Bi-articular Muscle Actuation Design for Robot Arms

Bi-Articular Muscle Actuation Designfor Robot Arms

V. Salvucci Y. Kimura S. Oh Y. Hori

Hori-Fujimoto Lab, The University of Tokyo

ICRA 2011 Workshop on Biologically-inspired Actuation, Shanghai

Bi-articularly Actuated Robot Arms Actuator Redundancy Problem Experimental Setup Experimental Results Conclusions References

Outline

1 Bi-articularly Actuated Robot Arms

2 Actuator Redundancy ProblemTraditional: Pseudo-inverse Matrix (2− norm)Our Solution: The ∞− norm Approach

3 Experimental SetupBiWi:Bi-Articularly Actuated & Wire Driven Robot ArmFeedforward Control Strategy

4 Experimental Results

5 Conclusions

V. Salvucci, Y. Kimura, S. Oh, Y. Hori Hori-Fujimoto Lab, The University of Tokyo

Bi-Articular Muscle Actuation Design for Robot Arms 2/24


Outline





5 Conclusions




What are Bi-articular Actuators?

Multi-articular actuators produce torque in 2 (or more) consecutive joints

Biceps brachii

Coracobrachialis Brachialis

Simplified model of human musculo-skeletal structure

f1− e1: antagonistic pair of mono-articular muscles

f2− e2: antagonistic pair of mono-articular muscles

f3 − e3: antagonistic pair of bi-articular muscles




Why Bi-Articular Actuators?

1 Homogeneous Maximum Force at End Effector [Fujikawa 1999]

2 Impedance control without FB [Hogan 1985]

3 Power transfer from proximal to distal joints [Schenau 1989]





2 actuatorsof 10 Nmeach

3 actuatorsof 6.6 Nmeach

Safety: smaller peak force (in case of controller failure)Vertical balance: greater ground horizontal force [Salvucci 2011b]








2 actuatorsof 10 Nmeach

3 actuatorsof 6.6 Nmeach

Safety: smaller peak force (in case of controller failure)Vertical balance: greater ground horizontal force [Salvucci 2011b]







Outline





5 Conclusions




Actuator Redundancy Problem

Model

{T1 = (f1 − e1)r + (f3 − e3)r

T2 = (f2 − e2)r + (f3 − e3)r

Statics

{T1 = τ1 + τ3

T2 = τ2 + τ3

Given desired T1 and T2 ⇒ τ1=?, τ2=?, τ3=?




Pseudo-inverse Matrix (2− norm)

Moore Penrose is the simplest pseudo inverse matrix = 2− norm [Klein 1983]

2− norm optimization criteria

minimize√τ 2

1 + τ 22 + τ 2

3 (1)

subject to

{T1 = τ1 + τ3

T2 = τ2 + τ3

(2)

Closed form solutionτ1 = 2

3T1 − 1

3T2

τ2 = − 13T1 + 2

3T2

τ3 = 13T1 + 1

3T2

(3)

T = [2.0, 1.5]⇒ τ = [1.66, 0.33, 0.83]

Given F ⇒ T =(JT)

FT ⇒ τ using (3)




Our Solution: The ∞− norm Approach [Salvucci 2010]

∞− norm optimization criteria

minimize max{|τ1|, |τ2|, |τ3|} (4)

subject to

{T1 = τ1 + τ3

T2 = τ2 + τ3

(5)

Closed form solution [Salvucci 2010]

if T1T2 ≤ 0 ⇒

τ1 =

T1−T22

τ2 =T2−T1

2

τ3 =T1+T2

2

(6)

if T1T2 > 0

and |T1| ≤ |T2|⇒

τ1 = T1 −

T22

τ2 =T22

τ3 =T22

(7)

if T1T2 > 0

and |T1| > |T2|⇒

τ1 =

T12

τ2 = T2 −T12

τ3 =T12

(8)

T = [2.0, 1.5]⇒ τ = [1.0, 0.5, 1.0]

Given F ⇒ T =(JT)

F

T ⇒ τ using (6), (7), or (8)




Outline





5 Conclusions




BiWi: Bi-Articularly Actuated & Wire Driven Robot Arm [Salvucci 2011a]

+ Human-like actuation structure

+ Wire Transmission ⇒ low linkinertia (safety, energy efficiency)

+ Mono-/bi- articular torquedecoupling (statics)

- Not intrinsically compliant, butsolvable with springs

- Transmission loss in the wires




Feedforward Control Strategy

F∗ = [F ∗x ,F∗y ]T and T∗ = [T ∗1 ,T

∗2 ]T : desired output forces and input

torque.

[τ∗1 ,τ∗2 , τ∗3 ]: desired actuator joint torques

[e∗1 , f ∗1 , e∗2 , f ∗2 , e∗3 , f ∗3 ]: motor reference torques calculated as:

e∗i =

{Ktliτ

∗i if τ∗i < 0

0 otherwisef ∗i =

{Kiτ∗i if τ∗i > 0

0 otherwise(9)

where Ktl2=1.33 (thrust wire transmission lost), Ktl1 = K3 = 0.

Fx and Fy : measured forces at the end effector.




Outline





5 Conclusions




Infinity Norm VS Pseudo-inverse matrix (2− norm) [Salvucci 2011c]

θ1 = −60◦

θ2 = 120◦

θ1 = −25◦

θ2 = 50◦

Measured maximum output force Relative difference in output force

F diff =|F∞−n| − |F 2−n|

|F 2−n| (10)




Outline





5 Conclusions




Conclusions

Bi-articular muscles key points

1 Homogeneous distribution of output force

2 Power transfer proximal to distal joints

3 FF impedance control

BiWi, Bi-articularly actuated and Wire driven Robot Arm

Human-like actuation structure

Low link-inertia ⇒ Safety, efficiency

Perfect decoupling between mono- and bi- articular actuator (statics)

The ∞− norm approach for actuator redundancy resolution

Closed form solution based on a piecewise linear function continuous inall the domain D = {T1,T2}Maximization of force at the end effector: +30% than 2− norm

Applicable to systems with 3 inputs and 2 outputs




Conclusions















Conclusions















Thank you for your kind attention

V. Salvucci Y. Kimura S. Oh Y. Hori

www.hori.k.u-tokyo.ac.jp www.valeriosalvucci.com



http://www.hori.k.u-tokyo.ac.jp

http://www.valeriosalvucci.com


2− norm Vs ∞− norm in 2D

Equation with infinite solutions

k = αx + βy

k, α and β are constant

x and y represent the motor torques ⇒ bounded

2− norm

minimize√

x2 + y 2

∞− norm

minimize max {|x |, |y |}

Comparison

Solutions comparison

max{y∞, x∞} ≤ max{y2, x2}

Smaller solution space for 2− norm

no solution for 2 norm!!




The Best Norm

Output Force for θ2 ∈ {30, 60, 90, 120, 150◦} |τ1|+ |τ2|+ |τ3| for θ2 = 90

norm 1 norm 2 norm ∞min (|τ1|+ |τ2|+ |τ3|) min (

√τ 2

1 + τ 22 + τ 2

3 ) min max{|τ1|, |τ2|, |τ3|}

|τ1|+ |τ2|+ |τ3| of ∞− norm > |τ1|+ |τ2|+ |τ3| of 2− norm|τ1|+ |τ2|+ |τ3| of 2− norm > |τ1|+ |τ2|+ |τ3| of 1− normThe best norm: switching between 1− norm, 2− norm and ∞− norm. . . but the system could not be stable due to discontinuity in torquepatterns




References

T. Fujikawa, T. Oshima, M. Kumamoto, and N. Yokoi. Output force at the endpointin human upper extremities and coordinating activities of each antagonistic pairs ofmuscles. Transactions of the Japan Society of Mechanical Engineers. C, 65(632):1557–1564, 1999.

N. Hogan. The mechanics of multi-joint posture and movement control. BiologicalCybernetics, 52(5):315–331, 1985.

V. Salvucci, S. Oh, and Y. Hori. Infinity norm approach for precise force control ofmanipulators driven by bi-articular actuators. In IECON 2010 - 36th AnnualConference on IEEE Industrial Electronics Society, pages 1908–1913, 2010.

V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. BiWi: Bi-Articularly actuated and wiredriven robot arm. In IEEE International Conference on Mechatronics (ICM), 2011a.

V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. Disturbance rejection improvement inNon-Redundant robot arms by bi-articular actuators. In Industrial Electronics(ISIE), IEEE International Symposium on, 2011b.

V. Salvucci, Y. Kimura, S. Oh, and Y. Hori. Experimental verification of infinity normapproach for force maximization of manipulators driven by bi-articular actuators. InAmerican Control Conference (ACC), 2011c.

G. J. V. I. Schenau. From rotation to translation: Constraints on multi-jointmovements and the unique action of bi-articular muscles. Human MovementScience, 8(4):301–337, Aug. 1989.

