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Related Concept Videos

Kinematic Equations for Rotation01:30

Kinematic Equations for Rotation

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In mechanics, when one observes a rigid body in rotational motion with constant angular acceleration, it is possible to establish equations for its rotational kinematics. This process resembles how linear kinematics are dealt with in simpler motion studies.
For instance, imagine a point A on a rigid body engaged in circular motion. The translational velocity of this particular point can be calculated by taking the time derivatives of the displacement equation, which essentially measures the...
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The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant velocity, and the distance traveled due to a change in velocity. Similar to the first kinematic equation, this equation is also only valid when the acceleration is constant throughout the motion of an object.
Suppose a car merges into freeway traffic on a 200 m long ramp. If its initial velocity is 10 m/s and it accelerates at 2 m/s2, then the...
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The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity as a function of the acceleration and distance over which it acts. The fourth kinematic equation does not have an acceleration term and provides the final position of the object at time t in terms of the initial and final velocities. This equation is useful when the value of the constant acceleration is unknown.
Using the kinematic equations,...
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Relative Motion Analysis using Rotating Axes01:25

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Consider a component AB undergoing a linear motion. Along with a linear motion, point B also rotates around point A. To comprehend this complex movement, position vectors for both points A and B are established using a stationary reference frame.
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When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of motions are derived for such cases where the acceleration of the object is constant. The first kinematic equation gives an insight into the relationship between velocity, acceleration, and time. We can see, for example:
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When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the appropriate kinematic equations to solve for the unknowns. Either one or two kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities. Generally, the number of equations required is the same as the number of unknown quantities in the given example. Two-body pursuit problems always require two...
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Constructing Kinematic Confidence Regions With Double Quaternions.

Q Jeffrey Ge1, Zihan Yu1, Anurag Purwar1

  • 1Department of Mechanical Engineering, Stony Brook University, Stony Brook, New York, USA.

Proceedings of Msr-Romansy 2024 : Combined Iftomm Symposium of Romansy and Usctomm Symposium on Mechanical Systems and Robotics. Msr-Romansy (Symposium) (2024 : Saint Petersburg, Fla.)
|September 9, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces a double-quaternion method for creating confidence regions for spatial displacements. This approach approximates 4D rotations, offering a new way to analyze uncertainty in kinematics.

Keywords:
Spatial displacementsco-variance matricesconfidence regionsdouble quaternionsdual quaternionsquaternions

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Area of Science:

  • Robotics
  • Kinematics
  • Computational Geometry

Background:

  • Spatial displacements are fundamental in kinematics.
  • Representing and analyzing uncertainty in these displacements is crucial for many applications.
  • Existing methods, like dual-quaternions, have limitations in preserving geometric properties.

Purpose of the Study:

  • To present a novel double-quaternion formulation for kinematic confidence regions.
  • To approximate spatial displacements using 4D rotations.
  • To compare the efficacy of this new approach with existing dual-quaternion methods.

Main Methods:

  • Approximating spatial displacements with 4D rotations.
  • Formulating kinematic confidence regions using double-quaternions.
  • Constructing confidence ellipsoids in a relevant mathematical space.
  • Comparative analysis with dual-quaternion formulations.

Main Results:

  • The double-quaternion formulation effectively approximates the geometry of spatial displacements.
  • This method provides a viable alternative for constructing confidence regions.
  • Demonstrated efficacy through comparative examples.

Conclusions:

  • Double-quaternions offer a promising approach for kinematic confidence regions.
  • The proposed method preserves geometric properties better than dual-quaternions.
  • This formulation enhances the analysis of uncertain spatial displacements.