Motion Profiling

*“Thread created automatically to discuss a document in CD-Media” does not allow attachments (I don’t know the purpose for that limitation. Brandon: would it be beneficial to change that setting?)

Determine how far you want the robot to move, how fast you want to allow it to accelerate at startup (and decelerate as it approaches the destination), and the max speed limit during that motion.

Run the calculator to get your distance vs time equations.

Each pass through TeleOp, chose the appropriate x(t) equation based on elapsed time “t” since start, and use that equation to compute a new position setpoint.

Use the setpoint from Step3 as the position command to a closed-loop controller controlling whatever it is you want to control (e.g. wheels of a drivetrain, position of an elevator, etc)

Seems like we would tune ‘amax’ until the robot drives the desired distance in the desired time.

You can always reduce amax and/or vmax if the computed time-to-destination is faster than required. And vice-versa.

For a given vmax, there will be an amax below which you can’t get a trapezoidal motion profile (it will be triangular instead). [strike]What I’ve presented here presently doesn’t handle that situation.[/strike] I just uploaded a Maxima script and plot for triangular motion profile.

*PS: The purpose of this thread is to raise awareness of and provide a focal point for conversation about the topic of motion profiling. I’m sure there are many motion profiling papers and apps out there in webland. If you have a favorite please feel free to posts links in this thread. *

I just uploaded a Maxima script and plot for triangular motion profile.

User enters amax and distance-to-destination. There is no speed limit. amax can be varied to get desired time-to-destination.

Paul Copioli wrote:

I am curious as to why you are proposing this equation based method when the filter based method posted from post #18 in this thread:

The filter method is so much more real time friendly and flexible as you can determine the motion acceleration type just by manipulating the filter lengths.

Hi Paul,

Thanks for that link. Since that thread is closed, I invite interested parties to respond here if they wish.

Please note that I am not advocating any particular method. I’d simply like to discuss the pros and cons.

Jared, that’s really interesting. We have a desire to do a trapezoidal profile on our tote lift in order to prevent sudden deceleration from jerking a tote off of the hooks. We’re doing something primitive right now, but I wonder if our programming lead thought about a lookup table for it.

It’s worth pointing out that the equation-based method is also real-time safe (deterministic runtime and does not allocate), although a microcontroller without an FPU would not be able to run it as quickly as a cascade of Boxcar filters.

The main advantage of polynomial-based motion profiling techniques is that you have a bit more flexibility in the constraints you can obey. For example, you could easily accommodate asymmetric forwards and backwards velocity/acceleration limits. It is also easy to change your constraints on the fly - as often as each control cycle. Doing this (for higher order constraints…velocity is easy) with FIR filters requires resizing the number of filter taps and pre-loading the new filter with the right initial conditions.

One other reason to favor the polynomial method is that relatively few high school students have had any exposure to linear filtering, but if they have taken a physics class they will know (and be able to apply) basic kinematics equations.

In general, though, either method works fine, and the filtering approach will be faster if you can get away with no asymmetries and constant constraints on higher order derivatives.

The only reason we used a lookup table last season is that (a) we wanted to be able to visualize the x/y/theta quintic spline paths we were generating before deploying to the robot to make sure they looked reasonable, and (b) we needed to re-parameterize our splines by arc length, which we did with naive (and costly) numerical integration that took a little too long on the cRIO. If we had more time we would have made some quick optimizations and just done it on the fly.

The lookup table method was the best way for our team, as all of our autonomous action is based on distance triggers. We would generate the desired motion profile on the roboRIO, generate a .csv file from the data, copy it to the laptop, and open it in excel. This let us sit in front of graphs and data and decide when and where each action in autonomous would happen. It takes some amount of time for our elevator and container swatter to move, so knowing how fast the robot was going at each point was very helpful to determine where we should set the distance trigger for starting elevator or swatter movement.

The table method is also super easy to explain. The polynomial method would require separate polynomials for acceleration ramp up, constant positive acceleration, acceleration ramp down, constant velocity, deceleration ramp, constant deceleration, and final deceleration ramp. That seems harder to keep track of.

The table calculation method also gave me an easy way to work in motor equations. You get more torque, and therefore more acceleration, from your motor at lower speeds. If your profiling takes this into account, you can go much faster. Without taking this into account, you must limit your max acceleration to less than the robot can actually achieve so that as it accelerates, it can still achieve the same acceleration.

The solution to the dc motor equations for a pulley driven lift is

v = (r * w_n * (a_g * r * m - T_s)/T_s) * (1 - e^(-(t * T_s)/(r^2 * w_n * m))))

where v is velocity, r is pulley radius, w_n is no load speed (rad/sec), a_g is acceleration of gravity, m is mass carried, T_s is stall torque, and t is time.

If you have lots of time, you could integrate this again and find the position equation, and use that to determine your motion profile.

However, that’s difficult, and you can come up with a simpler equation, v * a = k, where k is some constant specific to your setup. This means you either get a large acceleration or a large velocity, but not both at the same time. This equation is only valid when a is less than the acceleration achieved by stall torque and v is less than the no load speed. This is really easy to add into your calculations for the constant acceleration portion with

a = k/v_prev

If you have a large velocity, you can’t accelerate as fast.

This would let you achieve the most aggressive and fastest profile possible for when speed really matters.

*Just for fun and educational purposes, I wrote a Maxima script for sinusoidal accel profile and added it to the paper.

The user enters max allowable acceleration and time-to-destination, and the script computes the equations for distance, speed, acceleration, and jerk versus time… and plots them.

Attached is a sample graph generated with the script for max accel = 6.3 ft/sec2 and time-to-distance = 3 seconds.

The vehicle goes from x=0, v=0, a=0, and jerk=13.2 at t=0
to x=9, v=0, a=0, and jerk=13.2 at t=3 seconds.


For comparison, here’s a plot using trapezoidal accel, with the same max accel, jerk, and time-to-distance.

The vehicle goes from x=0, v=0, a=0, and jerk=13.2 at t=0
to x=9.7, v=0, a=0, and jerk=13.2 at t=3 seconds.

If you can experimentally (or analytically) determine the maximum speed vmax and the maximum acceleration amax, you could try using a linear relation:

acceleration = amax(1-speed/vmax)*

… and that would be easy to numerically integrate in a spreadsheet.

When speed=vmax the acceleration is zero, and when speed=0 the acceleration is amax.

EDIT: added screenshot and XLS for user-selectable accel & speed limiting

accel limited profile.xls (34 KB)

accel & speed limited profile.xls (34.5 KB)

accel limited profile.xls (34 KB)

accel & speed limited profile.xls (34.5 KB)

I’ve attached an excel spreadsheet with a few of the motion equations I’ve come up with so far. For inputs, they take motor constants. If you have a gear reduction before the motor constants, you need to come up with a new set of motor constants that match the characteristics of the output of the gearbox.

The first sheet is for some mechanism which is always fighting gravity in the same direction it is trying to move. The speed under load after a few seconds matches the loaded speed values calculated by the JVN design calculator.

The velocity is determined exactly, not by numerical integration. I don’t know how to integrate that velocity function (though I’m sure it’s possible), so I did what you suggested in your previous post.

The second sheet calculates velocity and position with explicit functions for each, but does not take into account gravity. The velocity function had a less complicated integral. This would work for a drive system, but not for an elevator. I like this better because the robot could simply plug in the time elapsed into the position function to determine its setpoint.

If you’re interested in seeing my work, I could take a picture of it.

Our 2015 elevator has these values
stall torque 25 nm
mass 35 kg
free speed 32 rad/sec
sprocket radius .022 meters

It’s way overpowered, and does achieve full speed in less than .25 seconds under load.

Elevator Motion Profile.xlsx (34.3 KB)

Elevator Motion Profile.xlsx (34.3 KB)

*I’ve encountered what appears to be a bug in the version of vBulletin® that CD is using (Version 3.6.4).

When I edited the text boxes for this whitepaper and added “center” tags (to center the text), it “locked” the text boxes so that I can no longer edit them: normally to edit a text box you just left-click on the box… but that no longer works (nothing happens).

Has anyone else encountered this? Better yet, does anyone know of a way to “unlock” the text boxes?

So, please ignore the hyperling in the text box; it is incorrect.

BTW: I just uploaded another short attachment providing details how to compute a sinusoidal accel motion profile from a given desired max accel and a given distance to target (starting from zero speed). Currently, it’s the one at the bottom of the list.


Someone just found a nasty typo in attachment “how to calculate sinusoidal acceleration motion profile revA”

The function x(t) should be K[sub]2[/sub]∙(t-K[sub]3[/sub]∙sin(K[sub]1[/sub]∙t))

I will make that correction and post revB


Has anyone ever tried a controller like this ?

How about with the velocity loop in the SRX and the position in the RIO ?

  • the diagram should have included separate gains for the velocity and acceleration feedforward.