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)
Task
-
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.
Advantages of Redundant Manipulators:
•
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:
Task
-
Oriented Kinematics
First order differential kinematics:
Task space
velocity
M
×
N
task Jacobian
matrix
Joint space
velocity
6
Configuration
of N
-
joint Serial
manipulator
Location of the
manipulator
end
-
effector
Redundant robot
举N
Inverse Differential Kinematics
The General Solution:
This solution provides all least
-
squares solution to the end
-
effector task constraint, it minimizes
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
Error in tracking the task.
(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
汯l
䍬潳Ct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
-
gradient of a
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
:
unsatisfactory performance over long tasks
-
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
extended by imaginary links
, 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
additional conditions to:
-
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
imaginary links
(each with 2 non
-
actuated spherical joints at both sides).
Each link is connected to:
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.
Planar manipulator (3 links
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:
Tasks
(T
1
, …
T
t
)
with an order of priority
(T
i
has priority over T
i+1
)
.
Task
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
achievable task
T
i
is satisfied, unless it conflicts with a higher
-
priority task
T
j
(j <
i
)
; in this case, its error has to be minimized.
Task
-
Priority Formulations
for the Kinematic Control
(
Baerlocher
, et al., 1998)
16
,
i i i
T J x
The first task
-
priority formulation (F1)
Case of Two tasks:
Generalization:
where
Amelioration:
The second task
-
priority formulation (F2)
17
Task
-
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):
inefficient task mapping that searches
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
Task
-
Priority Formulations
for the Kinematic Control
(
Baerlocher
, et al., 1998)
Secondary Task
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
task
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:
For any task function
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:
Potential Oscillations at Task
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
For any task function
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
minimum of the secondary task.
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
-
task error is accelerated by the secondary task. When it
reaches 0, the projection operator looses a rank.
24
Directional Redundancy
for Robot Control
(Mansard, et al., 2009)
Generally task is defined by
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:
Consider a task
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)
Desired Tasks:
-
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
StackOfTask
Approach for redundancy
resolution and tasks prioritizing.
Application on Simulation
OpenHRP
Application on HRP2 Robot in
LAAS Toulouse
29
Conclusion and
Prespectives
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