I’m working to improve the walking gait of the mjbots quad A1. In this iteration, I wanted to tackle an incremental step towards a more fully dynamic gait, but one that will still greatly increase the capability of the machine. As mentioned last time, the current walking gait cycles between all four legs, and then alternative opposing corner legs in order to move laterally. I’d like to keep that same basic structure, but be a bit smarter about what happens during the swing phase.

## Setup

First, the obvious: When the robot has legs on the ground, they support it. If all four legs are on the ground, they form a support polygon. If the center of mass of the robot is within this polygon when projected onto the ground, the robot will happily sit there forever without tipping over.

During the flight phase of the gait, only two legs are in contact with the ground. Now, we no longer have a support polygon, but a support line. The center of mass of the robot is either on one side of it or on the other. Well, I suppose it could be exactly over, but then the slightest breeze would push it to one side or the other. Whichever side the robot is on, gravity will exert a torque and cause it to continue to tip over in that direction.

The current gait sequencer is oblivious to this fact. It will happily pick up the legs when the robot is far away from the support polygon, and then move the center of mass even further away. Needless to say, this results in the robot tipping over rather rapidly.

## The “Idea”

Clearly the robot can’t move while keeping itself exactly over the support line, but can we arrange the gait such that it spends equal amounts of time on both sides (to be more precise, the integral of the distance from the support line over the time of the swing cycle is 0)? If we could, that would result in the minimum possible angular rotation during the swing phase, which would hopefully make things be a lot better without a whole lot more work.

In this post, I’ll consider the problem in 1 dimension, along a line as a simplification. Given that the robot can only move in one direction at a time, we should be able to generalize the 1 dimensional case to the 2 dimensional case later.

For a constant velocity and a fixed swing time, the answer is pretty clearly yes. If we start the swing phase right when the center of mass is 1/2 of the swing time from the opposing support, then the swing will be completed when it is on the other side spending exactly the same amount of time on each side of the support line.

In fact, this also holds for a more general case, as long as we don’t attempt to accelerate during the swing phase itself. All that is necessary is to start the swing when the center of mass is half a swing times away from the support point at the current velocity and not accelerate until the foot is down again.

Granted, in addition to the one dimensional simplification, in this model the feet are allowed to be infinitely far away from the center of mass. However, if we just apply a maximum time spent with all legs in stance, that places a limit on how far the legs can get away with only a slight decrease in performance.

The model also breaks down when the deceleration causes the robot to change directions. A simple heuristic is to change the leg order in that case.

## Summary

Well, that looks pretty good in 1 dimension. Next up I’ll extend it to 2 dimensions, and try it out on the robot.