fpga implementation of higher degree polynomial ... ?· these discontinuities can produce...
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Robotics and Computer-Integrated Manufacturing 25 (2009) 379392
FPGA implementation of higher degree polynomial acceleration profilesfor peak jerk reduction in servomotors
Roque Alfredo Osornio-Riosa,, Rene de Jesus Romero-Troncosob,Gilberto Herrera-Ruiza, Rodrigo Castaneda-Mirandaa
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
Received 14 December 2006; received in revised form 26 September 2007; accepted 29 January 2008
Acceleration profile generation for jerk limitation is a major issue in automated industrial applications like computer numerical
control (CNC) machinery and robotics. The automation machinery dynamics should be kept as smooth as possible with suitable
controllers where trajectory precision ensures quality while smoothness decreases machinery stress. During the operation of
commercially available CNC and robotics controllers, small discontinuities on the dynamics are generated due to the controller position
profiler which is generally based on a trapezoidal velocity profile. These discontinuities can produce undesirable high-frequency
harmonics on the position reference which consequentially can excite the natural frequencies of the mechanical structure and
servomotors. Previous works have developed jerk limited trajectories with higher degree polynomial-based profiles, but lack one or both
of computer efficiency for on-line operation and low-cost hardware implementation. The present work shows a low cost,
computationally efficient, on-line hardware implementation of a high-degree polynomial-based profile generator with limited jerk
dynamics for CNC machines and robotics applications to improve the machining process. The novelty of the paper is the development of
a multiplier-free recursive algorithm for computationally efficient polynomial evaluation in profile generation and a low-cost
implementation of the digital structure in field programmable gate array (FPGA). Two experimental setups were prepared in order to test
the polynomial profile generator: the first one with the servomotor at no load and the second one for the servomotor driving a CNC
milling machine axis. From experimental results it is shown that higher degree polynomial profiles, compared to the standard trapezoidal
speed profile improve the system dynamics by reducing peak jerk in more than one order of magnitude while precision is maintained the
same and on-line operation is guaranteed.
r 2008 Elsevier Ltd. All rights reserved.
Keywords: CNC machinery; Robotics; Jerk; Polynomial profile generator; FPGA
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
e front matter r 2008 Elsevier Ltd. All rights reserved.
ing author. Tel./fax: +52 427 2741244.
ess: email@example.com (R.A. Osornio-Rios).
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
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
ARTICLE IN PRESSR.A. Osornio-Rios et al. / Robotics and Computer-Integrated Manufacturing 25 (2009) 379392380
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.
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.
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.
According to Erkorkmaz and Altintas , 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.
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.
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).
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  developed an algorithm for
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  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  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. , where the positionprofile selection is stated as a priority in an adapti