The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory

Abstract

This paper presents planning of smooth trajectories using the asymmetric profiles of displacement, velocity, acceleration and jerk of described higher degree polynomials. The presented way of determination of polynomial and asymmetry coefficients as well as motion time includes the procedure at both constraints of single kinematic quantities, i.e., velocity, acceleration, jerk and introducing a few constraints simultaneously. Moreover, the paper presents a method of planning trajectories with intermediate points consisting in concatenation of the analyzed polynomials. The proposed method of trajectory planning was compared to that of using the S-curve. The results in the form of kinematic quantity courses are presented graphically. The discussed method guarantees continuity of displacement, velocity, acceleration and jerk. It can be used for planning trajectories of various technical objects (e.g., manipulators, mobile robots, CNC machine tools, optical disk drives, autonomous vehicles, etc.).

1. Introduction

Trajectory planning, e.g., of industrial manipulators, mobile robots, toolheads of treatment machines or optical disk drives, etc., is an important issue in the field of robotics, machines construction, automatic and control engineering being a subject of investigations for many researchers. Broader elaboration of the issues associated with planning trajectories of machines and robots executory systems can be found in the book by Biagiotti and Melchiorri [1].

The papers in scientific journals can be divided into several groups. Some of them focus on planning smooth trajectories with constraints of selected kinematic quantities. In papers [2,3], algebraic splines, B-splines and trigonometric splines were used for trajectory planning. The method of trajectory planning technique described by B-splines was presented in [4]. The method of trajectory planning with jerk and acceleration constraints that uses algebraic splines to ensure a smooth trajectory is presented in [5]. Kim et al. [6] used Catmull-Rom splines for trajectory planning. The S-curve and AS-curve for trajectory planning are described in many papers [7,8,9,10,11,12,13,14,15].

Some papers were on planning trajectory and its optimization applying various functions of purpose and various constraints. Planning of minimal-time trajectory by means of algebraic splines with kinematic and dynamic constraints is discussed in [16,17]. The S-curve was used for a planning trajectory in paper [18]. The author used constant values of jerk at the transition between the constant acceleration and deceleration periods. In this way, the minimal time of transition from the actual to assumed velocity on the motion sections with imposed jerk and acceleration constraints was obtained. Lambrechts et al. [19] used an S-curve with fifteen segment motion profile for trajectory planning. It is shown that these trajectories are time-optimal in the most relevant cases. Huang et al. [20] used algebraic splines for trajectory description and optimized them by means of GA (Genetic Algorithms). The aim of optimization was obtaining a minimal jerk, taking into account the additional constraints. The papers by Gasparetto and Zanotto present the way of planning smooth trajectories where B-splines and algebraic splines were applied for the description. The objective function takes into account the execution time as well as the integral squared jerk [21,22,23]. Nguyen et al. [24] used the general model of the S-curve for planning minimal-time trajectory. They presented also a strategy based on the trigonometric models of this curve. Biagiotti and Melchiorri [25,26] used the S-curve employing the filters FIR (Finite Impulse Response) for planning the minimum-time trajectories with kinematic and frequency constraints. Paper [27] applied the S-curve with the constraints of snap, jerk, acceleration and velocity as well as the optimization algorithm for reducing vibrations and developing the quality of machines CNC processing. Meligy et al. [28] employed B-splines for planning trajectories. They applied the algorithm SQP (Sequential Quadratic Programming) for the optimization of motion time taking into account the limits on the velocities and accelerations for each joint of the robot. Wang et al. [29] used the S-curve for planning the trajectory of the end-effector of the manipulator placed on a mobile platform. To obtain the time-optimal trajectory they made use of the MASA (Multi-Axis Synchronization Algorithm). For planning the trajectory, Wang et al. [30] proposed B-splines which were optimized by means of the hybrid algorithm WOA-GA (Whale Optimization Algorithm and Genetic Algorithm). As the objective function, the integral of the squared jerk along the entire trajectory and the total execution time were taken. Wu et al. [31] proposed the algorithm of planning the PTP (Point-To-Point) trajectory in the joint space. It is based on a locally improved asymmetrical jerk motion profile to obtain time-optimal and smooth joint trajectories.

Many papers in the literature discuss the application of polynomials for description of trajectories. However, it should be pointed out that the polynomials of a degree higher than seven were used for the displacement description in only a few papers. Zhao and Bai [32] presented the way of planning the trajectory for two robots, taking into account the deformations in the joints. The changes of the displacement of the manipulated object were described by applying 9th degree polynomials. Boryga and Graboś [33] used higher-degree polynomials for trajectory planning. They used 5th, 7th and 9th degree polynomials to describe the acceleration profile, with a constraint acceleration. Subsequent papers used polynomials of higher degrees to describe trajectories for various applications [34,35,36,37,38]. Mohamed et al. [39] used 6th, 5th, 4th and 3rd degree polynomials as well as cycloidal formula and elliptical formula to trajectory planning of a surgical micro-robot. Wu and Sun [40] used high degree polynomials for trajectories planning. They used the algorithm CS (Cuckoo Search) for the polynomial coefficients determination. Wang et al. [41] proposed a method for planning the trajectory in which the acceleration was described by means of the 4th degree polynomial. According to the authors, it is possible to obtain a near time-optimal trajectory by maximizing the constant velocity part under kinematical constraints. Zhang and Ming [42] presented planning of the trajectory for a parallel robot. In the end-effector operating space the rectangular transition of the pick-and-place trajectory was rounded with the Lamé curve. The algorithm GWO (Grey Wolf Optimizer) was suggested for the energy consumption optimization, whereby the reduction of residual vibration and precision of manipulator work were obtained using 5th and 6th degree polynomials.

Table 1. Collation of publications using typical parametrization of the trajectory.

Description of the Trajectory	Examples of Publications/Author and Year
Trygonometric splines	Visioli 2000 [2], Dyllong and Visioli 2003 [3].
Algebraic splines	Choi et al. 2000 [16], Constantinescu and Croft 2000 [17], Visioli 2000 [2], Dyllong and Visioli 2003 [3], Macfarlane and Croft 2003 [5], Huang et al. 2006 [20], Gasparetto i Zanotto 2008 [22], Gasparetto and Zanotto 2010 [23], Kim et al. 2020 [6].
B-splines	Saramago and Ceccarelli 2002 [4], Dyllong and Visioli 2003 [3], Gasparetto and Zanotto 2007 [21], Gasparetto and Zanotto 2010 [23], Meligy et al. 2013 [28], Wang et al., 2022 [30].
S-curve	Red 2000 [18], Lambrechts et al. 2005 [19], Nguyen et al. 2008 [24], Rew et al. 2009 [7], Rew and Kim 2010 [8], Chen et al. 2011 [9], Biagiotti and Melchiorri 2012 [25], Fan et al. 2012 [27], Ezair et al. 2014 [10], Lee and Choi 2015 [11], Li 2016 [12], Biagiotti and Melchiorri 2019 [26], Fang et al. 2019 [13], Wang et al. 2020 [29], Alpers 2021 [14], Alpers 2022 [15], Wu et al. 2022 [31].
Polynomials	Zhao and Bai 2000 [32], Boryga and Graboś 2009 [33], Graboś and Boryga 2013 [34], Boryga 2014 [35], Boryga et al. 2015 [36], Boryga, 2016 [37], Mohamed et al. 2018 [43], Wu and Sun 2019 [40], Wang et al. 2019 [41], Zhang and Ming 2019 [42], Boryga, 2020 [38].

presents the exemplary papers in which the most frequent methods of trajectories parameterization were used.

The objective of the paper is elaboration of trajectory planning algorithm using asymmetric polynomial profiles. Contrary to the previous papers, the asymmetry k coefficient was introduced to obtain the velocity asymmetric profile. Similar to the previous works [33,37,38], the most frequently used description of polynomial trajectory, in the form of a power expansion, was neglected and the root multiplicity was applied for building the polynomial defining the acceleration function. For the given trajectory with kinematic constraints, the problem consists in calculation of one coefficient of the polynomial, asymmetry coefficient as well as the time of realization independent of the polynomial degree. An important aspect of the proposed solution is that the calculated quantities can be considered the dependences describing velocity, position and jerk. This reduces the calculation complexity of parameterization in the time domain.

The paper arrangement is as follows. Section 2 presents the way of formulation of acceleration profile using the roots multiplicity of the polynomial and the description of the conditions imposed on the acceleration function. The root multiplicity was collated for the 5th–9th degree polynomials and the polynomials for the detailed analysis were chosen. A method of calculating the polynomial coefficient, asymmetry coefficient and motion time at the constraint of single kinematic quantity, i.e., velocity, acceleration and jerk as well as those of a few kinematic quantities at the same time is presented. Section 3 compares the profiles of displacement, velocity, acceleration and jerk for one or a few constraints based on the results of simulation made for the rectilinear trajectory of the end-effector. It also includes the results of simulation for the trajectory with two intermediate points and the comparison of polynomial trajectory with the asymmetric AS-curve. Chapter 4 presents a detailed discussion. The conclusions are given in the last chapter.

2. Materials and Methods

2.1. Asymmetric Acceleration Profile

The calculation of the polynomial coefficients in the form of power expansion comes across some difficulties [9]. Therefore, the polynomial product was applied with one coefficient, for which its profile can be easily formed using the properties of the root multiplicity [33,37,38].

The general form of the nth degree polynomial built exploiting the j zero places is as follows:

$$\mathrm{p}(\mathrm{t})=\mathrm{p}\cdot {(\mathrm{t}-{\mathrm{t}}_1)}^{{\mathrm{m}}_1}\cdot {(\mathrm{t}-{\mathrm{t}}_2)}^{{\mathrm{m}}_2}\cdot \dots \cdot {(\mathrm{t}-{\mathrm{t}}_{\mathrm{j}})}^{{\mathrm{m}}_{\mathrm{j}}}$$

(1)

whereby the polynomial degree:

$$\mathrm{n}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{j}}{\mathrm{m}}_{\mathrm{i}}}$$

(2)

where: p—the polynomial coefficient, t—the motion time, t₁, t₂, ..., t_j—the successive zero places, m₁, m₂, ..., m_j—the multiplicity of successive zero places, j—the number of polynomial zero places, n—the polynomial degree.

In this paper the product form of the polynomial was used for the formation of the acceleration profile. The displacement function was denoted s(t) and the dependences describing: velocity, acceleration, jerk and snap being successive derivatives in relation to the displacement function time were denoted: s⁽¹⁾(t), s⁽²⁾(t), s⁽³⁾(t), s⁽⁴⁾(t), respectively.

It was assumed that the acceleration profile s⁽²⁾(t):

• Should be in the interval t ∈ ⟨0,t_end⟩;
• Will be asymmetric;
• Should be composed of two motion phases—the acceleration phase, for which s⁽²⁾(t) > 0 and the braking phase, for which s⁽²⁾(t) < 0, whereby the sequence of acceleration and braking phases is optional. The first motion phase is in the interval t ∈ ⟨0,k·t_end⟩ but the other one is in the interval t ∈ (k·t_end,t_end⟩. The value of coefficient k, called the asymmetry coefficient, will be placed in the interval k ∈ (0,1);
• Should be tangent to the time axis for the first t = 0 and the last t = t_end of the zero place.

The form of the polynomial determining the acceleration profile meets the above requirements as follows:

$${\mathrm{s}}_{{\mathrm{m}}_1{\mathrm{m}}_2{\mathrm{m}}_3}^{(2)}(\mathrm{t})=\mathrm{p}\cdot {\mathrm{t}}^{{\mathrm{m}}_1}\cdot {(\mathrm{t}-\mathrm{k}\cdot {\mathrm{t}}_{\mathrm{e}\mathrm{n}\mathrm{d}})}^{{\mathrm{m}}_2}\cdot {(\mathrm{t}-{\mathrm{t}}_{\mathrm{e}\mathrm{n}\mathrm{d}})}^{{\mathrm{m}}_3}$$

(3)

The first requirement refers to planning the PTP (point-to-point) trajectory. In the case of concatenation during planning the trajectory with intermediate points, the shift in time of all profiles which determine the courses of kinematic quantities will be necessary. The second requirement enables obtaining various values of velocity at the beginning and end of motion. The third requirement confines the number of zero places up to three and determines the position of the second zero place. For the change of the acceleration phase into braking (or inversely) the multiplicity of the second root t = k·t_end should be an odd number. The fourth requirement causes elimination of the unfavourable phenomenon which is jerk in the initial and final periods of motion. This will allow for better positioning accuracy and will constrain the system vibrations at the beginning and end of the motion. For meeting this requirement, the multiplicity of the first and third roots must follow the inequality m ≥ 2. Considering the above requirements, the smallest degree of the polynomial determining the acceleration function is n = 5. In this case, the successive roots have the multiplicities m₁ = 2, m₂ = 1 and m₃ = 2. In the case of the polynomial of degree 6, there are two satisfying polynomials whose roots have the multiplicities m₁ = 3, m₂ = 1, m₃ = 2 and m₁ = 2, m₂ = 1, m₃ = 3.

Table 2. The compilation of the roots multiplicity of the acceleration function (n = 5, 6, 7, 8, 9).

											n
Degree of Polynomial		5	6	6	7	7	7	7	8	8	8	8	8	8	9	9	9	9	9	9	9	9	9
The roots multiplicity	m1	2	3	2	4	3	2	2	5	4	3	2	3	2	6	5	4	3	2	4	3	2	2
	m2	1	1	1	1	1	1	3	1	1	1	1	3	3	1	1	1	1	1	3	3	3	5
	m3	2	2	3	2	3	4	2	2	3	4	5	2	3	2	3	4	5	6	2	3	4	2

presents the roots multiplicity for the 5th–9th degree polynomials. The considerations were limited to the analysis of the polynomials whose roots multiplicity m_i ≤ 3. This reduces the polynomial degree and prevents formation of the Runge phenomenon [43]. The roots multiplicity included in Table 2 refers to the polynomials subjected to the analysis. The acceleration profiles were denoted with the symbols: s₂₁₂⁽²⁾(t), s₃₁₂⁽²⁾(t), s₂₁₃⁽²⁾(t), s₃₁₃⁽²⁾(t), s₂₃₂⁽²⁾(t), s₃₃₂⁽²⁾(t), s₂₃₃⁽²⁾(t) and s₃₃₃⁽²⁾(t). The notation of the roots multiplicity recorded in the bottom index of the acceleration function is preserved in the denotations of the displacement, velocity, jerk and snap profiles. Thus, these profiles for the exemplary 5th degree polynomial will be denoted: s₂₁₂(t), s₂₁₂⁽¹⁾(t), s₂₁₂⁽²⁾(t), s₂₁₂⁽³⁾(t), s₂₁₂⁽⁴⁾(t).

Formulas for the displacement, velocity, acceleration and jerk functions of the analyzed polynomials are included in the Appendix A.

2.2. Constraint of the Single Kinematic Quantity

It was assumed that the kinematic quantities which can be constrained are: velocity s⁽¹⁾(t), acceleration s⁽²⁾(t) or jerk s⁽³⁾(t) that is i = 1, 2, 3. The conditions which the function with constraints must satisfy are:

$$\underset{\mathrm{t}\in <0,{\mathrm{t}}_{\mathrm{e}}>}{\forall }\left|{\mathrm{s}}^{(\mathrm{i})}(\mathrm{t})\right|\le {\mathrm{s}}_{\max }^{(\mathrm{i})}$$

(4)

$$\underset{\mathrm{t}\in <0,{\mathrm{t}}_{\mathrm{e}}>}{\exists }\left|{\mathrm{s}}^{(\mathrm{i})}(\mathrm{t})\right|={\mathrm{s}}_{\max }^{(\mathrm{i})}$$

(5)

The use of the above constraints referring to the function describing velocity and acceleration (i = 1,2) results from the possibility of the drive system [5,7,28,44]. However, the jerk constraint (i = 3) provides: better accuracy of trajectory mapping, continuous and coordinated motion, reduction in the driving loads, limitation of activation of resonance systems resonant frequency [9,28,44,45].

In order to derive the dependences expressing: the coefficient of polynomial p and the asymmetry k as well as the time of realization of trajectory t_end at the optional constraint s_max⁽ⁱ⁾ and the assumed initial and terminal values: s_begin = s(0), s_end = s(t_end), v_begin = s⁽¹⁾(0), v_end = s⁽¹⁾(t_end) one should:

• Determine the zero place of the function s⁽ⁱ⁺¹⁾(t) in the interval t∈⟨0, t_end⟩;
• Determine the value of the function s⁽ⁱ⁾(t) for successive zero places and then choose the one in which the function s⁽ⁱ⁾(t) archives the largest absolute value;
• Create a system of three equations in which there are compared:
  • the dependence for the largest absolute value obtained in stage 2 with the given maximal value of the limited quantity s_max⁽ⁱ⁾;
  • the dependence for the terminal position with the given terminal position;
  • the dependence for the terminal velocity with the given terminal velocity;
• Solve the created system of equations by determining the coefficients of: polynomial p, asymmetry k and time of realization of the trajectory t_end.

2.3. Constraint of Several Kinematic Quantities Simultaneously

It is possible to introduce more than one constraint. In the case of a few constraints, one should calculate the time of realization of the trajectory t_end for each of them separately and then choose the one with the longest time of motion t_end. Then the function of the variable for which the time was the largest will satisfy Conditions (4) and (5). However, for the other variables, for which there was assumed longer time of motion than that from the calculations, the maximal value of the limited quantity will not be attained and thus only Condition (4) will be satisfied.

3. Results

3.1. Results of the Simulation for a Single Constraint

The simulation was performed for the rectilinear motion, whereby in all cases the initial and terminal values of displacement values were: s_begin = 0.3 m and s_end = 1.3 m, but the initial and terminal values of velocity were: v_begin = 0.2 m·s⁻¹ and v_end = 0.5 m·s⁻¹. Figure 1 presents the profiles of displacement s(t), velocity s⁽¹⁾(t), acceleration s⁽²⁾(t) and jerk s⁽³⁾(t) of the analyzed polynomials at the velocity constraint. The maximal velocity value was assumed to be v_max = 1 m·s⁻¹.

In the case of velocity constraint (Figure 1a–d), the time of trajectory realization t_end was in the range 1.344 to 1.659 s. The shortest time was achieved using the polynomial 2-3-2, but the longest time was for the polynomial 3-1-2. The value of the maximal velocity for all polynomials was equal to the assumed one being v_max = 1 m·s⁻¹. The largest, absolute value of acceleration (deceleration) was in the range from 1. 537 to 2.887 m·s⁻². The minimal value was obtained using the polynomial 3-1-2, and the largest one for the polynomial 2-3-3. The maximal absolute value of jerk was in the range from 6.359 to 22.633 m·s⁻³. The smallest value was obtained for the polynomial 2-1-2, and the largest one for the polynomial 2-3-3.

Figure 2 presents the profiles of displacement, velocity, acceleration and jerk for the analyzed polynomials with the acceleration constraint. The maximal absolute value of acceleration was assumed to be |a_max| = 2 m·s⁻².

In the case of the acceleration constraint (Figure 2a–d), the time of trajectory realization was in the range of 1.454 to 1.566 s. The shortest time was obtained using the polynomial 2-1-2, but the longest one in the case of the polynomial 3-1-3. The maximal value of velocity was in the range 0.884 to 1.138 m·s⁻¹. The smallest value was obtained for the polynomial 2-3-3, and the largest one for the polynomial 3-1-2. The value of maximal acceleration (deceleration) was equal to the assumed one |a_max| = 2 m·s⁻². The largest absolute value of jerk was in the range 8.412 to 13.683 m·s⁻³. The smallest value was in the case of the polynomial 2-1-2, but the maximal one for the polynomial 2-3-3.

Figure 3 presents the profiles of displacement, velocity, acceleration and jerk profiles for the analyzed polynomials with the jerk constraint. The maximal absolute value of jerk was assumed to be |j_max| = 10 m·s⁻³.

At the jerk constraint (Figure 3a–d), the time t_end of trajectory realization was in the range from 1.392 to 1.673 s. The shortest time was obtained using the polynomial 2-1-2, but the longest one for the polynomial 2-3-3. The maximal velocity value was in the range from 0.761 to 1.164 m·s⁻¹. The smallest value was obtained for the polynomial 2-3-3, but the largest one for the polynomial 2-1-2. The maximal acceleration was in the range from 1.605 to 2.253 m·s⁻². The smallest value was when the polynomial 2-3-3 was used but the largest one in the case for the polynomial 2-1-2. According to the assumption the maximal absolute jerk value being |j_max| = 10 m·s⁻³ was obtained for all polynomials.

3.2. Simulation Results for a Few Kinematic Constraints

Simulations were performed for the data presented in the previous subsection, i.e., s_begin = 0.3 m, s_end = 1.3 m, v_begin = 0.2 m·s⁻¹, v_end = 0.5 m·s⁻¹, with the simultaneous constraints of: velocity, acceleration and jerk (v_max = 1 m·s⁻¹,|a_max| = 2 m·s⁻², |j_max| = 10 m·s⁻³).

Table 3. The time of motion with the constraints of velocity, acceleration and jerk.

Polynomial Denotation	Time tend [s]
	With the Velocity Constraint	With the Acceleration Constraint	With the Jerk Constraint
2-1-2	1.557	1.454	1.392
3-1-2	1.659	1.510	1.504
2-1-3	1.558	1.535	1.478
3-1-3	1.640	1.566	1.539
2-3-2	1.344	1.467	1.543
3-3-2	1.412	1.467	1.552
2-3-3	1.357	1.547	1.673
3-3-3	1.412	1.533	1.542

presents the time of trajectory realization with the velocity, acceleration and jerk constraints for the analyzed polynomials.

In the case of the polynomial 2-1-2, the calculated time of trajectory realization with the velocity, acceleration and jerk constraints was: 1.557 s, 1.454 s, 1.392 s, respectively. As the longest motion time t_end = 1.557 s is obtained at the velocity constraint, the largest velocity is found for the polynomial 2-1-2 but at the acceleration and jerk constraints as well as t_end = 1.557 s the maximal values are not obtained. The situation is similar in the case of the polynomials: 3-1-2, 2-1-3, 3-1-3. However, for the polynomials 2-3-2, 3-3-2, 2-3-3 and 3-3-3, the longest motion time is obtained at the jerk constraint. Therefore, for these polynomials there will be obtained the maximal jerk but the velocity and the absolute acceleration value will not reach the assumed values.

The profiles of position, velocity, acceleration and jerk with the simultaneous constraints of velocity, acceleration and jerk for the analyzed polynomials are presented in Figure 4.

With the simultaneous velocity, acceleration and jerk constraints (Figure 4a–d), the time of trajectory realization was in the range 1.542 to 1.673 s. The shortest time was obtained when the polynomial 3-3-3, was used and the longest time was in the case of the polynomial 2-3-3. The largest value of velocity being v_max = 1 m·s⁻¹ was obtained for the polynomials 2-1-2, 3-1-2, 2-1-3 and 3-1-3 but in the case of the other analyzed polynomials the maximal value was not obtained. The largest absolute value of acceleration (deceleration) |a_max| = 2 m·s⁻² was not obtained. The maximal value being 1.971 m·s⁻² was obtained using the polynomial 3-3-3 and the smallest one 1.537 m·s⁻² in the case of the polynomial 3-1-2. The largest absolute jerk value |j_max| = 10 m·s⁻³ was obtained for the polynomials 2-3-2, 3-3-2, 2-3-3 and 3-3-3. However, it should be pointed out that this takes place in the acceleration phase for the polynomials 2-3-2, 2-3-3 and 3-3-3 but in the breaking phase for the polynomial 3-3-2. For the other analyzed polynomials, i.e., 2-1-2, 3-1-2, 2-1-3 and 3-1-3, the maximal absolute jerk value |j_max| = 10 m·s⁻³ was not obtained.

3.3. Concatenation of Polynomials

The chosen polynomials were used for generation of smooth trajectory with two intermediate points. The planned trajectory was formed combining three different polynomials. The polynomial 2-1-3 was taken in the time interval t∈⟨0,t₁) the polynomial of 3-1-3 in t∈⟨t₁,t₂⟩ and the polynomial 2-1-2 in t∈(t₂,t_end⟩. The polynomial coefficients, time of motion along individual sections as well as asymmetry coefficients were calculated based on: the distance between the assumed points, the velocities which must be reached in each point and the constraints of velocity, acceleration and jerk quantities. There were assumed the following data: s_begin = 0 m, s₁ = 0.5 m, s₂ = 1.5 m, s_end = 2 m, v_begin = 0 m·s⁻¹, v₁ = 0.2 m·s⁻¹, v₂ = 0.5 m·s⁻¹, v_end = 0 m·s⁻¹ as well as the constraints v_max = 1 m·s⁻¹, |a_max| = 2 m·s⁻², |j_max| = 10 m·s⁻³. Figure 5 presents the displacement, velocity, acceleration, jerk and snap profiles for the planned trajectory.

3.4. Comparison of the Polynomial Trajectory and the AS-Curve

The profiles displacement, velocity, acceleration, jerk and snap of the chosen polynomial were compared with the corresponding profiles obtained using the AS-curve. The polynomial 2-1-2 was chosen for comparison. The profile of AS-curve jerk was built based on the profile of jerk s₂₁₂⁽³⁾(t). It was assumed that phases of constant values of AS-curve jerk will occur in the surroundings of local profile jerk extremal s₂₁₂⁽³⁾(t)—j₁, j₂, and j₃, whereby the successive extreme jerk values for the polynomial and AS-curve profiles are equal. The end time of motion t_end for the jerk profile of the AS-curve was assumed to be equal to that of the motion for the polynomial profile s₂₁₂⁽³⁾(t). The time of beginning and end of individual motion phases t_i (i = 1, ..., 6) for the profile of the AS-curve was determined from the solution of the optimization task. In this task, the time t_i (i = 1, ..., 6) was taken as the decision-making variable but the coefficient of Pearson linear correlation between the polynomial jerk profile s₂₁₂⁽³⁾(t) and the AS-curve jerk was taken as the maximized objective. The courses of comparable jerk profiles with the denoted characteristic quantities are presented in Figure 6. As for the polynomial 2-1-2 the polynomial coefficient p, the asymmetry coefficient k and the motion time t_end were calculated for s_begin = 0 m, s_end = 1 m, v_begin = 0 m·s⁻¹ and v_end = 0.3 m·s⁻¹ whereby the jerk |j_max| = 10 m·s⁻³ was the constrained quantity. The correlation coefficient between the jerk courses was r = 0.998647.

Figure 7 presents the courses of displacement, velocity, acceleration, jerk and snap for the polynomial 2-1-2 and the AS-curve.

4. Discussion

Table 4. Literature references on the occurrence of extreme kinematic parameters for polynomial profiles with constraints.

Polynomial Denotation		Velocity Constraint			Acceleration Constraint			Jerk Constraint
		tend	a	j	tend	v	j	tend	v	a
2-1-2	min			[37,38],[*]	[*]		[37,38],[*]	[37,38],[*]
	max								[37,38],[*]	[37,38],[*]
3-1-2	min		[*]
	max	[*]				[*]
2-1-3	min
	max
3-1-3	min
	max				[*]
2-3-2	min	[*]			[37,38]
	max
3-3-2	min
	max
2-3-3	min					[*]			[*]	[*]
	max		[*]	[*]			[*]	[*]
3-3-3	min
	max
2-5-2	min	[37,38]				[37,38]			[37,38]	[37,38]
	max		[37,38]	[37,38]			[37,38]
4-1-4	min		[37,38]
	max	[37,38]			[37,38]	[37,38]		[37,38]

[*]—in this paper.

presents a comparison of the results included in papers [37,38] and in this paper [*] concerning the extreme values of the time realization of the trajectories modelled by means of higher degree polynomials. In papers [37,38], there were obtained the results of trajectories planning for which the initial velocity v_begin and the final velocity v_end were equal to zero. There were analyzed the polynomials: 2-1-2, 2-3-2, 3-1-3, 2-5-2, 3-3-3 and 4-1-4. In this paper, the velocities: initial v_begin and final v_end were different from zero and the analyzed polynomials were: 2-1-2, 3-1-2, 2-1-3, 3-1-3, 2-3-2, 3-3-2, 2-3-3, and 3-3-3. Based on the modelling results available in [37,38] as well as in this paper [*], it can be stated that at the constraint of both velocity and acceleration for the polynomial 2-1-2 there were obtained the minimal values of jerk. However, in the case of jerk as constraint there is obtained the shortest time t_end for the analyzed polynomial; however, then the extreme values of velocity and acceleration are greater (than those of the polynomials). Similar statements can be made for the other polynomials, whereby they can refer to the results presented in [37,38] or only in this paper [*].

In the case of polynomials concatenation during the trajectory planning with the intermediate points (Figure 5a–e), the maximal velocity v_max = 1 m·s⁻¹ was obtained in the interval t∈⟨t₁, t₂⟩ in which the polynomial 3-1-3 was used. The maximal value of acceleration (deceleration) |a_max| = 2 m·s⁻² was obtained in the interval t ∈ ⟨0, t₁) when the polynomial 2-1-3 was used and in the interval t ∈ (t₂, t_end⟩ when there was applied the polynomial 2-1-2. The largest, absolute jerk value |j_max| = 10 m·s⁻³ was not obtained. Its maximum was 9.858 m·s⁻³ being in the first-time interval t ∈ ⟨0, t₁). The displacement, velocity, acceleration profile is continuous in the whole motion range. However, the snap course is discontinuous for the time t = t₂. This is due to the use of the polynomial 2-1-2, whose multiplicity of the first root m₁ = 2, for t ∈ ⟨t₂, t_end⟩. For preservation of snap profile continuity, the polynomial was chosen from: 3-1-2, 3-1-3, 3-3-2 or 3-3-3 for which the multiplicity of the first root m₁ = 3. Moreover, in order to obtain a zero values snap at the motion beginning and end, the first root multiplicity of the polynomial used in the interval t ∈ ⟨0, t₁) should be equal to m₁ = 3 and the third root multiplicity of the polynomial used in the interval t ∈ ⟨t₂, t_end⟩ should be equal to m₃ = 3.

Comparing the corresponding courses of the polynomial 2-1-2 and AS-curve, there were observed slight differences in the course of displacement, velocity and acceleration (Figure 7a–c). The greatest differences were found during the change of motion phase or in the terminal motion stage. Application of the AS-curve linear function for the jerk profile resulted in the discontinuities in the snap course as its profile is the staircase function (Figure 7e).

5. Conclusions

The main advantage of the method of trajectory planning using root multiplicities of polynomials is the continuity of successive derivatives of displacement with respect to time, and obtaining zero values of acceleration and jerk for the time t = t_begin and t = t_end. Continuity of displacement, velocity, acceleration, jerk and even snap function is preserved when planning PTP trajectories and obtainable when planning trajectories with intermediate points. Based on the analysis, further conclusions can be drawn:

• The acceleration profiles presented in the paper, created using root multiplicities of polynomial, are simple in terms of mathematical description. Determination of other kinematic quantities based on them also does not cause computational difficulties.
• As a result of the use of asymmetric profiles, for which the assumed initial and final velocity values are arbitrary, it is possible to concatenate the analyzed polynomial profiles, allowing planning of trajectories with intermediate points.
• Application of all polynomials analyzed in the paper for the formation of motion trajectories passing through via-points, ensures continuity of acceleration and jerk in the whole motion range and using the polynomials 3-1-3 and 3-3-3, it is possible to obtain continuity of snap and at the same time its zero values for the time t = t_begin and t = t_end.
• Exploitation of the analyzed polynomials enables planning the motion paths for which there can be fulfilled (except for the assumed constraints) additional requirements, including, e.g., the shortest time, minimal acceleration or jerk, etc.
• The possibility of introducing constraints on speed, acceleration and jerk can facilitate, already at the initial planning stage, the adaptation of the trajectory to, for example, the known movement capabilities of the kinematic chain and the properties of the drive system.
• The acceleration function along with easily derived displacement, velocity and jerk functions are ready-made motion patterns and can be used for trajectory planning: effectors of manipulators, toolheads of the CNC machines as well as mobile robots and autonomous vehicles.

It is also advisable to conduct further work on the use of higher-degree polynomials for motion trajectory planning. In particular, simulation studies on the use of symmetric and asymmetric polynomial profiles should be undertaken to control the motion of real objects, taking into account their kinematic and dynamic properties, and the comparison of experimental and simulation results should be carried out.