FPGA implementation of higher degree polynomial ... ?· These discontinuities can produce undesirable…

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<ul><li><p>ARTICLE IN PRESS</p><p>0736-5845/$ - se</p><p>doi:10.1016/j.rc</p><p>CorrespondE-mail addr</p><p>Robotics and Computer-Integrated Manufacturing 25 (2009) 379392</p><p>www.elsevier.com/locate/rcim</p><p>FPGA implementation of higher degree polynomial acceleration profilesfor peak jerk reduction in servomotors</p><p>Roque Alfredo Osornio-Riosa,, Rene de Jesus Romero-Troncosob,Gilberto Herrera-Ruiza, Rodrigo Castaneda-Mirandaa</p><p>aFacultad de Ingeniera, Universidad Autonoma de Queretaro Cerro de las Campanas s/n, 76010 Queretaro, Qro., MexicobElectronics Department, FIMEE, Universidad de Guanajuato, Tampico 912, Col. Bellavista, 36720 Salamanca, Gto., Mexico</p><p>Received 14 December 2006; received in revised form 26 September 2007; accepted 29 January 2008</p><p>Abstract</p><p>Acceleration profile generation for jerk limitation is a major issue in automated industrial applications like computer numerical</p><p>control (CNC) machinery and robotics. The automation machinery dynamics should be kept as smooth as possible with suitable</p><p>controllers where trajectory precision ensures quality while smoothness decreases machinery stress. During the operation of</p><p>commercially available CNC and robotics controllers, small discontinuities on the dynamics are generated due to the controller position</p><p>profiler which is generally based on a trapezoidal velocity profile. These discontinuities can produce undesirable high-frequency</p><p>harmonics on the position reference which consequentially can excite the natural frequencies of the mechanical structure and</p><p>servomotors. Previous works have developed jerk limited trajectories with higher degree polynomial-based profiles, but lack one or both</p><p>of computer efficiency for on-line operation and low-cost hardware implementation. The present work shows a low cost,</p><p>computationally efficient, on-line hardware implementation of a high-degree polynomial-based profile generator with limited jerk</p><p>dynamics for CNC machines and robotics applications to improve the machining process. The novelty of the paper is the development of</p><p>a multiplier-free recursive algorithm for computationally efficient polynomial evaluation in profile generation and a low-cost</p><p>implementation of the digital structure in field programmable gate array (FPGA). Two experimental setups were prepared in order to test</p><p>the polynomial profile generator: the first one with the servomotor at no load and the second one for the servomotor driving a CNC</p><p>milling machine axis. From experimental results it is shown that higher degree polynomial profiles, compared to the standard trapezoidal</p><p>speed profile improve the system dynamics by reducing peak jerk in more than one order of magnitude while precision is maintained the</p><p>same and on-line operation is guaranteed.</p><p>r 2008 Elsevier Ltd. All rights reserved.</p><p>Keywords: CNC machinery; Robotics; Jerk; Polynomial profile generator; FPGA</p><p>1. Introduction</p><p>Nowadays, computer numerical control (CNC) ma-chines and robotics are widely spread in automatedindustrial applications, which improve end product quality,reduce production time and increase profits in the shortand long time, despite the high initial investment. In orderto guarantee an optimal relationship between productquality and production cost, the automation machinery</p><p>e front matter r 2008 Elsevier Ltd. All rights reserved.</p><p>im.2008.01.002</p><p>ing author. Tel./fax: +52 427 2741244.</p><p>ess: raor@uaq.mx (R.A. Osornio-Rios).</p><p>dynamics should be kept as smooth as possible withsuitable controllers where trajectory precision ensuresquality while smoothness decreases machinery stress.During the operation of commercially available CNC</p><p>and robotics controllers, small discontinuities on thedynamics are generated due to the controller positionprofile generator which is generally based on a trapezoidalvelocity profile. The discontinuities on the trajectorydynamics can produce undesirable high-frequency harmo-nics on the position reference, which consequentially canexcite the natural frequencies of the mechanical structureand servomotors. Such high-frequency harmonics can also</p><p>www.elsevier.com/locate/rcimdx.doi.org/10.1016/j.rcim.2008.01.002mailto:raor@uaq.mx</p></li><li><p>ARTICLE IN PRESSR.A. Osornio-Rios et al. / Robotics and Computer-Integrated Manufacturing 25 (2009) 379392380</p><p>saturate the actuators and therefore, decrease the positiontrajectory precision that affects the effective contouringprocess. Linear and circular interpolation techniques havebeen proposed to minimize the discontinuities; however,these techniques have limited efficiency when the geometryof the machining process is complex, which decreaseproductivity in CNC machines and do not provide therequired smooth dynamics in robotics.</p><p>The parameter that measures the discontinuities anddynamics smoothness is known as jerk and it is defined asthe acceleration change rate. Sudden control changes inacceleration result in higher jerk levels which can stimulatethe mechanical systems into resonance; therefore, it isnecessary to provide position profiles sufficiently smooth inorder to reduce jerk levels, minimizing discontinuities andavoiding system resonance.</p><p>Hardware and software digital signal processing (DSP)techniques have been proposed to develop algorithms forjerk reduction in industrial controllers. By minimizing jerk,two immediate benefits are granted: machinery stress andvibration reduction on the dynamics and smoother move-ment which allows speed increasing while error is reduced.In consequence, overall performance can be improvedwhen jerk is limited.</p><p>According to Erkorkmaz and Altintas [1], trapezoidalspeed profiles generate high-frequency harmonics in theacceleration dynamics which yields poor jerk behavior.They also showed that higher degree polynomial-basedprofiles give a smoother dynamics, making the resultingtrajectory easier to track by the limited bandwidth of theservo controller.</p><p>Higher degree polynomial profiles are difficult togenerate due to the computational load demands bothhardware resources and processing time, which seriouslycompromise on-line implementation for controllers inpractice. Direct polynomial evaluation algorithms demandhigh computational efforts to be made at the processorsystem, highly increased by the polynomial degree.</p><p>The present work shows a low cost, computationallyefficient, on-line hardware implementation of a high-degreepolynomial-based profile generator with limited jerkdynamics for CNC machines and robotics applications toimprove the machining process. The novelty of the paper isthe development of a multiplier-free recursive algorithm forcomputationally efficient polynomial evaluation in profilesgeneration, suited for any processing unit (DSP, micro-processor, etc.) and its special purpose digital structureoptimized for field programmable gate array (FPGA)which due to its reconfigurability allows a system on-a-chip(SOC) approach while its architecture freedom, from thedesigner point of view, improves the servo loop updatetime for conventional and high-speed machining with ahigh-resolution parallel processing structure. Considerablejerk reduction is obtained by the design of higher degreepolynomial profiles with overall dynamics smoothness (i.e.limited jerk polynomial profile that gives smooth accelera-tion, velocity and position profiles).</p><p>2. Background</p><p>Several research lines have been followed to improve thejerk limitation dynamics on CNC and robotics controllers.The reported algorithms deal with higher degree poly-nomial profile generators in software and hardware;however, all of them use the direct polynomial evaluationwhich is not computationally efficient because this techni-que requires floating-point operations to achieve the suitedprecision, besides being computational intensive, compro-mising on-line implementation. Personal computer (PC)-based software implementation of the direct polynomialevaluation, though achieves the required precision, lacksthe on-line restriction due to the computational intensivenature of the algorithm. On the other hand, for a generalpurpose microprocessor or DSP system, high resources arerequired for the hardware section in order to achieve on-line operation while keeping the error bounded, generallyrequiring expensive floating-point units or complicatedmultiple precision operations, making the algorithmunsuited for polynomials of 5 or higher degrees.Erkorkmaz and Altintas [1] developed an algorithm for</p><p>limited jerk trajectory generation which highlights thepolynomial approach advantages over the traditionaltrapezoidal profile approach. This work deals with a fifthdegree multiple polynomial generation and its implementa-tion into a TMS C32 commercially available DSP board,recognizing the computing resources demand for thesolution. Yih [2] presents the design of a third-order linearfilter for jerk limitation on CNC machines in order toimprove the servomotor position profile. This approachfilters the original trapezoidal profile for smoothing therelated discontinuities, but lacks the required precision dueto the filter side effects. The implementation is done into anADSP21060 commercially available DSP where the com-putation load is relaxed with the aid of three circularbuffers. Gasparetto and Zanotto [3] proposed a newmethodology for smooth trajectory planning in roboticsbased on a quadratic jerk profile for a fifth degreepolynomial spline. This work highlights the relevance onjerk limitation for robotics structure damage reductionalong with the avoidance of resonance frequencies thatincrease the error; no implementation is presented and onlysimulation results are shown. Another development forrobotics is presented by Yang et al. [4], where the positionprofile selection is stated as a priority in an adaptiveposition controller. The main goal for the trajectoryplanning is to smooth the displacement. The developedalgorithm is presented as software simulations underSimulink and Matlab. Heo et al. [5] presented a develop-ment on profile generation, related to the machiningprocess time for molds and dies. The errors in themachining process are reported as a function of theacceleration and deceleration profiles, implemented insoftware. Cheng et al. [6] proposed a code generatorfor a NURBS interpolator and states that a positionreference profile generator as a high-speed hardware unit is</p></li><li><p>ARTICLE IN PRESSR.A. Osornio-Rios et al. / Robotics and Computer-Integrated Manufacturing 25 (2009) 379392 381</p><p>mandatory in order to achieve on-line CNC control. Thisdevelopment is implemented with a PC and a TMS320C32DSP working together. Earlier polynomial-based NURBStrajectory planning algorithms were developed by Zhangand Greenway [7], and were programmed under C++ in aPC where it became obvious that on-line implementationof robotics trajectories with this approach were compro-mised due to the computational load, increasing thesampling period. Marchenko et al. [8] worked on aNURBS interpolator for CNC machines with adaptivecontrol for a constant feed rate. The algorithm wasprogrammed under Visual C++ and executed in a400MHz Pentium II-based PC with a Delta Tau PMACcommercial controller board. In order to implementthe algorithm, the sampling period 4 103 s, which isfour times higher than the typical 1 103 s servo loopcontrol time. Cervera and Trevelyan [9] developed anoptimization evolving structure for NURBS trajectorygeneration where its implementation in a PC takes 14.62 s,in order to perform the interpolation. Macfarlane andCroft [10] presented an on-line method for obtainingsmooth, jerk-bounded trajectories; their method describedherein uses a concatenation of fifth-order polynomials toprovide a smooth trajectory between two-way points. Thealgorithm was implemented in Matlab. All the cited worksclaim for an improvement in computation load whichdisables on-line application or increases the resourcerequirements.</p><p>On the other hand, FPGA implementation of specialpurpose hardware signal processing units for CNCapplications has became the state-of-the-art approachwhere low-cost, SOC and high computation power isrequired. Jeon and Kim [11] developed an FPGA-basedtrapezoidal acceleration and deceleration profile generatorfor industrial robotics applications, but the digital structureis not optimal, and although polynomial profile generationis mentioned in the work, no actual implementation ispresented because the authors estimated that the hardwareresources will increase exponentially with the polynomialdegree, based on a direct polynomial evaluation algorithm.Jimeno et al. [12] worked on an FPGA-based tool pathcomputation applied to shoe last machining on CNClathes, this work presents the implementation of the surfacevirtual digitalization for the specific tool geometry and itsdisplacement. The research focuses on the computationefficiency as well as precision and speed achieved by theFPGA application. Chen and Lin [13] developed anFPGA-based ultrasonic servomotor drive, highlightingthe speed performance efficiency, compared with othertechnologies, along with the device flexibility and thecapacity for non-linear real time system implementation.Girau and Boumaza [14] presented an embedded architec-ture to solve the navigation problem in robotic based onFPGA that computes trajectories along a harmonicpotential. Their architecture proposed allows very largeprecisions and computer resolution. Software implementa-tion of the harmonic function computation on a micro-</p><p>processor-based computer, Pentium 4, 2GHz required100 106 s per iteration, while in FPGA structurerequired around 193 109 s. These works are a fewexamples of FPGA applications for industrial CNC androbotics and its low-cost, flexibility and speed advantages.Previously cited works use several methodologies for</p><p>polynomial reference profile generation, and in some casesthe hardware implementation is presented as well, wherethe main identified problems are: computational load,execution time constraints and hardware requirements; nolow-cost computationally efficient solution is presented.This paper deals with two main goals: the development of amultiplier-free computationally efficient polynomial eva-luation algorithm and its digital structure to be implemen-ted into an FPGA, having in mind that low-cost on-lineoperation is mandatory for profile generation. Jerklimitation can be achieved with higher degree polynomials,while precision on position control is guaranteed, as shownby experimentation.</p><p>3. Hardware polynomial-based profile generator</p><p>The proposed profile generator is based in a multiplier-free polynomial evaluation algorithm which reconstructsthe polynomial by a parameterization process usingsuccessive discrete time integration. The algorithm can beeasily implemented into an FPGA, requiring simplestructures as adders and registers plus logic. Hardwarerequirements increase linearly with the polynomial degreewhile computation time is not sacrificed. Multiple precisionaccumulation is performed by the digital structure in orderto guarantee the desired reference accuracy....</p></li></ul>

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