CPG-based control of a turtle-like underwater vehicle

Transcription

1 Auton Robot (2010) 28: DOI /s CPG-based control of a turtle-like underwater vehicle Keehong Seo Soon-Jo Chung Jean-Jacques E. Slotine Received: 1 April 2009 / Accepted: 17 December 2009 / Published online: 22 January 2010 Springer Science+Business Media, LLC 2010 Abstract This paper presents biologically inspired control strategies for an autonomous underwater vehicle (AUV) propelled by flapping fins that resemble the paddle-like forelimbs of a sea turtle. Our proposed framework exploits limit cycle oscillators and diffusive couplings, thereby constructing coupled nonlinear oscillators, similar to the central pattern generators (CPGs) in animal spinal cords. This paper first presents rigorous stability analyses and experimental results of CPG-based control methods with and without actuator feedback to the CPG. In these methods, the CPG module generates synchronized oscillation patterns, which are sent to position-servoed flapping fin actuators as a reference input. In order to overcome the limitation of the open-loop CPG that the synchronization is occurring only between the reference signals, this paper introduces a new single-layered CPG method, where the CPG and the physical layers are combined as a single layer, to ensure the synchronization of the physical actuators in the presence of external disturbances. The key idea is to replace nonlinear oscillators in the conventional CPG models with physical actuators that oscillate due to nonlinear state feedback of the actuator states. Using contraction theory, a relatively new nonlinear stability tool, we show that coupled nonlinear oscillators globally synchronize to a specific pattern that can be stereotyped by an outer-loop controller. Results of experimentation with a turtle-like AUV show the feasibility of the proposed control laws. Keywords Biomimetic underwater vehicle Central pattern generator Synchronization Electronic supplementary material The online version of this article ( contains supplementary material, which is available to authorized users. K. Seo ( ) S.-J. Chung J.-J.E. Slotine Nonlinear Systems Lab., Massachusetts Institute of Technology, Cambridge, MA 02139, USA keehong.seo@samsung.com S.-J. Chung sjchung@illinois.edu J.-J.E. Slotine jjs@mit.edu Present address: K. Seo Mechatronics & Manufacturing Technology Center, Samsung Electronics CO., LTD., Suwon-City, Gyeonggi-do, , South Korea Present address: S.-J. Chung Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Champaign, IL, USA 1 Introduction Biologically inspired approaches to locomotion have been extensively studied in the robotics community. Examples include snake (Hirose 1993), fish (Triantafyllou and Triantafyllou 1995), salamander (Crespi and Ijspeert 2006), and turtle (Licht et al. 2004) robots. The flexibility and adaptability of bio-inspired mechanisms in dealing with dynamic environments have been discussed in the literature, especially in the context of an alternative to traditional means of locomotion such as wheels and propellers. However, a sophisticated control algorithm is required in order to control these biologically inspired types of robotic locomotion. One promising approach is to engineer the animal central pattern generators (CPGs) (Grillner et al. 1995, 1998), thus adopting modular designs. It is known that the CPGs can maintain rhythmic behaviors without external signals while it is also possible

2 248 Auton Robot (2010) 28: that such self-sustained behaviors are modulated by signals from brain or sensory organs (Stent et al. 1978; Rybak et al. 2006; Krouchev et al. 2006). In this regard, the CPGs, when applied to robotics, can significantly reduce the control bandwidth between the main controller and actuators. There have been noticeable efforts to design artificial CPGs for robotic systems. Lewis et al. (2005) have designed hardware implementation of the CPG for robots. Buchli et al. (2006) studied the categories and designs of limit cycle oscillators to be used with the CPGs. Arena et al. (2002) have developed a framework of cellular neural networks approximating chemical and electrical synapses in the model. Spatio-temporal algorithm was used to switch the locomotive patterns in a hexapod robot. Chen and Iwasaki (2008) and Iwasaki and Zheng (2006) interconnected two neurons to form a reciprocally inhibiting oscillator (RIO) that can be entrained to mechanical systems. RIOs were then used to construct their CPG model by using circulant matrices and as well to demonstrate the use of the CPG in a closed loop. Vogelstein et al. (2006) have developed a method of controlling locomotor patterns by applying electric pulses to the CPG module, which responds differently depending on the phase. Bandyopadhyay et al. (2008) demonstrated that synchronized inferior olive neuron model has rich enough dynamics to be used with their underwater vehicle by recording the simulation of the neuron model and then letting the vehicle replay the oscillating neuron trajectories. In Taga et al. (1991) and Taga (1998), the authors designed a CPG model by coupling flexor and extensor neurons for a simulated biped. The feedback from biped sensors to the oscillator helped the biped to walk stably in rough environments. As another underwater application, Wang and Yu (2008) proposed a systematic method to tune CPG parameters for a fish mechanism by fitting the locomotor pattern to a desired body-wave equation. Crespi and Ijspeert (2006) demonstrated that the CPGs can be used for an amphibian robot while successfully switching gaits between walking and swimming. Ekeberg (1993) proposed a combined model for neuronal and mechanical aspects for robot fish swimming. We have previously developed CPG models that can switch between swimming and crawling gaits by using coupled Hopf oscillators in our previous work (Seo and Slotine 2007). Its global stability was shown mathematically by extending the synchronization of nonlinear oscillators from Pham and Slotine (2007) and Wang and Slotine (2005). Although the model was able to respond to high control by changing a few parameters as commanded, the limitation was that the sensory feedback was not considered. If the CPG is used in a complex network comprised of many actuators and sensors, there might be issues of sensor timedelays. Hence, in order to fully benefit from the CPG, it would be desirable if the CPG adapts to the environment by itself without any external commands. There have been several studies (Taga et al. 1991; Iwasaki and Zheng 2006; Morimoto et al. 2006; Vogelstein et al. 2006) to accommodate direct sensory inputs from environments. Along this direction of research, we developed a novel CPG-based control and validated its performance with an underwater robot propelled by flapping-foils (Seo et al. 2008). In this approach, a limit cycle oscillator for the CPG emerges by the feedback of the physical actuator states. This is comparable to interconnecting non-oscillating neurons to obtain a limit cycle oscillator as in Chen and Iwasaki (2008), but in our case the result was with mechanical oscillators. Then by using the CPG framework developed in Seo and Slotine (2007), we could synchronize the mechanical oscillators thereby controlling locomotion patterns. In this paper, another CPG model is introduced in addition to those in Seo and Slotine (2007) and Seo et al. (2008). In this model, which we refer as the feedback CPG or the CPG with feedback couplings, by expanding the couplings within Hopf oscillators in the earlier CPG, the phase errors between the motor systems are fed back directly to the Hopf oscillators. As a result, the synchronization occurs between the motors rather than motor control signals. To compare the qualitative performance of our three CPG models: (1) the open-loop CPG (Seo and Slotine 2007), (2) the feedback CPG, (3) the single-layered CPG (Seo et al. 2008) provide an in-depth stability analysis as well as a study of the limit cycle property of the fin oscillation. Note that the name single-layered refers to the fact that there is only a mechanical layer without a neural layer in the control loop. The performance was tested with a real underwater platform for all the three CPGs. These are the main contributions of the paper compared to our previous works. For various types of robots, the actuation for locomotion essentially results from an oscillatory motion. This is also true for our testbed of interest, a turtle-like autonomous underwater vehicle (AUV) developed by the MIT Tow Tank Lab (Licht et al. 2004). Its swimming maneuvers are controlled by the roll and pitch motion of its four fins which mimic the forelimbs of a sea turtle. Note that the roll and pitch motions in the previous experiments in Licht et al. (2004) were generated by open-loop sinusoidal functions. The experimentation with the open-loop CPG and the single-layered CPG successfully maintained the attitudes with respect to a sequence of CPG commands. The test with the feedback CPG however indicates that there are more to investigate regarding the delayed feedback of nonlinear oscillators, which we leave as a future work. A benefit of using the CPGs in the control of underwater robot is that they reduce the control bandwidth the CPGs save main controllers from computing the time trajectories of all the fins. This point would be even more appealing if the

3 Auton Robot (2010) 28: CPGs were designed using electrical circuitry such as VLSI chips. Our proposed CPG algorithms were implemented on a microprocessor-based real-time controller that also serves as the main controller to govern the attitude of the vehicle. Let us emphasize the significance of the single-layered CPG, in which the actuators and the CPG dynamic model are combined in a single layer. The single-layered CPG preserves the advantage of the conventional CPGs that gait patterns can be switched by adjusting a few parameters, while ensuring the phase synchronization of the physical actuators. This is a significant improvement to CPG-based control theory because the open-loop CPG approach pursues only the synchronization of the reference inputs to the actuators. Unlike the open-loop or feedback CPGs, where the outputs from the CPGs are used as reference for position servos of fin actuators, the new CPG does not require reference trajectories because the fins are oscillating with the prescribed amplitudes and frequencies as if they are selfsustained, which results in some robustness. In the presence of external disturbances, no matter how much the actuators deviate from the desired limit cycle, there is no need for tracking control while ensuring convergence to the limit cycle. We believe that one of the reasons to pursue CPG-based control is that the CPGs eliminate the need to preplan motor trajectories for every instantaneous time and every degree of freedom, especially when the biomimetic robots have increased degrees of freedom. For this reason, the proposed single-layered CPG method presents a potential to realize truly adaptive locomotion in the presence of challenging dynamic environments. The paper is organized as follows. Section 2 describes the turtle-like AUV as our testbed for experimental validation. In Sect. 3, we introduce open-loop and feedback CPG-based control methods as well as experimental results with the AUV. In Sect. 4, the single-layered CPG-based controller is proposed and its performance of synchronization is discussed, followed by experimental results. 2 The bio-inspired underwater vehicle system For the test platform of our approach for biologically inspired control of locomotion, we use an autonomous underwater vehicle (AUV) propelled by flapping fins that resemble the flapping forelimbs of a sea turtle. As shown in Fig. 1, it has four flapping fins with two degrees of freedom per fin, whose roll and pitch directions are actuated by two independent electric motors. The motors are PD-controlled by a servo mechanism to follow reference inputs. The vehicle can be as large as 2 m 0.5 m 0.5 m. The top operating speed is near 2 m/s and the flapping fins provide all of the propulsion as well as control of attitude and position. A more detailed description of the vehicle can be found in Licht et al. (2004). Fig. 1 The biomimetic flapping foil autonomous underwater vehicle (BFFAUV) was developed as a test platform and proof of concept for the use of flapping foil fins as the sole source of propulsion and maneuvering forces in an underwater vehicle The experiments in the paper were performed at a water tank in MIT Towing Tank Lab. The tank is 33 m long and 2.6 m wide with an average depth of 1.4 m. The vehicle is set up neutrally buoyant and the center of the gravity is located higher than the center of buoyancy, hence the vehicle cannot maintain its attitude without a proper controller. In this paper, experimental results using the turtle-like AUV are presented to demonstrate the effectiveness of our CPG-based controllers. The performance is tested by controlling the roll, pitch, and yaw angles of the robot as well as the underwater height. The gait-switching ability of the proposed CPG model is also demonstrated. Gait patterns usually refer to the leg coordination patterns in legged animals but we also adopt the same definition to describe various coordination patterns of the four fins of the turtle robot. Figure 2 illustrates snapshots of pronk, walk, and bound gaits. In the pronk gait, all four fins are in phase. In the walk gait, the fins have a 90-degree phase difference with each other. In the bound gait, the front and hind fins are 180 degrees out of phase (Collins and Stewart 1993). 3 Conventional CPG-based control models In this section, we present a theoretical foundation for CPGbased control of the AUV, based on conventional CPG-based control approaches. In these approaches, the CPG exists as a separate module independent from the servos as in Fig. 3. Based on the architecture in the figure, we propose two CPG-based controllers one is an open-loop method without sensory feedback from the servos to the CPG and the other is a feedback approach where feedback couplings from the servos to the CPG exist. 3.1 Open-loop approach Hopf oscillator In the paper, a Hopf oscillator refers to the following dynamic system: ẋ = f H (x), x = (uv) T (1)

4 250 Auton Robot (2010) 28: Fig. 2 For each gait the relative phases of fins are shown for the period of 1. LF, RF, LH, and RH stand for left front, right front, left hind, and right hind fin, respectively Fig. 3 A conventional control loop containing a CPG module with feedback coupling is illustrated. If the direct sensory feedback from servos to CPG is omitted, then it forms an open-loop CPG approach instead where ( ( ωv λ u 2 +v 2 1 ) ) u ρ f H (x; ω,ρ,λ) = 2 ωu λ ( u 2 +v 2 1 ). v ρ 2 The terms ωv and ωu make the system state rotate around the origin on the u v plane and the parameter ω determines the frequency of the oscillation (i.e., the rotational velocity in the u v plane). The rest of the equation makes the system state attracted to a circular orbit of radius ρ>0 and the attraction rate depends on λ>0. Figure 4 shows the limit-cycle behavior of the oscillator and how the convergence depends on λ. In the sequel, we often refer to the phases and amplitudes of oscillators. On the u v plane, the phase φ of the oscillator is defined by the angle ( u + vi) of the oscillator and the amplitude is defined by the distance ( u + vi ) from the origin. Fig. 4 (a) Trajectories of the Hopf oscillators are plotted for ω = 1, ρ = 1 with different λ s on the state space along with the vector fields. (b) The time trajectories of v(t) are plotted for λ = 1 and 0.5 to illustrate the effect of λ on the convergence to the nominal limit cycle CPG model In the CPG literature, there exist various CPG models as well as various neuron models. In some CPG models, once neurons form reciprocally inhibiting relations, they then oscillate and spike periodically. In this case, the periodic behavior of each neuron and the spiking pattern of whole network both come from the coupled dynamics of nonperiodic systems. Other CPG models, as in Ijspeert et al. (2007), use limit-cycle oscillators to represent the periodicity of neurons although it actually comes from mutually inhibiting neurons. In these models, the couplings in the CPG contribute mainly to the pattern as a whole while in-

5 Auton Robot (2010) 28: dividual neuron models in the CPG is assumed to oscillate intrinsically. Hence, if we are interested in how a network of subsystems can generate various gait patterns, rather than how each neuron can spike periodically, it is a good idea to use limit cycle oscillators to represent neurons in CPG and assume they interact through particular couplings in a network. In the previous work (Seo and Slotine 2007) of the authors, Hopf oscillators were used to represent neuronal clusters that spike periodically. The oscillators were connected by diffusive couplings, which resemble Kuramoto couplings (Strogatz 2000), although we adopted one more degree of freedom: Kuramoto model has only phase variable but Hopf oscillator has amplitude variable as well. Since the turtle robot we consider has 4 fins with 2 DOF per fin, we can consider a network of 8 oscillators, where x i s are for roll motions and y i s for pitch motions, as follows: ẋ i = f H (x i ; ρ i ) + g w ij (r ij R(φ ij )x j x i ), j (2) ( ) ρyi ẏ i = f H (y i ; ρ yi ) + k R(φ yx )x i y i ρ i (3) for i = 1,...,4, where j should be determined from the CPG architecture; R(φ) is a 2-D rotational transformation of angle φ; g and k are constant coupling strengths; w ij is the weight of a coupling term. A constant r ij, which is the ratio of the desired oscillation amplitudes ρ i /ρ j, enables coupling between oscillators of different amplitudes. The desired phase offset φ ij guides x i to synchronize with x j after phase shift of φ ij. The couplings in (2) assume general architectures while y i receives signals exclusively from x i. For example, the CPG with a two-way ring structure looks like Fig. 5. Either element of each oscillator state x = (u, v) T in the CPG model above can be used as reference angular position for a fin motion controller as follows: r roll,i = x (2) i, r pitch,i = y (2) i. (4) Fig. 5 The turtle CPG with two-way ring architecture We chose the v-components, that is, the second elements (denoted by superscript in parentheses) of the state vectors x and y are selected for the reference angular position r roll and r pitch for roll and pitch servos, respectively. Both u and v from the Hopf oscillator should generate the sinusoidal waveforms after transient. For an initial condition of x = (ρ, 0) T, the solution of the oscillator is u = ρ cos ωt and v = ρ sin ωt. In this case, because the fin position is at zero initially, selecting v as the reference for the fin position is natural. For the initial condition of x = (0,ρ) T, one would choose the u-component as the reference for smooth startup. If one plots the position and velocity trajectory on a position-velocity plane with the horizontal axis being the position, then the trajectory will be circling clockwise. To make this comparable to the Hopf oscillator trajectory on the u v plane, it is reasonable to exchange u and v axis. Then v will correspond to the position of the fin and u to the velocity. There is no restriction on the topology when coupling the oscillators in (2) only if they form a connected graph. We tested one-way ring and two-way ring architectures and chose the two-way ring architecture as in Fig. 5 since the two-way ring architecture showed better gait transition performance. The one-way architecture is also stable in theory and causes no problem in practice Stability By using contraction theory we prove that the oscillators in the CPG model (2) (3) synchronize globally and exponentially for strong enough couplings. When they are synchronized, it is obvious that the coupling terms vanish and thus each oscillator shows stable limit cycle behavior of a Hopf oscillator. The framework we use was first introduced in Pham and Slotine (2007) and then further developed in Seo and Slotine (2007). In Pham and Slotine (2007), the authors discuss about coupled oscillators as an example to apply their framework. In Seo and Slotine (2007), an effort was made to adopt unbalanced couplings for oscillators of different sizes by devising generalized Jacobians for various CPG architectures. In this section, we focus on the open loop CPG model specialized for the turtle robot and present a simpler proof for the synchronization of oscillators with different amplitudes and phase offsets. To prove the synchronization, we first identify a flowinvariant linear subspace M 1 of the roll oscillators x i.after searching for a set of x i s that makes the couplings in (2) vanish, we have M 1 = { x R 8 x 1 = T 12 x 2 = T 13 x 3 = T 14 x 4 }, (5)

6 252 Auton Robot (2010) 28: where x = (x T 1,...,xT 4 )T and T ij = r ij R(φ ij ). If the system is in M 1, then the ẋ i s are also in M 1 due to the following property of Hopf oscillator: f H (rrx; ρ) = rrf H (x; ρ/r), (6) for an arbitrary rotational transformation R, and a scalar r>0. Hence, once the system is found in M 1 then it stays there for all future times verifying that M 1 is the equilibrium of the system. We can transform the system to be invariant to the phase offsets and radius differences using new vectors χ i s as χ i = T 1i x i. (7) As a result, (2) becomes χ i = f H (χ i,ρ)+ g j w ij (χ j χ i ) (8) for i = 1,...,4, where ρ = ρ 1. Consequently, M 1 is rewritten in terms of χ i s as M 1 ={χ R 8 χ 1 = χ 2 = χ 3 = χ 4 }. (9) According to the framework of the partial contraction introduced in Pham and Slotine (2007), the solution of the system (8) converges globally and exponentially toward M 1 if VJ 0 V T < 0, (10) where V is a 6 8 orthogonal matrix whose rows consist of a basis for M 1, and J 0 is the symmetric part of the Jacobian of the dynamics (8). V T V is a projection from the domain R 8 to M 1. Let us stack (8) fori = 1,...,4 in a brief form as χ = f H ( χ) gl χ, (11) where f H ( χ) is an 8 1 stack of vector functions f H (χ i,ρ); and L is a connection matrix representing the couplings between oscillators. The Jacobian of (11) is found as J = G gl, (12) where G(χ 1 ) G(χ G = 2 ) G(χ 3 ) 0 (13) G(χ 4 ) and G(χ) = f H χ, whose symmetric part is found for χ i = (u, v) T as G 0 = λ ( ρ 2 (3u 2 + v 2 ) ) 2uv ρ 2 2uv ρ 2 (u 2 + 3v 2 (14) ) whose eigenvalues are λ ρ 2 (ρ2 u 2 v 2 ); λ ρ 2 (ρ2 3u 2 3v 2 ). For the symmetric part G 0 of G, because V consists of orthonormal vectors, V G 0 ( χ)v T is upper-bounded as V G 0 V T < sup χ i λ max (G 0 (χ i ))I <λi. (15) Hence, a sufficient condition for global and exponential synchronization that satisfies (10) gλ 1 > sup λ max (G 0 ( χ))= λ (16) χ for g>0, where λ 1 is the smallest eigenvalue of VL 0 V T, i.e., (VL 0 V T ) λ 1 I and L 0 = 1 2 (LT + L). Such g>0exists if λ 1 > 0. Then, a solution χ(t)of (8) converges globally and exponentially toward M 1, which implies that the synchronization of x i s with the desired phase offsets for the heterogenous amplitudes as in (5) is globally and exponentially stable. Remark 1 (Olfati-Saber and Murray 2004; Pham and Slotine 2007) The condition for L to satisfy λ 1 > 0iswelldiscussed in Pham and Slotine (2007). If L is balanced, that is, i j w ij = i j w ji, since the couplings we use are diffusive, L 0 0 and VL 0 V T > 0. Remark 2 If L is not balanced, it is not guaranteed but possibletohavevl 0 V T > 0, which we can check easily for a given L. As a special case, if the network of oscillators form a twoway ring architecture in the order of and L is not balanced, we can find a transformation ψ = χ for an invertible such that the network of ψ i s is balanced if the following condition is satisfied by L: w 12 w 23 w 34 w 41 = w 14 w 43 w 32 w 21 (17) i.e., the products of coupling weights along the ring are equal for both directions. Then, it is guaranteed that the transformed connection matrix L = L 1 is symmetric and positive semidefinite; and there exists a projection V such that VL V T is positive definite. Notice that (17)issatisfied even if 7 weights are selected random as long as the 8th one follows the rule. Because this is an original claim, we provide its proof in Appendix A. Similar result was proved in Seo and Slotine (2007) for the network of Hopf oscillators with a chain architecture, e.g., , without requiring the constraint of (17). We can also prove that y i s in (3) synchronize exponentially with their paired x i s that are connected unilaterally

7 Auton Robot (2010) 28: from x i to y i. In fact, once x i is in synchrony with other x-oscillators, the diffusive coupling terms vanish and the system reduces as follows. ẋ i = f H (x i ; ρ i ), (18) ( ) ρy ẏ i = f H (y i ; ρ y ) + k R(φ yx )x i y i. (19) ρ i If we define a new variable w i = ρ y ρ i R(φ yx )x i, then we have ẇ i = f H (w i ; ρ y ), (20) ẏ i = f H (y i ; ρ y ) + k(w i y i ). (21) If we use the same approach with which we proved the stability of x-oscillators synchronization, we find V and L as V = 1 ( ) 0 0 (I I); L =. (22) 2 I I Hence we conclude that k>λis a sufficient condition such that w synchronizes with x, that is, y globally and exponentially converges to the flow-invariant subspace {( ) x y i M 2 = = R(φ yx ) x } i. y ρ y ρ i The oscillators so far discussed are centered at the origin (0, 0). To achieve full controllability of the attitude of the turtle robot, we need to modulate the center of oscillation for y-oscillators in addition to the amplitudes of y-oscillators. Hence, the model in (3) can be adjusted as ẏ i = f H (y i c i (t)) + k(rx i y i + c i (t)) (23) where c i (t) is the center of oscillation. For example, if we needed a 30-degree offset for r pitch,i at a certain time, then we could assign c i = (0, 30) T so that the pitch reference would oscillate around 30 degrees. Since c i (t) is independent of x and y, it does not affect the stability of synchronization discussed above Underwater experiments The purpose of this experiment is to demonstrate the feasibility of the proposed open-loop CPG-based controller and its ability to switch between different gaits. The overall control loop for the experiment is illustrated in Fig. 3 but in this experiment we do not have sensory feedback from servos into CPG. The robot senses its pitch, roll, yaw angles and underwater depth. The PID-controlled outer-loop controller computes amplitudes and offsets for fin oscillations, which change the thrust from each fin. The outer-loop controller then sends the desired amplitudes and offsets of fin oscillations to the CPG module. Accordingly, the CPG module generates oscillatory signals as commanded to form a specific patterns then yields r pitch and r roll as the reference for the fin servos. The servos control the fins to track the reference signals using built-in PD-type servo mechanism. This completes the overall control loop. The outer-loop controller and the position tracking servos we used are basically the same as the ones previously used in Licht et al. (2004) since the purpose was to demonstrate the feasibility of CPG itself as a pattern generating module. The test would be a success if the CPG responds to the low bandwidth signals from the outer-loop controller to govern the fin controllers, and the attitude is maintained underwater as commanded. If it fails to control the attitude, the vehicle immediately turns upside down since its center of buoyancy is below the center of gravity. The detailed performance measures may or may not outperform the one in Licht et al. (2004) but there could be various facts due to the circumstances that significantly affect the experiments. Hence, we evaluate the underwater result as a success or a failure from a qualitative point of view without providing quantitative comparison with Licht et al. s experiments. Figure 5 illustrates the architecture of CPG network used for the experiment. The x-oscillators in (2) are presented as roll oscillators in the figure while the y-oscillators are pitch oscillators. Roll oscillators are coupled with each other and the phase bias between them varies depending on the desired gait. Pitch oscillators are only coupled with the corresponding roll oscillators with a fixed phase bias of ±π/2, where the sign depends on whether the vehicle is commanded to hover or to proceed and whether the fin is the front one or rear. The signals r pitch and r roll from the oscillators are sent to the servos as reference inputs. We had the following coupling matrix L for (8) to represent the two-way ring architecture of roll-oscillators that we used in the experiment: 1.2I 0.6I 0 0.6I 0.7I 1.4I 0.7I 0 L =. (24) 0 I 2I I I 0 I 2I Notice that this L satisfies (17). Hence, we know this coupling will make the system synchronize globally and exponentially. In fact, for I /7I 0 0 =, (25) 0 0 6/10I /10I

8 254 Auton Robot (2010) 28: we have symmetric and positive definite L = L 1 > 0. Projection V is then found by orthogonalizing the row vectors of 6I V 7I 0 0 = 0 7I 10I 0 (26) I 10I to yield I I I VL V T = I I I, (27) I I I whose minimum eigenvalue is Hence, the trajectory of χ converges globally and exponentially to M 1 for g such that g>λ. (28) The convergence of χ to M 1 implies that x synchronizes globally and exponentially for any given phase offsets φ ij and desired amplitudes ρ i s in (5) since we did not impose conditions on them so far. The attraction rate λ of the Hopf oscillator was set to 0.75 in the experiment. We set λ at this range with the following reason. When the robot switches between hovering and going forward, two rear fins have to revert its facing direction by shifting the center of pitch oscillation between 0 and 180. An increased λ results in an increased speed in reverting the reference of fin directions, which may possibly damage the vehicle unless a smoothening filter is used. To test the performance of a controller with the CPG in the loop, we performed an underwater experiment where the main CPU controlled its attitude and depth at reference values that intermittently changed. To fit in the confined space of the test tank, we let the robot hover in one place by commanding the fore fins face forward and the rear fins backward. Hence the centers of pitch oscillation have 180 differences between the fore and rear fins as in Fig. 6, which shows the pitch oscillation of the fins during the underwater experiment. Due to the exponential stability of the synchronization and the stable limit cycle of the Hopf oscillator itself, the CPG oscillators quickly respond to the commanded amplitudes. In Fig. 6(b), the amplitude of the oscillatory motion is consistently beyond the commanded amplitudes (non-oscillatory lines). We attribute this difference to the slow update rate of the Hopf oscillator states within the microprocessor, which was 15 Hz. It should not cause a trouble in controlling the underwater vehicle because such errors show consistency. Figure 7 shows how the attitude (roll, yaw, and pitch) and depth (heave) of the turtle robot followed the reference values. The overall performance was reasonably good, especially considering limitations due to the following realworld factors: the overall vehicle response is slow in the Fig. 6 Pitch angles of the 4 fins are plotted (a) for the whole mission time and (b) magnified for t = 33 to 55 s. Piecewise constant values (thin lines) in(b) indicate the amplitude and offset of the oscillation commanded from the main controller. The command is sent down approximately every oscillation cycle. The time window shown here corresponds to the rise of the yaw angle in Fig. 7 Fig. 7 Attitude of the vehicle recorded from the underwater mission: the heave, yaw, pitch, and roll are tracking commanded values water; the dynamics of the vehicle is a multi-input-multioutput system, where the degrees of freedom are coupled,

9 Auton Robot (2010) 28: Remark 3 (Symmetry breaking) The reason we set uneven weights for couplings in (24) istobreakthesymmetryin the dynamics. Suppose we had a symmetric L, e.g., all the non-zero coupling weights are equal. The overall dynamics are then spatially symmetrical among the four oscillators, all the subspaces corresponding to the pronk, walk, and bound gaits are flow-invariant, even though only one gait can be globally stable for a given set of φ ij s. This fact makes it difficult for the CPG to switch between gaits. For example, suppose that the CPG was stably at walk subspace when a new gait pronk was commanded by changing the desired phase offset from 90 for walk to 0 for pronk. Walk then immediately becomes an unstable gait but, because it is a flow-invariant subspace, the CPG maintains walk gait until it is perturbed. 1 As soon as the CPG escapes from the unstable subspace, it exponentially converges to the pronk subspace, which is the globally stable flow-invariant subspace at the moment. When the CPG module is executed on a microprocessor, nothing perturbs the CPG dynamics from unstable synchrony unless random noise are added for example. In the experiment, perturbing the symmetry in the coupling weights broke the symmetry of the system dynamics and thus prevent the CPG from remaining at an unstable synchrony. 3.2 Feedback CPG Fig. 8 The figure shows how CPG changes its patterns. The commanded gaits over time were walk, bound and pronk. The roll anges of the fins follow the CPG outputs. The bottom plot shows relative phase of CPG oscillators during the experiment, where the phase offsets change from 0 to 90, 180, and back to 0 e.g., adjusting pitch affects heave, etc. From this experiment, we conclude that the CPG is feasible as a module that generates reference signals for fin motion while plugged in the overall vehicle control loop. To verify the CPG capability to generate various patterns of synchronization, i.e., various coordinations over fin oscillations and the capability to switch them continuously without extra computations, we performed another experiment. When we varied the phase difference φ ij between roll oscillators from 0 through 90, 180 and then back to 0, the gait changed in the following order: walk, bound, and pronk. These gaits are explained in Sect. 2 and illustrated in Fig. 2. Figure 8 shows CPG outputs, actual fin roll angles, and the relative phases of the CPG roll oscillators during the experiment. The experiment demonstrates that the CPG module can switch from one gait to another in a very smooth and quick fashion. We regard the approach in the previous section as an openloop CPG because there is no direct feedback from the fin states to the CPG although the CPG is located in a feedback loop via the main attitude controller. Generating synchronized CPG signals does not mean that the fin states will be also synchronized, particularly if fins are perturbed. In this section, we investigate if direct adjustment of CPG phase relation by the feedback of the mechanical fin states is possible. We simply modified the open-loop CPG in (2) and (3) so that the oscillator equations accept the fin positions and velocities. With the direct feedback from fins, the CPG can compensate for synchronization errors in servo systems. The overall system architecture in Fig. 3 shows the controller has an inner loop and an outer loop. The stability of feedback CPG was discussed under a steady-state assumption as well as simulations. The feedback CPG was also tested with the turtle robot Formulation In this section, we obtain a simplified CPG-fin model by regarding the fins as PD-controlled mass-spring-damper systems. In the steady state, the signals generated from CPGs are sinusoidal, and thus the resulting fin positions are also sinusoidal of the same frequency with a constant amplitude gain and phase lag. Suppose that the ith fin motion in roll axis is described by p i =[ẏ i /ω y i ] T, where y i is the angular position, and ω the oscillation frequency. One can then modify the original CPG model (2) by replacing the oscillator state x i in the coupling terms with the actual fin state p i as ẋ i = f H (x i ; ρ x ) + g(r(φ 1 )p i 1 p i ), (29) whose schematic is also illustrated in Fig. 9. Notice that we used one-way ring network with a common phase offset φ 1 for simple formulation. In the steady state, we can assume p i = A i (ω)r(φ lag (ω))x i, (30) 1 This is analogous to a pencil standing on its tip, which cannot really fall until it breaks its symmetry and chooses one specific direction to fall.

10 256 Auton Robot (2010) 28: We found an orthogonal projection matrix V to M 1 ={x 1 = =x 4 } as V = 1 I I I I 2I 2I (34) 0 0 2I 2I For the symmetric part L 0 of L, the eigenvalues of VL 0 V T are found as 2 cos φ lag and cos φ lag ± sin φ lag. To achieve the global synchronization the following condition should be satisfied: Fig. 9 The scheme for feedback coupling is illustrated. The synchronization errors of the actual physical states drive the Hopf oscillators so that the synchronization occurs between p i s where A i (ω) denotes the gain of p i to x i, and φ lag (ω) the phase difference between x i and p i. Hence, the CPG model in (29) can be rewritten as ẋ i = f H (x i ; ρ x ) + ga i R(φ lag )(R(φ 1 )x i 1 x i ). (31) The arguement ω is omitted since we assume constant ω and thus constant A i and φ lag. In this setting, we expect to see the synchronization of R(φ 1 )p i 1 = p i, that is, the synchronization of the physical system. If there is any perturbation on the phase of p i 1,we have φ lag + φ instead of φ lag, which affects (31) as ẋ i = f H (x i ; ρ x ) + ga i R(φ lag )(R(φ 1 + φ)x i 1 x i ) (32) and as a result x i will synchronize to R(φ 1 + φ)x i 1 compensating for the perturbation φ in p i 1. In this way, p i s, instead of x i s, synchronize with the desired phase difference of φ 1 when a perturbation exists Stability Let us assume A i = 1 and φ 1 = 0, without loss of generality g is always multiplied by A i and we can always use the transformed variable χ in (7) to forget about nonzero φ 1. Then, the only difference from the open-loop CPG (2)isthat (31) has R(φ lag ) representing added phase lag due to the PDcontrolled servos. To prove the stability of this system, we can apply the same stability analysis as in Sect with R(φ lag ) 0 0 R(φ lag ) R(φ L = lag ) R(φ lag ) R(φ lag ) R(φ lag ) R(φ lag ) R(φ lag ) (33) λ min (VL 0 V T )>0, (35) and thus φ lag must satisfy cos φ lag > 0 and cos φ lag > sin φ lag. Hence, the synchronization of delayed feedback coupling is exponentially stable if the constant coupling gain g satisfies g> λ (cos φ lag sin φ lag ) (36) for φ lag ( π/4,π/4). In general terms, if the fin controllers have phase lag less than 45, then the exponential synchronization is guaranteed for a sufficiently strong coupling gain g. The analysis extends to a two-way ring architecture and all-to-all architecture straightforwardly only by changing the coupling matrix L. The two-way ring architecture with 2R R 0 R R 2R R 0 L = (37) 0 R 2R R R 0 R 2R yields VL 0 V T > 0 with repeating eigenvalues of 2 cos φ lag and 4 cos φ lag. For the CPG with all-to-all coupling architecture with 3R R R R R 3R R R L =, (38) R R 3R R R R R 3R performing the same analysis shows that VL 0 V T has repeating eigenvalues 4 cos φ lag, where the global and exponential λ synchronization is achieved for g> 4cosφ lag with φ lag < π/ Simulation effect of φ lag We performed numerical experiments to verify the stability of the synchronization in coupled oscillators in (31). For the set of simulation parameters ρ = 1, ω = 1, g = 4 and λ = 1, the stability condition in (36) implies that the oscillators synchronize over φ lag ( 34.8, 34.8 ). During the

11 Auton Robot (2010) 28: Fig. 10 For 4 coupled Hopf oscillators with feedback delay φ lag,the phase lag was increased over time. Oscillators started to synchronize after φ lag had entered the stability zone φ < The synchronization was maintained for a while after φ lag had left the zone simulation, φ lag was increased from π to π over time, and we observed the oscillators began to synchronize as soon as φ lag gets larger than The synchronization was maintained until the delay increased past 34.8 to 60 or so as illustrated with the time trajectory of the second element x (2) i of the oscillator states in Fig Simulation with PD-controlled system We conducted another simulation to show that the CPG with feedback coupling in (29) can compensate for perturbation in PD-controlled mechanical systems. As depicted in Fig. 9, the second element x (2) i of the oscillator x i was the input r i to a PD-controlled mass-spring-damper system. The output y i of the PD-controlled system and its derivative ẏ i comprised p i =[ẏ i /ω y i ] T. The gains for PD-controllers were set to the same values of K P = 4 and K D = 16 but the mass, damping constant, and spring constant are different between the four mechanical systems: the masses ranged from 0.1 to 4, damping constants from 1 to 2, spring constants from 0.1 to 1 with MKS units. As a result, the four PD-controlled systems showed different phase lags from x i to p i in the steady state as plotted in Fig. 11(b), which appear as perturbation φ in the CPG model (32). In Fig. 11(a), the outputs y i s showed almost identical and phase-locked trajectories despite the difference in parameters. In contrast with the open-loop CPG method, the feedback CPG method synchronizes the states of the physical systems instead of synchronizing the reference signals only. The limitation of this method as proposed in this paper is obvious from its analysis, where we assumed steady state of the system and thus (30). As one can see from Fig. 11(b), the phase lag can be larger than 45 in the transient. Depending on the controller gains and coupling strength, it is possible that the system can become unstable so that the higher frequency oscillation in the transient time of Fig. 11(a) persists. From our experience which we Fig. 11 Simulation result of PD-controlled mass-spring-damper system governed by a feedback CPG. The controller gains were set identically to K P = 4andK D = 16 for the four controllers but system parameters are set differently: the masses ranged from 0.1 to 4, damping constants from 1 to 2, spring constants from 0.1 to 1. The coupling gain in the CPG was 3 do not present here, stronger coupling strength could invoke such a case. Performing more rigorous analysis on this matter without the assumption (30) is left as an open problem. 3.3 Turtle robot experiment We tested the feedback CPG to the turtle robot outside the tank. In the experiment, the roll motions were governed by the feedback CPG in (29): the state p i of the roll motion was compared with another fin state p j, then the sum of differences were added to the dynamics of governing Hopf oscillator x i to affect its amplitude and phase. The coupling schemes we tested for the experiment are one-way ring and all-to-all structures. Through the experiments, the governing oscillators for pitch motion unilaterally received coupling signals only from the roll Hopf oscillators that governed the same fin with a 90 phase offset. In the first experiment, all-to-all feedback coupling was used for roll CPG as ẋ i = f H (x i ; ρ x ) + g j i(p j p i ). (39) In Fig. 12, the fins synchronized despite initial errors. Disturbance was then applied by limiting the maximum torque of the second fin from t = 7 s over four cycles. Thanks to the feedback couplings from actual fin states, the CPG partially compensated for such disturbance by slightly increasing the amplitude of the second CPG oscillator while decreasing the

12 258 Auton Robot (2010) 28: Fig. 12 The feedback CPG approach was tested to synchronize the fins of the turtle robot. The second fin was limited in its torque. (a)the error in the second fin states affected the CPG oscillators to slightly adjust CPG amplitudes. (b) The synchronization recovered from disturbance demonstrating the robustness of the system Fig. 13 The feedback CPG approach with one-way ring architecture was tested. Despite an initial error, they synchronized. The phase error in the transient shows a chasing pattern, which could be due to the ring network amplitudes of others as shown in Fig. 12(a). The coupling strength g was set to 0.1. We also tested with one-way ring couplings of (29). From the plots in Fig. 13, one can observe that there was small initial error but soon they all synchronized. The relative phases during the transient time in Fig. 13(b) shows that they tend to chase each other head to tail in the order that they are coupled. It is probably due to the nature of the one-way ring network of the CPG. The coupling strength of g = 0.3 was used. In another experiment in Fig. 14, the CPG failed to synchronize. The only difference from the previous CPG was that the coupling was increased to g = 0.7. The reason of failure could be that the stability analysis, as discussed earlier, hold under the steady-state assumption. It seemed that growing the coupling strength over certain limit caused undesirable modes to emerge. Rigorous analysis for this phenomenon is out of our current scope. However, if one could find a way to control such bifurcation, then simply changing the coupling strength could lead to the change of gait patterns, which could be a powerful tool for engineering CPGs and perhaps for explaining biological systems. Fig. 14 When the feedback CPG with one-way ring architecture was tested, the roll motions failed to synchronize for increased g of Novel single-layered CPG approach The conventional CPG approaches in the previous sections show their feasibility through simulation and experiments. The CPG with feedback coupling was introduced to address perturbation in the mechanical layer of the overall control loop. Since p i is not perfectly circular as in a Hopf oscillator, the approach with feedback couplings has limitations in its performance. Actual synchronization of the fins is still affected to an extent by the identicalness of the fin mechanisms. In this section, we propose a control law such that the synchronization is truly extended beyond the CPG oscillators to the actual fins. The control law induces self-

13 Auton Robot (2010) 28: where I is the moment of inertia, B the nonlinear damping term as a function of ẋ, K the spring constant, and τ the input torque. x represents angular displacement, and ẋ angular velocity. The nonlinear damping of the fin in the fluid can be modeled as B(ẋ) = β 0 + β 1 ẋ, (42) Fig. 15 Control architecture using single-layered CPG sustained oscillation in the fin motions and also enforces direct synchronization of the fin oscillations. From the CPG framework, we replace Hopf oscillators in the CPG with oscillating fins, which also exhibit stable limit cycles. When considering the overall control architecture in Fig. 15 with the new control law, we realize that the CPG module and the servo module are merged in one and call it single-layered CPG. In this approach, we render oscillations with stable limit cycle to emerge from the servos by feeding a nonlinear function of velocity back to the motor torque. Even though the resulting oscillator is not circularly symmetric as in a Hopf oscillator, we use this mechanical oscillator to replace Hopf oscillators in the CPG and the coupling between mechanical oscillators is also implemented. In summary, the fin motions are synchronized without using synchronized signals from a CPG module while the system still preserves the advantage of the CPG that the gait patterns switch from one to another easily by changing the phase offsets among fins. 4.1 Simplified fin dynamic model One can model the servo-actuated fins by using Euler- Lagrange equations. The aim here is to apply nonlinear state feedback to construct a limit cycle oscillator. Consider the following second-order coupled roll-pitch actuator dynamics M(p, q) + C(p, q, ṗ, q) + K(p, q) = (τ roll,τ pitch ) T, (40) where p and q are the roll and pitch actuator states. In order to focus on verifying the feasibility of the proposed closedloop CPG method, let us assume that the coupling between p and q is relatively small. Then, the decoupled dynamics can be represented by Iẍ + B(ẋ)ẋ + Kx = τ, (41) where β 0,β 1 > 0. The term led by β 1 is justified by the experimental observation (Licht et al. 2004; Beal and Bandyopadhyay 2007), where the lift force of a single fin oscillating in the fluid is in phase with the angular velocity and its magnitude is proportional to the angular velocity squared. 4.2 Nonlinear feedback-induced oscillator Let us propose a torque control law as τ = Px+ γ 0 ẋ γ 1 ẋ ẋ + Is(t), (43) where P, γ 0 and γ 1 are positive constant, and s(t) is a synchronizing input to be discussed further below. The closed-loop dynamics of (41) and (43)is Iẍ + (β 0 γ 0 + β 1 ẋ +γ 1 ẋ )ẋ + (K + P)x= Is(t). (44) After dividing the equation by I and setting ω 0 = (K + P)/I and C(ẋ) = β 0 γ 0 +β 1 ẋ +γ 1 ẋ I,wehave ẍ + C(ẋ)ẋ + ω0 2 x = s(t). (45) By choosing γ 0 and γ 1 to satisfy β 0 γ 0 < 0 and β 1 + γ 1 > 0, the system shows limit cycle behavior. If we denote (γ 0 β 0 )/I = σ 0 and (β 1 + γ 1 )/I = σ 1, then ẍ + (σ 1 ẋ σ 0 )ẋ + ω0 2 x = s(t). (46) Since the resulting dynamic system has a stable limit cycle, we can deliberately select the values σ 1, σ 0, and ω 0 for the feedback controller to change the amplitude and frequency of the limit cycle as well as the shape of the orbit as discussed with details in Appendix B. 4.3 Bias To bias the oscillation of x with constant x 0, the feedback must be modified as τ = Px+ γ 0 ẋ γ 1 ẋ ẋ + Is(t)+ (K + P)x 0 (47) and the closed loop dynamics becomes ẍ + C(ẋ)ẋ + ω 2 0 (x x 0) = s(t). (48) Comparing with (46), we can predict this system will show the same dynamics with a biased center (x 0, 0) in the x ẋ plane.

14 260 Auton Robot (2010) 28: Coupled oscillators The synchronizing input for i-th system s i (t) can be designed as follows. First let us define a state vector x i = (u i,v i ) T, where u i = x i and v i = ω 1 0 ẋ i for the roll or pitch angle x i of oscillating fin about its roll or pitch axis. Then (46) can be written as ( 0 ω0 ω 0 h i (ω 0 v i ) ẋ i = ) ( x i + 0 ω 1 0 s i(t) ), (49) where h i (v) = σ i v σ 0. To exploit the CPG architectures that we discussed using Hopf oscillators and diffusive couplings, we propose to use the following coupling term: ( ) 0 ω0 ẋ i = x ω 0 h i (ω 0 v i ) i ( ) (T 0 g ij x j x i ), (50) j where T ij = r ij R(φ ij ) and r ij = σ i σ j. Comparing (49) and (50) yields s i (t) = g j (r ij (ω 0 sin φ ij x j + cos φ ij ẋ j ) ẋ i ). (51) Since the oscillator (46) does not have full circular symmetry, rigorously only phase offsets of 0 and π yield flowinvariant subspaces, which can represent such gaits as pronk, bound, and trot, but not walk. Coupling between different amplitudes is possible by using scaling term r ij in the couplings. For the synchronization with equal amplitudes and zero phase offsets, s(t) further simplifies to s i (t) = g j (ẋ j ẋ i ). (52) The dynamics of individual oscillator in (50) can be generalized using χ i = (u i,v i )T = T 1i x i as ( ω χ i = 0 v i ) + ω 0 u i (σ 1ω 0 v i σ 0)v i ( g and then stacked as ) (χ j χ i ) (53) χ = f( χ) gl χ. (54) Notice that as a result of the transform, σ i in each oscillator becomes σ Synchronization analysis In this section, we show synchronization by further developing partial contraction (Wang and Slotine 2005) and its extended theorems in Pham and Slotine (2007) forthe special cases where phase offsets are 0 or π. Also, after approximating the oscillator dynamics with weak nonlinearity as harmonic oscillator dynamics, we prove the synchronization with arbitrary phase offset. Omitting the approximation for arbitrary phase offset should be difficult because couplings change the shape of the limit cycle orbit when the phase offset is not 0 nor π. The flow-invariant subspace is not explicitly found for φ ij 0,π. It is still left as an open problem Special cases with φ = 0,π For the coupled oscillator model of (50), the phase offsets of 0 or π implies such flow-invariant subspaces as {x 1 = r 12 x 2 = r 13 x 3 = r 14 x 4 } and {x 1 = r 12 x 2 = r 13 x 3 = r 14 x 4 } and so on. When transformed using χ, these all correspond to M 1 ={χ 1 = =χ 4 }. Hence, without loss of generality we prove synchronization of (54) towardm 1. The network of the oscillators is represented by L and suppose that the graph representing the coupled oscillators is connected and balanced. For example, if we have a oneway ring architecture, then L = K K K K K K, (55) K 0 0 K where K = ( ) From this point on, we can apply almost same procedure as Sect The Jacobian J of the system has the following symmetric part as J 0 =[J i,0 ] 4 gl 0, (56) where J i,0 is the symmetric part of the Jacobian J i of each oscillator, L 0 is the symmetric part of L, and [] 4 denotes a block diagonal matrix repeating J i,0 over i = 1,...,4. J i,0 is found as J i,0 = ( σ 0 2σ 1 ω 0 y i Now we compute the projected Jacobian ). (57) P = VJ 0 V T, (58) where V can be obtained by orthogonalizing I I 0 0 V = 0 I I 0, (59) 0 0 I I whose rows are found from the constraining equations of M 1. There is another practical method to obtain V using eigenvectors of L as introduced in Chung and Slotine (2009).

15 Auton Robot (2010) 28: For some V that is found using either method, one can verify that P has zero rows and columns for i, j = 1, 3, 5. For example one can use V = 1 I I I I 2I 2I (60) 0 0 2I 2I Ruling out the zero columns and rows from P yields P = V[J i,0 ] 4 V T gvl 0 V T, (61) where ()denotes the remaining part after removing the zero columns and rows. The eigenvalues of VL 0 V T are all positive because, as we already discussed in Sect , L is diffusive and balanced. Let us denote the smallest of them as λ 1. Since V[J i,0 ]V T is upper-bounded by sup y i (σ 0 2ω 0 σ 1 y i ) = σ 0, we have P < 0if gλ 1 >σ 0. The negative definite P implies negative semidefinite P. By Barbalat s lemma, it is straightforward to show that the velocities ẋ i and ẋ j synchronize asymptotically (see also invariance-like theorems in Khalil 2002). Once the oscillators synchronize their velocities, i.e., ẋ i ẋ j 0, the coupling inputs s i vanish, resulting in a stable limit cycle. Let δy denote the virtual displacement in y = V χ. Since δy T δy tends to a lower limit asymptotically, its higher-order Taylor expansion, similarly to Wang and Slotine (2005), indicates that δy T δy 0 on the resulting limit cycle. Hence, the angular positions synchronize as well, i.e., x i x j 0. We can apply the analysis above to specific form of networks such as one-way or two-way ring and all-to-all networks to find specific values of λ 1 and also the threshold for coupling gain g. In fact, one-way ring network yields λ 1 = 1, two-way ring network λ 1 = 2, and all-to-all network λ 1 = Arbitrary phase differences For the model of (54), we prove that the synchronization occurs for arbitrary phase differences by approximating the oscillator as a harmonic one. The oscillator in (50) does not have circular symmetry, hence (54) is not flow-invariant for φ ij 0,π. Let us impose circular symmetry by assuming extremely weak nonlinearity, i.e., σ 0 = σ 1 = 0. Let us use the transform χ i = T 1i x i again. For φ ij 0,π, substituting χ i in (50) yields the following: ( ) 0 ω0 χ i = χ ω 0 0 i + gk i (χ j χ i ), (62) j where ( cos 2 ) φ 1i cos φ 1i sin φ 1i K i = cos φ 1i sin φ 1i sin 2 (63) φ 1i and φ 1i is the desired phase offset of x i with respect to x 1. After stacking the individual oscillators in the form of (54), the coupling matrix L is found for example as 2K 1 K 1 0 K 1 K L = 2 2K 2 K 2 0, 0 K 3 2K 3 K 3 K 4 0 K 4 2K 4 in case of a two-way ring network. For a balanced L, e.g., for one-way or two-way ring, and all-to-all connection with equal φ 1i for all i, VL 0 V T 0. For the rest of the proof, which should follow the special case of φ ij = 0,π,weprovide only a sketch of the proof. Using the same projection V as the special case, the projected Jacobian of the overall system has negative semidefinite symmetric part, which is actually VJ 0 V T = kvl 0 V T 0 (64) since [J i,0 ] 4 = 0. Applying the same semi-contraction result as the special case, the oscillators synchronize asymptotically. 4.6 Reference inputs Since the servo controller that was built in the turtle robot does not have torque servo, we let the motor generate the torque τ of (43) using an indirect method. In fact, we can compute the reference position r(t) that would let the motor position servo generate the desired torque. Because the servo is PD-controlled, we can emulate the desired torque as follows. The PD-controlled position servo has the following structure: P(r x) + D(ṙ ẋ) = τ, (65) which can be written for ṙ as ṙ = τ D +ẋ P (r x). (66) D We then find the reference input for the next CPU time by using linear interpolation as r(t + t) = r(t)+ṙ t. (67) This difference equation of r(t) actually generated the limit cycle oscillation of the fins in our implementation in Sect. 4.8.

16 262 Auton Robot (2010) 28: Fig. 16 Curve-fitting results for 1/γ 1 (vertical axis) versus amplitudes (horizontal axis). Using these relations, we could find values for control parameter γ 1 for the desired amplitudes 4.7 Integration with autonomous underwater vehicle control The turtle-like AUV is configured so that the first oscillator corresponds to the fore-left fin, the second to the hind-left fin, the third to the fore-right fin, and the fourth to the hindright fin; then the coupling with one-way ring structure is used. Such gaits as walk, bound and pronk are realized by setting φ i,i+1 = π 2,π and 0, respectively. For each cycle of the oscillation, the main CPU determines new oscillation parameters such as the amplitude and bias to achieve the desired attitude. Such updates of the oscillation parameters are performed at a much slower rate than the sampling rate of the feedback controller of the fins so that the control system can run at a substantially lower bandwidth. The main CPU is then required to compute the parameters γ 0 and γ 1 for the feedback controller (43) from the desired oscillation amplitude and bias. It is difficult in practice to measure all the system parameters and to precisely predict the oscillation amplitude and bias. It is also probable that there are some unmodeled dynamics missing in (41). Hence, for successful implementation of the proposed controller, we chose to determine the relationship of controller parameters γ 0 and γ 1 to oscillation amplitudes by collecting experimental data, and then applying the curve-fitting method as follows. The relationship of the parameters γ 0 and γ 1 with the amplitude of oscillation is determined after measuring the amplitudes for different values of γ 0 and γ 1.Forγ0 roll = 3 and γ pitch 0 = 4, Fig. 16 shows the curve-fitting results, where the x-axis represents the amplitude of fin oscillation and the y-axis represents 1/γ 1. The frequency was fixed at about 0.67 Hz and was not affected by γ 0 and γ 1. Using the linear relationship found, we could control the amplitude of oscillations. 4.8 Experiments We experimentally validated the feasibility of the novel single-layered CPG controller for synchronized fin motions with the turtle robot. The results indicate that the proposed Fig. 17 The states of the four fins are plotted for angles versus filtered angular rate with respect to the roll axis. The circular trajectory and slow convergence to the limit cycle are the characteristics of weak nonlinearity. (a) The oscillation grows from the origin. (b) The trajectories for t>25 s are plotted to illustrate limit cycle clearly approach is well suited for controlling the turtle robot. The self-sustaining fin oscillation due to state feedback is verified, and then they show synchronized oscillation when coupled. Changing the phase offsets caused the gait to change between pronk and bound. The approach also validates its capability to work in the overall control loop when the robot is operating underwater Self-sustaining fin oscillation By applying state feedback controller of (43), the fins actually oscillated around roll and pitch axis. Figure 17 shows the phase portrait of the fin oscillation about roll and pitch axis. The trajectories start near the origin, and the spirals grow outward to converge to the limit cycle. The shapes of the orbits are different between roll and pitch oscillations. In case of pitch, we used ±90 phase offsets with respect to the roll oscillations of the same fins. The shape of the pitch oscillation indicates that its parameters are in the stronger nonlinear region than the roll oscillation (see Appendix B). Also, the phase offset ±90 affected the resulting limit cycle of pitch oscillations. Only the phase offset of 0 and 180 can leave the limit cycle unchanged after coupling. This is why it is hard to find explicitly the flow-invariant subspace for φ ij 0,π as we mentioned in Sect The time series of the roll positions are plotted in Fig. 18, where one can see the motions were not synchronized because couplings were not activated. The roll motions and

17 Auton Robot (2010) 28: Fig. 18 Roll positions of the four fins are plotted when the synchronization is not applied. The phase differences persist Fig. 20 A fin was disturbed and then recovered back to its synchronized rolling motion Fig. 19 Unsynchronized behaviors of roll and pitch motions of the four fins when the coupling for synchronization is not applied pitch motions are plotted together without any couplings to indicate their unsynchronized behaviors in Fig. 19. Notice that their frequencies are also different Synchronized oscillations To achieve the synchronization of the fin motions, we applied one-way ring couplings for the roll controllers, whose schematic is illustrated in Fig. 15. For the pitch controllers, we added one-way diffusive coupling with a 90-degree phase lag from the corresponding roll controllers as in Fig. 5. Figure 21 shows the synchronized roll and pitch motions of all four fins. Compared to Fig. 19, the phase-locked behavior of pitch oscillations with respect to roll is remarkable because they had different frequencies when uncoupled. Such phase-locking to a common frequency is a dominant phenomenon in nature (Pikovsky et al. 2001) although our synchronization analysis did not deal with it. Fig. 21 Synchronized behaviors of roll and pitch motions of the four fins: all the roll motions are synchronized among themselves and the pitch motions are phase-locked to the roll motions with 90-degree phase lag Figure 20 demonstrates the robustness of the proposed controller in terms of both oscillation and synchronization. The rolling motion was disturbed by human intervention and the motion recovered to its limit cycle while all the four fins recovered to synchronization. One should also notice that while a fin was disturbed others we also affected in their amplitudes, which is the characteristic of the single-layered

18 264 Auton Robot (2010) 28: Fig. 23 Underwater mission to follow the reference yaw, pitch, roll angles and heave (depth) was performed using the single layered CPG-based controller Figure 23 shows results of underwater test, where the turtle robot followed the reference heave, pitch, roll, and yaw angles by synchronized oscillatory motions of the foil fins. 4.9 Comparison remark Fig. 22 Gait transition starts at t = 25 s. (a) The synchronization starts bifurcation. (b) The new pattern bound gait settles around t = 55 s approach. Similar behavior was observed in the experiment with the feedback CPG in Sect To demonstrate its ability to switch gaits, bound gait was commanded at t = 25 s by applying the common phase offset φ = π for roll oscillators. In Fig. 22(a), the pattern starts to shift from the pronk to the bound. Since the transition occurred slowly, Fig. 22(b) shows the correct bound gait about 30 seconds after the phase bias was changed to π. The transition speed can be adjusted by the coupling gain and topology Underwater experiment Finally, to demonstrate the feasibility of the single layered CPG controller for underwater missions, the attitude control of the vehicle was tested in the same setup as in Sect Judging from the experimental data, when the novel singlelayered approach was used in the control loop of the turtle robot, the performance was no less than the other two CPG methods in the paper. Furthermore, the single-layered approach showed stable responses in the overall coordinated behavior when a single fin was disturbed (Fig. 20). In case of open-loop CPG in Sect. 3.1, disturbance to a fin would not affect the CPG directly because it is one-way from the CPG to the fins. Depending on the applications, both CPGs should have their own benefits. In a sense, one may consider this single-layered approach as comparable to exploiting the reflex in animal nervous systems because the oscillation is induced via direct feedback connection. A possible and interesting extension of the single-layered CPG would be connecting the fin oscillators to another CPG layer that exists in software or in electrical circuits consisting of coupled Hopf oscillators. 5 Conclusion Based on the theoretical framework on the synchronization of limit cycle oscillators, we introduced an open-loop CPG that can generate reference trajectories for fin motions in a turtle-like underwater robot. The performance of synchronization is mathematically proven while its feasibility for the control of the underwater robot is demonstrated by the experiments performed in the MIT Tow Tank Lab. To enhance

19 Auton Robot (2010) 28: the function of the CPG module in controlling dynamical systems, we also presented the enhanced CPG models with feedback couplings. Its performance was proven for a simplified model and tested with simulations and the real robot. The experimental test showed that there are some unstabilizing effects that were not considered in analysis. An important contribution of the paper, other than introducing various CPGs and verifying them in theory and experiments, is introducing a novel single-layered CPGbased control method, where the oscillating fins themselves replaced the nonlinear oscillators in the CPG. When the nonlinear state feedback was applied, the fins oscillated along stable limit cycles with the desired amplitude without the need for reference positions from higher-level controls. Hence, they are more robust against disturbances than using position servos with reference signals from higher-level CPGs if we consider a scenario as this: momentary mechanical failure in a position-servo system would cause errors to build up, which may cause high-torque responses when recovered from the failure. Due to the coupling inputs in the feedback, the fin oscillations synchronize without referring to reference signals from main controllers or other CPGs, hence it will be simpler to implement it using analog electrical circuits than open-loop CPG and feedback CPG, which requires Hopf oscillators to be implemented. The method still preserves the original advantage of CPG-based methods in that the gait patterns can switch easily by changing the parameters in the couplings. The performance of the turtle-like robot using the new method was successfully demonstrated with the underwater tests. The proposed method, running at a much lower bandwidth and fewer degrees-of-freedom, efficiently controlled the attitude and altitude of the underwater vehicle by synchronizing the 8-DOF foil fins in the presence of external disturbances. The proposed methods should be applicable to a larger class of biomimetic robots. It should be directly applicable to fin-based underwater locomotion systems. If we focus on its aspects on synchronizing self-sustained oscillators, dynamical systems with limit cycle behaviors can be applications. Although such generalization is not simple, especially when such limit cycle behavior is a result of intelligent control over biomimetic systems, the idea can be used for coupling such oscillations. For example, our method by itself does not generate a stable walking motion of a biped, which is a complicated task. Suppose an intelligent control algorithm handled the task well. Resulting walking motion however should be a limit cycle system. Then, the torso and arm motion might be generated by using our method of inducing self-sustained oscillation. It is then possible to couple to the leg motion so that the arm motion can synchronize with leg motions without too much computational effort. Our possible future work includes improving the design to be adaptive to environments so that it can swim energy efficient under water, possibly by detecting resistance on the fins and adjust roll-to-pitch phase offsets as well as pitch amplitudes. It is desirable if such tuning can be optimized on-line. Another open problem is rigorous analysis of synchronization for more general settings. Finally, the implementation of the single-layered CPG in hardware level seems to be trackable. Acknowledgements The authors thank Prof. Michael Triantafyllou for allowing them to borrow the robotic turtle developed in his laboratory, as well as Stephen Licht for his extensive help with the implementation and inspiring discussions on the experiments at the MIT Tow Tank. The first author was supported by the Institute for Information Technology Advancement of South Korea. The first and second authors were partially supported by the Air Force Office of Scientific Research (AFOSR), and thank Prof. James Oliver at the Virtual Reality Application Center, Iowa State University. Appendix A: Coordinate Transform Suppose we have a two-way ring network with w 14 I + w 12 I w 12 I 0 w 14 I L = w 21 I w 21 I + w 23 I w 23 I 0 0 w 32 I w 32 I + w 34 I w 34 I w 41 I 0 w 43 I w 41 I + w 43 I (68) and a coordinate transformation matrix I ai 0 0 =, (69) 0 0 bi ci where w12 a = ; w 21 w23 b = a ; w 32 w34 c = b. (70) w 43 We then have L 1 = L 1, which is symmetric and positive semidefinite if w 12 w 23 w 34 w 41 = w 14 w 43 w 32 w 21.One can verify this by expanding L 1 as sum of coupling matrices between two oscillators to notice that x T L 1 x is actually a sum of squares for any x R 8. The projection matrix V is found as w21 I w 12 I 0 0 V = 0 w32 I w 23 I w43 I (71) w 34 I such that V L 1 V T > 0. (72) Similar result was derived in Seo and Slotine (2007) for bidirectional chain architecture.

20 266 Auton Robot (2010) 28: Appendix B: Limit cycle of ẍ + ( ẋ 1 2 )ẋ + x = 0 Let us discuss the behavior of the oscillator (46) in the absence of the synchronizing input s(t) in further detail. B.1 Existence of limit cycle In the literature (see a nonlinear systems textbook e.g., Strogatz (2001)), there is a nonlinear system called Liénard system, in the form of ẍ + f(x)ẋ + g(x) = 0, (73) for which Liénard s theorem states that there is a unique stable limit cycle around (0, 0) if following conditions are satisfied: 1. g( x) = g(x). 2. f( x) = f(x). 3. F(x)= x 0 f(s)ds has a positive root at a; F(x)<0for 0 <x<a; F(x)>0forx>0; nondecreasing for x>a; F(x) as x. The oscillator in (46), which we built using state feedback, can be written as follows without the synchronizing input s(t). To avoid any complication in notation the parameters are fixed. ẍ + ( ẋ 0.5)ẋ + x = 0. (74) If we differentiate the system and rewrite using z =ẋ, we then have z + (2 z 0.5)ż + z = 0, (75) which is a Liénard system. One can see that this system satisfies the Liénard s theorem and thus z has a limit cycle. Since the solution of x(t) is then obtained by integrating z(t) with an additional constant that can be uniquely fixed for an initial condition of x(0) and ẋ(0). Hence, the oscillator in (46) has a unique periodic solution or a stable limit cycle. Equation (74) can be rewritten into a vector differential equation using u = x and v =ẋ as u = v, (76) v = u + (0.5 v )v. (77) Its limit cycle with the vector field is illustrated in Fig. 24. The annular region M formed by ABCD is an invariant set found numerically. All the vectors in the field are pointing into M so that no trajectory can leave the region. Such region M is often required to be found to show the existence of a periodic solution when applying Poincaré-Bendixson theorem. The orbit of the limit cycle is drawn inside the region M. Fig. 24 For ẍ + ( ẋ 0.5)ẋ + x = 0, u = x and v =ẋ, the limit cycle orbit is drawn in uv-plane along with the vector field It is worth mentioning that the oscillator in (46) or(74) resembles Rayleigh equation, which was originally known from acoustics and widely studied in the nonlinear systems community e.g., Tuwankotta (2000). The following is the Rayleigh equation: ẍ + μ(ẋ 2 1)ẋ + x = 0. (78) B.2 Analysis of the oscillator Although it is difficult to find a closed form solution for the oscillator of (46), the literature can provide a useful discussion about quantitative aspects of such a limit cycle system. In a book by Strogatz (2001), approximations for the amplitude and frequency of a nonlinear oscillator is discussed. For each of strong and weak nonlinearity regimes, one can find approximations for amplitude and frequency. Let us apply the approximation technique to the following oscillator. We try to concise the derivation here because it is a straightforward application of Strogatz ẍ + (σ 1 ẋ σ 0 )ẋ + x = 0. (79) B.2.1 Weak nonlinearity By the weak nonlinearity we mean the parameters σ 0 and σ 1 are small enough to let us rewrite (46)forω 0 = 1as ẍ + ɛ(b 1 ẋ B 0 )ẋ + x = 0, (80) where 0 <ɛ 1 and B 1,B 0 O(1). In this regime, one can see there are two time scales going on for the dynamics. One is a slow dynamics of its amplitude changing over time and the other is a fast dynamics of the oscillation at a certain frequency.

21 Auton Robot (2010) 28: To apply two-timing, we first write x(t) into a series of ɛ: x(t,ɛ) = x 0 (τ, T ) + ɛx 1 (τ, T ) + O(ɛ 2 ), (81) where T = ɛt is the slow timing and τ = t the fast timing. After substituting ẋ = τ x 0 + ɛ( T x 0 + τ x 1 ), (82) ẍ = ττ x 0 + ɛ( ττ x Tτ x 0 ) (83) into (81) and collecting terms for O(1), wehave ττ x 0 + x 0 = 0 (84) whose general solution is x 0 = r(t)cos(τ + φ(t )). (85) Collecting terms for O(ɛ) and then applying the general solution of x 0 yields ττ x 1 + x 1 = 2 τt x 0 + (B 1 r sin θ B 0 )r sin θ = 2(r sin θ + rφ cos θ)+ (B 1 r sin θ B 0 )r sin θ, (86) where θ = τ + φ and denotes derivative with respect to T. To avoid resonant forcing in the solution, we set the coefficients of cos θ and sin θ equal to zeros: 2r b 1 = 0, (87) 2rφ a 1 = 0, (88) where a 1 and b 1 are Fourier coefficients of h h(θ) = (B 1 r sin θ B 0 )( r sin θ), (89) that is, a 1 = 1 π b 1 = 1 π 2π 0 2π 0 h(θ) cos θdθ, (90) h(θ) sin θdθ. (91) After computing the Fourier coefficients we have the following dynamics for the amplitude r and the phase φ: r = B 0 2 r 4B 1 3π r2, (92) φ = 0. (93) Solving the differential equation for r yields r(t) = 3πB 0 8B 1 e B0 2 (T +C) ( 1 + e B 0 2 (T +C) ) 1, (94) Fig. 25 Exact solution by numerical integral of ẍ +0.1( 3π 4 ẋ 2)ẋ + x = 0 to compare with its approximated envelope r(t) = e T (1 + e T ) 1 and approximated solution x(t) = r(t)cos(t) whose steady state value is r = 3B 0 16B 1. As an example, if B 0 = 2, B 1 = 3π/4, and r(0) = 1/2, then C = 0 and r(t) = e T (1 + e T ) 1. (95) To validate the approximation, we plot the exact solution x(t ) by numerical integration with ɛ = 0.1 to compare with predicted amplitude r(t) and approximated solution x(t) = r(t)cos(t) in Fig. 25, where the approximation is matching the exact solution without noticeable deviation. The frequency of oscillation is 1 rad/sec in this example, which can be controlled by adjusting the natural frequency ω 0 (rad/sec) in (46). B.2.2 Strong nonlinearity In the strong nonlinearity regime, we rewrite (46)forω 0 = 1 as ẍ + μ(b 1 ẋ B 0 )ẋ + x = 0, (96) where μ 1 and B 1,B 0 O(1). Foru = μ 1 x and v =ẋ, we have u = μ 1 v, (97) v = μu μ(σ 1 v σ 0 )v (98) and the trajectory on uv-plane is illustrated in Fig. 26(a) and its time profile in Fig. 26(b). The closed orbit with red line in Fig. 26(a) is its limit cycle orbit and the z-shaped curve is its nullcline, which is obtained from v = 0as u = (σ 1 v σ 0 )v. (99) When the distance between a trajectory and the nullcline is O(1), then v O(μ) hence moves very fast toward the nullcline. Once it closes to the curve, then it slows down and moves down along the null cline until it reaches the knee (point B in Fig. 26(a)) of the nullcline, and then it

22 268 Auton Robot (2010) 28: Fig. 26 Trajectories of ẍ + μ(b 1 ẋ B 0 )ẋ + x = 0, for μ = 20; B 1 = σ 1 μ = 1, B 0 = σ 0 μ = 0.5 zaps toward the other side of the nullcline. One can actually compute the period of oscillation by computing the time spent on the slow part of this cycle. From (99), the coordinate of points A and B are found as ( σ 0 2 ) and 4σ 1, ( 1 2)σ 0 2σ 1 ( σ 2 0 4σ 1, σ 0 2σ 1 ), respectively. The time T spent on the slow branches of the nullcline is computed as T = 2 B A vb dt = 2 v A μ(σ 0 2σ 1 v ) dv (100) v to yield 1 T = 2μσ 0 (ln ) 2, (101) which can be used as an approximate of the period. The amplitude of the oscillation is also approximated from the coordinate of A and B as μσ 0 2 4σ 1. References Arena, P., Fortuna, L., & Frasca, M. (2002). Multi-template approach to realize central pattern generators for artificial locomotion control. International Journal of Circuit Theory and Applications, 30(4), Bandyopadhyay, P., Singh, S., Thivierge, D., Annaswamy, A., Leinhos, H., Fredette, A., & Beal, D. (2008). Synchronization of animalinspired multiple high-lift fins in an underwater vehicle using olivo cerebellar dynamics. IEEE Journal of Oceanic, Engineering, 33(4), Beal, D. N., & Bandyopadhyay, P. R. (2007). A harmonic model of hydrodynamic forces produced by a flapping fin. Experiments in Fluids, 43(5), Buchli, J., Righetti, L., & Ijspeert, A. J. (2006). Engineering entrainment and adaptation in limit cycle systems. Biological Cybernetics, 95, Chen, Z., & Iwasaki, T. (2008). Circulant synthesis of central pattern generators with application to control of rectifier systems. IEEE Transactions on Automatic Control, 53(1), Chung, S.-J., & Slotine, J.-J. E. (2009). Cooperative robot control and concurrent synchronization of Lagrangian systems. IEEE Transactions on Robotics, 25(3), Collins, J. J., & Stewart, I. N. (1993). Coupled nonlinear oscillators and the symmetry of animal gaits. Journal of Nonlinear Science, 3, Crespi, A., & Ijspeert, A. (2006). Amphibot II: an amphibious snake robot that crawls and swims using a central pattern generator. In Proceedings of the 9th international conference on climbing and walking robots (CLAWAR 2006) (pp ). Brussels, Belgium. Ekeberg, Ö. (1993). A combined neuronal and mechanical model of fish swimming. Biological Cybernetics, 69, Grillner, S., Deliagina, T., Ekeberg, Ö., El Manira, A., Hill, R., Lansner, A., Orlovski, G., & Wallen, P. (1995). Neural networks that co-ordinate locomotion and body orientation in the lamprey. Trends in Neurosciences, 18, Grillner, S., Ekeberg, Ö., El Manira, A., Lansner, A., Parker, D., Tegnér, J., & Wallén, P. (1998). Intrinsic function of a neuronal network a vertebrate central pattern generator. Brain Research Reviews, 26, Hirose, S. (1993). Biologically inspired robots: snake-like locomotors and manipulators. Oxford: Oxford University Press. Ijspeert, A., Crespi, A., Ryczko, D., & Cabelguen, J.-M. (2007). From swimming to walking with a salamander robot driven by a spinal cord model. Science, 315(5817), Iwasaki, T., & Zheng, M. (2006). Sensory feedback mechanism underlying entertainment of central pattern generator to mechanical resonance. Biological Cybernetics, 94(4), Khalil, H. K. (2002). Nonlinear Systems (3rd edn.). New York: Prentice-Hall. Krouchev, N., Kalaska, J. F., & Drew, T. (2006). Sequential activation of muscle synergies during locomotion in the intact cat as revealed by cluster analysis and direct decomposition. Journal of Neurophysiology, 96, Lewis, M. A., Tenore, F., & Etienne-Cummings, R. (2005). CPG design using inhibitory networks. In Proceedings of the 2005 IEEE international conference on robotics and automation (pp ). Barcelona, Spain. Licht, S., Polidoro, V., Flores, M., Hover, F., & Triantafyllou, M. (2004). Design and projected performance of a flapping foil AUV. IEEE Journal of Oceanic Engineering, 29(3), Morimoto, J., Endo, G., Nakanishi, J., Hyon, S.-H., Cheng, G., Bentivegna, D., & Atkeson, C. G. (2006). Modulation of simple sinusoidal patterns by a coupled oscillator model for biped walking. In Proceedings of the 2006 IEEE international conference on robotics and automation (pp ). Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), Pham, Q.-C., & Slotine, J.-J. E. (2007). Stable concurrent synchronization in dynamic system networks. Neural Networks, 20(1),

23 Auton Robot (2010) 28: Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization: a universal concept in nonlinear sciences. Cambridge: Cambridge University Press. Rybak, I. A., Shevtsova, N. A., Lafreniere-Roula, M., & McCrea, D. A. (2006). Modelling spinal circuitry involved in locomotor pattern generation: insights from deletions during fictive locomotion. Journal of Physiology, 577(2), Seo, K., Chung, S.-J., & Slotine, J.-J. (2008). CPG-based control of a turtle-like underwater vehicle. In Proceedings of robotics: science and systems IV. Zurich, Switzerland. Seo, K., & Slotine, J.-J. E. (2007). Models for global synchronization in CPG-based locomotion. In Proceedings of 2007 IEEE international conference on robotics and automation (pp ). Rome, Italy. Stent, G. S., Kristan, W. B. Jr., Friesen, W. O., Ort, C. A., Poon, M., & Calabrese, R. L. (1978). Neuronal generation of the leech swimming movement. Science, New Series, 200(4348), Strogatz, S. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143, Strogatz, S. (2001). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (studies in nonlinearity). Cambridge: Perseus Books Group. Taga, G. (1998). A model of the neuro-musculo-skeletal system for anticipatory adjustment of human locomotion during obstacle avoidance. Biological Cybernetics, 78(1), Taga, G., Yamaguchi, Y., & Shimizu, H. (1991). Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biological Cybernetics, 65(3), Triantafyllou, M., & Triantafyllou, G. (1995). An efficient swimming machine. Scientific American, 272(3), Tuwankotta, J. M. (2000). Studies on Rayleigh equation. INTEGRAL, 5(1). Vogelstein, R. J., Tenore, F., Etienne-Cummings, R., Lewis, M. A., & Cohen, A. H. (2006). Dynamic control of the central pattern generator for locomotion. Biological Cybernetics, 95(6), Wang, W., & Slotine, J.-J. E. (2005). On partial contraction analysis for coupled nonlinear oscillators. Biological Cybernetics, 92(1), Wang, M., & Yu, J. (2008). Intelligent robotics and applications, Part I. In LNAI: Vol Parameter design for a central pattern generator based locomotion controller (pp ). Berlin: Springer. Soon-Jo Chung received the B.S. degree summa cum laude from Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 1998, and the S.M. degree in aeronautics and astronautics and the Sc.D. degree in estimation and control from Massachusetts Institute of Technology (MIT), Cambridge, in 2002 and 2007, respectively. He is currently Assistant Professor of aerospace engineering at the University of Illinois at Urbana- Champaign. During , he was Assistant Professor at Iowa State University. His current research interests include bio-inspired aerospace robotics, nonlinear control theory, unmanned aerial vehicles, formation flying spacecraft, multi-vehicle control, and vision-based navigation. Prof. Chung received an Air Force Office of Scientific Research Young Investigator Award in 2008 for his research on flapping flying microaerial vehicles. Jean-Jacques E. Slotine was born in Paris in 1959, and received his Ph.D. from MIT in After working in the Computer Research Department at Bell Labs, in 1984, he joined the faculty at MIT, where he is now Professor of Mechanical Engineering and Information Sciences, Professor of Brain and Cognitive Sciences, and Director of the Nonlinear Systems Laboratory. He is the coauthor of the textbooks Analysis and Control (Wiley, 1986) and Applied Nonlinear Control (Prentice-Hall, 1991). Prof. Slotine was a Member of the French National Science Council from 1997 to Keehong Seo received his Ph.D. degree in electrical engineering from POSTECH, Pohang, South Korea in After working at Nonlinear Systems Lab of MIT from 2005 to 2007 as a visiting scholar, he continued his post-doctoral study at the Department of Aerospace Engineering of Iowa State University in He is currently working in the Mechatronics and Manufacturing Technology Center at Samsung Electronics CO., LTD. His research interests include dynamic system control, robotics, and locomotion.