AnimationCS 551 / 651

KinematicsKinematicsLecture 09Lecture 09

Sarcos Humanoid

KinematicsThe study of object movements irrespective The study of object movements irrespective

of their speed or style of movementof their speed or style of movement

Degrees of Freedom(DOFs)The variables that affect an object’s The variables that affect an object’s

orientationorientationHow many degrees of How many degrees of

freedom when flying?freedom when flying?

• Six• x, y, and z positions• roll, pitch, and yaw

• So the kinematics of this airplane permit movement anywhere in three dimensions

Degrees of FreedomHow about this robot arm?How about this robot arm?

• Six again• 2-base, 1-shoulder, 1-elbow, 2-wrist

• The set of all possible positions (defined by kinematics) an object can attain

Configuration Space

Work Space vs. Configuration SpaceWork spaceWork space

• The space in which the object existsThe space in which the object exists• DimensionalityDimensionality

– RR33 for most things, R for most things, R22 for planar arms for planar arms

Configuration spaceConfiguration space• The space that defines the possible object configurationsThe space that defines the possible object configurations• Degrees of FreedomDegrees of Freedom

– The number of parameters that necessary and sufficient to The number of parameters that necessary and sufficient to define position in configurationdefine position in configuration

More examplesA point on a planeA point on a planeA point in spaceA point in spaceA point moving on a A point moving on a

line in spaceline in space

A matter of control

If your animation adds energy at a particular If your animation adds energy at a particular DOF, that is a DOF, that is a controlled DOFcontrolled DOF

High DOF, no controlLow DOF, high control

Hierarchical Kinematic ModelingA family of parent-child spatial relationships A family of parent-child spatial relationships

are functionally definedare functionally defined• Moon/Earth/Sun movementsMoon/Earth/Sun movements

• Articulations of a humanoidArticulations of a humanoid

• Limb connectivity is built into model (joints) and animation is easier

Robot Parts/TermsLinksLinksEnd effectorEnd effectorFrameFrameRevolute JointRevolute JointPrismatic JointPrismatic Joint

More Complex Joints

3 DOF joints3 DOF joints• GimbalGimbal

• Spherical Spherical (doesn’t possess (doesn’t possess singularity)singularity)

2 DOF joints2 DOF joints• UniversalUniversal

Hierarchy RepresentationModel bodies (links) as nodes of a treeModel bodies (links) as nodes of a treeAll body frames are local (relative to parent) All body frames are local (relative to parent)

• Transformations affecting root affect all childrenTransformations affecting root affect all children

• Transformations affecting any node affect all its Transformations affecting any node affect all its childrenchildren

ROOT

Forward vs. Inverse KinematicsForward KinematicsForward Kinematics

• Compute configuration (pose) given individual DOF Compute configuration (pose) given individual DOF valuesvalues

– Good for simulationGood for simulation

Inverse KinematicsInverse Kinematics• Compute individual DOF values that result in Compute individual DOF values that result in

specified end effector positionspecified end effector position

– Good for controlGood for control

Forward KinematicsTraverse kinematic tree and Traverse kinematic tree and

propagate transformations propagate transformations downwarddownward• Use stackUse stack

• Compose parent transformation with Compose parent transformation with child’schild’s

• Pop stack when leaf is reachedPop stack when leaf is reached

Denavit-Hartenberg (DH) NotationA kinematic representation (convention) A kinematic representation (convention)

inherited from roboticsinherited from robotics

Z-axis aligned with joint

X-axis aligned with outgoing limb

Y-axis is orthogonal

Joints are numbered to represent hierarchy Ui-1 is parent of Ui

Parameter ai-1 is outgoinglimb length of joint Ui-1

Joint angle, i, is rotation of xi-1 about zi-1 relative to xi

Link twist, i-1, is the rotation of ith z-axis about xi-1-axis relative to z-axis of i-1th frame

Link offset, di-1, specifies the distance along the zi-1-axis (rotated by i-1) of the ith frame from the i-1th x-axis to the ith x-axis

Inverse Kinematics (IK)Given end effector position, compute Given end effector position, compute

required joint anglesrequired joint anglesIn simple case, analytic solution existsIn simple case, analytic solution exists

• Use trig, geometry, and algebra to solveUse trig, geometry, and algebra to solve

What is Inverse Kinematics?Forward KinematicsForward Kinematics

Base

End Effector

?

What is Inverse Kinematics?Inverse KinematicsInverse Kinematics

Base

End Effector

What does look like?

?

Base

End Effector

Solution to Our exampleOur example

Number of equation : 2Unknown variables : 3

Infinite number of solutions !

Redundancy

System DOF > End Effector DOF

Our example

System DOF = 3End Effector DOF = 2

• Analytic solution of 2-link inverse kinematics

2

1

a1

a2

O2

O1

O0

x1

x0

x2

y1

y2

y0

(x,y)

2

22

221

22

22222

211

2

222

21

22

22222

21

22

21

2221

22

21

222122

21

22

21

22

2

22122

21

22

tan2

22

cos1cos1

2tan

accuracygreater for 2

cos

)cos(2

aayx

yxaa

aayxyxaa

aayxaaaayxaa

aaaayx

aaaayx

Failures of simple IKMultiple SolutionsMultiple Solutions

Failures of simple IKInfinite solutionsInfinite solutions

Failures of simple IKSolutions may not existSolutions may not exist

Iterative IK SolutionsFrequently analytic solution is infeasibleFrequently analytic solution is infeasibleUse Use JacobianJacobian

• Derivative of function output relative to each of its inputsDerivative of function output relative to each of its inputs

If y is function of three inputs and one outputIf y is function of three inputs and one output

33

22

11

321 ),,(

xxfx

xfx

xfy

xxxfy

• Represent Jacobian, J(X), as a 1x3 matrix of partial Represent Jacobian, J(X), as a 1x3 matrix of partial derivativesderivatives

Jacobian

In another situation, In another situation,

end effector has 6 end effector has 6 DOFs and robotic DOFs and robotic arm has 6 DOFsarm has 6 DOFs

f(xf(x11, …, x, …, x66) = (x, y, z, ) = (x, y, z, r, p, y)r, p, y)

Therefore J(X) = 6x6 Therefore J(X) = 6x6 matrixmatrix

1

1

1

1

1

654321

xfxfxfxfxf

xf

xf

xf

xf

xf

xf

y

p

r

z

y

xxxxxx

JacobianRelates velocities in parameter space to Relates velocities in parameter space to

velocities of outputsvelocities of outputs

If we know YIf we know Ycurrentcurrent and Y and Ydesireddesired, then we , then we subtract to compute Ysubtract to compute Ydotdot

Invert Jacobian and solve for XInvert Jacobian and solve for Xdotdot

XXJY )(

Turn to PDF slidesSlides from O’Brien and ForsythSlides from O’Brien and Forsyth• CS 294-3: Computer GraphicsCS 294-3: Computer Graphics

StanfordStanfordFall 2001Fall 2001

Differential KinematicsIs Is JJ always invertible? No! always invertible? No!

• Remedy : Pseudo InverseRemedy : Pseudo Inverse

Null space

The null space of J is the set of vectors The null space of J is the set of vectors which have no influence on the constraintswhich have no influence on the constraints

The pseudoinverse provides an operator The pseudoinverse provides an operator which projects any vector to the null space of Jwhich projects any vector to the null space of J

0)( JJnullspace

zJJIxJ

xJ

)(

Utility of Null SpaceThe null space can be used to reach The null space can be used to reach

secondary goalssecondary goals

Or to find comfortable positionsOr to find comfortable positions

)(min)(

f

zJJIxJ

z

i

comfort iif 2))()(()(

Calculating Pseudo InverseSingular Value DecompositionSingular Value Decomposition

RedundancyA redundant system has infinite number of A redundant system has infinite number of

solutionssolutions

Human skeleton has 70 DOFHuman skeleton has 70 DOF• Ultra-super redundantUltra-super redundant

How to solve highly redundant system?How to solve highly redundant system?

Redundancy Is BadMultiple choices for one goalMultiple choices for one goal

• What happens if we pick any of them?What happens if we pick any of them?

Redundancy Is GoodWe can exploit redundancyWe can exploit redundancy

Additional objectiveAdditional objective• Minimal ChangeMinimal Change

• Similarity to Given ExampleSimilarity to Given Example

• NaturalnessNaturalness

NaturalnessBased on observation of natural human Based on observation of natural human

posture posture Neurophysiological experimentsNeurophysiological experiments

Conflict Between Goals

base

ee 2ee 1

Conflict Between Goals

base

ee 2ee 1

Goal 1

Conflict Between Goals

base

ee 2

Goal 2

ee 1

Conflict Between Goals

base

ee 2ee 1

Goal 1Goal 2

Conflict Between Goals

base

ee 2ee 1

Goal 1Goal 2