Differential dynamic programming

Differential dynamic programming (DDP) is an optimal control algorithm of the trajectory optimization class. The algorithm was introduced in 1966 by Mayne ^[1] and subsequently analysed in Jacobson and Mayne's eponymous book. ^[2] The algorithm uses locally-quadratic models of the dynamics and cost functions, and displays quadratic convergence. It is closely related to Pantoja's step-wise Newton's method. ^{[3] [4]}

Finite-horizon discrete-time problems

The dynamics

${\displaystyle \mathbf {x} _{i+1}=\mathbf {f} (\mathbf {x} _{i},\mathbf {u} _{i})}$

1

describe the evolution of the state ${\displaystyle \textstyle \mathbf {x} }$ given the control ${\displaystyle \mathbf {u} }$ from time ${\displaystyle i}$ to time ${\displaystyle i+1}$. The total cost${\displaystyle J_{0}}$ is the sum of running costs ${\displaystyle \textstyle \ell }$ and final cost ${\displaystyle \ell _{f}}$, incurred when starting from state ${\displaystyle \mathbf {x} }$ and applying the control sequence ${\displaystyle \mathbf {U} \equiv \{\mathbf {u} _{0},\mathbf {u} _{1}\dots ,\mathbf {u} _{N-1}\}}$ until the horizon is reached:

${\displaystyle J_{0}(\mathbf {x} ,\mathbf {U} )=\sum _{i=0}^{N-1}\ell (\mathbf {x} _{i},\mathbf {u} _{i})+\ell _{f}(\mathbf {x} _{N}),}$

where ${\displaystyle \mathbf {x} _{0}\equiv \mathbf {x} }$, and the ${\displaystyle \mathbf {x} _{i}}$ for ${\displaystyle i>0}$ are given by Eq. 1 . The solution of the optimal control problem is the minimizing control sequence ${\displaystyle \mathbf {U} ^{*}(\mathbf {x} )\equiv \operatorname {argmin} _{\mathbf {U} }J_{0}(\mathbf {x} ,\mathbf {U} ).}$Trajectory optimization means finding ${\displaystyle \mathbf {U} ^{*}(\mathbf {x} )}$ for a particular ${\displaystyle \mathbf {x} _{0}}$, rather than for all possible initial states.

Dynamic programming

Let ${\displaystyle \mathbf {U} _{i}}$ be the partial control sequence ${\displaystyle \mathbf {U} _{i}\equiv \{\mathbf {u} _{i},\mathbf {u} _{i+1}\dots ,\mathbf {u} _{N-1}\}}$ and define the cost-to-go${\displaystyle J_{i}}$ as the partial sum of costs from ${\displaystyle i}$ to ${\displaystyle N}$:

${\displaystyle J_{i}(\mathbf {x} ,\mathbf {U} _{i})=\sum _{j=i}^{N-1}\ell (\mathbf {x} _{j},\mathbf {u} _{j})+\ell _{f}(\mathbf {x} _{N}).}$

The optimal cost-to-go or value function at time ${\displaystyle i}$ is the cost-to-go given the minimizing control sequence:

${\displaystyle V(\mathbf {x} ,i)\equiv \min _{\mathbf {U} _{i}}J_{i}(\mathbf {x} ,\mathbf {U} _{i}).}$

Setting ${\displaystyle V(\mathbf {x} ,N)\equiv \ell _{f}(\mathbf {x} _{N})}$, the dynamic programming principle reduces the minimization over an entire sequence of controls to a sequence of minimizations over a single control, proceeding backwards in time:

${\displaystyle V(\mathbf {x} ,i)=\min _{\mathbf {u} }[\ell (\mathbf {x} ,\mathbf {u} )+V(\mathbf {f} (\mathbf {x} ,\mathbf {u} ),i+1)].}$

2

This is the Bellman equation.

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

${\displaystyle \ell (\mathbf {x} ,\mathbf {u} )+V(\mathbf {f} (\mathbf {x} ,\mathbf {u} ),i+1)}$

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

${\displaystyle {\begin{aligned}Q(\delta \mathbf {x} ,\delta \mathbf {u} )\equiv &\ell (\mathbf {x} +\delta \mathbf {x} ,\mathbf {u} +\delta \mathbf {u} )&&{}+V(\mathbf {f} (\mathbf {x} +\delta \mathbf {x} ,\mathbf {u} +\delta \mathbf {u} ),i+1)\\-&\ell (\mathbf {x} ,\mathbf {u} )&&{}-V(\mathbf {f} (\mathbf {x} ,\mathbf {u} ),i+1)\end{aligned}}}$

and expand to second order

${\displaystyle \approx {\frac {1}{2}}{\begin{bmatrix}1\\\delta \mathbf {x} \\\delta \mathbf {u} \end{bmatrix}}^{\mathsf {T}}{\begin{bmatrix}0&Q_{\mathbf {x} }^{\mathsf {T}}&Q_{\mathbf {u} }^{\mathsf {T}}\\Q_{\mathbf {x} }&Q_{\mathbf {x} \mathbf {x} }&Q_{\mathbf {x} \mathbf {u} }\\Q_{\mathbf {u} }&Q_{\mathbf {u} \mathbf {x} }&Q_{\mathbf {u} \mathbf {u} }\end{bmatrix}}{\begin{bmatrix}1\\\delta \mathbf {x} \\\delta \mathbf {u} \end{bmatrix}}}$

3

The ${\displaystyle Q}$ notation used here is a variant of the notation of Morimoto where subscripts denote differentiation in denominator layout. ^[5] Dropping the index ${\displaystyle i}$ for readability, primes denoting the next time-step ${\displaystyle V'\equiv V(i+1)}$, the expansion coefficients are

${\displaystyle {\begin{alignedat}{2}Q_{\mathbf {x} }&=\ell _{\mathbf {x} }+\mathbf {f} _{\mathbf {x} }^{\mathsf {T}}V'_{\mathbf {x} }\\Q_{\mathbf {u} }&=\ell _{\mathbf {u} }+\mathbf {f} _{\mathbf {u} }^{\mathsf {T}}V'_{\mathbf {x} }\\Q_{\mathbf {x} \mathbf {x} }&=\ell _{\mathbf {x} \mathbf {x} }+\mathbf {f} _{\mathbf {x} }^{\mathsf {T}}V'_{\mathbf {x} \mathbf {x} }\mathbf {f} _{\mathbf {x} }+V_{\mathbf {x} }'\cdot \mathbf {f} _{\mathbf {x} \mathbf {x} }\\Q_{\mathbf {u} \mathbf {u} }&=\ell _{\mathbf {u} \mathbf {u} }+\mathbf {f} _{\mathbf {u} }^{\mathsf {T}}V'_{\mathbf {x} \mathbf {x} }\mathbf {f} _{\mathbf {u} }+{V'_{\mathbf {x} }}\cdot \mathbf {f} _{\mathbf {u} \mathbf {u} }\\Q_{\mathbf {u} \mathbf {x} }&=\ell _{\mathbf {u} \mathbf {x} }+\mathbf {f} _{\mathbf {u} }^{\mathsf {T}}V'_{\mathbf {x} \mathbf {x} }\mathbf {f} _{\mathbf {x} }+{V'_{\mathbf {x} }}\cdot \mathbf {f} _{\mathbf {u} \mathbf {x} }.\end{alignedat}}}$

The last terms in the last three equations denote contraction of a vector with a tensor. Minimizing the quadratic approximation (3) with respect to ${\displaystyle \delta \mathbf {u} }$ we have

${\displaystyle {\delta \mathbf {u} }^{*}=\operatorname {argmin} \limits _{\delta \mathbf {u} }Q(\delta \mathbf {x} ,\delta \mathbf {u} )=-Q_{\mathbf {u} \mathbf {u} }^{-1}(Q_{\mathbf {u} }+Q_{\mathbf {u} \mathbf {x} }\delta \mathbf {x} ),}$

4

giving an open-loop term ${\displaystyle \mathbf {k} =-Q_{\mathbf {u} \mathbf {u} }^{-1}Q_{\mathbf {u} }}$ and a feedback gain term ${\displaystyle \mathbf {K} =-Q_{\mathbf {u} \mathbf {u} }^{-1}Q_{\mathbf {u} \mathbf {x} }}$. Plugging the result back into (3) , we now have a quadratic model of the value at time ${\displaystyle i}$:

${\displaystyle {\begin{alignedat}{2}\Delta V(i)&=&{}-{\tfrac {1}{2}}Q_{\mathbf {u} }^{T}Q_{\mathbf {u} \mathbf {u} }^{-1}Q_{\mathbf {u} }\\V_{\mathbf {x} }(i)&=Q_{\mathbf {x} }&{}-Q_{\mathbf {xu} }Q_{\mathbf {u} \mathbf {u} }^{-1}Q_{\mathbf {u} }\\V_{\mathbf {x} \mathbf {x} }(i)&=Q_{\mathbf {x} \mathbf {x} }&{}-Q_{\mathbf {x} \mathbf {u} }Q_{\mathbf {u} \mathbf {u} }^{-1}Q_{\mathbf {u} \mathbf {x} }.\end{alignedat}}}$

Recursively computing the local quadratic models of ${\displaystyle V(i)}$ and the control modifications ${\displaystyle \{\mathbf {k} (i),\mathbf {K} (i)\}}$, from ${\displaystyle i=N-1}$ down to ${\displaystyle i=1}$, constitutes the backward pass. As above, the Value is initialized with ${\displaystyle V(\mathbf {x} ,N)\equiv \ell _{f}(\mathbf {x} _{N})}$. Once the backward pass is completed, a forward pass computes a new trajectory:

${\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}(1)&=\mathbf {x} (1)\\{\hat {\mathbf {u} }}(i)&=\mathbf {u} (i)+\mathbf {k} (i)+\mathbf {K} (i)({\hat {\mathbf {x} }}(i)-\mathbf {x} (i))\\{\hat {\mathbf {x} }}(i+1)&=\mathbf {f} ({\hat {\mathbf {x} }}(i),{\hat {\mathbf {u} }}(i))\end{aligned}}}$

The backward passes and forward passes are iterated until convergence. If the Hessians ${\displaystyle Q_{\mathbf {x} \mathbf {x} },Q_{\mathbf {u} \mathbf {u} },Q_{\mathbf {u} \mathbf {x} },Q_{\mathbf {x} \mathbf {u} }}$ are replaced by their Gauss-Newton approximation, the method reduces to the iterative Linear Quadratic Regulator (iLQR). ^[6]

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 and/or line-search to achieve convergence. ^{[7] [8]} Regularization in the DDP context means ensuring that the ${\displaystyle Q_{\mathbf {u} \mathbf {u} }}$ matrix in Eq. 4 is positive definite. Line-search in DDP amounts to scaling the open-loop control modification ${\displaystyle \mathbf {k} }$ by some ${\displaystyle 0<\alpha <1}$.

Monte Carlo version

Sampled differential dynamic programming (SaDDP) is a Monte Carlo variant of differential dynamic programming. ^{[9] [10] [11]} 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 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. ^[12] This creates a link between differential dynamic programming and path integral control, ^[13] 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. ^[14]

See also

• Optimal control

References

1. Mayne, D. Q. (1966). "A second-order gradient method of optimizing non-linear discrete time systems". Int J Control. 3: 85–95. doi:10.1080/00207176608921369.
2. Mayne, David Q.; Jacobson, David H. (1970). Differential dynamic programming. New York: American Elsevier Pub. Co. ISBN   978-0-444-00070-5.
3. de O. Pantoja, J. F. A. (1988). "Differential dynamic programming and Newton's method". International Journal of Control. 47 (5): 1539–1553. doi:10.1080/00207178808906114. ISSN   0020-7179.
4. Liao, L. Z.; C. A Shoemaker (1992). "Advantages of differential dynamic programming over Newton's method for discrete-time optimal control problems". Cornell University. hdl: 1813/5474 .
5. Morimoto, J.; G. Zeglin; C.G. Atkeson (2003). "Minimax differential dynamic programming: Application to a biped walking robot". Intelligent Robots and Systems, 2003.(IROS 2003). Proceedings. 2003 IEEE/RSJ International Conference on. Vol. 2. pp. 1927–1932.
6. Baumgärtner, K. (2023). A Unified Local Convergence Analysis of Differential Dynamic Programming, Direct Single Shooting, and Direct Multiple Shooting . 2023 European Control Conference (ECC). pp. 1–7. doi:10.23919/ECC57647.2023.10178367.
7. Liao, L. Z; C. A Shoemaker (1991). "Convergence in unconstrained discrete-time differential dynamic programming". IEEE Transactions on Automatic Control. 36 (6): 692. doi:10.1109/9.86943.
8. Tassa, Y. (2011). Theory and implementation of bio-mimetic motor controllers (PDF) (Thesis). Hebrew University. Archived from the original (PDF) on 2016-03-04. Retrieved 2012-02-27.
9. "Sampled differential dynamic programming". 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). doi:10.1109/IROS.2016.7759229. S2CID   1338737.
10. Rajamäki, Joose; Hämäläinen, Perttu (June 2018). Regularizing Sampled Differential Dynamic Programming - IEEE Conference Publication . 2018 Annual American Control Conference (ACC). pp. 2182–2189. doi:10.23919/ACC.2018.8430799. S2CID   243932441 . Retrieved 2018-10-19.
11. Rajamäki, Joose (2018). Random Search Algorithms for Optimal Control. Aalto University. ISBN   978-952-60-8156-4. ISSN   1799-4942.
12. Lefebvre, Tom; Crevecoeur, Guillaume (July 2019). "Path Integral Policy Improvement with Differential Dynamic Programming". 2019 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM). pp. 739–745. doi:10.1109/AIM.2019.8868359. hdl: 1854/LU-8623968 . ISBN   978-1-7281-2493-3. S2CID   204816072.
13. Theodorou, Evangelos; Buchli, Jonas; Schaal, Stefan (May 2010). "Reinforcement learning of motor skills in high dimensions: A path integral approach". 2010 IEEE International Conference on Robotics and Automation. pp. 2397–2403. doi:10.1109/ROBOT.2010.5509336. ISBN   978-1-4244-5038-1. S2CID   15116370.
14. Pavlov, Andrei; Shames, Iman; Manzie, Chris (2020). "Interior Point Differential Dynamic Programming". IEEE Transactions on Control Systems Technology. 29 (6): 2720. arXiv: 2004.12710 . Bibcode:2021ITCST..29.2720P. doi:10.1109/TCST.2021.3049416.

External links

• A Python implementation of DDP
• A MATLAB implementation of DDP

• The open-source software framework acados provides an efficient and embeddable implementation of DDP.