This document provides an overview of basic joints, kinematics concepts, and coordinate frame transformations using matrices. It defines spherical, revolute, and prismatic joints. It also defines forward and inverse kinematics. The document then reviews dot products, matrix operations, and basic transformations including translation, rotation, and combining them into homogeneous transformations using 4x4 matrices.
Monthly Economic Monitoring of Ukraine No 231, April 2024
Robot kinematics
1. Presented by-
Sameer Vishwakarma
M.Tech. (Production)
III SEM.
Submitted to-
Dr. Manoj Kumar Agrawal
Associate Professor
Institute of Engineering and Technology
Department of Mechanical Engineering
3. We are interested in two kinematics topics
Forward Kinematics (angles to position)
What you are given: The length of each link
The angle of each joint
What you can find: The position of any point
(i.e. it’s (x, y, z) coordinates
Inverse Kinematics (position to angles)
What you are given: The length of each link
The position of some point on the robot
What you can find: The angles of each joint needed to obtain
that position
4. Quick Math Review
Dot Product:
Geometric Representation:
A
B
θ
cosθBABA
Unit Vector
Vector in the direction of a chosen vector but whose magnitude is 1.
B
B
uB
y
x
a
a
y
x
b
b
Matrix Representation:
yyxx
y
x
y
x
baba
b
b
a
a
BA
B
Bu
5. Quick Matrix Review
Matrix Multiplication:
An (m x n) matrix A and an (n x p) matrix B, can be multiplied since the number of columns of A
is equal to the number of rows of B.
Non-Commutative Multiplication
AB is NOT equal to BA
dhcfdgce
bhafbgae
hg
fe
dc
ba
Matrix Addition:
hdgc
fbea
hg
fe
dc
ba
6. Basic Transformations
Moving Between Coordinate Frames
Translation Along the X-Axis
N
O
X
Y
Px
VN
VO
Px = distance between the XY and NO coordinate planes
Y
X
XY
V
V
V
O
N
NO
V
V
V
0
P
P x
P
(VN,VO)
Notation:
10. Rotation (around the Z-Axis)
X
Y
Z
X
Y
V
VX
VY
Y
X
XY
V
V
V
O
N
NO
V
V
V
= Angle of rotation between the XY and NO coordinate axis
11. X
Y
V
VX
VY
Unit vector along X-Axis
x
xVcosαVcosαVV NONOXYX
NOXY
VV
Can be considered with respect to
the XY coordinates or NO coordinates
V
x)oVn(VV ONX
(Substituting for VNO using the N and O components of the
vector)
)oxVnxVV ONX
()(
))
)
(sinθV(cosθV
90))(cos(θV(cosθV
ON
ON
12. Similarly….
yVα)cos(90VsinαVV NONONOY
y)oVn(VV ONY
)oy(V)ny(VV ONY
))
)
(cosθV(sinθV
(cosθVθ))(cos(90V
ON
ON
So….
)) (cosθV(sinθVV ONY
)) (sinθV(cosθVV ONX
Y
X
XY
V
V
V
Written in Matrix Form
O
N
Y
X
XY
V
V
cosθsinθ
sinθcosθ
V
V
V
Rotation Matrix about the z-axis
14.
O
N
y
x
Y
X
XY
V
V
cosθsinθ
sinθcosθ
P
P
V
V
V
HOMOGENEOUS REPRESENTATION
Putting it all into a Matrix
1
V
V
100
0cosθsinθ
0sinθcosθ
1
P
P
1
V
V
O
N
y
x
Y
X
1
V
V
100
Pcosθsinθ
Psinθcosθ
1
V
V
O
N
y
x
Y
X
What we found by doing a translation
and a rotation
Padding with 0’s and 1’s
Simplifying into a matrix form
100
Pcosθsinθ
Psinθcosθ
H y
x
Homogenous Matrix for a Translation in XY plane,
followed by a Rotation around the z-axis