sbrml part5 introduction to bipedal walking · 2 sensor based robotic manipulation and locomotion...
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1 Sensor Based Robotic Manipulation and Locomotion
Introduction to Bipedal Walking
Dr.-Ing. Christian OttDLR - Institute for Robotics and Mechatronics
For full lecture on humanoid systems see Christian‘s lecture at EIModeling and Control of Humanoid Walking Robots
http://www.robotic.dlr.de/chr
1
DLR 02/05/2012
2 Sensor Based Robotic Manipulation and Locomotion
Motivation
Honda Asimo
Bipedalism:• Stepping on different heights and over obstacles• Small region of support compared to wheeled robots• Humanoid body structure allows to act
in human environments• Smallest number of legs required for
standing, walking, running• Fundamental research:
Control, planning, mech. designUnderstanding (human) balance
• „Technological competition“: Sony, Honda, Toyota, Samsung,Aldebaran, Boston Dynamics, …
• …
3 Sensor Based Robotic Manipulation and Locomotion
Humanoid Balance
Vestibular system
Vision
Somatosensory System
Proprioceptivesensors
IMU
Vision
force sensors
joint sensing
“Balance” is a generic term describing the ability to control the body posturein order to prevent falling.
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Humanoid Balance
Small push:Ankle strategy
force controlZMP control
(Zero Moment Point)
angular momentum control
Medium push:Hip strategy
Large Push: Step strategy
Human
Robot
Strategies for human push recovery:
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Balancing Walking
Types of bipedal gaitStatic walking: Ground projection of center of motion (COM) never leaves support polygon
Dynamic walking:Def A: Ground projection leaves support polygon during motion„Def B: Walking with underactuation“ (e.g. point foot walking)
Running: includes flight phase
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Models
Multi-Body-Models Conceptual Models
Fixed Base Models(predefined contact state)
Floating Base Model Walking Running
Dynamical Models (Mechanical)
Complexity
Specialization
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Floating base model
)3(SEHb Qq
n
111 SSST n
Configuration Space: )3(SEQ
Using local coordinates: n6
6bx
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Free-Floating vs. Fixed Base Models
Fixed base modelsIn each contact state the model is different:
• Single support (right, left)• Double support• Heel Off• Toe Touch Down• …
Transition between contact states
double supportparallel kinematicsover-constrained
need for passive joints
single supportserial kin. chainor tree structure
Free-floating model
Components:• Lagrangian dynamics• Constraints due to contact forces• Transition equations (impacts)
underactuated
Planning & control must ensure that the considered contact state is valid! ground reaction force must fulfill constraints for balancing
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Zero Moment Point[Vukobratovic and Stepanenko,1972]
)(x1x 2x
F
ZMP as a ground reference point: Distributed ground reaction force under the supporting foot can be replaced by a single force F acting at the ZMP, called ground reaction force.
z
x
),( yx
y
ZMP = CoP (Center of Pressure)
p1x 2x
F0
ZMP
CoP
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Definition of the Zero Moment Point (ZMP)
Planar single support:
Distributed floor reaction force under the supporting foot can be replaced by a single force acting at the ZMP.
[Vukobratovic and Stepanenko,1972]
2
1
2
1
)(
)(0)( x
x
x
x
dxx
dxxxpp
z
x
)(x1x 2x
210)( xpxx
F
2
1
)()()(x
xdxxpxp
ZMP = CoP (Center of Pressure)
Extension to two dimensions is straight forward
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Some facts about the ZMP• Can ZMP leave the support polygon? NO• Can ZMP location be used as a stability criterion NO
• If ZMP reaches the border of the support polygon foot rotation possible.
• ZMP is defined on flat contact (no uneven surface).• ZMP gives no information about sliding.
)(x1x 2x
F
Measurement e.g. by Force/Torque Sensor
),,(0)()(!
ssssss fppfppp
z
sfs
spp
How to obtain ZMP in practice?
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First usage of the ZMP• Motion of the legs is predefined.• Upper body controls the ZMP in the center of the supporting foot
ensure proper foot contact during walking
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Conceptual Models for Walking
Cart table model Inverted pendulum model
Ground reaction force should stay within the stance area
Ground reaction force stays at the hinge point of the pendulum
Can be derived from the general model:
• Approximation of angular momentum
• Limited motion (no vertical COM motion)
Basis for many successful walking robots
NAO HRP-2 ASIMO
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Mass Concentrated Model for Linear Inverted Pendulum
Forces in the linear inverted pendulum (LIP) model
p
xc
zc
Mg
xcM
F
pccgc xz
x
pf p p
ppfp
c
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mass concentrated model
Forces in the LIP model
p
xc
zc
Mg
xcM
F
Effect of an additional hip torque
p
xcM
F
pccgc xz
x
z
xz
x Mcpc
cgc
Mg
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mass concentrated model
Strategies for gait stabilization: Effect of an additional hip torque
p
Mg
xcM
F
z
xz
x Mcpc
cgc
1. Controlling ZMP (constraints!)
2. Angular momentum control
3. Step adaptation
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Mass concentrated model for ZMP Control
p
c
gcccp xz
xx
xcInverted Pendulum Model [Sugihara]
xxz
x pccgc
p
c
xx pc xx cp
Cart Table Model [Kajita]
Simplifying assumptions• robot mass concentrated in the center of mass (CoM)• CoM height cz is kept constant
We have a simple relation between the motion of the CoM and the ZMP
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ZMP based walking pattern: basic scheme
Footstep planning
Walking Pattern
Generator
CoM Inverse
kinematics
Joint Position Control
dp dc dq
Image copied without explicit permission from Workshop material “Overview of ZMP-based Biped Walking” at Dynamic Walking conference 2008, by S. Kajita.
Simple solution: Use hip motion (+offset) instead of CoM
Here, CoM is controlled
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DLR Robot Control Based on Conceptual Models
Footstep Generation
Pattern Generation
x
pZMP-COMStabilizer
dxPos. Controlled
Robot
e.g. LQR Preview Control [Kajita, 2003]
Model Predictive Control [Wieber, 2006]
realtime
F
Automatica 2010
ZMP is controlled
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Torque Based BalancingA Unified Approach for
Grasping and Balancing
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Torque based balancingAssumptions:
Joint Torque Control
COM and hip orientation can be measured in a world-fixed frame (via inertia measurement unit – IMU - measurement)
extFMg
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Torque based balancingAssumptions:
Joint Torque Control
COM and hip orientation can be measured in a world-fixed frame (via inertia measurement unit – IMU - measurement)
Control Approach:
1. Compute desired force on the COM (according to compliant behavior)
COMF
extFMg
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Torque based balancingAssumptions:
Joint Torque Control
COM and hip orientation can be measured in a world-fixed frame (via inertia measurement unit – IMU - measurement)
Control Approach:
1. Compute desired force on the COM(according to compliant behavior)
2. Distribute COM force to contact points COMF
extFMg
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Torque based balancingAssumptions:
Joint Torque Control
COM and hip orientation can be measured in a world-fixed frame (via IMU measurement)
Control Approach:
1. Compute desired force on the COM(according to compliant behavior)
2. Distribute COM force to contact points
3. Realize contact forces via joint torques
COMF
extFMg
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Force distribution: grasping and balancing are very similar problems!
oFo
f1 f2
W
f1 f2
Fo
Grasping and Balancing
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Force Distribution in Grasping
F
FGGFGFGWO
1
111Net wrench acting on the object:
Grasp Map
if
)3(seFC
Well studied problem in grasping: Find contact wrenches such that a desired net wrench on the object is achieved.
FCFC
)3(se
friction cone
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Balancing & Posture Control
Compliant COM control [Hyon & Cheng, 2006]
Compliant trunk orientation Control =>
)()( dDdPCOM ccKccKMgF
Mg
extF
COMF
HIPT
)3(SOR
HIPT extT
),( HIPCOMd TFW Desired wrench:
Quaternion based orientation compliance control(see passivity based Cartesian impedance control)
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Force distribution
HIPT
COMF
f
fGGWd
1
1
ii
ii Rp
RG
ˆ
3if
Relation between balancing wrench & contact forces
Constraints:• Unilateral contact: (implicit handling of ZMP constraints)• Friction cone constraints
0, zif
Cf
T
F
GG
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Force distribution
HIPT
COMF
f
fGGWd
1
1
ii
ii Rp
RG
ˆ
3if
Relation between balancing wrench & contact forces
Constraints:• Unilateral contact: (implicit handling of ZMP constraints)• Friction cone constraints
0, zif
Formulation as a constraint optimization problem
Cf
23
22
21minarg CCTHIPCFCOMC ffGTfGFf
T
F
GG
321
30 Sensor Based Robotic Manipulation and Locomotion
ForceDistribution
Torque based balancing
Force Mapping
TorqueControl
RobotDynamics
Object ForceGeneration
IMU
cf
q
for orientation control and COM computation
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ForceDistribution
Torque based balancing
Force Mapping
TorqueControl
RobotDynamics
Object ForceGeneration
IMU
cf
q
for orientation control and COM computation
COM
Contact forces
32 Sensor Based Robotic Manipulation and Locomotion
Summary
1. Consistent treatment of COM and posture control (useful for manipulation, bipedal vehicles)
2. Implicit handling of ZMP via constraints in the force optimization
3. Utilizes a formulation from grasping theory: Allows for generalization to multi-contact situations
4. Controller is independent of precise knowledge about foot contact(however, IMU data is important!)
Outlook:- Extension to multi-contact interactions- Extension to walking
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Overview
1. Fundamentals about bipedal walking
2. Time based walking control ZMP based control
3. Limit cycle based walking Passive dynamic walking
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Passive Dynamic Walking„Passive Dynamic Walking“
Dynamics Control
• Careful mechanic design:knee retraction, foot shape, trunk, elastic elements
• Analysis: Limit cycle (Poincare Map), impacts
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Passive Dynamic Walking
„Dynamic Walking“
Actuation + Dynamics
„Passive Dynamic Walking“
Dynamics Control
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Conceptual Models: RunningTemplates
• Template for control
• Template for design
[Geyer, Seyfarth, Blikhan, 2006]
Role of compliance for human walking/running.
[Oscar Pistorius][A. Sato, Mc Gill Univ.]
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Legged Hopping Robots
[Raibert, MIT]
Three part control:
1. control of hopping height (during stance)
2. control forward velocity via foot positioning
3. control of body orientation by servoing the hip
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[Raibert, MIT]
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That's all Folks!