CRC Press - Robotics and Automation Handbook Episode 2 Part 8 ppsx

Flexible Robot Arms 24 -25 The generalized force corresponding to joint variable q i is the joint torque F i in those cases where the boundary conditions are clamped, as will be assumed here. d dt (∂ K/∂ ˙ q j ) −∂ K/∂q j + ∂V e ∂q j + ∂V g ∂q j = F j (24.89) d dt (∂ K/∂ ˙ δ jf ) −∂ K/∂δ jf + ∂V e ∂δ jf + ∂V g ∂δ jf = 0 (24.90) 24.2.6.1 Finite Element Representations Additional detail can be added to a model by using more exact geometry for the structural elements; however, a detailed finite element model will contain at least an order of magnitude more variables than a well-derived assumed modes model. Fortunately, it is possible to combine the best of both techniques. Examination of the above shows that the result of the specification of the geometry (mass and stiffness distribution) and all the analysis thatensuesareembodiedinamodalmassmatrix, a modal stiffness matrix, and similarparameter terms thataccount for the Coriolis andcentrifugal terms. Thesematrices can also be constructed from basis shapes resulting from a finite element representation of the components followed by a specification of approximate boundary conditions on each element separately and determination of the resulting mode shape. This process is fairly direct for the finite element expert but beyond the scope of this volume. 24.2.6.2 Approximations of This Method The reader should be aware that the above technique, while satisfactory for most robotic applications, is an approximation that will be inaccurate in some instances. Some of these instances have already been alluded to, such as the case of large elastic deformation. More subtle inaccuracies exist that bear names associated with the conditions of their occurrence. If the base of the arm is rotating rapidly about the Z axis, vibrations in the X −Y plane would not account for the position dependent body forces effectively stiffening the beam and leading to the phenomena of centrifugal stiffening. Hence, these equations are not recommended for modeling helicopter rotors or similar rapidly rotating structures. 24.2.7 Simulation Form of the Equations Twoformsoftheequationsofmotion are commonlyfound useful.For purposes ofsimulationthefollowing structure may be used:  M rr (q) M rf (q) M fr (q) M ff  ¨ q r ¨ q f  =−  00 0 K s  q r q f  + N(q, ˙ q) +G(q) = R(q, ˙ q, Q) (24.91) where q r =rigid coordinates or joint variables q f =flexible coordinates M ij =the “mass” matrix for rigid and flexible coordinates corresponding to the rigid (i, j = r)or flexible (i, j = f ) coordinates and equations N(q, ˙ q) =nonlinear Coriolis and centrifugal terms G(q) =gravity effects Q =externally applied forces R =effect of external and restoring forces and all other nonconservative forces including friction For serial arms it is generally more difficult to produce this form of the equations, because all second time derivatives must be collected. For simulation the second derivatives must besolved for, given the input forces Q, the coordinates, and their first derivatives. An integration scheme, often one capable of handling stiff systems of equations, is then used to integrate these equations, forward in time twice, given initial Copyright © 2005 by CRC Press LLC 24 -26 Robotics and Automation Handbook conditions and external forcing Q(t). The solution for ¨ q requires either the inversion of its coefficient matrix or, more, efficiently another means such as singular decomposition to solve these equations for known constant values (R is constant) of all other terms. This solution must be repeated at each time step, leading to the major computational cost of the process. 24.2.8 Inverse Dynamics Form of the Equations The inverse dynamics solution seeks the inputs Q that would yield a known time history of the coordinates q, ˙ q, and ¨ q. The inverse dynamics equations are less readily applied for flexible arms than for rigid arms, not due to the solutionof the equationas given below, but because the values of q f are not readily obtained, especially for some models. That situation is improved slightly if the rigid coordinates are chosen, not as joint angles, but as the angles connecting the preceding and following axes. In such a way the tip of the arm can be prescribed only in terms of q r . We know all velocities should come to zero after the arm reaches its final position. If gravity is to be considered, the static deflections at the initial and final times will also be needed. These can be solved for without knowing the dynamics. In the common case that Q = R Q =  M rr (q) M rf (q) M fr (q) M ff  ¨ q r ¨ q f  +  00 0 K s  q r q f  − N(q, ˙ q) −G(q) (24.92) In general Q is dependent on the vector T of motor torques or forces at the joints, and the distribution of these effects is such that Q =  B r B f  T (24.93) where T is the same dimension as q r . By rearranging the inverse dynamics equation we obtain  B r −M rf B f −M ff  T ¨ q f  =  M rr M rf  ¨ q r + N(q, ˙ q) +G(q) (24.94) which is arranged with avector of unknowns on the left side of theequation. Unknowns may also appear in the vector of nonlinearand gravitational terms which contain q f and ˙ q f , and strictly speaking, the solution must proceed simultaneously. In practice these are typically weak functions of the flexible variables and the solution can proceed as a differential-algebraic equation. It is worth noting what happens if we choose other rigid coordinates, such that the tip position is not described by q r alone. Now B f is zero, B r = I , and the tip position is unknown. We may be satisfied with specifying the joint coordinates, however, during the motion of the robot and accept the resulting deflection of the tip. Further note that the equations are likely to be nonminimum phase. Resorting to linear thinking forsimplicity,the transfer functionshave zeros in the right halfplane. The inverse dynamics effectively inverts the transfer function giving poles in the right half plane. Normal solution techniques produce an unstable solution which is causal. Another solution, the acausal, stable solution exists and will be found to be useful when we discuss control via inverse dynamics. 24.2.9 System Characteristic Behavior Nonminimum phase behavior will be observed at the tip of a uniform beam subject to a torque input at its root. In a flexible arm this manifests itself as an initial reverse action to a step torque input. In other words, the initial movement of the tip is in the opposite direction of its final displacement. The consequences of this behavior are pervasive if the tip position is measured and used for feedback control. To see this effect, a linear analysis is sufficient, and the transfer matrix frequency domain analysis is well suited for this purpose. The nonminimum phase in this case is described by zeros in the right half plane for the transfer function from torque to position. When a feedback loop is closed, the root locus analysis shows that the Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -27 positive zero must attract one branch of the root locus into the right, unstable region of the complex plane. The usual light damping of the flexible vibration modes allows this to happen with low values of feedback gain. A uniform beam shows these characteristics, and it has been shown (Girvin [14]) that tapering a beam reduces the severity of the nonminimum phase problem but does not eliminate it. 24.2.10 Reduction of Computational Complexity The equations presented above can result in very complex relationships that need to be solved in order to be used either for simulation or for inverse dynamics. It is worth enumerating several ways of reducing this complexity. The number of basis shapes can be reduced if assumed modes close to the true modes are used for basis shapes. Several terms in the equation involve integration of the product of two basis shapes. If orthogonal assumed modes are usedthe cross product of two different modeshapes is zero when integrated over the length of the beam. 24.3 Control The flexibility of a motion system becomes an issue when the controlled natural frequency becomes comparable to the vibrational frequencies of the arm in the direction of movement. One of the first things to know is when this might occur. If it is expected to be an issue, the next thing to consider is ways to ameliorate the problem. This involves a great number of possibilities, many of which involve control, but not exclusively control. Control should not be interpreted to exclusively mean feedback control, because open loop approaches are also very valuable. 24.3.1 Independent Proportional Plus Derivative Joint Control The vast majority of feedback controlled motion devices, including mechanical arms, use some of pro- portional plus integral plus derivative (PID) control. Velocity feedback may be an alternative to derivative feedback. The simplest approach here is to apply the PID algorithm to each joint independently. In this case the actuation of other joints can create a disturbance on each joint which could have been anticipated with a centralized approach. This problem was analyzed in the earliest work considering the flexibility of robot arms [11]. Because the parameters involved in a PD controller (no integral action because it has minimal effect) are only two per joint, the problem can be effectively studied by simply sweeping through the design space. The pole positions of the controlled flexible system provide a way to judge the controller effectiveness. PD control of a rigid degree-of-freedom enables arbitrary placement of the system poles. The dominant poles of the flexible arm behave as a rigid system when the lowest natural frequency of the system is about one-fifth the flexible natural frequency with the joints clamped. Critical damping on the dominant modes cannot be obtained by adding additional derivative gain when the servo frequency is above about one third of the structural natural frequency. Damping reaches a maximum and then de- creases with additional velocity or derivative feedback. The effect of derivative feedback for three values of proportional feedback is sketched in Figure 24.4. While more joints and links distort this conclusion to some extent, servo bandwidths above one third the system’s natural frequency are very hard to achieve. To the extent that true PD action can be obtained, the independent joint controlled arm will not go unstable but will become highly oscillatory with high gains. The root locus converges to the position of the clamped joint natural frequencies which make up the zeros of this system since this is the condition of zero joint motion. Higher modes that start at the pinned-free natural frequencies are seen to converge to the higher clamped-free natural frequencies. Zero joint motion means the joints are unable to move in response to vibration and thereby unable to remove the vibrational energy. Note that heavily geared joints or joints actuated by hydraulic actuators may not be back drivable in any event, and hence, the shaping of the commanded motion becomes the most effective way to damp vibration. Low gains still have the effect of smoothing out abrupt commands that would excite vibration although at the expense of rapid and precise response. Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -29 –1 –0.8 –0.6 –0.4 –0.2 0 Real Axis 0.2 0.4 0.6 0.8 1 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 X 3 X 2 X 1 + Image Axis FIGURE 24.5 Noncollocated feedback of dx 1 /dt to apply force at x 3 produces unstable (dashed) root locus. (Collo- cated feedback shown in solid lines.) To avert the limitations of flexibility a reasonable but na ¨ ıve approach to improve control performance, PID control based on measuring the end effector position fails because ofthenonminimumphase behavior (resulting from noncollocated sensors and actuators) of the system. Even simple systems of this type involving two masses connected by a spring illustrate the effect as shown in Figure 24.5 in a root locus plot. 24.3.2 Advanced Feedback Control Schemes Because the number of control algorithms that have been applied to flexible arms is extremely large, it is not possible to cover all of them in this treatment. Other texts and handbooks exist to present the details of these algorithms [15]. It is also impossible to verify that all of these approaches have practical merit, although there may be merit in specific cases to all of them. In this subsection the discussion will begin with the objectives and obstacles to practical advantages of these designs. 24.3.2.1 Obstacles and Objectives for Advanced Schemes The modeling of the flexible arm involves many assumptions and presumptions. We assume that joint friction is linear or at least known. We presume to know the mass, stiffness properties, and geometries of the links and payloads. We have assumed a serial structure or at least one that could be approximated as such. Some of the controls proposed use measurements of the system states or related variables. Sensors Copyright © 2005 by CRC Press LLC 24 -30 Robotics and Automation Handbook to measure these variables over the workspace may not exist, and actuators that apply desired (collocated) inputs are unavailable or inappropriate. Some traditional actuators are not back drivable and hence don’t serve as force or torque sources as the algorithms require. Computation to implement some algorithms is enormous, and hardware to perform these computations (typically done digitally) does not always exist at this time although the trend in computational speed is encouraging. Related to the computational complexity is the model order. Because a perfect model of the structure would contain an infinite number of modes, practical control design often violates perfection by working with a limited number of modes. Hence, the unmodeled modes may be excited by our attempts to control the modeled modes (control spillover), or our attempt to measure the modeled modes may be corrupted by the unmodeled modes (measurement spillover) that do affect the measurement. One of the objectives of a practical controller is to overcome these obstaclesand thereby move the flexible arm more rapidly, precisely, and reliably through its prescribed motions. 24.3.2.2 State Feedback with Observers The flexibility is modeled as a collection of second order equations that interact. In addition, the actuators will add dynamics that may be important. By converting this model to a state space model the control task is reduced to moving the arm states to a desired point in state space. That desired point is based on the desired position and velocity of the end effector, thus specifying the output of the system, plus the fact that all vibrations should quickly converge to zero. Very powerful techniques exist for doing this if the model of the system is linear. While the model is never perfectly linear, linearity may be a good approximation in the region around an operating point of larger or smaller size. To implement a regulator bringing the state to the origin of the state space the following model and feedback law can be used. The output is the vector y. The plant matrix A, the input matrix B, the output matrix C, and the feed through matrix D are all required to be of appropriate dimension. ˙ x = Ax + Bu u = Kx (24.95) y =Cx + Du Anextension ofthesebasicequationscanproduce acontrollerthattracksatimevarying referencetrajectory as well. The linear equations can be written so that the origin of the state space produces the desired value of y. There is no unique choice for states x. This allows one to transform to different representations for different purposes. In terms of the models derived above the state, variables may be joint variables and their derivatives and deflection variables and their derivatives. The first obstacle to this algorithm is measurement of all the needed states. In particular the deflection variables and their derivatives are not directly available. Strain gages have been used to effectively obtain this information and state feedback is then approximately achievable [16, 17, 18]. For each basis shape used to model a beam, one additional measurement will enable calculation of itsamplitude. The derivative of that amplitude can then be approximated with a filtered difference equation. Stateobservers aredynamicsystems that use presentandpastmeasurements of some outputs toconverge on estimates of the complete system states. Full order observers have the same number of states as the system they observe, while reduced order observers allow the measured variables to be used directly with a proportional reduction of theobserver order.Observers, that have been optimized according to a quadratic performance index assuming Gaussian noise is corrupting the measurements and the control input, are often referred to as Kalman filters after the pioneer in the use of this technique. Evaluation of the feedback matrix K and the gains of the observer system can be done in various ways. The optimization scheme referred to above is popular but requires some knowledge or assumption of the strengths of the noises corrupting inputs and measurements. Pole placement is another approach that allows the engineer more direct influence over the resulting dynamics. The choice of pole positions that are “natural” for the system and yet desirable for the application can be challenging. These approaches are discussed in appropriate references [15]. Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -31 Knowledge of the plant model is required to construct the observer and to design the feedback matrix. Sensitivity to error in this model is one of the major issues when seeking high performance from state feedback. Nonlinear models complicate this issue immensely. 24.3.2.3 Strain and Strain Rate Feedback Joint position is commonly differentiated numerically to approximate velocity andsimilarly link deflection can be differentiated numerically to complete the vector of state variables to be used in the state feedback equation. Thus, to control two modes, at least two strain gage readings are needed for effective flexible arm control which is not extremely sensitive to model parameters [16]. This approach has been implemented with good results as shown in Yuan [19]. Those experiments incorporated hydraulic actuation ofa6m long, two-link arm and roughly cut the settling time of PD control in half. Hence, the complexity of an observer was not required, although filtering of the noisy differentiated signals may be appropriate. One of the obstacles to the general application of strain feedback, or state feedback in general, is the lack of a credible trajectory for the flexible state variables. The end user knows where the tip should be, but the associated strain will not be zero during the move. See the section below on inverse dynamics for a solution to this dilemma. 24.3.2.4 Passive Controller Design with Tip Position Feedback While it is natural to want to feed back the position of the point of interest, the nonminimum phase nature of the flexible link arm results in instability. Nonminimum phase in linear cases results from zeros of the transfer function in the right half plane. The output can be modified to be passive and of minimum phase by closing an inner loop or by redefining the output as a modified function of the measured states. For example, if tip position is computed from ameasured jointangle θ and a measured deflection variable δ for a link of length l, the form of the equation will be (tip position) = lθ +δ and the system is nonminimum phase. If, on the other hand, the (reflected tip position) = lθ −δ is used for feedback control, the system is passive and can be readily controlled for moderate flexibility [20]. Obergfell [21] measured deflection of each link of a two link flexible arm and closed each inner loop using this measurement. Then a vision measurement of the tip position was used to control the passified system with a simple controller. He used classical root locus techniques to complete compensator design. The device and the nature of the results are shown in Figure 24.6 and Figure 24.7. 24.3.2.5 Sliding Mode Control Slidingmodecontroloperatesintwophases.First,the systemisdriven instatespacetowardaslidingsurface. On reaching the sliding surface, the control is switched in a manner to move along the surface toward the origin, set up to be the desired equilibrium point. The switching operation permits an extremely robust behavior for many systems, but if there are unmodeled dynamics, the abrupt changes may excite these modes more strongly than other controllers. Given the concern for robustness with model imperfections and changes, the sliding mode has a natural appeal. The implementation of sliding mode control is subject to some of the same problems as state feedback controllers; that is, knowledge of the state is difficult to obtain by measurement. Frame [22] has used observers based on a combination of joint position, tip acceleration, and tip position (via camera) measurements. The results are recent and, while promising, have not produced a clear advantage over state feedback. 24.3.3 Open Loop and Feedforward Control Three matters for discussion in this section are the initial generation of the motion trajectory for a flexible link system, either theend point orthe joints and the modification of anexisting trajectory to create amore compatible input. Finally, by observing the errors in tracking a desired trajectory, the learning control approach can improve the trajectory on successive iterations, reducing the demands on the feedback control. Copyright © 2005 by CRC Press LLC 24 -34 Robotics and Automation Handbook spaced around a circle, symmetric about the real axis, and no zeros. The stable poles are maintained in the filter design. The filter can then be converted to a digital filter by various mapping techniques. Chebyshev filters have poles similarly placed about an ellipse, and superior frequency domain specifications are achieved in the pass band but with some fluctuations in the magnitude of the response. For filtering the robot trajectory, we are also concerned about the time response of the filter itself and do not want the filter to insert oscillations of its own. Implementation of a low pass filter such as a Butterworth filter results in an infinite impulse response (IIR) filter as a result of the system poles. Theoretically, the response to an input never dies out. Artifacts of the filter include phase shift and an extension of the period of time over which the commanded motion continues. In the design of the low pass filter the engineer must base the cutoff frequency on the vibration frequencies of the arm. All components with frequencies above the cutoff are reduced in magnitude. Time delay filtering is readily implemented with a digital controller but not with an analog controller. Effectively, every finite impulse response (FIR) filter is a time delay filter. The input is received by the filter, and the output based on that input is given for only a finite time period thereafter. When implemented by a digital delay line with n delays of time T and a zero-order hold, a single input enduring for a period T would produce n outputs that were each piecewise constant for each period of time T. While low pass and other filters can be implemented in FIR form, our interest is with a specifictypeoffilter that places zeros at appropriate places to cancel arm vibrations. The placement of these zeros requires knowledge of the arm’s vibrational frequency. When contrasted with the IIR filter, the FIR filter stretches the input by only a known time, at most nT. The operation of this filter is explained by a combination of time response and frequency response. The early version of a command shaping filter [24, 25] suffered from excessive sensitivity to variations in the plant. Since then a revised form was proposed [26], and a number of researchers have explored variations on this theme [27, 28, 29, 30]. The problem of sensitivity has been addressed in several ways that can be grouped into two approaches: increase robustness or adapt the filter [31]. Increasing robustness allows the resonant frequencies to drift somewhat during operation while remaining in a broadened notch of low response. The notch is broadened by placement of multiple zeros. Unfortunately this requires additional delay in completing the filtered command. The time-delay filter in various forms has become a widely used approach to canceling vibration with minimal demand on controller complexity and design sophistication. Its effectiveness can be understood in terms of the superposition that linear systems obey. An impulse input is separated into several terms, which when combined by superposition cancel the vibration as shown in Figure 24.8. Single Impulse Shaped with a Three-Term Command Shaper 1st Impulse Response 2nd Impulse Response 3rd Impulse Response Resultant Response Single Impulse 0 time time 0 ∆ 2∆ FIGURE 24.8 Effect of three term OAT command shaping filter. Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -35 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 d T d e −ζω n T d )/M coefficient 3 =(e −2ζω n T d )/M (24.96) where M = 1 − 2cosω d T d e −ζω n T d + e −2ζω n T d ω d =ω n  1 −ζ 2 = damped natural frequency ω n =undamped natural frequency ζ =damping ratio T d =time delay selected, an integer number of samples =  The OAT filter and other time delay filtersaresimpleto implement as shown schematically in Figure 24.9. A delay line (memory) sufficient to hold 2π /ω d 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.Theobvious wayofidentifying vibrational frequencies from an 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. Copyright © 2005 by CRC Press LLC 24 -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) 2 –1 –2 1 0 0 Acceleration (Volt) 10.5 Time (s) 2 –1 –2 1 0 0 Acceleration (Volt) 10.5 Time (s) 2 –1 –2 1 0 8 Acceleration (Volt) 98.5 Time (s) 2 –1 –2 1 0 8 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, thetip remains stationary after thejoint 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 50 40 30 20 10 0 −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. Copyright © 2005 by CRC Press LLC Flexible Robot Arms 24 -39 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 (oneor 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 caneliminate 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]. Copyright © 2005 by CRC Press LLC [...]... Copyright © 20 05 by CRC Press LLC Flexible Robot Arms 2 4-4 3 [21 ] Obergfell, K and Book, W.J., Control of flexible manipulators using vision and modal feedback, Proceedings of the ICRAM, Istanbul, Turkey, 1995 [22 ] Frame, A and Book, W.J., Sliding mode control of a non-collocated flexible system, 20 03 ASME International Congress and Exposition, Washington D.C., Paper IMECE200 3-4 1 386 , November 16 22 , 20 03 [23 ]... Acceleration (g) 0 .2 0.1 0 – 0.1 –0 .2 –0.3 –0.4 –0.5 0 5 10 15 Time (sec) 20 25 FIGURE 24 .17 Inertial damping controller quenches oscillations of flexible base (Controller action started at vertical line.) Copyright © 20 05 by CRC Press LLC 2 4-4 2 Robotics and Automation Handbook 24 .4 Summary Elasticity of structural material is a pervasive and simple phenomenon that leads to unavoidable and complex undesirable... CISM-IFToMM Ro.Man.Sy Zaborow, Poland, pp 90–101, Sept 8 12, 1 981 [19] Yuan, B.S., Huggins, J.D., and Book, W.J., Small motion experiments with a large flexible arm with strain feedback, Proceedings of the 1 989 American Control Conference, Pittsburgh, PA, pp 20 91 20 95, June 21 23 , 1 989 [20 ] Wang, D and Vidyasagar, M., Passive control of a stiff flexible link, Int J Robotics Res., vol 11, pp 5 72 5 78, 19 92. .. no 1 pp 84 – 92, Jan 20 01 [29 ] Singer, N., Singhose, W., and Seering, W., Comparison of filtering methods for reducing residual vibrations, Eur J Control., vol 5, pp 20 8 21 8, 1999 [30] U.S Patent 6,0 78, 844, Optimal arbitrary time-delay (OAT) filter and method to minimize unwanted system dynamics, issued June 20 00 [31] Book, W.J., Magee, D.M., and Rhim, S., Time-delay command shaping filters: robust and/ or... Systems and Control Division, vol 58, Atlanta, GA, pp 781 – 788 , November 17 22 , 1996 [39] Bayo, E and Paden B., On trajectory generation for flexible robots, J Robotic Syst., vol 4, no 2, pp 22 9 23 5, 1 987 [40] Bayo, E and Moulin, H., An efficient computation of the inverse dynamics of flexible manipulators in the time domain, Proceedings of the IEEE Conference on Robotics and Automation, pp 710–715, 1 989 [41]... tool of weight up to 21 0 kg, with a maximum speed of 2 m/sec and a repeatability specification of 0 .2 mm This robot has a reach of 2. 5 m or more which makes it more than adequate for positioning the linear accelerator to precisely aim at the patient from various different approach angles Copyright © 20 05 by CRC Press LLC 2 5 -8 Robotics and Automation Handbook 25 .6.1.3.3 Stereo X-ray Imaging System The... Control Handbook, CRC Press, Boca Raton, FL, 1996 [16] Hastings, G.G and Book, W.J., Reconstruction and robust reduced-order observation of flexible variables, ASME Winter Annual Meeting, Anaheim, CA, December, 1 986 [17] Cannon, R.H and Schmitz, E., Initial experiments on the end-point control of a flexible one-link robot, Int J Robotics Res., vol 3, no 3, pp 62 75, 1 988 [ 18] Truckenbrot, A., Modeling and. .. York, 19 98 [3] Young, W.C and Budynas, R.G., Roark’s Formulas for Stress and Strain, McGraw-Hill, New York, 20 02 [4] Trahair, N.S., Flexural-Torsional Buckling of Structures, CRC Press, Boca Raton, FL, 1994 [5] Sciavicco, L and Siciliano, B., Modeling and Control of Robot Manipulators, 2nd ed., Springer-Verlag, London, 20 00 [6] Craig, J.J., Introduction to Robotics: Mechanics and Control, Addison-Wesley,... selected and the system assists the operator in aligning the patient in the center of the imaging system This is done by acquiring x-ray images using the diagnostic x-ray sources and cameras Once the patient is aligned the treatment begins FIGURE 25 .3 Treatment planning system for CyberKnife® (Source: Accuray, Inc.) Copyright © 20 05 by CRC Press LLC 2 5-1 0 Robotics and Automation Handbook FIGURE 25 .4 The... A.V., Schaefer, R.W., and Buck, J.R., Discrete-Time Signal Processing, 2nd ed., Prentice Hall, New York, 19 98 [24 ] Calvert, J.F and Gimpel, D.J., Method and apparatus for control of system output in response to system input, Patent 2, 80 1,351, July 30, 1957 [25 ] Smith, O.J.M., Feedback Control Systems, McGraw-Hill, New York, 19 58 [26 ] Singer, N and Seering, W.P., Preshaping command inputs to reduce system . of rapid and precise response. Copyright © 20 05 by CRC Press LLC Flexible Robot Arms 24 -2 9 –1 –0 .8 –0.6 –0.4 –0 .2 0 Real Axis 0 .2 0.4 0.6 0 .8 1 2 –1.5 –1 –0.5 0 0.5 1 1.5 2 X 3 X 2 X 1 + Image. PA, pp. 20 91 20 95, June 21 23 , 1 989 . [20 ] Wang, D. and Vidyasagar, M., Passive control of a stiff flexible link, Int. J. Robotics Res., vol. 11, pp. 5 72 5 78, 19 92. Copyright © 20 05 by CRC Press. different approach angles. Copyright © 20 05 by CRC Press LLC 25 -8 Robotics and Automation Handbook 25 .6.1.3.3 Stereo X-ray Imaging System The imaging system consists of two x-ray sources mounted to the