# Redundant Manipulators - IRCCyN

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Plan of the Presentation

Definition of Redundant Manipulators

Direct Differential Kinematics

Inverse Differential Kinematics :

-

General Pseudo
-
inverse Solution

-

Damped Least
-
Square Solution

Redundancy Resolution via Optimization

(Liégeois, 1977
Baillieul
, et al., 1984)

Analytical Method for redundancy resolution
(
Ivlev
, et al., 1997)

Heuristic Method for Redundancy resolution
(Marques, et al., 2009)

-
Priority Formulations for Kinematic Control
(
Baerlocher

et al,1998)

Directional Redundancy for Robot Control
Mansard, et al., 2009

Integrate Unilateral Constraints into
SoT

(Mansard, et al., 2009)

Conclusion and Perspectives

2

Kinematically Redundant Manipulators

(
Chiaverini
, et al., 2008)

Redundant Manipulators
: When a robotic manipulator has
more DOF than those strictly required to execute a given task.

Increase dexterity

Extend the applications workspace

Resolve the singularity, joint limits and workspace obstacles
problems

Increase robustness to faults

improve reliability

Allow
self
-
motions

(internal motions) of the manipulator

Minimize torque/energy

over a given task (manipulator can
achieve a higher degree of autonomy).

Imitate Human arm
: 7 DOF (3 in the shoulder, 1 in the elbow ,
3 in the wrist) without considering the DOFs in the fingers.

3

Kinematically Redundant Manipulators

7
-
DOF
DLR Lightweight
Robot

Institute of Robotics and
Mechatronics

7
-
DOF Mitsubishi

PA
-
10 Manipulator

4

Example of redundant robots:

Kinematically Redundant Manipulators

8 DOF
-

DEXTER robot

by
Scienzia

Machinale
.

Hyper
-
redundant robots

(
Snakey
)

large number of DOF (snake
-
like robots…)

5

Example of redundant robots:

-
Oriented Kinematics

First order differential kinematics:

velocity

M
×
N

matrix

Joint space

velocity

6

Configuration
of N
-
joint Serial
manipulator

Location of the
manipulator
end
-
effector

Redundant robot

Inverse Differential Kinematics

The General Solution:

This solution provides all least
-
squares solution to the end
-

Problem:

Near singular configurations

Excessive joint velocity

1
st

Solution:

Modify the planned trajectory

Real
-
time control!

2
nd

Solution
: Joint space interpolation when planned trajectories
is close to a singularity

(Taylor, 1979)

Orthogonal projection
matrix in the null space of
J
t

Arbitrary joint
-
space velocity

Pseudo
-

inverse of
J
t

7

Inverse Differential Kinematics

The Damped Least
-
Square Solution:

This Solution satisfies the condition:

Choice of

:

Small values of

:
Accurate solution, low robustness in the
singular and near
-
singular configurations.

High values of

:
Low tracking accuracy.

Use of varying

Ensures continuity and good shaping of the solution

Infinite solutions for the inverse kinematics problem

Should
consider more criterions

Damping factor

Far from singularity

䍬潳Ct漠獩s杵g慲楴礠

h楧i

8

(
Chiaverini
, et al., 1991)

Redundancy Resolution via

Optimization

Performance Criteria
:

Singularity avoidance

Avoid mechanical joint limits
Minimize the cost function

Local Optimization:

(
Baillieul
, et al., 1984)

Pseudo
-
inverse solution
: minimize

Another solution:
in the direction of the anti
-
scalar configuration dependent performance criteria
H(q)
.

Maximize the manipulability measure

Maximize the condition number

Maximize the smallest singular value.

q
0
q
Scalar step size

Gradient of H at the current joint configuration

9

Local Optimization:

(
Liégeois
, 1977)

Redundancy resolution scheme:

+
Simplicity, can be used in real
-
time kinematic inversion.

-

Local optimization
:
-

Choose the value of the scalar step size
k
H

-

Require a lot of knowledge about environment

-

Less reactive to change in the goal or environment.

Global Optimization:

Choose to minimize integral criteria of the form

Problem:
The solution of this problem may not exist

!!!

If
H(q)

is a quadratic form in the joint velocities or accelerations

Solution can be found.

0
q
10

Redundancy Resolution via

Optimization

Analytical Method

for redundancy resolution
(
Ivlev
, et al., 1997)

The redundant robot is
, which limit
the flexibility of the robot and adapt it to the working space,
without fixing some of the joints.

Forward kinematics

Inverse kinematics

Redundant kinematic structure (N>M):
r=N
-
M redundant joints
.

Introduce
r

-

Limit the robots flexibility

-

Allow the demanded end
-
effector movement

-

Restrict the actual robots work
-
space with respect of obstacles

Vector of joint
variables

End
-
effector

pose

11

Extend the redundant kinematic chain

with
r

(each with 2 non
-
actuated spherical joints at both sides).

An imaginary fixed point in Cartesian space (
Anchor point
A
k
).

The redundant kinematic chain at the other end.

Value of

and coordinates of
A
k
!

The additional equation can be then written:

Choose the length of

to avoid obstacles…

The selection of

and the coordinates of
A
k

are dependant.

with 3 rotational joints)

A goal point {
x
G
,y
G
}

12

Analytical Method

for redundancy resolution
(
Ivlev
, et al., 1997)

Pseudo
-
Inverse Jacobian Method:

The final position will be reached in
K

iterations,
with an approximately linear path.

For every iteration

Small variation of
q

Apply direct kinematics

Determine the
end
-
effector new position (until the target
position is intended).

Only
after obtaining the final configuration
,
robot joints are physically rotated between the
initial and the final configurations.

Heuristic Method

for Redundancy resolution
(Marques, et al., 2009)

Joints
variation

Displacement
discretized

in K intervals

Constant column
-
vector
obtained experimentally

13

Joint with larger influence first:

1
-

Compute

virtual positions of the end
-
effector for a variation

2
-

Compute

distance
d
i

between goal and each virtual position of 1

3
-

Determine

the joint rotation that minimizes the distance

4
-

Execute
in virtual model the previous rotation and returns to 1
while the end
-
effector doesn't arrive to the goal.

Last joints first:

-

In each iteration, it is first rotated by degrees the
joint
closer to the end
-
effector
.

-

If none of these two rotations approaches the end
-
effector to
the goal, the next joint is rotated (and so on).

-

When the rotation of degrees doesn't decrease the distance
of the end
-
effector to the goal, then the value of is reduced.


14

Heuristic Method

for Redundancy resolution
(Marques, et al., 2009)

i





Experimental results and
Comparaisons
:

-

The 2
nd

heuristic method (
last joints first
)
is faster than
the 1
st

one (
joint with the larger influence first
).

-

These two heuristic methods present
similar position errors
.

-
Different final conf.
but same final end
-
effector pose
.

-
Pseudo
-
inverse
jacobian approach
is generally
faster
but less accurate

than the heuristic
methods.

15

Heuristic Method

for Redundancy resolution
(Marques, et al., 2009)

Example:
the human kinematic control

Large number of DOF, tree structure, unstable equilibrium …

R
equires an approach for multiple tasks prioritization
(keeping balance is more important than reaching an object)

Formulation of the problem:

(T
1
, …
T
t
)

with an order of priority
(T
i

has priority over T
i+1
)
.

T
i

controls location and/or orientation of one or more end
-
effectors simultaneously, and is defined by

Find a joint velocity vector such that every kinematically
T
i

is satisfied, unless it conflicts with a higher
-
T
j

(j <
i
)
; in this case, its error has to be minimized.

-

Priority Formulations

for the Kinematic Control
(
Baerlocher
, et al., 1998)

16

,
i i i
T J x

-
priority formulation (F1)

Generalization:

where

Amelioration:

-
priority formulation (F2)

17

-

Priority Formulations

for the Kinematic Control
(
Baerlocher
, et al., 1998)

Comparison of the two formulations and Simulation results:

Computational cost of the formulation
: depends on the
computational cost of a single iteration, and the quality of the
convergence.
(F1) is slightly faster than (F2).

Priorities

among multiple cartesian tasks of diverse nature
have
been respected
.

Optimization of a desired criterion
in joint space may be applied
with lowest priority without affecting the progress of these tasks

Major problems of (F2):

the least
-
squares solution in the unconstrained subspace.

(F1) formulation
faces the algorithmic singularities problem, but
the artifacts can be reduced with proper damping.

18

-

Priority Formulations

for the Kinematic Control
(
Baerlocher
, et al., 1998)

helps the main task to be
completed faster

and
enhance its performance
by enlarging the
nb

of available DOF.

When the 2
ndary

task goes in the same direction than the main

Solution:

impose the secondary control law not to
increase the error of the main task.

Continuous approach:

Classical redundancy formalism (1):

The projection operator is used to transform any 2
ndary

vector
z
into a 2
ndary

control law that does not disturb the main task.

where

reference decrease speed

19

Directional Redundancy

for Robot Control
(Mansard, et al., 2009)

Continuous approach:

Extended projection (2) :

Search of a projection operator
P

(of a secondary task z) so that

respects the convergence condition:

Using SVD decomposition of J:

e

whose Jacobian
J

is full rank: If the
following control law is applied to the robotic
system, then the error
e

asymptotically converges to zero.

Problem:

N
ecessity to introduce
an upper bound on the value of the secondary term

20

Directional Redundancy

for Robot Control
(Mansard, et al., 2009)

Stability

theorem:

Discrete approach (3):

The solution of potential oscillations requires considering the
robot as a discrete system

Convergence condition:

Final control law:

where

e

whose Jacobian
J

is full rank, If the
following control law: is applied to the robotic
system, then, given that
∆t

is sufficiently small, the error
asymptotically converges to zero.

21

Directional Redundancy

for Robot Control
(Mansard, et al., 2009)

Stability

theorem:

Comparison and remarks:

The projection operator
in (2) has more non zero coefficients

(more free DOF) than that in case of classical redundancy (1).

(3) accelerates the decreasing
of each component
of the error
and takes the secondary task into account in the same way.

The system is globally stable, and asymptotically converges to
the main task completion and to the best reachable local

22

Directional Redundancy

for Robot Control
(Mansard, et al., 2009)

Application to visual servoing:

The robot has to move with respect to a visual target, and
simultaneously to take into account a secondary control law.

Cost function for avoidance:

Its gradient can be considered as an artificial force, pushing the
robot away from the undesirable configurations.

Joint
-
Limit Avoidance Law:

The cost function reaches its maximal value near the robot joint
limits, and the gradient is nearly zero far from the limits.

Occlusion Avoidance Law:

It should maximize the distance
d

between the occluding object
and the visual target that is used for the main task.

23

Directional Redundancy

for Robot Control
(Mansard, et al., 2009)

Experimental results:

Classical redundancy
: Rank(P) = 3 during the whole execution.
Perfect exponential decrease of the error.

Directional Redundancy
: Rank(P)> 3 while the error of the main
task is not null. The convergence of the 1
st

component of the
main
-
reaches 0, the projection operator looses a rank.

24

Directional Redundancy

for Robot Control
(Mansard, et al., 2009)

e = 0
, represents bilateral constraint.

Exist tasks described by a set of
unilateral constraints e
i
≤0.

Examples:

joints limits, collision avoidance, visibility loss,
avoidance of singularities.

Inverse
-
Kinematics

Control
:

Assume that
J is perfectly known
. The control law is then always
stable, and asymptotically stable if
J

is full rank.

Unilateral constraint :
e<0

Control law
(
Cheah
, et al., 2007):

where activation matrix

Discontinuity!

Integrate Unilateral Constraints into
SoT

(Mansard, et al., 2009)

25

Solution
: smooth H, by introducing an activation buffer
before the point of activation of the constraint.

Problem
: The activation buffer is not considered due to
inner mathematical simplification of the control law.

Keeping the Continuity at the Kinematics Level:

e
, its Jacobian

J

(
n
×
m

,
cte

rank
r
) and its
activation matrix
H

(diagonal matrix /
h
i

in the interval [0,1]).

The continuous inverse of J activated by H is defined by:

where

X
p

Coupling matrices of J :

26

all the subsets composed
of the m first integers

Integrate Unilateral Constraints into
SoT

(Mansard, et al., 2009)

where

Using this equation, each component of the task is:

• Perfectly realized if the corresponding
h
i

is equal to 1.

• Not taken into account if
h
i

is zero.

• Partially realized otherwise.

Extension of the Control law to k tasks:

where

+ Similar in shape to the classical control law.

+ Ensures the continuity whatever the evolution of the
activation of the features.

27

Integrate Unilateral Constraints into
SoT

(Mansard, et al., 2009)

+ Ensure the priority order of active features (when feature is
fully active, it is not disturbed by tasks of lower priority.

-

Like the classical pseudo inverse, it is sensitive to the
singularities of the Jacobian (& same set of singular points)

+ Points that are regularized by the continuous inverse are the
activation points of the unilateral constraints, which were
impossible to consider using the classical approach.

To smooth these singularities of the Jacobian:

-

Use the DLS instead of the classical pseudo inverse

priority order will not be ensured perfectly any more.

-

Set an explicit constraint to avoid the neighborhood of
singular points (
Gienger
, et al., 2006).

28

Integrate Unilateral Constraints into
SoT

(Mansard, et al., 2009)

-

Maintain the robot equilibrium

-

Grasp an object from a table

-

Avoid Environmental obstacles

-

Avoid Occlusion of the object

-

Manipulate objects using 2 hands

Use the

Approach for redundancy

Application on Simulation
OpenHRP

Application on HRP2 Robot in
LAAS Toulouse

29

Conclusion and
Prespectives