Robotics 2015 05 Dynamics - polito.it · Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW -2014/2015 3....

ROBOTICS

01PEEQW

Basilio Bona

DAUIN – Politecnico di Torino

Dynamics

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

Basilio Bona - DAUIN - PoliTo 3ROBOTICS 01PEEQW - 2014/2015

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


Newton-Euler approach – 1




ib

ic

Lagrange equations – 1


Kinetic Energy – 1




First form for the Kinetic Energy



Second form for the Kinetic Energy

Potential Energy – 1


Generalized forces – 1


Final equations – 1


Physical interpretation – 1


21 43 5

Physical interpretation – 2


1

2

3

4

5

Properties of the Lagrange Equations – 1


Dynamic calibration – 1


Dynamic calibration – 2

� Collecting all data one obtains


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

State equations – 1


State equations – 2


Direct and inverse dynamics


Numerical recursive algorithms – 1


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


NonlinearitiesProducts, squares, trigonometric functions

herehere here

Robotics 2015 05 Dynamics - polito.it · Basilio Bona -DAUIN -PoliTo ROBOTICS 01PEEQW -2014/2015 3....

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