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rli
robotics-codes-from-scratch
Commits
6d7ee83a
Commit
6d7ee83a
authored
2 years ago
by
Hakan GIRGIN
Browse files
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Script updated according to the template in iLQR_manipulator.py
parent
6bde513d
Branches
develop
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python/iLQR_manipulator_CP.py
+101
-110
101 additions, 110 deletions
python/iLQR_manipulator_CP.py
with
101 additions
and
110 deletions
python/iLQR_manipulator_CP.py
+
101
−
110
View file @
6d7ee83a
...
...
@@ -31,64 +31,62 @@ from math import factorial
# ===============================
# Logarithmic map for R^2 x S^1 manifold
def
logmap
(
f
,
f0
):
position_error
=
f
[:,:
2
]
-
f0
[:,:
2
]
orientation_error
=
np
.
imag
(
np
.
log
(
np
.
exp
(
f0
[:,
-
1
]
*
1j
).
conj
().
T
*
np
.
exp
(
f
[:,
-
1
]
*
1j
).
T
)).
conj
().
reshape
((
-
1
,
1
))
error
=
np
.
hstack
((
position_error
,
orientation_error
))
return
error
# Forward kinematics for E-E
def
fkin
(
param
,
x
):
x
=
x
.
T
A
=
np
.
tril
(
np
.
ones
([
param
.
nbVarX
,
param
.
nbVarX
]))
f
=
np
.
vstack
((
param
.
linkLengths
@
np
.
cos
(
A
@
x
),
param
.
linkLengths
@
np
.
sin
(
A
@
x
),
np
.
mod
(
np
.
sum
(
x
,
0
)
+
np
.
pi
,
2
*
np
.
pi
)
-
np
.
pi
))
#x1,x2,o (orientation as single Euler angle for planar robot)
return
f
.
T
# Forward Kinematics for all joints
def
fkin0
(
param
,
x
):
T
=
np
.
tril
(
np
.
ones
([
param
.
nbVarX
,
param
.
nbVarX
]))
T2
=
np
.
tril
(
np
.
matlib
.
repmat
(
param
.
linkLengths
,
len
(
x
),
1
))
f
=
np
.
vstack
((
T2
@
np
.
cos
(
T
@x
),
T2
@
np
.
sin
(
T
@x
)
)).
T
f
=
np
.
vstack
((
np
.
zeros
(
2
),
f
))
def
logmap
(
f
,
f0
):
position_error
=
f
[:
2
,:]
-
f0
[:
2
,:]
orientation_error
=
np
.
imag
(
np
.
log
(
np
.
exp
(
f0
[
-
1
,:]
*
1j
).
conj
().
T
*
np
.
exp
(
f
[
-
1
,:]
*
1j
).
T
)).
conj
()
diff
=
np
.
vstack
([
position_error
,
orientation_error
])
return
diff
# Forward kinematics for end-effector (in robot coordinate system)
def
fkin
(
x
,
param
):
L
=
np
.
tril
(
np
.
ones
([
param
.
nbVarX
,
param
.
nbVarX
]))
f
=
np
.
vstack
([
param
.
l
@
np
.
cos
(
L
@
x
),
param
.
l
@
np
.
sin
(
L
@
x
),
np
.
mod
(
np
.
sum
(
x
,
0
)
+
np
.
pi
,
2
*
np
.
pi
)
-
np
.
pi
])
# f1,f2,f3, where f3 is the orientation (single Euler angle for planar robot)
return
f
# Forward kinematics for all joints (in robot coordinate system)
def
fkin0
(
x
,
param
):
L
=
np
.
tril
(
np
.
ones
([
param
.
nbVarX
,
param
.
nbVarX
]))
f
=
np
.
vstack
([
L
@
np
.
diag
(
param
.
l
)
@
np
.
cos
(
L
@
x
),
L
@
np
.
diag
(
param
.
l
)
@
np
.
sin
(
L
@
x
)
])
f
=
np
.
hstack
([
np
.
zeros
([
2
,
1
]),
f
])
return
f
# Jacobian with analytical computation (for single time step)
def
j
kin
(
param
,
x
):
T
=
np
.
tril
(
np
.
ones
(
(
len
(
x
),
len
(
x
)))
)
J
=
np
.
vstack
(
(
-
np
.
sin
(
T
@
x
).
T
@
np
.
diag
(
param
.
l
inkLengths
)
@
T
,
np
.
cos
(
T
@
x
).
T
@
np
.
diag
(
param
.
l
inkLengths
)
@
T
,
np
.
ones
(
len
(
x
)
)
)
)
def
J
kin
(
x
,
param
):
L
=
np
.
tril
(
np
.
ones
(
[
param
.
nbVarX
,
param
.
nbVarX
])
)
J
=
np
.
vstack
(
[
-
np
.
sin
(
L
@
x
).
T
@
np
.
diag
(
param
.
l
)
@
L
,
np
.
cos
(
L
@
x
).
T
@
np
.
diag
(
param
.
l
)
@
L
,
np
.
ones
(
[
1
,
param
.
nbVarX
]
)
]
)
return
J
# Residual and Jacobian
def
f_reach
(
param
,
x
):
f
=
logmap
(
fkin
(
param
,
x
),
param
.
mu
)
J
=
np
.
zeros
((
len
(
x
)
*
param
.
nbVarF
,
len
(
x
)
*
param
.
nbVarX
))
for
t
in
range
(
x
.
shape
[
0
]):
f
[
t
,:
2
]
=
param
.
A
[
t
].
T
@
f
[
t
,:
2
]
# Object oriented fk
Jtmp
=
jkin
(
param
,
x
[
t
])
Jtmp
[:
2
]
=
param
.
A
[
t
].
T
@
Jtmp
[:
2
]
# Object centered jacobian
# Residual and Jacobian for a viapoints reaching task (in object coordinate system)
def
f_reach
(
x
,
param
):
f
=
logmap
(
fkin
(
x
,
param
),
param
.
Mu
)
J
=
np
.
zeros
([
param
.
nbPoints
*
param
.
nbVarF
,
param
.
nbPoints
*
param
.
nbVarX
])
for
t
in
range
(
param
.
nbPoints
):
f
[:
2
,
t
]
=
param
.
A
[:,
:,
t
].
T
@
f
[:
2
,
t
]
# Object oriented residual
Jtmp
=
Jkin
(
x
[:,
t
],
param
)
Jtmp
[:
2
]
=
param
.
A
[:,
:,
t
].
T
@
Jtmp
[:
2
]
# Object centered Jacobian
if
param
.
useBoundingBox
:
for
i
in
range
(
2
):
if
abs
(
f
[
t
,
i
])
<
param
.
s
izeObj
[
i
]:
f
[
t
,
i
]
=
0
Jtmp
[
i
]
=
0
if
abs
(
f
[
i
,
t
])
<
param
.
s
z
[
i
]:
f
[
i
,
t
]
=
0
Jtmp
[
i
]
=
0
else
:
f
[
t
,
i
]
-=
np
.
sign
(
f
[
t
,
i
])
*
param
.
sizeObj
[
i
]
J
[
t
*
param
.
nbVarF
:(
t
+
1
)
*
param
.
nbVarF
,
t
*
param
.
nbVarX
:(
t
+
1
)
*
param
.
nbVarX
]
=
Jtmp
return
f
,
J
f
[
i
,
t
]
-=
np
.
sign
(
f
[
i
,
t
])
*
param
.
sz
[
i
]
J
[
t
*
param
.
nbVarF
:(
t
+
1
)
*
param
.
nbVarF
,
t
*
param
.
nbVarX
:(
t
+
1
)
*
param
.
nbVarX
]
=
Jtmp
return
f
,
J
# Building piecewise constant basis functions
def
build_phi_piecewise
(
nb_data
,
nb_fct
):
...
...
@@ -131,37 +129,31 @@ param.nbPoints = 2 # Number of viapoints
param
.
nbVarX
=
3
# State space dimension (x1,x2,x3)
param
.
nbVarU
=
3
# Control space dimension (dx1,dx2,dx3)
param
.
nbVarF
=
3
# Objective function dimension (f1,f2,f3, with f3 as orientation)
param
.
l
inkLengths
=
[
2
,
2
,
1
]
# Robot links lengths
param
.
s
izeObj
=
[.
2
,.
3
]
# Size of objects
param
.
l
=
[
2
,
2
,
1
]
# Robot links lengths
param
.
s
z
=
[.
2
,.
3
]
# Size of objects
param
.
r
=
1e-6
# Control weight term
param
.
m
u
=
np
.
asarray
([[
2
,
1
,
-
np
.
pi
/
6
],
[
3
,
2
,
-
np
.
pi
/
3
]]).
T
# Viapoints
param
.
M
u
=
np
.
asarray
([[
2
,
1
,
-
np
.
pi
/
2
],
[
3
,
1
,
-
np
.
pi
/
2
]]).
T
# Viapoints
param
.
useBoundingBox
=
True
# Consider bounding boxes for reaching cost
param
.
nbFct
=
5
# Number of basis functions
param
.
A
=
np
.
zeros
([
2
,
2
,
param
.
nbPoints
])
# Object orientation matrices
param
.
basisName
=
"
PIECEWISE
"
#
Task parameters
#
Main program
# ===============================
# Targets
param
.
mu
=
np
.
asarray
([
[
2
,
1
,
-
np
.
pi
/
2
],
# x , y , orientation
[
3
,
1
,
-
np
.
pi
/
2
]
])
# Transformation matrices
param
.
A
=
np
.
zeros
(
(
param
.
nbPoints
,
2
,
2
)
)
for
i
in
range
(
param
.
nbPoints
):
orn_t
=
param
.
mu
[
i
,
-
1
]
param
.
A
[
i
,:,:]
=
np
.
asarray
([
[
np
.
cos
(
orn_t
)
,
-
np
.
sin
(
orn_t
)],
[
np
.
sin
(
orn_t
)
,
np
.
cos
(
orn_t
)]
# Object rotation matrices
for
t
in
range
(
param
.
nbPoints
):
orn_t
=
param
.
Mu
[
-
1
,
t
]
param
.
A
[:,:,
t
]
=
np
.
asarray
([
[
np
.
cos
(
orn_t
),
-
np
.
sin
(
orn_t
)],
[
np
.
sin
(
orn_t
),
np
.
cos
(
orn_t
)]
])
# Regularization matrix
R
=
np
.
identity
(
(
param
.
nbData
-
1
)
*
param
.
nbVarU
)
*
param
.
r
# Precision matrix
Q
=
np
.
identity
(
param
.
nbVarF
*
param
.
nbPoints
)
Q
=
np
.
identity
(
param
.
nbVarF
*
param
.
nbPoints
)
# Control weight matrix
R
=
np
.
identity
((
param
.
nbData
-
1
)
*
param
.
nbVarU
)
*
param
.
r
# System parameters
# ===============================
...
...
@@ -171,13 +163,13 @@ tl = np.linspace(0,param.nbData,param.nbPoints+1)
tl
=
np
.
rint
(
tl
[
1
:]).
astype
(
np
.
int64
)
-
1
idx
=
np
.
array
([
i
+
np
.
arange
(
0
,
param
.
nbVarX
,
1
)
for
i
in
(
tl
*
param
.
nbVarX
)])
u
=
np
.
zeros
(
param
.
nbVarU
*
(
param
.
nbData
-
1
)
)
# Initial control command
x0
=
np
.
array
(
[
3
*
np
.
pi
/
4
,
-
np
.
pi
/
2
,
-
np
.
pi
/
4
]
)
# Initial state (in joint space)
# Transfer matrices (for linear system as single integrator)
Su0
=
np
.
vstack
([
np
.
zeros
((
param
.
nbVarX
,
param
.
nbVarX
*
(
param
.
nbData
-
1
))),
np
.
tril
(
np
.
kron
(
np
.
ones
((
param
.
nbData
-
1
,
param
.
nbData
-
1
)),
np
.
eye
(
param
.
nbVarX
)
*
param
.
dt
))])
Sx0
=
np
.
kron
(
np
.
ones
(
param
.
nbData
)
,
np
.
identity
(
param
.
nbVarX
)
).
T
Su0
=
np
.
vstack
([
np
.
zeros
([
param
.
nbVarX
,
param
.
nbVarX
*
(
param
.
nbData
-
1
)]),
np
.
tril
(
np
.
kron
(
np
.
ones
([
param
.
nbData
-
1
,
param
.
nbData
-
1
]),
np
.
eye
(
param
.
nbVarX
)
*
param
.
dt
))
])
Sx0
=
np
.
kron
(
np
.
ones
(
param
.
nbData
),
np
.
identity
(
param
.
nbVarX
)).
T
Su
=
Su0
[
idx
.
flatten
()]
# We remove the lines that are out of interest
# Basis functions
...
...
@@ -195,33 +187,32 @@ PSI = np.kron(phi,np.identity(param.nbVarU))
# Solving iLQR
# ===============================
for
i
in
range
(
param
.
nbIter
):
x
=
np
.
real
(
Su0
@
u
+
Sx0
@
x0
)
x
=
x
.
reshape
(
(
param
.
nbData
,
param
.
nbVarX
)
)
u
=
np
.
zeros
(
param
.
nbVarU
*
(
param
.
nbData
-
1
)
)
# Initial control command
x0
=
np
.
array
(
[
3
*
np
.
pi
/
4
,
-
np
.
pi
/
2
,
-
np
.
pi
/
4
]
)
# Initial state (in joint space)
f
,
J
=
f_reach
(
param
,
x
[
tl
])
dw
=
np
.
linalg
.
inv
(
PSI
.
T
@
Su
.
T
@
J
.
T
@
Q
@
J
@
Su
@
PSI
+
PSI
.
T
@
R
@
PSI
)
@
(
-
PSI
.
T
@
Su
.
T
@
J
.
T
@
Q
@
f
.
flatten
()
-
PSI
.
T
@
u
*
param
.
r
)
for
i
in
range
(
param
.
nbIter
):
x
=
Su0
@
u
+
Sx0
@
x0
# System evolution
x
=
x
.
reshape
([
param
.
nbVarX
,
param
.
nbData
],
order
=
'
F
'
)
f
,
J
=
f_reach
(
x
[:,
tl
],
param
)
# Residuals and Jacobians
dw
=
np
.
linalg
.
inv
(
PSI
.
T
@
Su
.
T
@
J
.
T
@
Q
@
J
@
Su
@
PSI
+
PSI
.
T
@
R
@
PSI
)
@
\
(
-
PSI
.
T
@
Su
.
T
@
J
.
T
@
Q
@
f
.
flatten
(
'
F
'
)
-
PSI
.
T
@
u
*
param
.
r
)
du
=
PSI
@
dw
#
Perform
line search
#
Estimate step size with backtracking
line search
method
alpha
=
1
cost0
=
f
.
flatten
()
@
Q
@
f
.
flatten
()
+
np
.
linalg
.
norm
(
u
)
*
param
.
r
cost0
=
f
.
flatten
(
'
F
'
).
T
@
Q
@
f
.
flatten
(
'
F
'
)
+
np
.
linalg
.
norm
(
u
)
**
2
*
param
.
r
# Cost
while
True
:
utmp
=
u
+
du
*
alpha
xtmp
=
np
.
real
(
Su0
@
utmp
+
Sx0
@
x0
)
xtmp
=
xtmp
.
reshape
(
(
param
.
nbData
,
param
.
nbVarX
)
)
ftmp
,
_
=
f_reach
(
param
,
xtmp
[
tl
])
cost
=
ftmp
.
flatten
()
@
Q
@
ftmp
.
flatten
()
+
np
.
linalg
.
norm
(
utmp
)
*
param
.
r
xtmp
=
Su0
@
utmp
+
Sx0
@
x0
# System evolution
xtmp
=
xtmp
.
reshape
([
param
.
nbVarX
,
param
.
nbData
],
order
=
'
F
'
)
ftmp
,
_
=
f_reach
(
xtmp
[:,
tl
],
param
)
# Residuals
cost
=
ftmp
.
flatten
(
'
F
'
).
T
@
Q
@
ftmp
.
flatten
(
'
F
'
)
+
np
.
linalg
.
norm
(
utmp
)
**
2
*
param
.
r
# Cost
if
cost
<
cost0
or
alpha
<
1e-3
:
u
=
utmp
print
(
"
Iteration {}, cost:
{}, alpha:
{}
"
.
format
(
i
,
cost
,
alpha
))
print
(
"
Iteration {}, cost: {}
"
.
format
(
i
,
cost
))
break
alpha
/=
2
if
np
.
linalg
.
norm
(
alpha
*
du
)
<
1e-2
:
break
if
np
.
linalg
.
norm
(
du
*
alpha
)
<
1E-2
:
break
# Stop iLQR iterations when solution is reached
# Plotting
# ===============================
...
...
@@ -231,17 +222,17 @@ plt.axis("off")
plt
.
gca
().
set_aspect
(
'
equal
'
,
adjustable
=
'
box
'
)
# Get points of interest
f
=
fkin
(
param
,
x
)
f00
=
fkin0
(
param
,
x
[
0
]
)
f10
=
fkin0
(
param
,
x
[
tl
[
0
]]
)
fT0
=
fkin0
(
param
,
x
[
-
1
]
)
f
=
fkin
(
x
,
param
)
f00
=
fkin0
(
x
[:,
0
],
param
)
f10
=
fkin0
(
x
[:,
tl
[
0
]],
param
)
fT0
=
fkin0
(
x
[:,
-
1
],
param
)
u
=
u
.
reshape
((
-
1
,
param
.
nbVarU
))
plt
.
plot
(
f00
[
:,
0
]
,
f00
[
:,
1
],
c
=
'
black
'
,
linewidth
=
5
,
alpha
=
.
2
)
plt
.
plot
(
f10
[
:,
0
]
,
f10
[
:,
1
],
c
=
'
black
'
,
linewidth
=
5
,
alpha
=
.
4
)
plt
.
plot
(
fT0
[
:,
0
]
,
fT0
[
:,
1
],
c
=
'
black
'
,
linewidth
=
5
,
alpha
=
.
6
)
plt
.
plot
(
f00
[
0
,
:
]
,
f00
[
1
,:
],
c
=
'
black
'
,
linewidth
=
5
,
alpha
=
.
2
)
plt
.
plot
(
f10
[
0
,
:
]
,
f10
[
1
,:
],
c
=
'
black
'
,
linewidth
=
5
,
alpha
=
.
4
)
plt
.
plot
(
fT0
[
0
,
:
]
,
fT0
[
1
,:
],
c
=
'
black
'
,
linewidth
=
5
,
alpha
=
.
6
)
plt
.
plot
(
f
[
:,
0
],
f
[:,
1
],
c
=
"
black
"
,
marker
=
"
o
"
,
markevery
=
[
0
]
+
tl
.
tolist
())
#,label="Trajectory"2
plt
.
plot
(
f
[
0
,
:],
f
[
1
,
:
],
c
=
"
black
"
,
marker
=
"
o
"
,
markevery
=
[
0
]
+
tl
.
tolist
())
#,label="Trajectory"2
# Plot bounding box or via-points
ax
=
plt
.
gca
()
...
...
@@ -249,36 +240,36 @@ color_map = ["deepskyblue","darkorange"]
for
i
in
range
(
param
.
nbPoints
):
if
param
.
useBoundingBox
:
rect_origin
=
param
.
m
u
[
i
,:
2
]
-
param
.
A
[
i
]
@np.array
(
param
.
s
izeObj
)
rect_orn
=
param
.
m
u
[
i
,
-
1
]
rect_origin
=
param
.
M
u
[
i
,:
2
]
-
param
.
A
[
i
]
@np.array
(
param
.
s
z
)
rect_orn
=
param
.
M
u
[
i
,
-
1
]
rect
=
patches
.
Rectangle
(
rect_origin
,
param
.
s
izeObj
[
0
]
*
2
,
param
.
s
izeObj
[
1
]
*
2
,
np
.
degrees
(
rect_orn
),
color
=
color_map
[
i
])
rect
=
patches
.
Rectangle
(
rect_origin
,
param
.
s
z
[
0
]
*
2
,
param
.
s
z
[
1
]
*
2
,
np
.
degrees
(
rect_orn
),
color
=
color_map
[
i
])
ax
.
add_patch
(
rect
)
else
:
plt
.
scatter
(
param
.
m
u
[
i
,
0
],
param
.
m
u
[
i
,
1
],
s
=
100
,
marker
=
"
X
"
,
c
=
color_map
[
i
])
plt
.
scatter
(
param
.
M
u
[
i
,
0
],
param
.
M
u
[
i
,
1
],
s
=
100
,
marker
=
"
X
"
,
c
=
color_map
[
i
])
fig
,
axs
=
plt
.
subplots
(
7
,
1
)
axs
[
0
].
plot
(
f
[
:,
0
],
c
=
'
black
'
)
axs
[
0
].
plot
(
f
[
0
],
c
=
'
black
'
)
axs
[
0
].
set_ylabel
(
"
$f(x)_1$
"
)
axs
[
0
].
set_xticks
([
0
,
param
.
nbData
])
axs
[
0
].
set_xticklabels
([
"
0
"
,
"
T
"
])
for
i
in
range
(
param
.
nbPoints
):
axs
[
0
].
scatter
(
tl
[
i
],
param
.
m
u
[
i
,
0
],
c
=
'
blue
'
)
axs
[
0
].
scatter
(
tl
[
i
],
param
.
M
u
[
0
,
i
],
c
=
'
blue
'
)
axs
[
1
].
plot
(
f
[
:,
1
],
c
=
'
black
'
)
axs
[
1
].
plot
(
f
[
1
],
c
=
'
black
'
)
axs
[
1
].
set_ylabel
(
"
$f(x)_2$
"
)
axs
[
1
].
set_xticks
([
0
,
param
.
nbData
])
axs
[
1
].
set_xticklabels
([
"
0
"
,
"
T
"
])
for
i
in
range
(
param
.
nbPoints
):
axs
[
1
].
scatter
(
tl
[
i
],
param
.
m
u
[
i
,
1
],
c
=
'
blue
'
)
axs
[
1
].
scatter
(
tl
[
i
],
param
.
M
u
[
1
,
i
],
c
=
'
blue
'
)
axs
[
2
].
plot
(
f
[
:,
2
],
c
=
'
black
'
)
axs
[
2
].
plot
(
f
[
2
],
c
=
'
black
'
)
axs
[
2
].
set_ylabel
(
"
$f(x)_3$
"
)
axs
[
2
].
set_xticks
([
0
,
param
.
nbData
])
axs
[
2
].
set_xticklabels
([
"
0
"
,
"
T
"
])
for
i
in
range
(
param
.
nbPoints
):
axs
[
2
].
scatter
(
tl
[
i
],
param
.
m
u
[
i
,
2
],
c
=
'
blue
'
)
axs
[
2
].
scatter
(
tl
[
i
],
param
.
M
u
[
2
,
i
],
c
=
'
blue
'
)
axs
[
3
].
plot
(
u
[:,
0
],
c
=
'
black
'
)
axs
[
3
].
set_ylabel
(
"
$u_1$
"
)
...
...
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