Forward Dynamics Approach Research Articles

Dynamic production-pricing strategies for multi-generation products under uncertainty

Due to rapid advances in technology and design, firms periodically release new generations of electronic products such as mobile phones and computers. In order to increase product variability, firms may wish to develop a multiple-generation product line rather than replace the older versions with new ones. However, when multiple generations are available in the market, different generations compete with each other as well as other products in the market. Firms need to take joint decisions for the purpose of inventory management and dynamic pricing of multiple generations to tackle impact of uncertain demand and market competition. In this paper, we present a dynamic joint production-pricing decision model to obtain optimal strategies for a firm selling multiple generations of a product. We account for the internal competition among multiple generations by evaluating customer choices. In order to tackle a curse of dimensionality, we introduce a forward dynamic programming approach for approximately solving the joint production-pricing problem. We also propose a two-stage heuristic algorithm as an alternative solution approach. Different pricing rules determined from the abridged model and a price list derived by theoretical bounds are integrated to improve further computational performance of the solution approaches. We design computational experiments to illustrate effectiveness and efficiency of these approximate methods and show the benefit of joint decision-making process in multi-generation product line.

Task Space Control of Articulated Robot Near Kinematic Singularity: Forward Dynamics Approach

In this study, a forward dynamics-based control (FDC) framework is proposed for task space control of a non-redundant robot manipulator. The FDC framework utilizes forward dynamic robot simulation and an impedance controller to solve the inverse kinematics problem. For the practical use of the proposed control framework, the accuracy, robustness, and stability of robot motion are considered. Taking advantage of the stability of the implicit Euler method, a high-gain PD controller enables accurate end-effector pose tracking in the task space without losing stability even near the kinematic singularities. Also, the robustness of the controller is enhanced by borrowing the structure of the nonlinear robust internal-loop compensator. Lastly, the selective joint damping injection and spring force saturation are applied to the impedance controller so that the robot motion can always stay within the given dynamic constraints. This study suggests a new, effective solution for the kinematic singularity problem of non-redundant robot manipulators.

Optimal Control of Human Muscle‐Actuated Motion

AbstractThe question how the brain controls locomotion is a highly treated one. This topic is taken up in this study by means of human computational models, simulating the human musculoskeletal system actuated by advanced Hill‐type muscle models. A forward dynamic approach is considered allowing to simulate human motion independently from pre‐measured trajectories and enabling to resemble the control of the central nervous system – the motor command selection. It is suggested that movement paths arise implicitly through optimisation [1,4]. Hence, stimulation patterns passing through activation dynamics are determined by means of constrained optimal control problems with physiologically motivated objective functions. One advantage concerning the implementation within Neweul‐M2 is the possibility to apply well‐elaborated and tested optimisation algorithms available with Matlab to solve optimal control problems, enabling to employ the novel non‐linear model predictive control (NMPC). The introduced framework is applied to treat different biomechanical movement scenarios including reaching scenarios of the arm which are validated against experimental data from [2–4]. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A simple controller to overcome the lack of correlation between forward and inverse dynamic analysis of human motion tasks

The majority of biomechanical analyses of human motions, including those with musculoskeletal models, use inverse dynamic approaches due to its ability to deal with experimentally acquired kinematic and kinetic data. Yet, a forward dynamic approach can be more powerful and provide better insights on the transmission of forces in the internal biomechanical systems and structures of the human body. Although both approaches may use the same biomechanical model the results achieved do not necessarily correlate with each other. The aim of this study is to demonstrate the source of the lack of correlation between inverse and forward dynamics methodologies providing, in the process, insights on how to overcome such differences. Two types of problems involving the biomechanics of the spatial human motion are used to evaluate the correlation between the forward and inverse dynamic approaches: a gait analysis of a deterministic biomechanical model of the lower limbs, and, a full musculoskeletal model of the upper limb, which is characterised by the solution of a redundant muscle force sharing problem. For that purpose, an inverse dynamic model is applied to estimate the forces responsible for two experimentally acquired motions that are, afterwards, given as input to the forward dynamics model, which is used, in turn, to compute the kinematics of the biomechanical model. The comparison between the reference kinematics, acquired experimentally, and that resulting from the forward dynamic analysis supports that a lack of correlation between the inverse and forward dynamic analysis is always observed. It is proposed here, and demonstrated, that a controller implemented in a feedback loop is able to enhance numerical stability of the forward dynamics solution, leading to the ability of the forward dynamics approach to successfully simulate the acquired motions.

How muscle relaxation and laterotrusion resolve open locks of the temporomandibular joint. Forward dynamic 3D-modeling of the human masticatory system

Patients with symptomatic hypermobility of the temporomandibular joint report problems with the closing movement of their jaw. Some are even unable to close their mouth opening wide (open lock). Clinical experience suggests that relaxing the jaw muscles or performing a jaw movement to one side (laterotrusion) might be a solution. The aim of our study was to assess the potential of these strategies for resolving an open lock and we hypothesised that both strategies work equally well in resolving open locks. We assessed the interplay of muscle forces, joint reaction forces and their moments during closing of mouth, following maximal mouth opening. We used a 3D biomechanical model of the masticatory system with a joint shape and muscle orientation that predispose for an open lock. In a forward dynamics approach, the effect of relaxation and laterotrusion strategies was assessed. Performing a laterotrusion movement was predicted to release an open lock for a steeper anterior slope of the articular eminence than relaxing the jaw-closing muscles, herewith we rejected our hypothesis. Both strategies could provide a net jaw closing moment, but only the laterotrusion strategy was able to provide a net posterior force for steeper anterior slope angles. For both strategies, the temporalis muscle appeared pivotal to retrieve the mandibular condyles to the glenoid fossa, due to its’ more dorsally oriented working lines.

A forward dynamics simulation of human lumbar spine flexion predicting the load sharing of intervertebral discs, ligaments, and muscles.

Determining the internal dynamics of the human spine's biological structure is one essential step that allows enhanced understanding of spinal degeneration processes. The unavailability of internal load figures in other methods highlights the importance of the forward dynamics approach as the most powerful approach to examine the internal degeneration of spinal structures. Consequently, a forward dynamics full-body model of the human body with a detailed lumbar spine is introduced. The aim was to determine the internal dynamics and the contribution of different spinal structures to loading. The multi-body model consists of the lower extremities, two feet, shanks and thighs, the pelvis, five lumbar vertebrae, and a lumped upper body including the head and both arms. All segments are modelled as rigid bodies. 202 muscles (legs, back, abdomen) are included as Hill-type elements. 58 nonlinear force elements are included to represent all spinal ligaments. The lumbar intervertebral discs were modelled nonlinearly. As results, internal kinematics, muscle forces, and internal loads for each biological structure are presented. A comparison between the nonlinear (new, enhanced modelling approach) and linear (standard modelling approach, bushing) modelling approaches of the intervertebral disc is presented. The model is available to all researchers as ready-to-use C/C++ code within our in-house multi-body simulation code demoa with all relevant binaries included.

Appropriately complex modelling of healthy human walking

Modelling human gait has become an invaluable tool in a wide range of fields such as robotics and rehabilitation. With progress in computing, model complexity has advanced quickly but nevertheless, the contributions of incremental increases in model complexity are poorly understood. This thesis addresses this through a series of modelling studies. The first investigation examined the advantages and disadvantages of inverted pendulum (IP) models of walking, using a forward dynamics approach, by comparing to a normal set of experimental gait data. It was shown that the biggest failing of these models is their inability to adequately simulate double stance. The second investigation sought to highlight the effects of additional model complexities on the kinematics and kinetics, using optimisation. The additions, added one-by-one, were a knee joint, an ankle and static foot, a moving foot and a swing leg. The presence of a knee joint and an ankle moment were shown to be largely responsible for the initial peak in the vertical ground force reaction (GRF) curve. The second peak in this curve was achieved through a combination of heel rise and the presence of a swing leg. This gave mathematical evidence for the true determinants of human gait. A double support model was produced next, using a novel method to constrain both feet to the ground and calculate the GRF distribution. This was run in conjunction with the best single support model to simulate a whole gait cycle. Despite the problem of discontinuities at the transitions between double and single support, the whole gait cycle simulation had mean kinematic and mean GRF errors of less than a single standard deviation from the normal experimental data set. The final study collected gait and anthropometric data from ten subjects, which was then applied to the full gait cycle model. The model was shown to be adaptable to different people; a property that would be important for any computational model to be used in clinical assessment and diagnostics.

Biomechanical modeling of open locks of the human temporomandibular joint

BackgroundPatients with hypermobility of the temporomandibular joint may have problems closing their mouth after opening widely. In the worst case, the mandibular condyles become trapped in front of the articular eminences and the jaw muscles cannot reposition them into the fossae (open lock). The difference in ease of closing the jaw between patients and non-patients is presently not well understood. MethodsWide opening and subsequent jaw closing were simulated with a biomechanical model in a forward dynamics approach. The effect of anterior slope angle and orientation of jaw-closing muscles on condylar travel was determined. FindingsThe mandibular condyles traveled anterior of the eminences and back into the fossae uneventfully with backwardly oriented jaw closers and eminences with a gentle anterior slope. However, combinations of relatively forward oriented jaw closers and a steep anterior slope caused the condyles to continue traveling anteriorly upon jaw-closing attempts, ending in an open lock position. InterpretationOur results indicate that for the masticatory system to reach an open lock, various unfavorable combinations of jaw-closer orientation and anterior slope angle exist within normal physiological ranges. These findings could be relevant for maxillofacial surgeons, both for the diagnostic process and for clinical decisions, regarding patients suffering from open locks.

Effect of Load Positioning on the Kinematics and Kinetics of Weighted Vertical Jumps

One of the most popular exercises for developing lower-body muscular power is the weighted vertical jump. The present study sought to examine the effect of altering the position of the external load on the kinematics and kinetics of the movement. Twenty-nine resistance-trained rugby union athletes performed maximal effort jumps with 0, 20, 40, and 60% of their squat 1 repetition maximum (1RM) with the load positioned (a) on the posterior aspect of the shoulder using a straight barbell and (b) at arms' length using a hexagonal barbell. Kinematic and kinetic variables were calculated through integration of the vertical ground reaction force data using a forward dynamics approach. Performance of the hexagonal barbell jump resulted in significantly (p < 0.05) greater values for jump height, peak force, peak power, and peak rate of force development compared with the straight barbell jump. Significantly (p < 0.05) greater peak power was produced during the unloaded jump compared with all trials where the external load was positioned on the shoulder. In contrast, significantly (p < 0.05) greater peak power was produced when using the hexagonal barbell combined with a load of 20% 1RM compared with all other conditions investigated. The results suggest that weighted vertical jumps should be performed with the external load positioned at arms' length rather than on the shoulder when attempting to improve lower-body muscular performance.

The effect of bilateral asymmetry of muscle strength on the height of a squat jump: A computer simulation study

The aim of this study was to examine the effect of bilateral asymmetry of muscle strength on maximal height of the squat jump. A computer simulation technique was used to develop two kinds of 3D human lower limb musculoskeletal model (model-symmetry and model-asymmetry). The total muscle strength of the two models was set to be identical. Bilateral muscle strength was equal in the model-symmetry simulation, while the model-asymmetry simulation was performed with a 10% bilateral strength asymmetry. A forward dynamics approach was used to simulate squat jumps. The squat jumps were successfully generated, producing jump heights of 0.389 m for model-symmetry and 0.387 m for model-asymmetry. The small difference in height (0.5%) indicated that the effect of the 10% bilateral asymmetry of muscle strength on jump height is negligible. With model-asymmetry, the strong leg compensated for the muscle strength deficit of the weak leg. Importantly, the mono-articular and large extensor muscles of the hip and knee joint of the strong leg, including the gluteus maximus, adductor magnus, and vasti, compensated for the muscle strength deficit of the weak leg.

Supplier selection and order lot sizing using dynamic programming

In this paper, we consider a multi-period integrated supplier selection and order lot sizing problem where a single buyer plans to purchase a single product in multiple periods from several qualified suppliers who are able to provide the required product with the needed quality in a timely manner. Product price and order cost differs among different suppliers. Buyer’s demand for the product is deterministic and varies for different time periods. The problem is to determine how much product from which supplier must be ordered in each period such that buyer’s demand is satisfied without violating some side constraints. We have developed a mathematical programming model to deal with this problem, and proposed a forward dynamic programming approach to obtain optimal solutions in reasonable amount of time even for large scale problems. Finally, a numerical example is conducted in which solutions obtained from the proposed dynamic programming algorithm is compared with solutions from the branch-and-bound algorithm. Through the numerical example we have shown the efficiency of our algorithm.

The kinematic determinants of anuran swimming performance: an inverse and forward dynamics approach

The aims of this study were to explore the hydrodynamic mechanism of Xenopus laevis swimming and to describe how hind limb kinematics shift to control swimming performance. Kinematics of the joints, feet and body were obtained from high speed video of X. laevis frogs (N=4) during swimming over a range of speeds. A blade element approach was used to estimate thrust produced by both translational and rotational components of foot velocity. Peak thrust from the feet ranged from 0.09 to 0.69 N across speeds ranging from 0.28 to 1.2 m s(-1). Among 23 swimming strokes, net thrust impulse from rotational foot motion was significantly higher than net translational thrust impulse, ranging from 6.1 to 29.3 N ms, compared with a range of -7.0 to 4.1 N ms from foot translation. Additionally, X. laevis kinematics were used as a basis for a forward dynamic anuran swimming model. Input joint kinematics were modulated to independently vary the magnitudes of foot translational and rotational velocity. Simulations predicted that maximum swimming velocity (among all of the kinematics patterns tested) requires that maximal translational and maximal rotational foot velocity act in phase. However, consistent with experimental kinematics, translational and rotational motion contributed unequally to total thrust. The simulation powered purely by foot translation reached a lower peak stroke velocity than the pure rotational case (0.38 vs 0.54 m s(-1)). In all simulations, thrust from the foot was positive for the first half of the power stroke, but negative for the second half. Pure translational foot motion caused greater negative thrust (70% of peak positive thrust) compared with pure rotational simulation (35% peak positive thrust) suggesting that translational motion is propulsive only in the early stages of joint extension. Later in the power stroke, thrust produced by foot rotation overcomes negative thrust (due to translation). Hydrodynamic analysis from X. laevis as well as forward dynamics give insight into the differential roles of translational and rotational foot motion in the aquatic propulsion of anurans, providing a mechanistic link between joint kinematics and swimming performance.

Dynamics-based analysis and synthesis of human locomotion

One of the best ways to synthesize realistic human motions is to animate characters from captured motion data that inherently respect motion laws. Retargeting and interpolation methods are often used to adapt these motions to different representations of the character and to various environmental constraints but they may introduce physical inaccuracies, although the synthesized motions are natural looking. This paper presents a method for evaluating the physical correctness of retargeted and interpolated locomotions using an inverse dynamics analysis. Furthermore, we propose to improve an initial database with analysed motions that are synthesized again by using a forward dynamics approach. The analysis algorithm consists in determining the resulting forces and torques at joints. With this intention, we develop an automatic creation process of the mass/inertia model of the character. Then using support phase recognition, we compute resulting forces and torques by an inverse dynamics method. The retargeting and the interpolation methods change the physics of the motions. This change is evaluated by using the results of our analysis on artificial and real motions and by using literature results and experimental data from force plates. The evaluation relies on the study of several retargeting and interpolation parameters such as the global size of the character or the structure of the model. The output of this evaluation, the resulting forces and torques at joints, are used to produce physically valid motions by using forward dynamics simulation. With this purpose, we introduce forces and torques normalizations, and finally the synthesized motions may improve the initial database.

Ballistic muscle mechanisms determined using an EMG-driven model

The purpose of this study was to determine the mechanisms responsible for improving ballistic elbow extension. In doing so, an electromyography (EMG)-driven model was developed to predict the actual triceps torque so that the model parameters between subjects could be compared. Thirty-two subjects performed maximum isometric trials at 60°, 90° and 120° of elbow extension to determine torque–angle relations. Dynamic elbow extension trials were then performed against relative loads of 0%, 20%, 40%, 60% and 80% and absolute loads of 1.1 and 2.2 kg. These trials were used to determine the torque–angular velocity relation for each subject. The model predicted the triceps torque during the unloaded, 1.1 and 2.2 kg trials with an average r = 0.964 and an average root mean square error of 4.34 N m. As a result of the good predictions, a forward dynamics approach was used to substitute different neuro-muscular mechanisms of a poor performance with those from an individual that displayed a superior performance. Performance was shown to improve when these modifications were made. Therefore, the EMG-driven model was capable of modeling the actual muscle torque which allowed for the identification of areas of weakness of a poor performance. A prescription for improvement was identified, albeit artificially, on an individual basis. The next stage is to determine which specific interventions can accomplish those theoretically proposed.

Neural-Network-Based Design of Optimal Controllers for Nonlinear Systems

A neural-network-based methodology for the design of optimal controllers for nonlinear systems is presented. The overall architecture consists of two neural networks. The first neural network is a cost-to-go function approximator (CTGA), which is trained to predict the cost to go from the present state of the system. The second neural network converges to an optimal controller as it is trained to minimize the output of the first network. The CTGA can be trained using available simulation or experimental data. Hence an explicit analytical model of the system is not required. The key to the success of the approach is giving the CTGA a special decentralized structure that makes its training relatively straightforward and its prediction quality carefully controlled. The specific structure eliminates many of the uncertainties often involved in using artificial neural networks for this type of application. Validity of the approach is illustrated for the optimal attitude control of a spacecraft with reaction wheels. I. Introduction A RTIFICIAL neural networks have been investigated extensively in the optimal control of nonlinear systems. Control architectures known as adaptive critic designs (ACD) have been proposed for the optimal control problem. 1−3 ACD are based on the forward dynamic programming approach to optimization. The basic architecture consists of a critic that models the cost-to-go function and a controller. Both structures are parameterized using neural networks. These two functions are trained simultaneously and each of them depends on the fidelity of the other to get trained properly. This could make the training of the overall system particularly challenging. An interesting solution approach has been presented recently, where the critic and the controller are pretrained using linear models over the region of operation of the system and an algebraic training procedure is employed in the initialization of the neural networks. 4 In our previous work, a modified parametric optimization approach was developed to generate optimal controllers in both state feedback form and dynamic output feedback form for linear systems. 5 In the present work, we generalize the approach to nonlinear systems. Parametric optimization imposes the form of the controller in advance and the controller parameters to optimize the performance measure are found. If the controller parameterization is done via neural networks, then given their universal functionapproximating capability, the true optimal controller can in theory be captured. 6,7 One objective of our research is to remove many of the uncertainties associated with training a neural network architecture that results in an optimal controller. The key to the success of our method is to give the neural network a very special structure that permits tight control over its prediction quality during training. The special structure also makes it possible for each subsystem of the overall network to be trained independent of other subsystems. The issue of interdependency among various portions of the overall network during training, encountered in the basic adaptive

Problems and Techniques

With major elections in many countries during 2004, what could be more timely than the two papers in this SIAM Review issue about preferential voting---the Survey and Review paper by Roberto Serrano and our first Problems and Techniques article, "Single Transferable Votes with Tax Cuts," by Eivind Stensholt? In the Problems and Techniques paper, the author begins by explaining the philosophy and strategy of different ways to tally preferential votes, giving sometimes surprising illustrations of their strengths and weaknesses. Even using the "single transferable vote" (STV) systems described in the paper, which are designed to avoid certain defects of other systems, there will always be collections of individual preference relations that allow election manipulation---for example, under some conditions a voter can actually hurt candidate x by giving x the top rank. A further wrinkle is learning how a "tax cut" (not income tax...) can be included to address the "free ride" problem, in which some voters end up with more influence than others. The paper includes data from a real case of preference voting, analyzing (among several interesting insights) the situation of an "anti-establishment" voter. Moving from voting to fundamental numerical algorithms in scientific and engineering applications, our second paper, by Leslie Greengard and June-Yub Lee, concerns acceleration of the nonuniform fast Fourier transform (FFT). For applications with an irregular sampling of data in the frequency domain---for example, image reconstruction or radio astronomy---the standard FFT is inappropriate because it requires uniform sampling. Greengard and Lee stress that three separate issues arise in reconstructing functions from nonuniform data in the Fourier domain: acquisition of data, choice of a quadrature scheme, and existence of a fast algorithm for computing the discrete transform. Given that a quadrature approach has been chosen, the authors describe techniques for rapid and accurate computation of the needed discrete sums---in effect, combining an interpolation scheme with the standard FFT by replacing a Dirac delta function source with a sharply peaked Gaussian function, thereby "smearing" the source strength onto a regular grid. An especially interesting part of the paper is its characterization of fast Gaussian gridding, which relies on the fact that the values to be evaluated are negligible except at nearby grid points. Numerical examples are included that verify the speed and accuracy of the proposed method, illustrate the benefits of fast gridding, and display results on the ubiquitous Shepp--Logan phantom. The third paper, "Symplectic Rotational Geometry in Human Biomechanics," by V. Ivancevic, begins with informative background on the study of human motion, characterized by complex, multidimensional, highly nonlinear, and hierarchical dynamics---i.e., a very difficult problem! Classical biomechanics relies on vector geometry for the more than six hundred skeletal muscles, each generating forces and torques. With the widely used technique of inverse dynamics, segmental movements are measured using motion capture, often combined with other sensor data. For the more unusual forward dynamics approach, in which translational and rotational motion are simulated based on joint forces and torques, the author proposes the formalism of rotational symplectic geometry. A review of symplectic geometry in manifolds leads to an analysis of how different features of human motion (such as humanoid joint angles, "hinge" joints, and "ball-and-socket" joints) can be modeled in terms of constrained Lie groups, followed by a mathematical tour of (among other topics) segment dynamics, full body dynamics, joint dynamics, and muscular dynamics. We thank the authors of all three papers for proving by example the rich variety of the Problems and Techniques section.

Modelling of knee joint muscles during the swing phase of gait––a forward dynamics approach using MATLAB/Simulink

The present study primarily aims to predict the force (both active and passive) of eight main muscle–tendon actuators crossing the knee joint during the swing cycle. The knee-joint model was restricted and simplified to motion in the sagittal plane only. It was largely built in mathematical notation and was simulated using a high-performance computing and programming tool, MATLAB, along with Simulink. The eight muscles––Biceps Femoris Short Head (BFSH), three heads of Vasti (VL, VM and VI), Rectus Femoris (RF) and the Hamstrings (BFLH, SemiM and SemiT) were each modelled as a two-element, lumped-parameter entity, in series with tendon, which was assumed to be elastic with a known stress–strain relationship. The mechanical behaviour of muscle was described by a Hill-type contractile element which models the active force–length characteristic of muscle fascicles, and a parallel-elastic element which models its passive properties. The knee joint model was simulated for a swing period of 0.4 s and the knee flexed to a maximum of approximately 63°, providing safe toe clearance, and eventually extended to about 4° to prepare for the next heel strike. A comparison of the simulation results in this study with reported data warrants confidence in the prediction of muscle and tendon force. Based on further analysis and quantitative comparison of simulation results, functionally and anatomically similar muscles were eventually approximated with a muscle group and fusiform muscle fibres were favoured over pennate fibres in the muscle–tendon model.

Constructing the set of efficient objective values in linear multiple objective transportation problems

Abstract A forward dynamic programming approach is utilized to find an algebraic representation for a polyhedron in objective space associated with a multiple objective transportation problem having k linear objectives. This polyhedron has the same efficient structure as the set of all feasible objective values, and moreover all of its vertices are efficient. The algebraic representation of this polyhedron is of the form {y ϵ R k : Hy ≧ Ua + Vd} , where the matrices H, U and V are independent of the vector a of availabilities and vector d of demands. The procedure is illustrated by the numerical example of Isermann [H. Isermann, “The enumeration of all efficient solutions for a linear multiple-objective transportation problem”, Naval Research Logistics Quarterly 26 (1979) 123–139].

ROLLING SCHEDULES FOR A DYNAMIC LOT-SIZING PROBLEM WITH START-UP COST

This paper considers a dynamic lot size model with start-up cost where backlogging is not allowed. The model incorporates inventory holding cost and two kinds of fixed costs—a start-up cost incurred whenever the machine (production facility) is switched from “off” to “on”, and a reservation cost incurred in each production period. The optimal (cost-minimizing) solution over a finite horizon is found in a forward dynamic programming approach, upon which a planning horizon theorem is derived to treat the model in a rolling horizon environment. Accordingly, two procedures for selecting the run length of the first production block and the amount to meet the associated production requirement are developed. The cost effectivenesses of the procedures are demonstrated by a set of simulation experiments.

Reduced complexity dynamic programming based on policy iteration

In this paper, a forward method is introduced for solving the dynamic programming equations of Bellman. This is in contrast to most existing methods for dynamic programming which solve the problem backwards. A key advantage is that the forward dynamic programming approach can be systematically simplified to provide computation/optimality trade-offs. Such trade-offs are lacking in backwards iterative methods which tend to be “all or nothing” propositions. A second advantage is that the computation is independent of the state dimension. These properties together offer some promise for circumventing the “curse of dimensionality” on many problems of practical interest. Due to a strong connection with the work of Bellman on policy iteration, the method is denoted as the Iteration in Policy Space (IPS) algorithm. Several examples are given to demonstrate the general usefulness of the method.