Kurt Marti

Real-time Robust Optimal Trajectory Planning of Industrial Robots

The standard engineering approach for optimal control of industrial or service robots assumes that all model parameters are exactly known, but due to stochastic variations of the material, manufacturing errors, modeling errors and stochastic variations of the workspace environment (e.g. stochastic payload) they need to be modeled via stochastic methods instead. This leads to optimal control/ variational problems under stochastic disturbances, that can not be solved directly but have to be replaced by suitable substitute problems from Stochastic Optimization that can be solved approximately by means of a reduction to a finite nonlinear program. Due to the short cycle times of robots these nonlinear programs have to be solved in real-time in order to update the optimal controls, whenever new information about the uncertain parameters involved is available. A task impossible because of the complexity of the control problem. Hence, a sensitivity analysis based real-time solution strategy is presented, that takes advantage of an optimal reference solution calculated before-hand online and determines a neighboring optimal solution. Furthermore, a feed-back controller obtained by linearization of the robots dynamic equation is introduced to study the closed-loop robots performance based on deterministic and stochastic model descriptions.

Stochastic optimal open-loop feedback control

Considering a dynamic control system with random model parameters and using the stochastic Hamilton approach stochastic open-loop feedback controls can be determined by solving a two-point boundary value problem (BVP) that describes the optimal state and costate trajectory. In general an analytical solution of the BVP cannot be found. This paper presents two approaches for approximate solutions, each consisting of two independent approximation stages. One stage consists of an iteration process with linearized BVPs that will terminate when the optimal trajectories are represented. These linearized BVPs are then solved by either approximation fixed-point equations (first approach) or Taylor-Expansions in the underlying stochastic model parameters (second approach). This approximation results in a deterministic linear BVP, which can be handled by solving a matrix Riccati differential equation.

Adaptive Optimal Stochastic Trajectory Planning

In Optimal Stochastic Trajectory Planning of industrial or service robots the problem can be modelled by a variational problem under stochastic disturbances that compared to ordinary deterministic engineering techniques also accounts for stochastic model parameters. Using stochastic optimisation theory, this variational problem is transformed into a nonlinear mathematical program, that can be solved by means of standard optimisation routines like SQP. However, these methods are not applicable in the on-line control process of robots, since they are not capable of solving mathematical programs in real-time. Hence, Neural Networks are trained based on solutions obtained from a standard optimisation algorithm.

Expected Total Cost Minimum Design of Plane Frames

Yield stresses, allowable stresses, moment capacities (plastic moments with respect to compression, tension and rotation), applied loadings, cost factors, manufacturing errors, etc., are not given fixed quantities in structural analysis and optimal design problems, but must be modeled as random variables with a certain joint probability distribution. Problems from plastic analysis and optimal plastic design are based on the convex yield (feasibility) criterion and the linear equilibrium equation for the stress (state) vector.

Optimal Design of Regulators

The optimal design of regulators is often based on the use of given, fixed nominal values of initial conditions, pay loads and other model parameters.

Optimal Control Under Stochastic Uncertainty

Optimal control and regulator problems arise in many concrete applications (mechanical, electrical, thermodynamical, chemical, etc.) are modeled [5, 63, 136, 155] by dynamical control systems obtained from physical measurements and/or known physical laws.

Adaptive Optimal Stochastic Trajectory Planning and Control (AOSTPC)

An industrial, service or field robot is modeled mathematically by its dynamic equation, being a system of second order differential equations for the robot or configuration coordinates q = (q 1, …, q n )′ (rotation angles in case of revolute links, length of translations in case of prismatic links), and the kinematic equation, relating the space {q} of robot coordinates to the work space {x} of the robot.

Maximum Entropy Techniques

According to the considerations in the former chapters, in the following we suppose that v = v(ω, x) denotes the costs or the loss arising in a decision or design problem if the action or design x ∈ D is taken, and the elementary event \(\tilde{\omega }=\omega\) has been realized.

Special Issue: Civil-Comp

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Editorial of - "Special Issue: Civil-Comp"

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