Kinematics

The kinematics of the hexapod are explained here based on the hexapod_kin_calc.slx in ts_mt_hexRot_simulink. The inverse kinematics is used to transform the hexapod positions \((x, y, z, r_{x}, r_{y}, r_{z})\) to 6 delta strut lengths from the origin. The forward kinematics is to transform the 6 delta strut lengths to the hexapod positions.

Inverse Kinematics

As a typical hexapod with 6 struts, define their positions on the base are:

\[\vec{p}_{\text{base}} = (\vec{p}_{\text{base, 1}}, \vec{p}_{\text{base, 2}}, \cdots, \vec{p}_{\text{base, 6}})\]

, where \(\vec{p}_{\text{base, n}} = (x_{n}, y_{n}, z_{n})\) is the position of each strut \(n\). Define their positions on the mirror to be:

\[\vec{p}_{\text{mirror}} = (\vec{p}_{\text{mirror, 1}}, \vec{p}_{\text{mirror, 2}}, \cdots, \vec{p}_{\text{mirror, 6}})\]

Since the hexapod can do the translation \((x, y, z)\) and rotation \((r_{x}, r_{y}, r_{z})\), the related rotation center is defined to be the pivot, \(\vec{p}_{\text{v}} = (p_{\text{x}}, p_{\text{y}}, p_{\text{z}})\).

If the hexapod moves the position \((x, y, z)\) from the origin, the related strut change from the pivot is:

\[\Delta \vec{d}_{\text{trans}} = (x+p_{x}, y+p_{y}, z+p_{z})\]

For the rotation \((r_{x}, r_{y}, r_{z})\), define the rotationr matrix to be: \(R_{xyz} = R_{x}(r_{x})R_{y}(r_{y})R_{z}(r_{z})\). The related strut change is:

\[\Delta \vec{d}_{\text{rot}} = R_{xyz} (\vec{p}_{\text{base}} - \vec{p}_{\text{v}})\]

The strut change in the translation and rotation directions from the mirror is:

\[\begin{split}\begin{aligned} \Delta \vec{d} &= \Delta \vec{d}_{\text{trans}} + \Delta \vec{d}_{\text{rot}} - \vec{p}_{\text{mirror}} \\ &= \begin{pmatrix} \Delta d_{1, x}, \Delta d_{2, x}, \cdots, \Delta d_{6, x} \\ \Delta d_{1, y}, \Delta d_{2, y}, \cdots, \Delta d_{6, y} \\ \Delta d_{1, z}, \Delta d_{2, z}, \cdots, \Delta d_{6, z} \end{pmatrix} \end{aligned}\end{split}\]

Consider the strut change for each x, y, and z component, the total strut change is: \(\vec{s}=(s_{1}, s_{2}, \cdots, s_{6})\), where each component \(n\) is:

\[s_{n}=\sqrt{(\Delta d_{n,x}^{2} + \Delta d_{n,y}^{2} + \Delta d_{n,z}^{2})}\]

The norminal stuct change between the base and mirror is:

\[\vec{s}_{\text{norm}} = \|\vec{p}_{\text{base}} - \vec{p}_{\text{mirror}}\| = (\|s_{\text{norm, 1}}\|, \|s_{\text{norm, 2}}\|, \cdots, \|s_{\text{norm, 6}}\|)\]

Finally, the delta strut change is:

\[\Delta\vec{s} = \vec{s} - \vec{s}_{\text{norm}}\]

See the full calculation in SimpleHexapod.inverse_kinematics() of simple_hexapod.py.

Forward Kinematics

To calculate the hexapod positions, the Nelder-Mead method is used to minimize the difference between the estimated and current delta strut lengths with the initial guess of current hexapod positions. See the full calculation in SimpleHexapod.forward_kinematics() of simple_hexapod.py.

Hexapod Types

There are two hexapods on the main telescope: camera hexapod and M2 hexapod. See the following figures for these two hexapods:

Figure 1 Coordinate system and actuator locations/numbering for camera hexapod.

Figure 2 M2 hexapod coordinate system and actuator locations/numbering for the M2 hexapod.

The related base and mirror positions are recorded in the MockMTHexapodController of mock_controller.py as a reference.

You can see their mechanical structures are different. For the camera hexapod, 6 struts move together to do the z-axis movement. However, for the same movement, the M2 hexapod only uses the struts 1-3. In addition, for the x-axis movement, the M2 hexapod uses the struts 4-6. The strut 4 contributes most of this moving. For the y-axis movement, the M2 hexapod uses the struts 5 and 6 only, and the strut 4 is not used.