Parameter estimation gait synthesis and experiment design Sam Burden Shankar Sastry and Robert Full Optimization provides unified framework 2 Blickhan amp Full 1993 Srinivasan ID: 933376
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Slide1
Optimization for models of legged locomotion:Parameter estimation, gait synthesis, and experiment design
Sam Burden, Shankar Sastry, and Robert Full
Slide2Optimization provides unified framework
2
?
?
?
?
?
Blickhan
& Full 1993
Srinivasan
&
Ruina
2005, 2007
Vejdani
, Blum, Daley, & Hurst 2013
Seyfarth
, Geyer, Herr 2003
estimation
synthesis
design
Estimation
of unknown parameters for reduced-order models
Synthesis
of dynamic gaits to
extremize
performance criteria
Design
of experiments to distinguish competing hypotheses
Slide3Estimation of unknown parameters in simple models3
cockroach
human
m
L
,
k
Lumped parameters
r
= (
L
,
k
,
m
)
not known
a priori
leg length
L
and stiffness
k
; body mass
m
Model validity depends on parameter values
gait stability, parameter sensitivity, etc.
Estimate parameters
r
by minimizing prediction error
e
Full
,
Kubow
, Schmitt, Holmes, &
Koditschek
2002;
Seipel
& Holmes 2007;
Srinivasan
& Holmes 2008
model
Burden,
Revzen
, Moore,
Sastry
, & Full SICB 2013
Slide4Synthesis of optimal dynamic gaits & maneuvers4
Impulses u in idealized walking and running gaits minimize work W
Srinivasan
&
Ruina
2005, 2007
“Stutter
jump” s
inusoidal
input
u maximizes jumping height h
Aguilar, Lesov,
Wiesenfeld, & Goldman 2012, SICB 2013walking gait
running gaituu
Slide5Experiment design to maximally separate predictions5
Various extensions proposed to improve stability
Vejdani
, Blum, Daley, & Hurst 2013
Simple spring-mass unstable for high speeds or irregular terrain
H1
: leg retraction or reciprocation
H2
: axial leg actuation
Design treatment
t
to maximally distinguish
d
hypotheses
H1, H2
t
specifies, e.g., terrain height, inertial load, perturbation
Seyfarth
, Geyer, Herr 2003;
Seipel
& Holmes 2007
Slide6Optimization provides unified framework
Estimation
of unknown parameters for reduced-order models
Synthesis
of dynamic gaits to
extremize
performance criteria
Design
of experiments to distinguish competing hypotheses6
?
?
?
?
?
Blickhan
& Full 1993
Srinivasan
&
Ruina
2005, 2007
Vejdani
, Blum, Daley, & Hurst 2013
Seyfarth
, Geyer, Herr 2003
estimation
synthesis
design
Need tractable computational tool applicable to legged locomotion
Slide7Parameter estimation, gait synthesis, and experiment design posed as optimization problems
Existing techniques for optimization applicable to legged locomotionScalable algorithm based on computable first-order variationOptimization for models of legged locomotion
7
Slide8Simple illustrative model: jumping robot
8
Mass moves vertically in a gravitational field
Forces generated from leg spring and actuator when foot in contact with ground
Damping, impact losses yield discontinuous dynamics
This simple model contains essential challenges for optimization – approach generalizes to complex models
Slide9Translation to canonical optimization problem
9
Estimation
of lumped parameters
r
from experimental data
Synthesis
of inputs
u
for dynamic gaits that extremize performanceDesign of experimental treatments
t to distinguish hypotheses
Mathematically equivalent to extremizing generalized performance J at final condition x(T) by searching over initial conditions x(0) x(0) incorporates parameters r, inputs u, and treatments t J integrates error e, work W, or prediction difference
dH1,H2 along x(t)parameters r – (k,l,b,m,g)input u – (actuator input)treatment t – (e.g. spring law)
Slide1010
Estimation
of parameters
r
Design
of treatments
t
S
ynthesis
of inputs
u
Each of these optimization problems:Is equivalent to extremizing final performance J(x(T)) over initial conditions x(0):
Optimization
of initial state x(0) Translation to canonical optimization problemparameters r – (k,l,b,m,g)input u – (actuator input)treatment t – (e.g. spring law)
Slide1111
Typical jump: height, velocity, input versus time
g
Slide1212
Continuous optimization with fixed discrete sequence
Fix footfall sequence corresponding to particular trajectory
g
Define discrete event function
P
(e.g. apex) near
g
Optimize near
g using event function Pg
x(T)=P(x(0))
P
x(0)x(T)=P(x(0))
Slide1313
Continuous optimization with fixed discrete sequence
g
P
x(0)
x(T)=P(x(0))
Srinivasan
&
Ruina
2005, 2007; Phipps, Casey, &
Guckenheimer 2006; Remy 2011;Burden, Ohlsson, & Sastry
2012; Burden, Revzen, Moore,
Sastry, & Full SICB 2013
Tractable, but restricted to footfall sequence for ginappropriate for multi-legged gaits or irregular terrain
Slide14Discrete optimization of footfall sequence14
Golubitsky
, Stewart,
Buono
, & Collins 1999; Johnson &
Koditschek
2013
Naïvely, can optimize over all possible footfall sequences:
enumerate footfall sequences,
Sapply continuous optimization to each sequence s in Schoose sequence with best performance
x(0)x(T)
x(0)
x(T)
,, …single jumpdouble jump
Combinatorial explosion in number of sequencesintractable for multiple legs or irregular terrain
Slide15Parameter estimation, gait synthesis, and experiment design as optimization problems
Existing techniques for optimization applicable to legged locomotionScalable algorithm based on computable first-order variationOptimization for models of legged locomotion
15
Slide16Iteratively improve performance: initial trajectory
16
Slide17Iteratively improve performance: step 1
17
Slide18Iteratively improve performance: step 3
18
Slide19Iteratively improve performance: step 5
19
Slide20Key observation: performance criteria varies smoothly
discontinuous/non-smooth
smooth
Can apply gradient ascent using
dJ
/
dx(0)
to solve:
Elhamifar
, Burden, &
Sastry
2014
20
Slide21T =
100ms
T =
160ms
Key advantage
:
unnecessary to optimize footfall seq.
21
Initialize optimization from equilibrium
With final time
T =
100ms
, yields single jumpWith final time T = 160ms, yields “stutter” (double) jump
Slide22Continuous optimization can vary discrete sequence22
Scalable algorithm is applicable to optimization of:multi-legged gaitsaperiodic maneuvers
irregular terrain
multiple simultaneous models
?
?
Footfall sequence optimization is unnecessary
continuous initial condition implicitly determines discrete sequence
Enables
estimation
,
synthesis
, &
design
in unified framework applicable to terrestrial biomechanics
Slide23Provides unified framework for parameter estimation, gait synthesis, experiment design
Previous techniques impose restrictive assumptions, scale poorly with dimensionComputing first-order variation yields scalable algorithm applicable to hybrid models
Conclusions for optimization of legged locomotion
23
Slide24Provides unified framework for parameter estimation, gait synthesis, experiment design
Previous techniques impose restrictive assumptions, scale poorly with dimensionComputing first-order variation yields scalable algorithm applicable to hybrid models
Optimization provides practical link between model-based and data-driven studies
Conclusions for optimization of legged locomotion
24
Slide25Acknowledgements– PolyPEDAL
Lab – Biomechanics Group– Autonomous Systems Group– UC Berkeley25
Collaborators
Affiliations
Sponsors
–
NSF GRF
– ARL MAST
Thank you for your time!
–
Shankar
Sastry
– Robert Full
Slide26Open problems and future directionsempirical validation of reduced-order modelscontinuous parameterization of experimental treatments, outcomes
generating hypotheses from modelsdata-driven modelslocal vs global optimizationproperties of piecewise-defined models for multi-legged gaits
26
experimental biomechanics
dynamical sys & control theory
Elhamifar
, Burden, &
Sastry
, IFAC 2014
Burden,
Revzen
, &
Sastry
, 2013 (arXiv:1308.4158)Burden, Revzen, Moore, Sastry, & Full, SICB 2013Burden, Ohlsson, & Sastry, IFAC
SysID 2012
Slide2727
Technical assumption to enable scalable algorithm
Assume:
performance criteria
J
depends smoothly on final condition
x(T)
(i.e. derivative
dJ
/dx(T)
exists)
Optimization
of initial state x(0)