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Numerical analysis of constrained time-optimal satellite reorientation. Robert G. Melton Department of Aerospace Engineering Penn State University. 6 th International Workshop and Advanced School, “Spaceflight Dynamics and Control” Covilh ã , Portugal March 28-30, 2011.
Numerical analysis of constrained time-optimal satellite reorientation Robert G. Melton Department of Aerospace Engineering Penn State University 6th International Workshop and Advanced School, “Spaceflight Dynamics and Control” Covilhã, Portugal March 28-30, 2011
Unconstrained Time-Optimal Reorientation • Bilimoria and Wie (1993) unconstrained solution NOT eigenaxis rotation • spherically symmetric mass distribution • independently and equally limited control torques • bang-bang solution, switching is function of reorientation angle • Others examined different mass symmetries, control architectures • Bai and Junkins (2009) • discovered different switching structure, local optima • for magnitude-limited torque vector, solution IS eigenaxis rotation
Constrained Problem (multiple cones): No Boundary Arcs or Points Observed Example: 0.1 deg. gap between Sun and Moon cones
Constrained Problem (multiple cones) tf= 3.0659, 300 nodes, 8 switches
Keep-out Cone Constraint (cone axis for source A) (sensor axis)
Optimal Control Formulation • Resulting necessary conditions are analytically intractable
Numerical Studies Sensor axis constrained to follow the cone boundary (forced boundary arc) Sensor axis constrained not to enter the cone Entire s/c executes -rotation about A Legendre pseudospectral method used (DIDO software) Scaling: lie on constraint cone • I1 = I2 = I3and M1,max = M2,max = M3,max • lies along principal body axis b1 • final orientation of b2, b3generally unconstrained
Case BA-1 (forced boundary arc) • A = 45 deg. (approx. the Sun cone for Swift) • Sensor axis always lies on boundary • Transverse body axes are free • = 90 deg.
Case BA-1 (forced boundary arc) tf = 1.9480, 151 nodes
Case BA-2 (forced boundary arc) • A = 23 deg. (approx. the Moon cone for Swift) • Sensor axis always lies on boundary • Transverse body axes are free • = 70 deg. tf = 1.3020, 100 nodes
Case BP-1 • same geometry as BA-1 (A = 45 deg., = 90 deg.) • forced boundary points at initial and final times • sensor axis departs from constraint cone Angle between sensor axis and constraint cone tf= 1.9258 (1% faster than BA-1) 250 nodes
Case BP-2 • same geometry as BA-2 (A = 23 deg., = 70 deg.) • forced boundary points at initial and final times • sensor axis departs from constraint cone Angle between sensor axis and constraint cone tf= 1.2967 (0.4% faster than BA-2) 100 nodes
Sensor axis path along the constraint boundary Constrained Rotation Axis • Entire s/c executes -rotation • sensor axis on cone boundary • rotation axis along cone axis
Constrained Rotation Axis Problem now becomes one-dimensional, with bang-bang solution Applying to geometry of: BA-1 tf = 2.1078 (8% longer than BA-1) BA-2 tf = 2.0966 (37% longer than BA-2)
Practical Consideration • Pseudospectral code requires • 20 minutes < t < 12 hours • (if no initial guess provided) • Present research involves use of two-stage solution: • approx solnS (via particle swarm optimizer) • S = initial guess for pseudospectral code • (states, controls, node times at CGL points) • Successfully applied to 1-D slew maneuver
Dido No guess cpu time = 148 sec. With PSO guess cpu time = 76 sec,
Conclusions and Recommendations • For independently limited control torques, and initial and final sensor directions on the boundary: • trajectory immediately departs the boundary • no interior BP’s or BA’s observed • forced boundary arc yields suboptimal time • Need to conduct more accurate numerical studies • Bellman PS method • Interior boundary points? (indirect method) • Study magnitude-limited control torque case • Implementation • expand PSO+Dido to 3-D case