The paper provides a step-by-step tutorial on the Generalized Jacobian Matrix (GJM) approach for modeling and simulation of spacecraft-manipulator systems. The General Jacobian Matrix approach describes the motion of the end-effector of an underactuated manipulator system solely by the manipulator joint rotations, with the attitude and position of the base-spacecraft resulting from the manipulator motion. The coupling of the manipulator motion with the base-spacecraft are thus expressed in a generalized inertia matrix and a GJM. The focus of the paper lies on the complete analytic derivation of the generalized equations of motion of a free-floating spacecraft-manipulator system. This includes symbolic analytic expressions for all inertia property matrices of the system, including their time derivatives and joint-angle derivatives, as well as an expression for the generalized Jacobian of a generic point on any link of the spacecraft-manipulator system. The kinematics structure of the spacecraft-manipulator system is described both in terms of direction-cosine matrices and unit quaternions. An additional important contribution of this paper is to propose a new and more detailed definition for the modes of maneuvering of a spacecraft-manipulator. In particular, the two commonly used categories free-flying and free-floating are expanded by the introduction of five categories, namely floating, rotation-floating, rotation-flying, translation-flying, and flying. A fully-symbolic and a partially-symbolic option for the implementation of a numerical simulation model based on the proposed analytic approach are introduced and exemplary simulation results for a planar four-link spacecraft-manipulator system and a spatial six-link spacecraft manipulator system are presented.

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The performance of an inverse dynamics guidance and control strategy is experimentally evaluated for the planar maneuver of a "chaser" spacecraft docking with a rotating "target." The experiments were conducted on an airbearing proximity maneuver testbed. The chaser spacecraft simulator consists of a three-degree-of-freedom autonomous vehicle floating via air pads on a granite table and actuated by thrusters. The target consists of a docking interface mounted on a rotational stage with the rotation axis perpendicular to the plane of motion. Given a preassigned trajectory, the guidance and control strategy computes the required maneuver control forces and torque via an inverse dynamics operation. The recorded data of 150 experimental test runs were analyzed using two-way analysis of variance and post hoc Tukey tests. The metrics were maneuver success, vehicle mass change, maneuver duration, thruster duty cycle, and maneuver work. The results showed that the guidance and control algorithm provided robust performance over a range of target rotation rates from 1 to 4 deg ∕s. = maximum available torque, N · m T z = chaser torque about z axis of laboratory coordinate system, N · m t = maneuver time, s t D = total duration of docking maneuver, s t DA = duration of final docking approach, s t FF = duration of free-flight phase, s t f = final maneuver time, s W = maneuver work, J x C ; y C T = planar position vector of chaser center of mass in laboratory coordinate system, m x T ; y T T = planar position vector of the target center of rotation in laboratory coordinate system, m x; y; z = Cartesian coordinate system fixed in laboratory x 0 ; y 0 ; z 0 = Cartesian coordinate system fixed to chaser spacecraft simulator x 0 0 ; y 0 0 ; z 0 0 = Cartesian coordinate system fixed to target object γ = auxiliary maneuver targeting angle, rad Δm = chaser mass change, kg Δr = positioning tolerance for completion of docking maneuver, m Δt DA = extended maneuver time beyond nominal maneuver time, s Δt G = time step of the guidance algorithm, s ζ = auxiliary maneuver targeting angle, rad θ C = chaser orientation angle in relation to laboratory coordinate system, rad θ T = target orientation angle in relation to laboratory coordinate system, rad χ ij = thruster activity for thruster j at time step ∈ 0; 1, s ω T = target object angular rate, rad∕s