Forces

After kinematics, students can place a hand in space and still have no idea whether the hand can do anything there. Forces are the reason motors exist. A free-body diagram is the whole method: isolate a body, draw every contact and gravity and actuation, then refuse to hide a mystery arrow “so that it works.” If the diagram does not close, the robot will not either.

In the plane we balance \(\sum F_x\), \(\sum F_y\), and \(\sum M\). In space those become a wrench: a force and a moment packed together, transforming between frames the same way twists do. That packing is not pretension. It lets you ask whether a set of contact forces can resist an external wrench — the door into grasping.

Friction is where tidy statics meets the floor. Coulomb friction says a contact can resist shear only up to \(\mu N\). The set of allowed forces is a cone, not a line. If you assume “the wheel will not slip” you have smuggled in an unbounded lateral force. Sometimes that is a modeling choice. Sometimes it is how a demo leaves the table. Draw the cone. If the required force sits outside it, change the plan, the material, or the normal load — do not raise a PID gain.

Quasi-statics ignores acceleration: useful when motions are slow and you want to talk about holding, pressing, and sliding. Dynamics returns when you care about impact, swing-up, or a vehicle that leans into a turn. This course can stay quasi-static for grasping and still be honest, as long as you say the word “slow” out loud.

A good lab: hang a known mass from a gripper or press a scale with a robot fingertip. Predict the motor current (or the commanded torque) from the Jacobian transpose, \(\tau = J^\top F\). The residual is friction in the gearbox, a bad lever arm, or a student who used the wrong frame. That argument is the spine of the next two lessons.