Plotting Location w/ Accellerometer Project

[Ether]

Originally Posted by Invictus3593

We can use a time interval of about 100ms to calculate distance traveled with this equation: D=1/2at^2

You left out the second term...

D = ½at² + V_ot

... and that formula is valid only if "a" is constant over the time interval.

Hate to be a party pooper, but... Due to the double integration, tiny errors in "a" will rapidly accumulate so that the computed position will quickly diverge from the true position.

Trapezoidal integration will help somewhat: Given t, x, v, and a at some point in time, and a_new at some later point in time t_new, proceed as follows:

dt = t_new - t;
v_new = v + dt*(a_new+a)/2;
x_new = x + dt*(v_new+v)/2;

[Invictus3593]

Great idea,
but how is the kinect related to this?
Can't it be done with the KOP accelerometer too?

BTW in what units is the output of the accelerometer? m/s^2, etc..

We use the Kinect fro vision processing due to the depth capabilities. The kop acceleromater should work too! You can read about what data the kinect gives here.

Originally Posted by Ether

You left out the second term...

D = ½at² + V_ot

... and that formula is valid only if "a" is constant over the time interval.

Hate to be a party pooper, but... Due to the double integration, tiny errors in "a" will rapidly accumulate so that the computed position will quickly diverge from the true position.

Trapezoidal integration will help somewhat: Given t, x, v, and a at some point in time, and a_new at some later point in time t_new, proceed as follows:

dt = t_new - t;
v_new = v + dt*(a_new+a)/2;
x_new = x + dt*(v_new+v)/2;

So, theoretically, could you accurately plot a robot's location on the field using the equations you gave? and what other variables would I need get as inputs besides time and acceleration?

What if the time variable was such a very short amount of time, like 50ms, and you just took the average acceleration over that time period?

[Ether]

Originally Posted by Invictus3593

What if the time variable was such a very short amount of time, like 50ms,

Shorter integration times are generally better. You will still see some drift due to the double integration and the fact that the measured accel has error.

...and you just took the average acceleration over that time period?

That's what trapezoidal integration tries to do.


Play around with it in a spreadsheet or Maxima or Octave or SciLab and see for yourself.

[Greg McKaskle]

If you want to play with this, it may be worth checking to see if your laptop has an accelerometer in it. I know that my last Mac laptop did. It was used to park the hard drive before it landed during a fall.

If so, you can read that accelerometer in LV and play a bit.

I looked on the Apple app store and there are a number of free apps that let you see the accelerometer data and let see their interpretation of the data.

As for using this on the robot, the key cases to think about are the ones where the floor is uneven. Sitting still, the robot and its accelerometer are tilted. Gravity's 1g is no longer in the pure Z direction. It is imparting a force on the other axes. If your code doesn't calibrate and ignore this, it looks like the robot is constantly accelerating in the uphill direction. If you do calibrate it for that tilt, then drive the robot a foot and stop it again, the floor is likely tilted a new direction. Once again, sitting still, you will accelerate uphill.

It is an interesting problem, and accelerometers are certainly useful on the robot, but integrating them to identify speed or location without isolating them from the force of gravity is quite hard.

Greg McKaskle

[Ether]

Originally Posted by Greg McKaskle

..you can read that accelerometer in LV and play a bit.

Yes, thanks Greg, I should have included LV in my earlier post.

[Invictus3593]

Originally Posted by Greg McKaskle

If you want to play with this, it may be worth checking to see if your laptop has an accelerometer in it. I know that my last Mac laptop did. It was used to park the hard drive before it landed during a fall.

If so, you can read that accelerometer in LV and play a bit.

I looked on the Apple app store and there are a number of free apps that let you see the accelerometer data and let see their interpretation of the data.

As for using this on the robot, the key cases to think about are the ones where the floor is uneven. Sitting still, the robot and its accelerometer are tilted. Gravity's 1g is no longer in the pure Z direction. It is imparting a force on the other axes. If your code doesn't calibrate and ignore this, it looks like the robot is constantly accelerating in the uphill direction. If you do calibrate it for that tilt, then drive the robot a foot and stop it again, the floor is likely tilted a new direction. Once again, sitting still, you will accelerate uphill.

It is an interesting problem, and accelerometers are certainly useful on the robot, but integrating them to identify speed or location without isolating them from the force of gravity is quite hard.

Do you have any examples of how to get that built-in accelerometer in LV? we're not using any apple products, but it may be worth it to just check and see if we have one..

As for the tilt, if we had a gyro on there, and re-calculated the "down" direction whenever we weren't level, would that help fix the tilt problem? I mean, this year, the floor was totally level, but i guess it would go nuts if we get tipped or something.

[Invictus3593]

Also, I talked to my Physics teacher and he said that using this equation for each axis would work:
d=1/2at²+V_ot
But we would have to get "v_o" from the equation:
V_f=v_o+at

Also, we talked about the case where the robot is turning, putting acceleration on the axis tangent to the curve and he used these few equations:

a_c=(V_t²)/r
r=(V_t²)/a_c

w_f=w_o+xt
rx=a

I wasn't ale to make sense of it all, but if someone can explain, this may be able to, somewhat, work!

[Aaron.Graeve]

The use of a_c = (V_t²)/r and the other equations only apply if the robot is moving in a circle. I do not know the skill of your drivers, but most drivers I have seen do not drive in circles. The V_t refers to the velocity of the object tangential to the acceleration.

The w_f = w₀ + xt is similar to v_f = v₀ + at. It is the rotational velocity of a body under constant rotational acceleration. x is the rotational acceleration in radians/(s²).

In circular motion, velocity, acceleration, and position can be related to their rotational analogues by dividing by the radius.

Another interesting idea (that may be completely wrong) is to use a gyro with the forward position to create a set of vectors that might be used to find position in a polar system.

[Ether]

The code for trapezoidally integrating an acceleration to get distance was given in this thread in an earlier post.

If your acceleration is in a plane (the plane of the floor), use the same concept to get your position in the plane:

Given t, x, y, v_x, v_y, a_x, and a_y at some point in time, and a_xnew a_ynew at some later point in time t_new^*,
compute v_xnew v_ynew x_new and y_new as follows:

dt = t_new - t;

v_xnew = v_x + dt*(a_xnew+a_x)/2;
x_new = x + dt*(v_xnew+v_x)/2;

v_ynew = v_y + dt*(a_ynew+a_y)/2;
y_new = y + dt*(v_ynew+v_y)/2;

... where x,y is the location of the accelerometer in the fixed plane of the floor. Note that you will have to convert your accelerometer signal from the vehicle reference frame to the fixed x,y reference frame of the floor, using the gyro to do the coordinate rotation.

As stated earlier, the errors will accumulate quickly and the computed position will diverge from the true position.


^*Just to be absolutely clear for those who may be new to this: t_new is not one giant step from t. It is a very small integration time step (say 20ms) later than t. The repetition of this calculation over time is known as numerical integration.

[Invictus3593]

The use of ac = (Vt2)/r and the other equations only apply if the robot is moving in a circle. I do not know the skill of your drivers, but most drivers I have seen do not drive in circles. The Vt refers to the velocity of the object tangential to the acceleration.

The wf = w0 + xt is similar to vf = v0 + at. It is the rotational velocity of a body under constant rotational acceleration. x is the rotational acceleration in radians/(s2).

In circular motion, velocity, acceleration, and position can be related to their rotational analogues by dividing by the radius.

Another interesting idea (that may be completely wrong) is to use a gyro with the forward position to create a set of vectors that might be used to find position in a polar system.

Haha, he gave the ac = (Vt2)/r equation to use in the case that we were turning, which would throw off our real location; I guarantee you I don't drive in circles! If we updated the rotational acceleration every 10ms and took the average acceleration for that period, do you think would this be a short enough interval to be able to plot location semi-accurately?

Originally Posted by Ether

Given t, x, y, v_x, v_y, a_x, and a_y at some point in time, and a_xnew a_ynew at some later point in time t_new^*,
compute v_xnew v_ynew x_new and y_new as follows:

dt = t_new - t;

v_xnew = v_x + dt*(a_xnew+a_x)/2;
x_new = x + dt*(v_xnew+v_x)/2;

v_ynew = v_y + dt*(a_ynew+a_y)/2;
y_new = y + dt*(v_ynew+v_y)/2;

...the errors will accumulate quickly and the computed position will diverge from the true position.

I apoligize for not acknowledging, my friend; I appreciate your input!

Using trapezoidal integration, would that eliminate the errors? Or is there anther way to do it without the problems you describe? I've read that with robotic probes that go into caves and such, they use this kind of plotting system, an accelerometer and a gyro..

[Aaron.Graeve]

Originally Posted by Invictus3593

Haha, he gave the ac = (Vt2)/r equation to use in the case that we were turning, which would throw off our real location; I guarantee you I don't drive in circles! If we updated the rotational acceleration every 10ms and took the average acceleration for that period, do you think would this be a short enough interval to be able to plot location semi-accurately? ...

I suspect 10ms will be a short enough time, but that depends on a few hardware specific circumstances (gyro float and accelerometer responsiveness come to mind). Your team's driving style may also play a role in customizing the algorithm. I am interested to see how this will turn out. Also, have you considered encoders on your drivetrain? I hazard a guess that 2 encoders and a gyro can produce a position close enough for your needs.

[Invictus3593]

Originally Posted by Aaron.Graeve

I suspect 10ms will be a short enough time, but that depends on a few hardware specific circumstances (gyro float and accelerometer responsiveness come to mind). Your team's driving style may also play a role in customizing the algorithm. I am interested to see how this will turn out. Also, have you considered encoders on your drivetrain? I hazard a guess that 2 encoders and a gyro can produce a position close enough for your needs.

I was thinking the same thing, if the accelerometer proves to be unreliable (which is what it's looking to be), encoders + a definition of how wide the wheels are can give an accurate representation of how far the robot has moved, theoretically, and the gyro can give orientation.

[Greg McKaskle]

I was looking for a decent app on the iPhone that would work to experiment with. The closest I can come to is one called Vibration. I was using it to measure how the cell phone buzzed and compare that to an external accelerometer reading.

Anyway, the app will show the three axes and it calibrates to subtract out gravity at the initial orientation. If you leave the phone sitting still and run a five second recording, you should get relatively flat lines and that's expected. The integrated area should be zero.

If you run the app and move the phone to the left and right, you'll see similar cancellation. But it probably won't quite zero. Next, during a sample recording, walk from your chair to the front door. Each step looks like a heartbeat, on each axis. And yeah, they sort of cancel out, but where is my predictor of my acceleration that tells me how far I walked. It is a tiny bump at the beginning of that heartbeat signal.

Then run a sample and simply tilt the device a bit. You'll see that a five or ten degree tilt offsets the line quite a bit. And worse, it stays there for the entire sample. The integration of the tilt is huge.

Anyway, if you can find the app, or something similar, it is helpful in understanding why IMUs are hard. After all, if it was easy, the phone or Garmin would do this instead of or in addition to GPS.

Greg McKaskle

[Invictus3593]

Originally Posted by Greg McKaskle

I was looking for a decent app on the iPhone that would work to experiment with. The closest I can come to is one called Vibration. I was using it to measure how the cell phone buzzed and compare that to an external accelerometer reading.

Anyway, the app will show the three axes and it calibrates to subtract out gravity at the initial orientation. If you leave the phone sitting still and run a five second recording, you should get relatively flat lines and that's expected. The integrated area should be zero.

If you run the app and move the phone to the left and right, you'll see similar cancellation. But it probably won't quite zero. Next, during a sample recording, walk from your chair to the front door. Each step looks like a heartbeat, on each axis. And yeah, they sort of cancel out, but where is my predictor of my acceleration that tells me how far I walked. It is a tiny bump at the beginning of that heartbeat signal.

Then run a sample and simply tilt the device a bit. You'll see that a five or ten degree tilt offsets the line quite a bit. And worse, it stays there for the entire sample. The integration of the tilt is huge.

Anyway, if you can find the app, or something similar, it is helpful in understanding why IMUs are hard. After all, if it was easy, the phone or Garmin would do this instead of or in addition to GPS.

Greg McKaskle

Dang, I just tried it out. So i guess the accelerometer route is a dead end.

What do you think of the encoder idea? I mean, you don't have to worry about gravity or forces or anything with them, just rotations of the wheels.

[Ether]

Originally Posted by Invictus3593

Using trapezoidal integration, would that eliminate the errors?

Not with the accelerometer and gyro that come in the KoP. They're not accurate enough. The problem is the double integration (to get from accel to position). The small errors in the accel and gyro signal get integrated. The errors accumulate. After a short period of time, your computed position drifts away from your true position. The gravity problem Greg mentioned also contributes to errors in the accel signal in the plane of interest (the floor).

Haha, he gave the ac = (Vt2)/r equation to use in the case that we were turning, which would throw off our real location; I guarantee you I don't drive in circles!

The method I described applies to 2D motion in the plane of the floor, so it applies to turning (be it circular or not) as well as linear motion. The a=v²/r is not required.

If we updated the rotational acceleration every 10ms and took the average acceleration for that period, do you think would this be a short enough interval to be able to plot location semi-accurately?

You can answer this question by simulating a simple example in Excel or LabVIEW or Maxima or Octave or SciLab or any CAS tool or programming language of your choice. Assume you have a vehicle traveling in a perfect circle of radius R at a constant speed S. Then you know what the true ax and ay components of the acceleration are at any point in time. Do the numerical integration using those perfectly correct numbers. You should get almost perfect circular motion. Now introduce a small error into those ax and ay numbers, to reflect errors expected in the KoP gyro and accelerometer, and do the integration. You'll probably get something like a spiral instead of a circle.

I've read that with robotic probes that go into caves and such, they use this kind of plotting system, an accelerometer and a gyro

I don't know about the probes you are referring to, but if they use only an accelerometer and gyro to compute position they're probably much more expensive (and accurate) than those in the KoP.

Or is there anther way to do it without the problems you describe?

Placed properly, 3 unpowered omni follower wheels, each with an encoder, could theoretically be used to compute both position and rotational orientation -- without the need for a gyro or accelerometer. That would have a different set of problems.