legged robotics16311/current/schedule/ppp/...outline examples motivation design modeling for more...
TRANSCRIPT
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