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} = \frac{d_{a} v}{d t} = \lim_{Δ t \to 0} \frac{_{a} v (t + Δ t) -_{a} v (t)}{Δ t}

,

and:

_{a} \dot{ω} = \frac{d_{a} ω}{d t} = \lim_{Δ t \to 0} \frac{_{a} ω (t + Δ t) -_{a} ω (t)}{Δ 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} =_{a}^{b} R_{b} v_{q} +_{a} ω_{b} \times_{a}^{b} R_{b} q

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

_{a} {\dot{v}}_{q} = \frac{d}{d t}_{a}^{b} R_{b} v_{q} +_{a} {\dot{ω}}_{b} \times_{a}^{b} R_{b} q +_{a} ω_{b} \times \frac{d}{d t}_{a}^{b} R_{b} q

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

_{a} v_{q} = \frac{d}{d t}_{a}^{b} R_{b} q =_{a}^{b} R_{b} v_{q} +_{a} ω_{b} \times_{a}^{b} R_{b} q

Applying this result to the acceleration leads to:

_{a} {\dot{v}}_{q} =_{a}^{b} R_{b} {\dot{v}}_{q} +_{a} ω_{b} \times_{a}^{b} R_{b} v_{q} +_{a} {\dot{ω}}_{b} \times_{a}^{b} R_{b} q +_{a} ω_{b} \times (^{b}_{a} R_{b} v_{q} +_{a} ω_{b} \times_{a}^{b} R_{b} q)

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, o r g} +_{a}^{b} R_{b} {\dot{v}}_{q} +_{a} ω_{b} \times_{a}^{b} R_{b} v_{q} +_{a} {\dot{ω}}_{b} \times_{a}^{b} R_{b} q +_{a} ω_{b} \times (^{b}_{a} R_{b} v_{q} +_{a} ω_{b} \times_{a}^{b} R_{b} q)

For rotational joints, $_{b} q$ is constant, and the above expression simplifies to:

_{a} {\dot{v}}_{q} =_{a} {\dot{v}}_{b, o r g} +_{a} {\dot{ω}}_{b} \times_{a}^{b} R_{b} q +_{a} ω_{b} \times (_{a} ω_{b} \times_{a}^{b} R_{b} q)

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} ω_{c} =_{a} ω_{b} +_{a}^{b} R_{b} ω_{c}

Differentiating leads to:

_{a} {\dot{ω}}_{c} =_{a} {\dot{ω}}_{b} + \frac{d}{d t}_{a}^{b} R_{b} ω_{c}

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

_{a} {\dot{ω}}_{c} =_{a} {\dot{ω}}_{b} +_{a} ω_{b} \times_{a}^{b} R_{b} ω_{c}

Inertia Tensor

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

_{a} I = (\begin{matrix} I_{x x} & - I_{x y} & - I_{x z} \\ I_{x y} & I_{y y} & - I_{y z} \\ I_{x z} & - I_{y z} & I_{z z} \end{matrix})

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 ${\dot{v}}_{c o m}$ , equals:

F = m {\dot{v}}_{c o m}

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

N =_{c} I \dot{ω} + ω \times_{c} I ω

,

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.

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Robotics Kinematics and Dynamics/Serial Manipulator Dynamics

Acceleration of a Rigid Body

Inertia Tensor

Newton's and Euler's equation

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