Many concrete problems from engineering, economics, operations research, etc., can be formulated by an optimization problem of the type 

$$\displaystyle\begin{array}{rcl} \min f_{0}(a,x)& &{}\end{array}$$

(1.1a)



$$\displaystyle\begin{array}{rcl} \mbox{ s.t.}& & {}\\ & & {}\\ \end{array}$$



$$\displaystyle\begin{array}{rcl} f_{i}(a,x)& \leq & 0,i = 1,\ldots,m_{f}{}\end{array}$$

(1.1b)



$$\displaystyle\begin{array}{rcl} g_{i}(a,x)& =& 0,i = 1,\ldots,m_{g}{}\end{array}$$

(1.1c)



$$\displaystyle\begin{array}{rcl} x \in D_{0}.& &{}\end{array}$$

(1.1d)

Stochastic optimization methods

The problem of adaptive trajectory planning for robots under stochastic uncertainty is considered, where new information about the robots and their environment is presented on-line. Solving the problem numerically by means of spline approximation and by applying the method of neural networks, the optimal control can be calculated in real-time. Some numerical results are presented.

Path Planning for Robots by Stochastic Optimization Methods

In optimal control of robots, the standard procedure is to determine first off-line an optimal open-loop control, using some nominal or estimated values of the model parameters, and to correct then the resulting increasing deviation of the actual trajectory or the actual performance of the system from the prescribed trajectory, from the prescribed values of performance, resp., by on-line measurement and control actions. However, on-line measurement and control actions are in general very expensive and time consuming. By adaptive optimal stochastic trajectory planning and control (AOSTPC) i.e., by incorporating into the control design the available a priori and sample information about the unknown model parameters using stochastic optimization methods, the mean absolute deviation between the actual and prescribed trajectroy can be reduced cosiderably. Hence, robust optimal controls are obtained. The corresponding feedforward and feedback (PD-) controls are derived by means of stochastic optimzation methods and by using stability requirements. Morever, analytical estimates are given for the reduction of the tracking error, hence, for the reduction of the on-line correction expenses by (AOSTPC).

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Adaptive Optimal Stochastic Trajectory Planning and Control (AOSTPC) for Robots

In order to reduce large online measurement and correction expenses, the a priori informations on the random variations of the model parameters of a robot and its working environment are taken into account already at the planning stage. Thus, instead of solving a deterministic path planning problem with a fixed nominal parameter vector, here, the optimal velocity profile along a given trajectory in work space is determined by using a stochastic optimization approach. Especially, the standard polygon of constrained motion-depending on the nominal parameter vector-is replaced by a more general set of admissible motion determined by chance constraints or more general risk constraints. Robust values (with respect to stochastic parameter variations) of the maximum, minimum velocity, acceleration, deceleration, resp., can be obtained then by solving a univariate stochastic optimization problem Considering the fields of extremal trajectories, the minimum-time path planning problem under stochastic uncertainty ca...

Path planning for robots under stochastic uncertainty

Stochastic Optimization Methods in Robust Adaptive Control of Robots

Efficient control strategies of robots should cause only low on-line correction expenses. Hence, the mostly available statistical and a priori informations about the random parameters and disturbances of the underlying mechanical system and its environment should be considered already for off-line programming of robots. Measuring the violations of the basic mechanical conditions by means of expected penalty costs, a stochastic optimization problem is obtained for the computation of an optimal open-loop control. The stochastic optimization problem can be solved — after discretization — by parameter optimization.

Optimal Trajectory Planning for Robots under the Consideration of Stochastic Parameters and Disturbances

To obtain an efficient open-loop control strategy for robots under uncertainty, the statistical and a priori informations about the random parameters and disturbances of the underlying mechanical system and its environment are taken into account for off-line programming. The influence of our open-loop strategy on the online feed-back control is compared with that of other open-loop strategies. >

S. Qu

Papers

Optimal Trajectory Planning for Robots under the Consideration of Stochastic Parameters and Disturbances

Computation of an efficient open-loop strategy and its influence on feed-back control