Dynamics for Robotics

Dynamics

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

$$\mathbf{\tau} = \frac{d}{dt}[\mathbf{J}\omega] = \mathbf{J}\ddot{\theta}$$

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. $\begin{align*} ^A\mathbf{J} &= \begin{bmatrix} (y^2+z^2) &-xy &-xz\\ -yx &(x^2 + z^2) &-yz\\ -zx &-zy &(x^2+y^2) \end{bmatrix}\\ &= \begin{bmatrix} \mathbf{J}_{xx} &-\mathbf{J}_{xy} &-\mathbf{J}_{xz}\\ -\mathbf{J}_{yx} &\mathbf{J}_{yy} &-\mathbf{J}_{yz}\\ -\mathbf{J}_{zx} &-\mathbf{J}_{zy} &-\mathbf{J}_{zz} \end{bmatrix} \end{align*}$

The Parallel Axis Theorem

Given an arbitrary choice for frame $A$, the center of mass of the body is defined: $\mathbf{r}_{cm} = \frac{\sum m_i \mathbf{r}_i}{\sum m_i}$

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: $\begin{equation} ^A\mathbf{J} = ^{CM}J + \begin{bmatrix} m(r_y^2 + r_z^2) &-m(r_x r_y) &-m(r_xr_z)\\ -m(r_y r_x) &m(r_x^2+r_z^2) &-m(r_y r_z)\\ -m(r_z r_x) &-m(r_z r_y) &m(r_x^2 + r_y^2) \end{bmatrix} \end{equation}$

Rotating the Inertia Tensor

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

$$\begin{align} \mathbf{J}_1 \mathbf{\omega}_1 &=_1\mathbf{R}_0 (\mathbf{J}_0\mathbf{\omega}_0) \\ &=_1\mathbf{R}_0 \mathbf{J}_0 ({_1\mathbf{R}_0}^T _1\mathbf{R}_0)\mathbf{\omega}_0\\ &= (_1\mathbf{R}_0 \;\mathbf{J}_0 \;{_1\mathbf{R}_0}^T)\mathbf{\omega_1} \end{align}$$

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