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Stance Phase Control System of a Jumping Robot
Sinaia, Romania, September 9-11, 2019 7th and 7th IFAC IFAC Symposium Symposium on on System System Structure Structure and Control Control Available online at www.sciencedirect.com Sinaia, Romania, September 9-11,Structure 2019 7th IFAC Symposium on System System Structure and Control Control Sinaia, Romania, September 9-11, 2019 7th IFAC Symposium on and Sinaia, Romania, September 9-11, 2019 Sinaia, Romania, September 9-11, 2019
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IFAC PapersOnLine 52-17 (2019) 111–116 Stance Phase Control System of a Jumping Robot Stance System of Stance Phase Phase Control Control System of aa Jumping Jumping Robot Robot Mircea Ivanescu* Stance System of aa Jumping Jumping Robot Robot Stance Phase Phase Control Control System of
Mircea Ivanescu* Ivanescu* Mircea *University of Craiova, Craiova, Romania (e-mail: [email protected]). Mircea Ivanescu* Mircea Ivanescu* *University of Craiova, Craiova, Craiova, Craiova, Romania Romania (e-mail: [email protected]). *University of (e-mail: [email protected]). *University of of Craiova, Craiova, Craiova, Craiova, Romania Romania (e-mail: (e-mail: [email protected]). [email protected]). *University Abstract: The paper treats the control of the jumping robot during the stance phase using the fractal model of the system. Considering an of actuation system based on the Electro-Rheologic (ER) fluid Abstract: paper treats control the robot during the stance using the fractal Abstract: aThe The paper treatsis the the control oflinearized the jumping jumping robot during the models stance phase phase usingand the control fractal controller, fractal model inferred. The model and nonlinear are studied Abstract: The paper treats the control of the jumping robot during the stance phase using the fractal model Considering an of actuation system based on Electro-Rheologic fluid Abstract: The system. paper treats the control the jumping robot during the stance phase using (ER) the fractal model of of the the system. Considering actuation system based on Electro-Rheologic (ER) fluid frequential aremodel proposed usingan YKP criterions. Observer modelsmodels are proposed for and linear and model of system. Considering an actuation system based on Electro-Rheologic (ER) fluid controller, alaws fractal is inferred. inferred. The linearized model and nonlinear nonlinear are studied studied control model of the the system. Considering an actuation system based on the the models Electro-Rheologic (ER)control fluid controller, a fractal model is The linearized model and are and nonlinear systems and the global stability for “system-observer” is studied by Lyapunov techniques. controller, a fractal model is inferred. The linearized model and nonlinear models are studied and control frequential alaws laws aremodel proposed using The YKP criterions. Observer modelsmodels are proposed proposed for and linear and controller, fractalare is inferred. linearized model and nonlinear are studied control frequential proposed using YKP criterions. Observer models are for linear and Numerical simulations are presented. frequential laws using YKP criterions. Observer are proposed for linear nonlinear and the stability “system-observer” is Lyapunov frequential laws are are proposed using YKP for criterions. Observer models models are by proposed for techniques. linear and and nonlinear systems systems andproposed the global global stability for “system-observer” is studied studied by Lyapunov techniques. nonlinear the global stability for “system-observer” is Numerical simulations are nonlinear systems and thepresented. global stability for frequential “system-observer” is studied studied by by Lyapunov Lyapunov techniques. techniques. Copyright ©systems 2019. Theand Authors. Published by Elsevier Ltd. All rights reserved. Numerical simulations are presented. Keywords: robotics, control, stability, observer, criterion Numerical Numerical simulations simulations are are presented. presented. Keywords: stability, observer, observer, frequential frequential criterion criterion Keywords: robotics, robotics, control, control, stability, Keywords: robotics, control, stability, observer, frequential criterion Keywords: robotics, control, stability, observer, frequential criterion 1. INTRODUCTION 2. JUMPING MOTION CYCLE 1. INTRODUCTION MOTION This paper analyses the control problem of a class of robots, 2. A JUMPING bipedal motion cycle CYCLE is shown in Fig 1. (Ivanescu et all., 1. INTRODUCTION 2. JUMPING MOTION CYCLE 1. INTRODUCTION 2. JUMPING MOTION CYCLE the jumping biped robots, during the Stance Phase of a 2018) There are two main phases: Phase when the 1. INTRODUCTION 2. JUMPING MOTION CYCLE This paper paper analyses analyses the the control control problem problem of of aa class class of of robots, robots, A bipedal motion cycle shown in Stance Fig (Ivanescu et all., This A bipedal cycle is iswith shown Fig 1. 1. and (Ivanescu etPhase all., motion cycle. The motion study of these robots represented aa 2018) foot is There in motion direct contact the in ground Flight This paper analyses the control problem of a class of robots, A bipedal motion cycle is shown in Fig 1. (Ivanescu et all., the jumping biped robots, during the Stance Phase of are two main phases: Stance Phase when the This paper analyses control problem a class of robots, bipedal motion cyclemain is shown in Stance Fig 1. (Ivanescu et all., the jumping biped the robots, during the ofof Stance Phase of toa A 2018) There areleaves two phases: Phase when the fascinated attraction for a great number researchers due when the robot thewith ground. The motion isFlight determined the jumping biped robots, during the Stance Phase 2018) There are two phases: Stance Phase when the motion cycle. The motion motion study of these these robots represented is in direct the ground Phase the jumping biped robots, during the robots Stancerepresented Phase of of aa foot 2018) There are contact two main main phases: Stanceand Phase when the motion cycle. The study of foot is in direct contact with the ground and Flight Phase complexity control laws, the number of of constraints anddue large by two legs, each leg having aThe mechanical architecture motion cycle. The motion study of these robots represented a foot is in direct contact with the ground and Flight Phase fascinated attraction for a great number researchers to when the robot leaves the ground. motion is determined motion cycle. The motion study number of theseof robots represented a foot isthe in robot directleaves contact with the The ground and isFlight Phase fascinated attraction for a great researchers due to when the ground. motion determined possibilities to develop specific theories and technological consisting by an elastic lower configuration and an upper fascinated for great number researchers to when the robot leaves the ground. motion determined complexityattraction control laws, the number of of constraints anddue large two mechanical fascinated attraction for aathe great number of researchers due to by when thelegs, robot each leavesleg thehaving ground.aaThe The motion is is architecture determined complexity control laws, number of constraints and large by two legs, each leg having mechanical architecture achievements. A plethora of papers treats this topic and we configuration that ensures the actuation and damping effect complexity control laws, specific the number number of constraints constraints and large large consisting by two legs, legs, each leg having having mechanicaland architecture possibilities control to develop develop specific theories and technological technological by an elastic lower configuration an upper complexity laws, the of and by two each leg aa mechanical architecture possibilities to theories and consisting by ER an elastic lower configuration and an upper wish to mention (Masalkino et al.,2018, Szolt et al.,2018) for (Fig 2). An driving system allows the ER viscosity possibilities to develop specific theories and technological consisting by an elastic lower configuration and an upper achievements.to A Adevelop plethoraspecific of papers papers treats and this topic topic and we we configuration ensures the actuation and damping possibilities theories technological consisting by that an elastic lower configuration and an effect upper achievements. plethora of treats this and configuration that ensures the actuation and damping effect their contribution at the implementation of the bipedal modification and the implementation of the control strategies. achievements. A plethora of papers treats this topic and we configuration that ensures the actuation and damping effect wish to to mention mentionA(Masalkino (Masalkino etpapers al.,2018, Szolt ettopic al.,2018) for (Fig 2). An ER allows the ER viscosity achievements. plethora ofet treats thiset and we configuration that driving ensures system the actuation and damping effect wish al.,2018, Szolt al.,2018) for (Fig 2). An ER driving system allows the ER viscosity jumping and running strategies.ofet The servowish mention et for (Fig 2). ER driving system the viscosity their to contribution at the the motion implementation the bipedal bipedal implementation of wish tocontribution mention (Masalkino (Masalkino et al.,2018, al.,2018, Szolt Szoltofet al.,2018) al.,2018) for modification (Fig 2). An An and ER the driving system allows allows the ER ERstrategies. viscosity their implementation the modification and the implementation of the the control control strategies. technologies for running theat controllers thatstrategies. supervise the jumping their contribution at the implementation of the bipedal modification and the implementation of the control strategies. jumping and motion The servotheir contribution at the motion implementation of The the bipedal modification and the implementation of the control strategies. jumping and running strategies. servofunctions are analysed in (Taima et al., 2009, Sang-Ho, 2009) jumping and motion strategies. The servotechnologies for running the controllers controllers that supervise the jumping jumping jumping andfor running motionthat strategies. The servotechnologies the supervise the The principles of the hoping motion by using the equivalence technologies for that the functions are are analysed analysed in (Taima (Taima et et al.,supervise 2009, Sang-Ho, Sang-Ho, 2009) technologies for the the controllers controllers that supervise the jumping jumping functions in al., 2009, 2009) with the spring-loaded inverted pendulum model are studied functions are analysed in (Taima et al., 2009, Sang-Ho, 2009) The principles of the hoping motion by using the equivalence functions are analysed in (Taima et al., 2009) The principles of the hoping motion by 2009, using Sang-Ho, the equivalence in et the al.,hoping 2009 , motion Raibert, The The principles of the hoping motion by 1986). using the equivalence with(Takenaka the spring-loaded spring-loaded inverted pendulum model aredynamic studied The principles of by using the equivalence with the inverted pendulum model are studied analysis theetmotion cycles and the 1986). identification of the with the spring-loaded inverted pendulum model are studied ,, Raibert, The in (Takenaka (Takenaka al., 2009 2009 with the of spring-loaded inverted pendulum model studied in et al., Raibert, 1986). Thearedynamic dynamic control laws is presented in (Niiama et al., 2009). in (Takenaka al., ,, Raibert, The analysis of the cycles and the identification of in (Takenaka etmotion al., 2009 2009 Raibert, 1986). The dynamic dynamic analysis of treats theet motion cycles the 1986). identification of the the Thisr paper the control of and the jumping robot during the analysis of the motion cycles and the identification of control laws is presented in (Niiama et al., 2009). analysis of the motion cycles and et theal.,identification of the the control laws is presented in (Niiama 2009). stance phase using the fractal model of the system. control lawstreats is presented presented in (Niiama (Niiama et al., al., 2009). 2009). Thisr paper the of robot during the control laws is in et Thisr paper treats the control control of the the jumping jumping robot during the Considering anusing actuation system based on the ElectroThisr paper treats the control of the jumping robot during the stance phase the fractal model of the system. Thisr paper treats the control of the jumping robot during the stance phase using the fractal model of the system. Rheologic (ER) fluid controller, a fractal model is inferred. stance phase using the fractal model of the system. Considering actuation system based on stance phasean using the fractal model of the Electrosystem. Considering an actuation system based on ElectroThe linearized model and nonlinear modelson are studied and Considering an actuation system based ElectroRheologic (ER) controller, a fractal is Considering an fluid actuation system based model on the the ElectroRheologic (ER) fluid controller, fractal model is inferred. inferred. control laws are proposed. It aashould be noted that and the Rheologic (ER) fluid controller, fractal model is inferred. The linearized model and nonlinear models are studied Rheologic (ER)model fluid controller, a fractal model is inferred. The linearized and nonlinear models are with studied and Fig.1. Motion cycle literature (Dadras et al., 2017) propose solutions control The linearized model and models studied and control laws are proposed. It should be noted that the The linearized model and nonlinear nonlinear models are studied and Motion cycle control laws Itimplementable should be are noted that they the Fig.1. Fig.1. MotionPHASE cycle MODEL laws that are are notetproposed. practical because control laws are proposed. It should be noted that the 3. STANCE literature (Dadras al., 2017) propose solutions with control Motion cycle control laws are etproposed. Itpropose shouldsolutions be noted that the Fig.1. literature (Dadras al., 2017) with control Fig.1. Motion cycle require the measurement of all fractal variables that are literature (Dadras et al., 2017) propose solutions with control laws that are not practical implementable because they literature (Dadras et al., 2017) propose solutionsbecause with control laws that are not practical implementable they 3. STANCE PHASE 3. STANCE PHASE MODEL MODEL usually without anot physical meaning. To avoid these problems, laws that are practical implementable because they require the measurement of all fractal variables that are 3. STANCE PHASE MODEL laws that are not practical implementable because they require the measurement of all fractal variables that are The mechanical architecture is presented in Fig 3. The system 3. STANCE PHASE MODEL ausually class without of control laws defined on the fractal model with require the measurement of all fractal variables that are a physical meaning. To avoid these problems, require the measurement of all fractal variables that are usually without a physical meaning. To avoid these problems, is a two-leg system where the upper component is system a rigid respect to control direct observable variables are these proposed and The mechanical architecture is in 3. usually aa physical meaning. To avoid problems, a class class without of laws defined on the fractal model with The mechanical architecture is presented presented in Fig Fig 3. The The system usually without physical meaning. To avoid these problems, afrequential of control laws defined on the fractal model with beam and the lower component is an elastic curved structure. The mechanical architecture is presented in Fig Fig 3. 3. The The system are studied. A fractalfractal observer modeland is is a mechanical two-leg where the upper component is system aa rigid aarespect class control laws on model The architecture is in respect to criterions direct observable observable variables are proposed is two-leg system system where are thepresented upper in component rigid class of of control laws defined definedvariables on the the fractal model with with to direct are proposed and Theaa geometrical parameters shown Figcurved 3. Theis massless is two-leg system where the upper component is aa rigid proposed and the global stability “system-observer” is beam and the lower component is an elastic structure. respect to direct observable variables are proposed and is a two-leg system where the upper component is rigid frequential criterions are studied. studied. A fractal fractalare observer model is beam and the lower component is an elastic curved structure. respect to criterions direct observable variables proposed and frequential are A observer model is legs geometrical are the considered, all are gravitational components are beam and lower component is an elastic curved structure. studied by and Lyapunov techniques forfractal approximate linear and The parameters shown in Fig 3. The massless frequential criterions are studied. A observer model is beam and the lower component is an elastic curved structure. proposed the global stability “system-observer” The geometrical parameters are shown in Fig 3. The massless frequential criterions are studied. A fractal observer model is concentrated in the Centre of Gravity (COG). The dynamic proposed and the global stability “system-observer” The geometrical parameters are shown in Fig 3. The massless nonlinear model. Numerical simulations are presented. are considered, all gravitational are proposed and the stability “system-observer” is The geometrical parameters shown in Figcomponents 3. The massless studied by Lyapunov techniques for approximate linear and legs are considered, all are gravitational are proposed and the global global stability “system-observer” is legs studied by is Lyapunov techniques for approximate linear and model of the robot in the stance phase iscomponents obtained using legs are considered, all gravitational components are The paper structured as follows: section 2presented. presents motion concentrated in the Centre of Gravity (COG). The dynamic studied by Lyapunov techniques for approximate linear and legs are considered, all gravitational components are nonlinear model. Numerical simulations are concentrated in the Centre of Gravity (COG). The dynamic studied bymodel. Lyapunov techniques for approximate linear and Lagrange equations (Raibert, 1986). nonlinear Numerical simulations arethe presented. concentrated in the Centre of Gravity (COG). The dynamic cycle of a jumping robot, section 3 treats Stance Phase model of the robot in the stance phase is obtained using nonlinear model. Numerical simulations are presented. concentrated in the Centre of Gravity (COG). The dynamic The paper is structured as follows: section 2 presents motion model of the robot in the stance phase is obtained using nonlinear model. Numerical simulations are2presented. The paper is structured as follows: section presents model of the in Model,, section 4 describes the fractal model, section 5motion treats Lagrange 1986). The paper structured as 22the presents motion of equations the robot robot(Raibert, in the the stance stance phase is is obtained obtained using using cycle of aa is jumping robot, section 33section treats Stance Phase Lagrange equations (Raibert, 1986). phase The paper is structured as follows: follows: section presents motion cycle of jumping robot, section treats the Stance Phase model Lagrange equations (Raibert, 1986). the control system and section 6 verifies the control cycle of a jumping robot, section 3 treats the Stance Phase Lagrange equations (Raibert, 1986). Model,, 44 describes the fractal model, treats cycle of section a jumping robot, section 3 treats thesection Stance55 Phase Model,, section describes the fractal model, section treats techniques by system numerical simulations. Model,, section 4 fractal section treats the and section verifies the Model,, section 4 describes describes the fractal66 model, model, section 5control treats the control control system and the section verifies the 5 control the control system and section 6 verifies the control techniques numerical simulations. the controlby and section 6 verifies the control techniques by system numerical simulations. techniques numerical simulations. Copyright © by 2019 IFAC 207 techniques by numerical simulations. 2405-8963 Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 207Control. Copyright © 2019 IFAC 207 10.1016/j.ifacol.2019.11.036 Copyright © 2019 IFAC 207 Copyright © 2019 IFAC 207
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and fractional exponent coefficient strain relation is (Psaola et all., 2013)
(Fig 4). The stress-
(4) is the Caputo fractional derivative of order
where
(5) In this analysis, the viscoelastic components are characterized that correspond to an intermediate state by between elastic and viscous fluid components. Substituting these components (2-5) in (1) and considering the following , after simple calculations, results constraints
Fig. 2. Leg configuration
, (6) is determined by the admissible values of the where is the geometrical configuration during the motion, ,(the elastic equivalent coefficient, variable was omitted). The last term represents the nonlinear component determined by the gravitational forces. Assume that the viscoelastic exponent is =1/2. The initial conditions are defined by the initial position and velocity, , Fig. 3. Geometrical leg model
(7)
4. STATE MODEL For the stability analysis and the identification of the oscillation regime, the following state variables are defined, (8) (9) (10)
Fig. 4. ER fluid model (11) -
Substituting (7-10) into (6), yields
(1)
are gravitational and elastic potential, where is the respectively, is the equivalent inertial moment, actuator torque and represents the viscoelastic component,
(12) This equation can be rewritten as,
(2) (13) (3) is the fractal state vector of the where dynamic model. The output is defined by the direct measurable variable
The viscoelastic behaviour is determined by two components: a classical viscous friction in the rotational articulation C (Fig and the viscoelastic component created in 3) defined by ER damper. This component is analysed by using the fractional Kevin-Voigt model in which a pure elastic phase is connected in parallel defined by the elastic coefficient with a fractional viscoelastic phase that is characterized by
(14) The inequality 208
gravitational
nonlinear term is This component verifies the
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113
(23) (15) where
are obtained from Initial conditions for the variables the physical corresponding variables, angular position and velocity
In order to prove the stability, the following Lyapunov function is considered
(16) and , , that have not a physical meaning will have (Monie et all., 2010). ;
is a symmetrically, positive definite matrix. This where of function satisfies the condition (Li et all., 2010, Dadras et all., 20217) for The fractional derivative will be
(17)
From (8-14), we can easily infer that the fractal dynamic model (13), that satisfies (15)-(17), verify the conditions that ensure the existence and unicity of solutions (Diethelm., 2004) Remark 1 The linearized model of (13) is obtained if the is approximated as nonlinearity
(24) and substituting (19) yields
(18) (25) This relation can be rewritten
where is the new form of where the elastic coefficient . includes the term is Remark 2. The system defined by is controllable, respectively observable. The matrix fractional order stable ( ) but it is not Hurwitz stable. Remark 3. The classical transfer function can be obtained from the linearized model of (6) as (19)
(26) Considering the sector control law (20) and the condition (21), this relation becomes
(27)
5. CONTROL SYSTEM
Applying (22) and substituting the conditions of YKP Lemma (Khalil, 2002), results
5.1 Linearized Model a) Case 1. Consider the system (18) and the following control law is proposed
Using (20) and the condition (23), this inequality becomes
(20)
,
This control ensures the asymptotical stability if the following conditions are satisfied: a) There exists a positive definite matrix such that the matrix is Hurwitz stable b) control law (20) verifies the sector condition
where
5.2 Nonlinear Model
(21) c)
=
b) Case 2. Consider the nonlinear model (13), (14) with the initial conditions (15)-(18). The system (13)-(18) is asymptotically stable if the control law is
The frequency condition is satisfied (22)
There exists a gain
(28)
of the control low (20) that satisfies
and the following conditions are satisfied:
209
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“system-observer” defined by (19), (34) and the following control law is proposed:
a. The matrix is stable, b. (29) where and is the minimum The following Lyapunov function is eigenvalue of proposed to analyse the stability of the system,
(38) that satisfies the conditions: a) There exists a positive definite matrix such that the matrix is Hurwitz stable b) is Hurwitz stable (39)
(30) is a positive constant, . The where condition (i) of Theorem 5.4 (Li et all, 2010) is satisfied. For a control law, (31) we can infer that
c)
(40)
d)
(41)
e)
(42)
Consider the global system “observer-system”, (18), (34), and define the Lyapunov function This function satisfies the condition (i) of Theorem 5.4 (Li et all, 2010), , . By using the same procedure as in the previous section, yields
(32) that allow as , the Remark 3 For small values stability condition become the classical condition (Aguila et all., 2014) (33)
(43)
Remark 4 Control law (28) is a theoretical solution that ensures the stability of the system. This solution is not a technical solution because it requires the measurement of the fractal variables meaning.
that have not a physical By applying the Young inequality (Komaroff, 1994) it follows that
5.3 Observer Based Control c) Case 3. Consider the linearized model (18). The observer model is defined as (Dadras et all., 2011)
Substituting this result into (43) and considering YKP Lemma (Khalil, 2002) yields
(34) (35) is the observability vector and where output (14). The error is
(44)
+
is the system
Applying the control law (38), this inequality becomes (36) +
The error dynamics are obtained from (19), (34), (36), or
(37) with initial conditions
where minimum
We consider the global fractional-order system, 210
)) and eigenvalues .
of
are the ,
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d) Case 4. Consider the nonlinear model (13), (14) with initial conditions (16)-(18) and nonlinearity verifies the inequality (15). The following observer model is proposed
115
,The dynamic defined by (6) with initial conditions
model
is .
The eigenvalues of for the linearized model are shown in Fig 5. The autonomous system is stable, all the eigenvalues but the quality of verify the constraint
(45) (46) (47) The dynamics of the error (36) will be (48) We can prove that the global fractional-order system, “observer-system” is asymptotically stable if the following conditions are satisfied: a) The control law is (49) b) c)
Fig. 5. Eigenvalue distribution for system and observer
is Hurwitz stable is Hurwitz stable
d)
(50)
e)
(51)
f)
(52) Fig.6. Time evolution of the fractional-order variables autonomous system
The proof is similar as the previous cases. Consider a Lyapunov function,
motion does not correspond to the desired performances regarding setting time and oscillation regime. In Fig 6 are presented all trajectories of the fractional-order component and that represent position trajectories (Fig 6) but only and angular velocity, respectively, have a physical meaning. The initial conditions are:
(53) and developing the same technique, using the conditions (a)(c) of (49) and the inequality (15), yields
1.
(54)
+
Applying the conditions (51), (52), results
;
A control law (20) with is applied. This control -3 . The verifies the sector constraint (21) with was selected, and the vector matrix and a matrix was inferred with The polar plot of is shown in Fig 7. We remark that the closed-loop system satisfies the frequential criterion (22), the plot does not The condition (23) is encircle the critical point (The good evolution of verified for the system is remarked from the fractional-order variable trajectories that are shown in Fig 8.
where
6. NUMERICAL SIMULATIONS Consider the mechanical system of Fig 3 with the following parameters: bending
stiffness
,
viscosity
The linearized state model (18) will be:
coefficients
211
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Dadras, S., H. R. Momeni (2011), A New Fractional Order Observer Design for Fractional Order Nonlinear Systems, In: Proc of the ASME 2011, Washington DC, August, pp 1-8 Dadras, S., H. Malek, Y. Chen (2017) A Note on the Lyapunov Stability of Fractional Order Nonlinear Systems,In: Proc. of the ASME 2017, Cleveland, Orlando, August 2017, pp 123-129 Diethelm, K. (2004) The Analysis of Fractional Differential Equations, Springer- Verlag, London Ivanescu, M., M. Nitulescu, D.H. Nguyen, M. Florescu, C. Vladu (2018). In: Proc. of RAAD 2018, Springer-Verlag, pp 245-261. Li, Y., Y.Q. Chen, I. Podlubny (2010), Stability of Fractional Order Nonlinear Systems: Lyapunov direct method and generalized Mittag-Leffler stability, In: Comput. Math. Appl. vol. 59, pp. 1810-1821, 2010. Khalil, H. (2002) Nonlinear Systems, Prentice Hall, New Jersey. Komaroff, N. (1994) Diverse bounds for the eigenvalues of the continuous algebraic Riccati equation, IEEE Trans on Automatic Control, Vol 39, pp 532-534, 1994. Masalkino, S. S., R. Mototo, K. Matsusito, M. Sasaki (2018) Design and adaptive balance control of a biped robot with fewer actuators for slope walking. In: Mechatronics, vol 49, FEbr 2018, pp 56-66. Monje, A.C., Y. Q. Chen, B. Vinagre, D. Hue, V. Feliu (2010). Fractional-0rder Systems and Controls, Springer-Verlag London Limited. Niiyama, R., S. Nishikaya, Y. Kuniyoshi (2009). Athlete Robot with Applied Human Muscle Activation Pattern for Bipedal Running Robot, In : Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems 2009, pp 1092-1099 Paola, D. M., F.P. Pinola, M.Zingales (2013). Fractional differential equations and related exact mechanical models. In: Computers and Mathematics with Applications, 66 (2013) 608-620 Raibert, H. M. (1986) Legged Robots that Balance, The MIT Press 1986. Rivero, M., S. Rogosin, J. T. Machado, (2013), Stability of Fractional Fractional Order Systems. In: Mathematical Problems in Engineering, Vol 2013, ID 356235, pp 286293 Sang-Ho, H. (2009) Compliant Terrain Adaptation for Biped Humanoid without Measuring Ground Surface and Contact Forces. In: Proc of IEEE Trans on Robotics, 25(1):171-178 Szots, J., T. A. Feher, I.Hartmati (2018) Design and control of a low-cost bipedal robot. In: Proc of 19th Int Carpathian Control Conference ICCC 2018, pp 456-462. Tajma, R., D. Honda, K. Suga (2009) Fast running experiments involving a humanoid robot. In: Proc IEEE Int Conf on Robotics and Automation (ICRA 2009), pp 1571-1576 Takenaka, T., T. Matsumoto, T. Yoshiike, S. Shirokura (2009) Real Time Motion Generation and Control for biped Robots- Running Gate Pattern Generation, In: Proc IEEE-RSJ Int Conf on Intelligent Robots and Systems, pp 1092-1099.
Fig. 7. Polar plot of
Fig. 8. Fractional-order state trajectories 2. Consider the system (18), the observer (34) with , the control law (38) with that satisfies the sector condition for Initial conditions of the observer are:
Fig 9. Fractional order trajectories for the “system-observer” (the physical meaning variables) are selected as in the previous example. The matrices Also, the frequential condition (39) is satisfied (Fig 7). The conditions (41), (42) are verified for
In Fig 9 are shown the main parameters that have a physical meaning, position and velocity for system and observer. The good performances of the proposed control system is concluded from the graphics. REFERENCES Aguila-Camacho, N., M. Duarte-Mermoud, J. Callegos (2014). Lyapunov functions for fractional order systems. In: Commun Nonlinear Sci Numer Simulat 19, Elsevier, pp 2951-2957
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IFAC PapersOnLine 52-17 (2019) 111–116 Stance Phase Control System of a Jumping Robot Stance System of Stance Phase Phase Control Control System of aa Jumping Jumping Robot Robot Mircea Ivanescu* Stance System of aa Jumping Jumping Robot Robot Stance Phase Phase Control Control System of
Mircea Ivanescu* Ivanescu* Mircea *University of Craiova, Craiova, Romania (e-mail: [email protected]). Mircea Ivanescu* Mircea Ivanescu* *University of Craiova, Craiova, Craiova, Craiova, Romania Romania (e-mail: [email protected]). *University of (e-mail: [email protected]). *University of of Craiova, Craiova, Craiova, Craiova, Romania Romania (e-mail: (e-mail: [email protected]). [email protected]). *University Abstract: The paper treats the control of the jumping robot during the stance phase using the fractal model of the system. Considering an of actuation system based on the Electro-Rheologic (ER) fluid Abstract: paper treats control the robot during the stance using the fractal Abstract: aThe The paper treatsis the the control oflinearized the jumping jumping robot during the models stance phase phase usingand the control fractal controller, fractal model inferred. The model and nonlinear are studied Abstract: The paper treats the control of the jumping robot during the stance phase using the fractal model Considering an of actuation system based on Electro-Rheologic fluid Abstract: The system. paper treats the control the jumping robot during the stance phase using (ER) the fractal model of of the the system. Considering actuation system based on Electro-Rheologic (ER) fluid frequential aremodel proposed usingan YKP criterions. Observer modelsmodels are proposed for and linear and model of system. Considering an actuation system based on Electro-Rheologic (ER) fluid controller, alaws fractal is inferred. inferred. The linearized model and nonlinear nonlinear are studied studied control model of the the system. Considering an actuation system based on the the models Electro-Rheologic (ER)control fluid controller, a fractal model is The linearized model and are and nonlinear systems and the global stability for “system-observer” is studied by Lyapunov techniques. controller, a fractal model is inferred. The linearized model and nonlinear models are studied and control frequential alaws laws aremodel proposed using The YKP criterions. Observer modelsmodels are proposed proposed for and linear and controller, fractalare is inferred. linearized model and nonlinear are studied control frequential proposed using YKP criterions. Observer models are for linear and Numerical simulations are presented. frequential laws using YKP criterions. Observer are proposed for linear nonlinear and the stability “system-observer” is Lyapunov frequential laws are are proposed using YKP for criterions. Observer models models are by proposed for techniques. linear and and nonlinear systems systems andproposed the global global stability for “system-observer” is studied studied by Lyapunov techniques. nonlinear the global stability for “system-observer” is Numerical simulations are nonlinear systems and thepresented. global stability for frequential “system-observer” is studied studied by by Lyapunov Lyapunov techniques. techniques. Copyright ©systems 2019. Theand Authors. Published by Elsevier Ltd. All rights reserved. Numerical simulations are presented. Keywords: robotics, control, stability, observer, criterion Numerical Numerical simulations simulations are are presented. presented. Keywords: stability, observer, observer, frequential frequential criterion criterion Keywords: robotics, robotics, control, control, stability, Keywords: robotics, control, stability, observer, frequential criterion Keywords: robotics, control, stability, observer, frequential criterion 1. INTRODUCTION 2. JUMPING MOTION CYCLE 1. INTRODUCTION MOTION This paper analyses the control problem of a class of robots, 2. A JUMPING bipedal motion cycle CYCLE is shown in Fig 1. (Ivanescu et all., 1. INTRODUCTION 2. JUMPING MOTION CYCLE 1. INTRODUCTION 2. JUMPING MOTION CYCLE the jumping biped robots, during the Stance Phase of a 2018) There are two main phases: Phase when the 1. INTRODUCTION 2. JUMPING MOTION CYCLE This paper paper analyses analyses the the control control problem problem of of aa class class of of robots, robots, A bipedal motion cycle shown in Stance Fig (Ivanescu et all., This A bipedal cycle is iswith shown Fig 1. 1. and (Ivanescu etPhase all., motion cycle. The motion study of these robots represented aa 2018) foot is There in motion direct contact the in ground Flight This paper analyses the control problem of a class of robots, A bipedal motion cycle is shown in Fig 1. (Ivanescu et all., the jumping biped robots, during the Stance Phase of are two main phases: Stance Phase when the This paper analyses control problem a class of robots, bipedal motion cyclemain is shown in Stance Fig 1. (Ivanescu et all., the jumping biped the robots, during the ofof Stance Phase of toa A 2018) There areleaves two phases: Phase when the fascinated attraction for a great number researchers due when the robot thewith ground. The motion isFlight determined the jumping biped robots, during the Stance Phase 2018) There are two phases: Stance Phase when the motion cycle. The motion motion study of these these robots represented is in direct the ground Phase the jumping biped robots, during the robots Stancerepresented Phase of of aa foot 2018) There are contact two main main phases: Stanceand Phase when the motion cycle. The study of foot is in direct contact with the ground and Flight Phase complexity control laws, the number of of constraints anddue large by two legs, each leg having aThe mechanical architecture motion cycle. The motion study of these robots represented a foot is in direct contact with the ground and Flight Phase fascinated attraction for a great number researchers to when the robot leaves the ground. motion is determined motion cycle. The motion study number of theseof robots represented a foot isthe in robot directleaves contact with the The ground and isFlight Phase fascinated attraction for a great researchers due to when the ground. motion determined possibilities to develop specific theories and technological consisting by an elastic lower configuration and an upper fascinated for great number researchers to when the robot leaves the ground. motion determined complexityattraction control laws, the number of of constraints anddue large two mechanical fascinated attraction for aathe great number of researchers due to by when thelegs, robot each leavesleg thehaving ground.aaThe The motion is is architecture determined complexity control laws, number of constraints and large by two legs, each leg having mechanical architecture achievements. A plethora of papers treats this topic and we configuration that ensures the actuation and damping effect complexity control laws, specific the number number of constraints constraints and large large consisting by two legs, legs, each leg having having mechanicaland architecture possibilities control to develop develop specific theories and technological technological by an elastic lower configuration an upper complexity laws, the of and by two each leg aa mechanical architecture possibilities to theories and consisting by ER an elastic lower configuration and an upper wish to mention (Masalkino et al.,2018, Szolt et al.,2018) for (Fig 2). An driving system allows the ER viscosity possibilities to develop specific theories and technological consisting by an elastic lower configuration and an upper achievements.to A Adevelop plethoraspecific of papers papers treats and this topic topic and we we configuration ensures the actuation and damping possibilities theories technological consisting by that an elastic lower configuration and an effect upper achievements. plethora of treats this and configuration that ensures the actuation and damping effect their contribution at the implementation of the bipedal modification and the implementation of the control strategies. achievements. A plethora of papers treats this topic and we configuration that ensures the actuation and damping effect wish to to mention mentionA(Masalkino (Masalkino etpapers al.,2018, Szolt ettopic al.,2018) for (Fig 2). An ER allows the ER viscosity achievements. plethora ofet treats thiset and we configuration that driving ensures system the actuation and damping effect wish al.,2018, Szolt al.,2018) for (Fig 2). An ER driving system allows the ER viscosity jumping and running strategies.ofet The servowish mention et for (Fig 2). ER driving system the viscosity their to contribution at the the motion implementation the bipedal bipedal implementation of wish tocontribution mention (Masalkino (Masalkino et al.,2018, al.,2018, Szolt Szoltofet al.,2018) al.,2018) for modification (Fig 2). An An and ER the driving system allows allows the ER ERstrategies. viscosity their implementation the modification and the implementation of the the control control strategies. technologies for running theat controllers thatstrategies. supervise the jumping their contribution at the implementation of the bipedal modification and the implementation of the control strategies. jumping and motion The servotheir contribution at the motion implementation of The the bipedal modification and the implementation of the control strategies. jumping and running strategies. servofunctions are analysed in (Taima et al., 2009, Sang-Ho, 2009) jumping and motion strategies. The servotechnologies for running the controllers controllers that supervise the jumping jumping jumping andfor running motionthat strategies. The servotechnologies the supervise the The principles of the hoping motion by using the equivalence technologies for that the functions are are analysed analysed in (Taima (Taima et et al.,supervise 2009, Sang-Ho, Sang-Ho, 2009) technologies for the the controllers controllers that supervise the jumping jumping functions in al., 2009, 2009) with the spring-loaded inverted pendulum model are studied functions are analysed in (Taima et al., 2009, Sang-Ho, 2009) The principles of the hoping motion by using the equivalence functions are analysed in (Taima et al., 2009) The principles of the hoping motion by 2009, using Sang-Ho, the equivalence in et the al.,hoping 2009 , motion Raibert, The The principles of the hoping motion by 1986). using the equivalence with(Takenaka the spring-loaded spring-loaded inverted pendulum model aredynamic studied The principles of by using the equivalence with the inverted pendulum model are studied analysis theetmotion cycles and the 1986). identification of the with the spring-loaded inverted pendulum model are studied ,, Raibert, The in (Takenaka (Takenaka al., 2009 2009 with the of spring-loaded inverted pendulum model studied in et al., Raibert, 1986). Thearedynamic dynamic control laws is presented in (Niiama et al., 2009). in (Takenaka al., ,, Raibert, The analysis of the cycles and the identification of in (Takenaka etmotion al., 2009 2009 Raibert, 1986). The dynamic dynamic analysis of treats theet motion cycles the 1986). identification of the the Thisr paper the control of and the jumping robot during the analysis of the motion cycles and the identification of control laws is presented in (Niiama et al., 2009). analysis of the motion cycles and et theal.,identification of the the control laws is presented in (Niiama 2009). stance phase using the fractal model of the system. control lawstreats is presented presented in (Niiama (Niiama et al., al., 2009). 2009). Thisr paper the of robot during the control laws is in et Thisr paper treats the control control of the the jumping jumping robot during the Considering anusing actuation system based on the ElectroThisr paper treats the control of the jumping robot during the stance phase the fractal model of the system. Thisr paper treats the control of the jumping robot during the stance phase using the fractal model of the system. Rheologic (ER) fluid controller, a fractal model is inferred. stance phase using the fractal model of the system. Considering actuation system based on stance phasean using the fractal model of the Electrosystem. Considering an actuation system based on ElectroThe linearized model and nonlinear modelson are studied and Considering an actuation system based ElectroRheologic (ER) controller, a fractal is Considering an fluid actuation system based model on the the ElectroRheologic (ER) fluid controller, fractal model is inferred. inferred. control laws are proposed. It aashould be noted that and the Rheologic (ER) fluid controller, fractal model is inferred. The linearized model and nonlinear models are studied Rheologic (ER)model fluid controller, a fractal model is inferred. The linearized and nonlinear models are with studied and Fig.1. Motion cycle literature (Dadras et al., 2017) propose solutions control The linearized model and models studied and control laws are proposed. It should be noted that the The linearized model and nonlinear nonlinear models are studied and Motion cycle control laws Itimplementable should be are noted that they the Fig.1. Fig.1. MotionPHASE cycle MODEL laws that are are notetproposed. practical because control laws are proposed. It should be noted that the 3. STANCE literature (Dadras al., 2017) propose solutions with control Motion cycle control laws are etproposed. Itpropose shouldsolutions be noted that the Fig.1. literature (Dadras al., 2017) with control Fig.1. Motion cycle require the measurement of all fractal variables that are literature (Dadras et al., 2017) propose solutions with control laws that are not practical implementable because they literature (Dadras et al., 2017) propose solutionsbecause with control laws that are not practical implementable they 3. STANCE PHASE 3. STANCE PHASE MODEL MODEL usually without anot physical meaning. To avoid these problems, laws that are practical implementable because they require the measurement of all fractal variables that are 3. STANCE PHASE MODEL laws that are not practical implementable because they require the measurement of all fractal variables that are The mechanical architecture is presented in Fig 3. The system 3. STANCE PHASE MODEL ausually class without of control laws defined on the fractal model with require the measurement of all fractal variables that are a physical meaning. To avoid these problems, require the measurement of all fractal variables that are usually without a physical meaning. To avoid these problems, is a two-leg system where the upper component is system a rigid respect to control direct observable variables are these proposed and The mechanical architecture is in 3. usually aa physical meaning. To avoid problems, a class class without of laws defined on the fractal model with The mechanical architecture is presented presented in Fig Fig 3. The The system usually without physical meaning. To avoid these problems, afrequential of control laws defined on the fractal model with beam and the lower component is an elastic curved structure. The mechanical architecture is presented in Fig Fig 3. 3. The The system are studied. A fractalfractal observer modeland is is a mechanical two-leg where the upper component is system aa rigid aarespect class control laws on model The architecture is in respect to criterions direct observable observable variables are proposed is two-leg system system where are thepresented upper in component rigid class of of control laws defined definedvariables on the the fractal model with with to direct are proposed and Theaa geometrical parameters shown Figcurved 3. Theis massless is two-leg system where the upper component is aa rigid proposed and the global stability “system-observer” is beam and the lower component is an elastic structure. respect to direct observable variables are proposed and is a two-leg system where the upper component is rigid frequential criterions are studied. studied. A fractal fractalare observer model is beam and the lower component is an elastic curved structure. respect to criterions direct observable variables proposed and frequential are A observer model is legs geometrical are the considered, all are gravitational components are beam and lower component is an elastic curved structure. studied by and Lyapunov techniques forfractal approximate linear and The parameters shown in Fig 3. The massless frequential criterions are studied. A observer model is beam and the lower component is an elastic curved structure. proposed the global stability “system-observer” The geometrical parameters are shown in Fig 3. The massless frequential criterions are studied. A fractal observer model is concentrated in the Centre of Gravity (COG). The dynamic proposed and the global stability “system-observer” The geometrical parameters are shown in Fig 3. The massless nonlinear model. Numerical simulations are presented. are considered, all gravitational are proposed and the stability “system-observer” is The geometrical parameters shown in Figcomponents 3. The massless studied by Lyapunov techniques for approximate linear and legs are considered, all are gravitational are proposed and the global global stability “system-observer” is legs studied by is Lyapunov techniques for approximate linear and model of the robot in the stance phase iscomponents obtained using legs are considered, all gravitational components are The paper structured as follows: section 2presented. presents motion concentrated in the Centre of Gravity (COG). The dynamic studied by Lyapunov techniques for approximate linear and legs are considered, all gravitational components are nonlinear model. Numerical simulations are concentrated in the Centre of Gravity (COG). The dynamic studied bymodel. Lyapunov techniques for approximate linear and Lagrange equations (Raibert, 1986). nonlinear Numerical simulations arethe presented. concentrated in the Centre of Gravity (COG). The dynamic cycle of a jumping robot, section 3 treats Stance Phase model of the robot in the stance phase is obtained using nonlinear model. Numerical simulations are presented. concentrated in the Centre of Gravity (COG). The dynamic The paper is structured as follows: section 2 presents motion model of the robot in the stance phase is obtained using nonlinear model. Numerical simulations are2presented. The paper is structured as follows: section presents model of the in Model,, section 4 describes the fractal model, section 5motion treats Lagrange 1986). The paper structured as 22the presents motion of equations the robot robot(Raibert, in the the stance stance phase is is obtained obtained using using cycle of aa is jumping robot, section 33section treats Stance Phase Lagrange equations (Raibert, 1986). phase The paper is structured as follows: follows: section presents motion cycle of jumping robot, section treats the Stance Phase model Lagrange equations (Raibert, 1986). the control system and section 6 verifies the control cycle of a jumping robot, section 3 treats the Stance Phase Lagrange equations (Raibert, 1986). Model,, 44 describes the fractal model, treats cycle of section a jumping robot, section 3 treats thesection Stance55 Phase Model,, section describes the fractal model, section treats techniques by system numerical simulations. Model,, section 4 fractal section treats the and section verifies the Model,, section 4 describes describes the fractal66 model, model, section 5control treats the control control system and the section verifies the 5 control the control system and section 6 verifies the control techniques numerical simulations. the controlby and section 6 verifies the control techniques by system numerical simulations. techniques numerical simulations. Copyright © by 2019 IFAC 207 techniques by numerical simulations. 2405-8963 Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 207Control. Copyright © 2019 IFAC 207 10.1016/j.ifacol.2019.11.036 Copyright © 2019 IFAC 207 Copyright © 2019 IFAC 207
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and fractional exponent coefficient strain relation is (Psaola et all., 2013)
(Fig 4). The stress-
(4) is the Caputo fractional derivative of order
where
(5) In this analysis, the viscoelastic components are characterized that correspond to an intermediate state by between elastic and viscous fluid components. Substituting these components (2-5) in (1) and considering the following , after simple calculations, results constraints
Fig. 2. Leg configuration
, (6) is determined by the admissible values of the where is the geometrical configuration during the motion, ,(the elastic equivalent coefficient, variable was omitted). The last term represents the nonlinear component determined by the gravitational forces. Assume that the viscoelastic exponent is =1/2. The initial conditions are defined by the initial position and velocity, , Fig. 3. Geometrical leg model
(7)
4. STATE MODEL For the stability analysis and the identification of the oscillation regime, the following state variables are defined, (8) (9) (10)
Fig. 4. ER fluid model (11) -
Substituting (7-10) into (6), yields
(1)
are gravitational and elastic potential, where is the respectively, is the equivalent inertial moment, actuator torque and represents the viscoelastic component,
(12) This equation can be rewritten as,
(2) (13) (3) is the fractal state vector of the where dynamic model. The output is defined by the direct measurable variable
The viscoelastic behaviour is determined by two components: a classical viscous friction in the rotational articulation C (Fig and the viscoelastic component created in 3) defined by ER damper. This component is analysed by using the fractional Kevin-Voigt model in which a pure elastic phase is connected in parallel defined by the elastic coefficient with a fractional viscoelastic phase that is characterized by
(14) The inequality 208
gravitational
nonlinear term is This component verifies the
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113
(23) (15) where
are obtained from Initial conditions for the variables the physical corresponding variables, angular position and velocity
In order to prove the stability, the following Lyapunov function is considered
(16) and , , that have not a physical meaning will have (Monie et all., 2010). ;
is a symmetrically, positive definite matrix. This where of function satisfies the condition (Li et all., 2010, Dadras et all., 20217) for The fractional derivative will be
(17)
From (8-14), we can easily infer that the fractal dynamic model (13), that satisfies (15)-(17), verify the conditions that ensure the existence and unicity of solutions (Diethelm., 2004) Remark 1 The linearized model of (13) is obtained if the is approximated as nonlinearity
(24) and substituting (19) yields
(18) (25) This relation can be rewritten
where is the new form of where the elastic coefficient . includes the term is Remark 2. The system defined by is controllable, respectively observable. The matrix fractional order stable ( ) but it is not Hurwitz stable. Remark 3. The classical transfer function can be obtained from the linearized model of (6) as (19)
(26) Considering the sector control law (20) and the condition (21), this relation becomes
(27)
5. CONTROL SYSTEM
Applying (22) and substituting the conditions of YKP Lemma (Khalil, 2002), results
5.1 Linearized Model a) Case 1. Consider the system (18) and the following control law is proposed
Using (20) and the condition (23), this inequality becomes
(20)
,
This control ensures the asymptotical stability if the following conditions are satisfied: a) There exists a positive definite matrix such that the matrix is Hurwitz stable b) control law (20) verifies the sector condition
where
5.2 Nonlinear Model
(21) c)
=
b) Case 2. Consider the nonlinear model (13), (14) with the initial conditions (15)-(18). The system (13)-(18) is asymptotically stable if the control law is
The frequency condition is satisfied (22)
There exists a gain
(28)
of the control low (20) that satisfies
and the following conditions are satisfied:
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“system-observer” defined by (19), (34) and the following control law is proposed:
a. The matrix is stable, b. (29) where and is the minimum The following Lyapunov function is eigenvalue of proposed to analyse the stability of the system,
(38) that satisfies the conditions: a) There exists a positive definite matrix such that the matrix is Hurwitz stable b) is Hurwitz stable (39)
(30) is a positive constant, . The where condition (i) of Theorem 5.4 (Li et all, 2010) is satisfied. For a control law, (31) we can infer that
c)
(40)
d)
(41)
e)
(42)
Consider the global system “observer-system”, (18), (34), and define the Lyapunov function This function satisfies the condition (i) of Theorem 5.4 (Li et all, 2010), , . By using the same procedure as in the previous section, yields
(32) that allow as , the Remark 3 For small values stability condition become the classical condition (Aguila et all., 2014) (33)
(43)
Remark 4 Control law (28) is a theoretical solution that ensures the stability of the system. This solution is not a technical solution because it requires the measurement of the fractal variables meaning.
that have not a physical By applying the Young inequality (Komaroff, 1994) it follows that
5.3 Observer Based Control c) Case 3. Consider the linearized model (18). The observer model is defined as (Dadras et all., 2011)
Substituting this result into (43) and considering YKP Lemma (Khalil, 2002) yields
(34) (35) is the observability vector and where output (14). The error is
(44)
+
is the system
Applying the control law (38), this inequality becomes (36) +
The error dynamics are obtained from (19), (34), (36), or
(37) with initial conditions
where minimum
We consider the global fractional-order system, 210
)) and eigenvalues .
of
are the ,
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d) Case 4. Consider the nonlinear model (13), (14) with initial conditions (16)-(18) and nonlinearity verifies the inequality (15). The following observer model is proposed
115
,The dynamic defined by (6) with initial conditions
model
is .
The eigenvalues of for the linearized model are shown in Fig 5. The autonomous system is stable, all the eigenvalues but the quality of verify the constraint
(45) (46) (47) The dynamics of the error (36) will be (48) We can prove that the global fractional-order system, “observer-system” is asymptotically stable if the following conditions are satisfied: a) The control law is (49) b) c)
Fig. 5. Eigenvalue distribution for system and observer
is Hurwitz stable is Hurwitz stable
d)
(50)
e)
(51)
f)
(52) Fig.6. Time evolution of the fractional-order variables autonomous system
The proof is similar as the previous cases. Consider a Lyapunov function,
motion does not correspond to the desired performances regarding setting time and oscillation regime. In Fig 6 are presented all trajectories of the fractional-order component and that represent position trajectories (Fig 6) but only and angular velocity, respectively, have a physical meaning. The initial conditions are:
(53) and developing the same technique, using the conditions (a)(c) of (49) and the inequality (15), yields
1.
(54)
+
Applying the conditions (51), (52), results
;
A control law (20) with is applied. This control -3 . The verifies the sector constraint (21) with was selected, and the vector matrix and a matrix was inferred with The polar plot of is shown in Fig 7. We remark that the closed-loop system satisfies the frequential criterion (22), the plot does not The condition (23) is encircle the critical point (The good evolution of verified for the system is remarked from the fractional-order variable trajectories that are shown in Fig 8.
where
6. NUMERICAL SIMULATIONS Consider the mechanical system of Fig 3 with the following parameters: bending
stiffness
,
viscosity
The linearized state model (18) will be:
coefficients
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Dadras, S., H. R. Momeni (2011), A New Fractional Order Observer Design for Fractional Order Nonlinear Systems, In: Proc of the ASME 2011, Washington DC, August, pp 1-8 Dadras, S., H. Malek, Y. Chen (2017) A Note on the Lyapunov Stability of Fractional Order Nonlinear Systems,In: Proc. of the ASME 2017, Cleveland, Orlando, August 2017, pp 123-129 Diethelm, K. (2004) The Analysis of Fractional Differential Equations, Springer- Verlag, London Ivanescu, M., M. Nitulescu, D.H. Nguyen, M. Florescu, C. Vladu (2018). In: Proc. of RAAD 2018, Springer-Verlag, pp 245-261. Li, Y., Y.Q. Chen, I. Podlubny (2010), Stability of Fractional Order Nonlinear Systems: Lyapunov direct method and generalized Mittag-Leffler stability, In: Comput. Math. Appl. vol. 59, pp. 1810-1821, 2010. Khalil, H. (2002) Nonlinear Systems, Prentice Hall, New Jersey. Komaroff, N. (1994) Diverse bounds for the eigenvalues of the continuous algebraic Riccati equation, IEEE Trans on Automatic Control, Vol 39, pp 532-534, 1994. Masalkino, S. S., R. Mototo, K. Matsusito, M. Sasaki (2018) Design and adaptive balance control of a biped robot with fewer actuators for slope walking. In: Mechatronics, vol 49, FEbr 2018, pp 56-66. Monje, A.C., Y. Q. Chen, B. Vinagre, D. Hue, V. Feliu (2010). Fractional-0rder Systems and Controls, Springer-Verlag London Limited. Niiyama, R., S. Nishikaya, Y. Kuniyoshi (2009). Athlete Robot with Applied Human Muscle Activation Pattern for Bipedal Running Robot, In : Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems 2009, pp 1092-1099 Paola, D. M., F.P. Pinola, M.Zingales (2013). Fractional differential equations and related exact mechanical models. In: Computers and Mathematics with Applications, 66 (2013) 608-620 Raibert, H. M. (1986) Legged Robots that Balance, The MIT Press 1986. Rivero, M., S. Rogosin, J. T. Machado, (2013), Stability of Fractional Fractional Order Systems. In: Mathematical Problems in Engineering, Vol 2013, ID 356235, pp 286293 Sang-Ho, H. (2009) Compliant Terrain Adaptation for Biped Humanoid without Measuring Ground Surface and Contact Forces. In: Proc of IEEE Trans on Robotics, 25(1):171-178 Szots, J., T. A. Feher, I.Hartmati (2018) Design and control of a low-cost bipedal robot. In: Proc of 19th Int Carpathian Control Conference ICCC 2018, pp 456-462. Tajma, R., D. Honda, K. Suga (2009) Fast running experiments involving a humanoid robot. In: Proc IEEE Int Conf on Robotics and Automation (ICRA 2009), pp 1571-1576 Takenaka, T., T. Matsumoto, T. Yoshiike, S. Shirokura (2009) Real Time Motion Generation and Control for biped Robots- Running Gate Pattern Generation, In: Proc IEEE-RSJ Int Conf on Intelligent Robots and Systems, pp 1092-1099.
Fig. 7. Polar plot of
Fig. 8. Fractional-order state trajectories 2. Consider the system (18), the observer (34) with , the control law (38) with that satisfies the sector condition for Initial conditions of the observer are:
Fig 9. Fractional order trajectories for the “system-observer” (the physical meaning variables) are selected as in the previous example. The matrices Also, the frequential condition (39) is satisfied (Fig 7). The conditions (41), (42) are verified for
In Fig 9 are shown the main parameters that have a physical meaning, position and velocity for system and observer. The good performances of the proposed control system is concluded from the graphics. REFERENCES Aguila-Camacho, N., M. Duarte-Mermoud, J. Callegos (2014). Lyapunov functions for fractional order systems. In: Commun Nonlinear Sci Numer Simulat 19, Elsevier, pp 2951-2957
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