# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 3 Parameter Design for Sinusoidal Jerk Profile
### Optimization Process for Vibration Suppression  
From condition C1 to C3, the solutions for vibration cancellation are clearly infinite in number, as the values of \(k_{1}\), \(k_{2}\), and \(k_{3}\) can take any positive integer. Furthermore, the actuator limits represented by the \(V_{\max}\), \(A_{\max}\), and \(J_{\max}\) should be strictly adhered to for practical feasibility. To generate fast and smooth movements with low vibrations, it is crucial to find the optimal values of \(T_{1}\), \(T_{2}\), and \(T_{3}\) that minimize the motion duration under the actuator limits while satisfying the design constraints of vibration suppression and robustness. Algorithm 1 provides the parameter optimization procedure for the time-frequency-optimal motion profile. First, the parameters for a minimum-time motion profile with actuator limits are obtained using the formulas described in Section 3.1. Subsequently, these parameters are adjusted to accommodate the vibration characteristics. The number of vibration suppression conditions to be satisfied depends on the desired degree of robustness (\(r\)). It is worth noting that the selection of the vibration suppression conditions also impacts the motion duration. In essence, the parameters tuned in these conditions are \(t_{1}\), \(t_{2}\), and \(t_{4}\), and the total motion duration can be expressed as \(T_{f}=t_{1}+t_{2}+t_{4}\). Given that the original profile is time-optimal under physical limits, the parameters after modulation, in accordance with the selected conditions, should not be smaller than their original values in order to maintain respect for the physical limits. Tuning different parameters would  
Figure 6: Frequency spectra of acceleration profiles. result in varying time prolongations. Therefore, when selecting the vibration suppression conditions, the objective should be to minimize this prolongation. In addition, for the realization of the motion profile, \(t_{1}\) > 0, \(t_{2}\geq t_{1}\) and \(t_{4}\geq t_{1}+t_{2}\) must be fulfilled based on the pre-conditions \(T_{1}\) > 0, \(T_{2}\geq 0\), \(T_{3}\geq 0\). Clearly, if \(r\) is set to 1 or 2, then there are three possible choice options: if \(r=1\), then we can choose C1, C2, or C3; if \(r=2\), then we can choose (C1, C2), (C1, C3), or (C2, C3); if \(r=3\), then only one choice (C1, C2, C3) is available. We use a vector (**C**) to represent the selection made, where the conditions that are selected are marked as 1 and the conditions that are not selected are marked as 0. For instance, if only C1 is selected to be satisfied, then C = [1 0 0]. The set of possible options is denoted as S.  
```
Input: moving distance \(D\), actuator limits \(V_{\max}\), \(A_{\max}\), \(J_{\max}\), vibration period \(T_{d}\), robustness degree \(r\) Output: optimal profile parameters \(\hat{T}_{1}\), \(\hat{T}_{2}\), \(\hat{T}_{3}\), \(J_{peak}\)
1: Start
2: Compute the minimum time solution according to Table 1
3:if\(r=1\)
4:\(S=\{[1\ 0\ 0],[0\ 1\ 0],[0\ 0\ 1]];\(l=3\)
5:else if\(r=2\)
6:\(S=\{[1\ 1\ 0],[1,\ 0,\ 1],[0,\ 1,\ 1]];\(l=3\)
7:else
8:\(S=\{[1\ 1\ 1]];\(l=1\)
9:endif
10:for\(i=1\) to \(l\)do
11:\(\textbf{C}=S[i]\)
12: Modulate \(\hat{t}_{1}\), \(\hat{t}_{2}\) and \(\hat{t}_{4}\) using Equations (34)-(36)
13: Compute the total execution time \(T_{f}=\hat{t}_{1}+\hat{t}_{2}+\hat{t}_{4}\)
14:endfor
15: find \(\textbf{C}=S[i]\in S\) that minimizes \(T_{f}\)
16: Compute the optimal parameters \(\hat{T}_{1}\), \(\hat{T}_{2}\), \(\hat{T}_{3}\), \(\hat{J}_{peak}\)
17:end
```  
**Algorithm 1**Optimization of profile parameters  
In order to draw a distinction from the initial minimum-time solution, the caret symbol (\(\cdot\)) is added to indicate the variables associated with the final time-frequency-optimal motion profile. Thus, for a specific **C**, the parameters are modulated as follows:  
\[\hat{t}_{1}=\left\{ \begin{array}{ll}t_{1},&\textbf{C}(1)=0\\ \Big{(}\max\Big{\{}\text{ceil}\Big{(}\frac{t_{1}-T_{d}/2}{T_{d}}\Big{)},\ 1\Big{\}}+\frac{1}{2}\Big{)}T_{d},&\textbf{C}(1)=1\end{array}