CONDITIONSTO REALIZE HEXAROTOR STABLE FLIGHTS TO AVOID A CRASH UNDER COMPLETE PROPELLOR MOTOR FAILURES

Kaito Isogai and Hideaki Okazaki Graduate School of Electrical and Information Engineering, Shonan Institute of Technology, 1-1-25 Tsujido Nishikaigan, Fujisawa 251-8511, Japan. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 23 November 2019 Final Accepted: 25 December 2019 Published: January 2020

In this paper, in the case of complete propeller motor failures, a feedback-loop control for a multirotor to avoid a crash when some motors fail and stop, is presented, and verified by simulation results. First, the modeling foundation for a multirotor, and the flight states are summarized. Secondly, theorems of the stabilizability of the flight states for avoiding a crash, by nonlinear dynamic state equations with state variable feedbacks are provided, andalso verified by simulations.
In conclusion, the principal results are summarized, and the future research is described.

…………………………………………………………………………………………………….... Introduction:-
Multirotors using fixed motors for flight, have been widely used in many applications, such as agriculture, surveillance and search, entertainment, photography, and rescue missions. However, there are the risks resulting in injury or deaths and damages if multirotors crash. Therefore, multirotor flight crash avoidances in the case of complete motor failures have been studied [2], [4]- [7], [11], [12], [14]- [16], [18]- [20]. However, feedback-loop controls to avoid a crash for both configurations, have been seldom studied. Especially the establishment of the hexarotor feedback-loop controls to avoid a crash, is very important under general Euler angle rotational motions.On the other hand, the authors obtained the maneuver and flight states for multirotors (a quadrotor, a hexarotor, or an octorotor) with standard symmetrical configurations, based on both of the general Euler angle rotational motion and the translational motion of the state equations. They provided definitions of the operating points and equilibrium points of multirotors, and definitions of multirotor motor speed control signal vectors. Further they provided "add-on to underdetermined system equations" method of directly providing motor speed control signals for hexarotor or octorotor flights to avoid crashes (an open-loop control) when some motors fail and stop, and illustrated typical examples of hexarotor or octorotor flight states to avoid crashes obtained by the method [12]. Hence in this paper, to solve the problem to avoid a crash for the hexarotor in the case of complete failures, we expand on "add-on to underdetermined system equations" method based on [11], [12], [16], [18], and build a feedback-loop control to avoid a crash such that the hexarotor flight states to avoid crashes are stably realized when some motors fail and stop. First, we summarize the modeling foundation for a hexarotor, the maneuvers, and the flight states. Secondly, we provide theorems of the stabilizability of the flight states for avoiding a crash and stabilizing such flight states by nonlinear dynamic state equations with state variable feedbacks. Thirdly, we verify the theorems by simulations of the hexarotor stable flights to avoid a crash in the case of complete propeller motor failures. In Conclusion, we summarize the principal results and describe future research.

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Mathematical Description of a Multirotor:-In this section, we summarize the mathematical foundation for describing the motion of a multirotor as a rigid body, and multirotor body frame configurations. We also describe dynamical Euler angle state equations of rotations, dynamical system state equations of translations, and the maneuvers and flight states (or operating points) based on [12]. Table 1:-Describes the symbols for the motion of a multirotor as a rigid body based [1]. Table 1:-Mathematical description for the motion of a multirotor as a rigid body [12].  [12]. In addition, linear operator is also described by a matrix form:-

Rigid-body dynamics:-
The angles and are Tait-Bryan angles and are examples of the Euler angles [8], [13].

Symbol Description
Three-dimensional real vector space Time [15] Basis vectors of a right-handed Cartesian stationary coordinate system at the origin O ( Fig. 1) or ( ). [15] Basis vectors of a right-handed moving (or local) coordinate system connected to the body at the center of mass Linear operator, Radius vector of a point moving relative to the stationary system ( Radius vector of the point relative to the moving system such that

Description of multirotor body frame configuration:-
We assume that all rotors are the same, distributed evenly, and coplanar, and the distance from each rotor to the geometric center of the multirotor is equal [6]. Eachmultirotor has a standard symmetrical configuration with a clockwise-rotating rotor adjacent to a counterclockwise-rotating rotor as shown in Figs. 2-4.     On the basis of the above equations, we also describe Theorem 2 as an explicit form of dynamical system equations of translation for the multirotor as follows.
Theorem 2 [12]: Let ( ( ̇( ) be the solution for Eq. (2.13) in Theorem 1. Then the dynamical system state equation of translation for the multirotor is obtained in the following explicit form: We now summarize the dynamical system state equations of the multirotorthe symmetrical standard configurations ( ), as follows: Here, by using the state variables of rotational motion ( ̇ ( ̇ ̇ ̇) and translational motion ( ̇ ( ̇ ̇ ̇ (Fig. 5), we represent the multirotor maneuver and flight states in many applications as shown in Table 2. The multirotor uses fixed motors for flight. The angular velocities of the motors are directly used to achieve the flight states in   Operating points and equilibrium points of multirotors can be related to the flight states in Table 2. In the following, we describe Definition 1 for the operating points and equilibrium points of multirotors.

Problem of multirotor stable flights to avoid a crash:-
In this section, in the case of complete propeller motor failures, we describe a method of providing motor speed control signals to achieve flight states to avoid a crash [12]. We assume that the motors in the rotors ( ) have completely failed. Then the motor speed control signal vector of the remaining motors is determined by 267 The remaining moments of and the remaining translational forces of , , are respectively given as where is the matrix with the rows deleted ( ).
Here In [12] in the case of complete propeller motor failures, theflight operating points and equilibrium points of multirotorsare related to the two types of multirotor flight states in Table 3 to avoid a crash when some motors fail and stop. for the flight operating points and equilibrium points of multirotors to achieve the flight states in Table 3 to avoid a crashwhen some motors fail and stop, Definition 2 is given. Based on Definition 2, a methodto provide motor speed control signals to achieve the flight states in Table 3to avoid a crash is proposed and verified by several simulations. However, in this paper, we only consider the case of Type (I) in Table 3. We provide theorems of the stabilizability of the flight states for avoiding a crash and stabilizing such flight states by nonlinear dynamic state equations with state variable feedbacks. We alsoverify the therems by simulations. Table 3:-Two types of multirotor flight states to avoid a crash [12].
Type (I): Among the remaining motors, some must be stopped to achieve all states in Table 2 when certain motors fail and stop. Type (II): Among the remaining motors, some must be stopped to achieve all states (except for yaw control) in Table  2 when certain motors fail and stop. Type (I) and type (II) represent the severity levels of complete propeller motor failure. Type (I) and type (II) are determined by the equation forms of Eq. (41) in Method 1. Type (II) is more severe than type (I) because yaw angles cannot be controlled.  Table 2 when one motor fails and stops [12].
Type Achieved flight states in Table 2  (I) No Problem All (e.g., in Fig. 6).

(II) Admissible Problem
Yaw angles cannot be controlled. Yaw angles are freely changing (e.g., in Fig. 7).
268 Fig. 7:-Example of type (II) admissible problem where one of the remaining motors must be stopped to achieve all states(except for yaw control)of the quadrotor in Table 2 when one motor fails and stops [12].

Definition 2 [12]
: When some motors fail and stop, the flight operating points of multirotors to achieve the flight states in the case of Type (I) in Further, we provide Theorem 3 based on Definition 2 to provide motor speed control signals to achieve the flight states in the case of Type (I) in Table 3. Thus, ̃ are obtained as the motor speed control signals to achieve the flight states in the case of Type (I) in Table 3 to avoid a crash. As the number of failed motors increases, the rank of ( or ( and the dimensionality of decrease.Therefore, since ( ( is not a square matrix, it is necessary to remove arbitrary rows to obtain square matrices. The proof of Theorem 3 is provided in a similar way to that in the proof of either Theorem3 or Theorem4 in [12].

Linearized or nonlinear dynamic state equations to stabilize the flights for avoiding a crash
This subsection describes linearized or nonlinear dynamic state equations to stabilize thehexarotor flights for avoiding a crash the case of type (I)in Table 3.
In the case of type (I) in Table 3 (  ,  ,  ), we obtain the following variational differential equations with constant matrices at the operating points ( o ̇ ) T which can be used to stabilize the hexarotor flights in the case of type (I) in Table 3.    Proof: In the hexarotorcase of type (I) in Table 3 (  ,  ,  ), we utilize Hartman-Grobman theorem in [9], and for arbitrary constant matrix  Table 3 to avoid a crash in the case of complete propeller motor failures. We verify Theorem 3-5 of realizing the hexarotor stable flights to achieve the flight states in the case of Type (I) in Table 3 to avoid a crash. The following results are obtained by using Theorem 3-5 with Maple symbolic computations, MATLAB matrix calculations, and MATLAB numerical simulations with the ode45 solver [10].Notice that regarding the translational state equation of the hexarotor [Eq.  Table 4. Table 4:-Numerical parameters of a hexarotor ( ) [12].
In addition, we define the constant matrix of ( ( as follows: the hexarotor ( where ( in Eq. (3.13) in Theorem 3. In the following, we assume that under complete propeller motor failures, the hexarotor is in a hover control: maneuver (ii) in Table 2, as well as possible.As an example of type(I) in Table 3, we also assume the motor failure case such that the motor in the first rotor has completely failed (in Fig. 8 Table 3, maneuver (ii) in Table 2) as shown in Fig. 9 and Fig. 10. ( , ( , and ( in Fig. 9 completely overlap. ( and ( in Fig. 10 also completely overlap. In this case, the function ( ) in Theorem 3 is as follows: where , k = 1, , and . are the basis vectors of a right-handed twodimensional real vector space .
is the matrix with the th (or 1th) row deleted. is the matrix with the th (or 1th) row deleted ( = rot, tra). The example of (a) verifies that Theorem 3 achieves the hexarotor flight states to avoid a crashe (type (I) in Table 3) using the remaining motors of the hexarotor in the case of complete propeller motor failure, even without any feedbacks.
Theorem 4 is applied to the examples of (b).
(b) Under the following conditions in the case of Fig. 8      277 Fig. 19:-Simulation results for the remaining motor control signals for realizing a stable flight for avoiding a crash with state variable feedbacks by nonlinear state equations, when the motor of the first rotors has completely failed (example (c)).
This example of (c) verifies that Theorem 5 achieves the hexarotor stable flight states to avoid a crash (type (I) in Table 3) using the remaining motors of the hexarotor in the case of complete propeller motor failure.

Conclusion:-
The following results were obtained: 1. We have provided Theorem 3 based on Definition 2 to provide motor speed control signals to achieve the flight states in the case of Type (I) in Table 3. 2. We have provided Theorem 4 of stabilizability of the operating points for the hexarotor flights to avoid a crash in the case of Type (I) in Table 3, and Theorem 5 of nonlinear hexarotor dynamic state equations with state variable feedbacks to stabilize flight states to avoid a crash in the case of Type (I) in Table 3. 3. We have presented typical example of Type(I) of hexarotor stable flight states in Table 3 to avoid a crash, and verified Theorem 3-5. 4. we will build a state variable feedback control for stabilizing themultirotor flight statesin the case of Type (II) in Table 3 under the influence of disturbances such as wind. 5. We will verify theorems experimentally through several tests of actual multirotor flights to avoid a crash in the case of complete propeller failures using a commercial multirotor model with modifications based ontheorems.