Tendon Capstan Friction, Tension Loss and Cable Stretch

Sliding tension and axial compliance

For each ordered section, the friction exponent is $b_j=\mu_j\theta_j$, with angle in radians and all bends counted by magnitude. The cumulative exponent is $b(s)$ and $B=b(L)$. Euler–Eytelwein gives $T(s)=T_{in}e^{-b(s)}$ during pull toward the actuator; reverse sliding gives $T(s)=T_{in}e^{b(s)}$. The high-to-low ratio is $e^B$. Free spans have zero exponent.

Cable stretch is $\int T(s)/(EA_c)\,ds$. Optional sheath compression adds $\int_{sheath}T(s)/(EA_h)\,ds$ to relative endpoint take-up. Each section's exponential integral is evaluated analytically, using its straight-span limit when the exponent vanishes. Cable and housing axial compliances add, as in the measured stiffness example in Agrawal et al. (2010), p. 921.

Define $\chi(s)=1/(EA_c)+I_{sheath}/(EA_h)$ and $J_\pm=\int_0^L\chi(s)e^{\pm b(s)}ds$. Directional output tension stiffnesses are $k_{pull}=e^{-B}/J_-$ and $k_{release}=e^B/J_+$. They are $dT_{out}/d(x_{in}-x_{out})$ on fully sliding branches. The full-slip reversal lost motion at the operating point is $T_{in}(J_- -e^{-2B}J_+)$: the output stays still as the friction direction propagates down the cable.

Distributed reversal calculation

The single-tendon specialization of the elastic/Coulomb model in Agrawal et al. (2010), pp. 916–918, retains a tension value at every spatial station. On a monotonic drive increment, a trial inlet force updates the previous profile by $T_{new}(s)=\max[T_{old}(s),T_{in}e^{-b(s)}]$ on pulling, or the corresponding minimum with $T_{in}e^{b(s)}$ on releasing. Unchanged stations remain stuck; changed stations slide. These envelope updates follow directly from the capstan slope bound and strain memory.

A safeguarded Newton shooting iteration enforces $q=x_{out}+\int\chi(s)[T(s)-T_0]ds$, with $x_{out}=(T_{out}-T_0)/k_{load}$. Tension is never compressive; below the completely relaxed take-up the excess release is slack. Two cycles are solved from uniform pretension, and the second is shown. The cycle lost-motion result integrates the tension redistribution needed to start changing the distal tension after the upper reversal. Closed-cycle actuator work minus return-spring work gives frictional energy loss.

Sheath compression checks

Each enabled, nonzero-length sheath is checked separately. Its mean strain is $\delta_h/L$. The two sliding-limit mean strains are evaluated with the same exact section exponential integrals as the compliance calculation. Uniform pretension and the lower, upper, and final cycle tension profiles are also checked. The largest shortening across these cases is compared with that section's length; local peak strain is additionally bounded by $\max T(s)/(EA_h)$. A mean or local compression of 100% or more rejects the results. A local compression above 2% raises a model-range warning.

The route contains one tendon and a passive, preloaded linear return spring. Initial cable tension is uniform; the first cycle conditions this specified initial state. Contact geometry and coefficients stay fixed. Static and sliding friction share one coefficient, with no velocity dependence. Each sheath section distributes its absolute curvature uniformly over its entered path length.

EA is the effective axial rigidity of the assembled cable. Optional housing compression is linear, locally proportional to tendon tension, and applies only to sections marked as sheaths. The default effective modulus of 97 GPa and representative pretension of 3.2 N follow Wagner and Emmanouil (2018), Table 5, attached source p. 8. The 0.52 mm cable diameter also matches Agrawal's experimental cable. Routing dimensions, guide friction, load stiffness, and cycle stroke are illustrative engineering inputs rather than a material specification.

Full-slip tension, stretch, stiffness, and operating-point lost motion use exact piecewise exponential integrals. Reversal uses 64 intervals per section (257 stations) and 240 phase intervals per cycle. Composite trapezoidal compliance integration slightly rounds a slip front within one spatial cell; work integration resolves one phase interval. Doubling the spatial resolution at the defaults changed cycle energy by less than 0.001% and left the displayed reversal travel unchanged to four decimals. Drive position and inlet tension are linearly interpolated between solved phases; spatial tension is reconstructed from the stored reversal profile and the capstan envelope. The schematic symbols and motion amplitudes are illustrative.

The input range is limited to total friction exponent B ≤ 12 to keep shooting and discretization well conditioned. Accuracy deteriorates for a sharp localized curvature change inside one entered section, extremely small moving regions, or very high tension ratios. Split distinct contact regions between the available ordered sections. Check low-stroke cases against measured breakaway motion.

The two capstan equalities are sliding limits; a stationary tendon can retain intermediate, history-dependent tensions. The reversal result assumes the stated spring boundary, and does not reproduce antagonistic cable coupling. Material creep, relaxation, internal strand hysteresis, transverse cable stiffness, changing routing, inertia, temperature dependence, and freely rotating pulley bearing losses are excluded. A rolling pulley needs measured bearing-torque losses rather than a sliding capstan coefficient. Use effective EA and friction measured under the intended preload and environment; the linear model is flagged above 2% predicted cable strain. This calculation does not assess cable strength, wear, buckling of the sheath, or termination security.

Sheath compression is screened for pretension, forward sliding, reverse sliding, and the cycle extrema, in every enabled sheath section. Shortening equal to or greater than the entered sheath length, or 100% local compressive strain, is rejected. Peak local compression above 2% triggers a warning to increase sheath rigidity or reduce loading. The 2% level is a conservative calculator screening threshold, not a universal material limit from Agrawal et al.; use the actual housing's measured linear load–compression range, which may be narrower.