外文翻译---估计地形力和刚性轮式车辆参数

外文翻译---估计地形力和刚性轮式车辆参数 设计巴巴工作室www.88doc88.com 外文资料译文: 估计地形力和刚性轮式车辆参数 劳拉伊雷，电机及电子学 工程 路基工程安全技术交底工程项目施工成本控制工程量增项单年度零星工程技术标正投影法基本原理 师联合会，会员 摘要:抽象本文提供了一种估算方法阻力，推力，扭矩和电阻对每个车轮一个刚轮式车辆在车辆地形界面生成的，从这些力和力矩，一种方法，估计地形参数的思路。地形力的估算，这地形模型是一个独立的，可以推断的能力，加速，攀登，或拖负载的基本地形独立属性。当一个地形模型可用，参数这种模式，如土壤的凝聚力，摩擦角，最大正常压力，应力分布参数，决心从估计汽车越野势力利用多模型估计方法，提供相关的参数接受流动度量。该方法需要一个 标准 excel标准偏差excel标准偏差函数exl标准差函数国标检验抽样标准表免费下载红头文件格式标准下载 的本体传感器套房加速度，速率陀螺仪，车轮速度，力矩电机，和地面速度。下沉传感器不需要。仿真结果三个跨越地形展示了该方法的有效性一系列的土壤内聚力文献报到。 指数计算，移动机器人动力学，地形因素。 一(引言 移动在越野地形机器人自主取决于该机器人能够实时评估其流动性或近实时的时间。该机器人的信封运作，最高实现转化的速度，加速度，机动性在给定的地形取决于机器人的多体动力学并与地形的相互作用，从中外部力和力矩的机器人产生。多体动态是，在一般情况下，合理众所周知的。与此相反，车辆地面相互作用力的知识一般取决于对地形模型的有效性和大批这是难以衡量或利用地形参数推断实时算法。本文着重从本体上的地形特征传感器。直接测量地形的力量并时刻需要昂贵的扭矩传感器集成除了在每个车轮的速度，地面轮速度，加速度，速度传感器和表征车辆议案。有关地形耐半经验模型另外需要下沉下沉将传感器估计抵抗力量。 在本文中，我们引入估算阻力amethod，推力，扭矩和阻力，由于地形的每个车轮根据有关温和假设四轮驱动机器人机器人动力学与正常和剪应力分布沿轮地形接触。键的功能方法是汽车越野动力，力矩估计，与单轮没有经过本体传感器如实施了著名的汽车，地形模型，为植根于驱动和牵引轮半经验模型贝克理论[1] - [3]。车辆越野力和力矩与滑移行为是有价值的和自己的推断机器人的设计巴巴工作室www.88doc88.com 能力，加速，爬升，或拖负载。当地形模型可用，估计能力和力矩用于与该模型沿地形参数估计，从这些，车轮下的应力分布，可估计。在贝克刚性轮模型包括8个参数并涉及与正常压力和半经验关系下沉，和莫尔-库仑准则，涉及剪应力和正常的压力。 报告 软件系统测试报告下载sgs报告如何下载关于路面塌陷情况报告535n,sgs报告怎么下载竣工报告下载 不等土壤地形数据cohesiveless坚定粘土砂表明，一些参数几个数量相差ofmagnitude [2]，并没有明显之间的关系两个重要物理参数土凝聚力和摩擦角内的Mohr - Coulomb方程。在发展一个地形参数估计方法在本文中，我们考虑的唯一性问题，即之间是否有足够独特的一组映射地形参数和净部队和生成的时刻车辆。我们证明了这种映射弱的独特性，并在此基础上，我们提出了一个多模型估计(MME的)扎根在贝叶斯统计方法来估计地形参数。该方法利用地形参数设置等如[2]表列的地形，形成假说。地形假说，反过来，用在了部队的地形建模作为一个车轮打滑，这是从本体估计函数传感器。贝叶斯规则，然后用递归确定最可能的假设(地形参数设置)从在所有的假设或假说最佳组合代表地形。 地形特征有关的工作包括[4] - [7]。亚涅马等。[4]目前在线参数估计方法确定土壤凝聚力和内摩擦角对于一个刚轮行星探测车。这种方法使用一种简化和正常的剪切应力分布模型构成leastsquares估计他们的投入是正常的负载，车轮扭矩，下沉，车轮转速，线速度和轮。该简化模型假设沿对称应力分布轮地形接触补丁，这是一个假设，即取决于地形特性和车轮打滑。消除在这纸下沉传感器简化了仪器所需的地形参数估计。Hutangkabodee等人。[5]用牛顿迭代法，以确定内部摩擦角，剪切变形模量，并集中为轮式车辆穿越压力下沉系数未知的地形。Hutangkabodee等。[5]采取的办法类似[4]，但使用脱机识别和承担的平均值土壤凝聚力。奥赫达等。[6]地址地形通行表征与电机电流率的方法，转(偏航率)，从而评价指标实验一个小，差分指导商业机器人。奥赫达等人。[6]显示电流与电机之间的关系rateof -反过来，关于各种地形不同，例如，砾石，砂，土，和草。数据引起了执行控制，准稳态把演习。奥赫达等。[6]也发展的神经地形分类方法采用速率陀螺仪，加速度计，马达电流和电压。奥赫达等。[7]利用半经验汉尼贝克理论来检测和纠正车轮打滑时里程计。此方法假定土壤性质已知。对流动特性测量的传统方式使用牵引力直接测量拖着负荷，从东海岸滚动式电阻或在被测试为给拖走[8]，因此，不用于实时适合估计流动性。 参考文献[4] - [7]目前直接识别常见的主题执政地形参数指定的半经验地形模型和利用的净效果在观察地形，确定议案地形特征。在实践中，直接识别执设计巴巴工作室www.88doc88.com 政地形参数一个半经验模型，该模型假设良好的知识结构。贝克的理论，被广泛接受和验证在稳定，重型车辆纵向运动，打破了在高速瞬态和横向运动，它的适用性轻量(分- 500磅)的车辆已经不如从前knownthan重型车辆。瞬态模型，结合行为，土应变率的依赖，和其他特性可能在高速运动引起的尚未得到充分发展;因此，提取不承担车辆地形部队一车地形模型既具有应用自主地形地形模型力的估算和发展动力操纵。 在[4]提出的方法动机 - [8]，并提交从这里干需要预测车辆的流动性。该北大西洋公约组织(北约)参考移动模型(NRMM)[9]提供了一个全面的计算机模型来模拟和预测地面车辆的流动性关于和越野。利用地形特点，车辆动力学和表面状况，该模型可预测牵引力和阻力的流动与支路地图功能最高车速超过地形区域的生成。传统方法测量土壤性质的NRMM说明[10]，这些包括手工测量土壤强度使用圆锥贯入，板块下沉和剪切试验使用bevameter和剪切环，并使用车轮的仪器直接测量轮胎地形的力量。地形的能力来推断驾驶性能，同时对地形将有助于在实时移动地图的统计预测，不采用这些手段。审查对变形地形地势力的估算来自[11]提出了一种方法来提取压实电阻，总推力，以及相关的几何参数中心车辆沿地形联络这些部队的行动补丁。第三节考虑地形条件下的参数可估计从给定的一个刚性轮这些力量半经验车从地形模型[3]，并提出了MME的做法。第四节地形模拟结果参数估计的3种地形。 二(地形力的估算 A .净牵引力矩与阻力估计 汽车越野部队估计使用的是扩展卡尔曼- Bucy滤波(EKBF)以下的程序提出[11]。估计部队包括对每个车轮扭矩电阻，每方牵引力，和每轴的侧向力。该程 序这里是检讨的一个简单的情况下进行车辆纵向运动，从而忽略了横向地区吸收援引这是因为简化地形模型不变形充分解决纵向和招标相结合侧向力。一种四轮驱动，差异的刚体动力学指导机器人在体内固定坐标参照这里 设计巴巴工作室www.88doc88.com x = [vx r ωf l ωf r ωrl ωrr ]是系统状态，这是由纵向速度，偏航率和4车轮速度，Fxf l, Fxf r, Fxrl ,和Fxrr 是纵向网轮胎部队(减去总牵引阻力)在每个车轮，和和约电阻扭矩由于每个车轮Trf l, Trf r, Trrl Trrr 轮地形相互作用的旋转轴。图。 1(1)定义的身体固定坐标轴和净纵向力的方向;的Z轴是出页面。请注意，对于纵向模式，偏航率是零，因此，(2)提供了一个静态平衡方程。阿恢复时刻关于Z -轴通过每个车轮，例如，刚度为基础由于调整了车辆越野反应，是由一个模拟单总量在恢复momentMresr(2)withMres> 0。bwω(?)车轮阻尼条件，例如，由于机械阻尼在动力传动系统。Tf l, Tf r, , 和Trr 应用于变速箱损失的审议后车轮的扭矩，m是质量机器人，tw的Trl 是赛道的宽度，RW是车轮半径，是伊茨该机器人的惯性偏航时刻，信息作战是轮目前关于它的旋转轴转动惯量。体重和信息战可来自启动和与汽车滑行实验车轮离开地面。方程(1) - (6)，连同网对每个车轮的牵引力和阻力扭矩从地形模型，形成一个完全指定的系统或“真理模式”用于模拟在第四节的表现。虽然只有纵向运动是认为，(2)规定，每方限制纵向力量，因此，这个等式是保留。 设计巴巴工作室www.88doc88.com 射线:估计地形部队以及刚性四轮汽车参数 图一, a (与身体协调和固定力的方向)四个轮子的机器人定义。 b 作用力和力矩以及由此带动上强调，刚性车轮在变形的地形。 对未知的力和力矩设定在(1) - (6)包括Trf l, Trf r, Trrl ,和Trrr ,还有Fxf l, Fxf r, Fxrl ,以及Fxrr 构建一个EKBF是增强了车辆动力学与二阶每四个电阻扭矩为每方净牵引形式随机行走模型力量。假设每个车轮牵引力成正比网正常负荷在每个车轮，Fxf l, Fxf r, Fxrl ,和Fxrr是估计从每方净牵引和正常负荷。正常负荷预计从静态重量转移和测量加速度为给定的[12]和[13]。 设计巴巴工作室www.88doc88.com 图 2。车辆地面相互作用的结果表示为净(1)净牵引力和力矩Fx的电阻和Tr和(b)径向和切向力和神父署理θf英尺。 测量向量zm = [ax, ωf l, ωf r, ωrl, ωrr, vx, r],这是由纵向加速度在centerof -质量，车轮角速度，地面的速度为中心大规模，和偏航率，呈现增广状态，这是由X，4个电阻扭矩，和每边网牵引是观察。电机电流的测量提供了一个应用车轮扭矩。实施细则的EKBF有报道[11]对变形的地形和[12]和[13]阿克曼在刚性为指导地形车，因此，这些细节省略。 Ray和布兰德[11]目前的实验还结果估算纵向和横向力关于僵硬，变形为一个轻量级的地形机器人(13公斤)与非刚性车轮。 B .阻力和牵引总值估计从净牵引 图 1(b)显示了应用扭矩T和W对正常负荷一个驱动，刚性车轮产生正常的应力分布σ(θ)和剪应力分布τ(θ)和下沉z的一个轮子与纵向速度V。从这些应力分布，净在汽车开发力量，地形接口，通常被称为牵引力Fx的(推力减阻力)署理车轮悬架和电阻扭矩章。有效力量，在图所示。 2(1)，转换为一个沿接触点修补程序，如图所示。 2(b)项。这些部队是由此产生的积分σ(θ)和τ(θ)对接触面由θ1和界定θ2，以及相关的预计净扭矩牵引和电阻从EKBF由一个不知名的角度θf。有效径向从正常和剪应力产生的切向力分布假设在一个共同的行为角度θf。理由这一假设是从莫尔-库仑准则，在有关材料的最大剪应力正常的压力[2] 最大剪应力与正应力是通过两个材料常数，土壤凝聚力c和内部剪切角电阻或内部摩擦角υ[1]，因此，如果它的影响被忽视的剪切位移，最大剪和正常压设计巴巴工作室www.88doc88.com 力应该是大约一致。 由于正常的组成部分Fr和 Ft的图,2(b)必须平衡W,θf从EKBF所得估计电阻θf从EKBF所得估计电阻扭矩Tr和净牵引Fx的。切向力由与正常荷载和净力都是有关在何处Fr，Ft ,，θf是通过向Fr和解决 方案 气瓶 现场处置方案 .pdf气瓶 现场处置方案 .doc见习基地管理方案.doc关于群访事件的化解方案建筑工地扬尘治理专项方案下载 给Ft ,(9) - (11)。 ，,，θf为从其中，地形阻力Rc的，哪些行为对面的速度向量，给出了FrFt 代表的净效应标量变量σ(θ)和τ(θ)的车辆，但是，由于压力发行可不对称，这不是一般不错，θf=θm的角度来看，最大剪切和正常应力发生[见图 1(b)]。 C. .地形地形模型的参数估计 估计净部队和讨论的时刻第二节- B的假设没有先验知识的汽车越野模型和正常的剪切应力分布。然而，给予模型，力和力矩的估计可以用来确定该模型参数，假设一个足够独特的地形参数之间的映射和由此产生力和力矩存在。这种映射的唯一性问题参数之间的地形和净势力讨论第三节通知对地形参数选择方法估计。 最广为接受的地形模型是基于汉尼贝克理论，是从 总结 初级经济法重点总结下载党员个人总结TXt高中句型全总结.doc高中句型全总结.doc理论力学知识点总结pdf [1]在这里 - [3]硬性wheelmoving纵向以固定的速度水平，变形地形，如图所示。 1。该剪应力，剪切位移剪应力与执政的关系，给出了[2] 其中j是剪切位移，K是剪切变形弹性模量，而我是车轮打滑，其余几何参数定义图。 1。贝克[1]涉及正常强调要下沉通过经验确定压力下沉参数Kc，kυ和N 设计巴巴工作室www.88doc88.com 其中b是车轮宽度和z是下沉。最高正应力沿接触补丁的成就，给予了凭经 验确定的关系[3] 在C1和C2两个额外的经验引入地形参数。正常的压力转化为一个运作的θ在整合接触补丁。这在前面和后面θm地区，分别是正常的应力分布[3] 在恒定速度，下面的静力平衡条件按住[3]: 其中W是垂直力量平衡的正常负荷车轮，FX是牵引力或净力(毛额减去牵引电阻)，供车辆牵引负载，加速，或爬上山，TR是电阻扭矩。方程(13) - (20)提供地形投入变形地形模拟在第四节提出的结果。请注意在积分(18) - (20)没有封闭形式的解决方案。 地形参考估计 三( A. 地形模型行为 为了与地形强迫地形参数估计，我们首先考察了地形模型的行为一地形属性和功能的正常负荷。为此，我们选择3地形类型的约束范围的土壤内聚力[2]从一报道，以70千帕。表一报告这三个土壤和地形参数这些参数来源。我们评估设计巴巴工作室www.88doc88.com 的剪应力分布和正常而从大众100辆，而这些地形模型1000公斤。群众被假定为均匀分布在四车轮与刚性车轮(宽= 245和2450N，分别)直径0.508米(20)和宽0.15米(6)。通过举行轮子的大小不变，我们探讨了几何作为一个正常的压力作用应力分布。 图 3显示了剪应力场分布的正常轮滑比i = 0.15在表中的三个地形我和每个每个正常的压力。牵引力作为一个滑移率函数还为每地形和正常负荷。图。 3显示，在目前的0.15滑移率，我们有以下几点:1)最大剪应力发生在正常大致相同角θm，和2)正常和剪切应力分布有大约线性增加，并超过两个区域θ降低低凝聚力土压力。而最高剪应力和正常大约为精益同步黏土低滑比例，分布不是线性的。由于滑移的增加，最大剪应力和正常位置仍然大约一致，并采取正确的措施为低凝聚力的土壤。对于高凝聚力的土壤，最高剪应力不正常保持一致，但是，对于高凝聚力的土壤，下沉仍然很低，和应力分布几乎是在接触补丁持平。这些意见支持假设该行动的净角正常和切向部队从这些所产生的应力分布大约一致。 尽管这些意见似乎无动于衷正常负载时，净力或牵引力在很大程度上取决于正常负载，如预期。在图。 3，重型车辆不能培养正面的牵引力在干沙，因此，将在这个土壤不动，就证明了这牵引力与滑移曲线。它的流动性将是对沙地边缘壤土。轻型车辆上的一切积极的经验牵引网在足够高的3个滑带土。该牵引力与滑移特性而定，由与滑移的线性关系到饱和的关系。 这些意见建议力防滑反应映射可以从地形特征估计是有用的基本地形参数。 射线:估计地形部队以及刚性四轮汽车参数 图 3。剪应力分布和正常两三个正常荷载和滑移率0.15与挂钩，地形类型设计巴巴工作室www.88doc88.com 与滑拉每个地形和正常负荷。 (1)米= 100公斤。 (二)米= 1000公斤。 我地形参数表 B. 从地形参数的唯一性的映射汽车越野队 非独从地形参数映射到汽车越野部队直接体现在曲线相交为牵引力与滑图的比例 3，即对某些支路比，牵引力来自不同地形产生的是相同的。干砂，粘土精益目前类似牵引力在40,?轻滑的车辆，和沙质壤土和粘土目前瘦类似的?30,的重型车辆防滑牵引力。因此，围绕这些工作提出了滑移率估算挑战地形参数。 我们还观察的独特性问题直接从土壤力学模型(8)。无论是切线的摩擦角和土壤结构有直接关系的最大剪应力，从而净牵引力。一个具有给定的最大土壤正常的压力可以达到一定的最大剪应力或通过大摩擦角和凝聚力低或通过大的凝聚力和低摩擦角。因此，很难估计同时凝聚力和摩擦角。这个问题解决假设为一，土壤凝聚力平均值] [5估计摩擦角。在这里，我们的目标是估计的凝聚力与所有其他地形参数。 非独从地形参数映射到汽车越野部队还派生，部分来自非对称剪应力分布和正常的，当应力分布是不对称的，相同的力量可以从不同的压力通过发行积分方设计巴巴工作室www.88doc88.com 程(18) - (20)，即使最大的压力是不同的分布相同。亚涅马等人。 [4]提出了一种剪切和线性逼近正常，以近似封闭形式解应力分布(18) - (20)。这些解决方案是用来确定土壤凝聚力和摩擦角使用最小二乘方法。为方便封闭形式解，亚涅马等人。 [4]近似为对称的，即应力分布，θm发生在θ2之间的中点 3和[3]表明，应力分布不一定是线性的或对称的。黄和雷切[3]的然而，图。 实验表明从这些数据之间的线性关系和θmθ1，这是由于在(16)，推导。实验数据表明，在θm和θ1比率介于0.20.7作为一种低凝聚力土滑移功能，因此，应力分布是对称只在一个特定的值滑。实证地形参数c1and c2的捕捉不对称在应力分布，但很少在报文学。 对称性之间提供了一个σ(θ)和独特的关系τ(θ)，以及由此产生的力和力矩的(18) - (20)，即对称应力分布结果和Fx独特章图。 2(1)一c和υ给定值。当应力分布不对称，只有轻微的独特性，是保留，特别是对低凝聚力的土壤，即有可能存在一个以上的剪切和正常的应力分布，为相同的净牵引力地图，证明了图。 3。对于一个完全无粘性土，抗剪不对称应力可导致章相同的值(20)，即剪应力积分不会有所不同，如果最高重点是左或右之间θ1和中点θ2。虽然不是完全无粘性土的利益，移动机器人做运作非常低凝聚力的土壤，类似作者:章价值观会产生不同的应力分布在这样的土壤;测量传感器的不确定性土壤参数估计，因此，目前的一个挑战低凝聚力土壤地形参数估计。 C. 贝叶斯估计的地形参数 在选择一个贝叶斯参数的估计方法，我们寻求一种方法，是不敏感的独特性问题如前所述，不需要压力逼近分布是线性的，并且不需要应力分布是对称的。此外，作为未来实验的实施这种方法将使用不完善的，嘈杂的传感器和地形是不完全同质，地形参数估计方法必须拒绝测量噪声和过程噪音。最后，贝氏的方法可以利用现有的地面力学性能的地形数据库，限制参数空间很大. 克拉默和Sorenson [14]描述了贝叶斯参数估计的方法，其中包括未知参数作为系统状态的一部分，预计随着国家。贝叶斯估计是用来计算关节后密度的国家增广系统。由于有8个参数在这里，这个方法会增加大小由至少八个国家，需要的可观测性国家增强系统，这是由于独特性问题问题。此外，由于地形参数似乎隐含通过在国家，增广系统积分方程没有明确，一会要承担为国家，增广系统地形模型，并由此产生的非线性估计问题将难以计算的实质时间。 设计巴巴工作室www.88doc88.com MME的方法是一种描述[15]，其中一银行ñ卡尔曼滤波器是制定对N -假设参数集。每个过滤器是传播及时向方提供国家预算，以及贝氏规则是用来确定有条件概率的假设，由于估计状态每个过滤器。用最小的剩余过滤器应对应的最可能的假设。这种方法将更强大的独特性问题，因为竞争的假说可以采取的概率，但它是计算昂贵的，因为在相当长的卡尔曼滤波需要为每个假说。我们使用一个备用的贝叶斯方法，并在不增加国家的向量大小[14]和不没有规定的N繁殖扩展卡尔曼滤波在[15]。 皇[2]报道套汉尼贝克地形参数(c，υ值，九龙城区，kυ，n的地形)21。使用贝克刚性轮地形模型作为推进这些地形模型和参数作为假设集，我们提出以下建议MME的贝叶斯地形识别方法作为计算效率稳健的做法，应该到弱的独特性描述较早。对于每一个假设，参数显示的地形测绘设置为地形部队有决心作为一个先验的滑移功能和正常负荷。对地形参数设置最有可能从其中由递归执行决定的假说贝叶斯规则如下。让度Pj，对于j = 1到N，包括的N -虚拟地形参数vectorswith概率质量函数初始化为镨列(PJ)0 = 1 /注有条件的概率大规模的参数功能设置辟捷受一个向量从EKBF地形力量在时间K表估计FK型演变根据贝叶斯规则[16]在何处其中S是协方差的剩余财源(PJ)型矩阵=FK型(PJ)型- FK型，和FK列(PJ)是矢量地形部队映射参数设置为PJ和估计slipi。最可能的参数集赋予一个概率加权总和的假设方程(21) - (23)是递归执行，即每时间步钾，新势力和车轮估计单是用来更新每个假设的概率质量函数。注意，这个方法是成功在很大程度上取决于质量的假设，因此，该方法应被评估为凡是不正是代表着潜在的假设参数，但假设约束的基本参数设置。 设计巴巴工作室www.88doc88.com 射线:估计地形部队以及刚性四轮汽车参数 图 4。实际和估计(上)与滑移和净牵引(下)电阻扭矩与扭矩降低为一个应用随着时间的推移呈线性输入滑。 之间的EKBF和贝叶斯MME的分工一直在EKBF的增广状态大小最高12比原来的状态向量，即一个关于二阶多估计为6个部队每个随机漫步模型。这种方法还允许EKBF不执行假设地面力学模型，而该方法在[14]由于需要一个地面力学模型与地形地形参数未知的力量。因此，复杂性雅可比矩阵的计算，为国家扩展的系统并不比原来的状态，更多的增强线性输入国。计算所需的传统MME的[15]的方法是N倍，单一EKBF并会望而却步任何昂贵的合理数量假说。由于我们没有落实为每个假说EKBF，我们的计算成本适度规模与数量的假设，因为只有(21)- (23)取决于数量假说。 四 .评价地形参数估计 为了评估地形参数估计，我们模拟纵向加速车辆的massm轮= 100公斤直径0.508米(20)和宽0.15米(6)对均匀，变形的地形。外加扭矩在每个车轮吨= 0大，转矩随时间呈线性下降。这投入生产的100,，单轮在t = 0，它减少作为扭矩减小，征求净力与特色扭矩与滑和电阻在每个车轮滑移。 Zeromean，高斯过程噪声被注入到每个动态方程(1) - (6)。零均值，高斯噪声测量注射设计巴巴工作室www.88doc88.com 模拟测量加速度，车轮速度，偏航率，和地面的速度。选择测量噪声方差是根据二手传感器实验室测量值在实验中验证地形力估计在[11]给出。过程噪声和测量噪声的协方差用于据报道模拟表二。测量和处理噪声协方差被假定为已知。模拟的EKBF在采样频率为100赫兹。 该EKBF提供了国家和部队估计时间历程，历史和时间在每个车轮的滑移从派生车轮转动的速度和地面的速度。从这些时间的历史，地形部队和电阻转矩与滑推导。代表力与滑移的估计结果为每个地形表一载列于图。 4，真实，估计与滑移和真实，并预计在电阻转矩与滑左前方车轮。请注意，图中每个点。 4代表一个样本在时间的历史。图。 4表明，EKBF能够跟踪外力和力矩由于地形在由过程噪声，测量噪声施加限制，过滤瞬变。为此准稳态演习，过滤瞬变之间产生的实际和预计净牵引和错误阻转矩与滑;作为瞬态衰减，平均价值估计力和扭矩接近真实稳态工作点，而过程和测量噪声使出现“集群点”周围的准稳态运行条件。对于在模拟图。 4，线性下降扭矩适用于超过2沙和沙质壤土和4秒S对精益粘土;估计错误由于瞬变跌幅为投入是多方面的更慢。评价的地形力估计，包括身体测试，给出了[11]，因此，这里省略。 用估计牵引力，推力，电阻和电阻与滑力矩，地形参数的贝叶斯估计评价一个质量为m = 100公斤和3地形车表一，参数设置报告包括所有8地形模型参数p= [çυ架KCkυñ C1的C2的k]的。该首次报道了5个参数给出了21地形[2]，与这21个地形参数集的形式确定了假说贝叶斯MME的办法。参数的C1，C2，和K不报道[2这些地形]。K是报告[2]作为变从1厘米(公司沙地)至2.5厘米(沙地)，0.6厘米为在最大压实粘土，并为新鲜的雪2.5-5厘米，没有额外归属于地形参数的21套1。因此，如果没有数据存在，我们takeK = 0.025mfor沙地土壤，钾= 0.01米的沙地沃土，钾= 0.006米划为粘性土土壤及K = 0.05米的雪在21地形参数假说。C1和C2的变量在[3]总结了沙质地形，从个来源和范围从C1 = 0.43c2的紧凑型砂= 0.32 = 0.38至C1和C2 = 0.41干沙子。在给定的地形数据的情况下21 [2]，我们采取桑迪c1的沃土和粘质土和c2 = 0.43 = 0.32和C1和C2 = = 0.38为沙质土壤和雪0.41。的21假设，两个描述[2“沙]，”八描述作为“沙壤土，”三被称为“雪”六个形容为粘质土，各有1人被称为“粘土壤“和”沃土“，因此，假设代表了一个连续一个大范围的土壤类型。 鉴于21虚拟地形参数向量，其中如表一，贝叶斯multiplemodel三个地形估计是评估，这三个地形。 EKBF从单轮地形中使用武力的估计(21) - (23)，设计巴巴工作室www.88doc88.com 但这一方法不仅限于使用部队从singlewheel.Two候选人地形力向量进行了评价。候选1 FK型= [用FT升(十一)成绩单升(十一)]，即EKBF估计净牵引或牵引力(推力减阻力)与电阻扭矩在左前方的轮子，和候选2是FK型= [FTF的升(十一)余弦(θff升(k)段)区域合作框架升(K)的成绩单升(金)]， ) - (12)即总牵引，电阻，并在离开前的电阻扭矩车轮。候选2地形力向量员工(9分区估计牵引力和扭矩为总电阻牵引力和阻力。 我们还考虑的情况下地形的3套表参数我代表真正的基本地形特点是在假设没有确定。在确切的地点假设，这三个地形地形参数被修改，与随机与10,的标准偏差白噪声地形的名义值添加到每个参数。因此，在这种情况下，假设没有精确匹配的真正基础地形，但至少有一个近似的基本假设地形。在地形力估计的执行情况，我们拖延MME的20个样本(0.2秒)，以便初步EKBF瞬变腐烂后，我们从一开始就收敛措施的MME的。 仿真评估结果摘要如下,当表的准确地形参数集我都包含在21假设，贝叶斯估计集收敛到假设在最正确的两次迭代，或0.02对于所有3地形和候选人都力向量。在条件的支路收敛性高，贝叶斯估计仍然融合这一条正确的假设在整个模拟。当不精确表地形参数集我是代表假设中加入高斯白噪声等没有真正的假说相匹配的基本地形参数载体，地形参数估计收敛到一个单一的假说制作干沙和沙质壤土，即假设通过修改底层真实与高斯地形参数白噪声。收敛发生(0.02秒之内两次迭代)地形为候选人，并继续聚合力矢量在整个模拟。精益黏土，地形参数收敛一个在不到5反复单一的假说候选人一地形力向量。不过，这种假设是不通过注入高斯噪声产生的1。为地形假设thatMMEconverges的是描述[2]作为粘质土壤参数n = 0.13，架KC = 12.7(kPa/mn-1)，kυ=1556(千帕/分钟)和c = 68.95 kPa时，和υ= 34?，即一土参数相似，精益粘土参数报告表一. 为高黏性土结果归因于弱独特性净部队和力矩所产生的电阻粘质土壤不同的假说代表集。精益黏土和候选人2地形力向量，其中纳入而同时，净推力和纵向电阻元件力，融合参数是由一个线性两个或三个假设组合根据(23)。一个例如参数和概率密度函数的收敛精益粘土图给出了收敛。 5，这表明收敛假说在两年内两次迭代(0.02秒)。这两个假设是“重粘土”的[2]报道早些时候，从“贫粘土产生的假说”参数高斯噪声增加，因此，地形分类实现对所有三种类型的地形内两次迭代。可能性在t = 2.0，这些假设的群众职能是0.35和0.64，分别。 作为表现额外措施，图 6显示的例子正常和剪应力分布在0.15滑移率基本设计巴巴工作室www.88doc88.com 地形的实际，为参数的多模型估计的收敛，对候选人的基础2地形力向量，当假设集不包括实际的地形。结果图 6是代表的worstcase(干砂)和最佳情况(瘦土)应力估计分布的估计和实际参数，一滑比0.15。 图 7显示了估计和实际的牵引力和电阻扭矩与滑每个确定的地形特征。在 。该“真正的”牵引力和阻力转矩与滑代表这里，每一个点代表一滑价为0.05 在无噪声力与滑移曲线基本地形的实际参数。 “估计”曲线代表noisefree力与滑移曲线的地形参数值结果在t = 1秒的贝叶斯参数估计对照无花果。6图 7显示，即使底层应力分布不完全匹配的实际分布，净力与滑以及近似实际的力量与滑。精益粘土，小错误之间的真实和估计应力分布转化为细微的差别在真正的和估计的力量与滑移。为沙质壤土，干沙子，甚至在估计应力分布，较大的误差估计人口与滑移轨道内，测量和过程噪声设限的真正力量与滑注入模拟。 比较图。 7图 4，来自估计牵引力，阻力力矩和时间的历史轮单，显示，尽管不完善估计汽车越野力量的结果由扩展卡尔曼-布西滤波(由于测量噪声和过程所施加的限制噪音和过滤瞬变)，贝叶斯参数估计是能够选择一个参数向量的假设，表示基本地形与合理的准确性，因此，该方法提供了一些对噪声的鲁棒性。 射线:估计地形部队以及刚性四轮汽车参数 设计巴巴工作室www.88doc88.com 图 5。地形，以便就瘦粘土车辆模拟参数衔接候选人，2力向量在多模型估计使用。 (上)收敛历史的压力下沉参数和凝聚力。 (下)摩擦角的收敛性，氮的历史和条件概率，显示剩下的两个假设两个迭代后 图 6。剪应力分布比较正常，所描述的实际和估计的地形参数。 设计巴巴工作室www.88doc88.com 图 7。的牵引力和扭矩与滑电阻比较，所描述的实际和地形参数的估计。 五. 结论 本文开发了一个贝叶斯MME的识别方法从EKBF地形参数估计地形力量与滑移的特征。模拟评价方法显示1汉尼贝克刚性轮地形模型，它可以找出最佳的假说代表地形特征从具有良好的收敛性假设成立，它展示类似的竞争性假设插值性质。该方法不需要假设或近似基本剪应力分布正常。它构成一个低当计算负担从映射的假设地形地形特征参数集，以预先计算力量作为车轮打滑和正常负载的功能。计算适度规模与数量的假设，因此，额外的假设可从文献中无施加太大的额外计算。 在这个文件中，方法是显示一个基本地形模型驱动，刚性车轮，但是，方法不限制，使模型的结构构成每个假设必须一致，可以假设采取替代模型结构和参数的形式。此外，地形识别实例表明，毛牵引，阻力和阻力矩从单一车轮提供足够的信息识别地形，因此，地形参数可以独立地确定对每个车轮。最后，该方法也可以用来作为快速分类，因为它能够区分特征地形，如粘质土与沙质壤土与砂出色的收敛性。 设计巴巴工作室www.88doc88.com 参考 [1] M. G. Bekker, Theory of Land Locomotion. Ann Arbor, MI: Univ.Michigan Press, 1956. [2] J. Y. Wong, Theory of Ground Vehicles, 3rd ed. New York: Wiley– Interscience, 2001. [3] J. Y.Wong and A. R. Reece, “Prediction of rigid wheel performance based on the analysis of soil-wheel stresses part I. Performance of driven rigid wheels,” J.Terramech., vol. 4, no. 1, pp. 81–98, 1967. [4] K. Iagnemma, S. Kang, H. Shibly, and S. Dubowsky, “Online terrain parameter estimation for wheeled mobile robotswith application to planetary rovers,” IEEE Trans. Robot., vol. 20, no. 5, pp. 921–927, Oct. 2004. [5] S. Hutangkabodee, Y. H. Zweiri, L. D. Seneviratne, and K.Althoefer,“Soil parameter identification for wheel-terrain interaction dynamics and traversability prediction,” Int. J. Autom. Comput., vol. 3, pp. 244–251,2006. [6] L. Ojeda, J. Borenstein, G. Witus, and R. Karlsen, “Terrain characterization and classification with a mobile robot,” J. Field Robot., vol. 2, no. 2,pp. 103–122, 2006. [7] L. Ojeda, D. Cruz, G. Reina, and J. Borenstein, “Current-based Slippage detection and odometry correction for mobile robots and planetary rovers,”IEEE Trans. Robot., vol. 22, no. 2, pp. 366–377, Apr. 2006. [8] K. Wesson, M. Parker, B. Coutermarsh, S. Shoop, and J. Stanley, “Instrumenting an all-terrain vehicle for off-road mobility analysis,” ERDC/CRREL: TR-07-1, Jan. 2007. [9] R. B. Ahlvin and P. W. Haley, “NATO reference mobility model, Edition II, NRMMII user_s guide,” Geotech. Lab., USAEWES, Tech. Rep. GL- 92-19, 1992. [10] S. Shoop, “Terrain characterization for trafficability,” Cold 设计巴巴工作室www.88doc88.com Regions Res.Eng. Lab., Hanover, NH, CRREL Rep. 93-6, Jun. 1993. [11] L. R. Ray, D. Brande, and J. H. Lever, “Estimation of net traction for differential-steered wheeled robots,” J. Terramech., 2008. doi:10.1016/j.jterra.2008.03.003 [12] L. R. Ray, “Nonlinear state and Tire force estimation for advanced vehicle control,” IEEE Trans. Control Syst. Technol., vol. 3, no. 1, pp. 117–124,Mar. 1995. [13] L. R. Ray, “Nonlinear Tire force estimation and road friction identification:Simulation and experiments,” Automatica, vol. 33, no. 10, pp. 1819–1833,1997. [14] S. C. Kramer andH.W. Sorenson, “Bayesian parameter estimation,” IEEETrans. Autom. Control, vol. 33, no. 2, pp. 217–222, Feb. 1988. [15] P. S. Maybeck, Stochastic Modek, Estimation and Control, vol. 3. New York: Academic, 1982. [16] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering.Reading, MA: Addison-Wesley, 1994 劳拉伊雷(M'92)收到B.S.度(与最高荣誉)，机械和航空航天工程来自普 林斯顿大学，普林斯顿，新泽西州，在1984年，咪机械工程学士学位来自斯坦福 大学，斯坦福大学，加州，于1985年，在机械和航空航天工程博士学位来自普林 斯顿大学，于1991年。1996年，她加入了工程塞耶学院达特茅斯学院，汉诺威， 新罕布什尔州，在那里她一直是工程副教授自2002年以来的科学。她目前的研究 兴趣包括协同控制的移动机器人，机器人的流动性和车辆的地形互动，机器人技 术领域。 外文资料原文: 设计巴巴工作室www.88doc88.com Estimation of Terrain Forces and Parameters for Rigid-Wheeled Vehicles Laura E. Ray, Member, IEEE Abstract—This paper provides a methodology for the estimation of resistance, thrust, and resistive torques on each wheel of a rigid-wheeled vehicle generated at the vehicle–terrain interface,and from these forces and moments, a methodology to estimate terrain parameters is presented. Terrain force estimation, which is independent of a terrain model, can infer the ability to accelerate,climb, or tow a load independent of the underlying terrain properties. When a terrain model is available, parameters of that model, such as soil cohesion, friction angle, maximum normal stress, and stress distribution parameters, are determined from estimated vehicle–terrain forces using a multiple-model estimation approach, providing parameters that relate to accepted mobility metrics. The methodology requires a standard proprioceptive sensor suite—accelerometers, rate gyros,wheel speeds, motor torques, and ground speed. Sinkage sensors are not required. Simulation results demonstrate efficacy of the method on three terrains spanning a range of soil cohesions reported in the literature. Index Terms—Mobile robot dynamics, terrain factors. I. INTRODUCTION MOBILE robot autonomy in off-road terrain depends on the ability of the robot to assess its mobility in real time or near real time. The robot’s envelope of operation—maximum achievable translational velocities, accelerations, and maneuverability on a given terrain—depends on the robot’s multibody dynamics and its interaction with the terrain, from which external forces and moments on the robot are generated. The multibody dynamics are, in general, reasonably well known. In contrast, knowledge of vehicle–terrain interaction forces generally depends on the validity of a terrain model and a large number of terrain parameters that are difficult to measure or infer usingreal-time algorithms. This paper focuses on terrain characterization from proprioceptive 设计巴巴工作室www.88doc88.com sensors. Direct measurement of terrain forces and moments requires integrating expensive torque sensors on each wheel in addition to wheel speeds, ground speed, accelerometers, and rate sensors for characterizing vehicle motion. Semiempirical models relating terrain resistance to sinkage would additionally require sinkage sensors to estimate resistance forces. In this paper,we introduce amethod for estimating resistance,thrust, and resistive torque due to the terrain at each wheel of a four-wheel drive robot under mild assumptions regarding the dynamics of the robot and the normal and shear stress distribution along the wheel–terrain contact. The key feature of the methodology is the estimation of vehicle–terrain forces, moments, and wheel slips through proprioceptive sensors without imposing a vehicle–terrain model, such as the well-known semiempirical models for driven and towed wheels rooted in Bekker theory [1]–[3]. The vehicle–terrain forces and moments versus slip behavior are valuable in and of themselves to infer the robot’s capacity to accelerate, climb, or tow a load. When a terrain model is available, the estimated forces and moments can be used along with that model to estimate terrain parameters,and from these, the stress distributions under the wheels can be estimated. The Bekker rigid-wheel model includes eight parameters and involves semiempirical relations between normal stress and sinkage, and a Mohr–Coulomb criterion that relates shear stress and normal stress. Reported terrain data for soils ranging from cohesiveless sand to firm clay show that some parameters can vary by several orders ofmagnitude [2], and there is no apparent relationship between two important physical parameters—soil cohesion and friction angle—within the Mohr–Coulomb equation. In developing a methodology for terrain parameter estimation in this paper, we consider uniqueness issues, namely whether there is a sufficiently unique mapping between a set of terrain parameters and the net forces and moments generated on the vehicle. We demonstrate the weak uniqueness of this mapping, and based on this, we propose a multiple-model estimation (MME) method rooted in Bayesian statistics to estimate terrain parameters. This method uses sets of terrain parameters such as those tabulated in [2] to form terrain hypotheses. Terrain 设计巴巴工作室www.88doc88.com hypotheses, in turn, are used in forward modeling of terrain forces as a function of wheel slip, which is estimated from proprioceptive sensors. Bayes’ rule is then used recursively to identify the most likely hypothesis (set of terrain parameters) from among all hypotheses or the combination of hypotheses that best represents the terrain. Related work on terrain characterization includes [4]–[7]. Iagnemma et al. [4] present an online parameter estimation method to determine soil cohesion and internal friction angle for a rigid-wheel planetary rover. This method uses a simplified model of the shear and normal stress distribution to pose a leastsquares estimator whose inputs are normal load, wheel torque, sinkage, wheel rotational speed, and wheel linear speed. The simplified model assumes symmetric stress distributions along the wheel–terrain contact patch, which is an assumption that depends on the terrain properties and wheel slip. Elimination of the sinkage sensor in this paper simplifies instrumentation required for terrain parameter estimation. Hutangkabodee et al. [5] use a Newton–Raphson technique to identify internal friction angle, shear deformation modulus, and lumped pressure–sinkage coefficients for a wheeled vehicle traversing unknown terrain. Hutangkabodee et al. [5] take an approach similar to [4] but use offline identification and assume an average value for soil cohesion. Ojeda et al. [6] address terrain trafficability characterization by relating motor current to rate-of-turn (yaw rate), thus evaluating the metric experimentally with a small, differential-steered commercial robot. Ojeda et al. [6] show a relationship between motor current versus rateof- turn that differs on various terrain, e.g., gravel, sand, dirt, and grass. Data are elicited by performing a controlled, quasi-steady turning maneuver. Ojeda et al. [6] also develop a neural terrain classification approach using rate gyros, accelerometers, motor current, and voltage. Ojeda et al. [7] use semiempirical Bekker theory to detect and correct for wheel slip during odometry. This method assumes that the soil properties are already known. Traditional approaches to measurement of mobility characteristics use direct measurement of drawbar pull while towing a load, rolling resistance from coast-down tests or while being towed as given in [8] and, thus, are not suitable for real-time estimation of mobility. 设计巴巴工作室www.88doc88.com References [4]–[7] present common themes of direct identification of terrain parameters governing a specified semiempirical terrain model and of exploiting the net effect of the terrain on observed motion to determine terrain characteristics.In practice, direct identification of terrain parameters governing a semiempirical model assumes good knowledge of the model structure. Bekker theory, which is widely accepted and validated for heavy vehicles in steady, longitudinal motion, breaks down during high-speed transients and lateral motion, and its applicability to lightweight (sub-500 lb) vehicles is less well knownthan for heavy vehicles. Models that incorporate transient behavior,soil strain-rate dependency, and other characteristics that may be induced during high-speed motion have not yet been fully developed;hence, extracting vehicle–terrain forces without assuming a vehicle–terrain model has application both to autonomous terrain force estimation and terrain model development for dynamic maneuvering. Motivation for methods presented in [4]–[8] and that presented here stem from a need to predict vehicle mobility. The North Atlantic Treaty Organization (NATO) Reference Mobility Model (NRMM) [9] provides a comprehensive computer model used to simulate and predict the mobility of ground vehicles on- and off-road. Using terrain characteristics, vehicle dynamics, and surface conditions, the model predicts available traction and resistance versus slip from which mobility maps—maximum speed over a terrain region—are generated. Traditional approaches to measure soil properties for the NRMM are described in [10], and these include manual measurement of soil strength using a cone penetrometer, plate sinkage and shear testing using a bevameter and shear annulus, and use of instrumented wheels to directly measure tire–terrain forces. The ability to infer terrain properties while driving over terrain would aid in statistical prediction of mobility maps in real time,without employing these instruments. Section II reviews terrain force estimation on deformable terrain from [11] and presents a methodology to extract compaction resistance, gross thrust, and a geometric parameter related to center of action of these forces along the vehicle–terrain contact patch. Section III considers conditions under which terrain parameters can be estimated from these forces given a rigid-wheel semiempirical 设计巴巴工作室www.88doc88.com vehicle–terrain model from [3] and presents the MME approach. Section IV provides simulation results of terrain parameter estimation for three terrain types. II. TERRAIN FORCE ESTIMATION A. Estimation of Net Traction and Resistance Torques Vehicle–terrain forces are estimated using an extended Kalman–Bucy filter (EKBF) following the procedure presented in [11]. Estimated forces include resistive torques on each wheel,per-side drawbar pull, and per-axle lateral forces. The procedure is reviewed here for the simpler case of a vehicle undergoing longitudinal motion, and thus neglects lateral forces.We invoke this simplification because deformable terrain models do not adequately address solicitation of combined longitudinal and lateral forces. The rigid-body dynamics of a four-wheel drive, differentially steered robot are modeled in body-fixed coordinates as Here, x = [vx r ωf l ωf r ωrl ωrr ] is the system state,which is composed of longitudinal velocity, yaw rate, and four wheel velocities, Fxf l, Fxf r, Fxrl , and Fxrr are the net longitudinal tire forces (gross traction minus resistance) at each wheel,and Trf l, Trf r, Trrl , and Trrr are the resistive torques about the rotational axis of each wheel due to the wheel–terrain interaction. Fig. 1(a) defines the body-fixed coordinate axes and directions of net longitudinal forces; the z-axis is out of the page. Note that for the longitudinal model, yaw rate is zero, and thus, (2) provides a static equilibrium 设计巴巴工作室www.88doc88.com equation. A restoring moment about the z-axis through each wheel, e.g., stiffness-based realignment due to the vehicle–terrain response, is modeled by a single aggregate restoring momentMresr in (2) withMres > 0.bwω(?) are wheel damping terms, e.g., due to mechanical damping in the drivetrain. Tf l, Tf r, Trl , and Trr are applied wheel Fig. 1. (a) Four-wheeled robot with body-fixed coordinate and force direction definitions. (b) Applied forces and moment and resulting stresses on a driven,rigid wheel in deformable terrain. torques after consideration of gearbox losses, m is the mass of the robot, tw is the track width, Rw is the wheel radius, Izz is the yaw moment of inertia of the robot, and Iw is the wheel moment of inertia about its rotational axis. bw and Iw can be derived from spin-up and coast-down experiments with the vehicle wheels off the ground. Equations (1)–(6), together with net traction and resistive torque on each wheel from a 设计巴巴工作室www.88doc88.com terrain model, form a fully specified system or ―truth model‖ used to simulate performance in Section IV. While only longitudinal motion is considered, (2) imposes constraints on the per-side longitudinal forces, and thus, this equation is retained. The set of unknown forces and moments in (1)–(6) includes Trf l, Trf r, Trrl , and Trrr , and Fxf l, Fxf r, Fxrl , and Fxrr. An EKBF is constructed by augmenting the vehicle dynamics with second-order random walk models of the form for each of the four resistive torques and for the per-side net traction forces. Assuming that per-wheel net traction is proportional to normal loads at each wheel, Fxf l, Fxf r, Fxrl , and Fxrr are estimated from per-side net traction and normal load. Normal load is estimated from the static weight transfer and measured accelerations as given in [12] and [13]. forceFig. 2. Net result of vehicle–terrain interaction represented as (a) net traction and resistive torque Fx and Tr and (b) radial and tangential forces Fr and Ft acting at θf . A measurement vector zm = [ax, ωf l, ωf r, ωrl, ωrr, vx, r],which is composed of longitudinal acceleration at the centerofmass, wheel angular velocities, ground speed of the center of mass, and yaw rate, renders the augmented state, which is composed of x, four resistive torques, and per-side net traction is observable. Motor currents provide a measure of the applied wheel torques. Details regarding implementation of the EKBF are reported in [11] for deformable terrain and in [12] and [13] for Ackerman steered vehicles on rigid terrain, and thus, these details are omitted here. Ray and Brande [11] also present experimental results for estimating 设计巴巴工作室www.88doc88.com longitudinal and lateral forces on rigid and deformable terrain for a lightweight robot (13 kg) with nonrigid wheels. B. Estimation of Resistance and Gross Traction From Net Traction Fig. 1(b) shows the applied torque T and normal load W on a driven, rigid wheel that give rise to normal stress distribution σ(θ) and shear stress distribution τ (θ) and sinkage z for a wheel with longitudinal velocity V . From these stress distributions, net forces develop at the vehicle–terrain interface, which are commonly referred to as drawbar pull Fx (thrust minus resistance) acting at the wheel axle and a resistive torque Tr . The effective forces, shown in Fig. 2(a), are translated to a point along contact patch, as shown in Fig. 2(b). These forces are the resultant integrals of σ(θ) and τ (θ) over the contact patch defined by θ1 and θ2 , and are related to estimated net traction and resistive torque from the EKBF by an unknown angle θf . The effective radial and tangential forces resulting from the normal and shear stress distributions are assumed to act at a common angle θf . Justification of this assumption is taken from the Mohr–Coulomb criterion that relates the maximum shear stress in the material to the normal stress [2] Maximum shear stress is related to normal stress through two material constants, soil cohesion c and angle of internal shearing resistance or internal friction angle υ [1], and thus, if the effects of shear displacement are neglected, maximum shear and normal stress should be approximately coincident. Since the normal components of Fr and Ft in Fig. 2(b) must balance W, θf is derived from the EKBF estimated resistive torque Tr and net traction Fx . The tangential force is given by and the normal load and net force are related to Fr and Ft as 设计巴巴工作室www.88doc88.com where Fr, Ft , and θf are given by the solution to (9)–(11). From these, terrain resistance Rc , which acts opposite to the velocity vector, is given by Fr, Ft , and θf provide scalar variables representing the net effect of σ(θ) and τ (θ) on the vehicle; however, since the stress distributions can be asymmetric, it is not generally true that θf = θm, the angle at which the maximum shear and normal stress occurs [see Fig. 1(b)]. C. Terrain Model for Terrain Parameter Estimation Estimation of net forces and moments discussed in Section II-B assumes no a priori knowledge of a vehicle–terrain model of the shear and normal stress distributions. However,given a model, the force and moment estimates can be used to identify parameters of that model, assuming that a sufficiently unique mapping between the terrain parameters and resulting forces and moments exists. This issue of uniqueness of the mapping between terrain parameters and net forces is discussed in Section III to inform the approach selected for terrain parameter estimation. The most widely accepted terrain models are based on Bekker theory, which is summarized here from [1]–[3] for a rigid wheelmoving longitudinally at constant speed on horizontal, deformable terrain, as shown in Fig. 1. The shear stress–shear displacement relationship governs shear stress and is given by [2] where j is the shear displacement, K is the shear deformation modulus, and i is the wheel slip, and the remaining geometric parameters are defined in Fig. 1. Bekker [1] relates the normal stress to sinkage through empirically determined pressure–sinkage parameters kc, kυ , and n 设计巴巴工作室www.88doc88.com where b is the wheel width and z is the sinkage. The maximum normal stress achieved along the contact patch is given by the empirically determined relationship [3] where c1 and c2 introduce two additional empirical terrain parameters. The normal stress is transformed to a function of θ to integrate over the contact patch. For the region in front of and behind θm, respectively, the normal stress distribution is [3] At constant velocity, the following static equilibrium conditions hold [3] whereW is the vertical force that balances the normal load on the wheel, Fx is the drawbar pull or net force (gross traction minus resistance) available to the vehicle to tow a load, accelerate, or climb hills, and Tr is the resistive torque. Equations (13)–(20) provide the terrain inputs for deformable terrain simulation results presented in Section IV. Note that the integrals in (18)–(20) have no closed-form solution. III. TERRAIN PARAMETER ESTIMATION A. Terrain Model Behavior 设计巴巴工作室www.88doc88.com In order to relate terrain force estimates to terrain parameters,we first investigate the behavior of the terrain model as a function of terrain properties and normal load. To do so, we choose three terrain types that bound a range of soil cohesions reported in [2] from the order of 1 to 70 kPa. Table I reports terrain parameters for these three soils and sources for these parameters. We evaluate the shear and normal stress distributions arising from these terrain models for vehicles of mass 100 and 1000 kg. The mass is assumed to be distributed evenly over four wheels (W = 245 and 2450 N, respectively) with rigid wheels of diameter 0.508 m (20 in) and width 0.15 m (6 in). By holding the wheel size constant, we investigate the geometry of the stress distributions as a function of normal pressure. Fig. 3 shows shear and normal stress distributions for wheel slip ratio i = 0.15 for each of the three terrains in Table I and for each normal pressure. Drawbar pull as a function of slip ratio is also provided for each terrain and normal load. Fig. 3 shows that at a slip ratio of 0.15, we have the following: 1) The maximum shear and normal stress occur at approximately the same angle θm, and 2) normal and shear stress distributions are approximately linear with θ over the two regions of increasing and decreasing stress for low-cohesion soils. While the maximum shear and normal stress are approximately coincident for lean clay at low slip ratios, the distributions are not linear. As slip Fig. 3. Shear and normal stress distributions on three terrain types for two normal loads and slip ratio of 0.15, with drawbar pull versus slip for each terrain and normal load. (a) m = 100 kg. (b) m = 1000 kg. TABLE I 设计巴巴工作室www.88doc88.com TERRAIN PARAMETERS increases, the location of the maximum shear and normal stress remains approximately coincident and moves to the right for the low-cohesion soils. For high-cohesion soils, the maximum shear and normal stress do not remain coincident; however, for high-cohesion soils, sinkage remains low, and stress distributions are nearly flat over the contact patch. These observations support the assumption that the angle of action of the net normal and tangential forces arising from these stress distributions is approximately coincident. While these observations appear to be insensitive to normal load, the net force or drawbar pull depends significantly on normal load, as expected. In Fig. 3, the heavy vehicle cannot develop positive drawbar pull on the dry sand and, thus, would be immobile on this soil, as evidenced by the drawbar pull versus slip curve. Its mobility would be borderline on sandy loam. The light vehicle experiences positive net traction on all three soils at sufficiently high slip. The drawbar pull versus slip characteristics vary from a linear relationship with slip to a saturating relationship. These observations suggest that the force–slip response mapped from terrain characteristics could be useful in estimating underlying terrain parameters. B. Uniqueness of Mapping From Terrain Parameters to Vehicle–Terrain Forces Nonuniqueness of the mapping from terrain parameters to vehicle–terrain forces is 设计巴巴工作室www.88doc88.com directly evident in intersecting curves for drawbar pull versus slip ratio in Fig. 3, i.e., for certain slip ratios, drawbar pull resulting from different terrains is identical. Dry sand and lean clay present similar drawbar pull at ?40% slip for the lighter vehicle, and sandy loam and lean clay present similar drawbar pull at ?30% slip for the heavy vehicle. Thus,operating around these slip ratios presents a challenge for estimating terrain parameters. We also observe uniqueness issues directly from the soil mechanics model of (8). Both the tangent of the friction angle and soil cohesion are directly related to the maximum shear stress, and thus to net traction force. A soil with a given maximum normal stress can achieve a given maximum shear stress either through a large friction angle and low cohesion or through a large cohesion and low friction angle. Thus, it is difficult to estimate both cohesion and friction angle. This issue is addressed in [5] by assuming an average value for soil cohesion in order to estimate friction angle. Here, we aim to estimate cohesion along with all other terrain parameters. Nonuniqueness of the mapping from terrain parameters to vehicle–terrain forces also derives, in part, from asymmetric shear and normal stress distributions; when stress distributionsare asymmetric, identical forces can derive from different stress distributions through integral equations (18)–(20), even if the maximum stress is the same for different distributions. Iagnemma et al. [4] propose a linear approximation of shear and normal stress distributions in order to approximate closed-form solutions to (18)–(20). These solutions are then used to determine soil cohesion and friction angle using a least-squares approach. To facilitate the closed-form solution, Iagnemma et al. [4] approximate the stress distributions as symmetric, i.e., θm occurs at the midpoint between θ1 and θ2 . However, Fig. 3 and [3] show that the stress distributions may not necessarily be linear or symmetric. Wong and Reece [3] show experimental data from which the linear relationship between θm and θ1 , which is given in (16), is derived. The experimental data show that the ratio between θm and θ1 can lie between 0.2 and 0.7 as a function of slip for low-cohesion soils, and thus, the stress distributions are symmetric only at one particular value of slip. Empirical terrain parameters c1and c2 capture the asymmetry in the stress distribution but are rarely reported in the literature. 设计巴巴工作室www.88doc88.com Symmetry provides a unique relationship between σ(θ) andτ (θ), and resultant forces and moments in (18)–(20), i.e., symmetric stress distributions result in unique Tr and Fx in Fig. 2(a) for a given value of c and υ. When the stress distributions are asymmetric, only mild uniqueness is preserved, especially for low-cohesion soils, i.e., there can exist more than one shear and normal stress distribution that maps to the same net traction, as evidenced in Fig. 3. For a completely cohesionless soil, shear stress asymmetry can result in identical values of Tr in (20),i.e., the integral of shear stress would not differ if the maximum stress is to the left or right of the midpoint between θ1 and θ2 . Although a completely cohesionless soil is not of interest, mobile robots do operate on soils of very low cohesion, and similar values of Tr would result from different stress distributions on such soils; measurement uncertainty in sensors from which soil parameters are estimated, therefore, present a challenge for terrain parameter estimation in low-cohesion soils. C. Bayesian Terrain Parameter Estimation In selecting a Bayesian parameter-estimation approach, we seek an approach that is insensitive to the uniqueness issues described earlier, does not require approximation of the stress distributions to be linear, and does not require stress distributions to be symmetric. Moreover, as future experimental implementation of this methodology will use imperfect, noisy sensors, and terrain that is not perfectly homogeneous, the terrain parameter estimation method must reject measurement noise and process noise. Finally, a Bayesian approach can make use of the existing terramechanics database of terrain properties, limiting the parameter space considerably. Kramer and Sorenson [14] describe a Bayesian parameter estimation approach in which the unknown parameters are included as part of the system state and are estimated along with the state. Bayesian estimation is used to calculate the joint posterior density of the state-augmented system. Since there are eight parameters here, this method would increase the size of the state by at least eight and would require observability of the state-augmented system,which is problematic due to uniqueness issues. Moreover, since the terrain parameters appear implicitly through integral 设计巴巴工作室www.88doc88.com equations in the state-augmented system and not explicitly, one would need to assume a terrain model forthe state-augmented system, and the resulting nonlinear estimation problem would be computationally intractable in real time. An MME approach is described in [15], in which a bank of N Kalman filters is formulated for N-hypothesized sets of parameters. Each filter is propagated forward in time to provide state estimates, and Bayes’ rule is used to determine the conditional probability of the hypothesis, given the estimated state from each filter. The filter with the smallest residual should correspond to the most likely hypothesis. This method would be more robust to uniqueness issues, as competing hypotheses can take on probabilities; however, it is computationally costly, due to the need for an extended Kalman filter for each hypothesis. We use an alternate Bayesian approach that does not increase the size of the state vector as in [14] and does not require propagation of N extended Kalman filters as in[15]. Wong [2] reports sets of Bekker terrain parameters{c, υ, kc, kυ, n} for 21 terrains. Using the Bekker rigid-wheel terrain model as a forward model and these terrain parameters sets as hypotheses, we propose the following Bayesian MME approach to terrain identification as a computationally efficient approach that should be robust to weak uniqueness described earlier. For each hypothesis, a mapping of the terrain parameter set to terrain forces is determined a priori as a function of slip and normal load. The most likely set of terrain parameters from among the hypotheses is determined by recursive implementation of Bayes’ rule as follows. Let pj , for j = 1 to N, comprise N-hypothesized terrain parameter vectorswith probability mass function initialized as Pr(pj )0 = 1/N. The conditional probability mass function for parameter set pj subject to a vector ˆ Fk of terrain force estimates from the EKBF at time k evolves according to Bayes’ rule [16] as Where 设计巴巴工作室www.88doc88.com where S is the covariance matrix of the residual rk (pj) =Fk (pj ) ? ˆ Fk , and Fk (pj ) is the vector of terrain forces mapped for parameter set pj and estimated slipˆi. The most likely parameter set is given by a probability-weighted sum of the hypotheses Equations (21)–(23) are implemented recursively, i.e., at each time step k, new estimated forces and wheel slips are used to update the probability mass function for each hypothesis. Note that the success of this method depends heavily on the quality of the hypotheses, and thus, the method should be evaluated for cases where no hypothesis precisely represents the underlying parameters, but the hypotheses bound the underlying parameter set. RAY: ESTIMATION OF TERRAIN FORCES AND PARAMETERS FOR RIGID-WHEELED VEHICLES Fig. 4. Actual and estimated (top) net traction versus slip and (bottom) resistive torque versus slip for an applied torque input decreasing linearly over time The division of labor between the EKBF and Bayesian MME keeps the size of the augmented state in the EKBF to a maximum of 12 more than the original state vector, i.e., a secondorder random walk model for each of six forces estimated. This approach also allows implementation of the EKBF without assuming a terramechanics model, 设计巴巴工作室www.88doc88.com whereas the approach given in [14] would require a terramechanics model to relate unknown terrain parameters to terrain forces. Thus, the complexity of computing Jacobian matrices for the state-extended system is no more than that of the original state, as augmented states enter linearly. The computation required for the traditional MME approach of [15] is N times that for a single EKBF and would be prohibitively costly for any reasonable number of hypotheses. Since we do not implement an EKBF for each hypothesis, our computational cost scales modestly with the number of hypotheses, as only (21)–(23) depend on the number of hypotheses. IV. EVALUATION OF TERRAIN PARAMETER ESTIMATION To evaluate terrain parameter estimation, we simulate longitudinal acceleration of a vehicle of massm = 100 kg with wheel diameter 0.508 m (20 in) and width 0.15 m (6 in) over homogeneous, deformable terrain. The applied torque to each wheel at t = 0 is large, and the torque decreases linearly with time. This input produces wheel slips of 100% at t = 0, which decrease as torque decreases and solicits the characteristic net force versus slip and resistive torque versus slip at each wheel. Zeromean,Gaussian process noise is injected into each dynamic equation (1)–(6). Zero-mean, Gaussian measurement noise is njected to simulate measured acceleration, wheel speeds, yaw rate, and ground speed. Measurement noise variance is chosen TABLE II MEASUREMENT AND PROCESS NOISE COVARIANCE according to laboratory-measured values for transducers used in experiments to validate terrain force estimation given in [11].Process noise and measurement noise covariances used in the simulation are reported in Table II. Measurement and process 设计巴巴工作室www.88doc88.com noise covariances are assumed to be known. The EKBF is simulated at a sample frequency of 100 Hz. The EKBF provides time histories of the state and force estimates,and time histories slip at each wheel are derived from the wheel rotational speeds and ground speed. From these time histories, terrain forces and resistive torque versus slip are derived. Representative force versus slip estimation results for each terrain in Table I are given in Fig. 4 as true and estimated force versus slip and true and estimated resistive torque versus slip at the front left wheel. Note that each point in Fig. 4 represents one sample in the time history. Fig. 4 shows that the EKBF is able to track the external force and moments due to terrain within limits imposed by process noise,measurement noise, and filter transients. For this quasi-steady maneuver, filter transients generate error between the actual and estimated net traction and resistive torques versus slip; as transients decay, the mean values of the estimated force and torque approach the true steady-state operating points, while process and measurement noise gives rise to ―clusters‖ of points around the quasi-steady operating condition. For the simulation in Fig. 4, the linearly decreasing torque is applied over 2 s on sand and sandy loam and 4 s on lean clay; estimation error due to transients decreases as the inputs are varied more slowly. Evaluation of the terrain force estimation, including physical testing, is given in [11] and, thus,is omitted here. Using estimated drawbar pull, thrust, resistance, and resistive torque versus slip, Bayesian terrain parameter estimation is evaluated for a vehicle of mass m = 100 kg and three terrains reported in Table I. The parameter set includes all eight-terrain model parameters p = [c υ kc kυ n c1 c2 K]. The first five parameters are reported for 21 terrains given in [2], and these 21 terrain parameter sets form the hypothesis set for the Bayesian MME approach. Parameters c1, c2 , and K are not reported in [2] for these terrains. K is reported in [2] as varying from 1 cm (firm sandy terrain) to 2.5 cm (sandy terrain), 0.6 cm for clay at maximum compaction, and 2.5–5 cm for fresh snow, without additional attribution to 1 of the 21 sets of terrain parameters. Thus,where no data exist,we takeK = 0.025mfor sandy soils, K = 0.01 m for sandy loams, K = 0.006 m for clayey soils, and K = 0.05 m for snow within the 21 terrain parameter hypotheses. The variables c1 and c2 are summarized in [3] for sandy terrain from 设计巴巴工作室www.88doc88.com three sources and range from c1 = 0.43 and c2 = 0.32 for compact sand to c1 = 0.38 and c2 = 0.41 for dry sand. In the absence of data for the 21 terrains given in [2], we take c1 = 0.43 and c2 = 0.32 for sandy loams and clayey soils and c1 = 0.38 and c2 = 0.41 for sandy soils and snow. Of the 21 hypotheses, two are described in [2] as ―sand,‖ eight are described as ―sandy loam,‖ three are described as ―snow,‖ six are described as clayey soils, and one each are described as ―clayeybloam‖ and ―loam‖; thus, hypotheses represent a continuum over a large range of soil types. Given the 21 hypothesized terrain parameter vectors, among which are the three terrains in Table I, the Bayesian multiplemodel estimator is evaluated for these three terrains. EKBF terrain force estimates from a single wheel are used in (21)–(23), though the methodology is not limited to using forces from a singlewheel.Two candidate terrain force vectors are evaluated. Candidate 1 is ˆ Fk = [ ˆ Ff l(k) ˆ Trf l (k)], i.e., the EKBF estimated net traction or drawbar pull (thrust minus resistance) and resistive torque at the front left wheel, and candidate 2 is ˆ Fk = [ ˆ Ftf l (k) cos(θff l (k)) ˆRcf l(k) ˆ Trf l (k)], i.e., gross traction, resistance, and the resistive torque at the front left wheel. The candidate 2 terrain force vector employs (9)–(12) to partition estimated drawbar pull and resistive torque into gross traction and resistance. We also consider the case where the three sets of terrain parameters in Table I representing the true underlying terrain characteristics are not in the hypothesis set. In place of exact hypotheses, terrain parameters for these three terrains are modified, with random white noise with standard deviation of 10% of the nominal terrain value added to each parameter. Thus, in this case, no hypothesis precisely matches the true underlying terrain, but at least one hypothesis approximates the underlying terrain. In implementation of terrain force estimation, we delay MME for 20 samples (0.2 s) to allow initial EKBF transients to decay, and we measure convergence from the start of MME. Results of simulation evaluation are summarized as follows.When exact terrain parameter sets in Table I are included in the set of 21 hypotheses, the Bayesian estimator converges to the correct hypothesis in at most two iterations, or 0.02 s for all three terrains and both candidate force vectors. The slip condition at convergence is high, and the Bayesian estimator remains converged to this single correct 设计巴巴工作室www.88doc88.com hypothesis through the entire simulation. When inexact terrain parameter sets in Table I are represented among the hypotheses by adding Gaussian white noise so that no hypothesis matches the true underlying terrain parameter vector, terrain parameter estimates converge to a single hypothesis for dry sand and sandy loam, namely the hypothesis produced by modifying true underlying terrain parameters with Gaussian white noise. Convergence occurs within two iterations (0.02 s) for both candidate terrain force vectors and remains converged throughout the simulation. For lean clay, terrain parameters converge to a single hypothesis in fewer than five iterations for the candidate 1 terrain force vector. However, this hypothesis is not the one generated by injecting Gaussian noise. The terrain for the hypothesis thatMMEconverges to is described in [2] as a clayey soil with parameters n = 0.13, kc = 12.7(kPa/mn?1 ), kυ = 1556(kPa/mn ), c = 68.95 kPa, and υ = 34?, i.e., a soil with parameters similar to lean clay parameters reported in Table I. Results for the high-cohesion soil are attributed to weak uniqueness of the net forces and resistive torque resulting for the different clayey soils represented in the hypothesis set. For lean clay and candidate 2 terrain force vector, which incorporates both thrust and resistance components instead of net longitudinal force, the converged parameters are represented by a linear combination of two or three hypotheses according to (23). An example of parameter convergence and probability mass function convergence for lean clay is given in Fig. 5, which shows convergence to within two hypotheses in two iterations (0.02 s). These two hypotheses are those for ―heavy clay‖ [2] reported earlier and the hypothesis generated from the ―lean clay‖ parameters with Gaussian noise added; thus, classification of terrain is achieved for all three terrain types within two iterations. Probability mass functions at t = 2.0 s for these hypotheses are 0.35 and 0.64, respectively. As an additional measure of performance, Fig. 6 shows examples of normal and shear stress distributions at 0.15 slip ratio for the actual underlying terrain and for the parameters to which the multiple-model estimator converges, based on the candidate 2 terrain force vector, when the hypothesis set does not include the actual terrain. Results in Fig. 6 are representative of worstcase (dry sand) and best-case (lean clay) estimation of stress distributions for the estimated and actual parameters, for a slip 设计巴巴工作室www.88doc88.com ratio of 0.15. Fig. 7 shows the estimated and actual drawbar pull and resistive torque versus slip characteristics for each identified terrain. Here, each point represents a slip increment of 0.05. The ―true‖ drawbar pull and resistive torque versus slip represent the noise-free force versus slip curves for the actual underlying RAY: ESTIMATION OF TERRAIN FORCES AND PARAMETERS FOR RIGID-WHEELED VEHICLES Fig. 5. Terrain parameter convergence for simulation of vehicle on lean clay, with candidate 2 force vector used in multiple-model estimation. (Top) Convergence history of pressure-sinkage parameters and cohesion. (Bottom) Convergence history of friction angle, n, and conditional probabilities, showing two remaining hypotheses after two iterations. 设计巴巴工作室www.88doc88.com Fig. 6. Comparison of shear and normal stress distributions, as described by actual and estimated terrain parameters. terrain parameters. The ―estimated‖ curves represent the noisefree force versus slip curves for the terrain parameter values resulting from Bayesian parameter estimation at t = 1 s. Comparison of Fig. 6 with Fig. 7 shows that even though underlying stress distributions do not match the actual distributions exactly, the net forces versus slip well approximate the actual forces versus slip. For lean clay, small errors between the true and estimated stress distributions translate to imperceptible differences in true and estimated force versus slip. For sandy loam and dry sand, even for larger errors in estimated stress distributions, the estimated force versus slip tracks true force versus slip within limits imposed by measurement and process noise injected into the simulation. Comparison of Fig. 7 with Fig. 4, which is derived from time histories of estimated drawbar pull, resistive torques, and wheel slips, shows that even though imperfect estimation of vehicle–terrain forces results from extended Kalman–Bucy filtering (due to limits imposed by measurement noise and process noise, and filter transients), the Bayesian parameter estimator is able to select a parameter vector hypothesis that represents the underlying terrain with reasonable accuracy; hence, the method provides some robustness to noise. 设计巴巴工作室www.88doc88.com Fig. 7. Comparison of drawbar pull and resistive torque versus slip, as described by actual and estimated terrain parameters. V. CONCLUSION This paper develops a Bayesian MME methodology for identifying terrain parameters from EKBF estimated terrain force versus slip characteristics. Simulation evaluation of the methodology for a Bekker rigid-wheel terrain model shows that it can identify the hypothesis best representing terrain characteristics from a set of hypotheses with good convergence, and it exhibits interpolation properties for similar competing hypotheses. The methodology requires no assumptions or approximations of the underlying shear and normal stress distributions. It poses a low computational burden when mappings from the hypothesized terrain parameter sets to terrain forces characteristics are precomputed as functions of wheel slip and normal load. Computation scales modestly with the number of hypotheses; thus, additional hypothesis could be drawn from the literature without imposing undue additional computation. In this paper, the methodology is demonstrated for an underlying terrain model for driven, rigid wheels; however, the method is not constrained so that the model structure posed for each hypothesis needs to be consistent, and hypotheses can take the form of alternative model structures and parameters. Moreover, the terrain 设计巴巴工作室www.88doc88.com identification example indicates that gross traction, resistance force, and resistance torque from a single wheel provide sufficient information for terrain identification,and thus, terrain parameters could be identified independently for each wheel. Finally, the methodology can also be used as a rapid classifier as it is able to distinguish characteristics of terrain, e.g., clayey soil versus sandy loam versus sand with excellent convergence. REFERENCES [1] M. G. Bekker, Theory of Land Locomotion. Ann Arbor, MI: Univ. Michigan Press, 1956. [2] J. Y. Wong, Theory of Ground Vehicles, 3rd ed. New York: Wiley– Interscience, 2001. [3] J. Y.Wong and A. R. Reece, ―Prediction of rigid wheel performance based on the analysis of soil-wheel stresses part I. Performance of driven rigid wheels,‖ J. Terramech., vol. 4, no. 1, pp. 81–98, 1967. [4] K. Iagnemma, S. Kang, H. Shibly, and S. Dubowsky, ―Online terrain parameter estimation for wheeled mobile robotswith application to planetary rovers,‖ IEEE Trans. Robot., vol. 20, no. 5, pp. 921–927, Oct. 2004. [5] S. Hutangkabodee, Y. H. Zweiri, L. D. Seneviratne, and K. Althoefer,―Soil parameter identification for wheel-terrain interaction dynamics and traversability prediction,‖ Int. J. Autom. Comput., vol. 3, pp. 244–251, 2006. [6] L. Ojeda, J. Borenstein, G. Witus, and R. Karlsen, ―Terrain characterization and classification with a mobile robot,‖ J. Field Robot., vol. 2, no. 2, pp. 103–122, 2006. [7] L. Ojeda, D. Cruz, G. Reina, and J. Borenstein, ―Current-based Slippage detection and odometry correction for mobile robots and planetary rovers,‖IEEE Trans. Robot., vol. 22, no. 2, pp. 366–377, Apr. 2006. [8] K. Wesson, M. Parker, B. Coutermarsh, S. Shoop, and J. Stanley, ―Instrumenting an all-terrain vehicle for off-road mobility analysis,‖ ERDC/CRREL: TR-07-1, Jan. 2007. [9] R. B. Ahlvin and P. W. Haley, ―NATO reference mobility model, Edition II, NRMMII user_s guide,‖ Geotech. Lab., USAEWES, Tech. Rep. GL-92-19, 1992. 设计巴巴工作室www.88doc88.com [10] S. Shoop, ―Terrain characterization for trafficability,‖ Cold Regions Res.Eng. Lab., Hanover, NH, CRREL Rep. 93-6, Jun. 1993. [11] L. R. Ray, D. Brande, and J. H. Lever, ―Estimation of net traction for differential-steered wheeled robots,‖ J. Terramech., 2008. doi:10.1016/j.jterra.2008.03.003 [12] L. R. Ray, ―Nonlinear state and Tire force estimation for advanced vehicle control,‖ IEEE Trans. Control Syst. Technol., vol. 3, no. 1, pp. 117–124, Mar. 1995. [13] L. R. Ray, ―Nonlinear Tire force estimation and road friction identification:Simulation and experiments,‖ Automatica, vol. 33, no. 10, pp. 1819–1833,1997. [14] S. C. Kramer andH.W. Sorenson, ―Bayesian parameter estimation,‖ IEEE Trans. Autom. Control, vol. 33, no. 2, pp. 217–222, Feb. 1988. [15] P. S. Maybeck, Stochastic Modek, Estimation and Control, vol. 3. New York: Academic, 1982. [16] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering. Reading, MA: Addison-Wesley, 1994. Laura E. Ray (M’92) received the B.S. degree (withthe highest honors) in mechanical and aerospace engineering from Princeton University, Princeton, NJ,in 1984, the M.S. degree in mechanical engineering from Stanford University, Stanford, CA, in 1985, and the Ph.D degree in mechanical and aerospace engineeringfrom Princeton University, in 1991. In 1996, she joined the Thayer School of Engineering, Dartmouth College, Hanover, NH, where she has been an Associate Professor of engineering science since 2002. Her current research interests include cooperative control of mobile robots, robot mobility and vehicle–terrain interaction, and field robotics. 设计巴巴工作室www.88doc88.com 设计巴巴工作室www.88doc88.com