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 in Figure 9.5 shows 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 ΣΣ Feedforward Controller Disturbance Measurement FIGURE 9.21 Feedforward compensation of disturbances. Copyright © 2005 by CRC Press LLC 9 -16 Robotics and Automation Handbook 1 ms 2 + bs K Σ Feedforward Controller Σ e -sτ R Y U 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 246810 K = 1 K = 3 K = 7 Response Time FIGURE 9.23 Step response of time-delay system without feedforward compensation. Copyright © 2005 by CRC Press LLC Trajectory Planning for Flexible Robots 9 -17 −20 −15 −10 −5 0 5 10 15 20 0246810 K = 1 K = 3 K = 7 Control Effort 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: r (t) = 1 −cos(ωt) (9.23) The simple proportional feedback controller might be able to provide adequate tracking if the frequency ofthedesiredtrajectorywas very low.However,if thetrajectory isdemanding (relativetosystemfrequency), 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 in Figure 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 0246810 K = 1 K = 3 K = 7 Desired Path Response Time FIGURE 9.25 Tracking a smooth function without feedforward compensation. Copyright © 2005 by CRC Press LLC 9 -18 Robotics and Automation Handbook −0.5 0 0.5 1 1.5 2 2.5 3 0246810 Desired Path K = 1 K = 7 Response Time FIGURE 9.26 Tracking a smooth function with feedforward compensation. transfer function including the time delay is G p = e −sτ ms 2 + bs (9.24) The feedforward controller would then be G FF = ms 2 + bs e −sτ (9.25) 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 in Figure 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 0246810 Desired Path K = 1 K = 7 Response Time FIGURE 9.27 Effect of 5% model errors on feedforward compensation. Copyright © 2005 by CRC Press LLC 9 -20 Robotics and Automation Handbook −200 −150 −100 −50 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 0.5 Hz 1 Hz 2 Hz Control Effort Time FIGURE 9.29 Control effort tracking various frequencies with feedforward compensation. F applied F friction M B Y M A 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 9.32 shows 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 9.33 shows 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 in Figure 9.34. When the feedforward compensator is turned on, the tracking improves greatly, as shown in Figure 9.35. The control effort necessary to achieve this trajectory following is shown in Figure 9.36. The control effort Σ Feedforward Controller R Y G p S Friction F applied S G p 1 K Σ Σ Σ + − − FIGURE 9.31 Block diagram of friction system with feedforward compensation. Copyright © 2005 by CRC Press LLC Trajectory Planning for Flexible Robots 9 -23 0 0.5 1 1.5 2 0 5 10 15 20 Desired 5% Error 10% Error 15% Error 20% Error Response 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 ) A(z −1 ) (9.31) 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 in Figure 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 a c (z −1 )B u c (z −1 ) A c (z −1 ) (9.32) where B u c (z −1 ) = b u c0 + b u c1 z −1 +···+b u cs z −s .Thec 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 0 5 10 15 20 25 Desired Path 5% Error 10% Error 15% Error 20% Error Response Time FIGURE 9.38 Response of mass B with modeling errors and slower tr-ajectory. Copyright © 2005 by CRC Press LLC Trajectory Planning for Flexible Robots 9 -25 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 ofcoordinate measuring machines with shaped command signals, Precision Eng., 18, 138–146, 1996. [4] Smith, O.J.M., Posicast control of damped oscillatory systems, Proceedings of the IRE, 45, 1249–1255, 1957. [5] Smith, O.J.M., Feedback Control Systems, McGraw-Hill, New York, 1958. [6] Bolz, R.E. and Tuve, G.L., CRC Handbook of Tables for Applied Engineering Science, CRC Press, Boca Raton, FL, 1973. [7] Singer, N.C. and Seering, W.P., Preshaping command inputs to reduce system vibration, J. 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Copyright © 2005 by CRC Press LLC 9 -26 Robotics and Automation Handbook [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. 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[41] Singhose, W., Singh, T., and Seering, W., On-off control with specified fuel usage, J. Dynamic Syst., Meas., Control, 121, 206–212, 1999. Copyright © 2005 by CRC Press LLC Trajectory Planning for Flexible Robots 9 -27 [42] Wie, B., Sinha, R., Sunkel, J., andCox,K., Robustfuel- 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. 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Copyright © 2005 by CRC Press LLC 10 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 Copyright © 2005 by CRC Press LLC [...]... Equation (10 .15 ) is most often used in machine tool error budgets, but some linear joints are arranged the other way, so Equation (10 .16 ) is sometimes needed 1 ε z −ε y 0 Copyright © 2005 by CRC Press LLC −εz 1 εy X + δx −εx δ y + Xεz εx 1 0 0 δz + Xε y 1 (10 .16 ) 1 0 -1 0 Robotics and Automation Handbook The complete kinematic model is formed by multiplying all the joint and shape HTMs... Figure 10 .8 Let us include only six error motions in this model as described in Table 10 .2 The resulting HTM follows the form of Equation (10 . 17 ) with alternating shape and joint transformations 1 0 0 T3 = 0 0 cos( z1 + εz1 ) − sin z1 sin 1 0 0 mm cos( z1 + εz1 ) z1 · 0 1 1000 mm 0 0 0 0 1 0 0 0 0 0 mm 0 0 mm 0 0 mm 1 0 mm 0 1 0 0 500 mm cos( z2 + εz2 ) − sin z2 0 1. .. ) 2 z2 · · 0 xp2 1 60 mm 0 0 0 0 0 1 0 0 1 0 · 0 0 0 0 400 mm 1 0 0 1 0 0 1 0 ε y3 0 mm 0 0 0 0 0 mm 0 0 mm 1 0 mm 0 1 0 mm 0 1 −εx3 0 mm · 0 mm −ε y3 εx3 1 −Z − δz3 1 1 (10 .19 ) 1 The HTM can be used to compute the location of the tip of the end effector at any location in the working volume as a function of the joint motions and error motions ... and 2 are modeled using Equation (10 .14 ) Joint 3 is best modeled using Equation (10 .16 ) assuming the quill is supported by a set of bearings fixed to the arm of the robot rather than to the quill 500 mm 400 mm Θz2 Θz1 60 mm Z 300 mm 10 00 mm Point p z Base y x FIGURE 10 .8 A robot to be modeled kinematically Copyright © 2005 by CRC Press LLC 1 0 -1 1 Error Budgeting TABLE 10 .2 Error motions for the Example... motions and the commanded translations must be reversed This kind of joint is, therefore, modeled by the equation below Note the difference in Figure 10 .6 and Figure 10 .7 When the commanded motion X is zero, the effect of an error motion ε y is the same with either Equation (10 .15 ) or Equation (10 .16 ) But when X is non-zero, the error motion ε y causes a much larger z displacement using Equation (10 .16 )... variation present It is a useful measure W Lead >25%W Land FIGURE 10 .3 Specification for foot side overhang of electronic leads Copyright © 2005 by CRC Press LLC 1 0-6 Robotics and Automation Handbook TABLE 10 .1 Sources of Error Typically Included in an Error Budget Error Source Errors for linear axes Straightness Squareness Angular error motions (roll, pitch, and yaw) Drive related errors Errors for rotary... 0 0 0 0 1 0 0 0 1 (10 .13 ) where X, Y , and Z are displacements in the x, y, and z directions and x , y , and z are rotations about the x, y, and z axes This form of HTM allows for the large rotations required to model the commanded motions of the robot or machine tool HTMs can be used to model the joints in robots and machine tools For a rotary joint, let z represent the commanded motion of... represent the commanded motion of the joint in the x direction while δx , δ y , δz and εx , ε y , εz represent the translational and rotational error motions of the joint, respectively Entering these symbols into the general form of HTM (Equation (10 .13 )) and employing small angle approximations to simplify yields 1 ε z −ε y 0 −εz εy X + δx 1 −εx δy εx 1 δz 0 0 1 (10 .15 ) If the linear... all the joint and shape HTMs in sequence Thus, an HTM that models a kinematic chain of n LCSs from the base to the end effector is 0 Tn = 0 S1 · J1 · 1 S2 · J2 · Jn 1 n 1 Sn (10 . 17 ) With this HTM in Equation (10 . 17 ), one can carry out analysis of the position and orientation of any object held by the end effector by mapping local coordinates into the base (global) coordinate system If the coordinates... concept was e Copyright © 2005 by CRC Press LLC 1 0 -1 6 Robotics and Automation Handbook Initial Tool Position Final Tool Position Workpiece Stock (a) The cutting tool in its initial and final positions with respect to the workpiece Swept Volume of the Cutting Tool Machined Surface Machined Workpeice (b) The swept volume of the cutting tool as it moves between the initial and final positions Generator ^ . (10 .16 ) is sometimes needed. 1 −ε z ε y X + δ x ε z 1 −ε x δ y + Xε z −ε y ε x 1 δ z + Xε y 000 1 (10 .16 ) Copyright © 2005 by CRC Press LLC 10 -1 0 Robotics and Automation Handbook The. Control, 11 5, 3 41 3 47, 19 93. [ 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, 19 97. [18 ] Pao,. feedforward compensation. Copyright © 2005 by CRC Press LLC 9 -2 0 Robotics and Automation Handbook −200 15 0 10 0 −50 0 50 10 0 15 0 200 0 0.5 1 1.5 2 2.5 3 0.5 Hz 1 Hz 2 Hz Control Effort Time FIGURE