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First published online January 17, 2007
Journal of Experimental Biology 210, 533-540 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.02647

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Mechanics of dog walking compared with a passive, stiff-limbed, 4-bar linkage model, and their collisional implications

James R. Usherwood^*, Sarah B. Williams and Alan M. Wilson

The Royal Veterinary College, North Mymms, Hatfield, Herts, AL9 7TA, UK

View larger version (19K): [in this window] [in a new window]   Fig. 1. Model geometry. (A) A 4-bar linkage consisting of a stiff hindleg (red, length Lhind), back, foreleg (blue, length Lfore) and the ground between the hind and front feet. The path of the body centre of mass (COM) and its rotations are calculated from purely geometric considerations as the hindleg vaults over the vertical (B). The stride begins at the instant of hindfoot placement. Both hind- and foreleg angles are assumed to sweep an equal angle before and after vertical. The initial foreleg angle fore* is determined from the phase. The hindlimb angle is incremented, changing COM position (green dots) and body angle, and the appropriate time step for each increment throughout the stride is calculated such that the total mechanical energy is constant (although see the `powering strategies' section). Interchanges between mechanical energy components (potential, kinetic and rotational kinetic energies), velocities and accelerations were thus calculated from the 4-bar linkage geometry as a function of time. Total mechanical energy was adjusted so that the model and observed mean horizontal velocities corresponded.

View larger version (4K): [in this window] [in a new window]   Fig. 2. Collision geometry at foreleg contact. A component of the centre of mass (COM) velocity just prior to collision v can be maintained after the collision (v') despite an inelastic redirection through an angle determined from the 4-bar linkage geometry. The collisional energy loss calculated at each leg contact is dependent on both velocity and the angle with which the COM path is suddenly deflected.

View larger version (38K): [in this window] [in a new window]   Fig. 3. Observed and model energy fluctuations (A,B), accelerations (C,D) and powers (E) for three example walking strides ranging from very slow (i) through slow (ii) to moderate (iii). Model outcomes, derived using empirical kinematic inputs and the stiff-limbed, constant energy (the immediate energy recovery strategy) model are shown in blue; values derived from forceplate measurements and kinematics are shown in red. Model accelerations go off-scale at the instants of each foot initial contact. Foot contact timings (F) can be used to identify periods of double and triple support.

View larger version (10K): [in this window] [in a new window]   Fig. 4. Combined summary results for four walking dogs over a range of speeds. Solid outlines relate to the left axis [compression ratio (CR)] and represent the observed fluctuations in potential energy as a proportion of those calculated from stiff-limbed 4-bar linkage geometry. Symbols with grey outlines relate to the right-hand axis [energy recovery (ER)] and represent the effectiveness of `energy interchange' between kinetic and potential energies (1 being perfect). Lines show quadratic fits.

View larger version (25K): [in this window] [in a new window]   Fig. 5. Model collisional energy losses for a step with full ranges of forelimb–hindlimb phases. A ±25% limb phase suggests exactly even timing between hind- and forefoot contacts (H-F-H-F-, represented by vertical grey broken lines). Dotted lines represent energy lost at hind-foot contact; broken lines the energy lost at forefoot contact; solid lines the total energy lost for a step (one hind- and one forefoot contact). Three hypothetical powering strategies are shown: immediate recovery of energy loss (black); energy recovery only directly after forefoot contact (red) (most realistic); energy recovery only after hindfoot contact (blue). (A) Results for a fore-aft symmetrical walking dog with even fore- and hindleg lengths and a centre of mass (COM) mid-way along the body. (B) As for A, but with a realistic bias in leg lengths (foreleg = 0.47 m, hindleg = 0.46 m). (C) As for A, but with a realistic bias in mass position, with the COM towards the shoulders (p=0.6). (D) Results using realistic values of both limb length and mass bias. The underlying grey box represents the observed range of mean phases for walking dogs.

View larger version (8K): [in this window] [in a new window]   Fig. 6. Hypothetical propulsive impulses consistent with collision amelioration. If the mechanism for providing a powering impulse to the centre of mass (COM) also smoothes its path (A), powering is more efficient as collision losses are reduced. This may be achieved by two mechanisms (B): (i) hip torque while the hindleg is in early stance, or (ii) hindleg extension while the hindleg is late in stance. Either, or both, would have the effect of smoothing the path of the COM during forefoot placement, thus reducing the relatively high collision losses at this instant. Both of these mechanisms would be more effective, providing upward and forward impulses, with fore–hind phases of less than 25%.