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Stability Analysis of the Creep Gait
|> Introduction
|> Creep Step-Sequences

<| Introduction

Theoretically, the Creep Gait of the quadruped should have the best stability of all the gaits, since only 1 leg at a time is ever lifted to take a step, and the other 3 legs are all on the ground during this period. However, we wondered whether the creep could be implemented in different ways, with different step-sequencing, etc. Off-hand, with 4 legs taken one at a time for a step, there should be up to 24 ways to implement the Creep.

[cat on a fence] However, as indicated below, we found that the 24 possibles reduce immediately to only 6 different and unique step-sequences, and of these, only one step-sequence provides good stability at all phases of the gait - and this happens to be the same one that real animals use. As a result of evolution, nature discovered the one and only way to implement a stable working Creep.

Domestic cats are well-known for doing the Creep (as shown in the picture at the right), rather than just doing a regular alternating-diagonal walk. In our local neighborhood, they seem to be either slinking along with a crouch, or else running with a bound. This behavior is probably a genetic carry-over from that of lions and other predatory cats in the wild. We have also observed deer doing a similar gait, moving just one-leg-at-a-time, when grazing in the grass.

The picture of the cat illustrates the chief criterion necessary for having a stable slow-speed walk - namely that, when a leg is lifted, the other 3 legs need to form a stable tripod on the ground. Furthermore, this situation needs to obtain during all phases of the gait. The sections below deal with the issue of whether this can be realized in different ways.

We don't know if any other animals use the Creep, but it is a potentially useful gait for a quadruped walking robot, since robots typically don't have the leg-joint complexity, nor the wide array of sensors for touch, proprioception, balance, etc, that animals are endowed with. Trots and various walking gaits in quadrupeds involve dynamic stability situations, with 2 legs raised and 2 legs grounded, and from our experience, if a quadruped robot is not well-balanced, then there can be a lot of wobble during walking, and it's easy for it to flip over. A good stable gait, like the Creep, is a useful complement to other faster gaits, especially for situations where a robot might be negotiating rough or uneven terrain - or carrying or pulling a load.

<| Creep Step-Sequences

Shown below are the different unique possibilities for step sequencing in the Creep Gait. For a situation of 4 legs with only 1 taking a step at a time, there are potentially 24 different step-sequences, but as it turns out, only 6 of these 24 are unique. For each of these 6, there are 3 others that reduce to the same ultimate step-sequence as the gait is repeated over and over, with the only difference being which of the 4 legs took the first step (exercise/proof is left for the reader). Furthermore, of these 6 unique sequences, as we show below, only one of these is potentially stable for all positions of the legs. In the other cases, the stability is marginal at some positions, and/or the body/frame will pitch over either forwards or backwards or along a diagonal.

For a stable tripod position, the COG (center-of-gravity) of the body or frame must lie within the triangle drawn through the 3 grounded feet. See stability-triangle. Imbalance and tipping will occur if the COG is outside the stability-triangle, and stability will be marginal when the COG lies along, or close to, one of the edges of the stability-triangle. A slight imbalance in the load distribution, or uneven terrain, will likely cause the robot to tip.

Besides purely geometrical considerations, proper balance also involves having loads distributed evenly around the frame of the robot. An unbalanced load will always be a problem. Animals tend to be naturally symmetrical with relatively well-balanced proportions. Their bodies are mirror-imaged around the front-to-back midline axis, their weight tends to be fairly well apportioned front-to-back, and they use back-and-forth bobbing of their heads to trim the balance during locomotion. Many have a tail to help counter-balance the head, etc.

Stability Diagrams. In the diagrams shown here, the LF leg always takes the first step, followed by different sequencing of the 3 other legs. This gives a total of 6 different unique step-sequences. As mentioned above, 18 of the 24 total possible step-sequences are repeats of the 6 shown here, with one of the other legs taking the first step. The cartoon figures shown underneath the timing sequences represent the positions of the 4 legs for the time-quadrant directly above - these are defined as positions 1-4 for quadrants 1-4. The specific step-sequence is shown at the top of each diagram, eg LF-LR-RF-RR.

[sequence 1] [sequence 2]
In step sequence #1, all 4 positions are only marginally-stable. Assuming the loads are balanced evenly around the frame, the COG in every case falls along one of the diagonal-edges of the stability-triangle connecting the 3 grounded feet, when the 4th leg is lifted for its step. The frame will likely pitch over diagonally onto the lifted leg in every case, especially if there is any load-imbalance or unevenness in the terrain.

In step sequence #2, all 4 positions are also at best marginally-stable. The first 2 are similar to those in sequence #1 with pitch-over diagonally, while in the latter two cases, the frame will likely pitch over either backwards or forwards. Note in position 3, for instance, that the LR leg is pointed forwards when the RR leg takes its step, so the frame will fall onto its rear-end. For position 4, it will fall over on its front-end.

[sequence 3] [sequence 4]
For step sequences #3 and #4, all positions are similar to situations in sequences #1 and #2, and are only marginally-stable. The frame will pitch over either along a diagonal, or else onto the front-end or rear-end.
[sequence 5] [sequence 6]
Step sequence #5 is also bad, and just a variation on the 4 preceding.

Step sequence #6 is the only one of the 6 possibilities that actually has a good chance of being stable at all 4 leg positions, in all 4 time-quadrants. Positions 1 and 3 are similar to the other cases where the frame would pitch onto its front-end or rear-end, except that, here, the leg opposite to the lifted leg is directly vertical under the frame, so it can support the weight without pitching over. Eg, for position 1, the LR supports while the RR is stepping. In the previous step-sequences, the foot of the LR leg was farther forward, and couldn't support the rear-end as well.

Note that position 1 of sequence #6 is the same as in the picture of the cat on the fence, above right. The directly-opposite leg to the lifted-leg is squarely under the animal, and the other same-side leg to the lifted-leg is back towards the midline, forming a nice tripod. What makes position 1 here stable, while position 2 of sequence #5 is not stable, for instance, is that the same-side leg to the lifted-leg is positioned further back towards the midline here, which moves the one side of the stability-triangle rearward, to a position behind the COG of the frame.

Likewise, positions 2 and 4 of sequence #6 gain stability from pitching over diagonally onto the lifted-leg. For instance, position 2 works because the other leg (RR) on the same side as the lifted-leg (RF) is moved forward to a position near the midline, and the opposite leg (LF) to the lifted leg (RF) is vertical and directly supports the front-end of the frame. Again, a stable tripod is formed from the 3 grounded legs. A similar case holds for position 4, which is simply a mirror-image of position 2.

Summary. It is interesting that, of the 6 possible unique step-sequences, essentially all positions in all sequences are unstable, except for sequence #6, where all positions are relatively stable. Funny thing about nature and geometry.

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© Oricom Technologies, Dec 2004