Sliding Mode Variable Impedance Control for 4RRR Redundantly Actuated Parallel Robot

 Abstract: The workspace of parallel robot is limited, and the distribution characteristics of kinematics and force maneuverability are complex in the workspace. In addition, there are often operational singularities. The redundant actuation technology can not only overcome singularity problem of force maneuverability of parallel mechanism, but also eliminate the clearance of the mechanism, dynamically adjust the operating stiffness, realize the optimized operation of energy consumption, etc. In this paper, with the goal of realizing dexterous dynamic output control, the variable impedance control technology of redundantly actuated parallel robot based on operation space is studied. Compared with the traditional invariable impedance control, the variable impedance control can adapt to the change of unknown environment by time-varying controlling parameters, which provides the possibility to realize the polishing of complex curved surfaces and thin-walled parts, assembly task and improves the performance of the system and the machining quality of the parts. In order to avoid the influence of unmodeled errors and external interferences on the system, based on the sliding mode control (SMC) strategy, the variable impedance control (VIC) in the operating space of 4RRR redundantly actuated parallel robot (4RAPR) is proposed. The robust stability of the closed-loop system is analyzed and verified by Matlab programming simulation.


Introduction
Parallel robot is a closed-loop mechanism [1] that connects moving platform with static platform through at least two independent motion chains. It has many advantages, such as large structural stiffness, high motion precision, and relatively easy acquiring the inverse motion solution, which make it one of the hot spots in current research [2]. The main limitations of parallel mechanism are small workspace, anisotropy distribution of kinematics and force operation performance in workspace, singularities and inferior maneuverability area, which bring some challenges to the global motion planning and control of robot. A feasible method for solving these issues is to use redundantly actuated technology.
Redundantly actuated parallel robot means that the number of independent actuated units of the robot exceeds the number of degree of freedom (DOF) of the robot.
Consequently, this kind of robot system usually generates internal load. When one or more branches of the mechanism are in singular configuration, on the one hand, redundantly actuated units can be used to achieve active stiffness control or force optimization, thereby effectively improving the global dynamic characteristics of the robot's operating space [3]. On the other hand, the generation of internal force or internal torque makes the end effector (EE) has better force operation performance [4]. Controlling the internal force by selecting appropriate Novel 6-DOF Wearable Exoskeleton Arm with Pneumatic Force-Feedback for Bilateral Teleoperation ·3· force control algorithms can also improve the accuracy of the mechanism and adjust the operating stiffness of the robot [5]. These features that can improve the accuracy of the machine tool [6], the flexibility and force operation capability of the high-speed machine tool [7], and promote the development of human-computer interaction technology [8] have great value.
In recent years, redundantly actuated parallel robots have gradually attracted the attention of domestic and foreign scholars. For example, Chen Si [9] has studied the fault-tolerant performance of the over-constrained redundantly actuated spherical parallel mechanism, and improved the motion accuracy of parallel mechanism by removing transmission clearance. Redundantly actuated mechanism of actuation space dimension not only higher than that of configuration space dimension, also generally higher than its operating space dimension. Therefore, for specific operation space tasks, the driving force of redundantly actuated parallel mechanism has infinite sets of feasible solutions. By appropriate optimization strategies, the actuated forces can be minimized and the energy consumption can be reduced [10]. Redundantly actuated parallel mechanism has antagonistic actuation ability, which can take advantage of non-zero internal force of the mechanism to remove transmission clearance or alleviate impact vibration of the mechanism [11,12].
What is more attractive is that the introduction of redundant actuation provides a feasible way to eliminate the force singularity of parallel mechanism [13]. But the actuation space dimension of redundantly actuated parallel mechanism is higher than its configuration space dimension, which brings some challenges to precise dynamic modeling, inverse kinematics and dynamics analysis, and controller design of robot. Considering the errors of dynamic modeling, this paper studies the robust dynamic output control issues of redundantly actuated parallel robot based on SMC strategy. SMC is an important class of robust control method that is insensitive to the system's unmodeled errors and external disturbances, and ensures that the stability of the closed-loop system is not affected [14,15]. For the EE to implement operation tasks in contact with external environment, the commonly applied control methods are force/position hybrid control [16] and impedance control [17]. Force/position hybrid control can only be used for operating tasks where the operating force and the displacement vector are orthogonal (decoupled). While, based on the virtual concept of "mechanical impedance" and the principle of feedback equivalence, the essence of impedance control is to transform the original force adjustment target and displacement adjustment target into a second-order dynamic target system, so that a set of control inputs can achieve dynamic stability of two groups of different control tasks simultaneously. In practical applications, the EE of the robot has compliant output characteristic [18]. For practical engineering applications, the dynamic output properties of robot's EE should also be able to be adjusted, such as polishing and grinding of flexible thin-walled parts, friction welding of curved surfaces, assembly of crankshafts, etc., which is difficult to accomplish by using the invariant impedance control strategy. VIC technology is proposed for these high-level industrial application requirements, and is expected to promote the popularization and application of advanced industrial robot systems [19].
In this paper, we investigate the VIC approach for 4RAPR. The new contributions of this paper include that: (1) The proposed VIC strategy is to complete complex tasks of 4RAPR, such as polishing, grinding, assembling, and ensure the quality of parts processing. The control parameters of VIC adapt to the changes of unknown environment, as a result the EE has the characteristics of compliant output; (2) Using Lyapunov stability conditions, the relationship between the control parameters of VIC is established, which simplifies the selection of parameters; (3) For the purpose of enhancing the robust stability of the system, SMC strategy is combined with VIC strategy to effectively reduce the influence of unmodeled errors and external interferences. In SMC, the exponential approach term and the continuous function robust term are chosen to weaken the impact of chattering on the system. The remainder of this paper is organized as follows: In Section 2, the dynamic model of 4RAPR is established. The

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dynamic model based on joint space contains constraint force term, which is unknown and difficult to directly measure it. So, we can try to find null space of constraint matrix to eliminate constraint force term, and then establish the dynamic model in operation space. In Section 3, VIM of 4RAPR based on operation space is proposed. Moreover, SMVIC is designed, and the stability of closed-loop system is proved by Lyapunov theory. In Section 4, Matlab programming simulation confirms the performance of VIC and SMVIC. In Section 5, conclusions and prospects.

Dynamic modeling
The establishment of dynamic model is the basis of research on robot control technology [20]. Redundantly actuated parallel robot is an over-constrained system. No matter which modeling method [21][22][23] is adopted, the number of effective equations will be less than the number of unknowns. As a consequence, the solution of the dynamic model is not unique, which brings certain challenges to the control technology [24]. From the perspective of system energy, the Lagrange function is employed to build dynamic model, which is convenient for the analysis of the mechanism without calculation of internal constraint reaction. Hence, in this paper, we use Lagrange function to deal with dynamic model. 4RAPR is composed of four branch chains, each of which is a two-linkage mechanism. A point O represents the EE, and its Cartesian coordinate is ,  As can be seen in Figure 1, 4RAPR is symmetrical, thereby the mechanism can be divided into four series two-linkage mechanisms connected at one point O .The Lagrange function is used to establish dynamic model of four series two-linkage mechanisms, and the constraint is imposed at point O , and than we can obtain dynamic model of 4RAPR [25]. The series two-linkage mechanism is shown in Figure 2.
According to equations (1) and (2), the translational kinetic energy of the connecting rod on the horizontal plane can be expressed as: The rotational kinetic energy of the active linkage and the driven linkage around the center of mass is calculated by: The kinetic energy of series two-linkage is written as: Since 4RAPR is a planar mechanism, the potential energy of the mechanism is constant. Without loss of generality, this article assumes that the potential energy of planar 4RAPR is zero, such that the Lagrange function of the system satisfies  ii LT , the Lagrange function of the whole mechanism can be calculated as follows: Using Euler-Lagrange equation [26,27]: , the dynamic model of four series two-linkage mechanism can be obtained as follows: , is the inertia matrix in the joint space, is a coefficient matrix associated with the centrifugal and Coriolis forces in the joint space. According to the kinematic relationship of 4RAPR, the constraint equation of the closed chain mechanism can be calculated by:   (9) Where () Aq represents the set of velocity constraints and is the zero matrix of velocity vectors in the configuration space of the parallel mechanism.
Combining equations (7) with (9), we can acquire the dynamic model of 4RAPR, as shown in the following equation: R denotes the value of constraint force. Constraint internal force T A λ is an internal force for the closed-loop system, that is, this term is unknown. In practical applications, the constraint term can be removed by finding the zero space of the constraint matrix. For this purpose, we define the EE position coordinate as e     x y q .
The Jacobian matrix of q with respect to e q is S , i.e., . According to equation (11), we can further get ( ) Multiplying both sides of equation (10) by T S , we can get the following equation: Applying the Jacobian matrix S and the Jacobian matrix J of the active joint vector with respect to the EE e q , the dynamic model of 4RAPR based on operation space can be expressed as follows: S MS CS is a coefficient matrix associated with the centrifugal and Coriolis forces of operation space. Based on the above method, the joint space dynamics model (12) is equivalent to the operation space dynamics model (13). Equation (13) not only removes the unknown constraint term, reduces the scale of the system dynamics model, but also the structural characteristics of the model are consistent with the general parallel mechanism dynamic model and the open-chain mechanism dynamic model, which provides convenience for further research on the dynamic control and motion ·6· planning of redundantly actuated parallel robot based on operation space.

Variable Impedance Control
The essence of impedance control is to unify force control Thus, in complex environment, the dynamic characteristics of the system can be continuously changed and the control performance of the system can be improved by adjusting the control parameters of VIC.
In order to achieve the compliance of the EE, the position of the EE and the external forces are adjusted to reach the ideal state in real time, then the closed-loop dynamics model of VIC in operation space can be established as [28]: By substituting formula (14) into (13), the control law of VIC can be written as: e e e e a e e e e tt v q q H τ D q K q τ J M v C q J τ (15) Furthermore, we can prove the stability of the closed-loop system by using Lyapunov stability theory, and the detailed proof process can be found in literature [29]. The main disadvantage of the VIC closed-loop system shown in equation (15) is that it uses a simple PD controller. As we all know, PD controller is a kind of basic linear controller. The closed-loop control system based on PD controller has limited resistance to unmodeled errors and external disturbances. Subsequently, in this paper, we study the design method of SMVIC, and explore the effective method that enhances the robust stability of VIC system.

Sliding Mode Variable Impedance Control
Basically, SMC system is a class of discontinuous nonlinear control system which can make the system move purposefully in accordance with the desired trajectory [30], and is immune to external disturbances and parameters perturbation, etc. In practical applications, robot system usually has some uncertain factors, such as the inaccuracy of mathematical model, the interferences of the external environment and the influence of parameters perturbation, etc. By constructing a SMC law, these factors can be prevented from affecting the asymptotic stability of the closed-loop system, so as to ensure the robust stability of the closed-loop system [31]. Based on the characteristics of VIC and SMC, VIC and SMC are combined to design SMVIC. Define the sliding mode surface as: Λ  & ee s q q (16) where Λ is positive definite and symmetric matrix. Take the derivative of equation (16) with respect to time produces s q q q q q (17) Select the reaching law function as follows:  μ s is the switching term. The introduction of the switching term can eliminate the influence of model inaccuracy and external disturbances.  λs is the exponential approach term, which can ensure that the system can approach the sliding mode surface at a faster speed when the error is large. The value of μ and λ respectively determine the rate of the system approaching the sliding mode surface s and the convergence speed of the system. The appropriate values of μ and λ can ensure the stability of the system and make the system reach the sliding mode surface quickly in a short time.

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In the past, sgn μ s was usually used as a switching term in the approach law function to overcome the influence of disturbances and parameters perturbation, etc. However, the introduction of the switching term sgn μ  s would cause chattering. Larger chattering will affect the control performance of the system, and even cause damage to the mechanical structure. Therefore, in order to alleviate the chattering in the system, the continuous function tanh( ) s is taken instead of the sign function sgns .Where 0   , the larger the value of  is, the faster the change of the inflection point of tanh( ) s will be. When the value of  is infinitely increased, the double tangent smooth function tanh( ) s is close to sgns . In order to weaken the impact of chattering and ensure robustness of the system, 5   is taken here. Under equations (13), (14), (17), and (18), SMVIC law can be obtained as follows: Select the Lyapunov candidate function as: Then the closed-loop system is globally asymptotically stable at 0 e  τ . then   T  T  T   T  T  1  1   TT   T  1  T  1 (21) For the sake of removing the non-quadratic term of equation (21) and ensuring the positive definiteness of the Lyapunov candidate function, let  (21) can be rewritten as:  (22) To ensure  V0 & , the inequality group must be satisfied Equation (23) can be further simplified as: In equation (24), There are always Λ and , thereby the closed-loop system is globally asymptotically stable. The stability condition (20) of the closed-loop system (14) can be proved. should be further added.

MATLAB Simulation
For VIC method and SMVIC method of 4RAPR, a model is built in MATLAB's simulink for simulation analysis, where chap2_1ctrl is VIC or SMVIC, and chap2_1plant is the controlled objective. The simulink model diagram is shown in Figure 3.  Table 1.

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the stability of the interaction forces of the EE, so that the position error curves are fluctuating. Figure 6 shows the four drive torques of the parallel mechanism. It can be seen that the introduction of redundant actuation makes the distribution of drive torques more even. Figure 7 shows the interaction forces. From the curves, it can be seen that the interaction forces can be stabilized at desired value. According to the dynamic relationship between force and position of VIC, VIC can not only adjust the position, but also adjust the stiffness of the system through the given contact external forces, so as to realize the compliance control and make sure the safety of the processing objective and the robot system.  Table 2, where () t d are the unmodeled errors and external disturbance errors. The simulation results are shown in Figure 8-11. Figure 8 and 9 respectively show the position error curves and velocity error curves of the EE. It can be clearly seen from the curves that the state of the controlled system is stable. Figure 10 shows the driving torques. Due to external disturbances and unmodeled errors, the driving torques of the parallel mechanism are larger than that of VIC. Because the external disturbances imposed on the

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parameters and SMVIC parameters, so as to reduce energy consumption of the system as far as possible and improve the performance of trajectory tracking. In unknown and complex operating environment, such as the polishing and grinding of flexible thin-wall parts, friction welding of curved surfaces, etc., how to design control parameter () t K still requires further research.