Robotics and Automation Expert

Directed Search Constraint Tracking (page 3)

Figure 2. shows the hypercube for a robot with three degrees of redundancy. There are 2n points on the faces of the cube, ~ points at the vertices, and 3í~ points in all. Respectively, we call these the simple, 
factorial, and exhaustive exploration patterns. For more than four or five degrees of redundancy, the 
computational expense associated with current computing hardware prohibits the exhaustive pattern in real-
time applications. 

Annealing describes a process of heating a material to an elevated temperature and then cooling it very 
slowly. The slow cooling allows the material to reach a low energy state in which it is relatively ductile. 
With no intelligence or systematic strategy, some materials minimize energy state during the slow cooling. 
Simulated annealing is an approximation of this natural process carried out on a computer and is based on 
the Boltzmann probability distribution. 

In this equation, E is the energy of the system, k is Boltzmannís constant, and T is the temperature. 
Essentially, the Boltzmann probability distribution states that a systemís energy is probabilistically 
distributed depending upon the temperature. As the temperature increases, the probability of the system 
assuming a higher energy state increases. As the temperature is lowered, the odds of the system leaving a lower energy state decrease. Each configuration option corresponds to an energy state. Because simulated annealing algorithms sometimes leave lower energy states for higher ones, they can escape from local minima. Simulated annealing algorithms typically include a method of generating random changes in the systemís configuration. The random changes represent trial configurations evaluated using the Boltzmann probability distribution. If the distribution indicates, the system assumes the trial configuration; otherwise it is  discarded.

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