Proceedings of the ASME 2010 Dynamic Systems and Control Conference
DSCC2010
September 12-15, 2010, Cambridge, Massachusetts, USA
DSCC2010-
VARIABLE-VELOCITY EXPONENTIAL INPUT SHAPING FOR POSITION
CONTROLLED ROBOTIC SYSTEMS
P. Iravani
Department of Mechanical Engineering
University of Bath
Bath, BA2 7AY, UK
[email protected]
M. N. Sahinkaya
Department of Mechanical Engineering
University of Bath
Bath, BA2 7AY, UK
[email protected]
ABSTRACT
This paper demonstrates a new form of Input Shaping for vibration reduction applied to robotic systems that manipulate flexible loads. The method is based on using an exponential function
to define asymptotic and vibration-free trajectories for the flexible system. The required control input is calculated analytically
by using inverse dynamics which ensures the desired end-effector
trajectory. The method is demonstrated experimentally on the
control of point-to-point movements of a robotic manipulator.
INTRODUCTION
This paper focuses on Input Shaping (IS) strategies to minimize vibration in robotic systems that operate flexible systems
at their end-effector. The method defines trajectories in terms of
(position, velocity) pairs and thus does not interfere with lowlevel robot controllers.
The accuracy and settling time of lightly damped mechanical systems are constrained by their inherent flexible properties which introduce undesired vibrations during and after movements. Current robotic research aims at bringing robots closer
to humans. For these new environments, robots will have to be
compliant to ensure user safety. Light-weight links and flexible
joints are currently being designed to ensure mechanical compliance [1, 2]. These introduce new control challenges in relation to vibration minimization, as both introduce elasticity to the
robotic system. Traditional industrial robots have been designed
to be as rigid as possible to minimize vibrations at their endeffectors. Despite their rigid designs, flexible loads, compliant
force sensors and flexible couplers attached at the end-effector
of industrial robots result in unwanted vibrations during motions
in free space [3]. These challenges in robotics make IS an ideal
candidate for vibration-free positioning in robotic systems.
IS strategies have the benefit of being easily applicable
on existing machinery as they don’t require feedback measurements. The most popular approach to IS is based on the convolution of the input command with carefully calculated impulses [4].
In essence the method relies on compensating for the vibrations
associated with the system’s natural transient response. The resulting motion speed and thus settling time, are determined by
the system dynamics, i.e. the natural frequency and the damping
ratio. Therefore, conventional IS does not allow the control of
the robot’s settling time.
NOMENCLATURE
c System damping
f Force input
F Normalized force input
k System stiffness
m System mass
n Exponential order
t Time
Ts Settling time
x Desired system trajectory
X Normalized system trajectory
XE End position
XM Motion range
XO Initial position
y Control position input
Y Normalized system control input
u Normalized time
α Speed parameter
ωn Natural frequency
τ Time constant
ζ Damping ratio
1
Copyright © 2010 by ASME
In order to make IS applicable to industrial robots, Kamel
[3] argues that the following points must be satisfied:
n
x(t) = XO + XM (1 − e(−αt) )
1- Compensation of the delays introduced by the shapers
2- Digitization of the shapers
3- Desired end-effector trajectory definition
(1)
which describes a smooth asymptotic motion starting at position
XO to the end position XE , defined as XE = XO + XM , for any
n ≥ 1. The parameter α corresponds to the motion’s speed, i.e.
how fast the x(t) converges towards the final value. The order of
the equation n specifies the gradient of the function, therefore the
behavior of its derivatives.
In order to generalize the analysis a normalized time, u, is
defined as, u = αt, thus resulting in the normalized function for
XO = 0;
A novel IS techniques based on a third order exponential
function was introduced by Sahinkaya [5, 6]. The technique
is based on an inverse dynamic analysis using the exponential
function as the systems’ desired trajectory. The Exponential
IS method (EIS) inherently solves the problems associated with
conventional IS and offers the following benefits:
1- The speed of motions, thus the settling time can be controlled
2- Is based on a continuous trajectory, thus precise impulse
timings are not required
3- Uses inverse dynamics, thus end-effector trajectories are
ensured without delays
n
x(t)
= 1 − e−u
XM
X(u) =
(2)
Figure 1 illustrates the normalized function X(u) and its
derivatives for orders n = 2, 3, 4 i.e. normalized function, velocity, acceleration, and jerk. As X(u) defines a point-to-point
trajectory, it is essential that its first and second derivatives at the
start and end of the motion are zero, i.e. that the system starts
and reaches the desired end position with zero velocity and acceleration. Given the asymptotic characteristic of the exponential function presented, the end position will always meet these
constraints. It can be seen from Fig. 1(b) that n ≥ 3 will satisfy initial conditions of zero velocity and acceleration. It will be
shown later, that higher order functions are required for position
controlled systems.
This paper compares conventional and EIS methods, it derives a trajectory function for position controlled systems and
demonstrates the application of the new method on a robotic manipulator operating a flexible load.
INPUT SHAPING
Various IS techniques have been developed including posicast control [7], bang-bang control [8], convolution of impulses
[4, 9] and ramped sinusoids [10], all of which rely on compensating vibrations associated with the system natural transient response. For a second order system, the basic form of input shaping termed ZV shapers require two impulses giving zero vibration when the final position is reached. The ZV shapers are sensitive to errors in the natural frequency and the damping ratio
of the system. Robustness can be improved at the expense of a
slower speed by increasing the number of pulses and making the
derivative of the vibrations zero at the arrival of the final position
(ZVD shapers) and also the second derivatives (ZVDD shapers).
However, they can exercise very little control over the response
speed, which is governed mainly by the system’s modes.
An alternative approach was developed by [11] using a polynomial shaping function, which is formed from an inverse dynamic analysis. However, the polynomial input function is only
valid up to the point where the output reaches the final destination and it has to be switched to another function in order to keep
the system in its desired position. Switching introduces a discontinuity in the first and second derivatives, and can potentially
induce vibration.
3
n=1
0.8
0.6
Normalised Velocity
Normalised Position
1
n=2
0.4 n=3
0.2
0
0
n=4
0.5
1
1.5
2
Normalised Time α*t
n=4
2
n=3
n=2
n=1
1
0
0
2.5
0.5
1
1.5
2
Normalised Time α*t
4
(b) Normalized Velocity
n=4
n=3
n=2
2
Normalised Jerk
Normalised Acceleration
(a) Normalized Position
n=1
0
−2
0.5
1
1.5
2
Normalised Time α*t
Figure 1.
2
n=4
n=3
10
0
n=1
−10 n=2
−20
−4
0
2.5
(c) Normalized Acceleration
EXPONENTIAL INPUT SHAPING
EIS uses the following asymptotic function to define the desired motion x(t):
2.5
0
0.5
1
1.5
2
Normalised Time α*t
2.5
(d) Normalized Jerk
EXPONENTIAL FUNCTION CHARACTERISTICS
Copyright © 2010 by ASME
Inverse dynamics
In order to obtain the required input, an inverse dynamic
analysis is performed assuming that the system responds as a
linear second order system as the one illustrated in Fig. 2.
(a) Force controlled system
Figure 2.
For a force controlled system, a non-zero starting force
derivative is not an issue, as forces can be applied instantaneously. Thus, for a force controlled system a function of order
n = 3 satisfies all the motion constraints (zero starting velocity
and acceleration).
Position controlled systems should have a zero starting vef (t)
locity, given our previous simplification, y(t) ≈
, it follows
k
that Eq. (5) should have a zero starting derivative for position
controlled systems. In other words, higher order functions, with
n > 3, are all suitable candidates. As it can be observed in Fig. 1,
the higher the order the higher the maximum values of the function derivatives. In other words, higher order functions will require larger actuator effort for the same point-to-point motion.
Thus, in a position controlled systems it is important to balance
the zero-jerk constraint with that of a lower-order system to result in lower actuator effort.
A suitable order for the position controlled system is experimentally determined to be n = 7/2. The value of n does affect
the vibration suppression properties of EIS. The following equations are used to derive the normalized force F(u) for a desired
exponential trajectory of order n = 7/2
(b) Position controlled system
FLEXIBLE SECOND ORDER SYSTEM
The equation of motion for a force controlled (Fig. 2(a)) system can be written as:
ẍ(t) + 2ζωn ẋ(t) + ω2n x(t) =
ω2n f (t)
k
(3)
and for a position controlled system (Fig. 2(b)):
ẍ(t) + 2ζωn ẋ(t) + ω2n x(t) = 2ζωn ẏ(t) + ω2n y(t)
7/2
x(t)
= 1 − e(−u)
XM
7/2
7
ẋ(t)
= u5/2 e−u
Ẋ(u) =
αXM
2
¶
µ
ẍ(t)
35 3/2 49 5 −u7/2
=
u − u e
Ẍ(u) = 2
α XM
4
4
X(u) =
(4)
The relation between the force and position controlled sys2ζẏ(t)
tems, f (t) =
+ ky(t), is a low-pass filter with a time conωn
2ζ
. For flexible systems with a very low ζ the effect
stant of τ =
ωn
of this filter is negligible at low-movement frequencies, therefore
f (t)
.
allowing the following simplification, y(t) ≈
k
A normalized force F(u) can be derived from Eq. 3 as follows:
Substituting X(u), Ẋ(u), Ẍ(u) into Eq. (5) results in the following normalized force expression:
F(u) = 1 +
Ẍ(u) 2ζẊ(u)
F(u) = 2 +
+ X(u)
β
β
(6)
(5)
Ã
!
7/2
35u3/2 − 49u5 7ζu5/2
+
− 1 e−u
4β2
2β
(7)
Figure 3 illustrates the normalized force for different values
of ζ and β. These are arbitrary values to illustrate the effect of
the parameters on the resulting force.
As it can be seen in Fig. 3(a) increasing the value of ζ results in an increased normalized force. Contrarily, decreasing
β (Fig. 3(b)) increases the value of the normalized force. This
is expected as β = ωαn represents the relation between the system’s natural frequency ωn and the speed parameter α. Smaller
β represents a relatively (in relation to ωn ) faster motions, therefore requiring larger forces. It can be seen from Fig. 3(b) that at
relatively slow motions, the normalized force resembles the desired trajectory. In [5] a critical value of β > 8 for an exponential
function with n = 3 is defined, where above this threshold the demanded trajectory can be used directly as control input without
ωn
f (t)
and β =
.
where F(u) =
kXM
α
The normalized force F(u) can be easily calculated using
inverse dynamics by deriving the first and second derivatives of
X(u) and substituting in Eq. (5).
EXPONENTIAL INPUT SHAPING DESIGN
Following Eq. (5) the normalized force F(u) is a weighted
summation of the functions illustrated in Fig. 1. This summation
will have a non-zero starting gradient unless n > 3. This can
be observed in Fig. 1(d) where the normalized jerk starts at zero
only for n = 4 .
3
Copyright © 2010 by ASME
ζ=0.1
0
−2
0
0.5
1
1.5
2
Normalised Time α*t
(a) F(u) with changing ζ for β = 1
2
β=2
β=5−10
1
0
β=3
−1
−2
0
2.5
β=1
0.5
1
1.5
2
Normalised Time α*t
2.5
(b) F(u) with changing β for ζ = 0.1
10
5
0
−5
−10
−15
0
0.5
1
1.5
2
Normalised Time α*t
(a) First derivative
Figure 3.
2.5
Normalised Force Derivative F’’(u)
2
Normalised Force Derivative F’(u)
Normalised Force F(u)
Normalised Force F(u)
3
ζ=1
4
50
0
−50
0
0.5
1
1.5
2
Normalised Time α*t
2.5
(b) Second derivative
NORMALIZED FORCE FOR DIFFERENT ζ AND β VALUES
Figure 4.
the requirements of inverse dynamics. Hence, there is no need to
specify ωn and ζ for the shaper.
EXPERIMENTAL TESTS
This section presents experimental results using the EIS
method and the results are compared with conventional IS, ZV,
ZVD and ZVDD. In order to test the EIS method a flexible beam
was attached at the end-effector of a 6-DOF robotic manipulator
as illustrated in Fig. 5. The beam measures 840 mm, has a width
of 24 mm and a depth of 1.1 mm. A load of 200 grams is attached
at the end of the beam to decrease its natural frequency.
Variable velocity
One of the main advantages of EIS over conventional IS is
that it can achieve zero vibration motions for any desired motion
speed by selecting an appropriate value for the speed parameter
α.
Given that the exponential function is asymptotic, the end
trajectory will never be reached. In order to define a motion
speed, a settling time, Ts , is defined as the time it takes for the
system to reach 99% of the desired position. Therefore, the speed
parameter α is related to the motion’s settling time Ts as follows:
Ts (99%) =
p
n
−ln(0.01)
α
NORMALIZED DERIVATIVES OF FORCE F(u)
(8)
Motion speed increases with increasing α values. For example,
for n = 7/2 and Ts = 1 s, the speed parameter can be calculated
as α = 1.55
Actuator limits
In order to design a trajectory which falls within the limits
of the system’s actuators the maximum actuator values can be
calculated as function of the speed parameter α. As mentioned
previously, the following simplifications are made: Y (u) = F(u),
and thus, Ẏ (u) = Ḟ(u) and Ÿ (u) = F̈(u).
Figure 4 illustrates the normalized force derivatives Ḟ(u)
and F̈(u) for a system with β = 1 and ζ = 0.1 corresponding
to the force illustrated in Fig. 3(b) for β = 1.
The maximum normalized velocity Ẏ (u) occurs at u = 0.675
and has a value of Ḟ(0.675) = −13.07. The maximum normalized acceleration occurs at u = 0.85 and has a value of
F̈(0.85) = 68.47. As these maximum values depend on the system parameters ζ and ωn , and the speed parameter α, they should
be calculated for each particular system.
It was observed that the robustness of the EIS with respect
to errors in specifying ωn and ζ is a function of the motion
speed and presents similar characteristics to the conventional input shapers at their corresponding speeds.
Figure 5.
FLEXIBLE BEAM AND ROBOT TEST-BED
A LED was attached at the tip of the flexible beam and a vision system was used to track its position. The vision system runs
at about 30Hz which is sufficient to measure the vibrations in this
experiment. The LED and the vision system are constrained in a
way such that the camera can only measure the beam’s position
in the vicinity of the target position. This information is sufficient to measure the residual vibration of the system.
4
Copyright © 2010 by ASME
In order to estimate the natural frequency ωn and damping
ratio ζ of the flexible system a step input was introduced to the
beam and its response is approximated by a second order system.
The estimated values for the beam are ωn = 4.16 rad/s and ζ =
0.005. Figure 6 illustrates the response of the estimated system
with that of the measured system. The figure shows a good match
between the two responses.
inverse dynamics. In the third step, the computed control force
is transformed into an end-effector control trajectory. The endeffector trajectory is then approximated by 3rd order polynomial
Splines which are used to control the robot.
Residual vibrations
In order to generate the fastest possible motion with the
conventional IS technique the following method was used. The
maximum robot angular acceleration (10 rad/s2 ) was used to derive the fastest possible position command by double integration.
Figure 8 illustrates the acceleration and velocity command introduced to the conventional IS. Figure 9 illustrates the position input to the shaper and the simulated shaper output. Observe that
the output trajectory is not equal to the desired one. This is due
to the delays introduced by conventional IS.
0.5
0
−0.5
estimated system
measured response
−1
0
5
Figure 6.
10
Time [s]
15
10
Accleration
Velocity
2
Velocity [rad/s] Acceleration [rad/s ]
Normalised response
1
20
SYSTEM AND RESPONSE
The Katana 6-DOF manipulator [12] can be programmed to
follow trajectories using 3rd order Splines. For this experiments,
the 5th joint (as illustrated in Fig. 5) is used to control the rotational motion of the beam. The other joints are fixed.
5
0
−5
−10
0
Figure 8.
0.5
1
1.5
Time [s]
2
2.5
VELOCITY AND ACCELERATION INPUT TO THE SHAPERS
Conventional input shapers ZV, ZVD and ZVDD evaluated
against the EIS technique as shown in Fig. 10. EIS with a 1.2 s
settling time corresponds to ZV and 1.8 s settling time to ZVD.
Figure 7.
Figure 11 illustrates the relative residual vibration after a
π/5 rad rotational displacement input. The amplitude of the vibrations are relative to the maximum vibrations amplitudes of the
beam when the robot moves using its conventional point to point
motion (P2P) (using the typical robotic trapezoidal velocity profile). In Fig. 11, the time zero corresponds to the time when the
flexible beam crosses its desired position for the first time.
DIAGRAM OF THE PROPOSED EIS METHOD
Figure 7 summarizes the EIS method applied to the robotic
manipulator with a flexible load. The first step is to define the
desired exponential trajectory for the flexible system. The second
step involves calculating the force to achieve such trajectory by
It is clear that both, ZV and EIS drastically reduce the vibrations of the beam. The performance of both methods is comparable, thus this experiment shows that EIS, at the ZV equivalent
speed, performs as well as conventional IS.
5
Copyright © 2010 by ASME
0.7
1
Relative Vibration Amplitude
System Output [rad]
0.6
0.5
0.4
0.3
Traj. Demand
ZV
0.2
ZVD
0.1
ZVDD
0
0
0.5
Figure 9.
1
1.5
Time [s]
2
0.5
0
ZV
EIS 1s
−1
0
2.5
SIMULATED SHAPER OUTPUTS
2
4
6
Time [s]
Figure 11.
0.7
8
RESIDUAL VIBRATIONS
0.6
0.5
0.6
1s
2s
Motor Position
0.4
Motor Position
10
0.8
θ [rad]
θ [rad]
P2P
−0.5
0.3
ZVD
0.2
ZV
0.1
EIS 1.8 s
0.4
0.2
EIS 1.2 s
0
0
Figure 10.
0.5
1
1.5
Time [s]
2
0
0
2.5
1
2
3
Time [s]
MEASURED MOTOR POSITION VS TIME
Figure 12.
Variable Velocity in EIS
Figure 12 illustrate measured motor positions against time
to produce the vibration free positioning of the load. The figure
shows different EIS trajectories for settling times ranging from
1 s to 2 s (increments are of 0.2 s). EIS can generate faster trajectories than the ZV input shaper (e.g. Ts = 1 s trajectory in
the figure). The only restriction in the trajectory speed is the
physical actuator limits. This variable velocity characteristic is a
clear advantage over the conventional IS techniques in which the
trajectory velocity is determined by the system’s characteristics
(ωn , ζ).
In order to demonstrate that EIS can reduce vibrations at any
speed, the same π/5 rotational displacement was performed at
different speeds. Figure 13 illustrates the EIS method with three
different settling times, 1 s, 1.4 s and 1.8 s. As it can be observed
MEASURED MOTOR POSITION VS TIME
the method reduces vibrations at any desired speed. This is a
clear improvement over conventional IS where the motion speed
is determined by the physical system’s characteristics. This figure uses the same axis definition of Fig. 11.
CONCLUSIONS
A new method for input shaping has been demonstrated for
position controlled systems. The method is based on the inverse
dynamics analysis of an exponential function applied to a second
order flexible system.
The method has the following advantages over conventional
IS:
1. It is based on inverse dynamics, thus the actual trajectory is
6
Copyright © 2010 by ASME
Relative Vibration Amplitude
0.15
EIS 1s
EIS 1.4 s
EIS 1.8 s
0.1
[5]
0.05
[6]
0
−0.05
[7]
−0.1
[8]
0
2
4
6
8
10
Time [s]
Figure 13.
[9]
EIS RESIDUAL VIBRATIONS AT DIFFERENT SPEEDS
equal to the desired one
2. The motion’s velocity can be defined by the user, i.e. it is
not constrained by the physical parameters of the flexible
system
3. It is a continuous function, thus it does not require precise
digitization of the shaper
[10]
[11]
The experiments with the robotic test-bed illustrate the performance of the method. They show how the residual vibrations
of the EIS method are as good as the ones achieved by conventional IS with the advantage of continous variable velocity trajectories.
[12]
Systems Measurement and Control - Transactions of the
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and Control Engineering, 215(I5), pp. 467–481.
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Brouchure.
ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support of
the Engineering and Physical Science Research Council under
Grant RC-ME0436.
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