Lecture«Robot Dynamics»: Floating-base Systems€¦ · Newton-Euler: Free cut and conservation of impulse & angular momentum for each body Projected Newton-Euler (generalized coordinates)

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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

17.10.2017Robot Dynamics - Dynamics 1 1

Lecture «Robot Dynamics»: Floating-base Systems

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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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Description of “cause of motion” Input Force/Torque acting on system Output Motion of the system

3 methods to get the EoM Newton-Euler: Free cut and conservation of impulse & angular momentum for each body Projected Newton-Euler (generalized coordinates) Lagrange II (energy)

Introduction to dynamics of floating base systems External forces


Recapitulation of Introduction to Dynamics

, 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/JacobianExternal forcesC

forces

ontact Jac

c

qM q

b q q

g qτSFJ

cobian

τq

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Generalized coordinates

Generalized velocities and accelerations? Time derivatives depend on parameterization

Often

Linear mapping 10.10.2017Robot Dynamics - Kinematic Control 5

Floating Base SystemsKinematics

with

,q q Very often, people write but they mean q u

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Position of an arbitrary point on the robot

Velocity of this point

10.10.2017Robot Dynamics - Kinematic Control 6

Floating Base SystemsDifferential kinematics

BQ jr qB bC qIB IB br qI

QJ qI

TT

T

T

C C

C

ω

C CC

ω

CC BI BI

IB

I IB IB

I

B

IB I BIBB IB

with

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A contact point is not allowed to move:

Constraint as a function of generalized coordinates:

Stack of constraints


Contact Constraints

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Remember:

The system can be moved without violating the contact constraints!

!! However, the base is unactuated !! Which ones can be ACTIVELY controlled?


Last time: Null-space motion

c c 0 r J q

c c c 0 r J q J q

0 0c c c q J 0 N q N q

0c c c q J J q N q

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Contact constraints

Point on wheel

Jacobian

Contact constraints

=> Rolling condition10.10.2017Robot Dynamics - Kinematic Control 9

Contact ConstraintWheeled vehicle simple example

x

qUn-actuated base

Actuated joints

sincos0

IP

x rr r

rI

0x r

10 00 0

IP P

rx

r J q 0

I I

1 cos0 sin0 0

P

rr

JI

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Contact Jacobian tells us, how a system can move. Separate stacked Jacobian

Base is fully controllable if

Nr of kinematic constraints for joint actuators:

Generalized coordinates DON’T correspond to the degrees of freedom Contact constraints!

Minimal coordinates (= correspond to degrees of freedom) Require to switch the set of coordinates depending on contact state (=> never used)


Properties of Contact Jacobian

relation between base motion and constraints

-

c b jn n n

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Analyse the kinematic constraints of this example 1) P cannot move at all

2) P can only move horizontally

3) P can only move vertically


Stupid, simple exampleCart pendulum

g{I}

x

la

l

b

j

q xq

q

P

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


Quadrupedal Robot with Point Feet

Total constraints

Internal constraints

Uncontrollable DoFs


Dynamics of Floating Base Systems

EoM from last time

Not all joint are actuated

Selection matrix of actuated joints

Contact force acting on system

bq

jq

b

j

qq

qUn-actuated base

Actuated joints

• Quaternions• Euler angles• …

j Mq b g τ

T Mq b g τS

6n n n S 0 I j q Sq

sF

, exerted by robotTs s

T JM b Fq g τS

Manipulator: interaction forces at end-effectorLegged robot: ground contact forcesUAV: lift force

, acting on systems sT T M S J Fq b g τ

Note: for simplicity we don’t use here u but only time derivatives of q

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External forces from force elements or actuator

Aerodynamics

Contact Simple solution: soft contact model

no negative force


External Forces Some notes

12s v LF c Ac


Soft ContactPhysical accuracy vs. numerical stability?

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Stiff equation of motion Small time steps Can lead to instability

Contact behavior strongly depends on the robot parameters/configuration Contact parameters are not selected as physical parameters but as numerical Trade-off stability accuracy


Soft ContactPhysical accuracy vs. numerical stability?


ANYmal in Simulation


ANYmal in Simulation

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External forces from constraints Equation of motion (1)

Contact constraint (2)

Substitute in (2) from (1) (3)

Solve (3) for contact force

Back-substitute in (1),replace and use support null-space projection

Support consistent dynamics 17.10.2017Robot Dynamics - Dynamics 1 19

Hard Contact

T Tc c Mq b g J F S τ

c c c r J q J q 0 c c r J q 0

1 T Tc c c c c

r J M S τ b g J F J q 0

11 1T Tc c c c c

F J M J J M S τ b g J q

s s J q J q 11 1T T

c c c c c

N I M J J M J J

T T T Tc c c N Mq N b g N S τ

c c J N 0

This is only a projector,… we can use other ones


Some more insights into the EoM

,

,

Tbb bj b b b c b

cTjb jj j j j c j

M M u b g J 0F

M M u b g J τ

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Impulse transfer at contact Integration over a single point in time

Post impact condition

Impulsive

End-effector inertia

Change in generalized velocity

Post-impact velocity

Energy loss


Contact Dynamics

Documents

Lecture«Robot Dynamics»: Floating-base Systems€¦ · Newton-Euler: Free cut and conservation of impulse & angular momentum for each body Projected Newton-Euler (generalized coordinates)