Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk
Abstract
:1. Introduction
2. Related Work
3. Problem Formulation
4. Contributions
- We describe an innovative construction method for 5th order Bézier curves. The proposed parameterization is simple and intuitive, yet effective for generating smooth paths consisting of multiple splines (Section 5);
- The above smooth path generation basis is coupled with an algorithm that computes a minimum-time velocity profile with velocity, acceleration, and jerk constraints on a predefined path (see Ref. [5]). Together they form a powerful trajectory generation algorithm (Section 6). The resulting trajectories thus provide continuous velocity and acceleration profiles;
- To prove the applicability of our approach to trajectory optimization, we performed simulation experiments on a racetrack and in a warehouse environment (Section 6.1 and Section 6.2). In the warehouse simulation, we identified and analyzed realistic situations with different dynamic constraints to investigate and propose the most appropriate driving scenarios.
5. Curve Primitives
5.1. Construction of 5th Order Bézier Curves
- (1)
- Outline the first control point and mark it as . In the direction of , measure out the distance and mark the second point as .
- (2)
- In the direction , measure out the distance (from Equation (12)):
- (3)
- Measure in the perpendicular direction the distance (from Equation (10)):
- (4)
- All points away from for (Equation (11)) in the same direction (perpendicular to the line segment ) lie on the red dashed line.
- (5)
- Mark the last point as . Measure out the distance in the opposite direction from and mark the fifth point as .
- (6)
- (7)
- The fourth control point lies on the intersection of the red and green dashed lines. The Bézier curve is now completely defined.
6. Generation of Minimum-Time Trajectories
6.1. Racetrack Environment
6.2. Warehouse Environment
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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[s] | [s] | [s] | [s] | [s] | [s] | ||
---|---|---|---|---|---|---|---|
1 | 2 | 2.91 | 4.20 | 7.18 | 9.73 | 11.24 | 12.11 |
5 | 2.34 | 2.87 | 5.44 | 7.89 | 9.42 | 10.20 | |
2 | 2 | 2.87 | 4.15 | 7.14 | 9.69 | 11.19 | 12.03 |
5 | 2.34 | 2.88 | 5.54 | 7.90 | 9.27 | 9.66 | |
2 | 2 | 2.89 | 4.16 | 7.13 | 9.68 | 11.22 | 12.07 |
5 | 2.34 | 2.90 | 5.52 | 7.68 | 8.87 | 9.17 |
Load | |||||
---|---|---|---|---|---|
[m/s] | [m/s] | [m/s] | [m/s] | [m/s] | |
✓ | 1.0 | 2.0 | 1.0 | 4.0 | 4.0 |
× | 2.25 | 4.0 | 2.0 | 16 | 16 |
Circular Route Case | Pick Up Point | Drop off Point | Travel Time | ||
---|---|---|---|---|---|
PUP → DOP | DOP → PUP | (PUP) | (DOP) | [%] | |
A | 20.74 | 1.49 | |||
A | B | 19.08 | 1.62 | ||
A | C | 22.40 | 1.38 | ||
A | 20.65 | 1.04 | |||
A | B | 18.99 | 1.13 | ||
A | C | 22.25 | 0.66 | ||
A | 20.44 | 0 | |||
A | B | 18.78 | 0 | ||
A | C | 22.10 | 0 |
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Benko Loknar, M.; Klančar, G.; Blažič, S. Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk. Sensors 2023, 23, 1982. https://doi.org/10.3390/s23041982
Benko Loknar M, Klančar G, Blažič S. Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk. Sensors. 2023; 23(4):1982. https://doi.org/10.3390/s23041982
Chicago/Turabian StyleBenko Loknar, Martina, Gregor Klančar, and Sašo Blažič. 2023. "Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk" Sensors 23, no. 4: 1982. https://doi.org/10.3390/s23041982
APA StyleBenko Loknar, M., Klančar, G., & Blažič, S. (2023). Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk. Sensors, 23(4), 1982. https://doi.org/10.3390/s23041982