Torque-balancing algorithm for the redundantly actuated parallel mechanism
Introduction
Actuator redundancy has been employed to improve the performance of parallel mechanisms (PMs) in terms of kinematic characteristics, maximum payload, singularity avoidance, and fault tolerance [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. However, when the maximum (or minimum) torque limits of the actuators are exceeded, severe distortions of the desired output force/torque of the PM could occur, because some of the actual joint torque outputs are cutoff by the limit of the maximum (or minimum) joint torque. The capacity of the output force/torque of the redundantly actuated PM can also be limited when the actuator saturations are not avoided by distributing the torque appropriately.
A number of actuator redundancy resolution algorithms have been suggested, which can be employed to resolve the joint torque saturation-related problems. Such torque optimization methods as the minimum-norm (MN) torque method [2], [3], [11], the weighted MN (WMN) torque method [12], and the task-priority (TP) method [13] are typical methods belonging to them. However, the MN torque method becomes ineffective when more than one joint torque exceeds its joint torque limit, because it does not consider the joint torque limits of actuators. The WMN torque method may not be effective in that selecting the optimal weighting factors for the joint saturation conditions in real time is difficult. The TP methods suggested to date cannot effectively treat the joint torque saturation-related problems because they cannot include proper equality constraints as secondary constraints [14]. A numerical method, such as quadratic programming (QP) [15], [16], [17], [18], can be employed to solve the joint torque saturation-related problems, because it provides the best approximate solution to the given performance index. However, employing the QP method in real-time operations may be inadequate because of its heavy computation requirement. Thus, practical but efficient actuator redundancy resolution methods for handling the joint torque limit need to be studied.
In this work, a practical redundant actuation method is suggested, which can effectively avoid situations in which the maximum (or minimum) torque limits of the actuators are exceeded. The suggested method is tested and compared with other existing methods through simulations and actual experiments.
The remainder of the paper is structured as follows. Section 2 describes the actuator redundancy resolution problem subject to the joint torque saturations. Some of typical preexisting actuator redundancy resolution methods, which can be applied to the joint torque saturation-related problems, are then discussed. Section 3 introduces a torque-balancing (TB) actuator redundancy resolution method. Section 4 presents a modified planar 3-DOF PM with one redundant actuator, along with its kinematic model. A comparative simulation study of the suggested TB method with other preexisting redundancy resolution methods is conducted by implementing them on the modified planar 3-DOF PM. Section 5 indicates the comparative experiments of the suggested method with other methods through the prototype of the modified planar 3-DOF PM and summarizes their results. Finally, Section 6 concludes this study.
Section snippets
Actuator redundancy resolution problem subject to joint torque saturations
In this section, the actuator redundancy resolution problem subject to joint torque saturations is addressed. Typical strategies for resolving actuator redundancy, such as the MN method, the WMN method, the QP method, and the TP-based solution, are briefly discussed, along with their applicability to the actuator redundancy resolution problem subject to the actuator saturations.
Key concepts on the suggested TB method
In this section, an actuator redundancy torque resolution method that employs only one additional equality constraint is suggested. Its objective is to yield a uniform joint torque solution that is effective in avoiding actuator saturations by distributing the joint torque as uniformly as possible. Its key idea can be explained as follows.
For the PM with one redundant actuator, the column vectors of the Jacobian matrix in (2) and joint torques are rearranged, without loss of generality,
Comparative simulations on redundancy-resolving methods through a planar 3-DOF PM
In this section, a modified planar 3-DOF PM with one redundant actuator is briefly described, along with its screw theory-based velocity analysis. A comparative simulation study of the suggested TB method with other preexisting actuator redundancy resolution methods is conducted through the modified planar 3-DOF PM.
Experiments of the prototype with one redundant actuation
Fig. 7 shows the prototype of the planar 3-DOF PM with one redundant actuation. This prototype is implemented based on the optimal design parameters identified from the optimal design of considering the workspace size and the kinematic isotropy characteristics [9]. Its size is 360 × 360 × 160 mm, The radius of the base and the length of the moving link (or l33) are 160 mm and 48 mm, respectively. The other link lengths are , (for ), , and . Maxon
Conclusions
An actuator redundancy resolution method is suggested, which shows a promising performance of avoiding joint torque limit effectively. The suggested method modifies the actuator redundancy problem for the PM with one redundant actuator subject to actuator saturations into the problem of finding the intersections of linear lines representing the joint torque solutions. A planar 3-DOF PM with one actuator redundancy is selected in verifying the effectiveness of the suggested method. Comparative
Acknowledgments
This work was supported by the Korea University Research Fund.
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