NOn-holonomic system has been intensively studied in

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1 Trajectory Tracking Controls for Non-holonomic Systems Using Dynamic Feedback Linearization Based on Piecewise Multi-Linear Models Tadanari Taniguchi and Michio Sugeno Abstract This paper proposes a trajectory tracking control for non-holonomic systems using dynamic feedback linearization based on piecewise multi-linear PML) models The approximated model is fully parametric Input-output I/O) dynamic feedback linearization is applied to stabilize PML control system Although the controller is simpler than the conventional I/O feedback linearization controller the control performance based on PML model is the same as the conventional one The proposed methods are applied to a tricycle robot and a quadrotor helicopter Examples are shown to confirm the feasibility of our proposals by computer simulations Index Terms piecewise model tracking trajectory control dynamic feedback linearization non-holonomic system tricycle robot and quadrotor helicopter I INTRODUCTION NOn-holonomic system has been intensively studied in control engineering by many researchers But it is very difficult to control these systems because these systems cannot be asymptotically stabilized to an equilibrium point with smooth time-invariant state feedback control [] Therefore the non-holonomic system control is one of the challenging problems In control engineering a car robot dynamics a linked robot arm model hovercraft dynamics and helicopter dynamics are the typical non-holonomic systems This paper deals with a tracking trajectory control of a tricycle robot and a quadrotor helicopter using dynamic feedback linearization based on piecewise multi-linear PML) models Wheeled mobile robots are completely controllable However they cannot be stabilized to a desired position using time invariant continuous feedback control [] The wheeled mobile robot control systems have a non-holonomic constraint Non-holonomic systems are much more difficult to control than holonomic ones Many methods have been studied for the tracking control of wheeled robots The experimental approaches were proposed in eg [] [4]) The backstepping control methods were proposed in eg [] [6]) The sliding mode control methods were proposed in eg [7] [8]) and also the dynamic feedback linearization methods were in eg [9] [] []) For non-holonomic robots it is never possible to achieve exact linearization via static state feedback [] It was shown that the dynamic feedback Manuscript received August 7 This work was partially supported by the Grant-in-Aid for Scientific Research of Ministry of Education Culture Sports Science and Technology Japan MEXT KAKENHI Grant Number 68) T Taniguchi is with IT Education Center Tokai University Hiratsuka Kanagawa 99 Japan taniguchi@tokai-ujp M Sugeno is with Tokyo Institute of Technology linearization is an efficient design tool to solve the trajectory tracking and the setpoint regulation problem in [9] [] First reported quadrotor helicopter Gyroplane No was built in 97 by the Breguet Brothers [] A full-scale quadrotor helicopter was built by De Bothezat in 9 [4] Many methods have been applied to quadrotor helicopter control A visual feedback based on feedback linearization and backstepping-like control were used in [] Dynamics feedback linearization method was applied by [6] Sliding mode control methods were applied by [7] [8] [9] [] [7] and [9] designed the controller with the super twisting control algorithm based on sliding model control [] proposed a sliding mode controller to stabilize a cascaded underactuated system of helicopter [] applied a fuzzy controller and [] proposed a reinforcement learning method to the quadrotor helicopter In this paper we consider PML models as the piecewise approximation models of the tricycle robot and quadrotor dynamics The models are built on hyper cubes partitioned in state space and are found to be bilinear bi-affine) [] so the models have simple nonlinearity The model has the following features: ) The PML model is derived from fuzzy if-then rules with singleton consequents ) It has a general approximation capability for nonlinear systems ) It is a piecewise nonlinear model and second simplest after the piecewise linear PL) model 4) It is continuous and fully parametric The stabilizing conditions are represented by bilinear matrix inequalities BMIs) [] therefore it takes long computing time to obtain a stabilizing controller To overcome these difficulties we have derived stabilizing conditions [4] [] [6] based on feedback linearization [4] and [6] apply input-output linearization and [] applies full-state linearization We propose a dynamic feedback linearization for PML control system and design the tracking controllers to a tricycle robot and a quadrotor The control systems have the following features: ) Only partial knowledge of vertices in piecewise regions is necessary not overall knowledge of an objective plant ) These control systems are applicable to a wider class of nonlinear systems than conventional I/O linearization ) Although the controller is simpler than the conventional I/O feedback linearization controller the tracking performance based on PML model is the same as the conventional one This paper is organized as follows Section II introduces the canonical form of PML models Section III presents dynamic feedback linearizations of the car-like robot and the quadrotor helicopter Sections IV and V propose trajectory tracking controller designs using dynamic feedback lineariza-

2 tion based on PML models of the tricycle robot and the quadrotor helicopter Section VI shows examples demonstrating the feasibility of the proposed methods Finally section VII summarizes conclusions II CANONICAL FORMS OF PIECEWISE BILINEAR A Open-Loop Systems MODELS In this section we introduce PML models suggested in [] We deal with the two-dimensional case without loss of generality Define vector dσ τ) and rectangle R στ in twodimensional space as dσ τ) d σ) d τ)) T R στ [d σ) d σ )] [d τ) d τ )] ) σ and τ are integers: < σ τ < d σ) < d σ ) d τ) < d τ ) and d ) d ) d )) T Superscript T denotes a transpose operation For x R στ the PML system is expressed as σ τ ẋ ω x i )ω j x )f o i j) iσ jτ ) σ τ x ωx i )ω j x )di j) iσ jτ f o i j) is the vertex of nonlinear system ẋ f o x) ω σ x ) d σ ) x )/d σ ) d σ)) ω σ x ) x d σ))/d σ ) d σ)) ω τ ) x ) d τ ) x )/d τ ) d τ)) ω τ x ) x d τ))/d τ ) d τ)) and ωx i ) ω j x ) [ ] In the above we assume f ) and d ) to guarantee ẋ for x A key point in the system is that state variable x is also expressed by a convex combination of di j) for ωx i ) and ω j x ) just as in the case of ẋ As seen in equation ) x is located inside R στ which is a rectangle: a hypercube in general That is the expression of x is polytopic with four vertices di j) The model of ẋ fx) is built on a rectangle including x in state space it is also polytopic with four vertices fi j) We call this form of the canonical model ) parametric expression B Closed-Loop Systems We consider a two-dimensional nonlinear control system ẋ fo x) g o x)ux) 4) y h o x) The PML model ) is constructed from a nonlinear system 4) ẋ fx) gx)ux) ) y hx) σ τ fx) ωx i )ω j x )f o i j) iσ jτ σ τ gx) ω x i )ω j x )g o i j) iσ jτ σ τ hx) ωx i )ω j x )h o i j) iσ jτ σ τ x ωx i )ω j x )di j) iσ jτ and f o i j) g o i j) h o i j) and di j) are vertices of the nonlinear system 4) The modeling procedure in region R στ is as follows: ) Assign vertices di j) for x d σ) d σ) x d τ) d τ ) of state vector x then partition state space into piecewise regions see Fig ) ) Compute vertices f o i j) g o i j) and h o i j) in equation 6) by substituting values of x d σ) d σ) and x d τ) d τ ) into original nonlinear functions f o x) g o x) and h o x) in the system 4) Fig shows the expression of fx) and x R στ d τ ) f σ τ ) d τ) ω σ ω τ f σ τ) d σ) f x) f σ τ ) ω σ ω τ 6) f σ τ) d σ ) σ τ Fig Piecewise region f x) iσ jτ ωi ωj f i j) x R στ ) The overall PML model is obtained automatically when all vertices are assigned Note that fx) gx) and hx) in the PML model coincide with those in the original system at vertices of all regions III DYNAMIC FEEDBACK LINEARIZATIONS OF NON-HOLONOMIC SYSTEMS A Tricycle Robot Model We consider a tricycle robot model ẋ cos θ ẏ θ sin θ L tan ψ u u 7) ψ

3 x and y are the position coordinates of the center of the rear wheel axis θ is the angle between the center line of the car and the x axis ψ is the steering angle with respect to the car The control inputs are represented as u v s cos ψ u ψ v s is the driving speed Fig shows the kinematic model of tricycle robot The steering angle ψ is constrained y ψ The time derivative of η is calculated as ) ) L η f h ν cos θ u L f h L tanm tanh w) sin θ ν sin θ u L tanm tanh w) cos θ h h ) x y) Since the controller µ µ ) doesn t appear in the equation η we continue to calculate the time derivative of η Then we get η ) L f h L g L f hµ ) L f h Lg L L f h f h L g L f h ) ) µ L g L f h L g L f h ) µ L θ Equation ) shows clearly that the system is input-output linearizable because state feedback control µ L g L f h) L f h L g L f h) v Fig by x y) Kinematic model of tricycle robot ψ M < M < π/ The constraint [] is represented as ψ M tanh w w is an auxiliary variable Thus we get ψ Msech wµ u ẇ µ We substitute the equations of ψ and w into the tricycle robot model The model is obtained as ẋ cos θ ẏ θ sin θ L tanm tanh w) u µ 8) ẇ In this case we consider η x y) T as the output the time derivative of η is calculated as ẋ ) ) ) cos θ u η ẏ sin θ µ The linearized system of 8) at any points x y θ w) is clearly not controllable and the only u affects η To proceed we need to add some integrators of the input u Using dynamic compensators as u ν ν µ the tricycle robot model 8) can be dynamic feedback linearizable The extended model is obtained as ẋ u cos θ ẏ u sin θ θ ẇ u tanm tanh w) L µ µ 9) u ν ν x reduces the input-output map to y ) v The matrix L g L f h multiplying the modified input µ µ ) is non-singular if u Since the modified input is obtained as µ µ ) the integrator with respect to the input v is added to the original input u u ) Finally the stabilizing controller of the tricycle robot system 7) is presented as a dynamic feedback controller: B Quadrotor Helicopter Model u ν ν µ u Msech wµ ) We consider a quadrotor helicopter model [6] 4 ẋ f o x) g oi x)u oi i y h o x) ) x x y z ψ θ φ v v v p q r ) T h o x) x y z ψ ) T uo u o u o u o u o4 ) T v v v q sin φ sec θ r cos φ sec θ q cos φ r sin φ p q sin θ tan θ r cos φ tan θ f o x) A x /m g o A y /m A z /m g I y I z I x qr A p /I x I z I x I y rp A q /I y pq A r /I z I x I y I z G cos φ cos ψ sin θ sin φ sin ψ) m G cos φ sin θ sin ψ cos ψ sin θ) m G cos θ cos φ m G G G

4 g o g o g o4 d/i x d/i y /I z x y and z are the three coordinates of the absolute Fig Dynamic model of quadrotor z position of the quadrotor helicopter ψ θ and φ are the three Euler s angles of the attitude These angles are yaw angle π < ψ < π) pitch angle π/ < θ < π/) and roll angle π/ < φ < π/) respectively v v and v are the absolute velocities of the quadrotor helicopter with respect to an earth fixed inertial reference frame p q and r are the angular velocities expressed with respect to a body reference frame A x A y A z ) and A p A q A r ) are the resulting aerodynamics forces and moments acting on the quadrotor helicopter I x I y and I z are the moment of inertia with respect to the axes m is the mass g is the gravity constant and d is the distance from the center of mass to the rotors u is the resulting thrust of the four rotors u is the difference of thrust between the left rotor and the right rotor u is the difference of thrust between the front rotor and the back rotor u 4 is the difference of torque between the two clockwise turning rotors and the two counter-clockwise turning rotors We design a static state feedback controller [6] of the quadrotor helicopter model u o α o x) β o x)v o ) αx) A o x)b o x) βx) A o x) G A o x) G G dsin φ sec θ)/i y cos φ sec θ/i z B o x) L f o h o x) L f o h o x) L f o h o x) L f o h o4 x) ) T x y v o is a controller of the linearized system with respect to ) The linearized system of ) for all x is clearly not controllable since A o x) is singular To proceed we need to add some integrators of the input u o Using dynamic compensators with respect to u as ζ u o ξ ζ µ e ξ the quadrotor helicopter model ) can be dynamic feedback linearizable The other control inputs are also represented as µ e u o µ e u o µ e4 u o4 Then the extended model is obtained as 4 ẋ f e x) g ei µ ei i y h o x) x x y z ψ θ φ v v v p q r ζ ξ ) T v v v q sin φ sec θ r cos φ sec θ q cos φ r sin φ p q sin θ tan θ r cos φ tan θ A x /m G ζ f e x) A y /m G ζ A z /m g G ζ I y I z I x qr A p /I x I z I x I y rp A q /I y I x I y I z pq A r /I z ξ g e ) T g e d/i x ) T g e d/i y ) T g e4 /I z ) T 4) We apply the I/O feedback linearization to the extended system 4) Then the controller is obtained as µ e α e x) β e x)v e α e x) A e x)b e x) β e x) A e x) a a a L 4 f e h o x) A e x) a a a a a a B ex) L 4 f e h o x) L 4 f e h o x) a 4 a 44 L f e h o4 x) a ij L goj L f e h oi x) i j a 4k L gok L fe h o4 x) k 4 The coordinate is obtained as z e z e z e z e z e4 ) T

5 z e h o L fe h o x) L f e h o x) L f e h o x) ) T z e h o L fe h o x) L f e h o x) L f e h o x) ) T z e h o L fe h o x) L f e h o x) L f e h o x) ) T z e4 h o4 L fe h o4 x) ) T The I/O linearized system is formulated as ż e Az e Bv e ) A B A A A B B B A B C C C C A C ) A B ) T B ) T C ) C ) Stabilization linear controller v e F z e of the linearized system ) is designed so that transfer function CsI A) B is Hurwitz B Stabilizing Control Using Dynamic Feedback Linearization Based on PML Model We define the output as η x x ) T in the same manner as the previous section the time derivative of η is calculated as ) ) σ Lfp h η ) ẋ f d w i L fp h ẋ x ) i ))x f d i ))x i σ the vertices are f d i )) cos d i ) and f d i )) sin d i ) The time derivative of η doesn t contain the control inputs µ µ ) We calculate the time derivative of η We get η L h σ σ i σ w i x )f d i ))x 6 i σ w i x )f d i )) η L h σ σ σ 4 i σ w i x )f d i ))x 6 i σ w i x )f d i )) 4 x 4 )f d 4 i 4 ))x σ 4 4 x 4 )f d 4 i 4 ))x IV PML MODEL AND TRAJECTORY TRACKING CONTROLLER DESIGN OF THE TRICYCLE ROBOT MODEL A PML Model of the Tricycle Robot Model We construct PML model [7] of the tricycle robot system 9) The state spaces of θ and w in the tricycle robot model 9) are divided by the vertices x π π/6 π} and the vertices x 4 } The state variable is x x x x x 4 x x 6 ) T x y θ w u v ) T σ σ ẋ σ 4 i σ w i i σ w i i 4 σ 4 x )f d i ))x x )f d i ))x µ µ 4 x 4)f d 4i 4))x x 6 6) We can construct PML models with respect to f x) f x) and f x) The PML model structures are independent of the vertex positions x and x 6 since x and x 6 are the linear terms This paper constructs the PML models with respect to the nonlinear terms of x and x 4 Note that trigonometric functions of the tricycle robot 9) are smooth functions and are of class C The PML models are not of class C In the tricycle robot control we have to calculate the third derivatives of the output y Therefore the derivative PML models lose some dynamics In this paper we propose the derivative PML models of the trigonometric functions f d 4 i 4 )) tanm tanh d 4 i 4 ))/L We continue to calculate the time derivative of η We get η ) L h L g L h µ L g L h µ σ σ4 η ) x i σ w i x )f σ x x 6 x x σ d i )) i σ w i x )f d i )) i σ w i x )f d i ))µ σ i σ w i x )f d i )) 4 x 4 )f d 4 i 4 )) σ 4 σ 4 L h L g L h µ L g L h µ σ σ4 i σ w i x )f σ x x 6 x σ d i )) i σ w i x )f d i )) i σ w i x )f d i ))µ σ i σ w i x )f d i )) ) 4 x 4 )f d 4 i 4 )) 4 x 4 )f d 4 i 4 ))µ 4 x 4 )f d 4 i 4 )) σ 4 σ 4 ) 4 x 4 )f d 4 i 4 )) 4 x 4 )f d 4 i 4 ))µ

6 The vertices f d i )) f d i )) and The controller of 6) is designed as µ µ ) T L g L h) L h L g L h) v Lg L f p h L g L h L g L ) h L ) fp h L g L h L h Lg L f p h L g L h L g L ) h L g L v h v is the linear controller of the linear system 7) ż Az Bu 7) y Cz z h L fp h L h h L fp h L h ) T R 6 T A B C If x there exists a controller µ µ ) T of the tricycle robot model 6) since detl g L h) In this case the state space of the tricycle robot model is divided into vertices Therefore the system has local PML models Note that all the linearized systems of these PML models are the same as the linear system 7) In the same manner of ) the dynamic feedback linearizing controller of the PML system is designed as ü µ u Msech x 4 µ ) 8) µ L µ h L g L f hv The stabilizing linear controller v F z of the linearized system 7) is designed so that the transfer function CsI A) B is Hurwitz Note that the dynamic controller 8) based on PML model is simpler than the conventional one ) Since the nonlinear terms of controller 8) contain not the original nonlinear terms eg sin x cos x tanm tanh x 4 )) but the piecewise approximation models C Trajectory Tracking Controller Based on PML System We propose a tracking control [8] to the tricycle robot model 7) Consider the following reference signal model ẋr f r η r h r The controller is designed to make the error signal e e e ) T η η r as t The time derivative of e is obtained as ) Lfp h ė η η r p L fp h p ) Lfr h r L fr h r Furthermore the time derivative of ė is calculated as L ) ) ë η η r fp h p L L fr h r h p L f r h r Since the controller µ doesn t appear in the equation ë we calculate the time derivative of ë e ) η ) η r ) L ) fp h p L L h g L h p µ The tracking controller is designed as µ ) ) L fr h r L f r h r ü µ u Msech x 4 µ ) L µ h L f r h r L g L hv µ 9) The linearized system 7) and controller v F z are obtained in the same manners as the previous subsection The coordinate transformation vector is z e ė ë e ė ë ) T Note that the dynamic controller 9) based on PML model is simpler than the conventional one on the same reason of the previous subsection V PML MODEL AND TRAJECTORY TRACKING CONTROLLER DESIGN OF THE QUADROTOR HELICOPTER A PML Model of the Quadrotor Helicopter We construct PML model [9] of the quadrotor helicopter system 4) The state spaces of ψ θ and φ in the quadrotor helicopter model 4) are divided by the following vertices x 4 ψ π π/6 π/ π} x θ π/ δ δ / δ / δ π/} x 6 φ π/ δ δ / δ / δ π/} δ δ π/ We construct the following PML model of the quadrotor helicopter model 4) ẋ fx) y hx) 4 g i µ i i ) fx) x 7 x 8 x 9 f 4 f f 6 f 7 f 8 f 9 f f f x 4 ) T g ) T g d I x ) T T d g I y ) g 4 I z ) T

7 hx) h h h h 4 ) T x x x x 4 ) T σ f 4 x i σ σ x σ 6 f x i σ σ 6 i 6σ 6 x )ω i6 6 x 6)f 4 i i 6 ) σ 6 i 6σ 6 x )ω i6 6 x 6)f 4 i i 6 ) i 6σ 6 ω i6 6 x 6 )f i 6 ) x σ f 6 x x σ x i σ i σ σ 6 σ 4 f 7 A x /m x σ 6 σ 6 i 6σ 6 ω i6 6 x 6 )f i 6 ) i 6σ 6 x )ω i6 6 x 6)f 6 i i 6 ) i 6σ 6 x )ω i6 6 x 6)f 6 i i 6 ) σ i σ ω i6 6 x 6)f 7 i 4 i i 6 ) σ 4 σ f 8 A y /m x i σ σ 6 i 6σ 6 ω i4 4 x 4 ) σ 6 i 6σ 6 ω i4 4 x 4 ) x ) x ) ω i6 6 x 6)f 8 i 4 i i 6 ) σ σ 6 f 9 A z /m x x )ω i6 6 x 6)f 9 i i 6 ) i 6σ 6 i σ f I y I z x x A p I x f I x I y x x A r I z I z I x f I z I x I y x x A q I y In this paper we construct the PML models with respect to trigonometric functions Hence the state spaces of x 4 x and x 6 are divided into some vertices For example the vertices f 4 i i 6 ) and f 4 i i 6 ) are given by substituting the vertices x and x 6 for the nonlinear terms sin x 6 sec x sin φ sec θ) and cos x 6 sec x cos φ sec θ) Table I shows the vertices of PML model f 4 i i 6 ) In contrast nonlinear terms of f f and f are not transformed into the PML models Since the nonlinear terms of f f and f are multi-linear functions B Stabilizing Control Using Dynamic Feedback Linearization Based on PML Model The stabilizing conditions of PML system are represented by BMIs [] therefore it takes long computing time to obtain a stabilizing controller In this paper feedback linearization method is used for the controller design Using the feedback linearization method as the controller designs it is easy to design the continuous piecewise nonlinear controller for PML control system and the controller designs drastically reduce the time that it takes to find an optimal solution of the stabilizing condition by computer simulation Therefore PML modeling and feedback linearizations could be a powerful combination for the analysis and synthesis of nonlinear control systems We define the output as η η η η η 4 ) T x x x x 4 ) T the time derivative of η is calculated as L f h η L f h L f h L f h 4 ẋ ẋ ẋ ẋ 4 x 7 x 8 x 9 f 4 The time derivative of η doesn t contain the control inputs u u u u 4 ) We calculate the time derivative of η We get η L f h f 7 η L f h f 8 η L f h f 9 η 4 L f h 4 f 4 x f f 4 x 6 f 6 f 4 x f f 4 x f f 4 x x f 4 x d/i y µ f 4 x /I z µ 4 σ 6 x σ 6 f 4 x 6 x x σ ω i6 6 x 6 ) f 4 σ i 6 ) f 4 σ i 6 )) / i 6σ 6 ω i6 i 6σ 6 σ 6 x 6 ) f 4 σ i 6 ) f 4 σ i 6 )) / x ) f 4 i σ 6 ) f 4 i σ 6 )) / 6 i σ i σ x ) f 4 i σ 6 ) f 4 i σ 6 )) / 6 σ f 4 6 x x )ω i6 6 x 6)f 4 i i 6 ) i 6σ 6 σ f 4 6 x x )ω i6 6 x 6)f 4 i i 6 ) i 6σ 6 d σ ) d σ ) 6 d 6 σ 6 ) d 6 σ 6 ) We continue to calculate the time derivatives of η η and η We get η ) L f h f 7 x 4 f 4 f 7 x f f 7 x 6 f 6 f 7 x f η ) L f h f 8 x 4 f 4 f 8 x f f 8 x 6 f 6 f 8 x f η ) L f h f 9 x f f 9 x 6 f 6 f 9 x f f j x x 4 f j x x σ σ 6 x )ω i6 6 x 6) i σ i 6σ 6 f j σ 4 i i 6 ) f j σ 4 i i 6 )) / 4 σ 4 σ 6 ω i4 4 x 4 )ω i6 6 x 6) i 6σ 6 f j i 4 σ i 6 ) f j i 4 σ i 6 )) /

8 TABLE I VERTICES OF PML MODEL f 4 i i 6 ) f 4 x 6 π/ x 6 δ x 6 δ / x 6 x 6 δ / x 6 δ x 6 π/ x π/ x δ x δ / x x δ / x δ x π/ f j x x 6 σ f j 4 x σ 4 σ ω i4 4 x 4 ) x ) i σ f j i 4 i σ 6 ) f j i 4 i σ 6 )) / 6 σ σ 6 i σ i 6σ 6 ω i4 4 x 4) x )ω i6 6 x 6)f j i 4 i i 6 ) σ f 9 6 x ω i6 x 6 x 6 ) f 9 σ i 6 ) f 9 σ i 6 )) / i 6σ 6 σ f 9 x x x ) f 9 i σ 6 ) f 9 i σ 6 )) / 6 6 i σ σ f 9 x i σ σ 6 i 6σ 6 x )ω i6 6 x 6)f 9 i i 6 ) j 7 8 Since L f h L f h and L f h are independent of the controller u we continue to calculate the time derivatives of η ) η) and η ) η 4) L 4 f h L g L f h µ L g L f h µ L g L f h µ η 4) L 4 f h L g L f h µ L g L f h µ L g L f h µ η 4) L 4 f h L g L f h µ L g L f h µ L g L f h µ The details of the Lie derivatives are omitted due to lack of space The controller of ) is designed as µ αx) βx)v αx) A qr x)b qr x) βx) A qr x) L g L f h L g L f h L g L f h A qr x) L g L f h L g L f h L g L f h L g L f h L g L f h L g L f h L g L f h 4 L g4 L f h 4 B qr x) L 4 f h L 4 f h L 4 f h L f h 4) T v F z is the linear controller of the linear system ) z z z z z 4 ) T R 4 z h L f h L f h L f h ) T z h L f h L f h L f h ) T z h L f h L f h L f h ) T z 4 h 4 L f h 4 ) T A blockdiaga A A A 4 ) B ) B T B T B T B4 T T C blockdiagc C C C 4 ) ) A A A A A 4 B B ) B B 4 C ) C C C C 4 ) In this case the state space of the quadrotor helicopter robot model is divided into 7 7 vertices Therefore the system has 6 6 local PML models Note that all the linearized systems of these PML models are the same as the linear system ) The dynamic feedback linearizing controller of the PML system ) is designed as ü µ ) u i µ i i 4 The stabilizing linear controller v F z of the linearized system ) is designed so that the transfer function CsI A) B is Hurwitz Note that the dynamic controller ) based on PML model is simpler than the conventional one Since the nonlinear terms of controller ) contain not the original nonlinear terms eg sin φ cos φ sec θ) but the piecewise approximation models ż Az Bu y Cz ) C Trajectory Tracking Controller Based on PML System We propose a trajectory tracking control to the quadrotor helicopter robot model ) Consider the following reference

9 signal model ẋr f r η r h r The controller is designed to make the error signal e e e e e 4 ) T η η r as t The time derivative of e is obtained as L f h L fr h r ė η η r L f h L f h L fr h r L fr h r L f h 4 L fr h r Furthermore the time derivative of ė is calculated as L f h L ë L f h f r h r L f h L f r h r L L f h f r h r 4 L g L f h 4 u L g4 L f h 4 u 4 L f r h r4 Since the controller u doesn t appear in the equations ë ë and ë we calculate the time derivative of ë i i ) e ) L f h L L f h f r h r L L f h f r h r L f r h r e ) e ) We continue to calculate the time derivatives of e ) i i ) e 4) e 4) e 4) L 4 f h L 4 L 4 f h f r h r L 4 L 4 f h f r h r L 4 f r h r L g L f h µ L g L f h µ L g L f h µ L g L f h µ L g L f h µ L g L f h µ L g L f h µ L g L f h µ L g L f h µ The tracking controller is designed as ü µ u i µ i i 4 ) The linearized system ) and controller v F z are obtained in the same manners as the previous subsection The coordinate transformation vector is z e ė ë e ) e ė ë e ) e ė ë e ) e 4 ė 4 ) T Note that the dynamic controller ) based on PML model is simpler than the conventional one on the same reason of the previous subsection VI SIMULATION RESULTS We apply the trajectory tracking control to the tricycle robot model 7) and the quadrotor helicopter model ) Although the controller is simpler than the conventional I/O feedback linearization controller the tracking performance based on PML model is the same as the conventional one In addition the controller is capable to use a nonlinear system with chaotic behavior as the reference model A Tricycle Model In the following simulations the tricycle length L is [m] and the angle constrain M is π/ [rad] ) Ellipse-shaped Reference Signal: Consider an ellipse model as the reference trajectory ) ) xr R cos θ x r ) x r R sin θ x r ) R and R are the semiminor axes and x r ) x r )) is the center of the ellipse Fig 4 shows the simulation result The dotted line is the reference signal and the solid line is the tricycle tracking trajectory The semiminor parameters R and R are and The initial positions are set at x) y)) ) and x r ) y r )) ) Fig shows the control inputs u and ν of the tricycle Fig 6 shows the error signals of the tricycle position x y) ) Trajectory Tracking Control Using Some Ellipseshaped Reference Signals: Arbitrary tracking trajectory control can be realized using the ellipse-shaped tracking trajectory method The controller design procedure is as follows: ) Assign passing points p x i) p y i)) i n We consider the passing points: ) ) 6 ) 8 ) and 7) ) Construct some ellipses trajectories to connect the passing points smoothly From ) to ) the trajectory : ) ) xr cos θr 4) x r sin θ r π/ θ r π From ) to 6 ) the trajectory : ) ) xr 6 cos θr ) x r sin θ r π/ θ r From 6 ) to 8 ) the trajectory : ) ) xr 8 cos θr 8 6) x r sin θ r θ r π/ From 8 ) to 7) the trajectory 4: ) ) xr 6 cos θr 8 7) x r sin θ r 6 π/ θ r π/ ) Design the controllers 9) for the ellipse tracking trajectories 4)-7) We show a tracking trajectory control example for the tricycle robot system Fig 7 shows the reference signals 4)-7) and the tricycle tracking trajectory The dotted line is the reference signal and the solid line is the tricycle tracking trajectory Fig 8 shows the control inputs u and ν of the tricycle Fig 9 shows the error signals with respect to the tricycle position x y) B Quadrotor Helicopter Model ) Spiral descent reference trajectory: We consider two tracking trajectories as the reference signals Although the controllers are simpler than the conventional I/O feedback linearization controllers the tracking performance based on PML model is the same as the conventional one [6]

10 7 7) 6 Trajectory 4 8) y [m] ) ) x [m] y [m] 4 Trajectory 6) Trajectory ) Trajectory ) 4 6 x [m] Fig 4 Ellipse-shaped reference signal and the tricycle tracking trajectory Fig 7 Reference signals 4)-7) and the tricycle tracking trajectory u u [m] ν 4 4 ν [m] x [m] Fig Control inputs u and ν of the tricycle Fig 8 Control inputs u and ν of the tricycle error of x error of x [m] error of y error of y [m] t time Fig 6 Error signals of the tricycle position x y) Fig 9 Error signals of the tricycle position x y)

11 Consider a spiral-shaped reference trajectory [6] as the reference model x d cos t x d sin t x d t x d 4 π 8) The feedback gain is calculated as F F F F F F ) F 7 ) The helicopter is initially in hover flight and the initial positions are set at x x x x 4 ) ) and x r x r x r x r4 ) / π/) In this simulation the parameters are given by m 7 I x I y I z 4 d and g 98 Fig shows the trajectory signals The solid line is the state response x x x ) of the helicopter and the dotted line is the reference signal 8) ) Spiral ascending reference trajectory: Consider a spiral-shaped reference trajectory [6] as the reference model x d cos t x d sin t x d t x d 4 π 9) The feedback gain is the same as the previous example The helicopter is initially in hover flight and the initial positions are set at x x x x 4 ) ) and x r x r x r x r4 ) / π/) In this simulation the parameters are given by m 7 I x I y I z 4 d and g 98 Fig shows the trajectory signals The solid line is the state response x x x ) of the helicopter and the dotted line is the reference signal 9) These results confirm the feasibility of the tracking control performance Although the PML model ) and the controller 9) are simpler than the conventional dynamic feedback linearization controller and model [6] the control performance based on PML model is the same as the conventional one [6] x 4 Fig x Ascending reference trajectories ) Trajectory Tracking Control Using Some Ellipseshaped Reference Signals: Arbitrary tracking trajectory control can be realized using the ellipse-shaped tracking trajectory method The controller design procedure is as follows: ) Assign passing points p x i) p y i) p z i)) i n We consider the passing points: ) ) 4 ) and ) ) Construct these trajectories to connect the passing points smoothly From ) to ) the trajectory is x r x r sin θ r cos ϕ r sin θ r sin ϕ r ) cos θ r x r π/ θ r π and ϕ From ) to 4 ) the trajectory is x r x r x r x cos t 9 sin t t π ) From 4 ) to ) the trajectory is x r 4 sin θ r cos ϕ r 4 x r sin θ r sin ϕ r ) x r cos θ r π/ θ r and ϕ r ) Design the controllers ) for the ellipse tracking trajectories )-) We show the simulation result for the quadrotor helicopter system Fig shows the reference signals )-) and the quadrotor helicopter tracking trajectory The dotted line is the reference signal and the solid line is the tricycle tracking trajectory x 4 Fig x Decent reference trajectories x VII CONCLUSIONS We have proposed the trajectory tracking controller designs of a tricycle robot and a quadrotor as non-holonomic systems based on PML models The approximated model are fully parametric I/O dynamic feedback linearization is applied to stabilize PML control system PML modeling with feedback linearization is a very powerful tool for analyzing and synthesizing nonlinear control systems We also have designed the tracking controllers to the tricycle robot and the quadrotor Although the controllers are simpler than the conventional I/O feedback linearization controller the

12 x ) Trajectory Trajectory 4) ) x x 4 Trajectory Fig Reference signals )-) and the tracking trajectory of the quadrotor helicopter system tracking performance based on PML model is the same as the conventional one The examples have been shown to confirm the feasibility of our proposals by computer simulations REFERENCES [] R W Brockett Asymptotic stability and feedback stabilization Differential Geometric Control Theory Boston: Birkhäuser 98 pp 8 9 [] B d Andréa-Novel G Bastin and G Campion Modeling and control of non holonomic wheeled mobile robots in the 99 IEEE International Conference on Robotics and Automation 99 pp [] K Kimura S Nozawa Y Kakiuchi K Okada and M Inaba Tricycle manipulation strategy for humanoid robot based on active and passive manipulators control pp [4] F Kallasi D L Rizzni F Oleari M Magnani and S Caselli A novel calibration method for industrial agvs Robotics and Autonomouse Systems vol 94 pp [] R Fierro and F L Lewis Control of a nonholonomic mobile robot: backstepping kinematics into dynamics in the 4th Conference on Decision and Control 99 pp 8 8 [6] T-C Lee K-T Song C-H Lee and C-C Teng Tracking control of unicycle-modeled mobile robots using a saturation feedback controller IEEE Transactions on Control Systems Technology vol 9 no pp 8 [7] J Guldner and V I Utkin Stabilization of non-holonomic mobile robots using lyapunov functions for navigation and sliding mode control in the rd Conference on Decision and Control 994 pp [8] J Yang and J Kim Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots IEEE Transactions on Robotics and Automation pp [9] B d Andréa-Novel G Bastin and G Campion Dynamic feedback linearization of nonholonomic wheeled mobile robot in the 99 IEEE International Conference on Robotics and Automation 99 pp 7 [] G Oriolo A D Luca and M Vendittelli WMR control via dynamic feedback linearization: Design implementation and experimental validation IEEE Transaction on Control System Technology vol no 6 pp 8 8 [] E Yang D Gu T Mita and H Hu Nonlinear tracking control of a car-like mobile robot via dynamic feedback linearization in University of Bath UK no ID-8 4 [] A D Luca G Oriolo and M Vendittelli Stabilization of the unicycle via dynamic feedback linearization in the 6th IFAC Symposium on Robot Control [] R Xu and U Özgüner Sliding Mode Control of a Quadrotor Heilicopter in Proceedings of the 4th IEEE Conference on Decision & Control 6 pp [4] A Gessow and G Myers Aerodynamics of the helicopter third edition ed New York: Frederick Ungar Publishing Co 967 [] E Altug J P Ostrowski and R Mahony Control of a Quadrotor Helicopter Using Visual Feedback in Proceedings of the IEEE International Conference on Robotics & Automation pp 7 77 [6] V Mistler A Benallegue and N M Sirdi Exact linearization and noninteracting control of a 4 rotors helicopter via dynamic feedback in IEEE International Workshop on Robot and Human Interactive Communication pp 86 9 [7] M Bouchoucha S Seghour and M Tadjine Classical and Second Order Sliding Mode Control Solution to an Attitude Stabilization of a Four Rotors Helicopter : from Theory to Experiment in Proceedings of the IEEE International Conference on Mechatronics pp 6 69 [8] A Benallegue A Mokhtari and L Fridman Feedback Linearization and High Order Sliding Mode Observer For A Quadrotor UAV in Proceedings of the 6 International Workshop on Variable Structure Systems 6 pp 6 7 [9] L Derafa L Fridman A Benallegue and A Ouldali Super Twisting Control Algorithm for the Four Rotors Helicopter Attitude Tracking Problem in th International Workshop on Variable Structure Systems pp 6 67 [] C Zhang X Zhou H Zhao A Dai and H Zhou Three-dimensional fuzzy control of mini quadrotor uav trajectory tracking under impact of wind disturbance in Proceedings of the 6 International Conference on Advanced Mechatronic Systems 6 pp 7 77 [] J Hwangbo I Sa R Siegwart and M Hutter Control of a quadrotor with reinforcement learning IEEE Robotics and Automation Letters vol no 4 pp 96 7 [] M Sugeno On stability of fuzzy systems expressed by fuzzy rules with singleton consequents IEEE Trans Fuzzy Syst vol 7 no pp [] K-C Goh M G Safonov and G P Papavassilopoulos A global optimization approach for the BMI problem in Proc the rd IEEE CDC 994 pp 9 4 [4] T Taniguchi and M Sugeno Piecewise bilinear system control based on full-state feedback linearization in SCIS & ISIS pp 9 96 [] Stabilization of nonlinear systems with piecewise bilinear models derived from fuzzy if-then rules with singletons in FUZZ- IEEE pp 96 9 [6] Design of LUT-controllers for nonlinear systems with PB models based on I/O linearization in FUZZ-IEEE pp 997 [7] Trajectory tracking controller design for a tricycle robot using piecewise multi-linear models in Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 7-7 March 7 Hong Kong 7 pp 88 9 [8] T Taniguchi L Eciolaza and M Sugeno Look-Up-Table controller design for nonlinear servo systems with piecewise bilinear models in FUZZ-IEEE [9] Quadrotor control using dynamic feedback linearization based on piecewise bilinear models in 4 IEEE Symposium Series on Computational Intelligence 4 Tadanari Taniguchi receive the Doctor of Engineering degree from the University of Electro-Communications Tokyo Japan in He was born in Toyama Japan in 97 He graduated from the University of Electro-Communications His research interests include intelligent control and nonlinear control From to he was a research scientist at Brain Science Institute RIKEN Japan Then he has served Tokai University Kanagawa Japan since He was a Visiting Scientist with Universite de Valenciennes France in He is currently an Associate Professor of Tokai University He received a Best Paper Award of The 7 IAENG International Conference on Control and Automation

13 Michio Sugeno was born in Yokohama Japan in 94 He graduated from the Department of Physics The University of Tokyo Tokyo Japan He then worked with Mitsubishi Atomic Power Industry Then he served the Tokyo Institute of Technology as a Research Associate an Associate Professor and a Professor from 96 to After retiring from the Tokyo Institute of Technology he was a Laboratory Head at the Brain Science Institute RIKEN from to and then as a Distinguished Visiting Professor with Doshisha University from to He is currently an Emeritus Professor with the Tokyo Institute of Technology Tokyo Dr Sugeno was the President of the Japan Society for Fuzzy Theory and Systems from 99 to 99 and also the President of the International Fuzzy Systems Association from 997 to 999 He is the first recipient of the IEEE Pioneer Award in Fuzzy Systems with Zadeh in He also received the IEEE Frank Rosenblatt Award and Kampt de Ferit Award in