Table 1.
A list of symbols used in this paper and their meanings.
Fig 1.
The simple quadrupedal model used in this paper.
Two point masses sit on massless legs that can extend and contract. The point masses are connected with a rigid trunk. Five morphological parameters are inputs: the mass of the fore and hindquarters (mF and mH, respectively), the trunk length (lB), and the maximum length of the fore and hindlimbs (lFmax and lHmax, respectively). Two kinematic variables are also used as inputs: the stride length (D) and the average horizontal speed (U). From these seven constant inputs, trajectory optimization determines footfall positions (fijk, where i, j and k refer to Fore-Hind, Right-Left and {Trailing contact}-{Leading contact}, respectively) and over twenty temporal variables, including ground reaction forces and their rate of change (Fijk and ), center of mass position and velocity (x and
) and body pitch angle and angular velocity (θ and
). rF and rH are the displacement vectors from the COM to forequarters or hindquarters, respectively. Other notation conventions are described in Table 1.
Fig 2.
The strategy for determining stepping sequence.
(A-D) The configuration of the hindlimbs are shown at four points in the stride cycle, with right limbs in colour and the left (reference) in black. In this modelling approach, limbs are always “connected” to the body, but are either producing force (solid lines) or not (dashed lines) (A) At t = 0, the right limb acting through trailing contact (red) can produce an axial force FRHT(0) ≥ 0, but the reference (black) and limb acting through leading contact (blue) cannot produce force (FLH = FRHL = 0). (B) At some point in the cycle, the right limb is in swing, with FRHT = FRHL = 0. (C) The force through leading contact exceeds zero and causes FRHT = 0 for the remainder of stride. (D) The right hind limb starts and ends in the same configuration, but one stride length forward, acting through leading contact. (E) Forces shown during the stride. The right limb may start and end the stride in stance, so is modelled with “trailing contact” and “leading contact” forces. The reference limb starts and ends the stride in swing. (F) As soon as FRHL becomes positive, its time integral from t = 0 becomes positive for the rest of the stride (since Fijk(t) ≥ 0∀t). The constraint Eq 8 then ensures that FRHT cannot be nonzero again. The dotted line in E shows a violation of Eq 8 for some portion of stance (yellow line).
Table 2.
Empirical data used for validation and as input (bold) for the model.
Fig 3.
Empirical forelimb GRF (black) compared to model predictions for a range of force-rate penalty coefficients ().
(A) At lower , the solution becomes more impulsive, as expected from a work-minimizing “bang-coast-bang” solution [23]. As
increases, the force peaks become more shallow, but in all cases the optimal solution maintains the double-hump profile characteristic of walking bipeds and quadrupeds under most situations. (B) The actuation is seemingly elastic, as indicated by the negative linear relationship between force and leg length change, for small changes in leg length from resting. However, there are no springs in the model. The “pseudo-elastic” actuation occurs because of the compromise between work and force rate. High force rate penalties lead to only small deviations from pseudo-elastic actuation and low apparent stiffness (k′), and small force rate penalties deviate substantially from pseudo-elasticity with high apparent stiffness. The “hook” in the force-length curve occurs when the leg shifts to stance, modulating the force to ∼0.5 mg. Morphological data for the simulations, and empirical ground reaction forces, are derived from [37] for a Dalmation (see also Fig 5B).
Fig 4.
Optimal duty factors and pair lag vs magnitude of force rate cost coefficient () at
.
As force-rate penalties increase, duty factor also increases. Optimal duty factors in fore and hind remain close for all . Pair lag decreases slowly with force rate penalty. At no point do optimal duty factors and pair lags simultaneously match the empirical values (dotted lines) from [37]. The morphological data is based on a Dalmation from [37] (see also Fig 5B).
Fig 5.
Optimal solutions compared to empirical data for four test cases.
Left column shows an outline of the dog breed in question, with the approximate points used as morphological measurements. Middle column shows that gait diagrams for the empirical (grey) and optimal solutions (light blue) have substantial overlap (dark blue). Right two columns demonstrate that empirical ground reaction forces (GRF, black lines) qualitatively match optimization results. The optimization GRF from left (solid blue line) and right limbs (dashed red line) are very similar, reflecting the symmetrical solutions that are discovered. GRFs from double support are shown as grey lines (solid for empirical, dotted for simulation). At slow speeds (A-B), the simulation discovers a singlefoot walking gait, while at an intermediate speeds (C-D) it discovers trotting, matching natural gait choice in dogs. All optimizations use . Contours adapted with modification from (A) [55], (B) [56], and (C-D) [57]. Empirical data and morphological measurements from [41, 45] (A), [37] (B), [42, 44] (C) and [44, 48] (D). For left column, circles show shoulder and hip positions, with relative size indicating the relative mass carried at these points. “x” marker indicates centre of mass position. For middle column, LH = Left Hindlimb, RH = Right Hind, LF = Left Fore, RF = Right Fore. Error bars in gait diagrams are described in the methods.
Fig 6.
These locally optimal solutions are a sample of the diverse gaits possible with the model.
Each stick figure is a snapshot from a different part of the gait cycle. The first still (leftmost, purple) is at touchdown of the left hindlimb. Each still is separated from the next in time by one tenth of the stride time, culminating at the next left-hind touchdown. The “x” marker represents center of mass location. At both a walking speed of (A-D) and running speed of
(E-H), asymmetric (A,B,E,F) and symmetric gaits (C,D,G,H) are possible. y-axis and x-axis scales are equal.
Fig 7.
Actuation shows an elastic response to leg length change, despite a lack of elasticity in the model.
At both walking speeds (A,B), the apparent stiffness appears similar between fore and hind legs (when normalized to maximum leg length). In trotting, however (C) there is a substantial difference in apparent stiffness between the fore and hind legs. Data are from the same simulations shown in Fig 5.
Fig 8.
Pseudoglobal optimal solutions (markers) are compared to a large dataset for Belgian Malinois dogs.
(A) A slowly decaying duty factor with increasing speed is predicted well by the model for medium to fast speeds, using . At slow speeds, the model remains constant, slightly above impulsive predictions, while the empirical data approaches 1. A swing duration of half the natural pendular period (Tn) of the leg (Eq 31) does not explain the discrepancy. 95% CI for empirical mean duty factor is shown as a thin dotted line. (B) The optimal phase offset matches empirical values well for walking and trotting speeds, and the optimal walk-trot transition is very close to the natural transition speed. However, the model does not transition to a gallop after the natural trot-gallop transition speed (C) Hind Lag (HL) and Fore Lag (FL) is shown against speed. A perfectly symmetrical gait would have HL = FL = 0.5; walking and trotting in dogs is highly symmetrical both naturally [50] and in the model. Galloping is an asymmetrical gait, and past the trot-gallop transition, the lowest-cost gait discovered by the model becomes asymmetrical. Empirical phase offsets are mean values for each gait. All empirical data from [40].
Fig 9.
All local optima found for the model for each speed based on a Belgian Malinois morphology.
Each row corresponds to a given speed condition, and each data point is a feasible local optimum arising from energy optimization from one random guess, with relative cost (CoT—min CoT for a given ) indicated by color. Solutions are plotted in terms of Hind Lag (HL), Pair Lag (PL) and Fore Lag (FL), defined in Fig 5B. As speed increases, solutions become more variable. Pseudo-global optimal solutions for each case are shown in Fig 8. Bottom row: Some recognizable gaits and their positions in the plots. Dashed line: hind symmetry; dotted line: fore symmetry; dot-dash line: simultaneous right-hind and left-front contact (as in a canter). Black square: diagonal singlefoot; white square: lateral singlefoot; black diamond: pace; white diamond: trot; black triangle: rotary gallop; white triangle: transverse gallop; black circle: perfect symmetry.
Fig 10.
Changes in cost of transport with speed.
(A) Median Cost of Transport increases linearly for walking speeds (solid line), but exhibits a sharp change to a slower rate of increase after the walk-trot transition (dashed line), mirroring the response to speed of walking and running observed in human data [77]. The range of the costs of feasible solutions (whiskers) increases up to the walk-trot transition, and then settles into a near-constant range. The distribution of costs is heavily skewed, as indicated by the median value approaching the minimum at all speeds. This indicates that the solvers tended to discover solutions with costs close to the minimal value, but occasionally were “trapped” in local optima with costs far from the minimum. (B) As speed increases, the standard deviation of the distribution of costs of transport gets smaller, relative to the minimal cost, indicating that the variance in costs of local optima is relatively smaller at higher speeds than at lower speeds.