Linear Algebra Applied to Kinematic Control of Mobile Manipulators

Over the last 20 years, our learning environment for linear algebra has changed dramatically mathematical tools take an important role in our classes. Sage is popular mathematical software which was released in 2005. This software has efficient features to adapt the internet environment and it can cover most of mathematical problems, for example, algebra, combinatorics, numerical mathematics, calculus and linear algebra. Nowadays there are more mobile/smartphones than the number of personal computers in the world. Furthermore, the most sophisticated smartphones have almost the same processing power as personal computer and it can be connected to the internet. For example, we can connect from mobile phone to any Sage server through the internet. We have developed over the years on Mobile mathematics with Smartphone for teaching linear algebra (Ko et al. 2009; Lee and Kim 2009; Lee et al. 2001). In this article, we introduce Sage and how we can use it in our linear algebra classes. We aim to show the mobile infrastructure of the Sage and the mobile-learning environment. We shall also introduce mobile contents for linear algebra using Sage. In fact, almost all the concepts of linear algebra can be easily covered.

The complexity of the role of digital media in facilitation of learning mathematics can be approached by utilizing multiple theoretical frameworks. In this chapter, three theoretical frameworks have been applied with an aim to analyze the contribution of a created Dynamic Geometry Environment in developing deep understanding of concepts in linear algebra. The first one, which is in the main focus, refers to the integration of different description and thinking modes in linear algebra, such as synthetic-geometric, arithmetic and analytic-structural (Hillel in On the teaching of linear algebra. Springer, Netherlands, pp. 191–207, 2000; Sierpinska in On the teaching of linear algebra. Springer, Netherlands, pp. 209–246, 2000). The second one is related to the attributes of Dynamic Technological Environments, such as Recursive Exploration Space (Hegedus et al. in Proceedings of CERME5, WG 9. Tools and technologies in mathematical didactics 1331, pp. 1419–1428, 2007); and the third one is semiotic mediation (Bussi and Mariotti in Semiotic mediation in the mathematics classroom: artifacts and signs after a Vygotskian perspective handbook of international research in mathematics education. New York, pp. 746–783, 2008) of the dragging tool. A landscape of networking strategies for connecting theories (Prediger et al. in ZDM Math Educ 40(2):165–178, 2008) has been exploited as an attempt to ensure quality of the analysis.

This article describes an adaptive robust control algorithm of the mobile manipulator based on robust neural network to solve disturbances, nonlinearity, nonholonomic constraints between the mobile platform and the mounted manipulator. According to the nonlinear mapping ability of neural network and fast learning ability, neural network and robust control strategies are integrated into the adaptive control algorithm for the mobile manipulator with nonlinear and nonholonomic constraints. Basis on the mobile manipulator dynamics model and its modeling analysis, the controller can drive the trajectory tracking error of the mobile manipulator to converge to zero. The simulation results show that the algorithm has higher precision of trajectory tracking control and can achieve good control effect.

Mobile manipulators have attracted a lot of interest recently. This paper discusses the development of mobile manipulators in recent years in the aspect of motion planning and coordinated control between mobile platform and manipulator. An intelligent controller based on RBF neural network is introduced for the coordinating control of the mobile manipulator, which mimics human behavior. Simulation results was given out and the method has been applied on the Hebut-II mobile robot platform. Experimental results verified the validity of the method.

The linear algebra is an abstract mathematics course. In this paper, some basic and important concepts in linear algebra are explained from the geometrical point of view. Combining with the teaching practice, we explore the teaching method which unifies the linear algebra and the geometry. In this paper, several aspects are discussed, such as the determinant, the linear equations and the linear correlation of the vectors.

This paper discusses the modeling and control of a spatial mobile manipulator which consists of a robotic manipulator mounted upon a wheeled mobile platform. By neglecting slip of the wheeled platform’s tires, nonholonomic constraints are introduced into the equations of motion which complicates the control problem. The dynamic equations of the mobile manipulator are derived using the Lagrange-d’Alembert formulation. In this formulation, the equations of motion are projected onto the subspace of allowable motions. Then, the solvability of tracking problems for the mobile manipulator is investigated by using input-output feedback linearization. The feasibility of the control approach is demonstrated through computer simulation.

7 - LINEAR TRANSFORMATIONS, MATRICES AND DETERMINANTS

Deconvolutions based on singular value decomposition and the pseudoinverse: a guide for beginners.

13 - Experimental Studies

This paper introduces new planning and control methods for supermedia-enhanced real-time telerobotic operations via the Internet. Supermedia is the collection of video, audio, haptic information, temperature, and other sensory feedback. However, when the communication medium used, such as the Internet, introduces random communication time delay, several challenges and difficulties arise. Most importantly, random communication delay causes instability, loss of transparency, and desynchronization in real-time closed-loop telerobotic systems. Due to the complexity and diversity of such systems, the first challenge is to develop a general and efficient modeling and analysis tool. This paper proposes the use of Petri net modeling to capture the concurrency and complexity of Internet-based teleoperation. Combined with the event-based planning and control method, it also provides an efficient analysis and design tool to study the stability, transparency, and synchronization of such systems. In addition, the concepts of event transparency and event synchronization are introduced and analyzed. This modeling and control method has been applied to the design of several supermedia-enhanced Internet-based telerobotic systems, including the bilateral control of mobile robots and mobile manipulators. These systems have been experimentally implemented in three sites test bed consisting of robotic laboratories in the USA, Hong Kong, and Japan. The experimental results have verified the theoretical development and further demonstrated the stability, event transparency, and event synchronization of the systems.

In this paper the problem of trajectory tracking considers that the values of the control actions do not exceed a maximum allowable value and the zero convergence of tracking errors is demonstrated. The control law is based on a linear algebra approach. First, the desired trajectories of some state variables are determined by analyzing the conditions for a system of linear equations to have an exact solution. Therefore, the control signals are obtained by solving the system of linear equations. The optimal controller parameters are selected through nonlinear programming so as to prevent the saturation of the control actions. Experimental results are presented and discussed, demonstrating the controller's good performance. Finally, the performance of the proposed controller is compared with a fuzzy controller, and all the results are validated through experimental laboratory tests.

We study binomiality of the steady state ideals of chemical reaction networks. Considering rate constants as indeterminates, the concept of unconditional binomiality has been introduced and an algorithm based on linear algebra has been proposed in a recent work for reversible chemical reaction networks, which has a polynomial time complexity upper bound on the number of species and reactions. In this article, using a modified version of species-reaction graphs, we present an algorithm based on graph theory which performs by adding and deleting edges and changing the labels of the edges in order to test unconditional binomiality. We have implemented our graph theoretical algorithm as well as the linear algebra one in Maple and made experiments on biochemical models. Our experiments show that the performance of the graph theoretical approach is similar to or better than the linear algebra approach, while it is drastically faster than Grobner basis and quantifier elimination methods.

This paper investigates the problem of distributed control of multiple redundant mobile manipulators to collectively transport an object tracking a desired trajectory with energy and manipulability optimized.To solve this optimization problem, formation control tasks are introduced as equality constraints with the variables being the velocities. In this paper, we propose a distributed proximal gradient algorithm searching for the optimal solution, with which the stability of the closed-loop system is proved. Simulations demonstrate the effectiveness of the proposed distributed optimization scheme and proximal algorithm.

Fundamentals in modeling and control of mobile manipulators , Fundamentals in modeling and control of mobile manipulators , کتابخانه مرکزی دانشگاه علوم پزشکی ایران

Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. In this book we develop JCF and show how to apply it to solving systems of differential equations. We