Review Cedric Fischer and Michael Mattmann Institute of Robotics - - PowerPoint PPT Presentation

Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich

Review Cedric Fischer and Michael Mattmann

Mechanical Design

§ Robot Components: § Actuators § Sensors § End-effectors § … § Precision, Accuracy, Resolution

Rigid Body Motion

§ Degree of Freedom (DoF) § DoF in d-dimensional space § ! : translational DoF §

"("$%) '

: rotational DoF § SO(3) : Special Orthogonal group § () = ($% ∈ ,-(3) § Each column of R is a unit vector. § The columns of R are mutually orthogonal. § !/0( = 1 § SE(3) : Special Euclidean group § ( 2 1 , ( ∈ ,- 3 , 2 ∈ ℝ6

4

Rule of Composition of Rotations R = Ry,φRz,θ R = Ry,φ[Ry,−φRz,θRy,φ] = Rz,θRy,φ

Ry,−φRz,θRy,φ

z0 x0 y0, y1 z0 x0 y0 z1 x1 z1, z2 x1 x2 y1 y2 y2 x2 z2

ϕ θ

z0 x0 y0 z0 x0 y0 z0 x0 y0 z1 x1 x2 y2 x2 y2 z2

ϕ θ

Ry,φ Rz,θ Ry,φ § Postmultiply for rotations about the current frame § Premultiply for rotations about the original/fixed frame

5

D-H Convention

θi : joint angle di : link offset ai : link length αi : link twist

• angle from xi-1 to xi measured in a plane normal to zi-1
• distance from oi-1 to intersection of xi and zi-1 measured along zi-1
• distance between zi-1 and zi measured along xi
• angle between zi-1 and zi measured in a plane normal to xi

§ Assume two features: § DH1: The Z axis is pointing in the direction of movement § DH2: The axis Xi is perpendicular to Zi-1 § DH3: The axis Xi intersects the axis Zi-1

θi : joint angle di : link offset ai : link length αi : link twist

• angle from xi-1 to xi measured in a plane normal to zi-1
• distance from oi-1 to intersection of xi and zi-1 measured along zi-1
• distance between zi-1 and zi measured along xi
• angle between zi-1 and zi measured in a plane normal to xi

Question 1 Exam 2016

Forward/Inverse Kinematics

§ Kinematics: To describe the motion of the manipulator without consideration of the forces and torques causing the motion : A Geometric Description. § Forward Kinematics § To determine the position and orientation of the end effector with the given values for the joint variables. § D-H convention § Screw Theory § Inverse Kinematics § To determine the joint variables with the given the end effector’s position and orientation.

Screw Theory

§ Every rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis. § Screw parameters:

§ Pitch h § Ratio of translational motion to rotational motion § Axis l § Axis of rotation, line through a point § Direction of translation § Magnitude M § Amount of displacement § Net rotation and/or translation

h l M

Screw Theory: Mathematical Description

9

Derivation: page 39 – A Mathematical Introduction to Robotic Manipulation 1994

Twist Coordinates (6x1) Twist (4x4)

with

General description Revolute joint Prismatic joint

ξ = −ω ×q ω # $ % % & ' ( ( ξ = v ! " # $ % & ˆ ξ = ˆ ω −ω ×q # $ % % & ' ( ( ˆ ξ = v ! " # $ % &

ℎ = 0 à 8 = −:×< + > ℎ:

?

= −:×< ℎ = ∞ à 8 = 8

Rodrigues’ Formula

§ All rotation matrices can be written as a matrix exponential

• f a skew-symmetric matrix!

10

Homogeneous Transformation (4x4)

e

ˆ ωθ = I + ˆ

ωsinθ + ˆ ω 2 1−cosθ

( )

e

ˆ ξθ =

e

ˆ ωθ

I −e

ˆ ωθ

( ) ω ×v

( )+ hθω

1 # $ % % & ' ( (

Revolute joint Prismatic joint

e

ˆ ξθ =

e

ˆ ωθ

I −e

ˆ ωθ

( ) ω ×v

( )

1 # $ % % & ' ( ( e

ˆ ξθ =

I θv 1 ! " # $ % &

Rodrigues’ Formula:

Example

Exam 2017 § Find screw parameter

ξ = −ω ×q ω # $ % % & ' ( (

ξ = v ! " # $ % &

§ Find twist

Joint 1 à Revolute Joint 2 à Prismatic

Try different manipulators …

z x y

l0 l1 l2

S

AB AC

z x y T

DE

ω1 =   1   ω2 =   1   ν3 =   1   q1 =     q2 =   l0   q3 =   l1 l0   e

ˆ ξ1θ1 =

    c1 −s1 s1 c1 1 1     e

ˆ ξ2θ2 =

    1 c2 −s2 l0s2 s2 c2 l0(1 − c2) 1     e

ˆ ξ3θ3 =

    1 1 θ3 1 1     gst(0) =     1 1 l1 + l2 1 l0 1     gst =     c1 −s1c2 s1s2 −s1c2(θ3 + l1 + l2) s1 c1c2 −c1s2 c1c2(θ3 + l1 + l2) s2 c2 l0 + s2(θ3 + l1 + l2) 1    

Rigid Body Velocity

q : a point attached to the rigid body

gab(t) = Rab(t) pab(t) 1

• the rigid body motion of the frame B attached to the body,

y, relative to a fixed or inertial frame A

FG FH § Spatial velocity : 8IJ = :KL

M ×<K + 8KL M

§ Body velocity : 8IN = :KL

L ×<L + 8KL L

Spatial and body velocity – General case

• Spatial velocity
• Body velocity

(in twist form) (in twist form) (in twist coordinates)

• Spatial velocity
• Body velocity

(in twist coordinates)

16

As previously ̇ PKL(0) is not particularly useful But ̇ PKLPKL

$% and PKL $% ̇

PKLhave some special significance

• Rotation and translation

More detailed explanation p.54

Spatial and body velocity – General case

(in twist coordinates)

Spatial velocity Body velocity

(in twist coordinates)

17

• Transformation from body to spatial velocity:

More detailed explanation p.54

) 6 6 ( )

ˆ ,

x g

R R p R Ad ! " # $ % & =

Adjoint transformation

• The adjoint transformation is invertible

Rigid Body Velocity

18

Spatial velocity § Point of the body § Expressed in spatial coordinates § Velocity of that point written in spatial coordinates Body velocity § Point of the body § Expressed in body coordinates § Velocity of that point with respect to the spatial frame written in body coordinates vabs is the velocity of a point (possibly

imaginary) attached to the body frame and passing through the origin of the spatial frame, written in spatial coordinates

wabs is the instantaneous angular velocity

• f the body as viewed in the spatial frame

vabb is the velocity of the origin of the

body frame (relative to the spatial frame) written in the body coordinates

wabb is the angular velocity of the body

frame, written in the body coordinates

Jacobian

Velocity § Relates the joint velocities to the body velocity Inverse Kinematics § Relates Cartesian space to joint space Static force control § Defined by the virtual work principle Singularity identification § Rank loss implied a sigularity

20

̇ Q = R$% 8? :? S = R)T

U

VL = RL ̇ Q

Question 2 Exam 2016

§ Each column of the spatial manipulator Jacobian: the twist coordinates with respect to the spatial frame, when the manipulator is in an arbitrary configuration.

z x y

θ1 θ2 l0 l1

S T

AB′ AC′ θ1

z x y

l2 + θ3

l3 + θ4

XE′ XY′ θ2

ω0

1 =

  1   q0

1 =

   

ω0

2 =

  c1 s1   q0

2 =

  l0   ν0

3 =

  −s1c2 c1c2 s2   ν0

4 =

  −s1s2 c1s2 −c2   § Pure translation motion § Pure rotation motion

§ compare with the result using Adjoint