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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
17.10.2017Robot Dynamics - Dynamics 1 4
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
10.10.2017Robot Dynamics - Kinematic Control 7
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?
17.10.2017Robot Dynamics - Dynamics 1 8
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)
10.10.2017Robot Dynamics - Kinematic Control 10
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
17.10.2017Robot Dynamics - Dynamics 1 11
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)
10.10.2017Robot Dynamics - Kinematic Control 12
Quadrupedal Robot with Point Feet
Total constraints
Internal constraints
Uncontrollable DoFs
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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
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
17.10.2017Robot Dynamics - Dynamics 1 14
External Forces Some notes
12s v LF c Ac
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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
17.10.2017Robot Dynamics - Dynamics 1 16
Soft ContactPhysical accuracy vs. numerical stability?
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ANYmal in Simulation
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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
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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
17.10.2017Robot Dynamics - Dynamics 1 22
Contact Dynamics