Fiber-Normalized Manipulability and Determinant Proxies: Intrinsic Redundancy Optimization Across and Within Task Fibers

Antonio Franchi; Mirko Mizzoni

cs.ROSep 25, 2026

Fiber-Normalized Manipulability and Determinant Proxies: Intrinsic Redundancy Optimization Across and Within Task Fibers

Authors: Antonio Franchi, Mirko Mizzoni

Organizations: Robotics and Mechatronics Group, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, Enschede, The Netherlands · Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy

Abstract

This work establishes that determinant-based manipulability is an exact objective for fixed-task redundancy optimization, despite its dependence on task coordinates and the choice of task-space metric used for volume measurement. On every regular task fiber, the determinant proxy, its representation in any task chart, and every metric-completed manipulability differ only by positive constants. They consequently induce the same complete ordering, constrained extrema, gradient directions, critical points, and local optimality classifications. For comparisons and trajectory optimization across task fibers, this work introduces fiber-normalized manipulability: the capability attained at an internal state divided by the best capability available on the same fiber. The resulting dimensionless scalar is invariant under coordinate changes on the internal-state and task manifolds and independent of the task-space metric. Its associated loss provides an intrinsic objective for physically admissible cross-fiber trajectories, including problems with prescribed or free task evolution and temporal coupling. Planar-manipulator and redundant aerodynamic-allocation examples demonstrate fixed-fiber equivalence, task-dependent cross-fiber differences, and invariant fiber-normalized trajectory optimization.

Figures & tables

Panel of Figure 1 — Fig. 1: Geometric construction associated with the task map $f:\mathcal{V}\to\mathcal{W}$ , with $w=f(v)$ . The internal metric $g_{v}$ induces the musical isomorphisms $g_{v}^{\flat}$ and $g_{v}^{\sharp}$ and the co-metric $g_{v}^{*}$ on $T_{v}^{*}\mathcal{V}$ . The differential $\mathrm{d}f_{v}$ maps internal tangent vectors to task tangent vectors, whereas its pullback $\mathrm{d}f_{v}^{*}$ maps task covectors to internal covectors. Comparing these pullbacks through $g_{v}^{*}$ produces the state-induced task co-metric $d_{v}$ , whose sharp map is $d_{v}^{\sharp}=\mathrm{d}f_{v}\circ g_{v}^{\sharp}\circ\mathrm{d}f_{v}^{*}$ . Its inverse $\bar{d}_{v}^{\flat}=(d_{v}^{\sharp})^{-1}$ is the flat map associated with the state-induced inner product $\bar{d}_{v}$ on $T_{w}\mathcal{W}$ . The unit ball of $\bar{d}_{v}$ is the capability ellipsoid $\mathcal{E}_{v}$ . The independently selected task metric $h_{w}$ defines the reference geometry used to measure this ellipsoid. Starting from the input copy of $T_{w}\mathcal{W}$ , its flat map $h_{w}^{\flat}$ produces a task covector, which $d_{v}^{\sharp}$ maps to the output copy of $T_{w}\mathcal{W}$ . Their composition is the endomorphism $A_{v}=d_{v}^{\sharp}\circ h_{w}^{\flat}:T_{w}\mathcal{W}\to T_{w}\mathcal{W}$ . Its determinant defines the metric-completed manipulability $\rho_{H}(v)=\sqrt{\det A_{v}}$ . In the selected coordinate and dual bases, $g_{v}^{\sharp}$ , $d_{v}^{\sharp}$ , $\bar{d}_{v}^{\flat}$ , $h_{w}^{\flat}$ , and $A_{v}$ are represented by $G(v)^{-1}$ , $D(v)$ , $D(v)^{-1}$ , $H(w)$ , and $D(v)H(w)$ , respectively.

Change	Matrix relations	Manipulability identities
Configuration chart Exact cancellation	$\begin{aligned} \widehat{J}_{f}=J_{f}T_{B}&,\quad\widehat{G}=T_{B}^{\top}GT_{B},\\ \widehat{D}&=D\end{aligned}$	$\begin{aligned} \widehat{\rho}(v)&=\rho(v),\\ \widehat{\rho}_{H}(v)&=\rho_{H}(v)\end{aligned}$
Task chart Covariance and scaling	$\begin{aligned} \widetilde{J}_{f}&=T_{C}J_{f},\\ \widetilde{D}&=T_{C}DT_{C}^{\top}\end{aligned}$	$\begin{aligned} \widetilde{\rho}(v)&=\left\lvert\det T_{C}(f(v))\right\rvert\rho(v)\end{aligned}$
Task chart and reference metric Similarity and invariance	$\begin{aligned} \widetilde{H}&=T_{C}^{-\top}HT_{C}^{-1},\\ \widetilde{D}\,\widetilde{H}&=T_{C}(DH)T_{C}^{-1}\end{aligned}$	$\begin{aligned} \rho_{H}(v)&=\sqrt{\det H(f(v))}\,\rho(v),\\ \widetilde{\rho}_{H}(v)&=\sqrt{\det\widetilde{H}(f(v))}\,\widetilde{\rho}(v)\\ &=\sqrt{\det H(f(v))}\,\rho(v)=\rho_{H}(v)\end{aligned}$

TABLE I: Coordinate behavior of the manipulability constructions. In the matrix column, the arguments are suppressed according to

T_{B}=T_{B}(v)

T_{C}=T_{C}(f(v))

D=D(v)

, and

H=H(f(v))

Problem	Construction	Result
Fiber-normalized value Pointwise comparison	$\begin{aligned} \mu_{\mathcal{F}}(v)&=\frac{\rho_{H}(v)}{\displaystyle\sup_{u\in\mathcal{F}_{f(v)}}\rho_{H}(u)},\\ \ell_{\mathcal{F}}(v)&=1-\mu_{\mathcal{F}}(v)\end{aligned}$	$\begin{aligned} 0&<\mu_{\mathcal{F}}(v)\leq 1,\\ 0&\leq\ell_{\mathcal{F}}(v)<1\end{aligned}$ Coordinate invariant and independent of $h$ .
Static cross-fiber optimization $v\in\mathcal{S}\subseteq\mathcal{V}$	$\begin{aligned} \max_{v\in\mathcal{S}}\mu_{\mathcal{F}}(v)\quad\Longleftrightarrow\quad\min_{v\in\mathcal{S}}\ell_{\mathcal{F}}(v)\end{aligned}$	The complete optimizer set is coordinate invariant and independent of $h$ .
Intrinsic trajectory optimization $v\in\mathcal{A}\subset\mathrm{AC}([0,T],\mathcal{V})$	$\begin{aligned} \mathcal{J}_{\mathcal{F},\Phi}[v]&:=\int_{0}^{T}\Phi\!\left(\ell_{\mathcal{F}}(v(t)),\\|\dot{v}(t)\\|_{g_{v(t)}}^{2}\right)\,\mathrm{d}t\end{aligned}$	The functional and its complete minimizer set are coordinate invariant and independent of $h$ .
Prescribed task, uncoupled $v\in\mathcal{A}_{w}^{0}$	$\begin{aligned} f(v(t))&=w(t),\\ v(t)&\in\mathcal{M}(w(t))\quad\text{a.e.}\end{aligned}$	$\begin{aligned} \operatorname{arg\,min}\mathcal{J}_{\rho}&=\operatorname{arg\,min}\mathcal{J}_{\widetilde{\rho}}=\operatorname{arg\,min}\mathcal{J}_{\rho_{H}}\\ &=\operatorname{arg\,min}\mathcal{J}_{\mathcal{F}}\end{aligned}$ Pointwise selections may be discontinuous.
Cross-fiber motion with temporal coupling Prescribed or free task	$\begin{aligned} v&\in\mathrm{AC}([0,T],\mathcal{V}),\\ \\|\dot{v}(t)\\|_{g_{v(t)}}&\leq\bar{s}\quad\text{a.e.}\end{aligned}$ Endpoint, state, or task constraints may also be imposed.	Unnormalized objectives may select different paths. Fiber-normalized objectives remain coordinate invariant and independent of $h$ .

TABLE II: Summary of the optimization problems enabled by fiber-normalized manipulability. The admissible sets are physical sets and are therefore kept fixed when coordinates or the reference task metric are changed.

Fig. 2: Configuration-space logarithmic-cost landscapes for the planar 2R manipulator. The grayscale background represents the logarithmic cost, with darker gray indicating lower displayed values and white indicating higher displayed values. The same cost scale is used across all three panels. (a) Determinant-proxy cost $L=-\log\rho$ . (b) Cost $\widetilde{L}$ obtained after the nonlinear task-coordinate change. (c) Cost $L_{H}$ obtained using the reference task metric. Red curves show representative fixed-task fibers, and the common yellow branches form the fiberwise-optimal locus. The square, circle, and diamond identify global minimizers of $L$ , $\widetilde{L}$ , and $L_{H}$ , respectively; the adjacent annotations report their joint coordinates. The three landscapes differ across fibers, whereas their minimizers coincide on every prescribed fiber.

Fig. 3: Fiberwise and global comparisons for the planar 2R manipulator. (a) Restrictions of $L$ , $\widetilde{L}$ , and $L_{H}$ to $\mathcal{F}_{w_{0}}$ , with $w_{0}=0.60$ , parameterized by $q_{1}$ . The three curves differ only by constant vertical translations and have the same minimizers. (b) Common fiberwise-optimal locus, with the square, circle, and diamond identifying the global selections produced by $L$ , $\widetilde{L}$ , and $L_{H}$ , respectively. (c) Fiber-optimum profiles $L^{\star}$ , $\widetilde{L}^{\star}$ , and $L_{H}^{\star}$ . Task-side modifications preserve the complete ordering on each fixed fiber but can change the task value selected by a global comparison.

Fig. 4: Actuator-space logarithmic-cost landscapes for the dual-rotor system. The grayscale background represents the logarithmic cost, with darker gray indicating lower displayed values and white indicating higher displayed values. The same cost scale is used across all three panels. (a) Determinant-proxy DAAM cost $L=-\log\rho$ . (b) Cost $\widetilde{L}$ after the localized nonlinear task-coordinate stretching centered at $w=5$ . (c) Cost $L_{H}$ obtained using the reference task metric, whose weighting is centered at $w=-5$ . Red curves are fixed-task fibers, and the common yellow branches form the fiberwise-optimal locus. The four squares in panel (a) mark the symmetry-related global minima of $L$ . The circle and diamond mark selected global minima of $\widetilde{L}$ and $L_{H}$ , respectively; the annotations report the corresponding spin-rate coordinates.

Fig. 5: Fiberwise and global comparisons for the dual-rotor system. (a) Restrictions of $L$ , $\widetilde{L}$ , and $L_{H}$ to $\mathcal{F}_{w_{0}}$ , parameterized by $v_{1}$ ; constant vertical translations preserve the minimizers. (b) Common fiberwise-optimal locus, with the square, circle, and diamond marking the global selections obtained from $L$ , $\widetilde{L}$ , and $L_{H}$ , respectively. (c) Corresponding fiber-optimum profiles. The task-coordinate and task-metric constructions preserve the optimizer on every prescribed fiber but favor different fibers under global comparison.

Fig. 6: Pointwise-optimal lift of the prescribed task trajectory without temporal coupling. The grayscale background represents the raw logarithmic cost, ranging from medium gray at lower values to white at higher values. Thin gray curves are fixed-task fibers, while the gold curves form the common fiberwise-optimal locus. Dark-teal segments show a measurable selection from $\mathcal{V}^{\star}(w(t))$ ; dotted connectors and magenta-outlined markers identify the instantaneous jumps between disconnected branches. Because different time instants are uncoupled, this selection is simultaneously optimal for the instantaneous costs $-\rho$ , $-\widetilde{\rho}$ , $-\rho_{H}$ , and $\ell_{\mathcal{F}}$ .

Fig. 7: Optimal lifts of the same prescribed task trajectory under the intrinsic speed bound $\|\dot{v}\|_{g_{v}}\leq 0.56$ . In panels (a)–(c), the grayscale background represents the corresponding logarithmic running cost $L$ , $\widetilde{L}$ , or $L_{H}$ , while panel (d) displays $\ell_{\mathcal{F}}$ . Darker gray indicates lower displayed cost and white indicates higher displayed cost. Thin gray curves are fixed-task fibers, the gold curves form the common pointwise fiberwise-optimal locus, and open and filled circles identify $t=0$ and $t=T$ , respectively. Panels (a)–(c) show representative minimizers of $\mathcal{J}_{\rho}$ , $\mathcal{J}_{\widetilde{\rho}}$ , and $\mathcal{J}_{\rho_{H}}$ over their corresponding running-cost landscapes. Panel (d) superposes the independently computed lifts obtained by minimizing the functionals based on $\ell_{\mathcal{F}}^{\rho}$ , $\ell_{\mathcal{F}}^{\widetilde{\rho}}$ , and $\ell_{\mathcal{F}}^{\rho_{H}}$ . Their pointwise coincidence illustrates the theoretically predicted equality of the three fiber-normalized objectives and of their complete minimizer sets.

Fig. 8: Time-domain view of the prescribed-task allocation problem. (a) Prescribed task trajectory $w(t)$ . (b) Pointwise-optimal allocation without temporal coupling; dotted vertical lines mark instantaneous jumps between disconnected branches. (c)–(d) Spin-rate profiles of the speed-bounded optimal lifts. The intrinsic speed bound replaces the jumps with anticipatory transitions through configurations having positive fiber-normalized loss. The light-red solid, light-blue dashed, and black-marked profiles are obtained by independently minimizing $\ell_{\mathcal{F}}^{\rho}$ , $\ell_{\mathcal{F}}^{\widetilde{\rho}}$ , and $\ell_{\mathcal{F}}^{\rho_{H}}$ , respectively. Their pointwise coincidence confirms that the three task-side constructions define the same objective and select the same optimal lift. Any overlap between profiles produced by the unnormalized objectives is instead a numerical symmetry of this example and is not a general invariance property.

Fig. 9: Free-task planning for the planar 2R manipulator between the same prescribed endpoint configurations. All panels use the same periodic grid, horizon, and transition graph. The grayscale backgrounds show $L$ , $\widetilde{L}$ , $L_{H}$ , and $\ell_{\mathcal{F}}$ in panels (a)–(d), respectively, with darker gray denoting lower displayed values. Thin gray curves are fixed-task fibers and the gold curves form the common fiberwise- optimal locus. Open and filled circles identify the prescribed initial and final configurations. Panels (a)–(c) show representative paths minimizing the discrete functionals based on $-\rho$ , $-\widetilde{\rho}$ , and $-\rho_{H}$ , respectively. Panel (d) superposes the independently computed paths obtained from $\ell_{\mathcal{F}}^{\rho}$ , $\ell_{\mathcal{F}}^{\widetilde{\rho}}$ , and $\ell_{\mathcal{F}}^{\rho_{H}}$ . Their pointwise coincidence confirms that fiber normalization removes both the task-coordinate and task-metric dependence.

Fig. 10: Free-task planning for the dual-rotor system between the same prescribed endpoint internal states. All panels use the same horizon, feasible grid, and transition graph. The grayscale backgrounds show $L$ , $\widetilde{L}$ , $L_{H}$ , and $\ell_{\mathcal{F}}$ in panels (a)–(d), respectively, with darker gray denoting lower displayed values. Thin gray curves are fixed-task fibers and the gold curves form the common fiberwise- optimal locus. Open and filled circles identify the prescribed initial and final internal states. Panels (a)–(c) show representative paths minimizing the discrete functionals based on $-\rho$ , $-\widetilde{\rho}$ , and $-\rho_{H}$ , respectively. Panel (d) superposes the independently computed paths obtained from $\ell_{\mathcal{F}}^{\rho}$ , $\ell_{\mathcal{F}}^{\widetilde{\rho}}$ , and $\ell_{\mathcal{F}}^{\rho_{H}}$ . Their pointwise coincidence confirms that fiber normalization removes both the task-coordinate and task-metric dependence.

Fig. 11: Fiber-normalized loss $\ell_{\mathcal{F}}$ evaluated along the representative paths in Fig. 10 . The black, blue, and orange curves correspond to paths obtained by minimizing the unnormalized costs $-\rho$ , $-\widetilde{\rho}$ , and $-\rho_{H}$ , respectively. The light-red solid, light-blue dashed, and black-marked curves correspond to paths obtained by independently minimizing the functionals based on $\ell_{\mathcal{F}}^{\rho}$ , $\ell_{\mathcal{F}}^{\widetilde{\rho}}$ , and $\ell_{\mathcal{F}}^{\rho_{H}}$ . Their pointwise coincidence confirms that the three fiber-normalized constructions define the same objective and select the same optimal path.

Trajectory Optimization

Trajectory optimization computes motion or control sequences that minimize costs while satisfying system dynamics, safety limits, and task constraints.

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Complete Motion Planning using Workspace-Fibered Decomposition for nR-Planar Manipulator

Aayush Rath, Antony Thomas

Robotics Research Center, IIIT Hyderabad, Hyderabad 500032, India.

Constrained Motion Planning Hierarchical Planning Motion Planning Robot Motion Planning

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We propose a workspace-fibered decomposition framework for motion planning in nR planar redundant manipulators operating in cluttered environments. Rather than planning directly in the full n-dimensional configuration space, the method incrementally constructs obstacle-constrained reachable workspaces of lower-dimensional non-redundant sub-chains and recursively lifts them through redundant orientation fibers. This yields a sequence of reduced planning manifolds that preserve branch-consistent reachability structure while avoiding explicit construction of the full configuration-space obstacle geometry. We first establish that, for planar position-only manipulators, the obstacle-constrained reachable workspace induced by the minimal non-redundant sub-chain provides an exact characterization of feasibility with respect to the connected component of the start configuration, enabling early infeasibility detection prior to introducing redundant degrees of freedom (DOF). We then introduce an incremental fiber-lifting procedure that propagates reachable workspace structure through successive redundant links while enforcing local inverse-kinematic branch consistency using Jacobian determinant continuity constraints. The resulting representation admits efficient reduced-space planning directly on recursively-constructed workspace-fiber manifolds. Experimental results on redundant nR planar manipulators demonstrate that the proposed construction preserves collision-free connectivity structure across successive lifting stages while substantially reducing collision checking complexity relative to direct configuration space reasoning.

Abstract

Trajectory Optimization

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