An approximate stance map of the spring mass hopper with gravity correction for nonsymmetric locomotions

The Spring-Loaded Inverted Pendulum (SLIP) model has long been established as an effective and accurate descriptive model for running animals of widely differing sizes and morphologies, while also serving as a basis for several hopping robot designs. Further research on this model led to the discovery of several analytic approximations to its normally nonintegrable dynamics. However, these approximations mostly focus on steady-state running with symmetric trajectories due to their linearization of gravitational effects, an assumption that is quickly violated for locomotion on more complex terrain wherein transient, non-symmetric trajectories dominate. In this paper, we introduce a novel gravity correction scheme that extends on one of the more recent analytic approximations to the SLIP dynamics and achieves good accuracy even for highly non-symmetric trajectories. Our approach is based on incorporating the total effect of gravity on the angular momentum throughout a single stance phase and allows us to preserve the analytic simplicity of the approximation to support our longer term research on reactive footstep planning for dynamic legged locomotion. We compare the performance of our method in simulation to two other existing analytic approximations and show that it outperforms them for most physically realistic non-symmetric SLIP trajectories while maintaining the same accuracy for symmetric trajectories. © 2009 IEEE

Experimental Validation of a Feed-Forward Predictor for the Spring-Loaded Inverted Pendulum Template

Cataloged from PDF version of article.Widely accepted utility of simple spring-mass models for running behaviors as descriptive tools, as well as literal control targets, motivates accurate analytical approximations to their dynamics. Despite the availability of a number of such analytical predictors in the literature, their validation has mostly been done in simulation, and it is yet unclear how well they perform when applied to physical platforms. In this paper, we extend on one of the most recent approximations in the literature to ensure its accuracy and applicability to a physical monopedal platform. To this end, we present systematic experiments on a well-instrumented planar monopod robot, first to perform careful identification of system parameters and subsequently to assess predictor performance. Our results show that the approximate solutions to the spring-loaded inverted pendulum dynamics are capable of predicting physical robot position and velocity trajectories with average prediction errors of 2% and 7%, respectively. This predictive performance together with the simple analytic nature of the approximations shows their suitability as a basis for both state estimators and locomotion controllers. © 2004-2012 IEEE

Model based methods for the control and planning of running robots

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 2009.Thesis (Master's) -- Bilkent University, 2009.Includes bibliographical references leaves 115-122.The Spring-Loaded Inverted Pendulum (SLIP) model has long been established as an effective and accurate descriptive model for running animals of widely differing sizes and morphologies. Not surprisingly, the ability of such a simple spring-mass model to capture the essence of running motivated several hopping robot designs as well as the use of the SLIP model as a control target for more complex legged robot morphologies. Further research on the SLIP model led to the discovery of several analytic approximations to its normally nonintegrable dynamics. However, these approximations mostly focus on steady-state running with symmetric trajectories due to their linearization of gravitational effects, an assumption that is quickly violated for locomotion on more complex terrain wherein transient, non-symmetric trajectories dominate. In the first part of the thesis , we introduce a novel gravity correction scheme that extends on one of the more recent analytic approximations to the SLIP dynamics and achieves good accuracy even for highly non-symmetric trajectories. Our approach is based on incorporating the total effect of gravity on the angular momentum throughout a single stance phase and allows us to preserve the analytic simplicity of the approximation to support research on reactive footstep planning for dynamiclegged locomotion. We compare the performance of our method with two other existing analytic approximations by simulation and show that it outperforms them for most physically realistic non-symmetric SLIP trajectories while maintaining the same accuracy for symmetric trajectories. Additionally, this part of the thesis continues with analytical approximations for tunable stiffness control of the SLIP model and their motion prediction performance analysis. Similarly, we show performance improvement for the variable stiffness approximation with gravity correction method. Besides this, we illustrate a possible usage of approximate stance maps for the controlling of the SLIP model. Furthermore, the main driving force behind research on legged robots has always been their potential for high performance locomotion on rough terrain and the outdoors. Nevertheless, most existing control algorithms for such robots either make rigid assumptions about their environments (e.g flat ground), or rely on kinematic planning with very low speeds. Moreover, the traditional separation of planning from control often has negative impact on the robustness of the system against model uncertainty and environment noise. In the second part of the thesis, we introduce a new method for dynamic, fully reactive footstep planning for a simplified planar spring-mass hopper, a frequently used dynamic model for running behaviors. Our approach is based on a careful characterization of the model dynamics and an associated deadbeat controller, used within a sequential composition framework. This yields a purely reactive controller with a very large, nearly global domain of attraction that requires no explicit replanning during execution. Finally, we use a simplified hopper in simulation to illustrate the performance of the planner under different rough terrain scenarios and show that it is robust to both model uncertainty and measurement noise.Arslan, ÖmürM.S

An analytical solution to the stance dynamics of passive spring-loaded inverted pendulum with damping

The Spring-Loaded Inverted Pendulum (SLIP) model has been established both as a very accurate descriptive tool as well as a good basis for the design and control of running robots. In particular, approximate analytic solutions to the otherwise non integrable dynamics of t his model provide principled ways in which gait controllers can be built, yielding invaluable insight into their stability properties. However, most existing work on the SLIP model completely disregards the effects of damping, which often cannot be neglected for physical robot platforms. In this paper, we introduce a new approximate analytical solution to the dynamics of this system that also takes into account viscous damping in the leg. We compare both the predictive performance of our approximation as well as the tracking performance of an associated deadbeat gait controller to similar existing methods in the literature and show t hat it significantly outperforms them in the presence of damping in the leg

Adaptive control of a one-legged hopping robot through dynamically embedded spring-loaded inverted pendulum template

Ankara : The Department of Electrical and Electronics Engineering and the Graduate School of Engineering and Science of Bilkent University, 2011.Thesis (Master's) -- Bilkent University, 2011.Includes bibliographical references leaves 92-96.Practical realization of model-based dynamic legged behaviors is substantially more challenging than statically stable behaviors due to their heavy dependence on second-order system dynamics. This problem is further aggravated by the dif- ficulty of accurately measuring or estimating dynamic parameters such as spring and damping constants for associated models and the fact that such parameters are prone to change in time due to heavy use and associated material fatigue. In the first part of this thesis, we present an on-line, model-based adaptive control method for running with a planar spring-mass hopper based on a once-per-step parameter correction scheme. Our method can be used both as a system identifi- cation tool to determine possibly time-varying spring and damping constants of a miscalibrated system, or as an adaptive controller that can eliminate steady-state tracking errors through appropriate adjustments on dynamic system parameters. We use Spring-Loaded Inverted Pendulum (SLIP) model, which is the mostly used, effective and accurate descriptive tool for running animals of different sizes and morphologies, to evaluate our algorithm. We present systematic simulation studies to show that our method can successfully accomplish both accurate tracking and system identification tasks on this model. Additionally, we extend our simulations to Torque-Actuated Dissipative Spring-Loaded Inverted Pendulum (TD-SLIP) model towards its implementation on an actual robot platform. In the second part of the thesis, we present the design and construction of a onelegged hopping robot we built to test the practical applicability of our adaptive control algorithm. We summarize the mechanical, electronics and software design of our robot as well as the performed system identification studies to calibrate the unknown system parameters. Finally, we investigate the robot’s motion achieved by a simple torque-actuated open loop controller.Uyanık, İsmailM.S

Accurate Step Length Control Strategies for Underactuated and Realistic Series Elastic Actuated Hoppers via High Order PFL

Among the different types of legged robots, hopping robots, aka hoppers, can be classified as one of the simplest sufficient models that capture the important features encompassed in dynamic locomotion: underactuation, compliance, and hybrid features. There is an abundance of work regarding the implementation of highly simplified hopper models, the prevalent example being the spring loaded inverted pendulum (SLIP) model, with the hopes of extracting fundamental control ideas for running and hopping robots. However, real world systems cannot be fully described by such simple models, as real actuators have their own dynamics including additional inertia and non-linear frictional losses. Additionally, implementing feedback control for hopping systems with significant amounts of compliance is difficult as the input variable does not instantaneously change the leg length acceleration. The current state-of-the-art of step length control in the presence of non-steady state motions required for foothold placement is not precise enough for operation in the real world. Therefore, an important step towards demonstrating high controllability and robustness to real-world elements is in providing accurate higher order models of real-world hopper dynamics, along with compatible control strategies. Our modeling work is based on a series-elastic actuated (SEA) hopping robot prototype constructed by our lab group, and we provide verifying hardware results that high order partial feedback linearization (HOPFL) can be implemented directly on the leg state of the robot. Using HOPFL, we investigate two paths of compatible trajectory generation that can accomplish desirable tasks such as precise foothold planning. We investigate the practicality of using SLIP-based trajectory generation techniques on more realistic hopping robots, and show that by implementing HOPFL directly on the robot's leg, we can make use of computationally fast SLIP-based approximations, account for non-trivial pitch dynamics, and improve the state-of-the-art of precision step length control for SEA hoppers. We also consider control strategies towards hoppers for which SLIP-based trajectories may not be compatible, by planning all ground reaction force vector (GRF) components during the stance phase concurrently, using a lower order and very general model to construct trajectories for the system's center of mass (CoM), and maintain body stability by controlling the orientation of the GRF directly. While not purely analytical as our SLIP-based approaches, this method is general enough to work on a variety of hopping robots that are not necessarily kinematically structured resembling the classical SLIP model

Reachability and Real-Time Actuation Strategies for the Active SLIP Model

Running and hopping follow similar patterns for different animals, independent of the number of legs employed. An aerial phase alternates with a ground contact phase, during which the center of mass moves as if a spring were compressed and then extended to recover stored elastic energy. Hence, consisting of a point mass mounted on a massless spring leg, the Spring Loaded Inverted Pendulum (SLIP) is a prevalent model for analyzing running and hopping. In this work we consider an actuated version of the SLIP model, with a series elastic actuator added to the leg, serving the purposes of adding/removing energy to/from the system and of modifying dynamics during stance, toward achieving non-steady locomotion on varying terrain. While the SLIP model has been a topic of research in legged locomotion for several decades, studies on the effect of actuation on the system's behavior are still not complete.The goal of this thesis is to explore how a series elastic actuator applied to the SLIP model's leg can change the system's dynamics. This, in turn, enables a variety of long-term planning strategies for using limited footholds and design non-steady gaits while simultaneously recovering from unexpected perturbations, both sensorial and due to a limited knowledge of the terrain profile.We principally investigate how, through actuation, we can solve partially or completely the system's equations of motion, to enforce a desired trajectory and reach a desired state. We also determine the reachable state space of the model using several different actuation strategies, investigating the variation of the reachable set with respect to particular actuator motions and providing relationships between local actuator displacements throughout stance and location of the reached apex state. We then propose a control strategy based on graphical and numerical studies of the reachability space to drive the system to a desired state, with the ability to reduce the effects of sensing errors and disturbances happening at landing as well as during ground contact

Understanding preferred leg stiffness and layered control strategies for locomotion

Despite advancement in the field of robotics, current legged robots still cannot achieve the kind of locomotion stability animals and humans have. In order to develop legged robots with greater stability, we need to better understand general locomotion dynamics and control principles. Here we demonstrate that a mathematical modeling approach could greatly enable the discovery and understanding of general locomotion principles. ^ It is found that animal leg stiffness when scaled by its weight and leg length falls in a narrow region between 7 and 27. Rarely in biology does such a universal preference exist. It is not known completely why this preference exists. Here, through simulation of the simple actuated-SLIP model, we show that the biological relative leg stiffness corresponds to the theoretical minimum of mechanical cost of transport. This strongly implies that animals choose leg stiffness in this region to reduce energetic cost. In addition, it is found that the stability of center-of-mass motion is also optimal when biological relative leg stiffness values are selected for actuated-SLIP. Therefore, motion stability could be another reason why animals choose this particular relative leg stiffness range. ^ We then extended actuated-SLIP by including realistic trunk pitching dynamics. At first, to form the Trunk Spring-Loaded Inverted Pendulum (Trunk-SLIP) model, the point mass of actuated-SLIP is replaced by a rigid body trunk while the leg remains massless and springy. It is found that exproprioceptive feedback during the flight phase is essential to the overall motion stability including trunk pitching. Either proprioceptive or exproprioceptive feedback during stance could generate stable running motion provided that exproprioceptive feedback is used during flight. When both kinds of feedback are used during stance, the overall stability is improved. However, stability with respect to speed perturbations remains limited. ^ Built upon Trunk-SLIP, we develop a model called extended Trunk-SLIP with trunk and leg masses. We then develop a hierarchical control strategy where different layers of control are added and tuned. When each layer is added, the overall motion stability is improved. This layer by layer strategy is simple in nature and allows quick controller design and tuning as only a limited number of control parameters needs to be added and tuned at each step. In the end, we propose a future control layer where the commanded speed is controlled to achieve a higher level target such as might be needed during smooth walking to running transitions. ^ In summary, we show here that the simple actuated-SLIP model is able to predict animal center-of-mass translation stability and overall mechanical cost of transport. More advanced models are then developed based upon actuated-SLIP. With a simple layer by layer control strategy, robust running motion can be discovered. Overall, this knowledge could help better understand locomotion dynamics in general. In addition, the developed control strategy could, in principle be applied to future hip based legged robot design