advances in compliant and soft controlled structures for ... · 10.07.2014 mesrob 2014 - prof dr...

Advances in Compliant and Soft Controlled Structures for Eff i d S f R b E i I i Effective and Safe Robot - Environment Interaction

TUTORIAL

Prof. dr. Petar. B. PETROVIĆC b M f t i S t L b t (CMS L b)CyberManufacturing Systems Laboratory (CMSysLab)Faculty of Mechanical Engineering, Belgrade University

[email protected]

MESROB 2014International Workshop and Summer School on Medical and Service Robotics

EPFL La sanne CH J l 10 12 2014

10.07.2014 1MESROB 2014 - Prof dr Petar B. Petrovic, Faculty of Mechanical Engineering, University of Belgrade, SERBIA

EPFL Lausanne, CH, July 10-12 2014

Joint space stiffness 2Generalized stiffness 1Joint space stiffnessKinematic redundancy and null-space stiffness

23

Generalized virtual displacement 4Numerical examples and experiments 5p p


Generalized stiffness 1

Robot is not a rigid body! g yAny robot is flexible elastomechanical structure. It deforms like a spring!

From a modeling point of view, flexibility can be assumed as concentrated at the robot joints and / or distributed (in different ways) along the robot links.For the sake of simplicity, robots are classified into two classes:

Robots with flexible jointsRobots with flexible links


ROBOT ARM = GENERALIZED SPRING

xKF TCPxTCP δ=Force – displacement relationship:

Task space (Cartezian) stiffness matrix:Task space (Cartezian) stiffness matrix:

xFK TCPx

)(∂

∂=

Li l ti hi b t i fi it i l

xTCPx ∂

Linear relationship between infinitesimal displacements in task and joint space:

qqJxTCP δδ )(=Kineto-statical model of the robot arm


qqJxTCP δδ )(

ROBOT ARM = PROGRAMMABLE GENERALIZED SPRING

Robot arm behaves as a GENERALIZED SPRING with PROGRAMMABLE STIFFNESS parameters. We can change the stiffness parameters in time and adapt the robot behavior to the task at adapt the robot behavior to the task at hand.

SUPERVISOR specifies the DESIRED STIFFNESS d i th t h STIFFNESS and in that way morphs compliant behavior of the robot arm

The concept of SOFT ROBOTICS = pLightweight COMPLIANT robotic manipulators


WHY COMPLIANT BEHAVIOR?

Compliance can be exploited and is essential to:Compliance can be exploited and is essential to:

Constrained motion control / stability issues and adaptation

Force and Impedance control / robot interaction with environment

Impact control and energy storing

Human-robot interaction / robot in human environment

Safety / intrinsic safety

Humanoid robots that imitate human behavior / uncanny valley problem


An example of a manipulation task which requires complex complianace control patterns: Human performing a bimanual peg-in-hole task. Initial configuration (B), and realizedconfiguration as a result of compliance control motions (C). Human and robots’ endingg p ( ) gconfiguration coincide which probably gives evidence to the similarity in control principles.

AARobot controller should effectively

modulate of the size and directionality of the realized TCP

stiffness in task coordinates Since

Stiffness profiles of both robots should provide stiff behavior along the direction of the hole

d li t b h i fil stiffness in task coordinates. Since the hole axis are move in the system

task space TCP stiffness matrix should change in time too.

and a compliant behvioruprofile along other directions to avoid generation of high interaction forces.

6 component F / T sensor


B C

An example of the SOFT JOINT with integrated VARIABLE STIFFNESS ACTUATORMechatronic assembly with embedded control and sensory hardware

Link POSITION sensor Cross-roller bearing

Power converter unit

Joint and motor controller boardPower supply board

Mechanical and sensory interface to the nex link

dedicated force / position devicededicated force / position device

Frameless motor + brake + encoder

Harmonic drive gear unit

Link TORQUE sensor

Carbon fiber lightweight and rigid link


and rigid linkDLR Soft robot arm

OUR PHILOSOPHY IS CHANGING

Evolutionary stages in design and our understanding of robots:

Stiffer is better → Stiffness isn’t everything → Soft robotics

Robots like MACHINE TOOLS → Robots like HUMANS

adapt the ROBOT to the WORLDadapt the WORLD to the ROBOT


SOME COMPLIANT / SOFT ROBOTS

MIT DOMO RobotA Force Sensing Humanoid Robotgfor Manipulation Research (Edsinger and Weber, 2004)


ABB FRIDA concept robotDual-arm robot with 14 dof designed to work safely in human environmentto work safely in human environmentFriendly Robot for Industrial Dual-arm Assembly

Joint actuation system:


yNo reliable information!

DLR Robot ... KUKA IIWA Robot

Lightweight robot arm with 7 dof and advanced compliance control system

Joint actuation system:Joint actuation system:Custom-made frameles servo motor + hollow shaft harmonic drive reducer


UNIVERSAL ROBOTSUR10 Robot UR10 Robot

6 dof collaborative robot arm with inherent li d b kd i bilitcompliance, and backdrivability;

Extremely simple design and consequently cheep robot (towards of robot ubiquity)

Joint actuation system:Custom-made frameles servo motor + hollow shaft harmonic drive reducerhollow shaft harmonic drive reducer


BARRETT TECHNOLOGYWAM Arm

Weight of the arm beyond shoulder WAM Arm

7 dof cable drive lightweight robot arm ith i h t li d t l

is 5.8 kg / minimum kinetic and potential energy (less then weight of human arm)

with inherent compliance, and extremely low force backdrivability;

Joint act ation s stem

14 bits joint torque resolution!

Joint actuation system:Custom-made neodymium-iron servo-motor with embedded space-vector commutated current amplifier/control + commutated current amplifier/control + mechanical transmissions based on preloaded (zero-backlash) high-speed, near-frictionless cable transmission ( t t d bl d diff ti l )(patented cabled differentials)


WAM = Whole Aarm Manipulator

US Patent 4,903,536 / Feb. 27, 1990COMPACT CABLE TRANSMISSION WITH CABLE DIFFERENTIALInventnors: Kenneth Salisbury et. al.

Preloaded / zero-e oaded / e obacklash Near frictionlessNo vibrations

Entirely gearless driven!

Lightweight

A PERFECT MECHANICAL PLATFORM FOR HIGH PERFOMANCE COMPLIANCE


HIGH PERFOMANCE COMPLIANCE CONTROL (including backdrive)

CABLE DIFFERENTIAL DRIVE CONCEPT


YASKAWA SIA10 family of redundant articulated robot arms

7 dof anthropomophic robot arm

15 dof dual arm robot 15 dof dual-arm robot


JOINT SPACE STIFFNESS MATRIX

Joint space stiffness 2

Energy conservation principle:

Stiffness MAPPING from robot task space to the joint space

WWWW gF δδδδ μτ ++≡ Virtual works associated with joint torques and external forces are identical!0=+ μδδ WWg

xFq TCPT

TCPT δδτ =

μg

Di l t i f t k t

FqJ Tτ )(=qqJxTCP δδ )(= Displacements mapping from task to

joint space

JKJ

q

T δ)()(

)(xKF xδ=

Relationship between joint torques and robot TCP stiffness.

Relationship between joint torques


qqJKqJ x δτ )()(= p j qand robot TCP stiffness

qKq= δτ

)()( qqJKqJ T= δτ )()( qqJKqJ

T

x δτ

),()(),( RKqJKqJKqKK nnqx

Txqq ∈== ×

)( qKK qx →An effective interpretation of the above results can be achieved by regarding the robot arm as a MECHANICAL TRANSFORMER of displacements and forces from the joint space to the robot task space. Joint stiffness of the robot arm is a nonlinear function of the desired Cartesian stiffness i e


Joint stiffness of the robot arm is a nonlinear function of the desired Cartesian stiffness, i.e., the robot TCP stiffness in its task space and the robot configuration q.

PROBLEM!Off-diagonal entries of the joint space stiffnes matrix are NOT EQUAL to ZERO

∈= ×nnqxd

Tqd RKqJKqJK ),()(

≠∀≠= d jikKqfk 0)( ≠∀≠= ijqxdijqijq jikKqfk ___ ,0 ),,(

⎥⎥⎤

⎢⎢⎡

nqqq

nqqq

kkkkkk

22212

,1_2,1_1,1_

L

LStiffness of the cross-joint actuators

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

= nqqqq

kkk

K ,2_2,2_1,2_ MOMM

actuators


⎥⎦⎢⎣ nnqnqnq kkk ,_2,_1,_ L

Desired robot TCP stiffness as a process variable

Desired robot joint space stiffness as a function of the desired robot TCP stiffnes and also a function of the time

xdqdxdT tKqKqJtKqJ = ),,()()()( xdqdxd

KKKKKK

qqq

≠ΔΔ 0

),,()()()(

xxdxKqqqd KKKKKKq ≠Δ ≠−=Δ⎯⎯ →⎯Δ=− 0 0

mmT RKJKJtKK ×−− )()()( 1 mmxq

Tqx RKqJKqJtKqK ×∈= ,)()(),,( 1

Our ability to control of JOINT SPACE STIFFNESS MATRIX is essential for control of robot


ykineto-static behavior in the task space!

MORPHING TCP / ENDPOINT STIFFNESS

jik ijq ≠∀≠ ,0_

Technically TOO COMPLEX

REDUNDANCYSolution #1:

Joint actuation redundancy,polyarticular actuators, m = n

COMPLEX actuationSIMPLE kinematics

jik ∀0 0ΔK

kinematics

jik ijq ≠∀≈ ,0_

Solution #2:Kinematic redundancy

Solution #3:Joint actuation redundancy +

no limitations: 0=Δ xKSIMPLE actuationCOMPLEX

COMPLEX actuationCOMPLEX

monoarticular actuators, m < n kinematic redundancy

OPTIMAL FunctionalityTechnically and computationally complex

Technical COMPROMISEComputationally complex

COMPLEX kinematics

COMPLEX kinematics


Technically and computationally complexComputationally complex

Solution #1:Joint actuation redundancy

polyarticular actuators

Joint actuation redundancy generates off-diagonal entries of the joint space stiffness matrix, i.e., cross-joint stiffness. Generation of an arbitrary stiffness matrix in the robot task space requires polyarticular actuators

m = ny p q

redundant actuation of the 1 to N type - POLYARTICUALAR ACTUATION

Monoarticular soft drive q2Monoarticular soft drive q2

Monoarticular soft drive q1

⎥⎦

⎤⎢⎣

⎡= 1

00q

q kk

K


⎥⎦

⎢⎣ 20 q

q k

In case of redundantly actuated 2dof planar robot arm, the joint stiffness is the SUM of the monoarticular and the biarticular stiffnesses.monoarticular and the biarticular stiffnesses.

Biarticular cross-joint soft drive q12 Monoarticular soft drive q2soft drive q12 Monoarticular soft drive q2

Monoarticular soft drive q1

⎥⎦

⎤⎢⎣

⎡+

+= XXq

q kkkkkk

K 1


⎥⎦

⎢⎣ + XqX

q kkk 2

Actuation of the human arm is highly redundant! An extensive actuation redundancy in biological systems certainly has its purpose. This indicates that complex actuation system of human arm uses extensive actuation redundancy to effectively and precisely shape the endpoint stiffness.

The human musculoskeletal system has a spring-like y p gactuation, essential to our interaction with surroundings. These actuators are controlled by motor command and neural by motor command and neural feedback systems.

Th t k d d t diff i l ti ti tt i l l b d!


The task-dependent difference in muscle activation pattern is clearly observed!This kind of actuation is TECHNICALLY EXTREMELY COMPLEX to be effectively achieved!

BIARTICULAR VSA Solution with 6 or 12 servo motors and 2 output shafts

OUT #1

and 2 output shafts.

Bi ti l Biarticular coupling member


OUT #2

Output shaft and differential planetary gear assembly

BASIC BUILDING module of the bidirectional antagonistic module

Servo motorHarmonic driveWave generatorWave generator

Harmonic driveCircular splineHarmonic drive

Felx spline

Linear spring

Cam disc and cam follower mechanism for generating


Felx splinemechanism for generating nonlinear spring sharacteristics

Explicite stiffness control

Implicite stiffness control with embedded mechanical springs


Agonist Antagonist

Like in byological systems actuators work in pairs.This kind of actuation needs to have an agonist and an antagonist

Case #1: Linear spring joint stiffness

)(2 00 BA xxkkxF −+−= kdxdFkX 2−=−=

Case #1: Linear spring – joint stiffness

Case #2: Nonlinear spring joint stiffness

)()(2 20

2000 ABBA xxkxxkxF −+−−= )(2 00 BAX xxk

dxdFk −==

Case #2: Nonlinear spring – joint stiffness

002

02

0 BABA xxxxx +=

−=

Case #2: Nonlinear spring – equation of motion

Springs must be nonlinear


2)(2 00 BA xxx

−Springs must be nonlinear

Embodying Desired Behavior in Variable Stiffness Actuators

Mechanical subassembly with nonlinear springs dirven by two antagonist actuators that can variate the joint stiffnes – embodied desired behavior (developed in DLR Germany )


variate the joint stiffnes embodied desired behavior (developed in DLR Germany )

Biological solution of a Variable Stiffness Actuation


Solution #2:Kinematic redundancymonoarticular actuators

Position and orientation of the end-effector in redundant mechanism can be obtained by infinite number of postures / configurations of its links. The same holds for joint torques: ROBOT TCP FORCE CAN BE monoarticular actuators

m < n INDUCED BY INFINITE NUMBER OF ROBOT CONFIGURATIONS.

f )(nm

TCP

RR

qfx

→

= )(

Task space Joint space

A robot is kinematically redundant if: m < ni.e. if it has more degrees of freedom then those strictly necessary for performing the task asigned.

Redundancy is RELATIVE concept: the dimenzionality of the task space may be less then the dimenzionality of the robot operational space, i.e., the space in which robot can mechanically operate Task space is subspace of the operational space


operate. Task space is subspace of the operational space.

Monoarticular soft drive of the joint q2 and all other jointsjoint q2 and all other joints

⎤⎡ 00k

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

3

2

1

000000

q

q

q

kk

kK

⎦⎣ 300 qk

Only SUBOPTIMAL SOLUTION IS POSSIBLE with kinematically redundant robot!But we are engineers, we always look for a good compromise! Consequently, our task is to find g , y g p q y,a SUFFICIENTLY GOOD SOLUTION for the canonization problem of a joint stifness matrix by selecting the optimal robot configuration. With this approach we eliminate the needs for poliarticular actuation of the robot mechanism,


i.e., we are looking for the compromise where actuation redundancy will no longer be necessary!

PROBLEM!Only square Jacobian matrix is invertible

)( xqqJ δδ Given displacement of the )( TCPxqqJ δδ = probot TCP / Endpoint

)(1TCPxqJq δδ −=Joint displacement to be

calculated

)())(()()( 11 kTCPkkk txtqJtqtq δ−+ +=

Numerical integration


REDUNDANCY RESOLUTION AT THE DISPLACEMENT LEVEL

Two possible solutions: 1 ),( ×∈= mTCPTCP Rxqfx

1. Augmented (extended ) Jacobian method

qf ⎥⎤

⎢⎡ ∂

⎤⎡)(

1 ),(

),(×∈= r

TCPTCP

Ryqhy

qf

nrC

nmnn

CA RqJRqJR

qhq

qfq

qJqJ

qJ ××× ∈∈∈

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ ∂∂∂=⎥

⎦

⎤⎢⎣

⎡= )(,)(,

)(

)(

)()(

)(

As a result of extending the kinematics at the displacement / velocity level, Jacobian is no longer redundant. But there are two major drawbacks associated with this method:

The dimension of the additional constraint (secondary task constraint differentiable

q ⎥⎦⎢⎣ ∂

The dimension of the additional constraint (secondary task constraint differentiable function h(q)) should be equal to the degree of redundancy. This limitation makes the method not applicable for a wide class of additional / secondary tasks.The other problem is the occurrence of artificial singularities in addition to the primary p g p ytask kinematic singularities. There is a conflict between the primary and secondary tasks. Furthermore these additional singularities are task dependent which makes them hard to determine analytically.


2. Jacobian nullspace method

NONHOMOGENOUS SYSTEM

Kinematic redundancy and null-space stiffness 3

Redundant Jacobian matrix turns the robot inverse kinematics to the domain of underdetermined linear systems.

F t l t i t i f th i i t li d i i For rectangular matrices we use extension of the inversion concept - generalized inversion, or pseudo inversion. Generalized inversion requires additional conditions. Therefore, there is an infinite number of solutions.

qxqqJ →−)(min δδδThe least-norm of robot joint displacements condition leads to the Moore-Penrose pseudo inversion of Jacobian matrix

xqJq += )( δδ

Jacobian matrix.

The Moore-Penrose pseudoinverse exists for any matrix and is unique!

( ) mnTT RJJJJ ×−+ 1)()()()(

y q


( ) mnTT RqJqJqJqJ ×+ ∈= )()()()(

HOMOGENOUS SYSTEMEvery homogenous system has at least one solution, known as the zero, or trivial solution.

If the system has a square non-singular Jacobian matrix then the trivial solution is only l ti Th f l J bi t i f d d t b t h l ti t ith i fi it solution. Therefore, only Jacobian matrix of redundant robot has a solution set with an infinite

number of nontrivial solutions!

J 0)( δ mTCPxqqJ 0)( ==δ

Set of nontrivial solutions is called NULL-SPACE or KERNEL of JACOBIAN MATRIX

mq 0=δ Trivial solution

Set of nontrivial solutions is called NULL SPACE, or KERNEL of JACOBIAN MATRIX.

}0)(:{))(( mn qqJRqqJN =∈= δδ


})({))(( mqqqq

TWO FUNDAMENTAL SUBSPACES are associated with a linear transformation that is generated by Jacobian matrix of the kinematically redundant robot are its null space and its range space.

}:)({))(( nRqqqJqJR ∈= δδThe RANGE of the Jacobian matrix:

The NULL of the Jacobian matrix:

}0)(:{))(( qqJRqqJN mn =∈= δδ


))(())(( qJRqJN T=⊥ !!!

Null space calculation:

First, Jacobian matrix is transformed to reduced row echelon form (rref). This transformation always generates unique rref matrix. Jacobian matrix can be transformation in to rref by Gaussian elimination method.

m r = n - m

⎥⎥⎤

⎢⎢⎡ +

)()(00010)()(00001 ,11,1

qaqaqaqa nm

LL

LL

m( )⎥⎥⎥

⎦⎢⎢⎢

⎣

=

+

+

)()(10000

)()(00010)(

1

,21,2

qaqa

qaqaqJrref

nmmm

nm

LL

MMMMMMM

n

⎦⎣ + )()( ,1, qq nmmm


( ) 0)( = mqqJrref δ

0)()(0)()( ,111,11 =+++ ++ nnmm qqaqqaq

δδδδδδ L

0)()(

0)()( ,211,22

=+++

=+++ ++ nnmm

qqaqqaq

qqaqqaq

δδδ

δδδMMMM

L

0)()( ,11, =+++ ++ nnmmmmm qqaqqaq δδδ L

r = n - mr n mdimension of the nullspace

(degree of degeneration of the Jacobian matrix)


Generalized virtual displacement 4

1 )( xqJqq P δδδ == −Nonredundant robot kineto-statics

0)( )()( qqPxqJqqq JNc

NP δδδδδ +=+= +

PARTICULAR SOLUTION of the NONHOMOGENOUS SYSTEM

NULL SOLUTION for theHOMOGENOUS SYSTEM

Task-space displacementEnd-effector displacement, i.e., motion of the robot

mechanism that produces variation of the position and th i t ti f th d ff t

Null-space displacementSelf-motion displacement , i.e., internal motion of the robot mechanism that produces zero-variation of the

iti d th i t ti f th d ff t


the orientation of the end-effector position and the orientation of the end-effector

COMPLEMENTARY PROJECTOR Operator that projects an arbitrary n-dimensional vector to the null space of the JacobianPPP =

JNc qqP 0)()( ≡δ

vector to the null space of the Jacobian

)()( qJqJPPIP

PPPc

+=

−=

=

mJN qqP 0)( 0)( ≡δ)()( qJqJP =

0)( )()( qqPxqJqqq JNc

NP δδδδδ +=+= +

PARTICULAR SOLUTION of the NONHOMOGENOUS SYSTEM

NULL SOLUTION for theHOMOGENOUS SYSTEM

Task-space displacementEnd-effector displacement, i.e., motion of the robot

mechanism that produces variation of the position and th i t ti f th d ff t

Null-space displacementSelf-motion displacement , i.e., internal motion of the robot mechanism that produces zero-variation of the

iti d th i t ti f th d ff t


the orientation of the end-effector position and the orientation of the end-effector

0)]()([)( qqJqJIxqJq δδδ ++ −+= 0)]()([ )( qqJqJIxqJq n δδδ +

Displacement vector defined by the PRIMARY task

Displacement vector defined by the SECONDARY task

TASK FUSION MACHINEInference machine that provides formally consequent tool for SUPERPOSTION of the PRIMARY task: position and motion control of the robot arm and any other task of SECONDARY priority –NONCONFLINCTING INCLUSION OF SECONDARY TASK into primary control task.

SECONDARY task: Robot TCP STIFFNESS control / morphing

{ }qqdqqn KKKKqRqq −=ΔΔ=∈= ),min( 000 δδδ

S CO tas obot C S SS co t o / o p g


{ }qqdqqn KKKKqRqq −=ΔΔ=∈= ),min( 000 δδδ


Primary task: TCP position control Secondary task: TCP compliance control

0)]()([)( qqJqJIxqJq n δδδ ++ −+=

COST FUNCTION: scalar potential field defined over the robot configuration space

jiqkqunnijq ≠∀= ,)()(

)]1([defined over the robot configuration space - minimum perturbation condition

njq − )]1(2

[_

)min( qKΔ

⎤⎡

⎥⎥⎤

⎢⎢⎡

nqqq

nqqq

kkkkkk

KK ,22,21,2

,1_2,1_1,1_

L

L

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=≈ nqqqqqd

kkk

KK ,2_2,2_1,2_

L

MOMM


⎥⎦⎢⎣ nnqnqnq kkk ,_2,_1,_ L

Primary task: TCP position control Secondary task: TCP compliance control

0)]()([)( qqJqJIxqJq n δδδ ++ −+=

COST FUNCTION: scalar potential field defined over the robot configuration

,)()()]1([_ ≠∀=

−jiqkqu

nnijqdefined over the robot configuration space - minimum perturbation condition

)]1(2

[_ nj

0)min( qKΔ

0 ),()(0 >∂∂

−=∇−= λλλδ quq

quqSearching for the local minimumq

Gradient optimization method for selecting optimal null-space joint vector which minimizes influence of the cross-joint memmbers to the robot TCP stiffness.


Suboptimal solution! This solution always exists!

)( qqPq δδ =

Qasistatic Nullspace Projection

0)( )( qqPq JNN δδ =

DISPLACEMENT FORCE DISPLACEMENT-FORCE kineto-static duality cycle

0)( )( ττ qP TJNN =

Dynamic Nullspace Projection


SUGGESTED LITERATURE on

LINEAR SPACESLINEAR SPACES

H.Yanai, K. Takeuchi and Y. Takane, PROJECTION MATRICES, GENERALIZED INVERSE ,MATRICES, AND SINGULAR VALUE DECOMPOSITIONSpringer New York Dordrecht Heidelberg London, 2011, ISBN 978-1-4419-9886-6.


Example #1: MINIMAL r = 1 REDUNDANT ROBOT - MRR-R1

Numerical examples and experiments 5

+==+== 122111

)cos();cos()(

qqcqcclclqfx

m = 1 / n = 2 → r = n – m = 1

Example #1: MINIMAL r 1 REDUNDANT ROBOT MRR R1

+== 211211 )cos();cos( qqcqc

JACOBIAN MATRIX

[ ]( )[ ]−+−=

==

12212211

2,11,1 )()(slslsl

qJqJJ(q)( )[ ]

+==+

211211

12212211

)sin();sin(

qq sqsslslsl

( )[ ] ⎥⎤

⎢⎡

+ 1qlll

δδ

Direct kinematics


( )[ ] ⎥⎦

⎢⎣

−+−=2

122122111 qslslslx

δδ

Moore-Penrose PSEUDOINVERSION:

221

211

1,1

)()()(

)()( 11

q+ JqJqJ

qJJ ⎥

⎥⎤

⎢⎢⎡

⎥⎤

⎢⎡ +

+

22,1

21,1

2,1

2,11,1

)()()(

)()()(

)(1,2

1,1

q + JqJqJ

qqqJ

qJ

⎥⎥⎥

⎦⎢⎢⎢

⎣

=⎥⎥⎦⎢

⎢⎣

= ++

Inverse kinematics

)( xqJq δδ = +

( ) ( ) 1122

122112

1222

122112

1 1 xsl

slsl sl+ sls lq

qδ

δδ

⎥⎦

⎤⎢⎣

⎡ +

+−=⎥

⎦

⎤⎢⎣

⎡


⎤⎡⎤⎡

RANGE SPACE of Jacobian matrix

12

1

)()(

1,2

1,1 xqJqJ

qq

δδδ

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡+

+

2122

1221121 )(

)(

1,2

1,1 qsl

slslqqJqJ

q δδδ +== +

+

⎤⎡

Reduced row echelon form of Jacobian matrix .... NULLSPACE of Jacobian matrix

( )[ ]

[ ]2

112212211

1))(())((

0

JfJNqq

slslsl =⎥⎦

⎤⎢⎣

⎡+

δδ

2

1

12211

122 01qq

slslsl

δδ

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+[ ]

1222,1

)()(

)(

1))(())((

llsl

JqJ

qaa

aqJrrefqJN

xx

x

+===

==

212211

1222

1,1

2,11

212211

)()(

qslsl

slqqJqJ

q

q

δδδ+

−=−=

⎦⎣⎦⎣


122111,1 )( slslqJ + ,

PARTICULAR SOLUTION in accordance to the Moore-Penrose Jacobian inverse

2122

122111 q

slslslq δδ +

=212211

1221 q

slslslq δδ+

−=


COMPLEMENTARY PROJECTOR to the nullspace of the Moore-Penrose Jacobian inverse

[ ]⎥⎥⎥⎤

⎢⎢⎢⎡

−=−= +

22,1

21,1

2,11,12

2,12

1,1

2,1

))(( )()()()()(

)()()()(

)(

)()(1 qJqJqJq + JqJ

qJqJq + JqJ

qJ

qJqJP qJNc [ ]

⎥⎥

⎦⎢⎢

⎣− 2

2,12

1,1

1,12

2,12

1,1

2,11,1

)()()(

)()()()()()(

q + JqJqJ

q + JqJqJqJqq

( ) ( )( )

( ) ( ) ⎥⎤

⎢⎡ +

− 2212212211

22122

lllslslsl

lllsl

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ ++

++

−

++=

2122

212211

122112

1222

12211

12212211

2122

212211

2122

212211))((

sl+slslslsl

sl+slslslslsl

sl+ slslsl+ sls lP qJNc

( ) ( ) ( ) ( )

( ) ( )( )

( ) ⎥⎦

⎤⎢⎣

⎡++−+−

+=

⎦⎣ ++

1221112212211

1221221112222

1221221112212211

1 slslslslsl

slslslslsl+slsl

sl+ slslsl+ sls l


( ) ( ) ( ) ⎦⎣ +++ 122111221221112212211 slslslslslsl+ sls l

SECONDARY TASK DISPLACEMENT VECTORProjected arbitrary configuration vector to the nullspace of the Jacobian matrix:

[ ] [ ] TNNN qqqqJqJq )()(1 201000 δδδδ =−= +

0222,1

21,1

2,11,1012

2,12

1,1

2,110 )()(

)()()()(

)(q

q + JqJqJqJ

qq + JqJ

qJq N δδδ =−+=

( ) ( )( )

( ) ( ) 022122

212211

12212211012

1222

12211

122

,,,,

q sl+ sls l

slslslq sl+ sls l

sl δδ+

+−+

+=

021,1

012,11,1

20

)()()(q

qJq

qJqJq N δδδ =+−=

( )( ) ( ) ( ) ( ) 0222

122110122

12212211

0222,1

21,1

0122,1

21,1

20 )()()()(

qslslqslslsl

qq+ JqJ

qq+ JqJ

q N

δδ

δδδ

+−

+−=

+


( ) ( ) ( ) ( ) 022122

212211

012122

212211

qsl+ slsl

qsl+ sls l ++

⎤⎡ )()()(2 JJJ

⎥⎥⎦

⎤

⎢⎢⎣

⎡=→=

)()()()()()(

22,12,11,1

2,11,11,1

qJqJqJqJqJqJ

kKkK xqxx


Joint stiffness matrix

( ))()()( slslslkqJqJkqu +==

COST Function:

( )122111222,11,1 )()()( slslslkqJqJkqu xx +==

SCALAR POTENTIAL Field


⎥⎤

⎢⎡ +−+−∂

∇)2sin()22sin()()( 2211

2221 lqqlklqqkququq xxδ

OPTIMIZATION FUNCTION - Negative gradient of adopted cost function

( )⎥⎦⎢⎣ +++−

=∂

−=−∇=)sin()sin(2)cos(

)()(11212212

22112210 qlqqlqqlk

quq

quqx

xxδ

NEGATIVE GRADIENTS and top view of the scalar potential field created by the adopted cost functionthe adopted cost function.


Relative variation of generated TCP stiffness over the entire configuration space

,)()(),,( 1 ∈= ×−− mmxq

Tqx RKqJKqJtKqK

D i d tiff K d 1

0

,)()(),,(

0 ≠−=Δ⎯⎯ →⎯Δ=− ≠Δ xxdxKqqqd

xqqx

KKKKKK

qqq

q

Desired stiffness: Kxd = 1


An example of robot internal / nullspace motion for given x_d = (L1 + L2) / 2

Robot mechanism parameters:l_1 = 0.36 ml_2 = 0.7* l_1 = 0.252 m

Desired position:l 1 0 36

l_2 = 0.252 m

Nullspace motionp

x_d = (L1 + L2) / 2 = 0.306ml_1 = 0.36 m

q optimal:q_optimal:

q1 = - 0.7583 (rad)q2 = 2.1510 (rad)

x_d = 0.306m


Robot nullspace motion and corresponding value of generated TCP stiffness

Nullspace motion trajectory for given x_d = 0.306 m

Desired stiffness: Kxd = 1Desired stiffness: Kxd = 1


Desired stiffness: Kxd = 1

End of nullspace range End of nullspace range

q_optimal for x_d = 0.306mm :

q_1 = -0.7583 (rad)q_2 = 2.1510 (rad)


Example #2: MINIMAL r = 2 REDUNDANT ROBOT - MRR-R2

m = 1 / n = 2 → r = n – m = 2

Analytical model ...............


POTENTIAL FIELD of the COST FUNCTION u(q, KX)

Volume potential field over the

Orthogonal slice at q2 = pi/2

Volume potential field over the entire configuration space

q2, (rad)q1, (rad)

POTENTIAL FIELD SLICE for q_2 = pi/2Stream lines indicate negative ggradient of the potential field sliced surface. Stream lines smoothly converges to the nearest minimum of the potential field.


p

q1, (rad) q3, (rad)

POTENTIAL FIELD of the COST FUNCTION u(q, KX)

Volume potential field over the Volume potential field over the entire configuration space

ISOPOTENTIAL SURFACE for u (q)= 0.01*u_max

This is the surface by which we can This is the surface by which we can generate the norm of off-diagonal elements with jus 1% of the maximum value of the potential field generated b th d t d t f ti ( )


by the adopted cost function u(q)

EXPERIMENTAL SETUP USED AT CMSYS LAB FOR NULLSPACE STIFFNESS EVALUATION

7dof Yaskawa SIA 10F robot arm which is reduced to 3dof redundant SCARA configuration

CMSysLab at University of Belgrade


NULL-SPACE STIFFNESS MEASUREMENTInfluence of the robot self motion to the TCP stiffness parameters

Measuring setup for TCP excitation in X DIRECTION

Measuring setup for TCP excitation in Y DIRECTION

Mechanism for generation of external force with integrated

Laser sensor for TCP displacement



external force with integrated single axis force sensor

displacement measurement

Posture # 4Recorded displacements of the TCP in horizontal xy plane for 4 POSTURES in the robot nullspace induced by CONSTANT EXCITATION FORCE of 40N applied in x and y direction

Posture # 3 Posture # 4

direction.

Posture # 3 Posture # 4

Posture # 3

Posture # 2

Posture # 3

Posture # 2

Posture # 1Posture # 1


TCP stiffness is posture dependant and highly nonlinear!

Strain gaugesBonding of strain gauges to the link interface flange

Soft joint developed at CMSyLab

Servo motor

EncoderDistributed multiprocessor s stem for force/position


system for force/position control of the VSA actuator

ReducerSensorized link interfacing flange


Soft joint behaves as a LINEAR SPRINGVery soft Stiff

Joint angular displacementdisplacement

Joint reactive torqueq

Joint displacementfunction


Soft joint behaves as a SPRING WITH CONSTANT REACTIVE FORCEVery soft Stiff





Soft joint behaves as a PASSIVE, BACKDRIVABLE SYSTEMLight Heavy





ACCURACY OF THE NULL-SPACE MOTION MEASUREMENT‘motion leakage’ from the robot nullspace to the task space

Yaskawa SIA 10F robot R d d t ti l t d b t

Robot virtual TCP support / pivoting pin (no Redundant articulated robot

arm with 7 d.o.f.Payload : 10 kgArm reach: 720 mmT t l i ht 60k

pivoting pin (no motion is allowed theoretically)

Total arm weight: 60kg

Laser sensor (optical curtain)

Precise XY rotary table ith h ll t l (optical curtain)with hollow central area



Recorded variation of the robot TCP position while swinging the end link within the interval link within the interval of -40 to + 40 deg.

Pivoting pin is attached Pivoting pin is attached to the robot end-plate at the distance of 275mm from the third j i i (TCP joint axis (TCP location) l1 = 360 mmll2 = 360 mml3 = 275 mm


Th k f tt tiThank you for your attention

Self motion and null space stiffness control researchSelf motion and null-space stiffness control research


This research is supported by the Serbian Ministry for Science and Technology Development under the grant No. TR35007: Smart Robotic Systems for Customized Manufacturing

advances in compliant and soft controlled structures for ... · 10.07.2014 mesrob 2014 - prof dr...

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