Kinematic Redundancy A manipulator may have more DOFs than are - - PowerPoint PPT Presentation
Kinematic Redundancy A manipulator may have more DOFs than are necessary to control a desired variable What do you do w/ the extra DOFs? However, even if the manipulator has enough DOFs, it may still be unable to control
are necessary to control a desired variable
“enough” DOFs, it may still be unable to control some variables in some configurations…
Before we think about redundancy, let’s look at the range space of the Jacobian transform: The velocity Jacobian maps joint velocities onto end effector velocities: Space of joint velocities
J: Space of end effector velocities
space of J:
( )
v
( )
v
( )q
v
( )
In some configurations, the range space of the Jacobian may not span the entire space of the variable to be controlled:
( )
v
spans if ( ) ( )
( ) ( )
v
Example: a and b span this two dimensional space:
This is the case in the manipulator to the right:
the y direction (or the z direction)
v
Let’s calculate the velocity Jacobian:
π 2
( )
+ + + − − − − − − =
123 3 123 3 12 2 123 3 12 2 1 1 123 3 123 3 12 2 123 3 12 2 1 1
c l c l c l c l c l c l s l s l s l s l s l s l q Jv Joint configuration of manipulator:
( )
+ − + − − =
3 3 2 3 2 1
l l l l l l q Jv
( )q
v
There is no joint velocity, , that will produce a y velocity,
Therefore, you’re in a singularity.
In singular configurations:
velocities
Test for kinematic singularity:
a singular configuration
T
[ ] = + − + − − + − + − − = something det det ) ( ) ( det
3 3 2 3 2 1 3 3 2 3 2 1
l l l l l l l l l l l l q J q J
T
= Example:
The four singularities of the three-link planar arm:
Cartesian control involves calculating the inverse or pseudoinverse:
1 # −
T T JJ
However, in singular configurations, the pseudoinverse (or inverse) does not exist because is undefined.
1 − T
As you approach a singular configuration, joint velocities in the singular direction calculated by the pseudoinverse get very large:
1 #
− s T T s
In Jacobian transpose control, joint velocities in the singular direction (i.e. the gradient) go to zero:
s T x
Where is a singular direction.
s
pseudoinverse control where the pseudoinverse “blows up”.
manipulator gets “stuck” in a singular configuration because the direction of the goal is in a singular direction.
motion away from the singular configuration will allow the manipulator to continue on its way.
One way to get the “best of both worlds” is to use the “dampled least squares inverse” – aka the singularity robust (SR) inverse:
1 2 * −
T T
the SR inverse does not blow up.
velocities. BTW, another way to handle singularities is simply to avoid them – this method is preferred by many
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
A general-purpose robot arm frequently has more DOFs than are strictly necessary to perform a given function
two DOFs are strictly necessary
it is redundant w.r.t. the task of controlling two dimensional position.
effector position in 3-space, you need at least 3 DOFs
effector position and orientation, at least 6 DOFs are needed (they have to be configured right, too…)
1
q
2
q
3
q
x y z
1
z
1
x
1
y
2
z
2
x
2
y
3
z
3
y
3
x
1
l
2
l
3
l
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
23 22 21 13 12 11
The local redundancy of an arm can be understood in terms of the local Jacobian
Cartesian DOFs equal to the number of independent rows in the Jacobian Since there are two independent rows, you can control two Cartesian DOFs independently (m=2) You use three joints to control two Cartesian DOFs (n=3) Since the number of independent Cartesian directions is less than the number of joints, (m<n), this manipulator is redundant w.r.t. the task of controlling those Cartesian directions.
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
What does this redundant space look like?
linear because the Jacobian is linear
The dimension of the redundant space is the number of joints – the number of independent Cartesian DOFs: n-m.
space is a set of one dimensional curves traced through the three dimensional joint space.
configurations that place the end effector in the same position. Redundant manifolds in joint space
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
Joint velocities in redundant directions causes no motion at the end effector
manipulator. Redundant manifolds in joint space
Redundant joint velocities satisfy this equation: the null space of
( )
Compare to the range space of :
( )
( )
) (q J N
( )
) (q J R
( )
1 − ⊆ n SO Q
m
R X ⊆
Range space Null space
are internal motions Joint space Cartesian space
( )
( )
You can’t generate these motions
Motions in the redundant space do not affect the position of the end effector.
position, is there something we would like to do in this space?
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
# #
Null space projection matrix:
space of J:
space – just calculate what you would like to do and project it into the null space.
#
Zero end-effector velocities
Assume that you are given a joint velocity, , you would like to achieve while also achieving a desired end effector twist,
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
( ) ( )
T
d
Use lagrange multiplier method:
z z
Minimize subject to :
d
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
( )
T
z z
( )
T T
T
T
( )
1
−
( )
1
T T
−
# #
x y
1
q
z
2
q
3
q
x
1
y
1
y
2
x
2
x
3
y
3
1
l
2
l
3
l
Avoid kinematic singularities:
manipulability measure:
T
# #
Avoid joint limits:
squared distance from a joint limit:
# #
( )
m −
m
Avoid kinematic obstacles:
(nodes) on the manipulator:
joint space:
# #
i i
z
1
2
{ }
3 2 1
∈
nodes i i T i
d
Yes – imagine the possible instantaneous motions are described by an ellipsoid in Cartesian space. Can we characterize how close we are to a singularity?
Can’t move much this way Can move a lot this way
The manipulability ellipsoid is an ellipse in Cartesian space corresponding to the twists that unit joint velocities can generate:
# #
T
A unit sphere in joint velocity space
1 1
− −
T T T T T T
1
− −
T T T T T
1
− x
T T
Project the sphere into Cartesian space The space of feasible Cartesian velocities
You can calculate the directions and magnitudes
the eigenvalues and eigenvectors of
T
1
2
1
2
Yoshikawa’s manipulability measure:
T
Another characterization of the manipulability ellipsoid: the ratio of the largest eigenvalue to the smallest eigenvalue:
be the smallest.
ellipsoid is:
the more isotropic the ellispoid is.
1
2
1
2
1
n
n
Isotropic manipulability ellipsoid NOT isotropic manipulability ellipsoid
T T T
T T
A unit sphere in the space of joint torques The space of feasible Cartesian wrenches
Principle axes of the force manipulability ellipsoid: the eigenvalues and eigenvectors of:
are reciprocals:
ellipsoid are the longest principle axes of the force ellipsoid and vice versa…
1 − T
1 − T
T
f i v i
v i f i
Velocity ellipsoid Force ellipsoid This is known as force/velocity duality
directions that your max velocity is smallest
you can only apply the smallest forces
Solve for the principle axes of the manipulability ellipsoid for the planar two link manipulator with unit length links at
( )
+ − − − =
12 2 12 2 1 1 12 2 12 2 1 1
c l c l c l s l s l s l q J =
4 π
q
( )
+ − − =
2 1 2 1 2 1 2 1
1 q J = 0.1568
1v
λ
( ) ( )
− + + − + − − = λ λ 2 2 1 1 1
2 1 2 1 T
q J q J Principle axes: = 1.8411 0.9530
2v
λ
1 1v
λ
2 2v
λ
( )
) (q J N
( )
) (q J R
Degree of manipulability:
( ) ( ) ( ) ( )
( ) ( )
Degree of redundancy:
( ) ( )
( )
) (q J N
( )
) (q J R
As the manipulator moves to new configurations, the degree of manipulability may temporarily decrease – these are the singular configurations.
( )
) (q J N
( )
) (q J R
( )
T
⊥
( )
T
⊥
T
Remember the Jacobian’s application to statics:
( )
) (q J N
( )
) (q J R
( )
T
q J R ) (
( )
T
q J N ) (
T
( )
) (q J R
( )
) (q J N
( )
T
q J N ) (
( )
T
q J R ) (
( )
T
q J R ) (
( )
T
q J N ) (
T
( )
) (q J R
( )
) (q J N
( )
T
⊥
( )
T
⊥
velocity null space.