A Data Driven Vector Field Oscillator with Arbitrary Limit Cycle Shape
aa r X i v : . [ n li n . AO ] O c t A Data Driven Vector Field Oscillatorwith Arbitrary Limit Cycle Shape
Venus Pasandi a,b , Aiko Dinale b , Mehdi Keshmiri a , Daniele Pucci b ∗† October 11, 2019
Abstract
Cyclic motions in vertebrates, including heart beating, breathing andwalking, are derived by a network of biological oscillators having fascinat-ing features such as entrainment, environment adaptation and robustness.These features encouraged engineers to use oscillators for generating cyclicmotions. To this end, it is crucial to have oscillators capable of character-izing any periodic signal via a stable limit cycle. In this paper, we proposea 2-dimensional oscillator whose limit cycle can be matched to any peri-odic signal depicting a non-self-intersecting curve in the state space. Inparticular, the proposed oscillator is designed as an autonomous vectorfield directed toward the desired limit cycle. To this purpose, the desiredreference signal is parameterized with respect to a state-dependent phasevariable, then the oscillator’s states track the parameterized signal. Wealso present a state transformation technique to bound the oscillator’s out-put and its first time derivative. The soundness of the proposed oscillatorhas been verified by carrying out a few simulations.
Nonlinear oscillators have been widely used by the engineering community tomodel and control physical phenomena [1–3]. Their interesting features such asentrainment, synchronization and smooth modulation of the output signal makethem appropriate for robotics applications such as cyclic motions of manipula-tors or legged robot locomotion. Using oscillators for generating the referencetrajectory or control signal provides capabilities of smoothness, continuity, dis-turbance rejection and adaptation. The oscillator encodes the desired referencetrajectory or control signal via a stable limit cycle. The essence of well-knownoscillators, like the Matsuoka’s and Hopf’s, is their capability of generating limitcycles with a specific shape. However, a specific limit cycle shape constraintsthe types of signals that can be generated. In this paper, we propose a twodimensional oscillator which can generate any periodic trajectory, depicting anon-self-intersecting curve in the state space. ∗ a Isfahan University of Technology, Isfahan, Iran, [email protected], [email protected] † b Istituto Italiano di Tecnologia, Genoa, Italy, [email protected], [email protected] ssuming the desired limit cycle is defined by a Lyapunov function, theproblem of designing a dynamical system with a desired limit cycle is expressedas the problem of constructing a dynamical system for a desired Lyapunovfunction [4, 5]. Besides controlling the limit cycle, such framework has been ex-tended to non-autonomous dynamical systems to design transient trajectoriesand achieve the desired convergence [6]. Although these algorithms are inter-esting from the mathematics point of view, they can not be directly appliedfor engineering purposes because of their lack of analytical predictability. Infact, these algorithms generate different dynamic structures for different desiredlimit cycles and the properties of the dynamics, like domain of attraction andattracting rate, are not determined a priori.A widespread strategy for designing a dynamical system generating an ar-bitrary periodic signal is to transform a well-understood dynamical system intoan oscillating one with desired limit cycle. For instance, a linear spring-dampersystem can generate a variety of cyclic signals with the help of a forcing term.To design an autonomous system, the forcing term is defined by a nonlinearfunction of a phase variable and learned by standard machine learning tech-niques [7]. From a more general perspective, the limit cycle of a phase oscillatoris mapped to the desired periodic trajectory in the state space through a phase-dependent scaling function. Thus, a general family of nonlinear phase oscillatorswhich can track almost any continuous trajectory is constructed [8]. The phaseof the dynamical systems proposed in [7] and [8] is generated by an indepen-dent phase dynamics which results in trajectory tracking and not limit cycletracking. Furthermore, the desired trajectory is asymptotically stable and notasymptotically orbitally stable. To provide limit cycle tracking of a desired pe-riodic trajectory, a Hopf’s oscillator is altered by two nonlinear functions whichare determined such that the Poincar´e-Bendixson theorem is satisfied in a pre-defined neighborhood of the limit cycle [9]. The framework guarantees localstability and it is not straightforward to extend it for achieving global stability,since the Poincar´e-Bendixson theorem is no longer applicable.Another method for designing an oscillator is to use a data driven vectorfield which has been originally proposed for generating a discrete system withan arbitrary limit cycle [10]. The discrete vector field generated in the neighbor-hood of the limit cycle is approximated by a function, like a polynomial one, andhence, a continuous dynamical system with a locally stable desired limit cycleis created [11]. However, to the best of the authors’ knowledge, the possibilityof designing a continuous nonlinear vector field for ensuring the global stabilityof the desired limit cycle has not been explored yet.In the present paper, we propose a two dimensional continuous dynamicalsystem that can track any non-self-intersecting closed trajectory in the statespace. The main idea is to generate a data driven vector field directed towardthe desired trajectory. To this purpose, the desired trajectory is parameterizedwith respect to a state-dependent phase variable. Then, the oscillator dynamicsis designed to track the parameterized trajectory. Moreover, we propose a statetransformation method for generating a bounded output by using the proposedoscillator. The contribution of this paper is threefold. First, the proposedsystem provides asymptotic orbital stability of the desired trajectory. Second,the convergence to the desired trajectory is irrespective of the parameters of thesystem. Third, the proposed oscillator is capable of generating bounded output.The rest of the paper is organized as follows. Section 2 introduces the no-ations and definitions used in the paper. It also recalls the concepts of orbitalstability and transverse dynamics which will be used to prove the asymptoticorbital stability of the desired trajectory in the proposed oscillator. Section 3presents the model development of the proposed oscillator. In Section 4, wemodify the proposed oscillator to bound the output and its first time derivative.Section 5 reports simulation results. Finally, Section 6 concludes the paper witha few remarks and perspectives. • R and R + are the set of real and positive real numbers. • The i th component of a vector q ∈ R m is written as q i . • Given a function g ( x ( t )) : R → R where t represents the time, its first deriva-tives with respect to x and t are denoted as g ′ = dgdx and ˙ g = dgdt , respectively. • A C k -function is a function with k continuous derivatives. • A function f ( t ) : [0 , ∞ ) → R is a T -periodic function if for some positiveconstant p , we have f ( t + p ) = f ( t ) and T is the smallest p with such property. • A simple closed curve is a continuous closed curve that does not cross itself.In mathematical word, γ : [ a, b ] → R n is a simple closed curve if γ ( a ) = γ ( b )and additionally γ ( c ) = γ ( d ), ∀ c, d ∈ [ a, b ). Hence, the simple closed curve γ is a one-to-one mapping from [ a, b ) to R n . Given the dynamical system ˙ x = f ( x ), where x ∈ R n is the state vector, theperiodic trajectory x ∗ ( t ) is • stable if ∀ ǫ > , ∃ δ > k x ( t ) − x ∗ ( t ) k < δ ⇒ k x ( t ) − x ∗ ( t ) k < ǫ, • asymptotically stable (AS) if it is stable and ∃ η > k x ( t ) − x ∗ ( t ) k < η ⇒ lim t →∞ x ( t ) = x ∗ ( t ) , • orbitally stable (OS) if ∀ ǫ > , ∃ δ > ϕ k x ( t ) − x ∗ ( ϕ ) k < δ ⇒ inf ϕ k x ( t ) − x ∗ ( ϕ ) k < ǫ, • asymptotically orbitally stable (AOS), called also stable limit cycle, if it isorbitally stable and ∃ η > ϕ k x ( t ) − x ∗ ( ϕ ) k < η ⇒ lim t →∞ inf ϕ k x ( t ) − x ∗ ( ϕ ) k = 0 . he above stability definitions are stated from [12]. It is noteworthy that toverify either the OS or the AOS, we consider the time evolution of the distancebetween the system’s states and the closed set of the trajectory x ∗ . On theother hand, to prove the stability or the AS, we examine the time evolutionof the distance between the system’s states and a specific point of x ∗ whichchanges with respect to time. Therefore, the stable/AS conditions are stricterthan OS/AOS. The OS property of a periodic trajectory is usually investigated through twotechniques. In the first technique, the stability of a periodic trajectory of acontinuous system is attributed to the stability of the equilibrium point of thecorresponding discrete map, called Poincar´e map. A Poincar´e map, knownalso as first return map , is the intersection of the system’s trajectory in thestate space with a Poincar´e section, that is a lower-dimensional hypersurfacetransversal to the trajectory under study. Usually, it is not possible to findthe Poincar´e map analytically. Therefore, a linearization of the Poincar´e mapis often computed numerically, and its eigenvalues are used to verify local OS.In the second technique, the limit cycle is considered as a class of invariantsets and its stability is investigated through the LaSalle’s invariance principle.For this purpose, one uses a Lyapunov function that equals to zero along thetrajectory and is strictly positive elsewhere. An approach for constructing suchLyapunov function is the transverse dynamics , commonly called also movingPoincar´e sections . In this approach, a transversal hypersurface is defined inthe state space which moves along the trajectory under study. Consideringan n -dimensional T -periodic trajectory x ∗ , a hypersurface σ ( ϕ ) is defined for ϕ ∈ [0 , T ] where σ ( ϕ ) is transversal to x ∗ ( ϕ ), i . e . ˙ x ∗ ( ϕ ) / ∈ σ ( ϕ ) and σ (0) = σ ( T ). Then, a new coordinate system ( e, φ ) is established where the scalar φ represents which of the transversal surfaces σ is inhibited by the currentstate x , and the transverse coordinate e ∈ R n − determines the location of x within the hypersurface σ ( φ ), with e = 0 implying that x = x ∗ ( φ ). Thedynamics of the transverse coordinate is the transverse dynamics. The stabilityof the equilibrium point e = 0 of the transverse dynamics ensures the OS ofthe trajectory x ∗ . In this way, one can analyze the stability analytically andcharacterize the stability region. For more information, the reader can referto [13–15]. In this section, we design a continuous 2-dimensional dynamical system thatprovides asymptotic orbital stability of any desired T -periodic function f ( t ) :[0 , ∞ ) → R depicting a simple closed curve in the state space. From hereafter,we consider one period of f ( t ), i . e . we assume the T -periodic function f ( t ) as f ( t ) : [0 , T ) → R .Consider the 2-dimensional dynamical system described by the followingigure 1: Geometrical representation of the target point. For every state of thesystem (squares), there is a corresponding target point (circles) on the desiredtrajectory (black curve).differential equation¨ s = f ′′ ( ϕ ) − α ( ˙ s − f ′ ( ϕ )) − β ( s − f ( ϕ )) , (1)where s = ( s, ˙ s ) ∈ R represent the states, α, β ∈ R + are coefficients and ϕ isthe phase variable defined based on s as ϕ ( s ) = f − ( f l ) s ≤ f l { ϕ | f ( ϕ ) = s, f ′ ( ϕ ) ˙ s ≥ } f l ≤ s ≤ f u , ˙ s = 0 { ϕ | f ( ϕ ) = s, f ′ ( ϕ ) ≥ } f l ≤ s ≤ f u , ˙ s = 0 f − ( f u ) s ≥ f u , (2)where f l and f u are the lower and upper bounds of f .Conceptually, the proposed dynamics (1) follows the trajectory of the targetpoint , i . e . a point defined on the function f in the state space with coordinates( f ( ϕ ( s )) , f ′ ( ϕ ( s ))). So, the target point is associated with the states and rep-resents a parametrization of the function f with respect to ϕ . Fig. 1 showsthe relation between the states and the corresponding target points. The blackcurve is the function f . Squares represent states at different time instants whilecircles are the corresponding target points. Based on the phase variable defi-nition (2), the state space is divided into three regions. For all the states inRegion 1 ( R ), the target point is the red circle with coordinates ( f l , R ), the target point is the orange circle withcoordinates ( f u , R ), the target point is a pointon the function f with the same s coordinate as the state of the system. We callthe dynamics (1) along with the phase defined in (2) as Data driven Vector fieldOscillator (DVO). The remainder of this section is devoted to investigating theproperties of the DVO.
Remark 1.
As the function f ( t ) is a simple closed curve in the state space, thetarget point assigned to s is unique.In light of the above, we assume from now on that Assumption 1. f ( t ) is a T -periodic C -function as f ( t ) : [0 , T ) ⊆ R → R , (3)describing a simple closed orbit in the plane ( f, ˙ f ). emark 2. In addition to the limit cycle tracking, one can use DVO for pointtracking where the trajectory f is constant. In the case of point tracking, thedynamics (1) is simplified to the well-known PD control as¨ s = − α ˙ s − β ( s − f ) . (4) Theorem.
Given that Assumption 1 is satisfied, and α is a positive functionand β is a positive constant then the trajectory f ( t ) is the semi-stable limitcycle of the DVO expressed in (1)-(2). The region of attraction of f is D o = { ( s, ˙ s ) | ˙ s ≥ f ′ ( ϕ ) } . (5)The theorem states that a trajectory converges to f if it is initialized outsidethe closed curve of f in the state space. Proof:
Let us define a weighted error e as e = 12 ( ˙ s − f ′ ) + β s − f ) − f ′′ ( s − f ) . (6)If ( s, ˙ s ) = ( f, f ′ ), then we have e = 0. To show also that if e = 0 then ( s, ˙ s ) =( f, f ′ ), let us specify e in the three regions R , R and R , defined based on thephase definition (2). In the case s ∈ R , phase definition (2) results in s = f and the weighted error is simplified as e = 12 (cid:0) ˙ s − f ′ (cid:1) . (7)Thus, ˙ s = f ′ if e = 0. In the case s ∈ R , we have s < f l , f = f l , f ′ = 0 and f ′′ >
0. Thus, e is simplified as e = 12 ˙ s + β s − f ) − f ′′ ( s − f ) , (8)which is the sum of three positive terms. Thus ( s, ˙ s ) = ( f, f ′ ) if e = 0. Thesame results are true also if s ∈ R , but in this case s > f u , f = f u , f ′ = 0and f ′′ <
0. Consequently, e = 0 iff the states s coincide the trajectory of f in the state space, i.e. ( s, ˙ s ) = ( f, f ′ ). Thus, we consider v = e as thecandidate Lyapunov function for proving the semi-stability of the limit cycle f .To compute the time derivative of v , one needs to compute the time derivativeof the phase variable ϕ on which the trajectory f depends. Based on the phasedefinition (2), ϕ is continuous for s ∈ D o and thus, the time derivative of ϕ is˙ ϕ = ˙ sf ′ s ∈ R s ∈ R , , (9)where R , = R ∪ R . Therefore, the time derivative of the weighted erroralong the dynamics (1) is as follows˙ e = ( − α ˙ s ( ˙ s − f ′ ) s ∈ R − α ˙ s s ∈ R , . (10)he time derivative of v for x ∈ D o is obtained as˙ v = ( − α ˙ s ( ˙ s + f ′ ) ( ˙ s − f ′ ) s ∈ R − α ˙ s (cid:16) ˙ s + β ( s − f ) − f ′′ ( s − f ) (cid:17) s ∈ R , , (11)which is negative semi definite because ˙ sf ′ ≥ s ∈ R and f ′′ ( s − f ) ≤ s ∈ R , . Thus, e is bounded. This implies that the states s are bounded if thetrajectory f and its first derivative f ′ are bounded. For asymptotic results, itis sufficient to examine the largest invariant subset of the set Ω = { s : ˙ v = 0 } .Considering the dynamics (1), one verifies that { e = 0 } is the only invariant setof Ω. Therefore, asymptotic stability of e = 0 is concluded based on the LaSallelemma. The proof is completed by showing the radially unbounded property ofthe Lyapunov function v which is obvious from the definition of v . (cid:4) Remark 3.
If the set D = R − S where S := { ( s, ˙ s ) | s ∈ ( f l , f u ) , ˙ s = 0 } , (12)is a positive invariant set of the DVO then ϕ is also continuous in the inside ofthe closed curve of f . Thus, the proof is satisfied for s ∈ R . Consequently, onecan conclude that f is the globally stable limit cycle of the DVO.Assuring the positive invariancy of D is not possible as the matter of the conti-nuity of the dynamical system (1). Albeit, one can define the coefficient α suchthat D is almost an invariant set, i . e . for δ >
0, the set D δ = R − S δ where S δ := { ( s, ˙ s ) | s ∈ ( f l + δ, f u − δ ) , ˙ s = 0 } , (13)is a positive invariant set. Thus, f is almost global stable limit cycle of DVO i . e .the trajectory of the DVO converges to f from almost any initial condition. Thefollowing proposition suggests a definition for α which results in such a small δ that the trajectories converge to f from any initial condition in practice. Proposition 1.
Given that the following assumptions hold • Assumption 1 is satisfied, • β is positive constant, and • α is defined as α = α α b (cid:0) α b + tanh( f ′′ ) (cid:1) , (14)where α b = tanh (cid:0) α f ′ + α ˙ s + α ( s − f ) (cid:1) + ǫ, (15)and α , α , α , α , ǫ ∈ R + are constants and ǫ ≪ f is the almost globally stable limit cycle of the DVO. Considering Output Limits
The proposed DVO has been mainly conceived for performing cyclic motions inrobotics applications. If we define the output of the DVO as y ( t ) = s ( t ), thenwe can generate a cyclic signal, tracking a predefined desired trajectory, anduse it as the reference signal for the robot controller or directly as the controlsignal. Considering this scenario, it becomes necessary to provide the possibilityof generating a bounded output to avoid physical limitations of the robot suchas position, velocity or actuator limits. The rest of this section investigates theproblem of output limits in details.Assume that the feasible region of the output is as Q := { y ∈ R : y min < y < y max , | ˙ y | < δ ˙ y } , (16)where y min , y max , δ ˙ y ∈ R are constants denoting the minimum and maximumof the output y , and the maximum feasible magnitude of ˙ y . To preserve thefeasible region (16), we introduce the following output definition y = y avg + δ y tanh ( s ( τ )) , (17)where y avg = y min + y max , δ y = y max − y min and τ ( t ) is an exogenous state withthe following dynamics ˙ τ = δ ˙ y tanh ( s ′ ) J s s ′ , (18)where J s = δ y (cid:0) − tanh ( s ) (cid:1) .Given (17) and (18), the time derivative of the output y is˙ y = δ ˙ y tanh( s ′ ) . (19)Consequently, the output definition (17) guarantees that the output limits arepreserved, i . e . y ∈ Q .Now, we can write the DVO with respect to τ as s ′′ = g a ( ϕ ) − α ( s ′ − g v ( ϕ )) − β ( s − g p ( ϕ )) , (20)with g p ( ϕ ) = tanh − (cid:18) f ( ϕ ) − y avg δ y (cid:19) ,g v ( ϕ ) = tanh − (cid:18) f ′ ( ϕ ) δ ˙ y (cid:19) ,g a ( ϕ ) = δ y (cid:0) − tanh ( g p ) (cid:1) δ y (1 − tanh ( g v )) g v tanh( g v ) f ′′ , (21)and ϕ ( s ) = g − p ( g l ) s ≤ g l { ϕ | g p ( ϕ ) = s, g v ( ϕ ) s ′ ≥ } g l ≤ s ≤ g u , s ′ = 0 { ϕ | g p ( ϕ ) = s, g v ( ϕ ) ≥ } g l ≤ s ≤ g u , s ′ = 0 g − p ( g u ) s ≥ g u , (22)where g l and g u are the lower and upper bounds of g p .Integrating (20), one can compute s ( τ ), but we are still missing s ( t ) whichis required to compute the output y ( t ). To overcome this problem, let us defineew states ( s , s ) = ( s ( t ) , s ′ ( t )) and rewrite the dynamics (20) with respect tothe new states as following ( ˙ s = δ ˙ y tanh( s ) J s ˙ s = δ ˙ y tanh( s ) J s s ( g a − α ( s − g v ) − β ( s − g p )) . (23)Hence, one can integrate the dynamics (23) with respect to the time t andcalculate y ( t ).We call the dynamics (23) expressed with respect to the phase definition(22) as the Modified DVO (MDVO). In this section, we illustrate the DVO and MDVO performance when trackinga desired reference signal through a few numerical simulations.
We compared AS and AOS from a mathematical point of view in Section 2.Instead, to explore their difference from a practical point of view, we comparedthe response of the DMP, an autonomous system with AS trajectory proposedin [7], and the DVO, as an oscillator with AOS trajectory. To this purpose,we simulated these two systems when tracking the simple sinusoidal signal f =1 . t ) from the initial condition located on the desired trajectory. Fig. 2shows the behavior of the two systems in the state space (left plot) and in thetime domain (right plot). As it can be seen in the state space plot, the DVOremains on the desired trajectory but the DMP leaves the desired trajectory asthe initial condition is not equal to the desired initial value. The time domainplot shows that the steady state response of the DMP is in-phase with thedesired trajectory as the initial phase is chosen to be zero, while there is aphase difference between the steady state response of the DVO and the desiredtrajectory. In particular, the steady state phase difference of the DMP is alwaysequal to the chosen initial phase. However, the steady state phase difference of -2 -1 0 1 2-4-2024 0 2 4 6 8 10 Time (sec) -2-1012
DMPDVO Desired TrajectoryInitial Condition
Figure 2: The behavior of DVO vs. DMP when tracking a simple sinusoidaltrajectory.he DVO is not constant and is related to the initial conditions and convergencerate. This behavior is a consequence of the fact that the desired trajectory isan invariant set in the DVO but not in the DMP. So, we can say that the DMPimposes a time constraint on the system response, i . e . the system states mustassume the desired value at specific time instants. The DVO, instead, imposesa timing constraint , i . e . the system states replicate the desired trajectory whileguaranteeing the desired timing. More precisely, the system states assume thedesired value but not at specific time instants. For application such as leggedrobot locomotion where respecting the timing constraint is only required, anoscillator with AOS trajectory, as the DVO, is more appropriate than a systemwith AS trajectory in terms of tracking and control effort. To analyze the effect of different coefficients in the DVO structure, let us definetwo quantities: reaching phase and reaching time . The first one is the differencebetween the phase of the point at which the trajectory reaches the limit cycleand the initial phase, while the second one is the time required to reach thelimit cycle. Fig. 3 depicts the DVO response when tracking the sinusoidal signal f = 1 . t ) for five different values of the coefficients ( α , α , α , β ) whichmostly affect the motion in R , . As α increases and β decreases ( e . g . theblue and purple trajectories in Fig. 3), the reaching phase decreases. Though,as the time plot of the weighted error in Fig. 3 shows, these coefficients do notaffect much the reaching time. For α = 0, the system has a high dampingcoefficient when | s − f | and | f ′ | are small. Similarly, for α = 0, the dampingcoefficient is high when | ˙ s | and | f ′ | are small. In this way, the system convergesto the limit cycle with small velocity and acceleration ( e . g . the green and browntrajectories in Fig.3) which results in high reaching time. As Fig. 4 illustrates,the coefficient α influences the DVO behavior when the system is in R and | ˙ s | is small. In this case, the coefficient α increases and thus, the system experienceshigh acceleration which is unnecessary and also undesirable. -2 -1 0 1 2 3-4-2024 0 2 4 6 8 10 Time (sec)
Desired Trajectory = 2, = 1, = 1, = 1 = 4, = 1, = 1, = 1 = 2, = 0, = 1, = 1 = 2, = 1, = 0, = 1 = 2, = 1, = 1, = 4 Figure 3: The effect of coefficients ( α , α , α , β ) in the DVO structure. Thecoefficient α = 1 is constant. Time (sec) -6-4-20
Desired Trajectory = 1 Initial Condition = 0 Figure 4: The effect of coefficients α in DVO. All the remaining coefficients areconstants ( α = 2 , α , α , β = 1). Given the sinusoidal signal f = 1 . t ) as input, we simulated the MDVOwith output limits | y | < . | ˙ y | < . y , ˙ y ) = ( ± . , . e . g . the blue trajectory). Note that the two trajectories, withdifferent initial conditions, do not necessarily converge together because theMDVO provides limit cycle tracking not trajectory tracking. The response of the MDVO when changing the desired motion between threefunctions f , f and f is depicted in Fig. 6. In particular, f and f aretwo periodic functions with different amplitudes and frequencies, and f is aconstant function. The desired trajectory is changed every 20 seconds while thecoefficients of the oscillator are kept constant during the simulation. As can beseen, the output is smooth and its first time derivative ˙ y is continuous. As theMDVO is a second order differential equation, ˙ s and, consequently, the secondtime derivative of the output are not continuous. Time (sec) -2-1012 0 2 4 6 8 10
Time (sec) -4-2024
Figure 5: MDVO response for two different initial conditions. The red dashedlines are the output limits. The coefficients are chosen as α = β = 1, α = α = α = 2.
20 40 60 80 100
Time (sec)
Time (sec) -2-1012 0 0.5 1 1.5 2 2.5 3-2-1012
Figure 6: MDVO’s performances when changing the desired trajectory. The reddash lines are the output limits, yellow and green dashed curves are the desiredtrajectories f and f , and the blue point is the desired constant trajectory f .The coefficients are α = α = α = α = β = 2. The output limits are definedas 0 < y < . | ˙ y | < π . We presented a novel oscillator specifically designed for those robotic applica-tions where it is required to perform cyclic motions. The proposed oscillator isnamed DVO and it is a continuous 2-dimensional dynamical system which canconverge to any periodic trajectory depicting a non-self-intersecting curve inthe state space. Compared with existing results, our approach provides globalasymptotic orbital stability of the periodic function and, the stability property isirrespective of the parameters of the system. In addition, the proposed dynam-ical system can be used for tracking both periodic and constant functions. Thisproperty becomes important for those applications where both periodic motionsand constant posture are required. Using the proposed dynamics, one can alsogenerate a smooth modulation when switching from one desired trajectory toanother. Moreover, we proposed a modified version of the DVO, named MDVO,where we introduced a parameterization technique for satisfying the predefinedlimits on the output signal and its first time derivative. All the above mentionedroperties have been validated through simulations.The proposed dynamical system generates a one dimensional output, and soit can control only one degree of freedom of a robotic system. This means thatfor a robot with n degrees of freedom, we will need n DVOs (or MDVOs), i . e . onefor each degree of freedom. Thus, it becomes crucial to be able to synchronizemultiple systems of such kind to generate a multi-dimensional output. In ourfuture work, we will propose a technique to construct a synchronous networksof DVOs or MDVOs. References [1] S. H Strogatz.
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