Robotic Systems

Ch. 1: Introduction
Ch. 2 Craig & Ch. 1 Tsai
Description of position and orientation in 3D space
In order to describe the position and orientation of a body in space,
we will always attach a coordinate system, or frame, rigidly to the
object.

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Manipulators consists of nearly rigid links which are
connected by joints that allow relative motion of the
neighboring links.
These joint are usually
instrumented with position
sensors, which allow the
relative position of neighboring
links to be measured.
At the end of the chain of links
that make up the manipulator
is the end-effector.
End-effector could be:
Gripper
Welding torch Kinematic equation describes the tool frame relative
Electromagnet to the base frame as a function of the joint variables
Etc..
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𝑃𝑦 = 𝑃𝑣 cos 𝜃 + 𝑥 sin 𝜃 + 𝑃𝑢 − 𝑥 sin 𝜃
𝑃𝑦 = 𝑃𝑣 cos 𝜃 + 𝑃𝑢 sin 𝜃
𝑃𝑥 = 𝑃𝑢 cos 𝜃 − 𝑃𝑣 sin 𝜃

𝑃𝑥 cos 𝜃 − sin 𝜃 𝑃𝑢
𝑃𝑦 = sin 𝜃 cos 𝜃 𝑃𝑣

𝑃𝑢 cos 𝜃 − sin 𝜃 𝑃𝑥
=
𝑃𝑣 sin 𝜃 cos 𝜃 𝑃𝑦

𝑃𝑢 = 𝑃𝑥 cos 𝜃 + 𝑃𝑦 sin 𝜃
𝑃𝑣 = −𝑃𝑥 sin 𝜃 + 𝑃𝑦 cos 𝜃
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Description of an orientation
In order to describe the orientation
of a body, we will attach a
coordinate system to the body and
then give a description of this
coordinate system relative to the
reference system.
Rotation matrix

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The scalars 𝑟𝑖𝑗 can be obtained by noting that the components of any
vector are simply the projections of that vector onto the unit directions
of its reference frame.
Hence, each component of 𝐵𝐴𝑅 can be written as the dot product of a
pair of unit vectors:
𝐴 𝐴෠ 𝐴෠ 𝐴 መ
𝐵 𝑅 = 𝑋𝐵 𝑌𝐵 𝑍𝐵

𝑋෠𝐵 ∙ 𝑋෠𝐴 𝑌෠𝐵 ∙ 𝑋෠𝐴 𝑍መ𝐵 ∙ 𝑋෠𝐴

𝐴 ෠ ෠
𝐵𝑅 = 𝑋𝐵 ∙ 𝑌𝐴 𝑌෠𝐵 ∙ 𝑌෠𝐴 𝑍መ𝐵 ∙ 𝑌෠𝐴
𝑋෠𝐵 ∙ 𝑍መ𝐴 𝑌෠𝐵 ∙ 𝑍መ𝐴 𝑍መ𝐵 ∙ 𝑍መ𝐴

The dot product of two-unit vectors yields the cosine of the angle
between them, so it is clearly why the components of the rotation
matrices is often referred as direction cosines.

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Rotation matrix about the principal axes of the
reference coordinate system

1 0 0 1 0 0
𝑅𝑥,𝛼 = 0 𝐶𝛼 −𝑆𝛼 𝑇
𝑅𝑥,𝛼 = 𝑅𝑥,−𝛼
𝑅𝑥,−𝛼 = 0 𝐶𝛼 𝑆𝛼
0 𝑆𝛼 𝐶𝛼 0 −𝑆𝛼 𝐶𝛼

𝐶𝜑 0 𝑆𝜑 𝐶𝜑 0 −𝑆𝜑 𝑇
𝑅𝑦,𝜑 = 𝑅𝑦,−𝜑
𝑅𝑦,𝜑 = 0 1 0 𝑅𝑦,−𝜑 = 0 1 0
−𝑆𝜑 0 𝐶𝜑 𝑆𝜑 0 𝐶𝜑

𝐶𝜃 −𝑆𝜃 0 𝐶𝜃 𝑆𝜃 0
𝑇
𝑅𝑧,𝜃 = 𝑆𝜃 𝐶𝜃 0 𝑅𝑧,−𝜃 = −𝑆𝜃 𝐶𝜃 0 𝑅𝑧,𝜃 = 𝑅𝑧,−𝜃
0 0 1 0 0 1

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 Description of a frame

: is the vector that locates the

origin of frame B.

The description of a frame

can be thought of as a position
vector and a rotation matrix.

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 Mapping: changing description from frame to frame
Mapping involving translated frames (only translation)
Mapping involving
translated frames, only
translation, corresponding
axes of each frame are
parallel to each other.

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Mapping involving rotated frames (only rotation)

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Mapping involving general frames (translation and rotation)
Very often, we know the description of a vector with respect to some
frame B and we would like to know its description with respect to
another frame A.

: is called homogeneous
transform
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For the figure shown below, find the 4 x 4 homogeneous
transformation matrices 𝑖−1𝐴𝑖 , 0𝐴𝑖 ; for 𝑖 = 1, 2, 3, 4, 5.

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 Consider all transforms are
known except
Then,

Two possible description

of are

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The bolt frame relative
to the hand frame is
computed as:

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Rotation matrix

𝑢𝑥 𝑣𝑥 𝑤𝑥 𝑎11 𝑎12 𝑎13

𝐴 𝑤𝑦 = 𝑎21 𝑎22 𝑎23
𝑅𝐵 = 𝑢𝑦 𝑣𝑦
𝑢𝑧 𝑣𝑧 𝑤𝑧 𝑎31 𝑎32 𝑎33

𝐴
𝑅𝐵 is called the rotation matrix of the moving frame B with
respect to the fixed frame A.
The rotation matrix specifies the orientation of frame B completely
with respect to frame A.
The columns of a rotation matrix represents a three orthogonal unit
vectors of the moving coordinates expressed in the fixed
frame.
Therefore, the rotation matrix is orthogonal (orthonormal).

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The orthogonality conditions can be stated as follows:

𝒖∙𝒖=1 𝒖𝟐 = 1
𝒗∙𝒗=1 𝒗𝟐 = 1
𝒘∙𝒘=1 𝒘𝟐 = 1

𝒖∙𝒗=0 𝒖𝑻 𝒗 = 0
𝒗∙𝒘=0 𝒗𝑻 𝒘 = 0
𝒘∙𝒖=0 𝒘𝑻 𝒖 = 0

𝒖×𝒗=𝒘 𝒗×𝒘=𝒖 𝒘×𝒖=𝒗

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Using orthogonality conditions, it can be shown that:

𝐵
𝑅𝐴 = 𝐴𝑅𝐵𝑇 = 𝐴𝑅𝐵−1
𝐴
det 𝑅𝐵 = 1
𝐵 𝑇
𝒊𝐴
𝐴 𝐴 𝐴 𝐴 𝐵 𝑇
𝑅𝐵 = 𝒖 𝒗 𝒘 = 𝒋𝐴
𝐵 𝑇
𝒌𝐴

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Rotation matrices about the principal axes of the
reference coordinate system

𝑐𝜃 −𝑠𝜃 0
𝑅𝑧,𝜃 = 𝑠𝜃 𝑐𝜃 0
0 0 1

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 Rotation matrices about the principal axes of the
reference coordinate system

𝑐𝜃 −𝑠𝜃 0
𝑅𝑧,𝜃 = 𝑠𝜃 𝑐𝜃 0
0 0 1

1 0 0
𝑅𝑥,𝜓 = 0 𝑐𝜓 −𝑠𝜓
0 𝑠𝜓 𝑐𝜓

𝑐𝜑 0 𝑠𝜑
𝑅𝑦,𝜑 = 0 1 0
−𝑠𝜑 0 𝑐𝜑
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Euler Angles Representations
Roll – Pitch - Yaw Angles

𝑅 𝜓, 𝜃, 𝜙 = 𝑅 𝑧, 𝜙 𝑅 𝑦, 𝜃 𝑅 𝑥, 𝜓
𝑐𝜙 𝑐𝜃 𝑐𝜙𝑠𝜃𝑠𝜓 − 𝑠𝜙𝑐𝜓 𝑐𝜙𝑠𝜃𝑐𝜓 + 𝑠𝜙𝑠𝜓
= 𝑠𝜙 𝑐𝜃 𝑠𝜙𝑠𝜃𝑠𝜓 + 𝑐𝜙𝑐𝜓 𝑠𝜙𝑠𝜃𝑐𝜓 − 𝑐𝜙𝑠𝜓
−𝑠𝜃 𝑐𝜃 𝑠𝜓 𝑐𝜃 𝑐𝜓
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Roll – Pitch - Yaw Angles
Given the roll, pitch and yaw, we can compute the overall rotation
matrix from the above equation.
Also, given the rotation matrix, we can compute the equivalent roll-
pitch-yaw angles as follows:
𝜃 = sin−1 −𝑎31
𝜓 = Atan2 𝑎32 /𝑐𝜃, 𝑎33 /𝑐𝜃,

𝜙 = Atan2 𝑎21 /𝑐𝜃, 𝑎11 /𝑐𝜃,

Provided 𝑐𝜃 ≠ 0. sin−1 𝑥 is the arcsine function and atan2(𝑦, 𝑥) is a
two-argument function that yields one unique solution for the angle.
Hence, corresponding to a given rotation matrix, there are generally
two solutions of the roll-pitch-yaw angles.
If 𝜃 = ±90𝑜 , the solution degenerate, for this special case, only the
sum and the difference of 𝜑 and 𝜓 can be computed.
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w-u- w Euler Angles
Consider three successive rotations of the rigid body about the
coordinate axes of a moving frame.

The resulting rotation matrix can derived by a kinematic inversion,

that is, by considering the orientation of frame A with respect to
frame B. Under kinematic inversion, the coordinate axes of B are
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w-u- w Euler Angles (cont.)
𝐵
𝑅𝐴 −𝜙, −𝜃, −𝜓 = 𝑅 𝑤, −𝜓 𝑅 𝑢, −𝜃 𝑅 𝑤, −𝜙
𝐴 𝐵 −1
Since 𝑅𝐵 = 𝑅𝐴 and 𝑅−1 𝑤, −𝜙 = 𝑅 𝑤, 𝜙
𝐴 −1
𝑅𝐵 = 𝑅 𝑤, −𝜓 𝑅 𝑢, −𝜃 𝑅 𝑤, −𝜙
𝐴
𝑅𝐵 = 𝑅−1 𝑤, −𝜙 𝑅−1 𝑢, −𝜃 𝑅−1 𝑤, −𝜓
𝐴
𝑅𝐵 = 𝑅 𝑤, 𝜙 𝑅 𝑢, 𝜃 𝑅 𝑤, 𝜓

𝑐𝜙𝑐𝜓 − 𝑠𝜙𝑐𝜃𝑠𝜓 −𝑐𝜙𝑠𝜓 − 𝑠𝜙𝑐𝜃𝑐𝜓 𝑠𝜙𝑠𝜃

𝐴
𝑅𝐵 = 𝑠𝜙𝑐𝜓 + 𝑐𝜙𝑐𝜃𝑠𝜓 −𝑠𝜙𝑠𝜓 + 𝑐𝜙𝑐𝜃𝑐𝜓 −𝑐𝜙𝑠𝜃
𝑠𝜃𝑠𝜓 𝑠𝜃𝑐𝜓 𝑐𝜃

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w-u-w Euler Angles (cont.)
Given the w-u- w Euler angles , we can compute the overall rotation
matrix from the above equation.
Also, given the rotation matrix, we can compute the w-u- w Euler
angles as follows:
𝜃 = cos −1 𝑎33
𝜙 = Atan2 𝑎13 /𝑠𝜃, −𝑎23 /𝑠𝜃,

𝜓 = Atan2 𝑎31 /𝑠𝜃, 𝑎32 /𝑠𝜃,

Provided s𝜃 ≠ 0. When 𝜃 = 0 or 𝜃 = 180𝑜 , the solution of the
above equation degenerates. For this degenerated case, only the sum
or the difference ∅ and 𝜓 can be computed.

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Euler’s Theorem:
The general displacement of a rigid body with some point
fixed is a rotation about some axis.

This unique axis of rotation is called the screw axis.

−1
𝑎11 + 𝑎22 + 𝑎33 − 1
𝜃 = cos
2

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Also, if the elements of the rotation matrix are given, we can
calculate the angle of rotation by summing the diagonal elements:
𝑎11 + 𝑎22 + 𝑎33 − 1
𝜃= cos −1
2
The direction of the screw axis is obtained by taking the difference
between each pair of the two opposing off-diagonal elements:
𝑎32 − 𝑎23
𝑠𝑥 =
2 𝑠𝜃
𝑎13 − 𝑎31
𝑠𝑦 =
2 𝑠𝜃
𝑎21 − 𝑎12
𝑠𝑧 =
2 𝑠𝜃
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Rotation matrices about the principal axes of the
reference coordinate system

𝑎11 = 𝑠𝑥2 − 1 1 − 𝑐𝜃 + 1
𝑠𝑥 = 𝑠𝑦 = 0, 𝑠𝑧 = 1 𝑎12 = 𝑠𝑥 𝑠𝑦 1 − 𝑐𝜃 − 𝑠𝑧 𝑠𝜃
𝑎13 = 𝑠𝑥 𝑠𝑧 1 − 𝑐𝜃 + 𝑠𝑦 𝑠𝜃
𝑐𝜃 −𝑠𝜃 0
𝑅𝑧,𝜃 = 𝑠𝜃 𝑐𝜃 0 𝑎21 = 𝑠𝑦 𝑠𝑥 1 − 𝑐𝜃 + 𝑠𝑧 𝑠𝜃
0 0 1 𝑎22 = 𝑠𝑦2 − 1 1 − 𝑐𝜃 + 1
𝑎23 = 𝑠𝑦 𝑠𝑧 1 − 𝑐𝜃 − 𝑠𝑥 𝑠𝜃
𝑎31 = 𝑠𝑧 𝑠𝑥 1 − 𝑐𝜃 − 𝑠𝑦 𝑠𝜃
𝑎32 = 𝑠𝑧 𝑠𝑦 1 − 𝑐𝜃 + 𝑠𝑥 𝑠𝜃
𝑎33 = 𝑠𝑧2 − 1 1 − 𝑐𝜃 + 1
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 Rotation matrices about the principal axes of the
reference coordinate system

𝑐𝜃 −𝑠𝜃 0
𝑅𝑧,𝜃 = 𝑠𝜃 𝑐𝜃 0 𝑠𝑥 = 𝑠𝑦 = 0, 𝑠𝑧 = 1
0 0 1

1 0 0
𝑅𝑥,𝜓 = 0 𝑐𝜓 −𝑠𝜓 𝑠𝑦 = 𝑠𝑧 = 0, 𝑠𝑥 = 1
0 𝑠𝜓 𝑐𝜓

𝑐𝜑 0 𝑠𝜑
𝑅𝑦,𝜑 = 0 1 0 𝑠𝑥 = 𝑠𝑧 = 0, 𝑠𝑦 = 1
−𝑠𝜑 0 𝑐𝜑
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Description of a
Location of a rigid body
𝐴
𝑷 = 𝑂𝑃
𝐵
𝑷 = 𝑄𝑃
𝐴
𝒒 = 𝑂𝑄
𝐴
𝑷 = 𝑂𝑄 + 𝑄𝑃
𝐴 𝐴 𝐴 𝐵
𝑷 = 𝒒 + 𝑅𝐵 𝑷

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Homogeneous
Transformation
𝐴 𝐴 𝐴 𝐵
𝑷 = 𝒒 + 𝑅𝐵 𝑷
𝐴෡ 𝐴 𝐵෡
𝑷 = 𝑇𝐵 𝑷
𝐴෡ 𝑇
𝑷 = 𝑝𝑥 𝑝𝑥 𝑝𝑥 1
𝐵෡ 𝑇
𝑷 = 𝑝𝑢 𝑝𝑣 𝑝𝑤 1
𝐴 𝐴
𝐴
𝑇𝐵 = 𝑅𝐵 𝒒
000 1

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Homogeneous Transformation

𝐵෡ 𝐴 −1 𝐴 ෡
𝑷= 𝑇𝐵 𝑷

𝐴 𝑇 𝐴 𝑇𝐴
𝐴 −1
𝑇𝐵 = 𝑅𝐵 − 𝑅𝐵 𝒒
000 1

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The term line vector refers to a vector that is dependent on its line of
action along with direction and magnitude.
Example: the effects of a force vector depend upon its line of action
or point of application, so it would then be considered a line vector.
A free vector refers to a vector that may be positioned anywhere in
space without loss or change of meaning, provided that the
magnitude and direction are preserved.

Velocity is a free vector, then

ME 4641 Robotic and Automation 49

Then,

ME 4641 Robotic and Automation 50

Find 𝐵𝑇𝐶 and 𝐴𝑇𝐶

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 Homework
Chapter 1
Problems #: 4, 5, 7, 8, 10, 11.

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 Homework 1

2.1, 2.2, 2.4, 2.7, 2.12, 2.27, 2.28,

2.30, 2.32, 2.34, and 2.37

MATLAB Homework:
2A-a, 2B-a, 2B-b, 2B-e.

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