- #1

AbsoluteUnit

- 9

- 3

I have some issues understanding the inertial-frame (or global-frame, G-frame) versus the body-frame (B-frame) when it comes to simulating the motion of a rigid body in 2 dimensions (planar body mechanics) in a system of ODEs. I have been self-learning from textbooks on simulating rigid body motion, so I sometimes misunderstand concepts, since some books are a bit vague when they assume the reader already have the technical knowledge to make the necessary connections. I hope someone can clear up any misunderstandings that I have here.

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Given a state space, Q and its gradient Q' describing the planar mechanics of a rigid body:

Q = [ VBx(t), VBy(t), ψ(t) ]

^{T}... (1)

Q' = [ aBx(t), aBx(t), ψ'(t) ]

^{T}... (2)

Where

Bx := Quantity expressed in body-frame (B-frame) coordinates

V := Velocity

a(t) := Acceleration

ψ, ψ'(t) := Body angular velocity and acceleration (respectively)

The values of a(t) and ψ'(t) are being generated by some process in the B-frame at each time point (t) as forces and moments, e.g. vehicle accelerating/braking and steering.

Essentially,

- Some books (seem to) suggest that it is possible to perform integration directly
**in the B-frame**on Q'(t) (ref. (2)) to solve for Q(t) (ref. (2)) at every time point, t. Then, the G-frame positions of the rigid body, (X,Y,ψ) can be obtained by rotating Q to the G-frame coordinates via the transformation:

VGx = VBx*cos(ψ) - VBy*sin(ψ)

VGy = VBx*sin(ψ) + VBy*cos(ψ)

VGy = VBx*sin(ψ) + VBy*cos(ψ)

Then integrate VGx, VGy to get (X,Y) in G-frame.

In short:Fully solve the ODE in the B-frame first, then rotate the velocities to the G-frame and integrate to obtain the G-frame positions: (X,Y,ψ)

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Part of me thinks that this interpretation of mine doesn't really make sense, because Newton's First Law only applies directly in the G-frame, but not in the B-frame.

I suspect that for every time point, t, one should instead:

- Compute the forces and moments that give rise to aBx(t), aBx(t), and ψ'(t)
- Transform (a
_{Bx}(t), a_{Bx}(t), ψ'(t)) to G-frame coordinates - Solve for V
_{Gx}, V_{Gy}, ψ'(t)

Is that the correct way to think about this? Or is there a technicality that I am not getting here and both methods are actually equivalent?

Thank you for your insights,

AU