Kinematically Constrained Jerk–Continuous S-Curve Trajectory Planning in Joint Space for Industrial Robots
Abstract
:1. Introduction
2. Jerk–Continuous Trajectory Planning
2.1. From Third-Order to Fourth-Order S-Curve Trajectories
2.2. Calculation of the Time Parameter
2.2.1. Calculation of Time Interval ${T}_{s}$ with Varying Jerk
- Case 1:
- If ${T}_{s}={T}_{s}^{d}$, the displacement constraint is the only influencing factor that limits the motion time; thus, it is unnecessary to calculate the time intervals of the other trajectory segments, and there are only trajectory segments with varying jerks. In this case, the motion parameters, such as the jerk, acceleration, and velocity, cannot reach their maximum, and the real maximum jerk, acceleration, and velocity are equal to ${j}_{m}={s}_{max}{T}_{s}^{d}$, ${a}_{m}={j}_{m}{T}_{s}^{d}$, and ${v}_{m}=2{a}_{m}{T}_{s}^{d}$, respectively.
- Case 2:
- If ${T}_{s}={T}_{s}^{v}$, the maximum velocity can be reached without the maximum jerk and acceleration. In this case, the maximal reachable jerk and acceleration are ${j}_{m}={s}_{max}{T}_{s}^{v}$ and ${a}_{m}={j}_{m}{T}_{s}^{v}$, respectively. For the calculation of the remaining motion parameters, the reader can refer to Section 2.2.4.
- Case 3:
- If ${T}_{s}={T}_{s}^{a}$, the maximum acceleration reaches its maximum without the maximum jerk, and the maximum reachable jerk is ${j}_{m}={s}_{max}{T}_{s}^{v}$. For the calculation of the remaining motion parameters, the reader can refer to Section 2.2.3.
- Case 4:
- If ${T}_{s}={T}_{s}^{j}$, for the calculation of the remaining motion parameters, the reader can refer to Section 2.2.2.
2.2.2. Calculation of Time Interval ${T}_{j}$ with Constant Jerk
- Case 1:
- If ${T}_{j}={T}_{j}^{d}$, i.e., ${T}_{a}={T}_{v}=0$, only the trajectory segments with varying jerk and constant jerk exist, and the jerk can reach its maximum ${j}_{max}$; therefore, the real maximum acceleration and velocity are calculated as ${a}_{m}={j}_{max}({T}_{s}+{T}_{j}^{d})$ and ${v}_{m}={a}_{m}(2{T}_{s}+{T}_{j})$, respectively.
- Case 2:
- If ${T}_{j}={T}_{j}^{v}$, the maximum velocity can reach its maximum without the maximum acceleration, and the real maximum acceleration is calculated as ${a}_{m}={j}_{max}({T}_{s}+{T}_{j}^{v})$. For the calculation of the other motion parameters, the reader can refer to Section 2.2.4.
- Case 3:
- If ${T}_{j}={T}_{j}^{a}$, for the calculation of the time intervals, the reader can refer to Section 2.2.3.
2.2.3. Calculation of Time Interval ${T}_{a}$ with Constant Acceleration
- Case 1:
- If ${T}_{a}={T}_{a}^{d}$, i.e., ${T}_{v}=0$, due to the limitation on displacement, trajectory segments with a constant velocity do not exist. Although the acceleration can reach its maximum, the velocity cannot, and the real maximum velocity is ${v}_{m}={a}_{max}(2{T}_{s}+{T}_{a}^{d})$.
- Case 2:
- If ${T}_{a}={T}_{a}^{v}$, both the maximum acceleration and velocity can reach their maximums, and for the calculation of the motion parameters, the reader can refer to Section 2.2.4.
2.2.4. Calculation of Time Interval ${T}_{v}$ with Constant Velocity
2.3. Time Synchronization of Multi-Axis Motions
Algorithm 1 Trajectory planning algorithm for multi-axis synchronization |
Input: The initial ${\mathbf{q}}_{s}\left({t}_{0}\right)$ and final ${\mathbf{q}}_{e}\left({t}_{f}\right)$ positions of the robot and the kinematic constraints $\{{\mathbf{v}}_{max},\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{max},\phantom{\rule{0.166667em}{0ex}}{\mathbf{j}}_{max}\}$ |
Output: Time-synchronized motion trajectory |
1: Use of the single-joint trajectory planning algorithm to calculate the execution time of trajectory planning for each joint ${T}_{f,\phantom{\rule{0.166667em}{0ex}}k}$ |
2: Selection of the maximum execution time in step 1 as the synchronization time ${T}_{f}^{sync}=max\{{T}_{f,\phantom{\rule{0.166667em}{0ex}}1},\phantom{\rule{0.166667em}{0ex}}{T}_{f,\phantom{\rule{0.166667em}{0ex}}2},\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{T}_{f,\phantom{\rule{0.166667em}{0ex}}n}\}$ |
3: Calculation of the synchronization factor ${\lambda}_{k}=\frac{{T}_{f}^{sync}}{{T}_{f,\phantom{\rule{0.166667em}{0ex}}k}}$, $k=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}n$ |
4: Modification of the kinematic constraints of each joint $\{{\mathbf{v}}_{max}^{\prime}=\frac{{\mathbf{v}}_{max}}{{\lambda}_{k}},\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{max}^{\prime}=\frac{{\mathbf{a}}_{max}}{{\lambda}_{k}^{2}},\phantom{\rule{0.166667em}{0ex}}{\mathbf{j}}_{max}^{\prime}=\frac{{\mathbf{j}}_{max}}{{\lambda}_{k}^{3}}\}$ |
5: Calculation of the new motion trajectory according to the new kinematic constraint $\{{\mathbf{v}}_{max}^{\prime},\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{max}^{\prime},\phantom{\rule{0.166667em}{0ex}}{\mathbf{j}}_{max}^{\prime}\}$ |
3. Case Study: Trajectory Planning for a Five-Axis Manipulator
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Time | Function of Motion Profiles | Notations |
---|---|---|
$t\in [{t}_{0},\phantom{\rule{0.166667em}{0ex}}{t}_{1}]$ | $\begin{array}{c}j\left(t\right)=\frac{{j}_{max}}{{T}_{1}}{\tau}_{1}\hfill \\ a\left(t\right)=\frac{{j}_{max}}{2{T}_{1}}{\tau}_{1}^{2}\hfill \\ v\left(t\right)=\frac{{j}_{max}}{6{T}_{1}}{\tau}_{1}^{3}+{v}_{s}\hfill \\ q\left(t\right)=\frac{{j}_{max}}{24{T}_{1}}{\tau}_{1}^{4}+{v}_{s}{\tau}_{1}+{q}_{s}\hfill \end{array}$ | |
$t\in [{t}_{1},\phantom{\rule{0.166667em}{0ex}}{t}_{2}]$ | $\begin{array}{c}j\left(t\right)={j}_{max}\hfill \\ a\left(t\right)={j}_{max}{\tau}_{2}+\frac{{j}_{max}}{2}{T}_{1}\hfill \\ v\left(t\right)=\frac{{j}_{max}}{2}{\tau}_{2}^{2}+\frac{{j}_{max}}{2}{T}_{1}{\tau}_{2}+{v}_{1}\hfill \\ q\left(t\right)=\frac{{j}_{max}}{6}{\tau}_{2}^{3}+\frac{{j}_{max}}{4}{T}_{1}{\tau}_{2}^{2}+{v}_{1}{\tau}_{2}+{q}_{1}\hfill \end{array}$ | $\begin{array}{c}{v}_{1}={v}_{0}+\frac{{j}_{max}}{6}{T}_{1}^{2}\hfill \\ {q}_{1}={v}_{0}{T}_{1}+\frac{{j}_{max}}{24}{T}_{1}^{3}\hfill \end{array}$ |
$t\in [{t}_{2},\phantom{\rule{0.166667em}{0ex}}{t}_{3}]$ | $\begin{array}{c}j\left(t\right)={j}_{max}-\frac{{j}_{max}}{{T}_{3}}{\tau}_{3}\hfill \\ a\left(t\right)={j}_{max}{\tau}_{3}-\frac{{j}_{max}}{2{T}_{3}}{\tau}_{3}^{2}+\frac{{j}_{max}}{2}({T}_{1}+2{T}_{2})\hfill \\ v\left(t\right)=\frac{{j}_{max}}{2}{\tau}_{3}^{2}-\frac{{j}_{max}}{6{T}_{3}}{\tau}_{3}^{3}+\frac{{j}_{max}}{2}({T}_{1}+2{T}_{2}){\tau}_{3}+{v}_{2}\hfill \\ q\left(t\right)=\frac{{j}_{max}}{6}{\tau}_{2}^{3}-\frac{{j}_{max}}{6T3}{\tau}_{4}^{3}+\frac{{j}_{max}}{4}({T}_{1}+2{T}_{2}){\tau}_{3}^{2}+{v}_{2}{\tau}_{3}+{q}_{2}\hfill \end{array}$ | $\begin{array}{c}{v}_{2}={v}_{1}+\frac{{j}_{max}}{2}{T}_{2}({T}_{1}+{T}_{2})\hfill \\ {q}_{2}={q}_{1}+\frac{{j}_{max}}{12}{T}_{2}^{2}(2{T}_{2}+3{T}_{1})+{v}_{1}{T}_{2}\hfill \end{array}$ |
$t\in [{t}_{3},\phantom{\rule{0.166667em}{0ex}}{t}_{4}]$ | $\begin{array}{c}j\left(t\right)=0\hfill \\ a\left(t\right)={a}_{max}\hfill \\ v\left(t\right)={a}_{max}{\tau}_{4}-{v}_{3}\hfill \\ q\left(t\right)=\frac{{a}_{max}}{2}{\tau}_{4}^{2}+{v}_{3}{\tau}_{4}+{q}_{3}\hfill \end{array}$ | $\begin{array}{c}{v}_{3}={v}_{2}+\frac{{j}_{max}}{6}{T}_{3}(3{T}_{1}+6{T}_{2}+2{T}_{3})\hfill \\ {q}_{3}={q}_{2}+\frac{{j}_{max}}{8}{T}_{3}^{2}(2{T}_{1}+4{T}_{2}+{T}_{3})+{v}_{2}{T}_{3}\hfill \end{array}$ |
$t\in [{t}_{4},\phantom{\rule{0.166667em}{0ex}}{t}_{5}]$ | $\begin{array}{c}j\left(t\right)=-\frac{{j}_{max}}{{T}_{5}}{\tau}_{5}\hfill \\ a\left(t\right)=-\frac{{j}_{max}}{2{T}_{5}}{\tau}_{5}^{2}+{a}_{max}\hfill \\ v\left(t\right)=-\frac{{j}_{max}}{6{T}_{5}}{\tau}_{5}^{3}+{a}_{max}{\tau}_{5}+{v}_{4}\hfill \\ q\left(t\right)=-\frac{{j}_{max}}{24{T}_{5}}{\tau}_{5}^{4}+\frac{{a}_{max}}{2}{\tau}_{5}^{2}+{q}_{4}\hfill \end{array}$ | $\begin{array}{c}{v}_{4}={v}_{3}+{a}_{max}{T}_{4}\hfill \\ {q}_{4}={q}_{3}+\frac{{2}_{max}}{2}{T}_{4}^{2}+{v}_{3}{T}_{4}\hfill \end{array}$ |
$t\in [{t}_{5},\phantom{\rule{0.166667em}{0ex}}{t}_{6}]$ | $\begin{array}{c}j\left(t\right)=-{j}_{max}\hfill \\ a\left(t\right)=-{j}_{max}{\tau}_{6}+{a}_{max}-\frac{{j}_{max}}{2}{T}_{5}\hfill \\ v\left(t\right)=-\frac{{j}_{max}}{2}{\tau}_{6}^{2}+\left({a}_{max}-\frac{{j}_{max}}{2}{\tau}_{5}\right){\tau}_{6}+{v}_{5}\hfill \\ q\left(t\right)=-\frac{{j}_{max}}{6}{\tau}_{6}^{3}+\frac{1}{2}\left({a}_{max}-\frac{{j}_{max}}{2}{\tau}_{5}\right){\tau}_{6}^{2}+{v}_{5}{\tau}_{6}+{q}_{5}\hfill \end{array}$ | $\begin{array}{c}{v}_{5}={v}_{4}-\frac{{a}_{max}}{6}{T}_{5}^{2}+{a}_{max}{T}_{5}\hfill \\ {q}_{5}={q}_{4}-\frac{{j}_{max}}{24}{T}_{5}^{3}+\frac{{a}_{max}}{2}{T}_{5}^{2}+{v}_{4}{T}_{5}\hfill \end{array}$ |
$t\in [{t}_{6},\phantom{\rule{0.166667em}{0ex}}{t}_{7}]$ | $\begin{array}{c}j\left(t\right)=-{j}_{max}+\frac{{j}_{max}}{{T}_{7}}{\tau}_{7}\hfill \\ a\left(t\right)=-{j}_{max}{\tau}_{7}+\frac{{j}_{max}}{2{T}_{7}}{\tau}_{7}^{2}+{a}_{max}-\frac{{j}_{max}}{2}{T}_{5}-{j}_{max}{T}_{6}\hfill \\ v\left(t\right)=-\frac{{j}_{max}}{2}{\tau}_{7}^{2}+\frac{{j}_{max}}{6{T}_{7}}{\tau}_{7}^{3}+\left({a}_{max}-\frac{{j}_{max}}{2}{T}_{5}-{j}_{max}{T}_{7}\right){\tau}_{7}+{v}_{6}\hfill \\ q\left(t\right)=-\frac{{j}_{max}}{6}{\tau}_{7}^{3}-\frac{{j}_{max}}{24{T}_{7}}{\tau}_{7}^{4}+\frac{2{a}_{max}-{j}_{max}{T}_{5}-2{j}_{max}{T}_{1}}{4}{\tau}_{7}^{2}+{v}_{6}{\tau}_{7}+{q}_{6}\hfill \end{array}$ | $\begin{array}{c}{v}_{6}={v}_{5}-\frac{{j}_{max}}{2}{T}_{6}^{2}+\left({a}_{max}-\frac{{j}_{max}}{2}{T}_{5}\right){T}_{6}\hfill \\ {q}_{6}={q}_{5}+\frac{6{a}_{max}-3{j}_{max}{T}_{5}-2{j}_{max}{T}_{6}}{12}{T}_{6}^{2}+{v}_{5}{T}_{6}\hfill \end{array}$ |
$t\in [{t}_{7},\phantom{\rule{0.166667em}{0ex}}{t}_{8}]$ | $\begin{array}{c}j\left(t\right)=0\hfill \\ a\left(t\right)=0\hfill \\ v\left(t\right)={v}_{7}\hfill \\ q\left(t\right)={v}_{7}{\tau}_{8}+{q}_{7}\hfill \end{array}$ | $\begin{array}{c}{v}_{7}={v}_{6}-\frac{{j}_{max}}{3}{T}_{7}^{2}+\left({a}_{max}-\frac{{j}_{max}}{2}{T}_{5}-{j}_{max}{T}_{6}\right){T}_{7}\hfill \\ {q}_{7}={q}_{6}-\frac{{j}_{max}}{8}{T}_{7}^{3}+\frac{2{a}_{max}-{j}_{max}{T}_{5}-2{j}_{max}{T}_{6}}{4}{T}_{7}^{2}\hfill \end{array}$ |
Axis | Initial Position $\left[\mathbf{m}\right]$ | Final Position $\left[\mathbf{m}\right]$ | Velocity $[\mathbf{m}/\mathbf{s}]$ | Acceleration $[\mathbf{m}/{\mathbf{s}}^{2}]$ | Jerk $[\mathbf{m}/{\mathbf{s}}^{3}]$ |
---|---|---|---|---|---|
x | 0 | 3 | 2 | 4 | 40 |
y | 0 | 2 | 1.4 | 5 | 35 |
z | 0 | 1.5 | 1.2 | 3 | 35 |
Joint | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
Angular position$\left[\mathrm{rad}\right]$ | Initial | 0 | 0 | 0 | 0 | 0 |
Final | $\pi /6$ | $\pi /4$ | $\pi /3$ | $\pi /2$ | $\pi /3$ | |
Kinematic constraints | Velocity $[\mathrm{rad}/\mathrm{s}]$ | 1 | 1.4 | 1.4 | 2 | 3 |
Acceleration $[\mathrm{rad}/{\mathrm{s}}^{2}]$ | 3 | 5 | 5 | 7 | 8 | |
Jerk $[\mathrm{rad}/{\mathrm{s}}^{3}]$ | 25 | 35 | 40 | 40 | 40 |
Trajectory Model | Calculation Time $\left[\mathbf{s}\right]$ | Maximal Jerk $[\mathbf{rad}/{\mathbf{s}}^{3}]$ | Continuity Level |
---|---|---|---|
Quintic polynomial [25] | 1.4667 | 40 (Joint 4) | ${C}_{2}$ |
Cubic spline [26] | 2.1679 | 39.83 (Joint 4) | ${C}_{2}$ |
Third-order S-curve | 1.2734 | 39.06 (Joint 4) | ${C}_{2}$ |
Fourth-order S-curve | 1.3461 | 39.38 (Joint 4) | ${C}_{3}$ |
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Wu, G.; Zhang, N. Kinematically Constrained Jerk–Continuous S-Curve Trajectory Planning in Joint Space for Industrial Robots. Electronics 2023, 12, 1135. https://doi.org/10.3390/electronics12051135
Wu G, Zhang N. Kinematically Constrained Jerk–Continuous S-Curve Trajectory Planning in Joint Space for Industrial Robots. Electronics. 2023; 12(5):1135. https://doi.org/10.3390/electronics12051135
Chicago/Turabian StyleWu, Guanglei, and Ning Zhang. 2023. "Kinematically Constrained Jerk–Continuous S-Curve Trajectory Planning in Joint Space for Industrial Robots" Electronics 12, no. 5: 1135. https://doi.org/10.3390/electronics12051135