[1]
The planar 2-DOF redundantly actuated parallel manipulator can be cut into three 2R serial structure with common constraint. As fig.2 shows, generalized rate is , center coordinates are Fig.1 The architecture of 2-DOF mechanism Fig.2 2R serial architecture and , and are mass center distance, is the length of connecting rods. Then Kane equation can be expressed using Matrix as
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[3]
Then, dynamic model in joint space can be expressed as
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[4]
Considering the constraint at the end, define quasi-velocity of end-effector as , then dynamic model in task space can be written as
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[5]
If using generalized velocity to replace , it is easy to get Langrange equation as following
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[6]
For dynamic model in Eq. , several properties are presented as following[[] W. S. Mark, H. M. Seth and Vidyasagar, in: Robot Modeling and Control (John Wiley & Sons, Incorparation 2005). ]. Property 1: inclined symmetry Ifthen is inclined symmetry. Property 2: dynamic model linearization
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[7]
Where, is dimension regression matrix, is dimension vector parameter. Adaptive Sliding Mode Synchronous Controller Design Sliding Surface Design Based on Synchronous Error. Define trajectory tracking error vector of end-effector as
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[8]
Contour error of trajectory can be defined as synchronization error, and synchronization error is relevant to tracking error
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[9]
Where, is the relationship matrix between trajectory tracking error and synchronization error. If is tangent angle of desired trajectory, then
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[10]
Based on Eq. and Eq. , cross-coupling error is
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[11]
Where,is coupling positive definite matrix. Cross-coupling error contains the trajectory tracking error and the synchronization error. From , cross-coupling velocity and acceleration error vectors are
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[12]
Define the sliding surface as:
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[13]
Where, is positive definite. Sliding Mode Controller Design for Synchronization. Define reference velocity and acceleration as the following:
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[14]
From Eq. , the following process can be derived
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[15]
According to property 2 of dynamic model, Eq. can be transformed into
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[16]
Control law can be designed as
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[17]
Where, , are symmetric positive definite , is dimensional matrix, ,which can be calculated[12]65-66. System stability analysis. Before analysis, Barbalat lemma is shown as the following Barbalat Lemma If a differentiable function has a limit as , and if is uniformly continuous, then as . Theorem: the proposed controller which Eq. shows guarantees asymptotic convergence to zero both of the trajectory tracking errors and synchronization errors. Namely, the system is globally stable, i.e, when , , . Proof : Lyapunov function is defined as
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[18]
Differentiating with respect to time yields
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[19]
Considering property 1, then one can have
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[20]
Combining Eq. -Eq. , one can get
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[21]
It is easy to know,,, then . Hence, the system is globally stable. With the Barbalat Lemma, as, then one knows and as . Hence, and as . This control law could realize convergence to zero of the trajectory tracking error and synchronization error at the same time. Because a sliding model variable structure control has chattering phenomena, which affects the trajectory tracking precision. In order to eliminate chattering, the continuous function with relay characteristics is used to replace the function of symbols to restrict the trajectory in a boundary layer of ideal sliding mode.
DOI: 10.5220/0005019704920498
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[22]
Where, , and are small positive numbers. In addition to, a robust compensation term will be added to adaptive term. From Eq. and Eq., the adaptive robust sliding mode synchronous control law can be expressed as
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[23]
Where, , is a positive number. Matlab Simulations and Performances Comparison With Matlab2010 7.10, the end-effector is driven to track a four leaves rose curve as shown in Fig.3. To demonstrate that the proposed controller can improve trajectory tracking accuracy, simulation of using computed torque controller is implemented. In computed torque controller, the control law is[11]460
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[24]
In Eq. , define,. In control law Eq. ， we define, ，, , ，, . To verify the validity of the proposed controller, tracking errors and synchronization errors changing with time of two controllers are presented in Fig. 4. RASMAC means robust adaptive sliding mode synchronous control. Fig. 3: Desired trajectory (a)X-axis tracking error (b) Y-axis tracking error (c)X-axis synchronization error (d) Y-axis synchronization error Fig. 4: Errors comparison of two algorithms As Fig.4 shows, in the long term, errors of RASMSC are lower than that of computed torque control. Thus, the proposed controller is valid and can achieve better trajectory tracking. Conclusions In dynamic modeling, Kane equation doesn't require derivation and eases computer programming. The dynamic model of the planar 2-DOF parallel manipulator is derived easily. Based on the dynamic model, the adaptive robust sliding mode synchronous controller is designed to control the mechanism to track a desired trajectory. Theoretic analysis implies that both trajectory tracking errors and synchronization errors can converge to zero by using this controller. Compared with computed torque controller, the tracking errors decrease greatly with the proposed controller. Also the proposed controller can be used to other parallel manipulators to satisfy high speed and accuracy. Acknowledgements This work was financially supported by National natural science foundation of China (51105089). References
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