0
votes
0answers
16 views

Transforming euler rotations to other coordinate system

I have a global $3D$ coordinate system, and it's transform matrix is an identity matrix. I also have an object with it's own, local coordinate system and it's transform is not identity matrix. Now I ...
0
votes
2answers
25 views

Rotation between two vectors as a function of time (one parameter rational motion design)

Given a time varying vector: $\mathbf{w}(t) = \mathbf{u} + t\mathbf{v}$ I would like to find a rotation matrix $\mathbf{R}(t)$ that rotates the positive x-axis $[1, 0, 0]^T$ onto the vector ...
2
votes
2answers
96 views

What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + ...
0
votes
0answers
37 views

How to obtain relative rotation?

I have two rotations, each of which can be described as a roll, pitch, and yaw (in radians): $$ r_1 = (3.14159, 1.57080, 1.6) $$ $$ r_2 = (3.14159, 1.57080, 1.4) $$ I am interested in the relative ...
1
vote
0answers
109 views

Transform a vector to global frame and ignore rotation about one axis or Full tilt compensated magnetometer

Good day everyone. I would like to lock the rotation about one specified axis. For example, let`s imagine that we have a quaternion which desribes the orientation of our rigid body relative to the ...
0
votes
0answers
47 views

Quarternions from MPU and circumference of circles

First I should mention that my math skills are super basic. I do not understand formulas but I do understand pseudo code, C, C++, and other programming languages. I've been working on a electronics ...
0
votes
1answer
49 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
0
votes
2answers
895 views

Getting Euler (Tait-Bryan) Angles from Quaternion representation

Apologies if this has already been answered, but I haven't been able to get a clear answer from looking on Stack Exchange so-far. I'm trying to solve a camera stabilization problem. I have a 2-axis ...
0
votes
1answer
344 views

Quaternion Decomposition

I'm having trouble decomposing a unit quaternion into euler angles (or roll, pitch and yaw). The overall goal is to tell how a phone is rotated with respect to the world. I'm given a unit quaternion ...
1
vote
1answer
115 views

Unit quaternions as rotations

How would one represent the map $f$ such that $f(1) = i, f(i) = -1$ and keeping $j$ and $k$ fixed as a quaternion representation of rotations?
0
votes
0answers
31 views

Normalization of Euler angle data

I have head motion data for several speakers. Because not every speaker sat in the exact same position during recording I have to normalize the data. One option to do this, I think, would be to ...
0
votes
1answer
65 views

smooth orientation change with quaternions

My camera orientation is looking in the $v_1$ direction. Something happens on direction $v_2$ and I want the camera to move smoothly to look at that direction. So, to find the quaternion to go from ...
0
votes
1answer
66 views

Relative rotations using quaternions [duplicate]

I have a sensor at some arbitrary orientation (non-zero roll, yaw, pitch) given by quaternion $q_{0}$. I then pitch the sensor to orientation $q_{1}$. When I compute the relative rotation between the ...
1
vote
1answer
34 views

Quaternion techniques for a geometric description of the composition of two rotations

Let $q \in S^3$. Therefore $q$ can be represented as $q=\cos(\alpha/2) + \sin(\alpha/2)u$ for some $\alpha \in \mathbb{R}$ and some $u \in S^3$ with it's real part zero. Recall that the quaternions ...
1
vote
1answer
136 views

Magnitude of rotation between two quaternions

I have a quaternion for an object's starting rotation, and a quaternion for an object's ending rotation, and I am SLERPing the shortest rotation between the two. How can I figure out the magnitude of ...
0
votes
2answers
142 views

Difference between quaternions and rotation matrices

This is a really simple question, I guess. Do quaternions cover the same set of rotations as rotation matrices? I assume the answer is yes, they both can represent SO(3), but I'm unsure about the ...
1
vote
1answer
218 views

Rotation by quaternion conjugation and quaternion matrix

A rotation of vector $v$ can be done by matrix multiplication $Q^{*}Qv$ where $Q=\begin{pmatrix}w & -z & y & x \\ z & w & -x & y \\ -y &x &w& z\\ -x& -y ...
0
votes
1answer
18 views

Reference for the polar parameterization of quaternions

I would like to find the original reference in which the polar parameterization of quaternions was given (i.e. the relationship between the components of a unit quaternion and the polar angles of an ...
2
votes
2answers
1k views

How to rotate one vector about another?

Breif Having given 2 non-parallel vectors: a and b, is there any way by which I may rotate a about b such that b acts as the axis about which a is rotating. Question Given: vector a and b To find: ...
0
votes
0answers
125 views

Optimization of quaternion / Rotation of form Q*P*Qbar=R

I wish to find out a unit quaternion $Q$ (with complement $\bar Q$, s.t. $Q\bigotimes \bar Q = \bar Q\bigotimes Q$=[1 0 0 0]) that will satisfy the following equation $Q\bigotimes P \bigotimes \bar Q ...
1
vote
1answer
36 views

Set of rotations necessary to connect two points in R³ using a thin cylinder

I have been scratching my head for days trying to answer this question. Suppose i have 2 points on three-dimensional space, say, $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, and they are separated by ...
4
votes
1answer
127 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
1
vote
1answer
206 views

Small angular displacements with a quaternion representation

I have the orientation of a 3D spatial object represented by a unit quaternion: $$ q = a_1 + a_2 i + a_3 j + a_4 k $$ $$ \|q\| = (a_1^2 + a_2^2 + a_3^2 + a_4^2)^{1/2} = 1 $$ I'd like to perturb this ...
2
votes
0answers
58 views

The angle of an average rotation is $126.5^\circ$?

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
0
votes
0answers
36 views

Slicing Up Uniform Random Rotation Quaternions

I'm generating uniform random rotations using quaternions. I am using the method attributed to Shoemake, which is discussed in another post (Uniform Random Quaternion In a restricted angle range): ...
0
votes
0answers
88 views

Update rotation matrix

Imagine you have a two noded beam in space, defined by extreme nodes 1 and 2. Image is owned by Jean-Marc Battini. To ...
2
votes
1answer
241 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
1
vote
0answers
77 views

Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
2
votes
1answer
124 views

Algorithm for finding orientation of each face on a polyhedron?

I am working on making a dice rolling application and I need to find out how far in each of the three dimensions I must rotate each of the dice to make the correct side face the camera so the user can ...
0
votes
1answer
79 views

Quaternion Matrix Multiply

My question is about applying rules of quaternions to quaternion matrices. I know that for some rotation quaternion q = [w, x, y, z], I can find the rotation of ...
1
vote
1answer
182 views

Quaternion exponential map, rotations and interpolation

A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
2
votes
2answers
349 views

Rotate the axis of rotation of a quaternion by another quaternion

I have two quaternions, $q_1$ and $q_2$. I want to rotate the axis of rotation of $q_1$ by $q_2$. Is there any way of doing this directly, without extracting the axis of rotation from $q_1$, rotating ...
3
votes
3answers
193 views

Are there different conventions for representing rotations as quaternions?

I am trying to understand how quaternions are represented as rotations, in particular how to convert from a quaternion representation to a rotation matrix. The following paper by Diebel gives an ...
2
votes
3answers
805 views

Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
1
vote
1answer
176 views

Orthogonal procrustes problem using quaternions

Hello I'm trying solve orthogonal procrustes problem in 3d with a help of quaternions. Original problem is: For matrix $A$ find orthogonal matrix $Q$ that $$||A-Q||_F =\min_{\Omega \in SO(3)} ...
2
votes
0answers
112 views

Understanding quaternions and axis angle representations

I have a sensor that gives me a quaternion. I convert the quaternion to an axis-angle representation using http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToAngle/. When I ...
0
votes
1answer
72 views

Commutative applying rotations around three axis

Rotating an object in a 3 dimensional space by euler angles might be intuitive but comes with some problems. First, the order of applied rotations around the different axis matters. Second, there is ...
1
vote
3answers
2k views

Nullify (zero out, cancel) rotation in an arbitrary axis in a Quaternion

Question: How do you nullify (zero out) rotation around an arbitrary axis in a Quaternion? Example: Let's say you have an object with quaternion orientation $A$. You also have a rotation quaternion ...
1
vote
0answers
48 views

Optimimal rotation using non-linear conjugate gradient

The problem I'd like to ask is the following : let $M_1$ and $M_2$ two rigid bodies with a quadratic constraint function $f$ attached to its grid points. $M_2$ is always kept static while $M_1$ can be ...
0
votes
1answer
153 views

How do I calculate a position in front of a quaternion given the initial position and the quaternion?

Well I want to determine the position in front of, lets say 'an object'. 'An objects' has a position (pX,pY,pZ) and a quaternion rotation (qX,qY,qZ,qW) (where qW seems always to be ...
0
votes
1answer
153 views

How are these formulas for Quaternion -> Rotation Matrix related?

I'm trying to write a program to convert a quaternion to a rotation matrix. One source I found is: ...
1
vote
1answer
114 views

Quaternions, torque, and impulse.

In a physics simulation I have a solid ball of mass $m$ and moment of interia $M$ (which is a diagonal matrix with all entries equal to ${2\over5}mr^2=i$). Its instantaneous rotation is given by a ...
1
vote
2answers
507 views

half sine and half cosine quaternions

Something is a little bit unclear to me. In the image below you see that you need to divide the angle by a half. Acccording to wikipedia they say that this is so that I could rotate clockwise or ...
2
votes
2answers
272 views

Detail in the proof of the quaternion rotation identity

I am trying to understand the proof of the quaternion rotation identity illustrated in wikipedia ...
2
votes
1answer
76 views

How can I break down a rotation of known amount around a known axis into two rotations of unknown amounts around known axes?

I have a vector that's been rotated a known amount about a known axis. I would like to break this rotation down into two separate rotations around known, linearly independent axes where the amounts I ...
0
votes
2answers
468 views

Updating a quaternion orientation by a vector of euler angles

I'm trying to understand why this formula works to update an orientation with an angular velocity represented as a vector of rotations in $radians/{second}$. I understand that two quaternions ...
1
vote
1answer
69 views

How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from…

In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (colored in $\color{blue}{blue}$). ...
0
votes
1answer
426 views

3D points rotation to quaternions

For the simplicity, we'll consider two 3D points, that moves one relatively to other, in time. Let's say: ...
3
votes
2answers
94 views

Correspondence between rotation representations

I was wondering if there is a bijection between unit quaternions and other rotation representations such as vector of rotation, Euler angles or rotation matrices. It seems to me this is not the case ...
4
votes
1answer
321 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...