Forward Kinematics and Jacobians Kris Hauser CS B659: Principles of Intelligent Robot Motion Spring 2013

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) q1q1q1q1 q2q2q2q2

Forward Kinematics Given: A kinematic reference frame of the robot Joint angles q 1,…,q n Find rigid frames T 1,…,T n relative to T 0 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: T 1 ·T 2 indicates the transformation T 2 first, then T 1

Kinematic Model of Articulated Robots: Reference Frame T0T0 L0L0 L1L1 L2L2 L3L3 T 1 ref T 2 ref T 3 ref T 4 ref

Rotating the first joint T0T0 L0L0 T 1 ref q1q1 T 1 (q 1 ) T 1 (q 1 ) = T 1 ref ·R(q 1 )

Where is the second joint? T0T0 T 2 ref q1q1 T 2 (q 1 ) ?

Where is the second joint? T0T0 T 2 ref q1q1 T 2 parent (q 1 ) = T 1 (q 1 ) ·(T 1 ref ) -1 ·T 2 ref

After rotating joint 2 T0T0 T2RT2R q1q1 T 2 (q 1,q 2 ) = T 1 (q 1 ) ·(T 1 ref ) -1 ·T 2 ref ·R(q 2 ) q2q2

After rotating joint 2 T0T0 T2RT2R q1q1 Denote T 2->1 ref = (T 1 ref ) -1 ·T 2 ref (frame relative to parent) T 2 (q 1,q 2 ) = T 1 (q 1 ) ·T 2->1 ref ·R(q 2 ) q2q2

General Formula T0T0 L0L0 L1L1 L2L2 L3L3 T 1 (q 1 ) T 2 (q 1,q 2 ) T 3 (q 1,..,q 3 ) T 4 (q 1,…,q 4 )

Generalization to tree structures

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  f 1 /  q 1 …  f 1 /  q n …  f m /  q 1 …  f m /  q n

Aside on partial derivatives…

Single Joint Robot Example q (x,y) L

Single Joint Robot Example q (x,y) L

Single Joint Robot Example q (x,y) L

Significance q (x,y) L

Computing Jacobians in general

Derivative…

T0T0 L1L1 L2L2 L3L3 T 1 (q 1 ) T 2 (q 1,q 2 ) T 3 (q 1,..,q 3 ) T 4 (q 1,…,q 4 ) xkxk

Consequences… Column j of position Jacobian J x (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 T0T0 L1L1 L2L2 L3L3 T 1 (q 1 ) T 2 (q 1,q 2 ) T 3 (q 1,..,q 3 ) T 4 (q 1,…,q 4 ) xkxk

Orientation Jacobian Consider end effector orientation θ(q) in plane All entries of J θ (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 T0T0 L1L1 L2L2 L3L3 T 1 (q 1 ) T 2 (q 1,q 2 ) T 3 (q 1,..,q 3 ) T 4 (q 1,…,q 4 ) xkxk

Total Jacobian Total Jacobian J(q) is the matrix formed by stacking J x (q), J θ (q) 3 rows in 2D, 6 rows in 3D

Next class: Inverse Kinematics Readings on schedule: Wang and Chen (1991) Duindam et al (2008)