# Long Vehicle Turning

• Yong Chen
• Eng Sing Chua
• Daniel Thalmann
• Yiyu Cai
• Yi Gong
• Teng Sam Lim
• Peng Wong
Chapter
Part of the Gaming Media and Social Effects book series (GMSE)

## Abstract

Safety and efficiency are two major issues when it comes to long vehicle driving and operating, particularly those with very large lateral and longitudinal sizes. Trajectory planning for long vehicle tuning always plays an important role to secure safety and efficient operation, especially when turning in narrow spaces limited by surrounding objects such as buildings. This chapter discusses a trajectory calculation method for long vehicle turning, based on a set of differential equations. The solution can be numerically obtained for any trajectory under different circumstances. We develop a generalized and systematic mathematical approach to determine the trajectories swept by each wheel and other related components of the vehicle. The envelope of the trajectories of the vehicle can then be derived according to geometric relationships and characteristics. Based on numerical analysis results, a 3D simulation is developed in this work for different types of long vehicles along with different given turning roads surrounded by buildings and other objects. This way we are able to do trajectory planning for long vehicle turning.

### Keywords

Vehicle turning Simulation  Trajectory planning

### References

1. Baylis J (1973) The mathematics of a driving hazard. Math Gaz 57:22–26Google Scholar
2. Bender EA (1979) A driving hazard revisited. SIAM Rev 21:136–138
3. Battelle Team (1995) Comprehen-sive truck size and weight (TS & W) study. Phase 1—synthesis. Roadway geometry and truck size and weight regulations. Working paper 5. Vehicle characteristics affecting safety and truck size. Ohio, ColumbusGoogle Scholar
4. Bender EA (2000). An introduction to mathematical modeling. Dover Publications, New YorkGoogle Scholar
5. Craig Coulter R (1992) Implementation of the pure pursuit path tracking algorithm. Technical Report CMU-RI-TR-92–01. The Robotics Institute, Carnegie Mellon University, Pittsburg, USAGoogle Scholar
6. Dunbar SR, Bosman RJC, Nooij SEM (2001) The track of a bicycle back tire. Math Mag 74(4):273. doi: Google Scholar
7. Erkert TW, Sessions J, Layton RD (1989) A method for determining offtracking of multiple unit vehicle combinations. J For Eng 1(1):9–16Google Scholar
8. Fancher P, Balderas L (1987) Development of microcomputer models of truck braking and handling. Final Report UMTRI-87–37. The University of Michigan Transportation Research Institute, University of Michigan, USAGoogle Scholar
9. Fioretti L, Piacentini F, Liu L, Setekera R, Wang X (2008) Team three optimization of roundabouts. Report for 22nd ECMI modelling week. NetherlandsGoogle Scholar
10. Freedman HI, Riemenschneider SD (1983) Determining the path of the rear wheels of a bus. Soc Ind Appl Math Rev 25(4):561–567
11. Hellström T, Ringdahl O (2005) Autonomous path tracking using recorded orientation and steering commands. In: Proceedings of towards autonomous robotic systems 81–87. doi:
12. Jesse Lee Farmer (2008) Kinematic analysis of a two-body articulated robotic vehicle. Virginia Polytechnic Institute and State University, BlacksburgGoogle Scholar
13. Plass M, Stone M (1983) Curve-fitting with piecewise parametric cubics. Comput Graph 17(3):229–239
14. Prince GE, Dubois SP (2009) Mathematical models for motion of the rear ends of vehicles. Math Comput Model 49(9–10):2049–2060Google Scholar
15. Sayers MW (1986) Vehicle offtracking models. Transp Res Rec 1052:53–62Google Scholar
16. Sweatman P, George R, Tso Y, Ramsay E (1991) A study of heavy vehicle swept path performance. Australian Road Research Board Special Report No, Australia 48Google Scholar

## Authors and Affiliations

• Yong Chen
• 1
• Eng Sing Chua
• 1
• Daniel Thalmann
• 1
• Yiyu Cai
• 1
• Yi Gong
• 2
• Teng Sam Lim
• 2
• Peng Wong
• 2
1. 1.Nanyang Technological UniversityNanyang AvenueSingapore
2. 2.PEC LimitedJurongSingapore