Legged

Hartmut Geyer, Howie Choset, Hannah Lyness [email protected] Outline

Examples

Motivation

Design

Modeling

For more information: 16-868: Biomechanics and Motor Control, 16-665 Mobility on Air, Land, and Sea What are some examples of legs used in robotics? Humanoids

Boston Dynamics, Legged

EPFL’s six legged robot ANYmal from the DARPA SubT Challenge Prosthetic devices

Vanderbilt Bionic Leg [2010] iWalk BiOM Exoskeletons

Elastic Band with EMG Signal Pickup ReWalk 2012 Force Sensitive Resistor [HAL, Cyberdyne]

Vukobratovic [Yamamoto et al. 2002] [1970s] State of knowledge about legged dynamics and control probably compares to 1900s in aerodynamics

(Wright) (F-86) (Wostok)

(Cayley)

(DC-3) (Penaud) (X-1)

1800 1850 1900 1950

2015 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

Boston Dynamics Cheetah DARPA Robotics Challenge Legged systems more closely resemble biological systems

Robugtix T8X

HULC Exoskeleton Legged robot design considerations Actuators used in Legged Mobility

pneumatic: hydraulic: electric:

naturally compliant very strong quiet hard to control leakage rechargeable oil pump batteries noise Effect of Reflected Inertia in Geared Motors Series Elastic Actuation

Belt Drive Linkage Motor Battery Load 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

m

ll

x

lf Standing

y

mg

x Standing

y

mg

x

Fn=mg Standing

y

mg

mg

x Standing

y

mg

θ x Standing – ankle strategy

y

mg

x

Fn Standing – ankle strategy

y

mg

x

Fn

COP Standing – ankle strategy

y

mg

τ ankle x

Fn Standing – ankle strategy

y

mg

τ ankle Fl x

Fn Standing – ankle strategy

y Fl

Fτ

mg

τ ankle Fl x

Fn Standing?

y

mg

θ x 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

mg

θ x

COG’ COP

Fn 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

m

ll

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

y

vf=0

vi yf = ll

yi

x

xf=? Walking – IPM – Conservation of energy

+ = + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 vf=0

vi yf = ll

yi

x

xf Walking – IPM – Conservation of energy

+ = + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 vf=0 + = + 𝟐𝟐 𝟏𝟏 𝒊𝒊 𝒊𝒊 𝟏𝟏 𝒍𝒍 vi 𝒎𝒎𝒗𝒗 𝒎𝒎𝒎𝒎𝒚𝒚 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒍𝒍 yf = ll 𝟐𝟐 𝟐𝟐

yi

x

xf Walking – IPM – Divide by mg

+ = + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 vf=0 + = + 𝟐𝟐 𝟏𝟏 𝒊𝒊 𝒊𝒊 𝟏𝟏 𝒍𝒍 vi 𝒎𝒎𝒗𝒗 𝒎𝒎𝒎𝒎𝒚𝒚 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒍𝒍 yf = ll 𝟐𝟐 + 𝟐𝟐 = yi 𝟏𝟏 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒍𝒍𝒍𝒍 𝟐𝟐𝟐𝟐

x

xf Walking – IPM – Substitute for l using Pythagorean

+ = + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 vf=0 + = + 𝟐𝟐 𝟏𝟏 𝒊𝒊 𝒊𝒊 𝟏𝟏 𝒍𝒍 vi 𝒎𝒎𝒗𝒗 𝒎𝒎𝒎𝒎𝒚𝒚 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒍𝒍 yf = ll 𝟐𝟐 + 𝟐𝟐 = yi 𝟏𝟏 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒍𝒍𝒍𝒍 𝟐𝟐𝟐𝟐+ = + 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒙𝒙𝒇𝒇 𝒚𝒚𝒊𝒊 𝟐𝟐𝟐𝟐

x

xf Walking – IPM – Simplify

+ = + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 vf=0 + = + 𝟐𝟐 𝟏𝟏 𝒊𝒊 𝒊𝒊 𝟏𝟏 𝒍𝒍 vi 𝒎𝒎𝒗𝒗 𝒎𝒎𝒎𝒎𝒚𝒚 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒍𝒍 yf = ll 𝟐𝟐 + 𝟐𝟐 = yi 𝟏𝟏 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒍𝒍𝒍𝒍 𝟐𝟐𝟐𝟐+ = + 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒙𝒙𝒇𝒇 𝒚𝒚𝒊𝒊 𝟐𝟐𝟐𝟐 + + = + 𝟏𝟏 𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒚𝒚𝒊𝒊 𝒙𝒙𝒇𝒇 𝒚𝒚𝒊𝒊 𝟒𝟒𝒈𝒈 𝒈𝒈 = + x 𝟏𝟏 𝟐𝟐 𝟏𝟏 𝒙𝒙𝒇𝒇 𝒗𝒗𝒊𝒊 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝟒𝟒𝒈𝒈 𝒈𝒈

xf Walking – IPM – Simplify

+ = + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 vf=0 + = + 𝟐𝟐 𝟏𝟏 𝒊𝒊 𝒊𝒊 𝟏𝟏 𝒍𝒍 vi 𝒎𝒎𝒗𝒗 𝒎𝒎𝒎𝒎𝒚𝒚 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒍𝒍 yf = ll 𝟐𝟐 + 𝟐𝟐 = yi 𝟏𝟏 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒍𝒍𝒍𝒍 𝟐𝟐𝟐𝟐+ = + 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒙𝒙𝒇𝒇 𝒚𝒚𝒊𝒊 𝟐𝟐𝟐𝟐 + + = + 𝟏𝟏 𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝒚𝒚𝒊𝒊 𝒙𝒙𝒇𝒇 𝒚𝒚𝒊𝒊 𝟒𝟒𝒈𝒈 𝒈𝒈 = + x 𝟏𝟏 𝟐𝟐 𝟏𝟏 𝒙𝒙𝒇𝒇 𝒗𝒗𝒊𝒊 𝟐𝟐 𝒗𝒗𝒊𝒊 𝒚𝒚𝒊𝒊 𝟒𝟒𝒈𝒈 𝒈𝒈 Capture point xf Walking – Linear Inverted Pendulum Model (LIPM) for a single leg

y

y0 (constant)

ll (variable)

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

y

Fy Fy=mg y0 (constant)

Fx

mg Fx = Fy/tanθ=Fy*x/y0 ll (variable)

Fl θ x Walking – LIPM – Capture Point – Conservation of energy

+ = + + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 𝑾𝑾 + = + + 𝟎𝟎 𝟐𝟐 𝟏𝟏 𝒊𝒊 𝒐𝒐 𝟏𝟏 𝒐𝒐 𝒙𝒙 𝒎𝒎𝒗𝒗 𝒎𝒎𝒎𝒎𝒚𝒚 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒚𝒚 �−𝒙𝒙𝒙𝒙𝑭𝑭 𝒅𝒅𝒅𝒅 vi 𝟐𝟐vf=0 𝟐𝟐 y0

x

xf Walking – LIPM – Capture Point – Integrate Fx

+ = + + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 𝑾𝑾 + = + + 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟏𝟏 𝒎𝒎𝒗𝒗𝒊𝒊 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 � 𝑭𝑭𝒙𝒙𝒅𝒅𝒅𝒅 𝟐𝟐 𝟐𝟐 −𝒙𝒙𝒙𝒙 vi vf=0 = 𝟎𝟎 y0 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒎𝒎𝒎𝒎 𝒎𝒎𝒗𝒗𝒊𝒊 � 𝒅𝒅𝒅𝒅 𝟐𝟐 −𝒙𝒙𝒙𝒙 𝒚𝒚𝟎𝟎

= 𝟐𝟐 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒎𝒎𝒙𝒙𝒇𝒇 𝒎𝒎𝒗𝒗𝒊𝒊 𝟐𝟐 𝟐𝟐𝒚𝒚𝟎𝟎

x

xf Walking – LIPM – Capture Point – Simplify

+ = + + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 𝑾𝑾 + = + + 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟏𝟏 𝒎𝒎𝒗𝒗𝒊𝒊 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 𝒎𝒎 ∗ 𝟎𝟎 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 � 𝑭𝑭𝒙𝒙𝒅𝒅𝒅𝒅 𝟐𝟐 𝟐𝟐 −𝒙𝒙𝒙𝒙 vi vf=0 = 𝟎𝟎 y0 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒎𝒎𝒎𝒎 𝒎𝒎𝒗𝒗𝒊𝒊 � 𝒅𝒅𝒅𝒅 𝟐𝟐 −𝒙𝒙𝒙𝒙 𝒚𝒚𝟎𝟎

= 𝟐𝟐 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒎𝒎𝒙𝒙𝒇𝒇 𝒎𝒎𝒗𝒗𝒊𝒊 𝟐𝟐 𝟐𝟐𝒚𝒚𝟎𝟎 = 𝒚𝒚𝒐𝒐 𝒙𝒙𝒇𝒇 𝒗𝒗𝒊𝒊 𝒈𝒈 x

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

+ = + + y 𝑲𝑲𝑲𝑲𝒊𝒊 𝑷𝑷𝑷𝑷𝒊𝒊 𝑲𝑲𝑲𝑲𝒇𝒇 𝑷𝑷𝑷𝑷𝒇𝒇 𝑾𝑾 + = + + 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒗𝒗𝒊𝒊 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 𝒎𝒎𝒗𝒗𝒇𝒇 𝒎𝒎𝒎𝒎𝒚𝒚𝒐𝒐 � 𝑭𝑭𝒙𝒙𝒅𝒅𝒅𝒅 𝟐𝟐 𝟐𝟐 −𝒙𝒙𝒙𝒙 vi vf = 𝟎𝟎 y0 𝟏𝟏 𝟐𝟐 𝟏𝟏 𝟐𝟐 𝒎𝒎𝒎𝒎𝒎𝒎 𝒎𝒎𝒗𝒗𝒊𝒊 − 𝒎𝒎𝒗𝒗𝒇𝒇 � 𝒅𝒅𝒅𝒅 𝟐𝟐 𝟐𝟐 −𝒙𝒙𝒙𝒙 𝒚𝒚𝟎𝟎 ( ) = 𝟐𝟐 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝒎𝒎𝒎𝒎𝒙𝒙𝒇𝒇 𝒎𝒎 𝒗𝒗𝒊𝒊 −𝒗𝒗𝒇𝒇 𝟐𝟐 𝟐𝟐𝒚𝒚𝟎𝟎 = ( ) 𝒚𝒚𝒐𝒐 𝟐𝟐 𝟐𝟐 𝒙𝒙𝒇𝒇 𝒗𝒗𝒊𝒊 −𝒗𝒗𝒇𝒇 𝒈𝒈 x

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

BLIPM speed control and push recovery

implemented on walking robot model 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) Control – not covered in this class

3.1 Classical Approaches Reference Trajectory Control Scheme Zero Moment Point as Stability Measure Influence of Robot Motion on ZMP Reference Tracking with ZMP Stability Walking Pattern Generation

3.2 Optimization Approaches CoM Dynamics Control by MPC Instantaneous QP Tracking desired CoM

3.3 Synthesizing Functional Subunits Raibert Planar Hopper Control Subunits Extension to 3D Biped Virtual Model Control