Forward and Inverse Kinematics

Submitted To:Dr. B.S PablaProfessorM.E Department

Submitted By:Varinder Singh152227M.E(M.T)

ContentIntroductionMatrix RepresentationTransformationsStandard Robot coordinate SystemNumericals

Robot Kinematics: Position Analysis

INTRODUCTION

Forward Kinematics: to determine where the robot’s hand is? (If all joint variables are known)

Inverse Kinematics: to calculate what each joint variable is? (If we desire that the hand be located at a particular point)

Matrix Representation - Representation Of A Point In Space

Representation of a point in space

A point P in space : 3 coordinates relative to a reference frame

^^^kcjbiaP zyx

Representation of a vector in space

A Vector P in space : 3 coordinates of its tail and of its head ^^^__

kcjbiaP zyx

wzyx

P__

Matrix Representation -Representation of a Vector in Space

Where is Scale factor

w

It can Change overall size of vector similar to zooming function in computer graphics.

When w=1 , Size of components remain unchanged

When w=0, It represent a vector whose length is infinite

but it represents the direction so called as directional vector

Scale Factor w

Representation of a frame at the origin of the reference frame

Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector

zzz

yyy

xxx

aonaonaon

F

Matrix Representation -Representation of a Frame at the Origin of a Fixed-

Reference Frame

Representation of a frame in a frame

Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector

1000zzzz

yyyy

xxxx

PaonPaonPaon

F

Representation of a Frame in a Fixed Reference Frame

Representation of an object in space

An object can be represented in space by attaching a frame to it and representing the frame in space.

1000zzzz

yyyy

xxxx

objectPaonPaonPaon

F

Representation of a Rigid Body

Homogeneous Transformation Matrices

A transformation matrices must be in square form.

• It is much easier to calculate the inverse of square matrices.• To multiply two matrices, their dimensions must match.

1000zzzz

yyyy

xxxx

PaonPaonPaon

F

Transformations

A transformation is defined as making a movement in space.

Types of Transformation are:A pure translationA pure rotationA combination of translation and rotation

Representation of a Pure Translation

Representation of an pure translation in space

1000100010001

z

y

x

ddd

T

If a frame moves in space without any change in its orientation

Numerical Problem-1A frame F has been moved 10 units along y-axis and 5 units along z-axis of reference frame. Find new location of frame.

Answer:

Numerical Problem-1

Pure Rotation about an Axis

Coordinates of a point in a rotating frame before and after rotation.

Assumption : The frame is at the origin of the reference frame and parallel to it.

Pure Rotation about an Axis

Combined TransformationsCombined Transformation consist of a

number of successive translations and rotations about fixed reference frame axes.

The order of matrices written is the opposite of the order of transformations performed.

If order of matrices changes then final position of robot also changes

Numerical Problem (Forward Kinematics)-2A point p(7,3,1) is attached to frame and subjected to following transformations. Find coordinate of point relative to reference frame.

1.Rotation of 90° about z-axis2.Followed by rotation of 90 about y-axis3.Followed by translation of [4,-3,7].

Answer: The matrix equation is given as

Numerical Problem-2

Fig. 2.13 Effects of three successive transformations

A number of successive translations and rotations….

Numerical Problem-2

Forward Kinematics and Inverse Kinematics equation for position analysis and three types of standard robot coordinate system are: (a) Cartesian (gantry, rectangular) coordinates. (b) Cylindrical coordinates. (c) Spherical coordinates.

Forward and Inverse Kinematics Equations for Position

Cartesian (Gantry, Rectangular) Coordinates•All actuators are linear.•A Gantry robot is a Cartesian robot and used in pick and place applications like overhead cranes.

Cartesian Coordinates.

1000100010001

z

y

x

cartPR

PPP

TT

Cylindrical Coordinates

• 2 Linear translations and 1 rotation • translation of r along the x-axis • rotation of about the z-axis • translation of l along the z-axis

100010000

lrSCSrCSC

TT cylPR

,0,0))Trans(,)Rot(Trans(0,0,),,( rzllrTT cylPR

Suppose we desire to place the origin of hand frame of a cylindrical robot at [ 3,4,7]. Calculate the joint variables of robot.

Answer:

Numerical Problem (Inverse Kinematics)-3

100010000

lrSCSrCSC

TT cylPR

r= 5 units

Spherical Coordinates• 2 Linear translations and 1 rotation • translation of r along the z-axis • rotation of about the y-axis • rotation of along the z-axis

Spherical Coordinates.

10000

rCCSSrSSSCSCCrSCSSCC

TT sphPR

))Trans()Rot(Rot()( 0,0,,,,, yzlrsphPR TT

ReferencesSaeed B. Niku, “Introduction to Robotics –

Analysis, Control, Applications” , 2nd Edition, John Wiley & Sons, 2016