Robot arms equipped with an additional network of masses and springs enable a facilitation of control by the exploitation of morphological properties. Typically, these mass-spring networks are being generated randomly. This approach comes in its basic form unhandy when the network should obey specific constraints, such as spring lengths, angles between springs or, e.g., the physical expansion range of the network. We present an approach of emulating a growth process on the basis of L-systems. Therefore, we have developed a simulation environment to define L-systems and instructions to translate the produced strings into mass-spring networks

Bernhardsgrütter, Rene

Füchslin, Rudolf Marcel

Hauser, Helmut

Jaeger, Christian

Nakajima, Kohei

Senn, Christoph W.

ZHAW Zürcher Hochschule für Angewandte Wissenschaften

Employing L-systems to Generate Mass-Spring Networks forMorphological ComputingR. Bernhardsgru¨tter†, C. W. Senn†, R. M. Fu¨chslin†, C. Jaeger†, K. Nakajima‡ and H. Hauser]† Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Winterthur, Switzerland‡ Hakubi Center for Advanced Research, Kyoto University, Kyoto, Japan‡ Nonequilibrium Physics and Theoretical Neuroscience, Department of Applied Analysisand Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto, Japan] Artificial Intelligence Laboratory, University of Zurich, Zurich, SwitzerlandAbstract—Robot arms equipped with an addi-tional network of masses and springs enable a facili-tation of control by the exploitation of morphologicalproperties. Typically, these mass-spring networks arebeing generated randomly. This approach comes inits basic form unhandy when the network should obeyspecific constraints, such as spring lengths, angles be-tween springs or, e.g., the physical expansion range ofthe network. We present an approach of emulating agrowth process on the basis of L-systems. Therefore,we have developed a simulation environment to defineL-systems and instructions to translate the producedstrings into mass-spring networks.1. IntroductionContemporary robotics seems to experience limita-tions when it comes to the construction of systemsthat are compatible with requirements of environmentsand task schemes as they are found in complex envi-ronments, e.g., in a hospital. These limitations aremanifold, including adaptivity to new situations, ro-bustness and, last but not least, intrinsic safety. Thereis an apparent tension between contemporary designof robots and the way how natural evolution imple-ments its most versatile moving systems: The formerare most often stiff and equipped with a number of de-grees of freedom (DoF) as small as possible, the latterare made of rather soft materials and exhibit conse-quently a large number of DoF. Conventional robotdesigns aim at a small number of DoF as they implya facilitation in modelling and controlling the system(both with respect to computation as well as sensorybandwidth necessary for a sufficiently precise determi-nation of the state of the system). In recent years, theconcept of “soft robotics” has gained growing atten-tion [1, 2, 3]. There are several reasons for this trend,among them are the fact that the concept of soft robotsmatches well with the spirit of bio-inspired engineeringor the hope that soft robots can be made intrinsicallysafer than stiff and heavy systems. A main obsta-cle for the implementation of soft systems is the ap-parent need for more complex control protocols. Theconcept of morphological computation offers a way toovercome this difficulty: Instead of delegating controlto some control unit, the dynamics of the soft bodyitself is made a part of the control scheme, i.e., part ofthe control can be outsource to the physical body of arobotic platform.The term “morphological computation” can beloosely defined as the exploitation of the shape, ma-terial properties, and physical dynamics of a physi-cal system to improve the efficiency of a computation;e.g. the computation needed for the control of a robot.Thereby, the physical dynamics of the robot is madepart of the control process itself. A simple exampleis the exploitation of the self-stabilitzation propertyof limit cycles. For a detailed description of morpho-logical computation see [8, 9, 10]. The term “compu-tation” is justified, as demonstrated by Hauser et al.in [4]. was shown that a sufficiently large parameter-ized mass-spring system1 is in fact able of emulatinga large family of different computations, which could,e.g., be used for control.One way for the implementation of morphologicalcomputation uses the morphology of the robot as areservoir in the sense of reservoir computation. Com-pared to classical artificial neural network, reservoirshave improved learning speed as no recurrent weightshave to be learned [4]. Furthermore, reservoirs can beused in a multitasking context, i.e., multiple computa-tions can be carried out simultaneously with the samereservoir [5]. During the learning process, weights arecalculated with the help of linear regression. These pa-rameters are then used to calculate the output basedon a weighted sum of the (partial) inner state of thereservoir. Hauser et al. [4, 5, 6] have demonstratedthat a compliant body (a morphology consisting ofmass-spring systems) can be employed as such a reser-voir.Following this approach, our proposed setup con-sists of a compliant network of nonlinear springs and1By term mass-spring systems we refer physically realisticsystems, i.e., damped mass-springs.- 184 -2014 International Symposium on Nonlinear Theory and its ApplicationsNOLTA2014, Luzern, Switzerland, September 14-18, 2014masses attached to a rigid robot arm. This compli-ant physical body – the mass-spring network – servesas a reservoir [6]. Usually such networks are createdrandomly [5]. This task can be challenging, as ran-dom networks might not be suitable for a real-worldscenario, e.g. given shape constraints.In this work we used L-systems [7] in order to gener-ate networks in respect to a bio-inspired design. Thisallows to easily parameterize the creation process ofmass-spring networks and, therefore, enables one tocreate networks according to specific needs.2. MethodsWe used a simulation of a robot arm, based onthe Bullet physics engine,2 to explore the dynamicsof mass-spring networks. The task of the robot armwas to follow a given trajectory.2.1. Robot Arm and Mass-Spring NetworkNetworks qualify for reservoir computing when theyare dynamic, complex, nonlinear, and provide a fadingmemory [4]. Figure 1 shows a perpendicular robot armwith a small mass-spring network.3AAyxfixed massinput massmassend effectornon-linear springA actuatorsFigure 1 A robot arm with a mass-spring network, whichserves as a reservoir, consisting of a shoulder, an elbow, andan end effector. The network is attached to the arm viafour input nodes, located next to the joints.At the base, a fixed mass represents the shoulder. Itacts as joint with an actuator (A) so the arm can bemoved. The elbow, in the middle of the robot arm, is2http://www.bulletphysics.org3The mass-spring network shown in Figure 1 would proba-bly not perform well since it might have too few masses andsprings. Figure 1 has only the purpose of illustrating the con-cept schematically.also an active joint (A) and controls the angle betweenthe upper and the lower arm.4 The end effector isdesignated to follow a target trajectory. The mass-spring network is fixated to the arm segment by thegreen nodes called inputs. When the arm moves, forcesare transmitted via these nodes to the network and,thus, change the state of the reservoir. Furthermore,there are one or more, fixed nodes, which act as ananchor and give the network an orientation in space.2.1.1. Growing Networks Using L-systemsTypically, reservoirs of mass-spring networks arerandomly generated, i.e., that a random or given num-ber of masses are randomly placed around a robotarm and springs connect randomly selected masseswith each other [4, 5, 6]. This work uses a dif-ferent approach. We “grow” the networks using socalled L-systems. L-systems, introduced by A. Lin-denmayer [7], are based on formal languages (in itsmost fundamental form a parallel rewriting system inthe sense of a grammar in the theory of formal lan-guages). As being demonstrated by various authors(e.g. Prusinkiewicz and Lindenmayer [11]), the L-system approach is able of reproducing an astonish-ingly large variety of rather complex-looking artificialplants by using rather simple algorithmic rules. Thisgives the benefit of having a certain amount of controlover the network structure – like the size or certainproperties of the shape – and, furthermore, the stringrepresentation is fit for a usage with evolutionary algo-rithms. Typically, L-systems are purely deterministic.It is, however, easy to add some amount of randomnessto the growth process. We applied a similar approach,since it improved the usability of the grown networksas reservoirs.Growth Process As to use L-systems for the gen-eration of mass-spring networks, we need a grammarwith a set of production rules for terminal and non-terminal symbols, and we have to link these symbols toinstructions – for example “create a mass” or “create aspring” – in order to translate the produced L-systemstring to a mass-spring network.Let our grammar have the alphabetΣ = {A,B, a, b}, which translates the instruc-tions• a ⇒ mass creation [r1, r2] with inner radiusr1 = 2 and outer radius r2 = 4, and• b ⇒ spring creation [r1, r2] with inner radiusr1 = 2 and outer radius r2 = 4.4Upper arm is called the segment between the shoulder andthe elbow. Subsequently, the segment between elbow and endeffector is called lower arm. This notation does not depend onthe alignment of the robot arm.- 185 -Further, we provide the following production rules• A→ BAB and• B → abb.The parameters r1 and r2 allow to limit the area forrandom creation to a circle (r1 = 0) or a ring (r1 > 0)around the position of the interpreter, which reads thestring to translate symbol-by-symbol and executes it,thus creates new masses and springs.2.2. Learning PhaseThe learning follows the supervised procedure ex-plained in [4], which we summarize briefly. The pro-cedure is described for one actuator and must be re-peated for each actuator used. The desired trajectoryis discretised with a suitable time step. At each step,the angular velocity v(pre) of the actuator is calculatedand the spring lengths are recorded while moving alongthe trajectory. This leads to a matrix L = (lij) con-taining at position (i, j) the length of spring j at timeti. These spring lengths are superimposed with whitenoise, a procedure which has been shown to increaserobustness against perturbations [4]. The data froman initial period (the washout-time) is dismissed toexclude any transients from the start of the learningprocess. Based on the recorded settling times after afast 1◦ swiveling movement, the washout-time was setto tw = 300 simulation steps. The weights w = (wj)for each spring are then calculated by solving the lin-ear equation system Lw!= v(pre) in the least squaressense [4].After this learning procedure, the robot should beable to follow the learned trajectory without guidanceand controlled solely by the mass-spring network, us-ing the angular velocities v(morph) :=∑wj lj calcu-lated from the actual spring lengths lj measured dur-ing the motion and this movement is expected to berobust against perturbations.2.3. Exploitation PhaseWe call the stage, where the system runs freely andwithout guidance the exploitation phase. To startthis phase, we need an initialization as follows: therobot arm is guided along one complete cycle of thetarget trajectory. After this, at time t = 0, allinitial transients in the mass-spring network are ex-pected to have died out and control is smoothly trans-ferred to the network by applying the angular velocityv(trans) := λ v(pre) + (1 − λ) v(morph) to the actuatorsfor a suitable monotonous transition function λ withλ(t = 0) = 1 and λ(t ≥ t1) ≡ 0. Empirical evalua-tion has shown that a fade-over time t1 correspondingto half a cycle of our trajectory led to stable results.This smooth fade-over of the control to the morphol-ogy was necessary in all our simulations to guaranteestable morphological control.3. Trajectory ReproductionWe will show a successful trajectory reproduction,accomplished with a mass-spring network generatedby an L-system.By successful we mean, that the robot arm was ableto approximately follow a learned trajectory in a stablemanner.Mass-Spring Network We used the mass-springnetwork shown in Figure 2. This specific networkperformed best among other tested ones. It consistsof 19 masses and 43 springs. The illustration showsthe noteworthy airiness of the network – most of thesprings are longer than 7 length units, which equals to710 of a arm segment, and only a few network massesare connected via more than three springs.Figure 2 The mass-spring network grown using an L-system and used in this simulation.Exploitation Phase The robot arm managed tofollow the learned trajectory, although the whole circledrifted away over time, as shown in Figure 3.The dashed line represents the target trajectory.The robot arm was initialized with the end effectorlocated at the (x/y)-coordinates (0/21) and was thenguided through the warm-up sequence. Thus, the firstcycle is exactly the same as the learning trajectory.The control is then slowly taken over by the robot armand its morphology, again during one cycle. After fiveand a half cycles, the simulation was stopped.The reservoir was able to control the robot arm andcreate a stable approximation of the learned trajectory,if we ignore the drifting.55A detailed analysis of different mass-spring networks – alsorandomly generated ones – may be reasonable in order to evalu-ate performance varieties and estimate how different aspects ofa network effect its capability to control the robot arm.- 186 -Robustness Interestingly, despite the drift awayfrom the nominal trajectory the system was to acertain extent robust against external perturbations.This was shown in a simple robustness test, where weapplied a force of 5 N to all masses parallel to thex-axis for a short amount of time to simulate a blow.Figure 3 shows the corresponding experimental re-sults. The morphological computation setup was ableto recover from two applied blows, approximatelywithin one cycle.F2F1target trajectoryFigure 3 The mass-spring was able to handle two strongexternally applied perturbations and did not lose control.This is visualized by the forces ~F1 and ~F2, applied duringthe black colored periods next to the arrows.Less stable networks typically lost control after sucha perturbation. An often occurred behavior was thatthe end effector moved off the target trajectory andstopped there. Hence, the setup shown in Figure 3seems to be capable of handling externally influencedforces without losing control.4. OutlookFor future research we suggest the systematic eval-uation of the performance of grown mass-spring net-works in comparison to randomly generated ones. To-gether with more extensive research regarding thestructure of networks grown by L-systems could usprovide potentially with general growing rules and,hence, with implicit design rules.AcknowledgmentsThis paper is based on parts of the bachelor’s thesiswith the title “Exploitation of Morphological Proper-ties to Control a Robot Arm,” written by R. Bern-hardsgru¨tter and C. Senn at the Zurich University ofApplied Sciences under the supervision of R. Fu¨chslinand C. Jaeger and in scientific collaboration withH. Hauser and K. Nakajima.References[1] D. Trivedi et al., “Soft robotics: biological in-spiration, state of the art, and future research,”Appl. Bionics. Biomech., vol. 5, no. 3, pp. 99–117,Sept. 2008.[2] R. Pfeifer et al., “The challenges ahead for bio-inspired ‘soft’ robotics,” Commun. ACM., vol. 55,no. 11, pp. 76–87, Nov. 2012.[3] S. Kim et al., “Soft robotics: a bioinspired evo-lution in robotics,” Trends Biotechnol., vol. 31,no. 5, pp. 287–294, May 2013.[4] H. Hauser et al., “The role of feedback in mor-phological computation with compliant bodies,”Biol Cybern, vol. 106, p. 603, Feb. 2012.[5] H. Hauser et al., “Towards a theoretical foun-dation for morphological computation with com-pliant bodies,” Biol Cybern, vol. 105, p. 355,Apr. 2012.[6] H. Hauser et al., “Moving a Robot Arm by Ex-ploiting its Complex Compliant Morphology,” inProceedings of the 2nd International Conferenceon Morphological Computation, Venice (Italy),2011, p. 1.[7] A. Lindenmayer, “Mathematical Models for Cel-lular Interactions in Development, Parts I and II,”J. Theoret. Biol., vol. 18, pp. 280–315, 1968.[8] R. Pfeifer et al., How the Body Shapes the WayWe Think: A New View of Intelligence. Cam-bridge, MA: MIT Press, 2007.[9] R. Pfeifer el al., “Self-Organization, Embodi-ment, and Biologically Inspired Robotics,” Sci-ence, vol. 318, pp. 1088–1093, 2007.[10] R. Fu¨chslin et al., “Morphological Computationand Morphological Control: Steps Toward a For-mal Theory and Applications,” Artificial Life,vol. 19, pp. 9–34, 2013.[11] P. Prusinkiewicz, A. Lindenmayer, “The Algo-rithmic Beauty of Plants,” New York: Springer-Verlag, 1990.- 187 -

Employing L-systems to generate mass-spring networks for morphological computing

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Employing L-systems to generate mass-spring networks for morphological computing

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