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Parallel Parking Problem
The parallel parking problem is a motion planning problem in control theory and mechanics to determine the path a car must take to parallel park into a parking space. The front wheels of a car are permitted to turn, but the rear wheels must stay aligned. When a car is initially adjacent to a parking space, to move into the space it would need to move in a direction perpendicular to the allowed path of motion of the rear wheels. The admissible motions of the car in its configuration space are an example of a nonholonomic system. See also * Automatic parking * Bicycle and motorcycle dynamics * Falling cat problem * Moving sofa problem In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region ... References * . * . Control theory {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motion Planning
Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used in computational geometry, computer animation, robotics and computer games. For example, consider navigating a mobile robot inside a building to a distant waypoint. It should execute this task while avoiding walls and not falling down stairs. A motion planning algorithm would take a description of these tasks as input, and produce the speed and turning commands sent to the robot's wheels. Motion planning algorithms might address robots with a larger number of joints (e.g., industrial manipulators), more complex tasks (e.g. manipulation of objects), different constraints (e.g., a car that can only drive forward), and uncertainty (e.g. imperfect models of the environment or robot). Motion planning has several robotics applications, such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system eng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects result in displacements, or changes of an object's position relative to its environment. Theoretical expositions of this branch of physics has its origins in Ancient Greece, for instance, in the writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo, Kepler, Huygens, and Newton laid the foundation for what is now known as classical mechanics. As a branch of classical physics, mechanics deals with bodies that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Parking
250px, Parallel-parked cars in Washington, D.C. 250px, A motorist gets assistance parallel-parking 250px, Parallel parking animation Parallel parking is a method of parking a vehicle parallel to the road, in line with other parked vehicles. Parallel parking usually requires initially driving slightly past the parking space, parallel to the parked vehicle in front of that space, keeping a safe distance, then followed by reversing into that space. Subsequent position adjustment may require the use of forward and reverse gears. Techniques Parallel parking is considered to be one of the most stressful and difficult skills for new drivers to learn. While parallel parking is a required part of most driving tests, several states in the US have dropped it as a requirement. Parallel parking enables the driver to park a vehicle in a smaller space than would be true of forward parking. Driving forward into a parking space on the side of a road is typically not possible unless two or mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonholonomic System
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is autonomous division of Newtonian mechanics. Details More precisely, a nonholonomic system, also called an ''anholonomic'' system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conservativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Parking
Automatic parking is an autonomous car-maneuvering system that moves a vehicle from a traffic lane into a parking spot to perform parallel, perpendicular, or angle parking. The automatic parking system aims to enhance the comfort and safety of driving in constrained environments where much attention and experience is required to steer the car. The parking maneuver is achieved by means of coordinated control of the steering angle and speed which takes into account the actual situation in the environment to ensure collision-free motion within the available space. The car is an example of a nonholonomic system where the number of control commands available is less than the number of coordinates that represent its position and orientation. One of the first assistance systems for car parking was manual. It used four jacks with wheels to raise the car and then move it sideways into the available parking space. This mechanical system was proposed in 1934, but was never offered on any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicycle And Motorcycle Dynamics
Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles and their components, due to the forces acting on them. Dynamics falls under a branch of physics known as classical mechanics. Bike motions of interest include balancing, steering, braking, accelerating, suspension activation, and vibration. The study of these motions began in the late 19th century and continues today. Bicycles and motorcycles are both single-track vehicles and so their motions have many fundamental attributes in common and are fundamentally different from and more difficult to study than other wheeled vehicles such as dicycles, tricycles, and quadracycles. As with unicycles, bikes lack lateral stability when stationary, and under most circumstances can only remain upright when moving forward. Experimentation and mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Der ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falling Cat Problem
The falling cat problem is a problem that consists of explaining the underlying physics behind the observation of the cat righting reflex. Although amusing and trivial to pose, the solution of the problem is not as straightforward as its statement would suggest. The apparent contradiction with the law of conservation of angular momentum is resolved because the cat is not a rigid body, but instead is permitted to change its shape during the fall owing to the cat's flexible backbone and non-functional collar-bone. The behavior of the cat is thus typical of the mechanics of deformable bodies. Several explanations have been proposed for this phenomenon since the late 19th century: * Cats rely on conservation of angular momentum. * The rotation angle of the front body is larger than that of the rear body. * The dynamics of the falling cat have been explained using the Udwadia–Kalaba equation. History The falling cat problem has elicited interest from scientists including G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moving Sofa Problem
In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region with legs of unit width. The area thus obtained is referred to as the ''sofa constant''. The exact value of the sofa constant is an open problem. The currently leading solution, by Joseph L. Gerver, has a value of approximately 2.2195 and is thought to be close to the optimal, based upon subsequent study and theoretical bounds. History The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966, although there had been many informal mentions before that date. Bounds Work has been done on proving that the sofa constant (A) cannot be below or above certain values (lower bounds and upper bounds). Lower An obvious lower bound is A \geq \pi/2 \approx 1.57. This comes from a sofa that is a half-disk of unit ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
