Coordinate Frames in Smart-AHS

I. Introduction

Three different coordinate frames are used to describe the positions of components in Smart-AHS [1, 2, 3]. These are: coordinate (or reference) frames.

Vehicle coordinate frame is assumed to be attached to the vehicle at its center of gravity. The x-axis of this frame is aligned with the forward direction of motion. The y-axis points toward the left side of the vehicle, while the z-axis is directed away from the ground. This frame is in accordance with the "right hand rule." Vehicle movement is defined in this frame (e.g., lateral and longitudinal speeds).

Road coordinate frame has its origin at the point where the left edge and the line defining the beginning of a section intersect. The x-axis is the tangent to the left edge, and is pointing toward the the direction of traffic flow. The y-axis is parallel to the radius of curvature, pointing to the right when looking down the direction of vehicle movement. The z-axis is directed away from the ground. This frame is "left-handed." It is used to define the vehicle position on the road/lane (rxp, ryp, rzp, and lyp).

Global coordinate frame is assumed to be attached to a point on the ground. We assume this frame to be a "left-handed frame" with z-axis pointing away from the ground. This definition is in accordance with the description of the vehicle-to-global alignment matrix (vgam). The frame is used to define the vehicle's global position (gxp, gyp and gzp).

II. Relation Between Coordinate Frames

The following figure shows the global, road and vehicle coordinate frames for a (very simple) highway description. As seen in the figure, the road coordinate frame is aligned with the global coordinate frame for this specific example (The segment's orientation is 0 radian). Both coordinate frames are left-handed.


Figure 1. Reference frames in Smart-AHS

On the other hand, the vehicle coordinate frame is not completely aligned with the global, nor the road coordinate frames due to the definition of the transformation matrix (See type vrep in file vrep.hs. Vehicle coordinate frame (for a vehicle following the road segment as shown on the figure) is a left-handed frame where only the x and z-axes are aligned.
The transformation between the vehicle and global coordinate frames cannot be described by using "standard" matrices used in robotics applications, bacause the former is right-handed while the latter is left-handed. For a two-dimensional system, the transformation, given by the matrix vgam, can be written as: 
	 
                                        | cos(a)  sin(a) 0 |
    [gxp gyp gzp]' = [xDot yDot zDot] * | sin(a) -cos(a) 0 |
                                        |   0       0    1 |

                                                  ^
                                           vgam __|
where a is the angle between x-axes of the vehicle and global reference frames. The matrix vgam differs from a standard coordinate frame transformation matrix in two ways:
  1. The position/velocity vectors are written as row vectors resulting in a "transposed" transformation matrix definition, and
  2. The difference in the definition of the coordinate frames forces the second column of the matrix vgam to be the negative of the second row of a standard transformation matrix (*).

A Note on Visualization: The animation window on TkShift assumes a point of origin (0,0) at the upper left corner of the window. In other words, the unit vector (1,0) points to the right along the upper side of the animation window, while the unit vector (0,1) points down along the left side of the frame.

With the definitions of the reference frames given above, TkShift animation window will plot the "real" positions in the simulation environment. On the other hand, SmartPATH animations use right-handed coordinate frame.

III. References

  1. A. Deshpande, D. Godbole, A. Gollu, L. Semenzato, R. Sengupta, D. Swaroop and P. Varaiya, "Automated Highway Systems Tool Interface Format," PATH Report, 1996 [outdated].

  2. California PATH Smart AHS, http://www.path.berkeley.edu/smart-ahs, California PATH (June 1997).

  3. A. Deshpande, "AHS Components in SHIFT," California PATH Report (July 1997).
* The standard transformation matrix for a rotation of a radians around the z-axis is given as: 
		|  cos(a) sin(a) 0 |
	   	| -sin(a) cos(a) 0 |
		|   0       0    1 |
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Cem Ünsal and Aleks Göllü
Last modified on July 1997