Research Papers

Passive Dynamics Explain Quadrupedal Walking, Trotting, and Tölting

[+] Author and Article Information
Zhenyu Gan

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ganzheny@umich.edu

Thomas Wiestner

Equine Department,
Vetsuisse Faculty,
University of Zurich,
Zurich CH-8057, Switzerland
e-mail: twiestner@vetclinics.uzh.ch

Michael A. Weishaupt

Equine Department,
Vetsuisse Faculty,
University of Zurich,
Zurich CH-8057, Switzerland
e-mail: mweishaupt@vetclinics.uzh.ch

Nina M. Waldern

Equine Department,
Vetsuisse Faculty,
University of Zurich,
Zurich CH-8057, Switzerland
e-mail: nwaldern@vetclinics.uzh.ch

C. David Remy

Robotics and Motion Laboratory,
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: cdremy@umich.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 10, 2015; final manuscript received May 7, 2015; published online August 26, 2015. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 11(2), 021008 (Aug 26, 2015) (12 pages) Paper No: CND-15-1041; doi: 10.1115/1.4030622 History: Received February 10, 2015

This paper presents a simplistic passive dynamic model that is able to create realistic quadrupedal walking, tölting, and trotting motions. The model is inspired by the bipedal spring loaded inverted pendulum (SLIP) model and consists of a distributed mass on four massless legs. Each of the legs is either in ground contact, retracted for swing, or is ready for touch down with a predefined angle of attack. Different gaits, that is, periodic motions differing in interlimb coordination patterns, are generated by choosing different initial model states. Contact patterns and ground reaction forces (GRFs) evolve solely from these initial conditions. By identifying appropriate system parameters in an optimization framework, the model is able to closely match experimentally recorded vertical GRFs of walking and trotting of Warmblood horses, and of tölting of Icelandic horses. In a detailed study, we investigated the sensitivity of the obtained solutions with respect to all states and parameters and quantified the improvement in fitting GRF by including an additional head and neck segment. Our work suggests that quadrupedal gaits are merely different dynamic modes of the same structural system and that we can interpret different gaits as different nonlinear elastic oscillations that propel an animal forward.

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Grahic Jump Location
Fig. 1

A simplistic model that essentially consists of a single distributed mass on four massless springs is able to explain the dynamics of quadrupedal walking, trotting, and tölting (shown in (a)). In addition, we studied an extended version that includes a head–neck segment connected to the body by a rotational joint with a torsional spring (shown in (b)).

Grahic Jump Location
Fig. 2

Head–neck angle (top) and main body angle (bottom) of a single stride at walk, tölt, and trot for the headed model. Oscillation amplitudes change with gait and are most pronounced in the walk. Generally, the head and torso angles are 180 deg out of phase. That is, the head is low if the shoulders are high. A similar behavior is observed in horses.

Grahic Jump Location
Fig. 3

The discrete states of all four limbs are shown for a single stride of walking of the headless model. The different phases of each leg are ready for touch down (1), stance (2), and swing (3). At the beginning of swing, a timer is started (dotted line) that triggers the transition into ready for touch down once it reaches tswing. All four limbs share the same tswing value that is illustrated by dashed-dotted line. The associated waiting period prevents feet from striking the ground too early and allows for a coordinated motion of the model's legs. The absolute horizontal position of the foot on the ground (solid line) is only updated at touch down and remains constant throughout the other phases. At the terminal event, the forward motion is removed from this variable, making it periodic from step to step.

Grahic Jump Location
Fig. 4

Eight consecutive frames of a full stride of headless walking. The model is moving left to right. Uncompressed legs with open circles indicate legs that are ready for touch down (phasei = 1), filled circles indicate legs that are in stance (phasei = 2), and retreacted legs are in swing (phasei = 3).

Grahic Jump Location
Fig. 5

Experimentally recorded vertical GRFs (dotted lines ±1 std.) are compared to forces predicted by the headless model (solid lines, shown on the left) and to those predicted by the model with an articulated head and neck (solid lines, shown on the right). Shown are the results for walking (top), tölting (center), and trotting (bottom). Both models correctly predict the footfall pattern, timing, and the general shape of the force curves for all gaits. Quantitatively, a better fit is produced by the headed model, especially for the hind limbs at walk (see also Table 2). RH, RF, LH, and LF stand for right hind, right fore, left hind, and left fore, respectively.

Grahic Jump Location
Fig. 6

Shown is the increase in cost (quantifying the difference between model-predicted and experimentally measured vertical GRFs) as a function to variations in states (shown on the left) and parameter choices (shown on the right). Results are shown for walking (top), tölting (center), and trotting (bottom). This analysis highlights which states and parameters can be predicted well by matching of the vertical GRFs (indicated by a high sensitivity) and which cannot be predicted well (indicated by a low sensitivity).



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