Industrial Robots Industrial Robots Kinematic chains Kinematic - - PowerPoint PPT Presentation

Industrial Robots Industrial Robots

Kinematic chains Kinematic chains Kinematic chains Kinematic chains

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Readings & prerequisites

Chapter 2 (prerequisites)

Reference systems Vectors Matrices Rotations, translations, roto‐translations Homogeneous representation of vectors and matrices

Chapter 1

Introduction and definitions Robot classification

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Kinematic chains

Kinematics allows to represent positions, velocities and accelerations of specified points in a multi‐body structure, independently from the causes that may have generated the motion (i.e., forces and torques) In order to describe the kinematics of manipulators or In order to describe the kinematics of manipulators or mobile robots, it is necessary to define the concept of kinematic chains k h k h f l /l k A kinematic chain kinematic chain is a series of ideal arms/links connected by ideal joints

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Kinematic Chain – KC

A kinematic chain KC is composed by a variable number of

Arms/links (rigid and ideal) Arms/links (rigid and ideal) Joints (rigid and ideal)

I i d fi d l i i ( f i i It is defined only as a geometric entity (no mass, friction, elasticity, etc. is considered and modeled) It has a degree of motion (DOM) and may afford a degree

• f freedom (DOF)

One must define a reference frame (RF) on each arm → DH conventions are used (see later for definition) Then, one is able to describe in this RF every possible point

• f the arm
• f the arm

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KC

Links Links (or arms arms) are idealized geometrical bars connecting two or more joints Joints Joints are idealized physical components allowing a relative motion between the attached links Joints allow a single “degree of motion” (DOM) between connected links connected links Joints may be

Rotational (or revolute revolute); they allow a relative geometrical rotation between links Prismatic Prismatic or translation; they allow a relative geometrical translation between links

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Joint disambiguation

This is a joint joint This is a NOT a KC joint joint This is a joint joint

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Example

Revolute Prismatic

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Example

Joint k Link Joint

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Graphical representation

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Rotation joints

Rotation joints are drawn in 3D as small cylinders with axes aligned along each i i rotation axis

k j i j

Rotation joints are drawn in 2D as small circles or small hourglasses

j i k

axis is normal to the plane pointing toward the observer

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j k

p g

Example

This is called the end end effector effector or TCP TCP ff ff

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Prismatic joints

Prism atic joints are drawn in 3D as small Prism atic joints are drawn in 3D as small boxes with each axis aligned along the translation axis Prism atic joints are drawn in 2D as small j squares with a point in their centres or as small rectangles with a line showing the two successive links

j i k

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i

Example

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Example

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Example

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End effectors

End effector End effector – gripper – hand – end tool are synonymous

It identifies the structure at the end of the last link that is able to perform the required task or can hold a tool

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Tool center point – TCP

The TCP TCP (Tool Center Point) is the mathematical point on the end effector that the robot software moves through space.

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Example

This is the TCP

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Open and closed KC Open chains Open chains: when there is only one link Closed chains Closed chains: when there are more than one between any two joints. The KC has the tree‐like link between two joints. The KC has the cycle‐like structure structure

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Task space

The robot TCP moves in a 3D cartesian/euclidean space The Task space Task space is a subset of the cartesian space that can be reached by the TCP

Task space Task space

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Task space Task space

Joint space

The value of each joint variable qi is the component of a vector that

3

q is the component of a vector that belongs to the joint space joint space

2

q

4

q

5

q q6 q

Actuators TCP

1

q

When a joint is not actuated, it is called passive joint passive joint

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j p j p j

Joint space

The robot joints are moved by actuators (electric, hydraulic, pneumatic motors, etc.)

The joint motion produces a motion of h h k the TCP in the task space. One shall be able to describe the relation between the joint space

Actuators

between the joint space and the task space representations

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p

Tasks space – Joint space – kinematic functions This is called a pose pose

Joint space Task Space

z

6

( ) t ∈ p

• Joint space

3

q

Direct K function

( ) t ∈ p

• y

Direct K function

( )

n

t ∈ q

• x

y q

2

q

Inverse K function

1

q

Direct kinematic function is easier than inverse kinematic function

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Degrees of freedom – redundancy

• 1. Each joint adds one to the degree of motion

degree of motion (DOM) The robot DOM robot DOM is equal to n q

• 2. The number of independent variables that describe the TCP

reference frame is called the TCP degree of freedom (DOF). g ( ) The TCP DOF TCP DOF is equal to n’

• 3. The number of independent variables that characterize the

p task reference frame is called the task DOF The task DOF task DOF is equal to m q n can be as large as desired but m≤3 in the 2D plane m≤6 in n can be as large as desired, but m≤3 in the 2D plane, m≤6 in the 3D space

( ) ( ) t x y t x y z θ φ θ ψ ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ p p

T T

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2 3

( ) , , ( ) , , , , ,

D D

t x y t x y z θ φ θ ψ ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ p p

Degrees of freedom

N t l th b t DOM ll t bt i ’ DOF f th TCP Not always the n robot DOMs allow to obtain n’=n DOFs of the TCP Since the TCP DOF should be equal to the task DOF (otherwise the robot is useless for that task ) one can consider the following cases robot is useless for that task …) one can consider the following cases Case 1 Case 1 is the usual case; the robot is called non non redundant redundant It has as many Case 1 Case 1 is the usual case; the robot is called non non‐redundant

• redundant. It has as many

TCP DOF as required by the task Case 3 Case 3 is an unlikely case; the robot has less TCP DOF than required by the Case 3 Case 3 is an unlikely case; the robot has less TCP DOF than required by the

• task. Therefore it is useless

Case 4 Case 4 is another unlikely case. The KC has more joints than required (i.e.,

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Case 4 Case 4 is another unlikely case. The KC has more joints than required (i.e., more expensive than necessary and more complex to control)

Example of Case 4

This KC has three prismatic joints (all parallel) that allow only one This KC has three prismatic joints (all parallel) that allow only one DOF to the TCP This “robot” requires three motors, when only one would be sufficient for the same purpose (apart from other considerations

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related to redundancy )

Redundancy

Case 2 Case 2 characterize a class of kinematic chains called redundant chains redundant chains They have more TCP DOF that those required by the task Why redundant robots are important or useful ? They improve manipulability manipulability or dexterity dexterity, i.e., the ability to reach a desired pose avoiding obstacles, like the human arm does

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Redundancy of the human arm

W i t Wrist Arm

The human (arm + wrist) has 7 DOFs But it is not ideal, since it is composed by muscles, bones and other tissues; it is not a rigid body, the joint are elastic, etc.

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Redundancy of the human arm

This mechanical arm

Shoulder

This mechanical arm simulates the human arm

1 2

Shoulder = 4 DOM Wrist = 3 DOM

3

Wrist 3 DOM Industrial robots have a

4 7 5

Industrial robots have a shoulder with 3 DOM (joint 3 is missing), and a wrist

6

similar to this one with 3 DOM

6 Wrist

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Example of redundancy

TCP Joint 3 Joint 1 Joint 4 J i t 2 Joint 2 Base The KC has 4 DOM since there are 4 rotating joints; an object in a plane has only 3 DOF (two positions + one angle). Therefore this KC is redundant (redundancy degree 4 3 1) degree 4‐3 = 1). If the task requires only to position an object, with no particular constraint on the q y p j , p

• rientation, the DOF will reduce to 2 and the redundancy increases to 4‐2=2

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Robot types Robot types Robot types Robot types

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Types of robots

Industrial robots are usually composed by an arm and a wrist. The robot type is defined by the arm configuration, and depends on yp y g , p the type of joints in the arm. They are called P and R respectively P = prismatic joint P = prismatic joint p j p j R = R = rotoidal rotoidal joint joint R b t l ifi d di t th f ll i l Robots are classified according to the following classes

Cartesian = 3P Cylindrical = 1R‐2P Polar or Spherical = 2R‐1P SCARA = 2R‐1P; SCARA = Selective Compliance Assembly Robot Arm Articulated or Anthropomorphic = 3R There are also parallel parallel robots, but they do not follow this classification

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Cartesian

Cartesian Cartesian = 3P = P‐P‐P

The shoulder is composed by three prismatic joints, with The shoulder is composed by three prismatic joints, with mutually orthogonal axes Each DOM corresponds to a cartesian task variable Each DOM corresponds to a cartesian task variable The task space is a sort of parallelepiped They provide an accurate positioning in the whole task space, but have a limited dexterity The most common structures are lateral columns or suspended bridges

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Cartesian

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Cylindrical

Cylindrical Cylindrical = 1R‐2P = R‐P‐P

The shoulder has one revolute joint with vertical axis The shoulder has one revolute joint with vertical axis followed by two prismatic joints (one vertical the other horizontal) Each DOM corresponds to one cylindrical coordinate The task space is a cylindrical sector The task space is a cylindrical sector The horizontal prismatic joint allows to reach horizontal spaces but the accuracy decreases toward the arm ends spaces, but the accuracy decreases toward the arm ends They are used mainly to move large objects

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Cylindrical

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Polar or spherical

Polar Polar or spherical spherical = 2R‐1P = R‐R‐P

The shoulder has two revolute joints (one vertical the other The shoulder has two revolute joints (one vertical the other horizontal) followed by one prismatic joints (with its axis

• rthogonal to the last one)

Each DOM corresponds to one polar coordinate The task space is a spherical sector that may include part of The task space is a spherical sector that may include part of the floor, to allow the manipulation of objects there The structure is less rigid than the preceding ones and the The structure is less rigid than the preceding ones, and the accuracy decreases with the elongation of the prismatic arm

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Polar or spherical

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SCARA

SCARA SCARA = 2R‐1P = R‐R‐P

The shoulder has two revolute joints followed by one The shoulder has two revolute joints followed by one prismatic joints (all with parallel/vertical axes) The correspondence between DOM and cartesian The correspondence between DOM and cartesian coordinates is true only for the vertical component The effect of gravity is compensated by the structure itself The effect of gravity is compensated by the structure itself The structure is rigid in the vertical component and compliant in the horizontal components compliant in the horizontal components This robot is mainly used for small components manipulation and vertical soldering or assembly tasks (e g in electronic and vertical soldering or assembly tasks (e.g., in electronic boards assembly)

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SCARA

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Articulated/Antropomorphic

Articulated Articulated or Anthropomorphic Anthropomorphic = 3R = R‐R‐R

The shoulder has three revolute joints: the first one is The shoulder has three revolute joints: the first one is vertical, the other two are horizontal and parallel The structure is similar to the human body, with trunk, arm The structure is similar to the human body, with trunk, arm and forearm, with a final wrist No correspondence between joint and cartesian coordinates No correspondence between joint and cartesian coordinates Task space is a sort of sphere sector It is one of the most common structures in industry, since it provides the best dexterity Its accuracy is not constant inside the task space

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Articulated/Antropomorphic

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Parallel or closed chains

Parallel Parallel or closed chains

Closed chains are used to manipulate heavy payloads Closed chains are used to manipulate heavy payloads requiring a great rigidity of the structure Examples Examples

Articulated robots with parallelogram links between the second and the third link second and the third link Parallel geometry robots where the TCP is connected to the base through more kinematic chains g

Large structural rigidity with high TCP speed Reduced task space Reduced task space

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Parallel or closed chains

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Wrists Wrists Wrists Wrists

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Wrists

The main scope of the wrist wrist is to orient the TCP It can be said that the shoulder sets the TCP coordinates, It can be said that the shoulder sets the TCP coordinates, while the wrist orients it. Spherical wrists are the most common: a spherical wrist spherical wrist is Spherical wrists are the most common: a spherical wrist spherical wrist is a wrist that has the three axes always intersecting in a single point single point. A wrist (spherical or not) is composed by three consecutive rotational joints (prismatic wrist are uncommon); the rotational joints (prismatic wrist are uncommon); the mutual configuration of the three axis identifies two main types of wrists types of wrists

1. Eulerian wrist 2. Roll‐pitch‐yaw (RPY) wrist

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Examples: spherical wrist

A spherical wrist A non spherical wrist

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A non spherical wrist

Wrists types Eulerian Eulerian 3R RPY RPY (Roll‐Pitch‐Yaw) 3R

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Spherical wrist

Wrists types

An Eulerian wrist is a spherical spherical wrist A RPY wrist is considered spherical, although its three axes do not meet at a single point, due to physical volumes When computing or performing inverse kinematics, the f h i l i i ffi i di i f presence of a spherical wrists is a sufficient condition for the existence of a closed form solution

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