UNIVERSITY OF TORONTO

Faculty of Applied Science and Engineering

FINAL EXAMINATION Third Year - Program 5r

ROB3O1 F

Introduction to Robotics 17 December 2018

EXAMINER: G M T D'ELEUTERIO

Student Name: Last Name First Names

Student Number:

Instructions:

Attempt all questions.

The value of each question is indicated in the table opposite.

Write the final answers only in the boxed space provided for each question.

This is a closed-book test.

No calculator is permitted.

There are 20 pages and 6 problems in this test paper.

An aid sheet is provided.

For Examiner Only Problem Value Mark

A 1 10

B 2 10

C 3 20

0 4 15 5 25 6 20

Total 100

pr

A. Basics

Provide very short answers.

Define expected value for a continuous variable.

/2

State the regulator control problem.

/2

State two basic strategies for map building.

/2

State Lyapunov's theorem.

How many variables/parameters are required in the Dena vit-Hartenberg con-vention to describe each link in a mechanism?

/2

2

B. True or False

Determine if the following statements are true or false and indicate by "T" (for true) and "F" (for false) in the box beside the question. The value of each question is 2 marks.

If x, y E R are random variables then cov(x, y) = 0.

The Ziegler-Nichols method requires knowing the fundamental fre-quency of a system.

A particle filter has the advantage over a Kalman filter that it can operate with multimodal probability distributions.

The Kalman filter makes use only of the current and previous measure-ments.

All multijoint robotic manipulators possess a singularity.

3

C. Questions

Provide brief answers.

3(a). What is the basic idea behind the extended Kalman filter?

/4

= Akkk + Bkuk

Zk+lk = Dkk+1k

Sk+l = Zk+1 - Zk+lk

Xk+lk+l = Xk+1k + Wk+lSk+1

Pk+1k = AkPkIkA + Qk

Skf 1 = Dk+1Pk+lkD+l + Rk+1

T c'—1 k+lk k+1'k+1

PT

k+1k+1 = Pk+1k - Wk+lSk+lW+l

H

r05o 01 =

L

[vi [wj o 1]

X(t) = Xd + ale Alt

+ a2e

T= [C p1 LOT 1 j

= vsin0

= tan 'y

= v cos 0

University of Toronto - ENGINEERING SCIENCE Program 5r

ROB3O1 F AID SHEET Introduction to Robotics

1 0 0 cos 8 0

— Sin

8 cos 8 sin 8 0

C1 = 0 cos 8 sin 0 C2 0 1 0 C3 - sin 8 cos 0 0

0 -sin 0 cos 0 sin 8 0 cos 8 0 0 1

cos(x + y) = c.os x cos y - sin x sin y, sin(x + y) = sin x cos y + cos x sin y

P(xk+1Zo:k) = P(x+i, 'Uk)P(1kZO:k)

XkEA

p(zk+1xk+1)p(xk+1zo.k) p(xk+1zo:k+1)

= k+lEA P(Zk+1k+1)P(k+1o:k)

p(Xlkzok)

x111 kikeAklk

p(Zk+1 p(xlkzo.k+1)

XEA+1k P(Zk+1lk)p(X

ljl lk0:k)

nk k

P) = row [j]

.(v) I CO3_11p 1, if Joint i is revolute

Co, 11, if Joint i is prismatic

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

3(e). State two principles of the subsumption architecture.

/4

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(wkwT ) = 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 11

7

4(b). Show that

PT

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

011

(q)7

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 + e0w

= tlj Sin CO - ew

CO W — Wd

(C) Consider the candidate Lyapunov function

v(e1, e, e0) = + e) + (1 - COS CO)

and argue that v is positive-definite.

Now introduce the controller

—k1e - 'Ld COS e0, Li) = —k0 sin CO - ride0 + Wd

For what values of k, k0 is the robot (i.e., the solution e e = CO = 0)

stable. Prove.

Under the same conditions for k, k0, is the robot asymptotically stable?

Explain.

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

LI

5(c). Consider the candidate Lyapunov function

v(e, e, e9) = + e) + (1 - cos CO)

and argue that v is positive-definite.

5(d). Now introduce the controller

U = —k1e - Ud COS CO3 W = —k0 Sin co - UdCy + Wd

For what values of k, k is the robot (i.e., the solution e = 0) stable. Prove.

/51

MI

c

(p)

5(d) . . .

/10

5(e). Under the same conditions for k, k, is the robot asymptotically stable? Ex-plain.

16

6. Consider the robotic manipulator shown below. Its articulation consists of a three parallel revolute joints. The (relative) joint variables may be taken as 6, 62 and 63 . The link lengths are a1 , a2 , a3. The manipulator is (horizontally) planar so, in the following, consider only the translational position (:r, y) and orientation (ç) of the end-effector

Determine the x, y, 0 pose of the end-effector in terms of the joint variables.

Determine the manipulator Jacobian taking into account only the planar pose of the end-effector.

(C) Determine and describe any singulari-ties.

(d) A force f is applied on the end-effector in the x direction. What are the resulting torques at the joints to keep the manipulator stationary?

111

6(a). Determine the x, y, 0 pose of the end-effector in terms of the joint variables.

Is

17

6(b). Determine the manipulator Jacobian taking into account only the planar pose of the end-effector.

"S ii:

6(c). Determine and describe any singularities.

19

6(d). A force f is applied on the end-effector in the x direction. What are the resulting torques at the joints to keep the manipulator stationary?

Ic

I/s

FIE