Skip to main content
Top

2016 | OriginalPaper | Chapter

56. Integrated Trajectory Optimization

Author : Moritz Werling

Published in: Handbook of Driver Assistance Systems

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

New active driver assistance systems that work at the road and navigation level as well as automated driving face a challenging task. They have to permanently calculate the vehicle input commands (such as those for the steering, brakes, and the engine/powertrain) in order to realize a desired future vehicle movement, a driving trajectory. This trajectory has to be optimal in terms of some optimization criterion (in general a trade-off between comfort, safety, energy effort, and traveling time), needs to take the vehicular dynamics into account, and must incorporate lane boundaries or the predicted free space amid (possibly moving) obstacles. This kind of optimization can be mathematically formulated as a so-called optimal control problem. In order to limit the calculation effort, the optimal control problem is usually solved only on a limited prediction interval (starting with the current time) leading to a receding horizon optimization.
The chapter illustrates this practically proven approach in detail. Furthermore, the three general principles of dynamic optimization known from control theory and robotics are presented, namely, calculus of variations, direct optimization, and dynamic programming. Furthermore, their application to driver assistance systems and automated driving is exemplified and the high practical relevance supported by the given literature. Finally, the respective advantages and limitations of the optimization principles are discussed in detail proposing their combination for more involved system designs.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
go back to reference Althoff D, Werling M, Kaempchen N, Wollherr D, Buss M (2012) Lane-based safety assessment of road scenes using inevitable collision states. In: Intelligent vehicles symposium, Madrid (Spain), pp 31–36 Althoff D, Werling M, Kaempchen N, Wollherr D, Buss M (2012) Lane-based safety assessment of road scenes using inevitable collision states. In: Intelligent vehicles symposium, Madrid (Spain), pp 31–36
go back to reference Bellman R (1954) The theory of dynamic programming. RAND-P-550. Rand Corporation, Santa Monica Bellman R (1954) The theory of dynamic programming. RAND-P-550. Rand Corporation, Santa Monica
go back to reference Bertsekas DP (1995) Dynamic programming and optimal control. Athena Scientific, BelmontMATH Bertsekas DP (1995) Dynamic programming and optimal control. Athena Scientific, BelmontMATH
go back to reference Bock H, Plitt K (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: 9th IFAC world congress, Budapest (Hungary), pp 242–247 Bock H, Plitt K (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: 9th IFAC world congress, Budapest (Hungary), pp 242–247
go back to reference Boissonnat JD, Cerezo C, Leblond J (1994) A note on shortest paths in the plane subject to a constraint on the derivative of the curvature. Report 2160, INRIA Boissonnat JD, Cerezo C, Leblond J (1994) A note on shortest paths in the plane subject to a constraint on the derivative of the curvature. Report 2160, INRIA
go back to reference Brandt T (2008) A predictive potential field concept for shared vehicle guidance. PhD thesis, University of Paderborn Brandt T (2008) A predictive potential field concept for shared vehicle guidance. PhD thesis, University of Paderborn
go back to reference Bryson AE, Ho YC (1975) Applied optimal control. Wiley, New York Bryson AE, Ho YC (1975) Applied optimal control. Wiley, New York
go back to reference Carvalho A, Gao Y, Gray A, Tseng H, Borrelli F (2013) Predictive control of an autonomous ground vehicle using an iterative linearization approach. In: Conference on intelligent transportation system, IEEE, The Hague (Netherlands), pp 2335–2340 Carvalho A, Gao Y, Gray A, Tseng H, Borrelli F (2013) Predictive control of an autonomous ground vehicle using an iterative linearization approach. In: Conference on intelligent transportation system, IEEE, The Hague (Netherlands), pp 2335–2340
go back to reference Diehl M, Bock HG, Diedam H, Wieber PB (2006) Fast motions in biomechanics and robotics. Lecture notes in control and information sciences. Springer, Berlin, Heidelberg. pp 65–94 Diehl M, Bock HG, Diedam H, Wieber PB (2006) Fast motions in biomechanics and robotics. Lecture notes in control and information sciences. Springer, Berlin, Heidelberg. pp 65–94
go back to reference Dolgov D, Thrun S, Monteberlo M, Diebel J (2010) Path planning for autonomous vehicles in unknown semi-structured environments. Int J Robot Res 29(5):485–501CrossRef Dolgov D, Thrun S, Monteberlo M, Diebel J (2010) Path planning for autonomous vehicles in unknown semi-structured environments. Int J Robot Res 29(5):485–501CrossRef
go back to reference Dubins LE (1957) On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. Am J Math 79(3):497–516CrossRefMathSciNetMATH Dubins LE (1957) On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. Am J Math 79(3):497–516CrossRefMathSciNetMATH
go back to reference Eichhorn A, Werling M, Zahn P, Schramm D (2013) Maneuver prediction at intersections using cost-to-go gradients. In: International conference on intelligent transportation systems, IEEE, The Hague (Netherlands), pp 112–117 Eichhorn A, Werling M, Zahn P, Schramm D (2013) Maneuver prediction at intersections using cost-to-go gradients. In: International conference on intelligent transportation systems, IEEE, The Hague (Netherlands), pp 112–117
go back to reference Falcone P, Borelli F, Asgari J, Tseng E, Hrovat D (2007a) Predictive active steering control for autonomous vehicle systems. IEEE Trans Control Syst Technol 15(3):566–580 Falcone P, Borelli F, Asgari J, Tseng E, Hrovat D (2007a) Predictive active steering control for autonomous vehicle systems. IEEE Trans Control Syst Technol 15(3):566–580
go back to reference Falcone P, Tufo M, Borrelli F, Asgari J, Tsengz HE (2007b) A linear time varying model predictive control approach to the integrated vehicle dynamics control problem in autonomous systems. In: Conference on decision and control, IEEE, New Orleans (USA), pp 2980–2985 Falcone P, Tufo M, Borrelli F, Asgari J, Tsengz HE (2007b) A linear time varying model predictive control approach to the integrated vehicle dynamics control problem in autonomous systems. In: Conference on decision and control, IEEE, New Orleans (USA), pp 2980–2985
go back to reference Ferguson D, Howard TM, Likhachev M (2008a) Motion planning in urban environments: part I. In: International conference on intelligent robots and systems, IEEE/RSJ, Nice (France), pp 1063–1069 Ferguson D, Howard TM, Likhachev M (2008a) Motion planning in urban environments: part I. In: International conference on intelligent robots and systems, IEEE/RSJ, Nice (France), pp 1063–1069
go back to reference Ferguson D, Howard TM, Likhachev M (2008b) Motion planning in urban environments: part II. In: International conference on intelligent robots and systems, IEEE/RSJ, Nice (France), pp 1063–1069 Ferguson D, Howard TM, Likhachev M (2008b) Motion planning in urban environments: part II. In: International conference on intelligent robots and systems, IEEE/RSJ, Nice (France), pp 1063–1069
go back to reference Fraichard T (2007) A short paper about motion safety. In: International conference on robotics and automation, IEEE, Singapore, pp 1140–1145 Fraichard T (2007) A short paper about motion safety. In: International conference on robotics and automation, IEEE, Singapore, pp 1140–1145
go back to reference Gayko J (2012) Lane keeping support. In: Handbuch Fahrerassistenzsysteme. Vieweg + Teubner Verlag, Wiesbaden, pp 554–561 Gayko J (2012) Lane keeping support. In: Handbuch Fahrerassistenzsysteme. Vieweg + Teubner Verlag, Wiesbaden, pp 554–561
go back to reference Gerdts M, Karrenberg S, Müller-Bessler B, Stock G (2009) Generating locally optimal trajectories for an automatically driven car. Optim Eng 10(4):439–463CrossRefMATH Gerdts M, Karrenberg S, Müller-Bessler B, Stock G (2009) Generating locally optimal trajectories for an automatically driven car. Optim Eng 10(4):439–463CrossRefMATH
go back to reference Grüne L, Pannek J (2011) Nonlinear model predictive control: theory and algorithms. Springer, LondonCrossRef Grüne L, Pannek J (2011) Nonlinear model predictive control: theory and algorithms. Springer, LondonCrossRef
go back to reference Gu T, Dolan JM (2012) On-road motion planning for autonomous vehicles. In: Intelligent robotics and applications. Springer, Berlin, pp 588–597CrossRef Gu T, Dolan JM (2012) On-road motion planning for autonomous vehicles. In: Intelligent robotics and applications. Springer, Berlin, pp 588–597CrossRef
go back to reference Gu T, Snider J, Dolan JM, Lee J (2013) Focused trajectory planning for autonomous on-road driving. In: Intelligent vehicles symposium, IEEE, Gold Coast City (Australia), pp 547–552 Gu T, Snider J, Dolan JM, Lee J (2013) Focused trajectory planning for autonomous on-road driving. In: Intelligent vehicles symposium, IEEE, Gold Coast City (Australia), pp 547–552
go back to reference Gutjahr B, Werling M (2014) Automatic collision avoidance during parking and maneuvering – an optimal control approach. In: Intelligent vehicles symposium, IEEE, Dearborn (USA), pp 636–641 Gutjahr B, Werling M (2014) Automatic collision avoidance during parking and maneuvering – an optimal control approach. In: Intelligent vehicles symposium, IEEE, Dearborn (USA), pp 636–641
go back to reference Hargraves CR, Paris SW (1987) Direct trajectory optimization using nonlinear programming and collocation. J Guid Control Dyn AAS/AIAA Astrodyn Conf 10:338–342CrossRefMATH Hargraves CR, Paris SW (1987) Direct trajectory optimization using nonlinear programming and collocation. J Guid Control Dyn AAS/AIAA Astrodyn Conf 10:338–342CrossRefMATH
go back to reference Howard TM, Kelly A (2007) Optimal rough terrain trajectory generation for wheeled mobile robots. Int J Robot Res 26(2):141–166CrossRef Howard TM, Kelly A (2007) Optimal rough terrain trajectory generation for wheeled mobile robots. Int J Robot Res 26(2):141–166CrossRef
go back to reference Kang K (2012) Online optimal obstacle avoidance for rotary-wing autonomous unmanned aerial vehicles. PhD-thesis, Georgia Institute of Technology Kang K (2012) Online optimal obstacle avoidance for rotary-wing autonomous unmanned aerial vehicles. PhD-thesis, Georgia Institute of Technology
go back to reference Katzwinkel R, Auer R, Brosig S, Rohlfs M, Schöning V, Schroven F, Wuttke U (2012) Einparkassistenz. In: Handbuch Fahrerassistenzsysteme. Vieweg + Teubner Verlag, Wiesbaden, pp 471–477 Katzwinkel R, Auer R, Brosig S, Rohlfs M, Schöning V, Schroven F, Wuttke U (2012) Einparkassistenz. In: Handbuch Fahrerassistenzsysteme. Vieweg + Teubner Verlag, Wiesbaden, pp 471–477
go back to reference Kelly A, Nagy B (2003) Reactive nonholonomic trajectory generation via parametric optimal control. Int J Robot Res 22(7–8):583–601CrossRef Kelly A, Nagy B (2003) Reactive nonholonomic trajectory generation via parametric optimal control. Int J Robot Res 22(7–8):583–601CrossRef
go back to reference Koren Y, Borenstein J (1991) Potential field methods and their inherent limitations for mobile robot navigation. In: International conference on robotics and automation, IEEE, Sacramento (USA), pp 1398–1404 Koren Y, Borenstein J (1991) Potential field methods and their inherent limitations for mobile robot navigation. In: International conference on robotics and automation, IEEE, Sacramento (USA), pp 1398–1404
go back to reference Krogh BH (1984) A generalized potential field approach to obstacle avoidance control. In: International robotics research conference, Bethlehem Krogh BH (1984) A generalized potential field approach to obstacle avoidance control. In: International robotics research conference, Bethlehem
go back to reference Latombe JC (1990) Robot motion planning. Springer Science+Business Media New York Latombe JC (1990) Robot motion planning. Springer Science+Business Media New York
go back to reference Lawitzky A, Nicklas A, Wollherr D, Buss M (2014) Determining states of inevitable collision using reachability analysis. In: International conference on intelligent robots and systems, IEEE/RSJ, Chicago (USA), pp 4142–4147 Lawitzky A, Nicklas A, Wollherr D, Buss M (2014) Determining states of inevitable collision using reachability analysis. In: International conference on intelligent robots and systems, IEEE/RSJ, Chicago (USA), pp 4142–4147
go back to reference Lewis F, Syrmos V (1995) Optimal control. Wiley, New York Lewis F, Syrmos V (1995) Optimal control. Wiley, New York
go back to reference Likhachev M, Ferguson D (2009) Planning long dynamically feasible maneuvers for autonomous vehicles. Int J Robot Res 28(8):933–945CrossRef Likhachev M, Ferguson D (2009) Planning long dynamically feasible maneuvers for autonomous vehicles. Int J Robot Res 28(8):933–945CrossRef
go back to reference McNaughton M (2011) Parallel algorithms for real-time motion planning. PhD thesis, Carnegie Mellon University McNaughton M (2011) Parallel algorithms for real-time motion planning. PhD thesis, Carnegie Mellon University
go back to reference Montemerlo M et al (2008) Junior: the Stanford entry in the Urban Challenge. J Field Robot 25(9):569–597CrossRef Montemerlo M et al (2008) Junior: the Stanford entry in the Urban Challenge. J Field Robot 25(9):569–597CrossRef
go back to reference Nocedal J, Wright SJ (2006) Numerical optimization. Springer Science+Business Media New York Nocedal J, Wright SJ (2006) Numerical optimization. Springer Science+Business Media New York
go back to reference Park JM, Kim DW, Yoon YS, Kim HJ, Yi KS (2009) Obstacle avoidance of autonomous vehicles based on model predictive control. Proc Inst Mech Eng D J Automob Eng 223(12):1499–1516CrossRef Park JM, Kim DW, Yoon YS, Kim HJ, Yi KS (2009) Obstacle avoidance of autonomous vehicles based on model predictive control. Proc Inst Mech Eng D J Automob Eng 223(12):1499–1516CrossRef
go back to reference Pivtoraiko M, Knepper RA, Kelly A (2009) Differentially constrained mobile robot motion planning in state lattices. J Field Robot 26(3):308–333CrossRef Pivtoraiko M, Knepper RA, Kelly A (2009) Differentially constrained mobile robot motion planning in state lattices. J Field Robot 26(3):308–333CrossRef
go back to reference Reeds JA, Shepp LA (1990) Optimal paths for a car that goes both forwards and backwards. Pac J Math 145(2):367–393CrossRefMathSciNet Reeds JA, Shepp LA (1990) Optimal paths for a car that goes both forwards and backwards. Pac J Math 145(2):367–393CrossRefMathSciNet
go back to reference Rouchon P, Fliess M, Levine J, Martin P (1993) Flatness, motion planning and trailer systems. In: Conference on decision and control, IEEE, San Antonio (USA), pp 2700–2705 Rouchon P, Fliess M, Levine J, Martin P (1993) Flatness, motion planning and trailer systems. In: Conference on decision and control, IEEE, San Antonio (USA), pp 2700–2705
go back to reference Werling M, Liccardo D (2012) Automatic collision avoidance using model-predictive online optimization. In: Conference on decision and control, IEEE, Maui (USA), pp 6309–6314 Werling M, Liccardo D (2012) Automatic collision avoidance using model-predictive online optimization. In: Conference on decision and control, IEEE, Maui (USA), pp 6309–6314
go back to reference Werling M, Kammel S, Ziegler J, Gröll L (2012) Optimal trajectories for time-critical street scenarios using discretized terminal manifolds. Int J Robot Res 31(3):346–359CrossRef Werling M, Kammel S, Ziegler J, Gröll L (2012) Optimal trajectories for time-critical street scenarios using discretized terminal manifolds. Int J Robot Res 31(3):346–359CrossRef
go back to reference Yoon Y, Shin J, Kim HJ, Park Y, Sastry S (2009) Model-predictive active steering and obstacle avoidance for autonomous ground vehicles. Control Eng Pract 17(7):741–750CrossRef Yoon Y, Shin J, Kim HJ, Park Y, Sastry S (2009) Model-predictive active steering and obstacle avoidance for autonomous ground vehicles. Control Eng Pract 17(7):741–750CrossRef
go back to reference Ziegler J, Stiller C (2009) Spatiotemporal state lattices for fast trajectory planning in dynamic on-road driving scenarios. In: International conference on intelligent robots and systems, IEEE/RSJ, St. Louis (USA), pp 1879–1884 Ziegler J, Stiller C (2009) Spatiotemporal state lattices for fast trajectory planning in dynamic on-road driving scenarios. In: International conference on intelligent robots and systems, IEEE/RSJ, St. Louis (USA), pp 1879–1884
go back to reference Ziegler J, Werling M, Schröder J (2008) Navigating car-like robots in unstructured environments using an obstacle sensitive cost function. In: Intelligent vehicles symposium, IEEE, Eindhoven (Netherlands), pp 787–791 Ziegler J, Werling M, Schröder J (2008) Navigating car-like robots in unstructured environments using an obstacle sensitive cost function. In: Intelligent vehicles symposium, IEEE, Eindhoven (Netherlands), pp 787–791
go back to reference Ziegler J, Bender P, Dang T, Stiller C (2014) Trajectory planning for bertha – a local, continuous method. In: Intelligent vehicles symposium proceedings, IEEE, Dearborn (USA), pp 450–457 Ziegler J, Bender P, Dang T, Stiller C (2014) Trajectory planning for bertha – a local, continuous method. In: Intelligent vehicles symposium proceedings, IEEE, Dearborn (USA), pp 450–457
Metadata
Title
Integrated Trajectory Optimization
Author
Moritz Werling
Copyright Year
2016
Publisher
Springer International Publishing
DOI
https://doi.org/10.1007/978-3-319-12352-3_64

Premium Partner