On the Gait Robustness of Passive Dynamic Robots, anda Novel Variable Stiffness Series Elastic Actuator

February 15, 2008

Ivar Thorson

Nagoya University

– p. 1

Today’s Presentation

Outline:

1. Introduction to Passive-Dynamic Robots

2. Two Definitions of Gait Robustness

3. Custom Simulator

4. Data, Conclusions

5. Novel Variable Stiffness Actuator

6. Conclusions, Future Research

– p. 2

What is passive dynamic walking?

Stable limit cycle exists...but how stable?

Three hard problems: Desigining mechanics, controllers, andactuation

Today’s Questions:

1. どのように受動歩行ロボットのロバスト性を測るか

2. どのようなアクチュエータが受動歩行ロボットに適しているか

– p. 3

The difference between stability and robustness

Differential property vs. Disturbance Rejection:

Gait Stability means “It keeps walking."

Gait Robustness means “It can withstand this much disturbance andkeep walking"

Stability is often analyzed using the spectral radius of the Jacobian of thePoincare Map

This is fine for stability

But does not correlate well with robustness

– p. 4

Prior Research

“Real Robustness" is size of random disturbance per step such that itfalls in an average of 100 steps

Source: “A Disturbance Rejection Measure for Limit Cycle Walkers: The Gait Sensitivity

Norm" by D. Hobbelen, M. Wisse

– p. 5

What metric does this thesis propose?

Any momentary disturbance can be represented as a change ingeneralized momenta

We propose using the length of the smallest deterministicdisturbance of generalized momenta that moves the system from thelimit cycle to an unstable region.

Above: Post-heelstrike instant generalized momenta. Red dot is limitcycle, green is basin of attraction, and yellow circle represents maximumacceptable disturbances.

– p. 6

What does “length of the disturbance" mean?

We present two definitions of gait robustness using twodifferent definitions of length:

1. Length is measured as the magnitude of the impulse.(rIDR: Impulse Disturbance Rejection)

2. Length is measured using the metric tensor. (rEDR:Energy Disturbance Rejection)

– p. 7

Mathematical Definition of rIDR

The “Impulse Disturbance Rejection" radius rIDR of a systemwith generalized momenta p is defined as

(x∗, y∗) = arg min ‖px − py‖2, x ∈ QNR, y ∈ QLC

∆pIDR = x∗ − y∗

rIDR = min ‖∆pIDR‖2,

where Q is the configuration space of the system, QLC ⊆ Q

are states passed through during a circuit of the limit cycle,and QNR ⊆ Q are states which result in the system notreturning to the limit cycle. The notation (...)x meansevaluated at a point x.

– p. 8

Mathematical Definition of rEDR

The “Energy Disturbance Radius” rEDR is defined as thechange of kinetic energy resulting from an impulsedisturbance:

(x∗, y∗) = arg min(px − py)T M−1(px − py), x ∈ QNR, y ∈ QLC

∆pEDR = x∗ − y∗

rEDR = min ∆pTEDRM−1∆pEDR,

Here M is the inertial matrix (tensor) of the Lagrangiansystem. Since M is a metric tensor of a Riemannian spaceand p is a linear space, rEDR is a coordinate invariantquantity. We could also have written rEDR = ∆q̇M∆q̇ if wewished to express rEDR in terms of generalized velocities.

– p. 9

How do these correlate with real robustness to disturbances?

They appear to correlate well

Slightly underestimate real robustness because it uses theworst-case disturbance

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Arc Foot Radius

rIDR2 and rEDR vs. Arc Foot Radius

rIDR2 [x1000]rEDR[x10]

– p. 10

What are the advantages ofrIDR and rEDR?

rEDR is coordinate invariant

Both have clear physical meanings: “How hard can youbump the robot before it falls down?"

rIDR measures change in momentum from bump

rEDR measures change in kinetic energy from bump

Deterministic, not stochastic

Conservative, worst-case values that are useful forengineering and design

– p. 11

What are the disadvantages ofrIDR and rEDR?

Very difficult to analytically determine rIDR or rEDR.

However, we can measure it via simulation.

For 2DOF models, computing rIDR or rEDR requiresapproximately 3 min.

We will now introduce the simulator

– p. 12

The Simulation Environment

– p. 13

Custom Rigid Body Simulator

Simulator Features:

Simulates multiple robots simultaneously

Robots can be edited in real time

Measures rIDR, rEDR automatically

Draws PNG files of robots

Easy saving and loading of various parameters, models

Automatically logs and plots various quantities

Automatically discovers limit cycle

Plots limit cycle bifurcations

Fast simulation using Runge-Kutta numerical integration, secantmethod zero finding

Real-time debugging, compiling and interpretation of code via REPL– p. 14

The Simulation Environment

Automatically maps (p1, p2) plane, measures stability

Yellow is stable

0 2 4 6 8 10 12 14 16 18

GEN-MOM-F

GE

N-M

OM

-S

Cotangent Plane Map of biped-compass-a=0.520phi=03.000

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-1.4

-1.2

-1

-0.8

-0.6

– p. 15

Experiment: How do rIDR and rEDR vary with the slope?

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and

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Slope φ[Degrees]

rIDR2, rEDR, and Step Length vs. Downhill Slope

rIDR2 [x1000]

rEDR[x300]Step Length

1.5 deg 3.0 deg 4.5 deg Model– p. 16

What about using these metrics to design a robot?

Let’s consider the effects of several design parameters!

– p. 17

Summary of rIDR Data

ロバスト性　vs. 無次元化した歩行速度

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r IDR

2 [x10

00]

Forward Velocity (Froude Number Fr)

rIDR2 vs. Froude Number for many types of robots

Slope

Lower Leg Length

Hip Mass

Hip Spring

Ankle Spring

Arc Foot

Forward Foot

Torso

– p. 18

Conclusions from Data

We can make some simple conclusions

Parameters which affect natural leg swing period have agreat effect on robustness (e.g. hip springs)

Larger feet are always beneficial

Unlike what prior research has shown, torsos don’talways improve robustness

There seems to be an optimal hip spring stiffness for agiven forward speed

– p. 19

Is there an optimal hip spring stiffness for a given forward speed?

For different slopes, optimally robust hip spring stiffness are different

If we could change stiffness, we could maximize natural mechanicalgait robustness

Next, we will present such a variable stiffness mechanism

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Hip Spring Stiffness Khip

rIDR2 and rEDR vs. Hip Spring Stiffness (on a 1.0 degree slope)

rIDR2 [x1000]

rEDR[x200]

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Hip Spring Stiffness Khip

rIDR2 and rEDR vs. Hip Spring Stiffness

rIDR2 [x1000]

rEDR[x200]

1.0 deg斜面 3.0 deg斜面

– p. 20

Introducing a New Actuator: The VSSEA

VSSEA: Variable Stiffness Series Elastic Actuator

An actuatoor which does not destroy passive dynamic behavior

Two nonlinear springs act as variable linear spring

Two motors, (A) adjusts position, (D) adjusts stiffness

– p. 21

VSSEA photos

– p. 22

Conclusions

We have designed, implemented, and presented:

Two new definitions of gait robustness: rIDR and rEDR,applicable to systems with/without control. Latter iscoordinate invariant.

A new simulator to measure these quantities

A new actuator which can

Provide power without overwhelming natural dynamics

Adapt its stiffness to an operating environment tomaximize gait robustness

– p. 23

Future Research

Construct the proposed biped using the developedmethod

Investigate which has better correlation to real-liferobustness, rIDR or rEDR?

Improve the engineering of the VSSEA to make it lighter,have less friction

Consider using theory of manifolds and numericaloptimization to design controllers for these bipeds

– p. 24

Questions?

– p. 25

What is the state of the art for bipedal　robots?

Stiffly-actuated, position-controlled robots

Strengths: General method, easily understood

Weaknesses: Trying to constraining position via control is bad forefficiency, poor shock tolerance, dangerous, can’t run.

Asimo (Honda) HRP-2 (AIST) QRIO (Sony)– p. 26

How can we improve on existing robots?

We advocate basing robots on passive dynamic walking and running

Strengths: Energy efficiency, natural looking motion, good shocktolerance, safer

Weaknesses: Hard to analyze robustness to disturbances, hard todesign controllers, hard to actuate. (Can we solve these problems?)

Cornell Biped Monopod-II (McGill) Denise (Delft U.)– p. 27

How much more efficient are Passive Dynamic robots?

Cost of Transport: ct = energy

weight·distance.

cetはバッテリーあるいはmetabolicの消費したエネルギー, cmtは機械的な

仕事

Name Mfg cet cmt Passive-Dynamic?

Asimo Honda 3.2 1.6 no

Denise Delft 5.3 0.08 yes

Monopod II McGill 0.22 - yes

Cornell Biped Cornell 0.20 0.055 yes

Human Walking God 0.20 0.05 -

Dynamite McGeer - 0.04 yes

Reasons for high cet of passive-dynamic robots are thought to be mostlyengineering-related problems. – p. 28

Turning now to the other weaknesses of Passive Dynamic Bipeds

We can now calculate robustness and design theoreticalbipeds

But what about the practical requirements of control andactuation?

How do we actuate these robots without destroying theirpassive dynamics?

– p. 29

Design Concept for a Biped Based on Passive-Dynamics

Mechanical robustness can be examined separately bylocking the motors

Stiffness tunable to match forward speed

Mechanical robustness reduces control complexity

A

B

C

D

E

E

(b)(a)

– p. 30

The Simulation Environment

Automatically discovers limit cycle

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A-S

THETA-F,THETA-S

State Map for biped-compass-a=0.5.secs-05

D-THETA-FD-THETA-S

– p. 31

The Simulation Environment

Automatically logs and plots various quantities

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-1.5

-1

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0.5

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1.5

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TH

ET

A-F

,TH

ET

A-S

,D-T

HE

TA

-F,D

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ET

A-S

TIME

Time vs. State for biped-compass-a=0.5.secs-05

THETA-FTHETA-S

D-THETA-FD-THETA-S

– p. 32

The Simulation Environment

Plots limit cycle bifurcations

– p. 33

Effect of Varying Lower Leg Length

Greatly affects robustness. This is the only graph where rIDR

and rEDR do not agree.

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Froude Number Fr

rIDR2 and rEDR vs. Fr for Increasing Lower Leg Length a

rIDR2 [x1000]rEDR[x50]

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obus

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sLower Leg Length a

rIDR2 and rEDR vs. Lower Leg Length a

rIDR2 [x1000]rEDR[x50]

– p. 34

Effect of Mh

Little effect on robustness.

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Froude Number Fr

rIDR2 and rEDR vs. Fr for Increasing Hip Masses mH

rIDR2 [x1000]

rEDR[x300]

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Hip Mass mH

rIDR2 and rEDR vs. Hip Mass mH

rIDR2 [x1000]

rEDR[x300]

– p. 35

Effect of khip

Great effect on robustness, peaking behavior interesting.

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Froude Number Fr

rIDR2 and rEDR vs. Fr for Increasing Hip Spring Stiffnesses

rIDR2 [x1000]

rEDR[x200]

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Hip Spring Stiffness Khip

rIDR2 and rEDR vs. Hip Spring Stiffness

rIDR2 [x1000]

rEDR[x200]

– p. 36

Effect of kankle

Slightly unphysical, but improves robustness

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Froude Number Fr

rIDR2 and rEDR vs. Fr for Increasing Ankle Spring Stiffnesses kankle

rIDR2 [x1000]

rEDR[x200]

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Ankle Spring Stiffness Kankle

rIDR2 and rEDR vs. Ankle Spring Stiffness

rIDR2 [x1000]

rEDR[x200]

– p. 37

Effect of Arc Feet

Increasing arc radius improves speed and robustness

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Froude Number Fr

rIDR2 and rEDR vs. Fr for Increasing Arc Feet Radii

rIDR2 [x1000]

rEDR[x200]

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Arc Foot Radius r

rIDR2 and rEDR vs. Arc Feet Radius r

rIDR2 [x1000]

rEDR[x200]

– p. 38

Effect of Torso

Adding a torso made robot less robust

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Froude Number Fr

rIDR2 and rEDR vs. Fr for Increasing Torso Lengths

rIDR2 [x1000]

rEDR[x200]

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Torso Length d

rIDR2 and rEDR vs. Torso Length d

rIDR2 [x1000]

rEDR[x200]

– p. 39

What about varying more than one parameter?

Let’s pick some design parameters randomly and evolve abiped

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rIDR vs. Froude Number for three Generations of Compass Bipeds

A-BMH-MGen1Gen2

Gen3a-RGen3b-K

– p. 40

Summary of rEDR Data

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r ED

R2 [x

300]

Fr

rEDR vs. Froude Number for many types of robots

PhiA-B

MH-MKhip

KankleR

Psi(r=2.0)D

– p. 41

VSSEA schematics

– p. 42

VSSEA schematics

– p. 43

VSSEA schematics

– p. 44

The Simulation Environment

...or total kinetic energy.

– p. 45