Reliable solution to dynamic optimization problems using integrated residual regularized direct collocation

Direct collocation (DC) is a widely used method for solving dynamic optimization problems (DOPs), but its implementation simplicity and computational efficiency are limited for challenging problems. For DOPs involving singular arcs, DC solutions often exhibit significant fluctuations along the singular arc, accompanied by large residual errors between collocation points, where the dynamic constraints are enforced as equality constraints. In this paper, we introduce the direct transcription method of integrated residual regularized direct collocation (IRRDC). This approach enforces dynamic constraints using a combination of point-wise residual constraints (expressed as either equalities or inequalities) and a penalty term on the integrated residual error, which helps reduce errors between collocation points. IRRDC retains the implementation simplicity of DC while improving both solution accuracy and efficiency, particularly for challenging problem types. Through the examples, we demonstrate that for problems where traditional DC results in excessive fluctuations, IRRDC effectively suppresses fluctuations and yields solutions with greater accuracy — at least two orders of magnitude lower in various error measures in relation to the dynamic and path constraints.