تجزیه و تحلیل حساسیت از کنترل کننده تلفن همراه چرخ دار غیرهولونومیک انعطاف پذیر مشترک در پیکربندی منحصر به فرد

This paper proposes a method for decreasing jerk and increasing Maximum Allowable Load (MAL) of nonholonomic Wheeled Mobile Manipulator (WMM) considering flexibility of joints in singular conditions. The full dynamic model of nonholonomic WMM contains simultaneous operation of mobile base and manipulator with joint flexibility (in wheels and manipulator) which is presented here. The problem is formulated in terms of the optimal control which leads to a two point boundary value problem. Then Sobol's sensitivity analysis method is applied to determine the optimal values of flexible joint constants subject to the jerk minimization. To illustrate the proposed method, two categories of conditions are considered: conditions containing non-singular configuration and the singular conditions. An example is explained for non-singular condition of nonholonomic WMM in presence of obstacle in which a complex path is generated but there is no singularity in robot configuration. Some examples of occurring singular configuration in final point and moving boundary condition is also presented. The results show that flexibility of the joints near to singular configuration normalizes the sudden movement and jerk implied to actuators. That is why using a rotational spring with a low stiffness coefficient could be helpful to decrease the high jerk and increase the maximum allowable load in mobile robots.

Carrying the maximum allowable load according to bounded energy and actuator limitation in terms of torque and jerk in mobile manipulators is a vital factor, but nearing to singular condition must be avoided because of imposing high amounts of jerk to actuators. On the other hand, the robot performance changes in flexible joint robots. In addition, this property causes some differences in determination of maximum allowable load compared to a rigid model. For example, considering flexibility in joints will cause vibration, especially on fast turnings and in an environment with an obstacle, in which to prevent colliding, robot must move in complex paths. Determining maximum allowable load for a nonholonomic mobile robot with flexible joints in manipulator and wheels of the mobile base has applications in advanced trajectory planning, design and identification of motors size. Previous works in this area mostly considered the accuracy and deviation of the end effector [1]. The approaches used to solve the open loop optimal control problems are classified in direct and indirect methods. Direct methods are based on the conversion of the optimal control problem into a parameter optimization problem [2]. In a series of papers, some research effort has been paid to establish the maximum load carrying capacity of linear moving of rigid base holonomic manipulators by direct methods. Although the kinematic analysis of WMM has been considered a lot but the dynamic analysis is few [1]. Recently, MAL for a holonomic flexible arm mounted on a track with linear moving is dealt with, [3] but flexibility in wheels' joints was not considered. In addition, it has to be mentioned that the word “mobile” was referred to a linearly moving holonomic track. Important parameters which must be considered in path planning problem are torque and jerk, but none of the previous works considered the effect of joint flexibility on jerk. Although torque consideration in the most previous work has been done, but jerk limitation which has a considerable effect on deviation from desired path [4] and decreasing the actuator lifetime, is not considered in determination of MAL. Engelbrecht studied the minimum principles in path planning like the jerk but for complex systems such as WMM, it results in very large volume of equations [5]. Macfarlane and Croft proposed an algorithm for finding jerk bounded trajectories [4]. Mattmüller and Gisler determined smooth and near to time-optimal path-constrained trajectories, in which not only the velocity and the acceleration but also the jerk are explicitly constrained [6]. Beside, Zhu et al. proposed time-optimal and jerk-continuous trajectory planning algorithm to provide ideal trajectories for joint controller with the minimum traveling time [7]. Ider proposed a jerk bounded path equation for flexible joint fix robots but they did not study the WMMs and singular condition [8]. On the other hand, some researchers have focused on singularity for rigid joint WMM. Bayle discussed the manipulability ellipsoid for a rigid mobile manipulator regarding to internal configuration of mobile base [9]. Kim et al. studied the configuration dependent singularity in WMMs in geometrical viewpoint [10]. Korayem et al. described a path planning technique for obtaining the MAL of rigid joint nonholonomic wheeled mobile manipulator in presence of obstacles [11]. However no work has been reported on the effect of joint-flexibility on jerk and MAL in singular configuration. The subject of this paper is analysis of jerk and the MAL determination of nonholonomic mobile manipulator with joint flexibility in different environmental condition and configuration including the presence of obstacle in which WMM must track a complex path but there is no singularity in the configuration, obstacle free motion with singular configuration of the manipulator and moving end boundary condition in which optimality results in minimum motion of mobile base, extended configuration of manipulator and nearing to the singularity. The problem is stated as an optimal control problem and applying iterative algorithm to increase the payload, according to physical constraints and actuator limitations (torque and jerk). This paper is structured as follows. The configuration of the mobile manipulator with flexible joints (in wheels and manipulator) and its full dynamic model based on coordinated motion of vehicle and manipulator are described in Section 2. In Section 3, the algorithm and Sobol's sensitivity analysis method are introduced. In Section 4 the sensitivity analysis of jerk and following simulation results for various conditions of flexible and rigid joint (in both non-singular condition and singular condition) is described, Section 5 deals with the effect of joint flexibility on jerk and MAL in moving boundary problems. Finally the discussion is done for decreasing the jerk.

The effect of joint flexibility (in both wheels and manipulator) on maximum allowable load carrying of WMM in new viewpoint based on bounded torque and jerk is analyzed. Moreover benefits and problems caused by it are studied. Sobol's sensitivity analysis method is applied to determine the optimal values for stiffness coefficient of springs subject to the jerk minimization of the mobile manipulator. Comparing rigid and flexible joints of WMM in some conditions (free motion and moving in an environment with an obstacle) is done. Joint flexibility shows different behavior in path planning of WMMs. In singular configuration of mounted manipulator, it plays an important role to avoid imposing high jerk on the actuator of the wheels and manipulator since in this condition, actuators must impose sudden torque which results in high jerk. But existence of flexibility causes soft motion in singular conditions. In other word, the joint flexibility reduces the high jerk and this property increases the lifetime of the actuator and the maximum allowable load. The simulations show the following results: 1. High stiffness coefficient (such as k=50 Nm/rad and higher) causes observable increase in the amplitude envelope of the jerk by increasing frequency, but for a low value stiffness coefficient such as k=5 Nm/rad, there is no significant difference in maximum amount of the amplitude envelope in the jerk diagrams with rigid joint WMM, because of creating soft motion in the low value stiffness coefficient of flexible joint. 2. The configuration near to singularity produces high amount of jerk on the actuator in rigid joints WMM, but joint flexibility with low stiffness coefficient such as k=5 Nm/rad, prevents high jerk and normalizes it. Therefore flexible joint robot can carry a higher load near to singularity. 3. In moving boundary condition, the algorithm of finding MAL, results in minimum movement of mobile base and extended configuration of manipulator, which is near to singularity, but joint flexibility normalizes the jerk and so it provides higher load capacity and this is one of the advantages of flexibility in the joints. Furthermore, the comparison of flexible joint WMM with rigid joint in moving boundary condition is done and results showed that the flexible joint can carry a higher load than rigid joint one. According to the cost function, fewer amounts of torque and velocity, causes minimum movement of mobile base since it is heavier than manipulator. It causes the manipulator to open configuration and singularity, which results in high jerk in rigid joint robots, while flexibility reduces the high jerk and increases the life time of motor in all singularities. Hence for preventing high jerk and increasing MAL, in these configurations a low value stiffness coefficient of the spring in the joints can be helpful.