Optimal torque distribution for a redundant 3-RRR spherical parallel manipulator used as a haptic medical device
Introduction
Haptic devices are systems used to apply forces and/or torques to a user’s hand. These forces and torques are given by a virtual environment or by a force and torque sensor placed on another station. In recent years, many haptic devices have been proposed for different applications, such as gaming [1], medicine [2], [3] and virtual reality [4], [5]. Most haptic devices have a parallel structure because of their high stiffness, their load capability and their low weight.
Many haptic devices have a parallel spherical architecture. Spherical parallel manipulators (SPMs) are one class of parallel manipulator. These are mechanisms with three degrees of freedom of pure rotation (3-RRR). Gosselin and co-workers have proposed a haptic device for controlling the orientation of a camera [6], [7], and two haptic devices have been proposed for use in minimally invasive surgery (MIS) [8], [9].
The geometric parameters of the haptic device for MIS described in Ref. [9] were optimized to reach a prescribed workspace and to increase dexterity. In that paper, self-rotation was not taken into account during the optimization process. Consequently, parallel singularities located at the center of the workspace were not detected. This leads to unsatisfactory behavior of the structure, with errors being generated in the kinematic transformation. To solve this problem, we have proposed the addition of a redundant sensor [10]. The use of this sensor, first, eliminates the effect of parallel singularities on the forward kinematic model and, second, reduces the complexity of the model and allows it to work in real time. The model becomes more accurate close to the singular region.
Parallel singularities also amplify the required joint torques to values that exceed the limits of the actuators. To cope with this issue, we have proposed the use of a redundant actuator as well as a specific control scheme [11]. From the literature, the following three types of redundancy can be distinguished:
1. Kinematic redundancy: the number of legs exceeds the required number of degrees of freedom. This type of redundancy is used to increase the volume of the workspace [12], [13].
2. Actuation redundancy: the number of actuators exceeds the required number of degrees of freedom. This type of redundancy is mostly used to eliminate singularity issues [11], [14], [15].
3. Measurement redundancy: the number of sensors exceeds the required number of degrees of freedom. This type of redundancy is usually used to solve the forward kinematic problem as well as to reduce positioning errors when the robot is in a singular configuration [10], [16], [17].
In general, in most work in this area, an extra leg is added to the parallel mechanisms and the redundant actuator is placed in the base [18], [19], [20], [21]. In our previous work [11], we chose to replace one passive joint by an active one. In this case, a minor mechanical design change is needed. The redundant actuator is activated only in singular configurations, and it replaces the actuator in the same leg. This control strategy solves the singularity problem, but it generates discontinuities in the actuated joint torques. In this paper, an approach to an optimal torque distribution is proposed. This approach eliminates the effects of parallel singularities for torque control and guarantees the continuity of the actuated joint torques.
Most previous work [22], [23], [24] has been based on the null-space method to optimize the torque distribution. The method proposed in this paper is different. It is based on finding an optimal angle that gives a particular solution of the Jacobian matrix that leads to an optimal torque distribution.
The paper is organized as follows. In Section 2, the kinematic model of the SPM is presented. In Section 3, the kinematic model of the redundant SPM is developed. This model is validated in Section 4 using a SimMechanics model. In Section 5, an optimal distribution of joint torques is proposed and described. A validation of this approach using both a reference trajectory and reference torques is presented in Section 6. In Section 7, an experimental validation is carried out. Finally, Section 8 provides a summary.
Section snippets
Kinematic model of the SPM
The SPM is a robot with three identical legs. Each leg is made of two links and three revolute joints (Fig. 1). All axes of the revolute joints intersect at one point, called the center of rotations (CoR). Each link is characterized by the angle between its two revolute joints (Fig. 2). This angle is constant and represents the dimension of the link. The angle between the first two joint axes is denoted by and that between the last two joint axes by . The relative orientation of the axis
Redundant SPM kinematics
The redundancy is proposed to eliminate the effects of the parallel singularity on the force control of the haptic device. For a parallel manipulator, redundancy is usually introduced by adding a branch that connects the effector to the base. An actuator is then added to the additional branch. Our approach is different. It is based on the replacement of one passive joint of one leg of the SPM by an active joint through the addition of an actuator. In this case, there are an infinite number of
Validation of the kinematic model of the redundant SPM
In this section, the model of the redundant SPM developed above is validated using a SimMechanics model based on rigid-body dynamics according to the scenario presented in Fig. 8. The SPM is assumed to be a rigid body; therefore, deformations are not considered in this study. A function that calculates the actuated joint torques is constructed (Fig. 9). The two inputs to this function are the reference torque (which is a torque vector applied by the haptic device) and the coupling angle .
Optimal actuated joint torques for a given configuration and reference torques
Optimization is applied in many areas such as path planning [25], [26], parallel robot synthesis [9], kinematics performances [27]. In this paper, we look to optimize the torque distribution.
To choose the optimal actuated joint torques, we propose to minimize the quadratic sum of the actuator torques. The optimization is subject to constraints, which are the maximum torques given by the actuators placed on the haptic device. The three actuators placed on the base have a maximum torque of about
Optimal actuated joint torques for a given trajectory and reference torques
The motion of the haptic device is given by the trajectory of the user’s hand, and the reference torques are those applied by the haptic device to the user’s hand. The trajectory of the mobile platform is obtained experimentally by using the haptic device. It corresponds to a task in the workspace, when the SPM crosses singular configurations. The trajectory and the reference torques are shown in Fig. 18, Fig. 19, respectively.
The approach to determination of the optimal actuated joint torques
Experimental validation
An experimental test is carried out to validate the optimal force control approach. For this, the SPM is equipped with a redundant actuator placed on the joint with axis (Fig. 24). Then, a six-axes force/torque sensor is placed on the moving platform (Fig. 24) to measure the force applied to the user’s hand. The measured force is compared with the desired force applied by the actuators. An illustration of the experimental setup is shown in Fig. 25.
The SPM is placed in a singular
Conclusions
An optimal joint torque distribution approach for a redundant SPM has been investigated. The SPM is used as a haptic device to control a surgical robot. The workspace of the SPM contains singular configurations. In these configurations, the required joint torques exceed the actuator limits. Actuator redundancy has been used here to, first, eliminate the effect of parallel singularity and, second, minimize the actuated joint torques. A redundant actuator has been added to one joint of the SPM. A
Acknowledgments
This research is supported by the Poitou-Charentes Region 2007–2013 (Program Project 10 Images and Interactivities), in partnership with the European Union (FEDER/ERDF, European Regional Development Fund).
This research is supported by ROBOTEX, the French national network of robotics platforms (No. ANR-10-EQPX-44-01).
Houssem Saafi received his Ph.D. in Mechanics from University of Poitiers in 2015. He is currently a researcher with the Pprime Institute, Department GMSC. His current research interests include Robots design, Mechanism synthesis, Forward and Inverse Kinematics, Singularity, Optimal design, Haptic control and Redundant parallel robots.
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Houssem Saafi received his Ph.D. in Mechanics from University of Poitiers in 2015. He is currently a researcher with the Pprime Institute, Department GMSC. His current research interests include Robots design, Mechanism synthesis, Forward and Inverse Kinematics, Singularity, Optimal design, Haptic control and Redundant parallel robots.
Med Amine Laribi is an Associate Professor in the Fundamental an Applied Sciences Faculty of the Universit of Poitiers (UP), where he teaches robotics and mechanic. He has a Mechanical Engineer Degree (specialization on Mechanical Design) from École Nationale d’Ingénieurs de Monsatir (E.N.I.M.) in 2001. M.S. in Mechanical Design, 2002. He received his Ph.D. in Mechanics from University of Poitiers in 2005. His research interests include robots design and mechanism synthesis. His main research area in mechanism theory focuses on Forward and Inverse Kinematics, Singular configurations, Workspace determination and Optimal design.
Saïd Zeghloul received the Ph.D. degree in mechanics and the Doctorés Sciences degree in physics from the University of Poitiers, Poitiers, France, in 1983 and 1991, respectively. He is currently a Professor with the Faculty of Fundamental and Applied Sciences, University of Poitiers, where he teaches robotics and mechanics. His current research interests include robot design, computer-aided design for robotics, and object manipulation with mechanical hands and robots collision free path-planning.