Topic
Polar coordinate system
About: Polar coordinate system is a research topic. Over the lifetime, 2708 publications have been published within this topic receiving 27134 citations. The topic is also known as: polar coordinates system.
Papers published on a yearly basis
Papers
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TL;DR: In this paper, the authors derived exact state equations for the MP filter without imposing any restrictions on own-ship motion; thus, prediction accuracy inherent in the traditional Cartesian formulation is completely preserved.
Abstract: Previous studies have shown that the Cartesian coordinate extended Kalman filter exhibits unstable behavior characteristics when utilized for bearings-only target motion analysis (TMA). In contrast, formulating the TMA estimation problem in modified polar (MP) coordinates leads to an extended Kalman filter which is both stable and asymptotically unbiased. Exact state equations for the MP filter are derived without imposing any restrictions on own-ship motion; thus, prediction accuracy inherent in the traditional Cartesian formulation is completely preserved. In addition, these equations reveal that MP coordinates are well-suited for bearings-only TMA because they automatically decouple observable and unobservable components of the estimated state vector. Such decoupling is shown to prevent covariance matrix ill-conditioning, which is the primary cause of filter instability. Further investigation also confirms that the MP state estimates are asymptotically unbiased. Realistic simulation data are presented to support these findings and to compare algorithm performance with respect to the Cramer-Rao lower bound (ideal) as well as the Cartesian and pseudolinear filters.
477 citations
Book•
01 Jan 1974TL;DR: In this paper, the authors introduce concepts and mathematics of Elasticity in Cartesian Coordinates, and three-dimensional equations of elasticity in the plane and the polar coordinates.
Abstract: Introductory Concepts and Mathematics. Theory of Deformation. Theory of Stress. Three--Dimensional Equations of Elasticity. Plane Theory of Elasticity in Rectangular Cartesian Coordinates. Plane Elasticity in Polar Coordinates. Prismatic Bar Subjected to End Load. General Solutions of Elasticity. References. Bibliography. Index.
409 citations
TL;DR: Constraints on the desired linear and angular velocities as well as the posture of the mobile robot are eliminated unlike the previous studies based on kinematics expressed in polar coordinates, and arbitrary trajectories can be followed even with large initial tracking errors and bounded disturbances.
Abstract: This brief proposes a sliding-mode control method for wheeled-mobile robots in polar coordinates. A new sliding-mode control method is proposed for mobile robots with kinematics in two-dimensional polar coordinates. In the proposed method, two controllers are designed to asymptotically stabilize the tracking errors in position and heading direction, respectively. By combining these controllers together, both asymptotic posture (position and heading direction) stabilization and trajectory tracking are achieved for reference trajectories at global regions except the arbitrary small region around the origin. In particular, constraints on the desired linear and angular velocities as well as the posture of the mobile robot are eliminated unlike the previous studies based on kinematics expressed in polar coordinates. Accordingly, arbitrary trajectories including a circle and a straight line in various forms can be followed even with large initial tracking errors and bounded disturbances. The stability and performance analyzes are performed and also simulations are included to confirm the effectiveness of the proposed scheme.
366 citations
TL;DR: In this paper, a method for treating the coordinate singularity whereby singular coordinates are redefined so that data are differentiated smoothly through the pole, and avoiding placing a grid point directly at the pole is proposed.
313 citations
Book•
01 Jan 1936
TL;DR: In this article, the authors introduce a calculus of transcendental functions, and apply it to the Vector Calculus of Transcendental Functions (VCFL) and the Integral Calculus.
Abstract: 1. Functions and Graphs. 2. Prelude to Calculus. 3. The Derivative. 4. Additional Applications of the Derivative. 5. The Integral. 6. Applications of the Integral. 7. Exponential and Logarithmic Functions. 8. Further Calculus of Transcendental Functions. 9. Techniques of Integration. 10. Polar Coordinates and Plane Curves. 11. Infinite Series. 12. Vectors, Curves, and Surfaces in Space. 13. Partial Differentiation. 14. Multiple Integrals. 15. Vector Calculus. Appendices. Answers to Odd-Numbered Problems. References for Further Study. Teaching Outlines. Index.
242 citations