Patent Application: US-24331588-A

Abstract:
a sequence of impulses is determined which eliminates unwanted dynamics of a dynamic system . this impulse sequence is convolved with an arbitrary command input to drive the dynamic system to an output with a minimum of unwanted dynamics . the input sequence is time optimal in that it constitutes the shortest possible command input that results in no unwanted dynamics of the system subject to robustness and implementation constraints . the preshaping technique is robust under system parameter uncertainty and may be applied to both open and closed loop sy the government has rights in this invention pursuant to grant number n00014 - 86 - k - 0685 awarded by the department of the navy .

Description:
most researchers have examined transient vibration in terms of frequency content of the system inputs and outputs . this approach inherently assumes that the system inputs are not actually transient , but are one cycle of a repeating waveform . the approach taken in this invention is fourfold : first , the transient amplitude of a system &# 39 ; s unwanted dynamic response will be directly expressed as a function of its transient input . second , the input will be specified so that the system &# 39 ; s natural dynamic tendency is used to cancel unwanted dynamics . third , the input will be modified to include robustness to uncertainties . fourth , the case of arbitrary system inputs will be examined . the first step toward generating a system input which results in a system output without unwanted dynamics is to specify the system response to an impulse input . a linear , vibratory system of any order can be specified as a cascaded set of second - order poles each with the decaying sinusoidal response : ## equ1 ## where a is the amplitude of the impulse , ω is the undamped natural frequency of the plant , ζ is the damping ratio of the plant , t is time , and t o is the time of the impulse input . the impulse is usually a torque or velocity command to an actuator . equation 1 specifies the acceleration or velocity response , y ( t ), at some point of interest in the system . in what follows , only one system vibrational mode is assumed ( the general case is treated below ). fig1 demonstrates that two impulse responses can be superposed so that the system moves forward without vibration after the input has ended . the two impulse sequence is shown in fig2 . the same result can be obtained mathematically by adding two impulse responses and expressing the result for all times greater than the duration of the input . using the trigonometric relation : where ## equ2 ## the amplitude of vibration for a multi - impulse input is given by : ## equ3 ## the b j are the coefficients of the sine term in ( 1 ) for each of the n impulse inputs , and the t j are the times at which the impulses occur . elimination of vibration after the input has ended requires that the expression for a amp equal zero at the time at which the input ends , t end . this is true if both squared terms in 3 are independently zero , yielding : b . sub . 1 cos φ . sub . 1 + b . sub . 2 cos φ . sub . 2 +. . . + b . sub . n cos φ . sub . n = 0 ( 4 ) b . sub . 1 sin φ . sub . 1 + b . sub . 2 sin φ . sub . 2 +. . . + b . sub . n sin φ . sub . n = 0 ( 5 ) with ## equ4 ## where a j is the amplitude of the nth impulse , t j is the time of the nth impulse , and t end is the time at which the sequence ends ( time of the last impulse ). for the two - impulse case , only the first two terms exist in ( 4 ) and ( 5 ). by selecting 0 for the time of the first impulse ( t 1 ), and 1 for its amplitude ( a 1 ), two equations ( 4 ) and ( 5 ) with two unknowns ( a 2 and t 2 ) result . a 2 scales linearly for other values of a 1 . the two - impulse input , however , cancels vibration only if the system natural frequency and damping ratio are known exactly . in order to quantify the residual vibration level for a system , a vibration - error expression must be defined , here as the maximum amplitude of the residual vibration during a move as a percentage of the amplitude of the rigid body motion . fig3 shows a plot of the vibration error as a function of the system &# 39 ; s actual natural frequency . the input was designed for a system with a natural frequency of ω 0 . acceptable response is defined as less than 5 % of total move size residual vibration . fig3 shows that the two - impulse input is robust for a frequency variation of less than ≃± 5 %. in order to increase the robustness of the input under variations of the system natural frequency , a new constraint may be added . the derivatives of ( 4 ) and ( 5 ) with respect to frequency ( ω ) can be set equal to zero -- the mathematical equivalent of setting a goal of small changes in vibration error for changes in natural frequency . two equations are added to the system ; therefore , two more unknowns must be added by increasing the input from two to three impulses ( added unknowns : b 3 and t 3 ). the corresponding input and vibration error curves are shown in fig4 and 5 . in this case , the input is robust for system frequency variations of ≃± 20 %. the process of adding robustness can be further extended to include the second derivatives of ( 4 ) and ( 5 ) with respect to ω . setting the second derivatives to 0 requires that the vibration error be flat around the intended natural frequency . two more constraint equations are added , therefore , the impulse sequence is increased by one to a total of four impulses . the corresponding input and vibration error curves are shown in fig6 and 7 . in this case , the input is robust for system frequency variations of ≃- 30 %+ 40 %. in order for these system inputs to be insensitive to system parameter variation , uncertainty in damping ratio must also be considered . as with respect to natural frequency set forth above , the derivative of the amplitude of vibration with respect to damping ratio ( ζ ) can be computed . it can be shown that the same expressions that guarantee zero derivatives with respect to frequency also guarantee zero derivatives with respect to damping ratio . therefore , robustness to errors in damping has already been achieved by the addition of robustness to errors in frequency . fig8 shows the vibration - error expression for the same three sequences of impulses discussed above . note that extremely large variations in damping are tolerated . the previous discussion assumed only one vibrational mode present in the system . however , the impulse sequence can easily be generalized to handle higher modes . if an impulse or pulse sequence is designed for each of the first two modes of a system independently , they can be convolved to form a sequence which moves a two - mode system without vibration . the length of the resulting sequence is the sum of the lengths of the individual sequences . the sum , however , is an upper bound on the minimum length of the two - mode sequence which can be generated directly by simultaneously solving together the same equations that generated the two individual sequences . for example , if the four equations used to generate the sequence in fig4 were repeated for a different frequency , a system of eight equations would result and could be solved for four unknown impulse amplitudes and times ( plus the first , arbitrary impulse ), yielding a five - impulse sequence . the resulting sequence has one less impulse than the result of convolving the two independent sequences , and is always shorter in time . an arbitrary number of such sequences can be combined ( either by convolution or by direct solution ) to generate an input that will not cause vibration in any of the modes that have been included in the derivation . the sequences presented thus far have only assumed that the pulses or impulses are positive in magnitude . by using negative - going impulses or pulses , the sequences can be shortened . these sequences can be obtained by expressing the same ( or some other equivalent ) constraint equations as those presented above . in addition , the limitations on the magnitude of the impulses must be altered to enable negative going sequences . these new equations can be optimized thus yielding a shorter sequence . the shorter sequence provides the same level of robustness as the longer sequence , however , the delay time is reduced . unfortunately , there is a tradeoff . the sequences can be made arbitrarily short , but more and more actuator strength is required . in the limit that the sequences are infinitely short , infinite actuation is required . therefore , the minimum length of the sequence is limited by the physical system &# 39 ; s ability to respond . this can be shown to be consistent with system theory . as the input sequence becomes shorter , we are in effect asking the system to track a very short , high - frequency signal . since the system is rolling off in magnitude at these higher frequencies , a higher gain is required to get the system to respond . the actuator must supply this higher gain . if these combined negative and positive impulse sequences are to be normalized , it is important to set some constraints that never let the partial sum of the impulses to exceed a limit . this can be demonstrated by convolving a unit step with this sequence . if at any time during the convolution , the actuator limit is exceeded , the sequence is unacceptable . this constraint can be mathematically formulated by the following : ## equ5 ## where n is the number of impulses . we have presented above a method for obtaining the shortest system impulse input sequence which simultaneously eliminates vibration at the natural frequencies of interest , includes robustness to system varibility and meets other constraints . we now present a method for using these &# 34 ; time - optimal &# 34 ; sequences to generate arbitrary inputs with the same vibration - reducing properties . once the appropriate impulse sequence has been developed , it represents the shortest input that meets the desired design criteria . therefore , if the system is commanded to make an extremely short move ( an impulse motion ), the best that can be commanded in reality is the multiple - impulse sequence that was generated for the system . just as the single impulse is the building block from which any arbitrary function can be formed , this sequence can be used as a building block for arbitrary vibration - reducing inputs . the vibration reduction can be accomplished by convolving any arbitrary desired input to the system together with the impulse sequence in order to yield the shortest actual system input that makes the same motion without vibration . the sequence , therefore , becomes a prefilter for any input to be given to the system . the time penalty resulting from prefiltering the input equals the length of the impulse sequence ( on the order of one cycle of vibration for the sequences discussed above ). fig9 shows the convolution of an input ( for example , the signal from a joystick in a teleoperated system ) with a non - robust , two - impulse sequence . the impulse sequences set out above have been normalized to sum to one . this normalization guarantees that the convolved motor input never exceeds the maximum value of the commanded input . if the commanded input is completely known in advance for a particular move , the convolved motor input can be rescaled so that the maximum value of the function is the actuator limit of the system . for historical reference , the result of convolving a non - robust two - impulse sequence with a step input yields the posicast input developed by o . j . m . smith in 1958 . the robustness plot of fig3 demonstrates why posicast is not generally used . the shaped commands have been tested on a computer model of the space shuttle remote manipulator system ( rms ) which was developed for nasa by draper laboratories . this computer model includes many of the complicating features of the hardware shuttle manipulator such as stiction / friction in the joints ; non - linear gear box stiffness ; asynchronous communication timing ; joint freeplay ; saturation ; and digitization effects . the simulation has been verified with actual space shuttle flight data . excellent agreement was obtained for both steady state and transient behavior . fig1 shows a comparison between the current shuttle rms controller and a step input in velocity which was prefiltered by a three - impulse sequence . the residual vibration is reduced by more than one order of magnitude for the unloaded shuttle arm . comparable results were obtained for a variety of moves tested . the use of shaped inputs for commanding computer - controlled machines shows that significant reduction in unwanted dynamics can be achieved . the cost in extended move time is small ( on the order of one cycle of vibration ), especially when compared to the time saved in waiting for settling of the machine &# 39 ; s vibrations . a straightforward design approach for implementing this preshaping technique has been presented in this specification . although the above examples have utilized impulse sequences , it should be understood that finite duration pulses instead of impulses can form the building blocks for shaping command inputs . the pulses or impulses can be constrained to begin at specified intervals along with the additional constraint that the impulses or pulses are fixed in amplitude but are variable in duration . further , constraints can be applied that the pulses begin and / or end at specified intervals . fig1 illustrates the standard implementation of the invention discussed above . an input is applied to a shaping routine that scales the input and submits it to the system . at later times , it resubmits the input to the system scaled by different amounts , so that the output has a reduction in the unwanted dynamics with a specified level of robustness to system parameter changes . this approach applies to most computer controlled machines , chemical , and electronic systems . aspinwall , d . m ., &# 34 ; 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