Motion Planning of Uncertain Ordinary Differential Equation Systems

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and under-actuated systems, prescribes deterministic actuator inputs which yield uncertain state trajectories. The inverse dynamics formulation is the dual to the forward dynamic, and is only applicable to fully-actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to under-actuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories which yield uncertain unactuated states and actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case-studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space

Simultaneous Optimal Uncertainty Apportionment and Robust Design Optimization of Systems Governed by Ordinary Differential Equations

The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimal performance, and higher build costs. Treatment of small geometric uncertainty in the context of manufacturing tolerances is a well studied topic. Traditional sequential design methodologies have recently been replaced by concurrent optimal design methodologies where optimal system parameters are simultaneously determined along with optimally allocated tolerances; this allows to reduce manufacturing costs while increasing performance. However, the state of the art approaches remain limited in that they can only treat geometric related uncertainties restricted to be small in magnitude. This work proposes a novel framework to perform robust design optimization concurrently with optimal uncertainty apportionment for dynamical systems governed by ordinary differential equations. The proposed framework considerably expands the capabilities of contemporary methods by enabling the treatment of both geometric and non-geometric uncertainties in a unified manner. Additionally, uncertainties are allowed to be large in magnitude and the governing constitutive relations may be highly nonlinear. In the proposed framework, uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments of the uncertain system to be explicitly included in the optimization-based design process. The framework formulates design problems as constrained multi-objective optimization problems, thus enabling the characterization of a Pareto optimal trade-off curve that is off-set from the traditional deterministic optimal trade-off curve. The Pareto off-set is shown to be a result of the additional statistical moment information formulated in the objective and constraint relations that account for the system uncertainties. Therefore, the Pareto trade-off curve from the new framework characterizes the entire family of systems within the probability space; consequently, designers are able to produce robust and optimally performing systems at an optimal manufacturing cost. A kinematic tolerance analysis case-study is presented first to illustrate how the proposed methodology can be applied to treat geometric tolerances. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design at an optimal manufacturing cost, accounting for the entire family of systems within the associated probability space. This case-study highlights the general nature of the new framework which is capable of optimally allocating uncertainties of multiple types and with large magnitudes in a single calculation

Parameter Estimation for Mechanical Systems Using an Extended Kalman Filter

This paper proposes a new computational approach based on the Extended Kalman Filter (EKF) in order to apply the polynomial chaos theory to the problem of parameter estimation, using direct stochastic collocation. The Kalman filter formula is used at each time step in order to update the polynomial chaos of the uncertain states and the uncertain parameters. The main advantage of this method is that the estimation comes in the form of a probability density function rather than a deterministic value, combined with the fact that simulations using polynomial chaos methods are much faster than Monte Carlo simulations. The proposed method is applied to a nonlinear four degree of freedom roll plane model of a vehicle, in which an uncertain mass with an uncertain position is added on the roll bar. A major drawback was identified: the EKF can diverge when using a high sampling frequency, which might prevent the use of enough data to obtain accurate results when a low sampling frequency is necessary. When applying the polynomial chaos theory to the EKF, numerical errors can accumulate even faster than in the general case due to the truncation in the polynomial chaos expansions, which is illustrated on a simple example. An alternative EKF approach which consists of applying the filter formula on all the observations at once usually yields better results, but can still sometimes fail to produce very accurate results. Therefore, using different sampling rates in order to verify the coherence of the results and comparing the results to a different approach is strongly recommended

A Polynomial Chaos Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems - Part II: Applications to Vehicle Systems

This is the second part of a two-part article. In the first part, a new computational approach for parameter estimation was proposed based on the application of the polynomial chaos theory. The maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. In this part, the new parameter estimation method is illustrated on a nonlinear four-degree-of-freedom roll plane model of a vehicle in which an uncertain mass with an uncertain position is added on the roll bar. The value of the mass and its position are estimated from periodic observations of the displacements and velocities across the suspensions. Appropriate excitations are needed in order to obtain accurate results. For some excitations, different combinations of uncertain parameters lead to essentially the same time responses, and no estimation method can work without additional information. Regularization techniques can still yield most likely values among the possible combinations of uncertain parameters resulting in the same time responses than the ones observed. When using appropriate excitations, the results obtained with this approach are close to the actual values of the parameters. The accuracy of the estimations has been shown to be sensitive to the number of terms used in the polynomial expressions and to the number of collocation points, and thus it may become computationally expensive when a very high accuracy of the results is desired. However, the noise level in the measurements affects the accuracy of the estimations as well. Therefore, it is usually not necessary to use a large number of terms in the polynomial expressions and a very large number of collocation points since the addition of extra precision eventually affects the results less than the effect of the measurement noise. Possible applications of this theory to the field of vehicle dynamics simulations include the estimation of mass, inertia properties, as well as other parameters of interest

A Polynomial Chaos Based Bayesian Approach for Estimating Uncertain Parameters of Mechanical Systems – Part I: Theoretical Approach

This is the first part of a two-part article. A new computational approach for parameter estimation is proposed based on the application of the polynomial chaos theory. The polynomial chaos method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters. In the new approach presented in this paper, the maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. Direct stochastic collocation is used as a less computationally expensive alternative to the traditional Galerkin approach to propagate the uncertainties through the system in the polynomial chaos framework. This approach is applied to very simple mechanical systems in order to illustrate how the cost function can be affected by undersampling, non-identifiablily of the system, non-observability, and by excitation signals that are not rich enough. When the system is non-identifiable, regularization techniques can still yield most likely values among the possible combinations of uncertain parameters resulting in the same time responses than the ones observed. This is illustrated using a simple spring-mass system. Possible applications of this theory to the field of vehicle dynamics simulations include the estimation of mass, inertia properties, as well as other parameters of interest. In the second part of this article, this new parameter estimation method is illustrated on a nonlinear four-degree-of-freedom roll plane model of a vehicle in which an uncertain mass with an uncertain position is added on the roll bar

Parameter estimation method using an extended Kalman Filter

Fast parameter estimation is a non-trivial task, and it is critical when the system parameters evolve with time, as demanded in real-time control applications. In this study, a new computational approach for parameter identification is proposed based on the application of polynomial chaos theory. The polynomial chaos approach has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters. In the framework of this new approach, a (suboptimal) Extended Kalman Filter (EKF) is used to recalculate the polynomial chaos expansions for the uncertain states and the uncertain parameters. As a case study, the proposed parameter estimation method is applied to a four degree-of-freedom roll plane model of a vehicle for which the vertical stiffnesses of the tires are estimated from periodic observations of the displacements and velocities across the suspensions. The results obtained with this approach are close to the actual values of the parameters. In addition, the EKF approach gives more information about the parameters of interest than a simple estimated value: the estimation comes in the form of a probability density function. The approach presented in this paper has shown great promise for an improvement in the computational efficiency of current parameter estimation methods. Possible applications of this theory to the field of off-road vehicle simulations include the estimation of various vehicle parameters of interest, as well as the estimation of parameters related to the tire-terrain contact

Treating Uncertainties in Multibody Dynamic Systems using a Polynomial Chaos Spectral Decomposition

ABSTRACT This study addresses the critical need for computational tools to model complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. Polynomial chaos has been used extensively to model uncertainties in structural mechanics and in fluids, but to our knowledge they have yet to be applied to multibody dynamic simulations. We show that the method can be applied to quantify uncertainties in time domain and frequency domain

STATE-OF-THE-ART AND FUTURE DEVELOPMENTS IN INTEGRATED CHASSIS CONTROL FOR GROUND VEHICLES

Many modern ground vehicles feature state-of-the-art powertrain, braking and suspension control systems. These technologies are rapidly filtering through to heavy and off-road vehicles. The drawback of many of these systems is that their operation is largely considered only in stand-alone mode. The paper introduces up-to-date and coming ground vehicle technology related to the integration and advanced control of active chassis control systems. In particular, addressing the task of coordinated subsystems control can provide simultaneous enhancements in traction and braking performance, handling, off-road mobility, driving comfort and energy efficiency. A special focus in the paper is given to coordinated operation of brake control, active suspension, and dynamic tyre pressure management. The influence of each particular subsystem on the vehicle safety, off-road mobility and ride comfort is evaluated in simulation. It is further described and confirmed in simulation how the integrated chassis control (ICC) can simultaneously improve each of these vehicle characteristics. From the hardware viewpoint, a variant of ground vehicle architecture with aforementioned integrated active chassis systems is introduced. This architecture and its corresponding implementation on a sport utility vehicle are currently investigated within the European Union-funded Horizon 2020 consortium EVE. The work presented is a collaborative effort among several ISTVS members across the globe

Our initial experience with ventriculo-epiplooic shunt in treatment of hydrocephalus in two centers

Introduction Hydrocephalus represents impairment in cerebrospinal fluid (CSF) dynamics. If the treatment of hydrocephalus is considered difficult, the repeated revisions of ventriculo-peritoneal (VP) shunts are even more challenging. Objective The aim of this article is to evaluate the efficiency of ventriculo-epiplooic (VEp) shunt as a feasible alternative in hydrocephalic patients. Material and methods A technical modification regarding the insertion of peritoneal catheter was imagined: midline laparotomy 8–10cm long was performed in order to open the peritoneal cavity; the great omentum was dissected between its two layers; we placed the distal end of the catheter between the two epiplooic layers; a fenestration of 4cm in diameter into the visceral layer was also performed. A retrospective study of medical records of 15 consecutive patients with hydrocephalus treated with VEp shunt is also presented. Results Between 2008 and 2014 we performed VEp shunt in 15 patients: 5 with congenital hydrocephalus, 8 with secondary hydrocephalus and 2 with normal pressure hydrocephalus. There were 7 men and 8 women. VEp shunt was performed in 13 patients with multiple distal shunt failures and in 2 patients, with history of abdominal surgery, as de novo extracranial drainage procedure. The outcome was favorable in all cases, with no significant postoperative complications. Conclusions VEp shunt is a new, safe and efficient surgical technique for the treatment of hydrocephalus. VEp shunt is indicated in patients with history of recurrent distal shunt failures, and in patients with history of open abdominal surgery and high risk for developing abdominal complications