Forward kinematics and jacobians
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Forward Kinematics and Jacobians. Kris Hauser CS B659: Principles of Intelligent Robot Motion Spring 2013. q 2. q 1. Articulated Robot. Robot: usually a rigid articulated structure Geometric CAD models, relative to reference frames

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Forward kinematics and jacobians

Forward Kinematics and Jacobians

Kris Hauser

CS B659: Principles of Intelligent Robot Motion

Spring2013


Articulated robot

q2

q1

Articulated Robot

  • Robot: usually a rigid articulated structure

  • Geometric CAD models, relative to reference frames

  • A configuration specifies the placement of those frames (forward kinematics)


Forward kinematics

Forward Kinematics

  • Given:

    • A kinematic reference frame of the robot

    • Joint angles q1,…,qn

  • Find rigid frames T1,…,Tn relative to T0

  • A frame T=(R,t) consists of a rotation R and a translation t so that T·x = R·x + t

    • Make notation easy: use homogeneous coordinates

    • Transformation composition goes from right to left:T1·T2 indicates the transformation T2 first, then T1


Kinematic model of articulated robots reference frame

T2ref

T3ref

T1ref

T4ref

Kinematic Model of Articulated Robots: Reference Frame

L2

T0

L3

L1

L0


Rotating the first joint

T1(q1)

q1

Rotating the first joint

T1(q1) =T1ref·R(q1)

T0

T1ref

L0


Where is the second joint

Where is the second joint?

T2(q1) ?

T2ref

T0

q1


Where is the second joint1

Where is the second joint?

T2ref

T0

q1

T2parent(q1) = T1(q1) ·(T1ref)-1·T2ref


After rotating joint 2

After rotating joint 2

q2

T2R

T0

q1

T2(q1,q2) = T1(q1)·(T1ref)-1·T2ref·R(q2)


After rotating joint 21

After rotating joint 2

q2

T2R

T0

q1

Denote T2->1ref= (T1ref)-1·T2ref(frame relative to parent)

T2(q1,q2) = T1(q1) ·T2->1ref·R(q2)


General formula

T2(q1,q2)

T3(q1,..,q3)

T1(q1)

T4(q1,…,q4)

General Formula

Denote (ref frame relative to parent)

L2

T0

L3

L1

L0


Generalization to tree structures

Generalization to tree structures

  • Topological sort: p[k] = parent of link k

  • Denote (frame i relative to parent)

  • Let A(i) be the list of ancestors of i (sorted from root to i)


To 3d

To 3D…

  • Much the same, except joint axis must be defined (relative to parent)

  • Angle-axis parameterization


Generalizations

Generalizations

  • Prismatic joints

  • Ball joints

  • Prismatic joints

  • Spirals

  • Free-floating bases

From LaValle, Planning Algorithms


The jacobian matrix

The Jacobian Matrix

  • The Jacobian of a function x = f(q), with and is the m x n matrix of partial derivatives

  • Typically written J(q)

  • (Note the dependence on q)

f1/q1 … f1/qn

… …

fm/q1 … fm/qn


Aside on partial derivatives

Aside on partial derivatives…


Single joint robot example

Single Joint Robot Example

(x,y)

L

q


Single joint robot example1

Single Joint Robot Example

(x,y)

L

q


Single joint robot example2

Single Joint Robot Example

(x,y)

L

q


Significance

Significance

  • If x is an end effector, multiplying J(q) with a joint velocity vector gives the end effector velocity

(x,y)

L

q


Computing jacobians in general

Computing Jacobians in general

  • How do we compute it?

  • Consider derivative w.r.t. qj


Derivative

Derivative…


Derivative1

T2(q1,q2)

T3(q1,..,q3)

T1(q1)

T4(q1,…,q4)

Derivative…

xk

L2

T0

L3

L1


Consequences

T2(q1,q2)

T3(q1,..,q3)

T1(q1)

T4(q1,…,q4)

Consequences…

  • Column j of position JacobianJx(q) is the speed at which x would change if joint j rotated alone

  • Perpendicular and equal in magnitude to vector from x to joint axis

  • Larger when x is farther from the joint axis

xk

L2

T0

L3

L1


Orientation jacobian

T2(q1,q2)

T3(q1,..,q3)

T1(q1)

T4(q1,…,q4)

Orientation Jacobian

  • Consider end effector orientation θ(q) in plane

  • All entries of Jθ(q) corresponding to revolute joints are 1!

  • In 3D, column j is identical to the axis of rotation of joint j (in world space) at configuration q

xk

L2

T0

L3

L1


Total jacobian

Total Jacobian

  • Total Jacobian J(q) is the matrix formed by stacking Jx(q), Jθ(q)

  • 3 rows in 2D, 6 rows in 3D


Next class inverse kinematics

Next class: Inverse Kinematics

  • Readings on schedule:

  • Wang and Chen (1991)

  • Duindam et al (2008)


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