Passivity-based switching control of flexible-joint ... · Basic concepts Typical task representation Trajectories Xi,nc(·) denotes the desired trajectory of the unconstrained system

Passivity-based switching control of flexible-joint complementarity mechanical systems

Passivity-based switching control of flexible-jointcomplementarity mechanical systems

Constantin-Irinel Morarescu∗ and Bernard Brogliato†

∗ Laboratoire Jean Kuntzmann, Universite de Grenoble† INRIA Rhone-Alpes

E-mail: [email protected]

Reunion du groupe Systemes Dynamiques Hybrides


Basic conceptsTypical task representationNonsmooth dynamicsStability analysis criteriaMotivation

Main issuesController designDesired trajectories on transition and constraint phasesDesign of the desired contact force during constraint phasesStrategy for take-off at the end of constraint phases

Closed-loop stability analysis

Illustrative exampleMoreau’s time-stepping algorithm of the SICONOS platformSimulations

Conclusions


Basic concepts

Typical task representation


R+ =

[

k≥0

ΩBk2k ∪ I

Bkk ∪

mk[

i=1

ΩBk,i

2k+1

!!

, Bk ⊂ Bk,1; Bk+1 ⊂ Bk,mk⊂ Bk,mk−1 ⊂ . . .Bk,1

(1)

A

A’’

BA’

C

Φ

∂Φ

X ∗

d(t) = Xd(t)

Xd(t)X ∗

d(t)

X nc(t) = X ∗

d(t) = Xd(t)

X nc(t)


Basic concepts


Trajectories

X i,nc (·) denotes the desired trajectory of the unconstrained system. We supposethat F (X i,nc (t)) < 0 for some t (∈ Ω2k+1), otherwise the problem reduces to thetracking control of a system with no constraints.

X∗d (·) denotes the signal entering the control input and plays the role of the

desired trajectory during some parts of the motion.

Xd (·) represents the signal entering the Lyapunov function. This function is seton the boundary ∂Φ after the first impact.


Basic concepts


pǫ-impacts

ǫ

Figure: Illustration of 2ǫ-impacts


Basic concepts

Nonsmooth dynamics

Problem formulation

The class of complementarity Lagrangian systems encompassing flexible-jointmanipulators subject to frictionless unilateral constraints:

8

>

>

<

>

>

:

M(q)q + C(q, q)q + G(q) + K(q − θ) = D⊤λ

Jθ + K(θ − q) − KZ(ψ) = Uq1 ≥ 0, q1λ = 0, λ ≥ 0Collision rule

(2)

The admissible domain associated to the system (2): Φ = q | Dq ≥ 0 = q1 ≥ 0considering that a vector is positive if and only if all its components are positive.


Basic concepts

Stability analysis criteria


Consider x(·) the state of the closed-loop system in (2) with some feedback controllerU(X , X , t).

Definition (Weakly Stable System)The closed loop system is called weakly stable if for each ǫ > 0 there exists δ(ǫ) > 0such that ||x(0)|| ≤ δ(ǫ) ⇒ ||x(t)|| ≤ ǫ for all t ≥ 0, t ∈ Ω. The system isasymptotically weakly stable if it is weakly stable and lim

t∈Ω, t→∞x(t) = 0. Finally, the

practical weak stability holds if there exists 0 < R < +∞ and t∗ < +∞ such that||x(t)|| < R for all t > t∗, t ∈ Ω.

Consider Ik∆= [τk

0 , tkf ] and V (·) such that there exists strictly increasing functions α(·)

and β(·) satisfying the following conditions:

α(0) = 0, β(0) = 0.

α(||x ||) ≤ V (x , t) ≤ β(||x ||).


Basic concepts



Proposition (Weak Stability)Assume that the task admits the representation (1) and that

a) λ[IBkk ] < +∞, ∀k ∈ N,

b) outside the impact accumulation phases [tk0 , t

k∞] one has

V (x(t), t) ≤ −γV (x(t), t) for some constant γ > 0,

c) the system is initialized on Ω0 such that V (τ00 ) ≤ 1,

d) V (tk∞) ≤ ρ∗V (τk

0 ) + ξ where ρ∗, ξ ∈ R+.

Then V (τk0 ) ≤ δ(γ, ξ), ∀k ≥ 1 where δ(γ, ξ) is a function that can be made arbitrarily

small by increasing either the value of γ or the length of the time interval [t∞, tf ].Thus, the system is practically weakly stable with R = α−1(δ(γ, ξ)).


Basic concepts


The graph of the Lyapunov function

Figure: Example of evolution of the Lyapunov function during the first cycle of a weakly stablesystem


Basic concepts

Motivation

Motivation 1

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Y

X

Figure: The variation of the end-effector coordinates using the rigid controller when the stiffnessmatrix is defined by K = diag(5000N/m, 5000N/m).


Basic concepts

Motivation

Motivation 2

0.62

0.63

0.64

0.65

0.66

0.67

0.68

0.69

0.7

0.71

4 5 6 7 8 9 10

X

t

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7

Y

t

Figure: The variation of the end-effector coordinates using the rigid controller(K = diag(200N/m, 200N/m)).


Main issues

Controller design

Controller design 1

The controller is defined by

U = Jθr + K(θd − qd ) − γ1s2 − KZ(ψ)θd = qd + K−1Ur

(3)

where Ur is the switching controller designed for the rigid case

Ur =

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

U∅c , Unc = M(q)qr + C(q, q)qr + G(q) − γ1s1, for t ∈ Ω∅

2k

UBkc = Unc − Pd + Kf (Pq − Pd ), for t ∈ Ω

Bkk

UBkc , for t ∈ I

Bkk before the first impact

UBkt = M(q)qr + C(q, q)qr + G(q) − γ1s1, for t ∈ I

Bkk after the first impact

(4)where γ1 > 0 is a scalar gain, Kf > 0, Pq = DTλ and Pd = DTλd is the desiredcontact force during the persistently constrained motion.


Main issues

Controller design

Controller design 2

Constraint trajectory

Transient trajectory

Free motion trajectory

Desired trajectories generator

Nonlinearcontroller

MechanicalSystem

Force control

+

++

+_

_

(q∗

d , q∗

d)

Ω2k

(q, q)

Ik

Ω2k+1

(q∗

d , q∗

d)

(q∗

d , q∗

d)

Pq

Pd

Ω2k

⋃

IkΩ2k+1

Figure: Controller’s structure


Main issues

Desired trajectories on transition and constraint phases

Definition of desired signal entering the controllerWe consider the Lyapunov function

V (t, s, ψ) =1

2sT1 M(q)s1 +

1

2sT2 Js2 + γ1γ2q

T q + γ1γ2θT θ+

1

2(q − θ)T K(q − θ) (5)

O

B CA

ttk0 tk

1 tkd

(q∗

d)i(t)

qi1

qi1(t)

τ k0

tk0 > τ k

1

Ω2k Ik Ω2k+1 Ω2k+2

A′

tkf

τ k1

(q∗

d)i(t)

(q∗

d)i(t)

−ϕV 1/3(τ k0 )

Figure: The design of q∗1d on the transition phases Ik


Main issues

Design of the desired contact force during constraint phases

Design of the desired contact force during constraint phases

The dynamics on ΩBk2k+1 is:

8

>

>

<

>

>

:

M(q)q + F = (1 + Kf )D⊤p (λ − λd )

Js2 + γ1s2 + K(θ − q) = 0

0 ≤ qp ⊥ λp ≥ 0

(6)

The LCP monitoring the motion:

0 ≤ DpM−1(q)

ˆ

− F − (1 + Kf )D⊤p (λd )p

˜

+ (1 + Kf )DpM−1(q)D⊤p λp ⊥ λp ≥ 0 (7)

On Ω2k+1 the constraint motion of the closed-loop system (6)-(3) is assured if thedesired force is defined by

(λd )p , νp +Kp θp

1 + Kf−

Mp,p(q)

1 + Kf

“

[M−1(q)]p,pCp,n−p(q, q)

+ [M−1(q)]p,n−pCn−p,n−p(q, q) + γ1[M−1(q)]p,n−p

”

(s1)n−p

(8)

where Mp,p(q) =`

[M−1(q)]p,p´−1

=`

DpM−1(q)DTp

´−1is the inverse of the

so-called Delassus’ matrix and νp ∈ Rp , νp > 0.


Main issues

Strategy for take-off at the end of constraint phases

Strategy for take-offNecessary condition (λr (t

kd ) = 0)

On [tkf , t

kd )

(λd )h (tkd )

(λd )p−h (tkd )

!

=

0

@

“

A1 − A2A−13 AT

2

”−1 “

bh − A2A−13 bp−h

”

− C1(t − tkd )

C2 + A−13

`

bp−h − AT2 (λd )h

´

1

A

(9)

Sufficient condition (qr (tk+d ) > 0)

On [tkd , t

kd + ǫ)

q∗d (t) = qd (t) =

„

q∗r (t)

qncn−r (t)

«

,

where q∗r (·) is a twice differentiable function such that

q∗h (tk

d ) = 0, q∗h (tk

d + ǫ) = qnch (tk

d + ǫ),

q∗h (tk

d ) = 0, q∗h (tk

d + ǫ) = qnch (tk

d + ǫ)(10)

and q∗h (tk+

d ) = a > max`

0, −A1(q)(λd )h(tk−d )

´

.


Closed-loop stability analysis

Closed-loop stability analysis result

AssumptionThe controller U in (3) (4) assures that all the transition phases are finite.

TheoremLet Assumption 1 hold, e = 0 and q∗

d (·) defined as in Figure 7. The closed-loop

system (2)-(4) initialized on Ω0 such that V (τ00 ) ≤ 1, satisfies the requirements of

Proposition 1 and is therefore practically weakly stable with the closed-loop state

x(·) = [ψ(·), s(·)] and R =

q

e−γ(tkf−tk

∞)(ρ∗ + ξ)/ρ.


Illustrative example

Moreau’s time-stepping algorithm of the SICONOS platform

Moreau’s time-stepping algorithm of the SICONOS platform

it is based on the formulation of the Newton impact law and the unilateralconstraint in terms of velocity, together with an expression of the dynamics interms of measure

is a nonsmooth event-capturing method

the time-integration is performed with a time step that does not depend on theexact location of the nonsmooth events

the main advantage is that it converges and it is efficient even in the case offinite accumulation of impacts

it does not require any kinematic reformulations



Simulations

The trajectory in the XOY –plane

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Y

X

Figure: Left:The trajectory of the system during 6 rounds; Right: The variation of the Lyapunovfunction during the first round.



Simulations

Transition phases I 1k

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.206 0.208 0.21 0.212 0.214 0.216 0.218 0.22

Y

X

qd = q∗

d

−ν√

V (τ 20)

−ν√

V (τ 10)

q

Figure: Zoom on the transition phases I 1k .



Simulations

The variation of the control signal

-20

-15

-10

-5

0

5

10

15

20

0 2 4 6 8 10

U

t

2

4

6

8

10

12

14

16

18

2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.36 2.38 2.4

U

t

Figure: The control law applied to θ1 during the first round.



Simulations

The asymptotic dissipation of impacts 1

k V (τk0 )

1 1.4017 · 10−5

2 1.0623 · 10−8

3 3.9964 · 10−9

4 3.6527 · 10−9

5 2.4575 · 10−3

6 3.1765 · 10−7

Table: The behavior of V (τ k0 ) when k increases.



Simulations

The asymptotic dissipation of impacts 2

Figure: Left: The end-effector evolution with a perturbation introduced during the 5-th roundplotted with a dashed line, Right: The control signal magnitude decrease from one round to thenext one



Simulations

The importance of the impacting transition

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Y

X

0

2e-10

4e-10

6e-10

8e-10

1e-09

1.2e-09

1.4e-09

1.6e-09

1.8e-09

20 22 24 26 28 30

Y

t

Figure: Left: The trajectory of the end-effector when a tangential approach is imposed; Right:Zoom on the variation of the second coordinate of the end-effector in order to prove that the firstconstraint decreases to the desired value (y = 0).



Simulations

Compensation of flexibilities 1

Figure: Left: The rigid control applied to θ1 during the first round(K = diag(200N/m, 200N/m)); Right: The control law (3) (4) applied to θ1 during the firstround (K = diag(200N/m, 200N/m)).



Simulations

Compensation of flexibilities 2

K1 = K2 200 1000 2000H 0.304 0.058 0.024λ[I3] 9.2 · 10−2 3.9 · 10−2 1.9 · 10−2

Table: Higher flexibilities imply longer stabilization periods and more violent impacts.



Simulations

Numerical results for the restitution coefficient within (0, 1)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10

Y

t

Figure: The evolution of y (red) and y∗d (green) during the first round.



Simulations

Numerical results for the restitution coefficient within (0, 1)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Y

X

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30

Y

t

Figure: Left: The trajectory of the end-effector when the restitution coefficient is set to e = 0.9;Right: The variation of y when the restitution coefficient is set to e = 0.9.



Simulations

Methods to improve the tracking: larger P

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Y

X

Figure: Left: The trajectory of the end-effector when the restitution coefficient is set to e = 0.9and the duration of each round is 20 seconds; Right: The control law applied to θ1 during the firstround when the restitution coefficient is set to e = 0.9 and the duration of each round is 20seconds.



Simulations

Methods to improve the tracking: larger γ2

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.2 0.202 0.204 0.206 0.208 0.21 0.212 0.214 0.216 0.218 0.22

Y

X

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.2 0.202 0.204 0.206 0.208 0.21 0.212 0.214 0.216 0.218 0.22

Y

X

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.2 0.202 0.204 0.206 0.208 0.21 0.212 0.214 0.216 0.218 0.22

Y

X

Figure: Zoom on transition phases I 1k when γ2 = 2, γ2 = 3 and γ2 = 4, respectively.



Simulations

Methods to improve the tracking: larger γ2

Figure: The control signal applied to θ1 during the first round when γ2 = 2, γ2 = 3 and γ2 = 4,respectively.


Conclusions

Concluding remarks

Theoretical contributions

We have proposed a solution for the trajectory tracking control of complementaritynonsmooth Lagrangian systems with flexible joints

Complete stability analysis for the class of systems under consideration

The flexible-joint case is more difficult than the rigid-joint case since the backsteppingprocedure involves some exogenous trajectories that are defined as nonlinear functionsof states and other exogenous signals. Therefore, the ”passivity-based” Lyapunovfunction has jumps that are more difficult to characterize.

Conclusion regarding the simulations

Emphasize the qualitative performance of Moreau’s time-stepping algorithm of theSICONOS platform

Various numerical studies concerning the influence of different parameters

Thank You

Documents

Passivity-based switching control of flexible-joint ... · Basic concepts Typical task representation Trajectories Xi,nc(·) denotes the desired trajectory of the unconstrained system