Homogeneous Transformation
Matrices in Robotics
Lec-4
Presented By-
Nure Hafsa Shefa (20CSE018)
Rotation in 3D
Rotate over angle θ around x-as: Rotate over angle θ around y-as: Rotate over angle θ around z-as:
x′ = x x′ = x cos θ + z sin θ x′ = x cos θ - y sin θ
y′ = y cos θ - z sin θ y′ = y y′ = x sin θ + y cos θ
z′ = y sin θ + z cos θ z′ = z cos θ - x sin θ z′ = z
Composite Rotations: Yaw-Pitch-Roll
Imagine three lines running through an airplane and intersecting at right angles at the airplane's
center of gravity.
● Rotation around the front-to-back axis is called roll.
● Rotation around the side-to-side axis is called pitch.
● Rotation around the vertical axis is called yaw.
Homogeneous Coordinates (Observation)
● We need pure rotations and translations to characterize
the position and orientation of a point relative to the
coordinate frame attached to the base.
● While a rotation can be represented by a 3x3 matrix, it is
not possible to represent translation by the same dimension.
● So, we need to move to a higher dimensional space, like-
here we need four dimensional space of homogeneous
coordinates.
Homogeneous Coordinates (Definition)
Let q be a point in R³, and let F be an orthonormal coordinate
frame of R³. If σ is any non zero scale factor, then the
homogeneous coordinates of q with respect to F are denoted as
[q]F and defined:
[q]・F = [ σ・q1, σ・q2, σ・q3, σ]
In robotics we use σ = 1 for convenience So,
[q]F = [q1, q2, q3, 1]
Homogeneous Transformation Matrix
If a physical point in three dimensional space is expressed in terms of its homogeneous
coordinates and we want to change from one coordinate frame to another, we use a 4x4
homogeneous transformation matrix. In general T is —
● R is the 3x3 matrix rotation matrix
● P is a 3x1 translation vector
● η is a perspective vector, set to zero
● σ is 1 for robotics
Homogeneous Transformation Matrix
Using homogeneous coordinates, translations also can be represented by 4x4 matrices. In
terms of homogeneous coordinate frames, the translation of M can be represented by a 4x4
matrix, denoted Tran(p), where —
Tran(p) is known as the fundamental
homogeneous translation matrix
Inverse Homogeneous Transformation
If T be a homogeneous transformation matrix with rotation R and translation P between two
orthonormal coordinate frames and if η = 0, σ = 1, then the inverse transformation is:
Why Transformation Matrices are important?
● Solve forward and inverse kinematics problems.
● Facilitate coordinate transformations between different components of a robot.
● Enable accurate manipulator control and trajectory planning.
● Support workspace analysis and collision avoidance.
● Integrate sensor data into the robot's coordinate system.
● Assist in robot calibration for improved accuracy.
● Aid in robot simulation and visualization.
Components of an Industrial Robot
The four main parts of an industrial robot are
—
● Manipulator,
● Controller,
● Human interface device, and
● Power supply.
Components of an Industrial Robot
The four main parts of an industrial robot are
—
● Manipulator,
● Controller,
● Human interface device, and
● Power supply.
Degrees of Freedom (DoF)
● In general, degrees of freedom (DOF) are the set of
independent displacements that specify completely
the displaced or deformed position of the body or
system.
● In robotics, degrees of freedom is often used to
describe the number of directions that a robot can
move a joint.
Degrees of Freedom (DoF)
● A human arm is considered to have 7 DOF. A
shoulder gives pitch, yaw and roll, an elbow allows
for pitch, and a wrist allows necessary to move the
hand to any point in space, but people would lack the
ability to grasp things from different angles or
directions.
● A robot (or object) that has mechanisms to control all
6 physical DOF is said to be holonomic. An object
with fewer controllable DOF than total DOF is to be
non- holonomic and an object with more controllable
DOF than total DOF (such as human arm) is said to
be redundant.
Any Question???
