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λi=
[[image]]
Rip = rotation matrix that brings the components of a vector in the payload from Fp to those in Fi,

rip = vector from the tip of beam i to the payload centre of mass, expressed in Fi,

and the tilde (~) represents the vector cross-product operator.

The six natural frequencies computed from the above M and F provide good approximations for the frequencies of the SRMS arm loaded with heavy payloads (compared with the STARDYNE results) since these first six frequencies essentially correspond to the payload modes. This simplified model is used to study the variation of the arm natural frequencies with arm configurations at a much cheaper (200:1) rate than using STARDYNE.

Dynamics Simulations

Since the SRMS is designed for operation in the zero g environment, only a limited number of hardware tests can be done on the ground leading to the necessity of developing computer simulation programs to study the arm dynamics and the control system performance Two simulation programs, ONEJNT and ASAD, have been developed to simulate the singe joint and multi-joint arm/Orbiter/Payload dynamics respectively. they have been used throughout the design, development and verification phase of the SRMS. In both programs, the Orbiter and the payload are assumed to be rigid bodies

In ONEJNT, the arm is modelled as a uniform cantilever beam with a  payload attached at its tip (Figure 10). The payload mass, m, inertia, J, and centre of mass offset (from the End Effector) c are taken into consideration. The beam mass and stiffness are assumed to be uniformly distributed along its length. The beam bending stiffness is derived from the finite-element analysis described above. The arm only joint (and servo) is located at the beam root; the other five joints are assumed to be rigidly locked. The Orbiter rotation axis is assumed to be parallel to the joint axis. As a result, ignoring the swing-out joint angle, the Orbiter yaw motion is coupled with the arm yaw joint(s) while the Orbiter pitch motion is coupled with the arm pitch joint(s), etc.

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Figure 10 A Simplified Model for the Shuttle/Arm/Payload Dynamics

Let Sn = dimentionless nth bending mode shape of the beam,
η = dimensionless coordinate along the beam (η=0 at beam root and η=1 at beam tip),
Ωn = nth mode natural frequency,
[[q?]]n = nth mode generalized coordinates,

then considering no force/torque exerted on the payload, the beam motion equation can be shown to be:

[[image]]
(3)

Where a = linear acceleration at the beam root,

ώ = angular acceleration at the beam root,

l = beam length

[[image]]
(modal positional gain)

[[image]]
(modal angular gain)

ρ - beam mass per unit length.

The natural frequencies Ωn and mode shapes Sn are obtained from the beam continuum characteristic equation. An alternative approach is to use the cubic approximation for the mode shape and derive the 'natural' frequencies using Ritz method (finite element method).

Introducing modal damping factor ζn and combining the Orbiter and joint dynamics with the beam elastic motion, the accelerations a and ώ are eliminated and the system motion equation can be cast into the following form:

[[image]]
(4)

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