A Topological Approach to Globally-Optimal Redundancy Resolution with Dynamic Programming

Dynamic programming techniques have proven much more flexible than calculus of variations and other techniques in performing redundancy resolution through global optimization of performance indices. When the state and input spaces are discrete, and the time horizon is finite, they can easily accommodate generic constraints and objective functions and find Pareto-optimal sets. Several implementations have been proposed in previous works, but either they do not ensure the achievement of the globally optimal solution, or they have not been demonstrated on robots of practical relevance. In this communication, recent advances in dynamic programming redundancy resolution, so far only demonstrated on simple planar robots, are extended to be used with generic kinematic structures. This is done by expanding the Robot Operating System (ROS) and proposing a novel architecture meeting the requirements of maintainability, re-usability, modularity and flexibility that are usually required to robotic software libraries. The proposed ROS extension integrates seamlessly with the other software components of the ROS ecosystem, so as to encourage the reuse of the available visualization and analysis tools. The new architecture is demonstrated on a 7-DOF robot with a six-dimensional task, and topological analyses are carried out on both its state space and resulting joint-space solution.

The redundancy resolution schemes based on the optimization of an integral performance index are investigated from the topological point of view. The topological notions of self-motion manifold, C-path-homotopy and extended aspect are clarified in relation to the limitations of the necessary conditions of optimality provided by calculus of variations. On one hand, they do not guarantee the achievement of the optimal solution, and on the other hand, they translate into a two-point boundary value problem (TPBVP), whose resolution, under certain circumstances, may not lead to a feasible solution at all. In response to the limitations of calculus of variations, a dynamic-programming-inspired formalism is developed, which is based on the discretization of the state space and on its representation in the form of multiple grids. Building upon the topological analysis, effective algorithms are designed that are able to find the optimal solution in any condition, across all C-path homotopy classes and self-motion manifolds, with no limitation due to the passage through singularities. Moreover, if the grids are representative of the manipulator’s extended aspects, the topological notion of the transitional point can be used to reduce the computational complexity of the optimal redundancy resolution algorithm. The results are demonstrated on a canonical 4R planar robot in two different scenarios.

Problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals. In this way, methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems. The result is a necessary and sufficient condition of optimality that generalizes well-known sufficient conditions, referred to as verification theorems, in dynamic programming; as a byproduct, we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the Hamilton–Jacobi equation. It is a striking illustration of the wide range of problems to which convex analysis, and, in particular, convex duality, is applicable. The approach, applied to parametric problems in the calculus of variations, was pioneered by L. C. Young [Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, PA, 1969]. As recent work has shown, however, it is equally fruitful when applied in optimal control. This paper, which is expository, offers a self-contained treatment of the application of methods of convex duality to general nonlinear problems in deterministic optimal control. At the same time, it provides extensions of previously published results in several directions. A simple proof is given of the main “convex closure” theorem relating generalized flows and relaxed arcs; this is based on mollification techniques recently developed by Fleming and Vermes [SIAM J. Control Optim., 27 (1989), pp. 1136–1155] for constructing smooth subsolutions to the Hamilton–Jacobi equation.

The altimetry precision of conventional spaceborne Global Navigation Satellite Systems Reflectometry (GNSS-R) is limited, and the error models are complicated. To compensate for the shortcomings of conventional methods, we present a new grid search multihidden layer neural network feature optimization method (GSMHLFO) for sea surface height (SSH) retrieval. Firstly, the GSMHLFO is constructed by combining the multihidden layer neural network, feature engineering, and a grid search algorithm. Moreover, the retrieval performance of the GSMHLFO and its sensitivity to various features are analyzed. By analyzing 14 feature sets with different information details, we concluded that the elevation, signal-to-noise ratio (SNR), atmospheric delay, and ocean wind speed can provide essential contributions to the SSH retrieval based on GSMHLFO. Secondly, the Technical University of Denmark 18 mean sea surface (DTU18 MSS), which is corrected by the TPXO8 global tide model, was used to verify the GSMHLFO. The number of hidden layers and neurons was optimized using the grid search algorithm. The experimental results show that the proposed GSMHLFO with four hidden layers and 200 neurons per layer has a better retrieval performance. Compared with DTU18, the mean absolute difference (MAD), the root mean square error (RMSE), and the Pearson correlation coefficient (PCC) equal 4.23 m, 5.94 m, and 0.98, respectively. The retrieval precision obtained is significantly improved compared to that reported in the literature for the TDS-1 SSH retrieval. Finally, the retrieval performance of the GSMHLFO and the traditional HALF single-point retracking method were compared. The precision of GSMHLFO is higher than that of traditional retracking method according to MAD, RMSE, and PCC, which are increased by 32.86, 25.00, and 8.99%. The GSMHLFO will provide innovative theoretical and methodological support for the high-precision SSH retrieval of GNSS-R altimetry satellites in the future.

The optimal control problem Consider a fish stock which has some natural rate of growth and which is harvested. Too much harvesting could endanger the survival of the fish, too little and profits are forgone. Of course, harvesting takes place over time. The obvious question is: ‘what is the best harvesting rate, i.e., what is the optimal harvesting?’ The answer to this question requires an optimal path or trajectory to be identified. ‘Best’ itself requires us to specify a criterion by which to choose between alternative paths. Some policy implies there is a means to influence (control) the situation. If we take it that x ( t ) represents the state of the situation at time t and u ( t ) represents the control at time t , then the optimal control problem is to find a trajectory { x ( t )} by choosing a set { u ( t )} of controls so as to maximise or minimise some objective that has been set. There are a number of ways to solve such a control problem, of which the literature considers three: (1) Calculus of variations (2) Dynamic programming (3) Maximum principle. In this chapter we shall deal only with the third, which now is the dominant approach, especially in economics. This approach is based on the work of Pontryagin et al . (1962), and is therefore sometimes called the Pontryagin maximum principle .

In Chapter 5, we have shown how Optimal Control could be developed from the Calculus of Variations, presented Pontryagin’s Maximum Principle and discussed the Transversality conditions for the various cases, both in finite and infinite horizon problems. The second variations and sufficiency conditions have also been examined. In this chapter, we shall continue the exposition of the Optimal Control theory, concentrating on the point, differential equation and isoperimetric equality constraints as well as the control and state variables inequality constraints. Some economic applications will be presented. Finally, Dynamic Programming, Hamilton Jacobi equation and Euler equations will be briefly related to one another. We shall continue to use the results obtained from the Calculus of Variations to save lengthy discussions.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

Publication in the conference proceedings of EUSIPCO, Poznan, Poland, 2007

Sharia stocks are one of the investment instruments in the Islamic capital market. In the capital market, it is known that stock prices are very volatile. This makes investors need to carry out a strategy for making the right decision in investing, one of which can be done by predicting stock prices. In this study, predictions were made using historical data on the closing price of Islamic shares of PT. Telkom Indonesia Tbk with the Bidirectional Long Short Term Memory (BiLSTM) method. In building the best prediction model, it is necessary to choose the right parameters and one way to do this is to use the grid search algorithm. Based on the results of the test analysis, it was found that the smallest Mean Absolute Percentage Error (MAPE) value was found in the BiLSTM model in the distribution of data with a percentage of 90% training data and 10% testing data and parameter values obtained based on parameter tuning using grid search, including the number of neurons 25, 100 epochs, 4 batches, and 0.2 dropouts. The MAPE obtained in this study was 10.83% and based on the scale on the MAPE value criteria, this shows that the resulting prediction model is accurate. As for the test results from the comparisons made on the BiLSTM and LSTM models using grid search as a tuning parameter and models without using a grid search or it can be called a trial and error approach as a tuning parameter, it is found that the model with better predictive performance is found in BiLSTM using a grid search. compared to other models.

This study investigates data standardization methods based on the grid search (GS) algorithm for energy load forecasting, including zero-mean, min-max, max, decimal, sigmoid, softmax, median, and robust, to determine the hyperparameters of deep learning (DL) models. The considered DL models are the convolutional neural network (CNN) and long short-term memory network (LSTMN). The procedure is made over (i) setting the configuration for CNN and LSTMN, (ii) establishing the hyperparameter values of CNN and LSTMN models based on epoch, batch, optimizer, dropout, filters, and kernel, (iii) using eight data standardization methods to standardize the input data, and (iv) using the GS algorithm to search the optimal hyperparameters based on the mean absolute error (MAE) and mean absolute percent error (MAPE) indexes. The effectiveness of the proposed method is verified on the power load data of the Australian state of Queensland and Vietnamese Ho Chi Minh city. The simulation results show that the proposed data standardization methods are appropriate, except for the zero-mean and min-max methods.

A model to determine simultaneously the optimal thinning schedule and rotation is presented in two forms. If time is assumed to be continuous, the problem can be formulated in the calculus of variation form. In this form no closed form solution has been obtained. If the model is recastin dynamic programming form, a numeric solution can easily be obtained, without imposing any limitations on the form of the relevant revenue and cost functions. Thus the traditional Faustmann formula is generalized considerably. At the same time the close connection between the calculus ofvariations and dynamic programming is illustrated. Forest Sci. 17:333-339.

Due to a continuous change in people’s lifestyle and dietary habits, gastrointestinal diseases are on the increase, with dietary changes being a major contributor to a variety of bowel problems. Around two million people around the world die due to gastrointestinal (GI) diseases. Endoscopy is a medical imaging technology helpful in diagnosing gastrointestinal diseases like polyps and esophagitis. Its manual diagnosis is time-consuming; hence, computer-aided techniques are now widely used for accurate and fast GI disease diagnosis. In this paper, the Kvasir dataset of 4000 endoscopic images, comprising 500 images of each of the eight gastrointestinal tract disease classes have been classified using seven grid search fine-tuned transfer learning models. The fine-tuned transfer learning models employed in this paper are ResNet101, InceptionV3, InceptionResNetV2, Xception, DenseNet121, MobileNetV2, and ResNet50. The grid search algorithm has been used to determine the architectural and fine-tuning hyperparameters. The fine-tuned ResNet101 model performed the best, with a learning rate 0.001 and a batch size of 32 for the SGD optimizer at 40 epochs. These hyperparameters were optimized through grid search along with new set of layers added to the model. The newly added layers include one flatten layer, two dropout layers and five dense layers optimized using grid search. The grid search fine-tuned ResNet101 model obtained an accuracy of 0.90, a precision of 0.92, a recall of 0.92, and an f1-score of 0.91. Further, the grid search fine-tuned ResNet101 model was integrated with an attention mechanism to enhance performance by focusing on essential image features, notably in medical imaging where some regions may contain vital diagnostic information. The proposed grid search fine-tuned and attention mechanism integrated ResNet101 model achieved an accuracy of 0.935, precision of 0.93, recall of 0.94 and an f1-score of 0.93.

Active Contours represent a common Pattern Recognition technique. Classical active contours are based on different methodologies (variational calculus, dynamic programming and greedy algorithm). This paper reviews the most frequently used active contours in a practical application, comparing weights, manually obtained by food technology experts, to volumes, automatically achieved by computer vision results. An experiment has been designed to recognize muscles from Magnetic Resonance (MR) images of Iberian ham at different maturation stages in order to calculate their volume change, using different active contour approaches. The sets of results are compared with the physical data. The main conclusions of the paper are the excellent correlation established between the data obtained with these three non-destructive techniques and the results achieved using the traditional destructive methodologies, as well as the real viability of the active contours to recognize muscles in MR images.

An approach to robot-path planning is developed by considering both the traveling distance and the safety of the robot. A computationally-efficient algorithm is developed to find a near-optimal path with a weighted distance-safety criterion by using a variational calculus and dynamic programming (VCDP) method. The algorithm is readily applicable to any factory environment by representing the free workspace as channels. A method for deriving these channels is also proposed. Although it is developed mainly for two-dimensional problems, this method can be easily extended to a class of three-dimensional problems. Numerical examples are presented to demonstrate the utility and power of this method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An effective three-stage hybrid optimization method for source-network-load power generation of cascade hydropower reservoirs serving multiple interconnected power grids