A path generation for automated vehicle based on Bezier curve and via-points

https://doi.org/10.1016/j.robot.2015.08.001Get rights and content

Highlights

  • A path design method based on Bezier curve.

  • Path satisfies the endpoint restrictions and passes through designated via-points.

  • A base Bezier curve can be derived based on the endpoints restrictions.

  • Constrained optimization approach is introduced to modify the base Bezier curve to pass through the designated via-points.

  • Path generation results are verified by the computer simulations.

Abstract

This paper presents a path generation method for automated vehicle based on Bezier curve. Derivation of smooth path is one of important subjects for vehicle navigation system. Bezier curve is a smooth parametric curve, but adequate assignment of its control points that determine the shape of Bezier curve is not a simple problem. In the proposed method, first nth-order base Bezier curve in N-dimension is derived with the end point constraints (location and velocity). As a next step, path modification is done by using constrained optimization with designated via-points. The presented method is tested by several calculation experiments in case of 2-dimensional and 3-dimensional path. The results are shown and discussed in this paper.

Introduction

Automated vehicle technology is one of the key issues for supporting our daily life and economic activities. For example, new transportation service in the city  [1], [2], personal supporting service for elderly people and the persons with disabilities  [3], logistics for delivery service  [4] are typical significant applications. For realizing high quality services in such applications, automated vehicle technology becomes their basis and their working environment is usually not a guided one. Therefore, the automated mobile function in the real world and real-time must be introduced into such vehicles. Path generation is one of the important factors for realizing such autonomy. The path generation subject is to design a smooth curve based on the given conditions like the end point constrains and the via-points that are designated by the environment recognition process.

Path generation based on the polynomial is one of the major approaches to obtain the path  [5], [6], [7], [8]. The redundancy on the degree of the polynomial is a basis for being smooth and diverse. However, it also makes the parameter derivation of such polynomial complex. In  [5] cubic polynomial is selected as a curve primitive for the motion planning of automated vehicle. As the kinematic equations are adopted in this approach, the generated trajectory is relatively easily tracked by a real vehicle. However, the cubic polynomial primitive is difficult to compute and end state errors are hard to be overcome. Besides, cubic curve is hard to be modified when generated and a small noise in polynomial coefficients will cause a large curve shape change. Another curve primitive, cubic B-spline, is selected in  [8]. A collision-free path that is curvature continuous is generated by this approach. It requires strong restrictions between the number of the interpolation points and the number of the control points of the curve for deriving the path.

Bezier curve  [9] is also one of the major parametric polynomial curves that is utilized in various applications. Because Bezier curve has continuous curvature and can easily realize the shape modification by adjusting its control points. These are major advantages of the path based on Bezier curve, and the control point assignment is a main subject to generate the path by utilizing Bezier curve. Therefore, many previous works have been done and discussed for vehicle path generation utilizing Bezier curve  [10], [11], [12], [13]. An optimization algorithm is proposed in  [11], which can generate a 4th-order Bezier curve for three-wheel robots. 3rd-order and 4th-order Bezier are generated respectively in  [12] and [13] for automated vehicle. In their works, the order of Bezier curve is fixed and it is discussed how to set the control points for generating desired path under such fixed condition. In some cases, the solution for the path strongly depends on the order of the basis curve. Furthermore, these works are only considered in 2-dimension. Thus, these works mainly have discussed the path design utilizing Bezier curve with the restriction on the order and the dimension. Our motivation is the general discussion concerning nth-order Bezier curve in N-dimension which can be applied to the path generation of automated vehicle. Moreover, for utilizing Bezier curve such as a path generator of the vehicle, the discussion on the explicit via-point designation must be done. As mentioned above, the shape of Bezier curve can be modified easily by operating the control points, but it is difficult to treat directly how the control points should be maneuvered for passing through the designated via-points.

Thus we have discussed a vehicle path generation method based on Bezier curve with the end points and via-point constraints  [14]. The approach is to first derive the control points of N-dimensional nth-order Bezier curve as the base path based on the end point constraints. At the next step, it modifies the shape of the base path for designated via-points. Xu et al.  [15] and Wu et al.  [16] proposed a shape modification method for Bezier curve with constraint optimization approach. Hilario also proposed a Bezier trajectory deformation method for path planning based on the potential field approach  [17]. However, the precondition of their methods is that the shape of the base curve is given. Our method derives a base Bezier curve utilizing the end points constraints and realizes the modification of the path by constrained optimization with designated via-points. However, in  [14], we only discuss to derive the conditions for the control points of a Bezier curve and practically examine simple 2-dimensional test cases. Therefore, in this paper, we discuss a general solution of the control points of a Bezier curve that passes through designated via-points and conduct the calculations for deriving 2-dimensional and 3-dimensional Bezier curve based path. Moreover, we describe the performances of the computation time of the proposed method.

This paper is organized as follows. Section  2 describes the fundamental issue of Bezier curve. In Section  3, fundamental calculation scheme for base Bezier curve is discussed. Section  4 describes a path modification based on the restrictions for passing through via-points. In Section  5, we also show the results of the path generation by the proposed method and in Section  6, the performances of the proposed method are discussed. Finally, Section  7 summarizes this paper.

Section snippets

Bezier curve

Bezier curve is one of typical parametric curves for smooth path generation. When the order of Bezier curve is n in N-dimensional space, the curve defined by the parameters consists of (n+1) control points (P0,P1,P2,,PnRN). A N-dimensional nth-order Bezier curve with control input u is described in Eq. (1). nB(u)=i=0nJn,i(u)nPi(0u1,PiRN) where u and Jn,i(u) represent the intrinsic parameter and a nth order blending function with Bernstein basis polynomials  [18] respectively. The blending

Derivation scheme of base Bezier curve

As preparations for general discussions of derivation of the base Bezier curve, we introduce time variable: t and time at the destination: T to u as below. u=tT(0tT). Then, below the polynomial expansion of Eq. (1) can be derived by substituting Eq. (5) for Eq. (1) and expanding it. nB(tT)=nb0+nb1tT+nb2(tT)2+nb3(tT)3++nbn(tT)nRN where nbi denotes the coefficient vector for expanded nth-order Bezier curve.

Path modification for designated via-points

A solution of the control points of the base path can be presented in the last section, however, there is still no discussion about via-points in the path. In this section, path modification method with designated via-points is discussed. We take constraint optimal approach. Xu et al.  [15] constrained optimization method for modifying the Bezier curve based on control points. However, they did not discuss how to determine which points on the base path are utilized. Moreover, they did not

Simulation experiments

In this section, we conducted the simulation experiments to verify the proposed path generation method based on Bezier curve.

Discussions

When the results of aforementioned method are applied to a real system, some issues must be taken into account, which mainly include physical constraints and dynamic of the platform and real-time implementation. These two parts will be discussed in this section.

For the application of the proposed method to the real platform, especially the autonomous vehicle, its computation time must be considered first, because the perceived view and environmental information change as the vehicle traverses.

Conclusion

This paper described a Bezier curve based on path generation method for automated vehicle. First, as a general discussion, we described about a solution of base path based on N-dimensional (1N3) nth order Bezier curve with the end point constraints. Next, the nearest points on the base path from designated via-points are derived by Newton–Raphson method. By utilizing constrained optimization with designated via-points, the conditions for modifying the control points of the base curve are

Acknowledgments

This research was partially supported by the NSFC of China under Grant Nos. 91320301, 61273252 and by Grant-in-Aid for Scientific Research (B), MEXT of Japan (Nos. 23300074, 15H03953).

Kuniaki Kawabata received B.E., M.E. and Ph.D degrees in Electrical Engineering from Hosei University in 1992, 1994, and 1997, respectively. He joined Biochemical Systems Lab. at RIKEN (The Institute of Physical and Chemical Research) as a Special Postdoctoral Researcher from 1997 to 2000. In 2000, he joined Advanced Engineering Center at RIKEN as a Research Scientist. In 2002, he joined Distributed Adaptive Robotics Research Unit, RIKEN. In 2005, he became Unit leader of Distributed Adaptive

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      Bézier curves are parameterised curves controlled by handles. A finite number of handles can convey a trajectory containing an infinite number of points while accurately controlling the trend, continuity, and differential characteristics of the curves [39,40]. In complex scenarios, a vehicle is geometrised as a rigid body rather than a particle.

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    Kuniaki Kawabata received B.E., M.E. and Ph.D degrees in Electrical Engineering from Hosei University in 1992, 1994, and 1997, respectively. He joined Biochemical Systems Lab. at RIKEN (The Institute of Physical and Chemical Research) as a Special Postdoctoral Researcher from 1997 to 2000. In 2000, he joined Advanced Engineering Center at RIKEN as a Research Scientist. In 2002, he joined Distributed Adaptive Robotics Research Unit, RIKEN. In 2005, he became Unit leader of Distributed Adaptive Robotics Research Unit, RIKEEN. In 2007, he became Unit leader of Intelligent System Research Unit of RIKEN. From 2011, he became Team Leader of RIKEN–XJTU Joint Research Team and Foreign Expert of Shaanxi Province, China. Currently he is Unit Leader of RIKEN–XJTU Joint Research Unit. He is also a visiting researcher of RACE of the University of Tokyo, a Visiting Professor of Kagawa University and a Project Associate Professor of Keio University. His research interests cover mobile robotics, distributed autonomous systems, networked system, understanding social adaptability of the insect, etc. He is a member of IEEE, RSJ, SICE and JSME.

    Liang Ma received the B.S. degree in Electronic and Information Engineering and the M.S. degree in Communication and Information System in 2005 and 2008, respectively, from the China University of Geoscience, Wuhan, China. He was a Research Staff at Xi’an Institute of Optic and Precision Mechanics of Chinese Academy of Sciences, Xi’an, China, from 2008 to 2009. He is currently working toward the Ph.D. degree with the Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, Xi’an, China. His research interests cover motion planning, autonomous navigation, and computer vision.

    Jianru Xue received the B.S. degree from Xi’an University of Technology in 1994, and received the M.S. and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 1999 and 2003, respectively. He had worked in Fuji Xerox, Tokyo, Japan, from 2002 to 2003, and visited University of California, Los Angeles, from 2008 to 2009. He is currently a Professor of Institute of Artificial Intelligence and Robotic of Xi’an Jiaotong University. His research interest includes computer vision, visual navigation, and video coding based on analysis. He served as co-organization chair Asian conference on computer vision 2009 and Virtual System and Multimedia 2006. He also served as PC member of Pattern recognition 2012, Asian conference on Computer vision 2010 and 2012.

    Chengwei Zhu received the B.S. in Automation from Xi’an Jiaotong University in 2011. He is currently a postgraduate student in the Institute of Artificial Intelligence and Robotics in Xi’an Jiaotong University. His researches focus on motion planning and control methods of unmanned ground vehicle.

    Nanning Zheng graduated in 1975 from the Department of Electrical Engineering, Xi’an Jiaotong University, Xi’an, China, received the M.E. degree in Information and Control Engineering from Xi’an Jiaotong University, Xi’an, China in 1981, and a Ph.D. degree in Electrical Engineering from Keio University, Japan, in 1985. He is currently a Professor and the Director of the Institute of Artificial Intelligence and Robotics at Xi’an Jiaotong University. His research interests include computer vision, pattern recognition, computational intelligence, image processing, and hardware implementation of intelligent systems. Since 2000, he has been the Chinese representative on the Governing Board of the International Association for Pattern Recognition. He presently serves as Executive Editor of the Chinese Science Bulletin. He became a member of the Chinese Academy Engineering in 1999. He is a fellow of the IEEE.

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