The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
Abstract
:1. Introduction
2. Materials and Methods
2.1. Asymmetric Acceleration Profile
- Should be in the interval t ∈ ⟨0,tend⟩;
- 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·tend⟩ but the other one is in the interval t ∈ (k·tend,tend⟩. 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 = tend of the zero place.
2.2. Constraint of the Single Kinematic Quantity
- Determine the zero place of the function s(i+1)(t) in the interval t∈⟨0, tend⟩;
- 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 smax(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 tend.
2.3. Constraint of Several Kinematic Quantities Simultaneously
3. Results
3.1. Results of the Simulation for a Single Constraint
3.2. Simulation Results for a Few Kinematic Constraints
3.3. Concatenation of Polynomials
3.4. Comparison of the Polynomial Trajectory and the AS-Curve
4. Discussion
5. Conclusions
- 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 = tbegin and t = tend.
- 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.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
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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]. |
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 |
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 |
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] |
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Boryga, M.; Kołodziej, P.; Gołacki, K. The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory. Appl. Sci. 2022, 12, 12284. https://doi.org/10.3390/app122312284
Boryga M, Kołodziej P, Gołacki K. The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory. Applied Sciences. 2022; 12(23):12284. https://doi.org/10.3390/app122312284
Chicago/Turabian StyleBoryga, Marek, Paweł Kołodziej, and Krzysztof Gołacki. 2022. "The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory" Applied Sciences 12, no. 23: 12284. https://doi.org/10.3390/app122312284
APA StyleBoryga, M., Kołodziej, P., & Gołacki, K. (2022). The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory. Applied Sciences, 12(23), 12284. https://doi.org/10.3390/app122312284