Contrôle de la locomotion artificielle: Une approche par commande prédictive sans trajectoire de...

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


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


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


3. BIP: the anthropomorphic robot

6


Context

1. Biped robots



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


1. State of the art

ControlReference trajectories

Desired behaviour Real behaviour

Sensors information


8



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

1. Biped robots



11

1. no trajectory tracking

2. high adaptability + + no algorithm switches

3. robustness to disturbances

searching inspiration from human walking without mimicking


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]


1. Locomotion structure

1. Walking activity

14



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.

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1. Control process

musclesactuators

musclesactuators

skeletonsystem

skeletonsystem

SensorsSensors

CNScontroller

CNScontroller

intentionactivation force movement

environment

disturbances

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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





18

Unsuccessful approaches in exploiting movements invariants.


- Consider both stationary and transient walk - Optimal gaits / criteria adapted to goal (endurance, speed)- Consider both static and dynamic equilibrium


- No reference trajectory tracking- Perception- Anticipation and prediction- Consider internal and external constraints to ensure feasibility

and equilibrium.



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


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

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1. Modelling

1. Continuous dynamics (1)


cBG(q)q)qN(q,qM(q)

2. Ground contact

=(n,t)T

nn

tt

support force

21


1. Modelling



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


1. Modelling


t

Ttn

Tn λCλCBΓGqNqM

=(n,t)T

nn

tt

22


1. Modelling


t

Ttn

Tn λCλCBΓGqNqM

0qCqCΦ nnn unilateral constraint

=(n,t)T

nn

tt

22


1. Modelling


t

Ttn

Tn λCλCBΓGqNqM


0λn

=(n,t)T

nn

tt

22


1. Modelling


t

Ttn

Tn λCλCBΓGqNqM


0λn

0Φλ nTn

complementaritycondition

=(n,t)T

nn

tt

0)Φ 0(Φ nn

22


1. Modelling


t

Ttn

Tn λCλCBΓGqNqM


0λn

0Φλ nTn

complementaritycondition

0qCqCΦ ttt no-slipping assumption(friction cone )nt 0||||

=(n,t)T

nn

tt

0)Φ 0(Φ nn

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1. Modelling

2. Impact dynamics

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1. Modelling

2. Impact dynamics

Impact velocity jump: qq

=(n,t)T

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1. Modelling

2. Impact dynamics


nn

tt

=(n,t)T

Impulsive force

23


1. Modelling

2. Impact dynamics


tTtn

Tn ΛCΛC)qqM(q)(

nn

tt

=(n,t)T

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1. Modelling

2. Impact dynamics


tTtn

Tn ΛCΛC)qqM(q)(

0qCnn no take-off assumption

nn

tt

=(n,t)T

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1. Modelling

2. Impact dynamics


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


1. Modelling

2. Impact dynamics


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




5. Conclusions



25



1. Control without predictive horizon

25


time

inpu

tst

ate

k



25


time

inpu

tst

ate

k

Example: elevation of the swing ankle



25


time

inpu

tst

ate

k k+1




25


time

inpu

tst

ate

k k+1 k+2

?




25


time

inpu

tst

ate

k k+1 k+2




25


time

inpu

tst

ate

k k+1 k+2

?




25


time

inpu

tst

ate

k+1 k+2




25


time

inpu

tst

ate

k+1 k+2

?

Obstacledetection




25


time

inpu

tst

ate

k+2 k+3

?

ObstacleExample: elevation of the swing ankle



25


time

inpu

tst

ate

?

ObstacleExample: elevation of the swing ankle



25


time

inpu

tst

ate

Obstacle

?




25


time

inpu

tst

ate

Obstacle

?

No solution !!!




26


2. Control with predictive horizon


26


time

inpu

tst

ate

k




26


time

inpu

tst

ate

k k+1




26


time

inpu

tst

ate

k k+1 k+Nc




26


time

inpu

tst

ate

?

k k+1 k+Nc




26


time

inpu

tst

ate

?

k k+1 k+Nc

Obstacledetection




26


time

inpu

tst

ate

?

k k+1 k+Nc

Obstacledetection


sliding horizon



26


time

inpu

tst

ate

?

k+1 k+Nc+1

Obstacledetection




26


time

inpu

tst

ate

?

Obstacledetection

k+1 k+2 k+Nc+2




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3. Description

27



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

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4. Formal problem

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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]

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4. Formal problem

with:

]N , [0l X, x

]N , [0l U,u


)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

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5. State of the art

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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]

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1. Modelling




5. Conclusions



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1. Problem

}xxx/{xX

}uuu/u{U

maxkmink

maxkmink

n

m

0)g(x

]N , [0lX, x

]N , [0lU,u


)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)

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2. From human observation to problem specification


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2. From human observation to problem specification


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


3. Example of criteria and constraints specification


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3. Example of criteria and constraints specification


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

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1. Modelling



4. Some simulation results

5. Conclusions



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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

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1. One dynamic step

2D Dynamic walkingBIP - 6 actuators – 9 dofNc=3.Te= 30 ms

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1. One dynamic step

2D Dynamic walkingBIP - 6 actuators – 9 dofNc=3.Te= 30 ms

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1. One dynamic step

2D Dynamic walkingBIP - 6 actuators – 9 dof


Nc=3.Te= 30 ms

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2. Auto-adaptation to environment

2D Static walkingBIP - 6 actuators – 6 dof


Nc=5.Te= 50 ms

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3. Application to an under actuated robot structure

2D Dynamic walkingRABBIT – 4 actuators – 7dof


Nc=3.Te= 30 ms

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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

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