-
摘要:
相比于传统的刚性驱动器, 串联弹性驱动器(Series elastic actuator, SEA)具有被动柔顺性、阻抗低、抗冲击、力感知等诸多优点, 因而已被广泛应用于各种机器人系统中. 首先根据弹性和阻尼特性将串联弹性驱动器分为弹性型、阻尼型和弹性−阻尼型串联弹性驱动器, 介绍不同类型串联弹性驱动器的优缺点, 并详细概述弹性和阻尼特性的机械实现方式; 然后对各类串联弹性驱动器作为力传感器的建模方法进行介绍; 接着叙述串联弹性驱动器在机器人系统中的主要应用, 如力传感器、安全保护、降低能耗; 最后展望串联弹性驱动器未来的发展方向.
Abstract:Compared with traditional rigid actuators, series elastic actuator (SEA) has many advantages such as passive compliance, low impedance, impact resistance, and force sensing, and so they have been widely used in various robot systems. Firstly, according to the elasticity and damping characteristics, SEAs are divided into elastic SEAs, damped SEAs and elastic-damped SEAs. The advantages and disadvantages of each SEA are introduced, and the mechanical implementation methods of elasticity and damping characteristics are summarized in detail. Then, the modeling methods of SEAs as force sensors are presented. Next, the main applications of SEAs in the robot systems are described, such as force sensors, safety protection and energy consumption reduction. Finally, the future development directions of SEAs are recommended.
-
机动目标跟踪(Maneuvering target tracking, MTT)是状态估计领域的重要研究方向之一, 广泛应用于雷达跟踪、飞行目标监测、导航等领域. 目前机动目标跟踪方法的研究主要基于卡尔曼滤波(Kalman filter, KF). 卡尔曼滤波是一种基于先验模型的估计方法, 要求先验模型准确, 即目标运动模式已知. 然而, 机动目标的机动性就体现在其运动模式未知且剧烈变化, 因此单模型方法难以有效解决机动目标跟踪问题. 基于多模型的跟踪方法是目前机动目标跟踪的重要研究领域. 以交互式多模型(Interacting multiple model, IMM)[1]为代表的多模型机动目标跟踪方法结合隐马尔科夫模型(Hidden Markov models, HMM), 利用模型转移概率提高对机动目标的状态估计精度. IMM方法采用模型集, 但Li认为实际模式空间与模型集合不一定匹配, 且模型集合应适应外界条件变化, 并提出变结构多模型方法(Variable structure multiple model, VSMM)[2-5]. 由于其良好的状态估计效果和灵活性, VSMM方法被国内外学者广泛关注.
随着传感器、计算机和通信技术发展, 多传感器信息融合逐渐成为研究热点, 可分为集中式(Centralized)、分布式(Distributed)与混合式(Hierarchical) 三种融合架构[6]. 基于一致性的分布式融合架构无需融合中心, 具有通信带宽要求低、通信能量损耗低、且对复杂网络适应性强等优点, 日益受到国内外学者关注. 基于一致性的分布式状态估计包括多种实现形式, 例如卡尔曼一致滤波(Kalman consensus filter, KCF)[7-9]、信息一致滤波(Information consensus filter, ICF)[10-11]等.
目前对一致性滤波的研究主要基于单模型方法, 主要关注传感器网络内丢包[12]、时延[13]、动态网络拓扑[14]、自适应一致性滤波[15]、网络能量优化[16]以及带牵引控制[17]等问题. 近年来, 考虑到多模型方法比单模型方法有更好的机动目标跟踪效果, Chisci等学者结合多模型思想, 提出分布式交互式多模型估计方法 (Distributed interacting multiple model, DIMM)[18-20]. 虽然变结构交互式多模型比交互式多模型具有更好的跟踪精度, 但由于VSMM方法中模型集随时可能扩增或删减, 难以直接应用于基于一致性的分布式估计方法, 因此目前已发表的相关研究成果不多.
本文重点研究如何将变结构多模型方法有效地引入分布式非线性状态估计方法, 具体研究内容如下: 首先为了解决量测方程非线性的问题, 研究了一类无迹信息滤波方法(Unscented information filter, UIF); 通过对变结构多模型方法进行改进, 提出基于可能模型集的期望模式扩增方法 (Expected-mode augmentation based on likely model-set, EMA-LMS), 进而将VSMM应用于分布式状态估计, 提出分布式变结构多模型方法 (Distributed variable structure multiple model, DVSMM). 仿真实验结果验证了本文提出方法的有效性.
1. 问题模型
本节介绍分布式传感器网络的图论表示以及雷达和红外传感器的量测模型.
1.1 传感器网络模型
通常用图
$G = (V,E)$ 对传感器网络建模. 顶点集$V = \{ 1,2,\cdots,n\}$ 表示网络中的传感器节点. 如果传感器节点$i$ 和$j$ 可以通信, 则认为图中这两个节点之间存在边, 即$(i,j) \in E$ . 邻接矩阵${A}$ 是$n$ 行$n$ 列的布尔矩阵, 记${A} = [{a_{ij}}]$ , 如式1所示:$${a_{ij}} = \left\{ \begin{array}{l} 1,\hskip5mm{\rm{ (}}i,j) \in E\\ 0,\hskip5mm{\rm{ (}}i,j) \notin E{\rm{\ or\ }}i = j \end{array} \right.$$ (1) 记
${N_i} = \{ j:({v_i},{v_j}) \in E\} $ 为传感器节点$i$ 可以通通信的节点集,${J_i} = {N_i} \cup \{ i\} $ . 如图1所示: 包含6节点的分布式传感器网络.该传感器网络对应的邻接矩阵如式(2)所示:
$${A} = \left[ {\begin{array}{*{20}{c}} 0&1&0&1&1&1 \\ 1&0&1&0&0&1 \\ 0&1&0&1&0&0 \\ 1&0&1&0&1&0 \\ 1&0&0&1&0&1 \\ 1&1&0&0&1&0 \end{array}} \right]$$ (2) 1.2 量测模型
本文研究二维平面内直接坐标系中的机动目标跟踪问题. 记视线与直角坐标系
$x$ 轴正方向的夹角记为方位角$\theta, $ 传感器与目标的距离记为$\rho.$ 雷达可获得目标距离$\rho $ 与方位角$\theta $ 量测值, 而红外传感器仅获得目标方位角$\theta,$ 如图2所示.构造极坐标
$(\rho ,\theta )$ 与二维平面上直角坐标描述$(x,y)$ 一一映射, 如式(3)所示, 方位角的范围须为$\theta \in [0,2\pi )$ 或$\theta \in ( - \pi ,\pi ]$ :$$\left\{ \begin{array}{l} x = \rho \cos \theta \\ y = \rho \sin \theta \end{array} \right.$$ (3) 当
$\theta \in [0,2\pi )$ 时, 直角坐标$(x,y)$ 转换为极坐标$(\rho ,\theta )$ 的映射关系如式(4)所示.$$\begin{split} &\rho = \sqrt {{{({x_t} - {x_s})}^2} + {{({y_t} - {y_s})}^2}} \\ &\theta = \left\{ \begin{aligned} &\arctan \frac{{{y_t} - {y_s}}}{{{x_t} - {x_s}}},\ {\rm{ }}{x_t} - {x_s} > 0,\ {y_t} - {y_s} \ge 0\\ &\arctan \frac{{{y_t} - {y_s}}}{{{x_t} - {x_s}}} + 2\pi ,\ {\rm{ }}{x_t} - {x_s} > 0,\ {y_t} - {y_s} < 0\\ &\arctan \frac{{{y_t} - {y_s}}}{{{x_t} - {x_s}}} + \pi ,\ {\rm{ }}{x_t} - {x_s} < 0\\ &\frac{\pi }{2},\ {\rm{ }}{x_t} - {x_s} = 0{\rm{,\ }}{y_t} - {y_s} > 0\\ &\frac{{3\pi }}{2},\ {\rm{ }}{x_t} - {x_s} = 0{\rm{,\ }}{y_t} - {y_s} < 0 \end{aligned} \right. \end{split}$$ (4) 式中,
${x_t}$ 和${y_t}$ 表示目标位置;${x_s}$ 和${y_s}$ 表示传感器位置. 式(4)中的映射关系不具有连续性, 即存在一组由奇异点构成射线$y = 0,\ x > 0.$ 且目标方位角在该射线两侧发生突变, 导致目标方位角误差增大, 影响滤波器的状态估计结果.为避免由反正切函数不连续引起的误差, 需判断映射关系是否奇异.
图 3$ {\theta }_{1}-{\theta }_{2} $ 的定义Fig. 3 Definition of$ {\theta }_{1}-{\theta }_{2} $ 首先计算相邻两个时刻目标方位角的顺时针变量
$\Delta {\theta _{acw}}$ 与逆时针变量$\Delta {\theta _{cw}}$ , 构造具有连续性的映射关系, 如式(5)所示, 计算方位角的变化量.$$\begin{split} {\theta _1} - {\theta _2} =\;& \left\{ {\begin{aligned} &{0,} \qquad\qquad\;\; {{\theta _1} - {\theta _2} = 0}\\ &{\Delta {\theta _{acw}},} \;\qquad {\Delta {\theta _{cw}} > \Delta {\theta _{acw}}}\\ &{ - \Delta {\theta _{cw}},}\qquad {\Delta {\theta _{cw}} < \Delta {\theta _{acw}}} \end{aligned}} \right.=\\ \;&\left\{ {\begin{aligned} &{{\theta _1} - {\theta _2},}\;\;\;\qquad\qquad{\left| {{\theta _1} - {\theta _2}} \right| \le \pi }\\ &{2\pi - \left| {{\theta _1} - {\theta _2}} \right|,}\qquad{{\theta _1} - {\theta _2} < - \pi }\\ &{\left| {{\theta _1} - {\theta _2}} \right| - 2\pi ,}\;\qquad{{\theta _1} - {\theta _2} > \pi } \end{aligned}} \right. \end{split}$$ (5) 2. 无迹信息滤波
本节介绍下一节中DVSMM方法所采用的无迹信息滤波UIF[21]原理. 无迹信息滤波与无迹卡尔曼滤波 (Unscented Kalman filter, UKF) 均通过Sigma点采样计算状态向量的一阶矩与二阶矩, 区别在于UIF采用信息矩阵与信息状态向量进行量测更新.
设
$x$ 为n维随机向量, 其均值和协方差分别为$\bar x$ 和${{P}_x}$ .${{f}} ( \cdot )$ 为非线性函数:1)计算
$2n + 1$ 个Sigma点${\xi ^\iota }$ :$$\left\{ \begin{aligned} &{\xi ^i} = \bar x,{\rm{ }}\ i = 0\\ &{\xi ^i} = \bar x + {(\sqrt {(n + \lambda ){P_x}} )_i}{\rm{, }}\ i = 1,\cdots,n\\ &{\xi ^i} = \bar x - {(\sqrt {(n + \lambda ){P_x}} )_{i - n}}{\rm{, }}\ i = n + 1,\cdots,2n \end{aligned} \right.$$ (6) 式中,
$\lambda $ 为尺度参数;${(\sqrt {(n + \lambda ){P_x}} )_i}$ 表示用$(n + \lambda ){{P}_x}$ 平方根的第$i$ 行或第$i$ 列来构造Sigma点[22-24].2)每个Sigma点通过非线性函数传播, 得到
${y^i}$ :$${y^i} = f({\xi ^i}),\quad {\rm{ }}i = 0,\cdots,2n$$ (7) 3)计算
$y$ 的均值$\bar y$ 和协方差${{P}_y}$ .$$\begin{split} &\bar y = \sum\limits_{i = 0}^{2n} {W_s^i{y^i}} \\ &{{P}_y} = \sum\limits_{i = 0}^{2n} {W_c^i({y^i} - \bar y){{({y^i} - \bar y)}^{\rm{T}}}} \end{split}$$ (8) 式中,
$W_s^i$ 和$W_c^i$ 为加权系数[22-23].设离散时间非线性系统的状态方程和量测方程如式(9)所示:
$$\begin{split} &{x_k} = {f_{k - 1}}({x_{k - 1}}) + {w_k}\\ &{z_k} = {h_k}({x_k}) + {v_k} \end{split}$$ (9) 式中,
${x_k}$ 表示目标状态向量;${z_k}$ 表示传感器量测向量;${{f}}_{k}(\cdot )$ 和${{h}}_{k}(\cdot )$ 分别表示非线性的状态函数和量测函数;${w_k} \sim {\rm N}(0,{{Q}_k})$ 表示过程噪声;${v_k} \sim {\rm N}(0,{{R}_k})$ 表示量测噪声.假设上一时刻的状态估计
${\hat x_{k \!-\! 1|k \!-\! 1}}$ 和估计协方差矩阵${{P}_{k - 1|k - 1}}$ 已知, 状态向量维度为$L,$ 量测向量维度为$M.$ 则UIF的一步状态预测与量测更新过程如下:1)一步状态预测
由式(6), 计算
${\hat x_{k - 1|k - 1}}$ 周围的Sigma点$x_{k - 1|k - 1}^i$ ,$i = 0,1,\cdots,2L.$ 计算$x_{k - 1|k - 1}^i$ 经过状态转移函数${{f}}_{k-1}( \cdot )$ 传递后的$x_{k|k - 1}^i.$ 由式(10)计算状态预测${\hat x_{k|k - 1}}$ 和状态预测协方差${P_{k|k - 1}}:$ $$\begin{split} &{{\hat x}_{k|k - 1}} = \sum\limits_{i = 0}^{2n} {W_s^ix_{k|k - 1}^i} \\ &{{P}_{k|k - 1}} = \sum\limits_{i = 0}^{2n} W_c^i(x_{k|k - 1}^i - {{\hat x}_{k|k - 1}})\times\\ &\qquad\qquad{(x_{k|k - 1}^i - {\hat x}_{k|k - 1}})^{\rm T} + {{Q}_k} \end{split}$$ (10) 计算先验信息向量
${\hat y_{k|k - 1}}$ 和对应的信息矩阵${{Y}_{k|k - 1}}$ :$$\begin{split} &{{\hat y}_{k|k - 1}} = {P}_{k|k - 1}^{ - 1}{{\hat x}_{k|k - 1}}\\ &{{Y}_{k|k - 1}} = {P}_{k|k - 1}^{ - 1} \end{split}$$ (11) 2)量测更新
计算
$x_{k|k - 1}^i$ 在量测函数下的映射$g_k^i$ :$$g_k^i = {{h}_k}(x_{k|k - 1}^i),\;\;\;i = 0,1,\cdots,2L$$ (12) 计算量测预测
${\hat z_k}$ :$${\hat z_k} = \sum\limits_{i = 0}^{2L} {W_s^ig_k^i} $$ (13) 计算量测预测和状态−量测协方差矩阵:
$$\begin{split} &{{P}_{{z_k}{z_k}}} = \sum\limits_{i = 0}^{2L} {W_c^i(g_k^i - {{\hat z}_k}){{(g_k^i - {{\hat z}_k})}^{\rm{T}}}} + {{R}_k}\\ &{{P}_{{x_k}{z_k}}} = \sum\limits_{i = 0}^{2L} {W_c^i(x_{k|k - 1}^i - {{\hat x}_{k|k - 1}}){{(g_k^i - {{\hat z}_k})}^{\rm{T}}}} \end{split}$$ (14) 引入伪测量矩阵计算信息状态贡献
${i_k}$ 和对应的信息矩阵${I_k}$ [21]:$$\begin{split} {{I}_k} =\;& {P}_{k|k{\rm{ - 1}}}^{{\rm{ - 1}}}{{P}_{{x_k}{z_k}}}{R}_k^{{\rm{ - 1}}}{P}_{{x_k}{z_k}}^{\rm{T}}{P}_{k|k{\rm{ - 1}}}^{{\rm{ - 1}}}\\ {i_k} =\;& {P}_{k|k{\rm{ - 1}}}^{{\rm{ - 1}}}{{P}_{{x_k}{z_k}}}{R}_k^{{\rm{ - 1}}}{\rm{\bigg(}}{z_k} - {{\hat z}_k} +\\ &{P}_{{x_k}{z_k}}^{\rm{T}}{\left({P}_{k|k{\rm{ - 1}}}^{\rm{T}}\right)^{{\rm{ - 1}}}}{{\hat x}_{k|k{\rm{ - 1}}}}\bigg) \end{split}$$ (15) 通过
${i_k}$ 和${{I}_k}$ 计算后验信息向量${\hat y_{k|k}}$ 和对应的信息矩阵${{Y}_{k|k}}$ :$$\begin{split} &{{Y}_{k|k}} = {{Y}_{k|k - 1}} + {{I}_k}\\ &{{\hat y}_{k|k}} = {{\hat y}_{k|k - 1}} + {i_k} \end{split}$$ (16) 由式(17)计算状态估计
${\hat x_{k|k}}$ 和状态估计协方差${P_{k|k}}$ :$$\begin{split} &{{P}_{k|k}} = {Y}_{k|k}^{ - 1}\\ &{{\hat x}_{k|k}} = {Y}_{k|k}^{ - 1}{{\hat y}_{k|k}} \end{split}$$ (17) 考虑到第1.2节所述的方位角突变的问题, 需要按照如下两个步骤修改UIF:
将式(13)改为直接用
${\hat x_{k|k - 1}}$ 来计算${\hat z_k}$ :$${\hat z_k} = {{h}_k}({\hat x_{k|k - 1}})$$ (18) 将式(14)、(15)中
$g_k^i - {\hat z_k}$ 和${z_k} - {\hat z_k}$ 中的方位角相减都用式(5)的角度相减代替.3. 分布式变结构多模型方法
本节将分析变结构多模型方法应用在分布式状态估计所面临的关键问题. 通过结合期望模式扩增方法和可能模型集方法, 提出基于可能模型集的期望模式扩增方法 EMA-LMS与分布式变结构多模型跟踪方法DVSMM.
3.1 VSMM方法在分布式估计的关键问题
在DIMM[18-20]方法框架下, 将每个模型对应的预测信息或传感器后验估计信息与通信邻域中其他传感器对应模型中的信息进行一致性加权融合, 如图4所示:
图4中, 每个传感器具有相同的交互式多模型集, 且模型数量为
$M.$ 假设传感器$s$ 和$j$ 相邻, 本地传感器与相邻传感器进行一致性加权融合的变量可分为三类: 1)本地先验信息向量${\hat y_{k|k - 1}}$ 及其对应的信息矩阵${{Y}_{k|k - 1}};$ 2)本地信息状态贡献${i_k}$ 和对应的信息矩阵${{I}_k}$ [18]; 3)本地后验信息向量${\hat y_{k|k}}$ 和对应的信息矩阵${{Y}_{k|k}}$ [19]. 此外, 分布式交互式多模型方法将对每个模型下的模型似然对数与相邻传感器对应模型下的模型似然对数进行一致性加权融合[18].但上述DIMM方法框架并不适用于分布式变结构交互式多模型方法. VSMM方法中不同时刻模型集的模型种类与数量可能不同. 即在每个方法周期内, 每个传感器所使用的模型可能不一样. 因此实现分布式VSMM方法主要有面临两个难点:
1) 信息滤波器中先验及后验的信息向量
${y_k}$ 、对应的信息矩阵${{Y}_k}$ 和多模型方法中的模型似然都依赖于模型和传感器本地量测向量${z_k}$ 来计算. 由于VSMM方法每个时刻使用的模型种类和数量都在变化, 因此无法像分布式IMM方法那样对每个模型对应的这些信息使用一致性加权融合.2) 在线性系统中与非线性系统中, 信息状态贡献
${i_k}$ 和对应的信息矩阵${{I}_k}$ 的计算不但依赖于本地量测${z_k}$ , 也依赖于模型.如图5所示, 每个传感器每个时刻所交互的模型不同(VSMM方法核心特点), 因此无法采用DIMM方法的思路实现分布式状态估计.
3.2 分布式变结构多模型方法
由于VSMM方法在不同时刻选用不同的模型集进行交互, 因此难以在相邻传感器之间直接交互模型的信息向量和信息矩阵. 为解决这一问题, 本文对Li提出的VSMM方法结合无迹信息滤波UIF进行改进, 提出分布式变结构多模型跟踪方法(DVSMM). 通过在相邻传感器之间直接传递量测向量, 并在每个传感器内部平行计算采用不同模型的UIF对应的信息向量、信息矩阵和模型似然函数, 最后进行一致性加权融合. DVSMM具体方法如下:
假设本地传感器为传感器
$s$ , 通过MSA方法可得$k$ 时刻本地用于状态估计的新模型集合$M_k^s$ . 假设每个方法周期开始时, 每个传感器已经向相邻的传感器发送本时刻自身的本地量测${z_k}$ 和位置${p_k}$ , 且每个传感器可知其他传感器量测向量来自的传感器类型(雷达或红外). 记${J_s} = {N_s} \cup \{ s\} $ , 则传感器$s$ 在本方法周期可用的传感器量测为$\{ z_k^m\} ,m \in {J_s}$ .对模型
$i$ , 目标的状态转移方程为:$${x_k} = f_{k - 1}^{(i)}({x_{k - 1}}) + w_k^{(i)}$$ (19) 式中,
$w_k^{(i)}$ 为过程噪声,$w_k^{(i)} \sim {\rm N}(0,{Q}_k^{(i)})$ .传感器
$s$ 的量测方程为:$$z_k^s = {h}_k^s({x_k}) + v_k^s$$ (20) 式中,
$v_k^s$ 为量测噪声,$v_k^s \sim {\rm N}(0, R_k^s)$ .假设
$ k-1 $ 时刻基于$M_{k - 1}^s$ 的本地目标状态估计$\hat x_{k - 1|k - 1}^{s,(j)}$ 、状态估计误差协方差${P}_{k - 1|k - 1}^{s,(j)}$ 模型概率$\mu _{k - 1}^{s,(j)}$ ,${m^{(j)}} \in M_{k - 1}^s$ 均已知. 分布式变结构多模型方法的模型集合$[M_k^s,M_{k - 1}^s]$ 包括${J_s} = \{ {N_s} \cup s\} $ 中所有传感器的量测信息$\{ z_k^m\} ,m \in {J_s}$ , 以及一致性加权融合过程.在
$k$ 时刻, 传感器$ s $ 内模型集合$[M_k^s,M_{k - 1}^s]$ 的一步预测和量测更新方法流程如下$({\pi _{ij}}$ 为模型转移概率):1)模型交互(对
$\forall {m^{(i)}} \in M_k^s)$ 计算模型预测概率:
$$\mu _{k|k - 1}^{(i)} = \sum\limits_{{m^{(j)}} \in M_{k - 1}^s} {{\pi _{ji}}\mu _{k - 1|k - 1}^{(j)}} $$ (21) 计算交互权值:
$$\mu _{k - 1}^{j|i} = {\pi _{ji}}\frac{\mu _{k - 1|k - 1}^{(j)}}{\mu _{k|k - 1}^{(i)}}$$ (22) 计算交互估计和方差:
$$\begin{split} &\bar x_{k - 1|k - 1}^{s,(i)} = \sum\limits_{{m^{(j)}} \in M_{k - 1}^s} {\hat x_{k - 1|k - 1}^{s,(j)}\mu _{k - 1}^{s,j|i}}\times \\ &\qquad{{\bar P}}_{k - 1|k - 1}^{s,(i)} = \sum\limits_{{m^{(j)}} \in M_{k - 1}^s} {\mu _{k - 1}^{j|i}\bigg[{P}_{k - 1|k - 1}^{s,(j)} + } \\ &\qquad\left(\bar x_{k - 1|k - 1}^{s,(i)} - \hat x_{k - 1|k - 1}^{s,(j)}\right)\!{\left(\bar x_{k - 1|k - 1}^{s,(i)} - \hat x_{k - 1|k - 1}^{s,(j)}\!\right)^{\rm{T}}}\bigg] \end{split}$$ (23) 2)模型条件滤波(对
$\forall {m^{(i)}} \in M_k^s$ )分布式变结构多模型的模型集合
$[M_k^s,M_{k - 1}^s]$ 使用了${J_s}$ 中所有传感器的量测信息$\{ z_k^m\} ,m \in {J_s}.$ 状态预测:
由式(6), 计算
$\bar x_{k - 1|k - 1}^{s,(i)}$ 的Sigma点$x_{k - 1|k - 1}^{s,l,(i)}$ $(l = 0,1,\cdots,2L$ ,$L$ 为状态向量维度). 然后计算Sigma点$x_{k - 1|k - 1}^{s,l,(i)}$ 经过状态函数${{f}} _{k - 1}^{(i)}$ 传递后得到的$x_{k|k - 1}^{s,l,(i)}$ . 于是可以如式(10)计算得到模型$i$ 下的状态预测$\hat x_{k|k - 1}^{s,(i)}$ 和状态预测协方差${P}_{k|k - 1}^{s,(i)}$ . 然后得到模型$ i $ 下先验信息向量$\hat y_{k|k - 1}^{s,(i)}$ 和对应的信息矩阵$Y_{k|k - 1}^{s,(i)}$ .量测更新:
利用多个传感器量测
$\{ z_k^m\} ,m \in {J_s}$ 进行量测更新. 分别计算这些来自不同传感器的量测向量对应每个模型的信息状态贡献、信息矩阵以及模型似然函数, 然后进行一致性加权融合. 具体步骤如下:对每个
$\{ z_k^m\} ,m \in {J_s}$ 计算$x_{k|k - 1}^{s,l,(i)}$ 经过量测函数${{h}}_{k}^{m}( \cdot )$ 传播后的Sigma点$g_k^{s,m,l,(i)}$ , 由式(14) 得到${P}_{{z_k}{z_k}}^{s,m,(i)}$ 和${{P}}_{{x_k}{z_k}}^{s,m,(i)}$ . 然后计算量测预测$\hat z_{k|k - 1}^{s,m,(i)}$ 和残差$\tilde z_k^{s,m,(i)}$ :$$\begin{split} &{\hat z_{k|k - 1}^{s,m,(i)} = {{h}}_k^m(\hat x_{k|k - 1}^{s,(i)})}\\ &{\tilde z_k^{s,m,(i)} = z_k^{s,m} - \hat z_{k|k - 1}^{s,m,(i)}} \end{split}$$ (24) 计算可得量测
$z_k^m$ 和模型${m^{(i)}}$ 对应的信息状态贡献与信息矩阵:$$\begin{split} &i_k^{s,m,(i)} = {\left({P}_{k|k - 1}^{s,(i)}\right)^{ - 1}}{P}_{{x_k}{z_k}}^{s,m,(i)}{\left({R}_k^m\right)^{ - 1}} \times \\ &\qquad\qquad\left(\tilde z_k^{s,m,(i)} + {\left({P}_{{x_k}{z_k}}^{s,m,(i)}\right)^{\rm{T}}}{\left({P}_{k|k - 1}^{s,(i)}\right)^{ - {\rm{T}}}}\hat x_{k|k - 1}^{s,(i)}\right)\\ &I_k^{s,m,(i)} = {\left({P}_{k|k - 1}^{s,(i)}\right)^{ - 1}}{P}_{{x_k}{z_k}}^{s,m,(i)}{\left({R}_k^m\right)^{ - 1}}\times\\ &\qquad\qquad{\left({P}_{{x_k}{z_k}}^{s,m,(i)}\right)^{\rm{T}}}{\left({P}_{k|k - 1}^{s,(i)}\right)^{ - {\rm{T}}}}\\[-15pt] \end{split}$$ (25) 计算量测
$z_k^m$ 和模型${m^{(i)}}$ 下的模型似然函数:$$L_k^{s,m,(i)}{\rm{ = }}\frac{{\exp \left( { \frac{- {{\left(\tilde z_k^{s,m,(i)}\right)}^{\rm{T}}}{{\left({P}_{{z_k}{z_k}}^{s,m,(i)}\right)}^{ - 1}}\tilde z_k^{s,m,(i)}}{2}} \right)}}{{\sqrt {{{\left( {2\pi } \right)}^{{N^m}}}\left| {{P}_{{z_k}{z_k}}^{s,m,(i)}} \right|} }}$$ (26) 式中,
${N^m}$ 为传感器$m$ 的量测向量维度;$ \left|\cdot \right| $ 表示矩阵的行列式.对模型似然求对数
$$\Lambda _k^{s,m,(i)} = \ln \left(L_k^{s,m,(i)}\right)$$ (27) 至此, 获得模型
${m^{(i)}}$ 下, 关于${J_s}$ 内的所有量测数据$\{ z_k^m\} ,m \in {J_s}$ 的信息状态贡献、对应的信息矩阵和模型似然的对数${\{ {\pmb{i}}_k^{s,m,(i)},{I}_k^{s,m,(i)},\Lambda _k^{s,m,(i)}\} _{m \in {J_s}}}$ .进行一致性加权融合:
$$\begin{split} &i_k^{s,(i)} = i_k^{s,s,(i)} - \sum\limits_{n \in {N_s}} {{w_{sn}}\left(i_k^{s,s,(i)} - i_k^{s,n,(i)}\right)} \\ &{I}_k^{s,(i)} = {I}_k^{s,s,(i)} - \sum\limits_{n \in {N_s}} {{w_{sn}}\left({I}_k^{s,s,(i)} - {I}_k^{s,n,(i)}\right)} \\ &\Lambda _k^{s,(i)} = \Lambda _k^{s,s,(i)} - \sum\limits_{n \in {N_s}} {{w_{sn}}\left(\Lambda _k^{s,s,(i)} - \Lambda _k^{s,n,(i)}\right)} \end{split}$$ (28) 式中,
$w$ 为一致性加权系数. 常用的一致性加权系数有最大度加权和Metropolis加权[25], 本文采用Metropolis加权系数.恢复模型似然函数
$L_k^{s,(i)} = \exp (\Lambda _k^{s,(i)})$ .更新每个模型下的信息向量和信息矩阵更新:
$$\begin{split} &\hat y_{k|k}^{s,(i)} = \hat y_{k|k - 1}^{s,(i)} + i_k^{s,(i)}\\ &{Y}_{k|k}^{s,(i)} = {Y}_{k|k - 1}^{s,(i)} + {I}_k^{s,(i)} \end{split}$$ (29) 进而得到每个模型下的状态估计和状态估计协方差:
$$\begin{split} &{P}_{k|k}^{s,(i)} = {({Y}_{k|k}^{s,(i)})^{ - 1}}\\ &\hat x_{k|k}^{s,(i)} = {({Y}_{k|k}^{s,(i)})^{ - 1}}\hat y_{k|k}^{s,(i)} \end{split}$$ (30) 至此, DVSMM方法具有明确的输入和输出结构与递推公式:
$$\begin{split} &\left[M_k^s,M_{k - 1}^s\right]:\left\{ \hat x_{k|k}^{i|M_k^s},{P}_{k|k}^{i|M_k^s},L_k^{i|M_k^s},\mu _{k|k - 1}^{i|M_k^s}\right\} =\\ &\quad\;\;\left(\hat x_{k - 1|k - 1}^{i|M_{k - 1}^s},{P}_{k - 1|k - 1}^{i|M_{k - 1}^s},\mu _{k - 1|k - 1}^{i|M_{k - 1}^s},{\{ z_k^m\} _{m \in {J_s}}}\right) \end{split}$$ (31) 每个传感器通过与临近传感器交互量测信息及传感器位置, 通过计算
$[M_k^s,M_{k - 1}^s]$ 和$[M_k^{s,1},M_k^{s,2}; M_{k - 1}^s]$ [2], 即可将各种单传感器下的VSMM机动目标跟踪方法迁移到传感器网络中, 进行分布式状态估计.DVSMM更新模型集方法流程如图6所示.
3.3 基于可能模型集的期望模式扩增方法
VSMM方法所使用的模型集合随时可能扩增和删减, 其核心在于模型集自适应方法 (Model-set adaptation, MSA)[3] 和基于模型集序列状态估计方法 (Model-set sequence conditioned estimation, MSE)[2, 25] . 目前, 模型集自适应方法包括可能模型集 (Likely-model set, LMS) 方法[4]、期望模式扩增 (Expected-mode augmentation, EMA) 方法[5]等. 其中, LMS方法根据模型概率, 在一个包含较多模型的模型集中选择部分模型来参与滤波估计, 能够减少每个方法周期参与滤波的模型数量, 降低多模型方法的计算量. EMA方法适用于模型具有可加性, 模式空间连续的情况. 它在每个方法周期对已有的模型求加权和(权值为模型概率), 计算得到期望模型, 并把期望模型扩增到模型集中参与滤波估计. 当目标的运动模式不落在基础模型上时, 能够显著改善跟踪效果. 而当目标的运动模式恰好落在基础模型上时, 跟踪效果相较于IMM方法有所下降. EMA方法取决于模型集的准确程度, 若目标运动模式恰好符合模型集, EMA方法跟踪效果. 然而, 考虑到实际条件下目标真实运动模式未知且难以预测, 大部分情况下目标真实运动模式并不符合EMA模型集.
针对目标真实运动模式未知且难以预测的问题, 本节提出基于可能模型集的期望模式扩增方法EMA-LMS, 并通过仿真分析及仿真实验结果说明分布式DVSMM方法框架的通用性和易于实现的特点.
EMA-LMS方法的优点在于, 既能够达到EMA方法跟踪精度, 又能降低每个时刻参与滤波的模型数量, 即降低运算时间复杂度. 本文提出的DVSMM方法通过拓展VSMM的输入, 将本地传感器的量测信息拓展为通信邻域内其他传感器的所有量测信息, 并进行一致性融合估计.
EMA-LMS方法流程如下:
1) 当
$k + 1$ 时刻, 首先计算模型概率${\left\{ \mu _{k|k - 1}^{(i)}\right\} ^{_{{m^{(i)}} \in {M_{k - 1}}}}}$ 与${E_k} = [{M_{k - 1}};{M^1},\cdots,{M^q}]$ 扩展后的模型集, 并计算${\left\{ \hat x_{k|k}^{(i)},{P}_{k|k}^{(i)},\mu _{k|k}^{(i)}\right\} ^{{m^{(i)}} \in {E_k}}}.$ 令${M^f} = {M_{k - 1}} - {E_{k - 1}},$ 由$[{M^f},{M_{k - 1}}]$ 计算可得${\left\{ \hat x_{k|k}^{(i)},{P}_{k|k}^{(i)},\mu _{k|k}^{(i)}\right\} ^{_{{m^{(i)}} \in {M^f}}}}.$ 2) 根据
${\left\{ \mu _{k|k}^{(i)}\right\} ^{_{{m^{(i)}} \in {M^f}}}}$ , 将模型${M^f}$ 分为可能模型${M_p}$ $\left(\mu _{k|k}^{(i)} > {t_2}\right)$ 、重要模型${M_s}$ $\left({t_1} \leq \mu _{k|k}^{(i)} \leq {t_2}\right)$ 、不太可能模型${M_u}$ (${\mu _k} \leq {t_1}$ ). 统计与${M_p}$ 毗邻(转移概率不为0)的模型集合${M_a}$ , 令本时刻需要删除的候选基础模型为${M_d} = {M_u} \cup {\bar M_a}$ .3) 统计与
${M_p}$ 毗邻(转移概率不为0)的模型集合${M_a}$ , 得到本时刻需要添加的基础模型${M_n} = $ $ {M_a} \cap {\bar M_k}$ . 本时刻需要删除的候选基础模型${M_d} = $ $ {M_u} - {M_a}$ .4) 若
${M_n} = \emptyset $ , 转到第5)步. 否则计算$[{M_n}, $ $ {M_{k - 1}}]$ , 得到${M_n}$ 各模型状态估计值、协方差和模型概率:${\left\{ \hat x_{k|k}^{(i)},{P}_{k|k}^{(i)},\mu _{k|k}^{(i)}\right\} ^{_{{m^{(i)}} \in {M_n}}}}$ . 然后进行期望模型的再次更新, 计算估计融合$[{M^f},{M_n},{E_k};{M_{k - 1}}]$ , 由得到的模型概率计算新的期望模型${E'_k}$ . 再计算一致性融合估计$[{M^f},{M_n},{E'_k};{M_{k - 1}}]$ , 得到本算法周期的总体估计结果${\left\{ \hat x_{k|k}^{(i)},{P}_{k|k}^{(i)},\mu _{k|k}^{(i)}\right\} ^{_{{m^{(i)}} \in ({M^f} \cup {M_n} \cup {{E'}_k})}}}$ . 并令${M_k} = {M^f} \cup {M_n} \cup {E'_k}$ , 且记${E_k} = {E'_k}$ .5) 输出本时刻的估计融合结果
$\Big\{ \hat x_{k|k}^{(i)}, {P}_{k|k}^{(i)}, $ $ \mu _{k|k}^{(i)}\Big\} ^{{m^{(i)}} \in {M_k}}$ . 若${M_d} = \emptyset $ , 返回S1; 否则,令${M_{k + 1}} = $ $ {M_k}$ , 并从${M_{k + 1}}$ 中删掉${M_d}$ 中具有更小概率的那些模型, 直到${M_d}$ 中所有模型被删完或$\left| {{M_{k + 1}}} \right| = K$ .4. 仿真分析
本节通过仿真分析说明本文提出的DVSMM方法的有效性. 考虑一个雷达和红外传感器网络, 所有传感器在仿真过程中始终能观察到目标.
通过4种方法验证本文提出的分布式VSMM框架的有效性. DIMM1和DIMM2分别使用了文献[18]和[19]的分布式IMM方法框架. DIMM3表示用本文提出的DVSMM框架实现的分布式IMM方法. DEMA-LMS为用本文提出的DVSMM框架实现的分布式EMA-LMS.
假设目标为二维平面机动目标, 目标的状态变量为
$x = {\left[ x\;\;{\dot x}\;\;y\;\;{\dot y} \right]^{\rm{T}}}$ ,$x$ 与$y$ 分别表示目标在$x$ 轴、$y$ 轴方向上的位置,$\dot x$ 与$\dot y$ 分别表示目标在$x$ 轴、$y$ 轴方向上的速度. 目标状态转移方程如式(32)所示:$${x_{k{\rm{ + }}1}} = {{F}_k}{x_k} + {{G}_k}{u_k} + {{\varGamma }_{k + 1}}{w_{k + 1}}$$ (32) 式中,
${u_k} = {[ {{a_x}}\;\;{{a_y}} ]^{\rm{T}}}$ 为目标加速度, 可以进行阶跃变化;${w_k}$ 为过程噪声,${w_k} \sim {\rm N}(0,{Q_k}){Q_k})$ ;${{F}_k}$ 表示状态转移矩阵;${{G}_k}$ 为加速度输入矩阵;${{\varGamma }_k}$ 为噪声传递矩阵.$$\begin{split} &{{F}_k} = {{\pmb I}_{2 \times 2}} \otimes {F},\;\;\;\; {{G}_k} = {{\Gamma }_k} = {{\pmb I}_{2 \times 2}} \otimes {G}\\ &{F} = \left[ {\begin{array}{*{20}{c}} 1&{{T}}\\ 0&1 \end{array}} \right],{G} = \left[ {\begin{array}{*{20}{c}} {{{{T}}^2}/2}\\ {{T}} \end{array}} \right] \end{split}$$ (33) 式中,
${{T}}$ 为采样周期;${{\pmb I}_{2 \times 2}}$ 表示二阶单位矩阵;$ \otimes $ 表示矩阵的直积.目标初始状态
${x_0} = {\left[ {0\;\;\; 1500\;\;\; 0\;\;\; 1500} \right]^{\rm{T}}}$ , 过程噪声方差${Q_k} = {\rm diag}\{0.01,0.01\}$ . 仿真时长为300s,$ {{T}} = 1 $ s. 目标运动加速度输入如表1所示:表 1 目标运动模式的变化Table 1 Target mode switching时间k 1 ~ 50 50 ~ 100 100 ~ 150 150 ~ 200 200 ~ 250 250 ~ 300 加速度${u_k}$ $ {\left[\rm{0}, \rm{0}\right]}^{\rm{T}}$ $ {\left[\rm{0}, \rm{-20}\right]}^{\rm{T}}$ $ {\left[\rm{0}, \rm{0}\right]}^{\rm{T}}$ $ {\left[\rm{10}, \rm{10}\right]}^{\rm{T}}$ $ {\left[\rm{-10}, \rm{-10}\right]}^{\rm{T}}$ $ {\left[\rm{10}, \rm{10}\right]}^{\rm{T}}$ 仿真中使用的基础模型集均为文献[4]中包含13个模型的基础模型集, 是具有固定加速度输入的二维CV模型. 对于模型
$j$ , 目标状态转移方程为:$${x_{k + 1}} = {F}_k^{(j)}{x_k} + {G}_k^{(j)}u_k^{(j)} + {\varGamma }_k^{(j)}w_k^{(j)}$$ (34) 式中,
${F}_k^{(j)}$ 、${G}_k^{(j)}$ 、${\varGamma }_k^{(j)}$ 的含义与式(33)相同. 模型之间的区别只在于加速度输入$u_k^{(j)}$ 不同. 基础模型集中不同模型的加速度输入如式(35)和图7所示:仿真中在基础模型集中使用的模型转移概率矩阵
${{\pmb G}}_k^{(j)}$ 如式(36)所示:$$\left\{\!\! \begin{aligned} &{u^{(1)}} \!=\! {[0,0]^{\rm{T}}},\qquad{u^{(2)}} \!=\! {[20,0]^{\rm{T}}},\qquad\!\;\,{u^{(3)}} \!=\! {[0,20]^{\rm{T}}}\\ &{u^{(4)}} \!=\! {[ - 20,0]^{\rm{T}}},\;\;\,{u^{(5)}} \!=\! {[0, - 20]^{\rm{T}}},\;\;\quad{u^{(6)}} \!=\! {[20,20]^{\rm{T}}}\\ &{u^{(7)}} \!=\! {[ - 20,20]^{\rm{T}}},\;{u^{(8)}} \!=\! {[ - 20, - 20]^{\rm{T}}},\;{u^{(9)}} \!=\! {[20, - 20]^{\rm{T}}}\\ &{u^{(10)}} \!=\! {[40,0]^{\rm{T}}},\quad{\kern 1pt} {\rm{ }}{u^{(11)}}\!=\! {[0,40]^{\rm{T}}},\qquad{u^{(12)}} \!=\! {[ - 40,0]^{\rm{T}}}\\ &{u^{(13)}} \!=\!{[0, - 40]^{\rm{T}}} \end{aligned} \!\!\right.$$ (35) 雷达传感器位置量测误差标准差为50 m, 角度量测误差标准差为0.01°. 红外传感器角度量测误差标准差为0.01°. 雷达传感器共4个, 坐标分别为(1, 0.4), (2, 1.7), (3.7, 1.7), (5.5, 2). 红外传感器共8个, 坐标分别为(2, 1.2), (0.8, 1.4), (3, 1.4), (2.5, 1), (4.1, 3), (3, 2), (4.5, 1.8), (4, 2.5). 目标运动轨迹和传感器位置如图8所示.
为了比较一致性滤波的跟踪效果使用两类指标作为方法性能的衡量指标: 平均位置
${E_p}(k)$ 和速度估计误差${E_v}(k)$ 用来衡量传感器节点状态估计准确性; 平均位置估计一致性误差${D_p}(k)$ 和平均速度估计一致性误差${D_v}(k)$ 衡量每个传感器节点状态估计的一致程度. 评价指标计算见式(37)和(38).$$\begin{split} &{E_p}(k) = \sqrt \frac{1}{N}\sum\limits_{i \in V} ({{(\hat x_{k|k}^i - {x_k})}^2} + (\hat y_{k|k}^i - {y_k})) \\ &{E_v}(k) = \sqrt \frac{1}{N}\sum\limits_{i \in V} ({{({\hat {\dot x}}_{k|k}^i - {{\dot x}_k})}^2} + ({\hat {\dot y}}_{k|k}^i - {{\dot y}_k})) \end{split}\tag{37}$$ (37) $$\begin{split} &{D_p}(k) = \sqrt {\frac{1}{N}\sum\limits_{i \in V} {({{(\hat x_{k|k}^i - \hat x_{k|k}^{av})}^2} + (\hat y_{k|k}^i - \hat y_{k|k}^{av}))} } \\ &{D_v}(k) = \sqrt {\frac{1}{N}\sum\limits_{i \in V} {({{({\hat {\dot x}}_{k|k}^i - {\hat {\dot x}}_{k|k}^{av})}^2} + ({\hat {\dot y}}_{k|k}^i - {\hat {\dot y}}_{k|k}^{av}))} } \end{split}\tag{38}$$ (38) 式中,
$N$ 为传感器节点数量;$x$ 与$y$ 、$\dot x$ 与$\dot y$ 分别表示状态向量中的位置和速度;$\hat x_{k|k}^{av}$ ,$\hat y_{k|k}^{av}$ 和$\;{\hat {\dot x}}_{k|k}^{av}$ ,$\;{ \hat {\dot y}}_{k|k}^{av}$ 分别表示节点位置和速度估计的平均值:$${{G_k} =\left[\begin{array}{*{20}{c}} 308/360 & 2/360 & 2/360 & 2/360 & 2/360 & 1/360 & 1/360 & 1/360 & 1/360 & 0 & 0 & 0 & 0 & 1/9 \\ 1/70 & 3/4 & 1/140 & 0 & 1/140 & 1/140 & 0 & 0 & 1/140 & 1/140 & 0 & 0 & 0 & 1/5 \\ 1/70 & 1/140 & 3/4 & 1/140 & 0 & 1/140 & 1/140 & 0 & 0 & 0 & 1/140 & 0 & 0 & 1/5 \\ 1/70 & 0 & 1/140 & 3/4 & 1/140 & 0 & 1/140 & 1/140 & 0 & 0 & 0 & 1/140 & 0 & 1/5 \\ 1 /7 0 & 1/140 & 0 & 1/140 & 3/4 & 0 & 0 & 1/140 & 1/140 & 0 & 0 & 0 & 1/140 & 1/5 \\ 1/30 & 1/90 & 1/90 & 0 & 0 & 11/15 & 0 & 0 & 0 & 1/180 & 1/180 & 0 & 0 & 1/5 \\ 1/30 & 0 & 1/90 & 1/90 & 0 & 0 & 11/15 & 0 & 0 & 0 & 1/180 & 1/180 & 0 & 1/5 \\ 1/30 & 0 & 0 & 1/90 & 1/90 & 0 & 0 & 11/15 & 0 & 0 & 0 & 1/180 & 1/180 & 1/5 \\ 1/30 & 1/90 & 0 & 0 & 1/90 & 0 & 0 & 0 & 11/15 & 1/180 & 0 & 0 & 1/180 & 1/5 \\ 0 & 1/20 & 0 & 0 & 0 & 1/40 & 0 & 0 & 1/40 & 7/10 & 0 & 0 & 0 & 1/5\\ 0 & 0 & 1/20 & 0 & 0 & 1/40 & 1/40 & 0 & 0 & 0 & 7/10 & 0 & 0 & 1/5 \\ 0 & 0 & 0 & 1/20 & 0 & 0 & 1/40 & 1/40 & 0 & 0 & 0 & 7/10 & 0 & 1/5 \\ 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 1/4 0 & 1/40 & 0 & 0 & 0 & 7/1 0 & 1/5\\ 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 1/50 & 37/50\end{array}\right]} \tag{36}$$ (36) $$\begin{split} &\hat x_{k|k}^{av} = \frac{1}{N}\sum\limits_{i \in V} {\hat x_{k|k}^i} ,\hat{\dot x}_{k|k}^{av} = \frac{1}{N}\sum\limits_{i \in V} {\hat{\dot x}_{k|k}^i} \\ &\hat y_{k|k}^{av} = \frac{1}{N}\sum\limits_{i \in V} {\hat y_{k|k}^i} ,\hat{\dot y }_{k|k}^{av} = \frac{1}{N}\sum\limits_{i \in V} {\hat{\dot y}_{k|k}^i} \end{split} \tag{39}$$ 进行50次蒙特卡洛重复试验, 三种方法的一致性权值都使用Metropolis加权. 仿真结果如图9 ~ 图12所示:
在图9 ~ 12中所示仿真实验结果中, 当
$k < 150$ 时, 目标运动模式突变前后均符合EMA方法13个基础模型. 而当$150 < k < 300$ 时, 目标运动模式不符合EMA基础模型. 对比上述4种分布式跟踪方法, 结论如下:1)尽管EMA-LMS方法比较复杂, 包含很多的模型扩增和删除步骤, 但还是能非常方便地将其应用于分布式状态估计中, 说明了本文提出的分布式VSMM方法的有效性;
2)当目标的运动模式落在基础模型上时, 通过DVSMM实现的分布式IMM与信息状态贡献和对应的信息矩阵一致的分布式IMM方法效果类似;
3)当目标的运动模式落在基础模型间隙时, 使用DVSMM方法实现的分布式IMM在运动模式位于基础模型间隙时效果比两种分布式IMM方法更好;
4) EMA-LMS方法运用在分布式状态估计中, 效果显著, 体现在当目标的运动模式落在基础模型间隙时, 具有高于另外三种方法的状态估计准确性和一致性.
通过上述仿真实验结果与分析, 验证了本文提出的分布式VSMM方法的有效性. 相比于分布式IMM方法, 分布式VSMM能够根据需要灵活调整模型集结构, 具备更好的适应性和状态估计效果.
5. 结论
本文根据一致性理论, 对变结构交互式多模型方法进行改进, 与无迹信息滤波相结合, 提出基于一致性的分布式变结构多模型状态估计方法框架. 本文方法能够在基于一致性的分布式状态估计中引入各种已有的变结构多模型方法, 具有良好的跟踪精度和状态估计一致性.
-
图 8 弹性−阻尼型串联弹性驱动器示意简图
Fig. 8 Schematic diagram of elastic-damped series elastic actuator
图 10 基于弹性与阻尼元件并联组合装置的串联弹性驱动器
Fig. 10 Series elastic actuator based on parallel combination of elastic and damping elements
图 11 基于弹性元件放置位置的弹性-阻尼型串联弹性驱动器示意简图
Fig. 11 Schematic diagram of an elastic-damped series elastic actuator based on the placement position of the elastic element
图 14 具有预压缩的弹性元件模型特性曲线
Fig. 14 The characteristic curve of elastic element with pre-compressed
图 15 基于具有弹性和阻尼特性元件物理特性的建模方法
Fig. 15 The modeling method of element with elastic and damping characteristics by the physical characteristics
表 1 各类串联弹性驱动器机械实现方式及比较
Table 1 Mechanical realization and comparison of various series elastic actuators
类型 物理特性 机械实现方式 系统力带宽 能量特性 安全性 弹性型串联弹性驱动器 弹性 直线压缩弹簧 一般 效率高 高 直线拉伸弹簧 螺旋扭转弹簧 结构弹簧 阻尼型串联弹性驱动器 阻尼 磁流变液阻尼器 较高 效率低 低 弹性-阻尼型串联弹性驱动器 弹性和阻尼 粘弹性元件 高 效率一般 一般 弹性与阻尼元件并联组合装置 弹性元件放置位置 
下载: 导出CSV
表 2 各类串联弹性驱动器建模方法
Table 2 Modeling methods of various series elastic actuators
类型 建模方法 模型参数 复杂度 弹性型 胡克定律 1 简单 阻尼型 线性粘性阻尼 1 简单 弹性-阻尼型 粘弹性元件 Maxwell 模型、指数模型等 多 复杂 弹性和阻尼元件并联组合装置 线性粘性阻尼与胡克定律 弹性元件放置位置 动力学
下载: 导出CSV
-
[1] Li J, Li S Q, Tian G H, Shang H C. Muscle tension training method for series elastic actuator (SEA) based on gain-scheduled method. Robotics and Autonomous Systems, 2019, 121: Article No. 103253 [2] Pratt G A, Williamson M M. Series elastic actuators. In: Proceedings of the 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems. Pittsburgh, USA: IEEE, 1995. 399−406 [3] Chew C M, Hong G S, Zhou W. Series damper actuator: A novel force/torque control actuator. In: Proceedings of the 4th IEEE/RAS International Conference on Humanoid Robots. Santa Monica, USA: IEEE, 2004. 533−546 [4] Hurst J W, Rizzi A A, Hobbelen D. Series elastic actuation: Potential and pitfalls. In: Proceedings of the 2004 International Conference on Climbing and Walking Robots. 2004. [5] Vallery H, Veneman J, Van Asseldonk E, Ekkelenkamp R, Buss M, Van Der Kooij H. Compliant actuation of rehabilitation robots. IEEE Robotics & Automation Magazine, 2008, 15(3): 60−69 [6] Refour E R, Sebastian B, Chauhan R J, Ben-Tzvi P. A general purpose robotic hand exoskeleton with series elastic actuation. Journal of Mechanisms and Robotics, 2019, 11(6): Article No. 060902 [7] Vantilt J, Tanghe K, Afschrift M, Bruijnes A K B D, Junius K, Geeroms J, et al. Model-based control for exoskeletons with series elastic actuators evaluated on sit-to-stand movements. Journal of Neuro Engineering and Rehabilitation. 2019, 16: Article No. 65 [8] Trigili E, Crea S, Moise M, Baldoni A, Cempini M, Ercolini G, et al. Design and experimental characterization of a shoulder-elbow exoskeleton with compliant joints for post-stroke rehabilitation. IEEE/ASME Transactions on Mechatronics, 2019, 24(4): 1485−1496 doi: 10.1109/TMECH.2019.2907465 [9] Lee H D, Moon J I, Kang T H. Design of a series elastic tendon actuator based on gait analysis for a walking assistance exosuit. International Journal of Control, Automation and Systems, 2019, 17(11): 2940−2947 doi: 10.1007/s12555-018-0492-0 [10] Zhang T, Huang H. Design and control of a series elastic actuator with clutch for hip exoskeleton for precise assistive magnitude and timing control and improved mechanical safety. IEEE/ASME Transactions on Mechatronics, 2019, 24(5): 2215−2226 doi: 10.1109/TMECH.2019.2932312 [11] Guenther F, Quy Vu H, Lida F. Improving legged robot hopping by using coupling-based series elastic actuation. IEEE/ASME Transactions on Mechatronics, 2019, 24(2): 413−423 doi: 10.1109/TMECH.2019.2893235 [12] Verstraten T, Furnémont R, Beckerle P, Vanderborght B, Lefeber D. A hopping robot driven by a series elastic dual-motor actuator. IEEE Robotics and Automation Letters, 2019, 4(3): 2310−2316 doi: 10.1109/LRA.2019.2902071 [13] Lee C, Oh S. Development, analysis, and control of series elastic actuator-driven robot leg. Frontiers in Neurorobotics, 2019, 13: Article No. 17 [14] Werner A, Henze B, Keppler M, Loeffl F, Leyendecker S, Ott C. Structure preserving multi-contact balance control for series-elastic and visco-elastic humanoid robots. In: Proceedings of the 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Madrid, Spain: IEEE, 2018. 1233−1240 [15] Bellicoso C D, Jenelten F, Fankhauser P, Gehring C, Hwangbo J, Hutter M. Dynamic locomotion and whole-body control for quadrupedal robots. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, Canada: IEEE, 2017. 3359−3365 [16] Zhu Q G, Mao Y C, Xiong R, Wu J. Adaptive torque and position control for a legged robot based on a series elastic actuator. International Journal of Advanced Robotic Systems, 2016, 13(1): Article No. 26 [17] Werner A, Turlej W, Ott C. Generation of locomotion trajectories for series elastic and viscoelastic bipedal robots. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, BC, Canada: IEEE, 2017. 5853−5860 [18] Pfeifer S, Pagel A, Riener R, Vallery H. Actuator with angle-dependent elasticity for biomimetic transfemoral prostheses. IEEE/ASME Transactions on Mechatronics, 2015, 20(3): 1384−1394 doi: 10.1109/TMECH.2014.2337514 [19] Convens B, Dong D B, Furnemont R, Verstraten T, Cherelle P, Lefeber D, et al. Modeling, design and test-bench validation of a semi-active propulsive ankle prosthesis with a clutched series elastic actuator. IEEE Robotics and Automation Letters, 2019, 4(2): 1823−1830 doi: 10.1109/LRA.2019.2897993 [20] Sun X J, Sugai F, Okada K, Inaba M. Variable transmission series elastic actuator for robotic prosthesis. In: Proceedings of the 2018 IEEE International Conference on Robotics and Automation (ICRA). Brisbane, Australia: IEEE, 2018. 2796−2803 [21] Sun X J, Sugai F, Okada K, Inaba M. Design, control and preliminary test of robotic ankle prosthesis. In: Proceedings of the 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Madrid, Spain: IEEE, 2018. 2787−2793 [22] Zhu J Y, Wang Q N, Wang L. On the design of a powered transtibial prosthesis with stiffness adaptable ankle and toe joints. IEEE Transactions on Industrial Electronics, 2014, 61(9): 4797−4807 doi: 10.1109/TIE.2013.2293691 [23] Baccelliere L, Kashiri N, Muratore L, Laurenzi A, Kamedula M, Margan A, et al. Development of a human size and strength compliant bi-manual platform for realistic heavy manipulation tasks. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, BC, Canada: IEEE, 2017. 5594−5601 [24] Hu Y, Mombaur K. Optimal control based push recovery strategy for the iCub humanoid robot with series elastic actuators. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, BC, Canada: IEEE, 2017. 5846-5852 [25] Nava G, Pucci D, Nori F. Momentum control of humanoid robots with series elastic actuators. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, Canada: IEEE, 2017. 2185−2191 [26] Lee C, Lee J, Malzahn J, Tsagarakis N, Oh S. A two-staged residual for resilient external torque estimation with series elastic actuators. In: Proceedings of the IEEE-RAS 17th International Conference on Humanoid Robotics (Humanoids). Birmingham, England: IEEE, 2017. 817−823 [27] Tsagarakis N G, Laffranchi M, Vanderborght B, Caldwell D G. A compact soft actuator unit for small scale human friendly robots. In: Proceedings of the 2009 IEEE International Conference on Robotics and Automation. Kobe, Japan: IEEE, 2009. 4356−4362 [28] Lee C, Kwak S, Kwak J, Oh S. Generalization of series elastic actuator configurations and dynamic behavior comparison. Actuators, 2017, 6(3): Article No. 26 [29] Vanderborght B, Albu-Schaeffer A, Bicchi A, Burdet E, Caldwell D G, Carloni R, et al. Variable impedance actuators: A review. Robotics and Autonomous Systems, 2013, 61(12): 1601−1614 doi: 10.1016/j.robot.2013.06.009 [30] 魏敦文, 葛文杰, 高涛. 仿生灵感下的弹性驱动器的研究综述. 机器人, 2017, 39(4): 541−550Wei Dun-Wen, Ge Wen-Jie, Gao Tao. Review of elastic actuator research from bionic inspiration. Robot, 2017, 39(4): 541−550 [31] Hsieh H C, Chen D F, Chien L, Lan C C. Design of a parallel actuated exoskeleton for adaptive and safe robotic shoulder rehabilitation. IEEE/ASME Transactions on Mechatronics, 2017, 22(5): 2034−2045 doi: 10.1109/TMECH.2017.2717874 [32] Lee J W, Kim G. Design and control of a lifting assist device for preventing lower back injuries in industrial athletes. International Journal of Precision Engineering and Manufacturing, 2019, 20(10): 1825−1838 doi: 10.1007/s12541-019-00183-0 [33] 孙雷, 孙伟超, 王萌, 刘景泰. 基于RISE反馈的串联弹性驱动器最优控制方法. 自动化学报, 2018, 44(12): 2170−2178Sun Lei, Sun Wei-Chao, Wang Meng, Liu Jing-Tai. Optimal control for series elastic actuator using RISE feedback. Acta Automatica Sinica, 2018, 44(12): 2170−2178 [34] Rouse E J, Mooney L M, Herr H M. Clutchable series-elastic actuator: Implications for prosthetic knee design. International Journal of Robotics Research, 2014, 33(13): 1611−1625 doi: 10.1177/0278364914545673 [35] Agarwal P, Yun Y, Fox J, Madden K, Deshpande A D. Design, control, and testing of a thumb exoskeleton with series elastic actuation. International Journal of Robotics Research, 2017, 36(3): 355−375 doi: 10.1177/0278364917694428 [36] Marconi D, Baldoni A, McKinney Z, Cempini M, Crea S, Vitiello N. A novel hand exoskeleton with series elastic actuation for modulated torque transfer. Mechatronics, 2019, 61: 69−82 doi: 10.1016/j.mechatronics.2019.06.001 [37] Gim K G, Hong D W. Design of a series elastic resistance mechanism for exercise and rehabilitation. In: Proceedings of the IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids). Cancun, Mexico: IEEE, 2016. 1239−1244 [38] Yu N B, Zou W L, Tan W, Yang Z. Augmented virtual stiffness rendering of a cable-driven SEA for human-robot interaction. IEEE/CAA Journal of Automatica Sinica, 2017, 4(4): 714−723 doi: 10.1109/JAS.2017.7510637 [39] Yu N B, Zou W L, Sun Y B. Passivity guaranteed stiffness control with multiple frequency band specifications for a cable-driven series elastic actuator. Mechanical Systems and Signal Processing, 2019, 117: 709−722 doi: 10.1016/j.ymssp.2018.08.007 [40] Rao P, Deshpande A D. Analyzing and improving cartesian stiffness control stability of series elastic tendon-driven robotic hands. In: Proceedings of the 2018 IEEE International Conference on Robotics and Automation (ICRA). Brisbane, Australia: IEEE, 2018. 5415−5420 [41] Rao P, Thomas G C, Sentis L, Deshpande A D. Analyzing achievable stiffness control bounds of robotic hands with coupled finger joints. In: Proceedings of the 2017 IEEE International Conference on Robotics and Automation (ICRA). Singapore: IEEE, 2017. 3447−3452 [42] 尹伟, 孙雷, 王萌, 刘景泰. 针对串联弹性驱动器抖动抑制的轨迹规划和跟踪控制. 自动化学报, 2018, 44(8): 1436−1445Yin Wei, Sun Lei, Wang Meng, Liu Jing-Tai. Motion planning and tracking control of series elastic actuator for vibration suppression. Acta Automatica Sinica, 2018, 44(8): 1436−1445 [43] Yin W, Sun L, Wang M, Liu J T. Position control of a series elastic actuator based on global sliding mode controller design. IEEE/CAA Journal of Automatica Sinica, 2019, 6(3): 850−858 doi: 10.1109/JAS.2019.1911498 [44] Chen T Y, Casas R, Lum P S. An elbow exoskeleton for upper limb rehabilitation with series elastic actuator and cable-driven differential. IEEE Transactions on Robotics, 2019, 35(6): 1464−1474 doi: 10.1109/TRO.2019.2930915 [45] Jung Y, Bae J. An asymmetric cable-driven mechanism for force control of exoskeleton systems. Mechatronics, 2016, 40: 41−50 doi: 10.1016/j.mechatronics.2016.10.013 [46] DeBoon B, Nokleby S, La Delfa N, Rossa C. Differentially-clutched series elastic actuator for robot-aided musculoskeletal rehabilitation. In: Proceedings of the 2019 International Conference on Robotics and Automation (ICRA). Montreal, Canada: IEEE, 2019. 1507−1513 [47] Ates S, Sluiter V I, Lammertse P, Stienen A H A. ServoSEA concept: Cheap, miniature series-elastic actuators for orthotic, prosthetic and robotic hands. In: Proceedings of the 5th IEEE RAS/EMBS International Conference on Biomedical Robotics and Biomechatronics. Sao Paulo, Brazil: IEEE, 2014. 752−757 [48] Agarwal P, Deshpande A D. Series elastic actuators for small-scale robotic applications. Journal of Mechanisms and Robotics, 2017, 9(3): Article No. 031016 [49] Carpino G, Accoto D, Sergi F, Tagliamonte N L, Guglielmelli E. A novel compact torsional spring for series elastic actuators for assistive wearable robots. Journal of Mechanical Design, 2012, 134(12): Article No. 121002 [50] Irmscher C, Woschke E, May E, Daniel C. Design, optimisation and testing of a compact, inexpensive elastic element for series elastic actuators. Medical Engineering & Physics, 2018, 52: 84−89 [51] Ruiken D, Cummings J P, Savaria U R, Sup F C, Grupen R A. uBot-7: A dynamically balancing mobile manipulator with series elastic actuators. In: Proceedings of the IEEE-RAS 17th International Conference on Humanoid Robotics. Birmingham, England: IEEE, 2017. 676−682 [52] Lagoda C, Schouten A C, Stienen A H A, Hekman E E G, Van Der Kooij H. Design of an electric series elastic actuated joint for robotic gait rehabilitation training. In: Proceedings of the 3rd IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics. Tokyo, Japan: IEEE, 2010. 21−26 [53] Wang Y P, Chen Y L, Chen K B, Wu Y X, Huang Y J. A flat torsional spring with corrugated flexible units for series elastic actuators. In: Proceedings of the 2nd International Conference on Advanced Robotics and Mechatronics (ICARM). Hefei, China: IEEE, 2017. 138−143 [54] Dos Santos W M, Caurin G A P, Siqueira A A G. Torque control characterization of a rotary series elastic actuator for knee rehabilitation. In: Proceedings of the 16th International Conference on Advanced Robotics. Montevideo, Uruguay: IEEE, 2013. 1−6 [55] Yildirim M C, Sendur P, Bilgin O, Gulek B, Yapici G G, Ugurlu B. An integrated design approach for a series elastic actuator: Stiffness formulation, fatigue analysis, thermal management. In: Proceedings of the IEEE-RAS 17th International Conference on Humanoid Robotics. Birmingham, England: IEEE, 2017. 384−389 [56] Georgiev N, Burdick J. Design and analysis of planar rotary springs. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, BC, Canada: IEEE, 2017. 4777−4784 [57] Zhang T, Tran M, Huang H. Design and experimental verification of hip exoskeleton with balance capacities for walking assistance. IEEE/ASME Transactions on Mechatronics, 2018, 23(1): 274−285 doi: 10.1109/TMECH.2018.2790358 [58] Koopaee M J, Bal S, Pretty C, Chen X Q. Design and development of a wheel-less snake robot with active stiffness control for adaptive pedal wave locomotion. Journal of Bionic Engineering, 2019, 16(4): 593−607 doi: 10.1007/s42235-019-0048-x [59] Paine N, Mehling J S, Holley J, Radford N A, Johnson G, Fok C L, et al. Actuator control for the NASA-JSC valkyrie humanoid robot: A decoupled dynamics approach for torque control of series elastic robots. Journal of Field Robotics, 2015, 32(3): 378−396 doi: 10.1002/rob.21556 [60] Yasuda K, Ikeura R, Hayakawa S, Sawai H. Stability of impedance control of a series elastic actuator with a torque controlled actuator for improving the system performance. In: Proceedings of the 2015 IEEE International Conference on Robotics and Biomimetics (ROBIO). Zhuhai, China: IEEE, 2015. 2163−2168 [61] Lee Y F, Chu C Y, Xu J Y, Lan C C. A humanoid robotic wrist with two-dimensional series elastic actuation for accurate force/torque interaction. IEEE/ASME Transactions on Mechatronics, 2016, 21(3): 1315−1325 doi: 10.1109/TMECH.2016.2530746 [62] Bianchi M, Cempini M, Conti R, Meli E, Ridolfi A, Vitiello N, et al. Design of a series elastic transmission for hand exoskeletons. Mechatronics, 2018, 51: 8−18 doi: 10.1016/j.mechatronics.2018.02.010 [63] Zhou W, Chew C M, Hong G S. Inverse dynamics control for series damper actuator based on MR fluid damper. In: Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics. Monterey, California, USA: IEEE, 2005. 473−478 [64] Zhou W, Chew C M, Hong G S. Design of series damper actuator. Robotica, 2009, 27(3): 379−387 doi: 10.1017/S0263574708004797 [65] Westerveld A J, Aalderink B J, Hagedoorn W, Buijze M, Schouten A C, Van Der Kooij H. A damper driven robotic end-point manipulator for functional rehabilitation exercises after stroke. IEEE Transactions on Biomedical Engineering, 2014, 61(10): 2646−2654 doi: 10.1109/TBME.2014.2325532 [66] Kim D H, Oh J H. Hysteresis modeling for torque control of an elastomer series elastic actuator. IEEE/ASME Transactions on Mechatronics, 2019, 24(3): 1316−1324 doi: 10.1109/TMECH.2019.2906698 [67] Rollinson D. Control and Design of Snake Robots [Ph. D. dissertation], Carnegie Mellon University, Pittsburgh, PA, 2014 [68] Jarrett C, McDaid A. Modeling and feasibility of an elastomer-based series elastic actuator as a haptic interaction sensor for exoskeleton robotics. IEEE/ASME Transactions on Mechatronics, 2019, 24(3): 1325−1333 doi: 10.1109/TMECH.2019.2906918 [69] Austin J, Schepelmann A, Geyer H. Control and evaluation of series elastic actuators with nonlinear rubber springs. In: Proceedings of the 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Hamburg, Germany: IEEE, 2015. 6563−6568 [70] Shepherd M K, Rouse E J. Design and validation of a torque-controllable knee exoskeleton for sit-to-stand assistance. IEEE/ASME Transactions on Mechatronics, 2017, 22(4): 1695−1704 doi: 10.1109/TMECH.2017.2704521 [71] Van Dijk W, Meijneke C, Van Der Kooij H. Evaluation of the achilles ankle exoskeleton. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2017, 25(2): 151−160 doi: 10.1109/TNSRE.2016.2527780 [72] Kang I, Hsu H, Young A. The effect of hip assistance levels on human energetic cost using robotic hip exoskeletons. IEEE Robotics and Automation Letters, 2019, 4(2): 430−437 doi: 10.1109/LRA.2019.2890896 [73] Caputo J M, Collins S H. A universal ankle-foot prosthesis emulator for human locomotion experiments. Journal of Biomechanical Engineering, 2014, Article No. 65(3): Article No. 035002 [74] Senturk Y M, Patoglu V. MRI-VisAct: A Bowden-cable-driven MRI-compatible series viscoelastic actuator. Transactions of the Institute of Measurement and Control, 2018, 40(8): 2440−2453 doi: 10.1177/0142331217730429 [75] Oblak J, Matjačić Z. Design of a series visco-elastic actuator for multi-purpose rehabilitation haptic device. Journal of NeuroEngineering and Rehabilitation, 2011, 8: Article No. 3 [76] Mancisidor A, Zubizarreta A, Cabanes I, Bengoa P, Marcos M, Jung J H. Enhanced force control using force estimation and nonlinearity compensation for the universal haptic pantograph. In: Proceedings of the 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Hamburg, Germany: IEEE, 2015. 5599−5604 [77] Iwata H, Sugano S. Design of human symbiotic robot TWENDY-ONE. In: Proceedings of the 2009 IEEE International Conference on Robotics and Automation. Kobe, Japan: IEEE, 2009. 580−586 [78] Garcia E, Arevalo J C, Muñoz G, Gonzalez-De-Santos P. Combining series elastic actuation and magneto-rheological damping for the control of agile locomotion. Robotics and Autonomous Systems, 2011, 59(10): 827−839 doi: 10.1016/j.robot.2011.06.006 [79] Paine N, Oh S, Sentis L. Design and control considerations for high-performance series elastic actuators. IEEE/ASME Transactions on Mechatronics, 2014, 19(3): 1080−1091 doi: 10.1109/TMECH.2013.2270435 [80] Lee H, Kwak S, Oh S. Force control of series elastic actuators-driven parallel robot. In: Proceedings of the 2018 IEEE International Conference on Robotics and Automation (ICRA). Brisbane, Australia: IEEE, 2018. 5401−5406 [81] Isik K, He S D, Ho J, Sentis L. Re-engineering a high performance electrical series elastic actuator for low-cost industrial applications. Actuators, 2017, 6(1): Article No. 5 doi: 10.3390/act6010005 [82] Lens T, Von Stryk O. Design and dynamics model of a lightweight series elastic tendon-driven robot arm. In: Proceedings of the 2013 IEEE International Conference on Robotics and Automation. Karlsruhe, Germany: IEEE, 2013. 4512−4518 [83] Lee C, Oh S. Configuration and performance analysis of a compact planetary geared elastic actuator. In: Proceedings of the 42nd Annual Conference of the IEEE Industrial Electronics Society. Florence, Italy: IEEE, 2016. 6391−6396 [84] Lauria M, Legault M A, Lavoie M A, Michaud F. Differential elastic actuator for robotic interaction tasks. In: Proceedings of the 2008 IEEE International Conference on Robotics and Automation. Pasadena, CA, USA: IEEE, 2008. 3606−3611 [85] Kim S, Bae J. Force-mode control of rotary series elastic actuators in a lower extremity exoskeleton using model-inverse time delay control. IEEE/ASME Transactions on Mechatronics, 2017, 22(3): 1392−1400 doi: 10.1109/TMECH.2017.2687979 [86] Oh S, Kong K. High-precision robust force control of a series elastic actuator. IEEE/ASME Transactions on Mechatronics, 2017, 22(1): 71−80 doi: 10.1109/TMECH.2016.2614503 [87] Kim S, Bae J. Force-mode control of rotary series elastic actuators in a lower extremity exoskeleton using model-inverse time delay control (MiTDC). In: Proceedings of the 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Daejeon, South Korea: IEEE, 2016. 3836−3841 [88] Oh S, Lee C, Kong K. Force control and force observer design of series elastic actuator based on its dynamic characteristics. In: Proceedings of the 41st Annual Conference of the IEEE Industrial Electronics Society. Yokohama, Japan: IEEE, 2015. 4639−4644 [89] Calanca A, Fiorini P. A rationale for acceleration feedback in force control of series elastic actuators. IEEE Transactions on Robotics, 2018, 34(1): 48−61 doi: 10.1109/TRO.2017.2765667 [90] Zhao Y, Paine N, Jorgensen S J, Sentis L. Impedance control and performance measure of series elastic actuators. IEEE Transactions on Industrial Electronics, 2018, 65(3): 2817−2827 doi: 10.1109/TIE.2017.2745407 [91] Zhang J J, Collins S H. The passive series stiffness that optimizes torque tracking for a lower-limb exoskeleton in human walking. Frontiers in Neurorobotics, 2017, 11: Article No. 68 doi: 10.3389/fnbot.2017.00068 [92] Calanca A, Muradore R, Fiorini P. Impedance control of series elastic actuators: Passivity and acceleration-based control. Mechatronics, 2017, 47: 37−48 doi: 10.1016/j.mechatronics.2017.08.010 [93] Lee Y H, Lee Y H, Lee H, Phan L T, Kang H, Kim Y B, et al. Development of torque controllable leg for running robot, AiDIN-IV. In: Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Vancouver, BC, Canada: IEEE, 2017. 4125−4130 [94] 毛翊超. 采用串联弹性驱动器的仿生腿足式机器人跳跃与自适应平衡控制研究[博士学位论文], 浙江大学, 中国, 2018Mao Yi-Chao. Hopping and Adaptive Balance Control of Bionic Legged Robot Based on Series Elastic Actuators [Ph. D. dissertation], Zhejiang University, China, 2018 [95] 李占卫, 李治军. 磁流变阻尼器动力学模型的研究现状. 机械制造与自动化, 2012, 41(1): 142−145 doi: 10.3969/j.issn.1671-5276.2012.01.048Li Zhan-Wei, Li Zhi-Jun. Status of researching on dynamical models of MR damper. Machine Building & Automation, 2012, 41(1): 142−145 doi: 10.3969/j.issn.1671-5276.2012.01.048 [96] Parietti F, Baud-Bovy G, Gatti E, Riener R, Guzzella L, Vallery H. Series viscoelastic actuators can match human force perception. IEEE/ASME Transactions on Mechatronics, 2011, 16(5): 853−860 doi: 10.1109/TMECH.2011.2162076 [97] Karlsson F, Persson A. Modelling Non-linear Dynamics of Rubber Bushings-Parameter Identification and Validation [Master dissertation], Lund University, Sweden, 2003 [98] Abe K, Suga T, Fujimoto Y. Control of a biped robot driven by elastomer-based series elastic actuator. In: Proceedings of the 12th IEEE International Workshop on Advanced Motion Control. Sarajevo, Bosnia-Herzegovina: IEEE, 2012. 1−6 [99] Kikuchi M, Aiken I D. An analytical hysteresis model for elastomeric seismic isolation bearings. Earthquake Engineering and Structural Dynamics, 1997, 26(2): 215−231 doi: 10.1002/(SICI)1096-9845(199702)26:2<215::AID-EQE640>3.0.CO;2-9 [100] Choi W, Won J, Lee J, Park J. Low stiffness design and hysteresis compensation torque control of SEA for active exercise rehabilitation robots. Autonomous Robots, 2017, 41: 1221−1242 doi: 10.1007/s10514-016-9591-z [101] Schepelmann A, Geberth K A, Geyer H. Compact nonlinear springs with user defined torque-deflection profiles for series elastic actuators. In: Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA). Hong Kong, China: IEEE, 2014. 3411−3416 [102] Paskarbeit J, Annunziata S, Basa D, Schneider A. A self-contained, elastic joint drive for robotics applications based on a sensorized elastomer coupling-design and identification. Sensors and Actuators A: Physical, 2013, 199: 56−66 doi: 10.1016/j.sna.2013.04.028 [103] Laffranchi M, Chen L S, Kashiri N, Lee J, Tsagarakis N G, Caldwell D G. Development and control of a series elastic actuator equipped with a semi active friction damper for human friendly robots. Robotics and Autonomous Systems, 2014, 62(12): 1827−1836 doi: 10.1016/j.robot.2014.06.007 [104] Park Y, Oh S, Zoe H. Dynamic analysis of reaction force sensing series elastic actuator as unlumped two mass system. In: Proceedings of the 42nd Annual Conference of the IEEE Industrial Electronics Society. Florence, Italy: IEEE, 2016. 5784−5789 [105] Kong K, Bae J, Tomizuka M. A compact rotary series elastic actuator for human assistive systems. IEEE/ASME Transactions on Mechatronics, 2012, 17(2): 288−297 doi: 10.1109/TMECH.2010.2100046 [106] Sariyildiz E, Yu H Y, Nozaki T, Murakami T. Robust force control of series elastic actuators using sliding mode control and disturbance observer. In: Proceedings of the 42nd Annual Conference of the IEEE Industrial Electronics Society. Florence, Italy: IEEE, 2016. 619−624 [107] Sariyildiz E, Chen G, Yu H Y. An acceleration-based robust motion controller design for a novel series elastic actuator. IEEE Transactions on Industrial Electronics, 2016, 63(3): 1900−1910 doi: 10.1109/TIE.2015.2512228 [108] Losey D P, Erwin A, McDonald C G, Sergi F, O' Malley M K. A time-domain approach to control of series elastic actuators: Adaptive torque and passivity-based impedance control. IEEE/ASME Transactions on Mechatronics, 2016, 21(4): 2085−2096 doi: 10.1109/TMECH.2016.2557727 [109] Negrello F, Garabini M, Catalano M G, Malzahn J, Caldwell D G, Bicchi A, et al. A modular compliant actuator for emerging high performance and fall-resilient humanoids. In: Proceedings of the IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids). Seoul, South Korea: IEEE, 2015. 414−420 [110] Bicchi A, Tonietti G. Fast and “soft-arm” tactics. IEEE Robotics & Automation Magazine, 2004, 11(2): 22−33 [111] Nieto E A B, Rezazadeh S, Gregg R D. Minimizing energy consumption and peak power of series elastic actuators: A convex optimization framework for elastic element design. IEEE/ASME Transactions on Mechatronics, 2019, 24(3): 1334−1345 doi: 10.1109/TMECH.2019.2906887 [112] Roozing W, Malzahn J, Kashiri N, Caldwell D G, Tsagarkis N G. On the stiffness selection for torque-controlled series-elastic actuators. IEEE Robotics and Automation Letters, 2017, 2(4): 2255−2262 doi: 10.1109/LRA.2017.2726141 期刊类型引用(3)
1. 罗华福，罗胜利，朱禾祥，疏小龙，汪晓铭，喻洪流. 智能助力型髋离断假肢设计与实验验证. 国际生物医学工程杂志. 2024(02): 108-114 . 
百度学术2. 蒋建东，李明贵，乔欣. 渐开线齿面串联弹性器件传动机理研究. 农业机械学报. 2023(01): 425-431+458 . 百度学术3. 曹学鹏，鲁航，朱文锋，钱亚伟，田富元. 变负载下机器人液压串联弹性执行器动态位置控制方法. 西安交通大学学报. 2022(04): 91-100 . 百度学术其他类型引用(21)
-
下载:
计量
- 文章访问数: 3178
- HTML全文浏览量: 1746
- PDF下载量: 713
- 被引次数: 24
