Minimum time and minimum energy point-to-point trajectories for an industrial robot of the type Manutec r3 are computed subject to state constraints on the angular velocities. The numerical solutions of these optimal control problems are obtained in an efficient way by a combination of a direct collocation and an indirect multiple shooting method. This combination links the benefits of both approaches: A large domain of convergence and a highly accurate solution. The numerical results show that the constraints on the angular velocities become active during large parts of the time optimal motion. But the resulting stress on the links can be significantly reduced by a minimum energy trajectory that is only about ten percent slower than the minimum time trajectory. As a by-product, the reliability of the direct collocation method in estimating adjoint variables and the efficiency of the combination of direct collocation and multiple shooting is demonstrated. The highly accurate solutions reported in this paper may also serve as benchmark problems for other methods.
Abstract-We present a novel approach for analyzing two-dimensional (2D) flow field data based on the idea of invariant moments. Moment invariants have traditionally been used in computer vision applications, and we have adapted them for the purpose of interactive exploration of flow field data. The new class of moment invariants we have developed allows us to extract and visualize 2D flow patterns, invariant under translation, scaling, and rotation. With our approach one can study arbitrary flow patterns by searching a given 2D flow data set for any type of pattern as specified by a user. Further, our approach supports the computation of moments at multiple scales, facilitating fast pattern extraction and recognition. This can be done for critical point classification, but also for patterns with greater complexity. This multi-scale moment representation is also valuable for the comparative visualization of flow field data. The specific novel contributions of the work presented are the mathematical derivation of the new class of moment invariants, their analysis regarding critical point features, the efficient computation of a novel feature space representation, and based upon this the development of a fast pattern recognition algorithm for complex flow structures.
This paper discusses the optimal solution of Mayer’s problem for globally feedback linearizable time-invariant systems subject to general nonlinear path and actuator constraints. This class of problems includes the minimum time problem, important for engineering applications. Globally feedback linearizable nonlinear systems are diffeomorphic to linear systems that consist of blocks of integrators. Using this alternate form, it is proved that the optimal solution always lies on a constraint arc. As a result of this optimal structure of the solution, efficient numerical procedures can be developed. For a single input system, this result allows to characterize and build the optimal solution. The associated multi-point boundary value problem is then solved using direct solution techniques. [S0022-0434(00)02002-5]
Real-time collision-free trajectory control is dealt with by semi- infinite optimization techniques. This allows an optimal control problem incorporating a robotlobstacle distance function for col lision detection to be reduced to a finite-dimensional parameter- optimization problem. This reduced problem can be solved effi ciently by the numerical parameter-optimization method of sequen tial quadratic programming. In the case of a time-varying robot environment, a series of such optimization problems is solved in an iterative time frame, constituting a real-time optimization loop.
A method is addressed for real-time dynamics trajectory optimization of kinematically redundant manipulators. The considered task of trajectory planning is to teach interactively position and orientation of the tool-centerpoint frame, which is fixed in the manipulator hand. Thereby the manipulator has to autonomously preserve explicitly formulated kinematic constraints such as moving obstacle avoidance, singularity avoidance, and box-constraints on joint positions as well as dynamic constraints like box-constraints on joint velocities, accelerations and motor torques. The k e y idea is to transform the resulting overall motion planning problem into a timed series of point-to-point trajectory planning problems, which, in turn, may be formulated as parameter optimization problems, that can be eficiently solved in real-time b y the numerical method of sequential quadratic programming. Since the approach does not require an inverse kinematics formulation it is feasible for manipulators with redundant kinematics.
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