ee 451 - velocity kinematicsisl.ee.boun.edu.tr/courses/ee451/lectures/ch4_kin.pdf · angular vs...

Outline

Velocity Kinematics

EE 451 - Velocity Kinematics

H.I. Bozma

Electric Electronic Engineering

Bogazici University

November 11, 2019

H.I. Bozma EE 451 - Velocity Kinematics

Outline

Velocity Kinematics

Velocity KinematicsIntroductionAngular Velocity: Fixed AxisSkew-Symmetric MatricesJacobianAngular VelocityLinear VelocityTool VelocityAnalytic JacobianSingularitiesForce-Torque RelationshipsInverse Velocity


Outline

Velocity Kinematics

Introduction

Angular Velocity: Fixed Axis

Skew-Symmetric Matrices

Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities

Force-Torque Relationships

Inverse Velocity

Velocity Kinematics

◮ Velocity in Configuration space C ⇔ Velocity in Workspace W

◮ Representation of velocities◮ Revolute – angular◮ Prismatic – linear

◮ Angular velocity about a fixed axis◮ Rotation around a moving axis

◮ Instantaneous transformations btw n-vector joint velocities inC ⇔ 6-vector of angular and linear velocities in W →Jacobian (6-n matrix)


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Pure Rotation About Fixed Axis

◮ Pure rotation → Every point moves in a circle.

◮ Centers of circles – On the axis of rotation

◮ Perpendicular to the axis – θ

◮ Angular velocity ω = θk whereθ = dθ

dtand

k - unit vector in the axis of rotation

◮ Linear velocity v = ω × r

r - Vector from origin (axis of rotation) to the point


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Goal

◮ Goal – The motion of a moving frame.◮ The motion of the origin of the frame◮ The rotational motion of the frame’s axes


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Angular vs Linear Velocity

◮ Attach a frame rigidly to each object with an orientation

◮ Each point on the object – Same angular velocity!

◮ Angular velocity – Property of the frame attached to a body

◮ Linear velocity – Property of the point, but rather the frame


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Definition and Properties

◮ Linearity

◮ Relation to cross product

◮ Similarity transformation

◮ Quadratic form


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Derivative of a Rotation Matrix

ddθRk,θ = S(k)Rk,θ


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Jacobian

◮ n link robotic system – q1, . . . , qn

◮ T 0n =

[

R0n(t) o0n(t)0 1

]

◮ As robot moves around, qi , R0n and o0n – functions of time

◮ Angular velocity of end effector ω0n(t) – Defined by

S(ω0n(t)) = R0

n(t)(R0n(t))

T

◮ Linear velocity of end effector v0n = o0n

Goal: Find ξ =

[

v0nω0n

]

= Jq where J =

[

JvJω

]

⇐ Jacobian


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Angular Velocity about a Moving Axis

◮ Time varying rotation matrix R(t), R(t) ∈ SO(3)


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Addition of Angular Velocities

R0n = R0

1R12 . . .R

n−1n

R0n = S(ω0

0,n)R0n

ω00,n = ω0

0,1 + R01ω

11,2 + R0

2ω22,3 + . . .+ R0

n−1ωn−1n−1,n

= ω00,1 + ω0

1,2 + ω02,3 + . . .+ ω0

n−1,n


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Angular Velocity - Revolute Joint

If revolute joint, qi = θi with axis of rotation zi−1

Let ωi−1i – Angular velocity of joint i wrt oi−1xi−1yi−1zi−1 Note

that

ωi−1i = qiz

i−1i−1 = qik where k =

001


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Angular Velocity – Prismatic Joint

If prismatic joint, qi = di with axis of translation zi−1

ωi−1i = 0


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Angular Velocity - End Effector

Let ρi =

{

1 if revolute0 otherwise

ω0n = ω0

0,1 + ω01,2 + ω0

2,3 + . . .+ ω0n−1,n

= ω00,1 + R0

1ω11,2 + R0

2ω22,3 + . . .+ R0

n−1ωn−1n−1,n

= ρ1q1k + ρ2R01 q2k + . . .+ ρnR

0n−1qnk

= ρ1q1k + ρ2q2R01k + . . .+ ρnqnR

0n−1k

=

n∑

i=1

ρi qiz0i−1 =

n∑

i=1

ρiz0i−1qi


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Angular Velocity Jacobian

Note that Jω is a 3 × n matrix.

Jω =[

ρ1z00 . . . ρnz

0n−1

]

Equivalently,

Jω =[

ρ1k ρ2R01k . . . ρn−1R

0n−2k ρnR

0n−1k

]


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Linear Velocity of a Point p Attached to a Frame

Assume: p – Attached rigidly to o1x1y1z1

◮ Case 1: o1x1y1z1 is rotating wrt o0x0y0z0

◮ Case 2: Motion of o1x1y1z1 wrt o0x0y0z0 - Defined by

H01 (t) =

[

R01 (t) o01(t)0 1

]


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Linear Velocity - End effector

◮ Method 1: If kinematic transformation matrix T 0n (q) exists

o0n =n

∑

i=1

δo0nδqi

qi


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Linear Velocity - Individual Joints

◮ Prismatic joint:

◮ Revolute joint:

◮ Method 2:

Jvi =

{

z0i−1 Prismatic jointz0i−1 × (on − oi−1) Revolute joint


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Linear Velocity - Revolute Joint


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Jacobian – Summary

J =

[

Jv1 . . . JvnJω1

. . . Jωn

]

where

Jvi =

{

z0i−1 × (on − oi−1) if revolutez0i−1 if prismatic

Jωi=

{

z0i−1 if revolute0 if prismatic


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Using Velocities

◮ Linear velocity: v(t). After time δt:o0(t + δt) = o0(t) + v(t) ∗ δt

◮ Angular velocity: ω(t). After time δt: δθ = ω(t)δt

◮ Let δθ =[

δθx δθy δθz]T

◮ δR = Rz(δθz)Ry (δθy )Rx(δθx)◮ R(t + δt) = δRR(t)

Note R = S(ω)R is not an orthogonal matrix! Thus, we cannotuse it directly. However, recall that it involves a skew-symmetricmatrix that defines ω.


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

2 DOF RR Planar Robot


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

3 DOF RRR Robot


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Tool Velocity

◮ TI transformation T 6

tool =

[

R d

0 1

]

◮ ωtool = ω6 → ωtooltool = RTω66

◮ vtooltool = vtool6 + ωtool6 × r tool


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

End Effector Frame

◮ X =

[

d(q)α(q)

]

∈ R3 × SO(3)

◮ X = Ja(q)q ⇐ Analytic Jacobian

◮ Assuming Euler angles R = Rz,φRy ,θRz,ψ,

Ja(q) =

[

I 00 B−1(α)

]

J(q) =

[

I 00 B−1(α)

] [

d

ω

]

where B(α) =

cosψ sin θ − sinψ 0sinψ sin θ cosψ 0cos θ 0 1


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Singularities - Singular Configurations

◮ ξ = J(q)q

◮ J(q) =[

J1(q) J2(q) . . . Jn(q)]

→ ξ =∑n

i=1 Ji (q)qi◮ J(q) is a 6× n matrix → Rank(J(q)) ≤ min(6, n)

◮ End effector velocity ={

Arbitrary if Rank(J(q)) = 6

◮ Rank(J(q)) is time-varying!

◮ q ∈ C for which Rank(J(q)) < maxRank(J(q)) ← Singularity

◮ For 6× 6 matrix, det(J(q)) = 0


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Problems of Singularities?

◮ Certain velocities are not attainable

◮ Bounded end-effector velocities – Unbounded joint velocities

◮ Often correspond to points on the boundary of workspace

◮ Often correspond to points unreachable via smallperturbations of link parameters


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Decoupling of Singularities

◮ Arm singularities - Singularities resulting from the arm motion

◮ Wrist singularities - Singularities resulting from the wristmotion


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

RR Planar Robot


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity


◮ Interaction with the environment → Forces and moments atthe end effector F = [FxFyFznxnynz ]

T

◮ [FxFyFznxnynz ] → Joint torques τ where τ = J(q)TF


Outline

Velocity Kinematics

Introduction



Jacobian

Angular Velocity

Linear Velocity

Tool Velocity

Analytic Jacobian

Singularities


Inverse Velocity

Inverse Velocity Problem

◮ Problem Statement: Given ξ, find q such that ξ = J(q)q

◮ If J(q) is invertible (square and full rank), q = J(q)−1ξ

◮ If n 6= 6, J(q) is not invertible !

◮ If ξ ∈ Span(J(q)) ↔ Rank(J(q)) = Rank([J(q) | ξ])(Gaussian Elimination)

◮ If n > 6, Pseudoinverse J+ = (JT J)−1J → q = J+(q)−1ξ


ee 451 - velocity kinematicsisl.ee.boun.edu.tr/courses/ee451/lectures/ch4_kin.pdf · angular vs...

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