It is a type of direct collocation method used for solving optimal control problems.

The mains steps in Hermite Simpson collocation are summarized as follows.

  • time is divided into $N$ intervals
  • Simpson quadrature is used to evaluate the definite integrals
  • the derivative of the state vector ($x$) is approximated as a piecewise-quadratic polynomial
  • the states are piecewise cubic polynomials
  • the control inputs ($u$) can be parameterized as piecewise constant or linear polynomial
  • the integral in the objective function will use the composite form of the Simpson quadrature.
  • the path constrains which may arise from the optimal control problem must be enforced at the collocation point in addition to the interval boundaries.
  • results in a sparse optimization

Let the dynamics be given by

$\dot{x}=f(x,u)$

On the $i^{th}$ interval $t \in [t_i, t_{i+1}]$, integrate the dynamics. $i+\frac{1}{2}$ is the midpoint of the interval (the collocation point where the dynamics defect constraint is enforced) and $h=\dfrac{t_{i+1}-t_{i}}{N}$.

$\int_{t_i}^{t_{i+1}}\dot{x}dt=\int_{t_i}^{t_{i+1}}f~\mathrm{d}t$
$x_{i+1}-x_i=\int_{t_i}^{t_{i+1}}f~\mathrm{d}t$
$\int_{t_i}^{t_{i+1}}fdt=\dfrac{h}{6}\left(f_i+4f_{i+\frac{1}{2}}+f_{i+1}\right)$
$x_{i+1}-x_i=\dfrac{h}{6}\left(f_i+4f_{i+\frac{1}{2}}+f_{i+1}\right)$
where,

$f_i=f(x_i,u_i)$ and $f_{i+1}=f(x_{i+1},u_{i+1})$.

As the integrand is a quadratic polynomial, state is a piecewise cubic polynomial and on the $i^{th}$ interval let it be represented as

$x(\tau)=a\tau^3+b\tau^2+c\tau+d$

Note that for $\tau \in [0,h]$, the jacobian for this transformation is identity. The state and its derivative is evaluated at the boundaries on each interval to determine the coefficients of the cubic polynomial.

$x(0)=a0^3+b0^2+c0+d$
$x(h)=ah^3+bh^2+ch+d$
$\dot{x}(0)=3a0^2+2b0+c$
$\dot{x}(h)=3ah^2+2bh+c$

From the above equations, $c=\dot{x}(0)$ and $d=x(0)$.

\[\begin{bmatrix} h^3 &h^2\\ 3h^2 &2h \end{bmatrix} \begin{bmatrix} a\\ b \end{bmatrix} =\begin{bmatrix} x(h)-\dot{x}(0)h-x(0)\\ \dot{x}(h)-\dot{x}(0) \end{bmatrix}= \begin{bmatrix} x_{i+1}-f_ih-x_i\\ f_{i+1}-f_{i} \end{bmatrix}\] \[\begin{bmatrix} a\\ b \end{bmatrix}=\dfrac{-1}{h^4} \begin{bmatrix} 2h &-h^2\\ -3h^2 &h^3 \end{bmatrix} \begin{bmatrix} x_{i+1}-f_ih-x_i\\ f_{i+1}-f_{i} \end{bmatrix}\]

From this, the value of the state at the $i+\frac{1}{2}$ is computed.

\[\begin{aligned} x(\frac{h}{2}) &=a\dfrac{h^3}{8}+b\dfrac{h^2}{4}+c\dfrac{h}{2}+d\\ &= \begin{bmatrix} \dfrac{h^3}{8}&\dfrac{h^2}{4} \end{bmatrix} \begin{bmatrix} a\\ b \end{bmatrix} + f_i\dfrac{h}{2}+x_i\\ &= \begin{bmatrix} \dfrac{h^3}{8}&\dfrac{h^2}{4} \end{bmatrix} \dfrac{-1}{h^4} \begin{bmatrix} 2h &-h^2\\ -3h^2 &h^3 \end{bmatrix} \begin{bmatrix} x_{i+1}-f_ih-x_i\\ f_{i+1}-f_{i} \end{bmatrix} + f_i\dfrac{h}{2}+x_i\\ &= \dfrac{-h^4}{h^4} \begin{bmatrix} (\dfrac{1}{4}-\dfrac{3}{4})&(\dfrac{-1}{8}+\dfrac{1}{4}) \end{bmatrix} \begin{bmatrix} x_{i+1}-f_ih-x_i\\ f_{i+1}-f_{i} \end{bmatrix} + f_i\dfrac{h}{2}+x_i\\ &= \begin{bmatrix} \dfrac{1}{2}&\dfrac{h}{8} \end{bmatrix} \begin{bmatrix} x_{i+1}-f_ih-x_i\\ f_{i+1}-f_{i} \end{bmatrix} + f_i\dfrac{h}{2}+x_i\\ &= \dfrac{x_{i+1}}{2}-\dfrac{f_ih}{2}-\dfrac{x_i}{2}+ \dfrac{f_{i+1}h}{8}-\dfrac{f_{i}h}{8} + f_i\dfrac{h}{2}+x_i\\ &= \dfrac{1}{2}(x_{i+1}+x_i)+\dfrac{h}{8}(f_{i+1}-f_i) \end{aligned}\]

Depending on the control parameterization, the of control at $i+\dfrac{1}{2}$ can be computed using interpolation. Together, $f_{i+\frac{1}{2}}$ can be evaluated in the Simpson rule.

The equations corresponding to $x_{i+\frac{1}{2}}$ and $u_{i+\frac{1}{2}}$ can be substituted in the quadrature or enforced as an equality constraint with added decision variables respectively known as compressed and uncompressed form in literature.