Lecture 03 Feb. 06, 2019
21
Fundamentals Prof. S.K. Saha Prof. S.K. Saha Prof. S.K. Saha Prof. S.K. Saha Dept. of Mech. Eng. Dept. of Mech. Eng. Dept. of Mech. Eng. Dept. of Mech. Eng. IIT Delhi IIT Delhi IIT Delhi IIT Delhi Lecture 03 Feb. 06, 2019
Transcript of Lecture 03 Feb. 06, 2019
Fundamentals
Prof. S.K. SahaProf. S.K. SahaProf. S.K. SahaProf. S.K. SahaDept. of Mech. Eng.Dept. of Mech. Eng.Dept. of Mech. Eng.Dept. of Mech. Eng.
IIT DelhiIIT DelhiIIT DelhiIIT Delhi
Lecture 03 Feb. 06, 2019
Announcement
• Outlines of Lecture 01 and 02 are uploaded to
http://sksaha.com/courses
Review of Lecture 2
• More applications of robots
• Indian robots
• Robots by IIT Delhi students
Outline
• Mathematical Fundamentals
– Vectors and matrices
• Manipulator
– Links
– Joints
• Degrees of freedom
– Definition
– Formula
Vectors
• Array of n-numbers written column-wise
(not row-wise)
≡
na
a
⋮
1
a
• If a row-vector is needed, use transpose
[ ] T
naa a≡,,1 ⋯
Length and Direction
• Length or magnitude or norm of a vector
• For a Cartesian position vector, length
22
1 nT
aaa ++== ⋯aa
2
3
2
2
2
1 aaaaT
++== aa
• Direction (e.g., Angle with XY plane)
),(2tan2
2
2
13 aaaaAngle +=
Unit Vector
• A vector divided by its lengths,
• Examples of unit vectors
• Any vector can be represented as
a
aa =
a = a1 i + a2 j + a3 k
≡
0
0
1
i
≡
0
1
0
j
≡
1
0
0
k
Scalar and Dot Products
• Meaning projections
• Alternate way of calculations
nnT
baba ++= ⋯11ba
)( abba TT=
θcosabT
=≡⋅ baba
Vector- or Cross-product
• Definition
• Magnitude
321
321
bbb
aaa
kji
bac =×=
| | sinc ab θ= × =a b
kjic )()()( 122131132332 babababababa −+−+−=
Properties of Cross-product
• A is called cross-product matrix
cbabcacba )()()(TT
−=××
cbacba =× )(T
bab1a ×=× )(
−
−
−
=≡×
0
0
0
)(
12
13
23
aa
aa
aa
A1a
Differentiation of a Vector
• Chain rules of differentiation
T
naadt
d][
1ɺ⋯ɺɺ ≡≡ a
a
bababa ɺɺ TTT
dt
d+=)(
bababa ɺɺ ×+×=× )(dt
d
Linear Independence
• For a set of n independent vectors
0a =α∑=
n
i
ii
1
0=α ifor all i
Matrices
• For an m x n matrix
][ 1 naaA ⋯≡
≡Tm
T
a
a
A ⋮
1
Determinant
• For an n x n (square) matrix
∑ +−=== )det()1(||)det(
21
22221
11211
ijijji
nnnn
n
n
a
aaa
aaa
aaa
AAA
⋯
⋮⋱⋮⋮
⋯
⋯
)()()(
)det(
312232211331233321123223332211
3231
222113
3331
232112
3332
232211
333231
232221
131211
aaaaaaaaaaaaaaa
aa
aaa
aa
aaa
aa
aaa
aaa
aaa
aaa
−+−−−=
+−==A
• Example of a 3 x 3 (square) matrix
Inverse
• Generally the above is not used. Solution
of linear equations are used
• Use Gaussian Elimination (GE) to solve
Tij
jiAdj )]det()1[()( AA +
−=
)()det(
11A
AA Adj=
−
Manipulator
• It has a series of links connected by joints
� Kinematic Chain (KC)
• Simple: When each and every link is
coupled to at most two other links
– Open: If it contains only two links (end ones)
that are connected to only one link �
Manipulator
– Closed: If each and every link coupled to two
other links � Mechanism
Joints or Kinematic Pairs
• Lower Pair
– Surface contact: Hinge joint of a door
• Higher pair
– Line or point contact: Roller or ball rolling
• Several Lower Pair Joints
– Slides 5-12 of Chapter 5
Degrees of Freedom (DOF)
• No. of independent (or minimum)
coordinates required to fully describe its
pose or configuration
– A rigid body in 3D space has 6 DOF
• Grubler formula (1917) for planar
mechanisms
• Kutzbach formula (1929) for spatial
mechanisms
n = s (r − 1) − c, c ≡ . . . (5.1)i1
cp
i =
∑
Grubler-Kutzbach Criterion
s : dim. of working space
(Planar, s = 3; Spatial, s = 6);
r : no. of rigid bodies or links in the system;
p : no. of kinematic pairs or joints in the system;
ci : no. of constraints imposed by each joint;
c : total no. of constraints imposed by p joints;
ni : relative degree of freedom of each joint;
n : DOF of the whole system.
Basically, no. of parameters used to define free links –
no. of constraints (independent) by joints
Four-bar Mechanism,
n = 3 (4 − 4 − 1) + (1 + 1 + 1 + 1) = 1 . . . (5.4)
Six-DOF Manipulator
n = 6 (7 − 6 − 1) + 6 × 1 = 6 . . . (5.5)
Five-bar Mechanism
n = 3 (5 − 5 − 1) + 5 × 1 = 2 . . . (5.6)
Double Parallelogram
n = 3 (5 − 6 − 1) + 6 × 1 = 0 . . . (5.7)
Summary
• Vectors and matrices were defined
• Definitions of mechanisms, DOF, etc, were
explained.
