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I’ve helped develop the forward kinematic chain for a two-axis gimbal. The Denavit-Hartenberg parameter table for the kinematic chain is given below.

$\theta$ $\alpha$ $r$ $d$
$90^{\circ}$ $90^{\circ}$ $0$ $0$
$180^{\circ} + θ_{1}$ $90^{\circ}$ $0$ $0$
$180^{\circ} + θ_{2}$ $180^{\circ}$ $0$ $0$

If a known angular rate was applied to a particular joint in the kinematic chain, is there a way of determining the components of that angular rate which would be observed on the end effector frame? For example, if $\theta_{1}$ is rotated at $10$ deg/sec and $\theta_{2}$ equals $270^{\circ}$, what would be the rates observed on the x, y and z axes of the end effector?

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You can calculate that with the Angular Jacobian ($J_\omega$) of the gimbal (see details here).

Based on your D-H, your angular Jacobian should look like:

$$ {\ J_\omega= \left[ {\begin{array}{cc} 0 & 0 \\ -\sin(\theta_1) & \sin(\theta_1) \\ \cos(\theta_1) & -\cos(\theta_1) \\ \end{array} } \right] \ }$$

So to obtain the angular speeds of the end efector ${\omega_x, \omega_y, \omega_z}$ as seen from a grounded frame, you multiply $J_\omega$ by the vector of joint angular speeds $[\dot{\theta}_1, \dot{\theta}_2]^{T}$.

I ran quick calculations, so you need to verify if the Jacobian is correct. But I hope you got the idea.

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