Sub-optimal Solution of the Inverse Kinematic Task of Redundant Robots without Using Lagrange Multipliers

In the paper a novel approach is suggested for solving the inverse kinematic task of redundant open kinematic chains. Traditional approaches as the Moore-Penrose generalized inverse-based solutions minimize the sum of squares of the timederivative of the joint coordinates under the constraint that contains the task itself. In the vicinity of kinematic singularities where these solutions are possible the hard constraint terms produce high time-derivatives that can be reduced by the use of a deformation proposed by Levenberg and Marquardt. The novel approach uses the basic scheme of the Receding Horizon Controllers in which the Lagrange multipliers are eliminated by direct application of the kinematic model over the horizon in the role of the ”control force”, and no reduced gradient has to be computed. This fact considerably decreases the complexity of the solution. If the cost function contains penalty for high joint coordinate time-derivatives the kinematic singularities are ab ovo better handled. Simulation examples made for a 7 degree of freedom robot arm demonstrate the operation of the novel approach. The computational need of the method is still considerable but it can be further decreased by the application of complementary tricks.

Time-independent perturbation theory with Lagrange multipliers

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

Multi Time-Scale Engine and Powertrain Control for Autonomous Vehicles via Lagrange Multipliers

Connected and Autonomous vehicles (CAVs) have the ability to use information obtained via Vehicle-to-Infrastructure (V2I), Vehicle-to-Vehicle communication (V2V), and sensors to improve their fuel economy through predictive strategies, including velocity trajectory optimization and optimal traffic light arrival and departure. These powertrain control strategies operate on a slow timescale relative to the engine dynamics, hence assume that the engine torque production is instantaneous. This assumption results in a torque command profile that may lead to engine dynamics constraint violation, actuator saturation, poor tracking performance, decreased efficiency, poor drivability, and increased emissions. To address this issue, a supplemental controller based on an iterative hierarchical Model Predictive Control (MPC) is proposed in this paper. The constraint satisfaction is achieved through a novel two-way communication of the Lagrange multipliers. The proposed methodology is demonstrated on an autonomous Diesel semi-truck on two maneuvers. Compared to a traditional centralized approach, the proposed method achieves systematic constraints satisfaction with negligible effect on fuel economy, less than 1%, and significantly improved computation time, more than 10 times.

Utility Maximization of Bangladeshi Consumers within Their Budget: A Mathematical Procedure

A consumer is considered as a person or a group of people who uses purchased goods, products, or services only for personal use, and not for manufacturing or resale. Consumers usually purchase valuable and useful commodities or goods by spending all or partial of their income. The property of a commodity that enables it to satisfy human wants is called utility. Producers must be conscious to increase the utility among the consumers. This study has considered the maximization of utility problem of consumers of Bangladesh subject to two constraints; namely, budget constraint and coupon constraint. Consequently, in the study two Lagrange multipliers are used and interpreted these with mathematical analysis. Prediction of consumer behavior will help both producers and consumers to take decision of their future economic productions and consumptions, respectively. This article is ornamented with sufficient theorems and economic analyses. So that all the readers find interest when go through the economic analysis of utility maximization.