Hindawi International Journal of Aerospace Engineering Volume 2020, Article ID 1372052, 13 pages https://doi.org/10.1155/2020/1372052

Research Article Trajectory Planning Method Using Functional Tensor-Train-Based Dynamic Programming Algorithms

Yunjie Wang, Chen Jiang, Shuai Deng, and Haowen Wang

School of Aerospace, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Haowen Wang; [email protected]

Received 11 January 2020; Accepted 7 May 2020; Published 5 July 2020

Academic Editor: Luis E. González-Jiménez

Copyright © 2020 Yunjie Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Helicopter autorotation trajectory planning problems have been dealt within computationally expensive optimal control algorithms. This paper presents an efficient helicopter autorotation trajectory planning method, using functional tensor-train- (FT-) based dynamic programming (DP) algorithms. The autorotation trajectory planning method is shown real-time feasible, which involves general helicopter autorotation dynamics at the same time. To validate the dynamic feasibility of the trajectories, a trajectory-tracking controller using active disturbance rejection control (ADRC) is designed to ensure a helicopter model tracks the trajectories. Finally, a helicopter autorotation simulation with a six-degree-of-freedom high-fidelity multibody-based helicopter model is demonstrated for validation.

1. Introduction using algorithms such as sequential quadratic programming (SQP), as demonstrated in [6, 10, 11]. In general, these algo- Autorotation is a primary measure for , whether rithms are off-line, computationally expensive for current for manned or unmanned helicopters, to land safely after computing power, which is for now unrealistic for on-line engine power failure. In autorotation, the main rotor is usages. Furthermore, there is a lack of validations either driven by the upward airflow through the rotor, making the through high-fidelity simulations or experiments in the flight similar to a gliding fixed-wing aircraft. To achieve a safe above studies. autorotation landing, unique control strategies are needed Recently, Taamallah [12] proposed the first real-time fea- [1]. Generally speaking, the collective pitch should be care- sible, model-based trajectory planning method and designed fully handled to maintain a sufficient and steady rotor speed a model-based trajectory-tracking controller to ensure the before reaching the ground. helicopter tracks the trajectories. In specific, they employed Studies on means of achieving autorotation have been optimal planning based on differential flatness, assuming a focused on solving optimal control problems [2–4]. Tradi- helicopter as a rigid body. The trajectory planning problem tionally, simplified helicopter dynamic models are used, is solved, regarding all the rotor forces and moments as plant including 2-D point-mass models [4–6], three-degree-of- inputs of the rigid body. Taamallah’s work provides novel freedom rigid-body models [7, 8], and low-order six- directions for online trajectory planning of autorotation degree-of-freedom rigid-body models [3]. With the dynamic problems. However, as a consequence, forces and moments model formulated, optimal autorotation problems are solved are simplified and decoupled with helicopter dynamics in by numerical methods. Gradient algorithms, such as sequen- such methods. It is necessary to mention that, in autorotation tial gradient-restoration algorithms, are used in [5, 7, 9]. procedures, the main rotor’s ability to generate forces is Other methods, such as direct methods, discretize the prob- highly restrained by the states of helicopters, especially by lem first and turn it into a nonlinear programming problem. the main rotor’s rotating speed and the inflow state of the Then, the nonlinear programming problem can be solved rotor [13]. The rotor speed is the crucial factor that 2 International Journal of Aerospace Engineering determines whether a trajectory leads to a safe autorota- and bð⋅Þ and σð⋅Þ are generally nonlinear functions. The cost tion landing. Thus, the dynamics of autorotation is of functional is defined as great significance for trajectory planning and should be "#ð involved explicitly. τ x u −βs x u χ −βτψ τ x τ In this paper, we present a real-time feasible autorota- JðÞðÞt0 , = E e gsðÞ, ðÞs , ðÞs ds + τ<∞e J ðÞ, ðÞ , tion trajectory planning method using functional tensor- t0 train- (FT-) based dynamic programming (DP) algorithms, ð2Þ which ensures that general autorotation dynamics is satis- fied along the trajectory. For validation of the dynamic β χ where is a discount factor, τ<∞ is the indicator func- feasibility of the trajectories, we also present a trajectory- tion of the state boundary, and gð⋅Þ and ψ ð⋅Þ are stage tracking controller based on active disturbance rejection J cost function and terminal cost function. The SOC prob- control (ADRC) to make a helicopter model track the tra- lem is to find a control uðtÞ within a specified set on the jectories. The validation of the trajectories using the con- time interval ½t , τŠ, such that the cost Jðxðt Þ, uÞ is troller is then implemented on a six-degree-of-freedom, 0 0 minimized. high-fidelity, multibody-based helicopter simulation model. Next, using the MCA method [17], continuous SOC The trajectory planning method using functional ten- problems are discretized into the discrete Markov decision sor-train- (FT-) based dynamic programming (DP) algo- processes (MDPs). The discretized problem is turned into rithms will be presented in Section 2. The ADRC-based searching for a value function that satisfies the following trajectory-tracking controller is described in Section 3. In recursive equation: Section 4, we describe the six-degree-of-freedom, high- fidelity, multibody helicopter simulation model using the "#   Tsinghua Utility Simulation Tool (TRUST). In ′ ′ vðÞz = minμ gðÞz, u + γ〠p z, z u v z , ð3Þ Section 5, autorotation trajectories are demonstrated with z′ various initial conditions, and simulation results for valida- tion are demonstrated and discussed. Finally, conclusions z are presented in Section 6. where vð Þ is the optimal discretized value function and pðz, z′juÞ is the transition probability function. 2. Autorotation Trajectory Planning Using FT- Then, a discounted-cost infinite-horizon MDP is formu- Based DP Algorithms lated and can be solved by the FT-based DP algorithms, which are FT-based value iteration algorithm, FT-based pol- As mentioned in Section 1, traditional trajectory planning icy iteration algorithm, and FT-based multilevel algorithm, methods for autorotation have been dealt within computa- respectively. tionally expensive off-line algorithms. In 2017, the first For the traditional discrete-state Markov decision pro- real-time feasible trajectory planning method for autorota- cesses, computational requirements grow exponentially tion was demonstrated in [12], which is based on differen- with dimensionality. For example, if a MDP has 8 dimen- tial flatness of the rigid-body dynamics. Such methods sions, and each dimension has a discretization of 10 simplify forces as direct inputs, thus leaving the helicopter points, such a problem involves a search space of 108 autorotation dynamics not included during trajectory plan- points. In order to mitigate such curse of dimensionality, ning procedures. In this section, we introduce a real-time FT-based DP algorithms use low-rank-functions, namely feasible trajectory planning method, using functional ten- functional tensor-train, to represent value functions. The sor-train- (FT-) [14] based dynamic programming (DP) basic idea of function-train (FT) is to make a continuous algorithms, which guarantees a strict satisfaction of heli- analogue [14] of the tensor-train decomposition [18]. To copter dynamics along the trajectory. be specific, it is a continuous version of tensor-train cross-approximation (TT-cross-approximation) [19], with 2.1. FT-Based DP Algorithms. Functional tensor-train- the formulation as follows: based (FT-based) dynamic programming (DP) algorithms are newly proposed algorithms for solving high-dimensional Yd stochastic optimal control (SOC) problems, which are fxðÞ, ⋯, x = F ðÞx , ð4Þ fi 1 d i i mainly discounted-cost in nite-horizon Markov decision i=1 process (MDP) problems. Here, we give a brief review of the FT-based DP algorithms, and details can be found in ⋯ where f ðx1, , xdÞ is a d-dimensional multivariable func- [15, 16]. tion. F ðx Þ is a set of univariate functions, which are also ff i i Consider a system described by stochastic di erential called cores: equations (SDE) as follows: 2 3 ðÞi ðÞi x b x u σ x w f ðÞx ⋯ f ðÞx ðÞt = ðÞt, ðÞt , ðÞt dt + ðÞðÞt d ðÞt , ð1Þ 6 1,1 i 1,ri i 7 F 6 7 iðÞxi = 4 ⋮⋱⋮5, ð5Þ where xðtÞ ∈ Rn is the state vector, uðtÞ ∈ Rm is the control f ðÞi ðÞx ⋯ f ðÞk ðÞx input, wðtÞ is a vector of independent unit Wiener processes, ri−1,ri i ri−1,ri i International Journal of Aerospace Engineering 3

8 _ where ri are the FT ranks evaluated by a continuous ver- > z = vz ðkÞ > sion of TT-rounding [14] and f are univariate hat > i,j > _ − 1 θ θ α > vx = ðÞTB sin + HB cos + DF cos F functions. > m > With methods described above, the exponentially grow- > 1 ing computational complexity of OðndÞ for a typical dynamic > v_ = − ðÞT cos θ − H sin θ + D sin α + g < z m B B F F programming problem is compressed to a polynomially _ > θ = q growing complexity of > > > 1 > q_ = ðÞM − T ⋅ l + H ⋅ h − L cos ðÞθ − γ ⋅ l ÀÁÀÁ > I Y B R B R H H Odnr2κ n + d2nr2 + dnr3 , ð6Þ > FY op > > 1 : Ω_ = − Q, IR κ where is the operations during value evaluation, nop is the ð8Þ step of operations within each Bellman equation step. To summarize, this method formulates a SOC problem where TB and HB are rotor forces resolved in the body as a dynamic programming problem and implements a axes.Notethatalltheaxesaredefined in accordance continuous tensor decomposition method to compress with [13]. MY and Q are pitch moment and rotor tor- such a problem, resulting in significant improvements of que, respectively. DF and LH arethedragofthefuse- computing speed and storage savings. Such method has lage and the lift of the horizontal tail resolved in the been shown to be able to work in real-time [15]. α γ body axes. F and aretheangleofattackofthefuse- fl lage and the ight-path angle. hR and lR are vertical 2.2. Formulating Autorotation Trajectory Planning Problems and horizontal distance of rotor hub to the center of grav- within FT-Based DP Algorithms. In this part, we describe a ity. lH is the horizontal distance of the horizon tail from the general method for solving autorotation trajectory planning center of gravity. Details of forces generated by the fuselage problems in the framework of FT-based DP algorithms. First and the horizontal tail can be seen in [20]. of all, we need to describe the dynamics of a helicopter in a Next, the rotor performance during autorotation is given SOC form, as equation (1) shows. by the following procedures. The induced velocity ν is calcu- lated by [9]: 2.2.1. Helicopter Dynamic Model Formulation. The heli- ν κν copter dynamic function in the SOC equations is given = h f I f G, ð9Þ by a nonlinear three-degree-of-freedom rigid-body heli- copter autorotation dynamic model. Note that in this where κ is the empirical correction factor of nonuniform ff fl ν paper, two di erent helicopter dynamic models are in ow and h is the induced velocity at hover. f I is the described. The first one is the nonlinear three-degree-of- induced velocity parameter which has included the influence freedom rigid-body helicopter dynamic model, which is of the vortex-ring state, given by [9]: used in this section as the system dynamics in equation 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1). The second one is the six-degree-of-freedom, high- < 1:0/ v 2 + ðÞ−v + f 2,ifðÞ−2v +3 2 + v 2 ≥ 1:0 fidelity, multibody-based helicopter simulation model, xb zb I zb xb f I = : ÀÁ which will be used as a validation model in simulation −v 0:373v 2 +0:598v 2 − 1:991 , otherwise, and will be described in Section 5. The three-degree-of- zb zb xb freedom dynamic model is chosen here for computation ð10Þ considerations, and such model is capable of predicting   steady collective pitch manipulations and rotor power con- where vxb and vzb are normalized velocities resolved in body sumption [13]. axes: States variables for the nonlinear three-degree-of- "# "#"#  θ − θ freedom rigid-body helicopter dynamic model are cho- vxb 1 cos sin vx = : ð11Þ sen as  v θ θ vzb h sin cos vz

s θ Ω The ground effect factor f is expressed as [13] = ðÞz, vx, vz, , q, , ð7Þ G 1 f =1− , ð12Þ G − 2 which include height z, horizontal velocity vx,rateof 1 ðÞR/4h θ descent vz,pitchangle ,pitchrateq, and rotating Ω speed of the main rotor . The control variables are where h = z + z0 and z0 is the rotor height when the aircraft is θ θ fl collective pitch 0 and longitudinal cyclic pitch 1s. on the ground. Note that we ignore the in uence of horizon- The dynamic model equations are formulated as fol- tal speed on the ground effect, and such assumption is rea- lows [20]: sonable for a typical autorotation procedure. 4 International Journal of Aerospace Engineering

ffi σ The rotor thrust coe cient CT is expressed as [13] where a is the blade lift-curve slope, is the rotor solidity θ ratio, tw is the blade twist, and B is the blade tip-loss  factor. 1 σ 1 1  2 θ − 3 θ 3 The rotor drag force coefficient C is made up of the CT = a + vxb 0 tw B H 2 3 2 4 rotor profile drag C and the induced drag C . The rotor  ð13Þ H0 Hi 1 ÀÁ1 1 pitch moment coefficient is C . C and C are expressed + 1+v 2 B4 − λ 1+ v 2 B2 , MY H MY 4 xb 2 2 xb as follows:

8 > > CH = CH0 + CHi > ÀÁ > c σ : > C = d 3v +1:98v 2 7 <> H0 xb xb 8    aσ 1 1 1 1 1 3 1 1 ÀÁð14Þ > C = − β + v λ θ − θ v λ + − v β + λ θ + λβ + β β + β 2 + β 2 > Hi c xb 0 tw xb xb 1c 1s 1c 0 1c 0 1c > 2 3 2 8 4 4 4 6 4 > > 1 : C = − εn Ω2β S , MY 2 b 1c b

’ fi ffi ε fi fl β where cd is the blade s mean pro le drag coe cient, is the rst moment about the ap hinge. 0 is the coning angle of fl ff ’ β fi fl ap o set, nb is the number of blades, and Sb is the blade s rotor disk, and 1c is the rst harmonic cyclic ap, given by

8   ÀÁθ > β γ 1 θ − 3 θ  2 4 tw 5  2 5 − 1 λ −  θ 3 − 3 g <> = 0 tw 1+v B + 1+ v B ðÞv 1 B 0 8 4 xb 10 6 xb 6 xb s 2 Ω2R > ð15Þ > −8/3½Šθ − 3/4ðÞλ − v θ : β = 0 xb 1s − θ , 1c −  2 1s 1 1/2vxb

where λ is the normalized inflow velocity, expressed as by normalizing equations of (8). The normalized states and inputs are as follows: λ ν Ω −  : = / R vzb ð16Þ 8 < 1 100 100 1000 1 x = z, x = v , x = v , x =10θ, x = q, x = Ω ffi 1 10R 2 Ω R x 3 Ω R z 4 5 Ω 6 Ω The power coe cient required by the main rotor CP is : 0 0 0 0 ffi θ θ : equal to the rotor torque coe cient CQ, which is made up u1 =10 0, u2 =10 1s fi of the rotor induced power, the rotor pro le power, the rotor ð18Þ parasite power, and the rotor climb power. A general expres- sion is given by [13]

ν ÀÁ In addition, the Jacobian matrix of control inputs κ 1 σ :  2 1  3 f −  CP = CT + cd 1+46vxb + vxb vzbCT , ½∂x_ /∂u Š, i =1,⋯,6,j =1,2 is calculated by numerical ΩR 8 2 πR2 i j difference method for any state x and input u. The dif- ð17Þ fusion function σð⋅Þ is set to 10−9 for each term. Next, the cost functions are designed. The stage cost where f is the fuselage equivalent parasite drag area. gðx, uÞ is expressed by 2.2.2. Formulation of SOC Problem for Autorotation Trajectory Planning. Following the helicopter autorotation x u 2 2 2 2 gðÞ, = W1x1 + W2x2 + W3x3 + W4x4 dynamic model described above, and for numerical consider- ð19Þ − 2 2 2: ations [5], the stochastic differential equations are obtained + W6ðÞ1 x6 + W7u1 + W8u2 International Journal of Aerospace Engineering 5

ψ x The terminal cost J ð Þ is given by setting an absorbing region [15], which encourages a safe landing, as shown below:

( 0 if safely landed ψ ðÞx = ÀÁ ð20Þ J 2 − 2 2 2 2 WT1x1 + WT2 x2 x2glide + WT3ðÞx3 + WT4x4 + WT5x5 otherwise,

where W· and WT· denote weighting factors of stage cost and However, although such trajectories are generated by terminal cost, respectively. We add cost functions to control solving dynamic equations, the trajectories still need to be fi fi inputs in order to avoid erce manipulations. x2glide is the smoothed. There are two main reasons. The rst reason is average horizontal speed, estimated by empirical flight test that for real-time realizability, the integration time step (for results and numerical simulation results. A safe autorotation example, 0.05 s in our study) is not small enough for a con- landing is defined by troller to generate sufficient differential information. The second reason is that although we have cost functions on τ ≤ τ ≤ θτ control inputs, the control values still result in discontinu- vzðÞ vz max, jjvxðÞ vx max, jjðÞ ð21Þ ous variations, making the trajectory not smooth enough ≤ θ , q τ ≤ q , z =0: max jjðÞ max for our controller. Thus, we apply a fast interpolation method, i.e., cubic Hermite interpolation, to obtain smooth The design of cost functions is shown to be important for trajectories. such autorotation problems, and parameters are needed to be adjusted by numerical experiments. 3. Autorotation Trajectory-Tracking Controller 2.2.3. Trajectory Generation Using FT-Based Algorithms. Based on ADRC After the problem is formulated in the form of a SOC prob- lem in Section 2.2.2, solutions are then obtained using FT- In order to show the practicability of trajectories generated based algorithms. by FT-based DP methods, we demonstrate a trajectory- Although FT-based algorithms are proposed for sto- tracking controller based on active disturbance rejection con- chastic optimal control problems, we slightly alter such trol (ADRC) and use the controller to make a high-fidelity algorithms to generate a trajectory, rather than to obtain helicopter model track the trajectories. Tracking of position control inputs directly. The reason is that even though and velocity is vital for a successful autorotation landing, the nonlinear dynamics of helicopter described in Section and the dynamics during autorotation may be more compli- 2.2 gives nice results in terms of predicting steady collec- cated than the formulations described in Section 2.2. Besides, tive pitch manipulations and rotor power consumption, there also exist unexpected disturbances during real flights. such model is not able to make good predictions of the Thus, we use active disturbance rejection control (ADRC) helicopter dynamic response [21]. Therefore, the control methods to design the tracking controller. Active disturbance inputs generated by the SOC controller are not adopted, rejection control (ADRC) is proposed by Han [23], and the and we make use of the trajectory instead. As mentioned ADRC controller is capable of estimating inner modelling before, the results of rotor power and collective pitch errors or outer disturbances, thus making compensation along the trajectory can be regarded reasonable. accordingly. We employ the FT-based one-way multigrid algorithm In specific, we implement the ADRC-based trajectory- proposed in [15, 22]. For each discretization level, Niter = tracking controller described in [24], and such controller 100 steps of FT-based policy iterations are applied. The tra- has been successfully validated through flight tests [24, 25]. jectory of normalized states x is obtained by an integration The structure of the controller is shown in Figure 1: where x v procedure of given initial conditions to the controller: traj and traj are position vectors and velocity vectors of the u ()ð trajectory and r represents the reference velocities that ti needed to be tracked by the inner loop controller. x bx ux ∈ τ x x : fgðÞti = ðÞðÞti−1 , ðÞi−1 , ti ½Št0, , ðÞt0 = 0 The order of the ADRC controller differs with different ti−1 channels. For the autorotation application, we use a 3rd- ð22Þ order ADRC controller for forward, vertical, lateral channels, and a PI controller for yaw channel. In general, an ADRC In specific, a fourth-order order Runge-Kutta method is controller is made up of a tracking differentiator (TD), an implemented. Thus, preliminary trajectories are obtained extended state observer (ESO), and a nonlinear state error by making x unnormalized by equation (18). Because the tra- feedback (NLSEF). Disturbances or modelling errors of the jectories are generated by integrations based on the dynamic system are observed by the ESO, and then NLSEF is utilized equations of the helicopter, they satisfy the specified helicop- to restrain them. A typical architecture of an ADRC control- ter dynamics by nature. ler is shown in Figure 2. 6 International Journal of Aerospace Engineering

x u u Trajectory traj Multiorder r Inner loop Helicopter planning ADRC controller dynamics vtraj

� � � p q r (x y z vxb vyb vzb) ( )

Figure 1: Trajectory-tracking controller structure.

v e l + l N Note that because in this paper we focus on the longitudi- L u + u nal performance of the controller, the yaw controller can be v – 0 y · TD · · S Plant regarded as a yaw stabilizer. vn · en – + E The input signal of the ADRC is processed by the follow- l b n0 ing procedure: – F / 0 2 3 2 32 3 z l − n+ v cos θ 0 −sin θ vx traj vx 6 xb ref 7 6 76 7 6 7 6 ϕ 76 v − v 7 · 4 vyb ref 5 = 4 0 cos 0 54 y traj y 5, · ESO ÀÁ θ θ − − vzb ref sin 0 cos Kz ztraj z + vz traj vz Figure 2: ADRC architecture. ð26Þ

Where v is the input signal, b0 is the input gain factor, z· is the where xtraj, ztraj, vx traj, and vz traj are positions and velocities output of the ESO. For brevity, only the 3rd-order ADRC control- along the trajectory, Kz is the vertical factor, and ðvxb ref , ler is presented here. The TD of a 3rd-order is formulated as [25] vxb ref , vxb ref Þ is the input signal of the ADRC controller. 8 The inner loop controller is a PI controller with damping − > fh1 =fhanðÞv1 v0, v2, r, h0 feedbacks, and such controller is proven to be working well > > with ADRC in trajectory-tracking applications [25]. > v = v + h ⋅ v < 1 1 0 2 Following the above process, a trajectory controller is ⋅ > v2 = v2 + h0 fh1 ð23Þ obtained. It is needed to mention that when the helicopter > > is near the ground, the controller is reset to make the helicop- > fh2 =fhanðÞv2, v3, r, h0 :> ter descent slowly. Such techniques to ensure a stable landing ⋅ v3 = v3 + h0 fh2, is also reported by [12]. where fhanð⋅Þ is an optimal control function defined in [23] 4. The Tsinghua Rotorcraft Utility Simulation and r and h0 are controller parameters. The 3rd-order ESO Tools (TRUST) is formulated as 8 This section briefly introduces the Tsinghua Rotorcraft − > e = z1 y > Utility Simulation Tools (TRUST), with which we model > > z = z + hzðÞ− β e the S-58 helicopter and validate the trajectories using the < 1 1 2 01 ADRC controller. Tsinghua Rotorcraft Utility Simulation − β ⋅ : δ > z2 = z2 + hzðÞ3 02 falðÞe,05, ð24Þ Tools (TRUST) stems from [26], which is based on the > > − β ⋅ : δ framework of multibody dynamics. The TRUST simulator > z3 = z3 + hzðÞ4 03 falðÞe,025, + jjb0 u :> is able to simulate many kinds of , including −β ⋅ : δ z4 = z4 + hðÞ04 falðÞe,0125, , regular helicopters, tandem helicopters, and compound heli- copters, with articulated or flexible rotor hubs. Regular heli- β where h is the integration step of the system, · is the observer copter models consist of main rotor dynamics, tail rotor gain, and function falð⋅Þ is defined in [23]. The 3rd-order dynamics, rigid-body fuselage dynamics, horizontal tails, NLSEFisformulatedas and vertical tails. Such models have proven to be of good 8 agreement with various test data [26]. > − > e1 = v1 z1 > 4.1. Model Description. Considering both fidelity and simula- > − <> e2 = v2 z2 tion feasibility, we adopt the following modelling methods. e = v − z > 3 3 3 4.1.1. Main Rotor. Main rotor is modelled as an articulated > > u = β falðÞe , α , δ + β falðÞe , α , δ + β falðÞe , α , δ rotor with rigid blades of certain twist. Each blade is attached > 0 1 1 1 2 2 2 3 3 3 : to the hub through pitch, lag, and flap hinges. The inflow is u = u − z /jjb : 0 4 0 modelled as the three-state Pitt-Peters dynamic inflow [27]. ð25Þ Blade lift and drag forces are calculated using the blade International Journal of Aerospace Engineering 7

Table 1: S-58 specifications.

Parameter Value used Parameter Value used 2 Rotor radius, m 8.53 Fuselage inertia Iyy,kgm 48796 2 Blade chord, m 0.417 Fuselage inertia Ixx,kgm 16265 2 Blade twist, deg -8 Fuselage inertia Izz,kgm 48796 Number of blades 4 HT area, m2 1.15 Flap hinge offset, m 0.3048 Lift-curve slope of HT 3.73 Rotating inertia of the MR, kg m2 7101.2 Rotor hub height above ground, m 4.36 Tilting angle, deg 0 Fuselage flat plate area, m2 3.55 element theory, and lift-curve slope is obtained from a non- 5. Numerical Experiments linear tabulate of angle of attack and relative airflow speed. fi In addition, because the main rotor may traverse from In the rst part, we show the trajectory planning results of powered lift state into vortex-ring state (VRS) during autoro- various autorotation initial conditions based on the Sikorsky tation, a modification [28] of vortex-ring state is applied to S-58. In the second part, to validate the dynamic feasibility of the three-state Pitt-Peters model. the trajectory planning results, we implement the ADRC Due to the multibody dynamic framework, each blade’s controller and validate the trajectory planner by making a fi motion is solved individually by the Newton/Euler equations. 6-DOF nonlinear high- delity S-58 model track the Thus, flap angles of blades are not assumed to be small, nor trajectory. the blades motions are treated as a consequence of a period- The Sikorsky S-58 is a single-engined helicopter ical disk, which involves inevitable accuracy loss for blade equipped with a 4-bladed articulated rotor with a radius of ff motion responses [29]. 8.535 m. The take-o weight used in our research is 4500 kg. Detailed parameters can be found in [32, 33], and main parameters among which can be found in Table 1. 4.1.2. Tail Rotor. Tail rotor is modelled similar to the main rotor, except that the rotor hub is rigid and no VRS correc- 5.1. Autorotation Trajectory Planning Results. The parame- tion is applied. ters of the cost functions in equations (19) and (20) are set as follows: 4.1.3. Fuselage. Fuselage is regarded as a rigid body, with lin- ffi : : : : : : ear aerodynamic lift and drag coe cients. W1 =500, W2 =125, W3 =50, W4 =10, W5 =05, W6 =100 : : W7 =10, W8 =10 4.1.4. Horizontal and Vertical Tails. The aerodynamic forces : : : : : : of horizontal and vertical tails are calculated by flat plate WT1 = 400 0, WT2 =025, WT3 =025, WT4 =02, WT5 =003 models. ð27Þ For the purpose of validating autorotation trajectory planning and tracking, the nonlinear helicopter model Note that costs for control inputs are added to avoid described above is capable of predicting dynamic responses extreme manipulations. The empirical gliding speed is set : fi of the helicopter [28, 30]. as x2glide =50. The de nition of a successful autorotation landing is defined as 4.2. Dynamics of TRUST Model. As mentioned in Section 2, ff ≤ ≤ θ ≤ : there are two di erent helicopter dynamic models used in vz 2m/s, jjvx 3m/s, jj 12 deg ð28Þ our study. The nonlinear three-degree-of-freedom rigid- body helicopter dynamics is capable of calculating steady col- In addition, we also implement a start cost function lective pitch manipulations, rotor forces, and power, but not before the value iterations. A start cost function is used to ini- sufficient for dynamic response predictions. Due to the fact tialize the global value function before the iterations begin. that the rotor forces in such models are essentially zero- The start cost function can be expressed as follows: order processes, dynamic responses of the rotor cannot be 8 well described. ψ > J ifjjz > hp In the TRUST helicopter model, a more sophisticated < ! 2 dynamic inflow model is involved, and rotor blade motions Jstart = > jjx ÂÃ :> ψ + 1 − 1 15 2 +15 2 +20 2 otherwise, are accurately described in the multibody framework. Blade J x2 x3 x4 hp lift and drag forces are from nonlinear tabulates of experi- ment data. Furthermore, helicopter dynamics is well known ð29Þ for coupling effects between different channels [31]. Hence, for validation purposes, a high-order, six-degree-of-freedom, where hp is set to 4.0 m in this case. We use the one-way helicopter dynamic model should be considered. method of N = 20, 40 points for each dimension and set 8 International Journal of Aerospace Engineering

×104 2 100

10–1 1.5 || k 10–2 || k

1 –1||/|| v || v k –3 – v 10 k || v 0.5 10–4

0 0 100 200 0 100 200 Iteration steps (k) Iteration steps (k)

Figure 3: FT-based algorithms value iteration plots. The blue line denotes values of N =20, and the black line denotes values of N =40.

20 15

15 10

10 (m/s) (m/s) z z v v 5 5

0 0 0 2468 02468 Time (s) Time (s)

80 100

60 90

80 40 � (%)

Height (m) Height 70 20 60 0 02468 02468 Time (s) Time (s)

Figure 4: Autorotation trajectories. Black lines for Case 1, blue lines for Case 2, green lines for Case 3, and red lines for Case 4.

the max rank approximation of the core functions to Case 1. s0 = ð−55m, 20m/s, 0m/s,−3:7 deg, 0 deg/s, 235RPMÞ. rmax =15. Using the one-way method, the initial value function of N =40 is obtained from the solutions of N = Case 2. s0 = ð−70m, 10m/s, 0m/s,−2:4 deg, 0 deg/s, 235RPMÞ. 20. Convergence is plotted in Figure 3. The left panel shows when N =40, the value function Case 3. s0 = ð−70m, 20m/s, 0m/s,−3:7 deg, 0 deg/s, 235RPMÞ. converges to a certain value around 2000 quickly. The rel- ff ative error of the value function between di erent itera- Case 4. s0 = ð−80m, 10m/s, 0m/s,−2:4 deg, 0 deg/s, 235RPMÞ. tions is plotted in the right panel. It is shown that for the discretization of N =40, the relative error is small for Trajectories are obtained following procedures described − a value of about 3×10 5, which is near the converging in Section 2. We apply a Hermite interpolation of 20 points threshold of 10−5. sampled from the preliminary trajectories. Note that such Various autorotation cases using the trajectory planner number of points can be regarded as sufficient, considering are tested, with different initial height and horizontal there are 10 points in [11] and 16 points in [12]. Results are speed: shown in Figure 4. International Journal of Aerospace Engineering 9

20 10

15

10 5 (m/s) (m/s) z z v v 5

0 0 0246810 0 2 4 6 810 Time (s) Time (s)

60 100

90 40 80 � (%)

Height (m) Height 20 70

60 0 0246810 0 2 4 6 810 Time (s) Time (s)

Figure 5: Comparisons of trajectories with results from SQP methods. Black lines are trajectories using FT-based DP method, and blue lines are trajectories using SQP method.

Table 2: Time cost for each trajectory generation. engine failure, while the trajectory obtained by SQP Case number 1 2 3 4 Average method appears to be more agile since the rotor speed does not reach the lower limit of the path constraint. In Algorithm 1 time cost, s 0.62 0.59 0.61 0.59 0.60 this respect, the autorotation trajectory using the FT- Algorithm 2 time cost, s 9.12 11.54 11.21 13.74 11.40 based DP method is more conservative in terms of keeping a higher rotor speed. Both the above trajectory planning results are obtained As Figure 4 shows, the helicopter enters a steady descent using one core of a 3.20 GHz Inter i7-8700 CPU. Time costs quickly, and begins to decrease at about 20 meters above the for each trajectory generation using the FT-based DP method ground. Such behaviours agree generally with [9]. For high (shown as Algorithm 1) and SQP method (shown as Algo- initial speed cases, the forward speed is decreased immedi- rithm 2) are listed in Table 2. The average time cost of the ately after the engine fails, in order to get the helicopter pre- FT-based DP method is 0.60 s, which indicates that the pared for the final landing. Such behaviours are in method is real-time feasible. accordance with [5]. All the trajectories terminate within Before entering autorotation, the time delay of disenga- the landing constraints defined in (28). Thus, the trajectory ging the clutch is a key factor that determines whether the planning method using FT-based DP algorithms is able to subsequent maneuvers lead to a safe landing. A long time generate successful autorotation trajectories that satisfy the delay makes autorotation a tough task because of a low helicopter autorotation dynamics described in Section 2. Fur- rotor speed. Here, the influence of time delay is considered thermore, comparisons are made with results obtained from by setting lowered rotating speeds. Initial conditions are s − − : Ω Sequential Quadratic Programming (SQP) using the SNOPT given by 0 = ð 90m, 20m/s, 0m/s, 3 7 deg, 0 deg/s, Þ, software package [34]. In specific, besides the landing specifi- where Ω = f100%,95%,90%g ⋅ 235RPM for each trajectory. cations assumed in (28), path constraints are required, From Figure 6, we can see that for lowered rotor speeds ≤ ≤ − ≤ θ ≤ Ω ≥ ff mainly as 0m/s vz 9m/s, 30 deg 30 deg, 70%. caused by di erent time delays, the trajectory planner is still A typical result is demonstrated using initial conditions of capable of generating trajectories without modifying any cost Case 1 and is shown in Figure 5. function parameter. The rotating speed is recovered along Both trajectories in Figure 5 lead to safe autorotation the trajectory by certain maneuvers. landing, considering landing constraints are all satisfied. As for real-time applications, because the trajectory plan- Besides, the two trajectories show similarities in terms of ner is able to generate a global trajectory by any specified ini- the rate of descent. The main difference is that the trajectory tial condition, the computing speed shown in Table 2 can be using the FT-based DP method tends to maintain the rotat- considered real-time applicable with such hardware. The tra- ing speed by entering the steady autorotation right after the jectory computing time cost in our study is about 0.6 seconds, 10 International Journal of Aerospace Engineering

20 15

15 10

10 (m/s) (m/s) z z v v 5 5

0 0 0246810 0246810 Time (s) Time (s)

100 100

90

50 80 � (%) Height (m) Height 70

0 60 0246810 0246810 Time (s) Time (s)

Figure 6: Entering autorotation with different rotor speed. Black lines for 100%Ω, red lines for 95%Ω, and blue lines for 90%Ω. which is similar to the time of a fast human pilot reaction. the nose-up direction. The red lines denote the reference However, as mentioned above, the time delay is a key factor values that are needed to be tracked. The black lines of the autorotation procedure, which should be shortened denote the 6-DOF nonlinear helicopter model responses. with best efforts. Therefore, although the trajectory planner The blue line indicates that the helicopter is close to the is shown to be able to deal with situations of different time ground, and the controller is reset to make a slow landing. delays, an update frequency of at least 1.5 Hz is recom- The value of the reset height is chosen as 1.0 m by simu- mended for real-time applications, considering the time cost lation experiments. of other parts (such as detection of engine failure). Thus, we As the results shown in Figure 7, the velocities and atti- − : − recommend running such algorithms with CPU of at least tude angles of the helicopter are vxb = 0 14 m/s, vyb = 3.1 GHz or higher. : : ϕ : θ − : 0 42m/s, vz =115m/s, =321 deg, and = 2 33 deg when landing, which satisfy the safe autorotation landing specifi- 5.2. Trajectory-Tracking Simulation with the TRUST 6-DOF cations defined above. The height history and rotating Helicopter Dynamic Model. In this part, for validation of speed of the main rotor are shown in lower right and the dynamic feasibility of the trajectory, we demonstrate a the upper right panel of Figure 7, which indicate that six-degree-of-freedom flight simulation, with the trajectory the rotating speed is still beyond 75%, although the planner described in Section 2 and the trajectory-tracking ground effect is neglected in the six-degree-of-freedom controller described in Section 3. For brevity, control param- nonlinear helicopter model. Notice that the performance eters can be found in the Appendix. As mentioned before, the of horizontal speed is not as good as the vertical channel. simulation is based on a six-degree-of-freedom nonlinear However, we do not set horizontal distance as any kind of multibody-based helicopter model of S-58, with a VRS cor- target variable in this study. We also find that increasing rection of the Pitt-Peters dynamic inflow and an accurate horizontal controlling gains affects the vertical channel dynamic description of the blade’s flap motions. instead. Apparently, a more sophisticated model-based We demonstrate the simulation results of initial condi- controller should improve the tracking performance, but tions from Case 1. The initial conditions of states s are our purpose of validating the trajectories is achieved. Sim- given by ulation results show that following the trajectories generated s − − : : by the trajectory planner, the helicopter is successfully guided 0 = ðÞ55m,20m/s,0m/s, 0 7 deg, 0 deg/s, 235RPM ð30Þ to a safe landing. Thus the trajectory is both real-time feasible and dynamic feasible. We assume the engine failure starts at t =0sin the for- ward flight, and the simulations are terminated once the 6. Conclusions height descends to zero. Simulation results are shown in Figure 7. Remember that This paper demonstrates a trajectory planning method for θ vzb is positive downward in the body axes and is positive in autorotation, which is real-time feasible and guarantees a International Journal of Aerospace Engineering 11

30 1 100 10 0.5 95 20 5 0 90 0 10 –0.5 � (deg) � (deg) � (deg) � (deg) 85 –5 –1 0 –1.5 80 –10 –10 –2 75 0 510 0510 0510 0510 Time (s) Time (s) Time (s) Time (s)

12 20 3 60

10 15 2 50 8 1 40 10 6 0 30 (m/s) (m/s) (m/s) zb xb 5 yb V V V

4 –1 (m) Height 20

2 0 –2 10

0 –5 –3 0 0510 0510 0510 0510 Time (s) Time (s) Time (s) Time (s)

Figure 7: Height, rotor speed, attitude of Euler angles, and velocities in the body axes during simulation. Black lines denote simulation results, red lines denote reference values of the trajectory-tracking controller, and blue lines denote controller reset for landing.

Table 3: Parameters of the NLSEF. strict satisfaction of specified helicopter dynamics along the trajectory. A trajectory-tracking controller is demon- Controller channel ∣b ∣ β β β α α α 0 1 2 3 1 2 3 strated to ensure the helicopter fly along the trajectories. vxb 113 0.34 0.13 0.05 1.3 1.76 2.4 Successful autorotation trajectories with various initial vyb 158 0.20 0.08 0.02 0.8 1.1 1.5 conditions are shown and discussed, and the time cost of trajectory generation procedures is around 0.60 s. A com- v zb 1.6 0.12 0.04 0.01 0.2 0.58 0.8 prehensive validation is made based on a six-degree-of- ψ 1 0.15 ——0.8 —— freedom high-fidelity nonlinear S-58 model using the TRUST simulator. Simulation results show that although Table the horizontal velocities are not tracked as well as the ver- 4: Parameters of the ESO and TD. tical channel, the trajectories generated by the trajectory Controller channel r β β β β δ/h planner and the trajectory tracker are capable of guiding 01 02 03 04 0 a helicopter into a successful autorotation. Thus, the trajec- vxb 100 100 300 1000 1800 6 tory planner is real-time feasible, and the generated trajecto- vyb 100 100 300 1000 1800 6 ries are dynamic feasible. Further potential improvements for the trajectory planning method could be selecting the vzb 100 100 300 1000 1800 6 ψ —— — — — derivatives of the control as control inputs of the three- 0.5 degree-of-freedom helicopter dynamics, extending the two- —— — — — vxb ref 1 dimensional helicopter dynamics to a three-dimensional vyb ref 1 —— — — — helicopter dynamic model, and automatic tuning of the weight functions. vzb ref 0.8 —— — — —

Appendix Table 5: Parameters of the inner loop. A. Trajectory-Tracking Controller Parameters Controller channel KIDamping The controller parameters described in Section 3 are q 0.73 0.07 1.12 listed below. Parameters of the NLSEF are shown in p 0.23 0.019 0.32 Table 3. r 3.0 0.36 4.8 Parameters of the ESO and TD are listed in Table 4. Parameters of the inner loop are listed in Table 5. 12 International Journal of Aerospace Engineering

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