Hindawi International Journal of Aerospace Engineering Volume 2020, Article ID 1372052, 13 pages https://doi.org/10.1155/2020/1372052 Research Article Helicopter Autorotation 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 helicopters, 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 , τ, 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 Rotorcraft 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].