Differential dynamic programming: Difference between revisions

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{{Short description|Algorithm for trajectory optimization}}
'''Differential dynamic programming''' ('''DDP)''') is an [[optimal control]] algorithm of the [[trajectory optimization]] class. The algorithm was introduced in 1966 by [[David Mayne|Mayne]]<ref>{{Cite journal

| volume = 3

| pages = 85–95

| last = Mayne

| first = D. Q.

| author-link=David Mayne

| title = A second-order gradient method of optimizing non-linear discrete time systems

| journal = Int J Control

| year = 1966

| doi = 10.1080/00207176608921369

}}</ref> and subsequently analysed in Jacobson and Mayne's eponymous book.<ref>{{cite book|lastlast1=Mayne|firstfirst1= David HQ. and|last2= Jacobson,|first2= David QH.|title=Differential dynamic programming|year=1970|publisher=American Elsevier Pub. Co.|___location=New York|isbn=978-0-444-00070-45|url=httphttps://books.google.com/books/about/Differential_dynamic_programming.html?id=tA-oAAAAIAAJ}}</ref> The algorithm uses locally-quadratic models of the dynamics and cost functions, and displays [[Rate of convergence|quadratic convergence]]. It is closely related to Pantoja's step-wise Newton's method.<ref>{{Cite journal

| doi = 10.1080/00207178808906114

| issn = 0020-7179

Line 19 ⟶ 21:

| journal = International Journal of Control

| year = 1988

}}</ref><ref>{{Cite webjournal

| last = Liao

| first = L. Z.

| coauthors author2= C. A Shoemaker | author2-link = Christine Shoemaker

| title = Advantages of differential dynamic programming over Newton's method for discrete-time optimal control problems

| publisher website= Cornell University, Ithaca, NY

| year = 1992

| url = httphttps://hdl.handle.net/1813/5474

| hdl = 1813/5474 |hdl-access=free

}}</ref>

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where <math>\mathbf{x}_0\equiv\mathbf{x}</math>, and the <math>\mathbf{x}_i</math> for <math>i>0</math> are given by {{EquationNote|1|Eq. 1}}. The solution of the optimal control problem is the minimizing control sequence

<math>\mathbf{U}^*(\mathbf{x})\equiv \operatorname{argmin}_{\mathbf{U}} J_0(\mathbf{x},\mathbf{U}).</math>

''Trajectory optimization'' means finding <math>\mathbf{U}^*(\mathbf{x})</math> for a particular <math>\mathbf{x}_0</math>, rather than for all possible initial states.

== Dynamic programming ==

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== Differential dynamic programming ==

DDP proceeds by iteratively performing a backward pass on the nominal trajectory to generate a new control sequence, and then a forward-pass to compute and evaluate a new nominal trajectory. We begin with the backward pass. If

:<math>\ell(\mathbf{x},\mathbf{u}) + V(\mathbf{f}(\mathbf{x},\mathbf{u}),i+1)</math>

is the argument of the <math>\min[\cdot]</math> operator in {{EquationNote|2|Eq. 2}}, let <math>Q</math> be the variation of this quantity around the <math>i</math>-th <math>(\mathbf{x},\mathbf{u})</math> pair:

:<math>\begin{align}Q(\delta\mathbf{x},\delta\mathbf{u})\equiv &\ell(\mathbf{x}+\delta\mathbf{x},\mathbf{u}+\delta\mathbf{u})&&{}+V(\mathbf{f}(\mathbf{x}+\delta\mathbf{x},\mathbf{u}+\delta\mathbf{u}),i+1)

Line 91 ⟶ 94:

</math>|{{EquationRef|3}}}}

The <math>Q</math> notation used here is a variant of the notation of Morimoto where subscripts denote differentiation in denominator layout.<ref>{{Cite conference

| volume = 2

| pages = 1927–1932

| last = Morimoto

| first = J. |author2=G. Zeglin |author3=C.G. Atkeson

| coauthors = G. Zeglin, C.G. Atkeson

| title = Minimax differential dynamic programming: Application to a biped walking robot

| booktitlebook-title = Intelligent Robots and Systems, 2003.(IROS 2003). Proceedings. 2003 IEEE/RSJ International Conference on

| date = 2003

}}</ref>

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:<math>

\begin{alignat}{2}

\Delta V(i) &= &{} -\tfrac{1}{2}Q_Q^T_\mathbf{u} Q_{\mathbf{u}\mathbf{u}}^{-1}Q_\mathbf{u}\\

V_\mathbf{x}(i) &= Q_\mathbf{x} & {}- Q_\mathbf{uxu} Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}\mathbf{x}}\\

V_{\mathbf{x}\mathbf{x}}(i) &= Q_{\mathbf{x}\mathbf{x}} &{} - Q_{\mathbf{x}\mathbf{u}}Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}\mathbf{x}}.

\end{alignat}

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The backward passes and forward passes are iterated until convergence.

If the Hessians <math>Q_{\mathbf{x}\mathbf{x}}, Q_{\mathbf{u}\mathbf{u}}, Q_{\mathbf{u}\mathbf{x}}, Q_{\mathbf{x}\mathbf{u}}</math> are replaced by their Gauss-Newton approximation, the method reduces to the iterative Linear Quadratic Regulator (iLQR).<ref>{{Cite conference

| pages = 6921–7

| last = LiaoBaumgärtner

| first = LK. Z

| title = A Unified Local Convergence Analysis of Differential Dynamic Programming, Direct Single Shooting, and Direct Multiple Shooting

| conference = 2023 European Control Conference (ECC)

| year = 19912023

| doi = 10.23919/ECC57647.2023.10178367

| url = https://ieeexplore.ieee.org/document/10178367

| url-access = subscription

}}</ref>

== Regularization and line-search ==

Differential dynamic programming is a second-order algorithm like [[Newton's method]]. It therefore takes large steps toward the minimum and often requires [[regularization (mathematics)|regularization]] and/or [[line-search]] to achieve convergence.<ref>

{{Cite journal |last=Liao |first=L. Z |author2=C. A Shoemaker |author2-link=Christine Shoemaker |year=1991 |title=Convergence in unconstrained discrete-time differential dynamic programming |journal=IEEE Transactions on Automatic Control |volume=36 |issue=6 |page=692 |doi=10.1109/9.86943}}

<ref>

</ref><ref>{{Cite journalthesis

| volume = 36

| issue = 6

| pages = 692

| last = Liao

| first = L. Z

| coauthors = C. A Shoemaker

| title = Convergence in unconstrained discrete-time differential dynamic programming

| journal = IEEE Transactions on Automatic Control

| year = 1991

| doi = 10.1109/9.86943

}}

</ref>

.<ref>

{{Cite thesis

| publisher = Hebrew University

| last = Tassa

Line 167 ⟶ 166:

| date = 2011

| url = http://icnc.huji.ac.il/phd/theses/files/YuvalTassa.pdf

| access-date = 2012-02-27

}}

| archive-url = https://web.archive.org/web/20160304023026/http://icnc.huji.ac.il/phd/theses/files/YuvalTassa.pdf

</ref> Regularization in the DDP context means ensuring that the <math>Q_{\mathbf{u}\mathbf{u}}</math> matrix in {{EquationNote|4|Eq. 4}} is [[positive definite matrix|positive definite]]. Line-search in DDP amounts to scaling the open-loop control modification <math>\mathbf{k}</math> by some <math>0<\alpha<1</math>.

| archive-date = 2016-03-04

}}</ref> Regularization in the DDP context means ensuring that the <math>Q_{\mathbf{u}\mathbf{u}}</math> matrix in {{EquationNote|4|Eq. 4}} is [[positive definite matrix|positive definite]]. Line-search in DDP amounts to scaling the open-loop control modification <math>\mathbf{k}</math> by some <math>0<\alpha<1</math>.

== Monte Carlo version ==

Sampled differential dynamic programming (SaDDP) is a Monte Carlo variant of differential dynamic programming.<ref>{{Cite conference |title=Sampled differential dynamic programming |book-title=2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) |language=en-US|doi=10.1109/IROS.2016.7759229|s2cid=1338737}}</ref><ref>{{Cite conference|last1=Rajamäki|first1=Joose|first2=Perttu|last2=Hämäläinen|url=https://ieeexplore.ieee.org/document/8430799|title=Regularizing Sampled Differential Dynamic Programming - IEEE Conference Publication|conference=2018 Annual American Control Conference (ACC)|date=June 2018 |pages=2182–2189 |doi=10.23919/ACC.2018.8430799 |s2cid=243932441 |language=en-US|access-date=2018-10-19|url-access=subscription}}</ref><ref>{{Cite book|first=Joose|last=Rajamäki|date=2018|title=Random Search Algorithms for Optimal Control|url=http://urn.fi/URN:ISBN:978-952-60-8156-4|language=en|issn=1799-4942|isbn=978-952-60-8156-4|publisher=Aalto University}}</ref> It is based on treating the quadratic cost of differential dynamic programming as the energy of a [[Boltzmann distribution]]. This way the quantities of DDP can be matched to the statistics of a [[Multivariate normal distribution|multidimensional normal distribution]]. The statistics can be recomputed from sampled trajectories without differentiation.

Sampled differential dynamic programming has been extended to Path Integral Policy Improvement with Differential Dynamic Programming.<ref>{{Cite book|last1=Lefebvre|first1=Tom|last2=Crevecoeur|first2=Guillaume|title=2019 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM) |chapter=Path Integral Policy Improvement with Differential Dynamic Programming |date=July 2019|chapter-url=https://ieeexplore.ieee.org/document/8868359|pages=739–745|doi=10.1109/AIM.2019.8868359|hdl=1854/LU-8623968|isbn=978-1-7281-2493-3|s2cid=204816072|url=https://biblio.ugent.be/publication/8623968 |hdl-access=free}}</ref> This creates a link between differential dynamic programming and path integral control,<ref>{{Cite book|last1=Theodorou|first1=Evangelos|last2=Buchli|first2=Jonas|last3=Schaal|first3=Stefan|title=2010 IEEE International Conference on Robotics and Automation |chapter=Reinforcement learning of motor skills in high dimensions: A path integral approach |date=May 2010|pages=2397–2403|doi=10.1109/ROBOT.2010.5509336|isbn=978-1-4244-5038-1|s2cid=15116370}}</ref> which is a framework of stochastic optimal control.

== Constrained problems ==

Interior Point Differential dynamic programming (IPDDP) is an [[interior-point method]] generalization of DDP that can address the optimal control problem with nonlinear state and input constraints.<ref>{{cite journal |last1=Pavlov |first1=Andrei|last2=Shames|first2=Iman| last3=Manzie|first3=Chris|date=2020 |title=Interior Point Differential Dynamic Programming |journal=IEEE Transactions on Control Systems Technology |volume=29 |issue=6 |page=2720 |doi=10.1109/TCST.2021.3049416 |arxiv=2004.12710 |bibcode=2021ITCST..29.2720P }}</ref>

== See also ==

* [[optimalOptimal control]]

== References ==

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== External links ==

* [http://www.ros.org/wiki/color_DDP A Python implementation of DDP]

* [http://www.csmathworks.washington.edu/people/postdocscom/tassamatlabcentral/codefileexchange/52069-ilqg-ddp-trajectory-optimization A MATLAB implementation of DDP]

* The open-source software framework [https://github.com/acados/acados acados] provides an efficient and embeddable implementation of DDP.

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