# Estimating terrain slope

Last time I discussed the challenges when operating the mjbots quad A1 on sloped surfaces. While there are a number of possible means of tackling this, the approach I’ve gone with for now is to estimate the slope of the terrain under the robot, and use that to determine how to position the center of mass. Here’ll I’ll cover the estimation part of this solution.

On paper, the quad A1 has plenty of information to estimate the terrain under its feet. Between the IMU with attitude estimator, the proprioceptive feedback from the joints, and the ability to move the feet around, it would be obvious to a human whether the ground under them was sloped or level. The challenge here is to devise an algorithm to do so, despite the noise in the IMU, the fact that the feet are not always on the ground, and that as the robot moves, the terrain under it changes.

## Approach

My basic approach can be summarized in the follow flow chart / block diagram:

First, a brief description of the 3 pertinent reference frames:

• B Frame: The body frame (or B frame), is centered on the robot body, and rigidly fixed to the robot body. The proprioceptive system eventually calculates each of the 4 feet positions in this frame.
• M Frame: The CoM frame (or M frame), is centered at the robot’s idle center of mass and oriented such that positive Z points along gravity toward the ground with a heading that is arbitrary at start up, but that tracks the robot’s changing heading.
• T Frame: The terrain frame (or T frame), is referenced to the M frame at the average height of the legs with a slope that aligns with the average slope of the terrain under the robot.

The algorithm works in roughly the following steps:

1. First, project all the feet positions into the M frame.
2. For any in-flight legs, reset the Z value to one calculated from the current TM transform and a 0 T frame Z height.
3. Fit a plane to these “on-ground” M frame points.
4. Update the slope of the T frame using this plane with an exponential filter along the X and Y axes.

## Properties

This algorithm has the benefit that it will converge on the terrain underneath the robot as long as feet touch the ground with regularity, which is a somewhat necessary condition for a robot supported by its legs. The rate at which the estimate converges can be controlled by the filter constant. Selecting that to be the same order as the step frequency does a decent job of rejecting spurious noise while responding in a timely manner to updated terrain.

Next up we’ll see how to use this information to balance, and watch the results in simulation.