Sang-Bum Cho

We develop a mathematical model that describes the accelerating flight of a spacecraft in a fixed plane. The spacecraft is represented as a rigid body and fuel slosh dynamics are included using a common pendulum model. The control inputs are defined by a transverse body fixed force and a pitching moment about the center of mass of the spacecraft; the slosh dynamics are assumed to be unactuated. The model is placed in the form of a nonlinear control system that allows for the study of planar vehi

Abstract. We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in S 3 (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corr

The triaxial attitude control testbed has been developed as part of a research program on spacecraft multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models including: moment actuators that are fixed to the triaxial base body, as well as reaction wheel actuators and proof mass actuators that are fixed to the triaxial base body. The models also allow incorporation of unactuated

We develop equations of motion for multibody vehicles that can be used for nonlinear system analysis and control design. A multibody vehicle consists of a base body that can undergo general motion in three dimensions, as well as a finite number of body interconnections that can deform relative to the base body and hence define the shape of the multibody. A Lagrangian development leads to equations of motion that are expressed in terms of the locked inertia and the mechanical connection. We show

The Triaxial Attitude Control Testbed has been developed as part of a research program at the University of Michigan on multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models; these include fan actuators, reaction wheel actuators and proof mass actuators that are fixed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constr

The triaxial attitude control testbed (TACT) has been developed as part of a research program on spacecraft multibody rotational dynamics and control. In the paper, an attitude control problem for the TACT actuated by two proof mass devices is studied. Under the assumption of uniform gravity, proof masses generate control moments about the roll and pitch axes of the base body. The control objective is to accomplish attitude stabilization about all three axes using only two proof mass actuators.

Given a stabilized Heegaard splitting of a three-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus-2 Heegaard splitting of each lens space. In particular, we show that the complex for the genus-2 splitting for the lens space L(p,q) with 1≤q≤p/2 is connected if and only if p≡±1(modq), and descr

The genus-g Goeritz group is the group of isotopy classes of orientationpreserving homeomorphisms of a closed orientable 3-manifold that preserve a given genus-g Heegaard splitting of the manifold. In this work, we show that the genus-2 Goeritz group of S 2 S 1 is finitely presented, and give its explicit presentation.

Abstract. In previous work, we developed a theory of tunnels of tunnel number 1 knots in S 3. It yields a parameterization in which each tunnel is described uniquely by a finite sequence of rational parameters and a finite sequence of 0’s and 1’s, that together encode a procedure for constructing the knot and tunnel. In this paper we calculate these invariants for all tunnels of torus knots.

A $3$-manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb {S}^2 \times \mathbb {S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces and show that the mapping class group of the genus-$2$ Heegaard splitting is finitely presented for any lens space by giving its exp

In 2000, Goda, Scharlemann, and Thompson described a general construction of all tunnels of tunnel number $1$ knots using “tunnel moves”. The theory of tunnels introduced by Cho and McCullough provides a combinatorial approach to understanding tunnel moves. We use it to calculate the number of distinct minimal sequences of such moves that can produce a given tunnel. As a consequence, we see that for a sparse infinite set of tunnels, the minimal sequence is unique, but generically a tunnel wi

We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of S 2 S 1 , then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. We then construct infinitely many Heegaard splittings, including the

We use the theory of tunnel number $1$ knots introduced in an earlier paper to strengthen the Tunnel Leveling Theorem of Goda, Scharlemann, and Thompson. This yields considerable information about bridge numbers of tunnel number $1$ knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant $d$ of its tunnels. The growth of this value is on the order of $(1+\sqrt {2})^d$, which improves known estimates of the rate of growth of bridge numbe