Is it possible to stabilize a 2-wheel robotic vehicle?

This question refers to an interesting machine being developed by Team Frednet, which is one of a small number of teams competing to win the Google Lunar X-Prize for placing a robotic rover on the Moon, making some observations, and returning specified data. They have been working on a fairly conventional rover drive system, but in December 2008, they suddenly took a left turn into lightly explored territory by building a prototype with only 2 wheels. They named it JALURO (Just Another Lunar Robot).

JALURO consists of a chassis carrying the payload underslung between 2 wheels. Being underslung, it is in stable static equilibrium. Having independently controlled motors for each wheel, it is fairly simple to drive forward, backward, and to make turns in any direction with any radius from zero to infinity. It really is a neat concept!
 

JALURO’s payload mounts in a chassis underslung between two independently powered wheels. Click on the image to see a video of its first test drive. Click here for a higher-resolution version. It takes a while to download, so be patient. Source: Team Frednet

The problem is that the underslung chassis acts like a pendulum or, more precisely, a rocking chair. Notice that the rocking amplitude seems almost random, depending mainly on the shape of any disturbing impulses. The problem is most obvious when the vehicle comes to a stop. The chassis can be seen to rock forward and back for some time after the forward motion ceases. This will be worse on the Moon, where there is no damping from air resistance to slow the motion down.
While this rocking is especially noticeable at rest, it will be more of a nuisance when the vehicle is moving. How is an independent rover (and the X-Prize rules specify that it must be) supposed to navigate if its instruments are mounted on a platform that rocks back and forth while it’s moving forward, while it’s going over bumps and hollows, and while negotiating slopes? Stabilizing the chassis against rocking instability is critical.

To get a feel for the balance involved, I talked to Control Engineering chief editor and unicyclist, Mark Hoske. Starting and stopping a unicycle presents similar problems to those JALURO presents. Mark explained that to make a unicycle go forward, the rider has to do two things simultaneously: he or she must pitch weight forward to unbalance the cycle and apply a torque through the pedals to move the wheel forward to catch up with the weight. If both happen at the same time and to the same degree (also with attention side to side), then a wildly impossible-looking unicycle ride happens. If not, *SPLAT*!

JALURO does something similar. Unlike the unicycle, JALURO is statically stable. Like the unicycle, to move forward JALURO must unbalance its weight. It does so by applying a torque to rotate its chassis forward. As the chassis moves forward, a couple arises between the chassis’ weight and the ground’s reaction force. That couple produces a torque that rotates the drive wheels forward. That torque, of course, must equal the torque produced by the vehicle’s motor.

As the torque rotates the wheels, they roll across the ground. Some of the moment produced by moving the chassis forward goes into raising the wheels’ angular momentum, some goes into changing the chassis’ angular momentum around the wheel axis whenever the pitch angle is not constant, and the rest goes into driving force for the wheels. The driving force accelerates the mass of the wheels and chassis. A further complication arises if the pitch angle changes, as that modifies the acceleration of the chassis mass.

Putting this all together provides a differential equation of motion that is second order in both the forward (horizontal) dimension and the pitch angle. The pitch angle terms look particularly nasty because they include zeroth, first, and second order terms. Luckily, however, all coefficients are constant, and we can guess the solution:
QA19JAN09e1.jpg

where α is the pitch angle, φ is the target pitch angle to obtain the correct torque profile, A is an oscillation amplitude, ω is the system’s resonant frequency, and t is, of course, time. The second term on the right represents the rocking motion we’d rather do without.

Substituting this solution into the equation of motion and making various assumptions allows us to predict the motion under different conditions. In particular, the frequency depends only on constants, such as the wheel radius and various masses. The rocking motion amplitude is independent of everything else, so once it starts, something must be actively done to stop it.

Of course, banging the motor on and off is guaranteed to excite the rocking motion. To obtain smooth acceleration, the torque profile (as represented by pitch angle) must be shaped to be what we want (such as smooth acceleration) minus the rocking motion the acceleration will likely excite. The two terms together will thus sum to the desired profile without rocking instability.

For those who want to see the solution in detail, you’ll have to wait until I find time to transcribe 4 pages of math and post it online. Or, you could work it out for yourself. I’ve given you the broad outline, so, with additional information from the video, setting up the equation of motion is fairly straightforward.

For additional commentary on the Google Lunar X-Prize, see “Old Hat” and other entries in the AIMing for Automated Vehicles blog.