Forward Kinematics and Jacobians

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# Forward Kinematics and Jacobians - PowerPoint PPT Presentation

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

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

T1(q1)

q1

Rotating the first joint

T1(q1) =T1ref·R(q1)

T0

T1ref

L0

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

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)

Derivative…

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
Next class: Inverse Kinematics