Comments about the paper entitled ``A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity'' by A.J. Kassab and E. Divo.

An integral formulation for heat conduction problems in non-homogeneous media has recently been proposed by Kassab and Divo [Engineering Analysis with Boundary Elements 1996;18:273]. The goal of this communication is to revisit and clarify two key features of the formulation of Kassab and Divo. First, the contention that the integral equation formulation proposed by them does not possess the desired boundary-only character is made and substantiated; it is shown in particular that Eq. 10 therein does not hold owing to the fact that a crucial requirement for the fundamental solution, Eqs. (5c) and (5d) therein, is actually not met. Second, the limiting process associated with a vanishing neighbourhood in connection with the particular kernel function used therein is revisited

On the Identification of Machine Settings for Gear Surface Topography Corrections

In this paper we set out to investigate the performances of some algorithms proposed in the gear literature for identifying the machine-tool settings required to obtain predesigned gear tooth surface topographies, or needed to compensate for flank form deviations of real teeth. For ease of comparison, the problem is formulated as a nonlinear least squares problem, and the most widely employed algorithms are derived as special cases. The algorithms included in the analysis are: (i) one-step methods, (ii) iterative methods, (iii) iterative methods with step control. The performance index is devised in their ability of returning practical solutions in the presence of: (i) strong model nonlinearities, (ii) ill-conditioning of the sensitivity matrix, (iii) demanding topographic shapes. Instrumental here is an original classification of topographic modifications as either “simple” or “complex”, based on the SVD analysis of the sensitivity matrix. Some selected numerical examples demonstrate that iterative techniques with step control are the most convenient in terms of reliability and robustness of the obtained solutions. The generation process considered here is face-milling of hypoid gears, although the methodology is general enough to cope with any gear cutting/grinding method

Fixed-point implementation of a proximal Newton method for embedded model predictive control (I)

Extending the success of model predictive control (MPC) technologies in embedded applications heavily depends on the capability of improving quadratic programming (QP) solvers. Improvements can be done in two directions: better algorithms that reduce the number of arithmetic operations required to compute a solution, and more efficient architectures in terms of speed, power consumption, memory occupancy and cost. This paper proposes a fixed point implementation of a proximal Newton method to solve optimization problems arising in input-constrained MPC. The main advantages of the algorithm are its fast asymptotic convergence rate and its relatively low computational cost per iteration since it the solution of a small linear system is required. A detailed analysis on the effects of quantization errors is presented, showing the robustness of the algorithm with respect to finite-precision computations. A hardware implementation with specific optimizations to minimize computation times and memory footprint is also described, demonstrating the viability of low-cost, low-power controllers for high-bandwidth MPC applications. The algorithm is shown to be very effective for embedded MPC applications through a number of simulation experiments

Cautious NMPC with Gaussian Process Dynamics for Autonomous Miniature Race Cars

This paper presents an adaptive high performance control method for autonomous miniature race cars. Racing dynamics are notoriously hard to model from first principles, which is addressed by means of a cautious nonlinear model predictive control (NMPC) approach that learns to improve its dynamics model from data and safely increases racing performance. The approach makes use of a Gaussian Process (GP) and takes residual model uncertainty into account through a chance constrained formulation. We present a sparse GP approximation with dynamically adjusting inducing inputs, enabling a real-time implementable controller. The formulation is demonstrated in simulations, which show significant improvement with respect to both lap time and constraint satisfaction compared to an NMPC without model learning

Fixed-Point Constrained Model Predictive Control of Spacecraft Attitude

The paper develops a Model Predictive Controller for constrained control of spacecraft attitude with reaction wheel actuators. The controller exploits a special formulation of the cost with the reference governor like term, a low complexity addition of integral action to guarantee offset-free tracking of attitude set points, and an online optimization algorithm for the solution of the Quadratic Programming problem which is tailored to run in fixed-point arithmetic. Simulations on a nonlinear spacecraft model demonstrate that the MPC controller achieves good tracking performance while satisfying reaction wheel torque constraints. The controller also has low computational complexity and is suitable for implementation in spacecrafts with fixed-point processors

A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems

A reduced-order lie group-based race car model for systematic trajectory optimization on 3D tracks

This paper derives the dynamic equations of a reduced-order race-car model using Lie-group methods. While these methods are familiar to computational dynamicists and roboticists, their adoption in the vehicle dynamics community is limited. We address this gap by demonstrating how this framework integrates smoothly with the Articulated-Body Algorithm (ABA) and provides a fresh and systematic formulation of vehicle dynamics. For the first time, we model the car body as the end effector of a serial robot with a floating base connected to the track via virtual revolute and prismatic joints. Our formulation also accounts for the effects of 3D track geometry, providing a natural embedding of the car into the 3D track. We rigorously reconcile the ABA steps with key aspects of vehicle dynamics, including road-tire interactions, aerodynamic forces, and load transfers. The resulting model, simple yet accurate, is a powerful tool to efficiently solve Minimum-Lap-Time Planning problems. To demonstrate the effectiveness of our approach, we show numerical results obtained on the Nürburgring circuit. Our optimization problem is formulated via a direct collocation method and solved using the CasADi optimization suite. To validate the results, we test our reduced-order model against a full-fledged multi-body model recently developed by the same authors. The comparison confirms the validity of our reduced-order model, proving both the accuracy of the solution and the computational efficiency achieved