how force act on bodies to create accelerations

  1. the moment of inertia and how it depends on the geometry of motion
  2. computed-torque equation

Newton’s Laws

  • 1st Law: A particle remains in a state of constant rectilinear motion unless acted on by an external force
  • 2nd Law: The time-rate-of-change in the momentum(mv) of the particle is proportional to the externally applied forces, $\mathbf{f} = \frac{d}{dt}(m\mathbf{v})$
  • 3rd Law: Any force imposed on body $A$ by body $B$ is reciprocated by an equal and opposite reaction force on body $B$ by body $A$

Euler’s Equation written in the form of Newton’s 2nd Law

Construction of the inertia tensor

A inertia tensor is derived by summing the contribution of each lamina to the total moment of inertia of the body.

The Parallel Axis Theorem

Given an arbitrary choice for frame $A$, the center of mass of the body is defined:

For a body with a known inertia tensor at the center of mass $^{CM}\mathbf{I}$, the parallel axis theorem states that the inertia tensor at any other parallel frame is computed:

Rotating the Inertia Tensor

The angular momentum vector $\mathbf{J\omega}$ of a rotating body is conserved during the rotation of coordinates, therefore

Computed Torque Equation

[ \tau = M(q)\ddot{q} + V(q,\dot{q}) + G(q) + F ]

For a $n$ degree of freedom robot:

  • $q$: $n\times 1$ vector the configuration variables
  • $\tau$: $n\times1$ vector of forces and/or torques
  • $M(q)$: $n\times n$ configuration dependent generalized inertia matrix, it is positive definite and symmetric and is therefore, always invertible
  • $V(q,\dot{q})$: $n\times 1$ vector depends on both positions $q$ and velocities $\dot{q}$ which represents centripetal and Coriolis forces.
  • $G(q)$: $n\times 1$ vector of gravitational forces and/or torques.
  • $F$: the forces and torques in the mechanism from these types of “external” forces


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