# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 7 Freeflyer Manipulator Dynamics
We can express the Lagrangian (equating to kinetic energy) explicitly as a multibody system in space (ignoring any external forces):  
\[L=T=\sum_{i=0}^{n+1}\frac{1}{2}(r_{i}^{T}m_{i}\dot{r}_{i}+w_{i}^{T}R_{i}^{T}I_ {i}R_{i}w_{i}) \tag{51}\]  
The dynamically equivalent manipulator approach yields:  
\[T=\sum_{i=1}^{n+1}\frac{1}{2}({r^{\prime}}_{i}^{T}m^{\prime}_{i}\dot{r}_{i}+w ^{T}_{i}R^{\prime}_{i}^{T}I_{i}R^{\prime}_{i}w^{\prime}_{i}) \tag{52}\]  
where \({m^{\prime}}_{i}=\frac{{m_{i}(\sum_{j=0}^{n+1}m_{j})}^{2}}{{\sum_{j=0}^{n+1}m _{j}}}\), \(\lambda_{1}=\frac{{m_{1}}{s_{1}}}{{\sum_{j=0}^{n+1}m_{j}}}\), \(\lambda_{i}=-s_{i}\left(\frac{{\sum_{j=0}^{n}m_{j}}}{{\sum_{j=0}^{n+1}m_{j}}} \right)+r_{i}^{\left(\frac{{\sum_{j=0}^{n+1}m_{j}}}{{\sum_{j=0}^{n+1}m_{j}}} \right)}\)  
The Lagrangian must be converted into a form to yield joint torques. The Lagrange-Euler equation for manipulator torques in space may be expressed as (156):  
\[\tau=\left( \begin{array}{c}z_{i-1}^{T}p_{i-1}^{*}\times\\ z_{i-1}^{T}\end{array} \right)\left( \begin{array}{c}F_{0}\\ N_{0}-\sum_{k=0}^{n+1}p_{k}^{*}\end{array} \right)-D(\theta)\ddot{\theta}-C( \theta,\dot{\theta}) \tag{53}\]  
\[ \begin{array}{c}\text{where }D(\theta)=z_{i-1}^{T}\sum_{k=0}^{i-1}\left(m_{k}p_{i-1 }^{*}\times I_{k}^{L}+I_{k}I_{k}^{R}\right)\text{, }C(\theta,\dot{\theta})=z_{i-1}^{T}\sum_{k=0}^{i-1}(m_{k}p_{i-1}^{*}\times I _{k}^{L}+\\ I_{k}I_{k}^{R}+p_{k}^{*}\times I_{k}I_{k}^{R})\end{array}  \tag{54}\]  
This is a highly complex computation. The terrestrial Lagrange-Euler dynamics formulation has a computational complexity of \(\left(\frac{128}{3}\right)n^{4}+\left(\frac{512}{3}\right)n^{3}+\left(\frac{8 44}{3}\right)n^{2}+\left(\frac{76}{3}\right)n\) multiplies and \(\left(\frac{98}{3}\right)n^{4}+\left(\frac{781}{3}\right)n^{3}+\left(\frac{63 7}{3}\right)n^{2}+\left(\frac{107}{3}\right)n\) additions compared with the equivalent terrestrial Newton-Euler dynamics approach with complexity of \(132n\) multiplies and \((111n-4)\) additions. A space-based version will be even more complex. The Lagrange-Euler method also does not express constraint equations such as reaction forces between bodies which can be expressed explicitly in the Newton-Euler formulation--this is explicitly required for the compensation of reaction moments on the spacecraft base at the manipulator mount point to stabilise spacecraft attitude. The manipulator may be exploited to identify the spacecraft inertia properties through measurement of both spacecraft acceleration and velocity permitting the use of multibody Newton-Euler equations (157):  
\[H=Iw=H_{0}+\int_{0}^{n}(\sum\tau+\sum{(r_{i}\times F_{i})})dt \tag{55}\]  
where \(H_{0}=\) initial angular momentum. The Newton-Euler recursive dynamics equations for a manipulator progress from the base to the end effector for the velocity and acceleration for each link and then from the end effector to the base for the forces and torques of each link and joints. The forward recursion loop is given by:  
\[w_{i}=w_{i-1}+z_{i}\dot{\theta}_{i}\text{ where }z_{i}=\left( \begin{array}{ ccc}0&0&1\end{array}