Optimization of the Trajectory of a General Free-Flying Manipulator During the Rendezvous Maneuver
Authors: Seweryn, Banaszkiewicz · Year: 2008 · Venue: AIAA Guidance, Navigation and Control Conference and Exhibit
Raw: md
Summary
The paper treats the final, docking phase of a rendezvous-and-docking (RVD) maneuver in which a 6-DOF manipulator mounted on a thruster-controlled servicing satellite reaches for a passive, possibly tumbling target. The central contribution is an extension of the Generalized Jacobian Matrix (GJM) formalism to the case where the satellite-manipulator system’s linear and angular momenta are not conserved (because thrusters continuously exert force/torque to hold the chaser on a synchronous orbit), and a calculus-of-variations / Pontryagin trajectory optimization built on top of it. The cost functional is quadratic in joint torque; the resulting boundary value problem is solved by collocation (MATLAB bvp4c). A 6-DOF LBR-arm example yields a ~30% power saving (23.4 W vs 33 W) over a straight-line reference path.
Key Claims
- The chaser-manipulator system held on a synchronous orbit by thrusters is nonholonomic with non-conserved linear and angular momentum; the conservation-based equations of earlier authors (Umetani-Yoshida, Longman, Lindberg) cannot be used because forcing the chaser to a zero-momentum state before docking is infeasible for a tumbling target and is fuel-inefficient.
- The momentum balance Eq. (9) becomes a Pfaffian constraint with drift (explicitly time-dependent right-hand side {f_p}, {f_mp}), as opposed to the driftless Pfaffian form (RHS = 0) handled by prior work.
- Four manipulator regimes are unified under one GJM description: (i) fixed-base, (ii) free-floating, (iii) free-flying with momenta conserved, (iv) free-flying with external forces — this paper derives (iv).
- The optimized trajectory ends at different final joint angles than the straight-line path while still satisfying the same end-effector pose constraint — flagged as a characteristic signature of nonholonomy.
- Reported result: 30% energy gain (23.4 W vs 33 W) for the optimal versus straight-line trajectory; the BVP took ~24 h of processor time.
Method
Regime: free-FLYING (fully-actuated 6-DOF base) with external forces — this is the central distinction of the paper. Unlike free-floating dynamics (where base is uncontrolled and momentum is conserved at zero), here thrusters supply predefined force {F_s} and torque {H_s} to keep the chaser synchronous with the target, so momentum/angular momentum are governed by time-varying functions, not conserved.
End-effector position (Eq. 1, inertial frame U
{r_ee} = {r_s} + {r_q} + Σ_{i=1}^n {l_i}
End-effector twist (Eq. 4) decomposed into satellite and manipulator Jacobians:
{v_ee; ω_ee} = [J_s]{v_s; ω_s} + [J_M]{θ̇}
with [J_s] (Eq. 5) built from the skew matrix of r_ee_s = r_ee − r_s, and [J_M] (Eq. 6) the conventional manipulator Jacobian expressed in U
Kinetic energy (Eq. 7) uses the partitioned mass matrix with blocks [A],[B],[C],[E],[F],[N] (Eqs. 8). The momentum / angular-momentum balance (Eq. 9) is written
{P; L_0 + r_s×P} = [H_2]{v_s; ω_s} + [H_3]{θ̇} = {f_p; f_mp}
The generalized-Jacobian relation with drift (Eq. 12) solves for joint rates given a commanded end-effector twist:
{θ̇} = ([J_M] − [J_s][H_2]^{-1}[H_3])^{-1} ( {v_ee; ω_ee} − [J_s][H_2]^{-1}{f_p; f_mp} )
and the coupled base motion (Eq. 13): {v_s; ω_s} = [H_2]^{-1} ( {f_p; f_mp} − [H_3]{θ̇} ).
The full second-order dynamics use Lagrangian formalism (potential energy neglected), generalized coordinates q = {r, σ, θ} with Modified Rodrigues Parameters σ for attitude (Schaub & Junkins), giving (Eq. 15) [M(q)]{q̈} + [C(q̇,q)]{q̇} = {Q}, where [M(q)] = [R(σ)]^T · (mass matrix) (Eq. 16), [R(σ)] embeds the MRP kinematic map [S(σ)] (Eq. 17), and generalized forces {Q} = {F_s; [S(σ)]H_s; f} (Eq. 19).
Optimization (Sec. IV): augmented functional G (Eq. 21) with Lagrange multipliers λ(t), Hamiltonian H = L + λ^T f (Eq. 22), stationarity ∂H/∂u = 0 (Eq. 23). For Hamiltonian linear in u, a second-order condition (Eq. 24) gives u. The two-point BVP (Eqs. 26) couples 2(6+n) state and costate equations; terminal condition on costate (Eq. 27) λ(t_f) = (v^T ∂ψ/∂x)|_{t_f}. Cost L = 0.5 u^T u (Eq. 28). Terminal pose constraint ψ (Eq. 29). Solved with bvp4c (Kierzenka & Shampine collocation), seeded by the straight-line trajectory: 24 state ODEs + 24 costate ODEs + 6 algebraic torque equations + 54 boundary conditions.
Relevance to thesis
This is a direct precursor to our free-flying (fully-actuated base) regime: it explicitly rejects the conserved-momentum free-floating assumptions and builds a GJM with a drift term to account for the thruster forces/torques that keep the base on a prescribed trajectory. The Pfaffian-with-drift framing, the MRP-based Lagrangian dynamics, and the costate/Pontryagin BVP for minimum-torque (energy) trajectory planning all map onto our nominal guidance-and-control derivation. The treatment of base-manipulator dynamic coupling (manipulator motion perturbing the thruster force/torque needed to hold station) is exactly the coordinated-control problem we are formalizing before adding the risk layer.
Connections
Topics: generalized_jacobian · free_flying_vs_free_floating · dynamic_coupling · optimal_control_bvp
Key Equations / Quotes
“The system (satellite + manipulator) is nonholonomic and its linear momentum and angular momentum are not conserved.” (Abstract, p. 1)
“we have decided to cast our considerations in the form of GJM … a very general description that includes all four basic types of manipulators: (i) with a fixed base, (ii) free-floating, (iii) free-flying with moments conserved, (iv) free-flying with external forces included.” (p. 2, Sec. I)
“Equation (9) and (10) can be considered as Pfaff-type constraints that explicitly depend on time, therefore describe the Pfaff form with drift. The constraint in Eq. (9) with the right hand side equal to zero, considered by previous authors, describe the Pfaff form without drift that is much easier to handle analytically.” (p. 5, Sec. III)
Eq. (12), GJM with drift:
{θ̇} = ([J_M] − [J_s][H_2]^{-1}[H_3])^{-1} ( {v_ee; ω_ee} − [J_s][H_2]^{-1}{f_p; f_mp} )
“The gain in power (energy) of the optimal solution as compared with the straight solution is 30%, i.e. 23.4 W instead of 33 W.” (p. 11)
Open Questions
- No obstacle/collision constraints are included in the optimization; the authors list obstacle avoidance and a hybrid variational + path-planning method as future work.
- Reported maneuver time is “1 s” in Table I, which is implausibly short for a docking approach and likely an OCR/typo artifact — should be verified against the original.
- The 24-hour solve time for a single 6-DOF case raises tractability concerns; shooting methods are floated but not evaluated.
- Eq. (18) for [C(q̇,q)] (Christoffel-symbol construction) appears with index/derivative typos in the OCR (
d/dq_k m_ijetc.) and should be cross-checked against a standard Coriolis-matrix definition. - The {f_p}, {f_mp} drift terms depend on the thruster commands {F_s},{H_s}, which are themselves recomputed to hold station — the coupling/consistency of this inner loop is asserted but not fully derived in the excerpt.