# Assistance of a Person with Muscular Weakness Using a Joint-Torque-Assisting Exoskeletal Robot
## 4 Joint Torque Assistance Control
### Design of an Assistive Joint-Torque Profile  
Although human motion is not generated by a rotary actuation on a single axis joint, the sagittal joint torques (i.e., joint moments) can be estimated by using inverse dynamics. During stance phases, both hip and knee joints need extension torques to resist and propulsion, and flexion torques required to push forward the swing side leg. People with incomplete paraplegia have a lack of muscle strength, making it difficult to push the forelimbs forward, and have a short swing period. Therefore, once the initial swing is detected, the assistive torque should be provided for an appropriate amount of time so that the wearer can use the assistive torque through repeated practice. For individual gait training, it is necessary to easily modify the assistive torque profile without compromising the continuity of the trajectory function. Therefore, polynomial functions were used in this research to find the equations for joint torque trajectories. In addition to the conditions mentioned above, the derivatives at the beginning and end of the gait should be zero in order to smoothly switch to other motions such as standing. No other inflection points should exist between the main points. Therefore, we divided the trajectory to four sections based on the main points, and the third order polynomials were derived for each section. While \(p_{i}=\left[ \begin{smallmatrix}x_{i}\\ y_{i}\end{smallmatrix} \right]\) means the representative point of the torque trajectory, where \(x_{i}\) is the percentage of gait cycle and \(y_{i}\) is the reference torque, the coefficients of polynomials can be derived as below. \[ \begin{bmatrix}y_{i}\\ y_{i+1}\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}x_{i}^{3}&x_{i}^{2}&x_{i}&1\\ x_{i+1}^{2}&x_{i+1}&x_{i+1}&1\\ 3x_{i}^{2}&2x_{i}&1&0\\ 3x_{i+1}^{2}&2x_{i+1}&1&0\\ \end{bmatrix}\begin{bmatrix}a_{i,3}\\ a_{i,2}\\ a_{i,1}\\ a_{i,0}\end{bmatrix} , \tag{2}\]  
Figure 7: The overall control structure of Angel Legs. the inverse of which is  
\[ \begin{bmatrix}a_{i,3}\\ a_{i,2}\\ a_{i,1}\\ a_{i,0}\end{bmatrix} = \begin{bmatrix}x_{i}^{3}&x_{i}^{2}&x_{i}&1\\ x_{i+1}^{3}&x_{i+1}^{2}&x_{i+1}&1\\ 3x_{i}^{2}&2x_{i}&1&0\\ 3x_{i+1}^{2}&2x_{i+1}&1&0\end{bmatrix} ^{-1} \begin{bmatrix}y_{i}\\ y_{i+1}\\ 0\\ 0\end{bmatrix} . \tag{3}\]  
Therefore, the joint torque profile of hip joint can be expressed as  
\[f(x)= \begin{cases}a_{1,3}x^{3}+a_{1,2}x^{2}+a_{1,1}x+a_{1,0}&\text{if}&x_{1} \leq x<x_{2}\\ a_{2,3}x^{3}+a_{2,2}x^{2}+a_{2,1}x+a_{2,0}&\text{if}&x_{2}\leq x<x_{3}\\ a_{3,3}x^{3}+a_{3,2}x^{2}+a_{3,1}x+a_{3,0}&\text{if}&x_{3}\leq x<x_{4}\\ a_{4,3}x^{3}+a_{4,2}x^{2}+a_{4,1}x+a_{4,0}&\text{if}&x_{4}\leq x<x_{5}\end{cases}  \tag{4}\]  
Figure 8 shows the assistive torque profile based on the gait phases. The representative points \(h_{i}^{\prime}s\) and \(k_{i}^{\prime}s\), i.e., the maximum points of flexion and extension torques of hip and knee joints, can be individually set. The assistance instance is based on the users' gait phases. For the right leg, (a) in Figure 8 is the same as Gait index \(=1\) (i.e., right contact), and (b) is the same as Gait index \(=2\) (i.e., left toe-off). In this case, the time between (a) and (b) can not be predetermined since it truly depends on the users intent. When the user tries to move the hind leg for a swing motion, then Gait index \(=2\) is detected and the assistive torque corresponding to those between (b) and (c) are applied. Although the time of the (c) is not exactly the same as the actual Gait index \(=3\), which also depends on the user's intent, the same assistive torque of time (c) is applied until (d), i.e., Gait index \(=4\). Similarly, the torque of (e) (i.e., Gait index \(=1\)) continues until the next gait event Gait index \(=2\). The amplitudes of the peak torque can be adjusted depends on the patient's muscular strengths of each leg and desired assistance level.