Detection of gait perturbations based on proprioceptive information. Application to Limit Cycle Walkers

Walking on irregular surfaces and in the presence of unexpected events is a challenging problem for bipedal machines. Up to date, their ability to cope with gait disturbances is far less successful than humans': Neither trajectory controlled robots, nor dynamic walking machines (Limit CycleWalkers) are able to handle them satisfactorily. On the contrary, humans reject gait perturbations naturally and efficiently relying on their sensory organs that, if needed, elicit a recovery action. A similar approach may be envisioned for bipedal robots and exoskeletons: An algorithm continuously observes the state of the walker and, if an unexpected event happens, triggers an adequate reaction. This paper presents a monitoring algorithm that provides immediate detection of any type of perturbation based solely on a phase representation of the normal walking of the robot. The proposed method was evaluated in a Limit Cycle Walker prototype that suffered push and trip perturbations at different moments of the gait cycle, providing 100% successful detections for the current experimental apparatus and adequately tuned parameters, with no false positives when the robot is walking unperturbed.

Narrowband random excitation of a limit cycle system

The response of the Van der Pol oscillator to stationary narrowband Gaussian excitation is considered. The central frequency of excitation is taken to be in the neighborhood of the system limit cycle frequency. The solution is obtained using a non-Gaussian closure approximation on the probability density function of the response. The validity of the solution is examined with the help of a stochastic stability analysis. Solution based on Stratonovich''s quasistatic averaging technique is also obtained. The comparison of the theoretical solutions with the digital simulations shows that the theoretical estimates are reasonably good.

The Influence of Structural Modeshape Variability on Limit Cycle Oscillation Behaviour

Modal analysis is a popular approach used in structural dynamic and aeroelastic problems due to its efficiency. The response of a structure is compo
sed of the sum of orthogonal eigenvectors or modeshapes and corresponding modal frequencies. This paper investigates the importance of modeshapes on the aeroelastic response of the Goland wing subject to structural uncertainties. The wing undergoes limit cycle oscillations (LCO) as a result of the inclusion of polynomial stiffness nonlinearities. The LCO computations are performed using a Harmonic Balance approach for speed, the modal properties of the system are extracted from MSC NASTRAN. Variability in both the wing’s structure and the store centre of gravity location is investigated in two cases:- supercritical and subcritical type LCOs. Results show that the LCO behaviour is only sensitive to change in modeshapes when the nature of the modes are changing significantly.

Analysis of Transonic Limit Cycle Oscillations under Uncertainty

For the computation of limit cycle oscillations (LCO) at transonic speeds, CFD is required to capture the nonlinear flow features present. The Harmonic Balance method provides an effective means for the computation of LCOs and this paper exploits its efficiency to investigate the impact of variability (both structural a nd aerodynamic) on the aeroelastic behaviour of a 2 dof aerofoil. A Harmonic Balance inviscid CFD solver is coupled with the structural equations and is validated against time marching analyses. Polynomial chaos expansions are employed for the stochastic investiga tion as a faster alternative to Monte Carlo analysis. Adaptive sampling is employed when discontinuities are present. Uncertainties in aerodynamic parameters are looked at first followed by the inclusion of structural variability. Results show the nonlinear effect of Mach number and it’s interaction with the structural parameters on supercritical LCOs. The bifurcation boundaries are well captured by the polynomial chaos.

Prediction of Limit Cycle Oscillations Using an Implicit Aeroelastic-Harmonic Balance Method

This work proposes a extends a novel approach to compute tran sonic Limit Cycle Oscillations using high fidelity analysis. CFD based Harmonic Balance methods have proven to be efficient tools to predict periodic phenomena. This paper’s contribution is to present a methodology to determine the unknown frequency of oscillations using an implicit for- mulation of the HB method to accurately capture Limit Cycle Oscillations (LCOs); this is achieved by defining a frequency updating procedure based on a coupled CFD/CSD Harmonic Balance formulation to find the LCO condition. A pitch/plunge aerofoil and respective linear structural models is used to exercise the new method. Results show consistent agreement between the proposed and time-marching methods for both LCO amplitude and frequency.

Investigation of Limit Cycle Oscillations Using Aeroelastic-Harmonic Balance Method

This work investigates limit cycle oscillations in the transonic regime. A novel approach to predict Limit Cycle Oscillations using high fidelity analysis is exploited to accelerate calculations. The method used is an Aeroeasltic Harmonic Balance approach, which has been proven to be efficient and able to predict periodic phenomena. The behaviour of limit cycle oscillations is analysed using uncertainty quantification tools based on polynomial chaos expansions. To improve the efficiency of the sampling process for the polynomial-chaos expansions an adaptive sampling procedure is used. These methods are exercised using two problems: a pitch/plunge aerofoil and a delta-wing. Results indicate that Mach n. variability is determinant to the amplitude of the LCO for the 2D test case, whereas for the wing case analysed here, variability in the Mach n. has an almost negligible influence in amplitude variation and the LCO frequency variability has an almost linear relation with Mach number. Further test cases are required to understand the generality of these results.

Prediction of limit cycle oscillations under uncertainty using a Harmonic Balance method

The Harmonic Balance method is an attractive solution for computing periodic responses and can be an alternative to time domain methods, at a reduced computational cost. The current paper investigates using a Harmonic Balance method for simulating limit cycle oscillations under uncertainty. The Harmonic Balance method is used in conjunction with a non-intrusive polynomial-chaos approach to propagate variability and is validated against Monte Carlo analysis. Results show the potential of the approach for a range of nonlinear dynamical systems, including a full wing configuration exhibiting supercritical and subcritical bifurcations, at a fraction of the cost of performing time domain simulations.

Prediction of Transonic Limit-Cycle Oscillations Using an Aeroelastic Harmonic Balance Method

This work proposes a novel approach to compute transonic limit-cycle oscillations using high-fidelity analysis. Computational-Fluid-Dynamics based harmonic balance methods have proven to be efficient tools to predict periodic phenomena. This paper’s contribution is to present a new methodology to determine the unknown frequency of oscillations, enabling harmonic balance methods to accurately capture limit-cycle oscillations; this is achieved by defining a frequency-updating procedure based on a coupled computational-fluid-dynamics/computational-structural-dynamics harmonic balance formulation to find the limit-cycle oscillation condition. A pitch/plunge airfoil and delta wing aerodynamic and respective linear structural models are used to validate the new method against conventional time-domain simulations. Results show consistent agreement between the proposed and time-marching methods for both limit-cycle oscillation amplitude and frequency while producing at least a one-order-of-magnitude reduction in computational time.

Reducing Parametric Uncertainty in Limit Cycle Oscillations

The assimilation of discrete higher fidelity data points with model predictions can be used to achieve a reduction in the uncertainty of the model input parameters which generate accurate predictions. The problem investigated here involves the prediction of limit-cycle oscillations using a High-Dimensional Harmonic Balance method (HDHB). The efficiency of the HDHB method is exploited to enable calibration of structural input parameters using a Bayesian inference technique. Markov-chain Monte Carlo is employed to sample the posterior distributions. Parameter estimation is carried out on both a pitch/plunge aerofoil and Goland wing configuration. In both cases significant refinement was achieved in the distribution of possible structural parameters allowing better predictions of their
true deterministic values.

Frustrated limit cycle and irregular behaviour in a nonlinear pendulum

We discuss how the presence of frustration brings about irregular behaviour in a pendulum with nonlinear dissipation. Here frustration arises owing to particular choice of the dissipation. A preliminary numerical analysis is presented which indicates the transition to chaos at low frequencies of the driving force.

Control of Limit Cycle Oscillation in a Three Degrees of Freedom Airfoil Section Using Fuzzy Takagi-Sugeno Modeling

This work presents a strategy to control nonlinear responses of aeroelastic systems with control surface freeplay. The proposed methodology is developed for the three degrees of freedom typical section airfoil considering aerodynamic forces from Theodorsen's theory. The mathematical model is written in the state space representation using rational function approximation to write the aerodynamic forces in time domain. The control system is designed using the fuzzy Takagi-Sugeno modeling to compute a feedback control gain. It useds Lyapunov's stability function and linear matrix inequalities (LMIs) to solve a convex optimization problem. Time simulations with different initial conditions are performed using a modified Runge-Kutta algorithm to compare the system with and without control forces. It is shown that this approach can compute linear control gain able to stabilize aeroelastic systems with discontinuous nonlinearities.

Monte Carlo limit cycle characterization

The fixed point implementation of IIR digital filters usually leads to the appearance of zero-input limit cycles, which degrade the performance of the system. In this paper, we develop an efficient Monte Carlo algorithm to detect and characterize limit cycles in fixed-point IIR digital filters. The proposed approach considers filters formulated in the state space and is valid for any fixed point representation and quantization function. Numerical simulations on several high-order filters, where an exhaustive search is unfeasible, show the effectiveness of the proposed approach.