lecture«robot dynamics»: intro todynamics - eth z · robot dynamics -dynamics 1 117.10.2017...
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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»: Intro to Dynamics
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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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Kinematics = description of motions Translations and rotations Various representations (Euler, quaternions, etc.) Instantaneous/Differential kinematics Jacobians and geometric Jacobians Inverse kinematics and control Floating base systems (unactuated base and contacts)
17.10.2017Robot Dynamics - Dynamics 1 3
Recapitulation of Kinematics
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Dynamics in Robotics
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Dynamics in Robotics
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Description of “cause of motion” Input Force/Torque acting on system Output Motion of the system
Principle of virtual work Newton’s law for particles Conservation of impulse and angular momentum
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 6
DynamicsOutline
, Tc c M q q b q q g q τ J F
Generalized coordinates Mass matrix
, Centrifugal and Coriolis forces
Gravity forces Generalized forces
External forcesContact Jacobian
c
c
qM q
b q q
g qτFJ
τq
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Principle of virtual work (D’Alembert’s Principle) Dynamic equilibrium imposes zero virtual work
Newton’s law for every particle in direction it can move
17.10.2017Robot Dynamics - Dynamics 1 7
Principle of Virtual Work
d external forces acting on element acceleration of element
mass of element virtual displacement of element
ext ii
dm ii
Fr
r
dF
rdm
mF a
r
S
m r F r a
d mdt
p
F v p
Impulse orlinear momentum
force
d mdt
N
Γ r v N
angular momentummoment
variational parameter
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Rigid body Kinematics
Applied to principle of virtual work
17.10.2017Robot Dynamics - Dynamics 1 8
Virtual Displacements of Single Rigid Bodies
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Use the following definitions
Conservation of impulse and angular momentum
17.10.2017Robot Dynamics - Dynamics 1 9
Impulse and angular momentum
External forces and moments
Change in impulse and angular momentum
Newton
Euler
A free body can move In all directions
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Cut all bodies free Introduction of constraining force Apply conservation and to individual bodies
System of equations 6n equation Eliminate all constrained forces (5n)
Pros and Cons+ Intuitively clear + Direct access to constraining forces− Becomes a huge combinatorial problem for large MBS
17.10.2017Robot Dynamics - Dynamics 1 10
1st Method for EoMNewton-Euler for single bodies
{I}
iOSr
ivia
ΨΩ
gF
1iF
2iF
im
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Find the equation of motion
17.10.2017Robot Dynamics - Dynamics 1 11
Free CutCart pendulum example
g
,p pm
,c cm {I}
l
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For multi-body systems Express the impulse/angular momentum in generalized coordinates
Virtual displacement in generalized coordinates
With this, the principle of virtual work transforms to
17.10.2017 15
Newton-Euler in Generalized Motion Directions
M q ,b q q g q
0 W= T q q
Robot Dynamics - Dynamics 1
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Equation of motion Directly get the dynamic properties of a multi-body system with n bodies
For actuated systems, include actuation force as external force for each body If actuators act in the direction of generalized coordinates, corresponds to stacked actuator
commands
17.10.2017Robot Dynamics - Dynamics 1 16
Projected Newton-Euler
, M q q b q q g q 0
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Find the equation of motion
17.10.2017Robot Dynamics - Dynamics 1 17
Projected Newton-EulerCart pendulum example
g
,p pm
,c cm {I}
l
||
Lagrangian
Lagrangian equation
Since
17.10.2017Robot Dynamics - Dynamics 1 19
3rd Method for EoMLagrange II
ddt
τ
q q q T T U
inertial forces gravity vector 12
T q Mq T
11 ,
2
T
T
n
qb
q
g g
M
M q
q q
Mq q q qMq
q
q
M q τ
with
Mqq
T
,
q
q qU UT T
kinetic energy
potential energy
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Kinetic energy in joint space
Kinetic energy for all bodies
From kinematics we know that
Hence we get
17.10.2017Robot Dynamics - Dynamics 1 20
Lagrange IIKinetic energy
12
T q Mq T
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Two sources for potential forces
Gravitational forces
Spring forces
17.10.2017Robot Dynamics - Dynamics 1 21
Lagrange IIPotential energy
00 0
0E jk d
r rF r rr r
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Find the equation of motion
17.10.2017Robot Dynamics - Dynamics 1 22
Lagrange IICart pendulum example
g
,p pm
,c cm {I}
l
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Given:
Generalized forces are calculated as:
Given:
Generalized forces are calculated
For actuator torques:
17.10.2017Robot Dynamics - Dynamics 1 24
External Forces
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Equation of motion without actuation
Add actuator for the pendulum Action on pendulum Reaction on cart
Add spring to the pendulum (world attachment point P, zero length 0, stiffness k) Action on pendulum
17.10.2017Robot Dynamics - Dynamics 1 25
External Forces Cart pendulum example
g
,p pm
,c cm {I}
x
la
,s sP x y
l 2 sinx sF k x l x
2 cosy sF k l y
2
2
0cos sinsincos 0
c p p p
pp p p
m m lm lmm gllm m l
b gM
q 0
0 1Rp Ja
p aT c aT 0 0Rc J ,
0T T TR i i Rc c Rp p
a
τ J T J T J T
xs
y
FF
F
2 sin2 cos
1 2 cos0 2 sin
ss
x ll
ll
r
rJq
2 cos sinxT
sx y
F
l F F
τ J F
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What is the external force coming from the motor
17.10.2017Robot Dynamics - Dynamics 1 26
External Forces Cart pendulum example
g
,p pm
,c cm {I}
x
l
l
M
||
What is the external force coming from the motor
Action on cart
17.10.2017Robot Dynamics - Dynamics 1 27
External Forces Cart pendulum example
g
,p pm
,c cm {I}
x
l
l
actF
actF
c actF F 1 0Pc J 0actT
c
FF
τ J
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