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Orthogonal Trajectories01:26

Orthogonal Trajectories

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Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
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It isn't easy to measure a parameter such as the mean height or the mean weight of a population. So, we draw samples from the population and calculate the mean height or mean weight of the individuals in the sample. This sample data acts as a representative measure of the population parameter. These sample statistics are known as estimates. 
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The muscles that move the head are a dynamic and complex group of structures that work together to facilitate a wide range of head movements, including rotation, flexion, extension, and lateral bending.
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Aminoglycosides are a class of antibiotics used to treat various bacterial infections. Clinicians must determine the elimination rate constant (k) and volume of distribution (VD) to optimize therapeutic efficacy and minimize toxicity. The k value represents the rate at which the drug is removed from the body, and the VD reflects the degree to which the drug distributes into body tissues. Accurately estimating these parameters allows healthcare professionals to tailor drug dosing to individual...
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Integration by Parts: Indefinite Integrals01:26

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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Integration by Parts: Definite Integrals01:23

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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Related Experiment Video

Updated: Jan 29, 2026

Mapping Cortical Dynamics Using Simultaneous MEG/EEG and Anatomically-constrained Minimum-norm Estimates: an Auditory Attention Example
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Adaptive Trajectory-Constrained Heading Estimation for Tractor GNSS/SINS Integrated Navigation.

Shupeng Hu1, Song Chen2, Lihui Wang1

  • 1School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China.

Sensors (Basel, Switzerland)
|January 28, 2026
PubMed
Summary

This study introduces an adaptive heading estimation method for tractors using Global Navigation Satellite System (GNSS) and Strapdown Inertial Navigation System (SINS). The new approach significantly improves accuracy and reduces convergence time for autonomous navigation systems.

Keywords:
adaptive algorithmautonomous navigationheading measurementinertial navigation systemrobot tractorsensor fusiontrajectory constraint

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Area of Science:

  • Robotics and Automation
  • Navigation Systems
  • Agricultural Engineering

Background:

  • Accurate heading estimation is vital for autonomous tractor navigation.
  • Dual-antenna GNSS systems are precise but have installation and safety issues.
  • Single-antenna GNSS/SINS offers adaptability but struggles with slow convergence and low accuracy in low-speed farming.

Purpose of the Study:

  • To propose an adaptive trajectory-constrained heading estimation method for improved GNSS/SINS performance in tractors.
  • To address the limitations of existing methods in low-speed, low-dynamic agricultural environments.
  • To enhance the accuracy and convergence speed of heading estimation for autonomous tractor navigation.

Main Methods:

  • Developed a sliding-window adaptive extended Kalman filter (SWAEKF) with a heading constraint model using GNSS trajectory angle.
  • Employed an enhanced Sage-Husa algorithm for adaptive estimation of trajectory angle measurement variance.
  • Implemented a covariance initialization strategy based on trajectory angle increments to accelerate convergence.

Main Results:

  • The SWAEKF achieved rapid heading convergence (under 10s for straight lines, 14s for curves).
  • High accuracy was demonstrated with RMS heading errors below 0.15° (straight) and 0.25° (curved).
  • The SWAEKF improved accuracy by 23% and reduced convergence time by 62% compared to conventional adaptive EKF.

Conclusions:

  • The proposed adaptive trajectory-constrained method significantly enhances GNSS/SINS integrated navigation for tractors.
  • The algorithm effectively meets the demanding requirements for autonomous navigation systems in low-dynamic agricultural settings.
  • This method offers a robust and efficient solution for precise tractor guidance in complex farmland operations.