Port-based Modeling and Control for Efficient Bipedal ... · Port-based Modeling and Control for Efﬁcient Bipedal Walking Machines Vincent Duindam [email protected] Control

Outline Introduction Modeling Control Conclusions

Port-based Modeling and Control forEfficient Bipedal Walking Machines

Vincent [email protected]

Control Laboratory, EE-Math-CSUniversity of Twente, Netherlands

Joint work with Stefano Stramigioli, Arjan van der SchaftGijs van Oort, Edwin Dertien, Martijn Visser, Frank Groen

CHESS Seminar @ UC Berkeley — Oct 31 / Nov 7, 2006 1


IntroductionTwente University and GeoplexEfficient walking robots

Models of Mechanisms in ContactMechanism dynamicsCompliant and rigid contact

Efficient Model-Based Control of Walking RobotsControl by mechanical optimizationZero-Energy Control of Walking RobotsDribbel – Experimental walking robot

Conclusions



Introduction



University of Twente (UT)

Founded in 1961Only Dutch university with a real campusTypical engineering depts, but also biomedical technology,environmental sciences, educational policy



Control Engineering @ UT

UT → Dept. EE-Math-CS → Control Engineering

6 scientific staff, 10 PhD students

Focus on mechatronics, robotics, and embedded systems

Favorite sport: ‘klootschieten’



Geoplex

Geometric Network Modeling and Control of Complex PhysicalSystems (www.geoplex.cc)

European FP5 project, 2002 – 2006

Consortium: 9 universities + 1 company

Goal: use port-Hamiltonian systems theory to solvemodeling and control problems across different domains

My research: apply Geoplex philosophy to walking robots



Port-Hamiltonian Systems

Developed by A.J. van der Schaft and B.M. Maschke (1992)Idea: study energy flow instead of information (signal) flow

b

k

m b

k

1m

∫∫

--

Collocated power variables effort (force) and flow (velocity)





b

k

1m

∫

∫ --






b

k

1m

∫

∫

power

power

power

--





Graphical language: bond graphs

Mathematical language: structured differential equations

ddt

x = (J(x) − R(x))∂H∂x

+ G(x)u

y = GT (x)∂H∂x

with J(x) skew-symm, R(x) pos-def, H(x) energy function

ddt

H = yT u −∂T H∂x

R∂H∂x

≤ yT u



Walking Robots

Active static walking: versatile,robust, use traditional controlmethods, but not very efficient

Passive dynamic walking:efficient, natural, but gener-ally not robust or versatile



Passive Dynamic Walking



Goal: explain and extend passive walkers

Modeling goals:

Build physical models of walking robots usingport-Hamiltonian techniques

Analyze models and explain walking behavior

Control goals:

Use natural passive motions as much as possible

Use adaptive passive elements to change natural motions

Augment passive walking with active control if needed

In this way, combine efficiency, versatility, and robustness.



Models of Mechanisms in Contact



Model setup

Build up models as power-port interconnection of subsystemsMechanism + ground contact + actuation + other partsContinuous-time model with switching and state jumpsImplemented in project software 20sim (www.20sim.com)



Dynamics of general rigid mechanisms

Standard result for dynamics of rigid mechanisms

M(q)q + C(q, q)q + G(q) = τ

with coordinates q(t) and velocities q(t) as dynamic states.

Allows only joint spaces isomorphic to Rn → singularities

Not suitable for more general joints, e.g. ball joints withconfiguration R(t) ∈ SO(3) and velocity ω(t) ∈ R3




Proposed solution:1 Parameterize joints with local and global coordinates2 Write down Boltzmann-Hamel equations in local

coordinates, parameterized by global coordinates3 Evaluate results at origin of local coordinates4 Result: similar equations with extra coupling on joint-level

p1

p2

u1

v1

u2

v2




Generalized dynamics equations

Works for all Lie groups (take exponential coordinates aslocal coordinates)

Can be directly extended to nonholonomic constraints(e.g. rolling-but-no-slipping)

Has been implemented into 20sim 3d mechanics toolbox

Paper still to be written, so comments are very welcome!



Contact modeling

Contact models consist of two parts

Kinematics: what are the closest points between bodies?Dynamics: what forces are involved, how do bodies react?

Compliant: finite compression and contact forcesRigid: limit-case compression and contact forces



Contact kinematics

Extension and ‘geometrification’ of David Montana’s equations

αf = M−1f

(

Kf + Ko)−1

([

−ωy

ωx

]

− Ko

[

vx

vy

])

αo = M−1o Rψ

(

Kf + Ko)−1

([

−ωy

ωx

]

+ Kf

[

vx

vy

])

ψ = ωz + Tf Mf αf + ToMoαo

0 = vz

Limitations with this formulationOnly suitable when bodies are in contactNot symmetric in two bodiesDepends heavily on surface parameterization



Contact kinematics

f

f−1 g

D ⊂ R2

S ⊂ E3 S2 ⊂ E3



Contact kinematics

p1p2

Ψ1Ψ2 ∆

Basic observationsContact points are separated by line along normal vectors

p1 + ∆g1 = H12 p2

Normal vectors are equal in direction but opposite

g1 = −H12 g2



Contact kinematics

p1 + ∆g1 = H12 p2

g1 = −H12 g2

Time-differentiation and merging gives contact kinematics(

g1∗ + H12 g2∗H

21 (I + ∆g1∗)

)

p1 = T 1,12 g1 + H1

2 g2∗

(

∆g2 − T 2,21 p2

)

(

g2∗ + H21 g1∗H

12 (I + ∆g2∗)

)

p2 = T 2,21 g2 + H2

1 g1∗

(

∆g1 − T 1,12 p1

)

Geometric equations, add surface coordinates separately



Compliant contact dynamics

First model of contact dynamics:

Assumption: forces generated by spring and damperImplementation:

Upon impact, connect spring/damper at contact pointsDuring contact, keep connected between these fixed pointsRelease contact when distance ∆ > 0

Model only realistic for small deformations (reasonable formost walking robots)

Simple submodel, can be used at different locations

Stiff, fast contact → Stiff differential equations



Compliant contact dynamics



Rigid contact dynamics

Simpler rigid dynamics through limiting operationAssumptions: impact and deformation phaseinstantaneous, no slipping, impulsive contact forcesImpact = discontinuity in velocity/momentum on impactFor walking robots, results mostly in instantaneous releaseof rear leg on impact of front legInteresting detail: release of contact not determined byforce direction but by acceleration direction

f1

f2 u



Efficient Model-Based Control ofWalking Robots



Finding natural walking gaits

How to find the most efficient walking gait?

Purely passive: Search space of initial conditions that givestable cycle (Poincare mappings)

Efficient (almost-passive): Search space of all trajectories?




Make problem solvable by polynomial approximation

Parameterize joint trajectories over one step as

q(t) = α0 + α1t + α2t2 + . . . + αntn

Force cyclic behavior by matching initial to final conditions.

Minimize torque requirements

minα,T

∫ T

0‖τ(t)‖2 dt

with τ expressed in terms of α using model equations.

Does not say anything about gait stability.




Gait search as minimization problemRecovers passive gait if availableOtherwise, gives ‘most efficient’ gaitCan be extended with other parameters, e.g. formechanical structure.

xs

xs

xm

xm

k0

m0

m0



Constructing natural walking gaits

Optimization of mechanical structure for variable speed walkingExample: compass-gait walker with inter-leg spring andvariable mass distribution

0.6 0.8 1.0 1.2

10

1

0.1

0.01

0.001

v (m/s)

J

0.6 0.8 1.0 1.2

3.0

2.0

1.0

0.0

4

0

8

12

16

v (m/s)

m

k



Constructing natural walking gaits

Optimization of mechanical structure for variable speed walkingExample: compass-gait walker with inter-leg spring andvariable mass distribution

0.6 0.8 1.0 1.2

10

1

0.1

0.01

0.001

v (m/s)

J

0.6 0.8 1.0 1.2

3.0

2.0

1.0

0.0

4

0

8

12

16

v (m/s)

m

k



Power-continuous tracking

Given efficient trajectory, how to follow it?Time-synchronization not important, onlyjoint-synchronizationAssume full actuation → n degrees of freedom1 degree of freedom for energy/speed control(e.g. potential-energy shaping)n − 1 degrees of freedom for curve-tracking(zero-energy, or, power-continuous control)

Related workP.Y. Li and R. Horowitz (1995–2001)PVFC: Passive Velocity-Field Control

q1(t)

q2(t)

q3(t)

t




Given efficient trajectory, how to follow it?Time-synchronization not important, onlyjoint-synchronizationAssume full actuation → n degrees of freedom1 degree of freedom for energy/speed control(e.g. potential-energy shaping)n − 1 degrees of freedom for curve-tracking(zero-energy, or, power-continuous control)

Related workP.Y. Li and R. Horowitz (1995–2001)PVFC: Passive Velocity-Field Control

q1(t)

q2(t)

q3(t) q1

q2

q3

t




Given efficient trajectory, how to follow it?

Time-synchronization not important, onlyjoint-synchronization

Assume full actuation → n degrees of freedom

1 degree of freedom for energy/speed control(e.g. potential-energy shaping)

n − 1 degrees of freedom for curve-tracking(zero-energy, or, power-continuous control)

Related work

P.Y. Li and R. Horowitz (1995–2001)PVFC: Passive Velocity-Field Control



Step 1: encode desired behavior

q(t)

qd (t)

q(t)

curve

Encode the desired behavior1 Extend desired curve to vector field2 Split kinetic energy in M(q)-orthogonal

good and bad directions3 Transform into good/bad coordinates

Result: kinetic energy is sum of two parts, onedescribing desired behavior, the other describingundesired behavior.




q(t)

q(t)

curve







q(t)

q(t)

wcurve







q(t)

q(t)

wcurve







Example (2d point mass in a circular field)

Point mass dynamics

ddt

q1

q2

p1

p2

=

0 0 1 00 0 0 1−1 0 0 00 −1 0 0

00p1mp2m

+

0 00 01 00 1

[

u1

u2

]

Representation of desired vector field

w(q) =

[

−q2

q1

]

−2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

q1

q2p/m





Coordinate transformation (only for momentum)[

p1

p2

]

=1|q|

[

−q2 q1

q1 q2

] [

α1

α2

]

Dynamics equations in transformed coordinates

ddt

q1

q2

α1

α2

=

0 0 − q2|q|

q1|q|

0 0 q1|q|

q2|q|

q2|q| − q1

|q| 0 − α1|q|

− q1|q| − q2

|q|α1|q| 0

00α1mα2m

+

0 00 0

− q2|q|

q1|q|

q1|q|

q2|q|

[

u1

u2

]

and the energy equals H(q, α) = H(α) = 12m α2

1 + 12m α2

2.





input G J

desired energy

undesired energy

12m α2

2

12m α2

1



Step 2: design controller

Step 2: derive the controller almost trivially1 Keep energy in separate bins2 Create uni-directional flow from bad to good3 Take care of the potential energy4 Design 1D speed/energy controller

power- continuousinterconnection

desired(good)energy

undesired(bad)

energy

u

y





power- continuousinterconnection

desired(good)energy

undesired(bad)

energy

decouplingcontroller

u

y





desired(good)energy

undesired(bad)

energy





desired(good)energy

undesired(bad)

energy

asympt.controller






Dribbel – experimental walking robotFeatures

First walking robot at the University of Twente

Planar six link mechanism

One actuator in hip joint

Electro-magnetic knee lock

Design and controller fully tested insimulation, setup worked on first try

Very simple control algorithm

Low energy consumption (6 W)

Starting point and testbed for newresearch projects



Dribbel – experimental walking robot



Dribbel – experimental walking robot



Conclusions



Conclusions

Physical models can help in rigorous design and analysis ofwalking robots, to go beyond physical intuition

Benefit of port-based approach in mechanics usually a matterof taste, but sometimes gives just right level of abstraction

Theoretical results still need to be tested in experiments!


Documents

Port-based Modeling and Control for Efficient Bipedal ... · Port-based Modeling and Control for Efﬁcient Bipedal Walking Machines Vincent Duindam [email protected] Control