Lecture«Robot Dynamics»: Dynamics andControl · We learned how to get the equation of motion in...

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151-0851-00 Vlecture: CAB G11 Tuesday 10:15 – 12:00, every weekexercise: HG E1.2 Wednesday 8:15 – 10:00, according to schedule (about every 2nd week)

Marco Hutter, Roland Siegwart, and Thomas Stastny

24.10.2017Robot Dynamics - Dynamics 2 1

Lecture «Robot Dynamics»: Dynamics and Control

|| 31.10.2016Robot Dynamics - Dynamics 3 2

19.09.2017 Intro and Outline Course Introduction; Recapitulation Position, Linear Velocity26.09.2017 Kinematics 1 Rotation and Angular Velocity; Rigid Body Formulation, Transformation 26.09.2017 Exercise 1a Kinematics Modeling the ABB

arm

03.10.2017 Kinematics 2 Kinematics of Systems of Bodies; Jacobians 03.10.2017 Exercise 1b Differential Kinematics of the ABB arm

10.10.2017 Kinematics 3 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation Error; Multi-task Control

10.10.2017 Exercise 1c Kinematic Control of the ABB Arm

17.10.2017 Dynamics L1 Multi-body Dynamics 17.10.2017 Exercise 2a Dynamic Modeling of the ABB Arm

24.10.2017 Dynamics L2 Floating Base Dynamics 24.10.201731.10.2017 Dynamics L3 Dynamic Model Based Control Methods 31.10.2017 Exercise 2b Dynamic Control Methods

Applied to the ABB arm

07.11.2017 Legged Robot Dynamic Modeling of Legged Robots & Control 07.11.2017 Exercise 3 Legged robot14.11.2017 Case Studies 1 Legged Robotics Case Study 14.11.201721.11.2017 Rotorcraft Dynamic Modeling of Rotorcraft & Control 21.11.2017 Exercise 4 Modeling and Control of

Multicopter28.11.2017 Case Studies 2 Rotor Craft Case Study 28.11.201705.12.2017 Fixed-wing Dynamic Modeling of Fixed-wing & Control 05.12.2017 Exercise 5 Fixed-wing Control and

Simulation12.12.2017 Case Studies 3 Fixed-wing Case Study (Solar-powered UAVs - AtlantikSolar, Vertical

Take-off and Landing UAVs – Wingtra)19.12.2017 Summery and Outlook Summery; Wrap-up; Exam

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We learned how to get the equation of motion in joint space Newton-Euler Projected Newton-Euler Lagrange II

Introduction to floating base systems

Today: How can we use this information in order to control the robot


Recapitulation

, T Tc c M q q b q q g Sq τ J F

Generalized coordinates Mass matrix

, Centrifugal and Coriolis forces

Gravity forces Generalized

Selection matrix/JaExternal forces

fo

Contact Ja

cobiac

nr es

c

c

qM q

b q q

g qτ

FJ

S

cobian


Position vs. Torque Controlled Robot Arms


Setup of a Robot Arm

Actuator(Motor + Gear)

Robot Dynamics

ActuatorControl

( , )U I τ , ,q q q

System Control

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Position feedback loop on joint level Classical, position controlled robots don’t care about dynamics High-gain PID guarantees good joint level tracking Disturbances (load, etc) are compensated by PID => interaction force can only be controlled with compliant surface


Classical Position Control of a Robot Arm


Robot Dynamics

ActuatorControl

( , )U I τ

Position Sensor

, ,q q q ,q q

Robot

High gainPID

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Robot Dynamics

ActuatorControl

( , )U I τ

Position Sensor

Torque Sensor

, ,q q q ,q q

τ

Integrate force-feedback Active regulation of system dynamics Model-based load compensation Interaction force control


Joint Torque Control of a Robot Arm

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Modern robots have force sensors Dynamic control Interaction control Safety for collaboration


Setup of Modern Robot Arms


FRANKA – an example of a force controllable robot arm

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Special force controllable actuators Dynamic motion Safe interaction


ANYpulatorAn example for a robot that can interact

spring linkmotor gear

position force

Series Elastic Actuator

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Torque as function of position and velocity error

Closed loop behavior

Static offset due to gravity

Impedance control and gravity compensation


Joint Impedance Control

Estimated gravity term

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Compensate for system dynamics

In case of no modeling errors, the desired dynamics can be perfectly prescribed

PD-control law

Every joint behaves like a decoupled mass-spring-damper with unitary mass


Inverse Dynamics Control

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Motion in joint space is often hard to describe => use task space

A single task can be written as

In complex machines, we want to fulfill multiple tasks (As introduced already for velocity control)

Same priority, multi-task inversion

Hierarchical


Inverse Dynamics Control with Multiple Tasks

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Joint-space dynamics

Torque to force mapping

Kinematic relation

Substitute acceleration


Task Space Dynamics

{I}

{E}

eFew

End-effector dynamics

Inertia-ellipsoid

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Determine a desired end-effector acceleration

Determine the corresponding joint torque


End-effector Motion Control

{I}

{E}

eFew

e twTrajectory control R R RE χ χNote: a rotational error can be related

to differenced in representation by


Robots in InteractionThere is a long history in robots controlling motion and interaction

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Extend end-effector dynamics in contact with contact force

Introduce selection matrices to separate motion force directions


Operational Space ControlGeneralized framework to control motion and force

{I}

eFew

cFc F

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Given: Find , s.t. the end-effector accelerates with exerts the contact force


Operational Space Control2-link example

, 0T

e des yar

, 0 Tcontact des cFF

1

2 P

O

, 0c

contact des

F

F

,

0e des

ya

r

x

y

l

l

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Selection matrix in local frame

Rotation between contact force and world frame


How to Find a Selection Matrix

1: it can move0: it can apply a force

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Selection matrix in local frame

Rotation between contact force and world frame


How to Find a Selection Matrix

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Assume friction less contact surface


Sliding a Prismatic Object Along a Surface

1 0 00 1 00 0 0

Mp

Σ

0 0 00 0 00 0 1

Fp

Σ

0 0 00 0 00 0 1

Mr

Σ

1 0 00 1 00 0 0

Fr

Σ

MpΣ MrΣ

FpΣFrΣ

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Find the selection matrix (in local frame)


Inserting a Cylindrical Peg in a Hole

0 0 00 0 00 0 1

Mp

Σ0 0 00 0 00 0 1

Mr

Σ

1 0 00 1 00 0 0

Fp

Σ1 0 00 1 00 0 0

Fr

Σ

MpΣ MrΣ

FpΣFrΣ


Inverse Dynamics of Floating Base Systems

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Equation of motion (1) Cannot directly be used for control due to the occurrence of contact forces

Contact constraint

Contact force Back-substitute in (1),

replace and use support null-space projection

Support consistent dynamics

Inverse-dynamics

Multiple solutions


Recapitulation: Support Consistent Dynamics

s s J q J q


Some Examples of Using Internal Forces

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Floating base system with 12 actuated joint and 6 base coordinates (18DoF)


Recapitulation: Quadrupedal Robot with Point Feet

Total constraints

Internal constraints

Uncontrollable DoFs

0

0

6

3

0

3

6

1

1

9

3

0

12

6

0


Internal Forces extreme example


Least Square Optimizationsome notes on quadratic optimization

Ax b 0 2min x

Ax bx = A b 2min x

1 1 2 2 A x b A x 1 2

11 2,

2 2

min

x x

xA A b

x 1

1 22

x= A A b

x1

2 2

min

xx

1 1 A x b 0

2 2 A x b 01 1

2 2

A bx =

A b

1 1

2 2 2

min

x

A bx

A b 2min x

2 2 2min x

A x b

Equal priority

Hierarchy1 1 2

min x

A x b 1 1 1 0 x A b A xN

2 2 2 1 1 1 0 2 A x b A A b A x b 0N

0 2 1 2 2 1 1 x A A b A A bN 1 1 1. .s t c A x b


Least Square OptimizationApplication to Inverse Dynamics

Mq b g τ*

e e e J q J q w

*,2

mine e

q τ

M I b gqJ 0 Jq wτ

qM I b g 0

τ

*e e

qJ 0 Jq w

τ

*

,2

min e e

q τ

qJ 0 Jq w

τ

. .s t

qM I b g 0

τ

Single task

Priority

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We search for a solution that fulfills the equation of motion

Motion tasks:

Force tasks:

Torque min:


Operational Space Control as Quadratic ProgramA general problem

2min τ

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QPs need different priority!! Exploit Null-space of tasks with higher priority Every step = quadratic problem with constraints Use iterative null-space projection (formula in script)


Solving a Set of QPs


Behavior as Multiple Tasks

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No dynamic effects

Add virtual external forces to pull/support the robot Static equilibrium of forces and moments From principle of virtual work it follows directly that


Quasi-static: Virtual Model Control Pratt 2001

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Application of this technique for locomotion control of legged robots


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