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1R. Gregg and M. Spong CDC 2009, Shanghai, China 1

Reduction-Based Control of Branched Chains: Application to

3D Bipedal Torso Robots

Robert D. Gregg*Coordinated Science LaboratoryDepartment of Electrical and Computer EngineeringUniversity of Illinois at Urbana- Champaign

Mark W. Spong, DeanErik Jonsson School of Engineering and

Computer ScienceDepartment of Electrical Engineering

University of Texas at Dallas


Outline

1. Dynamic Walking Background

2. Symmetry and Reduction

3. Symmetries in Mechanical Systems

4. 3D Bipedal Walking

5. Closing Remarks


Human Bipedal Locomotion

Humanoid walking is:• dynamic, involving

“controlled falling”• completely 3D,

involving three planes-of-motion

• complex, involving a branching tree structure with many coupled DOF


Planar Compass-Gait Biped

• Many theoretical results consider a serial-chain biped constrained to the sagittal plane.

• Passive walking gaits (stable limit cycles) down shallow slopes [McGeer, 1990].

• Mapped to any slope using potential shaping [Spong & Bullo, 1995].


Decomposing Complex Motion• Most 3D bipeds do not

naturally have stable limit cycles for walking.

• Stable limit cycles may exist in the sagittal plane.

• Propose: Exploit symmetries to extend these gaits to 3D with reduction-based control, separating sagittal, lateral, and axial control problems.


Building Dynamic Gaits

[Gregg and Spong 2008-09]

• Goal: Method to construct straight-ahead and turning gaits for humanoid bipeds to quickly and efficiently navigate 3D space:


Lagrangian Mechanics

• Given config. space , a mechanical system is described by and Lagrangian function

• Integral curves satisfy E-L equations

n x ninertia/mass matrix

ddt∂L∂q̇ − ∂L∂q = u

L(q, q̇) = K(q, q̇)− V (q)

=1

2q̇TM(q)q̇ − V (q),

(q, q̇) ∈ TQQ

⇐⇒ M(q)q̈ + C(q, q̇)q̇ + ∂∂qV (q) = u


Example of Symmetry

• Letting , variable is cyclic if

• Lagrangian invariance under rotations of .• Generalized momentum

• Uncontrolled E-L equations show

• Conservation law: is constant.

0ddt

∂L∂q̇1− ∂L∂q1 =

ddtp1 = 0.

p1 = J1(q, q̇) :=∂L∂q̇1

p1

Q = Tn q1

S1 q1

∂L∂q1

= 0.


Symmetry-Based Reduction

• In Routhian reduction, a system with config. space has cyclic variables :

TQ

mod G

: phase space

momentum map surface

reduced phase space

J−1(μ) :

qi ∈ GiQ = G× S

pi = Ji(q, q̇) = μi

TS :

∂L∂qi

= 0


Divided Variables?

Likely to be unstable, e.g., yaw and lean for a biped…


Symmetry-Breaking Geometric Reduction

Stabilized cyclic coordinates

Controlled Reduction

Stabilized reduced subsystem

Ji(q, q̇) = λi(qi)Ji(q, q̇) 6= μi

Introduced in [Ames, Gregg, Wendel, Sastry, 2006]


Lagrangian Shaping• Lagrangian L with scalar cyclic and vector :

• Desire closed-loop system corresponding to an almost-cyclic Lagrangian:

where are special energy shaping terms based on momentum function

Kaugλ , Vaugλ

λ(q1).

q1

L(q2, q̇) = K(q2, q̇)− V (q2)

=1

2q̇TM(q2)q̇ − V (q2)

Lλ(q, q̇) = K(q2, q̇)− V (q2)+ Kaugλ (q, q̇2)− V augλ (q),

q2


Reduced Lagrangian

Given almost-cyclic Lagrangian

The Routhian (reduced Lagrangian):

J(q, q̇) = λ(q1)

Lred(q2, q̇2) = [Lλ(q, q̇)− λ(q1)q̇1]|J(q,q̇)=λ(q1)=

1

2q̇T2M2(q2)q̇2 − V (q2)

Lλ(q, q̇) =12

¡q̇1 q̇

T2

¢µ m1(q2) ??T M2(q2)

¶µq̇1q̇2

¶− V (q2)

+Kaugλ (q, q̇2)− Vaugλ (q)


Reduction Revisited

TQ: phase spacereduced phase spaceTS :

= λ(q1)

p1 = J1(q, q̇)

mod G

Lλ(q, q̇)

J−1(λ)

Lred(q2, q̇2)

(q1(t), q2(t), q̇1(t), q̇2(t)) ←→ (q2(t), q̇2(t))


Example: An matrix M is recursively cyclic in the first 3 coordinates if it has the form:

=

⎛⎝ m1(qn2 ) m12(qn2 ) M13(qn2 )m12(qn2 ) m2(qn3 ) M23(qn3 )MT13(q

n2 ) M

T23(q

n3 ) M3(q

n4 )

⎞⎠

Finding Symmetries

n× n

where qnj = (qj , qj+1, . . . , qn)T

(q2, . . . , qn)(q3, . . . , qn)(q4, . . . , qn)

M(q2, . . . , qn) =

µm1(qn2 ) M12(q

n2 )

MT12(qn2 ) M2(q

n3 )

¶


Extensive Symmetries

General case: An matrix M is recursively cyclic in k coordinates if it has the form:

n× n

for , where and qnn+1 = ∅.qnj = (qj , qj+1, . . . , qn)T

Cannot use one stage of reduction for all k variables!

Controlled reduction by stages.

M(q2, . . . , qn) =

⎛⎜⎜⎜⎝m1(qn2 ) –— M12(q

n2 ) –––

| . . ....

MT12(qn2 ) mk−1(q

nk ) Mk−1,k(q

nk )

| · · · MTk−1,k(qnk ) Mk(qnk+1)

⎞⎟⎟⎟⎠

k ≤ n


• For a branched chain, this holds for some .

• Related to kinetic energy invariance under rotations of inertial frame [Spong & Bullo, 1995].

Property of Mechanical Systems

z

xy

[Gregg and Spong, 2008]

Theorem: The inertia matrix of any n-DOF serialkinematic chain is recursively cyclic in n coordinates.

SO(3)

k ≤ n


• A hybrid control system has the form

• A hybrid system has no explicit input (e.g., closed-loop systems):

Hybrid Systems

H

HC :½ẋ = f(x) + g(x)u x ∈ D\Gx+ = ∆(x−) x− ∈ G

x∈ G ∆(x)

D

ẋ = f(x)

D = {x|h(x) ≥ 0} ⊆ TQ

x =

µqq̇

¶ P : G→ Gxj+2 = P 2(xj)δP 2 about x∗


Branched Biped Model

x00: θs

y0: ϕ

ZYX-Euler joint at the foot/ankle:

Yaw aboutRoll aboutPitch about

5-DOF config ,

with sagittal configθ = (θs, θt, θns)

T .

q = (ψ,ϕ, θT )T

z: ψ


Controlled Reduction by Stages

5-DOF 3D biped(no dynamic gaits)

5-DOF 3D biped with dynamic gaits

4-DOF 3D biped with dynamic gaits

3-DOF planar biped with dynamic gaits

energy shaping

Yaw

Lean

p1 = λ1(ψ) = −α1(ψ − ψ̄)

p2 = λ2(ϕ) = −α2ϕ


Straight-Ahead Gait

LES 2-periodic limit cycle along heading :ψ̄

x∗st = P 2st(x∗st)


Constant-Curvature Steering

• Constant steering angle induces LES periodic turning gaits modulo heading change:

s = ∆ψ̄

Hipless 4-DOF Hipless 5-DOF


CW-Turning Gait

LES 2-periodic limit cycle (mod yaw) for :s = π/14

modψ(x∗tu(s), s) = P 2tu(s)(x

∗tu(s))


Path Planning by Switching

Constant-curvaturewalking arcs

[Submitted ICRA10]

• A 3D biped is a discrete-time switched system where the switching signal from step-to- step determines the walking path:

σ(k)

xk+2 = P2σ(k)(xk)


Planned Walking Path


Locomotor Energeticscet = E/(mgd)


谢谢

(Thank You)!

Special thanks to Mark Spong, Tim Bretl, and Jessy Grizzle

Send comments to [email protected]


Select Publications•“Asymptotically Stable Gait Primitives for Planning Dynamic Bipedal Locomotion in Three Dimensions.” Gregg, Bretl, and Spong. Submitted to 2010 ICRA, Anchorage, AK.

•“Bringing the Compass-Gait Bipedal Walker to Three Dimensions.” Gregg and Spong. In 2009 IROS, St. Louis, MO.

•“Reduction-Based Control of 3D Bipedal Walking Robots.” Gregg and Spong. Int. J. of Robotics Research, Pre-print, Accepted 2009.

•“Reduction-based Control with Application to 3D Bipedal Walking Robots.” Gregg and Spong. In 2008 ACC, Seattle, WA.

•“A Geometric Approach to 3D Hipped Bipedal Robotic Walking.” Ames, Gregg, and Spong. In 2007 CDC, New Orleans, LA.

•“On the Geometric Reduction of Controlled 3-D Bipedal Walking Robots.” Ames, Gregg, Wendel, and Sastry. In the 2006 Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, Japan.


Reduction-Based Control

• Energy-shaping control

yields dynamics for controlled reduction.

• Does not cancel natural nonlinearities, only adds shaping terms.

• Recursively cyclic M allows multiple stages of controlled reduction!

u = C(q2, q̇)q̇ +N(q2)

+M(q2)Mλ(q)−1 (−Cλ(q, q̇)q̇ −Nλ(q) + v)


Mapping Branched Chains• Map branched chain to higher-

order redundant serial chain.

• Wrap around each radiating branch using redundant paths.

• Zero-mass redundant links, constrained redundant joints:

• Constrained redundant system (DAE) projects onto n-DOF dynamics of branched chain.

ir2 = π, ir6 = π, i

r4 = i5, i

r5 = i6


Branched Chain Symmetries

• An n-DOF branched chain is recursively cyclic down to the m-DOF submatrix, , of the irreducible tree structure.

• Proposition: The irreducible structure starts at the 2nd joint of the first non-simple branch.

reducible (gray) irreducible

1 ≤ m ≤ n

M(qn2 ) =

⎛⎜⎝ m1(qn2 ) M12(q

n2 )

. . ....

MT12(qn2 ) . . . M5(q

n5 )

⎞⎟⎠M5(q

n5 )

Reduction-Based Control of Branched Chains: Application to 3D Bipedal Torso RobotsOutlineHuman Bipedal LocomotionPlanar Compass-Gait BipedDecomposing Complex MotionBuilding Dynamic GaitsLagrangian MechanicsExample of SymmetrySymmetry-Based ReductionDivided Variables?Controlled ReductionLagrangian ShapingReduced LagrangianReduction RevisitedFinding SymmetriesExtensive SymmetriesProperty of Mechanical SystemsHybrid SystemsBranched Biped ModelControlled Reduction by StagesStraight-Ahead�GaitConstant-Curvature SteeringCW-Turning�GaitPath Planning by SwitchingPlanned Walking PathLocomotor Energetics谢谢 (Thank You)!Select PublicationsReduction-Based ControlMapping Branched ChainsBranched Chain Symmetries

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