Navigation Algorithm And Control Law Design Based On Geometric Algebra

The development of control science requires not only new control instruments toimprove control performances, but also new mathematic concepts and tools to developcontrol theories and methods. Geometric algebra (GA) is a general and powerful mathe-matic language, which can describe all conformal transformations, including rigid bodymotions by using orthogonal transformations. It is considered to be a potential mathemat-ic tool to substitute the vector algebra, which has been widely used in kinematics. Somepilot works on the application of GA to navigation and control fields have been done inthis thesis. The main work of this dissertation is as follows:1. The rotor kinematical equation, which represents the rotation, and the motor kine-matical equation, which represents the general rigid body motion, are derived, andthe relationships between the rotor/motor and the velocity/screw-velocity (a phys-ical quantity constituted of the velocity and the angular rate) are thus established.The GA is applied to coordinate transformation. The coordinate transformationsamong the frames with and without the same origin are proposed, and the kinemati-cal equations of the operators which represent these coordinate transformations arediscussed as well. The similarity between the rotor and the motor kinematical e-quations shows the universality of GA. These kinematic equations can be utilizedto solve kinematic and dynamic problems.2. A strapdown inertial navigation system (SDINS) algorithm based on GA is pro-posed. By introducing the thrust and the gravitational velocity frames, the SDINSequation in terms of GA, i.e., the relationship of the motors which represent therelative movement of the frames, and velocity, acceleration, as well as angular rate,is built on the basis of the Newton’s second kinematic law. The consistency of theGA-basedSDINSequationandtheconventionalnavigationequationisvalidatedaswell. The Bortz equation of the screw blade (a geometrical quantity constructed bythe rotation angle, the screw pitch, and the plane where the rotation locates, whichis similar to the rotation vector in the conventional attitude updating algorithm) isdeveloped on the ground of the trigonometric formula and the kinematic equationof the motor. And then, the screw blade algorithm, as well as its error analysis, is investigated. Numerical results reveal that the screw blade algorithm inherits theadvantages of the rotation vector algorithm, which has been commonly used in theconventional attitude updating. Moreover, the precision of velocity and positionin the screw blade algorithm is better than the precision of the conventional one,therefore, it is suitable for high-precision navigation systems.3. A Kalman filtering model for the GA-based SDINS/GPS integrated navigation sys-temisproposed. TheadditiveandthemultiplicativeGAerrormodelsforthemotionof the rigid body are derived. Further more, the additive and the multiplicative GAerror models of the SDINS are developed. According to the computation rules inGA, the multiplicative GA error model of the SDINS, together with the GPS er-ror model, are utilized to construct the state equation of the Kalman filter for theSDINS/GPS integrated navigation system Kalman filtering. The simulation resultsshow that the integrated navigation system can reduce the error accumulation, andimprove the precision remarkably.4. The attitude and position control problems are investigated by using GA. The kine-matical attitude-position control laws are proposed based on the motor kinematicalequation. And the dynamical attitude-position control laws are built on the groundof the motor kinematical equation and the dynamical equations of the rigid bodymotion. Their stabilities are proven by using the Lyapunov theorem. The experi-mental results show that both the kinematical and the dynamical attitude-positioncontrol laws are stable and convergent, which suggests that they can well track themaneuver of the target.