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.

References

References
1.
Schwartz
,
J. T.
, and
Sharir
,
M.
,
1986
, “
Motion Planning and Related Geometrical Algorithms in Robotics
,”
International Congress of Mathematicians
, Berkeley, CA, Aug. 3-11, pp.
1594
1611
.
2.
Likhachev
,
M.
,
Ferguson
,
D.
,
Gordon
,
G.
,
Stentz
,
A.
, and
Thrun
,
S.
,
2005
, “
Anytime Dynamic A*: An Anytime, Replanning Algorithm
,”
15th International Conference on Automated Planning and Scheduling (ICAPS 2005), Monterey, CA, June 5–10, pp. 262–271
.
3.
Hart
,
P. E.
,
Nilsson
,
N. J.
, and
Raphael
,
B.
,
1968
, “
A Formal Basis for the Heuristic Determination of Minimum Cost Paths
,”
IEEE Trans. Syst. Sci. Cybern.
,
4
(
2
), pp.
100
107
.10.1109/TSSC.1968.300136
4.
Khatib
,
O.
,
1986
, “
Real-Time Obstacle Avoidance for Manipulators and Mobile Robots
,”
Int. J. Rob. Res.
,
5
(
1
), pp.
90
98
.10.1177/027836498600500106
5.
Latombe
,
J.-C.
,
1991
,
Robot Motion Planning
,
Kluwer Academic Publishers
,
Norwell, MA
.
6.
Murray
,
R. M.
,
Li
,
Z.
, and
Sastry
,
S. S.
,
1994
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
,
Boca Raton, FL
.
7.
Laumond
,
J. P.
,
1998
, “
Nonholonomic Motion Planning for Mobile Robots
,” Laboratory for Analysis and Architecture of Systems–French National Centre for Scientific Research (LAAS-CNRS), Toulouse, France, last accessed Aug. 29, 2014, http://www.dis.uniroma1.it/~oriolo/eir/jpl/NonholonomicMotionPlanning_LAAS98211.pdf
8.
LaValle
,
S. M.
,
2006
,
Planning Algorithms
,
Cambridge University Press
,
New York
.
9.
Röschel
,
O.
,
1998
, “
Rational Motion Design—A Survey
,”
Comput.-Aided Des.
,
30
(
3
), pp.
169
178
.10.1016/S0010-4485(97)00056-0
10.
Lau
,
B.
,
Sprunk
,
C.
, and
Burgard
,
W.
,
2009
, “
Kinodynamic Motion Planning for Mobile Robots Using Splines
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS 2009
), St. Louis, MO, Oct. 10-15, pp.
2427
2433
10.1109/IROS.2009.5354805.
11.
Kobilarov
,
M. B.
, and
Marsden
,
J. E.
,
2011
, “
Discrete Geometric Optimal Control on Lie Groups
,”
IEEE Trans. Rob.
,
27
(
4
), pp.
641
655
.10.1109/TRO.2011.2139130
12.
Selig
,
J. M.
,
2007
, “
Cayley Maps for SE(3)
,” 12th International Federation of the Theory of Machines and Mechanisms World Congress (
IFToMM
), Besançon, France, June 18-21http://www.iftomm.org/iftomm/proceedings/proceedings_WorldCongress/WorldCongress07/articles/sessions/papers/A270.pdf.
13.
Leonard
,
N.
, and
Krishnaprasad
,
P. S.
,
1995
, “
Motion Control of Drift Free, Left-Invariant Systems on Lie Groups
,”
IEEE Trans. Autom. Control
,
40
(
9
), pp.
1539
1554
.10.1109/9.412625
14.
Justh
,
E. W.
, and
Krishnaprasad
,
P. S.
,
2004
, “
Equilibria and Steering Laws for Planar Formations
,”
Syst. Control Lett.
,
52
(
1
), pp.
25
38
.10.1016/j.sysconle.2003.10.004
15.
Maclean
,
C.
, and
Biggs
,
J. D.
,
2013
, “
Path Planning for Simple Wheeled Robots: Sub-Riemannian and Elastic Curves on SE(2)
,”
Robotica
,
31
(
8
), pp.
1285
1297
.10.1017/S0263574713000519
You do not currently have access to this content.