Yingge et al 2023a - Revision history

Rimni at 10:15, 8 September 2023

2023-09-08T10:15:02Z

Rimni: /* 4. Results */

2023-09-08T10:14:00Z

‎4. Results

Rimni at 10:08, 8 September 2023

2023-09-08T10:08:05Z

Rimni at 09:19, 8 September 2023

2023-09-08T09:19:42Z

Rimni at 09:05, 8 September 2023

2023-09-08T09:05:26Z

Rimni at 09:04, 8 September 2023

2023-09-08T09:04:43Z

Rimni: /* 4. Results */

2023-09-07T14:05:26Z

‎4. Results

Rimni at 13:53, 7 September 2023

2023-09-07T13:53:47Z

Show changes

Rimni: /* 4. Results */

2023-09-07T13:42:03Z

‎4. Results

Rimni at 13:40, 7 September 2023

2023-09-07T13:40:12Z

@@ Line 240: / Line 240: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>  >u\left(\tau \right)\approx U\left(\tau \right)=\sum_{i=0}^N{\Phi }_i\left(\tau \right)U_i </math>
+| style="text-align: center;" | <math>u\left(\tau \right)\approx U\left(\tau \right)=\sum_{i=0}^N{\Phi }_i\left(\tau \right)U_i </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(12)

@@ Line 474: / Line 474: @@
-'''1）The effect of initial angular momentum on hopping performance'''
+'''1) The effect of initial angular momentum on hopping performance'''
 In Haldane et al. [6], the stability of the hopping robot’s flight movement was investigated. It was proposed that in the flight stage, when the total angular momentum regarding the robot’s center of mass was very small, the flight movement was in a stable state. Since the angular momentum regarding the robot’s center of mass is conserved in the flight phase, the angular momentum of the robot in flight is determined by the initial angular momentum  <math>P_0</math>.  <math>P_0</math> has a significant impact on the robot’s movement during the flight phase.
@@ Line 498: / Line 498: @@
-'''2）The effect of flight time on hopping performance'''
+'''2) The effect of flight time on hopping performance'''
 In this section, the impact of flight time on hopping performance is discussed when initial angular momentum  <math>\mbox{P}_0</math> equals 0. [[#img-7|Figure 7]] shows the curve of performance indicator  <math>J</math> as a function of flight time. When the initial angular momentum is zero, the flight time can be arbitrarily selected. It can be seen that as the flight time increases, the performance indicator  <math>J</math> decreases, that is, the energy consumption decreases, and changes sharply within 0.5 seconds. As time increases, the robot has sufficient time to change the position of the joint angle, resulting in a smaller input and lower energy consumption. But at the same time, as time increases, the height of the hopping increases, and the vertical velocity of the center of mass at take-off also increases, requiring the robot to provide greater force . Therefore, it is necessary to comprehensively measure and select the appropriate time.

@@ Line 539: / Line 539: @@
 ==References==
 <span id='_Ref85812059'></span>
@@ Line 551: / Line 553: @@
 [5] Terry P., Piovan G., Byl K. Towards precise control of hoppers: Using high order partial feedback linearization to control the hopping robot FRANK. IEEE 55<sup>th</sup> Conference on Decision and Control (CDC) ARIA Resort & Casino, Las Vegas, USA, 6669-6675, 2016.
-<span id='_Ref85812063'></span> [6] Haldane D.W., Yim J.K., Fearing R.S. Repetitive extreme-acceleration (14-g) spatial jumping with Salto-1P. 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 3345-3351, 2017.
+<span id='_Ref85812063'></span>
-<span id='_Ref85812378'></span>[7] Vermeulen, J, Trajectory generation for planar hopping and walking robots: An objective parameter and angular momentum approach. McGill University, Department of Mechanical Engineering, 2004.
+<span id='_Ref85812378'></span>
-<span id='_Ref85812199'></span>[8] Li, Z,Huang, Q,Li, K, et al., Stability criterion and pattern planning for humanoid running. Proceedings of the 2004 IEEE International Conference on Robotics & Automation. 2004, 1059-1064.
+<span id='_Ref85812199'></span>
-<span id='_Ref85812961'></span>[9] Wu, T Y,Yeh, T J,Hsu, B H, Trajectory planning of a One-Legged robot performing a stable hop. The International Journal of Robotics Research, 2011, 30(8): 1072-1091.
+<span id='_Ref85812961'></span>
-[10] Shabestari, S S,Emami, M R, Gait planning for a hopping robot. Robotica, 2016, 34(8): 1822-1840.
+[10] Shabestari S.S., Emami M.R. Gait planning for a hopping robot. Robotica, 34(8):1822-1840, 2016.
-<span id='_Ref85812382'></span>[11] Ahn, D H,Cho, B K, Optimal periodic hopping trajectory generation for legged robots. IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM), Auckland, New Zealand: 2018, 1263-1268.
+<span id='_Ref85812382'></span>
-<span id='_Ref85814151'></span>[12] Rehman, F,Ahmed, M M,Memon, N Met al., Time-varying stabilizing control for a hopping robot model during the flight phase. Islamabad: 2006. 400-403.
+<span id='_Ref85814151'></span>
-<span id='_Ref85814567'></span>[13] Guo, Q,Macnab, C,Pieper, J, Hopping with nearly-passive flight phases. 2008 IEEE Conference on Robotics, Automation and Mechatronics, Chengdu, China: 2008, 743-748.
+<span id='_Ref85814567'></span>
-<span id='_Ref85817660'></span>[14] YONG Enmi, CHEN Lei, TANG Guojin. A survey of numerical methods for trajectory optimization of spacecraft. Journal of Astronautics, 2008, 29(2): 397-406. (In Chinese)
+<span id='_Ref85817660'></span>
-[15] XU Shaobing, LI Shengbo, CHENG Bo. Theory and application of Legendre pseudo-spectral method for solving optimal control problem. Control and Decision, 2014(12): 2113-2120. (In Chinese)
+[15]  Xu S.-B., Li S., Cheng B. Theory and application of Legendre pseudo-spectral method for solving optimal control problem (in Chinese). Control and Decision, (12):2113-2120 , 2014.
-<span id='_Ref85817664'></span>[16] GE Xingsheng, CHEN Kaijie. Path planning of free floating space robot using Legendre pseudospectral method. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(4): 823-831. (In Chinese)
+<span id='_Ref85817664'></span>

@@ Line 540: / Line 540: @@
 ==References==
-<span id='_Ref85812059'></span>[1] Raibet, M H, Legged robots that balance. Cambridge: the MIT Press. 1986.
+<span id='_Ref85812059'></span>
-[2] Ahmadi, M,Buehler, M, Controlled passive dynamic running experiments with the ARL-Monopod II. IEEE Transactions on Robotics, 2006, 22(5): 974-986.
+[2] Ahmadi M., Buehler M. Controlled passive dynamic running experiments with the ARL-Monopod II. IEEE Transactions on Robotics, 22(5):974-986, 2006.
-[3] Hyon, S H,Emura, T,Mita, T, Dynamics-based control of a one-legged hopping robot. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 2003, 217(2): 83-98.
+[3] Hyon S.H., Emura T., Mita T. Dynamics-based control of a one-legged hopping robot. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 217(2):83-98, 2003.
-[4] Haewon, P,Patrick, M W,Sangbae, K, High-speed bounding with the MIT Cheetah 2: Control design and experiments. International Journal of Robotics Research, 2017, 36(2): 167-192.
+[4] Park H.W., Wensing P.M., Kim S. High-speed bounding with the MIT Cheetah 2: Control design and experiments. International Journal of Robotics Research, 36(2):167-192, 2017.
-[5] Terry, P,Piovan, G,Byl, K, Towards precise control of hoppers: Using high order partial feedback linearization to control the hopping robot FRANK. IEEE 55<sup>th</sup> Conference on Decision and Control (CDC) ARIA Resort & Casino, Las Vegas, USA: 2016, 6669-6675.
+[5] Terry P., Piovan G., Byl K. Towards precise control of hoppers: Using high order partial feedback linearization to control the hopping robot FRANK. IEEE 55<sup>th</sup> Conference on Decision and Control (CDC) ARIA Resort & Casino, Las Vegas, USA, 6669-6675, 2016.
-<span id='_Ref85812063'></span> [6] Haldane, D W,Yim, J K,Fearing, R S, Repetitive extreme-acceleration (14-g) spatial jumping with Salto-1P. 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2017, 3345-3351.
+<span id='_Ref85812063'></span> [6] Haldane D.W., Yim J.K., Fearing R.S. Repetitive extreme-acceleration (14-g) spatial jumping with Salto-1P. 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 3345-3351, 2017.
 <span id='_Ref85812378'></span>[7] Vermeulen, J, Trajectory generation for planar hopping and walking robots: An objective parameter and angular momentum approach. McGill University, Department of Mechanical Engineering, 2004.

@@ Line 473: / Line 473: @@
 |}
-=====1）The effect of initial angular momentum on hopping performance=====
 In Haldane et al. [6], the stability of the hopping robot’s flight movement was investigated. It was proposed that in the flight stage, when the total angular momentum regarding the robot’s center of mass was very small, the flight movement was in a stable state. Since the angular momentum regarding the robot’s center of mass is conserved in the flight phase, the angular momentum of the robot in flight is determined by the initial angular momentum  <math>P_0</math>.  <math>P_0</math> has a significant impact on the robot’s movement during the flight phase.
@@ Line 497: / Line 498: @@
-====2）The effect of flight time on hopping performance====
+'''2）The effect of flight time on hopping performance'''
 In this section, the impact of flight time on hopping performance is discussed when initial angular momentum  <math>\mbox{P}_0</math> equals 0. [[#img-7|Figure 7]] shows the curve of performance indicator  <math>J</math> as a function of flight time. When the initial angular momentum is zero, the flight time can be arbitrarily selected. It can be seen that as the flight time increases, the performance indicator  <math>J</math> decreases, that is, the energy consumption decreases, and changes sharply within 0.5 seconds. As time increases, the robot has sufficient time to change the position of the joint angle, resulting in a smaller input and lower energy consumption. But at the same time, as time increases, the height of the hopping increases, and the vertical velocity of the center of mass at take-off also increases, requiring the robot to provide greater force . Therefore, it is necessary to comprehensively measure and select the appropriate time.

@@ Line 457: / Line 457: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_2716_ffig3.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_2716_ffig3.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 3'''. Joint angles of the hopping robot
@@ Line 468: / Line 468: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_8446_ffig4.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_8446_ffig4.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 4'''. Control input of the hip point
@@ Line 482: / Line 482: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_2565_ffig5.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_2565_ffig5.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 5'''. Maximum flight time vs. initial angular momentum
@@ Line 491: / Line 491: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_7008_ffig6.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_7008_ffig6.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 6'''. Performance index vs. initial angular momentum
@@ Line 504: / Line 504: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_6393_ffig7.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_6393_ffig7.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 7'''. Performance indicator <math>J</math> vs. flight time <math>T</math>
@@ Line 515: / Line 515: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_9139_ffig8.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_9139_ffig8.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 8'''. Trajectory of tip of the foot at different flight times
@@ Line 524: / Line 524: @@
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_3895_ffig9.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_3895_ffig9.png|300px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 9'''. Posture diagram during flight phase at  <math>T=0.{1}</math>
 |}
 ==5. Conclusions==

@@ Line 441: / Line 441: @@
-The initial and terminal positions of the given motion planning of the hopping robot system are <math>x_0={\left[-5^{\circ },-{25}^{\circ },-{18}^{\circ }\right]}^\mbox{T}</math> and  <math>x_\mbox{T}={\left[-5^{\circ },-{125}^{\circ },-{35}^{\circ }\right]}^{T}</math>, respectively. The flight time  <math>T=0.\{5}</math>,  <math>P_0=0.0{2}</math>, and the control constraints are  <math>u_{{1\max}}=\mbox{6rad}/{s}</math>,  <math>u_{{2\max}}=\mbox{6rad}/{s}</math>. The simulation results are shown in [[#img-2|Figures 2]]-[[#img-4|4]]. It can be seen that the hopping robot can complete the motion planning requirements and reach the preset goal position within the given time.
+The initial and terminal positions of the given motion planning of the hopping robot system are <math>x_0={\left[-5^{\circ },-{25}^{\circ },-{18}^{\circ }\right]}^\mbox{T}</math> and  <math>x_\mbox{T}={\left[-5^{\circ },-{125}^{\circ },-{35}^{\circ }\right]}^{T}</math>, respectively. The flight time  <math>T=0.{5}</math>,  <math>P_0=0.0{2}</math>, and the control constraints are  <math>u_{{1\max}}=\mbox{6rad}/{s}</math>,  <math>u_{{2\max}}=\mbox{6rad}/{s}</math>. The simulation results are shown in [[#img-2|Figures 2]]-[[#img-4|4]]. It can be seen that the hopping robot can complete the motion planning requirements and reach the preset goal position within the given time.
 <div id='img-2'></div>
 {| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
 |-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:Review_582677700696_2756_ffig2.png]]
+|style="text-align: center;padding:10px;"| [[File:Review_582677700696_2756_ffig2.png|500px]]
 |-
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 2'''. Posture diagram during flight phase

@@ Line 418: / Line 418: @@
 |  2
 |  3
-|-
+|-style="text-align:center"
 !  style="text-align:left;"|Length of rod  <math>l_i/\left(\mbox{m}\right)</math>
 |   0.11
 |  0.26
 | 0.174
-|-
+|-style="text-align:center"
 !  style="text-align:left;"| Centroid moment separation  <math>l_{Si}/\left(\mbox{m}\right)</math>
 |  0.11
 |  0.105
 |  0.082
-|-
+|-style="text-align:center"
 !  style="text-align:left;"|Mass of rod  <math>m_i/\left(\mbox{kg}\right)</math>
 |  4.24
 |  0.06
 |  0.14
-|-
+|-style="text-align:center"
 ! style="text-align:left;"|Moment of inertia  <math>J_i/(\mbox{kg}\cdot \mbox{m}^\mbox{2})</math>
 |  0.034
@@ Line 441: / Line 441: @@
-The initial and terminal positions of the given motion planning of the hopping robot system are <math>x_0={\left[-5^{\circ },-{25}^{\circ },-{18}^{\circ }\right]}^\mbox{T}</math> and  <math>x_\mbox{T}={\left[-5^{\circ },-{125}^{\circ },-{35}^{\circ }\right]}^\mbox{T}</math> , respectively. The flight time  <math>T=0.\mbox{5}</math> ,  <math>P_0=0.0\mbox{2}</math> , and the control constraints are  <math>u_{\mbox{1max}}=\mbox{6rad}/\mbox{s}</math> ,  <math>u_{\mbox{2max}}=\mbox{6rad}/\mbox{s}</math> . The simulation results are shown in Figs.2-4. It can be seen that the hopping robot can complete the motion planning requirements and reach the preset goal position within the given time.
+The initial and terminal positions of the given motion planning of the hopping robot system are <math>x_0={\left[-5^{\circ },-{25}^{\circ },-{18}^{\circ }\right]}^\mbox{T}</math> and  <math>x_\mbox{T}={\left[-5^{\circ },-{125}^{\circ },-{35}^{\circ }\right]}^{T}</math>, respectively. The flight time  <math>T=0.\{5}</math>,  <math>P_0=0.0{2}</math>, and the control constraints are  <math>u_{{1\max}}=\mbox{6rad}/{s}</math>,  <math>u_{{2\max}}=\mbox{6rad}/{s}</math>. The simulation results are shown in [[#img-2|Figures 2]]-[[#img-4|4]]. It can be seen that the hopping robot can complete the motion planning requirements and reach the preset goal position within the given time.
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div id='img-2'></div>
-[[File:Review_582677700696_2756_ffig2.png]]
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
- </div>
+|-style="background:white;"
-From Fig.3, it can be seen that the joint trajectory of the foot shows a trend of first rising and then falling, which means that the foot lifts inward and then quickly swings downwards during most of the flight phase, preparing to touch the ground. The range of changes in the ankle joint is the largest among the three curves, approximately from 48 ° to 205°. The range of body swing is relatively small, ranging from -16° to 5°, and the changes are relatively gentle.
+From [[#img-3|Figure 3]] it can be seen that the joint trajectory of the foot shows a trend of first rising and then falling, which means that the foot lifts inward and then quickly swings downwards during most of the flight phase, preparing to touch the ground. The range of changes in the ankle joint is the largest among the three curves, approximately from 48 ° to 205°. The range of body swing is relatively small, ranging from -16° to 5°, and the changes are relatively gentle.
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">