Robotics Lecture 1

Elective Course: Robotics and Object Tracking

KUMAR MAYANK

Assistant Professor

ECE Dept

Delhi Technological

University

02-02-2015 Kumar Mayank 1

Syllabus for Mid Semesters(26/02/2015)

Introduction to Robotics Mathematical Modelling

Rigid Motions and Homogeneous Transformations

Forward and Inverse Kinematics

Velocity Kinematics

Path and Trajectory Planning

Dynamics


Robot Definition

A robot is a reprogrammable multifunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks.

Kumar Mayank 302-02-2015

Example: Robotic Arm components


MATHEMATICAL MODELING OF ROBOTS

Symbolic Representation of Robots

Robot Manipulators are composed of links connected by joints to form a kinematic chain.

Revolute joint(R) = Joint is like a hinge and allows relative rotation between two links. (Joint variable = theta)

Prismatic joint(P) = linear relative motion between two links. (Joint variable = d)

Joint variable = Relative displacement between adjacent links


Controller ResolutionSmallest increment of motion that

the controller can sense

Resolution is computed as the total distance travelled by the tip divided by 2^n(where n is the number of bits of encoder accuracy)

Prismatic joints, typically have higher resolution than revolute joints, since the straight line distance traversed by the tip of a linear axis between two points is less than the corresponding arc length traced by the tip of a rotational link.


Problem 1Referring to right hand side Figure , suppose that the tip of a single link travels a distance d between two points. A linear axis would travel the distance d while a rotational link would travel through an arc length l as shown left hand side figure. Using the law of cosines show that the distance d(represented by c in the figure) is given by


which is of course less than l . With 10-bit accuracy and l = 1m, = 90 what is the resolution of the linear link? of the rotational link?

Problem 2 : A single-link revolute arm is shown in Figure . If the length of the link is 50 cm and the arm travels 180 degree? what is the control resolution obtained with an 8-bit encoder?


Wrists and End Effectors

Wrist (Revolute Joints) = Joints in the kinematic chain between the arm and end effector

Spherical wrists : Three joint axes intersect at a common point


Configuration Space

Configuration of a manipulator = Complete specification of the location of every point on the manipulator.

Configuration space = Set of all possible configurations

Degrees-of-freedom (DOF) = Configuration that can be minimally specified by n parameters.

Example: Arm = 6 DOF - three for positioning and three for orientation (e.g., roll, pitch and yaw angles)


Workspace Workspace of a manipulator is the total volume swept out by the

endeffector as the manipulator executes all possible motions.

Reachable workspace : Entire set of points reachable by the manipulator.

Dexterous workspace : Points that the manipulator can reach with an arbitrary orientation of the end-effector.


Classification of Robotic Manipulators1) Power Source

Electrically Powered = DC or AC Servo motors

Hydraulically Powered = for lifting heavy loads(Drawback: require more maintenance due to leakage of hydraulic fluid)

Pneumatic powered = Inexpensive and simple but cannot be controlled precisely.

2) Application Area

Assembly robots = small, electrically driven and either revolute or SCARA (Selective Compliant Articulated Robot for Assembly) in design

Non-Assembly robots = welding, spray painting, material handling, and machine loading and unloading


Classification of Robotic Manipulators(Cont)

3) Method of Control

Non-servo robots : Open-loop devices whose movement is limited to predetermined mechanical stops

(generally used for materials transfer)

Servo robots : closed-loop computer control to determine their

motion and are thus capable of being truly multifunctional, reprogrammable devices

Point-to-Point robot = no control on the path of the end-effector in between taught points

Continuous path robots = entire path of the end-effector can be controlled


Classification of Robotic Manipulators(Cont)

4) Geometry

Articulated (RRR)

Spherical (RRP)

SCARA (RRP)

Cylindrical (RPP)

Cartesian (PPP)

Parallel robot = links are arranged in a closed rather than open kinematic chain


Articulated manipulator (RRR)(Revolute, or Anthropomorphic manipulator)

Revolute manipulator provides for relatively large freedom of movement in a compact space

Figure : Elbow manipulator

Parallelogram linkage= common revolute joint design

Motoman SK16 manipulatorKumar Mayank 1602-02-2015

Articulated manipulator (RRR) ( Continued)(Revolute, or Anthropomorphic manipulator)

Parallelogram linkage= common revolute joint design

z2 is parallel to z1

z1 and z2 are perpendicular to z0


Spherical Manipulator (RRP)[Ex: Stanford Arm]


Spherical Manipulator (RRP) : Replace the third or elbow joint in the revolute manipulator by a prismatic joint to obtain the spherical manipulator

Revolute manipulator(RRR) Spherical manipulator(RRP)


SCARA(Selective Compliant Articulated Robot for Assembly) Manipulator (RRP)


Difference Between Spherical Manipulator (RRP)and SCARA Manipulator (RRP)

Workspace of the spherical manipulator

Workspace of the SCARA manipulator


Difference Between Spherical Manipulator (RRP)and SCARA Manipulator (RRP)

Spherical manipulator

z0 perpendicular to z1, and z1 perpendicular to z2

SCARA Manipulator

z0, z1, and z2 mutually

parallel


Cylindrical Manipulator (RPP)


Cartesian manipulator (PPP)

Generally used during transfer of materials or cargo in shipyards.


Parallel Manipulator

Some subset of the links form a closed chain.

It has two or more independent kinematic chains connecting the base to the end-effector

Closed chain kinematics of parallel robots can result in greater structural rigidity, and hence greater accuracy, than open chain robots.


Forward Kinematic Equations It helps to determine the position and orientation of the end-effector in

terms of the joint variables.

The coordinates (x, y) of the tool are expressed in this

Alpha 1 and 2 are the lengths of the two links

Orientation of the tool frame relative to the base frame

is given by the direction cosines of the x2 and y2 axes

relative to the x0 and y0 axes

(Equations for 2 link Manipulator)Kumar Mayank 2602-02-2015

Forward Kinematics Equations (Continued..) Orientation of the tool frame relative to the base frame can be rewritten in the

following matrix form


Find the direction cosines and direction angles of the vector -8i+3j+2k.


Inverse Kinematics


Inverse Kinematics Given the joint angles 1, 2 we can determine the end-effector

coordinates x and y(+veElbow Up, -veElbow Down) Find 1??

Ans: Using cosine formula solve for 1. Cos(90- 1)=()


Inverse Kinematics (Problem solution)


Velocity KinematicsThis tool helps us to know the relationship between the velocity of the tool and the joint velocities.

Differentiate eqn (1.1 and 1.2) to get

Using the vector notation


Velocity Kinematics(Continued.)

Joint velocities are found from the end-effector velocities via the inverse Jacobian

where inverse J is

Where cand sdenote respectively cos and sin .Kumar Mayank 3302-02-2015

Velocity Kinematics(Continued.)

When 2 =0 then the

manipulator is said to be in a singular configuration


Assignment Problem :For the two-link manipulator of Figure suppose 1 = 2 = 1.

Find the coordinates of the tool when

Find the joint angles 1, 2 when the tool is located at coordinates {Inverse Kine.}

If the joint velocities are constant at

what is the velocity of the tool ?

What is the instantaneous tool velocity when


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Robotics Lecture 1