 𝒳 𝑢 . 𝒽( 𝐼𝐷 𝑢 ‖ 𝒳 𝑢 )) ≟ 𝒞𝓇 𝑢 . 𝐷. • It attempts to verify the signature and to do so, it checks for the relation: (𝒞𝓇 𝑢 . 𝐷 + ( 𝛺 𝑢 + 𝒷 𝑢 ). 𝒽( 𝒳 𝑢 ‖ 𝛺 𝑢 ‖𝒞𝓇 𝑢 ‖𝒷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢 )) ≟ 𝛬 𝑢 . 𝐷 Suppose the intruder 𝒜 wants to generate the following session key: 𝒦 𝑢𝑣 = 𝒽( 𝐼𝐷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ‖ 𝜇 𝑢𝑣 ‖ 𝐼𝐷 𝑣 ) or 𝒦 𝑢𝑣 In order to do so, it is essential for 𝒜 to find out values of the following variables: 𝛶 𝑢𝑣 = 𝜀 𝑣 . 𝛺 𝑢 ; 𝜇 𝑢𝑣 = 𝒶 𝑣 . 𝒷 𝑢 ; 𝛶 𝑢𝑣 ⦁ = 𝜀 𝑢 . 𝛺 𝑣 and 𝜇 𝑢𝑣 ⦁ = 𝒶 𝑢 . 𝒷 𝑣 . However, an attempt aimed at finding such values further stipulate the values of 𝜀 𝑣 , 𝒶 𝑣 , 𝜀 𝑢 , and 𝒶 𝑢 , which is equivalent to computing the hyper elliptic curve discrete logarithm problem four times. Thus, it can be concluded that the proposed scheme is resistant to the Ephemeral Secret Leakage (ESL) attacks as well. 6) Malicious Drone Deployment Attack: In this case, let's assume that the intruder 𝒜, in an established network, makes an attempt to deploy a fake drone 𝒟ℛ𝒩 𝑓𝑎𝑘𝑒 . 𝒜 proceeds as follows: a. It chooses a fake identity, ID 𝑓𝑎𝑘𝑒 , and a random private key, 𝒶 𝑓𝑎𝑘𝑒 b. It computes the public identity, ID 𝑓𝑎𝑘𝑒 : 𝒷 𝑓𝑎𝑘𝑒 = 𝒶 𝑓𝑎𝑘𝑒 . 𝐷 c. It selects 𝒻 𝑓𝑎𝑘𝑒 and computes 𝒳 𝑓𝑎𝑘𝑒 : 𝒳 𝑓𝑎𝑘𝑒 = ( 𝒻 𝑓𝑎𝑘𝑒 + 𝒶 𝑓𝑎𝑘𝑒 ). 𝐷 d. It calculates the certificate for ID 𝑓𝑎𝑘𝑒 as: 𝒞𝓇 𝑓𝑎𝑘𝑒 = 𝓌 𝑓𝑎𝑘𝑒 + ( 𝒻 𝑓𝑎𝑘𝑒 + 𝒶 𝑓𝑎𝑘𝑒 )𝒽( ID 𝑓𝑎𝑘𝑒 ‖ 𝒳 𝑓𝑎𝑘𝑒 ) ID 𝑓𝑎𝑘𝑒 , 𝒞𝓇 𝑓𝑎𝑘𝑒 , 𝒷 𝑓𝑎𝑘𝑒 , 𝒶 𝑓𝑎𝑘𝑒 , 𝒳 𝑓𝑎𝑘𝑒 )

	In case of valid signature, the proposed scheme achieves
	mutual authentications.
	5) Ephemeral Secret Leakage (ESL) Attack: ⦁ 𝒽( 𝐼𝐷 𝑢 ‖ 𝑁𝑂𝑁 𝑒𝑢𝑣 ‖ 𝛶 𝑢𝑣 ⦁ ‖ 𝜇 𝑢𝑣 ⦁ ‖ 𝐼𝐷 𝑣 )	=
	e. Finally,	it	pre-loads	the	set
	(

TABLE II COMPARISON

 II

		COMPUTATIONAL COSTS
	Schemes	𝒟ℛ𝒩 𝑢 /device	𝒟ℛ𝒩 𝑣 /device	Total
		cost	cost
	Das et al [51]	6 𝑒𝑚	6 𝑒𝑚	12 𝑒𝑚
	Malani et al [53]	6 𝑒𝑚	6 𝑒𝑚	12 𝑒𝑚
	Semal et al [42]	2 𝒷𝓅𝓂	1𝒷𝓅+3𝓂𝑒𝓍𝓅	2 𝒷𝓅𝓂
				+1𝒷𝓅+3𝓂𝑒𝓍𝓅
	Odelu et al [54]	6 𝑒𝑚	6 𝑒𝑚	12 𝑒𝑚
	Our	6 𝒽𝓂	6 𝒽𝓂	12 𝒽𝓂

Where, 𝒽𝓂 = hyperelliptic curve divisor multiplication 𝑒𝑚 = elliptic curve scalar multiplication 𝒷𝓅 = bilinear pairing 𝒷𝓅𝓂 = pairing-based point multiplications 𝓂𝑒𝓍𝓅 = modular exponentiation

TABLE III COMPARISON

 III OF COMPUTATIONAL COSTS (IN MILLISECONDS)

	Schemes	𝒟ℛ𝒩 𝑢 /smart	𝒟ℛ𝒩 𝑣 /smart	Total (in
		IoT device	IoT device	milliseconds)
	Das et al [51]	5.28	5.28	10.56
	Malani et al [53]	5.28	5.28	10.56
	Semal et al [42]	8.62	18.65	27.27
	Odelu et al [54]	5.28	5.28	10.56
	Proposed	2.88	2.88	5.76

TABLE IV COMPARISON

 IV OF COMMUNICATION COSTS (IN BITS)

	Variable	Value
	|𝔖|	1024 bits
	|𝒵𝓆|	160 bits
	|𝒵n|	80 bits
	From Table V, it is evident that opting for our proposed
	scheme results in significant reduction in the associated
	communication costs.

TABLE V COMPARISON

 V

		OF COMMUNICATION COSTS
	Schemes	No. of	Total (in bits)
		messages
	Das et al [51]	16|𝒵𝓆|	2560
	Malani et al [53]	11|𝒵𝓆|	1760
	Semal et al [42]	3|𝔖|	3072
	Odelu et al [54]	11|𝒵𝓆|	1760
	Proposed	16|𝒵n|	1280

TABLE VI COMPARISON

 VI

OF STORAGE REQUIREMENT

  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2021.3055895, IEEE Transactions on Vehicular Technology

						11
	DIA	✓	✓		✓	✓
	DoSA	✓	✓	✓	✓	✓
	PCA	✓	✓		✓	✓
	FVA	✓	✓		✓	✓
	FSA	✓	✓	✓	✓	✓
	RCA					✓
	DoSA					✓
	DA					✓
	AP	✓	✓	✓	✓	✓
	ECA	✓	✓	✓	✓	✓
				TABLE VII
			COMPARISON OF SECURITY ATTRIBUTES
	Security	Das et al [51]	Malani et al [53]	Semal et al [42]	Odelu et al [54]	Proposed
	Attributes
	RA	✓	✓	✓	✓	✓
	MIA	✓	✓	✓	✓	✓
	MA	✓	✓	✓	✓	✓

0018-9545 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Legend: RA: replay attack; MIA: man-in-the-middle attack; MA: mutual authentication; DIA: device impersonation attack; MDA: malicious device deployment attack; DoSA: Denial-of-Service attack; FVA: formal security verification using AVISPA tool; FSA: formal security analysis; CA: Cloning Attack ; AP: anonymity preservation; DA: De-synchronization Attack; EAC: ESL attack under the CK-adversary model. Symbol: ✓ satisfy the security functionality, : does not satisfy the security functionality.

Authorized licensed use limited to: Consortium -Algeria (CERIST). Downloaded on March 19,2021 at 13:34:22 UTC from IEEE Xplore. Restrictions apply.


L A Blas

Moisés Bonilla
email: mbonilla@cinvestav.mx

Sergio Salazar
email: sergio.salazar.cruz@gmail.com

M Malabre
email: michel.malabre@ls2n.fr

V Azhmyakov

Luis Ángel Blas

Luis Angel
email: anghelblas@gmail.com

Blas Sanchez

Dr Moisés Bonilla Estrada

Dr Sergio Salazar Cruz

Robust tracking scheme for an experimental quadrotor

Keywords: unmanned aerial vehicle (UAV), trajectory tracking, non-standard feedback linearization, nonlinearities rejection

In this paper, we present a robust tracking scheme for an experimental quadrotor, with outdoor real-time implementation. This control scheme is based on the robust structural feedback linearization scheme, presented in [2] . This control scheme has the advantage of combining the classical linear control techniques with the sophisticated robust control techniques. This control scheme is specially ad hoc for unmanned aerial vehicles, where it is important not only to reject the actual nonlinearities and the unexpected changes of the structure, but also to look for the simplicity and effectiveness of the control scheme.

keywords Quadrotor aircraft,

The motion of the quadrotor is referred to a fixed orthogonal axis set (earth axes) (oxyz), where oz points vertically down along the gravity vector 1 0 0 g T , and the origin o is located at the desired height z, above the ground level. (2.1) In [START_REF] Blas | Structural feedback linearization based on nonlinearities rejec-tion[END_REF], we have shown that the quadrotor is represented by the state space representations:

dx i /dt = A i x i +B i u i +S i q oi , y i = C i x i , i ∈ {x, y, z, ψ}, (2.2)
A x =     0 1 0 0 0 0 -g 0 0 0 0 1 0 0 0 0     , A y =     0 1 0 0 0 0 g 0 0 0 0 1 0 0 0 0     , B x = I -1 yy B 4 , B y = I -1 xx B 4 , B 4 =     0 0 0 1     , S x = S y =     0 0 1 0 0 0 0 1     , C x = C y =     1 0 0 0     T , (2.3)
A z = A ψ = 0 1 0 0 , B z = M -1 q B 2 , B ψ = I -1 zz B 2 , S z = S ψ = B 2 , B 2 = 0 1 , C z = C ψ = 1 0 T .
(2.4) 1 We take the value: g = 9.81 [m s -2 ]. 2 Since the quadrotor is mechanically symmetric its cross inertia are zero.

where:

x x =
x dx/dt θ dθ/dt T , x y = y dy/dt φ dφ/dt T , x z = z dz/dt T , x ψ = ψ dψ/dt T , q ox = q x q θ T , q oy = q y q φ T , q oz = q z , q oψ = q ψ . The control actions, u x , u y , u z and u ψ , are related with the thrusters of the four rotors, f 1 , f 2 , f 3 and f 4 , by means of an invertible real constant matrix (cf. Section 5 of [START_REF] Blas | Structural feedback linearization based on nonlinearities rejec-tion[END_REF]). The nonlinear signals, q x , q y , q z , q φ , q θ and q ψ , are:

  q x q y q z   =   θ -q xx -φ -q yy -q zz   g + 1 M q   q xx q yy q zz   u z , (2.5)
  q φ q θ q ψ   = J -1 (η) -J -1 (0) τ -J -1 (η)C(η, dη/dt) dη dt , (2.6) where η = φ θ ψ T , τ = u y u x u ψ T , J
and C(η, dη/dt) are the inertial and the Coriolis matrices, defined in Appendix A of [START_REF] Blas | Structural feedback linearization based on nonlinearities rejec-tion[END_REF], and:

q xx =
cos(φ) sin(θ) cos(ψ) + sin(φ) sin(ψ) , q yy = cos(φ) sin(θ) sin(ψ)sin(φ) cos(ψ) , q zz = cos(φ) cos(θ) -1 .

(2.7)

B. Locally Stabilizing Feedback

In the fifth Section of [START_REF] Blas | Synthesis of a robust linear structural feedback linearization scheme for an experimental quadrotor[END_REF], we have used the LQR technique in order to locally stabilize (2.2) -(2.4) 3 . For this, we have solved the Algebraic Riccati Equations (ARE):

A T i P i + P i A i -P i B i (ρ i I) -1 B T i P i + Q i = 0 , (2.8
) where i ∈ {x, y, z, ψ}, and with:

Q x = Q y = 900     1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2.25     , Q z = 1 0 0 0.23 , Q ψ = 1 0 0 0.6 , (2.9)
(ρ x , ρ y , ρ z , ρ ψ ) = (1, 1, 1/19600, 1/12100) .

(2.10) Solving (2.8) for matrices (2.3) and (2.4), with the choices (2.9) and (2.10), we have obtained (i ∈ {x, y, z, ψ}): (2.13) With the state feedbacks (2.11), (2.12) and (2.13), we get the following spectra and characteristic polynomials: 3 Recall the celebrated Stability Principle of the First Approximation (see for example [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF]).

u i = F i x i + ūi , i ∈ {x, y}, (2.11)
u i = F i (x i -x i ) , i ∈ {z, ψ} , xz = z 0 T , xψ = 0 , ( 2
Λ A Fx = {-3214, -1.911, -0.9536 ± 1.585}, Λ A Fy = {-3358, -1.911, -0.9536 ± 1.585}, Λ A Fz = {-2.087, -49.32}, Λ A F ψ = {-1.291, -3328},
π xi (s) = det(sI -A Fi ) = s 4 + a i,1 s 3 + a i,2 s 2 + a i,3 s + a i,4 ,
(2.14) where:

A Fi . = A i + B i F i , i ∈ {x, y, z, ψ}.

C. Nonlinear Uncertainty Signal Estimator

In [START_REF] Blas | Structural feedback linearization based on nonlinearities rejec-tion[END_REF], we have shown that with the change of variable:4

ζ i = x i +M i   S i + n-1 j=1 M j i S i d j /dt j   q oi (x,
dζ i /dt = A Fi ζ i + B i (ū i + q * i (x i , u i )) , y i = C i ζ i , (2.16
) where the nonlinear uncertainty signals, q * i , are:

q * i ((x i , u i ) = X i   S i + n-1 j=1 M j i S i d j /dt j   q oi (x i , u i ).
(2.17) The matrices M i and X i , are solutions of the matrix equtions:

A Fi M i + B i X i = I . ( 2
dw i /dt = A Ki + B i G i C i w i -K i + B i G i y i , ūi = G i (C i w i -y i ) , ( 2

.19) where: A

Ki = A Fi + K i C i and G i = -C i A -1
Ki B i ∈ R 1×1 , i ∈ {x, y}, and (cf. [START_REF] Kailath | Linear systems[END_REF]):

A Fi =     0 0 0 -a i,4 1 0 0 -a i,3 0 1 0 -a i,2 0 0 1 -a i,1     , K i =     a i,4 -a o,4 a i,3 -a o,3 a i,2 -a o,2 a i,1 -a o,1     , C i = 0 0 0 1 , B i = ki bi 0 0 0 T , G i = -C i (A Fi + K i C i ) -1 B i = (a o,4 b i )/k i ,
(2.20) where: i ∈ {x, y}, (a i,1 , a i,2 , a i,3 , a i,4 ) are the coefficients of the polynomials (2.14), and:

(k x , b x ) = (-g, I yy ) and (k y , b y ) = (g, I xx ).
Let us note that the output signals of the nonlinear uncertainty signals observers (2.19) are the negative estimations of the nonlinear uncertainty signals q * i , namely: ūi = -q * i .

The transfer functions F CL i (s) of the closed loop system are (y i v.s. q * i ):5

F CL i (s) = F ζi (s)G HPF (s), (2.21)
where:

F ζi (s) = C i sI -A Fi -1 B i = k i b i π xi (s) , (2.22)
G HPF (s) = 1 -F e (s) = G i C i (sI -A Ki ) -1 B i = 1 -a o,4 π ei (s) = s πw (s) π e (s), (2.23)
and:

π e (s) = det (sI -A Ki ) = s 4 + a o,1 s 3 + a o,2 s 2 + a o,3 s + a o,4 , (2.24)
π w (s) = det sI -(A Ki + B i G i C i ) = s (s 3 + a o,1 s 2 + a o,2 s + a o,3 ), (2.25
)

π e (s) = s πw (s) + a o,4 . (2.26)
Following a root locus procedure, in Section V.B.1 of [START_REF] Blas | Synthesis of a robust linear structural feedback linearization scheme for an experimental quadrotor[END_REF] we have deduced:

s πw (s) = s (s + 4.75 c )(s + 4 c )(s + 3.5 c ), (2.27) π e (s) = (s + c ) 2 (s 2 + 10.25 c s + 28.125 2 c ), (2.28) a o,4 = 28.125 4 c , (2.29)
and that the cutoff frequency ω c of the high-pass filter

G HPF ( ω) = (ω 2 -a o,2 )ω 2 - (a o,1 ω 2 -a o,3 )ω ((ω 2 -a o,2 )ω 2 + a o,4 ) - (a o,1 ω 2 -a o,3 )ω (2.30) is: ω c = a o,2 /2 1 -1 -2 a o,4 /a 2 o,2 = 0.5339 c [rad/s] .
(2.31) We then have to adjust c sufficiently higher than the bandwidth B W of the Fourier transform of q * i , F{q * i }, i ∈ {x, y}. Experimentally, we have found that c = 18 is a good choice for our laboratory prototype, thus: π e (s) = s 4 + 220.5 s 3 + 16078.5 s 2 + 387828 s + 2952450 .

(2.32)

III. DRIFT-FREE ESTIMATOR

The nonlinear uncertainty signal estimator (2.19) has a good performance when the quadrotor is in hover flying.

However, when we wish that the quadrotor tracks some prescribed trajectory in the x-y plane, sometimes the quadrotor suffers a drift phenomenon. Doing a frequency analysis of the recorded x and y signals in hover flying (in outdoor realtime experiments), we have found that the signal disturbance is well approximated by:

d(t) = 0.009 sin(0.85t) + 0.0115η d (t) + 0.0073; (3.1) η d (t) = sin(2.1π(sin(2π t/T ) + 1)) , (3.2)
with 2π/T = = .4. Analyzing (3.1), we deduce: (i) The constant term 0.0073 is easily compensated by the integral term of the nonlinear uncertainty estimator, see (2.

25). (ii)

The sinusoidal term 0.009 sin(0.85t) is certainly attenuated by the high-pass filter G HPF (s), see (2.21) and (2.23). Then, the η d (t) term is the cause of the drift phenomenon.

In order to overcome the quadrotor drift due to the integration of wind noises like (3.2), we have proceeded as in [START_REF] Gavin | Drift-free integrators[END_REF] and [START_REF] Horowitz | The Art of Electronics[END_REF]; that is to say, we have shifted the pole at the origin of (2.19), slightly into the left-half complex plane (cf.

(2.25)). For this, we have reduced lightly the static gain G i of the nonlinear uncertainty signal estimator (2.19), namely:

dw df,i /dt = A Ki + (1 -)B i G i C i w df,i -K i + (1 -)B i G i y i , ūdf,i = (1 -) G i (C i w df,i -y i ) , (3.3)
where is a sufficiently small positive constant and i ∈ {x, y}.

Let us note that the output signals of the drift-free estimators (3.3) are -(1 -) times the nonlinear uncertainty signals q * i , namely: ūi = -(1 -)q * i . Now, the characteristic polynomial of

(A Ki + (1 - )B i G i C i )
has no roots at the origin (recall (2.20) and cf. (2.25)):

π w df (s) = det sI -(A Ki + (1 -)B i G i C i ) π w df (0) = det A Ki det -1 -(1 -)C i A -1 Ki B i G i = -= 0 .
A. Closed loop system Applying (3.3) to (2.16), we get the closed loop system represented by (cf. (4.16) and (4.17) of [START_REF] Blas | Synthesis of a robust linear structural feedback linearization scheme for an experimental quadrotor[END_REF]):

d dt e df,i ζ i = A CL df,i e df,i ζ i + B CL i q * i , y i = C CL i e df,i ζ i , (3.4)
A CL df,i = A Ki 0 (1 -)B i G i C i A Fi , B CL i = -B i B i ,
C CL i = 0 C i , (3.5
F CL df,i (s) = C CL i sI -C CL i -1 B CL i = F ζi (s) I -F e df,i (s) , (3.6) F ζi (s) = C i (sI -A Fi ) -1 B i , F e df,i (s) = (1 -) G i C i (sI -A Ki ) -1 B i , (3.7
π w df (s) = s πw (s) + a o,4 , (3.8) F e df,i (s) = (1 -) a o,4 /π e (s) , (3.9)

B. Frequency response

The closed loop transfer function (3.6) incorporates the high-pass filter (recall (3.9), (2.26) and (3.8)):

G HPF df (s) = 1 -F e df,i (s) = 1 - (1-)ao, 4 πe(s) = s πw (s)+ ao,4 πe(s) , (3.10)
G HPF df ( ω) = ((ω 2 -ao, 2 )ω 2 +ε ao, 4 )- (ao, 1 ω 2 -ao, 3 )ω ((ω 2 -ao, 2 )ω 2 +ao, 4 )- (ao, 1 ω 2 -ao, 3 )ω .
(3.11)

The cutoff frequency ω c df of the high-pass filter G HPF df ( ω) for = 1/50 is (recall (2.28) and cf. (2.31)): 7

ω c df = a o,2 /2 1 -1 -2 (1 + ) a o,4 /a 2 o,2 = 0.5392 c [rad/s] .
(3.12) In Figure 2, we compare the Bode plots of G HPF ( ω), (2.30) and (2.28), and G HPF df ( ω), (3.11) and (2.28), with c = 18 and = 1/50. From Figure 2, we realize that the high frequencies behaviors are very similar, and the low frequencies are attenuated by the factor . Now, there is no poles at the origin, avoiding in this way the drift phenomena due to the integration of wind noises like (3.2).

IV. TRAJECTORY TRACKING

In this section we present outdoor real-time experiments obtained when the quadrotor tracks a circular trajectory in the x-y plane. For this:

1) The quadrotor is locally stabilized with the LQR of Section II-B, and robustly linearized with the drift-free estimators of Section III. 2) An optimal state trajectory is synthesized for reaching a given local stationary point.

3) The circular trajectory is partitioned by a finite set of suitable local stationary points.

A. Control Scheme

In [START_REF] Bonilla | Robust structural feedback linearization based on the nonlinearities rejection[END_REF], we have shown that the nonlinear uncertainty signal estimator (2.19), has two interesting properties: it enlarges the stability neighborhood and it creates an attraction neighborhood B ρ * * , of radius ρ * * , around the stationary point (see Figure 3 of [START_REF] Bonilla | Robust structural feedback linearization based on the nonlinearities rejection[END_REF]). In consequence, we have partitioned the desired trajectory for tracking, x * i (t) (i ∈ {x, y}), t ∈ [0, T f ], by a set of N +1 stationary points:

Λ * SP i = x * 0,i , x * 1,i , . . . , x * N,i , x * i (kT s ) = x * k,i , (4.13
) where: k ∈ {0, 1, . . . , N }, N T s = T f and i ∈ {x, y}; T s is the trajectory sampling time and T f is the flying time. Then, the local stabilizing feedbacks (2.11) and the drift-free estimators (3.3) take the forms, respectively:

u i = F i (x i -x * i ) + ūdf,i , (4.14)
d dt w df,i = A Ki + (1 -)B i G i C i w df,i -K i + (1 -)B i G i C i (x i -x * i ), ūdf,i = (1 -) G i (C i w df,i -C i (x i -x * i )) , ( 4
dx * i /dt = A Fi x * i + B i ū * i , i ∈ {x, y} , (4.16)
where:

A Fi . = A i + B i F i , i ∈ {x
, y}, and with initial conditions:

x * i (0) = x o,i ; let us note that the pairs (A Fi , B i ) are controllable (cf. (2.3)).
We are interested in finding a minimal norm control input ū * i , such that: x * i (T s ) = 0, where T s is a given finite time, T s > 0.

This is a classical minimum norm problem of seeking the closest vector to the origin lying in a variety of finite codimension, in a Hilbert space, and it is solved with the help of the Projection Theorem. Indeed, according to Theorem 2 of Section 3.3 of [START_REF] Luenberger | Optimization by Vector Space Methods[END_REF], the control input ū * i solving the problem has the form (see also Theorem 1.1 of [START_REF] Wonham | Linear Multivariable Control: A Geometric Approach[END_REF]):

ū * i (t) = -F F i (T s -t) W -1 i Ts exp (A Fi T s ) x o,i , (4.17) i ∈ {x, y}, t ∈ [0, T s ], where: 9 F F i (t) = B T i exp A T Fi t , W i Ts = Ts 0 F T F i (T s -τ ) F F i (T s -τ )dτ. (4.18)
From (4.16), (4.17) and (4.18), we get for all the time intervals, kT s , (k + 1)T s , k ∈ 0, 1, . . . , (N -1) , the following optimal trajectories:

x * i (t) = exp (A Fi (t -kT s )) x * k,i + t kTs F T F i (t -τ ) β * i (t, τ )v * k,i dτ, (4.19)
8 For the ψ dynamics, we have chosen:

x * ψ = 0, A F ψ = 0 -a ψ,2 1 -a ψ,1 , B ψ = 1/Izz 0 , K ψ = a ψ,2 -a o,2 a ψ,1 -a o,1 , C ψ = 0 1 , G ψ = -C ψ (A F ψ + K o ψ C ψ ) -1 B ψ = a o,2 Izz,
where: πe(s) = s 2 + a o,1 s + a o,2 = s 2 + 208 s + 2704, and = 1/10. 9 Since the pairs (A F i , B i ) are controllable, the matrices W i Ts are invertible [START_REF] Wonham | Linear Multivariable Control: A Geometric Approach[END_REF].

β * i (t, τ ) = F F i (t -τ ) W -1 i Ts , v * k,i = x * (k+1),i -exp (A Fi T s ) x * k,i , (4.20)
where:

x * k,i ∈ Λ * SP i
, i ∈ {x, y} (recall (4.13)).

Stationary points set:

We are interested in following the circular trajectory:

x 2 * (t) + (y * (t) -r * ) 2 = r 2 * , (4.21) namely:
x * (t) = r * sin(ω s t) and y * (t) = r * (1cos(ω s t)) , (4.22) where:

ω s = 2π/T f . Partitioning T f , in N points we get (t = (k/N )T f , k ∈ {0, 1, . . . , N }): 10 x * k,x , x * k,y = r * sin (α k ) ω s cos (α k ) ω 2 s g sin (α k ) ω 3 s g cos (α k ) T , (1 -cos (α k )) ω s sin (α k ) ω 2 s g cos (α k ) - ω 3 s g sin (α k ) T , (4.23) where: α k = 2π k/N .

C. Outdoor real-time experiments

In this Section we present outdoor real-time experiments, when tracking a circular trajectory of a radius of 5 [m], in the x-y plane. For this, we have proceeded as follows:

• We have locally stabilized the Quadrotor with the state feedbacks (4.14) (for i ∈ {x, y, z}) and (2.12) (for i = z), together with (2.13). We have tracked the circular trajectory (4.21) with the radius r * = 5 [m], a sampling time of T s = 1 [s], and a partition of N = 20 points, namely: T f = 20 [s]. In Figure 3, we show the obtained results when applying the local stabilizing feedbacks (4.14), with and without the drift-free estimators (4.15). We compare the obtained path v.s. the goal path (4.21), and the quadrotor (x, y)-trajectories v.s. the references (4.22). We also show the tracking error,

e C (t) = (x(t) -x * (t)) 2 + (y(t) -y * (t)) 2 , (4.24)
when applying the local stabilizing feedbacks (4.14) together with the drift-free estimators (4.15), i ∈ {x, y}, we have 10 From (2.3) and (4.16), we have: The quadrotor was flown at an altitude around 14 3 and 6 [m]; see Figure 4. 13 Recall that: (1/T f )

x * 2,x = dx * 1,x /dt, x * 3,x = -g -1 dx * 2,x /dt, x * 4,x = dx * 3,x /dt, x * 2,y = dx * 1,y /dt, x * 3,y = g -1 dx * 2,
T f 0 sin 2 (2πt/T f )dt = 1/ √ 2.
14 This altitude was visually fixed with the only purpose of having a good sight. From these outdoor real-time experiments, we see that for the slow trajectory we have obtained an efficiency improvement around 10 %, when applying the drift-free estimators (4.15); and for the fast trajectory the efficiency improvement was around 20 %. Let us also note that the yaw error diminishes by a 1/3 factor, when applying the drift-free estimator (4.15); see Figure 4.

V. CONCLUSION

In this paper, we have tackled the problem of tracking some prescribed trajectory in the x-y plane. We have found that there exists a drift phenomenon due to the wind noise. In Section III, we have characterized the disturbance wind (see (3.1)), and we have checked that when the term (3.2), passes trough the integral action of the nonlinear uncertainty estimator (2.19), such a drift phenomenon occurs. Depending on the values of the scaling factors of (3.1) this drift phenomenon could arise, and when arising the nonlinear uncertainty signal estimator (2.19) will not have a good performance. Thus, we have to do some adjustments to the nonlinear uncertainty signal estimator (2.19) in order to ensure a good performance not mattering the scale factors of (3.1).

To overcome the quadrotor drift due to the integration of wind disturbance like (3.2), we have proceeded as in [START_REF] Gavin | Drift-free integrators[END_REF], [START_REF] Horowitz | The Art of Electronics[END_REF] shifting the pole at the origin of (2.19) slightly into the left-half complex plane, having now the drift-free estimator (3.3); see also the comparison shown in Figure 2.

We have done outdoor real-time experiments obtained when the quadrotor tracks a circular trajectory of a radius of 5 [m] in the x-y plane; we have first tracked a 5 [m] radius circular trajectory in 60 [s] (slow trajectory), and then a 5 [m] radius circular trajectory in 20 [s] (fast trajectory). In Section IV, we are showing the obtained results for the fast trajectory case. From these experiences we got an apreciable reduction on the tracking and yaw errors when applying the drift-free estimators (4.15). We have obtained an efficiency improvement around of the 10 %, in the slow trajectory experience, and an efficiency improvement around of the 20 %, in the fast trajectory experience, see Figure 3; with respect to the yaw error, it decreased by a 1/3 factor.

APPENDIX

For synthesizing the optimal state trajectory given by (4.19) and (4.20), we have first applied the change of bases: 3) The β * i (t, τ ) are: β * x (t, τ ) = F Fx (t -τ ) W -1

(A F i , Bi) = (T -1 i A F i Ti, T -1 i Bi), i ∈ {x,
x Ts T -1 x = k1,x e a 1,x (t-1) + k2,x e a 2,x (t-1) + k3,x cos(ωx (t -1)) + k4,x sin(ωx (t -1)) e bx (t-1) and β * y (t, τ ) = F Fy (tτ ) W -1 y Ts T -1 y = k1,y e a 1,y (t-1) + k2,y e a 2,y (t-1) + k3,y cos(ωy (t-1)) + k4,y sin(ωy (t-1)) e by (t-1) , where:

2 Figura 1 : 1 Fig. 1 .

 2111 Figura 1: Vista del arreglo experimental

  φ, θ and ψ are the Euler angles, roll, pitch and yaw, measured respectively over the axis o B x B , o B y B and o B z B ; where (o B x B y B z B ) is the body axis system, with its origin o B fixed at the centre of gravity of the quadrotor; see Figure 1. The total mass is M q , the moments of inertia about axis ox, oy and oz are I xx , I yy and I zz , and the distance of each rotor with respect to the centre of gravity of the quadrotor is denoted by L m : 2 M q = 1.36 [kg], I xx = 0.0134 [kg m 2 ], I yy = 0.0140 [kg m 2 ], I zz = 0.0256 [kg m 2 ], L m = 0.245 [m]

  .12) F x = 30 32.4264 -171.9158 -45.0535 , F y = -30 -32.4260 -171.9116 -45.0512 , F z = -140 -69.92 , F ψ = -110 -85.2387 .

( 4 .Figura 4 . 3 :Fig. 2 .

 4432 Figura 4.3: Gráfica de Bode de magnitud. |p1| = 74, |s2| = 351.5, |s3| = 296, |s4| = 259, ωn = 392.44 rad/s y ζ = 0.966.

7

  Doing: |GHPF df ( ω)| = 1, one gets: 2ao, 4 ω 4 -2ao, 2 ao, 4 ω 2 + (1 + ) a 2o, 4 = 0.

2 C

 2 obtained the maximum peak error: e C p = 0.5463 m and the root mean square error: e C rms = 0.2992 m, where:e C p = max t ∈ [0, T f ] e C (t), e C rms = (t)dt . (4.25) While, when applying only the local stabilizing feedbacks (4.14), i ∈ {x, y}, we have obtained the maximum peak error: e C p = 2.3582 m and the root mean square error: e C rms = 1.0277 m. Thus, with the combination, local stabilizing feedbacks (4.14) and drift-free estimators (4.15), we get the efficiencies: e ff p = 89.074 % and e ff rms = 91.537 %, and with only the local stabilizing feedbacks (4.14), we get the efficiencies: e ff p = 52.836 % and e ff rms = 70.932 %, where: 13 e ff p = 1e C p /r * × 100%, e ff rms = 1 -√ 2 e C rms /r * × 100% .

Fig. 3 .

 3 Fig. 3. Tracking of a circular trajectory in the plane x-y with a 5 [m] radius in 20 [s] (N = 20 and Ts = 1 [s]). (a, b) x [m] v.s. y [m]. (c, d) x [m] v.s. t [s] and y [m] v.s. t [s]. (e, f) e C (t) [m] v.s. t [s]. Applying the local stabilizing feedbacks (4.14) together with the drift-free estimators (4.15): (a) and (c). Applying only the local stabilizing feedbacks (4.14): (b) and (d). (a, b) Comparison of the obtained path v.s. the goal path (4.21). (c, d) Tracking error e C (t) (see (4.24)).

Fig. 4 .

 4 Fig. 4. Yaw trajectories ψ [ • ] v.s. t [s]. (a) Applying (4.14) and (4.15). (b) Applying only (4.14): (b) and.

1 )

 1 y}, in order to simplify the exponential matrix computation, where: The convolution integrals of (4.19) were synthesized with the discretization of the state space descriptions Σ(A F i , Bi), i ∈ {x, y}, in the time interval [0, Ts), Ts = 1, and with the initial conditions (4.23). 2) The exponential matrices of (4.19) are: exp (A Fx Ts) = Tx exp A Fx Ts T -1 Fy Ts = Ty exp A Fy Ts T -1

  a1,x = 3214, a2,x = 1.911, ωx = 1.585, bx = 0.9536, a1,y = 3358, a2,y = 1.911, ωy = 1.585, by = 0.9536, [ k1,x k2,x k3,x k4,x ] =

  ) where: e df,i = w df,iζ i . The transfer function of Σ(A CL df,i , B CL i , C CL i ) is (cf.

	(4.18) and (4.19) of [2]):

  + ax o,1 s 2 + ax o,2 s + ax o,3 ) s 4 + ax o,1 s 3 + ax o,2 s 2 + ax o,3 s + ax o,4 + ax o,1 s 2 + ax o,2 s + ax o,3 ) s 4 + ax o,1 s 3 + ax o,2 s 2 + ax o,3 s + ax o,4

	4.2. Simulación numérica					4.2. Simulación numérica
	del sistema en lazo cerrado ūx, ver figura 4.1.		del sistema en lazo cerrado ūx, ver figura 4.1.
	Para los valores obtenidos en la subsección 4.2.2 (cf. (4.19) y (4.18)) se tiene Para los valores obtenidos en la subsección 4.2.2 (cf. (4.19) y (4.18)) se tiene
	G(s) = 1 -s(s 3 = ax o,4 πe x (s) = s(s + 351.5)(s + 296)(s + 259) (s + 74) 2 (s 2 + 758.5 s + 154012.5)		(4.41) G(s) = 1 -ax o,4 πe x (s) s(s + 351.5)(s + 296)(s + 259) = s(s 3 = (s + 74) 2 (s 2 + 758.5 s + 154012.5)
	=	s (s 3 + 906.5 s 2 + 271746.5 s + 26947396) s 4 + 906.5 s 3 + 271746.5 s 2 + 26947396 s + 843372455	.	=	s (s 3 + 906.5 s 2 + 271746.5 s + 26947396) s 4 + 906.5 s 3 + 271746.5 s 2 + 26947396 s + 843372455	.
	En la figura 4.3 se muestra el diagrama de Bode de (4.41).
	10
	0				|p1| 6 ωc	ωn 6 |s2| ? 6 ? |s3| |s4|
	-10
	-20
	-30
	-40
	10 -2			10 0		10 2
		ωc =	ax o,2 2	-	a 2 xo,2 -2ax o,4 2	= 39.5 rad/s .	(4.42)
					43

? Figura 4.3: Gráfica de Bode de magnitud. |p1| = 74, |s2| = 351.5, |s3| = 296, |s4| = 259, ωn = 392.44 rad/s y ζ = 0.966. De (4.41) y de la figura 4.3, se tienen las siguientes observaciones 1. La frecuencia de corte del filtro es

  In , we give details on the numerical aspects of (4.19) and (4.20).

y /dt, x * 4,y = dx * 3,y /dt. 11 Recall footnote 8. 12

Cf. Lemma 1 and Theorem 1 of[START_REF] Blas | Synthesis of a robust linear structural feedback linearization scheme for an experimental quadrotor[END_REF].

See Section IV.D of[START_REF] Blas | Synthesis of a robust linear structural feedback linearization scheme for an experimental quadrotor[END_REF].

πw df,i (s) = det(s I 4 -((A F i + K i C i ) + (1 -)B i G i C i )).

In this section we present all the numerical values with four significant digits, but in the outdoor real-time experiments we have used eight significant digits.