


ROB301F 2018 INTRODUCTIONTOROBOTICS E.pdf
ROB301F_2018_INTRODUCTIONTOROBOTICS_E.pdf
Showing 613 out of 21
ROB301F 2018 INTRODUCTIONTOROBOTICS E.pdfUNIVERSI...
ROB301F_2018_INTRODUCTIONTOROBOTICS_E.pdfUNIVERSITY OF TORONTO Faculty of Applied
ROB301F 2018 INTRODUCTIONTOROBOTICS...
ROB301F_2018_INTRODUCTIONTOROBOTICS_E.pdfUNIVERSITY 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 positivedefinite.
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
Ace your assessments! Get Better Grades
Browse thousands of Study Materials & Solutions from your Favorite Schools
University of Toronto
University_of_Toronto
School:
Introduction_to_Robotics
Course:
Introducing Study Plan
Using AI Tools to Help you understand and remember your course concepts better and faster than any other resource.
Find the best videos to learn every concept in that course from Youtube and Tiktok without searching.
Save All Relavent Videos & Materials and access anytime and anywhere
Prepare Smart and Guarantee better grades
Students also viewed documents
lab 18.docx
lab_18.docx
Course
Course
3
Module5QuizSTA2023.d...
Module5QuizSTA2023.docx.docx
Course
Course
10
Week 7 Test Math302....
Week_7_Test_Math302.docx.docx
Course
Course
30
Chapter 1 Assigment ...
Chapter_1_Assigment_Questions.docx.docx
Course
Course
5
Week 4 tests.docx.do...
Week_4_tests.docx.docx
Course
Course
23
Week 6 tests.docx.do...
Week_6_tests.docx.docx
Course
Course
106