Multivariate Simplex Splines Research Articles

Reinforcement learning control with function approximation via multivariate simplex splines

SummaryIn the field of optimal control for continuous nonlinear systems, function approximation methods are often employed to overcome the curse of dimensionality. Compared to other global function approximators like neural networks, multivariate splines can be easily evaluated and adapted on a local basis with linearity in the parameters. In this work, a multivariate spline based reinforcement learning (RL) strategy is proposed for solving the continuous‐time nonlinear control problem. Based on the classic value iteration method, multivariate splines are integrated into RL algorithms to approximate continuous value functions and policy functions from discrete action and value samples. Hence, the determined splines with updated coefficients can be utilized in continuous control of nonlinear systems. In the simulation experiment, the performance of the spline‐based RL control is evaluated in controlling an under‐actuated inverted pendulum. The proposed method is compared with the value iteration based discrete control strategy and the neural network based continuous control strategy. The simulation results indicate that the proposed method based on multivariate splines has better control performance with less state oscillations, energy consumption and convergence time in comparison with discrete value iteration and neural network based RL, and the adoption of simplex splines improves the function approximation efficiency with less computation time than neural network optimization.

Modelling wing wake and tail aerodynamics of a flapping-wing micro aerial vehicle

Despite significant interest in tailless flapping-wing micro aerial vehicle designs, tailed configurations are often favoured, as they offer many benefits, such as static stability and a simpler control strategy, separating wing and tail control. However, the tail aerodynamics are highly complex due to the interaction between the unsteady wing wake and tail, which is generally not modelled explicitly. We propose an approach to model the flapping-wing wake and hence the tail aerodynamics of a tailed flapping-wing robot. First, the wake is modelled as a periodic function depending on wing flap phase and position with respect to the wings. The wake model is constructed out of six low-order sub-models representing the mean, amplitude and phase of the tangential and vertical velocity components. The parameters in each sub-model are estimated from stereo-particle image velocimetry measurements using an identification method based on multivariate simplex splines. The computed model represents the measured wake with high accuracy, is computationally manageable and is applicable to a range of different tail geometries. The wake model is then used within a quasi-steady aerodynamic model, and combined with the effect of free-stream velocity, to estimate the forces produced by the tail. The results provide a basis for further modelling, simulation and design work, and yield insight into the role of the tail and its interaction with the wing wake in flapping-wing vehicles. It was found that due to the effect of the wing wake, the velocity seen by the tail is of a similar magnitude as the free stream and that the tail is most effective at 50–70% of its span.

Nonlinear Multivariate Spline-Based Control Allocation for High-Performance Aircraft

High-performance flight control systems based on the nonlinear dynamic inversion principle require highly accurate models of aircraft aerodynamics. In general, the accuracy of the internal model determines to what degree the system nonlinearities can be canceled; the more accurate the model, the better the cancellation, and with that, the higher the performance of the controller. In this paper, a new control system is presented that combines nonlinear dynamic inversion with multivariate simplex spline-based control allocation. Three control allocation strategies that use novel expressions for the analytic Jacobian and Hessian of the multivariate spline models are presented. Multivariate simplex splines have a higher approximation power than ordinary polynomial models and are capable of accurately modeling nonlinear aerodynamics over the entire flight envelope of an aircraft. This nonlinear spline based controller is applied to control a high-performance aircraft (F-16) with a large flight envelope. The simulation results indicate that perfect feedback linearization can be achieved throughout the entire flight envelope, leading to a significant increase in tracking performance compared with ordinary polynomial-based nonlinear dynamic inversion.

Differential constraints for bounded recursive identification with multivariate splines

The ability to perform online model identification for nonlinear systems with unknown dynamics is essential to any adaptive model-based control system. In this paper, a new differential equality constrained recursive least squares estimator for multivariate simplex splines is presented that is able to perform online model identification and bounded model extrapolation in the framework of a model-based control system. A new type of linear constraints, the differential constraints, are used as differential boundary conditions within the recursive estimator which limit polynomial divergence when extrapolating data. The differential constraints are derived with a new, one-step matrix form of the de Casteljau algorithm, which reduces their formulation into a single matrix multiplication. The recursive estimator is demonstrated on a bivariate dataset, where it is shown to provide a speedup of two orders of magnitude over an ordinary least squares batch method. Additionally, it is demonstrated that inclusion of differential constraints in the least squares optimization scheme can prevent polynomial divergence close to edges of the model domain where local data coverage may be insufficient, a situation often encountered with global recursive data approximation.

A new approach to linear regression with multivariate splines

A new methodology for creating highly accurate, static nonlinear maps from scattered, multivariate data is presented. This new methodology uses the B-form polynomials of multivariate simplex splines in a new linear regression scheme. This allows the use of standard parameter estimation techniques for estimating the B-coefficients of the multivariate simplex splines. We present a generalized least squares estimator for the B-coefficients, and show how the estimated B-coefficient variances lead to a new model quality assessment measure in the form of the B-coefficient variance surface. The new modeling methodology is demonstrated on a nonlinear scattered bivariate dataset.

Fuzzy Simplex Splines

Multivariate simplex splines are capable of approximating non-linear input-output mappings with great accuracy, as can neural networks. A new method is proposed which combines the best qualities of simplex splines and polynomial neural networks while circumventing their drawbacks. The new method generates mappings which are continuously differentiable, use a stable polynomial basis, and are not restricted to contiguous configurations of simplices. The method is shown to perform equally well or better than both polynomial neural networks and standard simplex splines for a highly non-linear identification problem. An iterative polynomial order optimization scheme has been applied to all three methods to increase their performance. Finally the setup of the new method creates the opportunity for simplex vertex optimization using standard (gradient) methods.

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli[6] and extended to hold for multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box splines. To this end, we utilize a relation due to Dahmen and Micchelli[4] that connects box splines and cone splines and a degree reduction formula given by Cohen, Lyche, and Riesenfeld in [2].

Moments of Dirichlet splines and their applications to hypergeometric functions

Dirichlet averages of multivariate functions are employed for a derivation of basic recurrence formulas for the moments of multivariate Dirichlet splines. An algorithm for computing the moments of multivariate simplex splines is presented. Applications to hypergeometric functions of several variables are discussed.

Multivariate Divided Differences. I: Basic Properties

The notion of the univariate divided differences is generalized to the multivariate case. This generalization is based on a pointwise evaluation of a certain multivariate function. Several properties of the defined multivariate divided difference functional are derived, and a link with the multivariate simplex splines is established. This gives a new generalization of the so-called truncated power function, which is different from the one given in [Multivariate Approximation Theory, Birkhäuser Basel, Switzerland, 1979, pp. 69–82] and [SIAM J. Numer. Anal., 17 (1980), pp. 179–191].

On discrete simplex splines and subdivision

Discrete analogoues of multivariate simplex splines are introduced. Their study yields a subdivision scheme for simplex splines.

B-splines, hypergeometric functions, and Dirichlet averages

Properties of Dirichlet averages are used to derive some well-known and some new properties of univariate and multivariate simplex splines. A univariate B-spline is the jump discontinuity of a hypergeometric R-function, and the Fourier transform of a B-spline is a confluent hypergeometric S-function. A univariate or multivariate B-spline is a Dirichlet average of a Dirac delta function. Its dependence on the knots is governed by a system of Euler-Poisson partial differential equations.

The stable evaluation of multivariate simplex splines

This paper gives a general method for the stable evaluation of multivariate simplex splines, based on the well-known recurrence relation of Micchelli [12]. This paper deals with two problems which arise in the implementation of the recurrence relation. First, the coefficients in the recurrence are shown to be efficiently computable via the dual simplex method of linear programminig. Secondly, the problem of evaluation along mesh boundaries is discussed in detail.

The Evaluation of Inner Products of Multivariate Simplex Splines

This paper gives a general method for the stable evaluation of inner products of multivariate simplex splines. The method is based on a recurrence relation for these inner products. The base cases for this recurrence relation are handled by triangulating certain simploids into simplices.