||Type and size
||Title and Hyperlink(s)
||Appendices to the paper
||Text Appendices to the paper describe the simulation model parameters, simulation foot contact models used, and measurements of the bumpiness of the ramp used during motion-capture data acquisition experiments. |
||Stiesberg Simulation Code Package
||Computer code. The Stiesberg simulation system was written using C++ and open-source packages including
GiNaC, CPPLAPACK, and PyOpenGL. |
||Steinkamp Hopper at Cornell.
||Steinkamp hopper. Hopper at Cornell during data aquisition.
Many short clips. Some show hopping all the way down the ramp, some show failure modes.
At 3:02 a slow motion video emphasizes Steinkamp's launch technique. |
||Motion-capture data gallery
||Steinkamp hopper. A gallery of motion traces. Each plots the raw motion-capture data for one or more body parts during an attempt at hopping down the ramp. (Files in zip form, download and open index.html to view pictures) |
|| Effect of finite friction foot contact.
|| Animation from the simulations of GS shows what happens just below the lower practical limit of the Coulomb friction coefficient for the foot contact. At or below μ=0.3,
the landing of the hopper is less clumsy; the foot often slips forward causing a backwards fall.
On the other hand, for larger values of μ, the overall motion was close to indistinguishable from simulations assuming no slip. (file SteinLowMuFall.mp4)
||exotic periodic motions.
||GS simulations show a period-2 cycle (two hops per cycle) that also has two leg swings in each hop.
Double leg swings were occasionally observed in the physical device
when it was going much too fast, and just before falling, and sometimes when launched sufficiently poorly. (file: SteinPer2Swing2.mp4) |
||totter mode example.
||Animation from the GS simulations shows the totter mode. The variation in body angle causes is correlated with the variations in leg swing amplitude and and with variations in the forward speed of the device. However, the leg swing frequency is nearly constant.
The totter mode tends to decay both in simulations and in motion-capture data. However it is also excited by launch and by ramp-slope variations (bumps). |
||experimental device (left) vs. theoretical simulation (right).
||Comparison of experiment and theory. Side-by-side animations of the physical device and of an animation of the device. The time scale is not stretched, but is equal for simulation and experiment. The similarity indicates reasonable fidelity of the model to the device. (file: SteinDeviceVsSim)|
||videos 0:38, 1:10
||Stable period-2 motions; with
a point foot and
a round foot.
||Simulations of devices not like Steinkamp hopper. GS simulations show stable passive period-2 hopping for two nearly identical models;
one has a point foot (ColeStablePer2Point.mp4),
and one has a large round foot (ColeStablePer2Round.mp4).
These models are substantially different than the Steinkamp model in terms of parameters; most significantly, they have a much heaver leg.
All attempts to find stable period-1 motions in the neighborhoods of these models failed.
The parameter set used that led to these parameters was inspired by Working Model simulations from Mike Coleman (private communication). |
||videos 1:27, 1:14
||Stable period-1 motions;
a periodic motion and
a perturbed motion.
||Simulations of devices not like Steinkamp hopper. GS Simulations of a significantly different model that has stable period-1 solutions.
Although this model has a large round foot, it still posesses the desired statically unstable (can't stand up) property.
This model has the relative rotational degree of freedom between leg and body locked, so it has 4/2 DOF in flight/stance.
The periodic motions for this model (CirclePer1.mp4) are remarkably robust in the sense that the basin of attraction is large;
the animations (CirclePerturbed.mp4) show convergence to near the periodic motion within about 20 hops after a large initial perturbation.
Other models exist that also have stable period-1 hopping cycles, including models that do not have the relative leg/body degree of freedom locked.
Thus stable motions do exist for many parameter families of hopping models that can't stand up; the physical Steinkamp hopper is just in the wrong area of parameter space to have this property. |