Geometry - angle w.r.t. two planes to 3D pose?
I have a machine which rotates a tool about two axes, A and B. This is defined by two pivoting angles, $A$ and $B$. I would like to use $A$ and $B$, which rotate around a "virtual pivot point," to calculate the pose of the tool w.r.t. the origin.
A photo of the toolhead with the A and B axes labeled is here:
What is the sequence of steps I need to take?
geometry
asked Dec 4 '18 at 16:45
Brandon Dube
1086
add a comment |
I have a machine which rotates a tool about two axes, A and B. This is defined by two pivoting angles, $A$ and $B$. I would like to use $A$ and $B$, which rotate around a "virtual pivot point," to calculate the pose of the tool w.r.t. the origin.
A photo of the toolhead with the A and B axes labeled is here:
What is the sequence of steps I need to take?
geometry
asked Dec 4 '18 at 16:45
Brandon Dube
1086
1
Can you provide a picture? It's hard to understand what is given and what you want to find from the given description.
– Vasily Mitch
Dec 4 '18 at 16:50
1
You define $A$ to be two different things, and don't tell us what $I$, $J$, and $K$ are at all. A picture would be useful.
– user3482749
Dec 4 '18 at 16:52
What do you mean by “the axis of $A$” when you’ve described $A$ as an angle?
– amd
Dec 4 '18 at 17:24
Thanks for the comments -- I added a photo of the machine. Hopefully this clears up what I mean.
– Brandon Dube
Dec 4 '18 at 18:00
add a comment |
I have a machine which rotates a tool about two axes, A and B. This is defined by two pivoting angles, $A$ and $B$. I would like to use $A$ and $B$, which rotate around a "virtual pivot point," to calculate the pose of the tool w.r.t. the origin.
A photo of the toolhead with the A and B axes labeled is here:
What is the sequence of steps I need to take?
geometry
asked Dec 4 '18 at 16:45
Brandon Dube
1086
I have a machine which rotates a tool about two axes, A and B. This is defined by two pivoting angles, $A$ and $B$. I would like to use $A$ and $B$, which rotate around a "virtual pivot point," to calculate the pose of the tool w.r.t. the origin.
A photo of the toolhead with the A and B axes labeled is here:
What is the sequence of steps I need to take?
geometry
geometry
asked Dec 4 '18 at 16:45
Brandon Dube
1086
asked Dec 4 '18 at 16:45
Brandon Dube
1086
edited Dec 4 '18 at 17:59
asked Dec 4 '18 at 16:45
Brandon Dube
1086
asked Dec 4 '18 at 16:45
Brandon Dube
1086
asked Dec 4 '18 at 16:45
Brandon Dube
1086
1086
1
Can you provide a picture? It's hard to understand what is given and what you want to find from the given description.
– Vasily Mitch
Dec 4 '18 at 16:50
1
You define $A$ to be two different things, and don't tell us what $I$, $J$, and $K$ are at all. A picture would be useful.
– user3482749
Dec 4 '18 at 16:52
What do you mean by “the axis of $A$” when you’ve described $A$ as an angle?
– amd
Dec 4 '18 at 17:24
Thanks for the comments -- I added a photo of the machine. Hopefully this clears up what I mean.
– Brandon Dube
Dec 4 '18 at 18:00
add a comment |
1
Can you provide a picture? It's hard to understand what is given and what you want to find from the given description.
– Vasily Mitch
Dec 4 '18 at 16:50
1
You define $A$ to be two different things, and don't tell us what $I$, $J$, and $K$ are at all. A picture would be useful.
– user3482749
Dec 4 '18 at 16:52
What do you mean by “the axis of $A$” when you’ve described $A$ as an angle?
– amd
Dec 4 '18 at 17:24
Thanks for the comments -- I added a photo of the machine. Hopefully this clears up what I mean.
– Brandon Dube
Dec 4 '18 at 18:00
1
1
Can you provide a picture? It's hard to understand what is given and what you want to find from the given description.
– Vasily Mitch
Dec 4 '18 at 16:50
Can you provide a picture? It's hard to understand what is given and what you want to find from the given description.
– Vasily Mitch
Dec 4 '18 at 16:50
1
1
You define $A$ to be two different things, and don't tell us what $I$, $J$, and $K$ are at all. A picture would be useful.
– user3482749
Dec 4 '18 at 16:52
You define $A$ to be two different things, and don't tell us what $I$, $J$, and $K$ are at all. A picture would be useful.
– user3482749
Dec 4 '18 at 16:52
What do you mean by “the axis of $A$” when you’ve described $A$ as an angle?
– amd
Dec 4 '18 at 17:24
What do you mean by “the axis of $A$” when you’ve described $A$ as an angle?
– amd
Dec 4 '18 at 17:24
Thanks for the comments -- I added a photo of the machine. Hopefully this clears up what I mean.
– Brandon Dube
Dec 4 '18 at 18:00
Thanks for the comments -- I added a photo of the machine. Hopefully this clears up what I mean.
– Brandon Dube
Dec 4 '18 at 18:00
add a comment |
1 Answer
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You write down the matrices of rotations corresponding to axes B and A (axis-angle to rotation matrix formula), then composite rotation has matrix that is the product of A and B:
$$R=AB,$$
note here: matrix B has axis corresponding to the state prior to rotation around A.
After that you can transform the rotation matrix $R$ to whatever representation you like (to Euler angles, to axis-angle).
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
add a comment |
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1 Answer
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You write down the matrices of rotations corresponding to axes B and A (axis-angle to rotation matrix formula), then composite rotation has matrix that is the product of A and B:
$$R=AB,$$
note here: matrix B has axis corresponding to the state prior to rotation around A.
After that you can transform the rotation matrix $R$ to whatever representation you like (to Euler angles, to axis-angle).
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
add a comment |
You write down the matrices of rotations corresponding to axes B and A (axis-angle to rotation matrix formula), then composite rotation has matrix that is the product of A and B:
$$R=AB,$$
note here: matrix B has axis corresponding to the state prior to rotation around A.
After that you can transform the rotation matrix $R$ to whatever representation you like (to Euler angles, to axis-angle).
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
add a comment |
You write down the matrices of rotations corresponding to axes B and A (axis-angle to rotation matrix formula), then composite rotation has matrix that is the product of A and B:
$$R=AB,$$
note here: matrix B has axis corresponding to the state prior to rotation around A.
After that you can transform the rotation matrix $R$ to whatever representation you like (to Euler angles, to axis-angle).
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
You write down the matrices of rotations corresponding to axes B and A (axis-angle to rotation matrix formula), then composite rotation has matrix that is the product of A and B:
$$R=AB,$$
note here: matrix B has axis corresponding to the state prior to rotation around A.
After that you can transform the rotation matrix $R$ to whatever representation you like (to Euler angles, to axis-angle).
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
answered Dec 4 '18 at 19:50
Vasily Mitch
1,33837
1,33837
add a comment |
add a comment |
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1
Can you provide a picture? It's hard to understand what is given and what you want to find from the given description.
– Vasily Mitch
Dec 4 '18 at 16:50
1
You define $A$ to be two different things, and don't tell us what $I$, $J$, and $K$ are at all. A picture would be useful.
– user3482749
Dec 4 '18 at 16:52
What do you mean by “the axis of $A$” when you’ve described $A$ as an angle?
– amd
Dec 4 '18 at 17:24
Thanks for the comments -- I added a photo of the machine. Hopefully this clears up what I mean.
– Brandon Dube
Dec 4 '18 at 18:00