-
文中采用美国Sensable Technologies公司的PHANTOM系列触觉设备PHANTOM Omni对移动机器人进行控制。该控制器将力反馈设备手柄移动到合适的位置之后就可以按住按钮开始操控移动机器人,松开按钮会停止控制,如果环境中添加其他物体(比如障碍物),则还可以模拟环境中的力(接触力、重力、摩擦力、弹簧力等)让操控者能“感觉”到[15-16]。如果实际机器人上装有力传感器,则在用PHANTOM Omni控制机器人的同时也能读取力的信息,反馈给操控者。
-
PHANTOM Omni控制器的结构建模如图1所示。图中标示出的PHANTOM Omni控制器的1,2,3连杆用来控制移动机器人的位置和方向(X,Y,
$\theta $ ),使用4,5,6连杆来进行力反馈(Force Feedback)。图中${L_1}$ ,${L_2}$ ,${L_3}$ 表示连杆1,2,3的长度,${L_5}$ 表示从${L_1}$ 的起点到连杆4,5,6起点之间的距离。通过各个关节的角度$\theta $ 和连杆长度L可以得到末端装置的位置。经过对PHANTOM Omni控制器的结构分析,设定
${\theta _1}$ 来控制移动机器人的正面移动方向。这里${\theta _1}$ 为PHANTOM Omni控制器X轴的旋转角度。表1中,通过1,2,3连杆和关节进行建模得到的D-H模型。通过D-H模型可以推导出末端装置的位置公式为:
$$\left\{ {\begin{array}{*{20}{l}} {{Y_m} = [{L_2}\sin {\theta _2} + {L_3}\sin \left( {{\theta _2} + {\theta _3}} \right)]\cos {\theta _1}} \\ {{{\textit{Z}}_m} = [{L_2}\sin {\theta _2} + {L_3}\sin \left( {{\theta _2} + {\theta _3}} \right)]\cos {\theta _1}} \\ {{X_m} = {L_2}\sin {\theta _2} + {L_3}\sin \left( {{\theta _2} + {\theta _3}} \right) + {L_1}} \end{array}} \right.$$ (1) 表 1 PHANTOM Omni控制器D-H 参数表格
Table 1. D-H parameter table of the PHANTOM Omni structure
Number of joints Z axis rotation angle Distance between common perpendiculars Joint offset Joint torsional $\theta $ $d$ $a$ $\alpha $ 1 $\theta _1^{\rm{*}}$ ${d_1}$ 0 $ - {90^ \circ }$ 2 $\theta _2^{\rm{*}}$ 0 ${a_2}$ 0 3 $\theta _3^{\rm{*}}$ 0 ${a_3}$ 0 -
文中的移动机器人的移动装置由两个主动轮和一个辅助轮组成,动态模型由图2所示。移动机器人的参数见表2。
移动机器人的直线运动参数X,Y和旋转角度
$\theta $ 组成的三自由度运动方程用$\dot q = {[\begin{array}{*{20}{c}} {\dot x}&{\dot y}&{\dot \theta } \end{array}]^{\rm{T}}}$ 表示。则移动机器人的结构方程可以表示为:$$\dot q = J(\theta )\vec v$$ (2) 式中:
$J(\theta ) = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&0 \\ {\sin \theta }&0 \\ 0&1 \end{array}} \right]$ ;$\vec v = {\left[ {\begin{array}{*{20}{c}} v&\omega \end{array}} \right]^{\rm{T}}}$ ;v表示移动机器人直线运动的速度;$\omega $ 表示移动机器人转运动的速度。并且,移动机器人的动力学方程表示为:$$\tau = {E^{ - 1}} \cdot {\dot{ \vec v}}$$ (3) 式中:
$E = {M^{ - 1}}B$ ,${M^{ - 1}} = \dfrac{1}{{mI}}\left[ {\begin{array}{*{20}{c}} I&0 \\ 0&m \end{array}} \right]$ ,$B = \dfrac{1}{r}\left[ {\begin{array}{*{20}{c}} 1&1 \\ R&R \end{array}} \right]$ 。表 2 移动机器人的参数表
Table 2. Parameters table of the mobile robot
Parameter Meaning of parameters $m$ Mass of mobile robots($5.61 \;\rm kg$) $I$ Moment of inertia of mobile robot($0.05 \;\rm kg{n^2}$) $2R$ Distance between two driving wheels($0.290\;\rm m$) $r$ Driving wheel radius($0.075\;\rm m$) $O,X,Y$ Spatial coordinates $C,{X_c},{Y_c}$ Coordinates of the mobile robot $X,Y$ Center position of the mobile robot in Cartesian coordinates $\theta $ Angle between $X$ and ${X_c}$ -
史密斯预估器(Smith predictor)补偿方法的设计思想以及特点是,预先估计出在基本扰动下的动态响应,然后由预估器进行补偿,力图使被延迟了的被控量超前反馈到控制器,使控制器提前动作,从而明显地加速控制工程并减小超调量。
文中的史密斯预估器用于补偿PHANTOM Omni控制器与移动机器人之间信号的时间延迟。2.2节中移动机器人的动态模型参数E来表示对象除去纯迟延环节以后的传递函数。图3中,史密斯预估器使用的传递函数用
$\hat E$ 表示。参数$\tau $ 表示扭矩输入,${\dot{ \vec v}}$ 表示加速度输出值。基本模型公式化可表示为:$$\left[ {\tau - {{\rm{e}}^{ - s\frac{T}{2}}}{\dot{ \vec v}} - \left( {\hat Ew - {{\rm{e}}^{ - sT}}\hat Ew} \right)} \right]{{\rm{e}}^{ - s\frac{T}{2}}}CE = {\dot{ \vec v}}$$ (4) 整理上式,可以得到:
$${{\rm{e}}^{ - s\frac{T}{2}}}CE\tau - {{\rm{e}}^{ - sT}}CE{\dot{ \vec v}} - {{\rm{e}}^{ - s\frac{T}{2}}}CE\hat Ew + {{\rm{e}}^{ - s\frac{{3T}}{2}}}CE\hat Ew = {\dot{ \vec v}}$$ (5) 另外,史密斯预估器输出为:
$$w = \frac{{\dot {\vec {v}}}}{{{{\rm{e}}^{ - s\frac{T}{2}}}E}}$$ (6) 把公式(6)代入公式(5)得到:
$$\frac{{\dot {\vec {v}}}}{\tau } = \frac{{{{\rm{e}}^{ - s\frac{T}{2}}}CE}}{{1 + {{\rm{e}}^{ - sT}}CE + CE - {{\rm{e}}^{ - sT}}C\hat E}}$$ (7) 如果在公式(7)中
$\hat E = E$ ,整理公式可以得到:$$\frac{{\dot {\vec {v}}}}{\tau } = \frac{{CE}}{{1 + CE}} \cdot {{\rm{e}}^{ - s\frac{T}{2}}}$$ (8) 公式(8)表明系统将消除大迟延对系统过渡过程的影响,使控制过程的品质与过程无纯迟延环节的情况一样,只是在时间坐标上向后推迟了一个时间单位,此时的系统图见图4。
-
灰色预测(Grey prediction)是一种对具有不确定因素的系统进行预测的方法。灰色预测可以识别系统因素之间发展趋势的差异程度,即进行相关分析,并对原始数据进行生成处理来寻找系统变动的规律,并处理原始数据,找出系统变化的规律,生成规律性强的数据序列,然后建立相应的微分方程模型,从而预测事物未来的发展趋势。
文中使用的灰色预测模型是关于数列预测的一个变量、一阶微分GM(1,1)模型。利用GM(1,1)模型预测在移动机器人上传感器发出来的数值,从而预测因为时间延时得到的力反馈。GM(1,1)模型预测顺序如下:
首先,系统输出的原始数据表示为:
$${y^{(0)}} = \left\{ {{y^{(0)}}(1),{y^{(0)}}(2),{y^{(0)}}(3), \cdots, {y^{(0)}}(n)} \right\}$$ (9) 式中:
${y}^{(0)}(i)\ge 0,(i=1,2,\cdots, n)$ 。为了预测$n \geqslant 4$ 的值,上式最少需要4个测定值。再对数据做适当的预处理,预处理的数据平滑设计为三点平滑[9-10]。预处理后,取原始序列的第1个数据作为生成列的第1个数据,将原始序列的第2个数据加到原始序列的第1个数据中,两个数据之和作为生成列的第2个数据。按照此规则获取生成的列。根据以下公式:$${y^{(1)}}(k) = \sum\limits_{i = 1}^k {{y^{(0)}}(i)} $$ (10) 式中:
$k = 1,2,3, \cdots, n$ 。可以得到一个新的数列:$${y^{(1)}} = \left\{ {{y^{(1)}}(1),{y^{(1)}}(2),{y^{(1)}}(3), \cdots, {y^{(1)}}(n)} \right\}$$ 此时,为了可以建立GM(1,1)模型且可以进行灰色预测,新的数列必须满足下式:
$$0 \leqslant \frac{{{y^{(1)}}(k)}}{{{y^{(1)}}(k - 1)}} \leqslant 0.5:Quasi - smooth$$ (11) $$1 \leqslant \frac{{{y^{(1)}}(k)}}{{{y^{(1)}}(k - 1)}} \leqslant 1.5:Quasi - \exp $$ (12) 式中:
$k = 4,5, \cdots, n$ 。接下来用
${y^{(1)}}$ 序列的中间值来定义${z^{(1)}}$ :$${z^{(1)}}(k) = 0.5{y^{(1)}}(k) + 0.5{y^{(1)}}(k - 1)$$ (13) 式中:
$k = 2,3, \cdots, n$ 。新数列的变化趋势近似地用下面的微分方程描述:
$${y^{(0)}}(k) + a{z^{(1)}}(k) = b$$ (14) 式中:
$a$ 称为发展灰数;$b$ 称为内生控制灰数。设
$\hat \alpha $ 为待估参数向量,则$\hat \alpha = \left[ {\begin{array}{*{20}{c}} a \\ b \end{array}} \right]$ ,通过最小二乘法得到:$$\left[ {\begin{array}{*{20}{c}} a \\ b \end{array}} \right] = {\left( {{D^{\rm{T}}}D} \right)^{ - 1}}{D^{\rm{T}}}Y$$ (15) 上式中,序列
$D = \left[ {\begin{array}{*{20}{c}} { - {{\textit{z}}^{(1)}}(2)}&1 \\ { - {{\textit{z}}^{(1)}}(3)}&1 \\ \vdots & \vdots \\ { - {{\textit{z}}^{(1)}}(n)}&1 \end{array}} \right]$ ,序列$Y = \left[ {\begin{array}{*{20}{c}} {{y^{(0)}}(2)} \\ {{y^{(0)}}(3)} \\ \vdots \\ {{y^{(0)}}(n)} \end{array}} \right]$ 。求解微分方程,即可得灰色预测的离散时间响应函数,可以表示为:
$${\hat y^{(1)}}(k + 1) = \left( {{y^{(0)}}(1) - \frac{b}{a}} \right){{\rm{e}}^{ - ak}} + \frac{b}{a}$$ (16) 式中:
${\hat y^{(1)}}(k + 1)$ 为所得的累加的预测值,再利用公式${\hat y^{(0)}}(k + 1) = {\hat y^{(1)}}(k + 1) - {\hat y^{(1)}}(k)$ 将预测值还原,可求出${\hat y^{(0)}}(k + 1)$ 的值。利用以上灰度预测来预测来自于移动机器人上传感器的数值,把对应得到的力反馈传输给使用者。在实际实验之前通过仿真证明其系统效果的可行性。
Remote control robot system based on predictive algorithm
-
摘要: 为了解决时滞环节给远程遥控系统带来的巨大影响,在非可视环境下提出了一种的新的遥控控制移动机器人的预测系统。采用移动机器人的动力学模型作为基础,利用史密斯预估器补偿控制器与移动机器人之间信号延迟,减少因为延时所引起的定位误差。利用灰色预测模型来预测移动机器人上传感器得到的数值,从而减少因为时间延迟给操纵者带来的遥控误操作。通过仿真证明了算法的可行性,又通过力反馈遥控设备与移动机器人在非可视环境下的控制实验,证明了系统的可行性。尽管有操纵者参与了远程控制实验,但补偿效果已经得到了明显的证明。Abstract: In order to solve the influence of the time delay on remote control system, a new predictive system of remote-controlled mobile robot was proposed in the non-visual environment. The dynamic model of mobile robot was used as the basis, the Smith predictor was used to compensated the signal time delay between the controller and the mobile robot, reduced the positioning error caused by time delay. The grey predictor model was used to predict the value of the sensors on the mobile robot, thus reducing the remote control error caused by time delay to the operator. The feasibility of the algorithm was proved by simulation. And by the experiment of the force feedback device and the mobile robot in the non-visual environment, the feasibility of the system was proved. Although the human operator participated in the remote control experiment, the compensation effects had been clearly demonstrated.
-
Key words:
- Smith predictor /
- grey predictor /
- time delay system /
- mobile robot /
- force feedback
-
表 1 PHANTOM Omni控制器D-H 参数表格
Table 1. D-H parameter table of the PHANTOM Omni structure
Number of joints Z axis rotation angle Distance between common perpendiculars Joint offset Joint torsional $\theta $ $d$ $a$ $\alpha $ 1 $\theta _1^{\rm{*}}$ ${d_1}$ 0 $ - {90^ \circ }$ 2 $\theta _2^{\rm{*}}$ 0 ${a_2}$ 0 3 $\theta _3^{\rm{*}}$ 0 ${a_3}$ 0 表 2 移动机器人的参数表
Table 2. Parameters table of the mobile robot
Parameter Meaning of parameters $m$ Mass of mobile robots( $5.61 \;\rm kg$ )$I$ Moment of inertia of mobile robot( $0.05 \;\rm kg{n^2}$ )$2R$ Distance between two driving wheels( $0.290\;\rm m$ )$r$ Driving wheel radius( $0.075\;\rm m$ )$O,X,Y$ Spatial coordinates $C,{X_c},{Y_c}$ Coordinates of the mobile robot $X,Y$ Center position of the mobile robot in Cartesian coordinates $\theta $ Angle between $X$ and${X_c}$ -
[1] Ding Xiaodi, Cui Baotong. Adaptive control of smith predictor with uncertain parameters [J]. Computer Engineering and Design, 2016(11): 3007-3011. (in Chinese) [2] Saravanakumar G, Wahidabanu R S D, Arunraajesh K G. Performance analysis of modified smith predictor for integrating and time-delay processes [J]. Indian Chemical Engineer, 2011, 53(4): 261-270. doi: 10.1080/00194506.2011.670934 [3] Lei Z, Guo C. Disturbance rejection control solution for ship steering system with uncertain time delay [J]. Ocean Engineering, 2015, 95: 78-83. doi: 10.1016/j.oceaneng.2014.12.001 [4] Normey-Rico J E, Sartori R, Veronesi M, et al. An automatic tuning methodology for a unified dead-time compensator [J]. Control Engineering Practice, 2014, 27(5): 11-22. [5] Tsai M H, Tung P C. A robust disturbance reduction scheme for linear small delay systems with disturbances of unknown frequencies. [J]. Isa Transactions, 2012, 51(3): 362-372. doi: 10.1016/j.isatra.2011.12.002 [6] Rong H G, Hui Z, Li Z Q, et al. Tuning of fuzzy PID controller for smith predictor [J]. Journal of Central South University, 2010, 17(3): 566-571. doi: 10.1007/s11771-010-0524-2 [7] Zhang H, Hu J, Bu W. Research on fuzzy immune self-adaptive PID algorithm based on new smith predictor for networked control system [J]. Mathematical Problems in Engineering, 2015(10): 1-6. [8] Zhang Lei, Zhu Shuai, Liu Tianyu, et al. Target tracking of spatial dense group based on motion grouping [J]. Infrared and Laser Engineering, 2020, 49(11): 20200284. (in Chinese) [9] Xu Yunfei, Zhang Duzhou, Wang Li, et al. Design of lightweight feature fusion network for local feature recognition of non-cooperative target [J]. Infrared and Laser Engineering, 2020, 49(7): 20200170. (in Chinese) [10] Huo Ju, He Mingxuan, Li Yunhui, et al. Three-dimensional trajectory tracking and recognition for multiple aircraft with consistent displacement vectors [J]. Infrared and Laser Engineering, 2020, 49(10): 20200141. (in Chinese) [11] Xiong P P, Dang Y G, Wu X H, et al. Combined model based on optimized multi-variable grey model and multiple linear regression [J]. Journal of Systems Engineering and Electronics, 2011, 22(4): 615-620. [12] 曾波. 灰色预测建模技术研究[D]. 南京航空航天大学, 2012. Zeng Bo. Research on grey prediction modeling technology[D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2012. (in Chinese) [13] Zeng B, Liu S, Xie N. Prediction model of interval grey number based on DGM (1,1) [J]. Journal Systems Engineering and Electronics, 2010, 21(4): 598-603. [14] Li Mengwan, Sha Xiuyan. Improvement and application of grey prediction model based on GM(1,1) [J]. Computer Engineering and Applications, 2016, 52(4): 24-30. (in Chinese) doi: 10.3778/j.issn.1002-8331.1506-0257 [15] Wang Y, Dang Y, Li Y, et al. An approach to increase prediction precision of GM(1,1) model based on optimization of the initial condition [J]. Expert Systems with Applications An International Journal, 2010, 37(8): 5640-5644. doi: 10.1016/j.eswa.2010.02.048 [16] Wang Z X, Wang Z W, Li Q. Forecasting the industrial solar energy consumption using a novel seasonal GM(1,1) model with dynamic seasonal adjustment factors [J]. Energy, 2020, 200: 117460.