Kinematics & Workspace

Notation and Units

Symbol	Meaning	Unit
x, y, z	Cartesian position	mm
L	Effective link length	mm
a	Base offset	mm
theta1, theta2, theta3	Joint angles	deg (docs), rad (computation)

Mechanical Model

The physical arm uses shoulder/elbow 4-bar linkages. For kinematic computation, firmware uses an equivalent 3-DOF model:

• Base rotation joint: theta1

• Shoulder-equivalent joint: theta2

• Elbow-equivalent joint: theta3

• Model constants: L = 140.0 mm, a = 54.0 mm

Home Pose and Axis Frame

• theta1 = 0 deg (Robot facing middle/front)

• theta2 = 90 deg (Lower shank perpendicular to ground plane)

• theta3 = 0 deg (Upper shank parallel to ground plane)

Positive/Negative Rotation Directions

Base axis (theta1) Base rotation sign follows the base/top-view convention.

Shoulder and elbow (theta2, theta3) In side view (xy plane), positive theta2 and theta3 follow the arrow direction:

Joint Limits

Joint	Range
theta1	[-90, 90] deg
theta2	[0, 130] deg
theta3	[-17, 120] deg

Forward Kinematics

Let R denote the radial projection in the xz plane:

\[R = L\cos\theta_2 + L\cos\theta_3 + a\]

Then:

\[\begin{split}x = R\cos\theta_1 \\ y = L\sin\theta_2 - L\sin\theta_3 \\ z = R\sin\theta_1\end{split}\]

Inverse Kinematics

1. Solve theta1 from x and z

   \[R = \sqrt{x^2 + z^2}, \quad \theta_1 = \mathrm{atan2}(z, x)\]

2. Reduce to side-plane problem (theta2, theta3)

   Define:

   \[K_1 = \frac{R-a}{L}, \quad K_2 = \frac{y}{L}\]

   Reformulate as:

   \[A\cos\theta_3 + B\sin\theta_3 = C\]

   with \(A = -2K_1, \quad B = 2K_2, \quad C = -(K_1^2 + K_2^2)\)

   Then compute:

   \[\alpha = \mathrm{atan2}(B, A), \quad \phi = \arccos\left(\frac{C}{\sqrt{A^2 + B^2}}\right)\]
   \[\theta_{3,1} = \alpha + \phi, \quad \theta_{3,2} = \alpha - \phi\]

3. Recover theta2

   For each candidate value of theta3:

   \[\begin{split}\cos\theta_2 = K_1 - \cos\theta_3 \\ \sin\theta_2 = K_2 + \sin\theta_3 \\ \theta_2 = \mathrm{atan2}(\sin\theta_2, \cos\theta_2)\end{split}\]

Workspace Limits

Axis-aligned bounds: X: [0, 320], Z: [-320, 320]

Y bounds from extreme angles:

\[\begin{split}Y_{max} = L - L\sin(\theta_{3,min}) \\ Y_{min} = L\sin(\theta_{2,min}) - L\sin(\theta_{3,max})\end{split}\]

Radial reach bounds in xz projection:

\[\begin{split}R_{min} = L\cos(\theta_{2,max}) + L\cos(\theta_{3,min}) + a \\ R_{max} = 320\end{split}\]