I N T R O D U C T I O N TO
R O B OT I C S
(OEC-EE 703)
NIRMAL MURMU
D E PA RT M E N T O F A P P L I E D P H Y S IC S
U N I V E R S I TY O F C A LC U T TA
Department of Applied Physics, University of Calcutta
SYLLABUS
• Module 1: Basics of Robotics ( 10 Hours)
• Introduction, components and structure of robotics system.
• Module 2: Robotics Kinematics (10 Hours)
• Kinematics of manipulators, rotation translation and transformation, David –
Hastemberg Representation, Inverse Kinematics. Dyamics – modeling using Newton
Euler equation.
• Module 3: Robot Dynamics ( 10 Hours)
• Linearization of Robot Dynamics – State variable continuous and discrete models.
• Module 4: Robotic Motion (10 Hours)
• Different types of trajectories and introduction to their generation. Position Control:
Independent joint control. Introduction to advanced control for robot application.
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
INTRODUCTION TO ROBOTICS
• Industrial Manipulator
• Robot arm kinematics and dynamics
• Planning of manipulator trajectory
• Elementary steps for robot arm design
• Control of robot arm
• Force and Impedance Control
• Mobile Robot: Wheeled and legged robots, trajectory planning,
locomotion, SLAM.
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
BOOKS & REFERENCES
[1] J. Craig, Introduction to robotics. Upper Saddle River, N.J.:
Pearson/Prentice Hall, 2005.
[2] J. Selig, Introductory robotics. New York, NY: Prentice Hall, 1992.
[3] R. J. Schilling, Fundamentals of robotics analysis and control, 5th
ed. New Delhi: Prentice-Hall, 1990.
[4] M. Groover, M. Weiss, R. Nagel, N. Odrey and A.
Dutta, Industrial robotics. New Delhi: McGraw-Hill, 2012.ork, NY:
Prentice Hall, 1992.
[5] R. Murray, Z. Li and S. Sastry, A Mathematical Introduction to
Robotic Manipulation. CRC, 1994.
ODD SEMESTER, 2022
ODD SEMESTER, 2022
BOOKS & REFERENCES
Department of Applied Physics, University of Calcutta
Department of Applied Physics, University of Calcutta
OUTLINE
• Introductory overview
• Mathematical preliminaries
• Kinematics of serial manipulators
• Velocity analysis and statics of manipulator
• Dynamics of manipulators
• Trajectory planning
• Control of manipulators
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
D I R E C T & I N V E R S E K I N E M AT I C S
• Spatial description and transformation
• Direct kinematics
• Link description
• Link-connection description
• Affixing frames to links
• Manipulator kinematics
• Example
• Inverse kinematics
ODD SEMESTER, 2022
ODD SEMESTER, 2022
LINK DESCRIPTION
Department of Applied Physics, University of Calcutta
Department of Applied Physics, University of Calcutta
LINK DESCRIPTION
• Think of the manipulator as a chain of bodies (links)
connected by joints.
• Assume, manipulators constructed with joints of 1
degree of freedom (DOF): revolute and prismatic
joints.
• A mechanism is built with a joint having n degrees of
freedom is rare
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
LINK DESCRIPTION
• The links are numbered from 0 (immobile base) to n (free end of
the arm).
• An end-effector generally in 3-space, a minimum of 6 joints is
required.
• To obtain the kinematic equations of the mechanism
• a link is considered as a rigid body
• that defines the relationship between two neighboring joint
axes of a manipulator
• Joint axes are defined by lines in space
ODD SEMESTER, 2022
ODD SEMESTER, 2022
LINK DESCRIPTION
Department of Applied Physics, University of Calcutta
Department of Applied Physics, University of Calcutta
LINK DESCRIPTION
• Joint axis i :
• a line in space, or
• a vector direction, about which link i rotates relative to link (i-1)
• link i-1 can be specified by 2 numbers: link length ai-1 and link
twist αi-1
• Link length and twist are sufficient to define the relation between
any 2 axes in space
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
LINK DESCRIPTION
• Link length:
• distance is measured along a line that is mutually
perpendicular to both axes.
• mutual perpendicular always exists; it is unique except when
both axes are parallel
• Link twist:
• imagine a plane i.e. normal to the mutually perpendicular line
• project the axes (i-1) and i onto this plane
• measure the angle between them in the right-hand sense
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
D I R E C T & I N V E R S E K I N E M AT I C S
• Spatial description and transformation
• Direct kinematics
• Link description
• Link-connection description
• Affixing frames to links
• Manipulator kinematics
• Example
• Inverse kinematics
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
LINK-CONNECTION DESCRIPTION
• Neighboring links have a common axis
• 2 parameters define the link-connection:
• Link offset 𝑑𝑖 : the distance along the common axis from one
link to the next
• Joint angle 𝜃𝑖: amount of rotation about the common axis
• The link offset 𝑑𝑖 is variable if joint 𝑖 is prismatic
• The joint angle 𝜃𝑖 is variable if the joint is revolute
ODD SEMESTER, 2022
ODD SEMESTER, 2022
LINK-CONNECTION DESCRIPTION
Department of Applied Physics, University of Calcutta
Department of Applied Physics, University of Calcutta
LINK-CONNECTION DESCRIPTION
variable variable angle
offset di θi
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
FIRST AND LAST LINK IN THE
CHAIN
• The link length 𝑎𝑖, and the link twist 𝛼𝑖 depend on the joint
axis 𝑖 and 𝑖 + 1.
• Convention:
𝑎0 = 𝑎𝑛 = 0 and 𝛼0 = 𝛼𝑛 = 0
• Similar for the link offset di and the joint angle 𝜃𝑖 :
• if joint 1 is revolute, then 𝑑1 = 0.
• if joint 1 is prismatic, then 𝜃1 = 0.
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
D E N A V I T- H A R T E N B E R G N O TAT I O N
• Any robot can be described kinematically by 4 quantities for each
link:
• 2 for the link
• 2 to describe the link’s connection
• For revolute joints, 𝜃𝑖 is called the joint variable (the other 3
quantities are fixed).
• For prismatic joints, 𝑑𝑖 is the joint variable (the other 3 quantities
are fixed).
• The definition of mechanics by means of these quantities is called
the Denavit-Hartenberg notation.
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
D I R E C T & I N V E R S E K I N E M AT I C S
• Spatial description and transformation
• Direct kinematics
• Link description
• Link-connection description
• Affixing frames to links
• Manipulator kinematics
• Example
• Inverse kinematics
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
AFFIXING FRAMES TO LINKS
• We define a frame attached to each link:
• frame {𝑖} is attached rigidly to link {𝑖}
• Convention:
• the origin is located where the “link length line” 𝑎𝑖 intersects
the joint axis
• the 𝑍 axis is coincident with the joint axis
• the X axis points along ai, to the direction from joint 𝑖 to joint
𝑖+1
• The 𝑌 axis is formed by the right-hand rule
ODD SEMESTER, 2022
ODD SEMESTER, 2022
AFFIXING FRAMES TO LINKS
Department of Applied Physics, University of Calcutta
Department of Applied Physics, University of Calcutta
FIRST AND LAST LINK IN THE
CHAIN
• Frame {0} is the immobile base (link 0) of the robot. Thus 𝑎0 =
0 and α0=0.
• If joint 1 is revolute, then 𝑑1 = 0.
If joint 1 is prismatic, then 𝜃1 = 0.
• If joint 𝑛 is revolute, then 𝑋𝑛’s direction is the same as 𝑋𝑛 − 1’s (𝜃1 =
0), and {𝑛}’s origin is the intersection of 𝑋𝑛 − 1 and axis𝑛 when
𝑑𝑛 = 0.
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE
i αi-1 ai-1 di θi
1 0 0 0 θ1
2 0 L1 0 θ2
3 0 L2 0 θ3
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
D I R E C T & I N V E R S E K I N E M AT I C S
• Spatial description and transformation
• Direct kinematics
• Link description
• Link-connection description
• Affixing frames to links
• Manipulator kinematics
• Example
• Inverse kinematics
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
M A N I P U L ATO R K I N E M AT I C S
• We want to construct the transform that defines frame {i} relative
to frame {i-1}, as a function of the four link parameters
• Each transform will be a function of only 1 joint variable
• Each link has his frame, thus the kinematics problem has been
broken into n subproblems
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
M A N I P U L ATO R K I N E M AT I C S
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
M A N I P U L ATO R K I N E M AT I C S
• Each transform can be written as a combination of a
translation and a rotation
• The single transformation that relates frame {n} to
frame {0}:
0𝑇
𝑛 = 01𝑇 02𝑇 … 𝑛−1𝑛𝑇
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
M A N I P U L ATO R K I N E M AT I C S
• General form:
cos 𝜃𝑖 − sin 𝜃𝑖 0 𝑎𝑖−1
𝑖−1 sin 𝜃𝑖 cos 𝛼𝑖−1 cos 𝜃𝑖 cos 𝛼𝑖−1 − sin 𝛼𝑖−1 − sin 𝛼𝑖−1 𝑑𝑖
𝑖 𝑇 =
sin 𝜃𝑖 sin 𝛼𝑖−1 cos 𝜃𝑖 sin 𝛼𝑖−1 cos 𝛼𝑖−1 cos 𝛼𝑖−1 𝑑𝑖
0 0 0 1
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
F R A M E S W I T H S TA N DA R D
NAMES
Wrist Frame
Base Frame
Tool Frame
Station Frame
Goal Frame
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
A C T U ATO R S PA C E , J O I N T S PA C E
A N D C A R T E S I A N S PA C E
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
D I R E C T & I N V E R S E K I N E M AT I C S
• Spatial description and transformation
• Direct kinematics
• Link description
• Link-connection description
• Affixing frames to links
• Manipulator kinematics
• Example
• Inverse kinematics
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• PUMA 560 is a 6 DOFs industrial robot with all rotational joints
(6R mechanism)
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• PUMA 560 is a 6 DOFs industrial robot with all rotational joints
(6R mechanism)
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• PUMA 560 is a 6 DOFs industrial robot with all rotational joints
(6R mechanism)
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• Frame {0} and Frame {1} coincides when 𝜃1 = 0.
• The joint axes Z4, Z5 and Z6 (wrist’s joints) intersect at a common
point.
• Z4, Z5 and Z6 are mutually orthogonal.
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• Frames and link parameters:
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• Frames and link parameters:
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• Frames and link parameters:
αi-1 ai-1 di θi
1 0 0 0 θ1
2 -90º 0 0 θ2
3 0 a2 d3 θ3
4 -90º a3 d4 θ4
5 90º 0 0 θ5
6 -90º 0 0 θ6
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• Link transformations: using the manipulator kinematics
transformation matrix
𝑐𝜃1 −𝑠𝜃1 0 0 𝑐𝜃4 −𝑠𝜃4 0 𝑎3
0 𝑠𝜃1 𝑐𝜃1 0 0 3 0 0 1 𝑑4
1𝑇 = 4𝑇 =
0 0 1 0 −𝑠𝜃4 −𝑐𝜃4 0 0
0 0 0 1 0 0 0 1
𝑐𝜃2 −𝑠𝜃2 0 0 𝑐𝜃5 −𝑠𝜃5 0 0
1 0 0 1 0 4 0 0 −1 0
2𝑇 = 5𝑇 =
−𝑠𝜃2 −𝑐𝜃2 0 0 𝑠𝜃5 𝑐𝜃5 0 0
0 0 0 1 0 0 0 1
𝑐𝜃3 −𝑠𝜃3 0 𝑎2 𝑐𝜃6 −𝑠𝜃6 0 0
2 𝑠𝜃3 𝑐𝜃3 0 0 5 0 0 1 0
3𝑇 = 6𝑇 =
0 0 1 𝑑3 −𝑠𝜃6 −𝑐𝜃6 0 0
0 0 0 1 0 0 0 1
ODD SEMESTER, 2022
Department of Applied Physics, University of Calcutta
EXAMPLE: PUMA 560
• The kinematics equations of the PUMA 560:
&06 𝑇 =10 𝑇 ⋅12 𝑇 ⋅23 𝑇 ⋅34 𝑇 ⋅45 𝑇 ⋅56 𝑇 𝑟11 = 𝑐1 [𝑐23 (𝑐5 𝑐6 − 𝑠4 𝑠5 ) − 𝑠23 𝑠5 𝑐5 ] + 𝑠1 (𝑠4 𝑐5 𝑐6 + 𝑐4 𝑠6 ),
𝑟11 𝑟12 𝑟13 𝑝𝑋 𝑟21 = 𝑠1 [𝑐23 (𝑐4 𝑐5 𝑐6 − 𝑠4 𝑠6 ) − 𝑠23 𝑠5 𝑐6 ] + 𝑐1 (𝑠4 𝑐5 𝑐6 + 𝑐4 𝑠6 ),
𝑟 𝑟 𝑟 𝑝𝑌 𝑟31 = −𝑠23 (𝑐4 𝑐5 𝑐6 − 𝑠4 𝑠6 ) − 𝑠23 𝑠5 𝑐6 ,
= 21 22 23 𝑟12 = 𝑐1 [𝑐23 (−𝑐4 𝑐5 𝑐6 − 𝑠4 𝑐6 ) + 𝑠23 𝑠5 𝑐6 ] + 𝑠1 (𝑐4 𝑐6 − 𝑠4 𝑐5 𝑠6 ),
𝑟31 𝑟32 𝑟33 𝑝𝑍
0 0 0 1 𝑟22 = 𝑠1 [𝑐23 (−𝑐4 𝑐5 𝑠6 − 𝑠4 𝑐6 ) + 𝑠23 𝑠5 𝑠6 ] − 𝑐1 (𝑐4 𝑐6 − 𝑠4 𝑐5 𝑠6 ),
𝑊𝑖𝑡ℎ: 𝑟32 = −𝑠23 (−𝑐4 𝑐5 𝑠6 − 𝑠4 𝑐6 ) + 𝑐23 𝑠5 𝑠6 ,
𝑝𝑋 = 𝑐1 [𝑎2 𝑐2 + 𝑎3 𝑐23 − 𝑑4 𝑠23 ] − 𝑑3 𝑠1 , 𝑟13 = −𝑐1 (𝑐23 𝑐4 𝑠5 + 𝑠23 𝑐5 ) − 𝑠1 𝑠4 𝑠5 ,
𝑝𝑌 = 𝑠1 [𝑎2 𝑐2 + 𝑎3 𝑐23 − 𝑑4 𝑠23 ] − 𝑑3 𝑐1 , 𝑟23 = −𝑠1 (𝑐23 𝑐4 𝑠5 + 𝑠23 𝑐5 ) + 𝑐1 𝑠4 𝑠5 ,
𝑝𝑍 = −𝑎3 𝑠23 − 𝑎2 𝑠2 − 𝑑4 𝑐23 , 𝑟33 = 𝑠23 𝑐4 𝑠5 − 𝑠23 𝑐5 .
ODD SEMESTER, 2022
ODD SEMESTER, 2022
EXAMPLE: PUMA 560
Department of Applied Physics, University of Calcutta
