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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
rotating the first joint

T1(q1)

q1

Rotating the first joint

T1(q1) =T1ref·R(q1)

T0

T1ref

L0

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

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
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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