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Control of Full Body Humanoid Push Recovery Using Simple Models
Benjamin StephensThesis Proposal
Carnegie Mellon, Robotics Institute
November 23, 2009
Committee:Chris Atkeson (chair)
Jessica HodginsHartmut Geyer
Jerry Pratt (IHMC)
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Thesis Proposal Overview
Simple models can be used to simplify control of full-body push recovery for complex robots
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x
Lx
y
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LzF
RzF
LyF LxF
xy
z
LyRx
Ry
refp
fp
0fp
Strategy decisions and optimization over future
actions
Simple approximate
dynamics model with COM and two
feet
Reactive full-body force
control
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Motivations
• Improve the performance and usefulness of complex robots, simplifying controller design by focusing on simpler models that capture important features of the desired behavior
•Enabling dynamic robots to interact safely with people in everyday uncertain environments
•Modeling human balance sensing, planning and motor control to help people with balance disabilities
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Approaches to Humanoid Balance
Controls Complex
Robot
Utilizes Simple
Model(s)
Reactive to
Pushes
Optimizes Over the Future
ZMP Preview Control S. Kajita, et.al., ‘03
Reflexive ControlPratt, ‘98Yin, et. al., ’07Geyer ‘09
Passive Dynamic WalkingMcGeer ’90
Inverse-Dynamics-Based ControlHyon, et. al., ’07Sentis, ‘07
Proposed Work
Exa
mp
les
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Expected Contributions
•Analytically-derived bounds on balance stability defining unique recovery strategies
•Optimal control framework for planning step recovery and other behaviors involving balance
•Transfer of dynamic balance behaviors designed for simple models to complex humanoid through force control
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Outline
•Simple Models of Biped Balance
•Push Recovery Strategies
•Optimal Control Framework
•Humanoid Robot Control
•Proposed Work and Timeline
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fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Outline
•Simple Models of Biped Balance
•Push Recovery Strategies
•Optimal Control Framework
•Humanoid Robot Control
•Proposed Work and Timeline
refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Outline
•Simple Models of Biped Balance
•Push Recovery Strategies
•Optimal Control Framework
•Humanoid Robot Control
•Proposed Work and Timeline
refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Outline
•Simple Models of Biped Balance
•Push Recovery Strategies
•Optimal Control Framework
•Humanoid Robot Control
•Proposed Work and Timeline
refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Outline
•Simple Models of Biped Balance
•Push Recovery Strategies
•Optimal Control Framework
•Humanoid Robot Control
•Proposed Work and Timeline
refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Outline
•Simple Models of Biped Balance
•Push Recovery Strategies
•Optimal Control Framework
•Humanoid Robot Control
•Proposed Work and Timeline
refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Simple ModelsVery simple dynamic models approximate full body motion
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fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
•The sum of forces on the COM results in an acceleration of the COM
Simple Biped Dynamics
14
gF
RF LF
PF
gF
RF
LF
P~P Center of mass (COM)
PF
RP LP~, LR PP Foot locations
gPi FPmFF
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
•The COP is the origin point on the ground of the force that is equivalent to the contact forces
Simple Biped Dynamics
15
gF
PF
gF
P
eqF
CP
eqF
~CP Center of pressure (COP)
PF
RP LP
gPi FPmFF
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
•Ground torques can be used to move the COP or apply moments to the COM
Simple Biped Dynamics
16
gF
PF
gF
PF
PCP
eqF
HMFPP iii
~H Angular momentum
eqF
RP LP
gPi FPmFF
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
•The base of support defines the limits of the COP and, consequently, the maximumforce on the COM
Simple Biped Dynamics
17
gF
PF
gF
P
eqF
CP
gPi FPmFF PF
eqF
RP LP
HMFPP iii
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
•Instantaneous 3D biped dynamics form a linear system in contact forces.
Simple Biped Dynamics
18
gF
PF
PCP
eqF
RP LP
H
FPm
M
F
M
F
IPPIPP
II g
L
L
R
R
LR00
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Simple Biped Inverse Dynamics
•The contact forces can be solved for generally using constrained quadratic programming
WFFbAFbAFF TT
F minarg
dCF
Least squares problem
(quadratic programming)Linear Inequality Constraints•COP under the
feet•Friction
H
FPm
M
F
M
F
IPPIPP
II g
L
L
R
R
LR00
bAF
19
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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3D Linear Biped Model
•The Linear Biped Model is a special case derived by making a few additional assumptions:▫Zero vertical acceleration▫Sum of moments about COM is zero▫Forces/moments are distributed linearly
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F
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0
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P
P
LLRR
FPm
M
F
IPPPP
I
REFERENCE:Stephens, “3D Linear Biped Model for Dynamic Humanoid Balance,” Submitted to ICRA 2010
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Linear Double Support Region
•Using a fixed double support-phase transition policy, the weights can be defined by linear functions
Ry
Rx
Ly
Lx
y
x
yx
11
y
x
Ry
x
Rotated Coordinate Frame
D
yDL 2
D
yDR 2
Linear Weighting Functions
D2
REFERENCE:Stephens, “Modeling and Control of Periodic Humanoid
Balance using the Linear Biped Model,” Humanoids 2009
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Using Linear Biped Model
•Analytic solution of contact forces and phase transition allows for explicit modeling of balance control.
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Push Recovery Strategies For Simple ModelsSimple model dynamics define unique human-like recovery strategies
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fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Three Basic Strategies
•From simple models, we can describe three basic push recovery strategies that are also observed in humans
1. 2. 3.
24
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Ankle Strategy
Assumptions:▫Zero vertical acceleration▫No torque about COM
Constraints:▫COP within the base
of support
gF
PF
PCP
eqF
RP LP
REFERENCE:Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and application of the linear inverted pendulum mode," ICRA 1991
CP PPL
mgF
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Ankle Strategy
COM Position
CO
M
Velo
city
minmaxCC PP
L
gPPP
L
g
max2
minCC PP
PP
Linear constraints on the COP define a linear stability region for which the ankle strategy is stable
REFERENCE:Stephens, “Humanoid Push Recovery,” Humanoids 2007
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Hip Strategy
Assumptions:▫Zero vertical acceleration▫Treat COM as a flywheel
Constraints:▫Flywheel “angle” has limits
gF
PF
PCP
eqF
RP LP
L
PPL
mgF CP
Im,
REFERENCE:•Pratt J, Carff J., Drakunov S., Goswami A., “Capture Point: A Step toward Humanoid Push Recovery” Humanoids, 2006
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Hip Strategy
COM Position
CO
M
Velo
city
Linear bounds for the hip strategy are defined by assuming bang-bang control of the flywheel to maximum angle
Stephens, “Humanoid Push Recovery,” Humanoids 2007
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Stepping
• Stepping can move the base of support to recover from much larger pushes. Simple models can predict step time, step location and the number of steps required to recover balance.
1. 2. 3. 4.
11, xx
CxSxCOM Position
CO
M V
elo
city
0
29
22 , xx
33 , xx
REFERENCE:•Pratt J, Carff J., Drakunov S., Goswami A., “Capture Point: A Step toward Humanoid Push Recovery” Humanoids, 2006
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Stepping
• Stepping can move the base of support to recover from much larger pushes. Simple models can predict step time, step location and the number of steps required to recover balance.
1. 2. 3. 4.
11, xx
CxSxCOM Position
CO
M V
elo
city
0
30
22 , xx
33 , xx
REFERENCE:•Pratt J, Carff J., Drakunov S., Goswami A., “Capture Point: A Step toward Humanoid Push Recovery” Humanoids, 2006
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Stepping
• Stepping can move the base of support to recover from much larger pushes.
1. 2. 3. 4.
11, xx
CxSxCOM Position
CO
M V
elo
city
0
31
22 , xx
33 , xx
REFERENCE:•Pratt J, Carff J., Drakunov S., Goswami A., “Capture Point: A Step toward Humanoid Push Recovery” Humanoids, 2006
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Stepping
• Analytic models can predict step time, step location and the number of steps required to recover balance.
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-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x position
y p
osi
tion
Reaction RegionLocation of COP during capture swing phase
Capture RegionLocation of capture step that results in stable recovery
REFERENCE:•Pratt J, Carff J., Drakunov S., Goswami A., “Capture Point: A Step toward Humanoid Push Recovery” Humanoids, 2006
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Strategy State Machine
•Analytic push recovery strategies can be incorporated into a finite state machine framework that then generates appropriate responses.
? Ankle Strategy
HipStrategy
SteppingSimple Model Look-
up
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Optimal Control For Simple Model Push RecoveryEfficient optimal control performed on simple models approximates desired behavior of the full system. refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Optimal Control of Simple Model
• The dynamics of the simple model can be used to efficiently perform optimal control over an N-step horizon.
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N-step LIPM Dynamics
N-step COP Output
LIPM Dynamics
COP Output
35
REFERENCE: •Kajita, S., et. al., "Biped walking pattern generation by using preview control of zero-moment point," ICRA 2003
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Optimal Control of Simple Model• Given footstep location, optimal
control can solve for the optimal trajectory of the COM
Objective Function
222
1
2
2
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N
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02
1JUfHUUJ t
Tt
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t
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t 2
1minarg
reftz
tp
REFERENCE: •Wieber, P.-B., "Trajectory Free Linear Model Predictive Control for Stable Walking in the Presence of Strong Perturbations," Humanoid Robots 2006
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Optimal Control for Stepping
• Footstep location can be added to the optimization to determine optimal step location and COM trajectory.
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tp
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REFERENCE:•Diedam, H., et. al., "Online walking gait generation with adaptive foot positioning through Linear Model Predictive control," IROS 2008
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
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COM
ZMP
Optimal Step Recovery (Example)
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Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Optimization of Swing Trajectory•The optimization can be
augmented to generate natural swing foot trajectories.
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bp
fp
p
fF
fF
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y
COM
ZMPSwing Foot
0 5 10 15 20 250
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0.15
0.2
posi
tion
Swing Foot Trajectory
x
y
0 5 10 15 20 250
0.5
1
velo
city
0 5 10 15 20 25-10
-5
0
5
10
forc
e
timesteps
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Optimization of Torso Lean
• Similarly, a third mass corresponding to the torso can be added. This can be used to model small rotations of the torso and hip strategies.
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Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
41
Angular Momentum Regulation
•Large angular momentum about the COM must be dissipated quickly to regain balance
•There are two simple possibilities for dissipating angular momentum:
HKH Hdes
Asymptotically decrease angular momentum using a
fixed controller
TNTt
Tt
Tt HHH ,,, 1H
Include change of angular momentum in the optimization
REFERENCE:M. Popovic, A. Hofmann, and H. Herr, "Angular momentum regulation during human walking: biomechanics and control,“ ICRA 2004
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Minimum Variance Control
• As opposed to minimizing jerk trajectories, it has been suggested that a more human-like objective function minimizes the variance at the target.
REFERENCE:•Harris, Wolpert, “Signal-dependent noise determines motor planning” Nature 1998
1
0
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iit
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fp
0fp
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Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Humanoid Robot Control Using Simple ModelsDynamics, strategies and optimal control of simple models can be combined to control full-body push recovery refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Controlling a Complex Robot with a Simple Model
•Full body balance is achieved by controlling the COM using the policyfrom the simple model.
•The inverse dynamics chooses from the set of valid contact forces the forcesthat result in the desired COM motion.
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x
Lx
y
RyF
LzF
RzF
LyF LxF
xy
z
LyRx
Ry
Variable Fixed
Contact Force Selection
?
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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General Humanoid Robot Control
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Dynamics
Contact constraints
Desired COM Motion
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Control Objectives
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LyRx
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Variable Fixed
Contact Force Selection
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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General Humanoid Robot Control
Lx
RyF
LzF
RzF
LyF LxF
LyRx
Ry
Variable Fixed
Contact Force Selection
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Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
WxxbAxbAxx TT
x minarg
dCx
General Solution To Inverse Dynamics
•Fully general solution•Many “weights” to tune•May choose undesirable forces
Weighted least- squares solution
Linear Inequality Constraints:•COP under the feet•Friction
Variable Fixed
Contact Force Selection
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Feed-forward Force Inverse Dynamics
• Pre-compute contact forces using simple model and substitute into the dynamics
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Linear System
•Easier to solve•Less “weights” to tune•More model/task-specific•Pre-computing forces may be difficult
Variable Fixed
Contact Force Selection
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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Simple Model Policy-Weighted Inverse Dynamics•Automatically generate weights according
to the optimal controller.▫2nd order model of the value function
determines cost function for applying non-optimal controls.
tttttt uxfVuxLuxQ ,,,
uuQuuuu optuu
Toptoptu
2
1Q cost
Variable Fixed
Contact Force Selection
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
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des
gdes
L
L
R
R
LR H
FPm
M
F
M
F
IPPIPP
II00
Simple Model Policy-Weighted Inverse Dynamics•Using the simple model, the cost function
can be converted into weights on inverse dynamics.
Variable Fixed
Contact Force Selection
optuu
uuQuuuu optuu
Toptoptu
2
1Q cost
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
Task Control During Balance
•Modeled as a virtual external force/torque on the system
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Virtual COM Dynamics
Virtual Humanoid Dynamics
51
taskF
REFERENCE:•Pratt J., et.al., “Virtual Actuator Control," IROS 1996
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
52
Simulation of Full Body Push Recovery
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
53
Robot Push Recovery Experiments
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
54
Proposed Work
refp
fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
55
Proposed Work
•Implementation of human-like push recovery strategies on the Sarcos humanoid robot
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
56
Proposed Work
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fp
0fp
• Simple model dynamics• Simple model inverse dynamics
• Standing balance strategies• Stepping strategies• Strategy switching state machine
• Optimal control of stepping• Extensions to model (swing leg dynamics, hip strategy, etc.)• Sequential quadratic programming to determine optimal step time• 2nd order optimization generating local value function approximation• Full-body inverse kinematics tracking of optimal plan• Force feed-forward inverse dynamics for standing balance• Force feed-forward inverse dynamics for stepping• Policy-weighted inverse dynamics• Integral control for robustness
CompletedIn ProgressTo be completed
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
57
Receding Horizon Control of Simple Model
•The full body will not exactly agree with the simple model , but by re-optimizing over a receding horizon, control can be robust to small errors.
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
58
2nd Order Optimization of Simple Model
•A 2nd order optimization method produces a local approximation of the value function along the trajectory
Goal
Initial State
Local 2nd order model of value function
Optimal Trajectory
The 2nd order model describes the relative cost of applying an action other than the optimal action
Simple Model Policy-Weighted Inverse Dynamics
tttttt uxfVuxLuxQ ,,,
uuQuuuu optuu
Toptoptu
2
1Q cost
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
59
Sequential Quadratic Programming•SQP used to solve non-linear problems:
▫Step Time Optimization Existing optimal control framework is only
linear if a fixed step time is assumed.▫Double Support Constraints Because the step location is variable, the true
double support constraints are nonlinear.
•Analytic models can be used to estimate fixed values or provide good initial guesses.
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
60
Integral Balance Control
•Integral Balance Control, related to 2nd-order sliding mode control, was previously applied to control of humanoid balance.
•Can this method be used to transfer robust control of simple system to the full body?
REFERENCE:•Stephens, “Integral Control of Humanoid Balance," IROS 2007•Levant, “Sliding order and sliding accuracy in sliding mode control”, Journal of Control, 1993
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Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
61
Timeline
•November ‘09 – Thesis Proposal▫6 months – Controller theory/refinement 1 month – Open loop planning 2 months – Receding horizon planning 3 months – Policy-weighted inverse dynamics
▫4 months – Experiments 1 month - Step recovery robot experiments 2 month - Multiple strategy robot
experiments 1 month – Comparison to human experiments
▫2 months – Thesis writing•December ‘10 - Defense
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
62
Conclusion
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fp
0fp
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
63
Thesis Proposal Overview
Simple models can be used to simplify control of full-body push recovery for complex robots
RxF
x
Lx
y
RyF
LzF
RzF
LyF LxF
xy
z
LyRx
Ry
refp
fp
0fp
Strategy decisions and planning over future actions
Simple approximate
dynamics model with COM and two
feet
Reactive full-body force
control
Benjamin Stephens | Carnegie Mellon University | Control of Full-Body Humanoid Push Recovery Using Simple Models
64
Acknowledgements•Committee:
▫Chris Atkeson (Advisor/Chair)▫Jessica Hodgins▫Hartmut Geyer▫Jerry Pratt (IHMC/External)
•Stuart Anderson•People who helped with practice talk
Questions?