Possible to estimate the location of a measurement point?

[sharpedge]

Hi guys,

I am just wondering if it is possible to calculate/estimate the location of measurement point on a rigid body?

For example, let's say we have a rigid body that is in motion. We attach a sensor, say an accelerometer on the surface of the rigid body. Now can we estimate the location of the accelerometer by using motion equations or just mechanics? The location can be given as a vector from the center of the rigid body for instance. Things we know about the rigid boy are it mass, moment of inertia and rotation rate. Also, the axis of rotation is not at the center of the body.

Thanks

 

[FactChecker]

Suppose you have a set of 3 accelerometers and 3 gyros, for the 3 axis. Then you can keep track of position and orientation with some accuracy that deteriorates over time. You say that you know the rotation rates, so that eliminates the need for the gyros. The mathematics is Euler's equations of motion.

In many practical applications, the true positions are updated occasionally using GPS or known location "fixes". It is often necessary to take the rotation of the Earth into account.

 

[sharpedge]

Thanks for the answer. The rotation rate is actually being measured by a gyro. I looked at the Euler's equation of motion and I can calculate the Moment (torque) using the rate and moment of inertia, but how will this help in determining the position of the sensor? The sensor is placed at the surface of the body and it measures acceleration at that point.

 

[SlowThinker]

I'm pretty sure that the acceleration would be constant. Thus, there is clearly not enough information to figure out where the accelerometer is placed.
Try to close your eyes and guess your latitude and longitude on Earth.

 

[A.T.]

Thanks for the answer. The rotation rate is actually being measured by a gyro. I looked at the Euler's equation of motion and I can calculate the Moment (torque) using the rate and moment of inertia, but how will this help in determining the position of the sensor? The sensor is placed at the surface of the body and it measures acceleration at that point.

This is a common application. Have you done any research, like googling "position tracking accelerometer gyroscope"?

 

[Nidum]

Your problem as stated is purely geometric . The inertial properties of the body are irrelevant .

Given a sufficient number of accelerometers properly arranged the data coming back would be adequate for you to calculate at any time the position and orientation of the body relative to it's original position and orientation .

Can sometimes be an accurate method but as mentioned above drift errors are always a potential difficulty .

There are many other ways of determining the position and orientation of a moving body . .

If you want to predict where a body is going to be at any time then that is a much more complex problem .

 

[sharpedge]

This is a common application. Have you done any research, like googling "position tracking accelerometer gyroscope"?

Your problem as stated is purely geometric . The inertial properties of the body are irrelevant .

Given a sufficient number of accelerometers properly arranged the data coming back would be adequate for you to calculate at any time the position and orientation of the body relative to it's original position and orientation .

Can sometimes be an accurate method but as mentioned above drift errors are always a potential difficulty .

There are many other ways of determining the position and orientation of a moving body . .

If you want to predict where a body is going to be at any time then that is a much more complex problem .

I think you are talking about Dead Reckoning INS solution, which is not what I am trying to achieve. I am not moving the accelerometer alone but the rigid body to which it is attached, is moving with it. I want to find out the position of accelerometer (which is fixed) wrt to the rigid body, not the position and orientation of rigid body

 

[Nidum]

I don't understand . If at any time you know the position and orientation of the body then you must also know the position and orientation of the accelerometers fixed to the body ?

 

[sharpedge]

I don't understand . If at any time you know the position and orientation of the body then you must also know the position and orientation of the accelerometers fixed to the body ?

I want to find out the position of the accelerometer wrt to the center of mass of the body, which I do have the coordinates for. Think of it like this, I place a accelerometer on the surface of a rigid body but I don't know the location where I placed it. I also don't want to measure it from an external measurement system such as a Motion Capture etc (tried in the past). Given this, is there a way to determine the location using dynamics? Just by giving it some form of motion?

About what you asked, If I know the orientation and position of the body (center of mass), then to determine the position of accelerometer, i would need its position wrt to the COM. The position is what I am after.

 

[FactChecker]

Sorry, this is a completely different problem from what I (and I think others) had imagined.
Should we assume that the accelerometer can only measure acceleration in one direction and we do not know the orientation of that direction? If so, are there other accelerometers to measure other directions and do we know anything about their position and orientation, either relative to the COM or to each other?

 

[Nidum]

Apply accelerations of known magnitude to each axis in turn .

With three principle axes that means three translational accelerations and three rotary accelerations are needed .

Problems which are not fully 3D may be solvable with lesser number of applied accelerations .

 

[A.T.]

I want to find out the position of the accelerometer wrt to the center of mass of the body

Just by giving it some form of motion?

So you can apply any motion to the body?

Just spin it around different axes, which you align vertically (makes it easier to account for gravity). This will give you the centripetal acceleration, and thus the distance from each spin axis. So your solution will be the intersection of a bunch of cylinders (and eventually the body's surface, if that is a placement constraint).

 

[sharpedge]

Should we assume that the accelerometer can only measure acceleration in one direction and we do not know the orientation of that direction? If so, are there other accelerometers to measure other directions and do we know anything about their position and orientation, either relative to the COM or to each other?

The accelerometers have 3 orthogonal axes of measurements. The orientation of each accelerometer (and 3 axes) can be figured out. Their relative orientation can also be figured out. However, their relative position is not known. Maybe it can be figured out?

Apply accelerations of known magnitude to each axis in turn .

With three principle axes that means three translational accelerations and three rotary accelerations are needed .

Problems which are not fully 3D may be solvable with lesser number of applied accelerations .

Can you elaborate a bit. How will these accelerataions help?

So you can apply any motion to the body?

Just spin it around different axes, which you align vertically (makes it easier to account for gravity). This will give you the centripetal acceleration, and thus the distance from each spin axis. So your solution will be the intersection of a bunch of cylinders (and eventually the body's surface, if that is a placement constraint).

Won't this give me the distance of the accelerometer from the spin axis? The problem is that the spin axis is not the same as COM, which means that for this scheme to work, the distance between COM and spin axis must be known. right?

 

[FactChecker]

This is a challenging and (for me at least) interesting problem. Certainly we are not guaranteed to be able to solve it, since if the rigid body never spins, then we will never get any clue of the sensor's offsets from the COM.
I assume that you do not know the position of the rigid body center of mass, since you would have mentioned that. So it is a problem of sorting out the accelerations of the COM from the accelerations due to rotations to derive the offsets of the accelerometer from the COM. That will take some thought.

Am I stating the problem correctly?

 

[SlowThinker]

Am I stating the problem correctly?

Further,
- is this happening in the gravitational field or in free fall?
- is the axis of rotation known or not? (vertical/horizontal/general)
- can you explain why the rotation axis doesn't go through the COM? Is COM relevant at all? Would it be enough to give position relative to the axis of rotation?

 

[A.T.]

The problem is that the spin axis is not the same as COM

Can you apply any motion or not?

the distance between COM and spin axis must be known

You wrote that you know the COM, so what is the problem here?

 

[sharpedge]

This is a challenging and (for me at least) interesting problem. Certainly we are not guaranteed to be able to solve it, since if the rigid body never spins, then we will never get any clue of the sensor's offsets from the COM.
I assume that you do not know the position of the rigid body center of mass, since you would have mentioned that.

I can make the rigid body spin. I do know the location of the COM of the body in a reference frame attached to the body. The COM was determined when the body was freely moving, by geometric methods. The application, however, requires the body to be fixed to a support joint and then allow it to move. So, the position of the accelertometer is required in this scenario.

Further,
- is this happening in the gravitational field or in free fall?
- is the axis of rotation known or not? (vertical/horizontal/general)
- can you explain why the rotation axis doesn't go through the COM? Is COM relevant at all? Would it be enough to give position relative to the axis of rotation?

-Yes this is happening in Earth's gravitational field
-the axis of rotation in terms of coordinates in not known yet. I am thinking whether it can be measured.
-The COM is relevant. It's just due to the application I have at hand, the axis can't go through the COM. It would be enough, If I can figure out if the position of the spin axis wrt to the COM can be figured out, which is not known yet.

 

[A.T.]

The COM is relevant.

But based on your description completely irrelevant to the motion and the acceleration you measure. So how can there be any relation?

 

[Nidum]

Can you elaborate a bit. How will these accelerataions help?

The accelerometer has unknown coordinates relative to the axes zero location .

The component the accelerometer picks up from each applied acceleration is both known as output data and can be written as a function of the unknown location coordinates .

Should be possible to set up a system of simultaneous equations and solve for the coordinates .

Actually implementing this method would require detail knowledge of the type of accelerometer used .

 

[sharpedge]

But based on your description completely irrelevant to the motion and the acceleration you measure. So how can there be any relation?

Relevant in the sense that the position of the accelerometer needs to be known wrt to it.

The accelerometer has unknown coordinates relative to the axes zero location .

The component the accelerometer picks up from each applied acceleration is both known as output data and can be written as a function of the unknown location coordinates .

Should be possible to set up a system of simultaneous equations and solve for the coordinates .

Actually implementing this method would require detail knowledge of the type of accelerometer used .

It would have been a lot easier if the COM was at the spin axis, then measured accelereation would be directly a function of distance from the center and rotation rate. However, the problem is that it is not through the COM :( Even in the situation at hand, measured acceleration would still be a function of these two I suppose, but what that function would be, I have no clear idea...

 

[SlowThinker]

It would have been a lot easier if the COM was at the spin axis, then measured accelereation would be directly a function of distance from the center and rotation rate. However, the problem is that it is not through the COM :( Even in the situation at hand, measured acceleration would still be a function of these two I suppose, but what that function would be, I have no clear idea...

With some deep thought that I don't have the time to perform right now, it should be possible to find the distance of the accelerometer from the axis of rotation.
I'm also pretty sure that no other information can be recovered, at least in the scenario described so far.

Edit: Actually it should be also possible to find the angle of rotation axis w.r.t. vertical, and the direction of the axis in accelerometer's frame of reference.
If you could ensure that say z-axis of the accelerometer points towards COM, it might help.

 

[Nidum]

It doesn't matter where the CoM is . Method works for any set of orthogonal axes .

Please tell us what you are actually trying to do . Diagrams would be good .

 

[A.T.]

Relevant in the sense that the position of the accelerometer needs to be known wrt to it.

Then, you need some info that relates the COM to the motion, otherwise its pointless.