### Browsing by Author "Tsakiris, D.P."

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Item G-Snakes: Nonholonomic Kinematic Chains on Lie Groups(1994) Krishnaprasad, Perinkulam S.; Tsakiris, D.P.; ISRWe consider kinematic chains evolving on a finite-dimensional Lie group G under nonholonomic constraints, where snake-like global motion is induced by shape variations of the system. In particular, we consider the case when the evolution of the system is restricted to a subspace h of the corresponding Lie algebra g, where h is not a subalgebra of g and it can generate the whole algebra under Lie bracketing. Such systems are referred to as G- snakes. Away from certain singular configurations of the system, the constraints specify a (partial) connection on a principal fiber bundle, which in turn gives rise to a geometric phase under periodic shape variations. This geometric structure can be exploited in order to solve the nonholonomic motion planning problem for such systems.G-snakes generalize the concept of nonholonomic Variable Geometry Truss assemblies, which are kinematic chains evolving on the Special Euclidean group SE (2) under nonholonomic constraints imposed by idler wheels. We examine in detail the cases of 3-dimensional groups with real non-abelian Lie algebras such as the Heisenberg group H(3), the Special Orthogonal group SO (3) and the Special Linear group SL(2).

Item Motion Control and Planning for Nonholonomic Kinematic Chains(1995) Tsakiris, D.P.; Krishnaprasad, P.S.; ISRIn this dissertation we examine a class of systems where nonholonomic kinematic constraints are combined with periodic shape variations, giving rise to snake-like undulating motion of the system. Within this class, we distinguish two subclasses, one where the system possesses enough kinematic constraints to allow the control of its motion to be based entirely on kinematics and another which does not; in the latter case, the dynamics plays a crucial role in complementing the kinematics and in making motion control possible. An instance of these systems are the Nonholonomic Variable Geometry Truss (NVGT) assemblies, where shape changes are implemented by parallel manipulator modules, while the nonholonomic constraints are imposed by idler wheels attached to the assembly. We assume that the wheels roll without slipping on the ground, thus constraining the instantaneous motion of the assembly. These assemblies can be considered as land locomotion alternatives to systems based on legs or actuated wheels. Their propulsion combines features of both biological systems like skating humans and snakes, and of man-made systems like orbiting satellites with manipulator arms. The NVGT assemblies can be modeled in terms of the Special Euclidean group of rigid motions on the plane. Generalization to nonholonomic kinematic chains on other Lie groups (G) gives rise to the notion of G -Snakes.Moreover, we examine systems with parallel manipulator subsystems which can be used as sensor- carrying platforms, with potential applications in exploratory and active visual or haptic robotic tasks. We concentrate on specifying a class of configuration space path segments that are optimal in the sense of a curvature-squared cost functional, which can be specified analytically in terms of elliptic functions and can be used to synthesize a trajectory of the system.

In both cases, a setup of the problem which involves tools from differential geometry and the theory of Lie groups appears to be natural. In the case of G -Snakes, when the number of nonholonomic constraints equals the dimension of the group G, the constraints determine a principal fiber bundle connection. The geometric phase associated to this connection allows us to derive (kinematic) motion control strategies based on periodic shape variations of the system. When the G -Snakes assembly has one constraint less than the dimension of the group G, we are still able to synthesize a principal fiber bundle connection by taking into account the Lagrangian dynamics of the system through the so-called nonholonomic momentum. The symmetries of the system are captured by actions of non-abelian Lie groups that leave invariant both the constraints and the Lagrangian and play a significant role in the definition of the momentum and the specification of its evolution. The (dynamic) motion control is now based on periodic shape variations that build up momentum and allow propulsion and steering, as described by the geometric and dynamic phases of the system.

Item Nonholonomic Variable Geometry Truss Assemblies I: Motion Control(1993) Krishnaprasad, Perinkulam S.; Tsakiris, D.P.; ISRWe consider the nonholonomic motion planning problem for a novel class of snakelike modular mobile manipulators, where each module is implemented as a planar parallel manipulator with idler wheels. This assembly is actuated by shape changes of its modules, which, under the influence of the nonholonomic constraints on the wheels, induce a global motion of the assembly.We formulate the kinematics for a generic assembly of this type and specialize to the 2-module case in order to study the motion planning problem in greater detail.

Item Visual Tracking Strategies(1988) Tsakiris, D.P.; Krishnaprasad, P.S.; ISRVisual tracking is one of the most important applications of Computer Vision and several tracking systems have been developed, which, either focus mainly on the tracking of targets moving on a plane or attempt to reduce the 3-dimensional tracking problem to the tracking of a set of characteristic points of the target. These approaches are seriously handicapped in complex visual situations from segmentation and point correspondence problems. A mathematical theory for visual tracking of a three-dimensional target moving rigidly in 3-D is presented here and it is shown how a monocular observer can track an initially foveated object and keep it stationary in the center of his visual field. Our attempt is to develop correspondence-free tracking schemes and take advantage of the dynamic segmentation capabilities inherent in the optical flow formalism. Moreover, a general tracking criterion, the Tracking Constraint is derived, which reduces tracking to an appropriate optimization problem. The connection of our tracking strategies with the Active Vision Paradigm is shown to provide a solution to the Egomotion problem. In the first part of this work, tracking strategies based on the assumption that we know the optical flow field are examined and tracking is formulated as a constrained optimization and a penalized least-squares problem. In the second part, tracking strategies based on the recovery of the 3-D motion of the target are devised under the assumption that we know the shape of the target. A correspondence-free scheme is derived, which depends on global information about the scene (provided from linear features of the image) in order to bypass the ill-posed problem of computing the spatial derivatives of the image intensity function and amounts to the solution of a linear system of equations in order to estimate the 3-D motion of the target. An important feature of these tracking strategies is that they do not require continuous segmentation of the image in order to locate the target. Supposing that the target is sufficiently textured, dynamic segmentation using temporal derivatives of the linear features provides sufficient information for the tracking phase. Therefore, this approach is expected to perform best when previous ones perform worst, namely in a complex visual environment. Experimental results of the algorithms presented here demonstrate their robustness in the presence of noise.Item Visual Tracking Strategies.(1988) Tsakiris, D.P.; ISRVisual tracking is one of the most important applications of Computer Vision and several tracking systems have been developed, which, either focus mainly on the tracking of targets moving on a plane or attempt to reduce the 3-dimensional tracking problem to the tracking of a set of characteristic points of the target. These approaches are seriously handicapped in complex visual situations from segmentation and point correspondence problems. A mathematical theory for visual tracking of a three-dimensional target moving rigidly in 3-D is presented here and it is shown how a monocular observer can track an initially foveated object and keep it stationary in the center of his visual field. Our attempt is to develop correspondence-free tracking schemes and take advantage of the dynamic segmentation capabilities inherent in the optical flow formalism. Moreover, a general tracking criterion, the Tracking Constraint is derived, which reduces tracking to an appropriate optimization problem. The connection of our tracking strategies with the Active Vision Paradigm is shown to provide a solution to the Egomotion problem. In the first part of this work, tracking strategies based on the assumption that we know the optical flow field are examined and tracking is formulated as a constrained optimization and a penalized least-squares problem. In the second part, tracking strategies based on the recovery of the 3-D motion of the target are devised under the assumption that we know the shape of the target. A correspondence-free scheme is derived, which depends on global information about the scene (provided from linear features of the image) in order to bypass the ill-posed problem of computing the spatial derivatives of the image intensity function and amounts to the solution of a linear system of equations in order to estimate the 3-D motion of the target. An important feature of these tracking strategies is that they do not require continuous segmentation of the image in order to locate the target. Supposing that the target is sufficiently textured, dynamic segmentation using temporal derivatives of the linear features provides sufficient information for the tracking phase. Therefore, this approach is expected to perform best when previous ones perform worst, namely in a complex visual environment. Experimental results for the algorithms presented here demonstrate their robustness in the presence of noise.