## 4 Control and Actuation Strategies  
### Open-Loop Control  
One of the goals of the paper is to propose a pipeline, where a flexible manipulator will be modeled and fabricated to match the \"desired kinematics\" of a human leg. The approach is iterative, where the model and prototype inform each other and adapt to recapitulate the human leg geometry. Figure 4 shows an overview of the propose design strategy. We used the FD solver to provide the excitation pattern (Figure 5) that manipulates the hip and knee joints (Figure 6) throughout four the stages of gait in the sagittal plane shown in Figure 7. Stage 1: _Heel Lift_, starts from a neutral position, and contracts the hamster-inspired cable to initiate knee flexion backward. Stage 2: _Extension Forward_, contracts the iliopsoas-inspired muscle causing the entire tensegrity structure to move forward, while the gluteus maximus-inspired and hamstering-inspired cables release the posterior hip and hamstering. Stage 3: _Step Through_, implements the combination of \"leg-lift\" and \"swing\". When the iliopsoas flexes the lower limb at the hip, it drives the entire leg backward until it is positioned behind the body. Stage 4: _Resting Position_, restores the tensegrity structure to its neutral position. Once in the _Resting Position_, one cycle of gait has been completed, and if desired, the structure can continue the gait cycle repeatedly.  
Modeling the tensegrity hip and knee joint in OpenSim 3.0 provided analysis on both passive and active connections for our physical prototype. Forces imported into the FD tool generate motions for each model. Each stage of gait excites one active element to contract at a time while the others remain passive or release tension. Each actuator in our physical model correlates with one of the three prominent active muscles in a human leg. The tangential force generated by the cable-driven actuators (\(F_{T_{max}}\)) during each stage of gait is computed as:  
\[F_{T_{max}}=\frac{\tau}{r} \tag{6}\]  
where \(r\) is the radius of the spool attachment for the motors, and \(\tau\) is the stall-torque. Each cable-driven actuator contraction follows a sequential state machine.  
Figure 5: The applied activations by the three virtual “muscles” in the OpenSim 3.0 simulation environment (full line), and the intensity from the cable-driven actuators on the prototype (dotted lines). The varying levels of actuation (i.e., applied forces) input into the simulated models generalize an approximation for how to actuate the muscle-inspired cables in the prototype. Labels (a–c) indicate where the applied contracting muscles release.  
Figure 6: **(A)** Applied forces from all the active elements on the simulated model at varying sizes (50%, 100%, and 150% scales) show that a smaller model requires similar amounts of forces to replicate the four stages of gait. **(B)** Simulated moments (N-m) on the hip joint throughout the four stages of gait estimate the required actuators. **(C)** Power consumption (mW) of the structure is similar to the simulation and the physical prototype performing the four stages of gait. The average of the few trials (solid green line) for the physical prototype created the upper and lower error bars. As we enlarge the model size, the forces necessary to produce the same motion increase. Increasing the size of the simulated models demonstrated a significant spike in overall energy consumption. It is worth noting that the 100% scaled simulation model is the same size as the physical prototype.  
Figure 7: Outline of the active elements during the four stages of gait. Each pull (green) activates the cable-driven actuator to contract and shorten the muscle. The release of muscles (blue) reverses the direction of the actuator. In (Stage 4) the orange shows how the angles for the hip and knee were tracked.