Synthesis of tensegrity structures of desired shape using constrained minimization

There are two types of problems in tensegrity design: (i) form-finding when the tensegrity shape is not specified and (ii) synthesis when the tensegrity shape is specified. We address synthesis problems in this paper. We first formulated and solved an optimization problem to synthesize tensegrity structures of specified shape when the connectivity of the elements (bars and cables) is known a priori. We minimize the error in force-balance at the vertices in the desired equilibrium configuration by using force densities as the design variables. This constrained minimization problem enabled us to synthesize a known asymmetric tensegrity arch and a hitherto unknown tensegrity of biconcave shape similar to that of a healthy human red blood cell. We also extend the above method to a reduced order optimization problem for synthesizing complex symmetric tensegrity structures. Using this approach, we synthesized a truncated dodecahedron inside another truncated dodecahedron to emulate a nucleus inside a cell. We use a restricted global structure on an already available two-step mixed integer linear programming (MILP) topology optimization formulation to synthesize a non-convex tensegrity structure when only the coordinates are provided. We further improve this two-step MILP to a single-step MILP. We also present static analysis of a tensegrity structure by minimizing the potential energy with unilateral constraints on the lengths of the cables that cannot take compressive loads. Furthermore, we use this method to synthesize a tensegrity table of desired height and area under a predefined load. The prototypes of three synthesized tensegrities were made and validated.     

Exploring new tensegrity structures via mixed integer programming

A tensegrity structure is a prestressed pin-jointed structure consisting of continuously connected tensile members (cables) and disjoint compressive members (struts). Many classical tensegrity structures are prestress stable, i.e., they are kinematically indeterminate but stabilized by introducing prestresses. This paper presents a procedure for generating various prestress stable tensegrity structures. This method is based on truss topology optimization and does not require connectivity relation of cables and struts of a tensegrity structure to be known in advance. Unlike the conventional form-finding methods, the locations of nodes are fixed throughout optimization. The optimization problem with the constraints expressing the definition of tensegrity structure, kinematical indeterminacy, and symmetry of configurations is formulated as a mixed integer linear programming (MILP) problem. Numerical experiments demonstrate that various tensegrity structures can be generated from one given initial structure by solving the presented MILP problems by using a few control parameters.     

Modified iterated simulated annealing algorithm for structural synthesis

A new hybrid simulated annealing method is presented for the optimization of structural systems subjected to dynamic loads. The optimization problem is formulated as a structural weight minimization, with time-varying constraints on floor displacements, velocities, accelerations, or floor drifts, and structural member combined stresses. In addition, time-invariant constraints on structural frequencies and member sizes that will satisfy the strong column–weak beam philosophy of the building codes can be imposed. The method uses elements of existing simulated annealing algorithms and introduces certain new procedures. Firstly, the search range is automatically reduced, by using the updated information of the current design, at each iteration. Secondly, the inner and outer iteration loops are implemented. Thirdly, sensitivity analysis of the time-varying global displacements is performed with respect to the design variables that are the structural member cross-sectional areas. The results of the sensitivity analysis identify which design variables must be modified to decrease the global displacements in the most effective manner. However, once the variables are identified from the sensitivity analysis, the new values of these variables are determined in a random manner. The possibility of attaining a global minimum is thus maintained. The method is suited for structural optimization problems with time-varying constraints because the annealing is a random search technique and can locate global rather than local minima.     

Topology optimization of trusses with stress and local constraints on nodal stability and member intersection

A truss topology optimization problem under stress constraints is formulated as a Mixed Integer Programming (MIP) problem with variables indicating the existence of nodes and members. The local constraints on nodal stability and intersection of members are considered, and a moderately large lower bound is given for the cross-sectional area of an existing member. A lower-bound objective value is found by neglecting the compatibility conditions, where linear programming problems are successively solved based on a branch-and-bound method. An upper-bound solution is obtained as a solution of a Nonlinear Programming (NLP) problem for the topology satisfying the local constraints. It is shown in the examples that upper- and lower-bound solutions with a small gap in the objective value can be found by the branch-and-bound method, and the computational cost can be reduced by using the local constraints.     

Optimum cost design of reinforced concrete slabs using neural dynamics model

For structural optimization algorithms to find widespread usage among practicing engineering they must be formulated as cost optimization and applied to realistic structures subjected to the actual constraints of commonly used design codes such as the ACI code. In this article, a general formulation is presented for cost optimization of single- and multiple-span RC slabs with various end conditions (simply supported, one end continuous, both ends continuous, and cantilever) subjected to all the constraints of the ACI code. The problem is formulated as a mixed integer-discrete variable optimization problem with three design variables: thickness of slab, steel bar diameter, and bar spacing. The solution is obtained in two stages. In the first stage, the neural dynamics model of Adeli and Park is used to obtain an optimum solution assuming continuous variables. Next, the problem is formulated as a mixed integer-discrete optimization problem and solved using a perturbation technique in order to find practical values for the design variables. Practicality, robustness, and excellent convergence properties of the algorithm are demonstrated by application to four examples.     

基于混合整数线性规划无人机实时航迹规划

为了解决无人机实时航迹规划问题,特别足带动力学约束条件的实时航迹规划问题,给出了基于混合整数线性规划技术在模型预测控制框架下进行无人机实时航迹规划的方法.通过将威胁区、速度、加速度以及威胁规避等约束条件转化为能够直接应用在MILP中的形式,并结合模型预测控制方法来进行规划以满足实时性要求.在威胁区的规避上,使用了二进制变量进行逻辑判断,同时,利用松弛变量的方法将威胁规避条件转变为线性形式;在速度、加速度约束条件上,使用单位圆将其约束在圆内以满足速度约束的限制.最后根据仿真计算的验证和分析,得出基于混合整数线性规划的无人机实时航迹规划的有效性.     

A level set based topology optimization method using the discretized signed distance function as the design variables

This paper deals with a new topology optimization method based on the level set method. In the proposed method, the discretized signed distance function, a kind of level set function, is used as the design variables, and these are then updated using their sensitivities. The signed distance characteristic of the design variables are maintained by performing a re-initialization at every update during the iterated optimization procedure. In this paper, a minimum mean compliance problem and a compliant mechanism design problem are formulated based on the level set method. In the formulations of these design problems, a perimeter constraint is imposed to overcome the ill-posedness of the structural optimization problem. The sensitivity analysis for the above structural optimization problems is conducted based on the adjoint variable method. The augmented Lagrangian method is incorporated to deal with multiple constraints. Finally, several numerical examples that include multiple constraints are provided to confirm the validity of the method, and it is shown that appropriate optimal structures are obtained.     

Three-Dimensional Path Planning of a Climbing Robot Using Mixed Integer Linear Programming

The City-Climber robot is a novel wall-climbing robot developed at The City College of New York that has the capability to move on floors, climb walls, walk on ceilings and transit between them. In this paper, we first develop the dynamic model of the City-Climber robot when it travel on different surfaces, i.e., floors, walls and ceilings, respectively. Then, we present a path planning method for the City-Climber robot using mixed integer linear programming (MILP) in three-dimensional (3-D) building environments that consist of objects with primitive geometrical shapes. MILP provides an optimization framework that can directly incorporate dynamic constraints with logical constraints such as obstacle avoidance and waypoint selection. In order to use MILP to solve the obstacle avoidance problem, we simplify and decouple the robot dynamic model into a linear system by introducing a restricting admissible controller. The decoupled model and obstacle can be rewritten as a linear program with mixed-integer linear constraints that account for the collision avoidance. A key benefit of this approach is that the path optimization can be readily solved using the AMPL and CPLEX optimization software with a MATLAB interface. Simulation results show that the framework of MILP is well suited for path planning and obstacle avoidance problems for the wall-climbing robot in 3-D environments.     

航天器近距离相对运动的鲁棒约束模型预测控制

航天器在轨服务对近距离相对运动精确控制的需求越来越强.通过引入集合理论,采用鲁棒可变时域模型预测控制和混合整数线性规划,解决了航天器近距离相对运动的鲁棒控制问题,便于处理控制约束和状态约束,对未知有界干扰、推力误差和导航误差具有鲁棒性.首先,针对航天器近距离相对运动过程中向任意目标集的有限时间机动问题,采用离散化C-W(Clohessy-Wiltshire)动力学模型、时间一能量组合优化目标函数和线性约束表示建立了控制问题模型;其次,给出了基于约束压缩的鲁棒可变时域模型预测控制算法,可以确保鲁棒可行和鲁棒完成;引入i-步鲁棒可控集分析问题可行性,通过集合运算将导航误差处理成有界干扰,采用混合整数线性规划完成了控制器设计.最后,数值仿真验证了模型的有效性.     

Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.     

Non-parametric free-form optimization method for frame structures

In this paper, we propose a parameter-free shape optimization method based on the variational method for designing the smooth optimal free-form of a spatial frame structure. A stiffness design problem where the compliance is minimized under a volume constraint is solved as an example of shape design problems of frame structures. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and that the cross section is prismatic. The shape gradient function and the optimality conditions are then theoretically derived. The optimal curvature distribution is determined by applying the derived shape gradient function to each member as a fictitious distributed force both to vary the member in the optimum direction and to minimize the objective functional without shape parametrization, while maintaining the members’ smoothness. The validity and practical utility of this method were verified through several design examples. It was confirmed that axial-force-carrying structures were obtained by this method.     

An algebraic approach to shape-from-image problems

This paper presents a new method for recovering three-dimensional shapes of polyhedral objects from their single-view images. The problem of recovery is formulated in a constrained optimization problem, in which the constraints reflect the assumption that the scene is composed of polyhedral objects, and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture. For practical purpose it is decomposed into the two more tractable problems: a linear programming problem and an unconstrained optimization problem. In the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra, whereas similar constraints have usually been represented in terms of a gradient space. Moreover, superstrictness of the constraints can be circumvented by a new concept ‘position-free incidence structure’. For this reason the present method has several advantages: it can recover the polyhedral shape even if image data are incorrect due to vertex-position errors, it can deal with perspective projection as well as orthographic projection, the number of variables in the optimization problem is very small (three or a little greater than three), and any kinds of surface information can be incorporated in a unifying manner.     

A mixed GA-PSO-based approach for performance-based design optimization of 2D reinforced concrete special moment-resisting frames

A mixed genetic algorithm and particle swarm optimization in conjunction with nonlinear static and dynamic analyses as a smart and simple approach is introduced for performance-based design optimization of two-dimensional (2D) reinforced concrete special moment-resisting frames. The objective function of the problem is considered to be total cost of required steel and concrete in design of the frame. Dimensions and longitudinal reinforcement of the structural elements are considered to be design variables and serviceability, special moment-resisting and performance conditions of the frame are constraints of the problem. First, lower feasible bond of the design variables are obtained via analyzing the frame under service gravity loads. Then, the joint shear constraint has been considered to modify the obtained minimum design variables from the previous step. Based on these constraints, the initial population of the genetic algorithm (GA) is generated and by using the nonlinear static analysis, values of each population are calculated. Then, the particle swarm optimization (PSO) technique is employed to improve keeping percent of the badly fitted populations. This procedure is repeated until the optimum result that satisfies all constraints is obtained. Then, the nonlinear static analysis is replaced with the nonlinear dynamic analysis and optimization problem is solved again between obtained lower and upper bounds, which is considered to be optimum result of optimization solution with nonlinear static analysis. It has been found that by mixing the analyses and considering the hybrid GA-PSO method, the optimum result can be achieved with less computational efforts and lower usage of materials.     

A heuristic algorithm for optimal fleet composition with vehicle routing considerations

This paper proposes a fast heuristic algorithm for solving a combined optimal fleet composition and multi-period vehicle routing problem. The aim of the problem is to determine an optimal fleet mix, together with the corresponding vehicle routes, to minimize total cost subject to various customer delivery requirements and vehicle capacity constraints. The total cost includes not only the fixed, variable, and transportation costs associated with operating the fleet, but also the hiring costs incurred whenever vehicle requirements exceed fleet capacity. Although the problem under consideration can be formulated as a mixed-integer linear program (MILP), the MILP formulation for realistic problem instances is too large to solve using standard commercial solvers such as the IBM ILOG CPLEX optimization tool. Our proposed heuristic decomposes the problem into two tractable stages: in the first (outer) stage, the vehicle routes are optimized using cross entropy; in the second (inner) stage, the optimal fleet mix corresponding to a fixed set of routes is determined using dynamic programming and golden section search. Numerical results show that this heuristic approach generates high-quality solutions and significantly outperforms CPLEX in terms of computational speed.     

Optimum design of plano-milling machine structure using finite element analysis

The problem of optimum design of plano-milling machine structure is formulated as a nonlinear mathematical programming problem with the objective of minimizing the structural weight. The plano-milling machine structure is idealized with triangular plate elements and three dimensional frame elements based on finite element displacement method. Constraints are placed on static deflections and principal stresses in the problem formulation. The optimization problem is solved by using an interior penalty function method in which the Davidon-Fletcher-Powell variable metric unconstrained minimization technique and cubic interpolation method of one dimensional search are employed. A numerical example is presented for demonstrating the effectiveness of the procedure outlined. The results of sensitivity analysis conducted with respect to design variables and fixed parameters about the optimum point are also reported.     

单阶段多产品批处理过程的短期调度1. 基本模型的建立

具有并行设备的多产品单阶段批处理过程短期 调度问题需考虑订单发布时间、交货期，订单生产的顺序相关建立时间、禁止生产子序列， 及设备的准备时间等生产约束．本文在考虑上述约束的基础的上，利用时间间隙的概念和连 续时间表达，将设备、订单分配给时间间隙分别表达为两类0-1变量，建立了具有并行生产 线的多产品单阶段批处理过程的短期调度数学模型．模型表达为一个混合整数规划（MILP） 问题．该模型不但比已有的基于时间间隙描述的调度模型0-1变量少，而且能优 化多种目标函数．本文的第二部分将引入一些适当的启发性规则，减小了模型的规模，并应 用大量的计算实例说明该模型的有效性和适用性．     

On the formulation and implementation of geometric and manufacturing constraints in node–based shape optimization

We introduce a novel method to handle geometrical and manufacturing constraints in parameter–free shape optimization. Therefore the design node coordinates are split in two sets where one set is declared as new design variables and the other set is coupled to the new design variables such that the geometrical constraint is fulfilled. Thereby no additional equations are appended to the optimization problem. In contrast the implementation of a demolding constraint is presented by formulating inequality constraints which indeed have to be attached to the optimization problem. In the context of a sensitivity–based shape optimization approach all manufacturing constraints have to be formulated in terms of the finite element node coordinates such that first order gradients with respect to the design node coordinates can be derived.     

Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization

In this work, we explore simultaneous geometry design and material selection for statically determinate trusses by posing it as a continuous optimization problem. The underlying principles of our approach are structural optimization and Ashby’s procedure for material selection from a database. For simplicity and ease of initial implementation, only static loads are considered in this work with the intent of maximum stiffness, minimum weight/cost, and safety against failure. Safety of tensile and compression members in the truss is treated differently to prevent yield and buckling failures, respectively. Geometry variables such as lengths and orientations of members are taken to be the design variables in an assumed layout. Areas of cross-section of the members are determined to satisfy the failure constraints in each member. Along the lines of Ashby’s material indices, a new design index is derived for trusses. The design index helps in choosing the most suitable material for any geometry of the truss. Using the design index, both the design space and the material database are searched simultaneously using gradient-based optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous, although the material selection from a database is an inherently discrete problem. A few illustrative examples are included. It is observed that the method is capable of determining the optimal topology in addition to optimal geometry when the assumed layout contains more links than are necessary for optimality.     

Smooth time-optimal tool trajectory generation for CNC manufacturing systems

An optimization approach is proposed in this paper for generating smooth and time-optimal path constrained tool trajectory for Cartesian computer numerical control (CNC) manufacturing systems. The desired smooth time-optimal trajectory generation (STOTG) problem is formulated as a general optimal control problem. And axis jerk (derivative of acceleration with respect to time) constraints are introduced into this problem to remove discontinuities of the acceleration profiles. The desired smoothness of the trajectory can be accomplished by adjusting the values of jerk constraints. A control vector parameterization (CVP) method is applied to convert the optimal control problem into a nonlinear programming (NLP) problem which can be solved conveniently and effectively. The third derivative of the path parameter with respect to time (pseudo-jerk) and jerk act as optimization variables. The pseudo-jerk is approximated as piecewise constant, thus for at least second-order continuous parametric path, the resulted optimized trajectory with respect to time is also at least second-order continuous. Sequential quadratic programming (SQP) method is used to solve the NLP problem, through which numerical solution is obtained. Non-smooth (i.e. without considering jerk constraints) time-optimal trajectory generation (non-STOTG) problem is also considered in this paper for the purpose of comparison. Solutions of time-optimal trajectory generation (TOTG) problems for two test paths are performed to verify the effectiveness of the proposed approach.     

Optimization of singular problems

A new optimization method is presented that optimizes singular structures. An example of a singular problem is deleting an inefficient member from a structure. As the member is deleted, the stresses in the member may increase above the allowables. When a member is deleted the nature of the analysis changes because the member stiffness becomes zero. This causes a local optima because stress constraints prevent inefficient members from zeroing. The problem is reformulated using the percent method so that the appropriate stress constraints are deleted as the member is deleted. Several examples show that the global optimal design is reached. Other methods to reach the global optima are appropriate only if the optimal structure is statically determinate. The percent optimization is also useful for optimization of discrete problems.