We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length width ratio is small, the shortest path can be computed in polynomial time.
 We present here a new and efficient algorithm for planning collision-free motion of a line segment (a rod or a “ladder”) in two-dimensional space amidst polygonal obstacles. The algorithm uses a different approach than those used in previous motion-planning techniques, namely, it calculates the boundary of the (three-dimensional) space of free positions of the ladder, and then uses this boundary for determining the existence of required motions, and plans such motions whenever possible. The algorithm runs in timeO(K logn) =O(n 2 logn) wheren is the number of obstacle corners and whereK is the total number of pairs of obstacle walls or corners of distance less than or equal to the length of the ladder. The algorithm has thus the same complexity as the best previously known algorithm of Leven and Sharir [5], but if the obstacles are not too cluttered together it will run much more efficiently. The algorithm also serves as an initial demonstration of the viability of the technique it uses, which we expect to be useful in obtaining efficient motion-planning algorithms for other more complex robot systems. Many types of cellular motility, including muscle contraction, are driven by the cyclical interaction of the motor protein myosin with actin filaments, coupled to the breakdown of ATP. It is thought that myosin binds to actin and then produces force and movement as it ‘tilts’ or ‘rocks’ into one or more subsequent, stable conformations1,2. Here we use an optical-tweezers transducer to measure the mechanical transitions made by a single myosin head while it is attached to actin. We find that two members of the myosin-I family, rat liver myosin-I of relative molecular mass 130,000 (Mr 130K) and chick intestinal brush-border myosin-I, produce movement in two distinct steps. The initial movement (of roughly 6 nanometres) is produced within 10 milliseconds of actomyosin binding, and the second step (of roughly 5.5nanometres) occurs after a variable time delay. The duration of the period following the second step is also variable and depends on the concentration of ATP. At the highest time resolution possible (about 1 millisecond), we cannot detect this second step when studying the single-headed subfragment-1 of fast skelet al muscle myosin II. The slower kinetics of myosin-I have allowed us to observe the separate mechanical states that contribute to its working stroke. Theoretical modeling and computer simulations of molecular motors provide insight that engineers can exploit to design and control artificial nanomotors.For obvious reasons, the study of molecular motors has been a traditional area of research in molecular cell biology and biochemistry, but in recent years, it has attracted physicists' as well as engineers. Exploring the design and mechanisms of these motors from an engineering perspective requires investigating their structure and dynamics using the fundamental principles of physics at the subcellular level. The insights gained from such fundamental research could also find practical applications in designing and manufacturing artificial nanomotors - motors whose typical size is usually in the rage of a few nanometers to a few tens of nanometers. In contrast to man-made macroscopically large motors, natural nanomotors have evolved over billions of years. While discussing the design, mechanism, and control of molecular motors, this article also compares their macroscopic counterparts to emphasize common features as well as differences.   We present a new roadmap for a rod-shaped robot operating in a three-dimensional workspace, whose configuration space is diffeomorphic to R3 X S2. This roadmap is called the rod hierarchical generalized Voronoi graph (rod-HGVG) and can be used to find a path between any two points in an unknown configuration space using only the sensor data. More importantly, the rod-HGVG serves as a basis for an algorithm to explore an unknown configuration space without explicitly constructing it. Once the rod-HGVG is constructed, the planner can use it to plan a path between any two configurations. One of the challenges in defining the roadmap revolves around a homotopy theory result, which asserts that there cannot be a one-dimensional deformation retract of a non-contractible space with dimension greater than two. Instead, we define an exact cellular decomposition on the free configuration space and a deformation retract in each cell (each cell is contractible). Next, we "connect" the deformation retracts of each of the cells using a roadmap of the workspace. We call this roadmap a piecewise retract because it comprises many deformation retracts. Exploiting the fact that the rod-HGVG is defined in terms of workspace distance measurements, we prescribe an incremental procedure to construct the rod-HGVG that uses the distance information that can be obtained from conventional range sensors. 1 Introduction and Overview.- 2 Configuration Space of a Rigid Object.- 3 Obstacles in Configuration Space.- 4 Roadmap Methods.- 5 Exact Cell Decomposition.- 6 Approximate Cell Decomposition.- 7 Potential Field Methods.- 8 Multiple Moving Objects.- 9 Kinematic Constraints.- 10 Dealing with Uncertainty.- 11 Movable Objects.- Prospects.- Appendix A Basic Mathematics.- Appendix B Computational Complexity.- Appendix C Graph Searching.- Appendix D Sweep-Line Algorithm.- References.