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Contrôle de la locomotion artificielle:Une approche par commande prédictive
sans trajectoire de référence
Philippe Poignet (LIRMM, Montpellier) Christine Azevedo (INRIA, Grenoble)
2
Context | Human locomotion features | Control approach | Conclusions & perspectives
Context
1. Biped robots
2. Locomotion control
3. Guidelines of the research
3
ASIMO & P3Honda Motor Co
WabianWaseda University
M2MIT
JohnnieTUM
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Some realisations
1. Biped robots
Mobility environment perception & understandingadaptationautonomy
Biped stability skills (contacts, impacts)robustness to disturbancesfalls
para
digm
s
2. General issues
1. Biped robots
4
Context | Human locomotion features | Control approach | Conclusions & perspectives
Mobile robots: wheeled, caterpillar, legged
Legged robots: n-legs, biped
cluttered environments
human facilities (stairs, corridors…)
- trunk + pelvis + 2 legs
- 15 active joints:
7 sagittal: ankles, knees, hips, trunk 5 frontal: ankles, hips, trunk 3 horizontal: hips, trunk
- 105 kg - 180 cm
- human proportions
BIP was designed and built in collaboration between INRIA and LMS Poitiers
1. Biped robots
5
Context | Human locomotion features | Control approach | Conclusions & perspectives
3. BIP: the anthropomorphic robot
6
Context | Human locomotion features | Control approach | Conclusions & perspectives
Context
1. Biped robots
2. Locomotion control
3. Guidelines of the research
7
Pre-computed reference trajectory tracking
- anthropomorphic joint trajectories [vukobratovic et al 01]
- torque trajectories [goswami et al 96], [pratt & pratt et al 01]
- optimal trajectories [chevallereau et al 97], [chessé & bessonnet 01]
Pre-computed movements non-adaptable to environment and events changes
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. State of the art
ControlReference trajectories
Desired behaviour Real behaviour
Sensors information
2. Locomotion control
8
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Locomotion control
1. State of the art (2)
On-line walking adaptation
- ZMP compensation [park99]
- discrete set of trajectories [denk01]
large set of trajectories needed + switches
- continuous set of parameterized trajectories [[wieber00][chevallereau02]
defining the set
- learning techniques [kun96]
- neuro-fuzzy [meyret02]
no explicit model
9
Context | Human locomotion features | Control approach | Conclusions & perspectives
Context
1. Biped robots
2. Locomotion control
3. Guidelines of the research
11
1. no trajectory tracking
2. high adaptability + + no algorithm switches
3. robustness to disturbances
searching inspiration from human walking without mimicking
Context | Human locomotion features | Control approach | Conclusions & perspectives
ControlReference trajectories
Desired behaviour Real behaviour
Sensors information
New approach to biped locomotion control
12
1. Locomotion structure (biomechanics)
2. Locomotion control (neurosciences)
3. Conclusion: some principles
Human locomotion featuresContext | Human locomotion features | Control approach | Conclusions & perspectives
13
1. stationary / transient gait (stop, starting,…) 2. stationary walk: symmetric + cyclic3. phases : support and swing
4. supports: single support and double support5. variable patterns (tiredness, learning…)6. objective oriented optimization of displacements
(metabolic energy minimization in stationary walk)
[vaughan et al 92]
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Locomotion structure
1. Walking activity
14
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Locomotion structure
2. Equilibrium
Static equilibrium: CoM projection within support base (posture, difficult situations, working at a work station…)
Dynamic equilibrium: normal walking fall forward onto the foot receiving the body‘s weight. Definition remains an open problem for bipedal systems with unilateral constraints.
15
1. Locomotion structure (biomechanics)
2. Locomotion control (neurosciences)
3. Conclusion: some principles
Human locomotion featuresContext | Human locomotion features | Control approach | Conclusions & perspectives
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Locomotion control
1. Control process
musclesactuators
musclesactuators
skeletonsystem
skeletonsystem
SensorsSensors
CNScontroller
CNScontroller
intentionactivation force movement
environment
disturbances
16
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Locomotion control
1. Control process
musclesactuators
musclesactuators
skeletonsystem
skeletonsystem
SensorsSensors
CNScontroller
CNScontroller
intentionactivation force movement
environment
disturbances
16
2. Control properties
- No reference trajectory tracking- Anticipation and prediction: CNS internal models planning- Strategy: library of objective oriented solutions- Learning: taking lessons from past situations
17
1. Locomotion structure (biomechanics)
2. Locomotion control (neurosciences)
3. Conclusion: some principles
Human locomotion featuresContext | Human locomotion features | Control approach | Conclusions & perspectives
18
Unsuccessful approaches in exploiting movements invariants.
1. Locomotion structure
- Consider both stationary and transient walk - Optimal gaits / criteria adapted to goal (endurance, speed)- Consider both static and dynamic equilibrium
2. Locomotion control
- No reference trajectory tracking- Perception- Anticipation and prediction- Consider internal and external constraints to ensure feasibility
and equilibrium.
Context | Human locomotion features | Control approach | Conclusions & perspectives
3. Conclusion: some principles
idea: use a model predictive control approach
19
1. Modelling
2. Model predictive control
3. Application of MPC to locomotion control
4. Simulation results
5. Conclusions
Control approachContext | Human locomotion features | Control approach | Conclusions & perspectives
Use of a model predictive control (MPC) approach:
20
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics
q7
q8
n
2
1
q
q
q
q
joint positions
robot orientation and position in 3D space
1. Lagrange formulation
cBG(q)q)qN(q,qM(q)
[wieber00] [genot98] [pfeiffer96]
Depending on the contacts the system can be underactuated
n dof
21
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (1)
1. Lagrange formulation
cBG(q)q)qN(q,qM(q)
2. Ground contact
=(n,t)T
nn
tt
support force
21
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (1)
1. Lagrange formulation
cBG(q)q)qN(q,qM(q)
2. Ground contact
closure constraint: 0Φ
Φ
t
n
cΓ
t
Tt
n
Tn λ
q
Φλ
q
ΦBΓGqNqM
q
ΦC i
i
=(n,t)T
nn
tt
support force
22
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (2)
t
Ttn
Tn λCλCBΓGqNqM
=(n,t)T
nn
tt
22
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (2)
t
Ttn
Tn λCλCBΓGqNqM
0qCqCΦ nnn unilateral constraint
=(n,t)T
nn
tt
22
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (2)
t
Ttn
Tn λCλCBΓGqNqM
0qCqCΦ nnn unilateral constraint
0λn
=(n,t)T
nn
tt
22
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (2)
t
Ttn
Tn λCλCBΓGqNqM
0qCqCΦ nnn unilateral constraint
0λn
0Φλ nTn
complementaritycondition
=(n,t)T
nn
tt
0)Φ 0(Φ nn
22
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
1. Continuous dynamics (2)
t
Ttn
Tn λCλCBΓGqNqM
0qCqCΦ nnn unilateral constraint
0λn
0Φλ nTn
complementaritycondition
0qCqCΦ ttt no-slipping assumption(friction cone )nt 0||||
=(n,t)T
nn
tt
0)Φ 0(Φ nn
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
Impact velocity jump: qq
=(n,t)T
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
Impact velocity jump: qq
nn
tt
=(n,t)T
Impulsive force
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
Impact velocity jump: qq
tTtn
Tn ΛCΛC)qqM(q)(
nn
tt
=(n,t)T
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
Impact velocity jump: qq
tTtn
Tn ΛCΛC)qqM(q)(
0qCnn no take-off assumption
nn
tt
=(n,t)T
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
Impact velocity jump: qq
tTtn
Tn ΛCΛC)qqM(q)(
0qCnn
no-slipping assumption(friction cone )n0t Λμ||Λ||
no take-off assumption
0qC tt
nn
tt
=(n,t)T
23
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. Modelling
2. Impact dynamics
Impact velocity jump: qq
tTtn
Tn ΛCΛC)qqM(q)(
qC)CM(CΛ
qC)CM(CΛ
)ΛCΛ(CMqq
t1-T
t1
tt
n1-T
n1
nn
tTtn
Tn
1
0qCnn
no-slipping assumption(friction cone )n0t Λμ||Λ||
no take-off assumption
0qC tt
nn
tt
=(n,t)T
24
1. Modelling
2. Model predictive control
3. Application of MPC to locomotion control
4. Simulation results
5. Conclusions
Control approachContext | Human locomotion features | Control approach | Conclusions & perspectives
Use of a model predictive control (MPC) approach:
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1 k+2
?
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1 k+2
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1 k+2
?
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k+1 k+2
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k+1 k+2
?
Obstacledetection
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k+2 k+3
?
ObstacleExample: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
ObstacleExample: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
Obstacle
?
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
Obstacle
?
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
25
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
Obstacle
?
No solution !!!
Example: elevation of the swing ankle
2. Model predictive control
1. Control without predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Control with predictive horizon
2. Model predictive control
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
k k+1 k+Nc
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
k k+1 k+Nc
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
k k+1 k+Nc
Obstacledetection
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
k k+1 k+Nc
Obstacledetection
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
k k+1 k+Nc
Obstacledetection
Example: elevation of the swing ankle
sliding horizon
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
k+1 k+Nc+1
Obstacledetection
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
26
Context | Human locomotion features | Control approach | Conclusions & perspectives
time
inpu
tst
ate
?
Obstacledetection
k+1 k+2 k+Nc+2
Example: elevation of the swing ankle
2. Model predictive control
2. Control with predictive horizon
27
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
3. Description
27
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
3. Description
(k)
(k)
(k)
15
2
1
k
ikki|
k|Nk...|1k|0Nkk|Nk...|1k|0
Nk
u
]x,x,x[ x ]u,u,u[uP
P
C
C
Control horizon
time
k k+1 k+Nc k+Np
Predictive horizon
28
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
4. Formal problem
28
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
4. Formal problem
with:
]N , [0l X, x
]N , [0l U,u
xx )u,f(x x:tosubject
)u,J(xmin :solve
Pkl|
Ckl|
kk|0kl|kl|k|1l
Nkk
u
C
CNk
}xxx/{xX
}uuu/u{U
maxkmink
maxkmink
n
m
[allgöwer99]
28
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
4. Formal problem
with:
]N , [0l X, x
]N , [0l U,u
xx )u,f(x x:tosubject
)u,J(xmin :solve
Pkl|
Ckl|
kk|0kl|kl|k|1l
Nkk
u
C
CNk
}xxx/{xX
}uuu/u{U
maxkmink
maxkmink
n
m
function of inputand state (trajectory trackingor regulation)[allgöwer99]
controlhorizon
predictivehorizon
29
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
5. State of the art
29
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Model predictive control
5. State of the art
Linear systems:
- Widely used in linear slow systems (GPC, PFC) [richalet93]
- Many stability proofs results [garcia89][boucher96][rawlings93]
Nonlinear systems:
- Usually used in slow systems- Stability proofs / strong assumptions: infinite horizon [mayne90][meadow93], dual mode [michalska93][chisci96], terminal equality constraint [chen82][alamir94], quasi infinite horizon [garcia89][denicolao97]
30
1. Modelling
2. Model predictive control
3. Application of MPC to locomotion control
4. Simulation results
5. Conclusions
Control approachContext | Human locomotion features | Control approach | Conclusions & perspectives
Use of a model predictive control (MPC) approach:
31
Context | Human locomotion features | Control approach | Conclusions & perspectives
3. Application of MPC to locomotion control
1. Problem
}xxx/{xX
}uuu/u{U
maxkmink
maxkmink
n
m
0)g(x
]N , [0lX, x
]N , [0lU,u
xx )u,f(x x:tosubject
)u,J(xmin :solve
kl|
Ckl|
Ckl|
kk|0k|1lkl|k|1l
Nkk
u
C
CNk
Nc=Np
function of inputand state (trajectory trackingor regulation)
32
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. From human observation to problem specification
3. Application of MPC to locomotion control
32
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. From human observation to problem specification
3. Application of MPC to locomotion control
Walking = shift the body in a standing posture without falling
1) Criteria: gait optimization / objective of the walk
2) Constraints
i) Standing posture maintain the CoM height
ii) Locomotion rhythm forward moving of CoM
iii) static/dynamic equilibrium contact forces control
iv) adaptation to environment ground and obstacle avoidance
iinequality constraints expressions ?
[hurmuzlu93]
33
Context | Human locomotion features | Control approach | Conclusions & perspectives
3. Example of criteria and constraints specification
3. Application of MPC to locomotion control
33
Context | Human locomotion features | Control approach | Conclusions & perspectives
3. Example of criteria and constraints specification
3. Application of MPC to locomotion control
Criteria:
Constraints:
1) Dynamics: continuous + impacts
2) Actuator limits :
3) Joint limits:
4) Standing posture:
5) Forward progression:
6) Ground avoidance:
7) Dynamic balance:
maximin qqq maximin uuu
coeff) frict. (μ .λμ||λ|| 0n0t
h yCoM
)f(xy ankleankle vvxv maxCoMmin Expressed in
output space
kl|
N
0l
Tkl|k ΓΓJ
c
34
1. Modelling
2. Model predictive control
3. Application of MPC to locomotion control
4. Some simulation results
5. Conclusions
Control approachContext | Human locomotion features | Control approach | Conclusions & perspectives
Use of a model predictive control (MPC) approach:
35
Context | Human locomotion features | Control approach | Conclusions & perspectives
4. Some simulation results
Different simulation results have been tested, 3 of them are presented here:
1. One dynamic step with BIP2. Static walk on flat ground and stairs3. Dynamic steps with RABBIT
Simulation conditions:
sagittal plane sampling period: 10 ms algorithm: SQP software: matlab
36
Context | Human locomotion features | Control approach | Conclusions & perspectives
4. Some simulation results
1. One dynamic step
2D Dynamic walkingBIP - 6 actuators – 9 dofNc=3.Te= 30 ms
37
Context | Human locomotion features | Control approach | Conclusions & perspectives
4. Some simulation results
1. One dynamic step
2D Dynamic walkingBIP - 6 actuators – 9 dofNc=3.Te= 30 ms
38
Context | Human locomotion features | Control approach | Conclusions & perspectives
1. One dynamic step
2D Dynamic walkingBIP - 6 actuators – 9 dof
4. Some simulation results
Nc=3.Te= 30 ms
39
Context | Human locomotion features | Control approach | Conclusions & perspectives
2. Auto-adaptation to environment
2D Static walkingBIP - 6 actuators – 6 dof
4. Some simulation results
Nc=5.Te= 50 ms
40
Context | Human locomotion features | Control approach | Conclusions & perspectives
3. Application to an under actuated robot structure
2D Dynamic walkingRABBIT – 4 actuators – 7dof
4. Some simulation results
Nc=3.Te= 30 ms
41
Context | Human locomotion features | Control approach | Conclusions & perspectives
5. Conclusions
Exploration of a new approach to robot dynamic walking:
MPC + constraints
+ no reference trajectory generation and tracking
+ auto-adaptation to environment changes (no switches)
+ integration of internal and external constraints
+ adaptable to different robots structures
- computation times
- stability definition and proof
- walking activity translation into inequality constraints