loadrunner Controller运行碰到的--错误--转载 - lihangmei的个人空间 - 51Testing软件测试网 51Testing软件测试网-中国软件测试人的精神家园 - Powered by X-Space
loadrunner Controller运行碰到的--错误--转载
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中运行脚本报错-26628在Vuser脚本编译执行无误情况下，在controller中进行并发操作，并发数设为10，只通过5个，失败的报错信息是：&& Action.c(15): Error -26628: HTTP Status-Code=403 (Access Forbidden) for ""解决方案如下：分析原因可能是因为被测服务器用的是XP系统，本身有最大连接数的限制，就使用软件MtaEdt22.exe修改IIS最大连接数为1000，修改后重启再进行并发操作-27791运行Controller时遇到这个error：Action.c(7): Error -27791:
"10.10.0.88" has shut down the connection prematurely解决方案如下：1、应用服务器死掉。小用户时程序上的问题，程序上处理的问题2、应用服务没有死。应用服务参数设置问题。例如：在许多客户端weblogic应用服务器被拒绝，而在服务器端没有错误显示，则有可能是weblogic中的server元素的acceptbacklog属性值设得过低。如果连接时收到connection refused消息，说明应提高该值，每次增加25%。3、数据库的连接在应用服务的性能参数可能太小了数据库启动的最大连接数（跟硬件的内存有关）4、有时关闭卡巴斯基也会解决如上问题==========================脚本回放问题及解决发布时间:
11:21 && 作者: 未知 && 来源: 51Testing博客转载字体: || ||　　在运行脚本回放过程中，有时会出现错误，这在实际中是不可避免的，毕竟自动录制生成的脚本难免会有问题，需要运行脚本进行验证，把问题都解决后才加入到场景中进行负载。下面结合常用的协议（如、Web Services协议）录制的脚本进行回放时出现的问题介绍一下解决的方法。　　需要注意的是，回放脚本时出现的错误有时是程序自身的原因导致的，因此在解决脚本回放问题前必须保证程序录制出的脚本是正确的。　　1．超时错误：在录制Web协议脚本回放时超时情况经常出现，产生错误的原因也有很多，解决的方法也不同。　　错误现象1：Action.c(16): Error -27728: Step download timeout (120 seconds) has expired when downloading non-resource(s)。　　错误分析：对于HTTP协议，默认的超时时间是120秒（可以在LoadRunner中修改），客户端发送一个请求到服务器端，如果超过120秒服务器端还没有返回结果，则出现超时错误。　　解决办法：首先在运行环境中对超时进行设置，默认的超时时间可以设置长一些，再设置多次迭代运行，如果还有超时现象，需要在“Runtime Setting”&“Internet Protocol：Preferences”&“Advanced”区域中设置一个“winlnet replay instead of sockets”选项，再回放是否成功。　　错误现象 2：Action.c(81):Continuing after Error -27498: Timed out while processing URL=http://172.18.20.70:7001/workflow/bjtel/leasedline/ querystat/ subOrderQuery.do　　错误分析：这种错误常常是因为并发压力过大，服务器端太繁忙，无法及时响应客户端的请求而造成的，所以这个错误是正常现象，是压力过大造成的。　　如果压力很小就出现这个问题，可能是脚本某个地方有错误，要仔细查看脚本，提示的错误信息会定位某个具体问题发生的位置。　　解决办法：例如上面的错误现象问题定位在某个URL上，需要再次运行一下场景，同时在机器上访问此URL。如果不能访问或时间过长，可能是服务器或者此应用不能支撑如此之大的负载。分析一下服务器，最好对其性能进行优化。　　如果再次运行场景后还有超时现象，就要在各种图形中分析一下原因，例如可以查看是否服务器、DNS、网络等方面存在问题。　　最后，增加一下运行时的超时设置，在“Run-Time Settings”&“Internet Protocol:Preferences”中，单击“options”，增加“HTTP-request connect timeout” 或者“HTTP-request receive”的值。　　2．LoadRunner脚本中出现乱码：在录制Web协议脚本时出现中文乱码，在回放脚本时会使回放停止在乱码位置，脚本无法运行。　　错误现象：某个链接或者图片名称为中文乱码，脚本运行无法通过。　　错误分析：脚本录制可能采用的是URL-based script方式，如果程序定义的字符集合采用的是国际标准，脚本就会出现乱码现象。　　解决办法：重新录制脚本，在录制脚本前，打开录制选项配置对话框进行设置，在“Recording Options”的“Advanced”选项里先将“Surport Charset”选中，然后选中支持“UTF-8”的选项。　　3．LoadRunner HTTP服务器状态代码：在录制Web协议脚本回放脚本的过程中，会出现HTTP服务器状态代码，例如常见的页面-404错误提示、-500错误提示。　　错误现象1：-404 Not Found服务器没有找到与请求URI相符的资源，但还可以继续运行直到结束。　　错误分析：此处与请求URI相符的资源在录制脚本时已经被提交过一次，回放时不可再重复提交同样的资源，而需要更改提交资源的内容，每次回放一次脚本都要改变提交的数据，保证模拟实际环境，造成一定的负载压力。　　解决办法：在出现错误的位置进行脚本关联，在必要时插入相应的函数。　　错误现象2：-500 Internal Server Error服务器内部错误，脚本运行停止。　　错误分析：服务器碰到了意外情况，使其无法继续回应请求。　　解决办法：出现此错误是致命的，说明问题很严重，需要从问题的出现位置进行检查，此时需要此程序的开发人员配合来解决，而且产生的原因根据实际情况来定，测试人员无法单独解决问题，而且应该尽快解决，以便于后面的测试。　　4．LoadRunner请求无法找到：在录制Web协议脚本回放脚本的过程中，会出现请求无法找到的现象，而导致脚本运行停止。　　错误现象：Action.c(41): Error -27979: Requested form. not found [MsgId: MERR-27979]　　Action.c(41): web_submit_form. highest severity level was "ERROR",0 body bytes, 0 header bytes [MsgId: MMSG-27178]"　　这时在tree view中看不到此组件的相关URL。　　错误分析：所选择的录制脚本模式不正确，通常情况下，基于浏览器的Web应用会使用“HTML-based script”模式来录制脚本；而没有基于浏览器的Web应用、Web应用中包含了与服务器进行交互的Java Applet、基于浏览器的应用中包含了向服务器进行通信的JavaScript/VBScript代码、基于浏览器的应用中使用HTTPS安全协议，这时则使用“URL-based script”模式进行录制。　　解决办法：打开录制选项配置对话框进行设置，在“Recording Options”的“Internet Protocol”选项里的“Recording”中选择“Recording Level”为“HTML-based script”，单击“HTML Advanced”，选择“Script. Type”为“A script. containing explicit”。然后再选择使用“URL-based script”模式来录制脚本。　　5．LoadRunner不执行检查方法：在录制Web协议脚本中添加了检查方法Web_find，但是在脚本回放的过程中并没有执行。　　错误现象：在脚本中插入函数Web_find，在脚本中设置文本以及图像的检查点，但是在回放过程中并没有对设置的检查点进行检查，即Web_find失效。　　错误分析：由于检查功能会消耗一定的资源，因此LoadRunner默认关闭了对文本以及图像的检查，所以在设置检查点后，需要开启检查功能。　　解决办法：打开运行环境设置对话框进行设置，在“Run-time Settings”的“Internet Protocol”选项里的“Perference”中勾选“Check”下的“Enable Image and text check”选项。　　6．LoadRunner回放Web Services协议脚本错误：LoadRunner 8.0版本在录制Web Services协议的脚本时正常，但在回放时会出现错误，提示停止脚本运行。　　错误现象：利用LoadRunner 8.0版本来录制Web Services协议的脚本没有任何错误提示，回放脚本时会出现如下错误提示“Error：server returned an incorrectly formatted SOAP response”。　　错误分析：出现此错误的原因是LoadRunner8.0在录制Web Services协议的脚本时存在一个缺陷：如果服务器的是中文的，VuGen会自动将WSDL文件的头改为&?xml version="1.0"encoding="zh_cn" ?&，所以才会有此错误提示。　　解决办法：下载两个补丁，分别为“LR80WebServicesFPI_setup.exe”和“lrunner_web_ services_patch_1.exe”安装上即可。西屋公司OVATION系统-ov100系统培训_文档下载_文档资料库
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3秒自动关闭窗口Simultaneous design of optimal gait pattern and controller for a bipedal robot_文档下载_文档资料库
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Simultaneous design of optimal gait pattern and controller for a bipedal robot
Multibody Syst Dyn (1C429 DOI 10.-009-9185-zSimultaneous design of optimal gait pattern and controller for a bipedal robotA.R. Ghiasi ? G. Alizadeh ? M. MirzaeiReceived: 20 April 2009 / Accepted: 7 December 2009 / Published online: 19 December 2009 ? Springer Science+Business Media B.V. 2009Abstract Nowadays, biped robotics becomes an interesting topic for many control researchers. The biped robot is more adaptable than the other mobile robots in a varied environment and can have more diverse possibilities in planning the motion. However, it falls down easily and its control for stable walking is dif?cult. Therefore, generation of a desired walking pattern for the biped robot in the presence of some model uncertainties is an important problem. The proposed walking pattern should be also achievable by the designed controller. To achieve this aim and to reach the best control performance, the walking pattern and controller should be designed simultaneously rather than separately. In the present study, an optimal walking pattern is proposed to be tracked by a designed sliding mode controller. In this respect, a genetic algorithm (GA) is utilized to determine the walking pattern parameters and controller coef?cients simultaneously. Here, high stability, minimum energy consumption, good mobility properties, and actuator limitations are considered as the important indexes in optimization. Simulation results indicate the ef?ciency of the proposed scheme in walking the understudy biped robot. Keywords Biped robot ? Walking pattern ? Genetic algorithm ? Nonlinear control ? Optimization1 Introduction In recent years, the biped robot has been interested by many researchers. This type of robot has special abilities to move in various environments and therefore is more human-friendly than the other mobile robots. Despite the many advantages of the biped robot, it has high complexity and redundancy. It is easy to fall down. So, the stability of biped locomotion inA.R. Ghiasi ( ) ? G. Alizadeh Faculty of Electrical Engineering, University of Tabriz, Tabriz, Iran e-mail: agiasi@tabrizu.ac.ir M. Mirzaei Faculty of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran402A.R. Ghiasi et al.various terrains without falling down is very important. Furthermore, in reality, the biped robot encounters with uncertainties which should be considered in generation of motion pattern. There are two schemes for walking pattern generation. The ?rst one is a scheme in which the designer manually de?nes the appropriate parameters to generate the walking trajectory [1]. The extensive effort of trials and errors to get the better performance is the main shortcoming of this scheme. But, in the other one, the designer intelligently ?nds all parameters that satisfy the constraints required in walking [2C7]. So, the second scheme does not need to repeat some efforts and it has been considered by many researchers. In many studies, the design of gait pattern and robot controller was studied separately. For instance, in most of the researches, the walking pattern was described based on stability criteria without any attention to the ability of the designed controller [8C12]. In these works, the walking patterns were found based on either static stability or dynamic stability criteria. Static stability requirements used to determine the walking pattern [8, 9], are not enough to adapt the wide range of biped walking speeds. On the other hand, dynamic stability criterion is based on the ZMP which is de?ned as the point on the ground about which the sum of all the moments of the active forces equals zero. More explanations about the ZMP are given in Appendix 1. In the related works [10C12], a desired ZMP trajectory is ?rst proposed and then the hip and torso motions are generated to achieve this trajectory. The main problem in this approach is that all desired ZMP trajectories may not be achievable because the hip acceleration and the energy consumption caused by the relative massive torso might be too high [10C12]. This means that the ZMP trajectory should be described in a way that the designed controller can follow it exactly in the presence of some existing limitations. Regularly, the biped walking should be modeled in two phases: single support phase (SSP) and double support phase (DSP). Most papers concentrate on dynamic modeling and control of biped walking with only the SSP. The DSP is often neglected or assumed to be instantaneous [13C15]. Considering the DSP for the motion of a biped robot has some advantages: it is more convenient to realize the stable motion and can ful?ll more tasks than that considers only the SSP. However, dynamic modeling of biped walking during the DSP is more challenging due to the involvement of the constraints and the constraint forces. It is concluded from the above discussion that a complete dynamic model must be considered to describe the biped walking. Also, the optimal and stable motion of the biped robot depends on the proper choice of control structure and desired walking pattern. So, it seems that the walking pattern should be described according to the motion conditions, controller abilities, and actuator limitations. In the present study, for a complete dynamic model of the robot and actuators, an optimal walking pattern and controller are designed in the presence of some model uncertainties. In this way, by considering some practical and physical constraints, a new strategy is proposed to determine the walking pattern and controller parameters simultaneously. Since, beside the stability, the robustness of the walking dynamic is important, the sliding mode control technique is employed to compute the control inputs of the system. Also, in order to have no efforts of trials and errors manually, the genetic algorithm (GA) is used to ?nd all possible solutions for the optimal and robust behavior of the system. GA is a global search procedure widely applied in optimization of constrained multibody systems because its ?tness function has little mathematical restrictions. Also, no gradient information is required for the GA. At the rest of the paper, the biped walking models in both single support phase (SSP) and double support phase (DSP) together with the impact dynamics which describe collide behavior of the limbs with the ground are presented. Then considering the dynamics of aSimultaneous design of optimal gait pattern and controller403DC geared motor used as an actuator in the joints, the dynamics of the biped robot system is derived. After that, by de?ning the controller structure, an optimization problem is described to ?nd all walking pattern parameters and controller coef?cients simultaneously. Here, the important indexes such as high stability, minimum control effort, suitable mobility properties, and actuator limitations are considered to ?nd the free parameters by the GA. It should be noted that the effect of parameters in robot motion are often competing and it seems that the proposed method can ?nd the best solution for the speci?ed ?tness functions.2 Dynamic modeling of bipedal robot It is dif?cult to apply the human’s gait pattern to the biped robot model because it has a complicated mechanical structure. Therefore, the human’s model for the gait analysis needs to be as simple as possible. From this point of view, physiologists often show that the most biped walking take place on the sagittal plane [16]. In this paper, a 5-link biped locomotion model, shown in Fig. 1, is constructed to approximate the human’s complex mechanical structure in the image sequences. This model consists of ?ve links: a torso and two legs together with two knees between the thighs and the shanks, and two ankles at the tips of the two legs. All of the joints can only rotate on the sagittal plane and have no friction. Feet are not considered in this model but torques are applied to the ankles through the actuators installed in these points [17]. The model parameters are given as follows: mi : li : di : Ii : Mass of link i. Length of link i. Distance from joint i to the mass center of link i. Moment of inertia of link i about the axis passing though the mass center of link i and perpendicular to the sagittal plane. θi : Angle of link i from vertical line. τi : Torque of joint i. g: Gravitational acceleration.Fig. 1 5-Link bipedal model404A.R. Ghiasi et al.(xb , yb ): Coordinate of the supporting point (B). (xe , ye ): Coordinate of the tip of swing limb (E). In the following, the walking dynamics of the robot are brie?y presented. Then by considering the actuator for the joints, a complete dynamic model for the biped robot will be presented. 2.1 Walking dynamics Biped walking is a periodic phenomenon. A complete step cycle is divided into single support phase (SSP) and double support phase (DSP) as shown in Fig. 2. The SSP is a state that one leg of the biped robot swings and the other leg is in contact with the ground. But, on the DSP, both feet are in contact with the ground and the main torque for the robot motion is produced by the leading ankle actuator. In this case, the rear ankle possesses a minimum torque and can rotate by the effect of gravity, inertia forces, and front ankle actuator torque. The time periods of the SSP and the DSP are denoted as TS and TD , respectively. It should be noted that in the successive step, the role of the swing limb and stance limb are exchanged. In this case, the physical link displacements and velocities do not actually change but the members of the biped links, which are used in the dynamic modeling, need relabeling. For the SSP, by using Lagrange equations, the governing equation can be derived as follows: ¨ B B D(θ )θ + H θ, θ θ + G(θ ) = τ (1)B where D(θ ) is the 5 × 5 positive de?nite and symmetric inertia matrix, H (θ, θ ) is the 5 × 5 B ¨ centrifugal and Coriolis matrix, and G(θ ) is the 5 × 1 matrix of gravity terms. θ, θ , θ , τ are the 5 × 1 vectors of generalized coordinates, velocities, accelerations, and torques, respectively. The detailed forms of the matrices D, H , and G in (1) are presented as follows: ? ?Dij = pij cos(θi ? θj ) ? B H = pij sin(θi ? θj )θj ? ij ? Gi = gi sin(θi )for i, j = 1, 2, . . . , 5(2)Fig. 2 Full gait cycle of bipedal robotSimultaneous design of optimal gait pattern and controller405where5gi = mi di g + ai ? ?Ii + mi di2 + ai ? ? ? pij = ai mj dj li + ai aj ? ? ? ? pj i ak = 1 0k=i+1mk li g5 k=i+1 mk(3) li2 li lj j =i j &i j &i (5) (4)5 k=j +1 mkwhere k = 3, k = 1, . . . , 5 where k = 3On the DSP, the contact positions of tips of two limbs are ?xed on the ground. So, there exists a set of holonomic constraints as follows: (θ ) = xe ? xb ? L = ye ? yb5 i=1 (ai li sin θi ) ? L 5 i=1 (ai li cos θi )= 0,i = 1, 2; j = 1, . . . , 5(6)where (xe , ye ) and (xb , yb ) are the coordinate of the trailing limb and leading limb, respectively, and L is the distance between tips of two limbs. Therefore, the dynamic equation on the DSP is given as the following form: ¨ B B D(θ )θ + H θ, θ θ + G(θ ) = J T (θ )λ + τ (7)where J T (θ )λ represents the constraint forces and λ is a 2 × 1 vector of Lagrange multipliers. Here, J (θ ) is the Jacobian matrix as follows: J (θ ) = ? /?θ = l1 cos θ1 ?l1 sin θ1 l2 cos θ2 ?l2 sin θ2 0 0 l4 cos θ4 ?l4 sin θ4 l5 cos θ4 ?l5 sin θ5 (8)Referring to Goldstein [18], in order to describe the constrained system without using the terms of constraint forces, a set of independent generalized coordinates p = (xh yh θ3 )T including the hip position (xh , yh ), and the trunk orientation θ3 can fully describe the motion of the 5-link biped model system in the DSP. So, by considering R = ?p ∈ R 3×5 , the biped ?θ dynamic in the DSP in terms of the independent generalized coordinates can be derived as follows [19, 20]: p=B ¨ B B B where N = H (θ, θ )θ + G(θ ), B = R p B + C(τ ? N ) 02×1R ?1 J(9)R ?1 JB ? RD ?1 J T (J D ?1 J T )?1 J?1andC = RD ?1 ? RD ?1 J T J D ?1 J TJ D ?1Also, Lagrange multipliers can be computed as follows: λ = ? J D ?1 J T?1B R J J?1p B + J D ?1 (τ ? N ) 0(10)406A.R. Ghiasi et al.In the robot motion, when the swing limb collides with the ground, the impact phenomenon happens in an in?nitesimal period of time. In this study, it is assumed that the impact is perfectly plastic and the joint velocities have sudden jump [21C25]. Generally, the impact event is speci?ed as a single impact or a double impact. Single impact happens between two consecutive SSP. During single impact, the stance limb does not receive external impulses from environment. But in the double impact that occurs between the SSP and the DSP, both lower limbs receive external impulse forces from environment. The equations of double impact, used in the current study, are derived from Newtonian impulse theory which uses the principal of the conservation in linear and angular impulse and momentum. Note that, during double impact, the velocities of both contact points immediately after impact are zero and the two contact tips should neither lift up nor slip on the ground. The governing equations are as follows [17, 21, 25]: ? B ? W JT θ 05×1 ? = B (11) J 02×2 ?J θ ? Pl ? ? B ? Pl Pr = V θ where Pl and Pr are the impulses at the contact point of trailing limb and leading limb, respectively, and W is a 5 × 5 lower-triangular, invertible matrix with the following details: ? ?Ii ? mi di (ai li ? di ) j =i ? ? i?1 Wij = ? ai mj dj li + aj mi (ai li ? di )Ij + ai aj k=j +1 mk li lj cos(θi ? θj ) j & i ? ? ? 0 j &i (12) Also, V ∈ R 2×5 is de?ned with the following details:5Vij = mj dj +k=j +1mk (aj lj )bi cos θj + (bi ? 1) sin θj ,i = 1, 2; j = 1, . . . , 5(13) B? B B? B? In the above equation, bi = 1 if i = 1 and bi = 0 if i = 2. Since Xe = [xe , ye ]T = J θ ? denotes the velocity of swing tip immediately before impact, the dynamic equations describing the impact process can be expressed from (11): ? ?1 ? B B ?θ + = ?W ?1 J T J W ?1 J T Xe + θ ? ?B ? B? (14) P = J W ?1 J T Xe ? l ? ? ?1 T ?1 T ?1 B ? Pr = ? I + V W J J W J Xe According to (14), the validity of solutions depends on the following conditions [17]: PlY & 0 |PlX | &μ |PlY | |PrX | &μ |PrY | (15) (16) (17)where μ is the friction coef?cient between the contact points and the ground. Condition (15) assumes the trailing limb remains on the ground during and after double impact, andSimultaneous design of optimal gait pattern and controller407the next conditions (16) and (17) represent no slip in the contact points of the limbs with the ground. It should be noted that, during biped locomotion, when the swing limb contacts with the ground, the role of the swing and the stance side members will be exchanged and this leads to discontinuity in the mathematical model like hybrid systems. The generalized coordinates and velocities immediately before and after the switch can be easily written by the appropriate transformation equations [26]. 2.2 Biped robot model with actuator dynamics One of the common actuators installed in the robot joints to deliver the required torque is a DC servomotor, a fractional-horsepower motor with a stationary magnetic ?eld that is generated by a permanent magnet (PM motor). In these kinds of motors, the ?eld is constant over a wide range of armature current. High stall torque, small frame size, lighter weight, and linear speed-torque curve are the main advantages of these motors. The block diagram of a DC geared motor which represents a third-order system from input voltage va to output position θm is shown in Fig. 3 [27]. In this ?gure, L, R, Ia indicate inductance, resistance, and current of the armature, respectively. Km , Kb , and r denote torque constant, back emf constant and gear ratio, respectively. Also, Jm is the rotor inertia and Bm is the coef?cient of motor friction and includes friction in the brushes and gears. Since electrical time constant (L/R) is much smaller than the mechanical time constant (Jm /Bm ), the reduced order dynamic for the actuator of the joint i is derived as follows [27]: ¨ B Jm θmi (t) + (Bm + Kb Km /R)θmi (t) = (Km /R)vai (t) ? τi (t)/r, i = 1, . . . , 5 (18)Note that the characteristics of all ?ve actuators installed in the joints are the same. Now, by considering θmi = rθi , the dynamic equations of the robot system with the actuators in the SSP and the DSP are derived as follows: ? ?D(θ )θ + H (θ, θ )θ + G(θ ) = τ ¨ B B for the SSP (19) ?rJ θ + r(B + K K /R)θ = (K /R)V ? τ/r B ¨ m m b m m a ? ?D(θ )θ + H (θ, θ )θ + G(θ ) = J T (θ )λ + τ ¨ B B for the DSP (20) ?rJ θ + r(B + K K /R)θ = (K /R)V ? τ/r B ¨ m m b m m a where Va ∈ R 5×1 is a 5 × 1 vector of voltages of ?ve actuators.Fig. 3 Block diagram for a DC motor system408A.R. Ghiasi et al.3 Optimal design of gait pattern and controller It is clear that the optimal motion of the biped robot depends on the proper choice of walking pattern and controller. In this way, the proposed walking pattern must be achievable by the designed controller in the presence of some limitations in control energy and other physical constraints. In the present study, the walking pattern and controller are linked together and designed simultaneously. In this respect, some physical constraints in the robot motion and actuators are considered. In the following, the main constraints of the gait pattern are ?rstly introduced. Then by de?ning the controller structure and actuator limitations, an optimal problem is described to simultaneously ?nd the free parameters of the gait pattern and controller based on the satisfaction of some control indexes and walking constrains. 3.1 Gait pattern constraints According to the biped kinematic model, the joint angle pro?les can be uniquely determined by de?ning the hip and swing limb tip displacements. In this way, some constraints about continuity and repeatability of the motion together with the geometrical constraints are considered. The gait pattern parameters have to be chosen in a way that these constraints are satis?ed. Due to the important role of the DSP in keeping biped stability during walking with a wide range of speeds, both the SSP and DSP are synthesized to the gait pattern. For the sagittal plane with the origin of the coordinate system located at the tip of the supporting limb shown in Fig. 2, each foot trajectory is denoted by a vector Xa = (xa (t), ya (t)) where the (xa (t), ya (t)) is the coordinate of the swing limb tip position. Also, the hip trajectory is denoted by a vector Xh = (xh (t), yh (t)) in which (xh (t), yh (t)) is the coordinate of the hip position. The equality constrains for the swing limb tip of the robot are considered as follows: ? Geometrical constraints: ya (0) = 0, ? Maximum clearance of the swing limb: ya (Tm ) = Hm , ya (Tm ) = 0, B xa (Tm ) = Sm (22) ya (TS ) = 0 (21)? Repeatability and continuity of the gait: xa (0) = ?SL /2, xa (0) = 0, B xa (TS ) = SL /2 xa (Ts ) = 0, B ya (Ts ) = 0 B (23)ya (0) = 0, BIn the above equations, SL is the step length and Ts is the step period for the SSP. Hm is the maximum clearance of the swing limb and Sm is the x-coordinate of the swing limb tip corresponding to the maximum clearance. Tm is the time instant when the tip of the swing limb reach maximum clearance. Note that, the coordinates of the swing limb tip can be de?ned by the polynomial functions where the coef?cients of these functions can be found from the constraints (21) to (23) [28]. Also, the equality constrains for the hip motion in both the SSP and the DSP are considered as:Simultaneous design of optimal gait pattern and controller409? Vertical hip motion in the SSP and the DSP: yhS (t) = Hh , ? Repeatability conditions: xhS (0) = ?SS0 , xhS (0) = Vh1, B ? Continuity conditions: xhS (TS ) = SD0 , xhD (TS ) = SD0 , xhS (TS ) = Vh2, B xhD (TS ) = Vh2 B (26) xhD (TS + TD ) = SL /2 ? SS0 xhD (TS + TD ) = Vh1 B (25) yhD (t) = Hh (24)where SS0 and SD0 are positions of the hip at the beginning of the SSP and the DSP separately, TD is the step period of the DSP, Hh is the height of the hip, Vh1 is the hip velocity at the beginning of each step and Vh2 is the hip velocity at the beginning of each DSP. Again, the hip position in the SSP and DSP are described as polynomial functions so that their coef?cients can be found from the constrains (24) to (26). Another important issue which should be considered in the gait pattern design is the stability of the robot motion. In this respect, the ZMP must be exists within the boundary of stable region which is the line between the two tips of the supporting limbs in the sagittal plane. From this point of view, the stability margin is de?ned as the distance between the ZMP and the boundary of the stable region. It should be noted that the stability of the bipedal robot can be achieved by adjusting the initial and ?nal velocities of the hip motion, Vh1 and Vh2. Thus, a set of Vh1 and Vh2 can determine the largest stability margin. 3.2 Controller structure A sliding mode controller is employed for the biped robot system because of its potential to cope with model nonlinearities and intrinsic robustness. By combining the dynamic models of the actuator and robot walking described in (19) and (20) for the SSP and the DSP, respectively, the system equations in each phase can be written in the following general form: ¨ B Dθ θ + Hθ θ + Gθ = Vθ (27)Also, the time variant sliding surface s for each phase of the walking is de?ned as s(θ, t) = 0, wherets = e + K D e + KP B0e(τ ) dτ(28)Here, KP and KD are proportional and derivative gains, respectively. Also, eis the tracking error. By de?ning ¨ B ¨ B u = θd ? KD e ? KP e = θ ? s the dynamic equation (27) can be expressed as s Dθ (B + u) + Hθ (s + v) + Gθ = Vθ B B where v = θ ? s = θd ? KD e ? KPt 0(29)(30)e(τ ) dτ . Equation (30) can be written as (31)B Dθ s + Hθ s = Vθ ? (Dθ u + Hθ v + Gθ )410A.R. Ghiasi et al.Now, using symbol (∧) for the nominal values of the matrices D, H , and G, the equivalent controller can be constructed as ? ? ? ? Vθ = Dθ u + Hθ v + Gθ (32)The smoothened controller with less chattering for each actuator is chosen as follows [29, 30]: ? Vθi = Vθi ? (ηSi + σi ) tanh(γi si )5 5(33)where ηSi ∈ R represents the system uncertainty bounds and σi ∈ R is constant vector used to satisfy the sliding condition. By applying the smoothened controller to remove the chattering problem, the control algorithm maintains the trajectories close to the surface within a thin boundary layer. In this way, the positive parameter γi guarantees that for any arbitrary thickness boundary layer neighboring the sliding surface, the trajectories will be bounded. The veri?cation is given in Appendix 2. The important issue, which should be considered in the controller design, is that the control inputs calculated from (33) should be compatible with the actuation system capabilities. In other words, the calculated control inputs must remain below the maximum admissible values determined by the DC motor characteristics. Maximum input voltage, maximum current of the armature, maximum output power, and torque delivered by the motor are the main limitations which should be considered in selecting the parameters of the controller and gait pattern. 3.3 Description of the optimization problem The optimization problem is described to ?nd the free parameters of the gait pattern and controller simultaneously. In this problem, the stable walking must be provided for a speci?c biped model and the equality constraints (21) to (26) along with the actuator limitations have to be satis?ed. Therefore, the following criteria should be considered in designing the biped robot: C C C C C C Minimum torque. Minimum tracking errors. Minimum power consumption. Minimum input voltage. Maximum velocity. Maximum stability margin.In the described problem, beside the controller coef?cients in the SSP and the DSP phases, different parameters of the walking pattern should be selected in an optimal way. These parameters are listed as follows: Initial and ?nal velocities of the hip motion, Vh1 and Vh2, step length, SL , time interval in the SSP and the DSP, TS and TD , maximum clearance of the swing limb, Hm , the x-coordinate of the swing limb tip corresponding to the maximum clearance, Sm , the time instant when the tip of the swing limb reach maximum clearance, Tm , positions of the hip at the beginning of the SSP and the DSP, SS0 and SD0 , and ?nally the height of the hip, Hh . Since there is no clear relation between the parameters and walking performances, a genetic algorithm (GA) as an iterative procedure is used to ?nd the optimal parameters automatically. The GAs are stochastic optimization techniques based on the mechanism of natural evolution and survival of the ?ttest found in biological organisms. These algorithms work onSimultaneous design of optimal gait pattern and controller Fig. 4 Flowchart of the optimization procedure411a population of solutions and produced suitable and a better solution using the principle of surviving the best solution. The major advantages of the GAs over traditional optimization methods can be enumerated as follows: 1. It does not require derivative information. 2. Simultaneously searches in a wide area of the cost surface. 3. Works with numerical data or analytical functions. The ?owchart of the optimization procedure is shown in Fig. 4. The following ?tness functions are considered to penalize the violation of the basic constraints concerning stability, energy consumption, mobility properties, and actuator limitations during a walking cycle. FStability =0 TS +TD 2 EZMP dt(34) (35)FMobility =T S + TD 1 = velocity SL 1 55 0 TS +TD 2 Eerror dtFTracking error = FTorque = FVoltage = FPower =(36)k=111 5 TS +TD 5 k=1 0 2 Eτ dt(37) (38) (39)11 5 TS +TD 5 k=1 0 2 EV dt11 5 TS +TD 5 k=1 0 2 EP dtIn the above equations, FStability guarantees the maximum stability margin. FMobility is the mobility performance function. If the value of the mobility function is small, the mobility performance will be better and the motion will be faster. FTracking error provides minimum tracking errors. Minimizing the values of FTorque , FVoltage , and FPower contributes to the satisfaction of actuator limitations. In this way, the minimum energy consumption is also provided.412A.R. Ghiasi et al.Note that, Eτ , EV , EP represent the differences of the torque, voltage, and power of the each actuator from their maximum values determined from the motor speci?cations. These differences will be maximized by minimizing FTorque , FVoltage , and FPower according to (37) to (39). Therefore, by applying that the values of the torque, voltage, and power cannot exceed their maximum values, maximizing their differences from the maximum values lead to the minimum values of the torque, voltage, and power. It should be noted that all ?tness functions are normalized and the total ?tness function that should be minimized is a simple summation of normalized objective functions.4 Simulation results To demonstrate the effectiveness of the proposed method, the walking of the ?ve-link biped robot is simulated in the sagittal plane. The physical parameters of the understudy model are listed in Table 1 [20]. Also, speci?cations of the DC geared motor used for actuating the system are presented in Table 2. In this study, instead of Binary-Coded Genetic Algorithm (BCGA), a Real-Coded Genetic Algorithm (RCGA) is used to solve the optimization problem because BCGA has many dif?culties in the practical applications [31]. Generally, the GA are composed of three operators including reproduction, crossover, and mutation. In this simulation, the population consists of 35 chromosomes. After evaluating the selected population, the associated chromosomes are ranked from lower cost to higher cost. Then by selecting 0.5 for natural selection rate, the 50% of the primary population is selected to continue the algorithm. The weighted random paring approach is used to mating. In this approach, a chromosome with the lowest cost has the greatest probability of mating and vice versa. Also, the offspring are generated by using single point crossover procedure. This method is inspired from HeuristicTable 1 Parameters of the biped robot Link Torso Thigh Leg Ii (kg m2 ) 3.3 × 10?2 mi (kg) 14.79 5.28 2.23 Li (m) 0.486 0.302 0.332 di (m) 0.282 0.236 0.1893.3 × 10?2 3.3 × 10?2Table 2 Parameters of a geared DC motor Maximum power (W) Maximum voltage (V) Maximum torque (mN m) Maximum current (A) Rotor inertia (g cm2 ) Mechanical time constant (ms) Back emf constant (V/krpm) Motor constant (mN m/A) Inductance (mH) Resistance ( ) Gear ratio 100 24 845 16.3 105 17 5.45 52 1.15 0.44 99.52Simultaneous design of optimal gait pattern and controller Table 3 Walking pattern parameters extracted from the GA Vh1 (m/s2 ) 0.2346 Vh2 (m/s2 ) 0.1445 SL (m) 0.6853 TS (s) 0.79 TD (s) 0.12 SS0 (m) 0.159 SD0 (m) 0.161 Hm (m) 0.036 Sm (m) 0.342 Tm (s) 0.39413Hh (m) 0.59Table 4 Controller parameters in the SSP extracted from the GA Joint KP ,SSP KD,SSP 1 782.7 2 641.1 3 543.2 4 926.4 5 721.1Table 5 Controller parameters in the DSP extracted from the GA Joint KP ,SSP KD,SSP 1 237.5 2 219.0 3 212.9 4 272.5 5 347.1crossover method [32, 33]. In order to increase the algorithm’s freedom to search outside the current region of variable space, also to increase the algorithm convergence speed, the mutation technique with the mutation rate 0.15 is used. For simulation studies, it is ?rstly assumed that there is no uncertainty in the biped model. In this case, the optimal walking pattern and controller parameters extracted from the GA are presented in Tables 3, 4, and 5. According to the obtained results, the optimal velocity of the robot is calculated as 0.7490 m/s. Figure 5 illustrates the variation of the best cost function values versus the generation number which indicates the algorithm convergence. Depending on the initial values of the parameters which are selected randomly, different computational times may be achieved. For a selected run with 1500 generation, the computational time is about 80 minutes. The complete results of the simulation are shown in Figs. 6 to 17 for one cycle of walking. Figures 6 and 7 show the optimal joint angles and angular velocities, respectively, during both the SSP and the DSP. Figures 8 and 9 show the tracking errors in the SSP and the DSP, respectively. From the results, it can be observed that the average tracking errors are very small and the error trajectories are converged to zero. Especially in the presence of initial errors caused by the impact in the DSP, the trajectories of the controlled robot are approached to the reference trajectories. The horizontal displacement of the biped gravity center, the hip motion and the ZMP during the DSP, are shown in Fig. 10. It can be seen that the gravity center, the hip, and especially the ZMP trajectories remain approximately at the center of the stable region. This indicates that the robot walking is stable. Also, Fig. 11 shows the stick diagram of the biped walking for one cycle of motion. This illustrates the credibility of the results and the ef?ciency of the proposed scheme in walking the biped robot. Figures 12 and 13 show the optimal input voltages of the actuators. Also, Figs. 14 and 15 show the optimal powers consumed by the actuators in the SSP and the DSP. Simulation results indicate that the calculated voltages and powers remain below their maximum values414 Fig. 5 Convergence diagram of the GAA.R. Ghiasi et al.Fig. 6 Optimal joint angles in SSP and DSP for system without parametric uncertaintyFig. 7 Optimal joint angular velocities in SSP and DSP for system without parametric uncertaintydepicted in Table 2. The required torques delivered to the robot by the actuators in the SSP and the DSP are presented in Figs. 16 and 17.Simultaneous design of optimal gait pattern and controller Fig. 8 Joint angle errors in SSP for system without parametric uncertainty415Fig. 9 Joint angle errors in DSP for system without parametric uncertaintyFig. 10 Horizontal displacement for Hip, central gravity and ZMP in DSP phaseIn order to evaluate the robustness of the controlled system, 15% uncertainty for the mass and inertia moment of the links are considered. Figures 18 to 22 compare the tracking errors416 Fig. 11 Stick diagram for system without parametric uncertaintyA.R. Ghiasi et al.Fig. 12 Optimal actuator voltages in SSP for system without parametric uncertaintyFig. 13 Optimal actuator voltages in DSP for system without parametric uncertaintyin the DSP for the system with and without uncertainty. From the simulation results, it can be observed that the average tracking errors are still very small in the presence of uncertainty and the error trajectories are converged to zero. In this case, Fig. 23 shows the horizontal displacement of the gravity center of the biped, the hip motion, and the ZMP trajectories during the DSP. Figures 24 and 25 show the optimal voltages of the actuators. Also, Figs. 26Simultaneous design of optimal gait pattern and controller Fig. 14 Optimal actuator powers in SSP for system without parametric uncertainty417Fig. 15 Optimal actuator powers in DSP for system without parametric uncertaintyFig. 16 Optimal joint torques in SSP for system without parametric uncertaintyand 27 show the optimal powers consumed by actuators in the SSP and the DSP separately. Simulation results indicate that voltages and powers are increased due to the uncertainty but their values are still reasonable. The performance indexes of the system with and without uncertainty are compared in Tables 6 and 7 separately. In these tables, the joint torques extracted from the actuators are418 Fig. 17 Optimal joint torques in DSP for system without parametric uncertaintyA.R. Ghiasi et al.Fig. 18 Tracking error of joint 1 in DSP for system with & without uncertaintyFig. 19 Tracking error of joint 2 in DSP for system with & without uncertaintyconsidered as the system input. The results indicate an increase about 35.8% for the total energy consumption of the system owing to the uncertainty. Also, Table 7 indicates that when the parameter uncertainty exists in the model, the average tracking errors are slightlySimultaneous design of optimal gait pattern and controller Fig. 20 Tracking error of joint 3 in DSP for system with & without uncertainty419Fig. 21 Tracking error of joint 4 in DSP for system with & without uncertaintyFig. 22 Tracking error of joint 5 in DSP for system with & without uncertaintylarger than those without uncertainty, but they are still low and the controller is realizable. The comparison of stability margins for two cases show that the average of the ZMP error,420 Fig. 23 Horizontal displacement for Hip, central gravity, and ZMP in DSP for system with uncertaintyA.R. Ghiasi et al.Fig. 24 Optimal actuator voltages in SSP for system with parametric uncertaintyFig. 25 Optimal actuator voltages in DSP for system with parametric uncertaintymeasured from the center of the stable region in the DSP, is very small and the stability margin is large. Thus, the robot walking with uncertainties is still stable.Simultaneous design of optimal gait pattern and controller Fig. 26 Optimal actuator powers in SSP for system with parametric uncertainty421Fig. 27 Optimal actuator powers in DSP for system with parametric uncertaintyTable 6 Comparison of energy consumption of the system with and without parametric uncertaintyTs 2 5 i=1 0 τi dt TD +Ts 2 5 τi dt i=1 TsTotal Energy consumption in SSP and DSPSystem without uncertainties System with 15% uncertainties1..9095E+062..5084E+051.5.16034E+06Table 7 Comparison of tracking errors of the system with and without parametric uncertaintyTs 2 5 i=1 0 Ei dt TD +Ts 2 5 Ei dt i=1 TsTotal Error in SSP and DSPMean value of ZMP error 0.015 0.016System without uncertainties System with 15% uncertainties8..9042E?062..1976E?042.2.28664E?04In the next simulation, beside the previous parametric uncertainty, a constant disturbance with the amount of 1.8 volt is assumed to be applied to the input voltages after a third of the SSP time period. Figures 28 and 29 show joint angle errors and consumed powers in422 Fig. 28 Joint angle errors in SSP for system encountered with uncertainty and disturbanceA.R. Ghiasi et al.Fig. 29 Consumed power in SSP for system encountered with uncertainty and disturbancethe SSP. Simulation results indicate that the designed controller can successfully handle this internal disturbance. To show how the system reacts to an external disturbance, a torque with the amount of 5 N m is assumed to be applied to the torso joint in the middle of the SSP time period. As it is seen in Fig. 30, some tracking errors are found immediately after the disturbance, but they converge to zero rapidly. It is considered that the biped controller is able to cope with the eternally generated disturbances. In another analysis, the outcome of the proposed optimal method is compared with that of nonoptimal case considering 5% change in the optimal speed of the robot which was calculated as 0.7490 m/s before. Figures 31 to 35 show the joint torques of the robot in three different speeds during SSP. Also, the energy consumptions of the system for three cases are compared in Table 8. The results indicate that when the robot speed is taken as 0.7115 m/s or 0.7864 (5% less or greater than the optimal speed), the energy consumption is increased. This test shows the credibility of the optimality of the design procedure. It should be noted that all ?tness functions described in (34)C(39) are simply summarized to obtain the total ?tness function. In other words, all ?tness functions are equally weighted to design the gait pattern and controller. In general, these functions can be weighted differently. In order to evaluate the effects of the weights in total ?tness function, the weights of power, voltage, and torque are considered to be zero and another gait pattern is designed. Figure 36 shows horizontal displacement of the hip, gravity center, and the ZMP for the newSimultaneous design of optimal gait pattern and controller Fig. 30 Joint angle errors in SSP for the system encountered with an external disturbance in torso torque423Fig. 31 Comparison of the joint torque 1 in different velocitiesFig. 32 Comparison of the joint torque 2 in different velocitiesproblem. As it is seen, the stability margin is increased in this case. However, the actuator powers are increased as shown in Fig. 37.424 Fig. 33 Comparison of the joint torque 3 in different velocitiesA.R. Ghiasi et al.Fig. 34 Comparison of the joint torque 4 in different velocitiesFig. 35 Comparison of the joint torque 5 in different velocitiesIn order to compare the proposed optimal scheme with the previous works, which describe the gait pattern and controller separately, the controller designed by the feedback linearization method is employed to follow the proposed gait pattern [26]. This control scheme is referred to as second scheme. Note that both the ?rst scheme, proposed in the presentSimultaneous design of optimal gait pattern and controller425Table 8 Comparison of the energy consumption of the system without uncertainty for three different speeds Velocity (m/s) Optimal speed (0.7490) Non optimal speed (0.7115) Non optimal speed (0.7864) Total Energy consumption in SSP and DSP (TD +Ts 2 5 τi dt) i=1 01.5.7.93291E+06Fig. 36 Horizontal displacement for Hip, CG, and ZMP in DSP for not equally weighted ?tness functionFig. 37 Actuator powers in DSP for not equally weighted ?tness functionstudy, and the second scheme follow the same gait pattern, but their design processes are different. For the ?rst scheme, the gait pattern and the sliding controller are designed simultaneously but for the second scheme the feedback linearization controller in employed to track the given gait pattern. Table 9 compares the results of tracking errors and energy consumptions achieved by the two control schemes for the robot without uncertainty. As it is seen, the total energy consumptions and tracking errors of the second scheme are higher than those of the ?rst scheme proposed in this study. It can be concluded that the desired walking pattern and controller should be described simultaneously according to the motion conditions and control system abilities.426A.R. Ghiasi et al.Table 9 Comparison of performance indexes of the system without uncertainty extracted by two different control schemes Total Energy consumption in SSP and DSP First scheme proposed in this study Second scheme 1.5.0756E+06 Total Error in SSP and DSP 2.2.3824E?045 Conclusion This work contributes to design an optimal gait pattern and controller simultaneously for the speci?ed biped robot in the presence of some model uncertainties. In this study, against other most researches, a complete dynamic model including a complete gait cycle is used to describe a biped walking pattern in the sagittal plane. Then a nonlinear controller with the sliding mode structure is considered to track the walking pattern. A genetic algorithm (GA) is employed to determine the coef?cients of optimal walking pattern and controller simultaneously. Here, high stability, minimum energy consumption and good mobility properties are considered as important indexes in optimization. It is concluded that the simultaneous design of optimal gait pattern and controller for a bipedal robot leads to the better control performances compared with those schemes in which the gait pattern and controller are designed separately. In this scheme, the walking pattern can be achievable by the designed controller in the presence of some limitations of actuators and other motion conditions. Simulation results indicate the ef?ciency of the proposed optimal scheme in walking the robot.Appendix 1: Zero moment point de?nition The zero moment point (ZMP) is de?ned as the point on the ground about which the sum of all the moments of the active forces equals zero. If the ZMP exists inside the support polygon formed by the support points between the feet and the ground, a biped robot can walk stable without falling down [34, 35]. For a stable dynamic motion, the robot ZMP has to be in the convex region of all contact points between the robot feet and the ground. This convex region is also called the stable region. In three-dimensional spaces, the ZMP can be computed using the following equations [12]: xZMP = yZMP =n ¨ i=1 mi xi zi n mi (¨i + g) z i=1 n mi (¨i + g)yi ? n mi yi zi z ¨ i=1 i=1 n z i=1 mi (¨ i + g) n z i=1 mi (¨ i+ g)xi ?? ?n ¨ i=1 Iiy Ωiy n ¨ i=1 Iix Ωix(40)In the above equations, mi is the mass of link i, Iix and Iiy are the inertial components, ¨ ¨ Ωix and Ωiy are the absolute angular acceleration components around x-axis and y-axis at the gravity center of the link i, g is the gravitational acceleration, (xZMP , yZMP , 0) is the coordinates of the ZMP, and (xi , yi , zi ) is the coordinates of the mass center of link i on an absolute Cartesian coordinate system. It is clear that if the ZMP is designed near the center of the stable region, the stable margin can be large.Simultaneous design of optimal gait pattern and controller427Appendix 2: Stability analysis for the sliding mode controller If the quadratic Lyapunov function is considered as V = 0.5s T Dθ s & 0 for each phase, the derivation of the V is B B B B B V = 0.5s T Dθ s + s T Dθ s = 0.5s T Dθ s ? s T Hθ s + s T (Dθ s + Hθ s) Using (29) to (32), (41) can be written as B B ¨ B V = 0.5s T Dθ ? 2Hθ s + s T Dθ θ ? u + Hθ θ ? v = 0 + s T Vθ ? (Dθ u + Hθ v + Gθ ) = s Ts(41)? tanh(γ s)(ηs + σ )(42)? ? ? B where s = (Dθ ? Dθ )u + (Hθ ? Hθ )v + (Gθ ? Gθ ). Note that the term 0.5s T (Dθ ? 2Hθ )s B B is zero because (Dθ ? 2Hθ ) is a skew-symmetric matrix [36]. In order to guarantee V & 0 for all |si | ≥ εi (i = 1, . . . , 5), the following condition must be satis?ed: tanh γi |si | & where μi is upper error changing ratio. Since tanh γi |si | = Thus, eγi |si | ? e?γi |si | & μi eγi |si | + e?γi |si | which leads 1 ? μi & e?2γi |si | 1 + μi Or ln 1 ? μi 1 + μi & ?2γi |si | (47) (46) (45) eγi |si | ? e?γi |si | eγi |si | + e?γi |si | (44)siηsi + σi= μi(43)The above equation can be written as follows: γi & 1 1 ? μi ln 2|si | 1 + μi (48)So, for all |si | ≥ εi , the following inequality is held γi & 1 1 ? μi ln 2εi 1 + μi (49)Giving the width of the boundary layer εi , the coef?cients γi can be found from (49). As σi is a positive constant, ?1 + ? & μi & 1 ? ? , 0 & ? & 1, and thus 0 & ln( 1?μii ) & 1+μ +∞. This means that for any positive number εi denoting the width of the boundary layer, a positive γi can be always found. Thus, the continuous sliding controller is realizable for tracking the trajectory with the boundary layer width of ε.428A.R. Ghiasi et al.References1. Huang, Q., Yokoi, K., Kajita, S., Kaneko, K., Arai, H., Koyachi, N., Tanie, K.: Planning walking patterns for a biped robot. IEEE Trans. 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