The movement of a chase vehicle (the “trajectory”) during a berthing event is dominated by several factors. These include the mass properties of the two elements, forces acting on the in-coming element by the item surrendering control authority, forces exerted by the CBM on the in-coming elements, and constraints deriving from the alignment structures of the CBM itself.
Trajectories and loads were determined by analysis using the “method of soft constraints”1, where a pseudo-penetration of solid surfaces is “permitted” at their point(s) of contact. The depth of pseudo-penetration is used in concert with the local stiffness and coefficients of friction at each contact point to determine a combination of “rebound” and “sliding” forces. The net of the calculated forces is then applied to the bodies involved.
Failure modes that can be predicted by this method include jamming and wedging2, both of which follow from contact mechanics. Jamming occurs when the resultant force at each point of contact is inside the friction cone3 such that no “closing” motion is possible. Relieving the external load relieves the contact loads so that the faying surfaces are freed4.
Similarly, wedging occurs when the resultant at every point of contact is not only inside the friction cone, but aligned such that the resultants “self equilibrate”5, and the external loads cause significant elastic deformation. Under those circumstances, the contact loads don’t relieve when the external loads are relaxed. The two surfaces are stuck, and are likely to damage each other even if wrenched free.
The surrendering item adds some external dynamic influences over the chase vehicle during the event:
(1) When telerobotic devices are used, the constraints are imposed by6
(a) The orientation of the various telerobotic joints7
(b) Stiffnesses of the telerobotic joints, which might be both non-linear and non-conservative if friction is involved
(c) The lengths, inertial properties, and stiffnesses of the arms that
(i) Connect the joints of the device
(ii) Attach the ends of the device to the two vehicles
(2) In the case of an RCS, those constraints are imposed by the control laws of that system, which generally try to maintain a constant relative position between the chase and target vehicles8.
It is not reasonable to assume that the chase vehicle’s trajectory will monotonically converge during capture. It might, in fact, diverge for a while prior to a final convergence at “capture complete”. That is, angles and alignments might get worse before getting better9.
Footnotes- See Hall, D.P, M.M. Slone, and P.A. Tobbe, “Modeling and testing of docking and berthing mechanisms”, (2006). Document ID 20060046498 in the NTRS.[↩]
- See McKerrow, Phillip “Introduction to Robotics”, Addison-Wesley Publishing Company, New York (1991), Section 6.3.5.[↩]
- A resultant force within the “friction cone” means that the force results in no motion. The half-angle of the cone is equal to the coefficient of friction. The centerline of the cone is perpendicular to the surface at the point of contact.[↩]
- That is, separable with the application of force required to overcome inertial loads only.[↩]
- That is, taken in the aggregate, the resultant load is zero.[↩]
- McKerrow, op. cit., Chapter 4.[↩]
- Which might act in either rotation or translation.[↩]
- For elaboration and further conceptual development, see Wertz, James R. [Ed.], “Spacecraft Attitude Determination and Control” (Kluwer Academic Publishers, 1978) , Chapter 18.[↩]
- Not unlike a crab which, when threatened with danger from above, looks straight ahead and crawls off sidewise.[↩]