2015 L14 Summery - ETH Z

||Autonomous Systems Lab

151-0851-00 V

Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny

15.12.2015Robot Dynamics - Summary 1

Robot DynamicsSummary


Lecture 1 15.09.2015 Intro and OutlineLecture 2 22.09.2015 Kinematics 1 Lecture 3 29.09.2015 Kinematics 2 Lecture 4 06.10.2015 Multibody Dynamics Lecture 5 13.10.2015 Legged robots Lecture 6 20.10.2015 Introduction to Rotorcraft Lecture 7 27.10.2015 Dynamic Modeling of Rotorcraft I Lecture 8 03.11.2015 Dynamic Modeling of Rotorcraft II Lecture 9 10.11.2015 Control of Rotorcraft, Robots Case Study Lecture 10 17.11.2015 Fixed Wing General Introduction, Basics of Aerodynamics Lecture 11 24.11.2015 Stability and Derivation of a Dynamic Model Lecture 12 01.12.2015 Control and Solar Airplanes Lecture 13 08.12.2015 Some Aspects of Flight Dynamics and Flight Control,

Challenges of UAV AutoFlight System Design Lecture 14 15.12.2015 Backup, Summary


Lecture Program


… describe the relationship between forces/torques and motion (in joint space or workspace variables)

Two possible goals:1. Given joint torques ( ) or end-effector forces ( ), what

motions (e.g. or ), would result? (this is forward dynamics)

2. Given motion variables (e.g. or ), what joint torques ( ) or end-effector forces ( ) would have been the cause? (this is inverse dynamics)


Equations of Motion / Robot Dynamics

f, , , ,x x x

, , , ,x x x f

||Autonomous Systems Lab 15.12.2015Robot Dynamics - Summary 4

Multi-body Dynamics - General Formulation

, T Tex ex actq q b q q g q F M J S

mass matrix / inertial forces

coriolis / centrifugal

gravity

external forces and Jacobian

selection matrix and actuation torque


The main elements (general for all mechanical systems) Generalized coordinates Coordinate transformation Kinematics and Jacobian Multi-body dynamics

System dependent Actuator External forces (interaction, aerodynamics, …) Simplifications


Equation of Motion / Robot Dynamics


151-0851-00 V

Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas StastnyRobotic Systems Lab / Autonomous Systems Lab


Robot DynamicsSummary Kinematics, Dynamics and Legged robots

MarcoHutter

MichaelBloesch


Generalized coordinates, rotations, and transformations Different representations and descriptions of joint and task space Mapping from joint space to task space Exam: provide us with the homogeneous transformation matrix between

two points of a robot arm Differential kinematics, Jacobians Relation between joint and task space motion Exam: provide us with the end-effector Jacobian for a given robot arm

Inverse (differential) kinematics Given a task space motion/configuration, what is the joint

motion/configuration Singularity and redundancy Exam: show us how to determine the generalized coordinates for a given

end-effector position


Kinematics | Summary and possible exam questions


Equation of Motion (EoM) Impulse and angular momentum (for single bodies and multi-body-

systems) Concept of Newton-Euler (NE), projected NE, and Lagrange II Exam: given a robot system, outline how to determine the EoM Exam: given an external force, how does it influence the EoM

Inverse dynamics Inverse joint and task space dynamics as model-based control concept Ex: given the EoM and a desired contact force, determine the joint torque

Floating base systems (Legged robots) Non-actuated based and contact constraints Null-space motion and multi-task execution Ex: given a robot in contact with the environment, determine how it can

move


Dynamics | Summary and possible exam questions


151-0851-00 V

Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas StastnyAutonomous Systems Lab


Robot DynamicsSummary Rotorcrafts

KonradRudin

RolandSiegwart


Different configurations

Understand mechanical parts Rotor-head Control/reduce cyclic stress

Tail rotor Compensate rotor reaction torque

Swashplate Control collective/cyclic pitch angle

Robot Dynamics - Summary

Rotorcraft Mechanical Design Overview

Helicopter Gyrodyne Autogyro CoaxialTandem

0 1 1cos sinR c s

15.12.2015 10


Rotor TPP control Use blade flapping to control TPP Blade damped oscillator model with cyclic θR

Constant and cyclic output Constant blade coning Cyclic blade flapping to tilt TPP Phase delay between blade flapping and pitch angle

Example: Initiate forward motion Tilt TPP forward Minimum βFl at ξ=0° Minimum θR earlier aound ξ=300°


Rotorcraft Mechanical Design Overview

Lift

flflRRL

Gyroscopic

RRfl ),,,,,(),( vω

0 1 1( ) cos( ) sin( )Fl c s

βFlθR


Aerodynamic distribution along rotor/propeller blade Distribution of relative airflow

Distribution of lift force and induced velocity Lift proportional to speed squared Induced velocity increases with lift Reduces angle of attack

Robot Dynamics - Summary

Rotorcraft Aerodynamic Overview

ωR ωR

V

Hover Forward

Blade radius r

dL dvi

15.12.2015 12


Blade tip vortex General: high pressure above blade,

low pressure below blade Boundary: equalized pressure Generation of a vortex

Principle of autorotation Absorb energy from airflow to turn blades Vertical flow from bottom to top Aerodynamic force can have forward component


Rotorcraft Aerodynamic Overview

Driven region:Driving region:Stall region:


Conservation of fluid mass

Conservation of fluid momentum

Conservation of energy


Momentum Theory | applied to propeller

0 (1)V ndA

(2)p ndA VV ndA F

21 (3)2

dEV V ndA Pdt

0 0, , V A p

1 1, , RV u A p

2 2, , RV u A p

3 3 0, , V u A p


BEMT | Blade Elemental and Momentum Theory

Forces at each blade element

dT dL

dDdQ/r

V

VT = ωRrVP

uind θR αϕ

ωR

2

2 LdL C cdrV 2

2 LdD C cdrV


Modeling of Rotorcraft

ωR,cmd ωR F, M v, ω. .

, T Tex ex actb g F M J S

F1

1

F4

4

F3

3 F2

2


Linearize attitude dynamic at hover Equilibrium point:


Control of a Quadrotor | Linearization of Attitude Dynamics

This model has no coupling We use 3 individual controller

2

3

4

( )

( )xx yy zz

yy zz xx

zz

r

B

I p qr I I U

I q pr I I U

I r U

pJ q

r

21 U

I xx

31 U

I yy

41 UIzz

2 3 4 10 ; 0 ; p q r U U U U mg


151-0851-00 V



Robot DynamicsSummary Fixed Wing UAS

ThomasStastny


Aerodynamics


c.g.

2-D Analysis

3-D Effects• Induced drag• Fuselage drag• Interference drag• …

Aircraft Forces/Moments


Given: drag coeff. as a funcion of lift coeff.: ,

Required power:

Specific Excess Power: ⁄ ,


Power Required and Available for Level Flight

Drequired ACVVDP 3

21

Pavailable

Prequired

Pexcess

Vstall Vmax Vne

Pow

er

True Airspeed

Vemax Vrmax

min

L

D

CCmg

LDmgD

VP max

D

L

CC

Best glide ratio

max/

PEVTVs

min/ P

max2

3

D

L

CC

L

D

LL

D

CCmg

CAmg

CCmgVVDP

2

Minimum sink in gliding mode

Max. Range*(vrmax):

Max. Endurance*(vemax):

* Assuming constant propulsive efficiency η


c Assume…

L increases with

Vmin increases with

Demand for coordinated turn:

Turning in Level Stationary Flight00

V

constc

0Y

R

22

mRR

mV

D

≈FT

L

G

ccos1

ccos1


These forces are balanced!

T

cd V

g tan

Using the velocity and radius of the turn, a yawing rate can be derived which, as seen from the free-body diagrams, can subsequently achieved using a commanded roll angle:


Stability Example: LonditudinalDynamic stabilityStatic stability

Disturbance Aerodyn. reaction torque

Disturbance No reaction torque

Stable

Neutral

Disturbance Aerodyn.reactiontorqueUnstable

Stable

Neutral

Unstable

Trea

ted

with

aero

dyna

mic

deriv

ativ

a

Mod

elin

g of

the

dyna

mic

s re

quire

d



Modes of the linearized plant

Short Period Mode: oscillation of angle of attack

Phugoid mode: Exchange between kinetic and potential energy

Spiral Divergence Dutch Roll Mode:

combined yaw-roll oscillation

Grafics adapted from:http://history.nasa.gov/SP-367/chapt9.htm andhttp://www.fzt.haw-hamburg.de/pers/Scholz/Flugerprobung.html

15.12.2015 23Robot Dynamics - Summary

Δu, Δw; Δq;Δ

thr

elev

Longitudinal

Plant

Δv; Δp, Δr;Δ Δ

rudd

ail

Lateral

Plant

• Phugoid: Slow (long period)• Short Period: Fast (highly damped)

• Roll Subsidence: Damping of roll motion. Real Mode. Characteristics depend on aircraft configuration (e.g. wing shape).

• Dutch Roll: Complex mode. Important for directional stability.

• Spiral: Real and often (slightly) unstable, but easily controlled (long time-to-double amplitude)


AttitudeController

PI1: PI with anti-reset wind-upPD2: Gain scaled with 1/VT

2

Body Rate Controller

rd

qd

pd PD2

PD2

PD2

Control and Guidance

AirplaneDynamics

rudd

elev

ail

x

PI1

PI1d

d

Constrain tocoordinated turn:

V

gd

tan

d

d

d

Jr

thr

Bandwidths of inner Loops must be sufficiently larger!

TrajectoryGeneration &

Guidance Logic


HIGH LEVEL CONTROL

LOW LEVEL CONTROL

Control = low level part• Stabilize attitude and

speed

Guidance = high level part• Follow pathes or trajectory• Effect: Reject constant low frequency

perturbation (constant wind)

A popular concept: cascaded control loops:Guidance


Cascaded PID loops Low CPU cost, simple design Only effective within small deviations from the design point

Optimal Control Gains typically must be computed offline Techniques like LQR can achieve better performance as a MIMO controller than

PID, but depends on the selected weighting Robust Control Target and reject certain frequencies within signals More complex to design Without proper loop shaping, often must trade performance for wider reaching

stability Many more! The chosen control techniques determined according to: Computational Power Type of flight (aerobatics - level flight)

Control Concepts



151-0851-00 V



Robot DynamicsSummary Solar Airplane

PhilippOettershagen

Sebastian Verling


Solar-powered UAVs: Performance Metrics

If perpetual flight is possible, we define:

In the conceptual design phase, we optimize both Texc and Tcm togenerate sufficient safety margins for perpetual flight!

Ebat Battery energy [J]SoC Battery state of charge [%]Pout

nom Nominal required output power [W]


Solar-powered UAVs: Basic System Modeling

To derive the level flight power (constant altitude flight), we combine

v Airspeed [m/s]Awing Wing area [m²]CD, CL Drag / Lift coefficients [-]ρ Local air density [kg/m³]

with

and minimize the resulting expression w.r.t. the airspeed to yield

CD and CL are functions of the airspeed v! They areretrieved from airplane and airfoil analysis tools such asXFOIL or XFLR5.

Example (see image): AtlantikSolar UAV, MH139 airfoil, mtot=6.9kg, Awing=1.7m², vopt=7.6m/s, Plevel=21W.


Solar-powered UAVs: Basic System Modeling

Forward-integration of state equations:

Power modeling

I Solar irradiance [W/m²]Asm Solar module area [m²]ηsm, ηmppt Solar module and maximum power point

tracker efficiency [-]Pav, Ppld Avionics and payload power [W]ηprop Propulsion system efficiencymtot Total airplane mass

I = I (day,t,lat,h)


Saturday 13.02.2016, 9:00 – 11:00

4 A4-pages personal summary

Combination of Calculations, similar to exercises, but simpler and solvable without

computer Comprehension questions

More information about potential questions and example exam will be sent to you in the next days.


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Documents

2015 L14 Summery - ETH Z