Kurfess T.R. Robotics and Automation Handbook

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Trajectory Planning for Flexible Robots 9

-5

Payload

Trolley

Bridge

Control

Pendant

FIGURE 9.6 Overhead bridge crane.

0 5 10 15

Position

Trolley

Payload

Button On

Time

FIGURE 9.7 Crane response when operator presses move button.

100

150

−150 −100 −50 0

Y Position

X Position

Start

Finish

FIGURE 9.8 Payload response moving through an obstacle ﬁeld.

-6 Robotics and Automation Handbook

0 5 10 15

Trolley

Payload

Position

Time

Button On

FIGURE 9.9 Crane response when operator presses move button two times.

a bridge crane payload while it is being driven through an obstacle ﬁeld by a novice operator. There is

considerable payload sway both during the transport and at the ﬁnal position. These data were obtained

via an overhead camera that tracked the payload motion but was not capable of measuring the position for

feedback control purposes. An experienced crane operator can often produce the desired payload motion

with much less vibration by pressing the control buttons multiple times at the proper time instances.

This type of operator command and payload response for planar motion is shown in Figure 9.9. When

compared with the response shown in Figure 9.7, the beneﬁts of properly choosing the reference command

are obvious. This is the type of effect that the command generator block in Figure 9.5 strives to achieve.

9.2.2 Generating Zero Vibration Commands

As a ﬁrst step to understanding how to generate commands that move ﬂexible robots without vibration,

it is helpful to start with the simplest such command. A fundamental building block for all commands

is an impulse. This theoretical command is often a good approximation of a short burst of force such as

that from a hammer blow or from momentarily turning the actuator full on. Applying an impulse to a

ﬂexible robot will cause it to vibrate. However, if we apply a second impulse to the robot, we can cancel the

vibration induced by the ﬁrst impulse. This conceptis demonstrated in Figure 9.10. Impulse A

induces the

vibration indicated by the dashed line, while A

induces the dotted response. Combining the two responses

using superposition results in zero residual vibration. The second impulse must be applied at the correct

time and must have the appropriate magnitude. Note that this two-impulse sequence is analogous to the

two-pulse crane command shown in Figure 9.9.

−0.4

−0.2

0.2

0.4

0.6

0 0.5 1 1.5 2 2.5 3

A1 Response

A2 Response

Total Response

Position

Time

FIGURE 9.10 Two impulses can cancel vibration.

Trajectory Planning for Flexible Robots 9

-7

In order to derive the amplitudes and time locations of the two-impulse command shown in Figure 9.10,

a mathematical description of the residual vibration that results from a series of impulses must be utilized.

If the system’s natural frequency is ω and the damping ratio is ζ , then the residual vibration that results

from a sequence of impulses applied to a second-order system can be described by [5–7]

V(ω, ζ ) = e

−ζωt



[C(ω, ζ )]

+ [S(ω, ζ )]

(9.1)

where,

C(ω, ζ ) =



i=1

ζωt

cos(ω

) and S(ω, ζ ) =



i=1

ζωt

sin(ω

) (9.2)

and t

are the amplitudes and time locations of the impulses, n is the number of impulses in the impulse

sequence, and

= ω



1 −ζ

(9.3)

Note that Equation (9.1) is expressed in a nondimensional form. It is generated by taking the absolute

amplitude of residual vibration from an impulse series and then dividing by the vibration amplitude from

a single, unity-magnitude impulse. This expression predicts the percentage of residual vibration that will

remain after input shaping has been implemented. For example, if Equation (9.1) has a value of 0.05 when

the impulses and system parameters are entering into the expression, then input shaping with the impulse

sequence should reduce the residual vibration to 5% of the amplitude that occurs without input shaping.

Of course, this result applies to only underdamped systems, because overdamped systems do not have

residual vibration.

To generate an impulse sequence that causes no residual vibration, we set Equation (9.1) equal to zero

and solve for the impulse amplitudes and time locations. However, we must place a few more restrictions

on the impulses, because the solution can converge to zero-valued or inﬁnitely-valued impulses. To avoid

the trivial solution of all zero-valued impulses and to obtain a normalized result, we require the impulse

amplitudes to sum to one:



= 1 (9.4)

At this point, the impulses could satisfy Equation (9.1) by taking on very large positive and negative values.

These large impulses would saturate the actuators. One way to obtain a bounded solution is to limit the

impulse amplitudes to positive values:

> 0, i = 1, ..., n (9.5)

Limiting the impulses to positive values provides a good solution. However, performance can be pushed

even further by allowing a limited amount of negative impulses [8].

The problem we want to solve can now be stated explicitly: ﬁnd a sequence of impulses that makes Equa-

tion (9.1) equal to zero, while also satisfying Equation (9.4) and Equation (9.5).

Because we are looking

for the two-impulse sequence shown in Figure 9.10 that satisﬁes the above speciﬁcations, the problem has

four unknowns — the two impulse amplitudes (A

, A

) and the two impulse time locations (t

, t

Without loss of generality, we can set the time location of the ﬁrst impulse equal to zero:

= 0 (9.6)

The problem is now reduced to ﬁnding three unknowns (A

, A

, t

). In order for Equation (9.1) to equal

This problem statement and solution are similar to one ﬁrst published by Smith [4] in 1957.

-8 Robotics and Automation Handbook

zero, the expressions in Equation (9.2) must both equal zero independently because they are squared in

Equation (9.1). Therefore, the impulses must satisfy

0 =



i=1

ζωt

cos(ω

) = A

ζωt

cos(ω

) + A

ζωt

cos(ω

) (9.7)

0 =



i=1

ζωt

sin(ω

) = A

ζωt

sin(ω

) + A

ζωt

sin(ω

) (9.8)

Substituting Equation (9.6) into Equation (9.7) and Equation (9.8) reduces the equations to

0 = A

+ A

ζωt

cos(ω

) (9.9)

0 = A

ζωt

sin(ω

) (9.10)

In order for Equation (9.10) to be satisﬁed in a nontrivial manner, the sine term must equal zero. This

occurs when its argument is a multiple of π:

= pπ, p = 1, 2, ... (9.11)

In other words

pπ

, p = 1, 2, ...

(9.12)

where T

is the damped period of vibration. This result tells us that there are an inﬁnite number of possible

values for the location of the second impulse — they occur at multiples of the half period of vibration.

The solutions at odd multiples of the half period correspond to solutions with positive impulses, while

those at even multiples would require a negative impulse. Time considerations require using the smallest

value for t

, that is

(9.13)

The impulse time locations can be described very simply; the ﬁrst impulse is at time zero and the second

impulse is located at half the period of vibration.For this simple two-impulse case, the amplitude constraint

given in Equation (9.4) reduces to:

+ A

= 1

(9.14)

Using the expression for the damped natural frequency given in Equation (9.3) and substituting Equation

(9.13) and Equation (9.14) into Equation (9.9) gives

0 = A

− (1 − A



ζπ

√

1−ζ



(9.15)

Rearranging Equation (9.15) and solving for A

gives



ζπ

√

1−ζ



1 +e



ζπ

√

1−ζ



(9.16)

To simplify the expression, multiply top and bottom of the right-hand side by the inverse of the exponential

term to get

1 + K

(9.17)

Trajectory Planning for Flexible Robots 9

-9

02 0

∗

Initial Command Input Shaper

∆

2 + ∆

Shaped Command

FIGURE 9.11 Input shaping a short pulse command.

where the inverse of the exponential term is

K = e



−ζπ

√

1−ζ



(9.18)

substituting Equation (9.17) back into Equation (9.14), we get

1 + K

(9.19)

The sequence of two impulses that leads to zero vibration (ZV) can now be summarized in matrix

form as







1+K

00.5T



(9.20)

9.2.3 Using Zero-Vibration Impulse Sequences to Generate

Zero-Vibration Commands

Real systems cannot be moved around with impulses, so we need to convert the properties of the impulse

sequence given in Equation (9.20) into a usable command. This can be done by simply convolving the

impulse sequence with any desired command. The convolution product is then used as the command to the

system. If the impulse sequence, also known as an input shaper, causes no vibration, then the convolution

product will also cause no vibration [7, 9]. This command generation process, called input shaping, is

demonstrated in Figure 9.11 for an initial pulse function. This particular input shaper was designed for an

undamped system, so both impulses have the same amplitude. Note that the convolution product in this

case is the two-pulse command shown in Figure 9.9, which moved the crane with no residual vibration.

In this case, the shaper is longer than the initial command, but in most cases the impulse sequence will be

much shorter than the command proﬁle. This is especially true when the baseline command is generated

to move a robot through a complex trajectory and the periods of the system vibration are small compared

with the duration of the trajectories. When this is the case, the components of the shaped command

that arise from the individual impulses run together to form a smooth continuous function as shown in

Figure 9.12.

∗

Initial Command Input Shaper Shaped Command

FIGURE 9.12 Input shaping a generic trajectory command.

-10 Robotics and Automation Handbook

9.2.4 Robustness to Modeling Errors

The amplitudes and time locations of the impulses depend on the system parameters (ω and ζ ). If there

are errors in these values (and there always are), then the input shaper will not result in zero vibration. In

fact, when using the two-impulse sequence discussed above, there can be a noticeable amount of vibration

for a relatively small modeling error. This lack of robustness was a major stumbling block for the original

formulation of this idea that was developed in the 1950s [10].

This problem can be visualized by plotting a sensitivity curve for the input shaper. These curves show

the amplitude of residual vibration caused by the shaper as a function of the system frequency and/or

damping ratio. One such sensitivity curve for the zero-vibration (ZV) shaper given in Equation (9.20) is

shown in Figure 9.13 with a normalized frequency on the horizontal axis and the percentage vibration on

the vertical axis. Note that as the actual frequency ω deviates from the modeling frequency ω

, the amount

of vibration increases rapidly.

The ﬁrst input shaper designed to have robustness to modeling errors was developed by Singer and

Seering in the late 1980s [7, 11]. This shaper was designed by requiring the derivative of the residual

vibration, with respect to the frequency, to be equal to zero at the modeling frequency. Mathematically,

this can be stated as

∂V(ω, ζ )

∂ω

= 0

(9.21)

Including this constraint has the effect of keeping the vibration near zero as the actual frequency starts to

deviate from the modeling frequency. The sensitivity curve for this zero vibration and derivative (ZVD)

shaper is also shown in Figure 9.13. Note that this shaper keeps the vibration at a low level over a much

wider range of frequencies than the ZV shaper.

Since the development of the ZVD shaper, several other robust shapers have been developed. In fact,

shapers can now be designed to have any amount of robustness to modeling errors [12]. Any real robotic

system will havesome amount of tolerable vibration; real machines arealwaysvibrating at some level. Using

this tolerance, a shaper can be designed to suppress any frequency range. The sensitivity curve for a very

robust shaper is included in Figure 9.13 [12–14]. This shaper is created by establishing a tolerable vibration

limit V

tol

and then restricting the vibration to below this value over any desired range of frequency errors.

Input shaping robustness is not restricted to errors in the frequency value. Figure 9.14 shows a three-

dimensional sensitivity curve for a shaper that was designed to suppress vibration between 0.7 and 1.3 Hz

and also over the range of damping ratios between 0 and 0.2. Notice that the shaper is very robust to

changes in the damping ratio.

To achieve greater robustness, input shapers generally must contain more than two impulses and their

durations must increase. For example, the ZVD shaper [7] obtained by satisfying Equation (9.21) contains

0.5 0.75 1 1.25 1.5

ZV Shaper

Robust (ZVD) Shaper

Very Robust Shaper

Percentage Vibration

Normalized Frequency (w/w

)

tol

FIGURE 9.13 Sensitivity curves of several input shapers.

Trajectory Planning for Flexible Robots 9

-11

0.15

0.16

0.12

0.1

0.14

0.08

0.06

0.04

0.02

0.05

0.2

0.6

0.8

1.2

1.4

0.1

Residual

Vibration

Damping

Ratio, z

Frequency (Hz)

FIGURE 9.14 Three-dimensional sensitivity curve.

three impulses given by







(1+K )

00.5T



(9.22)

The increase in shaper duration means that theshaped commandwill also increasein duration.Fortunately,

even very robust shapers have fairly short durations. For example, the ZVD shaper has a duration of only

one period of the natural frequency. This time penalty is often a small cost in exchange for the improved

robustness to modeling errors. To demonstrate this tradeoff, Figure 9.15 shows the response of a spring-

mass system to step commands shaped with the three shapers used to generate Figure 9.13. Figure 9.15a

shows the response when the model is perfect, and Figure 9.15b shows the case when there is a 30% error

in the estimated system frequency. The increase in rise time caused by the robust shapers is apparent

in Figure 9.15a, while Figure 9.15b shows the vast improvement in vibration reduction that the robust

shapers provide in the presence of modeling errors. In this case the very robust shaper yields essentially

zero vibration, even with the 30% frequency error.

9.2.5 Multi-Mode Input Shaping

Many robotic systems will have more than one ﬂexible mode that can degrade trajectory tracking. There

have been several methods developed for generating input shapers to suppress multiple modes of vibration

[15–18]. These techniques can be used to solve the multiple vibration constraint equations sequentially,

or concurrently. Suppressing multiple modes can lead to a large increase in the number of impulses in the

input shaper, so methods have been developed to limit the impulses to a small number [16]. Furthermore,

the nature of multi-input systems can be exploited to reduce the complexity and duration of a shaper for

a multi-mode system [18].

A simple, yet slightly suboptimal, method is to design an input shaper independently for each mode

of problematic vibration. Then, the individual input shapers can simply be convolved together. This

straightforward process is shown in Figure 9.16.

-12 Robotics and Automation Handbook

0.5

1.5

012345

ZVD

Very Robust

Position

Time

(a)

(b)

0.5

1.5

012345

ZVD

Position

Time

Very Robust

FIGURE 9.15 Spring-mass response to shaped step commands: (a) Model is perfect; (b) 30% frequency error.

9.2.6 Real-Time Implementation

One of the strengths of command shaping is that it can often be implemented in a real-time control system.

Many command shaping methods can be thought of as ﬁltering the reference signal before it is fed to the

closed-loop control. For example, the input shaping technique requires only a simple convolution that

can usually be implemented with just a few multiplication and addition operations each time through the

control loop.

Many motion control boards and DSP chips have built in algorithms for performing the real-time

convolution that is necessary for input shaping. If these features are not available, then a very simple

algorithm can be added to the control system. The algorithm starts by creating a buffer, just a vector variable

of a ﬁnite length. This buffer is used to store the command values for each time step. For example, the ﬁrst

∗

Mode 2

Input Shaper

Two-Mode

Shaper

Mode 1

Input Shaper

FIGURE 9.16 Forming a two-mode shaper through convolution.

Trajectory Planning for Flexible Robots 9

-13

∗

1 5 10 15 1 5 10 1515

Send This Value

at Time Step 1

Acquire This Value

at Time Step 1

∗

15 15

1515

Send This Value

at Time Step 8

Acquire This Value

at Time Step 8

Store This Value

for Later

FIGURE 9.17 Real-time input shaping.

value in the buffer would be the shaped command at the ﬁrst time instance. A graphical representation

of such a buffer is shown in the upper right-hand corner of Figure 9.17. The upper left-hand portion of

the ﬁgure shows the unshaped baseline command in the digital domain. This baseline command can be

created in real time, for example, by reading a joystick position.

In order to ﬁll the buffer with the input-shaped command, the algorithm determines the baseline

command each time through the control loop. The algorithm multiplies the value of the baseline command

by the amplitude of the ﬁrst impulse in the input shaper. This value then gets added to the current time

location in the command buffer. The amplitude of the second impulse is then multiplied by the baseline

command value. However, this value is not sent directly out to the feedback control loop. Rather, it is

added to the future buffer slot that corresponds to the impulse’s location in time. For example, assuming

a 10 Hz sampling rate, if the time location of the second impulse were at 0.5 sec, then this second value

would be added to the buffer ﬁve slots ahead of the current position. This real-time process will build up

the shaped command as demonstrated in Figure 9.17. The ﬁgure indicates what the value of the shaped

command would be at the ﬁrst time step and at the eighth time step and how the index to the current

command value advances through the buffer. To avoid having the index exceed the size of the buffer, a

circular buffer is used where the index goes back to the beginning when it reaches the end of the buffer.

9.2.7 Applications

The robustness and ease of use of input shaping has enabledits implementation on a largevarietyof systems.

Shaping has been successfully implemented on a number of cranes and crane-like structures [19–22]. The

performance of both long-reach manipulators [23, 24] and coordinate measuring machines [3, 25, 26]

improves considerably withinput shaping.The multiple modes of a silicon-handling robot wereeliminated

with input shaping [27]. Shaping was an important part of a control system developed for a wafer stepper

[28, 29]. The throughput of a disk-drive-head tester was signiﬁcantly improved with shaping [8]. A series

of input-shaping experiments was even performed on board the space shuttle Endeavor [13, 30].

Although input shaping may not be speciﬁcally designed to optimize a desired trajectory, its ability

to reduce vibration allows the system to track a trajectory without continually oscillating around the

trajectory path. In this respect, input shaping relies on a good baseline command for the desired trajectory.

This trajectory needs to be nominally of the correct shape and needs to account for physical limitations of

the hardware such as workspace boundaries and actuator limits. This baseline trajectory would normally

be derived from these physical and kinematic requirements and this topic is addressed elsewhere.

As a demonstration of input shaping trajectories, consider the painting robot shown in Figure 9.18.

The machine has orthogonal decoupled modes arising from a two-stage beam attached vertically to an

-14 Robotics and Automation Handbook

Air

Brush

Recording

Surface

Compressed Air

FIGURE 9.18 Painting robot.

XY positioning stage moving in a horizontal plane. The end effector is a compressed-air paint brush that

paints on paper suspended above the airbrush. The stages are positionedby PD controllers. No information

about the paintbrush position is utilized in the control system. Experiments were conducted by turning

on the ﬂow of air to the paint brush, commencing the desired trajectory, and then shutting off the ﬂow of

paint at the end of the move. Figure 9.19 shows the result when a very fast 3 in. × 3 in. square trajectory

was commanded and input shaping was not used. When input shaping was enabled, the system followed

the desired trajectory much closer, as shown in Figure 9.20.

9.2.8 Extensions Beyond Vibration Reduction

At the core of command generation is usually the desire to eliminate vibration. However, many other

types of performance speciﬁcations can also be satisﬁed by properly constructing the reference command.

Methods for constructing commands that limit actuator effort have been developed [8, 31, 32]. If the

actuator limits are well known and the fastest move times are desired, then command generation can

be used to develop the time-optimal command proﬁles [33–37]. When the system is Multi-Input Multi-

Output, the shaped commands can be constructed to take this effect into account [18, 31, 38]. When

Desired Response

Direction

of Travel

FIGURE 9.19 Response to unshaped square trajectory.