2010 8th IEEE International Conference on Control and Automation Xiamen, China, June 9-11, 2010
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A Novel Decoupling Controller Design for Parallel Motion Platforms M. Aminzadeh, A. Mahmoodi, and M.B Menhaj, Member, IEEE
Abstract—In the present paper a novel classical control approach is proposed that provides an efficient and yet easy to implement control method for parallel-type motion platforms that are well known to have highly nonlinear coupled dynamics. It converts the MIMO nonlinear system to several decoupled reduced-order SISO systems that extremely simplifies the controller design using highly-developed classical control strategies for SISO systems. In addition, it provides direct control over operational variables that task is defined on. The method is applied to a 3-DoF platform. Simulation results reveal that the proposed controller considerably improves control specifications compared to the conventional classic controllers and is comparable to the nonlinear controllers that are of high computational load. It is also applied to a 6-DoF platform. The experimental results approve the applicability of the method.
INTRODUCTION
D
EMAND for high precision motion has been increasing in recent years. Since performance of today’s many mechanical systems requires high stiffness and accurate positioning capability, parallel manipulators gained popularity. As the direct result of their parallel structure they provide better load capacity and positioning accuracy compared to their serial counterparts [1]. Motion simulation is among the applications that have intensively been attracted by these merits. Extensive use of motion platforms with parallel structure for flight and driving simulators is an evident example. Despite the great advantages, parallel manipulators suffer from several disadvantages. The high coupling between the kinematic chains leads to highly-coupled nonlinear dynamics that has made control of these systems a matter of concern. Many control methodologies have been proposed and implemented by many researchers [2]-[4]. They vary from simple classical controllers to advanced nonlinear control strategies; each has its own disadvantages as well as the advantages it provides. Advanced control strategies ranging from nonlinear control methods to intelligent control such as adaptive, neural networks, fuzzy and etc. are difficult either to design or implement or both and utilizing them for real-time purposes requires a compromise to be Masoumeh Aminzadeh, Msc. stusent, Aerospace Department, Amirkabir University of Technology, Tehran, Iran. (corresponding author: +98-2166462210; fax: +98-21-66462210; e-mail:
[email protected]). Ali Mahmoodi, Phd student, Aerospace Engineering Department, Amirkabir University of Technology, Tehran, Iran (e-mail:
[email protected]). Mohammad Bagher Menhaj, professor, Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran (e-mail:
[email protected] ).
978-1-4244-5196-8/10/$26.00 ©2010 IEEE
made between precision and computational load. Additionally the use of a high speed computer to deal with computer arithmetic is unavoidable [5]. The conventional classical controllers, on the other hand, are easy to design and implement, but they are based on the assumption of a decentralized system. In other words, actuators’ motion effects on one another are assumed to be negligible despite the high coupling between the actuator motions due to the parallel structure of the mechanism. Inexact motion following is the inevitable consequence of this assumption. The controller determined in this way, to be implemented in the coupled system must be modified by trial and error [6], [7]. In addition, the control is performed in joint space which means that variables of operational space that the task is specified in, are controlled in an open-loop fashion through the manipulator mechanical structure. This results in a loss of accuracy in the presence of any uncertainty of the structure namely construction tolerance, lack of fabrication, gear backlash and elasticity [8]. Decoupling or non-interactive control has attracted considerable research attention since the 1960s when control engineers started to deal with multivariable systems [9]. Decoupling control owes its popularity to its ability to convert the system to several systems in which each input only affects one output. This property enables the controller designer to address the MIMO system as several SISO systems for which the control techniques have matured and are simple to apply. This decoupling can be achieved by several methods. One way is to design the first-level controller in form of a square matrix to convert the loop gain matrix to a diagonally dominant matrix. This method yields approximate decoupling and the final controller designed must have some degree of robustness to compensate for the neglected interaction. In addition, the mentioned controller gains become more and more difficult to determine as the order of the system increases [10]. For motion platforms with three or more degrees of freedom, this method does not imply to be suitable. Another method that completely decouples the system is the direct consequence of the fact that each matrix can be written in terms of a diagonal matrix with the main matrix singular values as its diagonal elements and two matrices containing the so-called left and right eigen vectors i.e SVD (Singular Value Decomposition) [11]. For a dynamic system as well as robotic systems, from the symmetric dynamic matrix when factorized in this way, it is seen the components of the state vector of the system along the eigen vectors behave independently [6]. Hoffman uses this decoupling approach to analyze a flight simulator motion system with a decentralized feedback [12]. Although this decoupling is seen to hold in practice, it is not seen to be applied after that time.
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FrD1.6 In the present paper, a new classical control approach is taken that decouples the system by input transformation rather than state transformation. It is seen to lead to a simpler structure and consequently simpler controller design compared to the previous decoupling methods. In addition, since the decoupled variables are the system’s operational variables rather than transformed states, the control specifications can be determined for each variable independent of the others. It is applicable to many parallel motion platforms of different degrees of freedom that consist of a moving platform connected to the base by up to six actuating legs. The transformation can be found from the geometric properties of the platform. In addition to simplifying the controller design and providing higher precision compared to the conventional classic controller, the proposed controller controls variables in operational space in which the task is specified. This, when measurement is also performed in operational space, eliminates the need for kinematic transformations that involve the error due to geometrical uncertainties. The proposed method is applied to a certain parallel-type of a heave-pitch-roll motion platform and a 6-DoF Stewart platform. A series of numerical simulations and experiments are carried out to test the effectiveness of the system, and the results verify the favorable tracking ability. DYNAMICS DECOUPLING METHOD Motion platforms of interest in this paper consist of a moving platform that is connected to a base platform by several parallel legs that are actuated by the same number of servomotors. The degrees of freedom of these systems are any combination of three position and three rotational coordinates. From the certain geometry of these platforms it is seen there can be found some linear relations between the way each generalized coordinate is affected by an input and the way it is affected by another input. In other words, due to the special arrangement of legs and the geometry of the platform, there can be found some linear relations between the transfer functions from different inputs to an output around an equilibrium point. By writing each output as a summation of the transfer functions’ outputs and applying the mentioned relations, it is seen that by a suitable input transformation, n decoupled transfer functions of the new inputs to the state variables can be found where n is the number of degrees of freedom. Therefore the multi-input-multi-output system is converted to n single-input-single-output reduced-order (second-order) systems that can now be controlled separately. If the transfer function of each input ( ) to a generalized coordinate ( ) i.e
represented as
as a linear term versus the other as below:
can be written
the generalized coordinate when written in terms of transfer functions’ outputs can be written as below:
By choosing generalized coordinate input as:
as a new input , the can be written in terms of the new
By applying the following transformation to all states the coupled system will be transformed to n SISO systems as above. The input transformation used is as below:
where for invertability of the transformation and maintaining the controllability, the rank of matrix R which appears in the control matrix must be n. Relations in (1) can be found for many parallel type motion platforms due to their special arrangement of legs. But as the platform deviates from the study point (equilibrium point), the constant coefficients ( ) gradually change due to the change in geometrical configuration. Decoupling is still possible but if the changes are too much for the coefficients to be assumed constant, it is not possible through a single transformation. For the case of parallel platforms, since the deviation in legs’ orientation is small as the platform sweeps the workspace, these relations hold in entire the workspace with constant ’s to a good degree of precision. Also, although the coefficients are geometry dependant so they might be affected by geometrical uncertainties, it is seen the effect of geometrical tolerances associated with these mechanical systems lead to completely negligible deviation, eliminating the need for high-precision system identification; especially for parallel motion platforms that are designed at high tolerances for ensuring smoothness of operation [13].
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FrD1.6 For determination of the coefficients, one can obtain the transfer functions by analytical or numerical methods for the study point. Another way is to determine the generalized forces caused by different inputs at the study point and calculating its partial derivative vs. each input as below:
where τi is the ith generalized force. The matrix formed by elements are seen to be the transpose of the semijacobian matrix of the platform that relates the actuator length rates to the parallel robot’s generalized velocities in operational space [6]. CASE STUDY 3-DoF parallel platform To evaluate the efficiency of the decoupling method and performance of the controller designed based on the method, a certain 3dof motion platform is considered. A schematic view of the platform is shown in Fig. 2. The kinematic and inertial parameters of the platform are given in Table 1.
where z, φ and θ are platform heave, roll and pitch coordinates respectively. Relations (6) can be obtained both by calculating transfer functions through linearizing the system or numerical methods or by taking partial derivatives of the generalized forces vs. the three inputs. In both procedures a simplified dynamic analysis of the system is required. It is done by deriving the equations of motion of the platform neglecting other dynamics such as legs’ inertia. It will later be shown that inclusion of the unmodeled dynamics leads to inappreciable difference. By applying the transformation of inputs as in (4) considering the relations (6), the following will be obtained:
where the following transformation has been used:
where
actuator. By calculating the system transfer functions at various equilibrium points throughout the workspace, it is seen for points within degrees deviation in roll and pitch, the relations (6) are seen to remain unchanged to three digit precision. Therefore it is reasonable to conclude that transfer functions in (7) can describe system behavior with acceptable precision. The transfer functions of (7) for the specific 3dof platform are obtained as follows:
Fig. 2. Schematic view of the 3DoF parallel motion platform
Table 1. 3 DoF platform characteristics
parameter Platform and payload mass (kg) Upper leg mass (kg) Lower leg mass (kg) The dimensions of the triangle connecting actuator joints:
is the force exerted to the platform by the ith
value 4000 80 55 Base (m) 1.48 Side (m) 1.33
For the studied 3dof platform the following relations hold between the transfer functions at the neutral point:
The sixth-order three-input-three-output system is then converted to the three above second-order transfer functions
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FrD1.6 for which controller can now be designed using simple classic controllers such as PID. To control z, a PD controller was designed as there is no need for integrator since the open-loop transfer function contains an integrating factor within. Two PID controllers were designed for and by use of the pole-placement method. They were then substituted by two PD controllers with feed forward to smooth the transient response, decrease the overshoot and avoid the unnecessary increase of system’s order. Controller gains are determined such that desirable transient response is achieved with zero steady state error. For simulation purposes the controllers are designed to yield a desired response specification of critical damping and 0.1 second settling time. The controller gains were determined using pole-placement method to achieve the above control specifications and are presented in Table 2.
Fig. 3. System response vs. desired trajectory for the system under decoupling controller
Fig. 4. Controller response applied to simplified and complete dynamic models
Table 2. Decoupling controller gains for the3 DoF platform
gains controller
z
feed forward gain -
To evaluate the effectiveness of the controller, it must be applied to a complete dynamic model of the system. Therefore a nonlinear dynamic model of the system was derived that considers platform dynamics, legs’ inertia and actuator frictions. It also considers dynamics of the DC servomotors used for actuation. Detailed information on the system and actuation dynamics can be found in [14], [15]. The controller must follow any desired trajectory in the desirable manner. Therefore the system response is studied for various trajectories such as step, ramp and sinusoidal functions. The responses are presented in Fig. 3 for an arbitrarily chosen desired trajectory given in (10).
From Fig. 3 it is seen although the relations (7) were derived for the simplified dynamics, the designed controller when applied to the complete system performs in the desired manner with completely negligible deviation which is better depicted in Fig. 4. Fig. 4 shows the controller response when applied to the complete dynamic model and to the simple dynamic model where the dynamics of the platform is only considered. It is obvious the control inputs are different in two cases but produced by the controller in such a way that both systems very well follow the desired trajectory. Considering that the legs form more than %10 of payload mass, correspondence of the results shows that the controller shows a good degree of robustness against unmodelled nondominant dynamics. Fig. 5 also shows the response error of the complete system under a PID controller designed based on conventional classic control compared to the decoupling controller’s response error. The improvement of the performance is clearly seen. The controller gains are obtained based on the designer’s intuition and experience and by trial and error and there is not usually an analytical method for determining the gains for this controller. The PID controller gains for the three joint variables i.e. the three actuator lengths to achieve , a favorable response, are obtained as and
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FrD1.6 through the inclusion of the inverse dynamics of the servomotors through the following linear relation [16]: (14) where is motor-torque constant, is rotor inertia, and are armature resistance and inductance respectively and is back emf constant. r is gears reduction ratio., and are the motor voltage and current. is the torque due to motor friction and is assumed to be equal to viscous friction that can be modeled as where b is viscous friction constant. and are the leg’s length velocity and acceleration that are obtained from measurements. Fig. 5. Controlled system response error for the system under conventional classic controller and decoupling controller
Table 3. 6-DoF Stewart platform characteristics
parameter Platform and payload mass (kg) Upper leg mass (kg) Lower leg mass (kg) Top radius (m) Offset angle from 120 degree spacing on top (deg) Base radius (m) Offset angle from 120 degree spacing on top (deg) Offset angle between top and bottom points (deg)
For this platform, it is seen that the error is in the order of which is nearly in the same order as that achieved by using more advanced control methods like nonlinear modelbased control [15]. 6-DoF Stewart platform The proposed controller is also applied to a 6-DoF highlynonlinear Stewart platform. The characteristics of the platform are presented in Table 3. Relations (1) must be found from the transfer functions of the simplified dynamics model. These relations for the considered parallel platform, for the transfer functions of x variable for example, are as below:
value 100 1 6 0.45 0.4
Table 4. Decoupling controller gains for the 6-DoF Stewart platform
gains controller
x, y, z φ, θ ψ The decoupling controller is applied to the complete dynamics of the Stewart platform considering legs’ inertia. The controller response to an arbitrary desired sinusoidal trajectory given by (15) is presented in Fig. 6.
x [m]
Applying the above transformation, the decoupled transfer functions are obtained as below:
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Controllers are then designed for the second-order systems to get a critically damped response with the settling time of 0.1 second. The PD controllers’ gains are obtained using pole-placement method as presented in Table 4. The controller provides the required forces to be exerted by servomotors. The required voltages are then obtained
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Command Response
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Fig. 6. System response vs. desired trajectory for the system under decoupling controller.
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Fig. 7. System response error for the system under decoupling controller
The ability of the controller to follow the desired trajectory is clearly depicted in Fig. 6 and 7. It is seen from Fig. 6 that the system reaches the command after the settling time of 0.1 second and closely follows it ever after. The controller is also implemented in a Stewart platform that is shown in Fig. 8 with the characteristics in Table 3. The measured actuator lengths that lead to the specified desired trajectory are shown in Fig. 9. The experimental results prove the feasibility of the controller.
control approaches for SISO systems. Compared to conventional classic controllers, the proposed controller provides higher precision and smoother response and it also allows control over operational variables i.e. in task space. Compared to previously-proposed decoupling control methods, this controller yields a simpler structure and also allows each operational variable to be controlled with its own particular control specifications. The method was applied to a 3-DoF motion platform and a 6-DoF Stewart platform. Simulation results for the 3-DoF platform prove high performance of the controller. Not only does the controller provide an easy mathematical approach for determining controller gains that enables direct control over operational variables with desirable control specifications for each variable, but also it considerably decreases command-following errors. For the case of 6-DoF Stewart platform, the method converts the system to six second-order SISO systems for which controller gains are determined using simple analytical classic control approaches. The applicability of the method is also verified by experimentally testing the controller on a Stewart platform. REFERENCES [1] [2]
[3]
[4]
[5] [6] Fig. 8. Photograph of experimental equipment
[7] [8]
Parallel Manipulators, Towards New Applications, ISBN 978-3902613-40-0, pp: 269-294, 2008. D. Zhao, S. Li, and F. Gao, “Fully Adaptive Feedforward Feedback Synchronized Tracking Control for Stewart Platform Systems”, International Journal of Control, Automation, and Systems, vol. 6, no. 5, pp. 689-701, 2008 N. Kim, C. Lee,“ High speed tracking control of Stewart platform manipulator via enhanced sliding mode control”, proceedings of international conference on robotics and automation, 1998 M. Idan and D. Sahar, "A Robust Controller for a Dynamic Six Degrees of Freedom Flight Simulator", AIAA Flight Simulation Technologies Conference, San Diego, CA, pp. 53-60 , 1996 L. Sciavicco and B. Siciliano, “Modeling and control of robot manipulators”, The McGraw-Hill Companies, Inc. , 1996 S. H. Koekkebakker, “Model-based control of a flight simulator motion system”, Phd. Thesis, University of Delf, 2001 O. Ulucay, “Design and control of a Stewart platform manipulator”, Msc thesis, Sabanci university, 2006 Z. Qu, D. M. Dawson, “Robust tracking control of robot manipulators”, IEEE Press, 1996 Q. Wang, Decoupling Control, Vol. 285 , Springer
[9] [10] J. M. Maciejowski., 1989. Multivariable feedback design [11] P. Albertos, A. Sala and M. Chadli., 2004. “Multivariable control systems—an engineering approach” [12] R. Hoffman. Dynamics and control of a flight simulator motion system. In Proc. Canadian, Conference of Automatic Control, 1979 [13] I. Davliakos and E. Papadopoulos,”Model-based control of a 6-dof electrohydraulic Stewart–Gough platform”, Mechanism and Machine Theory, 2008 [14] M. Aminzadeh, A. Mahmoodi, and M. Sabzehparavar, ”Dynamic analysis of a 3DoF Motion Platform”. International Journal of Robotics: Theory and Application, Vol. 1, No.1, IJR08, 2009. [15] M. Aminzadeh, and M. Sabzehparvar, “Model-based motion tracking control of an electric 3DoF parallel motion platform”, IEEE Aerospace Conference, Montana, USA, 2010 [16] R. Kelly, V. Santibanez and A. Loria, “Control of robot manipulators in joint space”, Springer, 2005
Fig. 9. Displacement of the six actuators of the Stewart platform CONCLUSION
In this paper, a novel classical control approach based on dynamics decoupling was proposed and effectively applied to parallel motion platforms. The method converts the coupled equations of motion of the MIMO system with n degrees of freedom to n second-order SISO systems that can simply be controlled by applying highly-developed classical
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