# A Human-Inspired Control Strategy for Improving Seamless Robot-To-Human Handovers
## 2 Design of the Robotic Human-Inspired Control Strategy for HRH
### Kinematic Model of the HSR  
This section briefly explains the kinematic model of the HSR, which relates to forward and inverse kinematics. In the case of the forward kinematics, the location and pose of the robot end-effector are controlled by the given links and joints of the HSR robot. The Denavit-Hartenberg parameters evaluation is used to achieve the kinematic chain of the robot rigid bodies. Based on the studies by researchers [39; 40; 41], the mechanical schematic diagram of a dual-wheel caster-drive robot base and the robot's joint configuration is shown in Figure 3a. The following notation is used in this analysis: the radius of each wheel (\(r\)), the angle of the pivot axis (\(\theta_{H}\)), the wheel tread (\(W\)), the displacement between the centre axis and the axle centre (\(H\)), the inverse kinematics (\(f\)), the reference value of the tip pose (\(T_{c}\)), the degree of freedom of the base (\(\theta_{2}\)), the torso lift height (\(l_{3}\)), the shoulder pitch angle (\(\theta_{4}\)), the length of arm (\(l_{52}\)), the length of the wrist(\(l_{82}\)) and finally, the joint angles of the arm (\(\theta_{5},\theta_{6},\theta_{7}\)). These are defined in Figure 3b. The robotic dynamic model is essential for describing the HSR robot's position, orientation and motions of the wheels. It can be used for analysing and synthesizing the overall dynamic behaviour of the robot. The robot itself has three degrees of freedom on the \(x\)-\(y\) plane and can perform forward-reverse movement and left-right movements. As reviewed [39; 40; 41], the kinematic equations and relations between robot velocity and various geometry specifications are expressed by the Transformation matrices from linear velocity on the global frame to linear on the robot frame as the following. \[\left[ \begin{array}{c}\omega_{R}\\ \omega_{L}\\ \omega_{H}\end{array} \right]=\left[ \begin{array}{cc}\frac{\xi cos\theta_{H}- \frac{rH}{W}sin\theta_{H}}{\beta}&\frac{\xi cos\theta_{H}+\frac{rH}{W}sin\theta _{H}}{\beta}&0\\ \frac{r}{W}sin\theta_{H}&\frac{rH}{W}cos\theta_{H}&0\\ \frac{r}{W}&\frac{r}{W}cos\theta_{H}&1\end{array} \right]^{-1}\left[ \begin{array} []{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{array} \right]. \tag{4}\]  
By assuming that translational and rotational transforms on \(x\)-\(y\)-\(z\) axes are represented as \(T_{t}(x,y,z),~{}R_{x},R_{y},R_{z}\) and \(R_{z}\), respectively, forward kinematics of the HSR arm (\(T_{HSR}\)) can be written as:  
\[T_{HSR}=R_{z}(\theta_{2})T_{e}(0,0,l_{3})R_{y}(\theta_{4})T_{e}(0,0,l_{52})R_{ z}(\theta_{5})R_{y}(\theta_{6})R_{z}(\theta_{7})T_{e}(0,0,l_{82}). \tag{5}\]  
According to the kinematic of the HSR arm, the base yaw angle (\(\theta_{2}\)) can be calculated using the relative wrist position (\(x\), \(y\) and \(z\)) as:  
\[\left[x~{}y~{}z~{}1\right]^{T}=fT_{e}(0,0,-l_{82})\left[0~{}0~{}0~{}1\right]^{T} \tag{6}\]  
\[\theta_{2}=atan2(y,x) \tag{7}\]  
Analysing torso lift height (\(l_{3}\)) and shoulder pitch angle (\(\theta_{4}\)) gives:  
\[\left[ \begin{array}{c}l_{3}\\ \theta_{4}\end{array} \right]=\left[ \begin{array}{c}z-\sqrt{l_{52}^{2}-x^{2} -y^{2}}\\ arscin\Big{(}\sqrt{x^{2}+y^{2}}/l_{52}\Big{)}\end{array} \right]\text{or}\left[ \begin{array}{c}l_{3}\\ \theta_{4}\end{array} \right]=\left[ \begin{array}{c}z-\sqrt{l_{52}^{2}-x^{2} -y^{2}}\\ \pi-arcsin\Big{(}\sqrt{x^{2}+y^{2}}/l_{52}\Big{)}\end{array} \right]. \tag{8}\]  
Investigating the joint angles of the arm (\(\theta_{5},\theta_{6},\theta_{7}\)) based on the matrix \(T_{R}\) offers:  
\[T_{R}=T_{e}(0,0,-l_{52})R_{y}(-\theta_{4})T_{e}(0,0,-l_{3})R_{z}(-\theta_{2}) fT_{e}(0,0,-l_{82}), \tag{9}\]  
\[\left[ \begin{array}{c}\theta_{5}\\ \theta_{6}\\ \theta_{7}\end{array} \right]=\left[ \begin{array}{c}atan2(R_{23},R_{13})\\ arccos(R_{33})\\ atan2(R_{32},-R_{31})\end{array} \right] \tag{10}\]  
However, to allow the HSR to interact naturally with a human and to facilitate the dexterous transfer of objects promptly, hybrid inverse kinematics proposed by [PERSON] et al. [41] was implemented.