Nonlinear Aerodynamics Research Articles

Nonlinear Oscillations of a Composite Stepped Piezoelectric Cantilever Plate with Aerodynamic Force and External Excitation

Axially moving wing aircraft can better adapt to the flight environment, improve flight performance, reduce flight resistance, and improve flight distance. This paper simplifies the fully unfolded axially moving wing into a stepped cantilever plate model, analyzes the structural nonlinearity of the system, and studies the influence of aerodynamic nonlinearity on system vibration. The model is affected by aerodynamic forces, piezoelectric excitation, and in-plane excitation. Due to Hamilton’s principle of least action, the mathematical model is established based on Reddy’s higher-order shear deformation theory, and using Galerkin’s method, the governing dimensionless partial differential equations of the system are simplified to two nonlinear ordinary differential equations, and then a study of the influence of the various engineering parameters on the nonlinear oscillations and frequency responses of this model is conducted by the method of multiple scales. It was found that the different values of a5, a6, b6 and b8 can change the shape of the amplitude–frequency response curve and size of the plate, while different symbols a7 and b7 can change the rigidity of the model. The excitations greatly impact the nonlinear dynamic responses of the plate.

Experimental and Numerical Nonlinear Stability Analysis of Wings Incorporating Flared Folding Wingtips

Recent studies have considered the use of wings incorporating flared folding wingtips (FFWTs) to enable higher aspect ratios (reducing overall induced drag) while also reducing gust loading and meeting airport operational requirements. This paper presents the first experimental research into the nonlinear dynamic behavior of a wing incorporating an FFWT. Wind-tunnel tests were conducted at a range of velocities below and beyond the linear flutter boundary. The experimental findings are compared with results obtained from continuation and bifurcation analyses on a representative low-fidelity numerical model. The results show that beyond the linear flutter boundary, stable limit cycle oscillations form, which is dependent on the flare angle, are bounded by either geometric or aerodynamic nonlinearities. Also presented is the effect of a wingtip trim tab on the stability boundary of a wing incorporating FFWTs. It is found that the tab angle can significantly alter the stability boundary of the system, indicating that the choice of camber is an important parameter when considering the stability boundary of FFWTs and that a moveable control surface on an FFWT could be used “in flight” to extend the stability boundary of an aircraft.

Dynamic responses of a conceptual two-dimensional airfoil in hypersonic flows with random perturbations

This paper investigates a conceptual pitch–plunge airfoil in hypersonic flows with both structural and aerodynamic nonlinearity, meanwhile the effects of irregular fluctuations in the flow and external random loads modeled as a Gaussian white noise are considered. The dynamic responses of the airfoil model especially the bifurcation behaviors are analyzed via the harmonic balance method for the deterministic case. We found the airfoil system undergo both subcritical and supercritical Hopf bifurcations. Subsequently, the effects of stochasticity on the dynamical behaviors of the hypersonic airfoil system are explored in the regimes of subcritical and supercritical Hopf bifurcations depending on the system parameters. Several interesting phenomena are triggered under random perturbations such as intermittency and stochastic transition between the low-amplitude oscillation state and the undesirable high-amplitude oscillation state. These dynamics are further characterized via the probability density function and time history. This work will provide new insights into the safety and reliability design of hypersonic aircraft.

Separated and vortical flow in aircraft aerodynamics: a CFD perspective

AbstractIn the early era of aviation, Frederick Lanchester was both an inventor and a theoretician driven by the need for a theory of flight that would reduce the guesswork in designing new aircraft. His book Aerodynamics in 1907 laid down the early foundations of such a theory. The theory with contributions from others, notably Ludwig Prandtl, was refined to become the basis for the sleek designs of WWII aircraft brought about with little guesswork. New technology changed aircraft design radically with the increased speed of jet propulsion reaching into the transonic range with nonlinear aerodynamics. In the late 1940s and early 1950s substantial guesswork returned to aircraft design. The legacy of Lanchester et al., however, lived on with the development of computational fluid dynamics (CFD) that could guide designers through nonlinear transonic effects. This article presents a historical sketch of how CFD developed, illustrated with examples explaining some of the difficulties overcome in the design of the first-generation swept-wing transonic fighters. The historical study is forensic CFD in search for the likely explanation of the designer’s choice for the wing shape that went into production a long time ago. The capability of current CFD applied to the aerodynamics of aircraft with slender wings is surveyed. The cases discussed involve flow patterns with coherent vortices over hybrid wings and wings of moderate sweep. Vortex-flow aerodynamics pertains to understanding the interaction of concentrated vortices with aircraft components. Modern Reynolds-Averaged Navier-Stokes (RANS) technology is useful to predict attached flow. But vortex interaction with other vortices and breakdown lead to unsteady, largely separated flow which has been found out of scope for RANS. Direct simulation of the Navier-Stokes equations is out of computational reach in the foreseeable future, and the need for better physical modeling is evident. Both cruise performance and stalling characteristics are influenced by strong interactions. Two important aspects of wing-flow physics are discussed: separation from a smooth surface that creates a vortex, and vortex bursting, the abrupt breakdown of a vortex with a subsequent loss of lift. Vortex aerodynamics of not-so-slender wings encounter particularly challenging problems, and it is shown how the design of early-generation operational aircraft surmounted these difficulties. Through use of forensic CFD, the article concludes with two case studies of aerodynamic design: how the Saab J29A wing maintains control authority near stall, and how the Saab J32 mitigates pitch-up instability at high incidence.

Oscillation quenching and physical explanation on freeplay-based aeroelastic airfoil in transonic viscous flow

Limit Cycle Oscillation (LCO) quenching of a supercritical airfoil (NLR 7301) considering freeplay is investigated in transonic viscous flow. Computational Fluid Dynamics (CFD) based on Navier-Stokes equations is implemented to calculate transonic aerodynamic forces. A loosely coupled scheme with steady CFD and an efficient graphic method are developed to obtain the aerodynamic preload. LCO quenching phenomenon is observed from the nonlinear dynamic aeroelastic response obtained by using time marching approach. As the airspeed increases, LCO appears then quenches, forming the first LCO branch. Following the quenching region, LCO occurs again and sustains until the divergence of the response, forming the second LCO branch. The quenching of LCOs was addressed physically based on the aerodynamic preload and the linear flutter characteristic. An “island” of stable region is observed in the flutter boundary, i.e. the flutter speed versus the mean Angle of Attack (AoA). The LCO quenches when the aerodynamic preload crosses this stable region with the increasing of airspeed. The LCO quenching of this model in transonic flow is essentially induced by destabilizing effect from aerodynamic preload, since the flutter speed is sensitive to AoA due to aerodynamic nonlinearity.

The influence and application of nonlinear aerodynamics on static derivatives in transonic regime

This paper details two static aeroelastic analysis methods applied to a passenger aircraft model with high aspect ratio wing. The influence of nonlinear aerodynamic force on static aeroelastic derivatives in the transonic regime is analysed. The traditional aerodynamic influence coefficient (AIC) matrix method can produce fast and reliable aerodynamic force and is widely used in aeroelastic analysis. However, the AIC matrix computed by linear aerodynamics will lead to some errors in transonic regime because of the nonlinear effect of aerodynamics. By generating the correction matrices, the AIC matrix is modified, and the accuracy of transonic static aeroelastic correction of aerodynamic data can be improved. The static derivatives are compared to the results of the computational fluid dynamics (CFD) / computational structural (CSD) interaction method.

Universal Maximum Power Extraction Controller for Wind Energy Conversion Systems Using Deep Belief Neural Network

Over the previous years, there has been a surge in need for wind turbines. Consequently, this paper aims to propose a Deep Belief Neural Network (DBNN) that preserves the Maximum Power Extraction's (MPE's) benefits while allowing the output power to be limited with substantially less complexity in the control loop. The suggested technique uses a deep belief neural network to learn the wind turbine's nonlinear aerodynamics. It is generalized to cover a wide range of wind turbine sizes and operating conditions to precisely track their maximum power trajectories. Also, the proposed method rewrites the machine model in order to run Wind Energy Conversion Systems (WECSs) under the MPE technique; which takes into account the radius and pitch angle of wind turbine blades, wind speed, air temperature, and power demand for any permanent magnet synchronous generators types (IPMSG or SPMSG). Moreover, the proposed methodology is considered a plug-and-play technique while it does not need any tuning measures. The suggested DBNN MPE controller's power tracking achievement is examined in various operating conditions using a set of experimental and simulation tests for different generator types and sizes. Finally, the findings are compared to the well-known verified technique to validate its generalized productiveness.

Nonlinear aeroelastic modeling and comparative studies of three degree of freedom wing-based systems

The effects of structural and aerodynamic nonlinearities are investigated on an aeroelastic system with three degrees of freedom, those being plunge, pitch, and flap motions using two different aerodynamic load representations. Stall effects are introduced using quasi-steady and unsteady aerodynamic representations. The effects of the unsteadiness of the flow and stall effects on the type of Hopf bifurcation and the limit cycle oscillations of the system are explored. A comparative study between the unsteady and quasi-steady representations from a nonlinear perspective is also carried out. Results from the linear analysis show that for this specific aeroelastic system, its linear characteristics are dependent on the aerodynamic representation and the system’s design. Additionally, the quasi-steady approximation of the aerodynamic loads results in an inaccurate prediction of the linear flutter speed. Furthermore, the nonlinear analysis shows the existence of an instability shift from the supercritical to the subcritical type, depending on the value of the stall coefficient, as well as that of the cubic nonlinearities, and is more dominantly for the case when the quasi-steady representation is considered. Results show that, even for small reduced frequencies and small angles of attack scenarios, the unsteady representation of aerodynamic loads is critical to correctly predict the flutter speed, instability type, and system’s dynamics.

Nonlinear dynamic analysis of galloping of a single iced conductor with four degrees of freedom based on curved beam theory

Based on the nonlinear strain-displacement relationship of spatial curved beam theory, a finite element model for galloping analysis of iced transmission lines is established, which involves three translational degrees of freedom (DOF) and one rotational degree of freedom for each node. Considering the aerodynamic nonlinearity and the geometric nonlinearity of large amplitude motions of the iced transmission line, the nonlinear dynamic finite element equation is presented by using the virtual work principle. The equation is transformed into the sub-space according to the mode superposition method and a time integration algorithm is also performed in the sub-space. Then, the element independence and modal convergence are researched. Finally, the influence of aerodynamic forces on structural frequency is analyzed. The numerical results show that aerodynamic forces will greatly affect the torsional frequency of the transmission line, which accurately reflects the dynamic characteristics of transmission lines. Furthermore, the galloping calculation of the transmission line has reliable accuracy by using 4-DOF (per node) elements for the iced conductor. On the other hand, the model can predict the actual galloping response of the transmission line, which is convenient for the implementation of subsequent control design.

Nonlinear dynamic response of sea-crossing bridges to 3D correlated wind and wave loads

Long span sea-crossing bridges are often slender and sensitive to wind and wave loads. Nonlinear dynamic response analysis of the bridges under three-dimensional (3D) correlated wind and wave loads is performed in this study in consideration of both geometric and aerodynamic nonlinearities. An optimized C-vine copula is first used to construct a 3D joint probability distribution and environmental contour of mean wind speed, significant wave height and peak wave period. Multi-point fluctuating wind loads with Davenport coherence function and random wave loads with pile group effect are then determined using wind and wave spectra respectively. The nonlinear wind-wave-bridge system considering geometric and aerodynamic nonlinearities is solved by the Newmark-β method with the 3D correlated wind and wave parameters as an input. The proposed approach is finally applied to a real sea-crossing cable-stayed bridge with the measured wind and wave data. The results show that the nonlinear response of the bridge is higher than its linear response with the same input. The bridge response is significantly reduced if the 3D correlated wind and wave loads other than conventional uncorrelated wind and wave loads are considered.

Numerical investigation of the nonlinear interaction between the sinusoidal motion-induced and gust-induced forces acting on bridge decks

With the increasing spans and complex deck shapes, aerodynamic nonlinearity becomes a crucial concern in the design of long-span bridges. This paper investigates the nonlinear interaction between the gust-induced and motion-induced forces acting on bridge decks using the Vortex Particle Method (VPM) as a Computational Fluid Dynamics (CFD) method. This nonlinear interaction is complex and intractable by the conventional linear semi-analytical models that employ the superposition principle. To excite such interaction, a sinusoidally oscillating bridge deck is subjected to sinusoidal vertical gusts. Two distinct aspects are studied: The influence of large-scale sinusoidal vertical gusts on the shear layer of a moving body and the nonlinear dependence of the aerodynamic forces on the effective angle of attack. For the latter, the resultant aerodynamic forces based on a linear semi-analytical and a CFD model are compared for the same effective angle of attack due to sinusoidal gust and motion. The methodology is first employed to verify the linear behavior of a flat plate and then study the nonlinear behavior of two bridge decks. The results show that the linear superposition principle holds for streamlined bridge decks with minimal shear layer instability. However, this principle may not be valid for bluff bridge decks due to strong separation vortices that dictate the shear layer, which induces nonlinearity in the aerodynamic forces manifested through amplitude difference in the first harmonic and emergence of higher-order harmonics. The outcome of this study aims to provide a deeper understanding of the complex nonlinear fluid–structure interaction occurring for bluff bodies subjected to motion and free-stream gusts.

Aerodynamic Nonlinear Energy Evolution Characteristics of Vertical Vortex-Induced Vibration of Open Bridge Girder

Compared with traditional streamlined box girders, girders with a typical open section have a highly bluff aerodynamic shape and are prone to cause vortex-induced vibration (VIV), which will affect the safety and serviceability of a bridge. In this study, the surface pressure distribution and aerodynamic nonlinearity characteristics of a typical open section are studied through wind tunnel tests and numerical simulations. Based on the synchronous measurement technology, the evolution characteristics of the surface pressure distribution in different periods of the VIV lock-in region are analyzed, and the numerical simulation of the section was performed by a user-defined function (UDF) program. The numerical simulation results are in good agreement with the wind tunnel test results. Furthermore, the multi-order components of aerodynamic force in the VIV process were decomposed by the Hilbert vibration decomposition (HVD) method. Our study found that the average pressure of the open section’s surface at the leading edge was negative, as this is where the air flow is separated. At the extreme point of the VIV lock-in region, the fluctuating pressure changes most drastically in the region ([Formula: see text]), indicating that the intensity of vortex shedding is the greatest in this region. High-order components of the aerodynamic forces acting on an open section girder can be identified by analysis. The aerodynamic damping of the open section girder is significantly reduced in the VIV lock-in region, and reaches its minimum near the extreme point of VIV. Due to the presence of high-order harmonic components, the degree of aerodynamic nonlinearity continues to increase. Moreover, the work performed by each aerodynamic component throughout the whole vibration period has different characteristics, and the influence of the third-order and above components on the VIV system can be ignored. This study can help us to further understand the aerodynamic nonlinear energy evolution characteristics of the open section, and further improve people’s understanding of the VIV system.

Nonlinear post-flutter bifurcation of a typical twin-box bridge deck: Experiment and empirical modeling

The post-flutter bifurcation behavior of a twin-box bridge deck was investigated through a series of elastically-supported section model tests. The experimental results suggested that the bridge deck will undergo limit cycle oscillations in post-flutter states via both subcritical and supercritical Hopf bifurcations due to aerodynamic nonlinearities. The stable and unstable branches were significantly affected by initial wind attack angle and mechanical nonlinearities. The range of unstable branch tended to decrease and stable branch tended to increase for larger initial attack angle. A novel wind-tunnel technique was employed to measure the aerodynamic force during post-flutter limit-cycle oscillations. A nonlinear single-degree-of-freedom (SDOF) model of nonlinear aerodynamic force was proposed based on the measured force signals. The model was able to reproduce the observed post-flutter limit-cycle oscillation with a satisfactory accuracy. The evolution mechanism of post-flutter bifurcations can be obtained via the diagrams of amplitude-dependent aerodynamic and mechanical damping.

Nonlinear aeroelastic analysis of a damped elastica-aerofoil system

This work formulates a comprehensive model of a nonlinear aeroelastic system developed for the analysis of complex aeroelastic phenomena related to structural and aerodynamic nonlinearities. The system is formulated as a two-dimensional cantilevered elastica with a rigid airfoil section firmly attached at its tip undergoing large displacements in the crosswind conditions. The system can demonstrate a wide range of domain specific as well as coupled nonlinear phenomena. The structural model is developed by means of the Rayleigh–Ritz approach, with shape functions discretizing both vertical and horizontal displacements and Lagrangian multipliers enforcing inextensibility. Damping is modeled based on a non-local strain-based mechanism in the Kelvin–Voigt arrangement. The resulting structural model is examined through studying the behavior under a follower load and with a tip-attached tendon under tension to study the shape convergence properties and the alignment of the results with known characteristics in the literature. The ONERA dynamic stall model is used to model the aerodynamics of the problem to accurately capture post-stall behavior at large deformations. The LCO responses of the aeroelastic problem are evaluated through time-marched simulations, and the combined airspeed–damping interactions are studied in this manner.

Finite wing effects on the energy harvesting from stall-induced oscillations

The modeling of aeroelastic harvesters usually employs methods for two-dimensional aerodynamics. Such convenient practice reflects the lack in the literature of efficient tools with low computational cost to assess unsteady and nonlinear aerodynamics loads accounting for three-dimensional wing effects. In this sense, the present work aims to fulfill this gap by developing a nonlinear unsteady lifting line to investigate finite wing effects on the power generation from stall-induced oscillations. The aerodynamic model links the Beddoes–Leishman dynamic stall method to a linear unsteady lifting line. The optimization of energy harvesters from stall-induced pitching oscillations is revisited considering the three-dimensional aerodynamic model. Results reveal that the three-dimensional harvester’s dynamic behavior may differ from the two-dimensional case, and the harvested power reduction is observed.

Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter

The numerical assessment of T-tail flutter requires a nonlinear description of the structural deformations when the unsteady aerodynamic forces comprise terms from lifting surface roll motion. For linear flutter, a linear deformation description of the vertical tail plane (VTP) out-of-plane bending results in a spurious stiffening proportional to the steady lift forces, which is corrected by incorporating second-order deformation terms in the equations of motion. While the effect of these nonlinear deformation components on the stiffness of the VTP out-of-plane bending mode shape is known from the literature, their impact on the aerodynamic coupling terms involved in T-tail flutter has not been studied so far, especially regarding amplitude-dependent characteristics. This term affects numerical results targeting common flutter analysis, as well as the study of amplitude-dependent dynamic aeroelastic stability phenomena, e.g., Limit Cycle Oscillations (LCOs). As LCOs might occur below the linear flutter boundary, fundamental knowledge about the structural and aerodynamic nonlinearities occurring in the dynamical system is essential. This paper gives an insight into the aerodynamic nonlinearities for representative structural deformations usually encountered in T-tail flutter mechanisms using a CFD approach in the time domain. It further outlines the impact of geometrically nonlinear deformations on the aerodynamic nonlinearities. For this, the horizontal tail plane (HTP) is considered in isolated form to exclude aerodynamic interference effects from the studies and subjected to rigid body roll and yaw motion as an approximation to the structural mode shapes. The complexity of the aerodynamics is increased successively from subsonic inviscid flow to transonic viscous flow. At a subsonic Mach number, a distinct aerodynamic nonlinearity in stiffness and damping in the aerodynamic coupling term HTP roll on yaw is shown. Geometric nonlinearities result in an almost entire cancellation of the stiffness nonlinearity and an increase in damping nonlinearity. The viscous forces result in a stiffness offset with respect to the inviscid results, but do not alter the observed nonlinearities, as well as the impact of geometric nonlinearities. At a transonic Mach number, the aerodynamic stiffness nonlinearity is amplified further and the damping nonlinearity is reduced considerably. Here, the geometrically nonlinear motion description reduces the aerodynamic stiffness nonlinearity as well, but does not cancel it.

Buffeting response of a suspension bridge based on the 2D rational function approximation model for self-excited forces

In the last thirty years, the aerodynamic nonlinearities related to the slow variation of the angle of attack produced by large-scale atmospheric turbulence and their impact on the buffeting response of long-span suspension bridges have been a hot topic in wind engineering research. Self-excited forces accounting for such an effect of turbulence have been crucial in predicting the dynamic response of bridge sectional models subjected to multi-harmonic gusts. In contrast, only few studies on full suspension bridges in realistic turbulent flows are available. To fill this gap, the 2D RFA model for self-excited forces, recently developed and experimentally validated by the authors, in this paper is incorporated into a stochastic time-variant state-space framework to assess the nonlinear buffeting response of a suspension bridge. The most important feature of this model is the modulation of the self-excited forces due to the spatio-temporal fluctuation of the angle of attack produced by low-frequency turbulence. Moreover, the nonlinear external buffeting forces are formulated to achieve a reasonable compromise between the conflicting needs of modelling both nonlinear and unsteady effects of wind velocity fluctuations. The model is applied to the Hardanger Bridge in Norway, considering different wind conditions. Indeed, the aerodynamic derivatives of this bridge deck cross-section present a strong dependence on the mean angle of attack. The results emphasise the significant impact on the buffeting response and flutter stability of considering time-variant self-excited forces, though in specific cases the classical time-invariant approach is found to provide accurate predictions of the bridge vibrations. Finally, the paper investigates the sensitivity of the response statistics to the model cut-offs used to separate the low-frequency and the high-frequency turbulence bands.

Energy harvesting from the gust response of membrane wing using piezoelectrics

Micro-air vehicles face challenges in aerodynamic and endurance. This paper presents a novel gust response-based membrane wing energy harvester to solve these issues. An aero-electro-mechanical model based on CFD/CSD coupling method is established to investigate the dynamic response and the energy harvesting performance of this energy harvester under harmonic gust loads. The geometric nonlinearity of structure and aerodynamic nonlinearity are considered in this paper. The harmonic gust loads were obtained from a Navier–Stokes CFD solver based on unstructured meshes. The large deformation electro-mechanical finite element model was built based on the updated Lagrangian formulation and the principle of virtual work. The mean power density is defined to evaluate the energy harvesting performance. The effects of the external load resistance, the PVDF patch location, the gust amplitude and frequency, and Reynold number are studied, respectively. The results show that the optimal resistance of this energy harvester is 1.6kΩ, and the flexibility of the membrane wing improves the aerodynamic and energy harvesting performance greatly through increases the amplitude of odd super harmonics. The PVDF location has a relatively small impact on the energy harvesting performance except near end surfaces. The PVDF patch located at 0.1c–0.9c all can get better energy harvesting performance. When the equivalent angle of attack of gust amplitude exceeds 4 deg, the energy harvesting performance decreases slightly due to the formation and convection of leading-edge vortex. A sustained output power density of 5.31mW/cm3 is harvested at Re=25,000. The present work provides an effective theoretical basis for further studying energy harvesting of Micro air vehicles.

A nonlinear aerothermoelastic model for slender composite beam-like wings with embedded shape memory alloys

This paper reports the formulation of an aerothermoelastic tool developed to investigate the behavior of flexible beam-like wings made of a hybrid adaptive material. Here, hybrid materials are defined as laminated composites additionally reinforced with embedded shape memory alloy wires. As main novelties, the proposed model couples geometrical, material, and aerodynamic nonlinearities to the thermal dynamics of SMA wires undergoing Joule’s effects, thereby establishing a multi-physical nonlinear problem. Geometrical nonlinearities were taken into account via an FE model of a 2D Timoshenko’s beam experiencing large deformations. Material nonlinearities were incorporated by a semi-empirical micro-mechanical model that computes the properties of hybrid laminates. To complement, nonlinear aerodynamic effects were introduced via an unsteady strip theory method in the time-domain, along with a nonlinear stall model and an assumption of follower aerodynamic forces. A set of numerical aerothermoelastic cases was performed by assuming various layups and SMA temperatures, with the objective of tailoring the aeroelastic response of hybrid wings. These cases were shown to lead to a considerable reduction in both post-flutter oscillations and post-divergence amplitudes as the SMA temperature increases. The outcomes have indicated compelling evidences on the applicability of embedded SMAs for structural, shape or aeroelastic control of flexible wings.

Reinforcement Learning-Based Composite Controller for Cable-Driven Parallel Suspension System at High Angles of Attack

This paper investigates an intelligent method for the motion control of a cable-driven parallel suspension system (CDPSS) in wind tunnel tests, especially at high angles of attack, which is characterized by unsteady and nonlinear aerodynamics. Considering the modeling uncertainties and the complex aerodynamic interference, a composite controller that combines deep deterministic policy gradient (DDPG) and computed-torque is proposed to improve the control performance. The tasks at hand consist in the construction of the training environment based on the dynamic equations and the Markov Decision Process (MDP) design. The supplementary computed-torque control is used to enhance the learning rate of the agent. Then a well-trained agent is applied in the high angles of attack maneuvers control with different examples, including single-DOF and multi-DOF coupled motion. The simulation results show the controller could fulfill the training tasks efficiently, and it turns out to be robust and maintain strong generalization ability despite handling the unlearned tasks.