Combination of terminal sliding mode and finite-time state-dependent Riccati equation: flapping-wing flying robot control

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering available at: 10.1177/09596518221138627 “© 2022 SAGE Journals. Personal use of this material is permitted. Permission from SAGE Journals must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 1 Combination of terminal sliding mode and finitetime state-dependent Riccati equation: Flappingwing flying robot control Saeed Rafee Nekoo, José Ángel Acosta, and Anibal Ollero GRVC Robotics Lab., Departamento de Ingeniería de Sistemas y Automática, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Seville, Spain [email protected], (jaar, aollero)@us.es Abstract A novel terminal sliding mode control (TSMC) is introduced to control a class of nonlinear uncertain systems in finite-time. Having command on the definition of the final time as an input control parameter is the goal of this work. TSMC is naturally a finite time controller though the time cannot be set as input, and the convergence time is not exactly known to the user before execution of the control loop. The sliding surface of the introduced controller is equipped with a finite-time gain that finishes the control task in the desired predefined time. The gain is found by partitioning the state-dependent differential Riccati equation (SDDRE) gain, then arranging the sub-blocks in a symmetric positive-definite structure. The SDDRE is a nonlinear optimal controller with a final boundary condition that penalizes the states at the final time. This guides the states to the desired condition by imposing extra force on the input control law. Here the gain is removed from standard SDDRE control law (partitioned and made symmetric positive-definite) and inserted into the nonlinear sliding surface to present a novel finite-time TSMC. The stability of the proposed TSMC is guaranteed by the definition of the adaptive gain of TSMC, which is limited by the Lyapunov stability condition. The proposed approach was validated and compared with SDDRE and conventional TSMC as independent controllers, applied on a Van der Pole oscillator. The capability of the proposed approach of controlling complex systems was checked by simulating a flapping-wing flying robot (FWFR). The FWFR possesses a highly nonlinear model with uncertainty and disturbance caused by flapping. The flight assumptions also limit the input law significantly. The proposed TSMC successfully controlled the illustrative example and FWFR model and has been compared with SDDRE and conventional TSMC. Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 2 Keywords: Terminal sliding mode control; SDRE; Optimal control; Nonlinear control; Unmanned aerial vehicle. 1. Introduction The sliding mode control (SMC) and variable structure systems are among the most popular nonlinear robust methods; they find the control law by defining a proper sliding surface. The SMC has so many branches such as a second-order, adaptive, terminal, and also in combination with other techniques including fuzzy, neural-network, optimal, optimalintegral, etc. [1]. The focus of this current research is on terminal sliding mode control (TSMC). The TSMC is an updated structure of the SMC, introduced in the 90s [2]. The terminal attractors in the new controller guarantee the finite-time state convergence. The nonlinear definition of the sliding surface makes the manifold an attractor [3]. Yu et al. used a continuous TSMC for controlling rigid robotic manipulators [4]. The TSMC was improved in many aspects: a nonsingular version of the controller [5, 6], fast terminal sliding mode control [7], fractional-order one [8], adaptive nonsingular [9], etc. Wang and Xu investigated the micropositioning control system via an adaptive terminal sliding mode control [10]. The micropositioning system used piezoelectric actuators that had resonance; the adaptive novel gain controlled the uncertain system with high performance. Xiong and Zhang employed global fast TSMC for quadrotor control in set-point regulation [11]. The system was divided into two fully- and under-actuated system and controlled successfully. This current work intends to find a finite-time controller with robust characteristics. The TSMC is a good selection though the final time cannot be set as a control input parameter [12]. Shiri et al. solved the problem by dividing the sliding surface into two linear (conventional SMC) and nonlinear sections (nonsingular TSMC) [12]. Then it was possible to define the final time as an input to the control system. The controller was applied for a Van der Pole oscillator and a robotic manipulator. Here in this current work, the time setting approach is used and the final time could be defined as an input to the control system. On the contrary to Ref. [12], here in this work, one nonlinear sliding surface is considered. To have a command over the final time, an optimal gain of the state-dependent differential Riccati equation (SDDRE) is used. The SDDRE is a nonlinear closed-loop suboptimal controller with the power of finite-time control by penalizing a terminal boundary condition [13, 14]. Tripathy used the state-dependent differential Riccati equation for manipulator control [15]. There are several solution methods Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 3 to the differential Riccati equation such as backward integration [16], Lyapunov-based method [17, 18], and state transition matrix approach [19]. Here we used the Lyapunov-based approach as the solution to the SDDRE with negative gain. The gain of the SDDRE could be partitioned and used in other controllers preserving the finite-time characteristics. Recently, the flapping-wing flying robot (FWFR) has been a challenging model for the assessment of the controllers in terms of robustness and performance. The model is highly nonlinear and coupled with force/moment components. One of the open problems in this field is the dynamic modeling of a flapping wing system. The six-degree-of-freedom (DoF) in modeling leads to complicated systems especially in terms of lift, drag, and aerodynamics calculations. The modeling of aircraft (Refs. [20, 21]) is the base of the modeling though the flapping wing adds complexity and changes to the system. The dynamic derivation of the gliding phase of the flapping systems is very similar to aircraft dynamic; however, the inertia and weight, and propulsion forces are significantly lower. The investigation began by looking into the bird’s flight dynamics [22, 23]. Taylor and Thomas studied the flapping flight dynamics, longitudinal stability of animal flights [22]. Deng et al. researched flapping flight dynamics for insect-sized systems; studied the lift, drag, and body dynamic [24]. Platzer et al. reviewed the progress and challenges of flapping-wing flight considering different wing configurations [25]. It was found that thrust, lift, and propulsive efficiency on flapping mode depend on the frequency and amplitude of the flapping. The insect-sized flapping-wing systems have been attractive since the actuation of them and the weight of the system provided the flight situation more comfortable than large-scale flapping systems [26-33]. In that case, significant load-carrying capacity is not expected. Increasing the size of the flapping-wing flying robot requires a complicated mechanical mechanism to obtain constant flapping and control over the frequency. Armanini et al. used flight data of experiments to define ornithopter dynamics through global system identification [34]. Hassan and Taha showed that the flight path of the flapping robot was subjected to vibration [35]. This result was found by relaxing the common assumption that neglected the wing inertial effects and directly averaged the dynamics over the flapping cycle. Bakhtiari et al. studied the modeling focusing on the aerodynamics effects on the edge of the tail, experimentally [36]. All the dynamics modeling (some works with the experimental investigation on lift, drag, and aerodynamics) and control implementations (mostly in theoretical schemes and simulations) lead the researchers towards carrying more weight to Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 4 apply the FWFR for a particular task [37-40]. One application is perception and visual inspection by FWFR which requires more lift capacity due to necessary external devices such as a camera, digital board, and heavier batteries [39]. The majority of the modeling used general dynamic modeling of a rigid body in gliding conditions with added lift, drag, and aerodynamics forces. The representations of the rigid body were similar though the aerodynamics modeling was different based on the situation and condition. In this work, six-DoF modeling of the FWFR using rigid body dynamics is considered. The aerodynamics of the system is added to the dynamic equation and the relation between the force/moment inputs and rudders and thrust actuators is presented using Taylor series expansion (linearized model). It should be emphasized that the model is nonlinear and coupled and the relation between force/moment and actuators is only linearized. The main contributions: C1. Introducing a terminal sliding mode to control a class of nonlinear uncertain systems in predefined finite-time; that is defined as an input parameter. C2. Stability proof of the TSMC gain for a predefined finite-time control using the partitioned SDDRE control gain. C3. Six-DoF fully-actuated SDDRE-based TSMC control of FWFR (modeling and simulation). The rest of the paper follows: Section 2 expresses the control structure of the proposed TSMC. Dynamic modeling of the FWFR is expressed in Section 3. State-space form and control implementation are reported in Section 4. Section 5 presents simulations: illustrative example, FWFR, and comparison with conventional SDDRE and TSMC. Concluding remarks are reported in Section 6. Notations: ℝ𝑛 denotes the 𝑛-dimensional Euclidean space; ℝ𝑛×𝑚 is the set of 𝑚×𝑛 real matrices; (∙)𝑇 is the transpose of a matrix or a vector; diag(∙) means a diagonal matrix; 𝐈𝑛×𝑛 and 𝟎𝑛×𝑛 denote 𝑛×𝑛 identity and zero matrices. |∙| defines absolute value and ‖∙‖2 is norm- 2 of a vector or a matrix. eig[𝐀] provides a diagonal matrix with eigenvalues of 𝐀. 2. Terminal Sliding Mode Control 2-1. System Definition Consider a second-order nonlinear differential equation Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 5 𝐪󰇘=𝐟(𝐪,𝐪󰇗)+𝐄(𝐪)𝐮+𝚿(𝐪,𝐪󰇗,𝑡), (1) subjected to unknown external disturbance 𝚿(𝐪,𝐪󰇗,𝑡):ℝ𝑛×ℝ𝑛×ℝ+→ℝ𝑛, which is bounded 𝚿(𝐪,𝐪󰇗,𝑡)≤𝚿c(𝐪,𝐪󰇗,𝑡). Equation (1) presents a fully actuated mechanical system. The generalized coordinates of system (1) are collected in 𝐪∈ℝ𝑛 vector and the input force or torque in 𝐮∈ℝ𝑛 in which 𝑛 denotes the degree-of-freedom (DoF). The system is limited to a common form that covers a wide range of mechanical systems by defining 𝐟(𝐪,𝐪󰇗)= −𝐌−1(𝐪)(𝐜(𝐪,𝐪󰇗)+𝐠(𝐪)+𝐃(𝐪,𝐪󰇗)𝐪󰇗) and 𝐄(𝐪)=𝐌−1(𝐪) where 𝐌(𝐪):ℝ𝑛→ℝ𝑛×𝑛 is inertia matrix, 𝐜(𝐪,𝐪󰇗):ℝ𝑛×ℝ𝑛→ℝ𝑛 is Coriolis and centrifugal vector, 𝐠(𝐪):ℝ𝑛→ℝ𝑛 is a vector that includes gravity terms, 𝐃(𝐪,𝐪󰇗):ℝ𝑛×ℝ𝑛→ℝ𝑛×𝑛 collects all terms related to friction, drag (underwater or flying systems), aerodynamics forces, etc. Selecting the state vector as 𝐱2𝑛×1=[𝐪𝑇,𝐪󰇗𝑇]𝑇, the reduced-order differential equation (state-space representation of (1)) is found: 𝐱󰇗=𝐚(𝐱)+𝐠(𝐱,𝐮)=[ 𝐪󰇗 𝐟(𝐪,𝐪󰇗)]+[𝟎𝑛×𝑛 𝐄(𝐪)]𝐮+[ 𝟎𝑛×1 𝚿(𝐪,𝐪󰇗,𝑡)], (2) where 𝐚(𝐱):ℝ2𝑛→ℝ2𝑛 and 𝐠(𝐱,𝐮):ℝ2𝑛×ℝ𝑛→ℝ2𝑛 are piecewise-continuous vectorvalued functions that satisfy the Lipschitz condition. 2-2. The Symmetric Gain of State-dependent Riccati Equation In this section, the finite-time suboptimal gain of the SDRE will be partitioned into four square blocks and will be used in the definition of the sliding surface in Section 2-3. The SDRE control law will not be used directly; however, the finite-time gain will be used in the definition of the TSMC sliding surface. The first step is to rewrite state-space representation (2) in the statedependent coefficient (SDC) parameterization: 𝐱󰇗=𝐀(𝐱)𝐱+𝐁(𝐱)𝐮=[𝟎𝑛×𝑛 | 𝐈𝑛×𝑛 −−− | −−−−−−−−−−−−− 𝟎𝑛×𝑛 | −𝐌−1(𝐪)(𝐂(𝐪,𝐪󰇗)+𝐃(𝐪,𝐪󰇗))]𝐱+[𝟎𝑛×𝑛 𝐄(𝐪)]𝐮, (3) where 𝐀(𝐱):ℝ2𝑛→ℝ2𝑛×2𝑛, 𝐁(𝐱):ℝ2𝑛→ℝ2𝑛×𝑛 and 𝐜(𝐪,𝐪󰇗)=𝐂(𝐪,𝐪󰇗)𝐪󰇗. Remark 1. The gravity vector usually contains trigonometric functions (robotic arms) or constant values (flying objects). The cos(∙) function cannot be factored after using Taylor series expansion since the first term is 1. Also dividing cos(∙) 𝑥𝑖 imposes singularities at equilibrium 𝑥𝑖=0. So, the gravity vector will be Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 6 removed from SDC matrices (3), and will be compensated in TSMC control law. Condition 1. {𝐀(𝐱),𝐁(𝐱)} is a completely controllable parameterization of system (2) for all 𝐱 in 𝑡∈[0,𝑡f] [41]. Property 1. 𝐌(𝐪) is a symmetric positive-definite matrix such that 𝐌1≤𝐌(𝐪(𝑡))≤𝐌2 where 𝐌1,𝐌2∈ℝ𝑛×𝑛 are constant symmetric positive definite matrices. Remark 2. Concerning Property 1, the diagonal elements of 𝐄(𝐪) are always positive even at 𝐪=𝟎. The pair of 𝐀(𝟎) and 𝐁(𝟎) is completely controllable and in the rest of the domain, the worst-case scenario is the pair of [𝟎 𝐈 𝟎 −𝐄(𝟎)𝐃] and [𝟎 𝐄(𝟎)]. It is clear that the case of [𝟎 𝐈 𝟎 𝟎] and [𝟎𝐈] is controllable. So, the SDC pair in (3), presents a completely controllable parameterization. The cost function is: 𝐽=12(∫[𝐮𝑇𝐑(𝐱)𝐮+𝐱𝑇𝐐(𝐱)𝐱]d𝑡 𝑡f 0+𝐱𝑇(𝑡f)𝐅𝐱(𝑡f)), (4) where 𝐑(𝐱):ℝ2𝑛→ℝ𝑛×𝑛 is input weighting matrix, symmetric positive-definite, and 𝐐(𝐱):ℝ2𝑛→ℝ2𝑛×2𝑛 is state weighting matrix, symmetric positive-semi-definite. 𝐅∈ℝ2𝑛×2𝑛 also penalizes the states at the terminal time 𝑡f; 𝐅=𝐅𝑇 and 𝐅≥𝟎. Condition 2. {𝐀(𝐱),𝐐12(𝐱)}, in (4), is a completely observable parameterization of system (2) for all 𝐱 in 𝑡∈[0,𝑡f], where 𝐐1/2(𝐱) is Cholesky decomposition of 𝐐(𝐱) [41]. Defining all matrices, 𝐀(𝐱), 𝐁(𝐱), 𝐐(𝐱), 𝐑(𝐱) and 𝐅, one could solve the state-dependent differential Riccati equation: −𝐊󰇗(𝐱)=𝐀𝑇(𝐱)𝐊(𝐱)+𝐊(𝐱)𝐀(𝐱)+𝐐(𝐱)−𝐊(𝐱)𝐁(𝐱)𝐑−1(𝐱)𝐁𝑇(𝐱)𝐊(𝐱), (5) with terminal boundary condition 𝐊(𝐱(𝑡f))=𝐅. Lemma 1. The nonlinear system (3), with SDC pair based on Conditions 1 and 2, can be stabilized using the standard SDDRE control law 𝐮=−𝐑−1(𝐱)𝐁𝑇(𝐱)𝐊(𝐱)𝐱, in which 𝐊(𝐱) is symmetric positive-definite solution to SDDRE (5) [42]. The proof of Lemma 1 could be found by using the Lyapunov candidate 𝑉(𝐱)= 𝐱𝑇𝐊(𝐱)𝐱. The details of solution methods to (5) could be visited in Ref. [42]; we suppose that Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 7 SDDRE could be solved with one of the methods such as the Lyapunov-based approach with negative gain. The control gain, the solution to the SDDRE, is partitioned into four-square blocks [43]: 𝐊(𝐱)=[𝐊11(𝐱)𝐊12(𝐱) 𝐊12 𝑇(𝐱)𝐊22(𝐱)]2𝑛×2𝑛, and is substituted in overall control gain 𝐗(𝐱)=𝐑−1(𝐱)𝐁𝑇(𝐱)𝐊(𝐱)=[𝐗1(𝐱)𝐗2(𝐱)]𝑛×2𝑛, in which 𝐗1(𝐱)=𝐑−1(𝐱)𝐌−1(𝐪)𝐊12 𝑇(𝐱), 𝐗2(𝐱)=𝐑−1(𝐱)𝐌−1(𝐪)𝐊22(𝐱). The gain 𝐗1(𝐱) is multiplied by the position vector, 𝐪(𝑡), and 𝐗2(𝐱) by the velocity vector, 𝐪󰇗(𝑡). From a general point of view 𝐗1(𝐱) acts as proportional gain and 𝐗2(𝐱) as a derivative one. Remark 3. The proportional and derivative gains 𝐗1(𝐱) and 𝐗2(𝐱) possess finite-time characteristics for the regulation of the error and velocity of the error vector to zero, tuned by 𝐅. The finite-time proportional gain of the SDDRE is required for the definition of the sliding surface in Section 2-3, though it is not a symmetric positive definite gain. To define the gain in that shape, and preserve the finite time characteristics, it will be transformed into [43]: 𝐊SP(𝐱)=𝐊12(𝐱)𝐌−1(𝐱)𝐑−1(𝐱)𝐌−1(𝐱)𝐊12 𝑇(𝐱) ‖𝐊12(𝐱)𝐌−1(𝐱)‖2, (6) where 𝐊SP(𝐱) is made symmetric by multiplication of 𝐊12(𝐱)𝐌−1(𝐱) from the left-hand side of the equation and leveled by division to the norm-2 of that ‖𝐊12(𝐱)𝐌−1(𝐱)‖2. If the weighting matrix of states (preserving symmetric positive-definite condition) at final time is partitioned into four submatrices: 𝐅=[𝐅11 𝐅12 𝐅12 𝑇𝐅22], (7) then, the 𝐅12 sub-block will define the final boundary condition of 𝐊12(𝐱) in (6), as 𝐊12(𝑡f)= 𝐅12. However, 𝐅12 is sign-indefinite and for bounding 𝐊SP(𝐱) in Section 2-3, a positive definite 𝐅12 is required. As a result, the following conditions are considered: 𝐅11, 𝐅12, and 𝐅22 are diagonal matrices with positive components, and 𝐅12<√𝐅11𝐅22. This provides a positive definite symmetric 𝐅 and also 𝐅12. Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 8 Assumption 1. It is assumed that 𝐅11, 𝐅12, and 𝐅22, in Eq. (7), are diagonal matrices with positive components, and 𝐅12<√𝐅11𝐅22, then it is also assumed that eig[𝐊SP(𝐱)]≤𝐅12 for all 𝐱∈ℝ2𝑛 in 𝑡∈[0,𝑡f]. Assumption 1 implies that the final weighting matrix must be chosen relatively bigger than the weighting matrix of states, 𝐅≫𝐐(𝐱), or in other words, 𝐅12≫[𝐐12(𝐱)]𝑛×𝑛 to force the solution to start from an initial condition and move to a large final boundary condition. 2-3. The SDRE-TSMC Design Consider the state-space representation of the general nonlinear affine-in-control timeinvariant system (2). The goal is to define a robust nonlinear control law to regulate the error of the system to the desired condition in predefined terminal time. The terminal sliding mode is a finite-time controller in nature though the user cannot set the final time as an input to the control problem using common standard forms. The terminal sliding mode controller is finite time in nature; however, the final time cannot be introduced as an input to the control system. The reason is that the total final time is the summation of rise time and settling time, 𝑡f=𝑡r+ 𝑡s. Calculation of 𝑡s needs the computation of 𝑒(𝑡r), and this value (error at rise time) is unknown before simulation of the system or implementation of the experiment. So, while the terminal time is finite, the user does not know the exact final time before the implementation. Shiri et al. used two sliding surfaces to present a structure accepting the final time as an input to a nonsingular terminal sliding mode controller [12]. A linear sliding surface was responsible to define the reaching time and the nonlinear one set the convergence time; together they could determine the total final time. The combination of terminal sliding mode and finite-time statedependent Riccati equation provides a solution for determining the final time before implementation of the controller and forces the system to finish its control task by weighting matrices of the SDRE. Here, one nonlinear sliding surface is selected with a finite-time gain to present a simpler structure. The nonlinear sliding surface is designed as 𝑠𝑖=𝑒󰇗𝑖+𝛽sign(∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 )|∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |𝑞𝑝, (8) for 𝑖=1,…,𝑛; 𝛽>0, 𝑞 and 𝑝 are positive odd integers following 𝑝>𝑞. ∑𝐾SP,𝑖𝑗(𝐱) 𝑛𝑗=1 denotes the 𝑖-th row of proportional SDDRE gain 𝐊SP(𝐱). Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 15 which results in [𝑡s,1 ⋮ 𝑡s,𝑛]≤ 1 𝛽(1−𝑞𝑝)𝐊C−1|𝐊C𝐞(𝑡r)|1−𝑞𝑝. (26) Considering Eq. (26) and if time tends to the final time of control task, 𝑡f, the control gain goes to final boundary value, lim 𝑡→𝑡f𝐊SP(𝐱(𝑡))=𝐅12. Each rows of right hand side of (26) define a settling time, 𝑡s,𝑖, for 𝑖-th error variable 𝑒(𝑡r,𝑖). 𝐅12 is a diagonal matrix with relatively large components, and it provides a situation where the 𝑖-th component dominates the rest of the arrays of 𝐊SP(𝐱(𝑡)) in each row of the matrix. The nature of the finite-time gain of the SDDRE includes three phases: transient, steady-state, and final oscillations (due to the final weighting condition 𝐅12), Fig. 2. Fig. 2. The behavior of the SDDRE control gain. Considering Fig. 2, the transient response is passed after reaching time, and after the steady-state value of the gain, the solution goes to the final boundary value. Finally, Eq. (26) is rewritten as [𝑡s,1 ⋮ 𝑡s,𝑛]≤ 1 𝛽(1−𝑞𝑝)𝐅12 −𝑞𝑝|𝐞(𝑡r)|1−𝑞𝑝, (27) which defines the terminal time 𝑇=max(𝑡r,𝑖+𝑡s,𝑖), and that concludes the proof. It perfectly makes sense; if one increases 𝐅12, it will force the system to a faster response, see Eq. (27), and results in a shorter 𝑡s,𝑖. ■ Remark 6. Theorem 2 expresses that the error of the system will converge to zero in finite time 𝑇, and this value is the shortest possible time, best case scenario. 𝑡 𝐾 SP (𝑡) steady-state 𝑡 r 𝑇 𝐾 SP (𝑡 r ) 𝑡 s 𝐹 12 Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 16 Considering [𝑡s,1 ⋮ 𝑡s,𝑛]≤ 1 𝛽(1−𝑞 𝑝)𝐊SP −𝑞 𝑝(𝑡r,𝑖)|𝐞(𝑡r)|1−𝑞 𝑝 computes a more realistic value, longer though finite, for the final time of error convergence. Remark 7. The role of control over final time is played by the SDDRE control gain 𝐊SP(𝐱) in Eq. (6), tuned by weighting matrices 𝐐(𝐱), 𝐑(𝐱) and 𝐅. Remark 8. The role of the chattering and elimination of that will be discussed in the simulation section. To keep the derivation intact based on the sign(∙) function, saturation function was not considered in the derivation. However, to remove the chattering, saturation function or other similar ones such as tanh(∙) 𝜀 could be used instead of sign(∙) in implementation. 3. Dynamic Modeling of Flapping-wing Flying Robot The weight of the body is light and the contribution of the aerodynamics of the wing is significant to the model. In reality, birds can perform various stable maneuvers during flight such as hovering, backward flight, diving, surging, etc. Such complete control over the motion expresses hybrid complex dynamics and continuous non-affine actuators. The mechanical design of a bird with all those properties might not be possible at the moment. This section intends to present a simplified six-DoF model of an FWFR with a minimum requirement of fully actuation mode. The coordinates and schematic model of the flapping-wing robot are presented in Fig. 3. The FWFR is not limited to a fully bioinspired design; so, a rudder was set on the tail for better control over the roll and yaw. Fig. 3. The reference frame and model of the FWFR. 𝛿LR 𝐹drag 𝛿RW 𝛿UD 𝐹lift 𝐹T(thrust) 𝑊 (weight) 𝛿LW 𝛿L 𝑤 𝑝 𝑟 𝑢 𝑣 𝑞 𝜃 𝜙 𝜓 (𝑥c,𝑦c,𝑧c) on CoM 𝑋 𝑌 𝑍 Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 17 To cover a fully actuated model, six inputs are necessary, defined in Fig. 3. Why six inputs? There are several flapping robots controlled experimentally by only two inputs on their tail and one input thrust (generated by constant flapping) [37, 38, 40]. These models presented underactuated systems subjected to many limitations with a lack of control over the motion. The landing was also like a fall by reducing the flapping frequency. Considering six actuators simplifies the control problem and extends the model to be used by so many methods, and closes the gap between the mechanical models and real birds. The three additional inputs are folding parts near the tip of the wings represented by 𝛿RW and 𝛿LW for the right and left wing; and a unified rudder on the back of the wings to enhance the lift. Observing the flapping birds, it can be realized that they are moving continuously possessing curved shapes and different parts for changing the lift force on the wing. So, the selected inputs are 𝐮(𝑡)= [𝛿RW,𝛿LW,𝛿L,𝐹T,𝛿UD,𝛿LR]𝑇 where 𝛿(∙) (rad) represents the rudders and folding blades on the tails and wings and 𝐹T (N) is thrust force caused by flapping of the wings. The following assumptions are held for the modeling and behavior of the system: 1. The FWFR moves only forward in 𝑥c axis direction. It can turn in a wide circular path; however, it cannot perform the lateral pure translation in 𝑦c. 2. There is always a positive initial velocity condition for 𝑢(0)=𝑢0 where 𝑢0>0. This represents throwing the FWFR to gain an initial lift and thrust for flight. 3. The flight path ought to be long enough for the FWFR to pass transient flight conditions. Specifically, 𝑥c(𝑡f) should be significantly greater than 𝑦c(𝑡f) and 𝑧c(𝑡f). 4. The FWFR motion avoids severe changes in orientation and the main changes are limited to small deviations due to the flapping. The fixed reference frame is set by (𝑋,𝑌,𝑍), known as Earth frame; and the body (moving) coordinate is also presented by (𝑥c,𝑦c,𝑧c) placed on the center-of-mass (CoM) of the system. Four main forces are acting on the robot, the total weight of the system 𝑊, lift, drag, and thrust force. The generalized coordinates of the system are translation and orientation of the system in six-DoF, presented by 𝐪(𝑡)={𝛏1(𝑡),𝛏2(𝑡)} where 𝛏1(𝑡)= [𝑥c(𝑡),𝑦c(𝑡),𝑧c(𝑡)]𝑇 (m) and 𝛏2(𝑡)=[𝜙(𝑡),𝜃(𝑡),𝜓(𝑡)]𝑇 (rad). On the body frame, local linear and angular velocities are named as {𝛖1(𝑡),𝛖2(𝑡)} where 𝛖1(𝑡)= Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 18 [𝑢(𝑡),𝑣(𝑡),𝑤(𝑡)]𝑇 (m/s) and 𝛖2(𝑡)=[𝑝(𝑡),𝑞(𝑡),𝑟(𝑡)]𝑇 (rad/s). The kinematics relation between the global and local frame, considering Flight-Assumption 4, is presented [50]: 𝛏󰇗1=𝐑𝑍𝑌𝑋(𝛏2)𝛖1, (28) 𝛏󰇗2=𝐓(𝛏2)𝛖2, (29) where 𝐑𝑍𝑌𝑋(𝛏2)=[𝑐𝜃𝑐𝜓𝑠𝜙𝑠𝜃𝑐𝜓−𝑐𝜙𝑠𝜓𝑠𝜙𝑠𝜓+𝑐𝜙𝑠𝜃𝑐𝜓 𝑐𝜃𝑠𝜓𝑠𝜙𝑠𝜃𝑠𝜓+𝑐𝜙𝑐𝜓−𝑠𝜙𝑐𝜓+𝑐𝜙𝑠𝜃𝑠𝜓 −𝑠𝜃𝑠𝜙𝑐𝜃𝑐𝜙𝑐𝜃], 𝐓(𝛏2)=[1 𝑠𝜙𝑡𝜃𝑐𝜙𝑡𝜃 0 𝑐𝜙−𝑠𝜙 0 𝑠𝜙𝑐𝜃 ⁄𝑐𝜙𝑐𝜃 ⁄], in which e.g. 𝑐𝜃=cos(𝜃(𝑡)) and 𝑡𝜃=tan(𝜃(𝑡)). The matrix 𝐓(𝛏2)=𝐖−1(𝛏2) holds in which 𝛖2=[𝐈3×3[𝜙󰇗00];𝐑𝑋(𝜙)[0𝜃󰇗0];𝐑𝑋(𝜙)𝐑𝑌(𝜃)[00𝜓󰇗]]=[1 0 −𝑠𝜃 0 𝑐𝜙𝑐𝜃𝑠𝜙 0 −𝑠𝜙𝑐𝜃𝑐𝜙]𝛏󰇗2=𝐖(𝛏2)𝛏󰇗2, and 𝐖(𝛏2) is invertible if det(𝐖(𝛏2))=cos𝜃≠0. The kinematics, Eqs. (28) and (29), and the reference frames are in the common forms of flying objects in the aerospace community such as aircraft, quadcopters, etc. The dynamics of the FWFR is 𝐌(𝐪)𝐪󰇘+𝐂(𝐪,𝐪󰇗)𝐪󰇗+𝐃(𝐪,𝐪󰇗)𝐪󰇗+𝐠(𝐪)=𝐔, (30) where 𝐌(𝐪):ℝ6→ℝ6×6 is the inertia matrix, [𝐂(𝐪,𝐪󰇗)𝐪󰇗]:ℝ6×ℝ6→ℝ6 is Coriolis and centrifugal forces, [𝐃(𝐪,𝐪󰇗)𝐪󰇗]:ℝ6×ℝ6→ℝ6 collects drag and aerodynamics terms, 𝐠(𝐪):ℝ6→ℝ6 is gravity vector and 𝐔∈ℝ6 is input vector: 𝐌(𝐪)=[𝑚×𝐈3×3 𝟎3×3 𝟎3×3 𝐉(𝛏2)],𝐂(𝐪,𝐪󰇗)=[𝟎3×3 𝟎3×3 𝟎3×3 𝐂(𝛏2,𝛏󰇗2)],𝐃(𝐪,𝐪󰇗)=[𝐃 (𝛏󰇗2) 𝟎3×3 𝟎3×3 𝟎3×3],𝐠(𝐪) =[0,0,𝑚𝑔,0,0,0]𝑇, in which 𝑚 (kg) is the total mass of the FWFR, 𝑔 (𝑚 𝑠2) is gravity constant, and: 𝐉(𝛏2)= 𝐖𝑇(𝛏2)𝐈𝐖(𝛏2), 𝐈=diag(𝐼𝑥𝑥,𝐼𝑦𝑦,𝐼𝑧𝑧), Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 19 𝐶11(𝛏2,𝛏󰇗2)=0,𝐶12(𝛏2,𝛏󰇗2)=(𝐼𝑦𝑦−𝐼𝑧𝑧)(𝜃󰇗𝑠2𝜙−𝜓󰇗𝑐2𝜙𝑐𝜃) 2,𝐶13(𝛏2,𝛏󰇗2) =−𝜃󰇗(𝐼𝑥𝑥𝑐𝜃+𝑐𝜃(𝐼𝑦𝑦−𝐼𝑧𝑧)𝑐2𝜙 2)−(𝐼𝑦𝑦−𝐼𝑧𝑧)𝜓󰇗𝑐𝜙𝑐𝜃2𝑠𝜙,𝐶21(𝛏2,𝛏󰇗2)=𝐼𝑥𝑥𝜓󰇗𝑐𝜃 2,𝐶22(𝛏2,𝛏󰇗2) =−(𝐼𝑦𝑦−𝐼𝑧𝑧)𝑠2𝜙(4𝜙󰇗−𝜓󰇗𝑠𝜃) 4,𝐶23(𝛏2,𝛏󰇗2) =𝜙󰇗(𝐼𝑥𝑥𝑐𝜃 2+(𝐼𝑦𝑦−𝐼𝑧𝑧)𝑐2𝜙𝑐𝜃)−(𝐼𝑦𝑦−𝐼𝑧𝑧)𝜃󰇗𝑠2𝜙𝑠𝜃 4 +𝜓󰇗𝑐𝜃𝑠𝜃(𝐼𝑦𝑦 2−𝐼𝑥𝑥+𝐼𝑧𝑧 2−𝐼𝑦𝑦𝑐2𝜙 2+𝐼𝑧𝑧𝑐2𝜙 2),𝐶31(𝛏2,𝛏󰇗2)=−𝐼𝑥𝑥𝜃󰇗𝑐𝜃,𝐶32(𝛏2,𝛏󰇗2) =(𝐼𝑦𝑦−𝐼𝑧𝑧)(2𝜙󰇗𝑐2𝜙𝑐𝜃−𝜃󰇗𝑠2𝜙𝑠𝜃) 2,𝐶33(𝛏2,𝛏󰇗2) =𝜙󰇗𝑠2𝜙(𝐼𝑦𝑦−𝐼𝑧𝑧)(𝑐2𝜃 2+12)−𝜃󰇗𝑠2𝜃(𝐼𝑦𝑦 2−𝐼𝑥𝑥+𝐼𝑧𝑧 2−𝐼𝑦𝑦𝑐2𝜙 2+𝐼𝑧𝑧𝑐2𝜙 2). The drag matrix in the translation part is defined as 𝐃 (𝛏󰇗2)= 𝜌2diag(𝐴𝑥𝐶𝐷𝑥𝑥󰇗c,𝐴𝑦𝐶𝐷𝑦𝑦󰇗c,𝐴𝑧𝐶𝐷𝑧𝑧󰇗c) where 𝐴𝑖 is the effective area of the system in 𝑖 axis and 𝐶𝐷𝑖 is the drag constant for 𝑖=𝑥,𝑦,𝑧. Based on Flight-Assumption 4, the drag and aerodynamics effect in orientation dynamics are neglected. The input vector includes two force and torque parts: 𝐔=[𝑋,𝑌,𝑍 ⏟ force ,𝐿,𝑀,𝑁 ⏟ torque]𝑇. The force/torque vector 𝐔 has a nonlinear complicated relation with ultimate input vector 𝐮, which is nonlinear non-affine and unknown: 𝐔=𝐅(𝐪,𝐪󰇗,𝐮). (31) Representation of the Eq. (31) using Taylor series expansion around zero is 𝐔=𝐅(𝐪,𝐪󰇗,𝟎)+𝜕𝐅(𝐪,𝐪󰇗,𝟎) 𝜕𝐮 𝐮+12𝜕2𝐅(𝐪,𝐪󰇗,𝟎) 𝜕𝐮2𝐮2+⋯. (32) All the rows of vector 𝐅(𝐪,𝐪󰇗,𝐮) are multiplied by 𝐮, or in other words, 𝐅(𝐪,𝐪󰇗,𝟎)=𝟎. Considering the equilibrium 𝐅(𝐪,𝐪󰇗,𝟎)=𝟎 and neglecting the higher terms of the Taylor series, Eq. (32) is approximated by 𝐔=𝜕𝐅(𝐪,𝐪󰇗,𝟎) 𝜕𝐮 𝐮= [ 𝑋𝛿RW 𝑋𝛿LW 𝑋𝛿L𝑋𝐹T𝑋𝛿UD 𝑋𝛿LR 𝑌𝛿RW 𝑌𝛿LW 𝑌𝛿L𝑌𝐹T𝑌𝛿UD 𝑌𝛿LR 𝑍𝛿RW 𝑍𝛿LW 𝑍𝛿L𝑍𝐹T𝑍𝛿UD 𝑍𝛿LR 𝐿𝛿RW 𝐿𝛿LW 𝐿𝛿L𝐿𝐹T𝐿𝛿UD 𝐿𝛿LR 𝑀𝛿RW 𝑀𝛿LW 𝑀𝛿L𝑀𝐹T𝑀𝛿UD 𝑀𝛿LR 𝑁𝛿RW 𝑁𝛿LW 𝑁𝛿L𝑁𝐹T𝑁𝛿UD 𝑁𝛿LR ] [ 𝛿RW 𝛿LW 𝛿L 𝐹T 𝛿UD 𝛿LR ] , (33) Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 20 where i.e. 𝑋𝛿RW=𝜕𝑋 𝜕𝛿RW. We call the coefficient matrix 𝐌x=𝜕𝐅(𝐪,𝐪󰇗,𝟎) 𝜕𝐮 a mixer matrix. In reality, for example, considering a real bird, it could be expected that all the inputs 𝐮 contribute to the force/torque vector 𝐔 and all the elements of the mixer have values; however, this model is simplified by several assumptions. To find the effective elements of the mixer, the effect of the inputs are considered in decoupled mode, then based on Fig. 3, one could rewrite Eq. (33) as: 𝐔= [ 𝑋𝛿RW −𝑋𝛿LW 0 𝑋𝐹Tcos𝜓 0 𝑋𝛿LR −𝑌𝛿RW 𝑌𝛿LW 0 𝑌𝐹Tsin𝜓 0 𝑌𝛿LR 0 0 𝑍𝛿L𝑍𝐹Tcos𝜃 𝑍𝛿UD 0 𝐿𝛿RW 𝐿𝛿LW 0 0 0 0 0 0 𝑀𝛿L0 𝑀𝛿UD 0 0 0 0 0 0 𝑁𝛿LR ] [ 𝛿RW 𝛿LW 𝛿L 𝐹T 𝛿UD 𝛿LR ] . The oscillation effect of flapping is more visible in 𝑧c axis and pitch angle 𝜃. Based on the flapping oscillation, a disturbance vector is applied to the dynamics, added to the system in state-space representation. The inputs are also bounded between 𝛿min≤𝛿(∙)≤𝛿max and 0≤ 𝐹T≤𝐹T,max. 4. State-space Form and Implementation The state variable of the system is selected as 𝐱=[𝛏1𝑇,𝛏2𝑇,𝛖1𝑇,𝛖2𝑇]𝑇=[𝑥c,𝑦c,𝑧c,𝜙,𝜃,𝜓,𝑢,𝑣,𝑤,𝑝,𝑞,𝑟]𝑇. (34) Considering the state vector (34) and the dynamics (30), the state-space form is obtained 𝐱󰇗=[ 𝐑𝑍𝑌𝑋(𝛏2)𝛖1 𝐓(𝛏2)𝛖2 𝐌−1(𝐪)(𝐔−𝐂(𝐪,𝐪󰇗)𝐪󰇗−𝐃(𝐪,𝐪󰇗)𝐪󰇗−𝐠(𝐪)−𝐃F)], (35) where 𝐃F=𝑑F×[0,0,cos𝜔𝑡,0,0,0]𝑇 is the disturbance caused by flapping in which 𝑑F is the amplitude of the oscillatory disturbance. Comparing Eq. (35) with (2), the values of 𝐟(𝐪,𝐪󰇗)=−𝐌−1(𝐪)(𝐂(𝐪,𝐪󰇗)𝐪󰇗+𝐃(𝐪,𝐪󰇗)𝐪󰇗+ 𝐠(𝐪)) and 𝐄(𝐪)=𝐌−1(𝐪) are defined. The SDC parameterization matrices are also shaped as Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 21 𝐀t(𝐱)=[𝟎3×3 | 𝐑𝑍𝑌𝑋(𝛏2) −−− | −−−−−−−− 𝟎3×3 | −1/𝑚𝐃 (𝛏󰇗2)],𝐀o(𝐱) =[𝟎3×3 | 𝐓(𝛏2) −−− | −−−−−−−− 𝟎3×3 | −𝐉−1(𝛏2)𝐂(𝐪,𝐪󰇗)],𝐁t(𝐱)=[𝟎3×3 1 𝑚𝐈3×3],𝐁o(𝐱)=[𝟎3×3 𝐉−1(𝛏2)], which results in the gain 𝐊SP(𝐱)=diag(𝐊SPt(𝐱),𝐊SPo(𝐱)). The nature of the orientation dynamic is faster than the translation. To solve the Riccati easier and avoid setting the eigenvalues of the dynamics near the imaginary axis, two separate SDDRE control is used. It should be noted that considering the smooth and small variation of orientation variables, 𝐑𝑍𝑌𝑋(𝛏2) and 𝐓(𝛏2) possess almost identity matrices. That is necessary to guarantee the controllability and observability conditions, Condition 1 and 2. The rest of the design is based on Section 2-3. 5. Simulations 5-1. Illustrative Example A Van der Pole oscillator is simulated and compared with two other controllers. The statespace representation of the disturbed Van der Pole equation is [12]: 𝐱󰇗=[0 1 −1 1−𝑥12]𝐱+[01]𝑢+[ 0 rand(1)×0.2sin100𝑡], where 𝛹(𝑡)=rand(1)×0.2sin100𝑡 is an external disturbance in which rand(1) generates a random number between 0 and 1 at each time-step of the simulation. The proposed method is finite-time and robust. It is good to compare with the SDDRE controller which is finite time though is not robust and also with the conventional TSMC which lacks optimality. The initial condition is 𝐱(0)=[2,1]𝑇 and the system is to be regulated to the equilibrium point. The control parameters are selected as 𝑅=1, 𝐐=𝐈2×2, 𝐅=10×𝐐, 𝜂=2, 𝛽=1.5, 𝑞=7, and 𝑝=9. We also consider 𝑓󰆹(𝐱)=1.2×(−𝑥1+(1−𝑥12)𝑥2), 𝐸=1.1, and 𝛹c(𝑡)=0.2sin100𝑡 which defines (for example here we considered rand(1)=0.501) the constants, based on Eqs. (10)-(12): 𝛹0=|max(𝛹)−𝐸𝐸−1max(𝛹c)|=|0.1060−𝐸𝐸−10.2|=0.0758, 𝑓0=|𝑓(𝐱)−𝐸𝐸−1𝑓󰆹(𝐱)|=|(1 1.2−𝐸𝐸−1)𝑓󰆹(𝐱)|=0.0758|𝑓󰆹(𝐱)|, Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 22 𝐸max=𝐸min=𝐸𝐸−1=0.9091. The parameters of the SDDRE and TSMC controllers are similar weighting matrices and parameters. The only change in the TSMC is the constant gain of 𝐾SP=1. The comparisons have been done in different terminal times to illustrate the role of predefined time-setting in the results. The error and input of the system in different final times are presented in Fig. 4 and the error details are in Table 1. The error of the proposed TSMC was obtained less than the TSMC and SDDRE which showed for a system with uncertainty, a combination of the robust and optimal control could score better results. For the 6s final time, the proposed TSMC improved 1 w.r.t. the TSMC 61% and SDDRE 61.3%; for the 4s final time, the proposed TSMC improved w.r.t. the TSMC 56.3% and SDDRE 87.3%; and for the 2s final time, the proposed TSMC improved w.r.t. the TSMC 82.4% and SDDRE 47.2%. Based on the presented data in Ref. [12], for 𝑡f=4s, the proposed TSMC response has been validated. Setting the final time as an input with two sliding surfaces was proposed in Ref. [12], though in this current work, a unified sliding surface is considered with additional sub-optimality in the gain. Table 1. The error details of the Van der Pole system in different final times. error at 𝑡f=6s 𝑡f=4s 𝑡f=2s TSMC 0.0138 0.0138 1.1773 SDDRE 0.0139 0.0478 0.3914 Proposed TSMC 0.0054 0.0060 0.2063 a, 𝑡f=6s b, 𝑡f=4s c, 𝑡f=2s d, 𝑡f=6s e, 𝑡f=4s f, 𝑡f=2s Fig. 4. The error and input of the system in different final times, Van der Pole oscillator. 1 Possible improvement (%): old data−new data old data ×100 Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 23 The chattering effect on the input is always an issue in the implementation of the different kinds of sliding mode controllers. Here we also observe the chattering in the input signals, Fig. 4 (d)- (f). For the sake of simplicity and derivations in the proof, it is better to consider the sign function in the process. Also, the sign function acts more precisely in finite-time regulation. For practical implementation and having an input signal without chattering, a saturation function or other similar ones such as tanh(∙) 𝜀 could be used. Here we considered tanh(∙) 𝜀 where 𝜀=0.1. The simulated Van der Pole with saturation function has resulted in Fig. 5. The error of the TSMC was gained 1.1817, SDDRE 0.3914, and proposed TSMC 0.2051 which outranked the other two. So, chattering is not a problem throughout this research and common functions for removing that show good performance as well. a b Fig. 5. Input signal and error of the Van der Pole using saturation function, chattering-free mode. Time analysis: Considering the final time of control problem 𝑡f=6s, the reaching time for conventional TSMC is found 𝑡r=1.78s, settling time 𝑡s=2.89s and the terminal time 𝑇= 4.67s. The states, error, and sliding surface are illustrated in Fig. 6-a. The phase plane of the systems is shown in Fig. 6-b. The error at the terminal time was gained 𝑒(𝑇=4.67)=0.0076 and at 𝑒(𝑡f=6)=0.0138, both of them in the steady-state range of error bound 0.02. Simulation of the proposed TSMC (with SDRE gain 𝐾SP(𝐱)) resulted in shorter reaching time 𝑡r=1.15s, longer settling time 𝑡s=6.67s and terminal time 𝑇=7.83s. However, error at final time was found 𝑒(𝑡𝑓=6)=0.0054, with 28.9% more precision. It should be noted that with this proposed design, the value of the error at the final time 𝑡f is important, not the settling time. Reducing the final time, results in 𝑒(𝑡f=5)=0.0067, and 𝑒(𝑡f=4)=0.0060, both of them less than 𝑒(𝑇=4.67) of conventional TSMC. Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 24 a b Fig. 6. (a) the state information; (b) phase plane. 5-2. Flapping-Wing Flying Robot The characteristics of the robot define the behavior of the system; the weight of an FWFR prototype is usually less than 1 (kg). The aerodynamics force and drag in the FWFR case play an important role. Unlike aircraft, during the regulation in 𝑋 axis, there are significant variations in other directions as well and therefore, aerodynamics force and drag should be included in three axes. The tuning of the controller is sensitive and dependent on the initial and final conditions. The physical characteristics of a sample FWFR are reported in Table 2. Table 2. Parameters and specifications of the FWFR. parameter value unit description 𝐼𝑥𝑥 0.0026 kgm2 moment of inertia around 𝑋 axis 𝐼𝑦𝑦 0.0010 kgm2 moment of inertia around 𝑌 axis 𝐼𝑧𝑧 0.0071 kgm2 moment of inertia around 𝑌 axis 𝑚 0.535 kg total mass of the system 𝑔 9.81 m/s2 gravity constant 𝜌 1.225 kg/m3 air density 𝐴𝑥 0.15 m2 effective area of system in 𝑋 axis 𝐴𝑦 0.24 m2 effective area of system in 𝑌 axis 𝐴𝑧 0.4 m2 effective area of system in 𝑍 axis 𝐶D,𝑥,𝐶D,𝑦,𝐶D,𝑧 0.045 − drag coefficient in 𝑋,𝑌,𝑍 axes The actuator limitations are applied by −𝜋6≤𝛿(∙)≤𝜋6 and −0.5248≤𝐹T≤7.8725. The elements of the mixer matrix are also considered as: 𝑋𝛿RW=0.05,𝑋𝛿LW=−𝑋𝛿𝑅𝑊,𝑋𝐹T=1.5,𝑋𝛿LR=2.45, 𝑌𝛿LR=𝑋𝛿LR,𝑌𝐹T=𝑋𝐹T,𝑌𝛿RW=0.75,𝑌𝛿LW=𝑌𝛿RW, 𝑍𝛿L=50.3,𝑍𝐹T=1.57,𝑍𝛿UD=1.75,𝐿𝛿RW=0.67, 𝐿𝛿LW=𝐿𝛿RW,𝑀𝛿L=1.74,𝑀𝛿UD=1.4,𝑁𝛿LR=0.00224. Then the inputs could be found by 𝐮=𝐌x−1𝐔, where the limits are applied by Preprint version of the paper: Nekoo, Saeed Rafee, JoséÁngel Acosta, and Anibal Ollero. "Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering (2022): 09596518221138627. DOI: https://doi.org/10.1177/09596518221138627 31 to compute the time derivative of the sliding surface, considering 𝑑 𝑑t|𝑓(𝑡)|=𝑓(𝑡)𝑓󰇗(𝑡) |𝑓(𝑡)|, as in the following: 𝑠󰇗𝑖 =𝑒󰇘𝑖+𝛽(∑𝐾󰇗SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 +∑𝐾SP,𝑖𝑗(𝐱)𝑒󰇗𝑗 𝑛 𝑗=1 )|∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |𝑞𝑝−1 +𝛽(𝑞𝑝−1)(∑𝐾󰇗SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 +∑𝐾SP,𝑖𝑗(𝐱)𝑒󰇗𝑗 𝑛 𝑗=1 )∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛𝑗=1 |∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛𝑗=1 ||∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |𝑞𝑝−2(∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 ). (36) Replacing ∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |=sign(∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛𝑗=1 ), Eq. (36) is rewritten as 𝑠󰇗𝑖=𝑒󰇘𝑖+𝛽(∑𝐾󰇗SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 +∑𝐾SP,𝑖𝑗(𝐱)𝑒󰇗𝑗 𝑛 𝑗=1 )|∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |𝑞𝑝−1 +𝛽(𝑞𝑝−1)(∑𝐾󰇗SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 +∑𝐾SP,𝑖𝑗(𝐱)𝑒󰇗𝑗 𝑛 𝑗=1 )[sign(∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 )]2|∑𝐾SP,𝑖𝑗(𝐱)𝑒𝑗 𝑛 𝑗=1 |𝑞𝑝−1. (37) Notice that [sign(∙)]2=1, then Eq. 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