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Swing-leg retraction: a simple control model for stable running

André Seyfarth^1,3,*, Hartmut Geyer^1,3 and Hugh Herr^1,2,4

¹ Artificial Intelligence Laboratory, Cambridge, MA 02139, USA
² Harvard/MIT Division of Health Sciences and Technology, Cambridge, MA 02139, USA
³ ParaCare Laboratory, Balgrist Hospital, University Zurich, CH-8008 Zurich, Switzerland
⁴ Department of Physical Medicine and Rehabilitation, Harvard Medical School, Spaulding Rehabilitation Hospital, Boston, MA 02114, USA

View larger version (57K): [in a new window]   Fig. 1. Spring-mass model with retraction. Swing-leg retraction in running, as indicated by the photographs of Muybridge (1955; reproduced with permission from Dover Publications), is modeled assuming a constant rotational velocity of the leg (retraction speed R), starting at the apex of the flight phase at retraction angle R. Depending on the duration of the flight phase, the landing angle of the leg (angle of attack 0) is a result of the model dynamics and has no predefined constant value in contrast to the previous model of Seyfarth et al. (2002). The axial leg operation during the stance phase is approximated by a linear spring of constant stiffness kLEG.

View larger version (36K): [in a new window]   Fig. 2. Center of mass trajectories (A) and leg kinematics (B,C) for spring-mass running with and without retraction (C, R=50 deg s-1; B, R=0 deg s-1). The bars in A indicate the change in centre of mass height between touch-down and take-off. B and C are expanded views of plot A from 0 to 6 m (boxed). For each simulated run, the same initial apex height was used (y0=1.25 m), and for the simulation with retraction, a retraction angle of R=60° was assumed (C). Here the model with retraction reached a steady state condition after two steps in contrast to approximately 8 steps for the model without retraction (A,B). The red dotted lines in B and C denote the steady state landing angle 0*.

View larger version (25K): [in a new window]   Fig. 3. Return maps yi+1(yi) of the apex height yAPEX of two consecutive flight phases (index i and i+1) for three different retraction speeds R (A, R=0 deg s-1; B, R=25 deg s-1; C, R=50 deg s-1).The system energy corresponds to a running speed of 5 m s-1 at an apex height yAPEX=1 m. (A–C) Three characteristic return maps represent the minimum, mean and maximum retraction angle R (see key in each panel) for stable fixed points (see text, Equation 2). With increasing retraction speed R, the range of retraction angles R with stable fixed points increases, and attraction of higher apex heights is observed (max. y01.3, 1.9, 2.2 for R=0, 25, 50 deg s-1, respectively) as shown by representative tracings (running sequences are indicated by stepped black lines with starting arrows).

View larger version (25K): [in a new window]   Fig. 4. Return maps yi+1(yi) of the apex height yAPEX are shown for different retraction speeds R (A, R=0 deg s-1; B, R=25 deg s-1; C, =50 deg s-1) but for a lower system energy corresponding to a slower running speed of 3 m s-1 at an apex height yAPEX=1 m. Stable fixed points require non-zero retraction velocities R>0 (B,C). As in Fig. 3, an increased retraction speed leads to an enlarged attraction of the stable fixed points with respect to a given initial (e.g. disturbed) apex height. Model parameters: m=80 kg, l0=1 m, kLEG=20 kN m-1.

View larger version (36K): [in a new window]   Fig. 5. The influence of retraction speed R on the robustness of running is shown with respect to a permanent change in leg stiffness kLEG. For each run (vx,0=5 m s-1, y0=1 m), the maximum and minimum leg stiffness (kMIN, kMAX) required to keep the system in a periodic running movement are depicted (A,C,E). The retraction angle R (denoted in A,C,E) is chosen according to the mean retraction angle for stable fixed points in Fig. 3 at kLEG=20 kN m-1. (B,D,F) The adaptation of the leg angle to the changed leg stiffness.

View larger version (28K): [in a new window]   Fig. 6. (A) A three-dimensional (3D) representation yi+1(yi, 0) of the return map yi+1(yi) characterizes spring-mass running (system energy corresponds to vX=5 m s-1 at yAPEX=1 m; m=80 kg, l0=1 m, k=20 kN m-1) for different angles of attack 0. For fixed angles of attack (slices in 3D), the corresponding return maps are shown on the left (yi, yi+1) plane. The red line depicts the return map for 0=68°. Different return maps are possible if the angle of attack 0 becomes dependent on the apex height yi. An `optimal' control model with respect to stability would be a direct projection of any initial apex height yi to a desired apex height yCONTROL in the next flight phase, or yi+1(yi)=yCONTROL=constant, as shown for apex heights of 1, 1.5 and 2 m (left plane). This corresponds to isolines on the 3D-surface yi+1(yi, 0) indicating a dependency between the angle of attack 0 and the initial apex height yI, as shown for yCONTROL=1, 1.5 and 2 m in (B). With careful selection of the retraction velocity R and the retraction angle R, the constant velocity leg retraction model can approximate the optimal control strategy.

View larger version (21K): [in a new window]   Fig. 7. Leg kinematics (leg length versus leg angle) during treadmill running at 3 m s-1. For the undisturbed condition, the mean ± S.D. of 35 running steps for one male subject (78 kg) are shown (A). The leg length lLEG was measured as the distance between hip and toe marker. The leg angle is defined as the projection angle with respect to the ground (Fig. 1). Swing-leg retraction is present between the onset angle R (length lR) and the angle of attack 0 (length l0) as shown magnified in (B). For the disturbed swing phase, the leg operation of the same experimental subject is plotted. Although only a single subject is depicted here, similar results were observed in all experimental subjects (see Table 1).