Hartmut Geyer, Howie Choset, Hannah Lynesshgeyer@cs.cmu.edu

Legged Robotics

Outline

Examples

Motivation

Design

Modeling

For more information: 16-868: Biomechanics and Motor Control, 16-665 Robot Mobility on Air, Land, and Sea

What are some examples of legs used in robotics?

Humanoids

Boston Dynamics, Atlas

Legged robots

ANYmal from the DARPA SubTChallenge

EPFL’s six legged robot

Prosthetic devices

iWalk BiOM

Vanderbilt Bionic Leg

[2010]

Exoskeletons

Elastic Band withForce Sensitive Resistor[Yamamoto et al. 2002]

EMG Signal Pickup[HAL, Cyberdyne]

Vukobratovic[1970s]

ReWalk 2012

State of knowledge about legged dynamics and control probably compares to 1900s in aerodynamics

1800 1850 1900 1950

(Cayley)

(Penaud)

(Wright)

(DC-3)

(F-86)

(X-1)

2015

(Wostok)

What are the benefits of legged robots?

Legged vehicles can overcome drastic obstacles

Boston Dynamics Atlas

Legged vehicles more seamlessly integrate into environments built for people

DARPA Robotics ChallengeBoston Dynamics Cheetah

Legged systems more closely resemble biological systems

Robugtix T8X

HULC Exoskeleton

Legged robot design considerations

Actuators used in Legged Mobility

pneumatic:

naturally complianthard to control

hydraulic:

very strongleakageoil pump

noise

electric:

quietrechargeable

batteries

Effect of Reflected Inertia in Geared Motors

Series Elastic Actuation

Belt DriveLinkageMotorBatteryLoad cell

zoom into knee actuator with laser-cut, custom torsional series springs

working principle

Series Elastic Actuation

HEBI X-Series actuator

Baxter robot

Legged Robot Modeling

Standing

Standing

y

x

m

ll

lf

Standing

y

x

mg

Standing

y

x

mg

Fn=mg

Standing

y

x

mg

mg

Standing

y

x

mg

θ

Standing – ankle strategy

y

x

mg

Fn

Standing – ankle strategy

y

x

mg

Fn

COP

Standing – ankle strategy

y

x

mg

Fn

τ ankle

Standing – ankle strategy

y

x

mg

Fn

τ ankleFl

Standing – ankle strategy

y

x

mg

Fn

τ ankleFl

Fτ

Fl

Standing?

y

x

mg

θ

For the ankle strategy, COP must be further from the ankle than projected COG, and COP is limited by foot length (polygon of support)

y

x

mg

θ

Fn

COPCOG’

Hip strategy or step strategy

https://www.researchgate.net/figure/The-fixed-support-strategies-the-ankle-and-hip-strategies-and-the-changeof-support-or_fig17_305223986

Walking

Walking – Inverted Pendulum Model (IPM)

y

x

m

ll

Walking – IPM – How far should I step so that I stop when I am at the apex of the step?

y

x

yf = llyi

vi

vf=0

xf=?

Walking – IPM – Conservation of energy

y

x

yf = llyi

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇

vi

vf=0

xf

Walking – IPM – Conservation of energy

y

x

yf = llyi

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒊𝒊 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒍𝒍𝒍𝒍

vi

vf=0

xf

Walking – IPM – Divide by mg

y

x

yf = llyi

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒊𝒊 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒍𝒍𝒍𝒍

vi

vf=0

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒍𝒍𝒍𝒍

xf

Walking – IPM – Substitute for l using Pythagorean

y

x

yf = llyi

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒊𝒊 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒍𝒍𝒍𝒍

vi

vf=0

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒍𝒍𝒍𝒍

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒙𝒙𝒇𝒇𝟐𝟐 + 𝒚𝒚𝒊𝒊𝟐𝟐

xf

Walking – IPM – Simplify

y

x

yf = llyi

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒊𝒊 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒍𝒍𝒍𝒍

vi

vf=0

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒍𝒍𝒍𝒍

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒙𝒙𝒇𝒇𝟐𝟐 + 𝒚𝒚𝒊𝒊𝟐𝟐

xf

𝟏𝟏𝟒𝟒𝒎𝒎𝟐𝟐

𝒗𝒗𝒊𝒊𝟒𝟒 +𝟏𝟏𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐𝒚𝒚𝒊𝒊 + 𝒚𝒚𝒊𝒊𝟐𝟐 = 𝒙𝒙𝒇𝒇𝟐𝟐 + 𝒚𝒚𝒊𝒊𝟐𝟐

𝒙𝒙𝒇𝒇 = 𝒗𝒗𝒊𝒊𝟏𝟏𝟒𝟒𝒎𝒎𝟐𝟐

𝒗𝒗𝒊𝒊𝟐𝟐 +𝟏𝟏𝒎𝒎𝒚𝒚𝒊𝒊

Walking – IPM – Simplify

y

x

yf = llyi

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒊𝒊 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒍𝒍𝒍𝒍

vi

vf=0

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒍𝒍𝒍𝒍

𝟏𝟏𝟐𝟐𝒎𝒎

𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒚𝒚𝒊𝒊 = 𝒙𝒙𝒇𝒇𝟐𝟐 + 𝒚𝒚𝒊𝒊𝟐𝟐

xf

𝟏𝟏𝟒𝟒𝒎𝒎𝟐𝟐

𝒗𝒗𝒊𝒊𝟒𝟒 +𝟏𝟏𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐𝒚𝒚𝒊𝒊 + 𝒚𝒚𝒊𝒊𝟐𝟐 = 𝒙𝒙𝒇𝒇𝟐𝟐 + 𝒚𝒚𝒊𝒊𝟐𝟐

𝒙𝒙𝒇𝒇 = 𝒗𝒗𝒊𝒊𝟏𝟏𝟒𝟒𝒎𝒎𝟐𝟐

𝒗𝒗𝒊𝒊𝟐𝟐 +𝟏𝟏𝒎𝒎𝒚𝒚𝒊𝒊

Capture point

Walking – Linear Inverted Pendulum Model (LIPM) for a single leg

y

x

ll (variable)

y0 (constant)

Walking – Linear Inverted Pendulum Model (LIPM) for a single leg

y

x

ll (variable)

y0 (constant)

Fy=mg

Fx = Fy/tanθ=Fy*x/y0

Fy

Fx

Fl θ

mg

Walking – LIPM – Capture Point – Conservation of energy

y

x

y0

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇 + 𝑾𝑾

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 + �

−𝒙𝒙𝒇𝒇

𝟎𝟎𝑭𝑭𝒙𝒙𝒅𝒅𝒙𝒙

vi vf=0

xf

Walking – LIPM – Capture Point – Integrate Fx

y

x

y0

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇 + 𝑾𝑾

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 + �

−𝒙𝒙𝒇𝒇

𝟎𝟎𝑭𝑭𝒙𝒙𝒅𝒅𝒙𝒙

vi vf=0

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 =

𝒎𝒎𝒎𝒎𝒙𝒙𝒇𝒇𝟐𝟐

𝟐𝟐𝒚𝒚𝟎𝟎

xf

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 = �

−𝒙𝒙𝒇𝒇

𝟎𝟎 𝒎𝒎𝒎𝒎𝒙𝒙𝒚𝒚𝟎𝟎

𝒅𝒅𝒙𝒙

Walking – LIPM – Capture Point – Simplify

y

x

y0

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇 + 𝑾𝑾

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 =

𝟏𝟏𝟐𝟐𝒎𝒎 ∗ 𝟎𝟎 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 + �

−𝒙𝒙𝒇𝒇

𝟎𝟎𝑭𝑭𝒙𝒙𝒅𝒅𝒙𝒙

vi vf=0

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 =

𝒎𝒎𝒎𝒎𝒙𝒙𝒇𝒇𝟐𝟐

𝟐𝟐𝒚𝒚𝟎𝟎

xf

𝒙𝒙𝒇𝒇 = 𝒗𝒗𝒊𝒊𝒚𝒚𝒐𝒐𝒎𝒎

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 = �

−𝒙𝒙𝒇𝒇

𝟎𝟎 𝒎𝒎𝒎𝒎𝒙𝒙𝒚𝒚𝟎𝟎

𝒅𝒅𝒙𝒙

Walking – LIPM – Arbitrary velocity (must be less than initial velocity)

y

x

y0

𝑲𝑲𝑲𝑲𝒊𝒊 + 𝑷𝑷𝑲𝑲𝒊𝒊 = 𝑲𝑲𝑲𝑲𝒇𝒇 + 𝑷𝑷𝑲𝑲𝒇𝒇 + 𝑾𝑾

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 =

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒇𝒇𝟐𝟐 + 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 + �

−𝒙𝒙𝒇𝒇

𝟎𝟎𝑭𝑭𝒙𝒙𝒅𝒅𝒙𝒙

vi vf

𝟏𝟏𝟐𝟐𝒎𝒎(𝒗𝒗𝒊𝒊𝟐𝟐−𝒗𝒗𝒇𝒇𝟐𝟐) =

𝒎𝒎𝒎𝒎𝒙𝒙𝒇𝒇𝟐𝟐

𝟐𝟐𝒚𝒚𝟎𝟎

xf

𝒙𝒙𝒇𝒇 =𝒚𝒚𝒐𝒐𝒎𝒎

(𝒗𝒗𝒊𝒊𝟐𝟐−𝒗𝒗𝒇𝒇𝟐𝟐)

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒊𝒊𝟐𝟐 −

𝟏𝟏𝟐𝟐𝒎𝒎𝒗𝒗𝒇𝒇𝟐𝟐 = �

−𝒙𝒙𝒇𝒇

𝟎𝟎 𝒎𝒎𝒎𝒎𝒙𝒙𝒚𝒚𝟎𝟎

𝒅𝒅𝒙𝒙

Speed changes and push recovery using (bipedal) linear inverted pendulum model

implemented on walking robot model

BLIPM speed control and push recovery

Running

Compliant legs can explain running dynamics,stiff legs cannot truly describe walking dynamics

A bipedal spring-mass model reveals that compliant leg behavior is fundamental to both run and walk

(right and left leg GRF)

Compliant legs integrate walking and running into large family of solutions to legged locomotion

(right and left leg GRF)

3.1 Classical ApproachesReference Trajectory Control SchemeZero Moment Point as Stability MeasureInfluence of Robot Motion on ZMPReference Tracking with ZMP StabilityWalking Pattern Generation

3.2 Optimization ApproachesCoM Dynamics Control by MPCInstantaneous QP Tracking desired CoM

3.3 Synthesizing Functional SubunitsRaibert Planar HopperControl SubunitsExtension to 3D BipedVirtual Model Control

Control – not covered in this class