Ph.D. Thesis Summary

FINITE ELEMENT TORQUE MODELING AND BACKSTEPPING CONTROL OF A SPHERICAL MOTOR

Raye Abdoulie Sosseh

January 2002

The increasing demand for multi-degree-of-freedom (DOF) actuators in a number of industries has motivated a flurry of research in the development of non-conventional actuators, one of which is a ball-joint-like variable reluctance (VR) spherical motor. This motor is capable of providing smooth and isotropic three-dimensional motion in a single joint. Compared to conventional robotic manipulators that offer the same motion capabilities, the innovative spherical motor possesses several advantages. Not only can the spherical motor combine 3-DOF motion in a single joint, it has a large range of motion with no singularities in its workspace. The VR spherical motor is much simpler and compact in design than most multiple single-axis robotic manipulators. The motor is also relatively easy to manufacture. These unique characteristics of a spherical motor have potential contributions to a wide range of applications such as coordinate measuring, object tracking, material handling, automated assembling, welding, and laser cutting.

Most of these features have been demonstrated in previous research efforts at Georgia Tech. The spherical motor, however, exhibits coupled, nonlinear and very complex dynamics that make the design and implementation of feedback controllers very challenging. The orientation-varying torque generated by the spherical motor also contributes to the challenges in controller design.

This thesis contributes to the on-going research effort by exploring alternate methods for controlling the motor. This thesis addresses three basic issues related to the control of a spherical motor; (1) formulation of an approximate but more appropriate (or tractable) form of the torque model for real-time control, (2) a more accurate representation of the dynamic model of an existing prototype, and (3) the design of a robust feedback controller. The robust backstepping controller proposed in this thesis is used to further demonstrate the appealing features exhibited by the spherical motor.

A complete formulation of the torque generated by the spherical motor requires the solution of the magnetic field distribution in the motor’s workspace. The magnetic field distribution of the spherical motor is governed by a set of partial differential equations commonly referred to as Maxwell’s equations. These equations relate the field quantities that are essential in formulating the electromagnetic interaction of the spherical motor components. The torque generated by the spherical motor is readily computed from the magnetic field variables that result from the solution of these equations. Closed-form solutions to Maxwell’s equations are available for only a few electromechanical devices with relatively simple structures. Due to the motor’s complex rotor structure, and the resulting boundary conditions, obtaining closed form solutions to Maxwell’s equations can be very challenging. The inherent three-dimensional nature of the magnetic field distribution also contributes to these challenges. With currently available computational hardware, numerical schemes such as finite element (FE) methods can solve the motor’s magnetic field distribution. The ANSYS package used in this thesis has the capacity to solve 3-D FE problems. The FE codes developed in this thesis can be used to investigate the sensitivity of key geometrical parameters on the torque performance. Thus, the FE codes serve as a potentially useful analysis tool in design of future spherical motors. The incorporation of FE formulations in the design process would significantly reduce design time while giving a closer starting point to a working prototype.

The second part of this thesis formulates the spherical motor dynamics, taking into account the effects of the orientation measurement system. The motor dynamics are derived using a constrained Lagrangian formulation (analytical method). With this method, the reaction forces from the orientation measurement system are implicitly accounted for. The motor dynamics are described in terms of five constrained generalized coordinates, and since the spherical motor offers 3-DOF motion, two Lagrange multipliers are introduced in the formulation. Therefore, in addition to the five Lagrange equations for each generalized coordinate, two more equations are needed to fully describe the spherical motor dynamics. These are derived from the kinematic relations of the orientation measurement system. The frictional effects that result due to the orientation measurement and transfer bearing systems are not accounted for in the derivation of the motor dynamics. These are treated as system perturbations and are compensated for by the robust controller designed in this thesis. The derived dynamics make up the nominal equations required to design and implement robust feedback controllers.
 

The third focus of this thesis involves the design and implementation of robust feedback controllers for the position control of the spherical motor. Using Lyapunov-type stability arguments, a robust backstepping controller is designed to achieve this objective. The controller developed in this thesis is designed in two steps. Firstly, a robust stabilizing torque is designed for the nominal spherical motor dynamics derived using the constrained Lagrangian formulation. Next, the solution to a static optimization problem is computed to determine the applied stator coil currents needed to generate the desired stabilizing torque. The static optimization problem is formulated to minimize the input energy subject to constraints imposed by the torque model. The eventual stability of the controller depends on the torque generating capabilities of the spherical motor.
 

Using the current experimental test-bed, the performance of this robust controller is compared to a PD controller.