6/26/2015
Pop Quiz for Tony
Bipedal Locomotion
on Small Feet
Jessy Grizzle
Elmer G. Gilbert Distinguished
University Professor
Levin Professor of Engineering
ECE and ME Departments
 Can you give the first name of either D’Alembert
or Lagrange?
 Jean le Rond d'Alembert and Joseph Louis Lagrange
 What is the air-speed velocity of an unladen
swallow?
 What do you mean? An African or European swallow?
 Whose birthday am I unable to celebrate today?
 My wife’s birthday!
Major Themes
Bipedal Locomotion
on Small Feet
Jessy Grizzle
Elmer G. Gilbert Distinguished
University Professor
Levin Professor of Engineering
ECE and ME Departments
From Simple  More Complex
 Walking may be easy for people, still
challenging for robots!
 Control of balance on small feet enables
agility and robustness on normal feet.
 The unreasonable effectiveness of
geometry + nonlinear control + parameter
optimization.
From Not So Simple  Very Complex
Bipedal robots @ Aaron Ames
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MABEL: Planar & Underactuated
Jonathan Hurst
Does the bar hold up the robot?
Control Algorithms
Large
Springs
K. Sreenath
H. W. Park
1m
65 Kg
Bipedal robots outdoors
J. Hurst: 3 ATRIAS robots built & delivered
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Ultimate Goal:
MARLO
Lateral leg motion
MARLO
Carnegie Mellon
Oregon State
Michigan
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13 DOF in SS and 6 Actuators
MARLO “Look ma, no bar”
Brian Buss
K. Akbari Hamed
A. Ramezani
Brent Griffin
Models are Hybrid
Models
• Hybrid
Lagrangian Dynamics
SS — Single Support
DS — Double Support
• Underactuated
Walking gait: alternating phases: SS, DS, SS, DS, …
Models are Hybrid
Models are Hybrid
Impact Dynamics
DS — Double Support
Walking gait: alternating phases: SS, DS, SS, DS, …
DS — Double Support
Walking gait: alternating phases: SS, DS, SS, DS, …
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Models are Hybrid
Models are Hybrid
Walking gait: alternating phases: SS, DS, SS, DS, …
SS
Models are Hybrid
SS
DS
Models are Hybrid
Hybrid System:
May have multiple phases
“Degree” of Actuation
joint angles
& velocities
motor torques
“Degree” of Actuation
joint angles
& velocities
motor torques
No longer fully actuated!
• Fully actuated: dim u = dim q
Common assumption, but dangerous.
• Fully actuated: dim u = dim q
severe limitations on ankle torque
(to prevent rotation of foot)
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6/26/2015
“Degree” of Actuation
State of the Art: DRC 6 & 7 June 2015
The bipeds in the following videos are
trying to stay fully actuated!
joint angles
& velocities
motor torques
No longer fully actuated!
• Fully actuated: dim u = dim q
severe limitations on ankle torque
(to prevent rotation of foot)
ZMP, M. Vukobratovic (1968)
Capture Point, Pratt et a. (2006)
State of the Art: DRC 6 & 7 June 2015
State of the Art: DRC 6 & 7 June 2015
“Degree” of Actuation
“Degree” of Actuation
joint angles
& velocities
• Fully actuated: dim u = dim q
• Underactuated: dim u < dim q
Passive
Pivot
motor torques
series
elastic
actuators
• Fully actuated: dim u = dim q
• Underactuated: dim u < dim q
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Periodic Orbit
Poincaré map (1854-1912)
Steady-state Walking  Periodic Solution
• How to find a periodic solution?
• How to check for stability?
Trajectory Tracking vs. Virtual Constraints
Feedback Design
• Virtual constraints
• Design via parameter optimization
• Initial 3D experiments
Physical  Virtual Constraints
Trajector
y
Gait
Generato
r
NL
PID
Cont.
Virtual constraints
y = (variables to be controlled) – (desired evolution)
Gait phase or timing variable
Feedback controller: y(t)0
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Virtual constraints
Virtual constraints
“holonomic constraint”
Gait phase or timing variable
Feedback controller: y(t)0
Virtual constraints
Virtual constraints
Virtual Constraints in ODE model
Switching Surface
Virtual constraints
Virtual constraints
Virtual Constraints in ODE model
Virtual Constraints in ODE model
Byrnes-Isidori Zero Dynamics (‘88)
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Geometric Interpretation
Geometric Interpretation
Hybrid Zero Dynamics
In general
Theorem: [TAC 2001, TAC 2009, Book]
Can design surface
such that
Hybrid zero dynamics
dim 2(N-k)
Hybrid zero dynamics
dim 2k
• Rendering Z hybrid invariant
• Rendering Z sufficiently rapidly attractive.
Hybrid zero dynamics
Design of the
Periodic Orbit
From invariance
reset map
is often
“expansive”
“Computation and Orbit Design”
Adjust feedback gains to achieve attractivity of Z
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Constrained Optimization (standard)
Desired Evolution
y = (variables to be controlled) – (desired evolution)
Design desired evolution
Cost Function
Integrate over
Hybrid Zero Dynamics
2(N-k) dim
Free parameters to be chosen
Constrained Optimization (standard)
Nonlinear-Hybrid Feedback Control
Design desired evolution
Cost Function
Equality Constraints
Inequality Constraints
•
•
•
•
• Swing leg impact at end of step
• Walking speed
• Periodicity or not
Ground reaction force positive
Friction coefficient < 0.6
Swing foot clearance
Torque bounds
Stability ???
Choosing What to Control?
Free Pivot
• For 1 DUA, STABILITY is INDEPENDENT of CHOICE of outputs!
angular
vs.
“Cartesian”
• Otherwise (such as in 3D), choice matters.
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3D: Output Choice Matters
Swing leg
3D: Output Choice Matters
Stance leg
Robot
Falls
Frontal Plane
Sagittal Plane
3D: Output Choice Matters
Swing leg
3D: Output Choice Matters
Stance leg
Eigenvalues now have
magnitude less than one
Robot
Walks
Frontal Plane
Sagittal Plane
3D: Output Choice Matters
Eigenvalues have
magnitude greater than one
Choosing What to Control
Systematic Selection of Outputs
Robot
Falls
Kaveh Akbari Hamed
Brian Buss
CDC 2014; ADHS 2015; IJRR [to appear]
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Systematic Virtual Constraint Selection
Systematic Virtual Constraint Selection
Systematic Selection of Outputs
• Parameterized family of controllers
Periodic Orbit
• Periodic orbit: independent of parameters
• Seek parameter values giving (exp.) stability
Systematic Virtual Constraint Selection
Orbit independent of parameter choice
but
changes the zero dynamics!
Systematic Virtual Constraint Selection
Applies Much More Broadly
Periodic Orbit
Feedforward term so that orbit
independent of parameter choice
Systematic Virtual Constraint Selection
Systematic Virtual Constraint Selection
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Systematic Virtual Constraint Selection
Systematic Virtual Constraint Selection
Seek parameter vector so that Jacobian is Hurwitz
Systematic Virtual Constraint Selection
Systematic Virtual Constraint Selection
Taylor Series Expansion
Taylor Series Expansion
Nominal parameter vector
Objective: Choose
such that
sum of matrices is Hurwitz
Choosing What to Control
Seek Lyapunov function and
satisfying
Bilinear Matrix Inequality
BMI Optimization Problem
Can be handled by the solver PENBMI.
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Systematic Virtual Constraint Selection
Systematic Virtual Constraint Selection
Giving new outputs that couple pitch and roll
in ways that we would not have found
through intuition
Giving new outputs that couple pitch and roll
in ways that we would not have found
through intuition
“standard”
“new”
Systematic Virtual Constraint Selection
Systematic Virtual Constraint Selection
Robot “falls” when we cut the power.
Can Include Disturbance Rejection
Goal
Outdoors
Rough Terrain
IJRR (to appear) & ADHS (submitted)
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Virtual Constraints and HZD
Westervelt
et al. 2004
Virtual Constraints and HZD
Martin
et al.
Chicago
Rehabilitation Institute
and
UT Dallas
Sreenath
et al. 2011
Robert D. Gregg et al.,
ICORR 2013
• Leg placement? No!
• ZMP? No!
Ames et al.
2015
Gregg et
al. 2014
Buss et
al. 2014
Vanderbilt leg being controlled with methods conceived
at Michigan for bipedal robots.
Virtual Constraints and HZD
Virtual Constraints and HZD
Robert Gregg
Aaron Ames (Ga Tech & TAMU)
Oregon State: Mikhail Jones and J. Hurst
Acknowledgments
University of Michigan
Brian Buss, Brent Griffin, Kaveh Akbari
Hamed, Ali Ramezani, Kevin Galloway,
Dennis Da, Ross Hartley, Omar Harib
Oregon State University
Jonathan Hurst, Dynamic Robotics
Laboratory, Jesse Grimes, Ross McCullough,
Soo-Hyun Yoos
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Concluding Remarks
 Great area for feedback control
 There is a lot going on …
 … and much more to do
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