Robot Manipulator 12 Instructor Prof ShihChung Kang 2008

Robot Manipulator (1/2) Instructor Prof. Shih-Chung Kang 2008 Spring

(I) Topics in Robot Manipulator – – – Spatial description Forward kinematics (FK) Inverse kinematics (IK) Dynamics Trajectory generation Force control

The scopes of robotics • Robotics is a study of the mechanics and control of manipulator of robots. • Strictly speaking, robotics is not a new science, but a collection of topics talekm from various fields, including – Mechanics engineering: methods for the study of machines in static and dynamic situations. – Mathematics: tools for describing spatial motions and other attributes of manipulators. – Computer science: basis for programming and simulations. – Electronic engineering: design of sensors and interface for industrial robots. 25 September 2020 S. C. Kang 3

Spatial description • In the study of robotics, we are constantly concerned with the location of objects in 3 D space. • These objects are the links of the manipulator, the parts and tools with which it deals, and other object in the manipulator’s environment. Object 2 Object 3, 4 Object 1 Object 5 25 September 2020 S. C. Kang 4

Forward kinematics of manipulators • Kinematics is the science of motion which treats motion without regard to the forces which cause it. • Forward kinematics is the geometrical problem of computing the position and orientation of the end-effector. • End-effector 25 September 2020 S. C. Kang 5

Forward kinematics of manipulators x y 25 September 2020 S. C. Kang z 6

Inverse kinematics of manipulators x y 25 September 2020 S. C. Kang z 7

Dynamics (won’t be covered in this course) Jacobian matrix v 25 September 2020 S. C. Kang 8

Trajectory generation (won’t be covered in this course) • To generate the trajectory for moving end-effector from position 1 to position 2. End-effector 2 End-effector 1 25 September 2020 S. C. Kang 9

Manipulator design and sensors (won’t be covered in this course) • The design of actuator, location, structural stiffness, sensor location. 10 kg 25 September 2020 S. C. Kang 10

Programming and simulation • A computer graphic interface to allow users to program and test the robots without using the real robots. real world computer 25 September 2020 S. C. Kang 11

(Il) Model tower crane numerically Part 1: Introduction to cranes Part 2: The description of position and orientation

Three type of typical cranes 25 September 2020 S. C. Kang 13

Horizontal jib crane 25 September 2020 S. C. Kang 14

Capacity of common horizontal jib crane (a) Jib length: 30 to 60 m (b) Section length: 3 to 6 m (c) Base dimension: 4 by 4 m to 6 by 6 m (d) Tower cross section: 1. 2 to 2. 4 m (e) Maximum lifting capacity (end of jib): 1 to 5 ton (f) Hoisting and trolleying speed: 2 to 3 m/minute 25 September 2020 S. C. Kang 15

Spatial description • To model a crane, we need an effective method to describe the position and orientation of each part of the crane. • Here we use a robot as an example. Object 2 Object 3, 4 Object 1 Object 5 25 September 2020 S. C. Kang 16

The description of position and orientation • Spatial description includes both position and orientation. • To describe a rigid body in space, we will always attach a coordinate system (also called frame) to objects. • The coordinate system (frame) can serve as a reference coordinate system of each object. • To present the position and orientation of the objects, we often think of transforming the coordinate systems the objects. 25 September 2020 S. C. Kang 17

Attach frames to all objects 1. describe objects = describe coordinate systems 2. We may use homogenous matrix to describe the coordinate system z y x 25 September 2020 S. C. Kang 18

Using frames to describe a robot 25 September 2020 S. C. Kang 19

Notations used in the geometrical model 25 September 2020 S. C. Kang 20

Now we are focusing on the description of a frame in space {B} {A} How to describe frame {B} with respect to reference frame {A}?

The derivation of ATB for frame description • First we would like to derive a transformation matrix that describes a coordinate system with respect to a reference coordinate system. • Assume we have two coordinate systems, {A} and {B}. • {B} denotes the local coordinate system we would like to describe, and { A} denotes the reference coordinate system. 25 September 2020 S. C. Kang 22

The derivation of ATB for frame description – AR represents the three principal axes of coordinate system { B}. – APBORG is the origin of {B} with respect to {A}. – XA, YA and ZA are three unit vectors that represent the X, Y and Z axis of the {A} coordinate system. – XB, YB and ZB are three unit vectors that describe the X, Y, Z axis of the {B} coordinate system. B 25 September 2020 S. C. Kang 23

The derivation of ATB for frame description – For presenting coordinate system {B} by a four by four matrix, we add [0 0 0 1] as the fourth row in [ARB APBORG]. – The columns of ARB are unit vectors defining the directions of the principal axes of {B} – AP BORG locates the position of the origin of {B}. 25 September 2020 S. C. Kang 24

Another purpose of ATB: transfer vectors • • Once the description of a coordinate system exists, we start explaining another purpose of the transportation matrix: – to transfer an arbitrary vector from respect to one coordinate system to another coordinate system. Here is a general case of mapping an arbitrary vector from relative to given coordinate system { B} to relative to {A}. 25 September 2020 S. C. Kang 25

Another purpose of ATB: transfer vectors • Assume we have a vector BP described with respect to the coordinate system {B} • And we would like to compute the description of vector AP that is the same vector as BP but with respect to the coordinate system {A}. AP {A} 25 September 2020 BP {B} S. C. Kang 26

Another purpose of ATB: transfer vectors • • • AP is the vector that represents the origin of {B} with respect to {A} AR is the rotation matrix that can rotate {A} to {B}. B AP can be found by BORG 25 September 2020 S. C. Kang 27

Another purpose of ATB: transfer vectors • We can also add [0 0 0 1] as the fourth row the matrix and obtain a 4 x 4 matrix AT B • We may transfer the vector from the reference frame {A} to reference frame {B} by following equation: 25 September 2020 S. C. Kang 28

Transformation matrix ATB • The four by four transformation matrix ATB serves two purposes 1. to describe the coordinate system {B} relative to the reference coordinate system {A} 2. to map the description of a vector with respect to a certain coordinate system {B} to the description of the same vector but with respect to another coordinate system {A}. 25 September 2020 S. C. Kang 29

Summary and next class • Today we have overviewed the robots and basic concepts of robotics. • We have learned the basic method to describe objects in space. We know the transformation matrix can be used for: – Describing frames in space – Transferring vector from one reference frame to another • In the next class, we will go into the derivation of forward manipulator kinematics. • I will use the tower crane as an example and explain how to develop its kinematics matrix step by step. 25 September 2020 S. C. Kang 30

(III) Forward kinematics

Forward kinematics • Kinematics of manipulators permit the study of the motion of manipulators without regard to the forces which cause them. Forward kinematics is one of the most important methods to deal with these motion problems. • The main purpose of forward kinematics is to compute the position and orientation of the end-effector of the manipulator from the motion in each rigid body of the robot/crane. • The end-effector is the free end of the chain of links which make up the manipulator. In the case of a tower crane, the end-effector is its hook. 25 September 2020 S. C. Kang 32

Forward kinematics • Given the joint angles or displacement of each link, forward kinematics can help us find geometrical status (including the position and orientation) of the hook. • We can also think forward kinematics as the transformation function which changes the representation of manipulator position from a joint space description into a global Cartesian space description. • Now we use a horizontal jib construction crane to illustrate the derivation of forward kinematics of manipulators. 25 September 2020 S. C. Kang 33

Links and joints • The first step in deriving the forward kinematics is to illustrate the schematics of a manipulator. • Manipulators consist of rigid or nearly rigid links (the parts of the crane are assumed to be rigid bodies) connected by joints which allow relative motion of neighboring rigid bodies (links). 25 September 2020 S. C. Kang 34

prismatic joints vs. revolute joints • Two types of joints are commonly used in manipulators. Revolute joint allows relative rotations between neighboring rigid links. 25 September 2020 S. C. Kang Prismatic joint allows relative displacement between links. 35

R-R-P manipulator • Consider the following mechanics, there are two revolute joints and one prismatic joints. • From the fixed end to the free end, the manipulator (the crane) has the following four types of joints: revolve, and prismatic. • To simplify the name, the crane can be referred to an R-R-P manipulator. 25 September 2020 S. C. Kang 36

Denavit-Hartenberg notation • Denavit-Hartenberg notation (Denavit and Hartenberg 1955) to find the geometrical relationship between links. • The Denavit-Hartenberg notation defines a coordinate system attached to each joint to describe the displacement of each link relative to its neighbors in a general form. • The coordinate system is named by a number corresponding to the link to which the joint is attached. • In other words, coordinate system {i} is attached to link i. 25 September 2020 S. C. Kang 37

Denavit-Hartenberg notation • What we would like to find the transformation matrix from frame i-1 to frame i. (i. e. i-1 Ti) 25 September 2020 S. C. Kang 38

Four parameters • The geometrical relationship between the joints at the two ends of Link i-1, i. e. Axis i-1 and Axis i, can be described by four parameters, ai-1, di, αi-1, and θi. – ai-1 denotes the length of Link i-1, which is the mutual perpendicular distance between Axis i -1 and Axis i. – di is called link offset. It is a signed distance measured along Axis i from the point where ai-1 intersects with the Axis i to the point where ai intersects with Axis i. – αi-1 denotes the twist angle between Axis i-1 and Axis i in the right hand sense with respect to the direction of ai-1 from Axis i-1 to Axis i. – θi is a parameter that describes the amount of rotation about this common axis between one link and its neighbor. 25 September 2020 S. C. Kang 39

Four parameters ai-1 the distance from Zi-1 to Zi measured about Xi-1 di the distance from Xi-1 to Xi measured along Zi αi-1 the angle between Zi-1 and Zi measured along Xi-1 θi the angle between Xi-1 and Xi measured about Zi 25 September 2020 S. C. Kang 40

Transformation matrix from frame i-1 to i • Using coordinate transformation matrices, we are able to compute the transformation matrix from frame axis i-1 to axis i. 25 September 2020 S. C. Kang 41

Six steps to determine the coordinate system Step 1: Identify the joint axes and imagine infinite lines along them. For steps 2 through steps 5 below, consider two of these neighboring lines at Axis i and Axis i+1. Step 2: Identify the common perpendicular between Axis i and Axis i+1 or point of intersection. At the point of intersection, or at the point where common perpendicular meets the Axis i, assign the origin of coordinate system. Step 3: Assign the zi axis pointing along Axis i. Step 4: Assign the xi axis pointing along the common perpendicular find in Step 2 or if Axis i and Axis i+1 intersect, assign xi to be normal to the plane containing the two axes. Step 5: Assign yi axis to complete a right-hand coordinate system. Step 6: Assign {0} and {1} when the first joint variable is zero. For {N} choose an origin location and xn direction freely, but generally so as to cause as many linkage parameters as possible to become zero. 25 September 2020 S. C. Kang 42