Two-Link Robot Arm Calculator: Inverse Kinematics & Torque

The kinematic convention follows Ben-Ari and Mondada, Elements of Robotics, §16.1–16.2, printed pp. 269–273 (attached PDF pp. 273–277). The shoulder is at the origin; +x is right, +y is up, and both joint coordinates are positive counterclockwise. Link 2 has absolute orientation $q_1+q_2$.

For inverse kinematics, the cosine law gives $C_2=(x^2+y^2-L_1^2-L_2^2)/(2L_1L_2)$. Directly using $\arccos C_2$ and $L_1+L_2\cos q_2$ loses significant digits near the equal-link folded origin. The implementation therefore uses the algebraically equivalent, factored half-angle construction derived here from the same triangle:
$h_f=\sqrt{(r_{\max}-r)(r_{\max}+r)},\qquad h_o=\sqrt{(r-r_{\min})(r+r_{\min})},$
$q_2=2\operatorname{atan2}(h_f,h_o),\qquad \delta=\operatorname{atan2}\!\left(h_fh_o,r^2+(L_1-L_2)r_{\max}\right),$
$q_{1,\mathrm{down}}=\operatorname{atan2}(y,x)-\delta,\qquad q_{1,\mathrm{up}}=\operatorname{atan2}(y,x)+\delta.$
The up-branch elbow angle is $-q_2$. Down is the positive-$q_2$ branch. The two square roots in each factor are evaluated separately; absolute values keep roundoff or rejected out-of-workspace inputs from generating complex values. The independent radius condition still rejects unreachable targets. $C_2$ remains an explanatory quantity, not the angle solver.

Shoulder angles can differ by a full turn without changing the pose. Only the exact equal-link folded origin uses $q_1=0$ as an arbitrary representative; nearby nonzero targets do not inherit that convention. Nonzero inverse targets nearer than $10^{-10}$ of total reach are rejected because degree-angle representation would otherwise limit relative precision. The radius condition is $|L_1-L_2|\le r\le L_1+L_2$, with a relative $10^{-12}$ allowance for roundoff at workspace boundaries.

Static torques are derived here from moment balance / potential-energy differentiation, consistent with Mohammadijoo, Gravity Compensation, §9 and Problem 5. With $c_i=f_iL_i$:

These are actuator holding torques, opposite to gravity's generalized torques. Unrestricted-workspace maxima are $A+B$ and $B$. At the selected elbow angle, the shoulder envelope is $\sqrt{A^2+B^2+2AB\cos q_2}$. The Jacobian determinant for angles in radians is $L_1L_2\sin q_2$; the normalized singularity indicator is $|\sin q_2|$.

Two rigid, mass-bearing links with ideal revolute joints in a vertical plane. Gravity acts along −y. Each center of mass lies on its link centerline between its joints; 0.5 represents a uniform link. The tool and payload form a point mass at the end of link 2. Link masses may include attached hardware if the combined mass and center of mass are represented correctly. There are no joint limits or obstacles. Both joint torques are measured at the joint output, with no transmission model. Animation is a geometric sweep of static equilibria, not a dynamic simulation. The moment equations and workspace bounds are derived here; no Panda parameters are substituted into a planar model.

Use for planar reach studies and gravity-only joint loading, with positive link lengths, nonnegative masses and gravity, and center-of-mass fractions between 0 and 1. The warning within about 1.7° of a collinear pose uses $|\sin q_2|<0.03$ as an educational indicator, not a robot controller limit. Unreachable targets are rejected rather than projected onto the workspace. At the exact equal-link folded origin the inverse shoulder angle is not unique. A nonzero inverse target must be at least 10⁻¹⁰ of total reach from the shoulder; smaller distances are rejected with a message instead of silently using the origin convention.

The displayed maximum torques cover unrestricted geometry and need not occur simultaneously with a specified target. Signed torques may cancel at particular poses. Friction, acceleration, Coriolis effects, motor/gear inertia, efficiency, backlash, compliance, tool offsets, cable forces, contact forces, thermal duty and safety margins are outside this model. Motor selection requires these additional loads and the manufacturer's continuous torque limits.

Gaz et al. (2019), DOI 10.1109/LRA.2019.2931248, identifies the full seven-joint Franka Panda model (attached PDF pp. 2–3 and 6–7); it is provided as further reading for a future dynamics calculation. This tool does not predict Panda dynamics. Diagrams keep SI annotations; numeric inputs, result cards, charts and tables use the catalog unit controls.