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The Spitzer space telescope (Spitzer) is currently NASA's largest and most sensitive infrared (IR) telescope in space. Spitzer's focal plane carries detectors from three science instruments, namely, the infrared array camera (IRAC), the infrared spectrograph (IRS), and the multiband imaging photometer for Spitzer (MIPS). In this article we discuss the instrument pointing frame (IPF) Kalman filter, which is used to calibrate Spitzer's telescope focal plane. The IPF filter is a high-order square-root iterated linearized Kalman filter that carries 37 states to estimate frame misalignments, while correcting for systematic errors due to optical distortions, scan-mirror errors, thermomechanically induced drift variations, and gyro bias and drift in all axes. The Spitzer application demonstrates that the integrated approach offers significant advantages with respect to optimality, time-efficiency, anomaly detection, and health monitoring compared to existing telescope-calibration approaches, where the parameters are artificially broken into subsets that are estimated by separate teams of analysts. Performance results for the IPF Kalman filter indicate that all Spitzer calibration requirements are satisfied, and are consistent with margins predicted by preflight error analysis. On a final note, after more than five-and-a-half years of probing the cool cosmos, Spitzer entered standby mode on May 15, 2009, as a result of running out of the liquid helium coolant that kept its infrared instruments chilled. This event marks the successful completion of the Spitzer's cold mission as originally commissioned by NASA. However, even though the telescope is warming up, the IRAC arrays continue to operate and provide useful scientific data. A new follow-on warm mission based on the IRAC arrays has been defined and initiated, so that Spitzer will remain in commission for several years to come.