Introduction to Robotics

Lecture 6

Lecture Contents – Manipulator Kinematics

• Introduction

• Kinematics Joints

• Link Description

• Denavit - Hartenberg Representation

Central Topic

Problem

• Given: The manipulator geometrical parameters

• Specify: The position and Orientation of Manipulator

Solution:

Coordinate System or “Frames” are attached to the manipulator and objects in environment according to Denavit Hartenberg notation

Introduction

Kinematics

• Kinematics is the science of motion that treats the subject without regard to the forces that cause it (e.g., the position, the velocity, the acceleration, and all higher order derivatives of the position)

Kinematics of Manipulators

• It refers to all geometrical and time based properties of the motion

Kinematic Chain

• Mechanisms of robots and other machines can be described with the help of a kinematic chain

• The definition of a kinematic chain is “ any assemblage of rigid members that are joined together”

• Consider as an example the piston and crank mechanism of an engine

• Its kinematic chain consists of four rigid members, also called as links

• The links are joined together via four kinematic joints

• Each link of kinematic chain has a special name, depending on its location and its role

Kinematic Chain • First, any kinematic chain in a machine has

a fixed link that does not move at all. Such link is also called a base or frame. It is the engine case in this example

• Second, each linkage has one or more input links. Input links are powered by an actuator or by other means. In this example, the piston is powered by the engine

• Third, each linkage has one or more output links whose power is used to do the desired job. In this example, the output link is the crank

• Finally, there are intermittent links whose role is to ensure power transmission

Kinematic Joints

• A kinematic joint is formed via direct contact between two links

• The German researcher Franz Reuleaux called kinematic joint a pair

The role of the joint is:

• To keep the links in contact

• To keep the links in relative motion along some direction(s)

• To constrain the motion along other directions

Kinematic Joints

From the viewpoint of contact condition, joints can be distinguished as:

1. Point contact

2. Line contact

3. Surface contact

These joints are shown as:

Kinematic Joints

• Also, a joint is characterized by its degree of freedom (DOF)

• Definition(Joint DOF): the number of independent coordinates needed to specify the position of one link with respect to the other

• The DOF of a joint can be obtained as:

Joint DOF = 6 – Number of constraints

Here, the number “6” means the DOF of a free body

Kinematic Joints

• From the viewpoint of joint DOF, there are lower pair joints and higher pair joints

• Lower pair joint are joints with surface contact only

There are only six types of lower pair joints as follows: • R-joint= Revolute joint (1 DOF) • P-joint= Prismatic joint (1 DOF) • H-joint= Helical joint ( Screw joint) (1 DOF) • C-joint= Cylindrical joint (2 DOF) • S-joint= Spherical Joint( Ball joint) (3 DOF) • PL-joint= Planar joint (3 DOF)

Kinematic Joints

Kinematic Joints • Higher pair joints, on the other hand, are joints with point and line contact

• There is an infinite number of such joints are shown in figure. Some of these

Link Description • A manipulator may be thought of as a set of bodies connected

in a chain by joints. These bodies are called links

• Joints form a connection between a neighboring pair of links

Link

• A rigid connection which defines the relationship between two neighboring joint axes of the manipulator

The links are numbered starting from

• The immobile base of the arm, which might be called link 0

• The first moving body is link 1, and so on, out to the free end of the arm, which is link n

Link Description

A single link of a typical robot has many attributes that a mechanical designer had to consider during its design:

• The type of material used

• The strength and stiffness of the link

• The location and type of the joint bearings

• The external shape

• The weight and inertia, etc.

Link Description

Joint-Axis • Joint axis are defined by

lines in space

• Joint axis i is defined by a line in space (or vector direction) about which link i rotates relative to the link i-1

Link Parameters Link Length

• a i-1 - Distance between axis i and axis i-1

• a i-1 is the length of perpendicular between the joint axes

• These two axes can be viewed as lines in space

• The common perpendicular is the shortest line between the two axis lines and is perpendicular to both axis lines

Link Parameters Link Twist

• α i-1 – Angle measured from axis i-1 to axis i

• It is the amount of rotation around the common perpendicular so that the joint axes are parallel

• It is taken positive when rotation is made counter-clockwise

Link Parameters Example

• Link length = 7 inches

• Link twist= 45 degrees

Joint Variables Link Offset

• di – signed distance measured along the common axis i (joint i) from point where ai-1 intersect axis i to point where ai intersect the axis i

• The link offset di is variable of joint i if joint is prismatic

Joint Variables

Joint Angle

• Θi – the signed angle made between an extension of ai-1 and ai

measured about the axis of the joint i

• The joint angle Θi is variable if joint i is revolute

• The sign of Θi is given by the right hand rule

Joint/Link Parameters and Variables – First and last links in the chain

• Link length and link twist, depend on joint axes i and i + 1.

• At the ends of the chain, it will be the convention to assign zero to these quantities

Joint/Link Parameters and Variables – First and last links in the chain

• Link offset and joint angle are well defined for joints 2 through n-1 according to the conventions

• If joint 1 is revolute, the zero position for Θi may be chosen arbitrarily and di = 0.0 will be the convention

• if joint 1 is prismatic, the zero position of di may be chosen arbitrarily and Θi = 0.0 will be the convention

• Exactly the same statements apply to joint n

Denavit – Hartenberg Representation

• In 1955, Denavit and Hartenberg published a paper in the ASME Journal of Applied Mechanics that was later used to represent and model robots and to derive their equations of motion

• This technique has become the standard way of representing robots and modeling their motions and thus is essential to learn

• The Denavit - Hartenberg (DH) model of representation is a simple way of modeling robot links and joints that can be used for any robot configuration

Denavit – Hartenberg Representation

Robot Coordinates

• Base coordinates ∑ 0

• Hand Coordinates ∑H

• Link Coordinates ∑1 to ∑n

• Construct i-1 i T

• Goal to find relation:

0 n T = 0 1 T 1 2 T …….. n-1 n T

Denavit – Hartenberg Representation

• To model the robot with DH representation, assign a local reference frame to each and every joint

• Thus, for each joint, will have to assign a Z-axis and X-axis

• Normally, no need to assign Y-axis, since Y-axis is mutually perpendicular to both X and Z axis

• In addition, DH representation does not use the Y-axis at all

• Procedure for assigning a local reference frame to each joint is discussed next

Denavit – Hartenberg Representation

Denavit – Hartenberg Rules Application

DH applicable to :

• Open link kinematic chain. Joints with one degree of freedom only. Joints are revolute or prismatic only

Affixing frames to Links – Intermediate Links in the chain

Origin of Frame {i}

• The origin of frame {i} is located where the distance ai perpendicular intersects the joint axis i

Z- axis

• The Z-axis of the frame {i} is coincident with the joint axis i

Affixing frames to Links – Intermediate Links in the chain

X-axis

• The Xi points along the distance ai in direction from joint i to joint i+1

• For ai=0, Xi is normal to the plane of Zi and Zi+1

• The link twist angle is αi is measure in a right hand sense about Xi

Affixing frames to Links – Intermediate Links in the chain

Y-axis

• The Yi is formed by the right hand rule to complete the ith frame

Frame {0}

• The frame attached to the base of the robot or link 0 is called frame{0}

• This frame does not move and for the problem of arm kinematics can be considered as the reference frame

Affixing frames to Links – Intermediate Links in the chain

• Frame {0} concides with frame {1}

a0 = 0

α0 = 0

• Joint 1-Revolute θi = 0 Arbitrary

di = 0 Convention

• Joint 1-Prismatic θi = 0 Convention

di = 0 Arbitrary

Link Frame Attachment Procedure- Summary

1. Identify the joint axes and imagine(or draw) infinite lines along them

2. For step to step 5 below, consider two of these neighboring lines( at axes i and i+1)

3. Identify the common perpendicular between them or point of intersection. At the point of intersection, or at the point where the common perpendicular meets the ith axis, assign the link frame origin

4. Assign the axis Zi axis pointing along the ith joint axis

Link Frame Attachment Procedure- Summary

5. Assign the Xi axis pointing along the common perpendicular or

if the axes intersect, assign Xi to be normal to the plane

containing the two axes

6. Assign Yi axis to complete the right hand coordinate system

7. Assign frame{0} to match frame {1} when the first joint

variable is zero

8. For frame{N} choose an origin location and Xn direction freely

but generally so as to cause as many linkage parameters as

possible to be zero

DH Parameters Summary

• 4 DH parameters

(αi, ai, di, θi)

• 3 fixed link parameters

• 1 joint variable

Θi = Revolute Joint

di = Prismatic Joint

• α i and ai describes link i

• di and θi describes the link connection

DH Parameters Summary • If the link frames have

been attached to the links according to our convention, the following definitions of the link parameters are valid