-- MarkCutkosky - 28 May 2003

Pasting in some email dialog among Trey (AMcClung), Noah (NoahCowan) and Mark (MarkCutkosky)

Date: Wed, 28 May 2003 10:59:42 -0700
Subject: Re: Developing reasoning for leg changes 
Cc: Mark Cutkosky <cutkosky@stanford.edu>
To: Arthur Joseph <nop>McClung <amcclung@stanford.edu>
From: Noah Cowan <ncowan@socrates.Berkeley.EDU>

Hi Trey,

You've asked a lot of nice, simple question, and your experimental outline looks good. I wonder though, do you think it makes sense to 'dive in' to such experiments, without first having a good prediction of how things will work? Without such, it will seem like a 'blind study' to the outside world, although I know that you've done a great deal of thinking and work, and have a lot of intuition as to what will happen.

My feeling is that for your thesis work to have a broader impact, you need a more "mechanistic" approach, i.e., what are the underlying physical mechanisms for turning? I know it may seem too complicated to get any theoretical headway, but I think there a few good paths that could work.

For example, you could think about turning as a steady-state phenomena, just as forward running is, after the initial acceleration period. We tend to think of turning as a non-steady-state phenomena, but as you know, once you get started and the initial transient dies out, it is no different in that regard to forward running. This quasi-steady-state approach could let you go in a similar direction as someone like Jorge went with his thesis, considering return maps on the angle of the body. This would be tricky, but would be a nice chapter even if you could just setup the problem.

As a second approach, you could think about turning in terms of the very first stride -- what is the net moment that you generate within one stride, when in transition from steady-state running to turning? Here, you might be able to look at simple things like the Jacobian, as a function of the leg angles to see what moment you can generate with the thrusters.

Either way, it is important to consider some principle from which you could generate specific hypotheses -- e.g. steady-state analysis or moment generation through kinematics or ... -- and then test those hypotheses out on the physical platform.

Take what you are doing below, and project it out to the best possible paper you could imagine. The abstract would read: "For Sprawlette, we examined the parameter space for turning, including leg angles and duty factor, and have determined the particular parameters that give the best turn. We have good heuristic rules that tell us how to combine the legs to get a graded turn, or a quick turn." This wouldn't be bad, but if you take an approach like above, it opens up the work to the broader field. For example, your abstract could read "We present a general approach to maneuvering for legged robots that can change their posture through a set of `shape' actuators, and generate thrust with a complimentary set of 'power' actuators. In general, the Jacobian at each shape maps the actuator force from the thrusters to turning moment. For our specific robot, Sprawlette, we have used this principle to parameterize a family a turning behaviors." This isn't yet all that great, but you can see that a hypothetical abstract like this can lead you to a much better conceived set of analyses and experiments. It is important to try to tell a story in advance, even though the details, and often even the underlying message, will change as you learn more!

I'll see you both on Friday.


Here's a summary of a what I came up with this weekend to tackle the question regarding the ways of changing legs for turning.


  • What roles do individual legs play in turning? (velocity, heading, pitch, roll; changes - leg angle, duty cycle)
  • What is the most effective 1 leg turning method?
  • How can legs be used together to affect turning? (2 legs, 3 legs, more?)
  • What combinations of legs are effective for turning methods?
  • How should these combinations be altered? (linear/exp, geometric/isometric)


  • Identify the effects of individual leg changes
  • Investigate the effects of 2 leg changes
  • Use results of 1 leg tests to guide 2 leg investigations
  • Investigate full realm of 2 leg changes, with more attention to certain areas
  • Vary by different values (geometric/isometric) for high potential combinations
  • Investigate effects of 3 leg combinations
  • Use results of 1 and 2 leg tests to guide 3 leg investigations


  1. Try LA and DC changes (2 levels, +ve/-ve) for individual ipsilateral legs
    • LAą, DCą 4 tests per leg, 3 legs =>12 tests
  2. Try LA and DC (2 levels) for pairs of ipsilateral legs
    • LAią & LAją, LAią & DCią, LAią & DCją, DCią & DCją, DCią & LAją (5 groups)
    • 3 pair combos, 4 per group, 5 groups => 60 tests
    • NOTE: i and j are legs, ie. LAią & LAją means leg angles for 2 separate legs
  3. Same as #2 for contralateral legs =>60 tests
  4. Same as #2 for 1st and 2nd diagonal legs => 60 tests
    • 1st diag (Leg #s: 1+2,3+4), 2nd diag (Leg #s: 1+4) - (3 pair combos)
    • NOTE: Legs numbered L to R from top to bottom, 0 to 5
  5. Try subset of 3 leg combinations based on individual and 2 leg results
  6. Investigate traditional 3 leg combinations (if not already high potential)
    • Geometric Ipsilateral - follow intersection point on grid (3x3, 4x4, 5x5)
    • NOTE: Use reasonable constraints on grid to provide stable forward motion
    • NOTE: Can use gradients along various grid lines to characterize turn performance
    • Geometric Tripod . same as geom. ips.
    • Isometric Ipsilateral . change angles by same amount (current method), characterize gradient behavior (+ve & -ve changes, 5 points each => 10 tests)
    • Isometric Tripod . same as iso. ips.
  7. Consider using more than 3 legs, if this seems beneficial at this point

  • More in depth investigations for high potential areas, as necessary

  • Any feedback would be greatly appreciated.



Noah J. Cowan
Department of Integrative Biology; 3060 VLSB, #3140
Berkeley, CA 94720-3140; Phone: (510) 643-5183

-- AMcClung? - 06 Aug 2003

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