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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

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## Forward Kinematics and Jacobians

Kris Hauser

CS B659: Principles of Intelligent Robot Motion

Spring2013

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

• 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

T2ref

T3ref

T1ref

T4ref

L2

T0

L3

L1

L0

T1(q1)

q1

### Rotating the first joint

T1(q1) =T1ref·R(q1)

T0

T1ref

L0

T2(q1) ?

T2ref

T0

q1

### Where is the second joint?

T2ref

T0

q1

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

### After rotating joint 2

q2

T2R

T0

q1

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

### 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)

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

• 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…

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

• Angle-axis parameterization

### Generalizations

• Prismatic joints

• Ball joints

• Prismatic joints

• Spirals

• Free-floating bases

From LaValle, Planning Algorithms

### 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

(x,y)

L

q

(x,y)

L

q

(x,y)

L

q

### 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

• How do we compute it?

• Consider derivative w.r.t. qj

T2(q1,q2)

T3(q1,..,q3)

T1(q1)

T4(q1,…,q4)

xk

L2

T0

L3

L1

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

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

• 3 rows in 2D, 6 rows in 3D