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Although animals and humans represent very complex dynamic systems, they show strikingly uniform and simple dynamics at the level of the center of mass (COM) when they walk or run. The COM dynamics can be measured using ground reaction force plates that record the forces with which the legs push against the ground during stance in locomotion. The figure below shows these leg forces for walking (left) and running (right). In the vertical direction (Fy) they are characterized by an Mshape pattern for walking and a bellshape pattern for running. Figure 1


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The spring biped is perhaps the simplest model that describes walking and running dynamics. It has only three independent parameters and one can characterize and understand its behavior with brute force parameter searches (compare figure 2). However, similar searches yield less meaningful results when extending this model into more complex ones. Mathematical solutions to the model dynamics can provide a deeper understanding. And I am interested in obtaining such solutions.
This task is not trivial. The springmass model describes a piecewise holonomic, nonlinear dynamic system whose equations of motion cannot be solved in closed form. Assuming that gravity corotates with the leg axis in stance, we derived an approximate solution to the model dynamics in running that is comparably simple yet surprisingly accurate.

The approximate solution can help to understand control strategies in running. For instance, we derived an explicit formula for how the leg parameters must be adjusted to use mechanical selfstability in legged systems. On the other hand, the solution provides a stepping stone toward a more general mathematical solution that includes running and walking.
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Because legged locomotion is a repetition of stance and swing phases, the behavior of the hybriddynamic springmass model can be investigated using Poincaré map analysis. This analysis shows a remarkable result. Legged locomotion is mechanically selfstable.
Poincaré map analysis is illustrated in figure 4 for
running. For each running step i, the model behavior is
uniquely characterized by the state variable `apex height'
(y_i). Mapping the change of this variable from one step (i)
to the next (i+1) identifies periodic solutions (y_i+1 = y_i =
y*). These solutions are stable if the slope of the map
stays within +/ 45 deg in the neighborhood of y*.

We showed that for wellchosen model parameters (example: swing leg orientation alpha_0=67deg in figure 4), running is mechanically selfstable. Without any feedback about (small) ground disturbances, the model stabilizes into a steady state running gait. The same holds true for the bipedal springmass model and walking.
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The Poincaré map analysis (figure 4) reveals a rather small tolerance of mechanical selfstability to ground disturbances and parameter adjustments. The required angular precision is ±1 deg. And the basin of attraction is narrow on the left side of the steady state solution y*, tolerating obstacle heights of a mere 5 percent of the leg length.
We showed that mechanical selfstability can be maximized in a canonical way. Maximum selfstability in running is equivalent to a deadbeat control that attracts any initial state y_i of the model into a desired state y_des within one step y_i+1=y_des=y*. In the language of Poincare maps this equals a horizontal map at the level y_des. Figure 5 illustrates that this imaginary return map can be realized if, for different initial apex heights y_i, different leg orientations alpha_0 in swing are selected.

The relationship alpha_0(y_i) represents a control that molds the basin of attraction into a desired shape, in a canonical way. It suffers however from a serious flaw. Continuous feedback about the ground clearance of the foot point is required to adapt alpha_0 with respect to the corresponding apex height. This control neither is elegant nor does it use mechanical selfstability inherent to legged dynamics.
A unique transformation changes the situation. Since any
alpha_0 defines a landing height y_l = L_0 * sin(alpha_0), the
pair y_i and alpha_0(y_i) corresponds to a defined drop height
Dy(y_i) = y_i  y_l. And this drop height uniquely
transforms into a flight time Dt(y_i) after the apex event in
running. In reverse this means that, for every time Dt,
there is a leg orientation alpha_0(Dt) in swing, which guarantees
that if the model hits ground at that moment, it gets directed
into the desired state y_des.

As a result the model rejects ground disturbances without sensing
them, within one step. This result is illustrated in figure
6 in a simulation and with a springleg robot.
I am interested in further developing the canonical expansion of
selfstability control to include design and environmental
constraints on springlegged robots, and to maximize the stability
of biped walking motions including gait transitions.
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