Design and Control of Variable Stiffness Actuation ... - Semantic Scholar

Design and Control of Variable Stiffness Actuation Systems Gianluca Palli, Claudio Melchiorri, Giovanni Berselli and Gabriele Vassura DEIS - DIEM - Universita` di Bologna LAR - Laboratory of Automation and Robotics Viale Risorgimento 2, 40136, Bologna

May 3, 2010

Gianluca Palli (Universita` di Bologna)

VIA Workshop @ ICRA 2010

May 3, 2010

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Outline

Variable Stiffness Joint Design VSJ Implementation VSJ Based on Compliant Flexures VSJ Robot Control Robot Dynamic Model Static Feedback Linearization Sliding Mode Controller Output-based Controller Conclusions



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Variable Stiffness Joint Design

VSJ Implementation

Characteristics of VSJ Drawbacks of the Variable Stiffness Actuation: ◮ ◮ ◮ ◮

A more complex mechanical design The number of actuators increases Non-linear transmission elements must be used High non-linear and cross coupled dynamic model

Design Approach

Open design issues: ◮

Mechanical implementation

◮

Physical dimensions Stiffness range

◮ ◮ ◮

Displacement range Torque/Displacement relation



◮

The joint structure and dimensions have been fixed

◮

A compliant transmission element has been designed to achieve the desired joint behavior May 3, 2010

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VSJ Implementation



Design Approach



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VSJ Implementation



Design Approach



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VSJ Implementation



Design Approach



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VSJ Implementation

Joint Frame Structure ◮

Antagonistic configuration τ{1,2}

= K (ǫ{1,2} )

ǫ{1,2} τJ

= θ{1,2} ± θJ = τ1 − τ2

SJ

=

∂K (ǫ1 ) ∂K (ǫ2 ) − ∂θJ ∂θJ

τe θj Motor 2

θ2

θ1 τ2

◮

◮

Compact and simple joint frame design The compliant element is changeable



Motor 1

τ1

Nonlinear Nonlinear Elastic Elastic Moving Link Transmission Transmission (CTE) (CTE)

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VSJ Implementation


τ{1,2}

= K (ǫ{1,2} )

ǫ{1,2} τJ

= θ{1,2} ± θJ = τ1 − τ2

SJ

◮

◮

Moving Link

Antagonistic configuration

=

Nonlinear Elastic Transmission (CTE)

Nonlinear Elastic Transmission (CTE)

∂K (ǫ1 ) ∂K (ǫ2 ) − ∂θJ ∂θJ


Base Link (Common Frame)

Motor 2 Gianluca Palli (Universita` di Bologna)


Motor 1 May 3, 2010

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VSJ Implementation


Antagonistic configuration τ{1,2}

= K (ǫ{1,2} )

ǫ{1,2} τJ

= θ{1,2} ± θJ = τ1 − τ2

SJ

◮

◮

=

Moving Link

Main Shaft

CTE Shaft Gear

Shaft Gear & CTE Support + Bearings

∂K (ǫ1 ) ∂K (ǫ2 ) − ∂θJ ∂θJ Joint Encoder


Base Link Motor Gear

Shaft Support + Bearing

DC Motors Motor Encoders



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VSJ Implementation

Design Considerations The selection of suitable displacement and stiffness ranges depend on: ◮ ◮ ◮ ◮

the inertia of the robot the robot cover (soft cover) the maximum velocity and load the adopted security criteria

Does an optimal torque/displacement (stiffness) relation exists? ◮ Several different torque/displacement characteristic can be found in current VSJ implementations ◮ ◮ ◮ ◮

◮

Quadratic Polynomial Exponential ...

Quadratic torque/displacement relation presents some interesting advantages from the control point of view



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VSJ Implementation




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VSJ Implementation




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VSJ Implementation




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VSJ Implementation

Selection of the torque/displacement relation τe

◮

◮

Quadratic stiffness relation of the coupling elements allows constant stiffness along joint displacement Actuators and Joint Torques τ{1,2} τJ

◮

θj Motor 2

θ2

θ1 τ2

Motor 1

τ1


= α2 ǫ2{1,2} + α1 ǫ{1,2} + α0 = [α2 (θ1 + θ2 ) + α1 ][θ1 − θ2 + 2θJ ]

Joint Stiffness SJ = 2 [α2 (θ1 + θ2 ) + α1 ]



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VSJ Implementation

Selection of the torque/displacement relation τe

◮

◮

Quadratic stiffness relation of the coupling elements allows constant stiffness along joint displacement Actuators and Joint Torques τ{1,2} τJ

◮

θj Motor 2

θ2

θ1 τ2

Motor 1

τ1


= α2 ǫ2{1,2} + α1 ǫ{1,2} + α0 = [α2 (θ1 + θ2 ) + α1 ][θ1 − θ2 + 2θJ ]

Joint Stiffness SJ = 2 [α2 (θ1 + θ2 ) + α1 ]



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VSJ Based on Compliant Flexures

Introducing Compliant Flexures ◮

Exploit compliant flexures enhances the compactness and the flexibility of the design ◮ ◮

◮

Concentrated deformation and with linear elastic deflection Continuous compliant mechanisms

Nonlinear torsional stiffness has been achieved by using compliant elements with linear elastic deflection (Palli et al., IDETC 2010 [PBMV10]) Ψ1

Outer Frame

Inner Frame k3 k2

r3

r2

r4

γ2

k3

Coupler

γ3 Crank

k2

γ3

k4

r4

φ

γ4

δ

r2 r1

Rocker

r3

γ2 r1

λ

k4

Inner Frame Outer Frame


γ4

β


Base

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Compliant Elements Design Procedure ◮ ◮

Selection of the material Optimization of the compliant elements design ◮ ◮ ◮

to obtain a quadratic torque/displacement relation with the desired displacement given the stiffness range 25 Optimized Desired FEM Analysis

Reaction Torque T1 [N m]

20

15

10

5

0

−5

−10 −0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Deflection Ψ1 [rad]



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VSJ Prototype ◮

Prototype of the VSJ with compliant flexures



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VSJ Prototype ◮

Prototype of the VSJ with compliant flexures



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VSJ Prototype ◮

Prototype of the VSJ with compliant flexures 30

Effective Desired 20

Stiffness [N m rad−1 ]

Joint Torque [N m]

10

94.4 83.0 71.5 60.0 48.5 37.0 25.5

0

−10

−20

−30 −0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Joint Deflection [rad]



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VSJ Robot Control

Robot Dynamic Model

Dynamic Model of Robots with Variable Joint Stiffness ◮

Dynamic equations of a robotic manipulators with elastic joints ˙ + K (q − θ) M(q) q¨ + N(q, q) B θ¨ + K (θ − q)

◮

The diagonal joint stiffness matrix K = diag{k1 , . . . , kn },

◮

= 0 = τ

K = K (t) > 0

The dynamics of the joint stiffness k is written as a generic nonlinear function of the system configuration (Palli et al., ICRA 2007 [PMW+ 07]) k¨ = β(q, θ) + γ(q, θ) τk ◮ ◮

The stiffness depends on the joint and actuators relative positions The dynamics of some mechanism used to adjust the stiffness is considered



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VSJ Robot Control

Robot Dynamic Model



◮

The diagonal joint stiffness matrix K = diag{k1 , . . . , kn },

◮

= 0 = τ

K = K (t) > 0





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VSJ Robot Control

Robot Dynamic Model



◮

The diagonal joint stiffness matrix is considered time-variant K = diag{k1 , . . . , kn },

◮

= 0 = τ

K = K (t) > 0





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VSJ Robot Control

Robot Dynamic Model



◮

The diagonal joint stiffness matrix is considered time-variant K = diag{k1 , . . . , kn },

◮

= 0 = τ

K = K (t) > 0





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VSJ Robot Control

Robot Dynamic Model


Complete robot dynamics with uncertainties ˙ + K (q − θ) + ηq (t) M(q) q¨ + N(q, q) B θ¨ + K (θ − q) + ηθ (t) h i γ(q, θ)−1 k¨ − β(q, θ)(t) + ηk (t) ◮

◮

◮

= 0 = τ = τk

All the effects due to frictions, dead-zones, non-modeled dynamics, parameters variability etc., can be introduced by the additive functions η{q,θ,k } (t) In the nominal case → η{q,θ,k } (t) = 0

The input u and the robot state x are: τ , x = q T q˙ T θT θ˙T u= τk



kT

k˙ T

T

,

y=

q k

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VSJ Robot Control

Robot Dynamic Model



◮

◮

= 0 = τ = τk





kT

k˙ T

T

,

y=

q k

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VSJ Robot Control

Robot Dynamic Model



◮

◮

= 0 = τ = τk





kT

k˙ T

T

,

y=

q k

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VSJ Robot Control

Robot Dynamic Model



◮

◮

= 0 = τ = τk





kT

k˙ T

T

,

y=

q k

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VSJ Robot Control

Static Feedback Linearization

Feedback Linearized Model ◮

The overall system can be written as [4] q α(x ) τ = + Q(x ) β(q, θ) τk k¨ where Q(x ) is the decoupling matrix: −1 −1 M KB Q(x ) = 0n×n

◮

Non-Singularity Conditions ◮

→

M −1 Φ γ(q, θ) γ(q, θ) ki > 0 γi (qi , θi ) 6= 0

∀ i = 1, . . . , n

By applying the static state feedback τ α(x ) vq = Q −1 (x ) − + vk τk β(q, θ)

the linearized model is obtained (Palli et al., ICRA 2008 [PMD08]) [4] q vq = vk k¨ Gianluca Palli (Universita` di Bologna)


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VSJ Robot Control




◮


→


∀ i = 1, . . . , n




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VSJ Robot Control




◮


→


∀ i = 1, . . . , n




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VSJ Robot Control




◮

Non-Singularity Conditions

M −1 Φ γ(q, θ) γ(q, θ)

Fully-decoupled linear model represented kby>two 0 indipendent i ∀ i = 1, . . . , n → chains γof(q integrators , θ ) 6= 0 i

◮

i

i




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VSJ Robot Control

Sliding Mode Controller

Sliding Mode Controller ◮

The time-varying sliding surface is defined in the linearized state space:   Z t [3] ¨q + Pq2 e˙ q + Pq1 eq + Pq0 eq dt   eq + Pq3 e 0  Z S= t   e˙ k + Pk 1 ek + Pk 0 ek dt 0

◮

If η{q,θ,k } (t) 6= 0: S˙ =

◮

[4]

eq ¨k e

+Pe =

"

[4]

qq k¨d

#

−

α′ (x , t) β ′ (q, θ, t)

− Q(x )

τ τk

+Pe

where α′ (x , t) and β ′ (q, θ, t) represent the dynamics of the system in the non-nominal case The actual control action is defined as (Palli & Melchiorri, ICAR 2009 [PM09]): ′ τ τ = + Q −1 (x )R sign(S) τk′ τk



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VSJ Robot Control




◮

If η{q,θ,k } (t) 6= 0: S˙ =

◮

[4]

eq ¨k e

+Pe =

"

[4]

qq k¨d

#

−

α′ (x , t) β ′ (q, θ, t)

− Q(x )

τ τk

+Pe




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VSJ Robot Control




◮

If η{q,θ,k } (t) 6= 0: S˙ =

◮

[4]

eq ¨k e

+Pe =

"

[4]

qq k¨d

#

−

α′ (x , t) β ′ (q, θ, t)

− Q(x )

τ τk

+Pe




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VSJ Robot Control


Lyapunov Stability ◮

Candidate Lyapunov function: V =

◮

1 T S S 2

Condition for the asymptotic trajectory tracking: V˙ = S T S˙ = S T {F − R sign(S)} < 0 where F =

α(x ) β(q, θ)

−

α′ (x , t) β ′ (q, θ, t)

represents the residual dynamics of the manipulator due to non-prefect knowledge of its model and to spurious effects ◮

The elements of the matrix R must be larger than the error in the robot dynamics: Rii > |Fi |

◮

Given the boundedness of the the robot dynamics, a matrix R that globally satisfies the required conditions can be found



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VSJ Robot Control




◮

1 T S S 2


α(x ) β(q, θ)

−

α′ (x , t) β ′ (q, θ, t)



◮




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VSJ Robot Control




◮

1 T S S 2


α(x ) β(q, θ)

−

α′ (x , t) β ′ (q, θ, t)



◮




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VSJ Robot Control




◮

1 T S S 2


α(x ) β(q, θ)

−

α′ (x , t) β ′ (q, θ, t)



◮




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VSJ Robot Control


Simulation of a two-links Planar Manipulator ◮

Nominal Model + Exact Feedback linearization Joint positions [rad] 1 Pos Joint 1 Pos Joint 2

1

0

−1

−0.5

2

4

6

8

x 10

0.5

0

−2 0

Position errors [rad]

−5

2

10

Err Joint 1 Err Joint 2

−1 0

Joint Stiffness [N m rad −1 ]

2

1

3

6

8

10

Stiffness errors [N m rad −1 ]

−5

4

4

x 10


0.5

2 0 1

−1 0

−0.5

Stiff Joint 1 Stiff Joint 2

0 2

4

6

Time [s]


8

10

−1 0


2

4

6

Time [s]

8

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VSJ Robot Control



Perturbed Model + Simplified Feedback linearization Position errors [rad]

Joint positions [rad] 3

0.5 Pos Joint 1 Pos Joint 2

2

0

(b)

1

−0.5 0 −1

−1 −2 0

2

4

6

8

10

−1.5 0

Err Joint 1 Err Joint 2 2

4

6

8

10


Joint Stiffness [N m rad −1 ] 4

0.1

3


0.05

2 0 1

−1 0

−0.05


0 2

4

6

Time [s]


8

10

−0.1 0


2

4

6

Time [s]

8

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VSJ Robot Control



Perturbed Model + Simplified Feedback linearization + Sliding mode Joint positions [rad] 1 Pos Joint 1 Pos Joint 2

1

0

−1

−0.5

2

4

6

8

x 10

0.5

0

−2 0

Position errors [rad]

−5

2

10


−1 0

Joint Stiffness [N m rad −1 ]

2

2

3

6

8

10


−3

4

4

x 10


1

2 0 1

−1 0

−1


0 2

4

6

Time [s]


8

10

−2 0


2

4

6

Time [s]

8

10 May 3, 2010

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VSJ Robot Control

Output-based Controller

Output-based controller ◮

Both the robot, the motors and the stiffness dynamics are in the form: A(v , w)v¨ + f (v˙ , v , w) + α = τ

◮

By using the general output-based dynamic compensator: τ

◮

= A(v , w) τ¯ + f (z2 , v , w)

τ¯ z˙ 1

= −Λ1 (v − vd ) − Λ2 z2 + L(z3 − z2 ) = −Γ1 (z1 − v ) + z2

z˙ 2 z˙ 3

= −Γ2 (z1 − v ) + τ¯ − L(z3 − z2 ) = τ¯ − L(z3 − z2 )

the dynamics of these systems can be decoupled (Palli & Melchiorri, NOLCOS 2010 [PM10]) The controller provides direct estimations of the disturbance acting on the motors, on the stiffness and on the robot arm



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VSJ Robot Control




◮


◮

= A(v , w) τ¯ + f (z2 , v , w)

τ¯ z˙ 1

= −Λ1 (v − vd ) − Λ2 z2 + L(z3 − z2 ) = −Γ1 (z1 − v ) + z2

z˙ 2 z˙ 3

= −Γ2 (z1 − v ) + τ¯ − L(z3 − z2 ) = τ¯ − L(z3 − z2 )




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VSJ Robot Control




◮


◮

= A(v , w) τ¯ + f (z2 , v , w)

τ¯ z˙ 1

= −Λ1 (v − vd ) − Λ2 z2 + L(z3 − z2 ) = −Γ1 (z1 − v ) + z2

z˙ 2 z˙ 3

= −Γ2 (z1 − v ) + τ¯ − L(z3 − z2 ) = τ¯ − L(z3 − z2 )




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VSJ Robot Control



Perturbed Model + Output-based controller End-effector positions [m]

Position errors [m]

−3

1

5

x 10

0.5 0 −0.5 −1 0

0 Pos x Pos y 2

Err x Err y 4

6

8

10

−5 0

2

4

6

8

10

Stiffness errors [N m rad−1 ]

Joint Stiffness [N m rad−1 ] 4

0.1

3

Err Stiff 1 Err Stiff 2

0.05

2 0 1

−1 0

−0.05


0 2

4

6

8

10

−0.1 0

Time [s] Gianluca Palli (Universita` di Bologna)

2

4

6

8

10

Time [s] VIA Workshop @ ICRA 2010

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Conclusions

Conclusions ◮

◮

◮

◮

◮

A flexible and compact solution for the implementation of variable stiffness joints has been shown Complaint flexures have been adopted to improve compactness and to achieve the desired stiffness profile The model of a robotic manipulator with variable joint stiffness has been defined taking into account model uncertainties In the non-ideal case, the asymptotic tracking performance have been recovered by means of a sliding mode controller that compensates for model uncertainties A robust output-based controller for the indipendent control of the robot trajectory and joint stiffness has been shown

Questions? Thank you for your attention...



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Conclusions

Conclusions ◮

◮

◮

◮

◮

A flexible and compact solution for the implementation of variable stiffness joints has been shown Complaint flexures have been adopted to improve compactness and to achieve the desired stiffness profile The model of a robotic manipulator with variable joint stiffness has been defined taking into account model uncertainties In the non-ideal case, the asymptotic tracking performance have been recovered by means of a sliding mode controller that compensates for model uncertainties A robust output-based controller for the indipendent control of the robot trajectory and joint stiffness has been shown

Questions? Thank you for your attention...



May 3, 2010

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Conclusions

Bibliography I G. Palli, G. Berselli, C. Melchiorri, and G. Vassura. Design and modeling of variable stiffness joints based on compliant flexures. In Proc. of the ASME 2010 Int. Design Engineering Technical Conf., Montreal, Canada, 2010. G. Palli and C. Melchiorri. Robust control of robots with variable joint stiffness. In Proc. Int. Conf. on Advanced Robotics, Munich, Germany, 2009. G. Palli and C. Melchiorri. Task space control of robots with variable stiffness actuation. In Proc. of the 8th IFAC Symp. on Nonlinear Control Systems, Bologna, Italy, 2010. G. Palli, C. Melchiorri, and A. De Luca. On the feedback linearization of robots with variable joint stiffness. In Proc. of the IEEE Int. Conf. on Robotics and Automation, 2008. Gianluca Palli (Universita` di Bologna)


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Conclusions

Bibliography II

¨ G. Palli, C. Melchiorri, T. Wimbock, M. Grebenstein, and G. Hirzinger. Feedback linearization and simultaneous stiffness-position control of robots with antagonistic actuated joints. In IEEE Int. Conf. on Robotics and Automation, 2007.



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