# How do I calculate components of an angular rate on end effector frame?

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?

You can calculate that with the Angular Jacobian ($$J_\omega$$) of the gimbal (see details here).
$${\ J_\omega= \left[ {\begin{array}{cc} 0 & 0 \\ -\sin(\theta_1) & \sin(\theta_1) \\ \cos(\theta_1) & -\cos(\theta_1) \\ \end{array} } \right] \ }$$
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 $$[\dot{\theta}_1, \dot{\theta}_2]^{T}$$.