# Flattening the Curve of Flexible Space Robotics [PERSON]
## 2 Materials and Methods
### Modeling and Analysis of Flexible Space Robots
#### 2.1.1 Rigid Body Dynamics  
_[PERSON]'s theorem [7] allows us to simply invoke [PERSON] [8] and [PERSON]'s equations [9] to fully describe the six degrees of freedom of mechanical motion [38]._  
These relationships were very recently used together with [PERSON]'s treatment of optimization [22] to reveal whiplash compensation for the flexible space robot depicted in Figure 5b [23], in addition to the development last year of the burgeoning field of deterministic optimization [24] that uses these relationships to formulate self-awareness statements that are augmented by either nonlinear adaptive feedback or optimum (in the 2-norm sense) learning, using a reparameterization of the same basic dynamic relationships. The basic relationships relating force to acceleration (per [PERSON]) and torques to angular acceleration (per [PERSON]) are illustrated in Equations (1) and (2), respectively, where the quantities are measured relative to a non-rotating reference frame (called \"inertial\"), while their expression in coordinates of the rotating reference frames are expressed in Equations (3) and (4), respectively. Simply invoking the two relationships comprise [PERSON]'s theorem. \[F|_{inertial}=\left.ma\right|_{inertial}=\left.m\dot{v}\right|_{inertial}=\left.m \ddot{x}\right|_{inertial} \tag{1}\]  
\[\tau_{inertial}=\left.J\alpha\right|_{inertial}=\left.m\dot{\omega}\right|_{ inertial} \tag{2}\]  
\[F=m\ddot{x}+m\frac{d\omega}{dt}\times rt+2m\omega\times vt+m\omega\times( \omega\times rt) \tag{3}\]  
\[\tau=J\dot{\omega}+\omega\times J\omega \tag{4}\]  
Equations (3) and (4) are implemented in this article as the rigid body mode of a flexible space robot designed for in situ replenishment or assembly operations on orbit where:  
\(F\) is the vector sum of the external forces acting on the body;  
\(\tau\) is the vector sum of the external torques acting on the body;  
\(\omega\) is the angular velocity;  
\(\
u t\) is the velocity relative to the rotating reference frames;  
\(rt\) is the position vector of the object relative to the rotating reference frame;  
\(at\) is the acceleration relative to the rotating reference frame;  
\(m\frac{d\omega}{dt}\times r\) is the Euler acceleration;  
\(2m\omega\times vt\) is the Coriolis acceleration;  
\(m\omega\times(\omega\times rt)\) is the centrifugal acceleration;  
\(J\) is the mass moments of inertia. Figure 5: (**a**) Simulations created in SIMULINK, to compare the disparate approaches. (**b**) Flexible space robot subsystem depicted in (**a**). Section 2.1.1 indicates the same results were obtained by [PERSON]. [PERSON]'s principle implies that the Lagrange method must be as follows:  
\[\text{Minimize }Cost[x(g),u(g),t]=\int_{t_{0}}^{t_{f}}L\big{(}\dot{x},x\big{)}dt \tag{5}\]  
\[\text{Subject to}\quad\dot{x}=u \tag{6}\]  
where  
\[L=\text{{Kinetics energy}}-\text{{potential energy}}=\frac{1}{2}mV^{2}+ mgh=\frac{1}{2}m\dot{x}^{2}+mgh \tag{7}\]  
\[\text{Writing the control Hamiltonian: }H(\lambda,x,u)\equiv L(x,u)+\lambda u \tag{8}\]  
\[\text{[PERSON]'s minimization condition: }\frac{\partial H(\lambda,x,u)}{\partial u}=\frac{ \partial L(x,u)}{\partial u}+\lambda=0 \tag{9}\]  
Differentiating leads to a minimization equation expression that may be combined with the Adjoint equation to produce the Euler-Lagrange equation, which by algebraic manipulation becomes Newton's Law. \[\tau_{inertial}=\left.J\alpha\right|_{inertial}=\left.m\dot{\omega}\right|_{ inertial} \tag{2}\]0  
\[\begin{array}{c}\frac{d}{dt}\bigg{(}\frac{\partial L(x,u)}{ \partial u}\bigg{)}-\frac{\partial L(x,u)}{\partial x}=\frac{d}{dt}\bigg{(} \frac{\partial L(x,u)}{\partial u}\bigg{)}-\frac{\partial L(x,u)}{\partial x} =\frac{d}{dt}\big{(}m\dot{x}\big{)}-\frac{\partial L(x,u)}{\partial x}\\ =m\dot{x}-\frac  
A higher number of integrators in the denominator is called a \"pole excess\" and the high frequency phase lag asymptotically approaches this pole excess with increasing frequency.