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[graph]
Figure 13 Bode Plot of Single Servo Open Loop Transfer Function
– Rigid Arm & Gearbox Loaded With the 14,615 Kg Payload
[graph] 
Figure 14 Bode Plot of Single Servo Open Loop Transfer Function 
– Flexible Arm and Gearbox Loaded With the 14,615 Kg Payload
[image]
Figure 15 Structural Flexibility Model for the Shuttle/Arm/Payload

(whose length is the distance between the two joints) and two massless, two-dimensional torsional springs attached to the two ends of the beam. The use of two springs across a joint allows for the variation of the joint housing stiffness with joint angle. The spring at the End Effector/Payload interface simulates the flexibility of the grapple fixture and the End Effector. The beam element plus the spring between the Orbiter and the shoulder yar joint simulate the combined stiffness of the Orbiter longeron and the Mechanical Positioning Mechanism. The Orbiter can have translation and rotation relative to its nominal orbital motion as a result of either Orbiter Reaction Control System jet firing, or interactive dynamics induced by arm and payload manoeuvring [[manoeuvering]]. The SRMS control system is simulated by the implementation of all six joint servos and a flight like SRMS control software. 

The arm dynamics equations in ASAD are based mainly on [8] which uses the Hooker-Marguiles type of approach to derive the motion equations for an open chain of elastic bodies. First, the translational and rotational equations of motion are derived for each free body in the chain, including the body elastic deformations. Next, the equilibrium equations are derived for each elastic body in terms of generalized coordinates. The boundary conditions are treated such that no time derivatives of forces/torques are required. Then, the hinge (internal) forces and torques are eliminated analytically. In this step, the technique in [8] differs from the Hooker-Margulies method in choosing joint-centric variable instead of the barycentric variable. For detailed derivation of the equations of motion, see [8]. 

Each beam element in Figure 15 is considered to be inextensible and allowed to twist as well as to bend along its two lateral axes. Rotary inertia and shear deformation of the beam element are ignored for simplicity. The beam bending equations are similar to equations (3) except that there exists a force and torque at the beam tip and that cantilever mode shapes are used. 




Where n denotes the nth mode, 
k denotes the kth link, 
x,y,z, denote components of a vector in the local beam fixed reference frame
P = beam mass/length = constant, 
l = beam length, 
a = linear acceleration at beam root, 
T^k+1,k = internal hinge torque on the kth beam from the (k+1)th beam, 
F^k+1,k = internal hinge force on the kth beam from the (k+1)th beam, 

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