Reference Command Shaping Using Specified ... - William Singhose

关15兴 Ishitobi, M., 2000, ‘‘A Stability Condition of Zeros of Sampled Multivariable Systems,’’ IEEE Trans. Autom. Control, AC-45共2兲, pp. 295–299. 关16兴 Laub, A. J., and Arnold, W. F., 1984, ‘‘Controllability and Observability Criteria for Multivariable Linear Second-Order Models,’’ IEEE Trans. Autom. Control, 29共2兲, pp. 163–165. 关17兴 M. Ikeda, 1990, ‘‘Zeros and Their Relevance to Control-关III兴; System Structure and Zeros 共in Japanese兲,’’ Journal of the Society of Instrument and Control Engineers, 29共5兲, pp. 441– 448. 关18兴 Gantmacher, F. R., 1959, The Theory of Matrices, Vols. I and II, Chelsea, New York. 关19兴 Rosenbrock, H. H., 1970, State-space and Multivariable Theory. Nelson, London. 关20兴 Suda, N., and Mutsuyoshi, E., 1978, ‘‘Invariant Zeros and Input-Output Structure of Linear, Time-Invariant Systems,’’ Int. J. Control, 28共4兲, pp. 525–535. 关21兴 Suda, N., 1993, Linear Systems Theory 共in Japanese兲. Asakura, Tokyo.

Reference Command Shaping Using Specified-Negative-Amplitude Input Shapers for Vibration Reduction William Singhose Erika Ooten Biediger Ye-Hwa Chen Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA

command. The shaping process increases the command rise time by the duration of the shaper, ⌬. This paper presents new input shapers that allow the designer to trade off rise time against robustness to modeling errors, actuator effort, and excitation of unmodeled high modes. Input shaping schemes first appeared in the 1950s 关1兴. Unfortunately, the methods were sensitive to modeling errors; small errors caused significant residual vibration. Input shaping became useful for many real systems when a robust method was developed 关2兴. Methods for further increasing the robustness have since been presented 关3兴. Given its robustness, input shaping has been implemented on a variety of systems including robotic manipulators 关4,5兴, cranes 关6,7兴, and wafer steppers 关8兴.

2 Previously Developed Input Shapers and Their Effects on Nth Order Systems The amplitudes and time locations of the impulses in an input shaper are determined by satisfying constraint equations. Most constraints can be categorized as residual vibration constraints, robustness constraints, constraints on the impulse amplitudes, or time optimality requirements. The constraint on residual vibration amplitude can be conveniently expressed as the ratio of residual vibration amplitude with input shaping to that without input shaping. For a mode of natural frequency, ␻, and damping ratio, ␨, this percentage residual vibration is given by 关2兴: V 共 ␻ , ␨ 兲 ⫽e ⫺ ␨␻ t n 冑关 C 共 ␻ , ␨ 兲兴 2 ⫹ 关 S 共 ␻ , ␨ 兲兴 2 where, n

C共 ␻,␨ 兲⫽

Bart Mills

(1)

兺 Ae i

␨␻ t i

cos共 ␻ 冑1⫺ ␨ 2 t i 兲

(2)

␨␻ t i

sin共 ␻ 冑1⫺ ␨ 2 t i 兲 ,

(3)

i⫽1

Sony Corporation, Springfield, OR 97477

n

S共 ␻,␨ 兲⫽

兺 Ae i

i⫽1

Residual vibrations can be greatly reduced by using speciallyshaped reference command signals. Input shaping is one such technique that reduces vibration by convolving a sequence of impulses with any desired reference command. Several types of useful impulse sequences have been developed. Most of these have contained only positively valued impulses. However, rise time can be improved by using some negative impulses in the sequence. Unfortunately, the use of negative impulses can excite unmodeled high modes. A new type of impulse sequence containing negative impulses is proposed. These sequences are designed to fill the performance gap between all-positive impulse sequences and the negative sequences previously developed. A proof governing the worst case scenario provides an upper bound on high-mode excitation. The resulting class of impulse sequences allows the designer to make a precise trade off between rise time and vibration reduction. 关DOI: 10.1115/1.1650385兴

1

A i and t i are the amplitudes and time locations of the ith impulse, and n is the number of impulses in the input shaper. If V is set

Introduction

Input shaping is a command generation scheme that reduces vibration of flexible systems. Input shaping is implemented by convolving a sequence of impulses, called the input shaper, with any system command. The convolved signal is then used as the reference command. Figure 1a demonstrates the process with an unshaped step input and an input shaper that consists of two positive impulses. The resulting shaped input is a two-step staircase Contributed by the Dynamic Systems, Measurement, and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division January 21, 2002; final revision, July 16, 2003. Associate Editor: R. Ghulil.

210 Õ Vol. 126, MARCH 2004

Fig. 1 Input shaping a step input

Copyright © 2004 by ASME

Transactions of the ASME

The ZVD shaper contains three impulses and the duration of the ZVD shaper is twice the duration of the ZV shaper 共one full period of the vibration兲. Any shaped command will have its rise time increased by the duration of the shaper; therefore, the ZVD shaper increases the rise time more than the ZV shaper. This increased rise time is the cost of the increased robustness to modeling errors.

Fig. 2 Input shaper sensitivity curves

equal to zero at the modeling parameters, ( ␻ m , ␨ m ) and the input shaper is required to satisfy 共1兲, then the result is called a Zero Vibration 共ZV兲 shaper. To ensure that the shaped command produces the same rigidbody motion as the unshaped command, the impulse amplitudes must sum to one:

兺 A ⫽1. i

(4)

The reason for this constraint is seen in Fig. 1. If the impulses do not sum to one, then the shaped command will not reach the same final value as the unshaped command. In practice, ZV shapers can be very sensitive to modeling errors. To demonstrate this effect, the amplitude of residual vibration can be plotted as a function of the modeling error. Figure 2 shows such a sensitivity curve for the ZV shaper. The vertical axis is the percentage vibration given by 共1兲, while the horizontal axis is the normalized frequency formed by dividing the actual frequency of the system, ␻ a , by the modeling frequency, ␻ m . Note that the residual vibration increases rapidly with modeling error. In order to increase robustness, the shaper must satisfy additional constraints. One such constraint takes the derivative of the residual vibration equation, 共1兲, with respect to frequency, and sets it equal to zero 关2兴. That is: 0⫽

d V共 ␻,␨ 兲. d␻

2.2 Effects of Positive Shapers on Nth-Order Systems. Although input shapers containing positive impulses eliminate the vibration at the frequencies they are designed to suppress, the question arises as to what their effect is on a complicated system with unmodeled modes. To address this question, consider an nthorder system whose transfer function is G(s). Suppose its unitmagnitude impulse response is given by c(t), t⭓0. Assume that the maximum of c(t) for all t⭓0 exists at a unique finite time, T p ⭓0. Denote this maximum by: max c 共 t 兲 ⫽c 共 T p 兲 t⭓0

Now consider the system subjected to a sequence of impulses: A 1 ␦ 共 t⫺T 1 兲 ,

A 2 ␦ 共 t⫺T 2 兲 , . . . ,A n ␦ 共 t⫺T n 兲 ,

where, T n ⬎T n⫺1 ⬎ . . . ⬎T 1 ⭓0 and, n

兺 A ⫽1, i

i⫽1

A i ⭓0

The response of the system is given by superposition as: n

cˆ 共 t 兲 ⫽

兺 A c 共 t⫺T 兲 h 共 t⫺T 兲 i

i⫽1

i

i

where h(t⫺T i ) is the Heavyside step function. ˜ )⭓0. Theorem 1: Assume there is a time ˜T ⭓0 such that c(T ˆ The maximum of the response to the impulse sequence, c (t), is less than the maximum of the unit-magnitude impulse responses: max cˆ 共 t 兲 ⬍c 共 T p 兲

(5)

When 共1兲, 共4兲, and 共5兲 are satisfied with V⫽0, the result is a Zero Vibration and Derivative 共ZVD兲 shaper 关2兴. Figure 2 shows the ZVD shaper is significantly more robust than the ZV shaper. The trade-off is that the ZVD shaper has a longer duration and, consequently, degrades rise time. Other types of robustness constraints have been proposed 关3兴; however, we will limit our discussion here to ZV and ZVD input shapers. Due to the transcendental nature of 共1兲 and 共5兲, there will be multiple solutions. To obtain the fastest rise time, the shaper duration must be made as short as possible therefore it must satisfy a minimum time constraint. It is necessary to place a limit on the magnitude of the impulses as minimizing the time duration will drive the impulse amplitudes to positive and negative infinity. The next subsections discuss previously proposed magnitude constraints. The remainder of the paper is then devoted to a new type of magnitude constraint that produces input shapers that span the performance gap between the previously proposed shapers. 2.1 Positive Input Shapers. One way to avoid the problem of large amplitude impulses is to require all of the impulses to have positive values 关9兴. If the amplitudes are all positive, then each individual impulse must be less than one so that 共4兲 can be satisfied. The positive ZV 关1,2兴 and ZVD 关2兴 shapers have been derived in the literature previously. The duration of the ZV shaper is equal to one-half period of the damped vibration. The ZV shaper is shown schematically in Fig. 1a. Journal of Dynamic Systems, Measurement, and Control

t⭓0

Proof: n

max cˆ 共 t 兲 ⫽max

兺 A c 共 t⫺T 兲 h 共 t⫺T 兲 i

t⭓0 i⫽1

t⭓0

i

i

n

⭐

兺

i⫽1

max兩 A i 兩 • 兩 c 共 t⫺T i 兲 兩 h 共 t⫺T i 兲 t⭓0

We first note that, under the assumption of the theorem: max A i c 共 t⫺T i 兲 h 共 t⫺T i 兲 ⫽max兩 A i 兩 • 兩 c 共 t⫺T i 兲 兩 t⭓0

t⭓T i

This is because: A i c 共 t⫺T i 兲 h 共 t⫺T i 兲 ⫽0 for all t⭐T i . Next, by the property of the translation functions and since A i ⭓0: max A i c 共 t⫺T i 兲 ⫽ 兩 A i 兩 max兩 c 共 t⫺T i 兲 兩 ⫽ 兩 A i 兩 • 兩 c 共 T p 兲 兩 t⭓T i

t⭓T i

This occurs when t⫽T p ⫹T i . This leads to: n

max cˆ 共 t 兲 ⭐ t⭓0

兺 兩 A 兩 • 兩 c共 T 兲兩 i⫽1

i

p

where the equality occurs as: MARCH 2004, Vol. 126 Õ 211

3

The larger the magnitude of the negative impulses, the shorter the shaper duration. However, the shaper induces more high-mode excitation. Given the tradeoffs that occur with negative impulses, it is desirable to develop an input shaper whose negative amplitudes can be set to any value. By specifying the negative impulse amplitudes, the designer can set the performance tradeoffs. The goal is to develop a shaper that spans the performance gap between positive shapers and UM shapers. Such a shaper is shown schematically in Fig. 1b. The duration is shorter than the positive ZV shaper, but longer than a UM ZV shaper. The key feature of the proposed specified-negative-amplitude 共SNA兲 shapers is that the amplitude of the negative impulses can be set to any negative value, b. The SNA ZV shaper can be determined in closed form if we assume an undamped system using the amplitude summation constraint, the zero vibration constraint, and a trigonometric identity. The resulting undamped SNA ZV shaper is given by:

Fig. 3 High-mode sensitivity curves

t⫽T p ⫹T 1

and

t⫽T p ⫹T 2

. . . t⫽T p ⫹T n

and

Specified-Negative-Amplitude Input Shapers

冋册

T 1 ⫽T 2 ⫽ . . . ⫽T n Therefore we conclude that n

max cˆ 共 t 兲 ⬍ t⭓0

兺

n

i⫽1

兩 A i兩 • 兩 c 共 T p 兲兩 ⫽ 兩 c 共 T p 兲兩 •

i⫽1

i

冉 冊

a⫽

(6)

These unity-magnitude 共UM兲 shapers are useful because they can be used with a wide variety of unshaped inputs without causing actuator saturation. Figure 1c shows a step input shaped with a UM ZV shaper. The duration of the UM ZV shaper is only one-third of the vibration period, as compared to one-half period for the positive ZV shaper. The time savings provided by using the negative impulse is not without penalty. Shapers containing negative impulses have a tendency to excite unmodeled high modes 关11兴. They also require larger swings in the actuator effort because the change in command at each new impulse is much larger. To see this, compare the shaped inputs shown in Figs. 1共a兲 and 1共c兲. To demonstrate the effect of high-mode excitation, the sensitivity curve for a shaper can be plotted over a range of high frequencies. The value of the curve at any high frequency indicates the degree to which the high frequency will be excited. Figure 3 compares the sensitivity curves for the positive ZV and UM ZV shapers for frequencies up to ten times the modeling frequency. Note that the curve for the positive ZV shaper never exceeds 100%. This indicates that the shaper will never cause high frequencies to be excited more than they are without input shaping, and demonstrates the key result of Theorem 1. On the other hand, the sensitivity curve for the UM ZV shaper reaches a maximum value of 300% at frequencies 3 and 9 times greater than the modeling frequency. This means that if the system happens to have unmodeled modes at these frequencies, then they will be excited to 3 times their level without input shaping. 212 Õ Vol. 126, MARCH 2004

冉 冊册 a

1 ⫺b cos⫺1 ␻ 2a

p

2.3 Unity-Magnitude „UM… Shapers. In an effort to improve rise time, the shaper duration can be shortened by allowing negative impulse amplitudes. The occurrence of negative impulses can occur during the process of designing shapers in the z-domain 关10兴. One method that allows negative impulses to be used in a systematic way requires the impulse amplitudes to switch between 1 and ⫺1 关11兴: i⫽1, . . . ,n

b

b2 1 cos⫺1 ⫺1 ␻ 2a 2

(7)

where,

兺 兩 A 兩 ⫽ 兩 c共 T 兲兩

and the theorem is proved. The main result of the theorem that concerns input shaping is that a shaper containing all positive impulses cannot increase the vibration of an nth-order system over what would exist without input shaping, even if there are modes that are completely ignored or unknown during the shaper design.

A i ⫽ 共 ⫺1 兲 i⫹1

冋

a Ai ⫽ ti 0

But, this never happens because:

1⫺b 2

(8)

The SNA ZVD input shaper must satisfy the additional constraint of 共5兲. Therefore, it must contain five impulses. In addition to specifying the amplitude of the negative impulses 共the second and fourth impulses兲, the amplitude of the middle impulse, c, must also be specified. The ZVD shaper cannot be solved in closed form, so a numerical optimization is performed to minimize the shaper duration. This is a relatively easy optimization and the shapers shown in this paper were generated using the MATLAB Optimization Toolbox. 3.1 Effects of SNA Shapers on nth-Order Systems. In Section 2.2 it was shown that shapers containing all positive impulses cannot increase vibration, even when there are unmodeled modes. The result for shapers containing negative impulses is not the same, as was demonstrated in Fig. 3. This section determines the worst case scenario. As in Sec. 2.2, consider an nth order system whose transfer function is G(s), and the response to an impulse response is given by c(t), t⭓0. The maximum of c(t) for all t⭓0 exists at a finite time, T p ⭓0. The maximum is the same as in Sec. 2.2, however, in this case A i can be negative, but the sum of the magnitudes is limited by: n

兺

i⫽1

n

A i⭐

兺 兩A 兩 i

i⫽1

The response of the system is given by superposition as: n

cˆ 共 t 兲 ⫽

兺 A c 共 t⫺T 兲 h 共 t⫺T 兲 i

i⫽1

i

i

where the Heavyside step function is: h 共 t⫺T i 兲 ˜ )⭓0. The Theorem 2: Assume there is a time ˜T ⭓0 such that cˆ (T maximum of the response to the sequence of impulses, is less than the maximum of the unit-magnitude impulse response multiplied by the sum of the absolute values of the amplitudes: n

max cˆ 共 t 兲 ⬍ t⭓0

兺 兩 A 兩 • 兩 c共 T 兲兩 i⫽1

i

p

Transactions of the ASME

Fig. 4 Duration of the SNA ZV shaper Fig. 7 Impulse time locations of the SNA ZVD shaper „bÄÀ1…

Proof: The proof for Theorem 2 follows the same logic as the proof for Theorem 1. However for Theorem 2, the sum of the maximum amplitudes must equal the absolute values of each amplitude as seem below. n

兺

i⫽1

n

max A i ⫽ t⭓0

兺 兩A 兩 i⫽1

i

3.2 SNA Shaper Properties. At this point it is of interest to examine the shaper properties as a function of the amplitude of the negative impulse 共and the middle impulse for the ZVD shaper兲. It can be seen from 共7兲 that as b becomes more negative the duration of the SNA ZV shaper decreases. When the negative impulse is nonexistent 共b⫽0兲, then the shaper is the same as the positive ZV shaper. When b⫽⫺1, then the shaper is equivalent to the UM ZV shaper. Figure 4 shows the duration of the SNA ZV shaper over a range of damping ratios. As the magnitude of the negative impulse in-

creases, the shaper duration decreases significantly. By using a shaper with a negative amplitude of ⫺1.0 the shaper duration is reduced by approximately 30%. Although this time savings is significant, it may not be justified if there are significant unmodeled high modes. These trade-offs are examined in the following paragraphs. Although Theorem 2 provides an upper bound on the vibration induced into any unmodeled modes, it is of interest to know how close to the upper bound a typical system will be. One way to characterize the effect on high modes is to determine the average value of the sensitivity curve over a range of high frequencies. High-mode excitation 共HME兲 will refer to the average value of the sensitivity curve from 2 to 10 times the modeling frequency. Figure 5 shows the HME as a function of the negative amplitude b. As b gets more negative, the HME increases as expected. Note, however, that these average values lie well below the upper bound provided by Theorem 2. Figure 6 shows the duration of the SNA ZVD shaper as a function of the magnitude of the negative impulses and the middle impulse for an undamped system. The ZVD shaper duration is minimized by some finite value of the middle impulse 共assuming a given value for the negative impulse兲. For a given value of b, the value of c that minimizes the shaper duration is: c⫽0.4980⫺ 共 0.6225兲 b⫹ 共 0.1019兲 b 2 .

(9)

Note that the middle impulse cannot be smaller than 0.5. When the middle amplitude exactly equals 0.5, the shaper duration is equal to one period and the shaper is equivalent to the positive ZVD shaper. This phenomenon can be seen in Fig. 7 where the impulse time locations are plotted as a function of the middle impulse when b⫽⫺1. Also noted on the figure is the case when the shaper is equivalent to the UM ZVD shaper 关11兴. Figure 8 shows the high-mode excitation for the SNA ZVD Fig. 5 SNA ZV high-mode excitation

Fig. 6 Duration of the SNA ZVD shaper

Journal of Dynamic Systems, Measurement, and Control

Fig. 8 SNA ZVD high-mode excitation

MARCH 2004, Vol. 126 Õ 213

Fig. 10 Experimental results from robotic manipulator

Fig. 9 Robotic manipulator

shapers. As the middle amplitudes increase, the HME tends to increase. However, it is clear there are some minima that would be prudent choices when selecting the middle amplitudes. For example, if a designer determines that a shaper duration of 0.75 is acceptable and then chooses b⫽⫺1.0, this would lead to a value of 1.788 for c and a value of 2.32 for the HME excitation. By examining Figs. 7 and 8 the engineer could intelligently refine the design by slightly reducing c, say to 1.68. This would decrease the shaper duration by 1.7% and also decrease the HME by 14.5%.

4

Experimental Results

It is possible that the best shaper for a given system does not always correspond to either a UM shaper or a positive shaper. By utilizing SNA shapers, better overall results can be obtained. This is especially true when the system has higher modes that can be easily excited by using a UM shaper. It is possible for the SNA shaper to have a much faster rise time than the positive shaper and yet not significantly excite the unmodeled high modes. To experimentally verify the key results in this paper, tests were conducted on a large robot arm used to move a suspended payload as seen in Fig. 9 关12兴. While the endpoint of the robot arm was moved in the horizontal direction, the position of the payload was captured using a video camera. The UM ZV, SNA ZV 共b⫽⫺0.5兲, and Positive ZV shapers were used to modify the commands used to drive the robot. The shapers were designed for the hanging pendulum’s frequency of 0.357 Hz, and the amplitudes and time locations for the shapers are:

冋册冋

1 Ai ⫽ ti 0

1

0.4669

0.9337

⫺0.5

0.75

0.5488

1.0976

冋册冋

0.75 Ai ⫽ ti 0

冋册冋

册

⫺1

0.5 Ai ⫽ ti 0

0.5 1.4006

册

共 UM ZV兲

册

共 SNA ZV兲

共 Positive ZV兲

(10)

(11)

(12)

The vibration modes of the robotic arm 共the dominant robot mode is at approximately 1.8 Hz兲 were not taken into account when designing the shapers. In this case, the unmodeled high frequency is approximately 5 times the pendulum frequency. The position of the payload for the three cases is shown in Fig. 10. All three shapers eliminate most of the residual vibration of the pendulum mode. In the unshaped case, the pendulum oscillation amplitude would be off the scale 共22 in兲. The ZV shaper does not excite the higher frequencies of the robotic arm and moves the hanging pendulum with a minimal 214 Õ Vol. 126, MARCH 2004

amount of residual vibration. On the other hand, the UM ZV excites the higher modes of the robot. However, it does produce the fastest rise time. The payload of the robot reaches the desired location much faster with the SNA ZV shaper than with the positive ZV shaper, and the robot modes are not excited. These results confirm that the flexibility of the SNA shaper provides a means for achieving excellent performance that is not possible with either the positive ZV of UM ZV shapers.

5

Conclusions

A new type of input shaper has been developed to allow the designer to specify the magnitude of the negative impulses. These shapers allow the designer to set the tradeoffs between rise time, robustness, and high-mode excitation. The shapers presented here provide a wide range of possible performance levels. A proof has been given that shows that positive input shapers cannot increase vibration even when there are unknown modes. This proof was extended to obtain an upper bound on the amount of vibration that can occur from the new type of negative input shaper. Both simulations and hardware experiments demonstrated the usefulness of the new input shapers.

References 关1兴 Smith, O. J. M., 1958, Feedback Control Systems, New York: McGraw-Hill Book Co., Inc. 关2兴 Singer, N. C., and Seering, W. P., 1990, ‘‘Preshaping Command Inputs to Reduce System Vibration,’’ J. Dyn. Sys., Meas. Control, 112, pp. 76 – 82. 关3兴 Singhose, W., Seering, W., and Singer, N., 1994, ‘‘Residual Vibration Reduction Using Vector Diagrams to Generate Shaped Inputs,’’ J. Mech. Des., 116, pp. 654 – 659. 关4兴 Drapeau, V., and Wang, D., 1993, ‘‘Verification of a Closed-Loop ShapedInput Controller for a Five-Bar-Linkage Manipulator,’’ presented at IEEE Int. Conf. on Robotics and Automation, Atlanta, GA. 关5兴 Magee, D. P., and Book, W. J., 1995, ‘‘Filtering Micro-Manipulator Wrist Commands to Prevent Flexible Base Motion,’’ presented at American Control Conf., Seattle, WA. 关6兴 Kress, R. L., Jansen, J. F., and Noakes, M. W., 1994, ‘‘Experimental Implementation of a Robust Damped-Oscillation Control Algorithm on a Full Sized, Two-DOF, AC Induction Motor-Driven Crane,’’ presented at 5th ISRAM, Maui, HA. 关7兴 Singer, N., Singhose, W., and Kriikku, E., 1997, ‘‘An Input Shaping Controller Enabling Cranes to Move Without Sway,’’ presented at ANS 7th Topical Meeting on Robotics and Remote Systems, Augusta, GA. 关8兴 deRoover, D., Sperling, F. B., and Bosgra, O. H., 1998, ‘‘Point-to-Point Control of a MIMO Servomechanism,’’ presented at American Control Conference, Philadelphia, PA. 关9兴 Tuttle, T. D., and Seering, W. P., 1994, ‘‘A Zero-Placement Technique for Designing Shaped Inputs to Suppress Multiple-Mode Vibration,’’ presented at American Control Conf., Baltimore, MD. 关10兴 Murphy, B. R., and Watanabe, I., 1992, ‘‘Digital Shaping Filters for Reducing Machine Vibration,’’ IEEE Trans. Rob. Autom., 8, pp. 285–289. 关11兴 Singhose, W., Singer, N., and Seering, W., 1997, ‘‘Time-Optimal Negative Input Shapers,’’ J. Dyn. Syst., Meas. Control, 119, pp. 198 –205. 关12兴 Obergfell, K., 1998, ‘‘End-Point Position Sensing and Control of Flexible Multi-Link Manipulators,’’ Atlanta: Georgia Institute of Technology.

Transactions of the ASME