ROB301F 2018 INTRODUCTIONTOROBOTICS E.pd...
ROB301F_2018_INTRODUCTIONTOROBOTICS_E.pdf-UNIVERSITY OF TORONTO Faculty of Applied
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ROB301F 2018 INTRODUCTIONTOROBOTICS E.pdf-UNIVERSI...
ROB301F_2018_INTRODUCTIONTOROBOTICS_E.pdf-UNIVERSITY OF TORONTO Faculty of Applied
ROB301F 2018 INTRODUCTIONTOROBOTICS...
ROB301F_2018_INTRODUCTIONTOROBOTICS_E.pdf-UNIVERSITY OF TORONTO Faculty of Applied
Page 6
In solving the tracking problem, it is common to use feedback control of the
form,
it
-
ke + ki
J
edt
where
e
=
-
xd)2
+ (y
-
-
d;
the set point
(x(j, Yd)
is progressively
moved along the desired path. What is the purpose of the offset
d>
0?
/4
To what does "integrator windup" refer?
/4


Page 7
3(e).
State two principles of the subsumption architecture.
/4


Page 8
D. Problems
4. Consider a system with the state model,
x
1
=
Akxk + Bkuk +
Vk
and the measurement model,
Zk
=
Dkxk +
Wk
where, as usual,
11k
is the control and
Vk,
Wk
are noise with E(Vk) =
E(wk) =
0
and
E(vkvfl
=
Qk,
E(wkw
T
)
=
Rk.
Assume the a
posteriori
estimate to be given as
Xk+1+1 =
k+1k
+
Wk+1(Zk+1
-
Zk+1k)
where
k+1k
is the measurement prediction based on the a
priori
state estimate
Xk+
1
Show that
=
Xk+1k
+
Wk+1[Dk+1(xk+1
-
Xk+1k)
+
Wk+11
Show that
P
1
/
T
k+1k+1
-
k+1 k+1)
k+1k
-
k+1 k+1)
+
k+1 k+1 k+1
where
Pk+lli
=
coy
(k+1j
-
Xk+1,
Xk+1j
-
xk+1)
is the a
priori
covariance matrix when
j
=
k
and the a
posteriori
covari-
ance matrix when
j
=
k +
1.
4(a).
Show that
+1k
+
Wk+1[Dk+1(xk+1
Xk+1)
+
w
1
1
7


Page 10
4(b). Show that
P
T
k+llk+l
=
(1
-
Wk+1Dk+1)Pk+1k(l
-
Wk+1Dk+1) + Wk+lRk+lW+l
where
=
Coy (+iij
-
Xk+1,Xk+1j
-
Xk+1)
is the a
priori
covariance matrix when
j
k
and the a
posteriori
covariance
matrix when
j
=
k +
1.
9


Page 11
011
(q)7


Page 12
5. Consider our typical robot kinematics model,
ti Cos 0
=
tt sin 0
0
w
where (x, y,
0)
is the pose of the robot in a global frame; it and
w
are the speed
and angular rate of the robot and these are the control inputs. The task is to track
a desired trajectory
(a:
j
(t),
yd(t), 0d(t))
at the desired speed
ud(t)
and angular rate
wd(t),
which must thus satisfy
'Lid
COS
0d
Yd
=
t1d Sill 0d
AY
Oil
Consider then a transformation to the robot's frame of
I
reference,
i.e.,
001
dl
-
Cos
sin
i
8
— Sill 0 Cos
Note that
0
is unaffected by the transformation. The same transformation matrix
(i.e.,
with
0
not
Od)
holds for
(Xd,
yd)
to
(d
rid).
Determine and
7)
in terms of x, y.
0
and their derivatives.
Defining the
errors,
A
A
A
y='L1'L1d,
COOOd
show that
E
'L1d
cos CO
+
u + e
0
w
=
tlj
S
i
n
CO
-
ew
CO
W
Wd
(C)
Consider the candidate Lyapunov function
v(e
1
,
e, e
0
)
=
+
e)
+
(1
-
COS CO
)
and argue that
v
is positive-definite.
Now introduce the controller
—k
1
e
-
'Ld
COS
e
0
,
Li) =
—k
0
sin
CO
-
ride0
+
Wd
For what values of
k,
k
0
is the robot
(i.e.,
the solution e
e
=
CO
=
0)
stable. Prove.
Under the same conditions for
k,
k
0
,
is the robot asymptotically stable?
Explain.


Page 13
5(a).
Determine and
i1
in terms of x. y,
8
and their derivatives.
5(b).
Defining the errors,
A
A
A
eoO — Od
show that
Ud
COS CO
+
U
+
CW
=
Ud
Sill co
-
CxU)
eo
Is
12


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