Robotics Kinematics and Dynamics/Serial Manipulator Dynamics

Acceleration of a Rigid Body

The linear and angular accelerations are the time derivatives of the linear and angular velocity vectors at any instant:

$_a \dot{v} = \dfrac{d \, _a v}{dt} = \lim_{\Delta t \rightarrow 0} \dfrac{_a v(t + \Delta t) - \,_a v(t)}{\Delta t}$,

and:

$_a \dot{\omega} = \dfrac{d \, _a \omega}{dt} = \lim_{\Delta t \rightarrow 0} \dfrac{_a \omega(t + \Delta t) - \,_a \omega(t)}{\Delta t}$

The linear velocity, as seen from a reference frame $\{a\}$, of a vector $q$, relative to frame $\{b\}$ of which the origin coincides with $\{a\}$, is given by:

$_a v_q = \, ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q$

Differentiating the above expression gives the acceleration of the vector $q$:

$_a \dot{v}_q = \dfrac{d}{dt} \, ^b _a R \, _b v_q + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \dfrac{d}{dt} \, ^b _a R \, _b q$

The equation for the linear velocity may also be written as:

$_a v_q = \dfrac{d}{dt} \, ^b _a R \, _b q = \, ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q$

Applying this result to the acceleration leads to:

$_a \dot{v}_q = ^b _a R \, _b \dot{v}_q + \, _a \omega_b \times \, ^b _a R \, _b v_q + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \left( ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q \right)$

In the case the origins of $\{a\}$ and $\{b\}$ do not coincide, a term for the linear acceleration of $\{b\}$, with respect to $\{a\}$, is added:

$_a \dot{v}_q = \, _a \dot{v}_{b,org} + \, ^b _a R \, _b \dot{v}_q + \, _a \omega_b \times \, ^b _a R \, _b v_q + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \left( ^b _a R \, _b v_q + \, _a \omega_b \times \, ^b _a R \, _b q \right)$

For rotational joints, $_b q$ is constant, and the above expression simplifies to:

$_a \dot{v}_q = \, _a \dot{v}_{b,org} + \, _a \dot{\omega}_b \times \, _a ^b R \, _b q + \, _a \omega_b \times \left( _a \omega_b \times \, ^b _a R \, _b q \right)$

The angular velocity of a frame $\{c\}$, rotating relative to a frame $\{b\}$, which in itself is rotating relative to the reference frame $\{a\}$, with respect to $\{a\}$, is given by:

$_a \omega_c = \, _a \omega_b + \, _a ^b R \, _b \omega_c$

Differentiating leads to:

$_a \dot{\omega}_c = \, _a \dot{\omega}_b + \dfrac{d}{dt} \, _a ^b R \, _b \omega_c$

Replacing the last term with one of the expressions derived earlier:

$_a \dot{\omega}_c = \, _a \dot{\omega}_b + \, _a \omega_b \times \, _a ^b R \, _b \omega_c$

Inertia Tensor

The inertia tensor can be thought of as a generalization of the scalar moment of inertia:

$_a I = \begin{pmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ I_{xy} & I_{zz} & -I_{yz} \\ I_{xz} & -I_{yz} & I_{zz} \\ \end{pmatrix}$

Newton's and Euler's equation

The force $F$, acting at the center of mass of a rigid body with total mass$m$, causing an acceleration $\dot{v}_{com}$, equals:

$F = m \dot{v}_{com}$

In a similar way, the moment $N$, causing an angular acceleration $\dot{\omega}$, is given by:

$N = \, _c I \dot{\omega} + \omega \, _c I \omega$,

where $_c I$ is the inertia tensor, expressed in a frame $\{c\}$ of which the origin coincides with the center of mass of the rigid body.