Mathematical Formulation

RR Reference Kinematics

For the planar two-link reference arm,

x = L_1\cos q_1 + L_2\cos(q_1+q_2)

z = z_0 - L_1\sin q_1 - L_2\sin(q_1+q_2).

The geometric Jacobian is

J(q)=
\begin{bmatrix}
-L_1\sin q_1-L_2\sin(q_1+q_2) & -L_2\sin(q_1+q_2)\\
-L_1\cos q_1-L_2\cos(q_1+q_2) & -L_2\cos(q_1+q_2)
\end{bmatrix}.

Rest-to-Rest Quintic Trajectory

With normalized time \tau=t/T,

s(\tau)=10\tau^3-15\tau^4+6\tau^5

q(t)=q_i+(q_f-q_i)s(\tau).

This interpolation enforces zero velocity and acceleration at both boundaries. The full workshop extends the method to piecewise splines across task waypoints.

Inverse Dynamics

The actuator torque structure is

\tau=M(q)\ddot q+C(q,\dot q)\dot q+g(q)+\tau_f-\tau_s.

The workshop evaluates this expression over the payload trajectory and then adds a tensile linear spring:

F_s=k\max(L-L_0,0).

Spring-Design Objective

The design search considers attachment point, fixed anchor, free length, and stiffness. A practical objective penalizes both peak and RMS torque:

\min_\theta
\sum_i
\left(
w_p \max_t |\tau_i(t;\theta)|
+
w_r \sqrt{\frac{1}{T}\int_0^T \tau_i^2(t;\theta)\,dt}
\right).

Experimental Validation

Measured servomotor current is analyzed as a practical proxy for actuator demand. The portable workflow trims inactive periods and reports mean absolute current, RMS current, peak current, and integrated absolute current. These measurements do not uniformly confirm the theoretical torque reduction, so the repository reports them as an experimental comparison rather than a completed validation claim.