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
3 Sensor Based Robotic Manipulation and Locomotion
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.
4 Sensor Based Robotic Manipulation and Locomotion
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.
5 Sensor Based Robotic Manipulation and Locomotion
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
6 Sensor Based Robotic Manipulation and Locomotion
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)(
7 Sensor Based Robotic Manipulation and Locomotion
Number of DOF in a LoopFor a serial robot:
for n joints, r constraints => n-r degrees of freedom at TCP
r
n
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
8 Sensor Based Robotic Manipulation and Locomotion
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 )()(
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?
9 Sensor Based Robotic Manipulation and Locomotion
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
10 Sensor Based Robotic Manipulation and Locomotion
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
11 Sensor Based Robotic Manipulation and Locomotion
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
13 Sensor Based Robotic Manipulation and Locomotion
Power graspPinch graspPrecision grasp
Grasp Types
14 Sensor Based Robotic Manipulation and Locomotion
Opposing Thumb
15 Sensor Based Robotic Manipulation and Locomotion
The Complete Finger System
16 Sensor Based Robotic Manipulation and Locomotion
Stiffness Control
17 Sensor Based Robotic Manipulation and Locomotion
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
18 Sensor Based Robotic Manipulation and Locomotion
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
19 Sensor Based Robotic Manipulation and Locomotion
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
20 Sensor Based Robotic Manipulation and Locomotion
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
21 Sensor Based Robotic Manipulation and Locomotion
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
22 Sensor Based Robotic Manipulation and Locomotion
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
23 Sensor Based Robotic Manipulation and Locomotion
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.)
24 Sensor Based Robotic Manipulation and Locomotion
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
25 Sensor Based Robotic Manipulation and Locomotion
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?)
26 Sensor Based Robotic Manipulation and Locomotion
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)
27 Sensor Based Robotic Manipulation and Locomotion
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)
28 Sensor Based Robotic Manipulation and Locomotion
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:
29 Sensor Based Robotic Manipulation and Locomotion
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
30 Sensor Based Robotic Manipulation and Locomotion
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:
31 Sensor Based Robotic Manipulation and Locomotion
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 #
32 Sensor Based Robotic Manipulation and Locomotion
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
33 Sensor Based Robotic Manipulation and Locomotion
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 -
35 Sensor Based Robotic Manipulation and Locomotion
• 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
36 Sensor Based Robotic Manipulation and Locomotion
)),(),((),),((
),),(()(
,
,
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
C. Ott & T. Wimböck
37 Sensor Based Robotic Manipulation and Locomotion
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
C. Ott & T. Wimböck
38 Sensor Based Robotic Manipulation and Locomotion - 38 -
39 Sensor Based Robotic Manipulation and Locomotion
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
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