# Acceleration vector of tipping object (robot)

I am about to develop a simulation application of a robot. For that reason, I would need to calculate the acceleration of the robot (centre of mass) when its centre of mass exceeds its feet (ie it will fall over/tip).

I will assume that it rotates around the edge of one of its feet (the pivot pivot / point of rotation) and that it will not slide, due to friction. I guess its motion (of the centre of mass) at any point in time will be perpendicular to the vector between the pivot point and the centre of mass, and so I have the direction of the vector (or am I wrong here), but how do I calculate the size of the acceleration vector, when I have given:

• the pivot point (location/coordinates)
• the point of centre of mass (location/coordinates)
• the angle of the vector between the two above points relative to horizontal (which can easily be calculated from the above)
• the accumulative mass of the robot/object
• the gravitational force vector g

I plan to model/simulate its motion in two dimensions only, but if it is relatively easy to add the third dimension, then I would like that as well :)

I hope this isn't a duplicate, but everything I could find assumed that the centre of mass was in the middle of the object, and they either calculated the speed or the time of falling, but not the acceleration at a given point in time.

• By "acceleration" do you mean "angular acceleration about the pivot point"? You can get the angular moment (i.e. torque) with simple geometry. The angular acceleration depends on the angular moment of inertia $I$. For that, you need to make an assumption, such as 1) all mass is concentrated at the COG, or 2) half the mass at the pivot point, half reflected through the COG, or 3) something else. – Mike Dunlavey Oct 18 '16 at 20:12