Kurfess T.R. Robotics and Automation Handbook

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Flexible Robot Arms 24

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Input series

coefficient 2

coefficient 3

coefficient 1

delay ∆

delay 2∆

Shaped

output series

FIGURE 24.9 Time delay ﬁlter implementation.

It can also be understood by examining the frequency response of the two- and three-term ﬁlters. The

two-term ﬁlter is effectively the original deadbeat ﬁlter with a narrow notch indicating low robustness

(high sensitivity). Singer et al. [26] termed this the zero vibration or ZV shaper. A three-term ﬁlter with the

delay between terms equal to half the vibration period was called the ZVD shaper because the derivative of

the magnitude is also zero. The optimal arbitrary time-delay or OAT ﬁlter of Magee [27] also shown allows

the delay to be selected at will according to the equations for the coefﬁcients given in Equations (24.96).

This is highly desirable if short time delay is needed, if delay needs to be matched to sampling time

multiples (usual for digital implementation), or if one desires to adapt the gains and thereby track known

or measured changes in the system behavior. Note, however, that robustness for the OAT ﬁlter is greatest

when the time delay is half the vibration period. When this is true, the OAT ﬁlter is equivalent to the ZVD

ﬁlter.

coefﬁcient 1 = 1/M

coefﬁcient 2 =−(2 cos ω

−ζω

)/M

coefﬁcient 3 = (e

−2ζω

)/M

(24.96)

where

M = 1 −2cosω

−ζω

+ e

−2ζω

=ω



1 −ζ

= damped natural frequency

=undamped natural frequency

ζ = damping ratio

=time delay selected, an integer number of samples = 

The OAT ﬁlter and other time delay ﬁlters are simple to implement as shown schematically in Figure24.9.

A delay line (memory) sufﬁcient to hold 2π /ω

T samples of the input is needed. After the appropriate

delay the sample is taken from memory and multiplied by an appropriate coefﬁcient as speciﬁed in

Equations (24.96), and the terms are added together to compose the output. If more than one mode of

vibration needs to be cancelled, two such ﬁlters are placed in series, resulting in a delay which is the sum

of the two periods. Note that multiple zeros result from a single ﬁlter. It may be possible (although maybe

not desirable) to cancel multiple modes with a single ﬁlter.

Adaptation of time delay ﬁlters based on measuring the system response has been proposed in many

ways.The obvious way of identifyingvibrational frequencies froman FFT and adjusting the gains according

to the design equation does not work as well as a direct adaptation of the OAT ﬁlter parameters based on

the measured residual vibration during periods when the arm should be at rest [32]. This does not work if

the arm does not have periods of rest or if the rest periods do not occur frequently relative to the variation

in parameters. Adaptation of the ﬁlter coefﬁcients occurs during rest and after computation the updated

parameters are transferred to the ﬁlter for the next motion as shown in Figure 24.10. Adaptation of this

form is shown in Figure 24.11 to be very effective after only one repetition of the path.

-36 Robotics and Automation Handbook

Input

Flexible

System

–

(

)

(

)

(

)

Transport the Updated

(

)

Copy of

(

)

ACS

(

)

FIGURE 24.10 Adaptive command shaping (ACS) modiﬁes the ﬁlter C(z, n).

24.3.3.3 Learning Control

Learning control, in particular repetitive learning control, has been shown to dramatically improve the

accuracy of rigid robot tracking [33]. Of special interest here is the use of learning control in conjunction

with OAT ﬁltering as described in the previous section [34]. Learning control remembers the error signal

that occurred on the last repetition of the motion and thereby supplements the feed forward signal based

solely on the input with a learned function appropriate for that corresponding time in the motion cycle.

As the motion is repeated the learned function progressively eliminates the need for an error correction.

Improvements of the accuracy of joint motion of more than an order of magnitude can result, effectively

eliminating all but the noise or unsystematic variation in the signal. While the joint movement becomes

more and more accurate, the tip motion may experience more vibration since the input trajectory is

followed more precisely, including all sudden changes in velocity. Also, vibration can no longer back drive

the joints and thereby dissipate the vibrational energy. If the input to the command has been preﬁltered

with an OAT or other shaping ﬁlter, however, vibration cancellation is shown to improve. This is due to

the ability of learning control to overcome the systematic component of such imperfections as friction and

cogging of the motor. Figure 24.12 shows a comparison of the tip acceleration of a gantry robot for four

cases: PID only, PID plus OAT preﬁlter, PD and learning, and PD, OAT, and learning. This combination

has been shown effective in commercial situations for the often found situation of repeated motions. The

referenced paper also shows how to reduce the amount of memory required for storing the learned feed

forward command by using multiple rates of sampling for different parts of the control loop. Adding vision

or other tip position measurement can further eliminate errors due to static deﬂection [35].

24.3.3.4 Inverse Dynamics for Trajectory Design

The prior discussion of modeling included the development and discussion of the inverse dynamics

equations. The speciﬁcation of tip motion is preferred or required since the tip carries the tool for the task.

Equations capable of doing this are presented above, but with the caveat about working with nonminimum

phase systems. For this important subset of ﬂexible arms, which includes arms with ﬂexible links, special

solution procedures must be considered. While complete discussion is beyond the scope of this article

and is a continuing research topic, an introduction is important to understand the limitations of current

techniques and the potential of future developments.

Consider for present purposes the linear form of the dynamic equations for a single joint moving a

single link which can be reduced to a transfer function. If the transfer function of the forward dynamics

has zeros in the right half plane, we do not want to apply the usual procedure to solve the inverse dynamics

for the causal form in which the response always follows the input. The inverse Laplace transform permits

another solution that is usually ignored but is appropriate here, the acausal solution. We effectively are

asking what the input should have been to get the response we want at time t. The input must precede the

output in the physical case, but for the computation, the input is the desired response and the output (of

the computation) is the necessary response. A practical solution procedure exists for separating the causal

and acausal solutions [36]. The rigid portion is speciﬁed to deﬁne the tip motion over time (using the

pinned-pinned mode shapes). The causal portion of the solution is based on initial conditions and solved

bymoving forward in time. The anticausal portionis solvedbackward in time startingwithﬁnal conditions.

The acausal solution for the driving torque is the sum of these three terms. By appropriate shaping of the

Flexible Robot Arms 24

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(a)

Tip Acceleration

= 25 ms)

0 1 2 3 4 5 6 7 8 9 10

0.6

0.4

0.2

–0.2

Time (s)

0 1 2 3 4 5 6 7 8 9 10

–10

–5

Acceleration (V)

Displacement (micron)

0 1 2 3 4 5 6 7 8 9 10

–1

–0.5

0.5

1.5

2.5

3.5

×4

Desired Joint

Trajectory

Coefficients C(n)

(b)

FIGURE 24.11 (a) Test bed for adaptive command shaping (dominant elastic mode at 16 Hz), (b) results of adaptive

command shaping. (Note reduced oscillation in acceleration after initial update at approximately 2 seconds as indicated

by arrow in acceleration plot. Delay time T

= 25 ms.)

-38 Robotics and Automation Handbook

Tip Acceleration (PID Only) Tip Acceleration (PID and 1 OATF)

Tip Acceleration (PD and Learning) Tip Acceleration (PD, Learning, and 1 OATF)

–1

–2

Acceleration (Volt)

10.5

Time (s)

–1

–2

Acceleration (Volt)

10.5

Time (s)

–1

–2

Acceleration (Volt)

98.5

Time (s)

–1

–2

Acceleration (Volt)

98.5

Time (s)

FIGURE 24.12 Comparison of joint PID, OAT ﬁltering, and repetitive learning combinations.

commanded joint motion of the nonminimum phase arm, the tip remains stationary after the joint moves,

then carries out the prescribed move and stops with no overshoot or vibration as shown in Figure 24.13.

As mentioned above under the feedback strategy of strain feedback, the desired history of strain or

other ﬂexible variables is needed to effectively create a tracking controller. The inverse dynamics solution

gives a desired strain proﬁle for the motion that can be applied to this end [37]. Neural networks have

Joint Angle (rad)

1.06

0.85

0.64

0.43

0.21

−0.00

−0.21

Generated Joint Trajectory

−10

Disp. (in)

Joint Angle

End Point Pos.

time (sec)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

FIGURE 24.13 Inverse dynamics trajectory moves joint before tip and leaves no vibration.

Flexible Robot Arms 24

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Optimum Section Length

Viscoelastic Layer

Flexible Beam

Sectioned Elastic Layer

FIGURE 24.14 Sectioned constraining layer for passive damping.

also been applied to learn the necessary inverse for relatively simple single link arms [38]. Extension to

multiple links has also been shown to be feasible [39, 40].

24.3.4 Design and Operational Strategies

24.3.4.1 Passive Damping Treatments

In the preceding discussion it has been apparent that we rely heavily on the truncation of the arm’s ﬂexible

response to a ﬁnite number (one or two) of degrees of freedom when a perfect representation of distributed

ﬂexibility would take an inﬁnite number. This assumption should not be taken for granted. It can be made

more credible if the damping on the ﬂexible modes is increased. Passive damping is an effective way to

do this. Constrained layer damping treatments have been shown to be effective in adding damping to all

modes of ﬂexible beams used in laboratory robots [41]. These treatments sandwich a viscoelastic damping

material between the structural member, typically a beam, and a sectioned elastic constraining layer. The

Figure 24.14 illustrates this construction and Figure 24.15 illustrates the improvement in the damping

that results. Optimization of the length of the constraining layer sections depends on the frequency to be

damped, but damping effect is not extremely sensitive to this choice so that a wide range of frequencies

will be treated [42]. These works also show that the treatment can eliminate instability in the higher modes

not treated by the active controller.

24.3.4.2 Augmentation of the Arm Degrees of Freedom

How an arm is designed must be based on how it is to be used. In the case of ﬂexible arms, this is also true,

with three examples presented below. By incorporating additional actuators on the arm with additional

degrees of freedom, a net gain in the performance of the arm can be achieved in spite of the physical

inevitability of elasticity.

24.3.4.2.1 Bracing Strategies

An anthropomorphic justiﬁcation may be the most effective way to introduce bracing. Fine motor skills

of the human are concentrated in the ﬁngers and gross motion capabilities in the arms, body, and even

the legs. We use them in a modular way on many occasions, bracing our wrists when typing, writing, or

threading needles. This permits a more stable base for ﬁne manipulation during these precise motions.

Sizing an arm’s structure without bracing becomes a tradeoff between gross and ﬁne motion. A short,

heavy, stiff, precise structure is best for ﬁne motion and a long, light, and consequently ﬂexible structure is

best for gross motion. Bracing enables one to have the second case when it is needed and then transition to

the use of the ﬁrst case for ﬁne motion. This concept was ﬁrst explored [43] without consideration of the

overall implementation technology. The complexity of the maneuvers to move, achieve bracing contact,

and then manipulate have been enabled by relevant research [44, 45].

-40 Robotics and Automation Handbook

10 dB/division

damped case

undamped case

0 Hz. 160

FIGURE 24.15 Resonance reduction from passive damping in the magnitude of the hub velocity/hub torque transfer

function.

24.3.4.2.2 Inertial Damping

Bracing requires a stable surface to brace against, but if additional degrees of freedom exist, one can use

them to advantage even without a bracing surface. The reaction to the movement of a “micro manipulator”

at the end of the “macro manipulator” can be coordinated to absorb vibrational energy as shown by several

researchers [46, 47, 48]. Because the micro manipulator is typically short and effectively rigid, it can have a

high bandwidth that is needed to damp relatively high frequency vibrations that are above the bandwidth

of the actuators of the macro manipulator. Furthermore, the micro manipulator is collocated with the

tip of the macro manipulator and avoids the nonminimum phase problems of the macro actuators. Six

actuators in the micromanipulator can provide three forces and three torques where the micromanipulator

is mounted to the macro manipulator that can couple to any bending mode and remove energy. The

complexity results from the fact that complex motions may be needed to generate the forces needed. The

inverse dynamics equations to solve for the joint motions and torques to generate the prescribed forces

have been generated, but the full equations are complex and must in general be solved in real time to create

the reaction to vibration as it occurs. Simpliﬁed equations to generate a portion (say, only the forces) of

the six components of force and torque can be implemented, but stability cannot be guaranteed in this

case because the act of damping one mode with a force can excite another mode with a torque that was not

desired. Very effective damping has been generated for vibration of a single beam in two dimensions [49]

and two beams in series in three dimensions [50]. Figure 24.16 shows the micromanipulator on a ﬂexible

mount that is representative of a macromanipulator. Figure 24.17 shows the effectiveness of the vibration

quenching on one of two axes. The other axis was similarly damped.

Flexible Robot Arms 24

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Link 3

Base Link

Link 1 Link 2

Beam (6 m long)

FIGURE 24.16 Three axis arm used as micromanipulator for inertial damping.

24.3.4.2.3 Piezoelectric Actuation for Damping

Thin layers of piezoelectric material (polymer, crystal, and ceramic forms exist) will change length upon

the application of a voltage. If the material is bonded to a beam the strain created by this action can be

used to deform the beam. By the same principle, the deformation of the beam creates a voltage across

the piezoelectric material. Vibrational energy can be dissipated by the current that results. In either a

passive or an active manner this transduction can be used for vibration control. The forces involved

are small but signiﬁcant in some cases and may provide a solution for some vibration problems as has

been known for many years [51]. The ability to place patches of the material in optimum location for

critical modes is one of the key advantages of this approach, but instability can also result from incorrect

placement.

X-Axis Acceleration

Time at Start of Controller

0 5

1510 20 25

–0.5

–0.4

–0.3

–0.2

–0.1

0.1

0.2

0.3

0.4

0.5

Time (sec)

Acceleration (g)

Link I & II Actuation: Theta 1 = –45deg, Theta 2 = 45deg

FIGURE 24.17 Inertial damping controller quenches oscillations of ﬂexible base. (Controller action started at vertical

line.)

-42 Robotics and Automation Handbook

24.4 Summary

Elasticity of structural material is a pervasive and simple phenomenon that leads to unavoidable and

complex undesirable behaviors of mechanical arms and other motion devices. It can be treated simply in

some cases by adding structural material and consequent mass. Better ways to treat this phenomenon do

exist, and some of those techniques are discussed above. The motivation for these alternative approaches

includes examination of when the simple approach is inadequate or undesirable. This implies modeling

of both static and dynamic phenomena and how they lead to failure of the arm device. The failure of a

structure may be due to its strength, stability, or ﬂexibility. If strength or stability (buckling) is the active

constraint, then a solution to ﬂexibility is not needed. For fast, precise; and long reach arms, ﬂexibility is

often the active constraint. Control, both feedback and feed forward, design strategies; and operational

strategies; should be considered to achieve a systematic solution.

References

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