# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 2 Formulation of Trapezoidal Velocity Profile  
The trapezoidal velocity profile, which is also referred to as the linear segment with parabolic blend (LSPB) trajectory, is characterized by three distinct phases: acceleration, cruise, and deceleration. The velocity curve over time resembles a trapezoid, as shown in Figure 1, where \(t_{0}\), \(t_{1}\), \(t_{2}\), and \(t_{3}\) indicate the switching times of each phase. In the acceleration phase [\(t_{0},t_{1}\)], the system starts from rest and accelerates at a constant rate, such that the velocity increases linearly over time until it reaches the maximum velocity. In the cruise phase [\(t_{1}\), \(t_{2}\)], the system moves at a constant maximum velocity. In the deceleration phase [\(t_{2}\), \(t_{3}\)], the system decelerates at the same rate as the acceleration phase until the system comes to a stop. The acceleration profile features a bang-off-bang type, which is described by the following piecewise function:  
\[\tilde{q}(t)=\left\{\begin{array}{ll}a,&t_{0}\leq t\leq t_{1}\\ 0,&\text{else}\\ -a,&t_{2}\leq t\leq t_{3}\end{array}\right. \tag{1}\]  
where \(a\) is the peak acceleration. Let \(T_{a}\) and \(T_{v}\) denote the time for the acceleration/deceleration phase and the time for the cruise phase, respectively. The total motion duration is, thus, \(T_{tot}=2T_{a}+T_{v}\). Let \(q_{0}\) and \(q_{1}\) denote the initial and the final position, respectively. The position profile \(q(t)\) and velocity profile \(\dot{q}(t)\) can be calculated through direct integration of Equation (1).  
It can be derived that the peak velocity \(v\) and the terminal position are:  
\[v=aT_{a} \tag{2}\]  
\[q(t_{3})=q_{0}+aT_{a}(T_{a}+T_{v}) \tag{3}\]  
It is necessary to determine \(a\), \(T_{a}\) and \(T_{v}\) such that the trajectory can achieve the target displacement while respecting the actuator limits of maximum velocity \(V_{\max}\) and maximum acceleration \(A_{\max}\). The constraint conditions imposed by the target displacement and actuator limits are expressed as follows:  
\[h=aT_{a}(T_{a}+T_{v}),\;a\leq A_{\max},\;v\leq V_{\max} \tag{4}\]  
where \(h=q_{1}-q_{0}\) denotes the total displacement. The time optimal solution is obtained by pushing the kinematic values to their physical limits. The peak acceleration can always attain its limit value, i.e., \(a=A_{\max}\). In contrast, if the target displacement is too short, the maximum velocity may not be attained. In this case, the whole profile consists of only the acceleration phase and the deceleration phase with \(h=A_{\max}T_{a}\)[2]; otherwise, the cruise phase is required for achieving the desired displacement, \(V_{\max}=A_{\max}T_{a}\).  
Without loss of generality, we assume that the target displacement and peak acceleration are positive values. The acceleration/deceleration time that satisfies all the constraints is calculated as follows:  
\[T_{a}=\min\left\{\frac{V_{\max}}{A_{\max}},\sqrt{\frac{h}{A_{\max}}}\right\} \tag{5}\]  
Two cases are distinguished here: if \(T_{a}=V_{\max}/A_{\max}\), the trajectory type belongs to the standard one, so the time of the cruise phase is calculated as follows:  
\[T_{v}=\frac{h-A_{\max}T_{a}{}^{2}}{V_{\max}} \tag{6}\]  
Figure 1: Standard trapezoidal velocity profile.  
Otherwise, if \(T_{a}=\sqrt{\hbar/A_{\max}}\), \(T_{v}=0\). The cruise phase disappears, and the trapezoidal velocity profile becomes a triangular one, as shown in Figure 2.  
The above solution minimizes the motion duration without violating the actuator limits. However, undesirable residual vibrations can occur at the end of the point-to-point motion due to the presence of infinite jerks. Motion-induced residual vibration negatively affects motion precision, particularly in flexible systems. It also generates unwanted noise and reduces the overall system lifespan. Therefore, it is necessary to design optimal trajectory parameters for residual vibration suppression.