A novel Fibonacci sequence-based motion profile

Abstract

In contrast to polynomial smooth S-curves and typical sine curves, this article presents a revolutionary curve design technique based on the Fibonacci sequence. This technique removes discontinuities that arise during the transition between acceleration segments by adding several fixed-form smooth transition segments into the original curve while maintaining the piecewise structure of the traditional S-curve. The proposed motion profile aims to improve control stability, positioning accuracy, and current performance, while maintaining the performance advantages of a multisegment S-curve design. This article elaborates on the design principles of the modified motion profile and derives its analytical models for acceleration, jerk, velocity, and displacement. Furthermore, it provides complete mathematical formulas for possible profile variants under different parameter settings, as well as profile type selection criteria based on hardware parameters. To evaluate the feasibility of the proposed approach on real control systems and its advantages in accuracy and current behavior, simulations are conducted. This article uses the proposed modified motion profile and the traditional S-curve as motion commands, comparing the speed and acceleration responses under motor models with different damping ratios to verify its feasibility and performance advantages. Furthermore, we conducted experiments on a brushless DC motor and recorded data through the motor's built-in feedback. The results show that, compared to the traditional S-curve, the proposed modified motion profile improves positioning accuracy and current stability.

Introduction

In motion control of the motor driven system, planning the input motion profile is crucial for achieving the desired control performance. The continuity of the curve improves motion smoothness and reduces vibration. Furthermore, the motion profile determines the overall trajectory of the motor driven system, which can be described by required information of position, velocity, and acceleration. The tasks of motion planning include selecting an appropriate curve type (e.g. trapezoidal curve, S-curve), determining parameters based on system requirements and constraints, and generating corresponding position, velocity, and acceleration commands for the controller to track. Traditional S-curves use jerk, acceleration, velocity, and displacement as designed parameters. Motion is typically divided into acceleration, constant velocity, and deceleration segments. By first defining the acceleration curve and then integrating it, smooth velocity and displacement trajectories can be obtained. Due to its simplicity, smooth transitions, and fast response, the S-curve method has been widely used in motion path planning for CNC machine tools and robotic arms, becoming one of the most commonly used motion profile design methods in control systems. However, discontinuities still exist in jerk and acceleration, which can lead to sudden current changes and overshoot. In view of the above limitations, this study proposes an improved design that retains the advantages of a multisegment S-curve while introducing a fixed smooth transition function at the boundaries of each segment. The aim is to suppress abrupt changes in jerk and impact force, thereby mitigating current fluctuations caused by discontinuities and improving positioning accuracy.

In the field of motion control, motion profiles have always been a key factor affecting control accuracy and stability. Motion profiles have evolved from the initial trapezoidal trajectories to the widely used S-curves. Due to their smoothness and design simplicity, the S-curve has become the most common method. However, with the increasing demands of motion control, the inherent acceleration discontinuity problem of the S-curve is becoming increasingly prominent and urgently needs to be addressed.

Fang et al. proposed a sinusoidal jerk-based motion profile design, ¹ aimed at mitigating vibration problems during high-speed motion and precision positioning. While the conventional S-curve can effectively alleviate abrupt changes in acceleration, its piecewise structure still introduces discontinuities in jerk, which can lead to residual vibration and resonance. Their work transformed the jerk into a sinusoidal form, ensuring continuity and differentiability in acceleration and velocity, and further introduced parameter optimization to minimize motion time and excitation frequency components under actuator constraints. Both simulation and experimental results demonstrated significant vibration reduction and improved positioning accuracy. Similarly, Li and Lin developed a motion profile planning method targeting jerk reduction and residual vibration suppression. ² They highlighted that conventional profiles under high-speed or heavy-load conditions often excite system resonances due to sudden changes in acceleration and jerk. Through mathematical modeling and experimental validation, they designed smooth acceleration transitions that reduce excitation energy imparted to the natural modes, effectively suppressing residual vibration and improving system response. Gonzalez-Villagomez et al. investigated Gaussian-based acceleration profiles, ³ generating smooth trajectories with jerk limitation. By leveraging Gaussian functions, their method avoided sharp changes in higher-order derivatives typical of polynomial trajectories, thus ensuring continuity of velocity and acceleration. Compared with conventional polynomial curves, Gaussian-based profiles were computationally simpler and reduced long-term vibration accumulation in mechanical systems. Lee and Ha introduced an optimization framework for polynomial motion profiles, ⁴ balancing high-speed performance with low vibration. Their study compared different polynomial orders, established selection criteria, and demonstrated advantages in wafer-handling robots, where efficiency was improved and vibrations reduced. Blejan and Blejan presented a real-time S-curve generator ⁵ capable of limiting velocity, acceleration, and jerk while efficiently producing motion profiles suitable for real-time control. MATLAB simulations and software validation confirmed its feasibility and stability. Wang et al. proposed an asymmetric corner smoothing method ⁶ for CNC machining and parallel scanning, addressing jerk spikes and vibrations at turning points. Their smoothing technique-maintained motion continuity while reducing symmetry disruption, significantly lowering speed and acceleration peaks, thereby improving path-following stability. This outcome shares the same motivation as the present study, which focuses on enhancing smoothness and reducing vibration through functional curve design. Other researchers have explored various advanced strategies, including: Chebyshev polynomial-based optimization for energy-efficient positioning ⁷ ; comparative evaluations of trapezoidal versus S-curves ⁸ ; sinusoidal displacement-shifted profiles for vibration suppression ⁹ ; nonlinear programming for time-optimal trajectories ¹⁰ ; global jerk-minimization strategies ¹¹ ; real-time bounded jerk methods ¹² ; quintic spline interpolation for CNC stability ¹³ ; asymmetric S-curves optimized by metaheuristics ¹⁴ ; parameterized jerk asymmetry for fast convergence ¹⁵ ; locally asymmetric jerk curves for industrial robots ¹⁶ ; decomposed jerk-limited planning to mitigate time-varying vibration ¹⁷ ; sigmoid-based time-optimal S-curves ¹⁸ ; fourth-order jerk-continuous S-curves ¹⁹ ; and the mathematical generalization of k-Fibonacci hyperbolic functions. ²⁰

Building upon previous work, this study aims to propose a novel motion profile design scheme that combines the hyperbolic Fibonacci sequence with the traditional S-curve. The core idea is to embed smooth segments of the hyperbolic Fibonacci sequence into the acceleration curve of the S-curve, thereby achieving a natural and continuous transition between stages. This design not only reduces discontinuities at segment connections but also simplifies the smoothing process and reduces computational complexity, providing a new perspective for motion profile design.

The content of this article is described as follows. In the introduction section, the research topic is introduced, the advantages and limitations of S-curve profiles are outlined, and a modified motion profile design based on traditional methods is proposed. The research methodology section elucidates the proposed improvement method and the transition function used. Furthermore, it derives the equation of motion for the modified motion profile, explains the parameter calculations for eight variations of the motion profile, and establishes criteria for selecting the appropriate curve under different operating conditions. This chapter also discusses time unit conversion, the calculation of the jerk coefficient of the transition function, and provides a comparative analysis of the jerk, acceleration, velocity, and displacement of the S-curve and the modified motion profile. The simulation and experimental verification section presents both simulation and experimental verification. The system response of a second-order motor model was simulated using MATLAB, with the S-curve and the modified motion profile used as speed commands to compare their speed and acceleration responses. The effect of changing the system damping ratio was observed, preliminarily verifying the advantages of the modified motion profile. Experimental verification used an RMD-X10-S2 brushless DC motor and a Teensy 4.1 microcontroller as the external controller. The S-curve and the modified motion profile were compared in terms of current response, positioning accuracy, execution time, and tracking error. Finally, the experimental data were analyzed and the research results were summarized. The results show that although the proposed modified motion profile requires a longer execution time due to the inclusion of a smooth transition section, it outperforms the traditional S-curve in terms of current stability and tracking accuracy.

Research methodology

This article presents the methodology of the proposed modified motion profile. First, the newly introduced smooth transition segments are described, and their mathematical formulations and practical implementation methods are derived. In addition, a jerk adjustment mechanism for the transition segments is introduced. Subsequently, the structural composition of the modified motion profile is explained. Based on different parameter conditions, the profile is categorized into eight variants. The corresponding time-parameter calculation methods for each variant are derived, along with a set of selection criteria that determine the most suitable motion profile according to the given hardware specifications.

Preliminary

Fibonacci sequence

The Fibonacci sequence is first proposed by the Italian mathematician Leonardo da Pisa in his 1202 work, “Liber Abaci,” and is widely known for its mathematical properties and extensive applications. One of its most prominent features is its association with the golden ratio (approximately 1.618), derived from the ratio of two consecutive terms in the sequence. The golden ratio is also frequently observed in nature, such as in the spiral shell of a nautilus, the branching structure of deer antlers, and the veins on insect wings, all of which demonstrate proportional relationships consistent with the Fibonacci sequence.

The golden ratio has long been used to create esthetically pleasing and harmonious curves and visual compositions. In engineering applications, it is used in numerical methods, such as the golden section search algorithm, which is an optimization algorithm that iteratively narrows the search interval based on the golden ratio. Beyond mathematics and engineering, the Fibonacci sequence also appears in fields such as music composition, antenna design, mechanical structures, and materials engineering.

The Fibonacci sequence is defined as a numerical sequence where each term is the sum of the two preceding terms, typically starting with 0 and 1. Therefore, the initial values of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, …. This recursive pattern can be observed in a wide range of natural phenomena and mathematical applications. The recurrence relation is expressed as $F(n)=F(n-1)+F(n-2)$ with the initial conditions: $n=0,1,2,\dots$ , $F(0)=0$ and $F(1)=1$ . From the recurrence relation, the limiting ratio between two successive terms of the Fibonacci sequence can be obtained as $\varphi =\underset{n\to \mathrm{\infty }}{lim}\frac{f(n+1)}{f(n)}=\frac{1+\sqrt{5}}{2}\approx 1.618$ , which is referred to as the golden ratio, while its conjugate is given by $\phi =\underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{f(n+1)}=\frac{\sqrt{5}-1}{2}\approx 0.618$ and satisfy the relation $\varphi \phi =1$ . According to Binet's formula, the closed-form expression of the Fibonacci sequence can be written as $F(n)=\frac{{\varphi }^n-{\phi }^n}{\varphi -\phi }$ .

Hyperbolic Fibonacci sequence

Hyperbolic Fibonacci functions are a class of functions inspired by the Fibonacci sequence and classical hyperbolic functions. They are based on the golden ratio $\varphi$ and its reciprocal $\phi$ , and exhibit properties similar to traditional hyperbolic functions. These functions can be viewed as a continuous extension of the discrete Fibonacci sequence into the continuous mathematical domain.

In the motion profile planning considered in this study, the discrete index variable n is replaced by a continuous variable (e.g. time $t$ ). Accordingly, the hyperbolic Fibonacci sine and cosine functions are defined by equation (1), and the Fibonacci hyperbolic tangent is defined as equation (2).

$$[\begin{array}{c}{F}_s(t)={\displaystyle \frac{2}{\sqrt{5}}{\displaystyle \frac{e^{2\ln (\varphi )t}-e^{-2\ln (\varphi )t}}{2}={\displaystyle \frac{2}{\sqrt{5}}\sinh (2\ln (\varphi )t)}}}\\ F_c(t)={\displaystyle \frac{2}{\sqrt{5}}{\displaystyle \frac{e^{(2t+1)\ln (\varphi )}+e^{-(2t+1)\ln (\varphi )}}{2}={\displaystyle \frac{2}{\sqrt{5}}\cosh ((2t+1)\ln (\varphi ))}}}\end{array}$$

(1)

$$F_t(t)={\displaystyle \frac{F_s(t)}{F_c(t)}={\displaystyle \frac{\sinh (2\ln (\varphi )t)}{\cosh ((2t+1)\ln (\varphi ))}={\displaystyle \frac{e^{2\ln (\varphi )t}-e^{-2\ln (\varphi )t}}{e^{(2t+1)\ln (\varphi )}+e^{-(2t+1)\ln (\varphi )}}}}}$$

(2)

To mitigate the discontinuities in the acceleration profile of the conventional S-curve at segment junctions, this study introduces the Fibonacci hyperbolic tangent $F_t(t)$ . Based on this function, four types of smoothing curves are further proposed to replace the discontinuous portions of the S-curve acceleration profile. These functions are denoted as $F_{t1}(t)$ , $F_{t2}(t)$ , $F_{t3}(t)$ , and $F_{t4}(t)$ , i.e.

$$[\begin{array}{cc}{F}_{t1}(t)=F_t(t)& 0\le t\le T\\ F_{t2}(t)=\phi -F_t(t)& 0\le t\le T\\ F_{t3}(t)=F_t(T-t)& 0\le t\le T\\ F_{t4}(t)=\phi -F_t(T-t)& 0\le t\le T\end{array}$$

(3)

The graphical representations in equation (3) are illustrated in Figure 1.

Figure 1.

Four types of Fibonacci hyperbolic transition functions.

Comparison of hyperbolic tangent Fibonacci sequence and conventional hyperbolic tangent function

The hyperbolic Fibonacci sequence can be seen as an extension of the Fibonacci sequence and the classical hyperbolic function. The hyperbolic tangent Fibonacci sequence and the traditional hyperbolic tangent function have similar function shapes and infinite differentiability; that is, both functions are continuous and smooth at every point on the curve. Therefore, they both have great potential as smooth transition functions in S-shaped motion profiles. Due to the geometric properties of the tangent function, its curve itself contains rising and converging segments. These segments naturally correspond to the accelerating rising phase and the uniformly accelerating phase in the S-shaped curve profile. Therefore, the hyperbolic tangent function can be directly embedded between different acceleration segments, eliminating corner discontinuities without additional calculations, thus achieving a smooth transition effect. The curvature of the function reflects the degree of curvature of its curve, i.e. the greater the curvature, the more drastic the change. In motion profile design, this is equivalent to the drastic change in acceleration, directly affecting system vibration and control force. To quantitatively compare the two functions, three metrics are used: (i) the root mean square (RMS) value of curvature, (ii) the maximum curvature, and (iii) the RMS value of the rate of change of curvature. Figure 2 shows a comparison of the curvature of the hyperbolic Fibonacci sequence and the hyperbolic tangent function, while Figure 3 shows their rates of change of curvature. As shown in Figure 2, the maximum curvature of the hyperbolic Fibonacci sequence and the hyperbolic tangent function are $0.247/s$ and $0.244/s$ respectively, while their RMS values of curvature are $0.248/s$ and $0.245/s$ . These results indicate that the two functions exhibit only minor differences in terms of curvature. As shown in Figure 3, the curvature variation rates of the two functions are $0.245/s^2$ and $0.338/s^2$ , respectively. It can be observed that the hyperbolic Fibonacci sequence possesses a smaller curvature variation rate, highlighting its superior higher-order smoothness and making it more suitable as a transition segment in the modified motion profile.

Figure 2.

Curvature comparison between the hyperbolic Fibonacci sequence and the hyperbolic tangent function.

Figure 3.

Comparison of curvature variation rates between the hyperbolic Fibonacci sequence and the hyperbolic tangent function.

Design of smooth transition curves

To generate hyperbolic Fibonacci sequence-based curves suitable for practical applications, the sampling interval and explicit convergence criteria must be specified. This ensures that the generated curves are continuous and have appropriate duration. This allows for a smooth transition when connected to acceleration curves from traditional S-curves, avoiding discontinuities or abrupt changes at switching points that could otherwise affect the motion stability of the control system. In this study, the sampling interval of the curve is set to the second level. The convergence condition was defined as follows: when the rate of change of the curve at a given time point is lower than a certain threshold, the curve is considered to have converged. This method ensures that the curve stabilizes before the calculation terminates, thereby avoiding unnecessary extension of motion duration and reducing the computational load on the hardware. This curve has both rising and converging characteristics and is infinitely differentiable, making it very suitable for smoothing discontinuities in S-shaped acceleration curves. However, under the convergence condition adopted, the duration of a single curve segment is on the second level. Since a complete S-shaped curve typically contains eight discontinuous switching points, smoothing all switching points using this method would result in a total transition time on the second level, which would significantly affect the overall motion efficiency and practical feasibility. To address this issue and improve efficiency, we applied time scaling adjustments in practice. Specifically, we shortened the sampling interval between consecutive data points from the second level to the second level and shortened the duration of a single curve segment from the second level to the second level. This scaling method preserves the monotonically increasing and convergent characteristics of the original function while maintaining its infinite differentiability. Therefore, this curve remains applicable to discontinuities in smooth S-shaped acceleration curves, while significantly shortening the transition time and improving practical applicability. Figure 4 shows a comparison between the original function based on the hyperbolic Fibonacci sequence and its time-scaled version.

Figure 4.

Comparison between the hyperbolic Fibonacci tangent function and its time-scaled version.

Jerk adjustment of smooth transition segments

The design of motion profiles must consider the driving capability of the motion platform to ensure that the controlled object completes its motion under safe conditions. The method proposed in this paper utilizes the hyperbolic Fibonacci sequence to smooth the discontinuities in the S-curve. Since the function used inherently possesses fixed variation characteristics, its rise and convergence rates cannot be directly adjusted through the function itself. Although the convergence time can be adjusted by changing the convergence conditions, the inherent rate of change determined by the function itself remains unchanged. Therefore, in order for the modified motion profile to fully utilize the maximum performance of the hardware, the jerk needs to be adjusted.

The jerk $j(t)$ of the hyperbolic Fibonacci-based function can then be expressed as

$$j(t)={\displaystyle \frac{d{F}_t(t)}{dt}}$$

(4)

Since differentiation is a linear operation, let a constant $c_j\in R^{+}$ be defined as the jerk coefficient. According to the principle of linearity, the following relation holds

$$c_j{\displaystyle \frac{d{F}_t(t)}{dt}={\displaystyle \frac{d{c}_j{F}_t(t)}{dt}}}$$

(5)

Equation (5) shows that if the original function is multiplied by a constant, its derivative will also be multiplied by the same constant. This property can be used to adjust the accelerometer of a function based on the hyperbolic Fibonacci sequence, thereby regulating the rate of change of acceleration of the motion profile. Through MATLAB calculations, under the aforementioned sampling interval and convergence conditions, it was found that the maximum acceleration of the curve based on the hyperbolic Fibonacci sequence is approximately $8.60417m/s^3$ . The jerk coefficient $c_j$ is defined as the ratio between the system jerk parameter and the maximum jerk of the hyperbolic Fibonacci-based curve. Assuming that a motion profile $T_j$ exists, the jerk coefficient $c_j$ can be expressed as

$$c_j={\displaystyle \frac{{\mathrm{J}}_{max}}{8.60417}}$$

(6)

This direct design approach ensures that the final motion profile meets the system's jerk limits while maintaining the smoothness and continuity of the original curve. However, if the modified motion profile does not contain a constant jerk segment $T_j$ , the above design method is no longer applicable. Design strategies for this situation can be discussed in subsequent sections.

Proposed modified motion profile design

Composition and derivation of the proposed modified motion profile equations

Traditional S-curve acceleration curves divide the motion process into several segments, including a uniform acceleration phase, a maximum acceleration phase, a deceleration phase, and a uniform velocity phase. This segmented design allows the system to operate at optimal performance, thereby shortening the overall motion duration. Simultaneously, it generates smooth velocity and displacement trajectories, improving motion smoothness, enhancing positioning accuracy, and suppressing unwanted mechanical vibrations and current fluctuations. Due to its simple design, the traditional S-curve has become the most widely used motion profile. However, discontinuities still exist in the acceleration and jerk curves of the traditional S-curve. These discontinuities can lead to adverse system reactions, such as sudden current changes and residual vibrations. Figure 5 illustrates the jerk, acceleration, velocity, and displacement curves of a traditional S-curve.

Figure 5.

Jerk, acceleration, velocity, and displacement curves of the S-curve motion profile.

The modified motion profile proposed in this study incorporates smooth transition segments generated by the hyperbolic Fibonacci sequence into the original acceleration curve. These transition segments replace the discontinuous transitions between acceleration stages in the traditional S-curve, resulting in a smoother overall acceleration profile, enhancing the curve's continuity, while preserving the multisegment structure and motion properties of the original S-curve. The improved acceleration curve can be divided into eight segments, which can be further classified according to the characteristics of acceleration and velocity in each segment. $T_s$ is the hyperbolic Fibonacci-based smooth transition segment, $T_j$ is the constant-jerk segment, $T_a$ is given as the constant-acceleration segment, and $T_v$ is the constant-velocity segment. Among these, the duration of the smooth transition segment is fixed at $0.5044$ seconds, while $T_j,T_a$ and $T_v$ are determined by the upper limits of jerk, acceleration, and velocity, as well as the specified displacement. A schematic illustration of the segmented modified motion profile is shown in Figure 6. To derive the equations of the modified motion profile, the complete jerk equation of the profile is first formulated, as shown in equations (7), (8), and (9).

$$[\begin{array}{cc}{\displaystyle \frac{d{F}_{t4}(t)}{dt}\left[=-{\displaystyle \frac{d{F}_t(T-t)}{dt}}\right]}& 0\le t\le T_1\\ J_{max}& T_1\le t\le T_2\\ {\displaystyle \frac{d{F}_{t1}(t-T_2)}{dt}\left[={\displaystyle \frac{d{F}_t(t-T_2)}{dt}}\right]}& T_2\le t\le T_3\\ 0& T_3\le t\le T_4\\ {\displaystyle \frac{d{F}_{t3}(t-T_4)}{dt}\left[={\displaystyle \frac{d{F}_t(T-t+T_4)}{dt}}\right]}& T_4\le t\le T_5\\ -J_{max}& T_5\le t\le T_6\\ {\displaystyle \frac{d{F}_{t2}(t-T_6)}{dt}\left[=-{\displaystyle \frac{d{F}_t(t-T_6)}{dt}}\right]}& T_6\le t\le T_7\\ 0& T_7\le t\le T_8\end{array}$$

(7)

$${\displaystyle \frac{d{F}_t(t)}{dt}={\displaystyle \frac{8\ln (\varphi )}{\varphi {[e^{(2t+1)\ln (\varphi )}+e^{-(2t+1)\ln (\varphi )}]}^2}}}$$

(8)

Figure 6.

Modified motion profile segments.

By integrating the jerk equation given above, the acceleration equation of the modified motion profile can be obtained by equation (10).

$$[\begin{array}{cc}{F}_{t4}(t)& 0\le t\le T_1\\ J_{max}(t-T_1)+F_{t4}(T_1)& T_1\le t\le T_2\\ F_{t1}(t-T_2)+J_{max}(T_2-T_1)+F_{t4}(T_1)& T_2\le t\le T_3\\ A_{max}& T_3\le t\le T_4\\ A_{max}-\phi +F_{t3}(t-T_4)& T_4\le t\le T_5\\ A_{max}-\phi +F_{t3}(T_5-T_4)-J_{max}(t-T_5)& T_5\le t\le T_6\\ A_{max}-2\phi +F_{t3}(T_5-T_4)-J_{max}(T_6-T_5)+F_{t2}(t)& T_6\le t\le T_7\\ 0& T_7\le t\le T_8\end{array}$$

(10)

Subsequently, by integrating the acceleration equation in equation (10), the velocity equation of the modified motion profile can be obtained, as expressed in equations (11) and (12).

$$[\begin{array}{cc}\int F_{t4}(t)dt=\int \phi -F_t(T-t)dt=\phi t-\int F_t(T-t)dt& 0\le t\le T_1\\ J_{max}\left({\displaystyle \frac{1}{2}{t}^2-T_{1}t}\right)+F_{t4}(T_1)t+c_1& T_1\le t\le T_2\\ \begin{array}{c}\int F_{t1}(t-T_2)dt+[J_{max}(T_2-T_1)+F_{t4}(T_1)]t\\ =\int F_t(t-T_2)dt+[J_{max}(T_2-T_1)+F_{t4}(T_1)]t\end{array}& T_2\le t\le T_3\\ A_{max}t+c_2& T_3\le t\le T_4\\ A_{max}t+\int F_{t3}(t-T_4)dt=A_{max}t+\int F_t(T-t+T_4)dt& T_4\le t\le T_5\\ -J_{max}\left({\displaystyle \frac{1}{2}{t}^2-T_{5}t}\right)+F_{t3}(T_5-T_4)t+A_{max}t+c_3& T_5\le t\le T_6\\ \begin{array}{c}\int F_{t2}(t-T_6)dt-J_{max}(T_6-T_5)t+F_{t3}(T_5-T_4)t+A_{max}t\\ =\phi t-\int F_t(t-T_6)dt-J_{max}(T_6-T_5)t+F_{t3}(T_5-T_4)t+A_{max}t\end{array}& T_6\le t\le T_7\\ V_{max}& T_7\le t\le T_8\end{array}$$

(11)

$$\begin{array}{rl}& \int F_t(t)dt=\int {\displaystyle \frac{e^{2\ln (\varphi )t}-e^{-2\ln (\varphi )t}}{e^{(2t+1)\ln (\varphi )}+e^{-(2t+1)\ln (\varphi )}}dt}\\ & =-\varphi t+{\displaystyle \frac{2\varphi -1}{4\ln \varphi }\ln ({\varphi }^{4t+2}+1)+c}\end{array}$$

(12)

Finally, by integrating the velocity equation in equations (11) and (12), the displacement equation of the modified motion profile can be obtained, as expressed in equations (13) and (14), where $\mathrm{S}$ represents half of the total displacement.

$$[\begin{array}{cc}{\displaystyle \frac{1}{2}\phi t^2-\int \int F_t(T-t)dtdt}& 0\le t\le T_1\\ J_{max}\left({\displaystyle \frac{1}{6}{t}^3-{\displaystyle \frac{1}{2}{T}_1{t}^2}}\right)+{\displaystyle \frac{1}{2}{F}_{t4}(T_1)t^2+c_{1}t+c_{11}}& T_1\le t\le T_2\\ {\displaystyle \int \int F_t(t-T_2)dtdt+{\displaystyle \frac{1}{2}[J_{max}(T_2-T_1)+F_{t4}(T_1)]t^2}}& T_2\le t\le T_3\\ {\displaystyle \frac{1}{2}{A}_{max}{t}^2+c_{2}t+c_{22}}& T_3\le t\le T_4\\ {\displaystyle \frac{1}{2}{A}_{max}{t}^2+\int \int F_t(T-t+T_4)dtdt}& T_4\le t\le T_5\\ -J_{max}\left({\displaystyle \frac{1}{6}{t}^3-{\displaystyle \frac{1}{2}{T}_5{t}^2}}\right)+{\displaystyle \frac{1}{2}[F_{t3}(T_5-T_4)+A_{max}]t^2+c_{3}t+c_{33}}& T_5\le t\le T_6\\ {\displaystyle \frac{1}{2}[\phi +A_{max}-J_{max}(T_6-T_5)+F_{t3}(T_5-T_4)]t^2-\int \int F_t(t-T_6)dtdt}& T_6\le t\le T_7\\ S& T_7\le t\le T_8\end{array}$$

(13)

$$\int \int F_t(t)dtdt=-{\displaystyle \frac{\varphi }{2}{t}^2-{\displaystyle \frac{\sqrt{5}}{16{(\ln \varphi )}^2}\mathrm{H}(-{\varphi }^{4t+2})+c\mathrm{H}(z)=-{\int }_0^z{\displaystyle \frac{\ln (1-t)}{t}dt}\approx \sum _{k=1}^{\mathrm{\infty }}{\displaystyle \frac{z^k}{k^2}}}}$$

(14)

After completing the mathematical derivation of the modified motion profile, the structural differences between the traditional S-curve and the modified motion profile are more intuitively displayed through graphical comparison. Figure 7 shows the changes in jerk, acceleration, velocity, and displacement curves, respectively. Through these comparison figures, the differences between the two curves at the transition points of each segment can be clearly observed. To facilitate the comparison between the modified motion profile and the traditional S-curve, the starting point of the modified motion profile in Figure 7 has been shifted.

Figure 7.

Comparison of the conventional S-curve and the modified motion profile: (a) jerk profiles, (b) acceleration profiles, (c) velocity profiles, and (d) displacement profiles.

Variants and parameter calculation of the modified motion profile

The S-curve motion profile can be classified into four types, depending on whether the motion process reaches the predefined limits of acceleration and velocity. In contrast, the modified motion profile proposed in this study always contains smooth transition segments $T_s$ , regardless of parameter settings. As a result, unlike the conventional S-curve, the modified motion profile can be categorized into eight variants according to whether the motion process reaches the specified limits of jerk, acceleration, and velocity. These variants are named based on the total number of segments, i.e. 8-Segments, 7-Segments A, 7-Segments B, 6-Segments A, 6-Segments B, 5-Segments A, 5-Segments B, and 4-Segments.

For a travel distance of 200 meters with the ${\mathrm{J}}_{max}=30m/s^3$ , ${\mathrm{A}}_{max}=$ $1000m/s^2$ , ${\mathrm{V}}_{max}=1500m/s$ , the resulting modified motion profile corresponds to the Segment-6A type, as illustrated in Figure 8 Under these parameter settings, neither the acceleration nor the velocity reaches their maximum value during the motion process. Therefore, the modified motion profile does not contain the $T_a$ and $T_v$ segments. Since the previous subsection derived the complete-form eight-segment modified motion profile, its equations cannot be directly applied to all parameter conditions. Consequently, it is necessary to perform mathematical derivations for all eight variants of the modified motion profile to ensure correct computation across different motion constraints. The categorization rules for all eight variants of the modified motion profile are summarized in Table 1.

Figure 8.

Segment 6A modified motion profile.

Table 1.

Modified motion profile categorize rules.

	Segment 8	Segment 7A	Segment 7B	Segment 6A
$T_s$	✓	✓	✓	✓
$T_j$	✓	✓	✓	✓
$T_a$	✓	✓	✗	✗
$T_v$	✓	✗	✓	✗

	Segment 6B	Segment 5A	Segment 5B	Segment 4
$T_s$	✓	✓	✓	✓
$T_j$	✗	✗	✗	✗
$T_a$	✓	✓	✗	✗
$T_v$	✓	✗	✓	✗

To calculate the time parameters of each variant of the modified motion profile, we need to refer to equations (7)–(10), which correspond to the complete expressions for acceleration, jerk, velocity, and displacement, respectively. Since the functions based on the hyperbolic Fibonacci sequence and their derivatives and integrals are quite complex, and the jerk coefficient has not yet been introduced into these formulas, this section will simplify the derivation process. Specifically, we can first simplify equation (7) and then introduce the jerk coefficient as the basis for subsequent derivations.

Let $\frac{d{\mathrm{F}}_{ti}(t)}{dt}=j_i(t)$ , then equation (7) can be rewritten as

$$[\begin{array}{cc}{j}_4(t)& 0<t\le T_1\\ J_{max}& T_1<t\le T_2\\ j_1(t)& T_2<t\le T_3\\ 0& T_3<t\le T_4\\ j_3(t)& T_4<t\le T_5\\ -J_{max}& T_5<t\le T_6\\ j_2(t)& T_6<t\le T_7\\ 0& T_7<t\le T_8\end{array}$$

(15)

$$[\begin{array}{c}\int j_i(t)dt=a_i(t)\\ \int a_i(t)dt=v_i(t)\\ \int v_i(t)dt=s_i(t)\end{array}$$

(16)

By integrating equation (15) according to the definition given in equation (16), the simplified acceleration equation can be obtained. The simplified acceleration equation is expressed by equation (17).

$$[\begin{array}{cc}{a}_4(t)& 0<t\le T_1\\ J_{max}t+\phi & T_1<t\le T_2\\ J_{max}{T}_j+\phi +a_1(t)& T_2<t\le T_3\\ A_{max}& T_3<t\le T_4\\ A_{max}-a_4(t)& T_4<t\le T_5\\ A_{max}-\phi -J_{max}t& T_5<t\le T_6\\ A_{max}-J_{max}{T}_j-\phi -a_1(t)& T_6<t\le T_7\\ 0& T_7<t\le T_8\end{array}$$

(17)

By integrating equation (17), the simplified velocity equation of the modified motion profile can be obtained, as expressed by equation (18).

$$[\begin{array}{cc}{v}_4(t)& 0<t\le T_1\\ {\displaystyle \frac{1}{2}{J}_{max}{t}^2+\phi t+v_4}& T_1<t\le T_2\\ T_j(J_{max}t+{\displaystyle \frac{1}{2}{A}_{max})+\phi +{\mathrm{v}}_4+{\mathrm{v}}_1(t)}& T_2<t\le T_3\\ A_{max}t+T_j(J_{max}{T}_s+{\displaystyle \frac{1}{2}{A}_{max})+2\phi T_s}& T_3<t\le T_4\\ A_{max}(T_a+{\displaystyle \frac{1}{2}{T}_j+T_s+t)-v_4(t)}& T_4<t\le T_5\\ A_{max}(2T_s+T_a+{\displaystyle \frac{1}{2}{T}_j+t)-{\displaystyle \frac{1}{2}{J}_{max}{t}^2-\phi t-v_4}}& T_5<t\le T_6\\ A_{max}(2T_s+T_a+T_j+t)-J_{max}{T}_{j}t+\phi t-v_4-v_1(t)& T_6<t\le T_7\\ V_{max}& T_7<t\le T_8\end{array}$$

(18)

Finally, by integrating equation (18), the simplified displacement equation of the modified motion profile can be obtained by equation (19).

$$[\begin{array}{cc}{s}_4(t)& 0<t\le T\\ {\displaystyle \frac{1}{6}{t}^2({\mathrm{J}}_{max}t+3\phi )+v_{4}t+s_4}& T_1<t\le T_2\\ {\displaystyle \frac{1}{2}{\mathrm{J}}_{max}{T}_j{t}^2+{\displaystyle \frac{1}{6}{A}_{max}(T_j^2+3T_{j}t)+{\displaystyle \frac{1}{6}\phi (T_j^2+3t^2)+v_4(T_j+t)+s_1(t)+s_4}}}& T_2<t\le T_3\\ \begin{array}{rl}{\displaystyle \frac{1}{2}{\mathrm{J}}_{max}(T_j{T}_s^2+2T_j{T}_{s}t)+{\displaystyle \frac{1}{6}{\mathrm{A}}_{max}(T_j^2+3T_j{T}_s+3T_{j}t+3t^2)+}}& \\ {\displaystyle \frac{1}{6}\phi (T_s^2+3T_s^2+12T_{s}t)+v_4(T_j+T_s)+s_2+s_1}\end{array}& T_3<t\le T_4\\ \begin{array}{rl}{\displaystyle \frac{1}{6}(V_{max}(3T_a+T_j+T_s)+A_{max}(T_s^2-T\phi (T_s+T_j-6t)+3t^2+t(3T_j+6T_s))+}& \\ \phi (T_j^2-3T_s^2))+v_4(T_s+T_j)+s_1+s_4-s_4(t)\end{array}& T_4<t\le T_5\\ \begin{array}{rl}{\displaystyle \frac{1}{6}(V_{max}(3T_a+T_j+6T_s)+A_{max}(-T_j(2T_s+T_a)+(12T_s+6T_a+3T_j+3t)t)+}& \\ \phi (T_j^2-3T_s^2-3t^2)-{\mathrm{J}}_{max}{t}^3)+v_4(T_s+T_j-t)+s_1\end{array}& T_5<t\le T_6\\ \begin{array}{r}{V}_{max}({\displaystyle \frac{1}{2}{T}_a+T_j+T_s)-{\displaystyle \frac{1}{2}\phi (T_s^2+t^2)+A_{max}(2T_s+T_a+T_j+{\displaystyle \frac{1}{2}t)t-}}}\\ {\displaystyle \frac{1}{2}{\mathrm{J}}_{max}{T}_j{t}^2+v_4(T_s-t)+s_1-s_1(t)}\end{array}& T_6<t\le T_7\\ V_{max}({\displaystyle \frac{1}{2}{T}_a+2T_s+T_j+t),}& T_7<t\le T_8\end{array}$$

(19)

As mentioned earlier, the classification of the modified acceleration motion profile depends on whether the curve reaches a preset limit during motion. Therefore, it is first necessary to derive the expressions for the maximum values of velocity, acceleration, and displacement of the modified motion profile. These expressions introduce an acceleration coefficient as the basis for subsequent derivations.

First, by substituting $t=T_s$ into the third expression of equation (17), the maximum acceleration of the modified motion profile can be obtained as

$$A_{max}=J_{max}{T}_j+2c_j\phi$$

(20)

Next, by substituting $t=T_s$ into the seventh expression of equation (18), the maximum velocity of the modified motion profile can be obtained as

$$V_{max}=A_{max}(2T_s+T_j+T_a)$$

(21)

Finally, by substituting $t=T_v$ into the eighth expression of equation (19), the displacement of the modified motion profile can be obtained as

$$\mathrm{S}={\mathrm{V}}_{max}\left({\displaystyle \frac{1}{2}{T}_a+2T_s+T_j+T_v}\right)$$

(22)

By substituting equations (20) and (21) into (22), the corresponding time durations can be obtained. The calculation methods for $T_j$ , $T_a$ , and $T_v$ are given as follows:

$$[\begin{array}{c}{T}_j={\displaystyle \frac{A_{max}-2c_j\phi }{J_{max}}}\\ T_a={\displaystyle \frac{V_{max}}{A_{max}}+{\displaystyle \frac{2c_j\phi -A_{max}}{J_{max}}-2T_s}}\\ T_v={\displaystyle \frac{S}{V_{max}}-{\displaystyle \frac{V_{max}}{2A_{max}}+{\displaystyle \frac{2c_j\phi -A_{max}}{2J_{max}}-T_s}}}\end{array}$$

(23)

Equations (20), (21), and (22) represent the relationship between the time interval and system parameters in the modified motion profiles and are used to derive the parameter calculation formulas for different types of modified motion profiles. First, the 8-segment, 7-segment A, 7-segment B, and 6-segment A curves are derived. Since these four types of modified motion profiles all reach maximum acceleration during motion, the acceleration coefficient is given by $c_j=\frac{8.60417}{J_{max}}$ .

For the 8-Segment modified motion profile, jerk, acceleration, and velocity can all reach their respective upper limits, i.e. $\mathrm{J}={\mathrm{J}}_{max}$ , $A=A_{max}$ , and $V=V_{max}$ . Thus $T_j>0$ , $T_a>0$ , and $T_v>0$ can be given. By substituting $T_j>0$ , $T_a>0$ , and $T_v>0$ into equations (20), (21), and (22), the time durations $T_j$ , $T_a$ , and $T_v$ of the 8-Segment modified motion profile can be obtained as

$$[\begin{array}{c}{T}_j={\displaystyle \frac{A_{max}-2c_j\phi }{J_{max}}}\\ T_a={\displaystyle \frac{V_{max}}{A_{max}}+{\displaystyle \frac{2c_j\phi -A_{max}}{J_{max}}-2T_s}}\\ T_v={\displaystyle \frac{S}{V_{max}}-{\displaystyle \frac{V_{max}}{2A_{max}}+{\displaystyle \frac{2c_j\phi -A_{max}}{2J_{max}}-T_s}}}\end{array}$$

(24)

For the 7-Segment A modified motion profile, the jerk and acceleration reach their respective upper limits, but the velocity does not reach its upper limit, i.e. $\mathrm{J}={\mathrm{J}}_{max}$ , $A=A_{max}$ and $V<V_{max}$ . Thus $T_j>0$ , $T_a>0$ and $T_v=0$ . By substituting $T_j>0$ , $T_a>0$ and $T_v=0$ into equations (20), (21), and (22), the time durations $T_j$ , $T_a$ , and V of the 7-Segment A modified motion profile can be obtained by

$$[\begin{array}{c}{T}_j={\displaystyle \frac{A_{max}-2c_j\phi }{J_{max}}}\\ T_a={\displaystyle \frac{-(6T_s+3T_j)+\sqrt{{(2T_s+T_j)}^2+{\displaystyle \frac{8S}{A_{max}}}}}{2}}\\ V=A_{max}(-{\displaystyle \frac{1}{2}(2T_s+T_j)+\sqrt{{\displaystyle \frac{{((2T_s+T_j)}^2}{4}+{\displaystyle \frac{2S}{A_{max}}}}})}\end{array}$$

(25)

For the 7-Segment B modified motion profile, the jerk and velocity reach their respective upper limits, but the acceleration does not reach its upper limit, i.e. $\mathrm{J}={\mathrm{J}}_{max}$ , $V=V_{max}$ , and $A<A_{max}$ . Thus $T_j>0$ , $T_v>0$ , and $T_a=0$ can be given. By substituting $T_j>0$ , $T_v>0$ , and $T_a=0$ into equations (20), (21), and (22), the time durations $T_j$ , $T_v$ , and A of the 7-Segment B modified motion profile can be obtained as

$$[\begin{array}{c}{T}_j={\displaystyle \frac{-(c_j\phi +T_s{J}_{max})+\sqrt{{(c_j\phi -T_s{J}_{max})}^2+V_{max}{J}_{max}}}{J_{max}}}\\ T_v={\displaystyle \frac{S}{V_{max}}-2T_s-T_j}\\ A=c_j\phi -T_s{J}_{max}+\sqrt{{(c_j\phi -T_s{J}_{max})}^2+V_{max}{J}_{max}}\end{array}$$

(26)

For the 6-Segment A modified motion profile, only the jerk reaches its upper limit, while the acceleration and velocity do not reaches its upper limit, i.e. $\mathrm{J}={\mathrm{J}}_{max}$ , $V<V_{max}$ , and $A<A_{max}$ . Thus $T_j>0$ , $T_v=0$ , and $T_a=0$ can be given. By substituting $T_j>0$ , $T_v=0$ , and $T_a=0$ into equations (20), (21), and (22), the time durations $T_j$ , A, and V of the 6-Segment A modified motion profile can be obtained as

$$[\begin{array}{c}{T}_j=-{\displaystyle \frac{b}{3a}+\sqrt[3]{-{\displaystyle \frac{q}{2}+\sqrt{\mathrm{\Delta }}}}+\sqrt[3]{-{\displaystyle \frac{q}{2}-\sqrt{\mathrm{\Delta }}}}}\\ A=J_{max}{T}_j+2c_j\phi \\ V=(J_{max}{T}_j+2c_j\phi )(2T_s+T_j)\\ q={\displaystyle \frac{2b^3-9abc+27a^{2}d}{27a^3}}\\ p={\displaystyle \frac{3ac-b^2}{3a^2}}\\ \mathrm{\Delta }={\left({\displaystyle \frac{q}{2}}\right)}^2+{\left({\displaystyle \frac{p}{3}}\right)}^3\\ a=J_{max}\\ b=4J_{max}{T}_s^2{c}_j\phi \\ c=4J_{max}{T}_s^2+8c_j\phi T_s\\ d=8T_s^2{c}_j\phi -S\end{array}$$

(27)

For the 6-Segment B, 5-Segment A, 5-Segment B, and 4-Segment modified motion profiles, the jerk limit is not reached during the motion process, i.e. $J<J_{max}$ and therefore, the jerk coefficient described earlier is not applicable to these four types of profiles. In the following, the corresponding expressions of the jerk coefficient suitable for each of these motion profile types can be derived individually.

For the 6-Segment B modified motion profile, the acceleration and velocity reach their respective upper limits, but the jerk does not reach the upper limit, i.e. $$\begin{gathered} \mathrm{J} \\ \mathrm{lt}{\mathrm{J}}_{max} \end{gathered}$$ , $A=A_{max}$ , and $V=V_{max}$ . Thus $T_j=0$ , $T_a>0$ , and $T_v>0$ can be given. By substituting $T_j=0$ , $T_a>0$ , and $T_v>0$ into equations (20), (21), and (22), the jerk coefficients $c_j$ , $T_a$ , and $T_v$ of the 6-Segment B modified motion profile can be obtained as

$$[\begin{array}{c}{c}_j={\displaystyle \frac{A_{max}}{2\phi }}\\ T_a={\displaystyle \frac{V_{max}}{A_{max}}-2T_s}\\ T_v={\displaystyle \frac{S}{V_{max}}-{\displaystyle \frac{V_{max}}{2A_{max}}-T_s}}\end{array}$$

(28)

For the 5-Segment A modified motion profile, only the acceleration reaches its upper limit, while the jerk and velocity do not reach its upper limit, i.e. $J<J_{max}$ , $A=A_{max}$ , and $V<V_{max}$ . Thus $T_j=0$ , $T_a=0$ , and $T_v>0$ can be given. By substituting $T_j=0$ , $T_a=0$ , and $T_v>0$ into equations (20), (21), and (22), the jerk coefficients $c_j$ , $T_a$ , and V of the 5-Segment A modified motion profile can be obtained as

$$[\begin{array}{c}{c}_j={\displaystyle \frac{A_{max}}{2\phi }}\\ T_a={\displaystyle \frac{-6T_s+\sqrt{4T_s^2+{\displaystyle \frac{8S}{A_{max}}}}}{2}}\\ V=A_{max}(2T_s+{\displaystyle \frac{-6T_s+\sqrt{4T_s^2+{\displaystyle \frac{8S}{A_{max}}}}}{2})}\end{array}$$

(29)

For the 5-Segment B modified motion profile, only the velocity reaches its upper limit, while the jerk and acceleration do not reach the upper limits, i.e. $J<J_{max}$ , $A<A_{max}$ , and $V=V_{max}$ . Thus $T_j=0$ , $T_a>0$ , and $T_v>0$ can be given. By substituting $T_j>0$ , $T_a>0$ , $T_v>0$ , and $T_v>0$ into equations (20), (21), and (22), the jerk coefficients $c_j$ , A, and $T_v$ of the 5-Segment B modified motion profile can be obtained as

$$[\begin{array}{c}{c}_j={\displaystyle \frac{V_{max}}{4\phi T_s}}\\ A={\displaystyle \frac{V_{max}}{2T_s}}\\ T_v={\displaystyle \frac{S}{V_{max}}-2T_s}\end{array}$$

(30)

For the 4-Segment modified motion profile, none of the motion limits are reached, i.e. $\mathrm{J}<{\mathrm{J}}_{max}$ , $A<A_{max}$ , and $V<V_{max}$ . Thus $T_j=0$ , $T_a=0$ , and $T_v=0$ can be given. By substituting $T_j=0$ , $T_a=0$ , and $T_v=0$ into equations (20), (21), and (22), the jerk coefficients $c_j$ , A, and V of the 4-Segment modified motion profile can be obtained as

$$[\begin{array}{c}{c}_j={\displaystyle \frac{S}{8\phi T_s^2}}\\ A={\displaystyle \frac{S}{4T_s^2}}\\ V={\displaystyle \frac{S}{8\phi T_s^2}}\end{array}$$

(31)

Determination method for the application of motion profiles

The parameter derivations of the modified motion profiles presented earlier are utilized to determine the appropriate profile type based on the system parameters and the required displacement.

For the 8-Segment modified motion profile, the validity conditions are $T_j>0$ , $T_a>0$ , and $T_v>0$ . By substituting these conditions into equation (24), the validity condition of the 8-Segment motion profile can be obtained as

$$[\begin{array}{c}0<A_{max}-2c_j\phi \\ 0<{\displaystyle \frac{V_{max}}{A_{max}}+{\displaystyle \frac{2c_j\phi -A_{max}}{J_{max}}-2T_s}}\\ 0<S-V_{max}(T_s+{\displaystyle \frac{V_{max}}{2A_{max}}+{\displaystyle \frac{2c_j\phi -A_{max}}{2J_{max}}}})\end{array}$$

(32)

For the 7-Segment A modified motion profile, the validity conditions are $T_j>0$ , $T_a>0$ , and $V<{\mathrm{V}}_{max}$ . By substituting these conditions into equation (25), the validity condition of the 7-Segment A motion profile can be obtained as:

$$[\begin{array}{c}0<A_{max}-2c_j\phi \\ 0<S-A_{max}{(2T_s+{\displaystyle \frac{A_{max}-2c_j\phi }{J_{max}}})}^2\\ 0<S-{\displaystyle \frac{V_{max}}{2}M-{\displaystyle \frac{V_{max}^2}{2A_{max}}+{\displaystyle \frac{\phi (1-2c_j)}{8}{M}^2}}}\\ M=2T_s+{\displaystyle \frac{\phi (1-2c_j)}{J_{max}}}\end{array}$$

(33)

For the 7-Segment B modified motion profile, the validity conditions are $T_j>0$ , $T_v>0$ , and $A<{\mathrm{A}}_{max}$ . By substituting these conditions into equation (26), the validity condition of the 7-Segment B motion profile can be obtained as:

$$[\begin{array}{c}0<S-{\mathrm{V}}_{max}{(2T_s+{\displaystyle \frac{-c_j\phi T_s{J}_{max}+\sqrt{{(c_j\phi T_s{J}_{max})}^2+V_{max}{J}_{max}}}{J_{max}}})}^2\\ 0<V_{max}-4c_j\phi T_s\\ 0>{\displaystyle \frac{-(c_j\phi +T_s{J}_{max})+\sqrt{{(c_j\phi -T_s{J}_{max})}^2+V_{max}{J}_{max}}}{J_{max}}-A_{max}}\end{array}$$

(34)

For the 6-Segment A modified motion profile, the validity conditions are $T_j>0$ , $V<{\mathrm{V}}_{max}$ , and $A<{\mathrm{A}}_{max}$ . By substituting these conditions into equation (27), the validity condition of the 6-Segment A motion profile can be obtained as:

$$[\begin{array}{c}0<N\\ 0<A_{max}-J_{max}N+2c_j\phi \\ 0>(J_{max}N+2c_j\phi )(2T_s+N)-V_{max}\\ N=-{\displaystyle \frac{b}{3a}+{(-{\displaystyle \frac{q}{2}+\sqrt{\mathrm{\Delta }}})}^{\frac{1}{3}}+{(-{\displaystyle \frac{q}{2}-\sqrt{\mathrm{\Delta }}})}^{\frac{1}{3}}}\end{array}$$

(35)

For the 6-Segment B modified motion profile, the validity conditions are $T_a>0$ and $T_v>0$ . By substituting these conditions into equation (28), the validity condition of the 6-Segment B motion profile can be obtained as:

$$[\begin{array}{c}0<V_{max}-2T_s{A}_{max}\\ 0<S-V_{max}{T}_s-{\displaystyle \frac{V_{max}^2}{2A_{max}}}\end{array}$$

(36)

For the 5-Segment A modified motion profile, the validity conditions are $T_a>0$ and $V<{\mathrm{V}}_{max}$ . By substituting these conditions into equation (29), the validity condition of the 5-Segment A motion profile can be obtained as:

$$[\begin{array}{c}0<S-4T_s^2{A}_{max}\\ 0>S-V_{max}{T}_s-{\displaystyle \frac{V_{max}^2}{2A_{max}}}\end{array}$$

(37)

For the 5-Segment B modified motion profile, the validity conditions are $T_v>0$ and $A<{\mathrm{A}}_{max}$ . By substituting these conditions into equation (30), the validity condition of the 5-Segment B motion profile can be obtained as:

$$[\begin{array}{c}0<S-2T_s{V}_{max}\\ 0<A_{max}-{\displaystyle \frac{V_{max}}{2T_s}}\end{array}$$

(38)

For the 4-Segment modified motion profile, the validity conditions are $$\begin{gathered} \mathrm{V} \\ \mathrm{lt}{\mathrm{V}}_{max} \end{gathered}$$ and $A<{\mathrm{A}}_{max}$ . By substituting these conditions into equation (31), the validity condition of the 4-Segment motion profile can be obtained as:

$$[\begin{array}{c}0>S-4T_s^2{A}_{max}\\ 0>S-2T_s{V}_{max}\end{array}$$

(39)

Of the eight motion profiles proposed in this study, the 8-segment, 7-segment A, 7-segment B, and 6-segment A curves all include a constant acceleration segment, thus their effectiveness requires meeting three conditions. In contrast, the remaining four curves (6-segment B, 5-segment A, 5-segment B, and 4-segment curves) do not include a constant acceleration segment, and therefore only need to meet two conditions. Because the first four curves require meeting more constraints simultaneously, they are more likely to be inapplicable in practical applications. Therefore, it is recommended to verify these curves first during the design process to quickly eliminate infeasible cases, thereby improving design efficiency. The flowchart of the decision-making process for selecting the appropriate motion profile type is shown in Figure 9.

Figure 9.

Flowchart of the motion profile design process.

Simulation and experimental verification

Motor simulation and response comparison

To verify the effectiveness and feasibility of the motion profile design method proposed in this paper, a unified closed-loop velocity control model was established in MATLAB/Simulink. Under the same dynamic constraints (i.e. the same jerk, acceleration, velocity, and displacement parameters), the traditional S-curve and the modified motion profile proposed in this article were used as motion commands, and their velocity and acceleration responses were compared. The controlled object was modeled as a second-order motor system, i.e. $G_w(s)=\frac{w_n^2}{s^2+2\xi w_n+w_n^2}$ .

In the simulation, the natural frequency was set to $w_n=50rad/s$ , the damping ratio to $\xi =0.15/0.3/0.5$ , and the velocity controller gains were configured as $K_p=6.5$ and $K_i=110$ , in order to achieve the target response while preserving observable damping behavior. The motion command parameters were defined as $J_{max}=1000m/s^3$ , $A_{max}=300m/s^2$ , $V_{max}=100m/s$ , and displacement $L=300m$ . Figures 10–12 show the simulation results of velocity and acceleration responses under different damping ratios. In the figures, the orange solid line represents the conventional S-curve, while the blue solid line represents the modified motion profile.

Figure 10.

(a) Velocity response comparison at damping ratio ζ = 0.5 and (b) acceleration response comparison at damping ratio ζ = 0.5.

Figure 11.

(a) Velocity response comparison at damping ratio ζ = 0.3 and (b) acceleration response comparison at damping ratio ζ = 0.3.

Figure 12.

(a) velocity response comparison at damping ratio ζ = 0.15 and (b) acceleration response comparison at damping ratio ζ = 0.15.

Simulation results show that at a larger damping ratio ( $\xi =0.5$ ), both commands converge monotonically with almost no overshoot. However, the S-curve still exhibits significant peaks at the beginning and end, while the modified motion profile shows smaller slope changes at the beginning and end, exhibiting stability in the middle section and avoiding significant peaks or overshoot. At a medium damping ratio, the peaks and overshoot of the S-curve are more pronounced, leading to a longer settling time. Although the modified motion profile also has slight fluctuations, its waveform is similar to that of the case with a larger damping ratio, and the increase in peak value and overshoot is limited. At a smaller damping ratio ( $\xi =0.15$ ), the S-curve produces significant oscillations and large amplitude fluctuations during acceleration and deceleration. The modified motion profile also exhibits oscillations, but its overall shape and peak level are similar to those of the case with a larger damping ratio, and the overshoot is significantly smaller than that of the S-curve, resulting in a smoother velocity trajectory.

Experimental verification

Experimental platform

The motor used in this study is an RMD-X10-S2 brushless DC motor, as shown in Figure 13. The motor is equipped with a built-in PI controller and provides feedback of velocity, acceleration, and current. An external Teensy 4.1 microcontroller was employed to generate the motion profiles and transmit the position commands to the motor via CAN bus. During operation, the motor sends feedback signals of velocity, position, and current back to the Teensy 4.1, which then transfers the data to a computer for further processing in MATLAB, where the results are plotted for comparison. The relevant specifications of the motor are listed in Table 2.

Figure 13.

Experiment setup.

Table 2.

Parameter specifications of the RMD-X10-S2 brushless DC motor.

Motor parameters	Value
Input voltage	24–48 V
Output nominal power	265 W
Nominal torque	50 N-M
Nominal current	6.7 A
Nominal output speed	110 dps
Reducer ratio	35:1
Encoder resolution	0.05 degree/LSB

Experimental method

The experimental method used in this study is as follows. Under the same motor and control environment, two motion commands were executed—the traditional S-curve and the modified motion profile proposed in this article. By adjusting the design parameters of the motion profiles, the responses of the two methods were observed and compared under the same conditions, including maximum current, current stability, positioning accuracy, and motion time. At the beginning of each experiment, the motor was initialized to the origin position. Four sets of experiments were conducted for both the S-curve and the modified motion profile, and the motion profile parameters for each set of experiments are listed in Table 3.

Table 3.

Experimental parameters.

Parameter	Experiment 1	Experiment 2	Experiment 3
Maximum jerk	$800\circ /{\mathrm{s}}^3$	$800\circ /{\mathrm{s}}^3$	$800\circ /{\mathrm{s}}^3$
Maximum acceleration	$50\circ /s^2$	$80\circ /s^2$	$200\circ /s^2$
Maximum velocity	$105\circ /s$	$105\circ /s$	$105\circ /s$
Displacement	$360\circ$	$360\circ$	$360\circ$

Figures 14–25 show the experimental results of motor speed and position responses in Experiments 1 to 3, respectively. In these figures, the horizontal axis represents the command step, where each step corresponds to one issued motor command and its instantaneous feedback. Since the command update rate of the controller is 80 Hz, each step reflects the motor response at successive time intervals. The experiments compared the RMS value of the motor tracking error, the maximum instantaneous current during motion, and the total motion time. The experimental results are summarized in Tables 4 and 5.

Figure 14.

Velocity and position responses of the S-curve in Experiment 1.

Figure 15.

Velocity and position responses of the modified motion profile in Experiment 1.

Figure 16.

Velocity and position responses of the S-curve in Experiment 2.

Figure 17.

Velocity and position responses of the modified motion profile in Experiment 2.

Figure 18.

Velocity and position responses of the S-curve in Experiment 3.

Figure 19.

Velocity and position responses of the modified motion profile in Experiment 3.

Figure 20.

Command and displacement response of the S-curve in Experiment 1.

Figure 21.

Command and displacement response of modified motion profile in Experiment 1.

Figure 22.

Command and displacement response of the S-curve in Experiment 2.

Figure 23.

Command and displacement response of the modified motion profile in Experiment 2.

Figure 24.

Command and displacement response of the S-curve in Experiment 3.

Figure 25.

Command and displacement response of the modified motion profile in Experiment 3.

Table 4.

Experimental results of S-curve.

Comparison parameters	Experiment 1	Experiment 2	Experiment 3
Tracking error RMS	$4.34\circ$	$5.12\circ$	$5.91\circ$
Maximum instantaneous current	1.32 A	1.33 A	1.568 A
Motion time	5.59 s	4.84 s	4.20 s

RMS: root mean square.

Table 5.

Experimental results of modified motion profile.

Comparison parameters	Experiment 1	Experiment 2	Experiment 3
Tracking error RMS	$4.00\circ$	$4.68\circ$	$4.99\circ$
Maximum instantaneous current	0.85 A	0.88 A	0.90 A
Motion time	6.53 s	5.74 s	5.44 s

RMS: root mean square.

Experimental results show that the modified motion profile performs better in terms of tracking error and instantaneous current. Specifically, from Experiments 1–3, the RMS value of the tracking error was reduced by approximately $7.8\text{\%}$ , $8.6\text{\%}$ , and $15.6\text{\%}$ , respectively. In terms of current performance, compared with the S-curve, the maximum instantaneous current during the motion process was reduced by $35.6\text{\%}$ , $33.8\text{\%}$ , and $42.7\text{\%}$ , respectively. These results indicate that the modified motion profile has significant advantages in positioning accuracy and current stability. However, because the modified motion profile introduces an additional smooth transition section, the overall motion duration is increased by $16.8\text{\%}$ , $18.3\text{\%}$ , and $29.5\text{\%}$ compared to the S-curve.

Conclusion

This study proposed a modified motion profile that incorporates infinitely differentiable transition segments to eliminate acceleration discontinuities. The unified mathematical formulation further enables the construction of eight variants of the motion profile according to whether the jerk, acceleration, and velocity limits are reached, thereby preserving design flexibility while ensuring smooth transitions under different motion constraints. Both simulation and experimental evaluations confirm the effectiveness of the proposed method. Compared with the conventional S-curve, the modified motion profile significantly improves motion stability and system performance. Experimental results show that the RMS tracking error was reduced by $15.6\text{\%}$ , while the peak current can be lowered by $37.5\text{\%}$ , indicating substantial benefits in precision and actuator load reduction. Although the modified motion profile slightly increases total motion time, the improvements in accuracy and current behavior demonstrate its practical value for applications requiring higher stability and smoother dynamic responses.