Cover sheet goes here

Answer FOUR questions only

MAY 2019

Page 2 DEN408 (2019)

Question 1

A typical three-link kinematic chain that is representative of a PUMA 560 manipulator

without its wrist is illustrated in Figure Q1. The first link in the chain is the capstan

which is free to rotate about a vertical axis in the plane of the paper while the other two

joints are revolute joints with their axes normal to the plane of the paper. The reference

frames are chosen according to the DH convention. Note that the first frame is fixed to

the base, while the other three frames are fixed to the manipulator links. The co-

ordinates of a point in a frame located at the end effector may be expressed in terms of

the base coordinates as,

3

0,31 1

=

0p pT

where pi is the position vector of the point in the ith joint frame, i= 0, 1, 2, 3.

Figure Q1 Three-link planar PUMA 560 manipulator; for all joints, the Z axes are aligned

with the joint axes.

Question 1 continues on the next page

DEN408 (2019) Page 3

Turn Over

a) Identify the four link parameters, the link length, ai, the link twist, i, the link offset, di,

and the joint angles, i, for the links when, i = 1, 2 and 3.

[9 marks]

b) Determine the homogeneous joint frame transformations, 1,0T , 2,1T and 3,2T .

[12 marks]

c) Determine the homogeneous transformation 0,3T in terms of 1,0T , 2,1T and 3,2T .

[4 marks]

Page 4 DEN408 (2019)

Question 2

Consider the 3 degree of freedom robot manipulator, the front view of which is as

shown in Figure Q2. In this model the capstan at the left is assumed to be fixed to the

base. The x and z axes are assumed to be in the plane of the paper, with z positive up

and the y axis being positive in the direction into the plane of the paper.

Figure Q2 Schematic drawing of the 3 degree of freedom robot manipulator

a) Assume that 2l and 3l are the lengths of the two horizontal links with the first link

pivoted to an extendable capstan by a revolute joint (joint 2). The links are pivoted to

each other by another revolute joint (joint 3). A prismatic joint (joint 1) at the base

permits the capstan to extend vertically in a sleeve. Let 0z be the unit vector in the

direction of the positive z0 axis, positive up, 2r be the vector distance of the joint 2

above the capstan relative to the end effector in the ( )000 ,, zyx frame and 1r be the

vector distance of joint 2, above the capstan, relative to the joint 3 in the ( )000 ,, zyx

frame.

Draw a neat sketch of the top-view of the robot manipulator and obtain expressions for,

the vector positions of joint 2, 2r relative to the end-effector and 1r relative to joint 3, in

terms of 2l , 3l and the quantities ic , is , ikc and iks which are defined as iic θcos = ,

iis θsin = , ( )kiikc θθ += cos and ( )kiiks θθ += sin .

[8 marks]

Question 2 continues on the next page

DEN408 (2019) Page 5

Turn Over

b) In the first instance consider the vertical joint at the robot base and the prismatic joint

above it. The Jacobian matrix J1 relating the angular and translation velocity vectors in

the base coordinates to the joint velocity vector is defined by the relationship,

1

=

J

v

ω θ .

Show that the corresponding 16× Jacobian matrix for the transformation from the end-

effector to the base co-ordinates is,

1

0

=

0J

z

,

where 0z is a unit vector in the direction of the positive z0 axis. What is 0z in terms of

the unit vectors of the ( )000 ,, zyx frame?

[7 marks]

c) Hence or otherwise obtain an expression for the 36× manipulator Jacobian matrix J ,

for the transformation from the end-effector to the base co-ordinates.

[10 marks]

Page 6 DEN408 (2019)

Question 3

Consider the two link planar arm which is a typical configuration representative of a planar

open-loop chain with only revolute joints as shown in Figure Q3. Apart from the inertias of

the two links, it is assumed that lumped mass 1M is attached to link 1L at the joint, as

shown. The end-effector and its payload are modelled as a lumped mass, 2M located at

the tip of link 2L of the arm.

Figure Q3 Two-link planar anthropomorphic manipulator; all Z axes are aligned normal to

the plane of the paper.

Question 3 continues on the next page

DEN408 (2019) Page 7

Turn Over

Question 3 continued

a) Obtain the total kinetic and potential energies of the arm in terms of the moment of

inertia and mass-moment components,

( ) ( ) 21222112121111 LMmLMkLmI cgcg ++++= ,

( ) 1221222221 LLLMLmI cg Γ=+= , ( ) 2222222222 LMkLmI cgcg ++= ,

( )1211121111 LMLMLmLm cg +++=Γ , ( )222222 LMLm cg +=Γ .

In the above expressions, 2M is the tip mass and im , iL , icgL and icgk are respectively

the ith link mass, the ith link length, the ith link position of the centre of mass with

reference to the ith joint and the ith link radius of gyration about its centre of mass.

[12 marks]

b) Apply the Lagrangian energy method and obtain the general equations of motion of the

two-link arm, in matrix form in terms of 1ω and 2ω , and in terms of 1θ and 2θ , where

1ω and 2ω are defined by the equations:

11 ωθ =

122 ωωθ −= .

[13 marks]

Page 8 DEN408 (2019)

Question 4

a) Explain with appropriate diagrams, the computed torque method with reference to serial

manipulators.

[7 marks]

b) Consider a typical inverted pendulum with a single revolute joint on a cart, which is

constrained to move in one direction.

The general non-linear equations of motion of a typical inverted simple pendulum with a

single revolute joint on the cart may be expressed in terms of the joint angle of the

pendulum θ , and the horizontal cart displacement x , as:

2

2

M m mLc x FmL s

mLc mL mgLs

θ

θ τ

+

− =

,

where M is the mass of the cart, m is mass of the simple pendulum, L is the length of

the simple pendulum, g is the acceleration due to gravity, F is the traction control

force applied to the cart, τ is the joint control torque applied to the pendulum, sins θ=

and cosc θ= .

Obtain an expression for the computed torque control traction force F , and joint control

torque τ , in terms of an auxiliary control input vector. Hence obtain the equations

governing the auxiliary control input vector in terms of the desired and actual motions of

the pendulum and the cart.

[10 marks]

c) Hence or otherwise obtain the equations governing the corresponding tracking errors of

the inverted pendulum and the cart considered in part b) and comment on the nature of

the error and the manner in which it is required to decay.

[8 marks]

DEN408 (2019) Page 9

Turn Over

Question 5

a) Describe with a diagram and the relevant equations, the principle of operation of an

accelerometer to measure translational motion.

[8 Marks]

b) Describe with diagrams the working principles of a DC servo motor. How does a DC

servo motor differ from a stepper motor?

[10 Marks]

c) Explain, with diagrams, the Pulse Width Modulation (PWM) technique in relation to the

position control of a DC servo motor?

[7 marks]

Page 10 DEN408 (2019)

Question 6

a) State and describe the primary factors that must be considered to select a sensor for

any robotic application.

[12 Marks]

b) Derive the input-output voltage transfer function of a wire-wound potentiometer with a

sliding wiper making contact with the wire and explain the conditions under which the

output voltage is linearly proportional to the linear position of the wiper.

[7 Marks]

c) Explain the need and application of touch and tactile sensors in robotics.

[6 Marks]

End of Paper

欢迎咨询51作业君