Determining Analytic Orientation from Angular Velocity

In summary, the angular velocity is a vector pointing in a particular direction, and the orientation of the body is given by the direction of the angular velocity.
  • #1
dgreenheck
23
0
I am doing an analysis concerning the torque-free motion of an axisymmetric body (J1 = J2 != J3).

The angular velocity is given

[itex]\omega(t) = [\omega_t\ sin(\Omega t), \omega_t\ cos(\Omega t), \omega_{z0}][/itex]

where [itex]\omega_t[/itex], [itex]\Omega[/itex] and [itex]\omega_{z0}[/itex] are constants. I would like to determine the orientation of the body at any time [itex]t[/itex] given an initial orientation at [itex]t = 0[/itex]. My end goal is to have an analytic representation of the orientation that I can use as "truth" to compare the errors of various numerical methods of estimating the orientation.

I know how to find numerical solutions to this problem using quaternions/direction cosine matrices/rotation vectors, etc., but am not sure how to approach this from an analytic point of view.
 
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  • #2
dgreenheck said:
I am doing an analysis concerning the torque-free motion of an axisymmetric body (J1 = J2 != J3).

The angular velocity is given

[itex]\omega(t) = [\omega_t\ sin(\Omega t), \omega_t\ cos(\Omega t), \omega_{z0}][/itex]

where [itex]\omega_t[/itex], [itex]\Omega[/itex] and [itex]\omega_{z0}[/itex] are constants. I would like to determine the orientation of the body at any time [itex]t[/itex] given an initial orientation at [itex]t = 0[/itex]. My end goal is to have an analytic representation of the orientation that I can use as "truth" to compare the errors of various numerical methods of estimating the orientation.

I know how to find numerical solutions to this problem using quaternions/direction cosine matrices/rotation vectors, etc., but am not sure how to approach this from an analytic point of view.
Unless I have not understood your question, it looks to me that you already have the answer. Since the constants are given, you already know the direction of the angular velocity as a function of time.
 
  • #3
Chandra Prayaga said:
Unless I have not understood your question, it looks to me that you already have the answer. Since the constants are given, you already know the direction of the angular velocity as a function of time.
Yes, but what is the orientation of the spinning body
 
  • #4
When you use the word orientation, you have to specify what you mean. Usually, orientation means that there is some vector property of the body, which is pointing in some direction. In the case of a spinning body, the angular velocity (or angular momentum) is that vector. The direction of the angular velocity IS the orientation of the body. What you are probably looking for is to specify the geometric shape of the body, such as a cube, and then you can ask a question like, how the edges of the cube are rotating as the object rotates about some axis. The axis in this case, is again, the direction of the angular velocity
 
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1. What is analytic orientation?

Analytic orientation refers to the ability to understand and interpret data or information in a logical and systematic manner. It involves breaking down complex problems or situations into smaller, more manageable parts in order to gain a deeper understanding of them.

2. How is angular velocity related to analytic orientation?

Angular velocity is a measure of the rate at which an object rotates around a fixed point. In the context of analytic orientation, angular velocity can be used to determine the orientation of an object or system in space, which is an important aspect of analyzing and interpreting data.

3. What methods can be used to determine analytic orientation from angular velocity?

There are several methods that can be used to determine analytic orientation from angular velocity, including using rotational matrices, quaternions, or Euler angles. Each method has its own advantages and limitations, and the choice of method will depend on the specific application and data being analyzed.

4. Are there any challenges in determining analytic orientation from angular velocity?

Yes, there can be challenges in accurately determining analytic orientation from angular velocity. One challenge is that small errors in measurements of angular velocity can lead to significant errors in the calculated orientation. Additionally, the choice of method and coordinate system can also affect the accuracy of the results.

5. How is determining analytic orientation from angular velocity used in scientific research?

Determining analytic orientation from angular velocity is used in a variety of scientific research fields, including robotics, aerospace engineering, and geology. It allows researchers to analyze and interpret data from sensors and other instruments to better understand the orientation and movement of objects and systems, which can provide valuable insights and applications in various fields.

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