Extended SLIP model

Module-Icon EMSLIP Extended SLIP model
Event none
Author Maziar A. Sharbafi
Requirements Module TMSLIP and EM
teaching time 90 min
Last modified 11.7.2017

This part and also the two other parts EM and EMIP are basically a summarized version of subchapter 3.6 of a book chapter recently published which is edited by Maziar Sharbafi and Andre Seyfarth titled: Bioinspired legged locomotion. The book can be found here and a complete description of these models can be found in the last subchapter of the third chapter of this book.

This lecture includes two 45 min sessions.

Extended SLIP models I (45 min)

The focus of SLIP models is on CoM movement, considering the stance as the first locomotion sub-function and partially leg swinging (the second locomotion sub-function). As aforementioned, to address the third locomotor subfunction, the upper body is required, but this is not the only extension of the SLIP model. Some of the extended SLIP models are summarized in the following:

  • stance leg: (a) adding leg mass, inertia and damping, (b) adaptation of leg parameters during motion and (c ) increasing number of segments
  • swing leg: (a) addition of one or more legs (b) increasing number of segments in swing leg, © adding leg mass
  • Balance: For modeling this subfunction, we add a rigid trunk and the focus will be on different approaches of hip torque control
Extending the number of limbs (B-SLIP, Q-SLIP)

The bipedal SLIP was already explained in TMSLIP in which a second leg is added to enable the model to describe walking. For walking at moderate speeds (around 1m/s) double humped GRF patterns are found while a running gait with single-peak GRF is observed at higher speeds. However, this model predicts further walking gaits with more than two peaks for lower energies. Such gaits (e.g. with three-humped force patterns) are observed in impaired gaits or during learning locomotion in early childhood (Gollhofer et al., 2013).

Inspired by the work on the SLIP model, Herr et al. (2002) developed a quadrupedal SLIP model to describe trotting and galloping in several animals (chipmunk, dog, goat, horse). The model was extended with a compliant trunk (described by neck and back stiffness). A very similar quadrupedal SLIP model with rigid trunk was created following the design of the Scout II robot (Poulakakis et al., 2005) as shown in Fig. 1. The predicted stable bounding gait is in close agreement with the behavior observed in the robot. Later on, the model and robot dynamics were extended to galloping (Smith and Poulakakis, 2004).

Figure 1. Extension of SLIP model for quadruped gaits (from Poulakakis 2004) of Scout II robot in the sagittal plane.

Stance leg adaptation (VLS and E-SLIP)

In the SLIP model, stance leg parameters like leg stiffness and angle of attack are often set to a specific value. This usually represents the steady-state (average) gait pattern during locomotion. However, leg function varies from step to step (e.g. in response to ground level changes, (Daley et al. 2007, Müller et al. 2010) and also during the stance phase. Such variations in leg parameters can be represented in extensions of the SLIP model in order to better match experimental data. With this also deviations from the conservative spring-like leg function can be described which may lead to also energetically stable gait patterns.

During human locomotion, there is a tendency towards higher leg stiffness during leg loading (leg shortening) compared to unloading (leg lengthening). Also, the leg length is often larger at takeoff compared to touchdown (Lipfert et al., 2012). Two simple approaches were introduced to address changes in leg parameters during stance phase: ESLIP and VLS. Extended SLIP models like ESLIP [35] or the variable leg spring (VLS) model [Riese et al. 2013] describe leg spring adjustments (stiffness, rest length) during stance phase.

  1. In the variable leg spring (VLS) model a continuous changes of leg parameters over time is assumed (Riese and Seyfarth, 2012). For stable hopping, a decrease in leg stiffness and a continuous increase in rest length of the leg spring (Figure 3A) was required in the model unless sufficient leg damping is provided. This is in line with experimental findings on changes in stance leg parameters during human locomotion (Lipfert et al., 2012).
  2. In the E-SLIP model a sudden change in leg parameters at midstance is considered (Figure 3B) without a sudden drop or increase in leg force. This model permits to consider step-to-step changes in system energy as found in human running (Ludwig et al., 2012).

Changes in leg parameters in steady-state movements were observed experimentally at global (leg) level (Lipfert et al, 2012, Lipfert, 2009 for walking and running, Riese et al., 2013 for hopping, Seyfarth et al., 1999 for take-off phase in long jump) as well as at local (joint or muscle) level (Peter et al., 2009 AMS for running). So far, it is still unclear whether and how limb stiffness is adjusted at global (leg) level or local (joint, muscle) level. It remains for future research to investigate in more detail how changes in state variables (angles, angular velocities) and environmental changes (e.g. changed ground properties) effect these adjustments of leg parameters during stance phase.

Addition of mass to SLIP leg (M-SLIP)

In [Peuker et al., 2012], the SLIP model was extended by adding leg mass and leg moment of inertia at a specific distance from the trunk CoM. They include a hip spring-damper (parallel combination of a linear spring and viscous damper) to control the leg during swing phase and to inject energy during stance phase. With these changes in SLIP model, both stance and swing locomotor subfunctions are influenced. Such an extended SLIP-based model with one leg is called M-SLIP, shown in Fig. 2. Running in SLIP model is simulated by one leg as the legs are massless. Here, single-legged (1L) and alternating bipedal running (2L) are introduced by M-SLIP model. The first model (1L) is mainly a hopping model instead of running. The knee spring-damper models a telescopic mass-attached leg in aligning leg segment and leg spring. During swing phase, the prismatic leg spring is bent such that the leg segment can freely swing forward. The leg angle is adjusted by setting the rest angle of the rotational hip spring and this leg angle switches depending on the state between two values (one for swing phase and another one for stance phase).

With leg masses, the gait dynamics is more realistic but also more complex (e.g. landing impacts). Compared to the SLIP model, the predicted solutions for stable running of a one-legged system with leg masses (M-SLIP) are shifted towards flatter angles of attack (Peuker et al., 2012). In an alternating, bipedal M-SLIP model, however, the inertial effects of both legs are compensating each other such that the region of stable running is similar to the one observed in the SLIP model. This indicates that also the model with leg masses can inherit solutions of the SLIP model. At the same time, leg inertia of the leg with mass permits creating swing-leg trajectories (e.g. by introducing a hip torque) that were not represented by the original SLIP model. This M-SLIP model can be also used for walking which was not investigated by Peuker et al.

Figure 2. Addition of leg mass to the SLIP model (from Peuker 2012)

<note tip>session 2: (45 min)</note>

Extended SLIP model II

Extending SLIP with leg segments (F-SLIP)

Biological limbs are designed as a serial arrangement of leg segments with muscles spanning the leg joints. There are a number of models extending the number of limbs in the SLIP model to better predict biological gaits and investigate the role of these segments. The main question that may be answered with these models is why humans/animals need segmented legs if prismatic legs like in SLIP provides the requirements. therefore, these extensions aim at analyzing the stance locomotor subfunction.

The prismatic leg in SLIP was extended to two or three segments. First we explain two models with 2-segment leg and then the advantages of biological 3-segment leg are shortly presented based on a modeling study. For extending the leg two options exist: i)adding foot like in the F-SLIP model (Maykranz et al. 2009), or ii) adding knee (Rummel 2008). In F-SLIP model the prismatic leg spring is extended distally by a rigid foot segment, which is attached by a rotational foot spring (similar to the ankle joint). This model enables SLIP to explain the center of pressure movement and variation in leg length resulting in an asymmetric arrangement at touchdown and takeoff (Maykranz 2014). In this model a rotational spring between the foot and the leg may partially support the nonlinear leg force-length relationship in walking. Surprisingly, the F-SLIP model is well able to predict running but has limited capability to generate walking patterns. For example, one of the main limitations of the BSLIP which is inability to model fast walking cannot be resolved with F-SLIP too.

The second group of extension to 2-segment leg considering knee joint were introduced with muscle-like joint function for describing jumping (Alexander 1990, 1992; Seyfarth 2000) and hopping tasks (Geyer et al., 2003). In (Rummel 2008), it was shown that running solutions are achievable for different rest angles of the knee spring with clear deformations in the predicted regions for stable locomotion. In this model knee stiffness needs to be increased for faster running which is in contrast to the SLIP model. This increase of knee stiffness with speed was also found experimentally (Lipfert, 2010 Kovac).

There are many detailed locomotion models with three segmented legs including foot, shank and thigh. However, few of them are developed based on the extension of the template & anchor concepts (e.g., extension of SLIP model). In [Seyfarth 2001], a three segmented model was utilized to investigate the adjustment of joint stiffness for spring-like leg function. This simulation study shows that a shared loading of knee and ankle requires not only a proper distribution of knee and ankle stiffness but also additional mean to avoid joint buckling or overextension. The transfer of this mechanical three-segment leg model to a muscle-skeletal model was presented by Geyer and Herr (2010).

Figure 3. Extension of SLIP model with 2 segmented leg from [Maykranz 2009] and [Rummel 2008].

Upper body modelling

For posture control, the third locomotion sub-function, we need to extend the template models by adding an upper body, e.g. by a rigid trunk. With this additional degree of freedom, developing a controller for balancing is required. TSLIP stands for Trunk+ SLIP and BTSLIP for bipedal TSLIP are shown in Figs. 4 and 5. In the MC1 and MC2, we explain bioinspired SLIP-based models for posture control based on by human/animal locomotion with VPP and FMCH models, respectively. In addition matlab codes for these models are also presented to better understand the concept and also for further applications. We have also extend these balance models by introducing muscle models and developing neuromuscular FMCH (nmF) model which is presented in MC3.

Figure 4. Upper body modelin with a rigind trunk in the TSLIP model.

Figure 5. BTSLIP model: addition of trunk to the BSLIP model for upper body modeling for walking.

Extension to 3d

In order to extend the models to 3d, in addition to increasing the system degrees of freedom and enlargement of the state space the lateral leg placement is the main challenge. In the following the 3d SLIP and IP models are presented.

Like in 2D SLIP stable gaits require a proper leg adjustment. Interestingly, stable gaits are not predicted with a given step length (or step frequency) but by adjusting the leg angle for landing (angle of attack) with respect to gravity. This indicates that the locomotion pattern is rather an outcome than a target of control. For instance, if a subject runs on a treadmill, the variability of the gait pattern increases when the very same preferred step length or step frequency is provided (markings on the belt, metronome) as targets for locomotion (Ludwig et al. 2010).

In 2005, Seipel and Holmes published a paper investigating running stability predicted by a SLIP model extended to 3D by including a lateral leg placement at touch-down (Seipel & Holmes 2005). The lateral leg angle was selected with alternating direction (left or right) with respect to a desired running direction (Fig. 8a). Surprisingly no stable running patterns were predicted by this novel 3D SLIP model. Later, Peuker et al. (2012) introduced a velocity-based leg adjustment. Here the leg angle was laterally adjusted within the plane spanned by the COM velocity vector and gravity vector. With this change in the coordinate frame for swing leg adjustment, stable running solutions were predicted for a huge range of angle of attack and lateral leg angles before landing.

In 2014, Maus and Seyfarth extended the model to a bipedal SLIP model for 3D walking. The simulation results of this 3D walking model reveals that changes in leg adjustments between the two legs can result in walking in curves (Maus & Seyfarth 2014). However, there are combinations of leg parameter adjustments between the two legs which still results in straight walking (with a fixed direction of progression) for even asymmetric leg configurations regarding leg stiffness and angle of attack. The predicted asymmetric walking patterns can be neutrally stable. This means that the direction of walking will change if a sudden lateral push is applied to the body, however after the perturbation the walking direction remains constant. This outcome is similar to the predictions of the lateral leg spring (LLS) model of Schmitt & Holmes (2000). and Full (2001) that operates in the horizontal plane only. This was the first paper predicting self-stable gaits in SLIP models.


  1. Considering the basics of BSLIP model, extend the model to have four legs coming out of CoM and generate stable quadrupedal waliking with this model. Note that for simplicity we do not consider a torso and all legs are connected to one point.
  2. How many symmetric gaits can you generate? For example considering every two legs moving together you have a BSLIP model, now with small time delay, you can generate rhythmic motion. How can you generate similar patterns in different legs.
  3. Based on basic SLIP model, generate the MSLIP 1l model. What are different in the stable gaits that you can achieve with these models.
  4. Are the model of Rummel 2008 in Fig. 3 and the TSLIP model different? explain the differences considering their contributions in explaining different locomotr subfunctions.
  5. What is your prediction about extending a model from 2d to 3d. For example assume that you have stable 2d model like BSLIP with appropriate controllers for different locomotr subfunctions. Can this model without changing anything work on 3d? If not, what do you need to consider?


Alexander, R. McN. (1976). Mechanics of bipedal locomotion. Perspectives in experimental biology, 1, 493-504.

Blickhan, R. (1989). The spring-mass model for running and hopping. Journal of biomechanics, 22(11-12), 1217-1227.

Blickhan, R., & Full, R. J. (1993). Similarity in multilegged locomotion: bouncing like a monopode. Journal of Comparative Physiology A, 173(5), 509-517.

Borelli, G.A. 1680. De Motu Animalium (English translation by P. Maquet, Springer-Verlag, Berlin, 1989).

Curtze, C., Hof, A. L., van Keeken, H. G., Halbertsma, J. P., Postema, K., & Otten, B. (2009). Comparative roll-over analysis of prosthetic feet. Journal of biomechanics, 42(11), 1746-1753.

Geyer, H., Seyfarth, A., & Blickhan, R. (2003). Positive force feedback in bouncing gaits?. Proceedings of the Royal Society of London B: Biological Sciences, 270(1529), 2173-2183.

Geyer, H., Seyfarth, A., & Blickhan, R. (2006). Compliant leg behaviour explains basic dynamics of walking and running. Proceedings of the Royal Society of London B: Biological Sciences, 273(1603), 2861-2867.

Geyer, H., & Herr, H. (2010). A muscle-reflex model that encodes principles of legged mechanics produces human walking dynamics and muscle activities. IEEE Transactions on neural systems and rehabilitation engineering, 18(3), 263-273.

Grizzle, J. W., Abba, G., & Plestan, F. (2001). Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transactions on automatic control, 46(1), 51-64.

Herr, H. M., Huang, G. T., & McMahon, T. A. (2002). A model of scale effects in mammalian quadrupedal running. Journal of Experimental Biology, 205(7), 959-967.

Ludwig, C., Grimmer, S., Seyfarth, A., & Maus, H. M. (2012). Multiple-step model-experiment matching allows precise definition of dynamical leg parameters in human running. Journal of biomechanics, 45(14), 2472-2475.

Ludwig, C., Lipfert, S., Seyfarth, A. (2010). Variableity in human running is not reduced by metronome signals. Proceedings International Society of Biomechanics (ISB) conference.

Maus, H. M., Lipfert, S. W., Gross, M., Rummel, J., & Seyfarth, A. (2010). Upright human gait did not provide a major mechanical challenge for our ancestors. Nature Communications, 1, 70.

Maus, H. M., & Seyfarth, A. (2014). Walking in circles: a modelling approach. Journal of The Royal Society Interface, 11(99), 20140594.

Maykranz, D., Grimmer, S., Lipfert, S., & Seyfarth, A. (2009). Foot function in spring mass running. In Autonome Mobile Systeme 2009 (pp. 81-88). Springer Berlin Heidelberg.

Maykranz, D., & Seyfarth, A. (2014). Compliant ankle function results in landing-take off asymmetry in legged locomotion. Journal of theoretical biology, 349, 44-49.

McMahon, T. A., & Cheng, G. C. (1990). The mechanics of running: how does stiffness couple with speed?. Journal of biomechanics, 23, 65-78.

Mochon, S., & McMahon, T. A. (1980). Ballistic walking. Journal of biomechanics, 13(1), 49-57.

Müller, R., Grimmer, S., & Blickhan, R. (2010). Running on uneven ground: leg adjustments by muscle pre-activation control. Human movement science, 29(2), 299-310.

Oehlke, J., Sharbafi, M. A., Beckerle, P., & Seyfarth, A. (2016). Template-based hopping control of a bio-inspired segmented robotic leg. In 2016 6th IEEE International Conference on Biomedical Robotics and Biomechatronics (BioRob) (pp. 35-40).

Owaki, D., Osuka, K., & Ishiguro, A. (2008). On the embodiment that enables passive dynamic bipedal running. In Robotics and Automation, 2008. ICRA 2008. IEEE International Conference on (pp. 341-346). IEEE.

Peuker, F., Seyfarth, A., & Grimmer, S. (2012). Inheritance of SLIP running stability to a single-legged and bipedal model with leg mass and damping. In IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob) 395-400.

Poulakakis, I., Smith, J. A., & Buehler, M. (2005). Modeling and experiments of untethered quadrupedal running with a bounding gait: The Scout II robot. The International Journal of Robotics Research, 24(4), 239-256.

Poulakakis, I., & Grizzle, J. W. (2009). The spring loaded inverted pendulum as the hybrid zero dynamics of an asymmetric hopper. IEEE Transactions on Automatic Control, 54(8), 1779-1793.

Rummel, J., & Seyfarth, A. (2008). Stable running with segmented legs. The International Journal of Robotics Research, 27(8), 919-934.

Rummel, J., & Seyfarth, A. (2010) “Passive stabilization of the trunk in walking,” in International Conference on Simulation, Modeling and Programming for Autonomous Robots, (SIMPAR).

Saranli, U., Buehler, M., & Koditschek, D. E. (2001). Rhex: A simple and highly mobile hexapod robot. The International Journal of Robotics Research, 20(7), 616-631.

Seipel, J., & Holmes, P. (2005). Running in Three Dimensions: Analysis of a Point-mass Sprung-leg Model. The International Journal of Robotics Research, 24(8), 657-674.

Seipel, J., & Holmes, P. (2007). A simple model for clock-actuated legged locomotion. Regular and chaotic dynamics, 12(5), 502-520.

Seyfarth, A., Friedrichs, A., Wank, V., & Blickhan, R. (1999). Dynamics of the long jump. Journal of biomechanics, 32(12), 1259-1267.

Seyfarth, A., Blickhan, R., & Van Leeuwen, J. L. (2000). Optimum take-off techniques and muscle design for long jump. Journal of Experimental Biology, 203(4), 741-750.

Seyfarth, A., Günther, M., & Blickhan, R. (2001). Stable operation of an elastic three-segment leg. Biological cybernetics, 84(5), 365-382.

Seyfarth, A., Geyer, H., & Herr, H. (2003). Swing-leg retraction: a simple control model for stable running. Journal of Experimental Biology, 206(15), 2547-2555.

Seyfarth, A., Geyer, H., Blickhan, R., Lipfert, S., Rummel, J., Minekawa, Y., & Iida, F. (2006). Running and walking with compliant legs. In Fast motions in biomechanics and robotics (pp. 383-401). Springer Berlin Heidelberg.

Smith, J. A., & Poulakakis, I. (2004). Rotary gallop in the untethered quadrupedal robot scout II, in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2556-2561.

Sharbafi, M. A., Maufroy, C., Ahmadabadi, M. N., Yazdanpanah, M. J., & Seyfarth, A. (2013a). Robust hopping based on virtual pendulum posture control. Bioinspiration & biomimetics, 8(3), 036002.

Sharbafi, M. A., Ahmadabadi, M. N., Yazdanpanah, M. J., Mohammadinejad, A., & Seyfarth, A., (2013b) “Compliant hip function simplifies control for hopping and running,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

Sharbafi, M. A., & Seyfarth, A. (2014). Stable running by leg force-modulated hip stiffness. In 5th IEEE RAS/EMBS International Conference on Biomedical Robotics and Biomechatronics, 204-210.

Sharbafi, M. A., & Seyfarth, A. (2015). FMCH: A new model for human-like postural control in walking. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 5742-5747.

Sharbafi, M.A., Rode, C., Kurowski, S., Scholz, D., Möckel, R., Radkhah, K., Zhao, G., Mohammadi, A., von Stryk, O. and Seyfarth, A., 2016. A new biarticular actuator design facilitates control of leg function in BioBiped3. Bioinspiration & Biomimetics, 11(4), p.046003.

Westervelt, E. R., Grizzle, J. W., Chevallereau, C., Choi, J. H., & Morris, B. (2007). Modeling, Analysis, and Control of Robots with Passive Point Feet, In Feedback control of dynamic bipedal robot locomotion, 43-79, CRC press.

Zijlstra, W., & Hof, A. L. (1997). Displacement of the pelvis during human walking: experimental data and model predictions. Gait & posture, 6(3), 249-262.

biomechanik/modellierung/mm4/emslip.txt · Zuletzt geändert: 28.11.2022 00:58 von

Warning: Undefined variable $orig_id in /is/htdocs/wp1019470_OPI92FFHXV/www/wikiLehre/lib/plugins/openas/action.php on line 232