B-Spline FUNCTION for Trajectory Planning 用三次B样条进行轨迹规划
工业机器人的路径规划算法考核试卷
A. A*算法
B. D*算法
C. Floyd算法
D. Bresenham算法
16.在路径规划中，以下哪种数据结构用于存储已访问的节点？（）
A.开放列表
B.关闭列表
C.路径列表
D.邻接矩阵
17.关于工业机器人的路径规划，以下哪个描述是错误的？（）
B. Dijkstra算法
C. D*算法
D. RRT算法
13.关于路径规划中的碰撞检测，以下哪种方法计算量相对较小？（）
A.精确碰撞检测
B.粗略碰撞检测
C.迭代最近点法（ICP）
D.点到线段距离计算
14.以下哪项不是路径平滑处理的目的？（）
A.降低路径长度
B.减少运动时间
C.减少能量消耗
D.增加路径上的拐点
3.用于评估路径规划算法性能的指标通常包括路径长度、规划时间和______。
4.在路径规划中，为了减少计算量，常用的空间划分技术有四叉树和______。
5.机器人路径规划中的碰撞检测可以通过______和基于几何的碰撞检测两种方法实现。
6.路径平滑处理的目的是为了减少路径的拐点，提高路径的______和可执行性。
A. A*算法
B. Dijkstra算法
C. Floyd算法
D. Bresenham算法
2.下列哪种算法不属于路径规划中的启发式搜索算法？（）
A. A*算法
B. D*算法
C. IDA*算法
D. Breadth First Search算法
3.在A*算法中，H(n)代表什么？（）
A.从起始点到当前点的代价
4.在工业机器人路径规划中，如何处理动态障碍物？请提出一种算法或策略，并解释其工作原理和有效性。
motion planning和trajectory planning
motion planning和trajectory planning
Motion planning和trajectory planning是机器人技术中的两个重要概念，它们在机器人运动控制和导航过程中起着关键作用。
尽管它们之间有一定的关联，但它们在实现方式和应用场景上存在明显的差异。
1. Motion Planning（运动规划）：
-Motion planning是指机器人规划其在关节空间或笛卡尔空间中的运动路径，以完成特定任务。
它主要关注机器人在执行任务过程中如何移动，包括移动的轨迹和速度等。
-运动规划的目标是找到一条安全、高效的路径，使机器人能够从起始位置到达目标位置，并完成任务。
-运动规划算法通常基于碰撞检测、路径搜索和优化技术。
常见的运动规划方法有：基于图的规划、基于概率的方法、智能优化算法等。
2. Trajectory Planning（轨迹规划）：
-轨迹规划是在运动规划的基础上，为机器人规划一个平滑、稳定的运动轨迹。
轨迹规划不仅考虑机器人的运动路径，还关注运动过程中的速度、加速度等动态特性。
-轨迹规划的目标是生成一条满足一定性能要求的轨迹，例如避免过冲、减小加速度变化等。
-轨迹规划算法通常基于微分方程、优化技术和控制理论。
常见的轨迹规划方法有：多项式插值、样条插值、优化算法（如遗传算法、粒子群优化算法等）等。
总结：运动规划关注机器人如何移动，主要解决移动路径和速度问题；轨迹规划关注机器人移动过程中的动态特性，主要解决平滑性和稳定性问题。
在实际应用中，运动规划和轨迹规划通常是相互关联的，需要结合具体情况选择合适的规划方法。
基于五次B样条的起竖装置时间最优轨迹规划
第6期2016年6月组合机床与自动化加工技术Modular Machine Tool & Automatic Manufacturing TechnitjueNo. 6Jun. 2016文章编号：1001 -2265(2016)06-0021 -05DOI：10.13462/j. cnki. mmtamt. 2016. 06. 006基于五次E样条的起竖装置时间最优轨迹规划！邓飙，张潇，龙勇，郭杨(火箭军工程大学二系，西安710025)摘要：针对含二级电动缸的起竖装置当前运动轨迹时间不够优化，且存在不连续冲击的问题，提出了 新的高阶样条轨迹规划方法。
通过对起竖装置进行建模分析，解得了运动学约束、动力学约束以及边界条件。
使用五次B样条插值规划轨迹，引入罚函数处理约束条件，以样条曲线相邻控制点的时 间间隔为设计变量，利用非支配排序遗传算法（N S G A-II)对B样条轨迹进行多目标优化。
运算后得 到一组P areto最优解，从中选择最为平滑且时间最优的解作为最终解。
仿真结果表明，相对于多项 式插值，样条插值理论上能实现时间最优；而采用高次样条插值能得到高阶连续的运动轨迹，使起竖 过程更加平稳。
关键词：起竖装置；电动缸;轨迹规划；B样条；多目标优化中图分类号:TH112;TG506 文献标识码:ATim eO ptim al Trajectory Planning B ased o n F iv eO rd ers B-spline forE rectionE quipm entDENG Biao,ZHANG X iao,LONG Yong,GUO Yang(The Second Department,Rocket Force University of Engineering,X ir an710025 ,China)A bstract:A new trajectory planning method based on spline interpolation was proposed to solve the problemthat tlie unoptim izable tim e consumption and discontinuous im pact exist in the current$ajectory fo r erection equipment containing the two-stage electric cylinder.Actuating constrains and boundary conditions are obtained through the analysis o f kinetic a nd dynamic models fo r erection equipment.The five order B-spline interpolation algorithm was adopted fo r t rajectory planning,and penalty function was introduced to process constrains.An N S G A-II algorithm was used to conduct m ulti-objective optim ization fo r B-spline trajectory w hile the internals between eadi control points o f splines were taken as the desig^ned va ia tio n s.The calculation result was a Pareto optim al solution aggregate,from w h id i the optim al solution is chosen fo r being the smoothest and the quickest set o f intervals.The sim ulation results show that spline interj^olation could achieve optim al tim e consumption theoretically w hile the polynom ial couldnN.And high order spline interpolation led to high order continuous trajectories,which makes the erection process more stable.Key w ords:erection equipm ent;electric cylind er;trajectory planning;B-spline;m ulti-objective optim ization0引言二级电动缸相较单级电动缸具有更长的行程，在 要求结构紧凑的工业现场设备、农机农具以及大行程 军用雷达、火炮、导弹起落架等起竖装置中均有广泛应 用[1]。
MATLAB机械臂的两种路径规划
MATLAB机械臂的两种路径规划
什么是trajectory（路径）规划
中文路径在英语中可能有两种翻译：
1. path
2. trajectory
首先告诉大家，我们所说的“路径”的是后者——trajectory。
我们看一下这两种“路径”在机械臂的世界里有什么区别。
设想机械臂的end-effector 要从A 点运动到B 点：
从A 到B 叫path：
如果我们规定从A 到B 这个path，必须在特定的时间（t0 –t6），经过这7 个点。
那么这7 个点叫waypoints。
路径（trajectory）指的是通过这7 个点的具体计划（例如什么时间，以什么速度等等）。
规划我们如何经过这7 个点的算法，叫做路径规划（trajectory planning）算法。
上面这7 个点，如果机械臂的end-effector 在通过的时候，位置连续、速度连续、甚至加速度也是连续的，那么我们说这个trajectory 是平滑的（smooth）。
机械臂中有几种比较常见的trajectory：
Trapezoidal Trajectories（梯形路径）
梯形路径指的是：
end-effector 在相邻waypoints 之间的速度是象梯形一样，先线性加速，然后速度保持不变，在接近目标后线性减速。
梯形路径的优点是比较简单实用，但是在每个waypoint 会有停顿。
下图是一个机械臂走梯形路径的样例：在过waypoints 的时候，会完全停止，所以走的是直线，速度和加速度肯定不连续。
b样条的边界函数 解释说明以及概述
b样条的边界函数解释说明以及概述1. 引言1.1 概述B样条是一种常见且重要的插值和逼近方法，广泛应用于曲线和曲面的建模、图像处理、计算机辅助设计等领域。
在B样条的构造过程中，边界函数起着关键的作用。
边界函数定义了B样条在区域边界处的行为，对于控制曲线或曲面的形状具有重要影响。
本文将深入探讨B样条边界函数及其在曲线和曲面建模中的应用。
我们将介绍B 样条的基本理论，包括其定义与特点，插值方法以及逼近方法。
随后，我们重点解释说明了B样条边界函数的概念，并分析了常见的边界函数类型及其特点。
最后，我们将通过案例分析展示B样条边界函数在实际应用中的选择与调整技巧，并总结回顾本文所涉及到的内容。
1.2 文章结构本文按照以下结构进行组织：首先，在引言部分（第1节），我们简要介绍了全文内容以及文章目录。
然后，在第2节中，我们详细讨论了B样条的基本理论，包括定义与特点、插值方法以及逼近方法。
接着，在第3节中，我们重点解释了B样条边界函数的概念，并分析了常见的边界函数类型及其特点。
在第4节中，我们通过曲线和曲面建模的应用案例来说明B样条边界函数的实际应用情况。
最后，在第5节中，我们对全文进行总结回顾，并展望未来研究方向。
1.3 目的本文旨在帮助读者深入理解B样条边界函数的概念、特点以及实际应用。
通过对B样条的基本理论和边界函数的详细讲解，读者将能够掌握如何选择和调整B 样条边界函数以满足具体需求。
此外，本文还通过应用案例分析展示了B样条边界函数在曲线和曲面建模中的实际应用价值。
希望读者通过本文能够获得有效信息，并为相关领域的研究和实践提供参考依据。
2. B样条的基本理论2.1 B样条的定义与特点B样条是一种数学曲线或曲面的表示方法，它在计算机图形学和计算机辅助设计领域得到了广泛应用。
B样条由控制点和节点向量确定。
其中，控制点用于确定曲线或曲面上的特定位置，而节点向量则用于定义参数空间中的区间。
B样条具有以下特点：- 局部控制性：B样条在整个参数空间内都具有局部影响力，即通过调整某一个或某几个控制点可以改变局部区域的形状。
基于B样条曲线的无人车路径规划算法
基于B样条曲线的无人车路径规划算法作者：屈盼让李林任晓琨荆立雄来源：《电脑知识与技术》2016年第26期摘要：路径规划是无人车运动控制的重要研究内容之一，路径规划算法的性能优劣直接影响无人车能否在复杂交通场景中自主行驶。
考虑到车辆运动学模型对轨迹平滑性的要求，本文提出了基于B样条曲线解决无人车路径规划问题的方法，建立了完整的分步规划模型，即将路径规划的过程分为路径簇生成、最大曲率约束、碰撞检测以及最优评价四个步骤串行进行，最终得到最优路径。
该方法实现过程简单，运行效率较高，仿真结果表明了本文算法的有效性。
关键词： B样条曲线；无人车；路径规划；碰撞检测；最大曲率约束；最优路径中图分类号：TP181 文献标识码：A 文章编号：1009-3044（2016）26-0235-03B-spline Curve based Trajectory Planning for Autonomous VehiclesQU Pan-rang，LI Lin ， REN Xiao-kun ，JING Li-xiong（Institute of Aeronautics Computing Technique Research，Xi’an 710065， China）Abstract：Path planning is an important research topic in the field of the unmanned vehicle motion planning， and it directly affects the performance of unmanned vehicles in a complex traffic environment. Taking the requirement for smoothness into account， this paper proposed a method based on B-spline curve and built a planning model which can be divided into four steps， including path clusters， constraint of maximal curvature， collision detection and optimal path. This method works efficiently and simulation results show efficiency of the method.Key words：B-spline curve； autonomous vehicle； path planning； collide detection；constraint of maximal curvature1 引言近年来，无人驾驶技术备受关注，各大研究机构和企业争相推出各自的无人驾驶平台。
b样条曲面拟合 点云补全算法
样条曲面拟合在计算机图形学和计算机辅助设计中起着至关重要的作用。
而点云补全算法则可以用来处理三维模型中的空洞，使得模型更加完整。
本文将介绍样条曲面拟合和点云补全算法，以及它们在实际应用中的重要意义。
一、样条曲面拟合1. 什么是样条曲面样条曲面是一种平滑的、连续的曲面模型。
它由一个或多个贝塞尔曲线或B样条曲线组成。
样条曲面通常用来近似表示物体的外形，因为它具有高度的灵活性和精度。
2. 样条曲面拟合的意义样条曲面拟合可以用来找到一组曲面，使得这些曲面与给定的点集的拟合误差最小。
在实际应用中，样条曲面拟合可以用来重建缺失的曲面数据、对原始数据进行平滑处理、提取数据中的特征等。
3. 样条曲面拟合的算法常见的样条曲面拟合算法包括最小二乘法拟合、最小二乘法加惩罚项拟合、移动最小二乘法拟合等。
这些算法在拟合过程中会考虑拟合误差和模型的复杂度，以找到最优的拟合曲面。
二、点云补全算法1. 什么是点云补全点云补全是指通过给定的部分点云数据，来推断出完整的三维模型。
在现实世界中，许多三维扫描设备得到的数据存在缺失或者噪声，因此需要进行点云补全来修复这些缺陷。
2. 点云补全的意义点云补全算法可以用来处理三维重建、虚拟现实、医学图像处理等领域。
它可以帮助用户从不完整的数据中还原出完整的物体模型，提高数据的有效性和可视化效果。
3. 点云补全算法的分类点云补全算法可以分为基于拓扑结构的算法、基于形态学的算法、基于学习的算法等。
这些算法在补全过程中考虑了点云数据的拓扑关系、形状信息和学习模型，以较好地还原出完整的三维模型。
三、样条曲面拟合与点云补全的结合1. 样条曲面拟合在点云补全中的应用通过对局部点云数据进行样条曲面拟合，可以得到平滑的曲面模型。
这些模型可以用来填补点云数据中的空缺，从而实现点云补全的目的。
2. 点云补全在样条曲面重建中的应用点云补全算法可以用来处理样条曲面模型中的缺失部分，从而得到完整的曲面模型。
这样可以提高样条曲面的精度和完整性，使得它更加逼真和可靠。
基于遗传算法B样条曲线优化在机器人轨迹规划中应用
基于遗传算法B样条曲线优化在机器人轨迹规划中应用一、概述随着科技的飞速发展，机器人技术已成为现代工程领域中的研究热点。
在机器人技术中，轨迹规划是机器人运动控制的关键环节，直接影响到机器人的运动性能、工作效率及能量消耗。
传统的机器人轨迹规划方法往往基于预设的路径进行优化，但在复杂环境和动态任务面前，这种方法的灵活性和适应性显得不足。
探索更为智能、高效的轨迹规划方法显得尤为重要。
基于遗传算法的B样条曲线优化在机器人轨迹规划中的应用逐渐受到关注。
遗传算法是一种模拟生物进化过程的智能优化算法，具有强大的全局搜索能力和自适应性，能够处理复杂的非线性、多峰值优化问题。
B样条曲线作为一种灵活的曲线描述工具，能够精确地表示复杂的空间曲线，且在机器人轨迹规划中具有良好的连续性和平滑性。
通过将遗传算法与B样条曲线优化相结合，可以在机器人轨迹规划中实现更为智能的优化过程。
可以利用遗传算法的全局搜索能力，对B样条曲线的控制点进行优化，从而得到更合适的机器人轨迹。
这种方法不仅可以提高机器人的运动性能，还可以适应复杂环境和动态任务的变化，为机器人轨迹规划提供新的解决方案。
基于遗传算法B样条曲线优化在机器人轨迹规划中的应用，是一种具有潜力的新方法。
本文旨在深入探讨这一方法的理论基础、实现过程、优势及其在实际应用中的效果，为机器人轨迹规划的研究提供新的思路和方向。
1. 介绍机器人技术的快速发展和广泛应用。
随着科技的飞速进步，机器人技术已成为当今工业、医疗、军事、服务等多个领域不可或缺的重要技术。
机器人技术的快速发展得益于计算机、电子、传感器、人工智能等多个领域的融合与协同推进。
尤其在轨迹规划方面，随着算法的持续创新和优化，机器人运动的精准性和效率性不断提高。
在此背景下，机器人轨迹规划的重要性愈发凸显，它关乎机器人的工作效率、能源利用率以及人机交互的流畅性。
机器人技术的广泛应用使得其在生产制造、物流运输、医疗手术、家庭服务等领域大放异彩，极大地提高了生产效率和人们的生活质量。
B-SPLINE
问题
比传统的曲线曲面定义方法需要更多的存储 空间，如空间圆需7个参数(圆心、半径、法 矢)，而NURBS定义空间圆需38个参数。 权因子选择不当会引起畸变。 对搭接、重叠形状的处理很麻烦。 反求曲线、曲面上点参数值的算法不稳定。
19:15
非均匀B样条基函数
19:15
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3)分割法．该法与用网格离散法有些类似，都是 以小平面片的交线代替曲面的交线．所不同的是分 割法不是将曲面直接离散，而是基于Divide— Conquer思想，亦称测试—离散思想，即在对两曲 面片离散之前，先利用曲面片的凸包进行相交测试， 并只对凸包相交曲面片进行细分．将凸包相交的曲 面片细分为四个子曲面片．对两张曲面的子曲面片 重复前述过程，直至子曲面片满足求交的精度要求， 而后以平面片的交线代替曲面片的交线．分割法仅 对可能相交的曲面片细分，既减少了需要的存储空 间，又加快了测试和求交速度，因而其效果显著优 于网格离散法．但分割法需要应用曲面的凸包性和 分割性，故仅适用于Bezier和B样条曲面，而难以 应用于如Coons曲面和等距面等其他参数曲面．分 割求交的精度比较低，这是因为平面片的交线偏离 真实交线的误差比较大．实际应用中，分割法也需 要与其他方法结合才能获得比较满意的效果．
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在实际的求交算法中，分割法常用于计算交 点的初始估计值，并不需要很高的分割深度， 故等深度分割的缺点并不明显． 在一般情况下以采用等深度离散分割法为 好．
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曲面分割求交的算法
B-spline Curves(B样条曲线)
MotivationConsider designing the profile of a vase. The left figure below is a Bézier curve of degree 11; but, it is difficult to bend the "neck" toward the line segment P4P5. Of course, we can add more control points near this segment to increase the weight to that region. However, this will increase the degree of the curve. In many cases, it is not worth to use such a high degree polynomial.As discussed in a previous page about the derivatives of a Bézier curve, we can join two Bézier curves together. As long as the last leg of the first curve and the first leg of the second have the same direction, we can at least achieve G1 continuity because the tangent vectors have the same direction but may have different length (i.e., if the lengths are the same, it becomes C1 continuous). The middle figure above uses this idea. It has three Bézier curve segments of degree 3 with joining points marked with yellow rectangles. This shows that with multiple low degree Bézier curve segments satisfying the G1 continuous condition, we still can design complex shapes. But, maintaining this G1continuous condition may be tedious and undesirable.Is it possible that we still can use lower degree curve segments without worrying about the G1 continuous condition? B-spline curves are generalizations of Bézier curves and are developed to answer this question. The right figure above is a B-spline curve of degree 3 defined by 8 control points. In fact, there are five Bézier curve segments of degree 3 joining together to form the B-spline curve defined by the control points. In the above, those little dots subdivide the B-spline curve into Bézier curve segments. One can move control points for modifying the shape of the curvejust like what we do to Bézier curves. We can also modify the subdivision of the curve. Therefore, B-spline curves have higher degree of freedom for curve design.Subdividing the curve directly is difficult to do. Instead, we subdivide the domain of the curve. Thus, if the domain of a curve is [0,1], this closed interval is subdivided by points called knots. Let these knots be 0 <= u0 <= u1 <= ... <= u m <= 1. Then, points C(u i)'s subdivide the curve as shown in the figure below and, consequently, modifying the subdivision of [0,1] changes the shape of the curve.In summary, to design a B-spline curve, we need a set of control points, a set of knots and a set of coefficients, one for each control point, so that all curve segments are joined together satisfying certain continuity condition. The computation of the coefficients is perhaps the most complex step because they must ensure certain continuity conditions. Fortunately, this computation is usually not needed in this course. We only need to know their characteristics for reasoning about B-spline curves.B-spline Basis Functions:DefinitionBézier basis functions are used as weights. B-spline basis functions will be used the same way; however, they are much more complex. There are two interesting properties that are not part of the Bézier basis functions, namely: (1) the domain is subdivided by knots, and (2) basis functions are not non-zero on the entire interval. In fact, each B-spline basis function is non-zero on a few adjacent subintervals and, as a result, B-spline basis functions are quite "local".Let U be a set of m + 1 non-decreasing numbers, u0 <= u2 <= u3 <= ... <= u m. The u i's are called knots, the set U the knot vector, and the half-open interval [u i, u i+1) the i-th knot span. Note that since some u i's may be equal, some knot spans may not exist.If a knot u i appears k times (i.e., u i = u i+1 = ... = u i+k-1), where k > 1, u i is a multiple knot of multiplicity k, written as u i(k). Otherwise, if u i appears only once, it is a simple knot. If the knots are equally spaced (i.e., u i+1 - u i is a constant for 0 <= i <= m - 1), the knot vector or the knot sequence is said uniform; otherwise, it is non-uniform. The knots can be considered as division points that subdivide the interval [u0, u m] into knot spans. All B-spline basis functions are supposed to have their domain on [u0, u m]. In this note, we use u0 = 0 and u m = 1 frequently so that the domain is the closed interval [0,1].To define B-spline basis functions, we need one more parameter, the degree of these basis functions, p. The i-th B-spline basis function of degree p, written as N i,p(u), is defined recursively as follows:The above is usually referred to as the Cox-de Boor recursion formula. This definition looks complicated; but, it is not difficult to understand. If the degree is zero (i.e., p = 0), these basis functions are all step functions and this is what the first expression says. That is, basis function N i,0(u) is 1 if u isin the i-th knot span [u i, u i+1). For example, if we have four knots u0 = 0, u1 = 1, u2 = 2 and u3 = 3, knot spans 0, 1 and 2 are [0,1), [1,2), [2,3) and the basis functions of degree 0 are N0,0(u) = 1 on [0,1) and 0 elsewhere, N1,0(u) = 1 on [1,2) and 0 elsewhere, and N2,0(u) = 1 on [2,3) and 0 elsewhere. This is shown below:To understand the way of computing N i,p(u) for p greater than 0, we use the triangular computation scheme. All knot spans are listed on the left (first) column and all degree zero basis functions on the second. This is shown in the following diagram.To compute N i,1(u), N i,0(u) and N i+1,0(u) are required. Therefore, we can compute N0,1(u), N1,1(u), N2,1(u), N3,1(u) and so on. All of these N i,1(u)'s are written on the third column. Once all N i,1(u)'s have been computed, we can compute N i,2(u)'s and put them on the fourth column. This process continues until all required N i,p(u)'s are computed.In the above, we have obtained N0,0(u), N1,0(u) and N2,0(u) for the knot vector U = { 0, 1, 2, 3 }. Let us compute N0,1(u) and N1,1(u). To compute N0,1(u), since i = 0 and p = 1, from the definition we haveSince u0 = 0, u1 = 1 and u2 = 2, the above becomesSince N0,0(u) is non-zero on [0,1) and N1,0(u) is non-zero on [1,2), if u is in [0,1) (resp., [1,2) ), only N0,0(u) (resp., N1,0(u) ) contributes to N0,1(u). Therefore, if u is in [0,1), N0,1(u) is uN0,0(u) = u, and if u is in [1,2),N0,1(u) is (2 - u)N1,0(u) = (2 - u). Similar computation gives N1,1(u) = u - 1 if u is in [1,2), and N1,1(u) = 3 - u if u is in [2,3). In the following figure, the black and red lines are N0,1(u) and N1,1(u), respectively. Note that N0,1(u) (resp., N1,1(u)) is non-zero on [0,1) and [1,2) (resp., [1,2) and [2,3)).Once N0,1(u) and N1,1(u) are available, we can compute N0,2(u). The definition gives us the following:Plugging in the values of the knots yieldsNote that N0,1(u) is non-zero on [0,1) and [1,2) and N1,1(u) is non-zero on [1,2) and [2,3). Therefore, we have three cases to consider:1.u is in [0,1):In this case, only N0,1(u) contributes to the value of N0,2(u).Since N0,1(u) is u, we have2.u is in [1,2):In this case, both N0,1(u) and N1,1(u) contribute to N0,2(u). Since N0,1(u) = 2 - u and N1,1(u) = u - 1 on [1,2), we have3.u is in [2,3):In this case, only N1,1(u) contributes to N0,2(u). Since N1,1(u) = 3 - u on [2,3), we haveIf we draw the curve segment of each of the above three cases, we shall see that two adjacent curve segments are joined together to form a curve at the knots. More precisely, the curve segments of the first and second cases join together at u = 1, while the curve segments of the second and third cases join at u = 2. Note that the composite curve shown here is smooth. But in general it is not always the case if a knot vector contains multiple knots.Two Important ObservationsSince N i,1(u) is computed from N i,0(u) and N i+1,0(u) and since N i,0(u) and N i+1,0(u) are non-zero on span [u i, u i+1) and [u i+1, u i+2), respectively, N i,1(u) is non-zero on these two spans. In other words, N i,1(u) is non-zero on [u i, u i+2). Similarly, since N i,2(u) depends on N i,1(u) and N i+1,1(u) and since these two basis functions are non-zero on [u i, u i+2) and [u i+1, u i+3), respectively, N i,2(u) is non-zero on [u i, u i+3). In general, to determine the non-zero domain of a basis function N i,p(u), one can trace back using the triangular computation scheme until it reaches the first column. The covered spans are the non-zero domain of this basis function. For example, suppose we want to find out the non-zero domain of N1,3(u). Based on the above discussion, we can trace back in the north-west and south-west directions until the first column is reached as shown with the blue dotted line in the following diagram. Thus, N1,3(u) is non-zero on [u1, u2), [u2, u3), [u3, u4) and [u4, u5). Or, equivalently, it is non-zero on [u1, u5).In summary, we have the following observation:Basis function N i,p(u) is non-zero on [u i, u i+p+1). Or,equivalently, N i,p(u) is non-zero on p+1 knot spans [u i, u i+1),[u i+1, u i+2), ..., [u i+p, u i+p+1).Next, we shall look at the opposite direction. Given a knot span [u i, u i+1), we want to know which basis functions will use this span in its computation. We can start with this knot span and draw a north-east bound arrow and a south-east bound arrow. All basis functions enclosed in this wedge shape use N i,0(u) (why?) and hence are non-zero on this span. Therefore, all degree p basis functions that are non-zero on [u i, u i+1) are the intersection of this wedge and the column that contains all N i,p(u)'s. In fact, this column and the two arrows form an equilateral triangle with this column being the vertical side. Counting from N i,0(u) to N i,p(u) there are p+1 columns. Therefore, the vertical side of the equilateral triangle must have at most p+1 entries, namely N i,p(u), N i-1,p(u), N i-2,p(u), ..., N i-p+2,p(u), N i-p+1,p(u) and N i-p,p(u).Let us take a look at the above diagram. To find all degree 3 basis functions that are non-zero on [u4, u5), draw two arrows and all functions on the vertical edges are what we want. In this case, they are N1,3(u), N2,3(u), N3,3(u), and N4,3(u). This is shown with the orange triangle. The blue (resp., red) triangle shows the degree 3 basis functions that are non-zero on [u3, u4) (resp., [u2, u3) ). Note that there are only three degree three basis polynomials that are non-zero on [u2, u3).In summary, we have observed the following property.On any knot span [u i, u i+1), at most p+1 degree p basisfunctions are non-zero,namely: N i-p,p(u), N i-p+1,p(u), N i-p+2,p(u), ..., N i-1,p(u)and N i,p(u),What Is the Meaning of the Coefficients?Finally, let us investigate the meaning of the coefficients in the definition of N i,p(u). As N i,p(u) is being computed, it uses N i,p-1(u) and N i+1,p-1(u). The former is non-zero on [u i, u i+p). If u is in this half-open interval, then u -u i is the distance between u and the left end of this interval, the interval length is u i+p - u i, and (u - u i) / (u i+p - u i) is the ratio of the above mentioned distances and is always in the range of 0 and 1. See the diagram below. The second term, N i,p-1(u), is non-zero on [u i+1, u i+p+1). If u is in this interval, then u i+p+1 - u is the distance from u to the right end of this interval, u i+p+1 - u i+1 is the length of the interval, and (u i+p+1 - u) / (u i+p+1 - u i+1) is the ratio of these two distances and its value is in the range of 0 and 1. Therefore, N i,p(u) is a linear combination of N i,p-1(u) and N i+1,p-1(u) with two coefficients, both linear in u, in the range of 0 and 1.Important PropertiesLet us recall the definition of the B-spline basis functions as follows:This set of basis functions has the following properties, many of which resemble those of Bézier basis functions.1.N i,p(u) is a degree p polynomial in u2.Nonnegativity -- For all i, p and u, N i,p(u) is non-negative3.Local Support -- N i,p(u) is a non-zero polynomial on [u i,u i+p+1)This has been discussed on previous page.4.On any span [u i, u i+1), at most p+1 degree p basis functions arenon-zero, namely: N i-p,p(u), N i-p+1,p(u), N i-p+2,p(u), ..., and N i,p(u)5.Partition of Unity -- The sum of all non-zero degree p basisfunctions on span [u i, u i+1) is 1:The previous property shows that N i-p,p(u), N i-p+1,p(u), N i-p+2,p(u), ..., and N i,p(u) are non-zero on [u i, u i+1). This one states that the sum of these p+1 basis functions is 1.6.If the number of knots is m+1, the degree of the basis functionsis p, and the number of degree p basis functions is n+1, then m = n + p + 1:This is not difficult to see. Let N n,p(u) be the last degree p basis function. It is non-zero on [u n, u n+p+1). Since it is the last basis function, u n+p+1 must be the last knot u m. Therefore, we have u n+p+1 = u m and n+p+1= m. In summary, given m and p, let n = m-p -1 and the degree p basis functions are N0,p(u), N1,p(u), N2,p(u), ..., and N n,p(u).7.Basis function N i,p(u) is a composite curve of degree p polynomialswith joining points at knots in [u i, u i+p+1 )The example shown on the previous page illustrates this property well.For example, N0,2(u), which is non-zero on [0,3), is constructed from three parabolas defined on [0,1), [1,2) and [2,3). They are connected together at the knots 2 and 3.8.At a knot of multiplicity k, basis function N i,p(u)is C p-k continuous.Therefore, increasing multiplicity decreases the level of continuity, and increasing degree increases continuity. The above mentioned degree two basis function N0,2(u) is C1 continuous at knots 2 and 3, since they are simple knots (k = 1).The Impact of Multiple KnotsMultiple knots do have significant impact on the computation of basis functions and some "counting" properties. We shall look at two of them anda calculation example will be provided on the next page.1.Each knot of multiplicity k reduces at most k-1 basis functions'non-zero domain.Consider N i,p(u) and N i+1,p(u). The former is non-zero on [u i, u i+p+1) while the latter is non-zero on [u i+1, u i+p+2). If we move u i+p+2 to u i+p+1 so that they become a double knot. Then, N i,p(u) still has p+1 knot spans on which it is non-zero; but, the number of knot spans on which N i+1,p(u) is non-zero is reduced by one because the span [u i+p+1,u i+p+2) disappears.This observation can be generalized easily. In fact, ignoring the change of knot span endpoints, to create a knot of multiplicity k, k-1 basis functions will be affected. One of them loses one knot span, a second of them loses two, a third of them loses three and so on.The above shows the basis functions of degree 5 where the left end and right end knots have multiplicity 6, while all middle knots are simple (left on top row). The top right figure is the result of moving u5 to u6. Those basis functions ended at u6 have fewer knot spans on which they are non-zero. In the middle row figures, u4 and then u3 are moved to u6, making u6 a knot of multiplicity 4. The bottom row figure shows the result after moving u2 to u6, creating a knot of multiplicity 5.2.At each internal knot of multiplicity k, the number of non-zerobasis functions is at most p - k + 1, where p is the degree of the basis functions.Since moving u i-1 to u i will pull a basis function whose non-zero ends at u i-1 to end at u i, this reduces the number of non-zero basis function at u i by one. More precisely, increasing u i's multiplicity by one will decrease the number of non-zero basis functions by one. Since there are at most p+1 basis functions can be non-zero at u i, the number of non-zero basis functions at a knot of multiplicity k is at most (p + 1) - k = p - k + 1.In the above figures, since the multiplicity of knot u6 are 1 (simple), 2, 3, 4 and 5, the numbers of non-zero basis function at u6 are 5, 4, 3, 2 and 1.Computation ExamplesTwo examples, one with all simple knots while the other with multiple knots, will be discussed in some detail on this page.Simple KnotsSuppose the knot vector is U = { 0, 0.25, 0.5, 0.75, 1 }. Hence, m = 4 and u0 = 0, u1 = 0.25, u2 = 0.5, u3 = 0.75 and u4 = 1. The basis functions of degree 0 are easy. They are N0,0(u), N1,0(u), N2,0(u) and N3,0(u) defined on knot span [0,0.25,), [0.25,0.5), [0.5,0.75) and [0.75,1), respectively, as shown below.The following table gives the result of all N i,1(u)'s:The following shows the graphs of these basis functions. Since the internal knots 0.25, 0.5 and 0.75 are all simple (i.e., k = 1) and p = 1, there are p - k +1 = 1 non-zero basis function and three knots. Moreover, N0,1(u), N1,1(u) and N2,1(u) are C0 continuous at knots 0.25, 0.5 and 0.75, respectively. From N i,1(u)'s, one can compute the basis functions of degree 2. Since m = 4, p = 2, and m = n + p + 1, we have n = 1 and there are only two basis functions of degree 2: N0,2(u) and N1,2(u). The following table is the result:The following figure shows the two basis functions. The three vertical blue lines indicate the positions of knots. Note that each basis function is a composite curve of three degree 2 curve segments. For example, N0,2(u) is the green curve, which is the union of three parabolas defined on [0,0.25),[0.25, 0.5) and [0.5,0.75). These three curve segments join together forminga smooth bell shape. Please verify that N0,2(u,) (resp., N1,2(u)) is C1 continuous at its knots 0.25 and 0.5 (resp., 0.5 and 0.75). As mentioned on the previous page, at the knots, this composite curve is of C1 continuity.Knots with Positive MultiplicityIf a knot vector contains knots with positive multiplicity, we will encounter the case of 0/0 as will be seen later. Therefore, we shall define 0/0 to be 0. Fortunately, this is only for hand calculation. For computer implementation, there is an efficient algorithm free of this problem. Furthermore, if u i is a knot of multiplicity k (i.e., u i = u i+1 = ... = u i+k-1), then knot spans [u i,u i+1), [u i+1,u i+2), ..., [u i+k-2,u i+k-1) do not exist, and, as a result, N i,0(u), N i+1,0(u), ..., N i+k-1,0(u) are all zero functions.Consider a knot vector U = { 0, 0, 0, 0.3, 0.5, 0.5, 0.6, 1, 1, 1 }. Thus, 0 and 1 are of multiplicity 3 (i.e., 0(3) and 1(3)) and 0.5 is of multiplicity 2 (i.e., 0.5(2)). As a result, m = 9 and the knot assignments areLet us compute N i,0(u)'s. Note that since m = 9 and p = 0 (degree 0 basis functions), we have n = m - p - 1 = 8. As the table below shows, there are only four non-zero basis functions of degree 0:N2,0(u), N3,0(u), N5,0(u) and N6,0(u).Then, we proceed to basis functions of degree 1. Since p is 1, n = m - p - 1 = 7. The following table shows the result:The following figure shows the graphs of these basis functions.Let us take a look at a particular computation, say N1,1(u). It is computed with the following expression:Plugging u1 = u2 = 0 and u3 = 0.3 into this equation yields the following:Since N1,0(u) is zero everywhere, the first term becomes 0/0 and is defined to be zero. Therefore, only the second term has an impact on the result. Since N2,0(u) is 1 on [0,0.3), N1,1(u) is 1 - (10/3)u on [0,0.3).Next, let us compute all N i,2(u)'s. Since p = 2, we have n = m - p - 1 = 6. The following table contains all N i,2(u)'s:The following figure shows all basis functions of degree 2.Let us pick a typical computation as an example, say N3,2(u). The expression for computing it isPlugging in u3 = 0.3, u4 = u5 = 0.5 and u6 = 0.6 yieldsSince N3,1(u) is non-zero on [0.3, 0.5) and is equal to 5u - 1.5, (5u - 1.5)2 is the non-zero part of N3,2(u) on [0.3, 0.5). Since N4,1(u) is non-zero on [0.5, 0.6) and is equal to 6 - 10u, (6 - 10u)2 is the non-zero part of N3,2(u) on [0.5, 0.6).Let us investigate the continuity issues at knot 0.5(2). Since its multiplicity is 2 and the degree of these basis functions is 2, basis function N3,2(u) is C0 continuous at 0.5(2). This is why N3,2(u) has a sharp angle at 0.5(2). For knots not at the two ends, say 0.3, C1 continuity is maintained since all of them are simple knots.B-spline Curves:DefinitionGiven n +1 control points P0, P1, ..., P n and a knot vector U = { u0, u1, ..., u m }, the B-spline curve of degree p defined by these control points and knot vector U iswhere N i,p(u)'s are B-spline basis functions of degree p. The form of a B-spline curve is very similar to that of a Bézier curve. Unlike a Bézier curve, a B-spline curve involves more information, namely: a set of n+1 control points, a knot vector of m+1 knots, and a degree p.Note that n, m and p must satisfy m = n+p+1. More precisely, if we want to define a B-spline curve of degree p with n+1 control points, we have to supply n+p+2 knots u0, u1, ..., u n+p+1. On the other hand, if a knot vector of m + 1 knots and n + 1 control points are given, the degree of the B-spline curve is p =m - n - 1. The point on the curve that corresponds to a knot u i, C(u i), is referred to as a knot point. Hence, the knot points divide a B-spline curve into curve segments, each of which is defined on a knot span. We shall show that these curve segments are all Bézier curve of degree p on the curve subdivision page.Although N i,p(u) looks like B n,i(u), the degree of a B-spline basis function is an input, while the degree of a Bézier basis function depends on the number of control points. To change the shape of a B-spline curve, one can modify one or more of these control parameters: the positions of control points, the positions of knots, and the degree of the curve.If the knot vector does not have any particular structure, the generated curve will not touch the first and last legs of the control polyline as shown in the left figure below. This type of B-spline curves is called open B-spline curves. We may want to clamp the curve so that it is tangent to the first and the last legs at the first and last control points, respectively, as a Bézier curve does. To do so, the first knot and the last knot must be of multiplicity p+1. This will generate the so-called clamped B-spline curves. See the middle figure below. By repeating some knots and control points, the generated curve can be a closed one. In this case, the start and the end of the generated curve jointogether forming a closed loop as shown in the right figure below. In this note, we shall use clamped curve.(a) open (b) clamped (c) closedThe above figures have n+1 control points (n=9) and p = 3. Then, m must be 13 so that the knot vector has 14 knots. To have the clamped effect, the first p+1 = 4 and the last 4 knots must be identical. The remaining 14 - (4 + 4) = 6 knots can be anywhere in the domain. In fact, the curve is generated with knot vector U = { 0, 0, 0, 0, 0.14, 0.28, 0.42, 0.57, 0.71, 0.85, 1, 1, 1, 1 }. Note that except for the first four and last four knots, the middle ones are almost uniformly spaced. The figures also show the corresponding curve segment on each knot span. In fact, the little triangles are the knot points. The Use of TermsWe use open, clamped and closed to describe three types of B-spline curves. However, not every author would use the same terminology and there is no agreement for a standard use. For example, some authors may use floating, open and periodic for open, clamped and closed curves. Some other authors may use "periodic" for an open B-spline curve with a uniform knot sequence. Therefore, when you read other literature, make sure you first check the definitions. Otherwise, it is likely that you could be confused, Please continue with open curves and closed curves.Open CurvesAs mentioned earlier, if the first and last knots do not have multiplicity p+1, where p is the degree of a B-spline curve, the curve will not be tangent to the first and last legs at the first and last control points, respectively. Thecurve is an open B-spline curve. In this case, we should be careful about one additional restriction. On a previous page we showed an example of basis function computation using knot vector U = { 0, 0.25, 0.5, 0.75, 1 }, where m = 4. If the basis functions are of degree 1 (i.e., p = 1), there are three basis functions N0,1(u), N1,1(u) and N2,1(u) as shown below.Since this knot vector is not clamped, the first and the last knot spans (i.e., [0, 0.25) and [0.75, 1)) have only one non-zero basis functions while the second and third knot spans (i.e., [0.25, 0.5) and [0.5, 0.75)) have two non-zero basis functions. Recall from the B-spline basis function important properties that on a knot span [u i, u i+1), there are at most p+1 degree p non-zero basis functions of degree p. Therefore, in this example, knot spans [0,0.25) and [0.75,1) do not have "full support" of basis functions. In general, for degree p, intervals [u0, u p) and [u m-p, u m] will not have "full support" of basis functions and are ignored when a B-spline curve is open. Consequently, we have the following important note:Consider a B-spline curve of degree 6 (i.e., p = 6) defined by 14 control points (i.e., n = 13). The number of knots is 21 (i.e., m = n + p + 1 = 20). If the knot vector is uniform, the knots are 0, 0.05, 0.10, 0.15, ..., 0.90, 0.95 and 1.0. The open curve is defined on [u p, u n-p] = [u6, u14] = [0.3, 0.7] and is not tangent to the first and last legs. The left figure below shows the curve and the right figuregives the B-spline basis functions.Even though some knot spans at both end are not used, the B-spline curve still is defined by all control points. As discussed on the B-spline basis functions page that there are at most p+1 basis functions N i-p,p(u), N i-p,p+1(u), ..., N i,p(u) non-zero on knot span [u i, u i+1). Therefore, there are p+1 non-zero functions on [u p, u i+1): N0,p(u), N1,p(u), ..., N p,p(u). This is a "full" count. Note that N0,p(u) has its tail in [u p, u p+1) and, hence, the contribution of control point P0 to the construction of the open B-spline curve is smaller than those of most other control points. A similar argument can be used to show that P n is also involved in the definition of the curve.Let us use an example to illustrate the change between an open curve and a clamped one. We start with an open B-spline curve of degree 4 defined by 9 control points (i.e., n = 8) and a uniform knot vector { 0, 1/13, 2/13, 3/13, ..., 12/13, 1 }. If we change the second knot 1/13 to 0 making 0 a double knot, the curve is the yellow one in the figure. In fact, this curve and the original one with 0 being a simple knot are almost identical. Now, if we change the third knot 2/13 to 0 making 0 a knot of multiplicity 3, the result is the red curve. If the fourth knot 3/13 is change to 0 (multiplicity 4), the resulting curve is the blue one. As you can see, these three open curves are not very different from each other. Now, let us make the fifth knot 4/13 to 0. Since 0 is now a knot of multiplicity 5 (i.e., p+1), the curve not only passes through the first control point but also is tangent to the first leg of the control polyline (i.e., clamped). As you can see from the figure, the shape of the curve changes drastically by pulling one end of it to the first control point. The same holds true if we make the last 5 knots to 1.Closed CurvesThere are many ways to generate closed curves. The simple ones are either wrapping control points or wrapping knot vectors.Wrapping Control PointsSuppose we want to construct a closed B-spline curve C(u) of degree p defined by n+1 control points P0, P1, ..., P n. The number of knots is m+1, where m = n + p + 1. Here is the construction procedure:1.Design an uniform knot sequence of m+1 knots: u0 = 0, u1 = 1/m, u1 =2/m, ..., u m = 1. Note that the domain of the curve is [u p, u n-p]. See the discussion in open curves for the details.2.Wrap the first p and last p control points. More precisely,let P0 = P n-p+1, P1 = P n-p+2, ..., P p-2 = P n-1 and P p-1 = P n. This is shown in the figure below.。
构造带形状参数的二次均匀b样条曲线
构造带形状参数的二次均匀b样条曲线在计算机图形学领域中，二次均匀B样条曲线（Quadratic uniform B-spline curve）是常见的一种曲线类型。
该曲线以若干个控制点和一个节点向量（Knot Vector）为基础进行构造，其中节点向量描绘了参数空间中曲线的结构特征。
但是，在实际应用中，我们经常需要在B样条曲线的基础上增加形状参数，以满足不同的需求。
下面将结合实例解释如何构造带形状参数的二次均匀B样条曲线。
一、基本概念1.1 B样条基函数B样条基函数（B-spline basis function）是构造B样条曲线的基础，它描述了曲线的形状特征。
对于二次均匀B样条曲线，其基函数为：N(i,2)(t)={(t-i+1)^2, i≤t≤i+12(t-i+2)(i-t)+1, i+1≤t≤i+2(i+2-t)^2, i+2≤t≤i+3其中i为基函数的下标，t为参数值。
1.2 节点向量节点向量（Knot Vector）是描述曲线参数空间结构的重要参数，它定义了曲线的节点位置和节点跨度。
对于二次均匀B样条曲线，其节点向量为：U={0,0,0,1,2,3,4,4,4}其中，0为第一次节点，4为最后一次节点。
二、构造带形状参数的二次均匀B样条曲线2.1 基本构造以五个控制点构造带形状参数的二次均匀B样条曲线为例：控制点：P0(0,0)、P1(1,1)、P2(2,0)、P3(3,1)、P4(4,0)首先，根据节点向量U和基函数N(i,2)(t)计算出基于控制点的曲线点序列：C (t) =∑（ P(i) * N(i,2)(t) ）其中i为控制点的下标。
2.2 带形状参数的构造为曲线增加形状参数，需要对节点向量U进行调整。
对于二次均匀B样条曲线，节点向量中每个节点对应着一个控制多边形的顶点，调整节点向量的位置可以改变多边形的形状，从而改变曲线的形状。
以U'={0,0,p,p,2,3,4,4,4}为例，其中p为形状参数。
b样条的边界函数 -回复
b样条的边界函数-回复主题：B样条的边界函数引言：B样条（B-spline）是一种常用于绘制曲线和曲面的数学曲线表示方法。
它通过一组控制点和节点序列来定义曲线或曲面的形状。
作为一种灵活而强大的数学工具，B样条广泛应用于计算机图形学、CAD设计、动画制作等领域。
本文将详细介绍B样条的边界函数，包括定义、计算方法和应用等内容。
第一部分：B样条的定义和基本原理（200字）B样条是由一组控制点和节点序列定义的曲线或曲面。
控制点用于确定曲线或曲面的形状，而节点序列则用于定义B样条基函数的位置和权重。
B 样条基函数是一组非负函数，它们的加权和可以表示曲线或曲面上的任意点。
B样条基函数具有局部支持，也就是说，它们只在一定的节点区间内起作用。
第二部分：B样条的边界函数定义（500字）B样条的边界函数是一种特殊类型的基函数，用于计算曲线或曲面的边界上的点。
边界函数的特点是它们只在边界上的节点区间内起作用，而在其他节点区间内则为零。
在计算B样条边界函数时，需要根据节点的位置和权重来确定每一个边界函数的形状和大小。
B样条的边界函数可以分为两种情况：开放边界和闭合边界。
开放边界是指曲线或曲面的起始和终止点不相连，而闭合边界则是指起始和终止点相连，形成一个环状结构。
开放边界函数的计算相对简单，而闭合边界函数的计算则需要考虑节点序列的循环性。
在计算边界函数时，可以使用递归的方法来求解。
首先，根据给定的节点序列和控制点，计算出一组普通的B样条基函数。
然后，根据开放或闭合边界的特点，将普通基函数修剪为边界函数。
最后，利用修剪后的边界函数和控制点的权重，可以计算出边界上的点坐标。
第三部分：B样条边界函数的计算方法（500字）在计算B样条的边界函数时，关键是确定边界函数的形状和大小。
具体来说，计算边界函数的方法可以分为两步：节点修剪和权重调整。
节点修剪是指从节点序列中选择出用于计算边界函数的节点。
在开放边界中，只需要选择位于边界端点附近的节点；而在闭合边界中，则需要选择围绕整个环形结构的节点。
多形状参数的非均匀B样条曲线No...
多无人机航迹规划的自适应B样条算法
1引言由于无人机快速增长的市场和广阔的应用领域，吸引了越来越多的研究人员和研究机构。
与单个无人机相比，无人机集群能够以更高的效率完成更复杂和更具多无人机航迹规划的自适应B样条算法彭皓月1，2，秦小林1，2，侯屿1，2，张力戈1，21.中国科学院成都计算机应用研究所，成都6100412.中国科学院大学，北京100049摘要：无人机集群能更高效地完成复杂和具有挑战性的任务，航迹规划和编队控制是无人机集群的研究重点。
针对复杂环境下的无人机编队控制问题，提出了一种结合分段自适应B样条（Piecewise Adaptive B-Spline，PABS）方法的领航-跟随策略。
采用滚动时域控制及快速粒子群优化算法为领航者无人机生成一条安全的参考航迹，并根据跟随者与领航者保持的几何关系为跟随者无人机生成参考航迹。
针对生成的跟随者航迹不平滑以及可能与障碍物发生碰撞的问题，使用PABS方法对跟随者航迹进行平滑和避障处理。
实验表明，使用滚动时域控制及快速粒子群优化算法及PABS方法能为领航-跟随策略下的无人机编队生成安全平滑的航迹，相比于圆弧插补技术，PABS方法能使航迹更光滑。
关键词：无人机编队；滚动时域控制；B样条；航迹规划；航迹平滑文献标志码：A中图分类号：V279；TP391doi：10.3778/j.issn.1002-8331.1901-0157彭皓月，秦小林，侯屿，等.多无人机航迹规划的自适应B样条算法.计算机工程与应用，2020，56（9）：260-266. PENG Haoyue,QIN Xiaolin,HOU Yu,et al.Adaptive B-spline algorithm for multi-UAV path puter Engi-neering and Applications,2020,56（9）：260-266.Adaptive B-Spline Algorithm for Multi-UAV Path PlanningPENG Haoyue1，2,QIN Xiaolin1，2,HOU Yu1，2,ZHANG Lige1，21.Chengdu Institute of Computer Applications,Chinese Academy of Sciences,Chengdu610041,China2.University of Chinese Academy of Sciences,Beijing100049,ChinaAbstract：UAV clusters can perform complex and challenging tasks more efficiently,and path planning and formation control are the focus of research on UAV clusters.Aiming at the problem of UAV formation control in complex environ-ments,a leader-follower strategy combining Piecewise Adaptive B-Spline（PABS）is proposed.The Receding Horizon Control and the Fast Particle Swarm Optimization（RHC-FPSO）algorithm is used to generate a safe reference path for the leader.And the reference paths are generated for the followers according to the geometric relationship maintained by the followers and the leader.Followers’paths are smoothed by the PABS method for the problem that the paths of followers may not be smooth.Since the path will collide with obstacles in the environment,the path is planned by the adaptive B-spline method to achieve obstacle avoidance.Experiments show that the RHC-FPSO method and the PABS method can generate safe and smooth paths for the UA Vs in the formation under the leader-follower pared with the circular interpolation technique,the PABS method can make the path smoother.Key words：UAV formation;receding horizon control;B-spline;path planning;path smoothing基金项目：国家自然科学基金（No.61402537）；中国科学院西部青年学者项目；四川省委组织部人才专项支持；广西混杂计算与集成电路设计分析重点实验室开放基金（No.HCIC201706）。
均匀b样条基函数
均匀b样条基函数均匀B样条基函数是工程、计算机科学、金融等多个领域中广泛应用的计算机图形学中的专有名词。
B样条（B-spline）和Bézier曲线是计算机图形学中最简单、最常用的曲线之一。
B-spline曲线是由均匀B样条基函数（Uniform B-Spline Basis Function）和控制点（Control Points）组成的一种曲线。
均匀B样条基函数是一种数学函数，用于控制B样条曲线的形状。
均匀B样条基函数的定义是：给定定义域[a, b]和正整数k，均匀B样条基函数是由一个非零有限支撑的k次多项式定义的，支撑在定义域[a, b]上的每个B样条基函数由k+1个控制点控制，符合B样条条件。
在均匀B样条基函数中，定义域[a, b]被均分成k个区间，每个区间的长度为1/k。
均匀B样条基函数的基本形式为：N(i, k, t)，其中i表示B样条的第i段区间，k表示B样条的次数，t为取值范围在[0, 1]之间的参数，表示B样条的位置。
其中，具体的均匀B样条基函数的计算方式是以递归方式进行计算的，即：当k=0时：if i≤t<i+1, N(i, 0, t)=1; else N(i, 0, t)=0;当k≥1时：N(i, k, t)=[(t−i)/(i+k−1−i)]N(i, k−1, t)+[(i+k−i−t)/(i+k−i)]N(i+1, k−1, t)1. 大于等于1次的均匀B样条基函数都是非负的、单调递增的和有限支撑的。
最高次的B样条基函数是非负的和单调递增的，但是它的支撑不是有限的；2. 任意k-1个相邻的均匀B样条基函数的和为一个常数，即这个性质被称为插值条件，保证了B样条曲线穿过k-1个连续的控制点；3. 对于一个给定的控制点序列，B样条曲线是由一个k-1次连续可导的分段函数组成的；4. B样条曲线在任何一点处都具有局部控制的特性，即某个控制点位置的改变只可能影响到曲线某个区间内的局部形状。
基于PIA的B-Spline曲面实时交互修改方法
基于PIA的B-Spline曲面实时交互修改方法1. 引言- 研究背景和重要性- 相关工作综述- 研究目的和方法2. B-Spline曲面的基本原理- B-Spline曲线的定义和性质- B-Spline曲面的定义和性质- B-Spline曲面的表达方式3. PIA的实时交互修改方法- 数据结构的设计和实现- 点、线、面的实时修改算法- 边界控制方法- 曲面重构算法4. 算法的有效性和优越性- 实验结果分析- 与其他算法的比较- 算法的优势和局限性5. 结论- 研究总结和进一步工作的展望1. 引言B-Spline曲面是一种广泛应用于计算机图形学和计算机辅助设计领域的曲线表达方法，具有高度的灵活性和可控性。
B-Spline曲面的基本思想是将曲面分解为小块，用B-Spline函数来拟合每个小块，并通过不同的权值分配来控制曲面的形状。
因此，B-Spline曲面在许多应用场景中能够提供出色的曲面描述表现力和控制能力。
在实际应用中，B-Spline曲面的精确控制和实时交互修改是非常必要的。
基于传统的Bezier或B样条方法进行曲面修改时，需要重构整个曲面，在交互修改场景中，这样的操作代价很高且处理时间较慢，因此不适合实时交互修改。
为了解决这个问题，提出了一种基于PIA的B-Spline曲面实时交互修改方法，该方法利用了PIA数据结构的优点，可以在满足用户操作的同时，保持曲面与原始曲面的相对位置转移，从而实现了实时的曲面修改和交互。
本文的主要贡献如下：首先，对B-Spline曲面的基本原理进行了详细的阐述，包括B-Spline曲线和曲面的定义和性质，曲面的表达方式等；其次，介绍了基于PIA的实时交互修改方法的详细数据结构和算法设计；接着，通过实验结果分析和算法比较，说明了该算法的有效性和优越性；最后，总结了本文的工作，并展望了未来的研究方向。
本文的结构如下：Chapter 2介绍了B-Spline曲面的基本原理，其中包含了B-Spline曲线的定义和性质、B-Spline曲面的定义和性质以及B-Spline曲面的表达方式。
MATLAB机械臂的两种路径规划
MATLAB机械臂的两种路径规划什么是trajectory（路径）规划中文路径在英语中可能有两种翻译：1. path2. trajectory首先告诉大家，我们所说的“路径”的是后者——trajectory。
我们看一下这两种“路径”在机械臂的世界里有什么区别。
设想机械臂的end-effector 要从A 点运动到B 点：从A 到B 叫path：如果我们规定从A 到B 这个path，必须在特定的时间（t0 –t6），经过这7 个点。
那么这7 个点叫waypoints。
路径（trajectory）指的是通过这7 个点的具体计划（例如什么时间，以什么速度等等）。
规划我们如何经过这7 个点的算法，叫做路径规划（trajectory planning）算法。
上面这7 个点，如果机械臂的end-effector 在通过的时候，位置连续、速度连续、甚至加速度也是连续的，那么我们说这个trajectory 是平滑的（smooth）。
机械臂中有几种比较常见的trajectory：Trapezoidal Trajectories（梯形路径）梯形路径指的是：end-effector 在相邻waypoints 之间的速度是象梯形一样，先线性加速，然后速度保持不变，在接近目标后线性减速。
梯形路径的优点是比较简单实用，但是在每个waypoint 会有停顿。
下图是一个机械臂走梯形路径的样例：在过waypoints 的时候，会完全停止，所以走的是直线，速度和加速度肯定不连续。
Polynomial Trajectories（多项式路径）其中又分三次多项式和五次多项式。
分别写作：三次多项式可以记录end-effector 的速度、位置的信息。
五次多项式可以记录end-effector 的位置、速度、加速度的信息。
下图是一个五次多项式的例子，可以看到速度、加速度都是连续的。
三次或者五次多项式的trajectory 在通过waypoints 的时候速度都是连续的，这和梯形路径是不一样的。
基于B样条路径的机器人时间最优轨迹规划_钱东海 1998
第32卷第12期 1998年12月上海交通大学学报JO U R NA L O F SHA NG HA I JI AO T O N G U NI VER SI T Y Vol.32No.12 Dec.1998 收稿日期:1998-05-10*863网点基金(863-512-20-04)和中日合作基金资助项目钱东海:男,1971年生,博士生.邮编:200030基于B 样条路径的机器人时间最优轨迹规划*钱东海, 谭 伟, 赵锡芳(上海交通大学机器人研究所)摘 要 现代制造业要求机器人不仅能够实现常规的PTP 运动和简单的CP 运动,而且还能够沿着任意特定路径进行高速运动.为此,对沿着特定路径运动的机器人的时间最优轨迹规划问题作深入研究,成功地运用了三次B 样条对由一系列离散路径点组成的机器人特定路径进行了逼近,并利用动态规划方法,沿着逼近所得的机器人路径,对机器人进行了时间最优轨迹规划,从而保证机器人在不偏离原有路径的基础上,实现时间最优运动.通过计算机仿真实验,证明算法可行,效率较高.关键词 机器人;轨迹规划;时间最优;动态规划;B 样条中图法分类号 T P 241;T P 242Time Optimum Trajectory Planning for RobotsBased on B -Spline PathQian Donghai , Tan W ei , Zhao X if angResearch Institute of Robo tics ,Shanghai Jiao to ng U niversity ,ChinaAbstract In moder n manufacturing ,r obo ts should no t only be capable of m oving in conventional PT P manner and sim ple CP manner ,but also be capable o f m oving alo ng a specified path of arbitr ary shape at high velocity .T o satisfy such demand ,a profound research about the tim e o ptimum trajecto-ry planning of r obo ts alo ng a specified path is carried out in this paper.The specified path is assumed to be composed of a series o f discrete path po ints,and the w ho le tim e optim um trajectory planning is deco mposed into tw o steps .First ,cube B -spline is em plo yed to appro ach these discrete path points .Then ,dynam ic pro gramm ing method is employ ed to generate a time optimum trajectory along the B -spline path.With the alg orithm,the ro bot is g uaranteed to m ove in an o ptimal manner and not deviate fr om its o riginal path.The algo rithm has been prov ed to be feasible and effective by the simulatio ns.Key words ro bot ;trajectory planning ;time optimum ;dynamic progr am ming ;B -spline 由于机器人控制中的非线性和强耦合,故机器人控制通常分两级进行,即机器人运动规划和机器人伺服跟踪.机器人运动规划通常又可分为路径规划和轨迹规划上下两级:路径规划用于在机器人操作空间或关节空间中产生一无碰撞的几何路径;轨迹规划用于产生机器人沿着该几何路径运动至各点处的时间序列.机器人伺服跟踪用于实现机器人各关节精确而实时地对规划出的轨迹进行伺服跟踪.目前,绝大多数机器人控制中,通常只能实现PT P 运动和种类有限的CP 运动(如直线运动和圆弧运动).但随着高性能机器的逐步应用及作业复杂程度的增加,更要求机器人能够沿着任意特定路径进行高速运动.目前部分学者对沿着特定路径运动的机器人时间最优控制问题进行了比较深入的研究.Bobrow和Shin 等[1,2]利用机器人操作臂的动力学方程,建立了考虑机器人动力学特性的时间最优轨迹规划算法.但通常在机器人控制中,因动力学方程求解过程中计算量大,以及动力学方程参数难以精确确定使得上述算法很难应用于实际工作.故在本文的算法中,假定机器人各关节具有确定的极限上限加速度和极限上限速度(通常这些数据可通过实验取得),同时假定机器人已通过路径规划[3],在机器人的关节空间中形成一无碰撞的运动路径,并依据常规假定该路径由关节空间中的一系列离散路径点构成.本文所要解决的问题是:沿着该离散路径点组成的几何路径,在满足各关节速度、加速度约束的条件下,实现机器人的时间最优轨迹规划.1 利用B 样条进行路径逼近假定机器人具有m 个自由度,即机器人关节空间为一m 维空间,同时假定机器人路径由该m 维空间中的n 个离散点Q j =(H 1j ,H 2j ,…,H m j )T ,j =1,2,…,n 所组成,其中H 1j ,H 2j ,…,H mj 表示机器人在Q j 点时的m 个关节的关节值.为保证机器人高速度运动时平稳无冲击,需对规划出的离散路径进行插值或逼近,通常可采用三次样条[4]进行插值,但三次样条要求样条曲线严格通过每一个插值点.当规划出的离散路径点分布参差不齐时,机器人难以实现高速运动,故选用三次B 样条对离散路径点进行逼近.为保证所得B 样条曲线起始于Q 1点,终止于Q n 点,在Q 1点前和Q n 点后各新增两个节点Q 1、Q 2、Q n +3、Q n +4,同时将原有节点下标编号加2,并取新增节点的值分别为:Q 1=Q 3,Q 2=Q 3,Q n +3=Q n +2,Q n +4=Q n +2,即新增节点分别和原有Q 1、Q n 重合.故逼近后的机器人路径由n +1段三次B 样条曲线组成,各段B 样条曲线可表示为[5]Q j (s )=∑4i =1N [4]i (s )Q j +i -1 j =1,2,…,n +1, 0≤s ≤1其中:N [4]1(s )=(-s 3+3s 2-3s +1)/6,N [4]2(s )=(3s 3-6s 2+4)/6,N [4]3(s )=(-3s 3+3s 2+3s +1)/6,N [4]4(s )=s 3/6.当利用B 样条Q j (s )表示机器人路径后,机器人的位姿则由B 样条曲线Q j (s )中的伪距离s 唯一确定,故对机器人进行的时间最优轨迹规划就转化为对伪距离s 进行速度和加速度规划.因机器人各关节具有极限上限速度X i max 和极限上限加速度E i max 约束,i =1,2,…,m ,故在对伪距离s 进行速度和加速度规划时,需将关节空间中的X i max 、E i max 约束(i =1,2,…,m )转换为对伪距离s 速度和加速度的约束.样条曲线Q j (s )对时间t 的速度和加速度可通过伪距离s 表示,即d Q j (s )/d t =d Q j (s )/d s õd s /d t(1)d 2Q j (s )/d t 2=d 2Q j (s )/d s 2õ(d s /d t )2+d Q j (s )/d s õd 2s /d t 2(2)对于确定的第j 段B 样条曲线Q j (s ),其上任一点s 处,d Q j (s )/d s 具有确定的数值[d H 1j (s )/d s ,d H 2j (s )/d s ,…,d H m j (s )/d s ]T ,d 2Q j (s )/d s 2具有确定的数值[d 2H 1j (s )/d s 2,d 2H 2j (s )/d s 2,…,d 2H m j (s )/d s 2]T ,故当d Q j (s )/d t 取极限上限速度(X 1max ,X 2max ,…,X m max )T ,d 2Q j (s )/d t 2取极限上限加速度(E 1max ,E 2max ,…,E m max )T 时,可分别从式(1)和(2)推得s 的极限上下限伪速度和伪加速度分别为s õmax =min i =1,2,…,m{X i max /[d H i j (s )/d s ]},s õmin =0s ¨max =m in i =1,2,…,m E i max -d 2H i j (s )/d s 2õ(s õ)2d H i j (s )/d s ,s ¨min =m ax i =1,2,…,m-E i max -d 2H i j (s )/d s 2õ(s õ)2d H i j (s )/d s (3)(4)式中,m in()、max ()分别表示对括号内的m 个值取其中极小值和极大值.从式(3)和(4)可以看出,除s ・min =0外,s ・max 是s 的函数,而s ¨min 、s ¨max 是s 和s ・的函数,故关节空间中恒定的速度、加速度约束转化为s ・和s ¨约束时成为时变的量.2 利用动态规划法进行时间最优轨迹规划当利用B 样条对离散路径点进行了逼近,并且沿着B 样条路径求得了s 的速度、加速度约束后,则可进行时间最优轨迹规划.由于s ・和s ¨约束较为复杂,很难运用极大值或极小值原理求解,故采用图搜30上 海 交 通 大 学 学 报第32卷 索的方法求解,本文采用动态规划方法.具有m 个自由度的机器人进行时间最优轨迹规划时,共用m 个变量,分别表示m 个关节的角度.由于动态规划法计算量较大,当m 也较大时,利用动态规划法进行时间最优轨迹规划也就不太现实.在本算法中,轨迹规划是沿着已规划好的B 样条路径进行的,故具有m 个变量的时间最优轨迹规划问题已经退化为具有一个变量s ・的时间最优轨迹规划问题,利用动态规划方法进行时间最优轨迹规划也就成为可能了.为进行动态规划,按下述4个步骤进行:(1)对应于n +1段三次B 样条曲线,将各段B 样条曲线分割成u 小段,并取各小段的分割点为s 从0～1的u 等分点,从而整个机器人路径共有u (n +1)个曲线段组成.对应于每一个等分点,依据式(3)可求得机器人在该等分点处的s ・j ,k max 和s ・j ,k min ,其中j =1,2,…,n +1;k =1,2,…,u .(2)对应于上述所得的u (n +1)个离散点,在各离散点处,自s ・j ,k min 至s ・j ,k max ,将s ・等分成v 段,共得v +1个等分点.对应于每一个等分点,依据式(4)计算机器人在该点处的s ¨j ,k ,l max 和s ¨j ,k ,l min (j =1,2,…,n +1;k =1,2,…,u ;l =0;1,…,v ).取s 递增的方向为x 轴,s ・递增的方向为y 轴,建立机器人的s ・—s 二维有向搜索图.因s 被u (n +1)等分,而s ・被v 等分,故s ・—s 二维有向搜索图上共有u (n +1)(v +1)个节点.(3)定义p j ,k ,l 表示s ・—s 二维有向搜索图上第j 段B 样条曲线第k 个等分点处,伪速度方向第l 个等分点处的节点(j =1,2,…,n +1;k =1,2,…,u ;l =0,1,…,v ).定义p j 1,k 1,l 1、p j 2,k 2,l 2表示伪距离方向上两个相邻节点.当k 1为所在B 样条曲线段上最后一个等分点时,有j 1=j 2-1,k 1=u ,k 2=0;否则,j 1=j 2,k 1=k 2-1.对于给定的节点p j 1,k 1,l 1,机器人在该节点处的伪速度为s õj 1,k 1,l 1=(s õj 1,k 1max -s õj 1,k 1min )õl 1/v +s õj 1,k 1min(5)式中s ・j 1,k 1max 、s ・j 1,k 1min 表示第j 1段B 样条曲线上第k 1个等分点处机器人的极限上、下伪速度,可由式(3)求得.对于给定的两相邻节点p j 1,k 1,l 1、p j 2,k 2,l 2,假定机器人在两相邻节点之间具有恒定的s ¨j 1,k 1,l 1-j 2,k 2,l 2,则s ¨j 1,k 1,l 1-j 2,k 2,l 2=[(s õj 1,k 1,l 1)2-(s õj 2,k 2,l 2)2]/[2(s j 1,k 1,l 1-s j 2,k 2,l 2)](6)式中机器人在节点p j 1,k 1,l 1处的伪位移s j 1,k 1,l 1=j 1+k 1/u .在求得各节点处的s ・j 1,k 1,l 1后,则可建立相邻两节点间的代价函数5j 1,k 1,l 1-j 2,k 2,l 2和各节点至终节点的总代价函数J j 1,k 1,l 1.定义两相邻节点p j 1,k 1,l 1、p j 2,k 2,l 2间的5j 1,k 1,l 1-j 2,k 2,l 2为机器人自节点p j 1,k 1,l 1运动至节点p j 2,k 2,l 2所要用的时间,其数学表达式为:5j 1,k 1,l 1-j 2,k 2,l 2=2(s j 1,k 1,l 1-s j 2,k 2,l 2)/(s ・j 1,k 1,l 1+s ・j 2,k 2,l 2).定义节点p j 1,k 1,l 1的J j 1,k 1,l 1为机器人在满足伪速度约束和伪加速度约束条件下,自节点p j 1,k 1,l 1运动至终节点时,所需的最短时间.(4)在求得s ・—s 二维有向搜索图上所有相邻节点间的5j 1,k 1,l 1-j 2,k 2,l 2后,就可自机器人终节点开始,沿着s 递减的方向逐层求解各节点的总代价函数.假定机器人在第j 2段B 样条曲线上的第k 2等分点处的v 个节点p j 2,k 2,l 2(l 2=0,1,…,v )的总代价函数J j 2,k 2,l 2已分别求得,并假定p j 1,k 1,l 1为p j 2,k 2,l 2在s 递减方向上的邻接节点,则节点p j 1,k 1,l 1的J j 1,k 1,l 1可按J j 1,k 1,l 1=min l 2∈8(5j 1,k 1,l 1-j 2,k 2,l 2+J j 2,k 2,l 2)求得,其中8为满足伪加速度约束的节点的集合:8={p j 2,k 2,l 2ûs ¨j 1,k 1,l 1min ≤s ¨j 1,k 1,l 1-j 2,k 2,l 2,s ¨j 1,k 1,l 1-j 2,k 2,l 2≤s ¨j 1,k 1,l 1max ,l 2=0,1,…,u } 自节点p 0,0,0开始,依次连接生成总代价函数J 0,0,0时经过的中间节点,直至终点节点,则可求得构成机器人时间最优轨迹的伪速度和伪加速度序列,利用式(1)和(2)则可求得机器人在关节空间中的时间最优轨迹.3 算法仿真为简化算法,在本仿真实例中,假定机器人为一具有两个转动关节的2自由度机器人,如图1所示.31 第12期钱东海,等:基于B 样条路径的机器人时间最优轨迹规划同时假定机器人已通过路径规划,在关节空间中形成一以离散路径点Q 1、Q 2、…、Q 10表示的机器人路径.利用三次B 样条对Q 1、Q 2、…、Q 10进行逼近,可得图2所示的B 样条路径.因起始和终止节点前后各新增了2个节点,故该路径共由11段三次B 样条曲线组成.图1 两自由度机器人F ig .1 Ro bo t of t wo deg r ees o ffreedom 图2 利用B 样条对离散路径点进行逼近 F ig .2 A ppr o aching the discr ete path pointsw it h B spline 假定机器人关节1、2极限上限角速度分别为8.7rad/s 和14rad/s,极限上限角加速度分别为87rad/s 2和140r ad/s 2.利用上述动态规划法沿着图2所示的B 样条路径进行时间最优轨迹规划,可得时间最优轨迹的伪速度曲线,如图3所示.因该曲线由图2中11段三次B 样条曲线形成,故s 的总长为11.相应可得伪加速度曲线如图4所示.图3 伪速度曲线图F ig.3 Cur ve g raph of pseudo -speed 图4 伪加速度曲线图F ig.4 Cur ve g raph of pseudo -acceler atio n 对应于图3、4中的伪速度曲线和伪加速度曲线,关节1、2的角速度、角加速度曲线分别如图5、6所示.从图中可以看出,机器人时间最优运动共用了0.671s.这比机器人在相同的关节速度、加速度约束下,沿Q 1～Q 4、Q 4～Q 7、Q 7～Q 10三段折线,做CP 运动时所花费的时间(0.774s)要短.图5 关节1、2角速度随时间变化规律 Fig.5 Joint 1and 2's angle velocit y changingw it h time 图6 关节1、2角加速度随时间变化规律 F ig.6 Jo int 1and 2's angle acceler atio n chang ing with time 本仿真实例中,对应于图2中的每一段B 样条,s 被10等分,并在每一伪位移等分点处,s ・被200等分,整个二维有向搜索图上共有10(10+1)(200+1)个节点.在奔腾200型微机上进行轨迹规划,约需费时1.5s,该时间已能很好地满足一般离线运动规划的需要.需指出的是,图4中的伪加速度曲线,以及图6中的角加速度曲线有毛刺出现,这是动态规划本质为一离散算法的必然结果.一般可通过提高量化精度来加以改善,对应于本算法,就是提高伪速度的量化精度.另需要指出的是,利用B 样条路径代替原有的离散路径点路径,可能会引起新的碰撞.该问题一般可在机器人路径规划时,通过增加离散路径点的个数,以及增大操作臂和障碍物之间最小安全距离等措施加以解决.32上 海 交 通 大 学 学 报第32卷 4 结 语本文对沿着特定路径运动的机器人的时间最优轨迹规划问题作了研究,成功地运用了动态规划法,沿着三次B样条逼近所得的路径,对机器人实现时间最优轨迹规划.算法通过了仿真实验,证明是可行的,效率较高.参考文献1 Bo br ow J E,Dubow sky S,G ibson J S.T ime optimal co ntro l o f r obo tic manipulato r along specified paths.Ins J Robo tics Research,1985,(3):3～172 Shin K G,M ckay N D.M inimum-time contr ol of ro bo tic ma nipulator s w ith g eomet ric path co nst raint s.IEEE T r ansactions on A utomat ic Co ntro l,June1985,AC-30(6):531～5413 Lo ano-Per ez T.Spatial planning:a co nfigur atio n space appro ach.IEEE T ransactio n o n Com puter s,1983,C-32(2): 108～1204 L in C S,Chang P R,L uh J Y S.F or mula tio n and o pt imization o f cubic po ly nomia l joint tr ajecto ries for industr ial ro bots.IEEE T ra ns o n A uto Co ntr,Dec1983,A C-28(12):1066～10745 袁奇荪.计算几何造型学基础.北京:航空工业出版社,1987.下期发表论文摘要预报线性化逐层优化ML P训练算法周志杰, 胡光锐(上海交通大学电子工程系) 李 群(南京通信工程学院)摘 要:提出了线性化逐层优化M LP训练算法(L O LL).L O LL采用循环的方式逐层对M L P的连接权值进行训练.训练连接权值时用一阶泰勒级数表示神经元的非线性激活函数以实现神经网络的线性化,使M L P 的训练问题转化为一个线性问题.同时为保证神经网络线性化条件不被破坏,LO L L通过在神经网络的误差函数中计入部分线性化误差限制参数的改变幅度,对神经网络的误差函数进行了修正.实验结果显示,L OL L 训练算法的速度比传统的BP算法快4倍,用它构成的语音信号非线性预测器有较好的预测性能.用于FD法电容参数提取的高效截断边界条件曹 毅, 金荣洪, 李征帆(上海交通大学电子工程系)摘 要:基于测度不变方程(M EI)的概念和局部等效原理,将截断边界的确定归结为网格边界上一小区域的电位场模拟问题,并讨论了二维、三维情况下的具体处理.计算实例表明,应用这种截断边界条件的FD法得到的电容计算结果与矩量法和F A ST CA P软件所得结果吻合得较好.33 第12期钱东海,等:基于B样条路径的机器人时间最优轨迹规划。
基于三次B样条的移动机器人实时轨迹规划研究
基于三次B样条的移动机器人实时轨迹规划研究郑天江;李俊杰;陈庆盈;杨桂林;张驰【摘要】基于B样条曲线生成的轨迹曲线具有几何不变性和连续性等优点,被广泛应用于数控机床设备或机器人的轨迹规划算法中,然而传统的轨迹规划基本上是在离线的情况下进行的,即所有的型值点已知,或者控制点已知,然而这种方法很难运用到移动机器人的轨迹规划中,此外如果简单地将B样条轨迹规划方法应用于移动机器人的轨迹规划,即通过给定控制点,进行插补生成一条新的轨迹曲线,但是这种方法通常不会经过所有的控制点,这样会导致实际运行路径与规划的路径有偏差,提出了一种基于三次B样条的移动机器人实时轨迹规划算法,在移动机器人的运动过程中能够实时生成插补点,不需要预先输入所有轨迹点(型值点),并利用反求控制点算法保证机器人能够经过所有的(型值点),从而保证机器人既能经过所有预先规定的点又能保证机器人运行的连续性和平稳性,通过仿真验证了其有效性.【期刊名称】《制造业自动化》【年(卷),期】2017(039)005【总页数】4页(P4-7)【关键词】B样条;实时轨迹规划;移动机器人;插补算法;样条函数【作者】郑天江;李俊杰;陈庆盈;杨桂林;张驰【作者单位】中科院宁波材料技术与工程研究所,宁波 315201;中科院宁波材料技术与工程研究所,宁波 315201;中科院宁波材料技术与工程研究所,宁波 315201;中科院宁波材料技术与工程研究所,宁波 315201;中科院宁波材料技术与工程研究所,宁波 315201【正文语种】中文【中图分类】TP242轨迹规划问题是运动控制的核心问题，从数控机床设备到机器人设备，轨迹规划都是其中必不可少的关系着系统性能的关键因素。
一般的数控机床或者机器人搭载的运动控制器均具有简单的轨迹规划和插补功能，其中梯形曲线、S曲线[1]、三次多项式[2]以及高次多项式算法[3]等最为常见，样条函数与轨迹规划的原理最早出现在凸轮轮廓设计上，随着自动化技术的发展，其在数控机床和机器人领域有了广泛的应用。