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ROBOTICS
01PEEQW
Basilio Bona
DAUIN – Politecnico di Torino
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Dynamics – 1
� Dynamics studies the relations between the task space
forces/torques and the joint forces/torques in non-static
equilibrium, i.e., when the robot moves
� The dynamic model equation can be obtained applying two main
approaches
� Lagrange equations based on energy functions
� Newton-Euler equations based on the equilibrium of the vector forces
� The first approach is conceptually simpler and will be adopted here
� The second approach is more efficient for implementation of
recursive computer algorithms; only a brief review of this approach
will be presented here
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Dynamics – 2
� The dynamic equations of the robot can be obtained adopting the
Lagrange approach
� The derived state-space differential equations represent the robot
dynamical model
� Why state equations are necessary?
� Used for control design
� Used for robot simulation
� Used to implement model identification or parameter estimation
algorithms
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Newton-Euler approach – 1
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Newton-Euler approach – 2
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Newton-Euler approach – 3
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Newton-Euler approach – 4
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ib
ic
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Newton-Euler approach – 5
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Newton-Euler approach – 6
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Newton-Euler approach – 7
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Lagrange equations – 1
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Lagrange equations – 2
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Lagrange equations – 3
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Lagrange equations – 4
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Kinetic Energy – 1
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Kinetic Energy – 2
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Kinetic Energy – 3
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First form for the Kinetic Energy
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Kinetic Energy – 1
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Second form for the Kinetic Energy
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Potential Energy – 1
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Potential Energy – 2
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Potential Energy – 3
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Generalized forces – 1
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Generalized forces – 2
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Generalized forces – 3
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Final equations – 1
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Final equations – 2
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Final equations – 3
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Physical interpretation – 1
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21 43 5
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Physical interpretation – 2
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1
2
3
4
5
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Properties of the Lagrange Equations – 1
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Properties of the Lagrange Equations – 2
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Properties of the Lagrange Equations – 3
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Dynamic calibration – 1
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Dynamic calibration – 2
� Collecting all data one obtains
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1 1( ) ( )
( ) ( )
c
c
c N N
t t
t t
= = =
τ Φ
τ θ Φθ
τ Φ
⋮ ⋮
� The linear least square solution is then computed, as
follows
( )1
ˆc
−
=θ Φ Φ Φ τT T
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State equations – 1
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State equations – 2
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Direct and inverse dynamics
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Numerical recursive algorithms – 1
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Numerical recursive algorithms – 2
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Numerical recursive algorithms – 3
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Numerical recursive algorithms – 4
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Numerical recursive algorithms – 5
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Numerical recursive algorithms – 6
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Conclusions
� Dynamics equations are essential for modeling and control purposes
� Modeling is easier to understand adopting the Lagrange energy
function
� Computer program are more efficient if they implement recursive
Newton-Euler approach
� Nonlinear state equations have this form
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NonlinearitiesProducts, squares, trigonometric functions
herehere here