Posture-Dependent Projection-Based Force Reflection Algorithms for Bilateral Teleoperators

It was previously established that the projection-based force reflection (PBFR) algorithms improve the overall stability of a force reflecting teleoperation system. The idea behind the PBFR algorithms is to identify the component of the reflected force which is compensated by interaction with the operator\u27s hand, and subsequently attenuate the residual component of the reflected force. If there is no a priori information regarding the behaviour of the human operator, the PBFR gain is selected equal to sufficiently small constant in order to guarantee stability for a wide range of human operator responses. Small PBRF gains, however, may deteriorate the transparency of a teleoperator system. In this thesis, a new method for selecting the PBFR gain is introduced which depends on human postures. Using an online human posture estimation, the introduced posture-dependent PBFR algorithm has been applied to a teleoperation system with force feedback. It is experimentally demonstrated that the developed method for selection of the PBFR gain based on human postures improves the transparency of the teleoperator system while the stability is preserved. Finally, preliminary results that deal with an extension of the developed methods towards a more realistic model of the human arm with 4 degrees of freedom and three dimensional movements are presented

Noncommutative complex geometry of quantum projective spaces

In this thesis, we study complex structures of quantum projectivespaces that was initiated in [19] for the quantum projective line, $\mathbb{C}P^1_q$. In Chapters 2 and 3, which are the main parts of this thesis, we generalize the the results of [19] to the spaces $\mathbb{C}P^{2}_q$ and $\mathbb{C}P^{\ell}_q$. We consider a natural holomorphic structure on the quantum projective space already presented in [11,9],and define holomorphic structures on its canonical quantum line bundles.The space of holomorphic sections of these line bundles then will determinethe quantum homogeneous coordinate ring of the quantum projective space as the space of twisted polynomials. We also introduce a twisted positiveHochschild cocycle $2 \ell$-cocycle on $\mathbb{C}P^{\ell}_q$, by using the complex structure of $\mathbb{C}P^{\ell}_q$, and show that it is cohomologous to its fundamental class which is representedby a twisted cyclic cocycle. This fits with the point of view of holomorphic structures in noncommutative geometry advocated in [4,5], that holomorphic structures in noncommutative geometry are represented by (extremal)positive Hochschild cocycles within the fundamental class. In Chapter 4, we directly verify that the main statements of Riemann-Roch formula andSerre duality theorem hold true for $\mathbb{C}P^1_q$ and $\mathbb{C}P^2_q$.In Chapter 5, a quantum version of the Borel-Weil theorem for $SU_q(3)$ is proved and is generalized to the case of $SU_q(n)$. Finally, in the last chapter the noncommutative complex structure of finite spaces is investigated. The space of holomorphic functions are determined and it is also proved that there is no holomorphic structure on the nontrivial vector bundle $\mathcal E_a\oplus \mathcal E_b$ over the space of two points $X=\{a,b\}$, where dim $\mathcal E_a=2$ and dim $\mathcal E_b=1$