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Microrobots with their promising applications are attracting a lot of attention currently. A microrobot with a triangular mechanism was previously proposed by scientists to overcome the motion limitations in a low-Reynolds number flow; however, the control of this swimmer for performing desired manoeuvres has not been studied yet. Here, we have proposed some strategies for controlling its position. Considering the constraints on arm lengths, we proposed an optimal controller based on quadratic programming. The simulation results demonstrate that the proposed optimal controller can steer the microrobot along the desired trajectory as well as minimize fluctuations of the actuators length.
Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved in applying deep learning: most deep learning methods require the solution of hard optimisation problems, and a good understanding of the trade-off between computational effort, amount of data and model complexity is required to successfully design a deep learning approach for a given problem.. A large amount of progress made in deep learning has been based on heuristic explorations, but there is a growing effort to mathematically understand the structure in existing deep learning methods and to systematically design new deep learning methods to preserve certain types of structure in deep learning. In this article, we review a number of these directions: some deep neural networks can be understood as discretisations of dynamical systems, neural networks can be designed to have desirable properties such as invertibility or group equivariance and new algorithmic frameworks based on conformal Hamiltonian systems and Riemannian manifolds to solve the optimisation problems have been proposed. We conclude our review of each of these topics by discussing some open problems that we consider to be interesting directions for future research.
A multi-agent engagement scenario is considered in which a high-value aircraft launches two defenders to intercept two homing missiles aimed at the aircraft. Under the assumption that all aircrafts have first-order linear dynamic characteristics, a combined multiple-mode adaptive estimation (MMAE) and a two-way cooperative optimal guidance law are proposed for the target–defenders team. Considering the full cooperation of the target and both the two defenders, the two-way cooperative strategies provide the analytical expressions for their optimal control input, enabling the target–defenders team to intercept the missiles with minimal control effort. To successfully intercept the missiles, MMAE is used to identify the guidance laws adopted by the missiles and estimate their states. The simulation results show that the target cooperating with the defenders to perform lure manoeuvres for the missiles can improve the guidance performance of the defenders as well as reduce the control effort of the defenders for intercepting the missiles.
Although wearable robotic systems are designed to reduce the risk of low-back injury, it is unclear how effective assistance is, compared to improvements in lifting technique. We use a two-factor block study design to simulate how effective exoskeleton assistance and technical improvements are at reducing the risk of low-back injury when compared to a typical adult lifting a box. The effects of assistance are examined by simulating two different models: a model of just the human participant, and a model of the human participant wearing the SPEXOR exoskeleton. The effects of lifting technique are investigated by formulating two different types of optimal control problems: a least-squares problem which tracks the human participant’s lifting technique, and a minimization problem where the model is free to use a different movement. Different lifting techniques are considered using three different cost functions related to risk factors for low-back injury: cumulative low-back load (CLBL), peak low-back load (PLBL), and a combination of both CLBL and PLBL (HYB). The results of our simulations indicate that an exoskeleton alone can make modest reductions in both CLBL and PLBL. In contrast, technical improvements alone are effective at reducing CLBL, but not PLBL. The largest reductions in both CLBL and PLBL occur when both an exoskeleton and technical improvements are used. While all three of the lifting technique cost functions reduce both CLBL and PLBL, the HYB cost function offers the most balanced reduction in both CLBL and PLBL.
In this paper, a new approach is presented for perfect torque compensation of the robot in point-to-point motions. The proposed method is formulated as an open-loop optimal control problem. The problem is defined as optimal trajectory planning with adjustable design parameters to compensate applied torques of a planar 5R parallel robot for a given task, perfectly. To illustrate the effectiveness of the approach, the obtained optimal path is used as the reference command in the experiment. The experimental outputs show that the performance index has been reduced by over 80% compared to the typical design of the robot.
Quadrotors are unmanned aerial vehicles with many potential applications ranging from mapping to supporting rescue operations. A key feature required for the use of these vehicles under complex conditions is a technique to analytically solve the problem of trajectory planning. Hence, this paper presents a heuristic approach for optimal path planning that the optimization strategy is based on the indirect solution of the open-loop optimal control problem. Firstly, an adequate dynamic system modeling is considered with respect to a configuration of a commercial quadrotor helicopter. The model predicts the effect of the thrust and torques induced by the four propellers on the quadrotor motion. Quadcopter dynamics is described by differential equations that have been derived by using the Newton–Euler method. Then, a path planning algorithm is developed to find the optimal trajectories that meet various objective functions, such as fuel efficiency, and guarantee the flight stability and high-speed operation. Typically, the necessary condition of optimality for a constrained optimal control problem is formulated as a standard form of a two-point boundary-value problem using Pontryagin’s minimum principle. One advantage of the proposed method can solve a wide range of optimal maneuvers for arbitrary initial and final states relevant to every considered cost function. In order to verify the effectiveness of the presented algorithm, several simulation and experiment studies are carried out for finding the optimal path between two points with different objective functions by using MATLAB software. The results clearly show the effect of the proposed approach on the quadrotor systems.
We determine the optimal asset allocation to bonds and stocks using an annually recalculated virtual annuity (ARVA) spending rule for DC pension plan decumulation. Our objective function minimizes downside withdrawal variability for a given fixed value of total expected withdrawals. The optimal asset allocation is found using optimal stochastic control methods. We formulate the strategy as a solution to a Hamilton–Jacobi–Bellman (HJB) Partial Integro Differential Equation (PIDE). We impose realistic constraints on the controls (no-shorting, no-leverage, discrete rebalancing) and solve the HJB PIDEs numerically. Compared to a fixed-weight strategy which has the same expected total withdrawals, the optimal strategy has a much smaller average allocation to stocks and tends to de-risk rapidly over time. This conclusion holds in the case of a parametric model based on historical data and also in a bootstrapped market based on the historical data.
The necessity of using subject-specific data analysis of nonergodic psychological processes is explained while emphasizing the difference between interindividual and intraindividual variation. The chapter argues that subject-specific data analysis not only matches the principles underlying developmental systems theory, which is relevant to obtaining a comprehensive understanding of change in human psychopathology, but also enables testing of all principles of person-oriented theory, which is fundamental to the formation and implementation of individualized treatments. A new generalized perspective on measurement equivalence in subject-specific data analysis is introduced. The importance of adaptive optimal control of psychological processes within the context of subject-specific data analysis is emphasized. In addition, some broader aims of subject-specific data analysis are considered, including principled ways to bridge the nomothetic and idiographic levels of analysis.
The present article presents novel results on the Ramsey–Cass–Koopmans growth model. It is shown that the shadow price of capital goes to infinity as the capital stock goes to zero even if all functions are bounded with finite derivatives and that imposing the Inada condition of infinite derivative of the per capita production function at zero stock is irrelevant. It is also shown that unless marginal utility at zero consumption is infinity, there will be a non-empty interval where the Keynes–Ramsey rule does not hold. The paper also shows that the stable saddle path in a phase diagram with the state variable and the shadow price has an unrecognized economic interpretation that enables us to illustrate the value function as the integral of the stable saddle path.
In this paper, two strategies are proposed to optimize the energy consumption of a new screw in-pipe inspection robot which is steerable. In the first method, optimization is performed using the optimal path planning and implementing the Hamilton–Jacobi–Bellman (HJB) method. Since the number of actuators is more than the number of degrees of freedom of the system for the proposed steerable case, it is possible to minimize the energy consumption by the aid of the dynamics of the system. In the second method, the mechanics of the robot is modified by installing some turbine blades through which the drag force of the pipeline fluid can be employed to decrease the required propulsion force of the robot. It is shown that using both of the mentioned improvements, that is, using HJB formulation for the steerable robot and installing the turbine blades can significantly save power and energy. However, it will be shown that for the latter case this improvement is extremely dependent on the alignment of the fluid stream direction with respect to the direction of the robot velocity, while this optimization is independent of this case for the former strategy. On the other hand, the path planning dictates a special pattern of speed functionality while for the robot equipped by blades, saving the energy is possible for any desired input path. The correctness of the modeling is verified by comparing the results of MATLAB and ADAMS, while the efficiency of the proposed optimization algorithms is checked by the aid of some analytic and comparative simulations.
We consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.
In this paper, optimal control of a 3PRS robot is performed, and its related optimal path is extracted accordingly. This robot is a kind of parallel spatial robot with six DOFs which can be controlled using three active prismatic joints and three passive rotary ones. Carrying a load between two initial and final positions is the main application of this robot. Therefore, extracting the optimal path is a valuable study for maximizing the load capacity of the robot. First of all, the complete kinematic and kinetic modeling of the robot is extracted to control and optimize the robot. As the robot is categorized as a constrained robot, its kinematics is studied using a Jacobian matrix and its pseudo inverse whereas its kinetics is studied using Lagrange multipliers. The robot is then controlled using feedforward term of the inverse dynamics. Afterward, the extracted dynamics equation of the robot is transferred to state space to be employed for calculus of variations. Considering the constrained entity of the robot, null space of the robot is employed to eliminate the Lagrange multipliers to provide the applicability of indirect variation algorithm for the robot. As a result, not only are the optimal controlling signals calculated but also the corresponding optimal path of the robot between two boundary conditions is extracted. All the modeling, controlling, and optimization process are verified using MATLAB simulation. The profiles are then double-checked by comparing the results with SimMechanics. It is proved that with the aid of the proposed controlling and optimization method of this article, the robot can be controlled along its optimal path through which the maximum load can be carried.
During visual servoing space activities, the attitude of free-floating space robot may be disturbed due to dynamics coupling between the satellite base and the manipulator. And the disturbance may cause communication interruption between space robot and control center on earth. However, it often happens that the redundancy of manipulator is not enough to fully eliminate this disturbance. In this paper, a method named off-line optimizing visual servoing algorithm is innovatively proposed to minimize the base disturbance during the visual servoing process where the degrees-of-freedom of the manipulator is not enough for a zero-reaction control. Based on the characteristic of visual servoing process and the robot system modeling, the optimal control method is applied to achieve the optimization, and a pose planning method is presented to achieve a second-order continuity of quaternion getting rid of the interruption caused by ambiguity. Then simulations are carried out to verify the method, and the results show that the robot is controlled with optimized results during visual servoing process and the joint trajectories are smooth.
Linear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement. The aim is to derive transformation of the LQR weighting matrix for fixed weighting factor, using the discrete algebraic Riccati equation (DARE) to design a discrete time optimal PID controller producing similar time response to its continuous time counterpart. Continuous time LQR-based PID controller can be transformed to discrete time by establishing a relation between the respective LQR weighting matrices that will produce similar closed loop response, independent of the chosen sampling time. Simulation examples of first/second order and first-order integrating processes exhibiting stable/unstable and marginally stable open loop dynamics are provided, using the transformation of LQR weights. Time responses for set-point and disturbance inputs are compared for different sampling times as fraction of the desired closed loop time constant.
This work presents a systematic design selection methodology that utilizes a co-design strategy for system-level optimization of compliantly actuated robots that are known for their advantages over robotic systems driven by rigid actuators. The introduced methodology facilitates a decision-making strategy that is instrumental in making selections among system-optimal robot designs actuated by various degrees of variable or fixed compliance. While the simultaneous co-design method that is utilized throughout guarantees systems performing at their full potential, a homotopy technique is used to maintain integrity via generation of a continuum of robot designs actuated with varying degrees of variable and fixed compliance. Fairness of the selection methodology is ensured via utilization of common underlying (variable) compliant actuation principle and dynamical task requirements throughout the generated system designs. The direct consequence of the developed methodology is that it allows robot designers make informed selections among a variety of systems which are guaranteed to perform at their best. Applicability of the introduced methodology has been validated using a case study for system-optimal design of an active knee prosthesis that is driven by a mechanically adjustable compliance and controllable equilibrium position actuator (MACCEPA) under a periodic/real-life dynamical task.
The purpose of this study is to determine the dynamic load carrying capacity (DLCC) of a manipulator that moves on the specified path using a new closed loop optimal control method. Solution methods for designing nonlinear optimal controllers in a closed-loop form are usually based on indirect methods, but the proposed method is a combination of direct and indirect methods. Optimal control law is given by solving the nonlinear Hamilton–Jacobi–Bellman (HJB) partial differential equation. This equation is complex to solve exactly for complex dynamics, so it is solved numerically using the Galerkin procedure combined with a nonlinear optimization algorithm. To check the performance of the proposed algorithm, the simulation is performed for a fixed manipulator. The results represent the efficiency of the method for tracking the pre-determined path and determining the DLCC. Finally, an experimental test has been done for a two-link manipulator and compare with simulation results.
Optimal growth theory as it stands today does not work. Using strictly concave utility functions systematically inflicts on the economy distortions that are either historically unobserved or unacceptable by society. Moreover, we show that the traditional approach is incompatible with competitive equilibrium: Any economy initially in such equilibrium will always veer away into unwanted trajectories if its investment is planned using a concave utility function. We then propose a rule for the optimal savings-investment rate based on competitive equilibrium that simultaneously generates three intertemporal optima for society. The rule always leads to reasonable time paths for all central economic variables, even under very different hypotheses about the future evolution of population and technical progress.
In this paper we develop the nonlinear motion equations in terms of the true anomaly varying Tschauner–Hempel equations relative to a notional orbiting particle in a Keplerian orbit, relatively close to an orbiting primary satellite to estimate the position of a spacecraft. A second orbiting body in Earth orbit relatively close to the first is similarly modelled. The dynamic relative motion models of the satellite and the second orbiting body, both of which are modelled in terms of independent relative motion equations, include several perturbing effects, such as the asymmetry of the Earth gravitational field resulting in the Earth's oblateness effect and the third body accelerations due to the Moon and the Sun. Linear control laws are synthesised for the primary satellite using the transition matrix so it can rendezvous with the second orbiting body. The control laws are then implemented using the state estimates obtained earlier to validate the feedback controller. Thus, we demonstrate the application of a Linear Quadratic Nonlinear Gaussian (LQNG) design methodology to the satellite rendezvous control design problem and validate it.
In this paper, we consider an optimal control problem governed by Stokes equations with H1-norm state constraint. The control problem is approximated by spectral method, which provides very accurate approximation with a relatively small number of unknowns. Choosing appropriate basis functions leads to discrete system with sparse matrices. We first present the optimality conditions of the exact and the discrete optimal control systems, then derive both a priori and a posteriori error estimates. Finally, an illustrative numerical experiment indicates that the proposed method is competitive, and the estimator can indicate the errors very well.