# Hardware Development and Safety Control Strategy Design for a Mobile Rehabilitation Robot
## 3 Dynamic Models
### The Dynamic Model of AMBSS  
The medical harness is used to connect the AMBSS frame to the waist of the human body and can be mathematically modeled as a mass-spring-damping model. From the viewpoint of dynamics, the AMBSS shown in Figure 1 can be approximately modeled as depicted in Figure 2. The center of gravity trajectory in the vertical direction of the sagittal plane can be expressed as a general function of time \(y_{p}(t)\) during gait training on the MRR. Let \(f_{r}\) be the total vertical reaction force exerted on the human foot by the left and right sides and \(a(t)=\ddot{y}_{p}\) be the body acceleration in the vertical direction; then, the dynamics equation for the AMBSS can be represented as:  
\[m_{p}\ddot{y}_{p}+b_{p}(\dot{y}_{p}-\dot{y}_{p0})+k_{p}(y_{p}-y_{p0})+m_{p}g=f_ {r}+f_{pc}, \tag{6}\]  
where \(m_{p}\) is the mass of the human body, \(b_{p}\) and \(k_{p}\) denote the damping coefficient and stiffness of the human body, \(y_{p0}\) is the original position of the center of gravity in the vertical direction of the sagittal plane, \(g\) is the gravity acceleration, \(f_{r}\) is the total vertical reaction force exerted to the human foot by the left and right sides, and \(f_{pc}\) is the support force provided by the AMBSS to the human in the vertical direction. Figure 2: Modeling of the active mass-offload AMBSS. In this study, a linear actuator comprising a ball screw and a DC motor is used to lift the AMBSS mechanism. The linear actuator used by AMBSS is a one-dimensional device; therefore, the force control target was simplified to:  
\[f_{pc}=m_{p1}(\ddot{y}_{p}+g), \tag{7}\]  
where \(m_{p1}\) is the expected reduction in the weight. Substituting (7) into (6) yields:  
\[m_{p}\ddot{y}_{p}+b_{p}(\dot{y}_{p}-\dot{y}_{p0})+k_{p}(y_{p}-y_{p0})+m_{p}g+2 \mu m_{p2}(g+\ddot{y}_{m})-2\mu f_{pm}=f_{r}, \tag{8}\]  
where \(m_{p2}\) is the mass of the AMBSS unilateral mechanism, \(f_{pm}=\big{[}\frac{f_{pc}}{2}\div(\frac{y_{m}-y_{p}}{\sqrt{x_{p}^{2}+(y_{m}- y_{p})^{2}}})^{2}+m_{p2}(g+\ddot{y}_{m})\big{]}\) is the force provided by the DC motors, \(\mu=1/(\frac{y_{m}-y_{p}}{\sqrt{x_{p}^{2}+(y_{m}-y_{p})^{2}}})^{2}\), \(y_{m}\) is the height on either side of the AMBSS, \(x_{p}\) is the horizontal distance from the two support columns on both sides of the human body center. According to an earlier study [35], the dynamic equations of a DC motor can be expressed as:  
\[\left\{ \begin{array}{l}V_{a}=k_{a}w+iR_{a}+L_{a}\frac{\ddot{m}}{dt}\\ \tau_{m}=K_{T}i\end{array} \right., \tag{9}\]  
where \(V_{a}\) is the applied voltage, \(w\) is the angular velocity, \(i\) is the current flowing through the windings, \(L_{a}\) represents the actuator circuit inductance, \(R_{a}\) is the total armature circuit resistance, \(k_{a}\) is the electrical constant, \(K_{T}\) is a proportionality constant of torque and \(\tau_{m}\) is the electromagnetic torque of the motor. Concerning the external torque \(\tau_{l}\) from a payload, the mechanical behavior is further obtained as:  
\[J_{m}\dot{w}+B_{m}w=\tau_{m}-\tau_{l}, \tag{10}\]  
where \(J_{m}\) is the rotor moment of inertia and \(B_{m}\) is the frictional coefficient. The objective of AMBSS is to keep the unloading force precise and provide feedback on the measured unloading force. Therefore, if the unloading force required for AMBSS is \(f_{pc}\), the torque for the unilateral payload is provided as:  
\[\tau_{l}=\frac{l_{0}}{2\pi}\bigg{[}\frac{f_{pc}}{2}\mu^{2}+m_{p2}(g+\ddot{y}_{ m})\bigg{]}, \tag{11}\]  
where \(l_{0}\) is the length of the ball screw lead. Generally, linear actuators are used for load torque control to reduce the angular velocity as the load torque increases.