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Progettazione Funzionale di SistemiMeccanici e Meccatronici
Motion Laws
prof. Paolo Righettini
Dipartimento di Progettazione e Tecnologie
Mechatronics and Mechanical Dynamics Laboratories
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 1/34
Payload position
Linear or rotary movements
position pay-load y as a function as time t
y = y(t)
external forces
dynamic actions due to pay-load acceleration
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 2/34
Periodic motion
the problem is how to describe the pay-load motionrequest
rotary motion may hasn’t limits, while linear motion has
a typical representation of a linear movement request maybe
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 3/34
Periodic motion
T total motion time or period
h motion rise (stroke)
four main parts
I rise
II dwell
III return
IV dwell
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 4/34
Periodic motion
rise part, for a total pay-load displacement h
dwell part, holding the position y = h
return part, return of the pay-load to the initial positiony = 0
dwell part, holding the position y = 0
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 5/34
Rise part
analysing the rise motion
the pay-load reaches the final position y = h at the time td
at the starting point A we have null velocity → y(0) = 0
at the ending point B we have null velocity → y(td) = 0
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 6/34
Rise part
four constraints, two for t = 0, two for t = td
t = 0 → y(0) = 0 , y(0) = 0
t = td → y(td) = h , y(td) = 0
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 7/34
Rise part
null velocities at the starting and ending motion points (A,B) are required to avoid shock
the movement profile y(t) must be continuous for thederivatives of order 0 and 1 (c1 function)
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 8/34
Motion Law - displacement
the conditions at the motion starting and ending points,lead to an ∞ of solutions of the displacement problem
which is the best way to choose a motion profile that respects t heboundary conditions?
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 9/34
Motion Law
kinematics: any motion displacement request leads toacceleration request
the acceleration of the mechanical part is the cause ofarising of inertial actions, moreover acceleration in themotion profile shows a positive and a negative part
inertial forces may cause vibration or action overload ofthe mechanical parts
therefore we study the motion law at the accelerationlevel, to control the acceleration amplitude and so theinertial forces
how to configure an acceleration profile y(t) that leads to therequired motion (stroke h in the time td)?
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 10/34
Adimensional approach
given a movement as a function y(t), stroke h and drivetime td
the acceleration profile may be expressed as
y(t) =h
t2d
f(t/td)
dimensional term h
t2d
depends on the two main motionparameters
adimensional function f(x), x = t/td, 0 ≤ x ≤ 1 represents theacceleration shape profile
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 11/34
Velocity profile
starting from the expression
y(t) = h/t2df(t/td)
of the acceleration profile, the velocity profile results
y(t) =
∫y(t)dt
remembering that x = t/td and therefore dt = dxtd we get
y(xtd) =
∫h
t2d
f(x)dxtd =h
td
∫f(x)dx =
h
tdF (x)
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 12/34
Velocity profile
y(t) =h
tdF (t/td)
dimensional term h
tdrepresents the mean velocity of the
motion required
adimensional function F (x) is the primitive of the functionf(x), with x = t/td, 0 ≤ x ≤ 1
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 13/34
Position profile
starting from the expression
y(t) = h/tdF (t/td)
of the velocity profile, the position profile results
y(t) =
∫y(t)dt
remembering that x = t/td and therefore dt = dxtd we get
y(xtd) =
∫h
tdF (x)dxtd = h
∫F (x)dx = hG(x)
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 14/34
Position profile
y(t) = hG(t/td)
dimensional term h represents the stroke of the requiredmotion
adimensional function G(x) is the primitive of the functionF (x), with x = t/td, 0 ≤ x ≤ 1
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 15/34
Motion law description
positiony(t) = hG(t/td)
velocity
y(t) =h
tdF (t/td)
acceleration
y(t) =h
t2d
f(t/td)
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 16/34
Characteristics of f (x)
the function f(x) has some useful characteristics
∫ td
0
y(t)dt = [y(t)]td0= y(td)− y(0) = 0
the velocity at the starting and ending points is null toavoid shock
taking into account the variable x, it results
∫ td
0
y(t)dt =
∫ 1
0
h
t2d
f(x)dx = 0 ⇒
∫ 1
0
f(x)dx = 0
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 17/34
Characteristics of f (x)
therefore the acceleration profile y(t) and theadimensional function f(x) have a null mean value
∫ 1
0
f(x)dx = 0
the two area A and B must have the same surface
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 18/34
Characteristics of f (x)
it is necessary to calculate the scale factor of the functionf(x)
we must relate the amplitude of the function f(x) to therise h ∫ ∫
td
0
y(t)dt = h
remembering that
y(t) = hG(t/td) and y(td) = h
it resultsy(td) = hG(td/td) = h ⇒ G(1) = 1
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 19/34
Characteristics of f (x)
the twice integration of the function f(x) from 0 to 1 leadsto 1
the function f(x) is an adimensional acceleration profile,defined in the interval 0− 1, that leads to unitary stroke
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 20/34
Characteristics of f (x)
the scale factor of the function f(x) may be more easilydetermined by means of the momentum of the functionf(x)
∫ 1
0
f(x)xdx
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 21/34
Characteristics of f (x)
∫ 1
0
f(x)xdx = [F (x)x]10−
∫ 1
0
F (x)dx
but F (0) = 0 and F (1) = 0 and∫ 1
0
F (x)dx = [G(x)]10= G(1)−G(0) = 1
unless the minus sign the integration results∫ 1
0
f(x)xdx = 1
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 22/34
Characteristics of f (x)
the resultants of the areas, positive and negative, havethe same modulus (f(x) has a null mean value)
R = R1 = R2
the momentum of f(x) may be calculated as∫ 1
0
f(x)xdx = Rδ = 1 ⇒ Rδ = 1
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 23/34
Characteristics of f (x)
the momentum of f(x) may be calculated as∫ 1
0
f(x)xdx = Rδ = 1 ⇒ Rδ = 1
this relation allows an easy evaluation of the scale factor
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 24/34
Constant acceleration motion law
symmetric law
two zones at constant acceleration, one positive and onenegative
due to symmetric profile, the positive and negativeaccelerations have the same modulus (A)
one parameter xv allows shape changing of the profile
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 25/34
Constant acceleration motion law
scale factor calculation
remembering that must results Rδ = 1
R = Axv , δ = 1− xv
results
Axv(1− xv) = 1 ⇒ A =1
xv(1− xv)
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 26/34
Constant acceleration motion law
the amplitude of the constant piece function f(x) dependson the parameter xv
A =1
xv(1− xv)
each value of the parameter xv leads to a differentposition profile but with the same boundary condition
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 27/34
Acceleration coefficientCa
it is useful a motion low index that gives indications on themaximum value of the acceleration
y(t)max =h
t2d
f(t/td)max
we introduce a kinematic coefficient that represents themaximum value of the function f(x)
Ca = max (|f(x)|)
we also introduce others two acceleration coefficientsregarding the positive and the negative part of thefunction f(x)
C+a = max(f(x)) , C−
a = |min(f(x))|
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 28/34
Acceleration coefficientCa
Ca = max(C+a , C
−
a )
maximum value of the acceleration
y(t)max =h
t2d
Ca
moreover these coefficients make more easy to comparetwo acceleration profile expressed as f1(x) and f2(x).
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 29/34
Constant acceleration motion law
for symmetric constant acceleration law, it results
Ca = C+a = C−
a =1
xv(1− xv)
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 30/34
Constant acceleration motion law
the minimum value of Ca occur for xv = 1/2
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 31/34
Constant acceleration motion law
this configuration leads to the minimum value of maximumacceleration
for a half of the time drive the pay-load has a positiveacceleration, for the other half the pay-load has a negativeacceleration
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 32/34
Constant acceleration motion law
remembering that
y(t) =h
t2d
f(t/td)
it results
y(t)max =h
t2d
max (f(t/td)) =h
t2d
Ca
it is impossible to realize a movement with stroke h in thetime interval td with a maximum acceleration lower thanh
t2d
4
y(t)max ≥h24
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 33/34
Constant acceleration motion law
it is a good practice to limit the acceleration coefficient to 8
twice the value of the minimum value of Ca
what occurs to the velocity profile while xv changes from 1/2 to0?
prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini – p. 34/34