# Tutorial Review on Space Manipulators for Space Debris Mitigation
## 5 Space Manipulator Operations--Evolution from Teleoperation to Autonomy
Adding a dissipative term to both master and slave PD controllers introduces passivity to guarantee stability [99]:  
\[\tau_{m}(t)=-k_{p}e_{m}(t)-k_{v}\dot{e}_{m}(t)-(k_{d}+\Delta k_{d})\dot{q}_{m}(t) \tag{9}\]  
\[\tau_{s}(t)=-k_{p}e_{s}(t)-k_{v}\dot{e}_{s}(t)-(k_{d}+\Delta k_{d})\dot{q}_{s}(t) \tag{10}\]  
where \(e_{m}(t)=q_{s}(t-\Delta t)-q_{m}(t)\) = master position error, \(\dot{e}_{m}(t)=\dot{q}_{s}(t-\Delta t)-\dot{q}_{m}(t)\) = master velocity error, \(e_{s}(t)=q_{m}(t-\Delta t)-q_{s}(t)\) = slave position error, \(\dot{e}_{s}(t)=\dot{q}_{m}(t-\Delta t)-\dot{q}_{s}(t)\) = slave velocity error, \(k_{d}=\Delta tk_{p}\) = dissipative gain, \(\Delta k_{d}\) = additional damping, \(\Delta t\) = time delay. There is also a considerably more complex adaptive version of passivity stable to positional drift between master and slave [100]. The master/slave joint torques are determined by the adaptive regressor and synchronised coordinating joint torques \(\tau_{m}^{c}\) and \(\tau_{s}^{c}\) for master and slave arms respectively:  
\[\tau_{m}(t)=\tau_{m}^{c}-Y_{m}(\theta_{m},\dot{\theta}_{m})\dot{\phi}_{m} \tag{11}\]  
\[\tau_{s}(t)=\tau_{s}^{c}-Y_{s}(\theta_{s},\dot{\theta}_{s})\dot{\phi}_{s} \tag{12}\]  
where \(\tau_{m}^{c}(t)=K(r_{s}(t-\Delta t)-r_{m}(t))\), \(\tau_{s}^{c}(t)=K(r_{m}(t-\Delta t)-r_{s}(t))\), \(r_{l}(t)=\dot{\theta}_{l}+\lambda\theta_{l}\) = reference position tracking, Y\({}_{l}\)(.) = known dynamic parameters, \(\dot{\phi}_{l}\) = estimated unknown dynamic parameters, \(l\) = master or slave, \(\lambda\) = scale constant \(>\) 0, \(K\) = gain constant \(>\) 0. Adaptive bilateral hybrid position/force control synchronisation of master-slave teleoperation may be enforced through the imposition of holonomic constraints [101]. These schemes require a constant time delay \(\Delta t\). For varying time delays, wave variables compute energy flows to determine excess energy and eliminate force reflection waves between the remote and operator robotic devices by matching their characteristic impedances [102; 103]. Velocity and force at the master and slave are encoded into time-delay robust wave variables which are subsequently transmitted between master and slave manipulators:  
\[u_{m}(t)=\frac{Z\dot{q}_{m}(t)+F_{m}(t)}{\sqrt{2Z}} \tag{13}\]  
\[v_{s}(t)=\frac{Z\dot{q}_{s}(t)-F_{s}(t)}{\sqrt{2Z}} \tag{14}\]  
where \(Z\) = wave impedance, which tunes the degree of stiffness/compliance. Wave-based communications occurs at a natural frequency \(w=\frac{1}{\Delta t}=\sqrt{\frac{k}{D(0)}}\), where \(\Delta t\) = time delay. As wave variables encode velocity and force, so wave integrals encode position x and momentum p:  
\[x=\frac{fx_{c}}{f-z_{c}}\text{ and }y=\frac{fy_{c}}{f-z_{c}} \tag{6}\]0  
\[x=\frac{fx_{c}}{f-z_{c}}\text{ and }y=\frac{fy_{c}}{f-z_{c}} \tag{6}\]1  
Wave variables automatically exhibit passivity. Wave variables can be combined with Smith predictors to impart stability to the Smith predictor. More recent approaches have incorporated sliding mode controllers and generalised predictive controllers [104]. It has been suggested that passivity-based methods offer guaranteed stability to time delays over shared compliance control which can only tolerate maximum time delays of the order of human reaction speed of ~0.1-0.2 s, but experimental results dispute this limitation [105]. The tele-impedance approach is an alternative to bilateral force-reflecting teleoperation which combines position reference with a stiffness reference, the latter being determined from non-intrusive EMG (electromyographic) measurements from the human operator's arm [106]. Spring-like compliance at the remote end effector in contact-and-grasp tasks on hard environmental objects dramatically improves teleoperation performance and alleviates the problem of rapid buildup of transient forces on collision [107; 108]. Specific applications of teleoperation with time delays of 5-10 s are discussed in [109], which concludes that predictive simulation is essential in time-delayed teleoperation.