Industrial Robots Industrial Robots Kinematic chains Kinematic chains Kinematic chains Kinematic chains Basilio Bona 1 ROBOTICA 03CFIOR
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 Basilio Bona 2 ROBOTICA 03CFIOR
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 A kinematic chain kinematic chain is a series of ideal arms/links k k h h f l /l k connected by ideal joints Basilio Bona 3 ROBOTICA 03CFIOR
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) � It is defined only as a geometric entity (no mass, friction, � I i d fi d l i i ( f i i elasticity, etc. is considered and modeled) � It has a degree of motion (DOM) and may afford a degree of 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 of the arm of the arm Basilio Bona 4 ROBOTICA 03CFIOR
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 Basilio Bona 5 ROBOTICA 03CFIOR
Joint disambiguation This is a NOT a KC joint joint This is a joint This is a joint joint joint Basilio Bona 6 ROBOTICA 03CFIOR
Example Prismatic Revolute Basilio Bona 7 ROBOTICA 03CFIOR
Example Joint Link k Joint Basilio Bona 8 ROBOTICA 03CFIOR
Graphical representation Basilio Bona 9 ROBOTICA 03CFIOR
Rotation joints Rotation joints are drawn in 3D as small cylinders with axes aligned along each k rotation axis i i j j i Rotation joints are drawn in 2D as small circles or small hourglasses axis is normal to the plane i j j p pointing toward the observer g k k Basilio Bona 10 ROBOTICA 03CFIOR
Example This is called the end end effector effector or TCP ff ff TCP Basilio Bona 11 ROBOTICA 03CFIOR
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 k i i Basilio Bona 12 ROBOTICA 03CFIOR
Example Basilio Bona 13 ROBOTICA 03CFIOR
Example Basilio Bona 14 ROBOTICA 03CFIOR
Example Basilio Bona 15 ROBOTICA 03CFIOR
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 Basilio Bona 16 ROBOTICA 03CFIOR
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. Basilio Bona 17 ROBOTICA 03CFIOR
Example This is the TCP Basilio Bona 18 ROBOTICA 03CFIOR
Open and closed KC � Open chains � Closed chains Open chains : when Closed chains : when there is only one link there are more than one between any two joints. link between two joints. The KC has the tree ‐ like The KC has the cycle ‐ like structure structure Basilio Bona 19 ROBOTICA 03CFIOR
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 Task space Task space Basilio Bona 20 ROBOTICA 03CFIOR
Joint space q 3 The value of each joint variable q i is the component of a vector that is the component of a vector that q q 4 belongs to the joint space joint space 2 q 5 q q 6 Actuators TCP q 1 When a joint is not actuated, it is called passive joint j passive joint p p j j Basilio Bona 21 ROBOTICA 03CFIOR
Joint space The robot joints are moved by actuators (electric, hydraulic, pneumatic motors, etc.) The joint motion produces a motion of the TCP in the task h h k space. One shall be able to describe the relation Actuators between the joint space between the joint space and the task space representations p Basilio Bona 22 ROBOTICA 03CFIOR
Tasks space – Joint space – kinematic functions This is called a pose pose Task Space z 6 t ∈ t ∈ p p ( ) ( ) � � Joint space Joint space q 3 Direct K function Direct K function n t ∈ q ( ) � y y x Inverse K function q 2 q q 1 Direct kinematic function is easier than inverse kinematic function Basilio Bona 23 ROBOTICA 03CFIOR
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 T = = θ θ = = φ θ ψ φ θ ψ p p ( ) ( ) t t x y x y , , p p ( ) ( ) t t x y z x y z , , , , , ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎣ ⎦ ⎣ ⎦ 2 D 3 D Basilio Bona 24 ROBOTICA 03CFIOR
Degrees of freedom N t l Not always the n robot DOMs allow to obtain n ’ = n DOFs of the TCP th b t DOM ll t bt i ’ DOF f th 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 Case 1 Case 1 is the usual case; the robot is called non non redundant non ‐ redundant redundant It has as many 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., 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) Basilio Bona ROBOTICA 03CFIOR 25
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 related to redundancy ) Basilio Bona 26 ROBOTICA 03CFIOR
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 Basilio Bona 27 ROBOTICA 03CFIOR
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. Basilio Bona 28 ROBOTICA 03CFIOR
Redundancy of the human arm Shoulder This mechanical arm This mechanical arm simulates the human arm 1 2 Shoulder = 4 DOM 3 Wrist = 3 DOM Wrist 3 DOM 5 7 4 Industrial robots have a Industrial robots have a shoulder with 3 DOM (joint 3 is missing), and a wrist 6 6 Wrist similar to this one with 3 DOM Basilio Bona 29 ROBOTICA 03CFIOR
Example of redundancy Joint 3 TCP 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 orientation, the DOF will reduce to 2 and the redundancy increases to 4 ‐ 2=2 Basilio Bona 30 ROBOTICA 03CFIOR
Robot types Robot types Robot types Robot types Basilio Bona 31 ROBOTICA 03CFIOR
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