Body rates from Euler angles....

In summary: So$$p=D^{-1} \dot{D}=D^{\mathrm{T}}\dot{D}$$andq=D^{-1} \dot{\phi}=D^{\mathrm{T}}\dot{\phi}$$andr=D^{-1} \dot{\vartheta}=D^{\mathrm{T}}\dot{\vartheta}$$which are all self-consistent and so can be substituted into the Euler angle equation for ψ:ψ=φ_dot θ_dot ψ_dot.
  • #1
refrigerator
15
0
Referring to slides 3-4 (page 2) of this link: https://www.princeton.edu/~stengel/MAE331Lecture9.pdf

The author states the relationship between body rates [p q r] and Euler angle rates [φ_dot θ_dot ψ_dot]. I want to verify this but have been failing miserably...

My reasoning:
1) p, q, and r are angular rates about the body axes xB, yB, and zB.
2) φ, θ, and ψ are Euler angles about the intermediate axes x2, y1, and zI, respectively.

3) OK, so I just need to express xB, yB, and zB in terms of x2, y1, and zI, right?

4) p is trivial because xB already coincides with x2. Therefore, p = φdot.
5) When I do the same for yB, I get
yB = cosΦy1 + sinΦcosθzI + sinΦsinθx1.

This is awfully similar to the solution given, which is

q = cosΦ * θdot + sinΦcosθ * ψdot

What am I doing wrong? I've tried to find other sources online but they all just gloss over the derivation or present the result only.

Thank you in advance... I am completely at a loss and have been thinking unproductively about this for hours.[/SUB]
 
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  • #2
I just saw the stickied thread about posting homework-type questions here... I hope this doesn't qualify as a homework-type question. Although it does involve debugging my thought process, it is also a conceptual question about why my reasoning is wrong. If this counts as a homework-type question, my apologies.
 
  • #3
refrigerator said:
Referring to slides 3-4 (page 2) of this link: https://www.princeton.edu/~stengel/MAE331Lecture9.pdf

The author states the relationship between body rates [p q r] and Euler angle rates [φ_dot θ_dot ψ_dot]. I want to verify this but have been failing miserably...

My reasoning:
1) p, q, and r are angular rates about the body axes xB, yB, and zB.
2) φ, θ, and ψ are Euler angles about the intermediate axes x2, y1, and zI, respectively.

3) OK, so I just need to express xB, yB, and zB in terms of x2, y1, and zI, right?
No. x,y,z refer to positions in the axis systems, not rotation rates. If any x,y,z is in your answer, that is wrong. You want to express p, q, r in terms of φ, θ, ψ, φdot, θdot, and ψdot.
 
  • #4
You can derive the angular momentum vector from
$$\omega_j=1/2 \epsilon_{jkl} (D^{-1} \dot{D})_{kl}=1/2 \epsilon_{jkl} (D^{\mathrm{T}}\dot{D})_{kl},$$
where ##D=D(\varphi,\vartheta,\psi)## is the SO(3) matrix in terms of Euler angles to be read as function of time, ##t##.
 

FAQ: Body rates from Euler angles....

What are Euler angles?

Euler angles are a set of three angles that describe the orientation of a rigid body in three-dimensional space. They are often used in aerospace and robotics to represent the orientation of an object in relation to a fixed coordinate system.

How are Euler angles calculated?

Euler angles can be calculated using a variety of methods, but the most common is the Euler rotation sequence, where the object is rotated around three axes (usually x, y, and z) in a specific order. The resulting angles represent the amount of rotation around each axis.

What is the difference between body rates and Euler angles?

Body rates refer to the angular velocities of a body, while Euler angles represent the orientation of the body. In other words, body rates describe how fast the body is rotating, while Euler angles describe the direction and amount of rotation.

How are body rates from Euler angles calculated?

Body rates can be calculated from Euler angles by taking the derivative of the angles with respect to time. This provides the angular velocities around each axis, which can then be used in equations of motion to determine the body's rotational dynamics.

What are some applications of using body rates from Euler angles?

Body rates from Euler angles are commonly used in aerospace and robotics for navigation, control, and simulation of aircraft, spacecraft, and robotic systems. They can also be used in virtual reality and video games to accurately depict the orientation of objects in a three-dimensional environment.

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