Navigation function

Navigation function usually refers to a function of position, velocity, acceleration and time which is used to plan robot trajectories through the environment. Generally, the goal of a navigation function is to create feasible, safe paths that avoid obstacles while allowing a robot to move from its starting configuration to its goal configuration.

Navigation function usually refers to a function of position, velocity, acceleration and time which is used to plan robot trajectories through the environment. Generally, the goal of a navigation function is to create feasible, safe paths that avoid obstacles while allowing a robot to move from its starting configuration to its goal configuration. \phi(x)=0\ \forall x \in X_g \phi(x) = \infty if and only if no point in {X_{g}} is reachable from x . For every reachable state, x \in X \setminus {X_{g}} , the local operator produces a state x' for which \phi(x') . {\varphi}(x) = \frac{\gamma_d (x)}{{{{\left[ {\gamma_d^{K}(x) + \beta \left( x \right)} \right]}^{\frac{1}{K}}}}} where K is a predefined constant like in the classical navigation function, which ensures the Morse nature of the function. \gamma_d(x) is the distance to the target position , and \beta \left( x \right) takes into account all obstacles, defined as \beta \left( x \right) = \prod\limits_{i = 0}^ {{\beta _i}\left( x \right)} where \beta _i(x) is based on the probability for a collision at location x . The probability for a collision is limited by a predetermined value \Delta , meaning: \beta _i (x) = \Delta- p^i\left( x \right) and, \beta _0 (x)= - \Delta + p^0\left( x \right) where p^i(x) is the probability to collide with the i-th obstacle. A map \varphi is said to be a probabilistic navigation function if it satisfies the following conditions: It is a navigation function. The probability for a collision is bounded by a predefined probability \Delta . \text{minimize } J(x_{1:T},u_{1:T})=\int\limits_T L(x_t,u_t,t) dt \text{subject to } \dot{x_t} = f(x_t,u_t) whereby x is the state, u is the control to apply, L is a cost at a certain state x if we apply a control u , and f models the transition dynamics of the system. Applying Bellman's principle of optimality the optimal cost-to-go function is defined as \displaystyle \phi(x_t) = \min_{u_t \in U(x_t)} \Big\{ L(x_t,u_t) + \phi(f(x_t,u_t)) \Big\} Together with the above defined axioms we can define the optimal navigation function as \phi(x)=0\ \forall x \in X_g \phi(x) = \infty if and only if no point in {X_{G}} is reachable from x . For every reachable state, x \in X \setminus {X_{G}} , the local operator produces a state x' for which \phi(x') . \displaystyle \phi(x_t) = \min_{u_t \in U(x_t)} \Big\{ L(x_t,u_t) + \phi(f(x_t,u_t)) \Big\} Even if a navigation function is an example for reactive control, it can be utilized for optimal control problems as well which includes planning capabilities. ==References== Sources ==External links== NFsim: MATLAB Toolbox for motion planning using Navigation Functions.

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