CRC Press - Robotics and Automation Handbook Episode 1 Part 7 pot

By including an anticipatory corrective action before an error shows up in the response, a feedforward controller can provide much better trajectory tracking than with feedback control a

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Trajectory Planning for Flexible Robots 9-15

Desired Response

FIGURE 9.20 Response to shaped square trajectory.

constructing reaction jet commands, the amount of fuel can be limited or set to a specific amount [39–44]

It is also possible to limit the transient deflection [45, 46] Furthermore, vibration reduction and slewing can be completed simultaneously with momentum dumping operations [47]

The sections above are not intended to be complete, but they are an attempt to give an introduction to and a reference for using command generation in the area of trajectory following The list of successful applications and extensions of command generation will undoubtedly increase substantially in the years

to come

9.3 Feedforward Control Action

Feedforward control is concerned with directly generating a control action (force, torque, voltage, etc.), rather than generating a reference command By including an anticipatory corrective action before an error shows up in the response, a feedforward controller can provide much better trajectory tracking than with feedback control alone It can be used for a variety of cases such as systems with time delays, nonlinear friction [48], or systems performing repeated motions [49] Most feedforward control methods require

an accurate system model, so robustness is an important issue to consider Feedforward control can also

be used to compensate for a disturbance if the disturbance itself can be measured before the effect of the disturbance shows up in the system response A block diagram for such a case is shown in Figure 9.21 The generic control system diagram that was first shown inFigure 9.5shows the feedforward block injecting control effort directly into the plant, as an auxiliary to the effort of the feedback controller Decoupling this type of control from the action of the command generator, which creates an appropri-ate reference command, makes analysis and design of the overall control system simpler However, this nomenclature is not universal There are numerous papers and books that refer to command generation

Plant Feedback

Controller

Feedforward Controller

Disturbance Measurement

FIGURE 9.21 Feedforward compensation of disturbances.

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9-16 Robotics and Automation Handbook

1

ms 2 + bs K

Σ

Σ e-sτ

FIGURE 9.22 Feedforward compensation of a system with a time delay.

as feedforward control In order to establish clarity between these two fundamentally different control techniques, the following nomenclature will be used here:

Command Generation attempts to produce an appropriate command signal to a system The

system could be open or closed loop In an open loop system, the command would be a force acting directly on the plant In a closed-loop system, the command would be a reference signal

to the feedback controller

Feedforward Control produces a force acting directly on the plant that is auxiliary to the feedback

control force Without a feedback control loop, there cannot be feedforward control action

One reason for the inconsistent use of the term feedforward is because some techniques can be employed in

either a feedforward manner or in the role of a command generator However, the strengths and weakness

of the techniques change when their role changes, so this effect should be noted in the nomenclature

9.3.1 Feedforward Control of a Simple System with Time Delay

To demonstrate a very simple feedforward control scheme, consider a system under proportional feedback control that can be modeled as a mass-damper system with a time delay The block diagram for this case

is shown in Figure 9.22 The desired output is represented by the reference signal R The actions of the feedforward control and the feedback control combine to produce the actuator effort U that then produces the actual output Y.

Let us first examine the response of the system without feedforward compensation Suppose that the

feedback control system is running at 10 Hz, the time delay is 0.1 sec, m = 1, and b = 0.5 The dynamic response of the system can be adjusted to some degree by varying the proportional gain K Figure 9.23 shows the oscillatory step response to a variety of K values This is a case where the system flexibility results from the feedback control rather than from the physical plant Note that for low values of K , the system is

− 0.5 0 0.5 1 1.5 2 2.5 3

K = 1 K = 3 K = 7

Time

FIGURE 9.23 Step response of time-delay system without feedforward compensation.

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− 20

− 15

− 10

− 5 0 5 10 15 20

K = 1 K = 3 K = 7

Time

FIGURE 9.24 Control effort without feedforward compensation.

sluggish The system rise time can be improved by increasing K , but that strategy soon drives the system

unstable The corresponding control effort is shown in Figure 9.24

Rather than performing a step motion, suppose the desired motion was a smooth trajectory function such as:

The simple proportional feedback controller might be able to provide adequate tracking if the frequency

of the desired trajectory was very low However, if the trajectory is demanding (relative to system frequency), then the feedback controller will provide poor tracking as shown in Figure 9.25 To improve performance,

we have several options ranging from redesigning the physical system to improve the dynamics and reduce the time delay, adding additional sensors, improving the feedback controller, using command shaping, or adding feedforward compensation Given the system delay, feedforward compensation is a natural choice

A simple feedforward control strategy would place the inverse of the plant in the feedforward block shown inFigure 9.22.If this controller can be implemented, then the overall transfer function from desired response to system response would be unity That is, the plant would respond exactly in the desired manner There are, or course, limitations to what can be requested of the system But, let us proceed with this example and discuss the limitations after the basic concept is demonstrated In this case, the plant

0 1 2 3 4 5

K = 1

K = 3

K = 7 Desired Path

Time

FIGURE 9.25 Tracking a smooth function without feedforward compensation.

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− 0.5 0 0.5 1 1.5 2 2.5 3

Desired Path K = 1 K = 7

Time

FIGURE 9.26 Tracking a smooth function with feedforward compensation.

transfer function including the time delay is

G p= e −s τ

The feedforward controller would then be

G FF = ms2+ bs

Note that this would be implemented in the digital domain, so the time delay in the denominator becomes a time shift in the numerator This time shift would be accomplished by essentially looking ahead

at the desired trajectory Without knowing the future desired trajectory for at least the amount of time corresponding to the delay, this process cannot be implemented

Figure 9.26 shows that under feedforward compensation, the system perfectly tracks the desired trajec-tory for various values of the feedback gain This perfect result will not apply to real systems because there will always be modeling errors Figure 9.27 shows the responses when there is a 5% error in the system mass and damping parameters With a low proportional gain, the tracking is still fairly good, but the system goes unstable for the higher gain If the error is increased to 10%, then the tracking performance with the low gain controller also starts to degrade as shown inFigure 9.28

One important issue to always consider with feedforward control is the resulting control effort Given that the feedback controller generates some effort and the feedforward adds to this effort, the result might

− 0.5 0 0.5 1 1.5 2 2.5 3

Desired Path K = 1 K = 7

Time

FIGURE 9.27 Effect of 5% model errors on feedforward compensation.

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− 200

− 150

− 100

− 50 0 50 100 150 200

Time

FIGURE 9.29 Control effort tracking various frequencies with feedforward compensation.

Fapplied

Ffriction

MB

Y

MA

FIGURE 9.30 Model of system with Coulomb friction.

If a reasonable model of the friction dynamics exists, then a feedforward compensator could be useful

to improve trajectory tracking A block diagram of such a control system is shown in Figure 9.31 Note again that the feedforward compensator contains an inverse of the plant dynamics

Let us first examine the performance of the system without the feedforward compensation To do this,

we start with a baseline system where Mass A is 1, Mass B is 0.5, the spring constant is 15, the damping constant is 0.3, and the coefficient of friction is 0.3.Figure 9.32shows the response of Mass A for various values of the feedback control gain when the desired path is a cosine function in Equation (9.23) For a gain

of 1, the control effort is too small to overcome the friction, so the system does not move Larger values

of gain are able to break the system free, but the trajectory following is very poor.Figure 9.33shows the response of Mass B for the same range of feedback gains This part of the system responds with additional flexible dynamics The corresponding control effort is shown inFigure 9.34

When the feedforward compensator is turned on, the tracking improves greatly, as shown inFigure 9.35

The control effort necessary to achieve this trajectory following is shown inFigure 9.36.The control effort

Σ

Gp

S Friction

Fapplied

S

G1p

Σ +

−

FIGURE 9.31 Block diagram of friction system with feedforward compensation.

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0 0.5 1 1.5 2

Desired 5% Error 10% Error 15% Error 20% Error

Time

FIGURE 9.37 Response of mass B with modeling errors.

this controller, the plant transfer function is written in the following form:

G (z−1)= z −d B a (z−1)B u (z−1)

The numerator is broken down into three parts — a pure time delay z −d, a part that is acceptable for

inversion B a , and a part that should not be inverted B u

9.3.4 Conversion of Feedforward Control to Command Shaping

Consider the control structure using a ZPETC controller shown inFigure 9.39.The physical plant and the feedback controller have been reduced to a single transfer function described by

G c (z−1)= z −d B c a (z−1)B u

c (z−1)

where B u

c (z−1)= b u

c 0 + b u

c 1 z−1+ · · · + b u

c s z −s The c subscripts have been added to denote the transfer

function now represents the entire closed-loop system dynamics The superscripts on the numerator again refer to parts that are acceptable and unacceptable for inversion In this case, the output of the ZPETC is

a reference signal for the closed-loop controller It does not directly apply a force to the plant It therefore

0 0.5 1 1.5 2

Desired Path 5% Error 10% Error 15% Error 20% Error

Time

FIGURE 9.38 Response of mass B with modeling errors and slower tr-ajectory.

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Feedforward control uses a system model to create auxiliary forces that are added to the force generated

by the feedback controller In this way, it can produce a corrective action before the feedback controller has sensed the problem With knowledge of the intended trajectory, feedforward control can drive the system along the trajectory better than when only feedback control is utilized Because feedforward control can dominate the forces from the feedback control, unexpected disturbances and errors in the system model may be problematic when an aggressive feedforward control system is operating

A highly successful robotic system for trajectory following would likely be composed of four well-designed components: hardware, feedback control, feedforward control, and command shaping Each of the components has its strengths and weaknesses Luckily, they are all very compatible with each other and, thus, a good solution will make use of all of them when necessary

References

[1] Magee, D.P and Book, W.J., Eliminating multiple modes of vibration in a flexible manipulator, presented at IEEE International Conference on Robotics and Automation, Atlanta, GA, 1993 [2] Honegger, M., Codourey, A., and Burdet, E., Adaptive control of the hexaglide, a 6 DOF parallel ma-nipulator, presented at IEEE International Conference on Robotics and Automation, Albuquerque, 1997

[3] Singhose, W., Singer, N., and Seering, W., Improving repeatability of coordinate measuring machines

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[11] Singer, N.C., Seering, W.P., and Pasch, K.A., Shaping command inputs to minimize unwanted dynamics, MIT, Ed.: U.S Patent 4,916,635, 1990

[12] Singhose, W.E., Seering, W.P., and Singer, N.C., Input shaping for vibration reduction with specified insensitivity to modeling errors, presented at Japan-U.S.A Symposium on Flexible Automation, Boston, MA, 1996

[13] Singhose, W.E., Porter, L.J., Tuttle, T.D., and Singer, N.C., Vibration reduction using multi-hump

input shapers, ASME J Dynamic Syst., Meas., Control, 119, 320–326, 1997.

[14] Singhose, W., Singer, N., Rappole, W., Derezinski, S., and Pasch, K., Methods and appara-tus for minimizing unwanted dynamics in a physical system, June 10: U.S Patent 5,638,267, 1997

[15] Hyde, J.M and Seering, W.P., Using input command pre-shaping to suppress multiple mode vibra-tion, presented at IEEE International Conference on Robotics and Automavibra-tion, Sacramento, CA, 1991

[16] Singh, T and Heppler, G.R., Shaped input control of a system with multiple modes, ASME J Dynamic

[17] Singhose, W.E., Crain, E.A., and Seering, W.P., Convolved and simultaneous two-mode input

shapers, IEE Control Theory and Applications, 515–520, 1997.

[18] Pao, L.Y., Multi-input shaping design for vibration reduction, Automatica, 35, 81–89, 1999.

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[19] Feddema, J.T., Digital filter control of remotely operated flexible robotic structures, presented at American Control Conference, San Francisco, CA, 1993

[20] Kress, R.L., Jansen, J.F., and Noakes, M.W., 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, 1994

[21] Singer, N., Singhose, W., and Kriikku, E., An input shaping controller enabling cranes to move without sway, presented at ANS 7th Topical Meeting on Robotics and Remote Systems, Augusta,

GA, 1997

[22] Singhose, W., Porter, L., Kenison, M., and Kriikku, E., Effects of hoisting on the input shaping

control of gantry cranes, Control Eng Pract., 8, 1159–1165, 2000.

[23] Jansen, J.F., Control and analysis of a single-link flexible beam with experimental verification, Oak Ridge National Laboratory ORNL/TM-12198, December 1992

[24] Magee, D.P and Book, W.J., Filtering micro-manipulator wrist commands to prevent flexible base motion, presented at American Control Conference, Seattle, WA, 1995

[25] Seth, N., Rattan, K., and Brandstetter, R., Vibration, control of a coordinate measuring machine, presented at IEEE Conference on Control Apps., Dayton, OH, 1993

[26] Jones, S and Ulsoy, A.G., An approach to control input shaping with application to coordinate

measuring machines, J Dynamics, Meas., Control, 121, 242–247, 1999.

[27] Rappole, B.W., Singer, N.C., and Seering, W.P., Multiple-mode impulse shaping sequences for re-ducing residual vibrations, presented at 23rd Biennial Mechanisms Conference, Minneapolis, MN, 1994

[28] deRoover, D., Sperling, F.B., and Bosgra, O.H., Point-to-point control of a MIMO servomechanism, presented at American Control Conference, Philadelphia, PA, 1998

[29] deRoover, D., Bosgra, O.H., Sperling, F.B., and Steinbuch, M., High-performance motion control

of a wafer stage, presented at Philips Conference on Applications of Control Technology, Epe, The Netherlands, 1996

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on MACE: Mission Results, presented at IFAC World Congress, San Francisco, CA, 1996

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Dynamic Sys., Meas., Control, 121, 443–447, 1999.

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229–236, 1998

[33] Pao, L.Y., Minimum-time control characteristics of flexible structures, J Guidance, Control,

Dynam-ics, 19, 123–29, 1996.

[34] Pao, L.Y and Singhose, W.E., Verifying robust time-optimal commands for multi-mode flexible

spacecraft, AIAA J Guidance, Control, Dynamics, 20, 831–833, 1997.

[35] Tuttle, T and Seering, W., Creating time optimal commands with practical constraints, J Guidance,

Control, Dynamics, 22, 241–250, 1999.

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[37] Singh, T and Vadali, S.R., Robust time-optimal control: a frequency domain approach, J Guidance,

[38] Cutforth, C.F and Pao, L.Y., A modified method for multiple actuator input shaping, presented at American Control Conference, San Diego, CA, 1999

[39] Meyer, J.L and Silverberg, L., Fuel optimal propulsive maneuver of an experimental structure

exhibiting spacelike dynamics, J Guidance, Control, Dynamics, 19, 141–149, 1996.

[40] Singhose, W., Bohlke, K., and Seering, W., Fuel-efficient pulse command profiles for flexible

space-craft, AIAA J Guidance, Control, Dynamics, 19, 954–960, 1996.

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Meas., Control, 121, 206–212, 1999.

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[42] Wie, B., Sinha, R., Sunkel, J., and Cox, K., Robust fuel- and time-optimal control of uncertain flexible space structures, presented at AIAA Guidance, Navigation, and Control Conference, Monterey, CA, 1993

[43] Lau, M and Pao, L., Characteristics of time-optimal commands for flexible structures with limited

fuel usage, J Guidance, Control, Dynamics, 25, 2002.

[44] Singh, T., Fuel/time optimal control of the benchmark problem, J Guidance, Control, Dynamics, 18,

1225–31, 1995

[45] Singhose, W., Banerjee, A., and Seering, W., Slewing flexible spacecraft with deflection-limiting

input shaping, AIAA J Guidance, Control, Dynamics, 20, 291–298, 1997.

[46] Kojima, H and Nakajima, N., Multi-objective trajectory optimization by a hierarchical gradient algorithm with fuzzy decision Logic, presented at AIAA Guidance, Navigation, and Control Con-ference, Austin, TX, 2003

[47] Banerjee, A., Pedreiro, N., and Singhose, W., Vibration reduction for flexible spacecraft following

momentum dumping with/without slewing, AIAA J Guidance, Control, Dynamics, 24, 417–428,

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Error Budgeting

Daniel D Frey

Massachusetts Institute of Technology

10.1 Introduction 10.2 Probability in Error Budgets 10.3 Tolerances

10.4 Error Sources 10.5 Kinematic Modeling 10.6 Assessing Accuracy and Process Capability

Sensitive Directions • Correlation among Multiple Criteria

• Interactions among Processing Steps • Spatial Distribution

of Errors

10.7 Modeling Material Removal Processes 10.8 Summary

10.1 Introduction

An error budget is a tool for predicting and managing variability in an engineering system Error budgets are particularly important for systems with stringent accuracy requirements such as robots, machine tools, coordinate measuring machines, and industrial automation systems To provide reasonable predictions of accuracy, an error budget must include a systematic account of error sources that may affect a machine Error sources include structural compliance, thermally induced deflections, and imperfections in machine components An error budget must transmit these error sources through the system and determine the effects on the position of end effectors, work-pieces, and/or cutting tools Ultimately, an error budget should combine the effects of multiple error sources to determine overall system performance which is often determined by the tolerances held on finished goods

Error budgeting was first comprehensively applied to the design of an industrial machine by Donaldson [1] Over the past two decades error budgeting methods have been improved and extended by several researchers [2–7] and applied to a wide variety of manufacturing-related systems [1, 8–11] Error budgets are also frequently applied to optical systems, weapons systems, satellites, and so on, but this article will focus on robotics, manufacturing, and industrial automation

An appropriately formulated error budget allows the designer to make better decisions throughout the design cycle When bidding on a machine contract, an error budget provides evidence of feasibility During conceptual design, an error budget aids in selection among different machine configurations During system design the error budget can be used to allocate allowable error among subsystems to balance the level of difficulty across design teams At the detailed design stage, an error budget can inform selection of components, materials, and manufacturing process steps Error models of machine tools can also be used to enhance machine accuracy [12] During operation, an error model can be used for diagnosis if system accuracy degrades [13] or to help schedule routine maintenance [5] Error budgets can

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