# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 11 Dual-Manipulator Control
This will require compliant control to control the object motion and the internal stresses, e.g., Dual-Arm Generalised Compliant Motion (DAGCM) [253]. An example of a cooperative task is the use of multiple mobile platforms, each mounting a manipulator--this requires a combination of vehicle-manipulator coordination and multiple manipulator coordination for which the operational space formulation, virtual null space internal forces, and a multi-manipulator object grasp matrix have been combined to yield solutions [250]. A similar space application example involves multiple (minimum of three to provide full control of an object) spacecraft with single manipulators handling a common object for assembly tasks, in this case involving coordination between the manipulators and thrusters [254]. Although there are similarities, these are different to the case of multi-manipulators on a single spacecraft that concerns us here [255]. Post grappling, a closed kinematic chain reduces the number of DOF so the excess actuators permit minimisation of a weighted function of actuator torques while a fifth order polynomial reference trajectory for manipulator motions reduces reaction forces on the spacecraft [256]. Multiple arms on a spacecraft comes at a cost of vastly increased computational complexity of coordination. The Jacobian for a dual manipulator system is an \(18\times 18\) matrix assuming 6 DOF manipulators because, unlike terrestrial dual manipulators, they are coupled dynamically through the base. For closed chain configurations, constraints may be applied kinematically through the Jacobians [257]. Nevertheless, dual arms introduce greater challenges in avoiding dynamic singularities [258]. This reinforces the necessity of simplifying controllers for space-based manipulators using dedicated attitude control permitting decoupling of linear and angular momentum aspects [259]. The use of dual or more arms introduces considerable complexities to path planning, particularly for cooperative tasks. The potential field method is particularly promising in this regard to prevent collisions. Nevertheless, there are several simplifying assumptions that can be imposed: (i) during grappling, the arms exhibit mirroring motion through the bisecting plane; (ii) during grappling and subsequent to grappling, objects may be passed between arms while ensuring that no manipulator crosses the bisecting midplane (Zambesi bridge configuration); (iii) a fixed arm can maintain docking with the target while the other performs servicing tasks. Multiple manipulators grappling a single object are characterised by the grasp matrix which defines the object internal forces [260]:  
\[\left( \begin{array}{c}F_{ext}\\ F_{int}\end{array} \right)=G\left( \begin{array}{c}f_{1}\\ \vdots\\ f_{n}\end{array} \right)\text{where }G=\left( \begin{array}{ccc}G_{1}^{ext}&\cdots&G_{n}^{ ext}\\ G_{1}^{int}&\cdots&G_{n}^{ext}\end{array} \right)\text{, }G_{i}=\left( \begin{array}{c}I_{3}&O_{3}\\ -p_{i}\times&I_{3}\end{array} \right) \tag{138}\]  
Each manipulator grip is defined by its own grasp submatrix. Internal grasp forces defined by the constraint polygon formed by the fingers determine the stability of the grasp. For two manipulators grasping one object, the grasp matrix defines the transformation of forces at the grasp points (g\({}_{i}\)) to the centre of mass (o):  
\[G=\left( \begin{array}{ccc}I_{3}&O_{3}&I_{3}&O_{3}\\ -S(p_{\alpha 1})&I_{3}&-S(p_{\alpha 2})&I_{3}\end{array} \right) \tag{139}\]  
where \(O_{3}=3\times 3\) null matrix, \(I_{3}=3\times 3\) identity matrix, \(p_{\alpha i}=\) position vector from grasp point g\({}_{i}\) to o (virtual stick), \(S=\) cross product matrix. Hybrid position/force control of dual arms afford direct control of internal forces with cooperative load distribution. Adaptive versions using derivatives of position and force are feasible. For soft contact, impedance control and passivity-based control are suitable. An observer-based approach to impedance control in compliant capture of a spacecraft has been explored [261].