An improved "sensor-integrating" algorithm has been developed for use in extracting partial information on the time-varying attitude of a balloon-borne robotic instrumentation system from the outputs of accelerometers that measure accelerations along three Cartesian axes fixed in the instrument package. The partial attitude information in question comprises two coordinates, representing angles of rotation of the instrument package about two orthogonal horizontal axes. If a single additional vector measurement (e.g., the direction to the Sun) is available, then all three angular coordinates are known; that is, the attitude is fully characterized. Because the algorithm could make it unnecessary to carry a complement of conventional high-precision attitude-measuring instrumentation, it affords potential for reducing the sizes, weights, and costs of meteorological, military, planetary exploration, and other balloon-borne robotic instrumentation systems.

While the outputs of the accelerometers in the instrument package are useful for estimating changes in velocity and position, they do not directly provide attitude information. However, the outputs the accelerometers contain indirect, partial information about the attitude of the instrument package in the following sense: Each accelerometer responds to the projection, onto its axis of sensitivity, of a g, where g is the gravitational acceleration and a is the inertial acceleration. If one knows |g| and can estimate a, then one can estimate the projections of g onto the accelerometer axes and, from these projections, deduce the orientation of the instrument package relative to the local vertical axis.

The key to estimating a lies in recognizing that the motion of the balloon-borne instrumentation system is dominated by swaying like that of a pendulum with damping plus random force and torque excitations from wind gusts. For the purpose of mathematically modeling the pendulum dynamics to estimate the state of the system (the state includes the angular coordinates that one seeks), the system is approximated as a rigid body suspended below a pivot. Among the parameters of the dynamical model are the resonance frequencies of pendulum oscillations about the horizontal axes and the rates of damping of motions about all three axes. The resonance frequencies can be measured prior to flight. The rates of damping can be estimated and, even if not precise, help to ensure that, statistically, the pendulum oscillates about a vertical orientation and settles gradually to a vertical orientation when excitation is removed. The excitations from wind gusts are represented statistically, by use of first-order low-pass processes estimated from a wind power spectrum.

The dynamical model described above is converted to a nonlinear state-estimating model via an intermediate mathematical model that features a quaternion representation of attitude. The model integrates data from both accelerometers and simple gyroscopes to provide estimates of both position and attitude. The equations of the model are solved and state estimates updated by an algorithm that includes a continuous/discrete extended-Kalman-filter.

The model and algorithm were applied in a test case in which the dynamics of a balloon-borne system with representative parameters were simulated computationally. Among the results obtained in the test were covariances of the angular coordinates about the two horizontal (*x* and* y*) body axes and about the vertical (*z*) axis. As shown in the figure, the covariances for the *x* and *y* axes were found to be bounded (signifying that the angular coordinates in question can be known to within a specified accuracy), while the covariance for the* z* axis was found to increase approximately linearly with time (signifying that the estimate of the third angular coordinate deteriorates with time in the absence of further information).

*This work was done by David Bayard, Robert Scheid, Donald Gennery, and Jayarao Balaram of Caltech for *NASA's Jet Propulsion Laboratory*. **NPO-20315*

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This article first appeared in the November, 1998 issue of *NASA Tech Briefs* Magazine.

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