Standard Quadratic Programming Research Articles

Two Acceleration-Layer Configuration Amendment Schemes of Redundant Robot Arms Based on Zhang Neurodynamics Equivalency.

Two innovative acceleration-layer configuration amendment (CA) schemes are proposed to achieve the CA of constrained redundant robot arms. Specifically, by applying the Zhang neurodynamics equivalency (ZNE) method, an acceleration-layer CA performance indicator is derived theoretically. To obtain a unified-layer inequality constraint by transforming from angle-layer and velocity-layer constraints to acceleration-layer constraints, five theorems and three corollaries are theoretically derived and rigorously proved. Then, together with the unified acceleration-layer bound constraint, an enhanced acceleration-layer CA scheme specially considering three-layer time-variant physical limits is proposed, and a simplified acceleration-layer CA scheme considering three-layer time-invariant physical limits is also proposed. The proposed CA schemes are finally formulated in the form of standard quadratic programming and are solved by a projection neurodynamics solver. Moreover, comparative simulative experiments based on a four-link planar arm and a UR3 spatial arm are performed to verify the efficacy and superiority of the proposed CA schemes. At last, physical experiments are conducted on a real Kinova Jaco2 arm to substantiate the practicability of the proposed CA schemes.

Convex distributed robust model predictive control for collision and obstacle avoidance

AbstractThe article proposes a convex distributed robust model predictive control algorithm for collision avoidance in multi‐agent systems with additive disturbances, which utilizes the concept of tube model predictive control to address the disturbances. To tackle the coupled collision avoidance constraints, each agent integrates assumed nominal position and input trajectories from its neighbors, rather than relying on actual ones. Compatibility constraints are then designed using the normal vector of the constructed separating hyperplanes for collision avoidance and the residual collision avoidance margin of the optimal solutions at the last time step to restrict deviations between assumed and actual trajectories and ensure consistency among the agents. The nonconvex collision and obstacle avoidance constraints are convexified using the concept of safe sets, transforming them into time‐varying closed polyhedral constraints while accounting for the impact of disturbances. Particularly, the residual collision avoidance margin is incorporated into the construction of safe sets for the satisfaction of coupled collision avoidance constraints. Consequently, the original collision avoidance optimal control problems can be efficiently and simultaneously solved for all agents using standard quadratic programming techniques. Under the design of local terminal constraint sets for each agent, the proposed algorithm ensures a rigorous analysis of robust constraint satisfaction for collision avoidance constraints, as well as an assessment of recursive feasibility and closed‐loop stability. The effectiveness of the algorithm is demonstrated through an illustrative example, and simulation results validate its ability to successfully achieve collision and obstacle avoidance even in the presence of disturbances.

Shape restricted additive hazards models: Monotone, unimodal, and U-shape hazard functions.

We consider estimation of the semiparametric additive hazards model with an unspecified baseline hazard function where the effect of a continuous covariate has a specific shape but otherwise unspecified. Such estimation is particularly useful for a unimodal hazard function, where the hazard is monotone increasing and monotone decreasing with an unknown mode. A popular approach of the proportional hazards model is limited in such setting due to the complicated structure of the partial likelihood. Our model defines a quadratic loss function, and its simple structure allows a global Hessian matrix that does not involve parameters. Thus, once the global Hessian matrix is computed, a standard quadratic programming method can be applicable by profiling all possible locations of the mode. However, the quadratic programming method may be inefficient to handle a large global Hessian matrix in the profiling algorithm due to a large dimensionality, where the dimension of the global Hessian matrix and number of hypothetical modes are the same order as the sample size. We propose the quadratic pool adjacent violators algorithm to reduce computational costs. The proposed algorithm is extended to the model with a time-dependent covariate with monotone or U-shape hazard function. In simulation studies, our proposed method improves computational speed compared to the quadratic programming method, with bias and mean square error reductions. We analyze data from a recent cardiovascular study.

Novel Snap-Layer MMPC Scheme via Neural Dynamics Equivalency and Solver for Redundant Robot Arms With Five-Layer Physical Limits.

To obtain smoother kinematic control of minimum motion, a novel snap-layer minimum motion scheme, otherwise known as the minimum motion planning and control (MMPC) scheme for redundant robot arms, is proposed for the first time in this study. With the primary task of tracking planned paths and the consideration of satisfying five-layer physical limits, the snap-layer MMPC problem is transformed into a quadratic programming (QP) problem. Five-layer physical limits include angle-layer, velocity-layer, acceleration-layer, jerk-layer, and snap-layer limits, which are all considered and then transformed into a unified-layer bounded constraint through Zhang neural dynamics (ZND) equivalency. Furthermore, the snap-layer performance index and equation constraint are derived by utilizing the ZND formula. Therefore, the proposed snap-layer MMPC scheme is formulated as a standard QP that can avoid the potential physical damage of redundant robot arms. The snap-layer projection neural dynamics (PND) solver is presented and used to acquire the neural solution of the QP. Simulation results on a 6-degrees-of-freedom (DOF) planar redundant robot arm are presented to substantiate the effectiveness and superiority of the proposed snap-layer MMPC scheme by comparing it with the jerk-layer MMPC scheme and the minimum snap norm (MSN) scheme.

A Dynamic Path-Planning Method for Obstacle Avoidance Based on the Driving Safety Field.

Establishing an accurate and computationally efficient model for driving risk assessment, considering the influence of vehicle motion state and kinematic characteristics on path planning, is crucial for generating safe, comfortable, and easily trackable obstacle avoidance paths. To address this topic, this paper proposes a novel dual-layered dynamic path-planning method for obstacle avoidance based on the driving safety field (DSF). The contributions of the proposed approach lie in its ability to address the challenges of accurately modeling driving risk, efficient path smoothing and adaptability to vehicle kinematic characteristics, and providing collision-free, curvature-continuous, and adaptable obstacle avoidance paths. In the upper layer, a comprehensive driving safety field is constructed, composed of a potential field generated by static obstacles, a kinetic field generated by dynamic obstacles, a potential field generated by lane boundaries, and a driving field generated by the target position. By analyzing the virtual field forces exerted on the ego vehicle within the comprehensive driving safety field, the resultant force direction is utilized as guidance for the vehicle's forward motion. This generates an initial obstacle avoidance path that satisfies the vehicle's kinematic and dynamic constraints. In the lower layer, the problem of path smoothing is transformed into a standard quadratic programming (QP) form. By optimizing discrete waypoints and fitting polynomial curves, a curvature-continuous and smooth path is obtained. Simulation results demonstrate that our proposed path-planning algorithm outperforms the method based on the improved artificial potential field (APF). It not only generates collision-free and curvature-continuous paths but also significantly reduces parameters such as path curvature (reduced by 62.29% to 87.32%), curvature variation rate, and heading angle (reduced by 34.11% to 72.06%). Furthermore, our algorithm dynamically adjusts the starting position of the obstacle avoidance maneuver based on the vehicle's motion state. As the relative velocity between the ego vehicle and the obstacle vehicle increases, the starting position of the obstacle avoidance path is adjusted accordingly, enabling the proactive avoidance of stationary or moving single and multiple obstacles. The proposed method satisfies the requirements of obstacle avoidance safety, comfort, and stability for intelligent vehicles in complex environments.

Identification and high-precision trajectory tracking control for space robotic manipulator

In this paper, an effective control and parameters identification scheme was proposed for high-precision trajectory tracking of space robotic manipulators. Our proposed control method employed a linear-extended-state-observer (LESO) based model predictive control (MPC) scheme. The feedback linearization approach was utilized to develop a model predictive controller. The optimization objective function was simplified to a standard quadratic programming (QP) problem for solution. To minimize the impact of the inaccuracy in the dynamics model, a LESO was designed to estimate the disturbance, and its convergence was demonstrated. In order to improve the accuracy of dynamic model, the physical consistency of inertial parameters was constrained using linear matrix inequality (LMI) techniques. Moreover, the dynamics parameters were estimated through semidefinite programming (SDP) techniques. To validate the effectiveness of the proposed control method, it was compared with existing controllers through simulations and experiments. The results demonstrated the superior performance of the proposed scheme, highlighting its potential for achieving high-precision trajectory tracking control in space robotic manipulators.

Quadratic Programming for Continuous Control of Safety-Critical Multiagent Systems Under Uncertainty

This paper studies the control problem for safety-critical multi-agent systems based on quadratic programming (QP). Each controlled agent is modeled as a cascade connection of an integrator and an uncertain nonlinear actuation system. In particular, the integrator represents the position-velocity relation, and the actuation system describes the dynamic response of the actual velocity to the velocity reference signal. The notion of input-to-output stability (IOS) is employed to characterize the essential velocity-tracking capability of the actuation system. The standard QP algorithms for collision avoidance may be infeasible due to uncertain actuator dynamics. Even if feasible, the solutions may be non-Lipschitz because of possible violation of the full rank condition of the active constraints. Also, the interaction between the controlled integrator and the uncertain actuator dynamics may lead to significant robustness issues. Based on the current development of nonlinear control theory and numerical optimization methods, this paper first contributes a new feasible-set reshaping technique and a refined QP algorithm for feasibility, robustness, and local Lipschitz continuity. Then, we present a nonlinear small-gain analysis to handle the inherent interaction for guaranteed safety of the closed-loop multi-agent system. The proposed method is illustrated by numerical simulation and a physical experiment.

Parameter identification and model prediction path following control of underactuated AUV: Methodology and experimental verification

This paper explores the integrated optimal control scheme from the identification of underactuated autonomous underwater vehicle (AUV) system dynamics model parameters to the application of Model Predictive Control (MPC) technology in underactuated AUV motion control, and presents the simulation and field test results. First, the AUV kinematics and dynamics models are introduced to represent the horizontal plane motion characteristics. Based on Computational Fluid Dynamics (CFD) numerical simulation and overset mesh technology, AUV plane motion is simulated to determine the unknown hydrodynamic coefficients in the model, and the accuracy of the mathematical model is validated by comparing the maneuverability simulation and field test results. After that, an adaptive line-of-sight (ALOS) guiding algorithm with exponentially variable forward distance is utilized in the guidance layer to reconcile the rapidity and stability of tracking. The path following problem is formulated as an MPC optimization problem that takes into account multiple realistic constraints and then converted into a standard convex quadratic programming structure, permitting for online real-time optimization. Simulation and field test results indicate that the control algorithm proposed in this paper provides a more effective control strategy than the traditional LOS+PID controller in tracking effect and robustness.

A classification method of fuzzy semi-supervised support vector machines for the problems of imbalance

Positive instances are often significantly less than negative instances in real-world classification problems. However, positive categories are typically more relevant to the primary focus of categorization tasks. Moreover, obtaining labeled data is often expensive, and the majority of real-life data is unlabeled. Therefore, semi-supervised learning has become a popular approach for addressing imbalanced problems. Traditional support vector machines (SVMs) treat all samples equally and are not suitable for semi-supervised learning. To address this issue, a semi-supervised model called the fuzzy semi-supervised SVM (FS3VM) has been proposed. The FS3VM model uses the degree of entropy-based fuzzy membership to ensure the materiality of positive classes by assigning positive instances to relatively large degrees of fuzzy membership. After introducing the mainstream FS3VM model, the fundamental theory and methods of the model are discussed and expanded upon, including the FS3VM algorithm, which applies the Sequential Minimal Optimization (SMO) algorithm to the dual problem. The proposed FS3VM model is a smooth and continuous optimization problem, and its dual is a standard quadratic programming. Experimental results demonstrate that the proposed FS3VM model outperforms other compared learning algorithms.

Research on Trajectory Planning and Tracking Methods for Coal Mine Mobile Robots

Coal Mine Mobile Robots (CMRRs) are generally large in size and inertia, while narrow laneway space and bumpy terrain pose great challenges to CMRR’s planning and control. Aiming at the trajectory planning and tracking problems of CMRR, a new trajectory class MINCO is derived in detail based on the properties of differential flat systems. A trajectory planning method based on MINCO trajectory and safety corridor constraints constructed with underground environmental constraints is further proposed. A trajectory tracking method based on model predictive control (MPC) is further proposed. The prediction model of MPC is constructed by a kinematics model and transformed into a standard quadratic programming problem according to the cost function of a trajectory tracking target. Finally, large quantities of field tests were carried out for the proposed approaches. The results show that the proposed planning algorithm based on MINCO trajectory can achieve good avoidance effects within 10 planning attempts in different obstacle scenarios, and the trajectory is smoother compared to the Fast-Planner algorithm, with shorter trajectory length and less planning time. The tracking error of MPC is always less than 0.05 m in different underground scenarios, having a more adaptable trajectory tracking effect than PID.

An active swing suppression control scheme of overhead cranes based on input shaping model predictive control

An input shaper-based model predictive control scheme is proposed for an overhead crane to suppress the swing of the payload. A control-oriented mathematical model of the crane is firstly derived based on the small angle hypothesis. Furthermore, three kinds of commonly used shapers are discussed and a zero vibration derivative shaper is selected for the crane. Then, a model predictive swing suppression controller is designed for the crane based on the selected shaper. Constraints on the control input and its slew rate are respected. Besides, the constraint on the swing angle of the payload is also respected as an output constraint. The constrained solution is given in the form of standard quadratic programming. The designed control scheme combines open-loop and closed-loop control techniques, achieving accurate positioning of the trolley and good swing suppression of the payload. A series of simulation demonstrates the effectiveness and advantages of the proposed scheme.

A controller management scheme for active fault-tolerant tracking: Linear discrete-time systems

This paper proposes an active fault-tolerant control (FTC) approach based on the controller management and virtual actuator idea for linear discrete-time systems subject to unknown L2-bounded disturbances, input constraint, and time-varying additive actuator faults. The closed-loop faulty system, which includes the modified nominal controller, the fault and state estimator, and the virtual actuator, suppresses the effects of disturbances and faults, while ensuring input-constraint satisfaction. The management of the nominal controller is performed through an online optimization method – in the form of a standard quadratic programming problem – by manipulating the reference input and intervening in the nominal controller evolution. The proposed method proves the input-to-state stability (ISS) criterion of the overall closed-loop faulty system. The problem of minimizing the ultimate bound of the ISS criterion is formulated in terms of tractable linear matrix inequality (LMI) conditions that allow the fault and state estimation errors to converge to a small neighborhood of the origin. To illustrate the capabilities and advantages of the proposed control strategy, comparative simulation results are presented for a flexible joint robotic system, tracking control of a DC motor’s angular velocity, and the multivariable VTOL aircraft.

Incorporation of Data-Mined Knowledge into Black-Box SVM for Interpretability

The lack of interpretability often makes black-box models challenging to be applied in many practical domains. For this reason, the current work, from the black-box model input port, proposes to incorporate data-mined knowledge into the black-box soft-margin SVM model to enhance accuracy and interpretability. The concept and incorporation mechanism of data-mined knowledge are successively developed, based on which a partially interpretable soft-margin SVM ( pTsm -SVM) optimization model is designed and then solved through reformulating the optimization problem as standard quadratic programming. An algorithm for mining linear positive (negative) class knowledge from general data sets is also proposed, which generates a linear two-dimensional discriminative rule with specificity (sensitivity) equal to 1 and the highest possible sensitivity (specificity) among all two-dimensional feature spaces. The knowledge-integrated pTsm -SVM works by achieving a good trade-off among the “large margin”, “high specificity”, and “high sensitivity”. Our experimental results on eight UCI datasets demonstrate the superiority of the proposed pTsm -SVM over the standard soft-margin SVM both in terms of accuracy and interpretability.

Ensemble Support Vector Recurrent Neural Network for Brain Signal Detection.

The brain-computer interface (BCI) P300 speller analyzes the P300 signals from the brain to achieve direct communication between humans and machines, which can assist patients with severe disabilities to control external machines or robots to complete expected tasks. Therefore, the classification method of P300 signals plays an important role in the development of BCI systems and technologies. In this article, a novel ensemble support vector recurrent neural network (E-SVRNN) framework is proposed and developed to acquire more accurate and efficient electroencephalogram (EEG) signal classification results. First, we construct a support vector machine (SVM) to formulate EEG signals recognizing model. Second, the SVM formulation is transformed into a standard convex quadratic programming (QP) problem. Third, the convex QP problem is solved by combining a varying parameter recurrent neural network (VPRNN) with a penalty function. Experimental results on BCI competition II and BCI competition III datasets demonstrate that the proposed E-SVRNN framework can achieve accuracy rates as high as 100% and 99%, respectively. In addition, the results of comparison experiments verify that the proposed E-SVRNN possesses the best recognition accuracy and information transfer rate (ITR) compared with most of the state-of-the-art algorithms.

Linear programming with nonparametric penalty programs and iterated thresholding

It is known [Mangasarian, A Newton method for linear programming, J. Optim. Theory Appl. 121 (2004), pp. 1–18] that every linear program can be solved exactly by minimizing an unconstrained quadratic penalty program. The penalty program is parameterized by a scalar t>0, and one is able to solve the original linear program in this manner when t is selected larger than a finite, but unknown . In this paper, we show that every linear program can be solved using the solution to a parameter-free penalty program. We also characterize the solutions to the quadratic penalty programs using fixed points of certain nonexpansive maps. This leads to an iterative thresholding algorithm that converges to a desired limit point. We show in numerical experiments that this iterative method can outperform a variety of standard quadratic program solvers. Finally, we show that for every , the solution one obtains by solving a parameterized penalty program is guaranteed to lie in the feasible set of the original linear program.

New bounds for nonconvex quadratically constrained quadratic programming

In this paper, we study some bounds for nonconvex quadratically constrained quadratic programs (QCQPs). We propose two types of bounds for QCQPs, quadratic and cubic bounds. We use affine functions as Lagrange multipliers for quadratic bounds. We demonstrate that most semidefinite relaxations can be obtained as the dual of a quadratic bound. In addition, we study bounds obtained by changing the ground set. For cubic bounds, in addition to affine multipliers we employ quadratic functions. We provide a comparison between the proposed cubic bound and typical bounds for standard quadratic programs. Moreover, we report comparison results of some quadratic and cubic bounds.

A Hierarchy of Standard Polynomial Programming Formulations for the Maximum Clique Problem

The maximum clique problem is a classical optimization problem which finds many important applications. Some of the most theoretically interesting and computationally successful approaches to this problem are based on the Motzkin--Straus formulation, expressing the clique number in terms of the maximal value of a standard quadratic program. This intriguing relationship has also motivated significant developments in quadratic optimization, copositive programming, and complexity in nonlinear optimization. Inspired by such developments, this paper establishes a hierarchy of polynomial programming formulations (\textbfP$^k$), $k\in\{2,\ldots, \omega\}$ for the maximum clique problem, relating the clique number $\omega$ of a given graph to the global maximal value of a multilinear polynomial function $f_k(x)$ of degree $k$ over the standard simplex in $\mathbb{R}^{|V|}$. The case of $k=2$ corresponds to the Motzkin--Straus formulation, and the hierarchical feature is in the fact that the set of local maxima of (\textbfP$^{k+1}$) is a subset of the set of local maxima of (\textbfP$^{k}$), $k\in\{2,\ldots, \omega-1\}$. In particular, every local maximum of (\textbfP$^{\omega}$) is global.

Real-time power optimization for application server clusters based on Mixed-Integer Programming

In the environment of peer competition and energy conservation, optimizing the deployment of application server clusters in real time according to actual workload conditions to reduce operating costs and energy consumption is an important issue that must be urgently addressed. In this paper, we propose a real-time power optimization strategy for application server clusters, and the optimization measures include CPU dynamic voltage/frequency scaling and server dynamic switching. First, the feasibility of defining variables for server types is proved, and appropriate variables are defined to describe the cluster power optimization as a mixed-integer programming (MIP) problem. Then, two solution methods are proposed: the exact method based on the Gurobi optimizer and the approximate method based on primary–secondary optimization and differential evolution with two mutations (PSODE). The former turns the MIP problem into a standard mixed-integer quadratic programming form by introducing intermediate variables and solves it using the Gurobi optimizer. The latter rewrites the MIP problem as a primary–secondary optimization problem and proposes a differential evolutionary-based solution algorithm for the primary optimization problem. The evolutionary process consists of two mutation operations, inter- and intraindividual mutations, which both use a heuristic policy to accelerate the evolution convergence. The test results reveal that the Gurobi-based method can quickly determine the global optimal deployment when the cluster size is small. The PSODE-based method can quickly determine the global optimal deployment or high-quality suboptimal deployment when applied to large-scale clusters.

Finite Convergence of Sum-of-Squares Hierarchies for the Stability Number of a Graph

We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp.875--892], who conjectured convergence to $\alpha(G)$ in $r=\alpha(G)-1$ steps. Even the weaker conjecture claiming finite convergence is still open. We establish links between this hierarchy and sum-of-squares hierarchies based on the Motzkin-Straus formulation of $\alpha(G)$, which we use to show finite convergence when $G$ is acritical, i.e., when $\alpha(G\setminus e)=\alpha(G)$ for all edges $e$ of $G$. This relies, in particular, on understanding the structure of the minimizers of Motzkin-Straus formulation and showing that their number is finite precisely when $G$ is acritical. As a byproduct we show that deciding whether a standard quadratic program has finitely many minimizers does not admit a polynomial-time algorithm unless P=NP.

Position and Orientation Tunnel-Following NMPC of Robot Manipulators Based on Symbolic Linearization in Sequential Convex Quadratic Programming

The tunnel-following nonlinear model predictive control (NMPC) scheme allows to exploit acceptable deviations around a path reference. This is done by using convex-over-nonlinear functions as objective and constraints in the underlying optimal control problem (OCP). The convex-over-nonlinear structure is exploited by algorithms such as the generalized Gauss-Newton (GGN) method or the sequential convex quadratic programming (SCQP) method to reduce the computational complexity of the OCP solution. However, the modeling effort and engineering time required to implement these methods is high. We address the problem of reducing the modeling effort in the implementation of SCQP, focusing on a standard sequential quadratic programming (SQP) implementation where symbolic linearization is applied to the nonlinear part of the convex-over-nonlinear functions in the objective and constraints. The novelty of this letter is twofold. It introduces a novel operator that applies symbolic linearization in a transparent and easy way to solve nonconvex OCPs with the SCQP method, and introduces a meaningful representation of an orientation-tunnel for robotic applications by means of a convex-over-nonlinear constraint, which preserves the convexity exploitation by the SCQP method. The proposed technique is demonstrated in a tunnel-following task for a 7-degrees-of-freedom manipulator.