3D Independent Actuators Control Based On A Proportional Derivative Active Force Control

Transcription

1 Proceeding of Ocean, Mecanical and Aerospace October 21, 215 D Independent Actuators Control Based On A Proportional Derivative Active Force Control Kamarudin, a and Endra Pitowarno, b a Electronic Engineering, Electronic Engineering Polytecnic Institute of Surabaya, Indonesia b Mecatronic Engineering, Electronic Engineering Polytecnic Institute of Surabaya, Indonesia a kamarudin@polibatam.ac.id, b epit@eepis-its.edu Paper History Received: 29-September-215 Received in revised form: 18-October-215 Accepted: 2-October-215 ABSTRACT β γ W1,W2, W K τ IN I m Pitc Roll Reion weel Constant of motor torque Actual motor torque Massa of te inertia Current of motor Tis paper presents a control metod for D independent uators using PDAFC (Proportional Derivative Active Force Control. PD is used to stabilize te uators, were as AFC is used to reect disturbance uncertainty by estimating disturbance torque value of uator. Simulation result sows tat PDAFC can minimize disturbance uncertainly effect. To test te performances of a control system uator given disorder constant and sine. To researc it uses cube as an uator. wen give te uators disturbance t te constant value of 45, to acieve setpoint uators takes 1 second. Tis condition is in all uators weter it was te yaw, pitc and roll. Wen give te uators disturbance t te sine, to acieve setpoint uators takes 1.25 second. Tis condition is in all uators weter it was te yaw, pitc and roll. KEY WORDS: Inertia, Reion Weel, PID, AFC NOMENCLATURE AFC PD PID PDAFC LQR TTE Q D α Active Force Control Proportional Derivative Proportional Integral Derivative Proportional Derivative Active Force Control Linear Quadratic Regulator Traectory Tracking Error Disturbance Tree dimention Yaw 1. INTRODUCTION Tis paper presents a metod of control uators independent D. On tis experiment as an uator using cube mounted reion weel, as in a figure 1. Reion weel on te cube is used to produce te moment of inertia so tat te cube can be controlled. In tis researc control algoritm used is PDAFC (Proportional Derivative Active Force Control. Tere are many researc pertaining to te reion weel as uator t a variety of control algoritms. Moanaraa. from te Sss Federal Institute of Tecnology controls cube using algoritms LQR (Linear Quadratic Regulator for 2D and D uator [1,2]. Snider from te Air Force Institute of Tecnology, USA conducting researc controlling attitude on te satellite using reion weel and Control algoritms used is PID. Controlled system as a cange in attitude at ig speed, by using reion weel a system can be controlled properly []. Sirazi from Toosi University of Tecnology, Iran Conducting researc draws up reion weel saped like a pyramid, used to evaluate te performance of te satellite attitude control, t te goal of minimizing te total consumption of power from te system [4]. Pranaaya from University of Toronto, Canada researcing on te nano a satellite t te orbit of low to detect and identify a signal transmitted by a sip (Automatic Identification System. Tis satellite known as nano satellites Sip Tracking [5]. Mayer from Osaka University, Japan conduct te researc uses 12 Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering

2 Proceeding of Ocean, Mecanical and Aerospace October 21, 215 a reion weel as an uator to influence te movement of te robot in two ways: (1 Te roll and yaw stabilized by te rotation of te rotor, te iger te speed of te rotor ten slows te robot to re to disturbances. (2 His area in stabilizing by te acceleration of te rotor control using te principle of reion te [6]. Pitowarno from te Electronics Engineering Polytecnic Institute of Surabaya ad designed Active Force Control and Knowledge-Based System for planar two-oint robot arm to improve performance of Active Force Control [7]. Aorkar from te Amirkabir University of Tecnology, Teran Conducting researc to develop adaptive algoritms, neural network control of te systems. Actuators are used for satellite attitude control using 4 configuration reion weels [8]. Muelebac from te Sss Federal Institute of Tecnology nonlinear analysis and control of a reion weel based d invert pendulum [9]. To control uator needed algoritms control, te more simple an algoritm is getting better. Wit te control algoritm, uator stability is controlled despite getting disturbingly from te environment. Te more complicated algoritms need a computing performance ig. On tis researc control PD (Proportional Derivative serves to stabilize uator, but it still is not enoug if tere is any disturbance of te environment, Teore, added te AFC (Active Force Control control as te ability to cancel te disturbances tat occurs on te uator. Te purpose of tis researc is controlling uator D independently use algoritms control PD and AFC. Tis paper compiled divided into several capters. Capter 1 introduction, capter 2 presenting modeling a system uator, capter uator controller designs. Capter 4 provides te performance of te controller sown in te simulation numerical, and capter 5 conclusion of te researc. 2. ACTUATOR MODELING Before designing te controller, in tis section te matematical model of te uator ll be presented : 2.1. Kinematic Equation Consider te Cube balancing on its corner, as sown in Figure. 1. Let ω R denote te angular velocity of te cube relative to te inertial frame {I} expressed in te Cube s body fixed frame {B}, ω R, i = 1,2, te angular velocities of te weels around te rotational axis e Bi I and R SO ( denote B te attitude of te Cube relative to te inertial frame {I}. Note tat attitude of te Cube can also be represented by te xyz angle Euler ( Yaw (, Pitc (, and Roll ( α β γ.te relationsip between te rotation matrix and Euler angle representation is given by : I R e eα e e2b e e1γ B = % % % (1 Figure 1: Cube Actuator were e is te matrix exponential. Te nonlinear system dynamics are derived using Kane s equation for multi bodies given by : T T T T a T a δ v J p J n J F J T = P R P R (2 ( 6 v : = ω, ω, ω, ω R ( w1 w2 w denotes generalized velocity, denotes a particular rigid body of te multi-body system, p te linear momentum of te rigid body, n te angular momentum, F te external ive force, T a te a external ive torque, and J te Jacobian matrix. * Geometrically, Kane s equation can be interpreted as te proection of te Newton-Euler equation on to te configuration manifold s tangent space. Consider te Cube as a multi-body system consisting of four rigid bodies: Te Cube ousing and tree reion weels W1 (yaw as in a figure 2, W2 (pitc as in a figure and W (roll as in a figure 4. consider te Cube ousing and let and let γ denote te position of te center of mass of te Cube frame expressed in Cube body fixed frame {B} now, using : Υ & = ω xυ (4 Υ && = & ω xυ ω x ( ω xυ (5 Te time derivative of te ousing s linear momentum is given by p& m ( & ω xυ ω x ( ω xυ (6 Similarly, te time derivative of te ousing s angular momentum n = Θ ω is given by n& = Θ & ω ( Θ ω xω (7 x were Θ R is te inertia tensor of te Cube ousing Next, consider te i t weel i = 1,2, t angular velocity given by ω e ω. Te time derivative of te weel s linear momentum i is given by p& = m ( & ω xυ ω x w ( ω xυ (8 Were Υ te position of weel center of mass. Te angular momentum of te weel and its time derivative are given by 1 Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering

3 Proceeding of Ocean, Mecanical and Aerospace October 21, 215 ω ( ω ( ( ( Θ ( ii, ω e xω n = Θ Θ i, i e i n& = Θ & ω Θ ω xω Θ i, i & ω e i i x Were Θ R is te inertia tensor of te reion weel w i Te Jacobian matrices related to te Cube ousing are given by x6 J = Υ & ( = ( Y,,, P R (1 v x6 J = ( I,,, R v ω = & R (11 and Jacobian matrices related to te weels are given by ( x6 J P = Υ &,,, = Υ R (12 v x6 J = ( e ( I, R i i v ω ω = δ R (1 t Were δ Rx as all zero elements except for te i i Diagonal element, c is one. Te ive torque on te Cube ousing and te weels are given by T = ( T, T, T w1 w2 w = K u C ω, ω, ω (14 ( m w w1 w2 w (9 Figure 4: Actuator t te roll rotation Were K motor constant u: = m ( u, u, u 1 2 R is te current input of eac motor driving te weels, and C w is te damping constant. Finally, gravity g leads to an ive force on all bodies. Note tat. g is expressed in te body fixed frame { B } and given by : g( = g ( s( β, s ( γ c( β, c( γ c( β (15 2 Were g = 9.82 ms. Now, inserting (24 to (2 into (27 te follong equations of motion Θ & ω =Θ ω xω Mg Θ ω xω w w ( K u C ω m w w Θ & ω = K u C ω Θ & ω, = 1,2, (16 m w w w M= m Υ % m Υ% i = Θ= Θ m Υ % Θ m Υ% w i = 1 ( ( w w1 w2 w Θ = diag(θ 1,1,Θ 2,2,Θ (, Figure 2: Actuator t te yaw rotation ˆΘ=Θ Θ w Finally, te kinematic equation of te Cube is given by[1] s( β 1 & α ω = s( γ c( β c( γ & β c( γ c( β s( γ γ & (17 Figure : Actuator t te pitc rotation 2.2. Dynamic Equation Let φ and ψ describe te positions of te 1D inverted pendulum as sown in Figure 5. Next, let Θ denote te reion w weel s moment of inertia, Θ denote te system s total moment of inertia around te pivot point in te body fixed coordinate frame, and m tot and l represent te total mass and distance between te pivot point to te center of gravity of te wole system. 14 Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering

4 Proceeding of Ocean, Mecanical and Aerospace October 21, 215 purpose is to control te stability of uator by combining algoritms control te PD and AFC. Figure 6 is block a diagram of te uator stability control. Te controller design is focused to stabilize te uator toward disturbance. PD controller is used to stabilize uator and AFC to reect uncertainty disturbance from environment. In tis simulation, uator get constant and fluctuated. [ α, βγ, ] [ α, β, γ] error Reion Weel Controller PD [ && α βγ&& ],, IN/K I m Figure 5: Reion weel based 1D Te Lagrangian of te system is given by L = Θ ˆ & ϕ Θ (& ϕψ& mgcosϕ ( w were Θ ˆ =Θ Θ >, m= m l and g is te constant w tot gravitational acceleration. Te generalized moment are defined by : p : L ϕ = =Θ & ϕ Θ ψ& Figure (19 6: Diagram block PDAFC w & ϕ L p : = ψ =Θ & w ψ & ψ& ( ϕ (2 Let T denote te torque applied to te reion weel by te motor. Now, te equations of motion can be derived using te Euler-Lagrange equations t te torque T as a non potential force. Tis yield L p& = = mgsinϕ (21 ϕ ϕ L p& : = T = T (22 ψ ψ Note tat te introduction of te generalized moment in (19 and (2 leads to a simplified representation of te system, were (21 resembles an inverted pendulum augmented by an integrator in (22. Since te ual position of te reion weel is not of interest, we introduce x: = ( ϕ, p, p to represent te reduced ϕ ψ set of states and describe te dynamics of te mecanical system as follows[1]: ˆ 1 & ϕ Θ ( p p ϕ ψ x& = p& = f( x, T = mgsinϕ ϕ (2 p T & ψ. CONTROLLER DESIGN In tis capter presents te algoritm control for uator. Te.1 AFC Control Figure 6 sows te Active Force Control (AFC t IN is mass of te inertia moment of estimator matrix, K is constant of I is current of motor, τ is ual motor torque, motor torque, m τ is used ual motor torque, τ is measured motor torque, Q is disturbance, and Q is disturbance estimation. τ can be measured by torque sensor, considering (measuring te ual motor current multiplied t te motor torque constant. Wit erence to Figure 6 AFC, te simplified dynamic model of te system can be written as τ = τ Q= I( θ[ && αβγ && &&] (24 Were I (θ is te mass of te moment of inertia of reion weel and θ is angle of eac reion weel, && α, && βγ,&& is acceleration angular of body moving. From figure, te measurement of Q (i.e., an estimate of te disturbance, Q can be obtained suc tat Q = τ [&& α, && β, && γ] IN (25 Using te torque sensor or vice versa, if te current sensor is used ten te equation becomes. [ α, β, γ ] [ α, β, γ] Q = I K && && && IN m (26 Were te superscript ( denotes a measured or estimated quantity. Te torque τ can be measured directly using a torque sensor or indirectly by means of a current sensor. Te ual motor torque can be written as [ && α, && β, && γ] IN Q τ = K (27 K K 1/K 15 Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering

5 Proceeding of Ocean, Mecanical and Aerospace October 21, 215 Substitute equation (26 into equation (27 and (24, ten results a new equation tat is. [ && α, && β, && γ] IN I K [ && α, && β, && γ] IN m τ = K K K = [&& α, && β, && γ ] IN I K [ && α, && β, && γ] IN (28 m ' = IN && α, && β, && γ && α, && β, && γ I K ([ ] [ ] ( α, && β, α, && β, τ = IN && && γ && && γ I K Q (29 ' m Equation (28 is known as te equation of AFC controller output. It as been confirmed tat if te measured values or estimation of te parameters in te equation obtained accurately, it ll be a very solid system to reect te disturbances[8]..2 PDAFC Control Referring to Figure and equations (28 and (29, were te superscript mark ( indicates a measured or estimated number. Equation (28 sows a proportional form (P controller in connection t te use of errors acceleration signal. In tis context, IN can be regarded as a proportional constant. Tis is te known f tat te proportional controller works tout an additional dynamic parameter and as sufficient ability to increase te error steady state from te system because of te system dynamics an uncertainty. An addition of derivative component (D can enance te system performance by enforcing te occurred oscillation to te control in a minimum condition. Assuming tat te equation (28 is an acceleration local proportional control and as given tat te disturbances are igly nonlinear, varied and unpredictable, modification AFC sceme by incorporating derivative components of te inertia matrix estimator. Tis is named PDAFC. Te equation of derivative controller feedback can be written as follows G ( t = [ e( t c ] IN ( P de( t Gc( t = IN (1 D dt Wic of G c is control signal, If te error e (t is relatively G t ll become large and ll opefully correct te constant ( c error. By letting e ( t = ( α, && β, γ α, && β, γ m && && && && and ten incorporating equation ( into equation (28 to include te additional integral element and ten incorporating equation (1 into equation (29 to include te additional derivative element, te proposed algoritm is given by[8] : τ = IN (,,,, PDAFC P && α && β && γ && α && β && γ d ( α&, && β, & γ α, β, γ && && & (2 ' IN I K D m dt Wic of IN D is a derivative constant. Equation (2 is for PDAFC. 4. SIMULATION RESULT Te simulation test was performed using simulink to evaluate te performance of te controller. Te simulation model was used in S-Function block. In tis simulation te model contain disturbance tat as been modeled. PD coefficients tat used for simulations were derived by trial and error to get best performance, te PD parameter are listed in Table 1. Te first simulation is done t PDAFC system tout disturbance. Te second simulation performed PDAFC system t constant disturbance, by setting te value of 45. Te tird simulation performed PDAFC system t sine disturbance, by setting te value of te frequency of disturbance and amplitude of disturbance. Table 1. PD coefficients simulation parameter Parameter Value Kp 5 Ki 7 Kd.15 Kpafc.2 Kdafc Frequency sine 1 Hz Amplitude sine Vp-p Constant 45 Massa cube 2kg Gravitation 8.9 m/s 2 Angle ( Angle ( Cube Yaw Position( PDAFC(.2&2 PDAFC(.2&2Constant Disturbance Constant Disturbance Time (s Figure 7: Yaw Position constant disturbance Cube Pitc Position( PDAFC(.2&2 PDAFC(.2&2Constant Disturbance Constant Disturbance Time (s Figure 8: Pitc Position constant disturbance 16 Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering

6 Proceeding of Ocean, Mecanical and Aerospace October 21, Cube Roll Position( PDAFC(.2&2 PDAFC(.2&2Constant Disturbance Constant Disturbance Cube Roll Position( PDAFC(.2&2 PDAFC(.2&2sine Disturbance sine Disturbance Angle ( 8 6 Angle ( Angle ( Time (s Figure 9: Roll Position constant disturbance Cube Yaw Position( PDAFC(.2&2 PDAFC(.2&2sine Disturbance sine Disturbance Time (s Figure 12: Roll Position sine disturbance From te attempt to test performance control PDAFC by giving constant disturbance and sine disturbance on te uators. On experiment t constant disturbance, te time it takes of eac uator to steady 1 seconds can be seen in figure 7, 8 and 9. On experiment t sine disturbance, te time it takes of eac uator to steady 1.25 seconds can be seen in figure 1, 11 and 12. Angle ( Time (s Figure 1: Yaw Position sine disturbance Cube Pitc Position( PDAFC(.2&2 PDAFC(.2&2sine Disturbance sine Disturbance Time (s Figure 11: Pitc Position sine disturbance 5. CONCLUSION Te simulation results ave been presented to sow te performance of te controller. Altoug tere are a disturbance in uators, uators can remain stable briefly.1 seconds to constant disturbance and 1.25 seconds to sine disturbance. If uator observe time to steady, wen uator t disturbance and tout disturbance little te difference. Tis sows control working very well. ACKNOWLEDGEMENTS Te autor would like to tank te Electronic Engineering Polytecnic Institute of Surabaya for support REFERENCE [1] Gaamoan, M., Merz, M., Tommen, I., & D Andrea, R. (21. Te Culbi : A Reion Weel Based D Inveted Pendulum. IEEE, [2] Gaamoan, M., Merz, M., Tommen, I., & D Andrea, R. (212. Te Culbi : A Cube tat can Jump Up and Balance. IEEE/RSJ, [] Ray, S.E.(21. Tesis :Attitude Control of a Satellite Simulator Using Reion Weel and a PID Controller. US AIR Force Institute of Tecnology [4] Abolfazl, S., & Meran, M. (June 214. Pyramidal Reion Arrangement Optimization of Satellite Attitude Control Subsystem for minimizing Power Consumption. International ournal of Aeronautical and Space Sciences, Vol. 15 (2, pp [5] Freddy, M.P., & Robert, E.Z. (April 29. Nanosatellite Tracking Sips : Responsive Seven mont Nanosatellite Construction for a Rapid on Orbit Automatic Identification 17 Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering

7 Proceeding of Ocean, Mecanical and Aerospace October 21, 215 System Experien. 7t Responsive space Conference (pp.1-7. Los Angeles, CA : 7t Responsive Space. [6] Mayer, N. M., Masui, K., Browne, M., Asada, M., & Ogino, M.(26. Using a gyro as a tool for Continuously Variable lateral Stabilization of Dynamic Bipeds. ABBI, [7] Pitowarno, Endra. "An implementation of a knowledge-based system metod to an ive force control robotic sceme." Master Tesis, Universiti Teknologi Malaysia, 22. [8] Abbas, A., Alireza, F., Farad, F. S., & Mansour, K. (214. Design of an Adaptive-Neural Network Attitude Controller of a Satellite using Reion Weels. Applied and Computational Mecanics, [9] Muelebac, M., Moanaraa, G., & D Andrea, R. (21. Nonlinear Analysis and Control of a Reion Weel-based D Inverted Pendulum. American Control Conference, Publised by International Society of Ocean, Mecanical and Aerospace -scientists and engineers- Mecanical Engineering