3 · Analysis · Sec. V

What the tendons can transmit, and why the tip error does not grow

Section V characterizes the deformation that null-space motion can transmit through the tendons under an EE constraint, and the convergence of the nominal controller on the constraint manifold. The workspace evaluates both on the simulated plant, in three steps.

Step 1 · Deformation available under the EE constraint (Sec. V-A)

To separate the deformation transmitted by the tendons from the transport, the tendon-dependent part of the Jacobian is restricted to the null space of the EE constraint:

Definition 1 (Constrained deformation Jacobian). \Jdef:=\JDt C_\tau\Jlq N_c (20) maps the null-space component of a joint velocity to the twist of \{D\} relative to \{B\} produced by the tendons.

By (7), (9), and (17), the constrained tip Jacobian separates into transport at fixed shape and tendon-transmitted deformation:

A=S\Jtr N_c+S\Rbar\Jdef

Proposition 1 (Local bound on the deformation rank). With m_a taut, unsaturated tendons in a fixed active set,

\begin{aligned}&\rank\Jdef\\&\ \le\min\{\rank\JDt,\,m_a,\,\rank(\Jlq N_c)\}\\&\ \le\min\{\rank\JDt,\,m_a,\,\rank\Jlq,\,n-r_c\}\end{aligned}

The mechanical factors are the rank of the tension-to-twist map and the number of taut, unsaturated tendons; the kinematic factors are the span sensitivity \Jlq, which depends on the anchor placement, and the null-space dimension n-r_c. The smallest factor limits the rank.

Corollary 1 (Transmission-null configuration). If \Jlq N_c=0, then \Jdef=0, so null-space motion produces no tendon-transmitted deformation there. An anchor on the EE moves with the entry point and gives a zero row of \Jlq.

The bound is local: it holds at a configuration and does not establish global controllability (Remark 1).

Step 2 · Convergence of the nominal controller (Sec. V-B)

The analysis covers the control laws in which \nuv is computed with a spectrally filtered transpose of A, where AA^{\T}=U\diag(\lambda_1,\ldots,\lambda_{n_t})U^{\T}:

\nuv=A^{\T}f(AA^{\T})W\vres,\qquad f(AA^{\T})=U\diag\big(f(\lambda_1),\ldots,f(\lambda_{n_t})\big)U^{\T}

The damped inverse (18) is f(\lambda)=(\lambda+\xi^2)^{-1}, and f(\lambda)=1 gives the Jacobian-transpose control law.

Assumption 1 (Nominal on-manifold operation). (i) Fixed target; (ii) exact tip Jacobian; (iii) fixed tendon active set; (iv) no active safety constraint; (v) \ec=0.

Then \qdc=0, and with J_tN_cA^{\T}=AA^{\T} the tip error evolves as

\qd=N_cA^{\T}f(AA^{\T})W(k_t\e),\qquad \dot{\e}=-k_tAA^{\T}f(AA^{\T})W\e.

Proposition 2 (Output-error descent). For V(\e)=\tfrac12\e^{\T}W\e, W\succ0 (27), and a positive filter,

\dot V=-k_t(W\e)^{\T}AA^{\T}f(AA^{\T})(W\e)\le0,\qquad \dot V=0\ \Leftrightarrow\ A^{\T}W\e=0.

Every positive filter has the same stationary set \mathcal E=\{\q:A^{\T}W\e=0\} (Corollary 2, (30)), at which the tip error can remain nonzero.

Corollary 3 (Exponential convergence). Where A has full row rank and \min_i\lambda_if(\lambda_i)\ge\beta>0,

\dot V\le-2k_t\beta\lambda_{\min}(W)V.

For W=I, the error decays along the i-th eigenvector of AA^{\T} at the rate

\rho_i^{\mathrm{dls}}=k_t\frac{\lambda_i}{\lambda_i+\xi^2},

so the descent rate -\dot V/V of the damped inverse lies in the band [2k_t\beta,\,2k_t], whereas the transpose gives rates proportional to the eigenvalues, \rho_i^{\mathrm{tr}}=k_t\lambda_i.

Fig. 7 of the paper. Convergence of the nominal controller (25) on the simulated plant for the five targets of S2 under the fulcrum, with the exact Jacobian (solid) and the identified model (dashed). Step 2 of the workspace reproduces it.

Step 3 · Scope of the analysis (Sec. V-C)

Three conditions of operation fall outside these hypotheses. Away from the constraint manifold, the compensation -J_t\qdc cancels the tip motion of the restoring velocity only if

J_t\qdc\in\range A.

When a tendon becomes slack or saturates, its entry of K_a changes, and the map from the joints to the tip is piecewise smooth; within each interval of fixed active set, Propositions 1 and 2 apply unchanged. When an inequality of the QP (19) is active, the closed loop is no longer the analyzed control law, and the tip error can settle at a nonzero value, as for the infeasible targets of S3.

Try this

In Step 1, compare the three EE constraints: the null-space dimension n-r_c drops from 6 to 3, but with two taut tendons the rank of \Jdef stays 2.
Move both anchors onto the EE (link 6): \Jlq=0, and the soft extension can only be transported (Corollary 1). Move one anchor back to the forearm: the rank returns to 1.
Make tendon 2 slack: m_a=1 limits the rank to 1 under every EE constraint.
In Step 2, run the five targets of Fig. 7: with the exact Jacobian, V never increases, and -\dot V/V stays in the band [2k_t\beta,2k_t] up to the time discretization. With the identified model, the rate leaves the band slightly, and V still decreases.
Switch to the transpose filter: the stationary set is the same, but the rates follow the eigenvalues \lambda_i, which are small, and convergence is much slower. Increase \xi to slow the damped inverse in the weakest direction.
In Step 3, run S3 and follow V at T3 and T5: it settles at a nonzero value while a barrier is active, and decreases again after the run moves on.

The tendons transmit at most as many deformation directions as the smallest of the four factors in (22). On the constraint manifold, every positive spectral filter makes the weighted tip error non-increasing, and the damped inverse converges exponentially where A is well conditioned.

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