Dynamic Modeling and Control System Design of Stewart Platform for Vibration Isolation

Transcription

1 3B22 Dynamic Modeling and Control System Design of Stewart Platform for Vibration Isolation Jinjun SHAN, Ziliang KANG and Ulrich GABBERT Department of Earth and Space Science and Engineering, York University 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada Institut fr Mechanik, Otto-von-Guericke-Universitt Magdeburg Universittsplatz 2, Magdeburg, Germany Abstract Stewart platform in cubic configuration is of great application in the prospect of vibration control. However, the twist and torque between the conjunction of the strut and plate and the special characteristics of piezo would enhance the nonlinearity of cubic configuration, and influence the tracking accuracy and efficiency of the platform. Therefore, in this paper, an adaptive control based on least-square identification method of the dynamics model of piezoelectric Stewart Platform is proposed. Regarding that the force matrix can represent all the nonlinear influence and the real model can be identified accurately, the controller is designed to replace the original force matrix of the linear model of the cubic configuration with the identification outcome. Then simulation is implemented to confirm the feasibility of this method. Results show that the adaptive control law is able to remedy the influence of nonlinearity of the dynamic model and be efficient in isolation vibration. Key words : Vibration isolation; piezoelectric Stewart platform; adaptive control; parameter identification 1. Introduction With deep investigation to the space, recent years, there is increasing interest in the vibration isolation applied to aerospace missions. While researcher are dedicated in developing various spacial mission like Terrestrial Planet Finder (Traub et al., 2007) (1) of deep space exploration and Lunar Laser Communications Demonstration (Boroson et al, 2009) (2) in space laser communication, disturbance from usage of mechanical devices as reaction wheels, gyroscopes, solar array drives and other devices, as well as from outside spacial disturbance like space shuttle docking, gradually become severe and significant problems to consider. Diversified attempts are implemented to solve this problems, such as vibration isolation tab, miniature vibration isolation system (Quenon et al., 2001) (3) and application of stewart platform. As a popular parallel mechanism invented by Gough and summarized by Stewart (Stewart, 1965) (4), stewart platforms have wide-range application such as flight simulator, vehicle simulator, vibration table and spatial dock mechanism (Dasgupta and Mruthyunjaya, 2000) (5). Because stewart platform has multiple freedom and precise position accuracy while maintaining high force-weight ratio, it becomes a ideal device in dealing with the wide-band and high frequency vibration in space structure. Actuators equipped in the stewart platform range from voice coil to piezo, categorizing the stewart platform as soft and stiff according to the stiffness of actuators (Hanieh et al., 2002) (6). To satisfied the needs of high accuracy while eliminating high frequency disturbance, plenty of control laws are proposed as sliding mode control (Kim and Lee, 1998) (7), adaptive control (Geng and Haynes, 1994) (8), decentralized integral force feedback control (Hanieh et al, 2001) (9) (Preumont et al., 2007) (10) and fuzzy control (Bahrami et al., 2013) (11). However, the twist and torque between the conjunction of the strut and plate and the special characteristics of piezo would enhance the nonlinearity of cubic configuration, and influence the tracking accuracy and efficacy of the platform. In this paper, an adaptive PD control is adopted as an endeavor in control piezoelectric stewart platform., in order to get better isolation result on account of the real complex dynamic model of stewart platform. 1

2 2. Modeling of Stewart Platform 2.1. Cubic Configuration While usually the 6-DOF characteristic of the stewart platform is use to achieve large deformation in robotic simulators; the specific application considered in vibration isolation, since the displacements of the struts are rather small, is fairly different from most applications in robotic manipulations. (a) 6-DOF Stewart platform developed at SDCNLab, York University (b) Cubic Configuration Fig. 1 Stewart plateform in cubic configuration Different from general stewart platform, the Cubic Configuration was invented by Z.Geng and L.Haynes from Intelligent Automation Inc.(IAI) in (8) The cubic configuration is obtained by cutting a cube through surface diagonals to two planes as base plate and moving plate, illustrated in Fig.1. The edges of the cube are the six struts connecting the two plates as a platform. In such construction, the cubic configuration has several unique characteristics, such as its uniformity in control capability in all directions and minimum cross-coupling amongst actuators. These characters enabled by the special geometric of cube make such stewart platform capable of decoupled control action, since adjacent legs are orthogonal to each other, as well as using the symmetry of cube to provide identical legs. All these features mentioned would be very useful in vibration isolation thus this paper adopted such configuration for stewart platform with piezo stacks serving as the active legs of the system. As a kind of stiff stewart platform, piezoelectric stewart platform can be used in positioning with high resolution to accomplish missions of fine pointing of optics, telescopes and other precision devices. (Rahman et al., 1998) (12) (Hanieh, 2003) (13) 2.2. Dynamics Model for Controller Design To simplify the design of the adaptive controller, linear dynamic model of stewart platform is adopted in the controller. The linear model is as following illustrated Based on the structure characteristics of cubic configuration and considering the disturbance force and moments on the payload plate as F = [F x, F y, F z, M x, M y, M z ], the dynamic model equation of the stewart platform can be illustrated as MẌ + KX = Bu + F (1) where M and K is the mass inertia matrix and the stiffness of stewart platform. B is the force jacobian matrix which transfers the piezo force u in the strut axis to the coordinates in accordance with the plate attitude. Since the active legs are made of piezo stacks, the active force u can be illustrated as u = kδ, where k is the axial stiffness of strut and δ is the elongation of the piezo actuator. In terms of convenience for attitude illustration, the inertia frame x,y,z on the base plate and the reference frame x r,y r,z r on the moving plate both originate from their mass center, and their axes aligned with the principle inertia axes of each plate. X = [x, y, z, ψ, θ, φ] T is the attitude of the moving plate which depicts the translational displacement x, y, z and rotational displacement ψ, θ, φ according to the inertia frame. The 2

3 mass matrix of the stewart platform can be defined as m M = (2) I x I y I z where m is the mass of the plate, I x, I y, I z are the principle moments of inertia of the plate. Meanwhile, the stiffness matrix K of the Stewart platform can be calculated as K = B diag(k 1, k 2, k 3, k 4, k 5, k 6 ) B T (3) where k i, i = 1,..., 6 is the stiffness of each strut connecting the upper and lower platform. The stiffness matrix K is asymmetric because of the effect of force jacobian matrix B. B transfers the force aligned with strut axes to the inertia frame. To get the matrix B, the jacobian matrix boldsymbolj should be obtained first. According to the virtue work theorem, then B T = J can be obtained. Turn to the configuration of the stewart platform as shown in Fig.2, we can define the vector q i aligned with each of the six struts in the inertia frame as q i = x 0 + R p i r i (4) where x 0 is the vector connecting the mass center of the lower and upper plate, r i is the vector on the base plate from the origin O to the strut joint A i. Both of the two vector are expressed in the inertia frame. Meanwhile, p i is the vector on the moving plate from the origin o to the strut joint B i expressed in the reference frame. Fig. 2 Vector form of the platform R is the transfer matrix from the reference frame to the inertia frame according to the roll-yaw-pitch rotation sequence, and can be defined as cφcθ sφcψ + cφsθsψ sφsψ + cφsθcψ R = sφcθ cφcψ + sφsθsψ cφsψ + sφsθcψ (5) sθ cθsψ cθcψ Then the unit length l i of each strut would be l i = q i (6) q i From the virtue work theorem, using Eq. (4), Eq. (5)and Eq. (6), the force jacobian matrix B can be obtained as... B = q i q i R p i q i q i... (7) 3

4 3. Adaptive Control Algorithm Since adaptive control has already enhance the difficulty of system operation, PD controller is adopted to balance complexity. Considering expect attitudes as input, vibration sources as disturbance, then the close-loop control system should be structured as Fig.3 shows, where u is the force vector of the piezo. Fig. 3 Adaptive PD Control System 3.1. PD Controller PD controller is simply designed to adopt basic PD control law according to the kinematics of the cubic configuration. Then the force vector y on the six struts can be defined as y = k(δ B T X) (8) where k and δ are the stiffness and elongation of the piezo, X is the attitude of the moving plate and B T = J is the jacobian matrix which transfers the attitudes from the reference frame to the inertia frame. This force equation expresses the fact that the total force on strut is the sum of the control force kδ and passive force caused by attitude changes. In this respect, if the stewart platform is exclusively vibrated and getting an attitude as X by disturbing force, then the control force or elongation for vibration isolation can be calculated from the Eq. (8). Combined with the simple PD control law, then the modified PD control law, can be defined as u = kb T ( K d ė K p e) (9) where u and e = X X d are still the piezo control force vector and attitude error, and the rest of parameters have same definition as in the Eq. (8). From the PD control law Eq. (9), we assume that the nonlinearity of cubic configuration can be expressed through the force jacobian matrix B, therefore the adaptive controller is designed to replace the original force jacobian matrix of the linear model of the cubic configuration with an identification outcome of real model Prove of Lyapunov Stability Before investigating the efficiency of this control law, analysis about this modified PD controller stability should be considered. Since the expect attitude X should be a constant for vibration isolation, then Ẋ 0 would always be tenable. Provided that K d,k p >0, then the closed-loop system can be get from dynamic Eq. (1) and control law Eq. (9) as MẌ + KX + kbb T K d Ẋ + kbb T K p (X X d ) = 0 (10) If only in order to isolate the vibration, the expect attitude should be assumed as X d = 0(if not, redefine X = X X d ). Then the Lyapunov function can be illustrated from the energy of the system as V(X, Ẋ) = 1 2 ẊT MẊ XT KX XT kbb T K p X (11) Lyapunov energy V should always be positive. Deriving the energy V and take into the Eq. (10), we can get that V(X, Ẋ) = Ẋ T (MẌ + KX + kbb T K p X) = Ẋ T BkB T K d Ẋ (12) 4

5 We get that the derivative function V is negative semi-definite, since BkB T and K d are positive definite. Then we can conclude that the PD control law provides the equilibrium point X = X d with asymptotic setpoint stability Identification Algorithm In order to get efficient identification of the force jacobian Matrix B, the identification algorithm needs to have preferable convergent characteristics. Meanwhile, on account that the dynamic model of the stewart platform is a MIMO (Multi-Input-Multi-Output) system, the algorithm as well needs to be simple for calculation. In this respect, Least- Square(LS) method is adopted for identification, since its simplicity and efficiency to small noise. For computer disposition, the difference equation of the linear dynamic model of cubic configuration can be derived from the Eq. (1) as y(n) + km 1 BB T y(n 2) = M 1 Bu(n 2) + ξ (13) where y is the output of attitude and u is the input of strut force, and ξ is the disturbance force or illustrated as noise on the payload platform. Derive Eq. (13) to standard LS form, it can be obtained that Y = Φθ + ξ where y(0) T Y = Φ =. y(n) T y( 2) T, ξ =. y(n 2) T ξ(0) T. ξ(n) T u( 2) T. u(n 2) T, θ = (km 1 BB T ) T (M 1 B) T (14) (15) (16) θ is the parameter matrix which needs to be identification. To simplify the identification, Mass inertia matrix M and the stiffness k of each strut can be accurately calibrated or calculated from equipment information. Hence, matrix B remains to be the only matrix to be identified. Considering that the noise ξ is mean zero-value random excitation, then the recurrence LS formula (Zhang and Li, 2002) (14) for the identification of force jacobian matrix B would be ˆθ N+1 = ˆθ N + G N+1 (y N+1 φ T N+1 ˆθ N ) G N+1 = P N φ N+1 (1 + φ T N+1 P Nφ N+1 ) 1 P N+1 = P N P N φ N+1 (1 + φ T N+1 P Nφ N+1 ) 1 φ T N+1 P N (17) where to start calculation, the initial value can be calculated as P N0 = (Φ T N0 Φ N0) 1, ˆθ N0 = P N0 Φ T N0 Y N0 (18) 4. Simulation of Adaptive Control System In order to model the real stewart platform, nonlinear dynamics model of stewart platform served as real model in simulation, of which derivation can be seen from the appendix. For simulation, the linear dynamics model Eq. (1) for designing PD controller, nonlinear model Eq. (A.6) for imitating the real model and adaptive LS identification algorithm Eq. (17) are taken to build control system in MATLAB\SIMULINK. The stewart platform of cubic configuration taken into simulation is a 6-DOF stewart platform developed in SDCN- Lab at York University, which is showed in Fig.1(a). The base platform and payload form are exactly designed the same; the material of whole structure is aluminum and joint points in each plate distribute according to a equilateral triangle. The following Table 1 shows the specific parameters of the plate. Assume that the initial attitude elements, velocity and acceleration of the moving plate are all zero; M-squence and white noise from 10Hz 500Hz are taken into simulation as disturbance. Then the efficiency of vibration isolation can be 5

6 Density of structure(aluminum) [kg/m 3 ] Plate thickness [m] Plate radius [m] Stiffness of piezo strut(p ) [N/µm] Table 1 Parameters of stewart platform built in SDCNLab, York University get once set the expect attitude as zero. After the simulation outcome become steady, the typical result of identification of force jacobian matrix B in disturbance of 100Hz can be obtained as below B = (19) Following Fig.4 is a typical simulation result in 100Hz, which is similar to other simulation results. The background lines in each figure is the white noise. (a) Axial Displacement (b) Rotation about axis Fig. 4 Typical Response to 100Hz Noise of Adaptive Control System To the figures above in Fig.4, it can be seen that the vibrating of isolation results are far smaller than the original vibration caused by disturbance. To better see the isolation result, result in Fig.4 is magnified, and the figure below is a typical result of axial displacement in Z axe. Since the cubic stewart platform is not flexible than general stewart platform, in order to maintain its configuration. Hence it obvious that the biggest displacement would always be the axial displacement in Z axis, of which result is typical in the 6 altitudes. According to the magnifying result shown in Fig.5, the control result is desired with a fast stabilization. 6

7 Fig. 5 Magnified vibration isolation result of displacement in Z axis The following picture are the velocity result after vibration isolation of six altitudes under a disturbance of 100Hz, which are as well ideal to become stable in a short period of time (a) Axial velocity (b) Angular velocity about axis Fig. 6 Axial velocity and angular velocity result in disturbance of 100Hz Similar as the typical result of vibration isolation in disturbance of 100Hz, other results according to the disturbances in different frequency also obtain ideal characteristic. In Fig.7, the motion of centroid after vibration isolation of the payload platform in the typical disturbance of 10Hz, 50Hz, 100Hz and 400Hz is obtained, though the stable points are different under different extent of disturbances. Since the centroid motion trails in different disturbance are similar, for clarity, only centroid motion in 100Hz disturbance before isolation vibration is shown below. (a) Centroid motion in disturbance of 100Hz before control (b) Centroid motion in disturbance of 10Hz, 50Hz, 100Hz and 400Hz after control Fig. 7 Typical centroid motion under different disturbance. For clarity, the motion trails after vibration isolation of 10Hz, 100Hz and 400Hz only show the stable results, while the motion trail of 50Hz(green) as well shows the stable process. According to the all of the results above, it can be observed that the adaptive PD control law is efficient in attitude tracking, of which trajectory is in accordance with the attitude expectation. The result confirms that identification and control law is fast in calculation and as well as render ideal isolation result to different vibration source in simulation. 7

8 5. Conclusion Concentration of this paper is to analyze dynamic characters of piezoelectric stewart platform and design a proper controller that can remit the uncertain characteristics in real nonlinear dynamic system. Therefore, linear dynamics model as well as nonlinear dynamics model of six piezo strut and the whole stewart platform are derived. Furthermore, an adaptive PD controller based on LS method law are proposed, grounded on the special kinetic relation between attitudes of the moving plate and elongation of the piezo actuators. Then, simulation in MATLAB\SIMULINK are conducted with white noise and M-sequence from 10Hz to 500Hz serving as disturbance. From the results, identification is confirmed efficiency and the adaptive control laws is ensured of capability of isolating vibration and attitudes tracking, which has paved the way for real model experiment. Appendix Nonlinear Dynamics Model of Stewart Platform for Simulation Dynamics of Single Strut Fig. 8 Model of single strut Similar as the linear model, considering six strut share the same physical characteristics, the matrix contained the six strut vector still defined as q. From the strut dynamic model shown as Fig.8, we can obtained that m s ẍ 2 + (c 1 + c 2 )ẋ 2 + (k 1 + k 2 )x 2 k 2 x 3 = c 1 ẋ 1 + k 1 x 1 + u (A.1) where m s is the mass matrix of six vector and u is the matrix of the driving force in six strut; k 1 and k 2 are stiffness parameters, while c 1 and c 2 are damping parameters. Since x 3 = q + x 1, then the single strut dynamic model can be derived from above equation as m s ẍ 2 + (c 1 + c 2 )ẋ 2 + (k 1 + k 2 )x 2 k 2 q = c 1 ẋ 1 + (k 1 + k 2 )x 1 + u (A.2) Dynamics of Payload platform Define the complex dynamic model of the payload platform in standard form as MẌ + c(ω) = B f (A.3) where M is the mass inertia matrix and B is the force jacobian matrix, whose definition are similar as they are in the linear dynamic model of stewart platform. f is the force on the payload platform from struts. c(ω) is the coriolis component. From the Fig., it can be obtained that the force f from the strut to the payload platform is f = k 2 x 3 c 2 ẋ 3 + k 2 x 2 + c 2 ẋ 2 u (A.4) 8

9 Since as well x 3 = q + x 1, then the force f can be derived from above equation as f = k 2 (q + x 1 ) c 2 ( q + ẋ 1 ) + k 2 x 2 + c 2 ẋ 2 u (A.5) Taking the force above into the complex dynamic model of stewart platform, Eq. (A.3), then the complex dynamic model become MẌ + c(ω) = B( k 2 (q + x 1 ) c 2 ( q + ẋ 1 ) + k 2 x 2 + c 2 ẋ 2 u) (A.6) To remit the simulation complexity of the model, the velocity vector of the strut connect to the base platform x 1 can be restrained by the boundary conditions derived as follows. From Fig.2, it can be obtained that the differential coefficient of strut joint to the base platform A i is Ȧ i = v B Ã i ω B + B Ȧ i (A.7) where means to get antisymmetric matrix, v B is the velocity and ω B is angular velocity of the centroid of the base platform, B Ȧ i is the differentiation of A i in the coordinates according to the base platform. Since l i is the strut unit vector, the derivation of Ȧ i can be obtained as following when define ẍ i 1 = li T Ä i as that.. ẍ 1 = ł T i li T B b. T i ẍ B + ł T i ( ω B ω B B b i + 2 ω B B ḃ i + B b i (A.8)... where still means to get antisymmetric matrix and x B is the attitude matrix of the base platform. Hence the boundary of ẍ 1 can be get and the simulation of the complex dynamic model of stewart platform can be simplified. References Traub, W., Shaklan, S., and Lawson, P. Terrestrial planet finder. University of California-Berkeley 4 No.8 (2007). Boroson, D. M., Scozzafava, J. J., Murphy, D. V., Robinson, B. S., and Shaw, H. The lunar laser communications demonstration (llcd). In Space Mission Challenges for Information Technology, SMC-IT Third IEEE International Conference on (2009), IEEE, pp Quenon, D., Boyd, J., Buchele, P., Self, R., Davis, T., Hintz, T. L., and Jacobs, J. H. Miniature vibration isolation system for space applications. In SPIE s 8th Annual International Symposium on Smart Structures and Materials (2001), International Society for Optics and Photonics, pp D.Stewart. A platform with six degrees of freedom. Proceedings of the institution of mechanical engineers Vol.180, No.1 (1965), pp Dasgupta, B., and Mruthyunjaya, T. The stewart platform manipulator: a review. Mechanism and machine theory Vol.35, No.1 (2000), pp Hanieh, A. A., Horodinca, M., Preumont, A., Loix, N., and Verschueren, J. P. Stiff and soft stewart platforms for active damping and active isolation of vibrations. In Proc. of ACTUATOR, eighth international conference on new actuators. Bremen (2002), pp Kim, N.-I., and Lee, C.-W. High speed tracking control of stewart platform manipulator via enhanced sliding mode control. In Robotics and Automation, Proceedings IEEE International Conference onvol. 3, (1998), IEEE, pp Geng, Z. J., and Haynes, L. S. Six degree-of-freedom active vibration control using the stewart platforms. Control Systems Technology, IEEE Transactions on Vol.2, No.1 (1994), pp Hanieh, A. A., Preumont, A., and Loix, N. Piezoelectric stewart platform for general purpose active damping interface and precision control. EUROPEAN SPACE AGENCY-PUBLICATIONS-ESA SP 480 (2001), pp

10 Preumont, A., Horodinca, M., Romanescu, I., De Marneffe, B., Avraam, M., Deraemaeker, A., Bossens, F., and Abu Hanieh, A. A six-axis single-stage active vibration isolator based on stewart platform. Journal of sound and vibration Vol.300, No.3 (2007), pp Bahrami, A., Tafaoli-Masoule, M., and Bahrami, M. N. Active vibration control of piezoelectric stewart platform based on fuzzy control. International Journal of Material and Mechanical Engineering (IJMME) Vol.2, No.1 (2013). Rahman, Z. H., Spanos, J. T., and Laskin, R. A. Multiaxis vibration isolation, suppression, and steering system for space observational applications. In Astronomical Telescopes & Instrumentation (1998), International Society for Optics and Photonics, pp Hanieh, A. A. Active isolation and damping of vibrations via stewart platform. Unpublished doctoral dissertation, Active Structures Laboratory, Universit e Libre de Bruxelles, Brussels, Belgium (2003). Zhang, Y. and Li, K. System Identification Theory and Application. National Defense Industry Press, (In Chinese) 10