Research Papers

Rational Interpolation of Car Motions

[+] Author and Article Information
J. M. Selig

School of Engineering,
London South Bank University,
London SE1 0AA, UK
e-mail: seligjm@lsbu.ac.uk

Manuscript received August 28, 2013; final manuscript received March 16, 2015; published online June 24, 2015. Assoc. Editor: Qiaode Jeffrey Ge.

J. Mechanisms Robotics 7(3), 031023 (Aug 01, 2015) (10 pages) Paper No: JMR-13-1167; doi: 10.1115/1.4030298 History: Received August 28, 2013; Revised March 16, 2015; Online June 24, 2015

This work introduces a general approach to the interpolation of the rigid-body motions of cars by rational motions. A key feature of the approach is that the motions produced automatically satisfy the kinematic constraints imposed by the car wheels, that is, cars cannot instantaneously translate sideways. This is achieved by using a Cayley map to project a polynomial curve in the Lie algebra se(2) to SE(2) the group of rigid displacements in the plane. The differential constraint on se(2), which expresses the kinematic constraint on the car, is easily solved for one coordinate if the other two are given, in this case as polynomial functions. In this way, families of motions obeying the constraint can be found. Several families are found here and examples of their use are shown. It is shown how rest-to-rest motions can be generated in this way and also how these motions can be joined so that the motion is continuous and differentiable across the join. A final section discusses the optimization of these motions. For some cost functions, the optimal motions are known but can be rather impractical to use. By optimizing over a family of motions which satisfy the boundary conditions for the motion, it is shown that rational motions can be found simply and are close to the overall optimal motion.

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Fig. 1

Differential drive (left) and front-wheel steering cars (right)

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Fig. 3

Interpolated motion for a translation in the y-direction

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Fig. 2

Interpolated motion between two given poses. The black, gray, black dashed, and gray dashed curves have b1 = -1,1/3,1(2/3), and 3, respectively.

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Fig. 4

Interpolated motion between two poses with zero generalized velocity at either end. In (a) the black, gray, and dashed curves have b2 = 0, 2, and 4, respectively. In (b) the parameter values are b2 = −2, −4, and −6 for the black, gray and dashed curves, respectively.

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Fig. 5

Rest-to-rest motion in the y-direction. The two segments of each motion are shown solid and dashed, respectively. The values of γ used are, from left to right 3, 4, 5, and 7.5.

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Fig. 6

Minimal path-length. On the left is a contour plot for values of the integral J1 defined in the text. The right hand figure shows the motion given by the optimal values of the parameters a1 ≈ 5.665 and b1 ≈ 0.092.

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Fig. 7

Minimal control effort. The left hand graph shows the optimal path dashed and the rational approximation in black. On the right is a detail from the graph where the difference between the two can be seen more clearly. The value of the constant c is 1 in the appropriate units.




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