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72
The dynamics of legged locomotion: Models, analyses, and challenges
- SIAM Review
, 2006
"... Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we ..."
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Cited by 115 (22 self)
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Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects, and highlighting achievements and challenges that remain. Newtonian body-limb dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons, and simplified phase oscillator descriptions of ensembles of them. A full neuro-mechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goal-oriented sensing, planning and learning. We outline relevant background material from neurobiology and biomechanics, explain key properties of the hybrid dynamical systems that 1 underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble ‘peg-leg walker ’ to complex neuro-muscular subsystems that are yet to be assembled into models of behaving animals. 1
A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities
- IEEE Trans. Neural Syst. Rehabil. Eng
"... Abstract — While neuroscientists identify increasingly complex neural circuits that control animal and human gait, biomechanists find that locomotion requires little control if principles of legged mechanics are heeded that shape and exploit the dynamics of legged systems. Here we show that muscle r ..."
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Cited by 44 (11 self)
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Abstract — While neuroscientists identify increasingly complex neural circuits that control animal and human gait, biomechanists find that locomotion requires little control if principles of legged mechanics are heeded that shape and exploit the dynamics of legged systems. Here we show that muscle reflexes could be vital to link these two observations. We develop a model of human locomotion that is controlled by muscle reflexes which encode principles of legged mechanics. Equipped with this reflex control, we find this model to stabilize into a walking gait from its dynamic interplay with the ground, reproduce human walking dynamics and leg kinematics, tolerate ground disturbances, and adapt to slopes without parameter interventions. In addition, we find this model to predict some individual muscle activation patterns known from walking experiments. The results suggest not only that the interplay between mechanics and motor control is essential to human locomotion, but also that human motor output could for some muscles be dominated by neural circuits that encode principles of legged mechanics. Index Terms — legged locomotion, feedback control, balance. I.
Stability analysis of legged locomotion models by symmetry-factored return maps
- International Journal of Robotics Research
, 2004
"... We present a new stability analysis for hybrid legged locomotion systems based on the “symmetric” factorization of return maps. We apply this analysis to 2 and 3 degree of freedom (DOF) models of the Spring Loaded Inverted Pendulum (SLIP) with different leg recirculation strategies. Despite the non- ..."
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Cited by 32 (8 self)
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We present a new stability analysis for hybrid legged locomotion systems based on the “symmetric” factorization of return maps. We apply this analysis to 2 and 3 degree of freedom (DOF) models of the Spring Loaded Inverted Pendulum (SLIP) with different leg recirculation strategies. Despite the non-integrability of the SLIP dynamics, we obtain a necessary condition for asymptotic stability (and a sufficient condition for instability) at a fixed point, formulated as an exact algebraic expression in the physical parameters. We use this expression to study a variety of 2 DOF SLIP models that have previously been posited as low dimensional representations of running, focusing on the sensory “cost” required to achieve “fast ” transients as measured by the degree of singularity of the linearized dynamics. We introduce a new 3 DOF SLIP model with pitching dynamics whose stability properties, revealed by this analysis, provide for the first time the beginnings of a formal explanation for the surprisingly stable gaits of the open loop controlled robot, RHex.
Design of a Bio-inspired Dynamical Vertical Climbing Robot
"... Abstract — This paper reviews a template for dynamical climbing originating in biology, explores its stability properties in a numerical model, and presents empirical data from a physical prototype as evidence of the feasibility of adapting the dynamics of the template to robot that runs vertically ..."
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Cited by 30 (17 self)
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Abstract — This paper reviews a template for dynamical climbing originating in biology, explores its stability properties in a numerical model, and presents empirical data from a physical prototype as evidence of the feasibility of adapting the dynamics of the template to robot that runs vertically upward. The recently proposed pendulous climbing model abstracts remarkable similarities in dynamic wall scaling behavior exhibited by radically different animal species. The present paper’s first contribution summarizes a numerical study of this model to hypothesize that these animals ’ apparently wasteful commitments to lateral oscillations may be justified by a significant gain in the dynamical stability and, hence, the robustness of their resulting climbing capability. The paper’s second contribution documents the design and offers preliminary empirical data arising from a physical instantiation of this model. Notwithstanding the substantial differences between the proposed bio-inspired template and this physical manifestation, initial data suggest the mechanical climber may be capable of reproducing both the motions and ground reaction forces characteristic of dynamical climbing animals. Even without proper tuning, the robot’s steady state trajectories manifest a substantial exchange of kinetic and potential energy, resulting in vertical speeds of 0.30 m/s (0.75 bl/s) and claiming its place as the first bio-inspired dynamical legged climbing platform. I.
On the Stability of the Passive Dynamics of Quadrupedal Running with a Bounding Gait
, 2006
"... all correspondence related to this paper to the first author. This paper examines the passive dynamics of quadrupedal bounding. First, an unexpected difference between local and global behavior of the forward speed versus touchdown angle in the self-stabilized Spring Loaded Inverted Pendulum (SLIP) ..."
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Cited by 28 (14 self)
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all correspondence related to this paper to the first author. This paper examines the passive dynamics of quadrupedal bounding. First, an unexpected difference between local and global behavior of the forward speed versus touchdown angle in the self-stabilized Spring Loaded Inverted Pendulum (SLIP) model is exposed and discussed. Next, the stability properties of a simplified sagittal plane model of our Scout II quadrupedal robot are investigated. Despite its simplicity, this model captures the targeted steady state behavior of Scout II without dependence on the fine details of the robot structure. Two variations of the bounding gait, which are observed experimentally in Scout II, are considered. Surprisingly, numerical return map studies reveal that passive generation of a large variety of cyclic bounding motion is possible. Most strikingly, local stability analysis shows that the dynamics of the open loop passive system alone can confer stability of the motion! These results can be used in developing a general control methodology for legged robots, resulting from the synthesis of feedforward and feedback models that take advantage of the mechanical system, and might explain the success of simple, open loop bounding controllers on our experimental robot. KEY WORDS – Passive dynamics, bounding gait, dynamic running, quadrupedal robot. 1
The Spring Loaded Inverted Pendulum as the Hybrid Zero Dynamics of an Asymmetric Hopper
- SUBMITTED TO THE IEEE TRANSACTIONS ON AUTOMATIC CONTROL AS A REGULAR PAPER
, 2007
"... A hybrid controller that induces provably stable running gaits on an Asymmetric Spring Loaded Inverted Pendulum (ASLIP) is developed. The controller acts on two levels. On the first level, continuous within-stride control asymptotically imposes a (virtual) holonomic constraint corresponding to a de ..."
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Cited by 25 (7 self)
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A hybrid controller that induces provably stable running gaits on an Asymmetric Spring Loaded Inverted Pendulum (ASLIP) is developed. The controller acts on two levels. On the first level, continuous within-stride control asymptotically imposes a (virtual) holonomic constraint corresponding to a desired torso posture, and creates an invariant surface on which the two-degree-of-freedom restriction dynamics of the closed-loop system (i.e., the hybrid zero dynamics) is diffeomorphic to the center-of-mass dynamics of a Spring Loaded Inverted Pendulum (SLIP). On the second level, event-based control stabilizes the closed-loop hybrid system along a periodic orbit of the SLIP dynamics. The controller’s performance is discussed through comparison with a second control law that creates a one-degreeof-freedom non-compliant hybrid zero dynamics. Both controllers induce identical steady-state behaviors (i.e. periodic solutions). Under transient conditions, however, the controller inducing a compliant hybrid zero dynamics based on the SLIP accommodates significantly larger disturbances, with less actuator effort, and without violation of the unilateral ground force constraints.
On the dynamics of bounding and extensions towards the half-bound and the gallop gaits
- In Proceedings of the 2nd International Symposium on Adaptive Motion of Animals and Machines (AMAM
, 2003
"... Abstract. This paper examines how simple control laws stabilize complex run-ning behaviors such as bounding. First, we discuss the unexpectedly different local and global forward speed versus touchdown angle relationships in the self stabilized SLIP. Then we show that, even for a more complex, energ ..."
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Cited by 20 (2 self)
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Abstract. This paper examines how simple control laws stabilize complex run-ning behaviors such as bounding. First, we discuss the unexpectedly different local and global forward speed versus touchdown angle relationships in the self stabilized SLIP. Then we show that, even for a more complex, energy conserving, unactuated quadrupedal model, many bounding motions exist, which can be locally open loop stable! The success of simple bounding controllers motivated the use of similar con-trol laws for asymmetric gaits resulting in the first experimental implementations of the half-bound and the rotary gallop gaits on Scout II. 1
Stability analysis of a simple walking model driven by a rhythmic signal
- Proc. of the IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS2004
, 2004
"... Abstract – Walking motion of human is achieved through the interaction between the dynamics of the human mechanical system and the rhythmic signals of the central pattern generator (CPG). In this paper, we analyze dynamic properties of a simple walking model of a biped robot driven by a nonlinear os ..."
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Cited by 13 (4 self)
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Abstract – Walking motion of human is achieved through the interaction between the dynamics of the human mechanical system and the rhythmic signals of the central pattern generator (CPG). In this paper, we analyze dynamic properties of a simple walking model of a biped robot driven by a nonlinear oscillator. The simple model consists of a hip and two legs which are connected at the hip and has touch sensors at the tips of the legs. The swing leg is controlled by the rhythmic signal of the oscillator. The oscillator receives feedback signals from the touch sensors and modifies the walking motion according to the signals. From the analysis of the stability of the periodic walking motion using a Poincaré map, it is revealed that the simple model has the stabilization property and moreover the stability region is enlarged due to the signal feedbacks.
Experimental Investigations into the Role of Passive Variable Compliant Legs for Dynamic Robotic Locomotion
- in IEEE Int. Conf. on Robotics and Automation
"... ISSN={1050-4729},} ..."
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A self-exciting controller for high-speed vertical running
- in Proceedings of the International Conference on Intelligent Robots and Systems
"... Abstract — Traditional legged runners and climbers have relied heavily on gait generators in the form of internal clocks or reference trajectories. In contrast, here we present physical experiments with a fast, dynamical, vertical wall climbing robot accompanying a stability proof for the controller ..."
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Cited by 11 (8 self)
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Abstract — Traditional legged runners and climbers have relied heavily on gait generators in the form of internal clocks or reference trajectories. In contrast, here we present physical experiments with a fast, dynamical, vertical wall climbing robot accompanying a stability proof for the controller that generates it without any need for an additional internal clock or reference signal. Specifically, we show that this “self-exciting ” controller does indeed generate an “almost ” globally asymptotically stable limit cycle: the attractor basin is as large as topologically possible and includes all the state space excluding a set with empty interior. We offer an empirical comparison of the resulting climbing behavior to that achieved by a more conventional clock-generated gait trajectory tracker. The new, self-exciting gait generator exhibits a marked improvement in vertical climbing speed, in fact setting a new benchmark in dynamic climbing by achieving a vertical speed of 1.5 body lengths per second. I.
