1. A point in space can be represented by a coordinate vector relative to a reference coordinate system, while a rigid object can be represented by a single coordinate frame with its constituent points described by displacements from that frame. 2. The position and orientation of an object's coordinate frame is called its pose, which can be described relative to another frame using a relative pose variable ξ. 3. We can represent and manipulate points and poses algebraically, including representing a point with respect to a different frame by applying the relative pose using the · operator and combining relative poses through composition.

1. SBE 403 B: Bioelectronic Systems (Biomedical Robotics)
Lecture 02
Representing Position and
Orientation 1Orientation 1
Muhammad Rushdi
mrushdi@eng1.cu.edu.eg

2. Main Problems in Robotics
• What are the basic issues to be resolved and what must we
learn in order to be able to program a robot to perform its
tasks?
• Problem 1: Forward Kinematics
• Problem 2: Inverse Kinematics
• Problem 3: Velocity Kinematics Represent• Problem 3: Velocity Kinematics
• Problem 4: Path Planning
• Problem 5: Vision
• Problem 6: Dynamics
• Problem 7: Position Control
• Problem 8: Force Control
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the position and orientation of
objects (e.g. robots, cameras,
workpieces, obstacles and
paths) in an environment.

3. Textbook
• Ch. 2 Representing
Position and
Orientation
Peter Corke:
Robotics, Vision and
Control - Fundamental
Algorithms in MATLAB®.
http://www.petercorke.com/RVC/
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4. Point and Rigid Object Representation
• A point in space can be described by a coordinate
vector (bound vector).
• An object is rigid if its constituent points maintain a
constant relative position with respect to the object’s
coordinate frame.
A vector with fixed initial and
terminal points.
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5. Rigid Object Pose
• The position and orientation of a coordinate frame is
known as its pose.
• The relative pose of a frame with respect to a
reference coordinate frame is denoted by the symbol
ξ – pronounced ksi.
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6. Relative Pose
• The point P can be described with respect to either
coordinate frame: {A} or {B}. Formally we express this
as
The motion from {A} to {B}
and then to P.
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The motion from {A} to P.

7. Pose Composition (Compounding)
• If one frame can be described in terms of another by
a relative pose then they can be applied sequentially
The pose of {C}
relative to {A}
Compounding
the relative poses from {A}
to {B} and {B} to {C}
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8. Pose in 3D Space
The fixed camera observes
the object from its fixed
viewpoint and estimates the
object’s pose relative to
itself.
The other camera is not
fixed, it is attached to the
robot at some constant
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robot at some constant
relative pose and estimates
the object’s pose relative to
itself.

9. Pose in 3D Space: Graph Description
Each node in the graph
represents a pose
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Each edge represents a
relative pose.
Equivalent
graph
paths:

10. Pose in 3D Space: Algebraic Operations
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The pose of
the robot
relative to
the fixed
camera
The pose of
the robot
relative to
the world
coordinate
frame

11. Pose in 3D Space: Algebraic Operations
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12. Pose in 3D Space: Group Theory Description
In mathematical object terms, poses
constitute a group – a set of objects that
supports:
an associative binary operator
(composition) whose result belongs to the
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(composition) whose result belongs to the
group,
an inverse operation and
an identity element. In this case the group
is the special Euclidean group in either 2 or
3 dimensions which are commonly referred
to as SE(2) or SE(3) respectively.

13. Summary
1. A point is described by a coordinate vector that represents its
displacement from a reference coordinate system;
2. A set of points that represent a rigid object can be described by a single
coordinate frame, and its constituent points are described by
displacements from that coordinate frame;
3. The position and orientation of an object’s coordinate frame is referred to
as its pose;as its pose;
4. A relative pose describes the pose of one coordinate frame with respect
to another and is denoted by an algebraic variable ξ;
5. A coordinate vector describing a point can be represented with respect to
a different coordinate frame by applying the relative pose to the vector
using the · operator;
6. We can perform algebraic manipulation of expressions written in terms of
relative poses.
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14. Representing Pose in 2D
• A 3-vector representation , is not
convenient for compounding since
is a complex trigonometric function of both
poses.poses.
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15. Representing Pose in 2D
• Instead, represent the relation using a rotation
matrix
then a translation.
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16. Properties of 2D Rotation Matrices
Orthonormal since each of its columns is a unit
vector and the columns are orthogonal
Determinant = +1 The length of a vector is
unchanged after transformation
R−1= RT The inverse rotation is the same as theR = R The inverse rotation is the same as the
transpose
• 2D Rotation belong to the special orthogonal
group of dimension 2 or
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17. 2D Rotation Matrices:
Non-minimal Representation
• A rotation matrix is an example of a non-
minimum representation:
• Instead of representing an angle, which is a scalar, we have used a 2 ×
2 matrix that comprises four elements.
• However these elements are not independent. Each column has a unit
magnitude which provides two constraints. The columns are orthogonal
which provides another constraint. Four elements and three constraints
are effectively one independent value.
• Disadvantages (e.g. increased memory) <<
Advantages (e.g. composability)
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18. 2D Rotation and Translation
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19. Homogeneous Transformation
is homogeneous transformation matrix
that has a very specific structure and belongs
to the special Euclidean group of dimension
2 or
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20. A homogeneous transformation
represents relative pose ξ
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21. MATLAB Demo (pp. 22-24, RVC)
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