In this technical brief, we focus on solving trajectory optimization problems that have nonlinear system dynamics and that include high-order derivatives in the objective function. This type of problem comes up in robotics—for example, when computing minimum-snap reference trajectories for a quadrotor or computing minimum-jerk trajectories for a robot arm. DirCol5i is a transcription method that is specialized for solving this type of problem. It uses the fifth-order splines and analytic differentiation to compute higher-derivatives, rather than using a chain-integrator as would be required by traditional methods. We compare DirCol5i to traditional transcription methods. Although it is slower for some simple optimization problems, when solving problems with high-order derivatives DirCol5i is faster, more numerically robust, and does not require setting up a chain integrator.

References

References
1.
Westervelt
,
E. R.
,
Grizzle
,
J. W.
, and
Koditschek
,
D. E.
,
2003
, “
Hybrid Zero Dynamics of Planar Biped Walkers
,”
IEEE Trans. Autom. Control
,
48
(
1
), pp.
42
56
.
2.
Dai
,
H.
,
Valenzuela
,
A.
, and
Tedrake
,
R.
,
2014
, “
Whole-Body Motion Planning With Simple Dynamics and Full Kinematics
,”
International Conference on Humanoid Robots
, pp.
295
302
.
3.
Gasparetto
,
A.
, and
Zanotto
,
V.
,
2007
, “
A New Method for Smooth Trajectory Planning of Robot Manipulators
,”
Mech. Mach. Theory
,
42
(
4
), pp.
455
471
.
4.
Saramago
,
S. F. P.
, and
Steffen
,
V.
, Jr
,
1998
, “
Optimization of the Trajectory Plannung of Robot Manipulators Taking Into Account the Dynamics of the System
,”
Mech. Mach. Theory
,
33
(
7
), pp.
883
894
.
5.
Mellinger
,
D.
, and
Kumar
,
V.
,
2011
, “
Minimum Snap Trajectory Generation and Control for Quadrotors
,”
IEEE
International Conference on Robotics and Automation
, Shanghai, China, May 9–13, pp.
2520
2525
.
6.
Mueller
,
M. W.
,
Hehn
,
M.
, and
D'Andrea
,
R.
,
2015
, “
A Computationally Efficient Motion Primitive for Quadrocopter Trajectory Generation
,”
IEEE Trans. Rob.
,
31
(
6
), pp. 1294–1310.
7.
Richter, C.
,
Bry, A.
, and
Roy, N.
, 2016, “
Polynomial Trajectory Planning for Aggressive Quadrotor Flight in Dense Indoor Environments
,”
Robotics Research
(Springer Tracts in Advanced Robotics, Vol. 114), M. Inaba and P. Corke, eds., Springer, Cham, Switzerland.
8.
Müller
,
M. W.
,
Hehn
,
M.
, and
D'Andrea
,
R.
,
2013
, “
A Computationally Efficient Algorithm for State-to-State Quadrocopter Trajectory Generation and Feasibility Verification
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems
(
IROS
), Tokyo, Japan, Nov. 3–7, pp.
3480
3486
.
9.
Piazzi
,
A.
, and
Visioli
,
A.
,
2000
, “
Global Minimum-Jerk Trajectory Planning of Robot Manipulators
,”
IEEE Trans. Ind. Electron.
,
47
(
1
), pp.
140
149
.
10.
Gasparetto
,
A.
, and
Zanotto
,
V.
,
2008
, “
A Technique for Time-Jerk Optimal Planning of Robot Trajectories
,”
Rob. Comput.-Integr. Manuf.
,
24
(
3
), pp.
415
426
.
11.
Tedrake
,
R.
,
2009
, “
Underactuated Robotics: Learning, Planning, and Control for Efficient and Agile Machines Course Notes for MIT 6.832
,” Massachusetts Institute of Technology, Cambridge, MA, Technical Report.
12.
Biegler
,
L. T.
, and
Zavala
,
V. M.
,
2009
, “
Large-Scale Nonlinear Programming Using IPOPT: An Integrating Framework for Enterprise-Wide Dynamic Optimization
,”
Comput. Chem. Eng.
,
33
(
3
), pp.
575
582
.
13.
Gill
,
P. E.
,
Murray
,
W.
, and
Saunders
,
M. A.
,
2006
, “
User's Guide for SNOPT Version 7: Software for Large-Scale Nonlinear Programming
,” Stanford University, Stanford, CA, pp.
1
116
.
14.
Mathworks
,
2014
, “Matlab Optimization Toolbox,” Mathworks, Natick, MA.
15.
Betts
,
J. T.
,
1998
, “
A Survey of Numerical Methods for Trajectory Optimization
,”
J. Guid. Control Dyn.
,
21(
2), pp. 193–207.
16.
Rao
,
A.
,
2009
, “
A Survey of Numerical Methods for Optimal Control
,” AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, PA, Aug. 10–13, AAS Paper No. 09-334.
17.
Kelly
,
M. P.
,
2017
, “
An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation
,”
SIAM Rev.
,
59
(
4
), pp.
849
904
.
18.
Betts
,
J. T.
,
2010
,
Practical Methods for Optimal Control and Estimation Using Nonlinear Programming
,
SIAM
,
Philadelphia, PA
.
19.
Bryson
,
A. E.
, and
Ho
,
Y.-C.
,
1975
,
Applied Optimal Control
,
Taylor & Francis
, Hemisphere, NY.
20.
Williams
,
P.
,
2009
, “
Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization
,”
J. Guid. Control Dyn.
,
32
(
4
), pp.
1392
1395
.
21.
Patterson
,
M. A.
, and
Rao
,
A. V.
,
2013
, “
GPOPS II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp Adaptive Gaussian Quadrature Collocation Methods and Spa and Rse Nonlinear Programming
,”
ACM Trans. Math. Soft.
,
39
(3), pp.
1
41
.
22.
Herman
,
A. L.
, and
Conway
,
B. A.
,
1996
, “
Direct Optimization Using Collocation Based on High-Order Gauss-Lobatto Quadrature Rules
,”
J. Guid. Control Dyn.
,
19
(
3
), pp.
522
529
.
23.
Chevallereau
,
B. C.
,
Abba
,
G.
,
Aoustin
,
Y.
,
Plestan
,
F.
,
Westervelt
,
E. R.
,
Canudas-de wit
,
C.
, and
Grizzle
,
J. W.
,
2003
, “
RABBITA: Testbed for Advanced Control Theory
,”
IEEE Control Syst. Mag.
,
23
(5), pp.
57
79
.
24.
Garg
,
D.
,
Patterson
,
M.
,
Hager
,
W.
,
Rao
,
A.
,
Benson
,
D.
, and
Huntington
,
G.
,
2017
, “
An Overview of Three Pseudospectral Methods for the Numerical Solution of Optimal Control Problems
,” accessed Oct. 8, 2018, https://hal.archives-ouvertes.fr/hal-01615132/document
25.
Finn
,
D. L.
,
2004
, “
MA 323 Geometric Modelling Course Notes: Day 09 Quintic Hermite Interpolation
,” Rose-Hulman Institute of Technology, Terre Haute, IN.
26.
Meir
,
A.
, and
Sharma
,
A.
,
1965
, “
On the Method of Romberg Quadrature
,”
J. SIAM Numer. Anal.
,
2
(
2
), pp.
250
259
.
27.
Lambers
,
J.
,
2009
, “
Romberg Integration
,” University of Southern Mississippi, Hattiesburg, MS, Lecture Notes
MAT 460/560
.http://www.math.usm.edu/lambers/mat460/fall09/lecture29.pdf
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