Ph.D.
Thesis Summary
FINITE ELEMENT TORQUE MODELING AND BACKSTEPPING
CONTROL OF A SPHERICAL MOTOR
Raye Abdoulie Sosseh
January 2002
The increasing demand for multidegreeoffreedom (DOF) actuators in a number of
industries has motivated a flurry of research in the development of
nonconventional actuators, one of which is a balljointlike variable
reluctance (VR) spherical motor. This motor is capable of providing smooth and
isotropic threedimensional 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 3DOF 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 singleaxis 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 orientationvarying torque generated by the spherical
motor also contributes to the challenges in controller design.
This thesis contributes to the ongoing 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 realtime
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. Closedform
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
threedimensional 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
3D 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 3DOF 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 Lyapunovtype
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 testbed, the performance of this robust controller is compared to
a PD controller.
