virgili-llop2019convex

A convex-programming-based guidance algorithm to capture a tumbling object on orbit using a spacecraft equipped with a robotic manipulator

Authors: Virgili-Llop, Zagaris, Zappulla, Bradstreet, Romano · Year: 2019 · Venue: The International Journal of Robotics Research, 38(1), 40–72
Raw: md

Summary

The paper presents a real-time-capable guidance algorithm for a chaser spacecraft equipped with a robotic manipulator capturing a tumbling resident space object (RSO). The full nonlinear optimal control problem is rendered tractable by decomposing the capture into two simultaneously executed but sequentially optimized sub-maneuvers: a system-wide translation of the chaser’s center of mass, and an internal re-configuration (manipulator motion plus base re-orientation about the CoM). Each step is solved by a sequential convex programming (SCP) procedure, yielding a collection of convex programs solvable in polynomial time by interior-point methods. A convergence proof to a KKT point is offered for Step 1 (extended to non-convex keep-out zones), while Step 2 relies on trust-region heuristics; hardware-in-the-loop experiments on a planar air-bearing testbed substantiate the real-time claim.

Key Claims

• The coupled capture problem can be split into a translation sub-maneuver (CoM dynamics, thruster-actuated) and an internal re-configuration sub-maneuver (attitude + joints), optimized consecutively, with the Step 1 solution restoring the dynamic coupling in Step 2 via inertial reaction forces.
• For Step 1, if the RSO keep-out zone is approximated by its convex hull (or a union of overlapping convex sets), an admissible seed trajectory is used, and the grapple fixture lies on the boundary of the keep-out set, the SCP iterates form a recursively feasible, non-increasing-cost descent that converges to a KKT point of the non-convex problem (Propositions 2 and 3).
• Every Step-1 SCP iterate is itself an admissible point of the original non-convex Problem 3, so the iteration may be stopped at any time and still yield a collision-free control/trajectory (Remark 7) — valuable for an anytime onboard scheduler.
• An explicitly convex line-of-sight (and attitude keep-out) constraint is derived as a quadratic form in the attitude quaternion, convexified by eigenvalue shifting using the analytically known eigenvalues λ_A = ±‖v‖‖r‖ of the constraint matrix A (Eqs. 64–73).
• The optimization ordering (translation first) implicitly prioritizes minimizing propellant-consuming translational cost over internal re-configuration cost, which can be borne by momentum-exchange devices and joint actuators (Remark 1).

Method

Regime: free-FLYING. The chaser has a fully actuated base: base generalized forces τ₀ = [f₀; n₀] include both thruster forces f₀ ∈ ℝ³ and base torques n₀ ∈ ℝ³, and the translation sub-maneuver is driven solely by base thrusters (f₀ = f_c). This is explicitly not a free-floating model — although note the keep-out radius R_KO is taken “equivalent, in the absence of any large chaser appendages, to the free-floating reachable workspace radius (Umetani and Yoshida, 2001),” borrowing a free-floating workspace bound to size the bounding sphere.

Dynamics in canonical form (Eq. 1): H u̇ + C u = τ, with generalized velocities partitioned u = [u₀; u_m], u₀ = [ṙ₀; ω₀]. Quaternion kinematics (Eq. 4): q̇₀ = ½ ω₀ ⊗ q₀. Cost is quadratic in generalized forces (Eq. 5).

Step 1 (system-wide translation): whole-body CoM dynamics f₀ = m r̈_c (Eq. 16), direct transcription with piecewise-constant f₀, discrete state-space x^[n+1] = Φ_r x^[n] + Ψ_r f₀^[n]. The non-convex signed-distance keep-out constraint d ≥ 0 (Schulman et al. 2014) is linearized about the prior iterate (Eq. 36); the non-convex terminal equality ‖r_g − r_c‖ = R_f is relaxed to the convex inequality ‖r_g − r_c‖ ≤ R_f (Eq. 31), shown equivalent when r_g ∈ ∂S_RSO. Convergence to a KKT point proven via convex-hull / union-of-convex-sets modeling (Eqs. 38–41).

Step 2 (internal re-configuration): EoM rewritten in a CoM-attached, inertially-aligned frame 𝒞 (Eq. 43–51). The Step-1 force solution f₀* enters as a generalized inertial reaction force via the kineto-static duality τ_𝒾 = −J_c^T f₀* (Eq. 48), with the CoM Jacobian J_c = (Σ J_i m_i)/(Σ m_i) (Eq. 49). Nonlinear kinematics/dynamics linearized about a reference trajectory and solved with SCP under trust regions (no convergence guarantee). The line-of-sight terminal-pointing constraint is convexified to the quadratic forms of Eqs. 70–73.

Six modeling assumptions (A.1–A.6): rigid bodies; environmental/orbital effects neglected so an orbiting frame is treated as inertial; states and inertias known; designated grapple fixture; pre-set manipulator motion for t ≥ t_ps; constant chaser mass.

Relevance to thesis

This is a directly on-target reference for free-flying space manipulator guidance. The translation/re-configuration decomposition is a concrete coordinated-control architecture for a fully actuated base, and the kineto-static-duality coupling (τ_𝒾 = −J_c^T f₀*) is a clean way to feed translational decisions into the attitude/joint layer. The anytime KKT-feasible SCP (Remark 7) and deterministic convergence are exactly the properties wanted before layering risk/uncertainty on top. The explicitly convex quaternion LoS constraint is reusable for inspection/pointing constraints in the planning layer.

Connections

Topics: sequential_convex_programming · keep_out_zone · dynamic_coupling · center_of_mass_jacobian · free_flying_vs_free_floating

Key Equations / Quotes

“Despite being dynamically coupled through Equation (1), the two concurrent sub-maneuvers can be decoupled, allowing one of them to be optimized first and independently from the other.” (Sec. 2.2)

System-wide translation dynamics (Eq. 16):
${\mathbf{f}}_0=m{\ddot{\mathbf{r}}}_c$

Linearized keep-out constraint (Eq. 36):
${\tilde{d}}^{[n]}+{\hat{n}}_{\tilde{d}}^{[n]T}\left({\mathbf{r}}_c^{[n]}-{\tilde{\mathbf{r}}}_c^{[n]}\right)\ge 0$

Inertial reaction via kineto-static duality (Eqs. 48, 50):
${\mathbf{\tau }}_{\mathcal{I}}=-{\mathbf{J}}_c^T{\mathbf{f}}_0^{\star },f_{0,\tau }=-f_0^{\star }$

Explicitly convex LoS constraint (Eq. 70):
${\mathbf{q}}_0^T{A}^{+}{\mathbf{q}}_0+\Vert v\Vert \Vert r\Vert (\cos \varphi -1)\le 0,A^{+}=A+\Vert v\Vert \Vert r\Vert I_4$

“All successive solutions to the convex Problem 4 are admissible points of the non-convex Problem 3. Therefore, the iterative process can stop at any time…” (Remark 7)

Open Questions

• Step 2’s SCP has no convergence or recursive-feasibility guarantee (relies on trust-region heuristics). What conditions, if any, would extend the Step-1-style KKT proof to the coupled re-configuration?
• The decomposition prioritizes translation cost first; is the consecutive (lexicographic) ordering ever far from the jointly optimal coupled solution, and can a bound on the suboptimality gap be given?
• The bounding-sphere keep-out is conservative (target protrusions penetrating the final sphere render the problem infeasible). How much admissible set is lost on realistic appendage-laden targets?
• Assumption A.3 (states and inertias known) and A.2 (orbital dynamics neglected) define the nominal regime — the natural entry point for a risk-aware layer the paper does not address.