RiSE Project

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RiSE Project
Toward a Template for Dynamical Climbing
Bob Full Dan Goldman Dan Koditschek Hal Komsuoglu
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Outline

• Templates SLIP
• In animals RHex
• Active SLIP Control
• Active SLIP Anchoring
• Self-stabilized Templates LLS SLIP
• LLS Full-Kubow Holmes-Schmitt Full-Jindrich
• SLIP Seyfarth et al. Ghigliazza et al.
• ASLIP Altendorfer et al.
• Control of Self-stabilized Templates Clocks
• Centralized Feedforward Clock
• Adaptation of Centralized Feedforward Clock
• Template for Climbing? (Komsuoglu et al.)

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Legged Animal COM TrajectoriesFullFarley.98
Vertical Forces
Fore-Aft Forces
Vertical Forces
Fore-Aft Forces
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Spring Loaded Inverted Pendulum (SLIP)

• No gravity during stance
• Conservation of Energy
• Conservation of Angular Momentum
• Integrable (Kepler) Problem
• (complicated elliptic integrals!)
• Gravity during stance
• Energy is the only integral invariant (Poincare)
• (restricted 3 body problem)

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RHex and SLIP COM Trajectories

Altendorfer, et al. Autonomous Robots 2001 11
207-213
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Fitting and Cross Validation of SLIP Human
Running SchwindUM98
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Details of the Liftoff Return Map

• Closed form integration of the air spring,
  unperturbed stance dynamics SchwindKod ICRA
  95 yields the liftoff map Saranli, Schwind
  Kod ICRA 98
• ,
• where

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Controllers for Nonlinear Plants

• Deadbeat Control
• Gives exact solution (with respect to plant
  model) to the control problem
• Potentially sensitive to model mismatches
• PD Control
• Controllers based on local linearizations
• Global Raibert-style decoupled controllers
• Integral Term
• Eliminates steady state errors
• Can be incorporated into any of these controllers

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Control of a Simulated SLIP legSaranli et al.
ICRA98

• Comparison of deadbeat and Raibert-style
• Step and sinusoid apex velocity trajectories
• Results are averaged over several runs

Tracking error step references
Tracking error sinusoid references
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Attracting Invariant Submanifolds
Buehler, et al., Rizzi et al., 1990 - 2000

• Vertical Batting Template
• The robot should impart energy to the 1 DOF ball
• in the right time, place, and magnitude
• so that the balls low energy state is a
  periodic vertical orbit
• Anchored Template
• The 2 DOF balls horizontal energy is damped out
• The balls 1 DOF vertical component has an
  oscillation that resembles that of the template

• Control Alternatives

• Formal Results
• Either can stabilize if properly tuned
• Mirrors typically have far larger domain of
  attraction
• Clocks typically require far less sensory overhead

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Anchoring SLIP in AKH
Saranli, et al., ICRA98
Q

• Idea eliminate 2 degrees of freedom by
• imposing virtual leg coordinates, g Q ! B
• choosing a posture, gy B ! Q, such that g
  gyidB
• Map virtual spring to joint torques
• Force joint work to equal SLIP work
  ?SLIP D gy FSLIP
• Enforce posture constraints around the error
  ePOS(q) gy g(q) - q
• Control via Superposition of Force Terms
• ?control D gy FSLIP N (KP ePOS KD dePOS
  /dt)
• where N projects into the orthogonal complement
  of D gy
• Sinusoidal Tracking Errors
• Two Different Postures
• Two Different Template
• Controllers
• x deadbeat
• o Raibert

B
?
FSLIP
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Anchoring SLIP in RHex
Template
Anchor
Heavy Reliance on Dynamical Model
Proprioceptive Feedback

• Simsect
• Complete 24 DOF Hybrid (foot contact) Dynamics
• Physical model at joint level
• Control joint torque motors

• SLIP
• Simple 2 DOF spring loaded inverted pendulum
  dynamics
• Abstract model relating COM world states
• Control virtual leg angle and leg spring
• Backwards, forwards
• Faster, slower
• Higher, lower

Saranli, et al. (in Preparation)
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Outline

• Templates SLIP
• In animals RHex
• Active SLIP Control
• Active SLIP Anchoring
• Self-stabilized Templates LLS SLIP
• LLS Full-Kubow Holmes-Schmitt Full-Jindrich
• SLIP Seyfarth et al. Ghigliazza et al.
• ASLIP Altendorfer et al.
• Control of Self-stabilized Templates Clocks
• Centralized Feedforward Clock
• Adaptation of Centralized Feedforward Clock
• Template for Climbing? (Komsuoglu et al.)

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Dynamical Effects of Horizontal Sprawled Posture
Full Kubow, Tr.Roy.Soc. B99
Horizontal Plane
Hypothesis Mechanical Preflexive
Self-stabilization in the horizontal plane
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LLS Template Schmitt Holmes, Biol. Cyb.00

• Pre-Template Peg-leg cartoons
• Analytically soluble models
• Complete equilibrium steady state analysis

• LLS Template Single flung leg
• 3 dof model with springy massless legs
• Constant touchdown angle control policy
• Analytical computation of return maps in special
  cases
• Numerical derivations and simulations in more
  general cases

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Self-Stability of a Hybrid Hamiltonian System

• Complete return map analysis
• 3dof model with leg attachment at mass center
• stable gaits (in heading) for a range of forward
  velocities.

• Local return map analysis
• 3dof model with leg attachment at distance d
  from mass center
• Attachment ahead of mass center (d gt 0) yields
  instability
• Attachment behind mass center (d lt 0) yields
  stability
• Bifurcation Analysis
• Self-stable regime persists over specific ranges
  in state and parameters

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LLS Theory Experiments

• Experimentation Self-stabilization in Horizontal
  Plane
• Hypothesis Three legs act as single virtual leg
  to stabilize yaw by purely mechanical action of
  fixed leg Kubow Full (1999) Phil. Trans. R.
  Soc. Lond. B, 354 849-862

Anchor

• Fact Speed of recovery exceeds fastest neural
  time constants Jindrich Full (2002) J Exp
  Biol. 205(18)2803-23

• Analysis Stability of Hybrid (piecewise
  holonomic) Hamiltonian Systems
• Fact LLS model with fixed leg placement angle
  exhibits (partial) asymptotic stability
  Schmitt Holmes (2000) Biol. Cyb. 83(6)501-515
  

Template

• Observation LLS model matches overall features
  of running cockroach data Schmitt et al. (2002)
  Biol. Cyb. 83(6)501-515

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Self-stability of SLIP templateGhigliazza et
al. SIADS03

• Physical parameters

• Dimensional analysis equations of motion are
  completely specified by 3 dimensionless quantities

• Rich dynamics period doubling and chaotic
  behavior observed before the gap opens
  (see also Schmitt Holmes, Seyfarth et al.)

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Outline

• Templates SLIP
• In animals RHex
• Active SLIP Control
• Active SLIP Anchoring
• Self-stabilized Templates LLS SLIP
• LLS Full-Kubow Holmes-Schmitt Full-Jindrich
• SLIP Seyfarth et al. Ghigliazza et al.
• ASLIP Altendorfer et al.
• Control of Self-stabilized Templates Clocks
• Centralized Feedforward Clock
• Adaptation of Centralized Feedforward Clock
• Template for Climbing? (Komsuoglu et al.)

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Template and Task Level Control
Physical Model
Animal
Template
Thoracic Ganglia
Control Language Symbols
Neural Oscillators (CPG Ist Order clocks)

Mechanical Oscillators (Leg Springs 2nd Order)
Pogo Stick
Cockroach
Robot
Time
Time
Time
Altendorfer, Moore, Komsuoglu, Buehler, Brown,
McMordie, Saranli, Full and Koditschek. 2001
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RHex 1.0 Clock DrivenCentralized Feedforward
Control
RHex
about its legs or body
knows nothing
yet maintains dynamic
stability nonetheless
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Adjustment of Self-Stabilized SLIP Gait
Altendorfer et al., (in preparation)
At constant energy?
(computationally tractable approximate closed
form available special cases)
Considerably more DvA affordance by varying
energy ? Need feedback controller (deadbeat)
to reach energy level and basin of
attraction
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Clock Excited SLIP
Altendorfer, et al. IJRR04

• Constant Touchdown Angle
• Fixed (in inertial frame) at b

• Solvable (1D) Return Map

• Complete Stability Analysis
• Too Primitive to Model RHex

• Constant Leg Recirculation
• ?2(t) ?2start ?2(t tLO)

• Triggered by previous liftoff

• ?2start h2 h1-1 (?1LO)
• Partially Solvable (2D) Return Map
• Partial Stability Analysis

• Partially Predicts Simsect Steady State!

Altendorfer, et al. (2003) ICRA
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Outline

• Templates SLIP
• In animals RHex
• Active SLIP Control
• Active SLIP Anchoring
• Self-stabilized Templates LLS SLIP
• LLS Full-Kubow Holmes-Schmitt Full-Jindrich
• SLIP Seyfarth et al. Ghigliazza et al.
• ASLIP Altendorfer et al.
• Control of Self-stabilized Templates Clocks
• Centralized Feedforward Clock
• Adaptation of Centralized Feedforward Clock
• Template for Climbing? (Komsuoglu et al.)