Some new results for a classical minimum time, rest-to-rest maneuver problem are presented. An inertially symmetric rigid body is considered. For the case that the magnitude of the control is constrained while the control direction is left free, we analytically prove that the eigenaxis maneuver is the time minimum solution by using the Pontryagin's principle. For the case that the three components of the control are independently constrained, we discover six switch and seven switch solutions for reorientation angles less than 72 degrees by using a hybrid numerical approach. The seven switch solutions are consistent with the classical results and the six switch solutions are reported here for the first time. We find that the two sets of solutions are widely separated in state and control spaces. However, the two locally optimum maneuver times are very close to each other although the six switch control always has a slightly shorter time. Simulation results are summarized that illustrate and validate the new *

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Modified Chebyshev-Picard iteration methods are presented for solving boundary value problems. Chebyshev polynomials are used to approximate the state trajectory in Picard iterations, while the boundary conditions are maintained by constraining the coefficients of the Chebyshev polynomials. Using Picard iteration and Clenshaw-Curtis quadrature, the presented methods iteratively refine an orthogonal function approximation ofthe entire state trajectory, in contrast to step-wise, forward integration approaches, which render the methods well-suited for parallel computation because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. The presented methods solve optimal control problems through Pontryagin's principle without requiring shooting methods or gradient information. The methods are demonstrated to be computationally efficient and strikingly accurate when compared with Battin's method for a classical Lambert's problem and with a Chebyshev pseudo spectral method for an optimal trajectory design problem. The reported simulation results obtained on a serial machine suggest a strong basis for optimism of 'using the presented methods for solving more challenging boundary value problems, especially when highly parallel architectures are fully exploited.