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# How do I find the "effective" yaw, pitch and roll from a rotated Inertial Measurement Unit?

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Consider a right-hand coordinate system, N with basisvectors $${hat{x_N},hat{y_N},hat{z_N}}$$ where:

• the x-axis is pointing north
• the y-axis is pointing east

Consider an airplane with a right-hand coordinate system, B with basis vectors $${hat{x_B},hat{y_B},hat{z_B}}$$ where:

• the x-axis is pointing towards the nose of the plane
• the y-axis is pointing towards the right wing

Looking at an online Euler Angle Visualization Tool
it seems that the definition of roll, pitch and yaw of the plane is ?:

• the N basisvectors are first rotated yaw around the the z-axis
• the new basisvectors are rotated pitch around the new y-axis
• the new basisvectors are rotated roll around the new x-axis and the result is the B basis vectors:

$$begin{bmatrix} hat{x_B} \ hat{y_B} \ hat{z_B} end{bmatrix} =R_{BN}*begin{bmatrix} hat{x_N} \ hat{y_N} \ hat{z_N} end{bmatrix}$$
where
$$R_{BN}=RX_{BN}(roll)*RY_{BN}(pitch)*RZ_{BN}(yaw)$$

I find that
$$RZ_{BN} = RZ^T$$
$$RY_{BN} = RY^T$$
$$RX_{BN} = RX^T$$
$$R=RZ*RY*RX=R_{BN}^T$$
where RX, RY RZ and R are the standard rotation matrices used to convert a position from a rotated coordinate system to an unrotated:

Is this correct?

Next let us assume that roll, pitch and yaw is measured by an instrument (an Inertial Measurement Unit).
We introduce a coordinate system for the instrument, I with basisvectors $${hat{x_I},hat{y_I},hat{z_I}}$$.

The instrument is rotated wrt to the airplane:
$$begin{bmatrix} hat{x_I} \ hat{y_I} \ hat{z_I} end{bmatrix} =R_{IB}*begin{bmatrix} hat{x_B} \ hat{y_B} \ hat{z_B} end{bmatrix}$$

Given a known $$R_{IB}$$ and a measurement from the instrument: $$(yaw_i, pitch_i,roll_i)$$ how can I find $$(yaw, pitch,roll)$$ of the airplane?

## Which is the real Euler’s Rotation theorem?

There are two quite different, but useful, theorems related to 3D rotation that are referred to as “Euler’s Rotation Theorem“.

The first definition is given as a quotation in the book by Kuipers

• Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis.

There is a similar, but more pithy definition, from Wolfram MathWorld

• An arbitrary rotation may be described by only three parameters.

This is related to the three-angle representation of an arbitrary rotation by Euler, roll-pitch-yaw, Tati-Bryan angles etc.

The second definition is from Wikipedia

• When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position. (Also given in latin)

This related to the angle-axis representation of an arbitrary rotation. This version of the Rotation Theorem is found in many other places as well, including here on Math Overflow.

Both definitions are related to rotation, and the second at least has a reference to Euler 1776. The first is related to Euler axes and Euler angles, seems like the sort of thing Euler might have figured out, but Kuipers does not provide a reference.

Euler was notorious for not publishing much. Somebody, post Euler, has declared these theorems as Euler’s Rotation Theorem. Which one is it? Is there a more nuanced way to refer to them?

## Using Overpass to find intersections with 3 or more links

I’m trying to find all the intersections (nodes) with 3 or more links (ways) in an area as an indicator for walkability. Complete beginner to the Overpass API and this has me pretty stumped. OSM clearly shows which ways are part of a node, suggesting that this should be very doable. Can’t be much more specific as I don’t even know where to start, so thanks in advance for any help!

## USN-5931-1 Why is this security patch only for ESM/ Ubuntu Pro

18.04 LTS still has general support until April 2023, the patch for USN-5931-1 was released March 7th 2023. As march is before April, and this is a security patch, this should have been included in support offered via LTS. Why was this decided to be only offered for ESM?

## A riddle involving the axiom of choice

So i found a riddle on reddit which is a consequence of the axiom of choice.

There is a house with 100 rooms, and each room contains countably many boxes indexed with the natural numbers. Each box contains a random real number, which is the same over all the rooms (that is, box n contains the same real number in every room).

100 set theorists play a game. Each person will go into a unique room and open as many boxes as they like (perhaps countably many) as long as they leave at least one box in their room unopened. Then, each of them need to pick an unopened box in their room, and guess what real number is inside of it.

In order to win, 99 of them need to guess correctly.

The mathematicians can discuss a strategy beforehand, but after they go into their respective rooms, no more communication is allowed. What is a 100% winning strategy for this seemingly impossible task?

My question is why can the mathematicians agree on a specific representative for each equivalence class? I thought that the axiom of choice guaranteed a choice function existed, but didnt specify it. But for the solution to the riddle, they would need to know what the choice function is. Can someone explain?

## problem with Ubuntu Pro blocked my pc

I am user of Bodhi 18.04 LTS Bionic 32 bit wich based on Ubuntu.I update to nvidia-304 and now i can’t enter on my Bodhi os. because it’s show me Ubuntu Pro at the beginning.After insert my password and Token, he show me the error:

WARNING:root:failed to process /proc/version_signature.

Anyone have solution to help me ?

## Exponentials of stochastic band matrices

This may be a duplicate, but I’ve done some searching and I can’t find exactly this problem setting, probably due to not knowing the right terminology for how to refer to the transition matrix.

I’m considering absorbing random walks $${X_t}$$ which have symmetric transition probabilities inside a window of size $$k$$. For example, on $${0, 1, 2, dots, 100}$$, with $$k=3$$ if $$X_t = 5$$
begin{align*} P[X_{t+1} = 4] &= P[X_{t+1} = 6]\ P[X_{t+1} = 3] &= P[X_{t+1} = 7]\ P[X_{t+1} = 2] &= P[X_{t+1} = 8]\ end{align*}

When the walk is within $$k$$ of the endpoint, the allowable transitions are truncated on both sides, e.g. if $$X_t = 1$$, $$P[X_t = 3] = 0$$.

I wanted to figure out the probability of the walk being absorbed at the upper endpoint, so I did some numerical experiments with walks on the integers from 0-100.
I calculated $$T^n$$ for the transition matrix $$T$$ and a large $$n$$. I then looked at $$(T^n)_{i, 100}$$ to see the probability of being absorbed at 100. It appears that the value is always equal to $$i/100$$, regardless of how I set the transition distribution or window-size.

The transition matrix for a walk on the integers in $$[0, N]$$ has the following properties:

1. $$T$$ is (row) stochastic
2. $$T$$ is a band matrix with upper and lower band size $$k$$
3. $$T_{i, j} = 0$$ if $$|i – j| > max (i, N – i)$$
4. $$T_{i,i + r} = T_{i, i – r}$$ (this is the main property I don’t know the name of)
5. $$T_{0, 0} = T_{1, 1} = 1$$

I would like a proof or counterexample of the following statement:
$$begin{equation*} limlimits_{n to infty} (T^n)_{i, N} = i/N end{equation*}$$

If the statement above is true, is it still true if the transition distributions are no longer left-right symmetric, but it’s still the case that $$E[X_{t + 1}] = X_t$$?

## trouble with writing negative fractional exponential

Code:

``````begin{equation*}
v_o = (v_i - V_D) e^-frac{ t}{tau}
end{equation*}
``````

What I want:

What I am getting:

## Erro in Ubuntu Server for service bind9

Error in service bind9 when I’m starting

## If \$varphi\$ is a normal faithful semifinite weight, is \$eta_varphi(mathfrak{n}_varphicapmathfrak{n}_varphi^*)\$ dense in \$mathfrak{H}_varphi\$

Let $$M$$ be a von Neumann algebra and $$varphi: M_+to [0, infty]$$ be a normal, faithful semifinite weight. Consider its associated semi-cyclic representation
$$pi_varphi: Mto B(mathfrak{H}_varphi)$$
(see Takesaki’s second book, chapter VII for details).

Consider the associated map
$$eta_varphi: mathfrak{n}_varphito mathfrak{H}_varphi.$$

Is it true that $$eta_varphi(mathfrak{n}_varphicap mathfrak{n}_varphi^*)$$ is norm-dense in $$mathfrak{H}_varphi$$?

Attempt: If $$xi perp eta_varphi(mathfrak{n}_varphicap mathfrak{n}_varphi^*)$$, then for $$x,yin mathfrak{n}_varphi$$
$$0 = langle xi, eta_varphi(x^*y) rangle = langle pi_varphi(x)xi, eta_varphi(y)rangle$$
so that $$pi_varphi(mathfrak{n}_varphi)xi=0$$. Maybe this is sufficient to conclude that $$xi= 0$$? I feel like non-degeneracy of $$pi_varphi$$ is relevant.

## Can’t install any Nvidia driver

When i run:

``````sudo ubuntu-drivers devices
== /sys/devices/pci0000:00/0000:00:01.0/0000:01:00.0 ==
modalias : pci:v000010DEd00002520sv00001025sd00001547bc03sc00i00
vendor   : NVIDIA Corporation
model    : GA106M [GeForce RTX 3060 Mobile / Max-Q]
driver   : nvidia-driver-525-open - third-party non-free recommended
driver   : nvidia-driver-515-server - distro non-free
driver   : nvidia-driver-510 - distro non-free
driver   : nvidia-driver-470-server - distro non-free
driver   : nvidia-driver-515 - distro non-free
driver   : nvidia-driver-470 - distro non-free
driver   : nvidia-driver-515-open - distro non-free
driver   : nvidia-driver-525 - third-party non-free
driver   : nvidia-driver-525-server - distro non-free
driver   : xserver-xorg-video-nouveau - distro free builtin
``````

When i install the recommended driver (nvidia-driver-525-open), i get a black screen.

When i install nvidia-driver-515 (or any below version) i get stuck at login screen (keyboard, mouse, nothing is working… even ctrl+alt+f1 to open another tty).

This is the bug report log for the nvidia-driver-525: