The Dynamics of Legged Locomotion: Models, Analyses, and Challenges; Holmes, Full, Koditschek, Guckenheimer

This was the 100 pager, which i managed in one sitting, at the expense of section 5 (the meat) and lots of biophysics (boring). It was better than I was expecting, lots of leads and good insight.

This is a very impressive review consolidating work in modeling, analysis, robotics, biophysics and neuroethology. They cover a ridiculous array of methods and models, drawing from everywhere. It isn't quite my one stop reference shop, but it will provide many useful leads, and after I look closer at section 5, some tools too. The bias is towards running insects.

• Section 1 is introduction
• Section 2 reviews earlier work on locomotion and movement modeling, relevant machanical, biomechanical, neurobiologaical and robotics background.
• Section 3 summarizes experimental work on running and walking animcals
  • The Meat: "Although short of listing concrete experiements, these propositions nonetheless suggest how a unified framework can give rise to new ways of asking questions about the structure, organization , and function of locomotion" Hypoheses:

1. 1. Stable Dynamic System (goal of insect locomotion is stability, emergent. This hypothesis is complicated by my above observation that the more complicated control is used on the simpler task. Insects will need fewer states to be stable in simpler environments. slow locomotion implies a simpler environment, but we see more complex (and bottom up) behavior. This suggests to me that the slow gait fills other purposes besides stability. Specific to running, i can't dispute this hypothesis.
   2. Collapse of dimension. Reread all this! My vision is failing right now, i need to stop. but i won't
   3. Tunable Coordinate Architecture (context dependant shift on sensation/pattern based locomotion)
   4. task level control and its identification (can this only be centralized?)
   5. hybrid systems

• Section 4 introduces analytic methods to be employed in section 5
• Section 5 solves some problems and provides procedures for answering some tricky questions.

• • Hyp
• Section 6 has lots of great conclusions and directions

Comments

"Ultimate goal is to produce a "behaving insect"" That is without any reference to Brooks that I can determine.

Acknowledging role of engineer's optimization and reduction

Contrast "Completely actuated and sensed machines..whose stability can be established and tuned analytically...[requiring] a very high deegree of control authority" with "The analytically messier, "low-affordance" ... autonomous, dynamically stable ... preflexively stabilized...[with] centralized/decentralized feedforward/feedback lovomotion control architectures". Further: "a gulf remains between the performance we can elicit empirically and what mathmatical analyses or numberical simulations can explain." Brings up the interesting personal question. Is my role to work towards filling the gap? Right now, I think my goal will be to find tools that make researchers comfortable leaving the other ones behind.a

simulations are allowing empirical results to be trustworthy enough to replace theoretical results, in many cases.

A worried question: "Are coordination patterns unique within species?" Even if the answer is yes, the procedure each species followed to find the pattern is the same. The problem of designing distributed systems is making them design themselves.

There was a subtle point that seemed important that I didn't understand (and can't find). Animal limbs have many many degrees of freedom, which is a pain for analysis, so they must have methods for constraining the space of conformations. Mention was made that the constraint is goal based? I have to determine the possibilities of that for myself. Oh, here is the mention:

"collapse of dimension: the emergence of a low-dimensional attractive incariant submanifold in a much larger state space. This dynamical collapse appears to be associated with a posture principle: the restiction of motion to a low-dimensional subspace within a high-dimensional joint space. A kinematic posture principle has been discovered in mammalian walking[citation], as demonstrated by planar sovarianation of limb elebation angles which persists in the face of large variations in steady state loading conditions[citation]"

Collapse of dimension is a great concept and section 5 may provide a tool useful to me. What is the 'posture principle' and can I abstract it or generalize it? I don't know what submanifolds are. 'dynamical collapse'? I wonder if randy has anything to say about that.

My notes on this quote say "many constraints" and "patterns". As for the first, the quote validates the intuition behind something I wrote elsewhere, about needing a procedure by which we can engineer how modules locally constrain each other in the same way that we can engineer how a central module constrains its parts in a central architecture. Regarding "patterns", it may be that the best way to collapse the search space will be to create behavior patterns like in Brook's Attila or Hannibal. Related to this is collapsing the search space by making irreducible modules like in conventional engineering and cockroach legs, where each leg can influence the other, but the actual motion of a leg is largely locally controlled.

"We also believe that the rapid runing regime pushes animals close to limits of feasible neuromuscular activity and hence constrains the space of activations and dynamical forces available, much as in the case of static force production [citations], making it more likely that lower-dimensional behavior will emerge" Great quote! Important insight! I just don't know in what way.

Also: "A model that leaves nothing out is not a model" hear hear!

"growing consensus with the animal neuromotor community that control is organized in a distributed modular hierarchy" Distributed implies (only) local communication. Modular implies modules, or functional black boxes. hierarchy implies multiple scales and modules that control action at scales below them. It also, by my definitions, contrasts with the word distributed (though ther aren't necessarily mutually exclusive in this context). What are their definitions? Well, there is a chart on p230 with a 'decentralized centralized' axis and a 'feed-forward feedback' axis. Modules are assumed. It is possible that for them decentralized and distributed are synonymous and centralized and heirarchical are synonymous.

Regardless, I've emailed the man.

To return to earlier ideas: I was confused earlier because my inclination from my time at NECSI was to believe centralized control schemes are best suited for simple environments while decentralized schemes are best suited to complex environments. This was nuanced by the observation that roaches use decentralized schemes for running and top down control for slow walking (less complex). Huh? My first attempt to reconcile these two ideas is that slow walking serves a completely different purpose (foraging, exploration) than running (escape) and therefore the measure of 'complexity' is necessarily different for each and they can't be compared solely on, say, stability or 'complexity' or landscape.

Tough quote, potentially valuable: "The observation that certain DOFs exhibit significantly higher variability that others can be interpreted in the framework of stochastic optimal feedback control as a hedge against noise [citation]" Huh? Does this imply a heirarchical relationship between high and low variability DOFs?

Continuing: "Dependending on environmental demands, the full range from pure feedback to pure feedforward control poicies is porbably employed in animcal motion. Indeed, the suggestion, based on linear systems theory that feedback should be preferred when internal models are uncertain or unabailable, while feedforward stratevies should be more appropriate in the presence of significant sonsor noise [citation], seems very reasonable. The extremes of this coninuum are exemplified, respectively, by "mirror laws" developed for juggling machines [citation] and legged robots[citations], and passive stabilization based on preflexes, as exhibited by the SLIP and LLS models described in this paper. Overall, since centralized feedback circuits imply greater time delays, as running speeds increase, we expect control to emphasize decentralized modes, and increasingly to rely on feedforward strategies."

Some less relevant interesting results for comparative animal studies:

• largest variations in patterns of locomotion come from comparing animals that differ greatly in size.
• within an individual, metabolic costs of locomotion over the range of possible conditions vary less than tenfold
• by contrast, metabolic costs vary by over five orders of magnitude when all legged animals are compared
• Re historical assistance to comparative approaches:
  • "If the process of interest has severe functional or structureal constraints of nearly complete adaptation has taken place, then the potentially confounding effects of historical differences may be of little consequence. If, however, constraint and adaptation have been less than completely dominant, than the most parsimonious assumption is that the process operates as it did the the ancestor"
• recoil in ankle extensor tendons reduces total work by 45%
• in qudrupeds, stride requency increases linearly with speed during trotting, but becomes nearly independent of speed as mammals seitch to a gallop, higher speeds obtained by increasding stride length.
• tripod gait in insect has the front and middle legs pushing in the wrong direction. Don't know why that makes sense, but one effect is to put some yaw into the torso.

Insects are statically unstable at their highest speeds. That is a possibility an engineer would never allow.

"Simple feedforward motor output appears to be effective in the negotiation of such rough terrains when used in concert with a mechanical system tuden to stabilitze passibly" is preflex the passive stability?

Can I engineer a terrain that would slow/trip an insect basec on an analysis of the different scales on which the different leg parts act and interact? What scales of pebble are insects slowest over. How does that correspond with their physiology?

(skipped forty pages)

distinction made between 'logical' and dynamical critera (for liftoff and touchdown)

"Paraphrasing T.S.Elliot[citation], we will then have removed a little more of the shadow between intent and action.?

Clarification from Dr. Holmes

• On page 229, halfway down the first paragraph of section 2.4.2 is this

sentence where you mention the "growing concensus within the animal neuromotor community that control is organized in a distributed modular heirarchy". You then provide some valuable clues clarifying your vocabulary. On the next page you provide a very handy chart with feed-forward/feedback on one axis and centralized/decentralized on the other.

• My confusion is about the relationship of the sentence to the chart:

Can I take it to mean that distributed and decentralized are synonymous in this context? How about in your personal understanding?

• • No. A distributed modular heirarchy implies the whole 2-d plane

sketched in Fig 7. Distributed is mostly along the left-right axis, and heirarchy is up-down, but in order to collect & process and effect centralized control, feedback signals may need to go "higher" in the CNS than local reflexes. After a "centralized" decision is made, the commands are then distributed to the separate components. A decentralized feedback module (bottom left) might be a load sensor on one leg modulating motoneurons activating extensors and flexors for a particular joint. A brain area involved in motion planning, taking exteroceptive antennal and visual input and modulating the CPG and motoneurons to steer the animal, would be a centralized feedback module (bottom right).

• • I also prefer decentralized in describing overall

control strategies, since "distributed systems" has the technical meaning of continua modeled by partial differential equations in (mathematical) control theory, and here we are also thinking of discrete elements distributed among, e.g. sets of legs.

• Does 'heirarchical' place a system on the feedforward part of the

vertical axis or does it shift the system over a bit from being completely decentralized on the x axis? Or both? To phrase it differently, is 'heirarchical' synonymous with either of 'feed-forward' or 'centralized'? Some combination?

• • Heirarchical often (but not always) implies centralized, but it may

operate purely feed forward, or with varying degrees of feedback. See ** above & below.

• And a bit more concretely; insofar as roaches can be shift between

pattern or sensation governed locomotion (correlating with speed) do you think of them as shifting only along the feedforward/feedback axis or do you also see a 'centralized' 'control signal' kicking in to change the locomotor regime?

• In other words, should I think 1) of the range of roach locomotion

and, separately, 2) of models of the range of roach locomotion, as moving vertically or diagonally along the two axes of your p 230 diagram?

• • Our 2-d chart is an idealization of a multidimensional control space

in which roaches (or any animals) can move, depending on environment and task demands. In a simple case like running toward a preset goal (dark recess under refrigerator), they may mostly move diagonally along bottom right (smooth floor, no obstacles) to top left (obstacles, cats), but in exploring, sensing for food, they'll leave the plane of our idealized scheme, e.g. using sensors on their front legs to gather info, not to walk. We're currently beginning to build & test models with realistic proprioceptive sensory feedback time constants to see if feedback is "automatically" disabled as they run faster.

To Look Into

• Hutchinson
• autopoesis, Manchuera? Bolera?
• Look into RHex
• juggling machines [citation] and legged robots[citations]
• Are their nets of neural nets?
• Differential algebraic equations?