robot kinematics and linkage configurations 2

Robot Kinematics andlinkage configurations 2

cmput 412cmput 412

M. JagersandM. JagersandWith slides from A. Shademan With slides from A. Shademan

and A Casalsand A Casals

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. it’s (x, y, z) coordinates

Inverse Kinematics (position to angles)What you are given: The length of each link

The position of some point on the robot

What you can find: The angles of each joint needed to obtain that position

Review Kinematics

Numerical Inverse Kinematics

•Cartesian Location Cartesian Location

•Motor joint angles:Motor joint angles:

•Local linear model: Local linear model:

•Numerical steps: 1 Solve:Numerical steps: 1 Solve:

2 Update:2 Update:

y = x;y;z[ ]T = f (x)

x = x1; x2; . . . xn[ ]T

É y = J (x)É x

yãà yk = J É x

xk+1= xk+É x

y0

yã

Cartesian Trajectory Motion

yã =xyz

" #Desired goal:

Current Cartesian positiony0


yã

Goal

Line with sub goals

y0

yk

y1

y2


Move the robot to each subgoal in sequence


Iterate until convergence at final goal


•Cartesian Location Cartesian Location

•Motor joint angles:Motor joint angles:

•Local linear model: Local linear model:

•Visual servoing steps: 1 Solve:Visual servoing steps: 1 Solve:

2 Update:2 Update:

y = x;y;z[ ]T = f (x)

x = x1; x2; . . . xn[ ]T

É y = J (x)É x

yãà yk = J É x

xk+1= xk+É x

y0

yã

How do we find the Jacobian J(x)?J(x)?

Jacobian

• We can analytically derive the Jacobian from the We can analytically derive the Jacobian from the forward kinematicsforward kinematics

• EX: two link manipulatorEX: two link manipulator– Analytic Jacobian Analytic Jacobian JJ is: is:

11

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0012212211

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sasasa

qJ

Jacobian

•We can also numerically compute the Jacobian in We can also numerically compute the Jacobian in various ways based on the Secant constraint various ways based on the Secant constraint

1.1. Move to joint anglesMove to joint angles

2.2. Fwd kin gives Eucl pos Fwd kin gives Eucl pos

3.3. Relative movementsRelative movements

4.4. Secant constraint for Secant constraint for JJ: :

y = x;y;z[ ]T = f (x)

x = x1; x2; . . . xn[ ]T

É y = J (x)É x

y0y1

É y;É x

Find J Method 1: Test movements along basis

•Remember: Remember: J J is unknown is unknown m m by by n n matrixmatrix– For position only: 3x3, position and orientation 6x6 or mx6For position only: 3x3, position and orientation 6x6 or mx6

•Do test movements Do test movements

•Finite difference:Finite difference:

J =@x1@f1 ááá @xn

@f1

.... . .

@x1@fm

@xn@fm

0

B@

1

CA

É x1= [1;0;. . .;0]T

É x2= [0;1;. . .;0]T...É xn = [0;0;. . .;1]T

J t

...É y1...

2

4

3

5

...É y2...

2

4

3

5 ááá

...É yn...

2

4

3

5

0

@

1

A

Find J Method 2:Secant Constraints

•Constraint along a line:Constraint along a line:

•Defines Defines m m equations equations

•Collect Collect n n arbitrary, but different measuresarbitrary, but different measures y y

•Solve for Solve for JJ

É y = J É x

ááá É yT1 áááâ ã

ááá É yT2 áááâ ã

...ááá É yTn áááâ ã

0

BB@

1

CCA =

ááá É xT1 áááâ ã

ááá É xT2 áááâ ã

...ááá É xTn áááâ ã

0

BB@

1

CCA J T

Find J Method 3:Recursive Secant Constraints

Broydens method• Based on initial Based on initial J J and one measure pairand one measure pair

• Adjust Adjust J J s.t. s.t.

• Rank 1 update:Rank 1 update:

• Consider rotated coordinates: Consider rotated coordinates: – Update same as finite difference for Update same as finite difference for nn orthogonal moves orthogonal moves

É y;É x

É y = J k+1É x

Jêk+1= Jêk+É xTÉ x

(É y à JêkÉ x)É xT

É x


1.1. Solve for motion:Solve for motion:

2.2. Move robot joints:Move robot joints:

3.3. Read actual Cartesian move Read actual Cartesian move

4.4. Update Jacobian:Update Jacobian:

[yãà yk]= J kÉ xxk+1= xk+É x

Jêk+1= Jêk+É xTÉ x

(É y à JêkÉ x)É xT

É yMove the robot to each subgoal in sequence

yãGoal

y0yky1

y2

É y

Iterate until convergence at final goal

Singularities

• J singular J singular cannot solve eqs cannot solve eqs• Definition: we say that any configuration in which the Definition: we say that any configuration in which the

rank of rank of JJ is less than its maximum is a singular is less than its maximum is a singular configurationconfiguration– i.e. any configuration that causes i.e. any configuration that causes JJ to lose rank is a singular configuration to lose rank is a singular configuration

• Characteristics of singularities:Characteristics of singularities:– At a singularity, motion in some directions will not be possibleAt a singularity, motion in some directions will not be possible– At and near singularities, bounded end effector velocities would require At and near singularities, bounded end effector velocities would require

unbounded joint velocitiesunbounded joint velocities– At and near singularities, bounded joint torques may produce unbounded At and near singularities, bounded joint torques may produce unbounded

end effector forces and torquesend effector forces and torques– Singularities often occur along the workspace boundary (i.e. when the arm Singularities often occur along the workspace boundary (i.e. when the arm

is fully extended)is fully extended)

[yãà yk] = J É x

Singularities

• How do we determine singularities?How do we determine singularities?– Simple: construct the Jacobian and observe when it will lose rankSimple: construct the Jacobian and observe when it will lose rank

• EX: two link manipulatorEX: two link manipulator– Analytic Jacobian Analytic Jacobian JJ is: is:

– This loses rank if we can find some This loses rank if we can find some such that such thatcolumns are linearly dependentcolumns are linearly dependent

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qJ

R for 21 JJ

Singularities

• This is equivalent to the following:This is equivalent to the following:

• Thus if Thus if ss1212 = = ss11, we can always find an , we can always find an

that will reduce the rank of that will reduce the rank of JJ

• Thus Thus 22 = 0, = 0, are two singularities are two singularities

12212211

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2

21

a

aa

2

12

a

aa

Determining Singular Configurations

• In general, all we need to do is observe how the In general, all we need to do is observe how the rank of the Jacobian changes as the configuration rank of the Jacobian changes as the configuration changeschanges

•Can study analyticallyCan study analytically

•Or numerically: Singular if eigenvalues of square Or numerically: Singular if eigenvalues of square matrix 0, or singular values of rectangular matrix matrix 0, or singular values of rectangular matrix zero. (Compute with SVD), or condition number zero. (Compute with SVD), or condition number tends to infinity.tends to infinity.

How accurate is the movement?

•Does the robot always reach the goal?Does the robot always reach the goal?

Final position error dy

Accuracy, Repeatability and Resolution

Accuracy:Accuracy:

•Start far awayStart far away

•Move a long distanceMove a long distance

•Measure error Measure error dacc

dacc


Repeatability:Repeatability:

•Start at goalStart at goal

•Move away a long distanceMove away a long distance

•Move back to goalMove back to goal

•Measure error Measure error drp

drp


Resolution:Resolution:

• The smallest incremental The smallest incremental distance a robot can move.distance a robot can move.

•Typically limited by joint Typically limited by joint encoder resolution. encoder resolution.

dres

Points potentially weak in mechanical designPoints potentially weak in mechanical design

Weak points Mechanical correction

Permanent deformation of the whole structure and the components

Increase rigidness

Weight reduction

Counterweight

Dynamic deformation

Reduction of the mass to move

Weight distribution

BacklashReduce gear clearances

Use more rigid transmission elements

Increase rigidness

Axes clearance

Use pre stressed axes

Friction Improve clearance in axes

Increase lubrication

Thermal effects Isolate heat source

Bad transducers connection

Improve mechanical connection Search for a better location Protect the environment

Points potentially weak in the mechanical designPoints potentially weak in the mechanical design

Weak points Mechanical correction

Robot Morphology:

An classic arm - The PUMA 560

• AccuracyAccuracy– Factory about 10mmFactory about 10mm

– Calibrated 1-5mmCalibrated 1-5mm

• Repeatability 1mmRepeatability 1mm• Resolution 0.1mmResolution 0.1mm

An modern arm - The Barrett WAM

• Resolution and Resolution and repeatability similar or repeatability similar or better than PUMAbetter than PUMA

• Accuracy worse due to Accuracy worse due to lighter, more flexible lighter, more flexible linkagelinkage

• But can move fasterBut can move faster• Needs external feedback Needs external feedback

(e.g. camera vision) for (e.g. camera vision) for accurate motionsaccurate motions

Can build this with LEGO!

5 – Bar linkage

Parallel linkageSensable Phantom

Workspace of a typical serial arm

Cartesian robot

Typical performanceTypical performance• Accuracy 0.1-1mmAccuracy 0.1-1mm• Repeatability: 0.01mmRepeatability: 0.01mm• Resolution: 0.01mmResolution: 0.01mm• DrawbacksDrawbacks

– Large, heavyLarge, heavy

– Workspace “inside” the robotWorkspace “inside” the robot

x

y

Coordinates of the target

R

offset

AccuracyAccuracy

• Capacity to place the end effector into a given position and orientation (pose) within the robot working volume, from a random initial position.

increases with the motiondistance

Measuring Accuracy

Measure:Measure:• Sample many start and end Sample many start and end

positionspositions• Characterize errors:Characterize errors:

– systematic systematic (offset)

– randomrandom

Example of Map of

admitted loads, in function

of the distance to the main

axis

Dynamic CharacteristicsDynamic Characteristics

Payload:Payload:• The load (in Kg) the robot is able to transport in a

continuous and precise way (stable) to the most distance point

• The values usually used are the maximum load and nominal at acceleration = 0

• The load of the End-Effector is not included.

VelocityVelocity

• Maximum speed (mm/sec.) to which the robot can move the End-Effector.

• It has to be considered that more than a joint is involved.

• If a joint is slow, all the movements in which it takes part will be slowed down.

• For shorts movements acceleration matters more.

Vmax

time

speed

Short movements

Long movement

Dynamic CharacteristicsDynamic Characteristics

Path/trajectory planning

•Split the workspace into areas of fast and slow Split the workspace into areas of fast and slow guarded motion:guarded motion:

• In practice robot has to smoothly accelerate.In practice robot has to smoothly accelerate.

•Can create position/velocity profiles with linear Can create position/velocity profiles with linear or higher order polynomials/splines.or higher order polynomials/splines.

y0y1

y2

Path/trajectory planning

• With many obstacles may need motion planning With many obstacles may need motion planning algorithms (Potential fields, RRT etc -- later)algorithms (Potential fields, RRT etc -- later)

Conclusions:

•Different architectures have different kinematics Different architectures have different kinematics and different accuracy.and different accuracy.– Serial arms: Slender, agile but somewhat inaccurateSerial arms: Slender, agile but somewhat inaccurate

– Cartesian (x,y,z- table): Very accurate, but bulkyCartesian (x,y,z- table): Very accurate, but bulky

•Accuracy can be improved by:Accuracy can be improved by:– Cartesian calibrationCartesian calibration

– Visual or other sensory feedback (next in course)Visual or other sensory feedback (next in course)

robot kinematics and linkage configurations 2

Documents

initial j

update jacobian

jfind j method

measure pairadjust j

jacobian jx

robot joints

unbounded joint velocitiesat

joint anglesfwd kin