Adaptive Hierarchical Decoupling Sliding-Mode Control of a Electric Unicycle. T. C. Kuo 1/30

Transcription

1 Adaptive Hierarchical Decoupling Sliding-Mode Control of a Electric Unicycle T. C. Kuo 1/30

2 Outline Introduction System Modeling Adaptive Hierarchical Decoupling Sliding-Mode Control Simulation Results Conclusions /30

3 Outline Introduction System Modeling Adaptive Hierarchical Decoupling Sliding-Mode Control Simulation Results Conclusions 3/30

4 Introduction (1/) Riding unicycle has been shown to gain many benefits for human health, establishing his or her psychological confidence on learning various sports. In assisting the development of the cerebellum, unicycle exercise also can help people promote their intelligence. Aware of the shortage of the resource of the Earth, more and more technology has been attempted to use unicycles for transportation between short distance and commute. 4/30

5 Introduction (/) Dr. David Vos, 90 s, MIT Trevor Blackwell 007 Twente University 008, MuRuta 5/30

6 Outline Introduction System Modeling Adaptive Hierarchical Decoupling Sliding-Mode Control Simulation Results Conclusions 6/30

7 System Modeling (1/6) Electric unicycle and its components 7/30

8 Fx y FMθφ VM H System Modeling (/6) The simple draw of rider and free-body diagrams θ j θ i j k i 8/30

9 System Modeling (3/6) The kinetic energy and the potential energy of the wheel 1 1 Twheel = mx& + I & mφ, Vwheel = 0 = mr + I & φ ( m ) / I : moment of inertia, m : mass, x& : velocity, & φ : angle velocity m no slip between the wheel and the floor The velocity of the center of gravity (COG) for the body v v v v v = xi & + l & θθ = r & φi + l & θθ G i i l : the distance between COG of the body and the center of the wheel 9/30

10 System Modeling (4/6) The kinetic energy and the potential energy of the body 1 v v 1 T M I & V Mgl body = ( G G ) + Mθ, body = cos( θ ) v = [ r & φ l & v θcos( θ)] i l & v θsin( θ) j G & & & & = IMθ /+ M{[ rφ+ lθcos( θ)] + l θ sin ( θ)}/ I M : moment of inertia, M : mass The total kinetic energy of the electric unicycle with rider T = T + T T wheel body = & φ /+ & θ /+ && θφcos( θ) Iφ Iθ Mrl I = mr + I + Mr, I = I + Ml φ m θ M 10/30

11 System Modeling (5/6) Define the state vector of the electric unicycle as v = [ θ φ] T s Define the viscous and the Coulomb frictions as Dv ( & ) = [ μ ( & θ & φ) + c sgn( & θ & φ) μ & φ+ c sgn( & φ)] T s The Lagrangian function of the electric unicycle with rider L = T V T θ θ φ φ μ and μ are respectively the viscous damping coefficients; θ φ c and c are respectively the coefficients of the Coulomb frictions. θ φ body The Euler-Lagrange equations of motion for electric unicycle d L L T ( ) = D( vs ) dt vs v s T & & T : the torque applied between the body and the wheel 11/30

12 System Modeling (6/6) The equations of motion I && θθ = T Mlr&& φcos( θ) + Mlgsin( θ) μθ( & θ & φ) cθsgn( & θ & φ) (1) I && T Mlr&& Mrl & & c & φφ = θ cos( θ) + θ sin( θ) μφφ φsgn( φ) () The substitution of (1) into () and () into (1) give two second-order dynamic equations of the inclination and rotation angles controlled by torque T && θ = A( θ) T + B( θ, & θ, & φ) && φ = C( θ) T + D( θ, & θ, & φ) where A( θ ) = [ I + Mlrcos( θ )] φ { II θ φ [ Mlrcos( θ)] } B & & 1 I & & c & & Mlg Mlr & c & Mlr & ( θ, θφ, ) = { [ ( ) sgn( ) sin( )] cos( )[ sgn( ) sin( )]} φ μθ θ φ + θ θ φ θ + θ μφ φ + φ φ θ θ { II θ φ [ Mlrcos( θ )] } C( θ) = [ I + Mlrcos( θ)] θ { II θ φ [ Mlrcos( θ)] } D 1 θ & θ & φ = Mrl θ μ & θ & φ + c & θ & φ Mlg θ I μ & φ + c & φ Mrl & θ θ (,, ) { cos( )[ ( ) sgn( ) sin( )] [ sgn( ) sin( )]} θ θ θ φ φ { II θ φ [ Mlrcos( θ )] } 1/30

13 Outline Introduction System Modeling Adaptive Hierarchical Decoupling Sliding-Mode Control Simulation Results Conclusions 13/30

14 Adaptive Hierarchical Decoupling Sliding-Mode Control (1/4) A. Controller Synthesis Only one torque T Two control goals 1. balancing ( θ θ desired ). velocity control ( & & ) φ φ desired Hierarchical Sliding Surface function Two first-layer sliding surface functions S ( θ ) = & θ + k ( θ θ ) = & θ + k θ TH TH desired TH S (& φ) = & φ & φ PH desired The second-layer sliding surface function S = S ( θ ) + αs (& φ) α: real and positive constant T TH PH k TH : real and positive constant 14/30

15 Adaptive Hierarchical Decoupling Sliding-Mode Control (/4) Differentiating S T, and using (1) obtain S& = A( θ) T + B( θ, & θ, & φ) + k & θ + αs& (& φ) (3) T TH PH T T eq = T eq β S T A( θ ) (4) B( θθφ, &, &) k & THθ αs& PH (& φ) = A( θ ) Hierarchical Decoupling Sliding-Mode Control β : real and positive constant (5) Adaptive Hierarchical Decoupling Sliding-Mode Control T βs ˆ sat( ) ˆ T + kv ST + kcsgn( S & T φ ) = Teq A( θ ) (6) sat( ): the saturation function : the absolute function kˆ v : estimate of the compensation for the viscous friction kˆ : estimate of the compensation for the Coulomb friction c 15/30

16 Adaptive Hierarchical Decoupling Sliding-Mode Control (3/4) The parameter updating laws ˆ& k = γ S sat( S ) (7) v v T T ˆ& k = γ S sgn( S & φ ) (8) c c T T B. Stability Analysis Lyapunov function Vt () where ST k% v k% c = + + γ γ k% = k kˆ v v v k% = k kˆ c c c v c γ v, γ : two positive and real adaptation gains c 16/30

17 Adaptive Hierarchical Decoupling Sliding-Mode Control (4/4) 1 ˆ 1 V& () t = SS& kk % & & kk % ˆ γ γ T T v v c c v c 1 ˆ 1 = S[ βs ( k k% )sat( S ) ( k k% )sgn( S & φ)] kk % & & kk % ˆ γ γ T T v v T c c T v v c c c 1 ˆ 1 = βs ks sat( S ) + ks % sat( S ) ks sgn( S & φ) + ks % sgn( S & & & φ) kk % kk % ˆ T v T T v T T c T T c T T v v c c γ γc = βs k S sat( S ) k S T v T T c = βs k S sat( S ) k S sgn( S & φ) 0 T v T T c T T Using (3)-(7) obtain 1 ˆ 1 sgn( S & φ) + k% & [ S sat( S ) k ] + k% [ S sgn( S & & φ) kˆ ] γv γc & kˆ v = γ vst sat( ST) & kˆ = γ S sgn( S & φ ) T T v T T v c T T c c c T T Since V& ( t) is negative definite, it is easy to prove via Lyapunov stability theory that the second-layer sliding function, S T ( t) converge to zero as time approaches infinity. 17/30

18 Outline Introduction System Modeling Adaptive Hierarchical Decoupling Sliding-Mode Control Simulation Results Conclusions 009/08/06, 07 18/30

19 Simulation Results (1/9) The first simulation : the electric unicycle for self-balancing Computer simulation parameters: The weights : M = 98, m =. The moments of inertia : I = , I = θ The half diameter of the wheel : r = 0.5. The distance between COG of the body and the center : l = 1. φ The friction coefficients between the body and the wheel : μ = 1, μ = 5, c = 1, c = 10. The initial conditions : θ = (rad) = 10(deg), & φ = 0. θ φ θ φ The parameters of the proposed controller : α = 0.1, β = 1, γ = 1, γ = 1, μ = 1, μ = 5, c = 1, c = 10. θ desired = 0, & φ = 0. desired v c θ φ θ φ 19/30

20 Simulation Results (/9) S T S TH S PH converge to zero time (sec) Time behavior of the sliding functions ST, STH and SPH for self-balancing. 0/30

21 Simulation Results (3/9) theta 1.0 k v k c angle (rad) time (sec) angle velocity (rad/sec) converge to zero d phi/dt time (sec) Simulation results of θ and & φ for self-balancing time (sec) Simulation results of kˆ and kˆ for self-balancing. v c 1/30

22 Simulation Results (4/9) The second simulation : robustness against uncertain frictions Computer simulation parameters: The weights : M = 98, m =. The moments of inertia : I = , I = θ The half diameter of the wheel : r = 0.5. The distance between COG of the body and the center : l = 1. The initial conditions : θ = (rad) = 10(deg), & φ = 0. φ The friction coefficients between the body and the wheel : μ = 1, μ = 5, c = 1, c = 10. θ φ θ φ The parameters of the proposed controller : α = 0.1, β = 1, γ = 1, γ = 1, μ = 0, μ = 0, c = 0, c = 0. θ desired = 0, & φ = 0. desired v c θ φ θ φ /30

23 Simulation Results (5/9) 3 S T.5 S TH S PH converge to zero time (sec) Time behavior of the sliding functions ST, STH and SPH for self-balancing with uncertain frictions. 3/30

24 Simulation Results (6/9) angle (rad) theta k v k c time (sec) angle velocity (rad/sec) converge to zero d phi/dt time (sec) time (sec) Simulation results of θ and & Simulation results of kˆ ˆ v and kc for self-balancing φ for self-balancing with uncertain frictions. with uncertain frictions /30

25 Simulation Results (7/9) The second simulation : angle velocity control with uncertain frictions Computer simulation parameters: The weights : M = 98, m =. The moments of inertia : I = , I = θ The half diameter of the wheel : r = 0.5. The distance between COG of the body and the center : l = 1. The initial conditions : θ = (rad) = 10(deg), & φ = 0. φ The friction coefficients between the body and the wheel : μ = 1, μ = 5, c = 1, c = 10. θ φ θ φ The parameters of the proposed controller : α = 0.1, β = 1, γ = 1, γ = 1, μ = 0, μ = 0, c = 0, c = 0. θ desired = 0, & φ = 5. desired v c θ φ θ φ 5/30

26 Simulation Results (8/9) S T S TH S PH Can t converge to zero Time(sec) Time behavior of the sliding functions ST, STH and SPH for angle velocity control with uncertain frictions. 6/30

27 Simulation Results (9/9) theta 4 k v angle(rad) k c time(sec) angle velocity (rad/sec) Steady state error d phi/dt time(sec) time(sec) Simulation results of θ and & φ for angle velocity Simulation results of kˆ and kˆ for angle velocity control with uncertain frictions. control with uncertain frictions. v c 7/30

28 Outline Introduction System Modeling Adaptive Hierarchical Decoupling Sliding-Mode Control Simulation Results Conclusions 8/30

29 Conclusions The adaptive hierarchical decoupling sliding mode controller has been proposed to accomplish robust balancing and angle velocity control (0 rad/sec) of the electric unicycle with neglected all of the friction coefficients. The reason for steady state error in final simulation is that θ is not a steady state. desired =0 an d & φ =5 desired 9/30

30 Thanks for your attention 30/30