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  "body": "\n## Summary\n\nDirect collocation is a method of [[trajectory optimization|58895A7T4H]] that turns an optimal control problem into a finite nonlinear program.\nIt represents the state and the control by their values on a time grid.\nIt imposes the differential equation only at chosen collocation points, through a polynomial model of the state on each interval.\nThe cost integral becomes a quadrature sum, and the resulting sparse nonlinear program is solved by a general optimiser.\nIt is \"direct\" because it discretises first and does not use the [[maximum principle|7RPV2T29CF]].\n\n## Explanations\n\n### L1 human\n\nInstead of solving the equations of motion exactly, direct collocation draws the trajectory as a chain of short polynomial pieces through a few grid points.\nIt asks only that each piece obey the equations of motion at a few chosen times.\nThe grid values become the unknowns of an ordinary finite optimisation problem.\nWith the trapezoidal rule, for example, the condition on one step of length $h$ is $x_{k+1} - x_k = \\frac{h}{2} (f_k + f_{k+1})$, where $f_k$ is the velocity at grid point $k$.\nConstraints such as speed limits are easy to add, because they are just conditions on the grid values.\n",
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