## 3 Nonlinear Error Dynamics and Control Strategy  
The error dynamics are derived in the following for the difference between the orthogonal projection of the trajectory foot point and the dynamic model reference point (see Figure 1) [8; 10]:  
\[\left(\vec{\mathbf{r}}^{FR}\right)_{T}=\left(\vec{\mathbf{r}}^{R}\right)_{T}- \left(\vec{\mathbf{r}}^{F}\right)_{T}=\left[\Delta\,x_{T}\quad\Delta\,y_{T} \quad\Delta\,z_{T}\right]^{T} \tag{4}\]  
Based on Equation (4), the velocity deviation can be calculated by deriving Equation (4) with respect to the trajectory frame and applying the Euler differentiation rule (it should be noted that we do not need to derive with respect to an inertial frame in this case, as the position deviation is not based on Newtonian mechanics. Furthermore, the velocity is specified kinematically, and thus has an additional index \(K\) added inside the bracket) [8; 10]:  
\[\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^{T}\left(\vec{\mathbf{r}}^{FR} \right)_{T}=\left(\vec{\mathbf{v}}_{K}^{FR}\right)_{T}^{T}=\left[\Delta\,u_{ T}\quad\Delta\,v_{T}\quad\Delta\,w_{T}\right]^{T}=\left(\vec{\mathbf{v}}_{K}^{FR} \right)_{T}^{N}+\left(\vec{\mathbf{\omega}}_{K}^{TN}\right)_{T}\times\left( \vec{\mathbf{r}}^{FR}\right)_{T} \tag{5}\]  
Then, the acceleration deviation is consequently given by deriving Equation (5) once more in the trajectory frame as well as applying the Euler differentiation rule to introduce measurable quantities [8; 10]:  
\[\left(\vec{\mathbf{a}}_{K}^{FR}\right)_{T}^{TT} =\left[\Delta\,a_{x,T}\quad\Delta\,a_{y,T}\quad\Delta\,a_{z,T} \right]^{T}=\underbrace{\underbrace{\left(\vec{\mathbf{a}}_{K}^{R}\right)_{T}^ {NK}+\left(\vec{\mathbf{\omega}}_{K}^{TK}\right)_{T}\times\left(\vec{\mathbf{ v}}_{K}^{R}\right)_{T}^{N}-\left(\vec{\mathbf{a}}_{\mathrm{des},K}^{F}\right)_{T}^{NT}} _{\left(\vec{\mathbf{a}}_{K}^{R}\right)_{T}^{NT}}-\left(\vec{\mathbf{a}}_{ \mathrm{des},K}^{F}\right)_{T}^{NT}} \tag{6}\] \[+\left(\vec{\mathbf{\omega}}_{K}^{TN}\right)_{T}\times\left(\vec{ \mathbf{v}}_{K}^{FR}\right)_{T}^{T}+\left(\vec{\mathbf{\alpha}}_{K}^{TN} \right)_{T}^{T}\times\left(\vec{\mathbf{r}}^{FR}\right)_{T}\]  
Here, the lead acceleration \(\left(\vec{\mathbf{a}}_{K}^{FR}\right)_{T}^{NT}\) is split (see Equation (4)) into the a part with respect to the reference point, \(\left(\vec{\mathbf{a}}_{K}^{R}\right)_{T}^{NT}\), and a part with respect to the foot point (the \"desired\" acceleration), \(\left(\vec{\mathbf{a}}_{\mathrm{des},K}^{F}\right)_{T}^{NT}\). This is used in the following to introduce the dynamic equations from Section 2 into the jerk dynamics derived next. Additionally, the desired acceleration/jerk is an additional design variable in the commanded trajectory specification that can be used to achieve a desired behavior. Finally, the jerk error can be calculated by deriving Equation (6) one final time with respect to the trajectory frame:  
\[\left(\vec{\mathbf{\eta}}_{K}^{FR}\right)_{T}^{TTT} =\left(\vec{\mathbf{\eta}}_{K}^{R}\right)_{T}^{NKT}+\left(\vec{ \mathbf{\alpha}}_{K}^{TK}\right)_{T}^{T}\times\left(\vec{\mathbf{v}}_{K}^{R} \right)_{T}^{N}+\left(\vec{\mathbf{\omega}}_{K}^{TK}\right)_{T}\times\left( \vec{\mathbf{a}}_{K}^{R}\right)_{T}^{NT}-\left(\vec{\mathbf{\eta}}_{\mathrm{ des},K}^{F}\right)_{T}^{NTT} \tag{7}\] \[+\left(\vec{\mathbf{\alpha}}_{K}^{TN}\right)_{T}^{T}\times\left( \vec{\mathbf{r}}^{FR}\right)_{T}^{T}+\left(\vec{\mathbf{\omega}}_{K}^{TN} \right)_{T}^{T}\times\left(\vec{\mathbf{\omega}}_{K}^{FR}\right)_{T}^{TT}\]  
Now, the Euler differentiation rule may be applied once more to transform the derivatives with respect to the trajectory frame to a more suitable representation for analysis and implementation:  
dynamics by commanding a stabilizing pseudocontrol. Thus, a proper choice of these matrices ensures not only the stability of the trajectory controller, but also provides disturbance rejection. It should be noted that the nonlinear error controller in Equations (9)-(11) is derived for the case of trajectory control here.