## 2 Method  
### Control Algorithm  
Let us consider a human model and an assistive robot (e.g., prosthesis, avatar, supernumerary limb) as part of a unified kinematic chain, with \(n_{h}\) human joints and \(n_{r}\) robotic joints. The user's intent is defined in terms of reaching a desired goal \(\bar{x}_{e}\) with the robotic end-effector \(x_{e}\), whose pose depends on the joint configuration \(q= \begin{bmatrix}q_{h}&q_{r}\end{bmatrix} ^{T}\), where \(q_{h}\) and \(q_{r}\) represent the human and robotic joint variables, respectively. The reaching error is defined as  
\[e_{e}=\bar{x}_{e}-x_{e}.\]  
Humans often attempt to reach the goal without leveraging the assistance provided by the robotic device. This results in unnatural body movements, which can lead to fatigue for the user [18]. To mitigate this, building on the work presented in [PERSON] et al. [19], the study in [PERSON] et al. [17] introduces a reference frame, \(x_{c}\), attached to a selected body segment (e.g., the shoulder). The target pose of this frame, when the user's body is in a relaxed configuration, is denoted as \(\bar{x}_{c}\), which makes it possible to evaluate the relative displacement of user from their relaxed configuration. The displacement error, expressed in terms of both translation and orientation, is defined as  
\[e_{c}=\bar{x}_{c}-x_{c}.\]  
As outlined in [17], the human and robotic Jacobian matrices for the reaching task are denoted as \(J_{he}=J_{he}(q)=\frac{\partial Q_{c}}{\partial q_{h}}\) and \(J_{re}=J_{re}(q)=\frac{\partial Q_{c}}{\partial q_{e}}\), respectively. Similarly, for the body motions, we have \(J_{hc}=J_{hc}(q)=\frac{\partial Q_{c}}{\partial q_{h}}\) and \(J_{rc}=J_{rc}(q)=\frac{\partial Q_{c}}{\partial q_{r}}\). The overall system kinematics can then be expressed as  
\[ \begin{bmatrix}\dot{e}_{e}\\ \dot{e}_{c}\end{bmatrix} = \begin{bmatrix}\dot{\pi}_{e}\\ \dot{\pi}_{c}\end{bmatrix} - \begin{bmatrix}J_{hc}(q)&J_{rc}(q)\\ J_{hc}(q)&J_{rc}(q)\end{bmatrix}\begin{bmatrix}\dot{q}_{h}\\ \dot{\eta}_{r}\end{bmatrix} . \tag{1}\]  
The control objective is to compute \(\dot{\eta}_{r}\) such that \(e_{e}\) asymptotically approaches zero. This is particularly challenging because the desired target \(\tilde{\pi}_{e}\), and consequently \(e_{e}\), is unknown to the robotic controller. To address this, the solution proposed in [17] integrates both the human and robotic components into a unified dynamic system with state \(x= \begin{bmatrix}e_{e}&e_{c}\end{bmatrix} ^{T}\), control input \(u=\dot{q}_{r}\), and measurable output \(y=e_{c}\). This results in the following time-variant dynamic system:  
\[ \begin{cases}\dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)+Du(t)\end{cases} . \tag{2}\]  
The solution proposed in [17] involves implementing an asymptotic observer to estimate \(e_{e}\). This estimate is then used in a linear quadratic Gaussian (LQG) regulator, where the gain matrix \(K\) is iteratively recomputed during the motion. This control framework is adaptable to a variety of human-robot systems. Indeed, this approach translates user-performed residual motions into the necessary robotic commands for reaching targets. In this way, even slight user motions are sufficient to activate the controller. Once the user observes the robot beginning to move toward the desired target, they can return to their resting position. As highlighted in [17], residual motions are significantly more pronounced when the control is not activated--leaving the user fully responsible for the reaching task--compared to when the control is active, with the robotic device assisting the user in completing the task. This system effectively leverages the user's residual motions without actively amplifying them, as excessive residual motion could lead to pain and poor posture over time.