Chap 4 - Robot Kinematics

Abd Razak Ahmad Ext 134Electrical Engineering [email protected] KINEMATICS CHAPTER 4

The Robot System

ManipulatorsCartesian: PPPCylindrical: RPPSpherical: RRPSCARA: RRP(Selective Compliance Assembly Robot Arm)Articulated: RRRHand coordinate:n: normal vector; s: sliding vector; a: approach vector, normal to thetool mounting plateRobot Configuration:

ManipulatorsRobot SpecificationsNumber of AxesMajor axes, (1-3) => Position the wristMinor axes, (4-6) => Orient the toolRedundant, (7-n) => reaching around obstacles, avoiding undesirable configurationDegree of Freedom (DOF)WorkspacePayload (load capacity)Precision v.s. RepeatabilityWhich one is more important?

An Example - The PUMA 560The PUMA 560 has SIX revolute jointsONE revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle1234There are two more joints on the end effector (the gripper)56

Kinematic ConsiderationsUsing kinematics to describe the spatial configuration of a robot gives us two approaches:Forward Kinematics. (direct)Given the joint angles for the robot, what is the orientation and position of the end effector?Inverse Kinematics.Given a desired end effector position what are the joint angles to achieve this?

What is KinematicsForward kinematics

Given joint variables (angles)

End-effector position and orientation, - Formula?

What is KinematicsInverse kinematicsEnd effector position and orientation

Joint variables -Formula?

Example 1Coordinate position (x, y) in terms of joint angle, Joint angle in terms of coordinate position

PreliminaryRobot FramesWorld frame - referenceJoint frameTool framework pieceWP

*Co-ordinate FrameszyxRight-handedCo-ordinate frameBase FrameTool FrameGoal FrameLink Frame

Kinematic RelationshipBetween two frames we have a kinematic relationship - basically a translation and a rotation.

This relationship is mathematically represented by a 4 4 Homogeneous Transformation Matrix.

Forward Kinematics (angles to position)What you are given: The length of each link The angle of each joint What you can find: The position of any point (i.e. its (x, y, z) coordinates

Inverse Kinematics (position to angles)What you are given:The length of each linkThe position of some point on the robot

What you can find:The angles of each joint needed to obtain that positionWe are interested in two kinematics topics

Axis and Orientation Terminology :

Reference Origin = O and Derived Origin = O

Global/ Reference CoordinateMoving/ Derived/ local CoordinateTools Orientation - ITools Orientation - IIToolsOrientation - IIIXUn: normalY: Yawn: normalYVs: slidingP: Pitcho: orientation

zWa: approachR: Rolla: approach

Orientation Convention

In three dimensions, the six DOFs of a rigid body are sometimes described using these nautical names:Moving up and down (heaving);Moving left and right (swaying);Moving forward and backward (surging);Tilting forward and backward (pitching);Turning left and right (yawing);Tilting side to side (rolling).

Orientation Convention

Orientation ConventionEnd-effector or Tool Front ViewTry to move your hand accordingly !

PreliminaryCoordinate TransformationReference coordinate frame oxyzOPoint represented in OXYZ:P = 2x + 5y + 7z Vector representation in OXYZ :2=px x=ix 5=py y=jy 7=pz z=kzi, j, k are unit vector along x, y, z2575

Preliminary

PreliminaryCoordinate TransformationReference coordinate frame OxyzBody-attached frame Ouvw Point represented in OXYZ:Point represented in Ouvw:Two frames coincide ==>Therefore Pxyz = Puvw

Preliminaryi, j, k are unit vector along x, y, z and u, v, w directionsOxyz and Ouvw are coordinate systemsO,OP is the point in the space

PreliminaryMutually perpendicular

Unit vectorsProperties of orthonormal coordinate frameProperties: Dot Product Let and be arbitrary vectors in and be the angle from to , then

PreliminaryCoordinate TransformationRotation onlyHow to relate the coordinate in these two frames? But now Pxyz = Puvw

PreliminaryBasic Rotation , , and represent the projections of onto OX, OY, OZ axes, respectively

Since

PreliminaryBasic Rotation Matrix


Rotation about x-axis with

Rotation about z-axis

Preliminary

Rotation about x axis with

Basic Rotation MatricesRotation about x-axis with

Rotation about y-axis with

Rotation about z-axis with

Rotation Matrices in 3D OK,lets return from homogenous representationRotation around the Z, W, R, a -AxisRotation around the Y, V, P, s-AxisRotation around the X, U, Y, n-Axis


Obtain the coordinate of from the coordinate of

Example 2A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.

Example 3A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis. - Theta - Phi Psi

MATLAB - Robotic Toolbox>> rotz(pi/3) ans = 0.5000 -0.8660 0 0 0.8660 0.5000 0 0 0 0 1.0000 0 0 0 0 1.0000

Composite Rotation MatrixA sequence of finite rotations matrix multiplications do not commuterules:if rotating coordinate OUVW is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrixif rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

Example 4Find the rotation matrix for the following operations:

Homogeneous Transformationsx

y 3 1 Translation

z1 Global Scale

Homogeneous Matrices in 3DH is a 4x4 matrix that can describe a translation, rotation, or both in one matrixTranslation without rotationYXZONAONARotation without translationRotation part: Could be rotation around z-axis, x-axis, y-axis or a combination of the three.

Example 5Translation along Z-axis with h:

Example 6Rotation about the X-axis by

Homogeneous TransformationComposite Homogeneous Transformation MatrixRules:Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre-multiplicationTransformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication

Example 7Find the homogeneous transformation matrix (T) for the following operations:

Example 8Find the homogeneous transformation matrix (T) for the following operations:

Homogeneous RepresentationA frame in space (Geometric Interpretation)

Principal axis n w.r.t. the reference coordinate system(X)(y)(z)

Homogeneous TransformationTranslation

>> a=rotx(0)a = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1>> plotT1(a)

Homogeneous TransformationComposite Homogeneous Transformation MatrixTransformation matrix for adjacent coordinate framesChain product of successive coordinate transformation matrices

Rotate x, 600 (Rotx(600)

Translate x=2 (transl(2,0,0)

Further translate (2,3,2)

Rotation and Translation

Matlab codes for rotation and translation Robot Toolbox>>a=rotx(0)plotT1(a)b=rotx(pi/3)plotT1(b)c=transl(2,0,0)*bplotT1(c)d=transl(2,3,2)*cplotT1(d)plotT21(a,d)plotT21(b,c)plotT21(c,d)

Example 8For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?

Examples :Theoretical and graphical examples

Orientation RepresentationRotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body. Any easy way? Euler Angles Representation

Orientation RepresentationEuler Angles Representation ( , , )Many different typesDescription of Euler angle representations

xyzu'v'fv "w"w'==u"v'"u'"w'"=Euler Angle I, Animated

Orientation RepresentationEuler Angle I

Euler Angle I

Resultant Eulerian rotation matrix:

Euler Angle II, Animatedxyzu'v'=v"w"w'=u"v"'u"'w"'=Note the opposite (clockwise) sense of the third rotation, f.

Orientation RepresentationMatrix with Euler Angle IIQuiz: How to get this matrix ?

Orientation RepresentationDescription of Roll Pitch YawXYZQuiz: How to get rotation matrix ? - Phi roll angle Theta pitch angle Psi yaw angle

ReviewComposite Homogeneous Transformation MatrixRules:Transformation (rotation/translation) w.r.t. (X,Y,Z) (OLD FRAME), using pre-multiplicationTransformation (rotation/translation) w.r.t. (U,V,W) (NEW FRAME), using post-multiplication

ReviewHomogeneous Representation A point in space

A frame in space

Homogeneous coordinate of P w.r.t. OXYZ

ReviewOrientation Representation (Roll-Pitch-Yaw) Description of Yaw, Pitch, Roll premultiply !A rotation of about the OX axis ( ) -- yawA rotation of about the OY axis ( ) -- pitchA rotation of about the OZ axis ( ) -- roll

Quiz 1How to get the resultant rotation matrix for YPR?XYZ

Quiz 2Geometric Interpretation?

Position of the origin of OUVW coordinate frame w.r.t. OXYZ frameInverse of the rotation submatrix is equivalent to its transposePosition of the origin of OXYZ reference frame w.r.t. OUVW frameOrientation of OUVW coordinate frame w.r.t. OXYZ frameInverse Homogeneous Matrix?

Inverse Rotation Matrix R and Transformation Matrix TThe inverse of rotation matrix R and Homogeneous Transformation Matrix T

Kinematics ModelForward (direct) Kinematics

Inverse Kinematics

Exercise : Prove that

Robot Links and Joints

Denavit-Hartenberg ConventionNumber the joints from 1 to n starting with the base and ending with the end-effector. Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1.Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1.

Denavit-Hartenberg Convention ctd.Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis. Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel.Establish Yi axis. Assign to complete the right-handed coordinate system. Find the link and joint parameters

Link and Joint ParametersJoint angle : the angle of rotation from the Xi-1 axis to the Xi axis about the Zi-1 axis. It is the joint variable if joint i is rotary.

Joint distance : the distance from the origin of the (i-1) coordinate system to the intersection of the Zi-1 axis and the Xi axis along the Zi-1 axis. It is the joint variable if joint i is prismatic.

Link length : the distance from the intersection of the Zi-1 axis and the Xi axis to the origin of the ith coordinate system along the Xi axis.

Link twist angle : the angle of rotation from the Zi-1 axis to the Zi axis about the Xi axis.

Transformation between i-1 and i

More detail

Example I 3 Revolute JointsLink 1Link 2

Link Coordinate FramesAssign Link Coordinate Frames:To describe the geometry of robot motion, we assign a Cartesian coordinate frame (Oi, Xi,Yi,Zi) to each link, as follows: establish a right-handed orthonormal coordinate frame O0 at the supporting base with Z0 lying along joint 1 motion axis.the Zi axis is directed along the axis of motion of joint (i + 1), that is, link (i + 1) rotates about or translates along Zi;Link 1Link 2

Link Coordinate FramesLocate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis. the Xi axis lies along the common normal from the Zi-1 axis to the Zi axis , (if Zi-1 is parallel to Zi, then Xi is specified arbitrarily, subject only to Xi being perpendicular to Zi);

Link Coordinate FramesAssign to complete the right-handed coordinate system.The hand coordinate frame is specified by the geometry of the end-effector. Normally, establish Zn along the direction of Zn-1 axis and pointing away from the robot; establish Xn such that it is normal to both Zn-1 and Zn axes. Assign Yn to complete the right-handed coordinate system.

Example ID-H Link Parameter Table : rotation angle from Xi-1 to Xi about Zi-1 : distance from origin of (i-1) coordinate to intersection of Zi-1 & Xi along Zi-1 : distance from intersection of Zi-1 & Xi to origin of i coordinate along Xi : rotation angle from Zi-1 to Zi about Xi

Joint i

(i

ai

di

(i

1

0

a0

0

(0

2

-90

a1

0

(1

3

0

0

d2

(2

- angle twist for z axisa - distance traveled by z-axis along x-axisd - distance traveled by x-axis along z-axis - angle at z-axis

*3 Revolute JointsDenavit-Hartenberg Link Parameter TableNotice that the table has two uses:1) To describe the robot with its variables and parameters.2) To describe some state of the robot by having a numerical values for the variables.

i

((i-1)

a(i-1)

di

(i

0

0

0

0

(0

1

0

a0

0

(1

2

-90

a1

d2

(2

Transformation between i-1 and iFour successive elementary transformations are required to relate the i-th coordinate frame to the (i-1)-th coordinate frame:Rotate about the Z i-1 axis an angle of i to align the X i-1 axis with the X i axis.Translate along the Z i-1 axis a distance of di, to bring Xi-1 and Xi axes into coincidence.Translate along the Xi axis a distance of ai to bring the two origins Oi-1 and Oi as well as the X axis into coincidence.Rotate about the Xi axis an angle of i ( in the right-handed sense), to bring the two coordinates into coincidence.

In Short : Rotate z-axis to align with the x-axis iSlide along z-axis diSlide along x-axis aiRotate x-axis to align with z-axis - i

Develop a D-H Parameter Table

Matrix transform :

Transformation between i-1 and iD-H transformation matrix for adjacent coordinate frames, i and i-1. The position and orientation of the i-th frame coordinate can be expressed in the (i-1)th frame by the following homogeneous transformation matrix:

Source coordinateReferenceCoordinate

Matlab codes Robot Toolbox% create robot object R with the DH parameter below :% alpha A Theta D% 0 a0 0 0 % -90 a1 0 0 % 0 0 0 d2 % L = LINK([alpha A theta D])L1 = LINK([0 1 0 0]) % a0 = 1L2 = LINK([-pi/2 1 0 0]) % a1 = 1L3 = LINK([0 0 0 1.5]) % d2 = 1.5R=robot({L1 L2 L3}) % create IBot01 objectdrivebot(R)plot(R)

Try this ?

Example II: PUMA 260Number the jointsEstablish base frameEstablish joint axis ZiLocate origin, (intersect. of Zi & Zi-1) OR (intersect of common normal & Zi ) Establish Xi,Yi PUMA 260 t

Link Parameters : angle from Zi-1 to Zi about Xi : distance from intersectionof Zi-1 & Xi to Oi along Xi Joint distance : distance from Oi-1 to intersection of Zi-1 & Xi along Zi-1 : angle from Xi-1 to Xiabout Zi-1

PUMA ROBOT : OPEN CHAIN

Stanford Arm : Old Design

Robotic Toolbox

Announcement !Next week (W9) is for mini-Project weekNo need to attend tutorial classDo your own work on mini-projectMini project check on Assembly will be done together with programming check on week 11.Laboratory II will start on week 10.

Kinematic Equations Forward KinematicsGiven joint variablesTo get end-effector position & orientationHomogeneous matrix specifies the location of the ith coordinate frame w.r.t. the base coordinate systemchain product of successive coordinate transformation matrices ofOrientation matrixPosition vector

Kinematics EquationsOther representationsreference from, tool frame

Yaw-Pitch-Roll representation for orientation

Solving forward kinematicsForward kinematics

Transformation Matrix - Phi : roll angle Theta : pitch angle Psi : yaw angle

Solving forward kinematicsYaw-Pitch-Roll representation for orientationProblem?Solution is inconsistent and ill-conditioned!!

atan2(y,x)

Yaw-Pitch-Roll Representation

Yaw-Pitch-Roll Representation(Equation A)

Yaw-Pitch-Roll Representation Compare LHS and RHS of Equation A, we have:

Kinematic ModelSteps to derive kinematics model:Assign D-H coordinates framesFind link parametersTransformation matrices of adjacent jointsCalculate Kinematics MatrixWhen necessary, Euler angle representation

Example - angle twist for z axisa - distance traveled by z-axis along x-axisd - distance traveled by x-axis along z-axis - angle at z-axis

Joint i

(i

ai

di

(i

1

0

a0

0

(0

2

-90

a1

0

(1

3

0

0

d2

(2

Transformation between i-1 and iD-H transformation matrix for adjacent coordinate frames, i and i-1. The position and orientation of the i-th frame coordinate can be expressed in the (i-1)th frame by the following homogeneous transformation matrix:

Source coordinateReferenceCoordinate

More detail

Example

Joint i

(i

ai

di

(i

1

0

a0

0

(0

2

-90

a1

0

(1

3

0

0

d2

(2

Exercise Find the Homogeneous Transformation Matrix of

T03 = T01 T12 T23

Example: Puma 560

Link Coordinate ParametersPUMA 560 robot arm link coordinate parameters

Joint i

(i

(i

ai(mm)

di(mm)

1

(1

-90

0

0

2

(2

0

431.8

149.09

3

(3

90

-20.32

0

4

(4

-90

0

433.07

5

(5

90

0

0

6

(6

0

0

56.25

Example: Puma 560

Exercise :Specify the parameter table for IGBot-01 from the data given.Find the D-H transformation matrix for adjacent coordinate frames, i and i-1.Finally, find the Homogeneous Transformation matrix, T, for the IGBot-01

IGBot-01

Work Envelope of IGBot-01

Link Configuration of IGBot-01Link 1 = 9.5 cm

Link 2 = 11.5 cm

Link 3 = 8.5 cm

Link 4 = 15.0 cm

IGBot-01 specifications :

Link #Joint TypeLink DimensionJoint Angle Range BaseRevolute--177.50 to +177.50 ShoulderRevolute11.5 cm-450 to +900ElbowRevolute8.5 cm-600 to +900WristRevolute5.0 cm-1800 to +1800 Gripper---

IGBot-01Link parameter table for IGBot-01

Joint #idiaii11d10-900220a10330a2-90044d200

Joint #idiaii11d10-900220a10330a2-90044d200

OR - IGBot-01Link parameter table for IGBot-01

Joint #idiaii11d10900220a10330a290044d200

IGBot01 - simulation% create robot object IGBot01 with the DH% parameter below :% alpha A Theta D% -90 0 0 d1 % 0 a1 0 0 % -90 a2 0 0% 0 0 0 d2%L = LINK([alpha A theta D])L1 = LINK([-pi/2 0 0 1]) % d1 = 1L2 = LINK([0 1 0 0]) % a1 = 1L3 = LINK([-pi/2 0 0 0]) % a2 = 0L4 = LINK([0 0 0 1])% d2 = 1IGBot01=robot({L1 L2 L3 L4})% create robot IGBot01drivebot(IGBot01)%simulateplot(IGBot01)%plotT=fkine(IGBot01, [0 0 pi/2 0])%derive forward kinematicsQ=ikine(IGBot01, T)%inverse kinematicsrpy=tr2rpy(T)%derive roll/pitch/yaw

Inverse KinematicsGiven a desired position (P) & orientation (R) of the end-effector

Find the joint variables which can bring the robot the desired configuration

Inverse KinematicsMore difficultSystematic closed-form solution in general is not availableSolution not uniqueRedundant robotElbow-up/elbow-down configuration Robot dependent

Inverse Kinematics Transformation MatrixSpecial cases make the closed-form arm solution possible: Three adjacent joint axes intersecting (PUMA, Stanford) Three adjacent joint axes parallel to one another (MINIMOVER)

*The Denavit-Hartenberg MatrixJust like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame.

A Simple Example of Inverse KinematicsXYSRevolute and Prismatic Joints Combined(x , y)Finding :More Specifically:arctan2() specifies that its in the first quadrantFinding S:- Finding angles given coordinate x and y

Thank you!

*The Situation:You have a robotic arm that starts out aligned with the xo-axis.You tell the first link to move by 1 and the second link to move by 2.

The Quest:What is the position of the end of the robotic arm? Solution:1. Geometric ApproachThis might be the easiest solution for the simple situation. However, notice that the angles are measured relative to the direction of the previous link. (The first link is the exception. The angle is measured relative to its initial position.) For robots with more links and whose arm extends into 3 dimensions the geometry gets much more tedious.

2. Algebraic Approach Involves coordinate transformations.

Geometry of planar robot

Link #Joint TypeLink DimensionJoint Range 1Revolutel1-900 - +9002Revolutel2-1800 - +18003Revolutel3-1800 - +1800

*X2X3Y2Y3123123Example Problem: You have a three link arm that starts out aligned in the x-axis. Each link has lengths l1, l2, l3, respectively. You tell the first one to move by 1 , and so on as the diagram suggests. Find the Homogeneous matrix to get the position of the yellow dot in the X0Y0 frame.

H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 ) i.e. Rotating by 1 will put you in the X1Y1 frame. Translate in the along the X1 axis by l1. Rotating by 2 will put you in the X2Y2 frame. and so on until you are in the X3Y3 frame.

The position of the yellow dot relative to the X3Y3 frame is(l3, 0). Multiplying H by that position vector will give you the coordinates of the yellow point relative the the X0Y0 frame. X1Y1X0Y0

Slight variation on the last solution:Make the yellow dot the origin of a new coordinate X4Y4 frame X2X3Y2Y3123123X1Y1X0Y0X4Y4H = Rz(1 ) * Tx1(l1) * Rz(2 ) * Tx2(l2) * Rz(3 ) * Tx3(l3)

This takes you from the X0Y0 frame to the X4Y4 frame.

The position of the yellow dot relative to the X4Y4 frame is (0,0). Notice that multiplying by the (0,0,0,1) vector will equal the last column of the H matrix.

*The Denavit-Hartenberg MatrixJust like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame.

A Simple Example of Inverse KinematicsXYSRevolute and Prismatic Joints Combined(x , y)Finding :More Specifically:arctan2() specifies that its in the first quadrantFinding S:- Finding angles given coordinate x and y

*21(x , y)l2l1Inverse Kinematics of a Two Link ManipulatorGiven:l1, l2 , x , y

Find: 1, 2

Redundancy:A unique solution to this problem does not exist. Notice, that using the givens two solutions are possible. Sometimes no solution is possible.(x , y)l2l1

*The Geometric Solutionl1l221(x , y)Using the Law of Cosines:Using the Law of Cosines:Redundant since 2 could be in the first or fourth quadrant.Redundancy caused since 2 has two possible values

*The Algebraic Solutionl1l221(x , y)Only Unknown

We know what 2 is from the previous slide. We need to solve for 1 . Now we have two equations and two unknowns (sin 1 and cos 1 )Substituting for c1 and simplifying many timesNotice this is the law of cosines and can be replaced by x2+ y2

***Precision (validity) is defined as how accurately a specified point can be reached. It is a function of the resolution of the actuators, as well as its feedback devices. Most industrial robots can have precision of 0.001 inch or better.

Repeatability (variability) is how accurately the same position can be reached if the motion is repeated many times. The robot may not reach the same point every time, but will be within a certain radius from the desired point. The radius of a circle that is formed by this repeated motion is called repeatability. It is much more important than precision. If a robot is not precise, it will generally show a consistent error, which can be predicted and thus corrected through programming. However, if the error is random, it cannot be predicted and thus cannot be eliminated. Repeatability defines the extent of this random error. Manufactures must specify repeatability in conjunction with the number of tests, the applied payload during the tests, and the orientation of the arm. Have repeatability in the 0.001 inch range.

Depending on their configuration and the size of their links and wrist joints, robots can reach a collection of points called a workspace.

Payload is the weight a robot can carry and still remain within its other specification.*

***WF is a universal coordinate frame.

JF is used to specify movements of each individual joint of the robot.

HF specifies movements of the robots hand relative to a frame attached to the hand**************************************

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Chap 4 - Robot Kinematics

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Transcript of Chap 4 - Robot Kinematics