Novel Gaussian Acceleration Profile for Smooth Jerk-Bounded Trajectories

Industrial machines regularly work at their limits causing excessive long-term vibrations that deteriorate their movement, stability, and precision. In this sense, reference profiles are able to reduce the detrimental vibration effects by manipulating the machine motion dynamics using predefined movement trajectories. Most of the approaches for lessening damages due to long-term machine vibrations are based on polynomial functions with high computational complexity and high resources demand. Hence, in this work, an innovative acceleration outline based on a Gaussian function is proposed for machine motion trajectories. The introduced strategy simplifies the position-profile estimation that works as reference for restraining the machine movements through the parameters that define its motion dynamics; thus, a smooth and continuous jerk contour is produced, which reduces vibrations and improves the machine stability. Exhaustive computer-based and real-time experimental comparisons of the introduced scheme produces a significantly lower maximum jerk value than any of the others. The assessment of the presented approach was performed utilizing the software Matlab (R2020a) on a PC with an Intel Core i7-6500U microprocessor at 2.5 GHz, with 16 GB in RAM and a 64-bit operating system.


I. INTRODUCTION
Industrial equipment, such as robots and computer numerical control (CNC) machines, carries out many distinct movements in a wide range of applications that require high speed, precision, and motion stability to maintain product quality keeping down production time and cost. Therefore, industrial equipment usually works at its limits, causing excessive long-term vibrations that worsen their movement stability and precision. This issue has attracted researchers attention, who have proposed a variety of strategies in literature, mainly focusing on the design and implementation The associate editor coordinating the review of this manuscript and approving it for publication was Shihong Ding . of control laws or the proposal of reference motion profiles. The former has been widely studied [1], [2]; however, the latter is an open subject for new suggestions that allow diminishing the long-term vibration effects by manipulating the industrial-machine motion dynamics through the definition of templates for movement trajectories [3].
A machine motion dynamics is defined by its position, velocity, acceleration, and jerk profiles [4], [5], [6], [7]. It is well known that high jerk values produce high amplitude vibrations in a machine component. This has motivated researchers to look for new control approaches to limit jerk effects by producing smooth motion profiles [8], [9], [10]. The trapezoidal velocity profile is the most-common movement scheme applied to industrial machines [11], [12].
It consists of sudden acceleration changes that involve very high amplitude jerk. In recent approaches, researchers have proposed different motion profiles that alleviate some of the trapezoidal-speed-scheme issues by utilizing a variety of motion references like the one known as the S curve profile, which requires 3rd-degree polynomial functions for obtaining a smooth jerk pattern [13], [14], [15], [16]. This is achieved by making the start and the end of the reference scheme uniform. A smooth jerk profile brings physical and mechanical benefits to motors, as described in [17], where a trajectory, based on a defined jerk track and exponential-function filters, is proposed for reducing motion-system vibrations. A control system, which decreases undesirable output-force ripples using jerk minimization as the base model for obtaining a velocity profile by implementing a 5th-degree polynomial, is proposed in [18]. In [19], a trigonometric function is used for obtaining smooth acceleration and jerk profiles. An analytical algorithm is introduced in [20] for continuous corner smoothing during CNC machining. This approach is aimed to reduce jerk effects and decrease trajectory errors by obtaining a smoother piece. The look-ahead window technique is used in [21] and [22] for smoothing the path-geometry corners to reduce the jerk magnitude in a CNC machine through the jerk-limited acceleration profile, which consists of three phases: acceleration increment at continuous rate, followed by a constant acceleration, and a final deceleration at a sustained ratio. In [23], a method for characterizing a machine-tool trajectory movement is proposed based on the analysis and manipulation of velocity, acceleration, and jerk profiles. The approach is used for improving the manufacturing-process efficacy and productivity by lessening consumed resources as materials and energy during iterative machining tests; however, some discontinuities are noticed in the acceleration profile. A study in the movement trajectory of a robot is introduced in [24] based on an S-shaped acceleration and deceleration algorithm that employs complex polynomial functions for its implementation. The approach yields a smooth velocity curve reducing the execution time and guaranteeing soft movements in industrial robots. From the described above, it can be inferred that motion-profile implementation for industrial machinery has become a notorious subject in recent years. For instance, performance comparison and evaluation among four motion profiles are performed in [25] by assessing vibration decreasing and machine wear. The studied profiles are applied in a gripper arm and its drive train. In [26], the implementation and performance of different motion profiles are compared in a linear platform, with a flexible beam system used for verifying obtained results from a theoretical analysis. From reviewed literature, it is clear that a machine motion dynamics is improved by lessening jerk effects; unfortunately, 3rd-or higher-degree polynomial functions must be implemented in order to reach this outcome, making difficult their integration because of their computational complexity and high resources demand; furthermore, there are some trajectory discontinuities in the obtained profiles that provoke high jerk effects.
As mentioned previously, a smooth jerk profile in the motion dynamics allows reducing vibrations and possible damages in an industrial machine, as well as improving its movement precision and performance. Therefore, in this paper, an innovative acceleration profile based on a Gaussian function is proposed. This scheme is used for generating the position-reference outline to achieve a smooth motion trajectory of industrial machines. A jerk contour with smooth and continuous behavior is accomplished by using the Gaussian function as a basis to build the acceleration profile, from the desired values for maximum position, velocity, and acceleration, together with the estimation of all remainder-parameter outlines, which define the machine motion dynamics.
The remainder of the document is organized as follows: Section II presents the design of the proposed acceleration profile based on the suggested Gaussian function scheme. Section III describes the computer-based simulation and real-time experimentation carried out, as well as the results obtained from each treated case of study, and their contrast against the state of the art in the subject. Section IV provides some conclusions.

II. MATHEMATICAL FRAMEWORK
This section describes the mathematical fundament for the proposed Gaussian acceleration profile by defining motion dynamics, the Gaussian function behavior and its implementation as acceleration profile for motion dynamics reconstruction.

A. MOTION DYNAMICS
Motion dynamics has an essential impact on the design of displacement trajectories since it provides smoothness and precision during industrial equipment operation. Motion dynamics consist of consecutive derivative operations on a machine-movement function with respect to time, obtaining its corresponding mathematical expressions of speed, acceleration, and jerk. It is worth it to notice that motion dynamics can be obtained conversely by successive integration of the movement functions defining it. These expressions are known as motion dynamics profiles that delineate the corresponding position, velocity, acceleration, and jerk layouts. The relation among the distinct profiles, making up motion dynamics, is described from (1) to (3): where p(t), a(t), v(t), and j(t) represent the position, velocity, acceleration, and jerk through time t, respectively.

B. GAUSSIAN FUNCTION
The Gaussian function in (4) is used for manipulating the acceleration profile to obtain smooth dynamics; hence, VOLUME 10, 2022 the Gaussian function amplitude and width can be modified by adjusting the standard deviation σ , as depicted in Fig. 1.
In (4), µ represents the mean of the discrete random variable x, and it can be used for shifting Gaussian bell in Fig. 1.
Therefore, different motion-dynamics parameters as position, velocity, and acceleration can be steered through a Gaussian function to obtain a smooth jerk pattern.

C. ACCELERATION PROFILE DESIGN
The proposed Gaussian acceleration profile produces a smooth machine movement that allows reducing vibrations and improving its motion precision, increasing the useful life of the machine mechanical parts. In this Gaussian layout, the maximum desired values for position (P max ), velocity (V max ) and acceleration (A max ), with corresponding units mm, mm/s and mm/s 2 , are assumed. Therefore, the discrete values for these parameters are defined as follows, according to [27]: where n is the sample number, r is the ratio between the number of millimeters and the corresponding linear displacement generated in the machine, and f s represents the sampling frequency.
The proposed Gaussian acceleration profile is divided into two segments, Gaussian acceleration, and Gaussian deceleration, each one of size 2k samples, as shown in Fig. 2.
The proposed acceleration profile starts with a 0 amplitude, and it reaches its maximum at the k-th sample; then, at the 2k-th sample, the acceleration amplitude is zero again, reaching its minimum value at the 3k-th sample. Finally, at the end of the proposed profile, the acceleration amplitude is 0 at the 4k-th sample. The discrete acceleration Gaussian profile is defined by (8).
From (8), it can be observed that the acceleration-profile amplitude is defined through its maximum value A max scaled by an exponential, i.e., Gaussian, factor, plus an adjustment value d. (Please refer to Appendix IV for a thorough description).
The maximum standard deviation σ max can be defined as a function of the maximum velocity and the maximum acceleration, as show in (9). (Please refer to Appendix IV for a thorough description).
The Gaussian-function double integral represents half of the maximum displacement in the motion dynamics, as described in (10). (Please refer to Appendix IV for a thorough description).
In the 2k-th sample of the proposed Gaussian profile, it is assumed that the machine has covered half of its maximum displacement, reaching its maximum velocity with an acceleration equal to zero at this point. Therefore, at the 2k-th sample, the machine position is defined by (11).
From (11), the function delay (n-µ) provides the value of k, so that, the acceleration profile starts with zero amplitude at sample zero, as defined in (12).
Analyzing the proposed Gaussian acceleration-profile behavior, a discontinuity is observed at the junction of the acceleration and deceleration outlines; therefore, the adjustment value d is worked out by estimating k in (4), such that the Gaussian acceleration-profile starts and ends with 0 amplitude, as defined in (13).
It should be noticed that to preserve the desired design parameters for the proposed acceleration profile, it would be necessary to use the maximum standard deviation defined in (9). If the used standard deviation value is selected within the range 0 < σ max , the maximum acceleration and jerk values increase, while the maximum velocity and position remain unchanged; under this condition the jerk profile remains smooth and continuous.

D. PROFILE RECONSTRUCTION
The profiles defining motion dynamics through the proposed acceleration profile are defined in (14) for reconstructing the jerk contour, in (15) for retrieving the velocity curve, and in (16) for recovering the controller position or reference outline.
where N is the total number of samples. The motion dynamics profiles obtained through the proposed technique are depicted in Fig. 3, where the jerk profile shows a continuous and smooth behavior that guarantees the performance improvement of any motor or machine, obtaining additional benefits as system stability and precision by reducing vibrations in the motor or machine. Once the design parameters of the proposed Gaussian acceleration profile have been defined, the maximum value of jerk can be obtained by (17).

III. RESULTS
This section describes obtained motion-dynamics results from extensive software simulation, utilizing Matlab, and real-time experimentation. These results are compared and discussed against those from recent literature about the subject.

A. COMPUTER SIMULATION RESULTS
To demonstrate the smoothness and continuity of the jerk profile obtained through the introduced Gaussian acceleration outline, a computer-based modeling is carried out by modifying the design parameters described in Section II. Fig. 3 shows the determined shapes that define a machine motion dynamics when the introduced acceleration profile is designed utilizing the following desired values: acceleration = 2 mm/s 2 , velocity = 3.2 mm/s and position = 20mm; hence, a σ max = 0.6383 is obtained applying (9) that yields a maximum jerk peak of 1.9 mm/s 3 . However, if the standard deviation (σ ) is decreased, the acceleration and jerk profiles are modified; in both cases, the maximum peak values increase as shown in Fig. 3 (c) and Fig. 3 (d), but the maximum values obtained for the velocity and position profiles are not changed, as depicted in Fig. 3 (b) and Fig. 3 (a), respectively. Similarly, if the standard deviation σ is decreased, the acceleration and jerk profiles are altered; in both cases, the maximum peak values increase, too, as shown in Fig. 3 (c) and Fig. 3 (d), but the top values obtained for the velocity and position profiles are continuous and smooth, without any alteration, as depicted in Fig. 3 (b) and Fig. 3 (a), respectively. Hence, if the standard deviation value is the maximum one that can be obtained utilizing the desired parameters; then, the minimum peak value of jerk will be produced.
On the other hand, if the desired position of the reference profile changes to 25 mm from 20 mm, and the maximum acceleration and maximum velocity are kept constant to 2 mm/s 2 and 3.2 mm/s, respectively, the maximum peak value of the jerk profile is sustained too. This is shown in Fig. 4; where, it can be seen that the velocity, acceleration, and jerk maximum peak value from their corresponding profiles are kept without change, as depicted in Fig. 4 (b), Fig. 4 (c), and Fig. 4 (d), respectively, and the jerk profile maintains a continuous and smooth shape. However, the execution time increases to achieve the desired position with these design parameters provided by the user.
The results obtained from the computer-based simulations demonstrate that a Gaussian acceleration profile designed through the proposed technique can regulate the motion-dynamics shapes by utilizing the configuration parameters defined in Section II, maintaining a continuous and smooth jerk profile over time, which will improve the performance of any industrial machine.
From Fig. 5, it is evident the profile-behavior improvement by showing continuous and smooth paths with a remarkable maximum jerk reduction of more than 98% taking as reference that one from the trapezoidal-velocity profile. Table 1 shows the maximum jerk values from the commonly used profiles in literature and the two associated variants of the proposed acceleration profile. From this table it is worth it to notice that the first derived form of the proposed acceleration profile takes longer time than the others to reach its final displacement, but has the lowest maximum-jerk value; in contrast, the second variant of the proposed acceleration profile requires higher acceleration and velocity to keep the same execution, i.e. displacement, time than the commonly used profiles in literature, but still with a highly outstanding maximum jerk reduction.
The assessment of the presented approach was performed utilizing the software Matlab (R2020a) on a PC with an Intel Core i7-6500U microprocessor at 2.5 GHz, with 16 GB in RAM and a 64-bit operating system.

B. EXPERIMENTAL ASSESSMENT
The performance of the proposed technique is evaluated experimentally by designing an acceleration profile,  according to the corresponding theoretical framework described in Section II of this work, for controlling the motion dynamics on the x-axis of a cartesian robot, which is a fundamental element of many production and industrial processes for carrying out high precision tasks.
The experimental setup is shown in Fig. 6 and consists of a servomotor (model GYB41D5-RC2), which has a 20-bit, serial encoder, a servo driver (RYH401F5-VV2), and a Terasic DE0-CV development board with an on-board Cyclone-IV field programmable gate array (FPGA) from Altera. The digital controller and the reference profile generator are embedded into the FPGA device.
Similarly to the computational simulation in the previous section, two variants of the proposed acceleration profile are used during the experimental validation utilizing the design parameters described in Table 2. The commonly used profiles in literature, i.e., trapezoidal-velocity, trapezoidalacceleration, and polynomial profiles, are generated using the same configuration parameters as those in the first derived form of the proposed acceleration profile in Table 2, for comparison purposes. The obtained position profile (control reference) is sent to x-axis servomotor in the cartesian robot.   acquiring the position signal provided by the encoder and applying the estimation technique proposed in [28] to retrieve the system motion dynamics. Table 3 shows the maximum jerk values and RMSE from all evaluated movement profiles.

C. DISCUSSION
The computer-based simulation of the proposed technique based on a Gaussian acceleration profile demonstrates that it is possible to handle the shape of the parameters that define a system motion dynamics through the maximum desired values of position, velocity and acceleration, maintaining a smooth and continuous jerk contour over time by taking into account the directives put forward in Section II. On the other hand, experimentally obtained results demonstrate that position can be tracked with a RMSE and maximum peak of jerk less than other commonly used motion profiles in literature, as described in Table 3; validating the high precision obtained by means of the proposed method, which, as described before, produces a smooth and continuous jerk profile.
From the results obtained through meticulous experimentations, it is clear that the first derived form of the proposed Gaussian acceleration profile produces a maximum jerk value of 4.821 × 10 4 mm/s 3 , which is significantly lower than any other of the analyzed motion profiles. On the other hand, the second alternative form of the proposed Gaussian profile reaches a jerk peak value of 5.4374 × 10 4 mm/s 3 , being just above the first configuration, but below all the commonly used motion profiles in literature, demonstrating the advantages of using the introduced strategy for diminishing the long-term vibration effects on motors or machines.

IV. CONCLUSION
High velocity, precision, and motion stability are required in industrial machines to maintain product quality and keep down production time and cost. Hence, a variety of schemes mainly based on the definition of control laws have been proposed for diminishing long-term vibration effects on industrial equipment. On the other hand, motion reference profiles have attracted researches interest in the recent years for tackling this issue by defining movement trajectories. Therefore, in this work, the design and implementation of a novel acceleration profile based on a Gaussian function is presented, which can be used for planning a robotic manipulator trajectory, and speed optimization in multi-axis industrial machines, such as lathes, CNC, and programmable universal machine for assembly, among many others. This strategy simplifies the position-profile estimation used as reference, and all-remainder parameters that define motion dynamics, producing a smooth and continuous jerk contour that reduces vibrations and improves the machine stability. On the other hand, by reducing the standard deviation for the Gaussian acceleration profile proposed in this work, the peak acceleration and jerk values increase, but the jerk outline remains smooth and continuous over time. Results obtained through exhaustive experimentation and comparison against commonly used profiles in literature show a remarkable lessening on the maximum jerk value of more than 98% taking as reference that one produced by the trapezoidal-velocity profile. Furthermore, computer-based simulations and real-time experimentation validate the proposed technique since a maximum jerk value of just 4.821 × 10 3 mm/s 3 and an RMSE on position profile of just 97.2 µm are obtained during the x-axis servomotor-trajectory tracking, outperforming all previously proposed approaches in the related subject.
Future Work: The proposed Gaussian acceleration profile will be implemented for assessing jerk effects on distinct industrial machines with multiple-axis trajectories.

APPENDIX A GAUSSIAN FUNCTION FIRST INTEGRATION
Integrating (4)  where G represents The Fig. 8 shows the graphical representation of the Gaussian function with its integral When x = 0 and µ = 0 the maximum value of the Gaussian function is obtained, while in the integral half the maximum value of the function is obtained, therefore Isolating C and sigma in (A. 3) gives Substituting (A.5) into (A.1) and Fig. 9 shows the comparison of (4) and (A.6)

APPENDIX B GAUSSIAN FUNCTION SECOND INTEGRATION
Taking and integrating (A.6) Changing of variable p = x − µ in (B.1) and takign (A.5) Solving the integral by parts gives Factoring (B.3) and substituting (4)  Professor at the Multidisciplinary Studies Department. He has been author/coauthor of more than 95 research papers published in scientific journals and in conference proceedings. His current research interests include digital instrumentation, digital image and signal processing, artificial intelligence, smart sensors, real-time processing, FPGAs, power quality, power electronics, and embedded systems. He is currently an Active Member of the IEEE Instrumentation and Measurement Society and the IEEE Industrial Electronics Society. He is also a member of the National System of Researchers, level 2, the National Council for Science and Technology (CONACYT), Mexico. PALILLERO-SANDOVAL OMAR received the B.S. degree from the Instituto Nacional de Astrofísica, Óptica y Electrónica INAOE, Mexico, the M.S. degree in engineering electronics from the University of Guanajuato, Mexico, and the Ph.D. degree in optics science from INAOE. He was trained in optical design using phase-space optics. He is currently a Professor with the Center for Research in Engineering and Applied Science (CIICAp), Mexico. His current research interests include digital signal processing, digital image processing, applied optical imaging systems in spectral imaging, microscopy, and optoelectronic systems. VOLUME 10, 2022