SBRML Part4 Motion of Complex Kinematic Robotic Structures · 1 3 2 1 2, J J q q q v J J ... f : f...

Motion of Complex Kinematic Robotic Structures

2 Sensor Based Robotic Manipulation and Locomotion

Example: Complex Kinematic Chain• Tree structure• Closed kinematic loops

So far in this course: only serial robotic structures


Tree Structures

^

Kinematics and dynamics can be computed in principle identically as for serial manipulators (for example using Euler-Lagrange or Newton-Euler formalism for the dynamics).

)(),()( qgqqqCqqM

4

1

1

1

F

F

A

A

q

qqq

q

Desired values can be assigned for all joints independently.


Closed Kinematic Loops

p

aqq

q

^

O

active joints

passive joints

Assumptions: • Point contact• The contact point between finger and object does not change.

Under the given assumptions, the contact can be modelled as a virtual passive joint.

The loop introduces mathematical constraintsbetween the joints.

Passive joints are necessary in order to minimize the constraint forces which would appear if all joints were active due to positioning inaccuracies.


Forward Kinematics for Closed Loops

active

passive

active

passive

21 TCPTCP

2´TCP1TCP

TCP

p

aqq

q

1A

2A

4A

3A

3A

Remark: inverse kinematics (given TCP, find q) is computed in the known manner for both serial manipulators resulted by breaking the loop. It is generally much easier than direct kinematics, since the two manipulators have simple structures.

1A

2A 3A

4A

(attached to )32AA

General procedure:• Breaking the link at a certain joint (typically the output)

=> a tree structure results• Computation of the geometric model for the tree structure

• From the loop condition (constraint)the passive joints are expressed.

• The TCP is expressed by substituting

),(),,(21 paTCP

OpaTCP

O qqTqqT

pq))(,(1 apa qqqTCP ))(,(2 apa qqqTCP

or

),(),(21 paTCP

OpaTCP

O qqTqqT

OOusually, only the position of active joints is measured by encoders

pq


Example: Four Bar Linkage

active

passive

active

passive

2´TCP1TCP

TCP

p

aqq

q

1A

2A

4A

3A

3A

1l

2l

Generally: ITTTT '32

21

14

43

Three constraint equations result:

1q

2q

4q

3q

14 qq 12 qq 13 qq

4 joints3 constraints

=>The robot has 1 DoF at the TCP

=> it follows that only one joint can be actively positioned.

In the simple case of the parallelogram, the solution can be written directly :

corresponding to two translationsand one rotation in the plane

3q R ,0)(


Number of DOF in a LoopFor a serial robot:

for n joints, r constraints => n-r degrees of freedom at TCP

r

n

qq

R

R

,0)( constraints

N links, n 1dof joints

For an arbitrary closed structure with N links, n joints, k DOF/joint:

• Planar case: 3 DOF/Link, 3-k constraints/joint

n

iiknNDoF

1)(3

• Spatial case: 6 DOF/Link, 6-k constraints/joint

n

iiknNDoF

1)(6

Example at the whiteboard

(known asGrübler formula)

is valid only for independent constraints


Differential Kinematic Model

active

passive

3v

1A

2A

4A

3A

3A

1l

2l3v

p

aqq

q

By equating the Cartesian velocities at the separated joint one obtains the differential constraints:

4

34,33

2

12,1333 )()(

qq

qqJqq

qqJvv

0)(),(

4

3

2

1

),,,(

4,332,13

43210

qqqq

qqJqqJqqqqJc

Together:

0)( 1 qqJc

By introducing the constraints:

differential constraints

"velocity constraints"

33 vv

How do I compute the Jacobian?


Differential Kinematic ModelOne can obtain the equations also by differentiating the constraints:

0),( pa qq

0)(

,)(

0

,,

p

a

J

p

pa

a

paqq

qqq

qqq

c

0)( qqJ ac

differential constraints

Or: 0)(),(

p

aacpaca qq

qJqJ

since is quadratic (number of constraints = number of passive joints) )( acp qJ

aacaacpp qqJqJq )()(1 (if is not singular))( acp qJ

The Cartesian velocity of the open kinematic structure can nowbe computed in al classical way.

e.g., for the four bar linkage: 1331

1333 2121

, qJJqq

JJv


1A

2A

4A

3A3A

3

3

)(),()( qgqqqCqqMB

Bp

BaB

First, the dynamics of the opened chain (tree structure)is computed with following additional assumptions:• All joints are actuated.• The torques are chosen such that the robot performsthe same motions as for the closed loop.qqq ,,

In reality, we have and the closed chain is held together by

at the virtual separation point the constraint force .

0a

The forces are dual to the constraintvelocities at the separation point:

v

T

qqq

qqv

)()(

Since the real and the virtually separated robot must move identically,it follows: ),,( qqqB ),,( qqqB

Dynamics of Closed Kinematic Structures


Remark: In der classical mechanics one obtains this relation through the constrained Euler-Lagrange equation:

)(),(),()(),(),( qUqqTqqLqq

dqqqdL

qdqqdL

dtd T

mit

),,( qqqB

Tcp

Tca

Bp

BaaJJ

0from the second equation:

BpTcpJ

BpTcp

TcaBaa JJ

or simply withBTBa J TcpT

caTB JJIJ ,

The dynamics of the closed loop can be computed from the dynamics of the tree structure and from the constraint conditions.

Dynamics of Closed Kinematic Structures

Dexterous Hands


Power graspPinch graspPrecision grasp

Grasp Types


Opposing Thumb


The Complete Finger System


Stiffness Control


Kinematics of Dexterous Hands

Grasp planning: selection of a grasp, which should be able to:

• Withstand forces acting on the object in arbitrary directions- Force closure grasp

• Move the object in arbitrary directions: - Manipulable grasp

Assumptions:• Contact point on the object is known (or can be measured).• Contact point does not change during motion.• Object and fingers are rigid bodies• Exact geometric model of the object and the hand are available

In the following, precision grasps are treated


Contact Models

OiC

• Point contact without friction:

iz 1iz1iC

normal

ciF

Rcicici ffF ,

000100

only force in normal direction

• Point contact with Coulomb friction:

cicicici FCffF

,

000000000100010001

33ci ffff:fFC 2

22

1,RR3

friction coefficient

friction cone


Contact Models

OiC

Soft finger model:

iz 1iz1iC

normal

ciFcicicici FCffF

,

100000000000010000100001

333ci ffffff:fFC 42

22

1 ,,RR4

friction coefficienttranslation

friction coefficientrotation

The contact can transfer also rotations around the z-axis

generally:

cicicicici FCffBF ,

selection matrix(wrench basis) friction cone


Grasp Matrix

OiCiz 1iz

1iC

Normal

ciFof om

ciici

G

ciToci

ci

G

cioci

ocioci

oci

o

ooi

fGfBAd

fBRRp

Rmf

F

ci

i

ˆ

0

Contribution of contact forceat the coordinate system O:

ciF

c

fkG

kk

iciio Gf

f

fGGfGF

c

1

11

,,

The total wrench in object coordinates is

FCfGfF cco ,or simply:

ocip̂

grasp matrix

ociAd is called theadjoint matrix


Example: Point Contact without Friction

Rcicio

ciocioci

oci

oi ffRRp

RF ,

000100

ˆ0 iii

oci zyxR ,,

c

kG

kock

k

oco Gf

f

f

zpz

zpz

F

1

11

1ˆˆ

The grasp matrix results as:

ciioci

ioi f

zpz

F

ˆ

By performing the multiplication one obtains:

ioci zp lever arm

normal


Force Closure GraspA grasp is force closure if it can resist any externally applied wrench eF

FCfFGf cec ,

Definition: A grasp force is called internal force if it generates no net object force cN ff

FCfGf NN ,0

O1C1z 2z

2C1cfExample: 2cf

Theorem (without proof – Murray & al.):A grasp is force closure if and only if G is surjective and if the grasp admitsinternal forces


Evaluation of Force Closure GraspsGenerally a difficult problem, depending on the friction cone.For point contact without friction, simple solution:

R

icc

kG

kock

k

oco fGf

f

f

zpz

zpz

F ,ˆˆ

1

11

1

Theorem ( Murray & al.):Assume point contacts without friction. are the columns of the grasp matrix.Following statements are equivalent:1. The grasp is force closure.2. Each object force can be generated by a positive linear

combination of

3. the positive convex hull of all contains a vicinity of the origin

iG

pRoFiG

pR ociio FfGF ,

iGp=3 in the planar and 6 in the spatial case

(Schlegel, Buss & al.)


ExampleIn the planar case we have

z

y

x

m

ff

Fo

000

or simply

z

y

x

mff

Fo3

and correspondingly the grasp matrix

zoczoc

yky

xkx

zpzpzzzz

G

1111

1

1

3ˆ,,ˆ

,,,,

babaG 1010

0101

000010100101

G

grasp convex hull

a

b


Overview: Generation of Optimal Grasps

Problem statement:

1. What can the grasp achieve?Grasp Wrench Space

2. What should the grasp achieve?Task Wrench Space

3. Can it fulfil the task?Quality measure

The available fingertip forces are limited. Under these constraints, one wouldlike to generate an "optimal" grasp with respect to possible disturbances.

(Where should the fingers best be placed?)


Repetition

Collision testhand configuration

Grasp evaluation

Input Data

Grasp generation

Rejection of obviously

invalid grasps

Force closureQuality measure

Output of optimal grasp

Overview: Grasp Planning(C. Borst et al. 2003)


Differential Finger Kinematics

O1C 2C1cx

2cx

hand surfaceHqJxc 11 o

Tc xGx 1

ox

Duality vector - covector:

cio GfF

oT

c xGx

By computing finger tip velocity over the finger and the object: cx

oT xGqJ

A grasp is completely described by . FCGJ ,,

Remark: the velocities of the passive joints are already eliminated by using the selection matrix B. One can therefore compute the object velocity directly from thejoint velocity of the fingers.

(in contrast to a general parallel kinematics)


Manipulability

oT xGqJ

Velocity reachable bythe fingers:

Velocities required for omni-directional object motions

}Im{ Tc Gx }Im{Jxc

Manipulability: }Im{}Im{ JGT

J

TJ G

TGq cx ox

cf of

Transformation of forces and velocities:


Remark: The passive degrees of freedom at the contacts (e.g. rotations for point contact)are removed by the selection matrix in the grasp matrix.

Hand Dynamics

O1C1z 2z

2C1cf 2cf

hand surface

virtual 6 DoFpassive joint

Opening the loop at the contact points,the constraint forces are .cifPassive joints: the virtual object jointsActive joints: finger joints

oT xGJq 1

T

ffff JqgqqqCqqMBa

)(),()(

GxgxxxCxxMBp

ooo

)(),()(0

with differential constraint

By expressing from the first equation and insertion in the second, the object dynamics results:

oFwgxwCxwM )()()(

To GJF without proof

oxq

w


Zusatzfolie: HanddynamikBemerkung: Die hier abgeleitete Handdynamik gilt unter folgenden Annahmen:1. Der Griff ist kraftschlüssig und manipulierbar.2. Die Kontaktkräfte sind innerhalb des Reibkegels.3. J ist invertierbar (nichtsingulär und nichtredundant)

AdtdMAACACwwC

gAgwg

AF

AMAMGJJMGMwM

fT

fT

o

fT

o

To

fT

oTT

fo

),(

)(

)(11

mit

#TT GJA 111# )( T

fT

f JJMJMJ dynamisch konsistente Pseudoinverse

oFwgxwCxwM )()()( allgemein:


Hand Control

oFwgxwCxwM )()()(

All controllers, which have been developed for redundant systems can alsobe applied for hand control.

For example, the feedback linearization controller:

)()()( 21 wgxwCececxwMF do xxe d

Closed loop dynamics:021 ecece

To ensure that the contact forces stay within the friction cone, the controller must contain additional internal forces:

cf

The joint torques in the fingers are then:

NT

oT fJFGJ #

oT FGJ #


Generation of Internal Forces

• alternative passivity based approach: Generation of internal forces and object forces using potentials and virtual springs

The problem can be formulated as a convex optimization problem and can therefore be solved easily numerically.

Alternative problem formulation:Given the external forces , find the minimal contact forces such that the contact forces lie in the friction cone:

oF cf

FCfFGf coc ,

=> unique global minimum! (Schlegl, Buss & al.,’97)

• If the grasp matrix is not quadratic, the question of distributingthe object force over the finger has to be answered:oF

oco FGf # not unique, depending on the choice ofcof

• The internal forces have to be chosen such thatNf FCfff Ncoc

#G


Passivity-Based Object-Level Impedance Control for a Multi-fingered Hand T. Wimböck, Ch. Ott and G. Hirzinger, IROS 2006.

qqDqqVqgτT

)()()(

)),(),((),),(()( ,

hchohck

hodhohoS

qHqHVHqHVqV

Passivity Based Object Impedance Control for Hands

The force distribution problem is solved here implicitly, by the choiceof the springs and their rest lengths.

Springs for grasping

Spring for object movement

34 Sensor Based Robotic Manipulation and Locomotion - 34 -


• Control the “grasping” forces via an additional virtual spring.

• Stiffness matrices must be compatible!

)),(),((),),((

),),(()(

,

,

crlS

rdrrS

ldllS

qHqHVHqHVHqHVqV

)),(),((),),(()( ,

crlS

odooS

qHqHVHqHVqV

• Introduce an virtual object frame [Natale 2003]

• moving the two arms with only one spring connected to this virtual frame.

Generalization of the Single Arm Impedance Control

Object-Impedance Control

))(),(( qHqHH lro

qqDqqVqgτT

)()()(

C. Ott & T. Wimböck

Extension: Two Arm Manipulation


)),(),((),),((

),),(()(

,

,

crlS

rdrrS

ldllS

qHqHVHqHVHqHVqV

Generalization of the Single Arm Impedance Control

• Control the “grasping” forces via an additional virtual spring.

• Stiffness matrices must be compatible!

Two-Armed Manipulation



Connect the two-arm impedance with the virtual object frames of the hands, rather than to the end effector frames.

Intuitive combination of the two-armed impedance behavior and the object level impedance for the hands

qqDqqVqgτT

)()()(

Two-handed Manipulation


38 Sensor Based Robotic Manipulation and Locomotion - 38 -


M

D

Impe

danc

e co

ntro

lA

dmitt

ance

con

trol

w

vf

Compliant control of the entire robot

• 53 active dof• 150 kg

Rollin’ Justin

Human-Robot-Interaction

Date post:	03-Sep-2019
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SBRML Part4 Motion of Complex Kinematic Robotic Structures · 1 3 2 1 2, J J q q q v J J ... f : f...

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