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]

The dynamics

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

1

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

J 0 ( x , U ) = ∑ i = 0 N − 1 ℓ ( x i , u i ) + ℓ f ( x N ) , {\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 x 0 ≡ x {\displaystyle \mathbf {x} _{0}\equiv \mathbf {x} } , and the x i {\displaystyle \mathbf {x} _{i}} for i > 0 {\displaystyle i>0} are given by Eq. 1. The solution of the optimal control problem is the minimizing control sequence U ∗ ( x ) ≡ argmin U ⁡ J 0 ( x , U ) . {\displaystyle \mathbf {U} ^{*}(\mathbf {x} )\equiv \operatorname {argmin} _{\mathbf {U} }J_{0}(\mathbf {x} ,\mathbf {U} ).} Trajectory optimization means finding U ∗ ( x ) {\displaystyle \mathbf {U} ^{*}(\mathbf {x} )} for a particular x 0 {\displaystyle \mathbf {x} _{0}} , rather than for all possible initial states.

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

J i ( x , U i ) = ∑ j = i N − 1 ℓ ( x j , u j ) + ℓ f ( x 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 i {\displaystyle i} is the cost-to-go given the minimizing control sequence:

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

Setting V ( x , N ) ≡ ℓ f ( x N ) {\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:

V ( x , i ) = min u [ ℓ ( x , u ) + V ( f ( x , u ) , i + 1 ) ] . {\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.

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

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

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

Q ( δ x , δ u ) ≡ ℓ ( x + δ x , u + δ u ) + V ( f ( x + δ x , u + δ u ) , i + 1 ) − ℓ ( x , u ) − V ( f ( x , u ) , i + 1 ) {\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

≈ 1 2 [ 1 δ x δ u ] T [ 0 Q x T Q u T Q x Q x x Q x u Q u Q u x Q u u ] [ 1 δ x δ u ] {\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 Q {\displaystyle Q} notation used here is a variant of the notation of Morimoto where subscripts denote differentiation in denominator layout.^[5] Dropping the index i {\displaystyle i} for readability, primes denoting the next time-step V ′ ≡ V ( i + 1 ) {\displaystyle V'\equiv V(i+1)} , the expansion coefficients are

Q x = ℓ x + f x T V x ′ Q u = ℓ u + f u T V x ′ Q x x = ℓ x x + f x T V x x ′ f x + V x ′ ⋅ f x x Q u u = ℓ u u + f u T V x x ′ f u + V x ′ ⋅ f u u Q u x = ℓ u x + f u T V x x ′ f x + V x ′ ⋅ f u x . {\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 δ u {\displaystyle \delta \mathbf {u} } we have

δ u ∗ = argmin δ u ⁡ Q ( δ x , δ u ) = − Q u u − 1 ( Q u + Q u x δ x ) , {\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 k = − Q u u − 1 Q u {\displaystyle \mathbf {k} =-Q_{\mathbf {u} \mathbf {u} }^{-1}Q_{\mathbf {u} }} and a feedback gain term K = − Q u u − 1 Q u x {\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 i {\displaystyle i} :

Δ V ( i ) = − 1 2 Q u T Q u u − 1 Q u V x ( i ) = Q x − Q x u Q u u − 1 Q u V x x ( i ) = Q x x − Q x u Q u u − 1 Q u x . {\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 V ( i ) {\displaystyle V(i)} and the control modifications { k ( i ) , K ( i ) } {\displaystyle \{\mathbf {k} (i),\mathbf {K} (i)\}} , from i = N − 1 {\displaystyle i=N-1} down to i = 1 {\displaystyle i=1} , constitutes the backward pass. As above, the Value is initialized with V ( x , N ) ≡ ℓ f ( x N ) {\displaystyle V(\mathbf {x} ,N)\equiv \ell _{f}(\mathbf {x} _{N})} . Once the backward pass is completed, a forward pass computes a new trajectory:

x ^ ( 1 ) = x ( 1 ) u ^ ( i ) = u ( i ) + k ( i ) + K ( i ) ( x ^ ( i ) − x ( i ) ) x ^ ( i + 1 ) = f ( x ^ ( i ) , u ^ ( i ) ) {\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 Q x x , Q u u , Q u x , Q x u {\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]

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 Q u u {\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 k {\displaystyle \mathbf {k} } by some 0 < α < 1 {\displaystyle 0<\alpha <1} .

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.

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]

• Optimal control

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

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