On the Gait Robustness of Passive Dynamic Robots, and a Novel - - PowerPoint PPT Presentation

On the Gait Robustness of Passive Dynamic Robots, and a 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, and actuation 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

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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 and keep walking" Stability is often analyzed using the spectral radius of the Jacobian of the Poincare Map This is fine for stability But does not correlate well with robustness

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Prior Research

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

Source: “A Disturbance Rejection Measure for Limit Cycle Walkers: The Gait Sensitivity Norm" by D. Hobbelen, M. Wisse

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What metric does this thesis propose?

Any momentary disturbance can be represented as a change in generalized momenta We propose using the length of the smallest deterministic disturbance of generalized momenta that moves the system from the limit cycle to an unstable region. Above: Post-heelstrike instant generalized momenta. Red dot is limit cycle, green is basin of attraction, and yellow circle represents maximum acceptable disturbances.

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What does “length of the disturbance" mean? We present two definitions of gait robustness using two different 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 system with generalized momenta p is defined as (x∗, y∗) = arg min px − py2 , x ∈ QNR, y ∈ QLC ∆pIDR = x∗ − y∗ rIDR = min ∆pIDR2 , 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 not returning to the limit cycle. The notation (...)x means evaluated at a point x.

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Mathematical Definition of rEDR The “Energy Disturbance Radius” rEDR is defined as the change of kinetic energy resulting from an impulse disturbance: (x∗, y∗) = arg min(px − py)TM −1(px − py), x ∈ QNR, y ∈ QLC ∆pEDR = x∗ − y∗ rEDR = min ∆pT

EDRM −1∆pEDR,

Here M is the inertial matrix (tensor) of the Lagrangian

• system. Since M is a metric tensor of a Riemannian space

and p is a linear space, rEDR is a coordinate invariant

• quantity. We could also have written rEDR = ∆ ˙

qM∆ ˙ q if we wished to express rEDR in terms of generalized velocities.

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How do these correlate with real robustness to disturbances?

They appear to correlate well Slightly underestimate real robustness because it uses the worst-case disturbance

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Robustness Arc Foot Radius rIDR

2 and rEDR vs. Arc Foot Radius

rIDR

2 [x1000]

rEDR[x10]

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What are the advantages of rIDR and rEDR? rEDR is coordinate invariant Both have clear physical meanings: “How hard can you bump 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 for engineering and design

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What are the disadvantages of rIDR and rEDR? Very difficult to analytically determine rIDR or rEDR. However, we can measure it via simulation. For 2DOF models, computing rIDR or rEDR requires approximately 3 min. We will now introduce the simulator

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The Simulation Environment

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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, secant method zero finding Real-time debugging, compiling and interpretation of code via REPL

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The Simulation Environment

Automatically maps (p1, p2) plane, measures stability Yellow is stable

2 4 6 8 10 12 14 16 18 GEN-MOM-F GEN-MOM-S Cotangent Plane Map of biped-compass-a=0.520phi=03.000 0.1 0.2 0.3 0.4 0.5 0.6

• 1.4
• 1.2
• 1
• 0.8
• 0.6

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

0.2 0.4 0.6 0.8 1 1 2 3 4 5 Robustness and Step Length Slope φ[Degrees] rIDR

2, rEDR, and Step Length vs. Downhill Slope

rIDR

2 [x1000]

rEDR[x300] Step Length

1.5 deg 3.0 deg 4.5 deg Model

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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

1 2 3 4 5 0.05 0.1 0.15 0.2 Robustness rIDR

2 [x1000]

Forward Velocity (Froude Number Fr) rIDR

2 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 a great effect on robustness (e.g. hip springs) Larger feet are always beneficial Unlike what prior research has shown, torsos don’t always improve robustness There seems to be an optimal hip spring stiffness for a given forward speed

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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 mechanical gait robustness Next, we will present such a variable stiffness mechanism

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 Robustness Hip Spring Stiffness Khip rIDR

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

rIDR

2 [x1000]

rEDR[x200] 1 2 3 4 5 6 2 4 6 8 10 Robustness Hip Spring Stiffness Khip rIDR

2 and rEDR vs. Hip Spring Stiffness

rIDR

2 [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

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VSSEA photos

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Conclusions We have designed, implemented, and presented: Two new definitions of gait robustness: rIDR and rEDR, applicable to systems with/without control. Latter is coordinate 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 to maximize gait robustness

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Future Research Construct the proposed biped using the developed method Investigate which has better correlation to real-life robustness, rIDR or rEDR? Improve the engineering of the VSSEA to make it lighter, have less friction Consider using theory of manifolds and numerical

• ptimization 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 for efficiency, 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 shock tolerance, safer Weaknesses: Hard to analyze robustness to disturbances, hard to design 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 mostly engineering-related problems.

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Turning now to the other weaknesses of Passive Dynamic Bipeds We can now calculate robustness and design theoretical bipeds But what about the practical requirements of control and actuation? How do we actuate these robots without destroying their passive dynamics?

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Design Concept for a Biped Based on Passive-Dynamics Mechanical robustness can be examined separately by locking the motors Stiffness tunable to match forward speed Mechanical robustness reduces control complexity

A B C D E E

(b) (a)

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The Simulation Environment Automatically discovers limit cycle

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 D-THETA-F,D-THETA-S THETA-F,THETA-S State Map for biped-compass-a=0.5.secs-05 D-THETA-F D-THETA-S

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The Simulation Environment Automatically logs and plots various quantities

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 1 2 3 4 5 THETA-F,THETA-S,D-THETA-F,D-THETA-S TIME Time vs. State for biped-compass-a=0.5.secs-05 THETA-F THETA-S D-THETA-F D-THETA-S

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The Simulation Environment Plots limit cycle bifurcations

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Effect of Varying Lower Leg Length Greatly affects robustness. This is the only graph where rIDR and rEDR do not agree.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Robustness Froude Number Fr rIDR

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

rIDR

2 [x1000]

rEDR[x50] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.4 0.5 0.6 0.7 0.8 0.9 Robustness Lower Leg Length a rIDR

2 and rEDR vs. Lower Leg Length a

rIDR

2 [x1000]

rEDR[x50]

– p. 34

Effect of Mh Little effect on robustness.

0.5 1 1.5 2 2.5 3 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 Robustness Froude Number Fr rIDR

2 and rEDR vs. Fr for Increasing Hip Masses mH

rIDR

2 [x1000]

rEDR[x300] 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 Robustness Hip Mass mH rIDR

2 and rEDR vs. Hip Mass mH

rIDR

2 [x1000]

rEDR[x300]

– p. 35

Effect of khip Great effect on robustness, peaking behavior interesting.

1 2 3 4 5 6 7 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Robustness Froude Number Fr rIDR

2 and rEDR vs. Fr for Increasing Hip Spring Stiffnesses

rIDR

2 [x1000]

rEDR[x200] 1 2 3 4 5 6 2 4 6 8 10 Robustness Hip Spring Stiffness Khip rIDR

2 and rEDR vs. Hip Spring Stiffness

rIDR

2 [x1000]

rEDR[x200]

– p. 36

Effect of kankle Slightly unphysical, but improves robustness

0.5 1 1.5 2 2.5 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 Robustness Froude Number Fr rIDR

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

rIDR

2 [x1000]

rEDR[x200] 0.5 1 1.5 2 1 2 3 4 5 6 7 8 Robustness Ankle Spring Stiffness Kankle rIDR

2 and rEDR vs. Ankle Spring Stiffness

rIDR

2 [x1000]

rEDR[x200]

– p. 37

Effect of Arc Feet Increasing arc radius improves speed and robustness

10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Robustness Froude Number Fr rIDR

2 and rEDR vs. Fr for Increasing Arc Feet Radii

rIDR

2 [x1000]

rEDR[x200] 5 10 15 20 25 30 35 40 45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Robustness Arc Foot Radius r rIDR

2 and rEDR vs. Arc Feet Radius r

rIDR

2 [x1000]

rEDR[x200]

– p. 38

Effect of Torso Adding a torso made robot less robust

0.2 0.4 0.6 0.8 1 0.046 0.048 0.05 0.052 0.054 0.056 Robustness Froude Number Fr rIDR

2 and rEDR vs. Fr for Increasing Torso Lengths

rIDR

2 [x1000]

rEDR[x200] 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 Robustness Torso Length d rIDR

2 and rEDR vs. Torso Length d

rIDR

2 [x1000]

rEDR[x200]

– p. 39

What about varying more than one parameter? Let’s pick some design parameters randomly and evolve a biped

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 rIDR Froude Number rIDR vs. Froude Number for three Generations of Compass Bipeds A-B MH-M Gen1 Gen2 Gen3a-R Gen3b-K

– p. 40

Summary of rEDR Data

5 10 15 20 0.05 0.1 0.15 0.2 rEDR

2 [x300]

Fr rEDR vs. Froude Number for many types of robots Phi A-B MH-M Khip Kankle R Psi(r=2.0) D

– p. 41

VSSEA schematics

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VSSEA schematics

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VSSEA schematics

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The Simulation Environment ...or total kinetic energy.

– p. 45