Differential Kinematics and Statics

Ref: 2001

Incremental MotionWhat small (incremental) motions at the end-effector (Dx, Dy, Dz) result from small motions of the joints (Dq1, Dq2, , Dqn )? Alternatively, what velocities at the end-effector (vx, vy, vz) result from velocities at the joints (w1, w2, wn)?

Some DefinitionsLinear Velocity: The instantaneous rate-of-change in linear position of a point relative to some frame.v=(vx, vy, vz)TAngular Velocity: The instantaneous rate-of-change in the orientation of one frame relative to another.Angular Velocity depends on the way to represent orientation (Euler Angles, Rotation Matrix, etc.)Angular Velocity Vector and the Angular Velocity Matrix.

Some DefinitionsAngular Velocity Vector: A vector whose direction is the instantaneous axis of rotation of one frame relative to another and whose magnitude is the rate of rotation about that axis.

Free VectorLinear velocity are insensitive to shifts in origin but are sensitive to orientation.{D}xx

Free VectorAngular velocity are insensitive to shifts in origin but are sensitive to orientation.{D}xxxx

Velocity Framesframe of reference: this is the frame used to measure the objects velocityframe of representation.: this is the frame in which the velocity is expressed.

X0Y0x0y00q1Y1X10x2a1vvvvq2Ra2y2Figure 2.13: Two-Link Planar Robot

X0Y0x0y00w10vvvvEnd-effector velocity for w1r0n

X0Y0x0y00w20vvvvEnd-effector velocity for w2r1n

Two-Link Planar RobotDirect kinematics equation

Incremental Motiontaking derivatives of the position equation w.r.t. time we have

note that

Incremental Motionwritten in the more common matrix form,

or in terms of incremental motion,

Differential KinematicsFind the relationship between the joint velocities and the end-effector linear and angular velocities.Linear velocityAngular velocityfor a revolute jointfor a prismatic joint

Differential KinematicsDifferential kinematics equation

Geometric Jacobian

Relationship with T(q)Direct kinematics equation

Linear velocity

Angular velocity?

Vector (Cross) ProductVector product of x and y

Skew-symmetric matrix

Vector (Cross) ProductSkew-symmetric matrix

Derivative of a Rotation MatrixdefineS(t) is skew-symmetric

Interpretation of S(t)

Interpretation of S(t)Given R(t)

Example 3.1: Rotation about Z

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