Kinematics: position, velocity, acceleration

What is it?

Formulas

$$v(t) = \dot x(t), \qquad a(t) = \ddot x(t), \qquad x(t) = x_0 + \int_0^t v(s)\,\dd s$$

$$x(t) = x_0 + (x_1 - x_0)\big(10\tau^3 - 15\tau^4 + 6\tau^5\big), \quad \tau = t/T$$

minimum-jerk point-to-point motion (zero velocity and acceleration at both ends)

Why does it matter?

The mathematics behind it

• Derivative★★★★★fundamental

  Velocity is the derivative of position, acceleration the derivative of velocity.

• Higher-order derivatives★★★★★fundamental

  Acceleration is the second derivative of position; Newton's law $F = ma$ is about it.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Classical mechanics
• Robot Jacobian (velocity kinematics)
• Robot dynamics