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content/zettels/isotropy_of_parallel_manipulator.md
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@@ -10,18 +10,18 @@ Tags
Here are some notes on the literature about the isotropy of parallel manipulators.
## ([Tsai and Huang 2003](#org0724ed5)) {#tsai-and-huang-2003--org0724ed5}
## ([Tsai and Huang 2003](#orgfdcbc5f)) {#tsai-and-huang-2003--orgfdcbc5f}
## ([Fassi, Legnani, and Tosi 2005](#orgd63d03a)) {#fassi-legnani-and-tosi-2005--orgd63d03a}
## ([Fassi, Legnani, and Tosi 2005](#org420bcfa)) {#fassi-legnani-and-tosi-2005--org420bcfa}
## ([Bandyopadhyay and Ghosal 2008](#orgf824a14)) {#bandyopadhyay-and-ghosal-2008--orgf824a14}
## ([Bandyopadhyay and Ghosal 2008](#org403a5a5)) {#bandyopadhyay-and-ghosal-2008--org403a5a5}
Uses `mathematica` to inverse analytical Jacobian matrix and obtain conditions for isotropy.
## ([Legnani et al. 2010](#org513282b)) {#legnani-et-al-dot-2010--org513282b}
## ([Legnani et al. 2010](#orgf42f367)) {#legnani-et-al-dot-2010--orgf42f367}
### Abstract {#abstract}
@@ -115,7 +115,7 @@ Then conditions are given to find an isotropic TCP.
Conditions can be applied to the Stewart platform and isotropy points can be found.
## ([Tong et al. 2011](#org6ea337f)) {#tong-et-al-dot-2011--org6ea337f}
## ([Tong et al. 2011](#org3d4f33e)) {#tong-et-al-dot-2011--org3d4f33e}
A parallel manipulator consists of a movable platform, a fixed base, and six struts, each with a linear actuator.
The struts are partitioned into two groups: the first group with strut 1,3,5 and the second group with strut 2,4,6.
@@ -123,7 +123,7 @@ The attached points of each strut are uniformly spaced on the circumferences of
The three struts in each group are rotational symmetry and repeat every 120 deg.
This parallel manipulator with this kind of configurations are defined as generalized symmetric Gough-Stewart parallel manipulators (GSGSPMs).
<a id="orgf6e6061"></a>
<a id="org1222642"></a>
{{< figure src="/ox-hugo/tong11_architecture_gsgspm.png" caption="Figure 1: Architecture of a GSGSPM" >}}
@@ -131,7 +131,7 @@ A compliance center exists consequentially for any GSGSPMs.
At the compliance center, a GSGSPM is uncoupled.
## ([Legnani et al. 2012](#org0747a45)) {#legnani-et-al-dot-2012--org0747a45}
## ([Legnani et al. 2012](#orgac23b06)) {#legnani-et-al-dot-2012--orgac23b06}
A manipulator is called partially of totally decoupled if the general movements of the robot can be subdivided in elementary tasks, each actuated by one or a group of actuators.
Decoupling may be referred to the end effector coordinate or to local kinetostatic properties related to the Jacobian.
@@ -140,7 +140,7 @@ Decoupling may be referred to the end effector coordinate or to local kinetostat
- Partial decoupling is when the Jacobian is triangular
- Block decoupling is when the Jacobian is block diagonal
<a id="org0738057"></a>
<a id="orga046465"></a>
{{< figure src="/ox-hugo/legnani12_isotropic_pkm.png" caption="Figure 2: An isotropic PKM" >}}
@@ -151,26 +151,32 @@ It is highlighted how isotropy and decoupling may be achieved for pure translati
</summary>
## ([Ding et al. 2014](#orgbeecf44)) {#ding-et-al-dot-2014--orgbeecf44}
## ([Ding et al. 2014](#orga0fa269)) {#ding-et-al-dot-2014--orga0fa269}
## ([Wu et al. 2018](#orgcb2f4d0)) {#wu-et-al-dot-2018--orgcb2f4d0}
## ([Afzali-Far 2016](#orgc42aa83)) {#afzali-far-2016--orgc42aa83}
> The problem of dynamic isotropy, as an optimal design solution for hexapods, is also addressed in this dissertation.
> **Dynamic isotropy is a condition in which all eigenfrequencies of a robot are equal**.
## ([Wu et al. 2018](#orgbacd7c7)) {#wu-et-al-dot-2018--orgbacd7c7}
Isotropy => J\*J' = a\*I
- Stiffness isotropy = static isotropy
- velocity isotropy = kinematic isotropy
They also proved that the symmetric generalized Stewart platform at a neutral position could be fully decoupled by adjusting the payload's center of mass to coincide with its **compliance center**. ([Tong et al. 2011](#org6ea337f))
They also proved that the symmetric generalized Stewart platform at a neutral position could be fully decoupled by adjusting the payload's center of mass to coincide with its **compliance center**. ([Tong et al. 2011](#org3d4f33e))
Dynamic isotropy => same resonance frequency for all suspension modes.
<a id="org171ed4c"></a>
<a id="orge4ddb31"></a>
{{< figure src="/ox-hugo/wu18_stewart_picture.png" caption="Figure 3: Optimized Stewart platform" >}}
## ([Yang et al. 2020](#orgda6537c)) {#yang-et-al-dot-2020--orgda6537c}
## ([Yang et al. 2020](#orge144852)) {#yang-et-al-dot-2020--orge144852}
<summary>
This paper proposes a novel concept, namely _isotropic control_ to solve the problem of having identical performance in all DoF.
@@ -182,28 +188,30 @@ An identical corner frequency, active damping, and rate of low-frequency transmi
</summary>
## ([Kang et al. 2020](#org460918c)) {#kang-et-al-dot-2020--org460918c}
## ([Kang et al. 2020](#orgb043f30)) {#kang-et-al-dot-2020--orgb043f30}
## Bibliography {#bibliography}
<a id="orgf824a14"></a>Bandyopadhyay, Sandipan, and Ashitava Ghosal. 2008. “An Algebraic Formulation of Kinematic Isotropy and Design of Isotropic 6-6 Stewart Platform Manipulators.” _Mechanism and Machine Theory_ 43 (5):591–616. <https://doi.org/10.1016/j.mechmachtheory.2007.05.003>.
<a id="orgc42aa83"></a>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.
<a id="orgbeecf44"></a>Ding, Boyin, Benjamin S. Cazzolato, Richard M. Stanley, Steven Grainger, and John J. Costi. 2014. “Stiffness Analysis and Control of a Stewart Platform-Based Manipulator with Decoupled Sensor-Actuator Locations for Ultrahigh Accuracy Positioning under Large External Loads.” _Journal of Dynamic Systems, Measurement, and Control_ 136 (6):nil. <https://doi.org/10.1115/1.4027945>.
<a id="org403a5a5"></a>Bandyopadhyay, Sandipan, and Ashitava Ghosal. 2008. “An Algebraic Formulation of Kinematic Isotropy and Design of Isotropic 6-6 Stewart Platform Manipulators.” _Mechanism and Machine Theory_ 43 (5):591–616. <https://doi.org/10.1016/j.mechmachtheory.2007.05.003>.
<a id="orgd63d03a"></a>Fassi, Irene, Giovanni Legnani, and Diego Tosi. 2005. “Geometrical Conditions for the Design of Partial or Full Isotropic Hexapods.” _Journal of Robotic Systems_ 22 (10):507–18. <https://doi.org/10.1002/rob.20074>.
<a id="orga0fa269"></a>Ding, Boyin, Benjamin S. Cazzolato, Richard M. Stanley, Steven Grainger, and John J. Costi. 2014. “Stiffness Analysis and Control of a Stewart Platform-Based Manipulator with Decoupled Sensor-Actuator Locations for Ultrahigh Accuracy Positioning under Large External Loads.” _Journal of Dynamic Systems, Measurement, and Control_ 136 (6):nil. <https://doi.org/10.1115/1.4027945>.
<a id="org460918c"></a>Kang, Shengzheng, Hongtao Wu, Shengdong Yu, Yao Li, Xiaolong Yang, and Jiafeng Yao. 2020. “Modeling and Control of a Six-Axis Parallel Piezo-Flexural Micropositioning Stage with Cross-Coupling Hysteresis Nonlinearities.” In _2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)_, 1350–55. IEEE.
<a id="org420bcfa"></a>Fassi, Irene, Giovanni Legnani, and Diego Tosi. 2005. “Geometrical Conditions for the Design of Partial or Full Isotropic Hexapods.” _Journal of Robotic Systems_ 22 (10):507–18. <https://doi.org/10.1002/rob.20074>.
<a id="org0747a45"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. <https://doi.org/10.1016/j.mechmachtheory.2012.07.008>.
<a id="orgb043f30"></a>Kang, Shengzheng, Hongtao Wu, Shengdong Yu, Yao Li, Xiaolong Yang, and Jiafeng Yao. 2020. “Modeling and Control of a Six-Axis Parallel Piezo-Flexural Micropositioning Stage with Cross-Coupling Hysteresis Nonlinearities.” In _2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)_, 1350–55. IEEE.
<a id="org513282b"></a>Legnani, Giovanni, D Tosi, I Fassi, Hermes Giberti, and Simone Cinquemani. 2010. “The ‘Point of Isotropy’ and Other Properties of Serial and Parallel Manipulators.” _Mechanism and Machine Theory_ 45 (10). Elsevier:1407–23.
<a id="orgac23b06"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. <https://doi.org/10.1016/j.mechmachtheory.2012.07.008>.
<a id="org6ea337f"></a>Tong, Zhizhong, Jingfeng He, Hongzhou Jiang, and Guangren Duan. 2011. “Optimal Design of a Class of Generalized Symmetric Gough-Stewart Parallel Manipulators with Dynamic Isotropy and Singularity-Free Workspace.” _Robotica_ 30 (2):305–14. <https://doi.org/10.1017/s0263574711000531>.
<a id="orgf42f367"></a>Legnani, Giovanni, D Tosi, I Fassi, Hermes Giberti, and Simone Cinquemani. 2010. “The ‘Point of Isotropy’ and Other Properties of Serial and Parallel Manipulators.” _Mechanism and Machine Theory_ 45 (10). Elsevier:1407–23.
<a id="org0724ed5"></a>Tsai, K.Y., and K.D. Huang. 2003. “The Design of Isotropic 6-Dof Parallel Manipulators Using Isotropy Generators.” _Mechanism and Machine Theory_ 38 (11):1199–1214. <https://doi.org/10.1016/s0094-114x(03)00067-3>.
<a id="org3d4f33e"></a>Tong, Zhizhong, Jingfeng He, Hongzhou Jiang, and Guangren Duan. 2011. “Optimal Design of a Class of Generalized Symmetric Gough-Stewart Parallel Manipulators with Dynamic Isotropy and Singularity-Free Workspace.” _Robotica_ 30 (2):305–14. <https://doi.org/10.1017/s0263574711000531>.
<a id="orgcb2f4d0"></a>Wu, Ying, Kaiping Yu, Jian Jiao, Dengqing Cao, Weichao Chi, and Jie Tang. 2018. “Dynamic Isotropy Design and Analysis of a Six-Dof Active Micro-Vibration Isolation Manipulator on Satellites.” _Robotics and Computer-Integrated Manufacturing_ 49 (nil):408–25. <https://doi.org/10.1016/j.rcim.2017.08.003>.
<a id="orgfdcbc5f"></a>Tsai, K.Y., and K.D. Huang. 2003. “The Design of Isotropic 6-Dof Parallel Manipulators Using Isotropy Generators.” _Mechanism and Machine Theory_ 38 (11):1199–1214. <https://doi.org/10.1016/s0094-114x(03)00067-3>.
<a id="orgda6537c"></a>Yang, Xiaolong, Hongtao Wu, Yao Li, Shengzheng Kang, Bai Chen, Huimin Lu, Carman K. M. Lee, and Ping Ji. 2020. “Dynamics and Isotropic Control of Parallel Mechanisms for Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 25 (4):2027–34. <https://doi.org/10.1109/tmech.2020.2996641>.
<a id="orgbacd7c7"></a>Wu, Ying, Kaiping Yu, Jian Jiao, Dengqing Cao, Weichao Chi, and Jie Tang. 2018. “Dynamic Isotropy Design and Analysis of a Six-Dof Active Micro-Vibration Isolation Manipulator on Satellites.” _Robotics and Computer-Integrated Manufacturing_ 49 (nil):408–25. <https://doi.org/10.1016/j.rcim.2017.08.003>.
<a id="orge144852"></a>Yang, Xiaolong, Hongtao Wu, Yao Li, Shengzheng Kang, Bai Chen, Huimin Lu, Carman K. M. Lee, and Ping Ji. 2020. “Dynamics and Isotropic Control of Parallel Mechanisms for Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 25 (4):2027–34. <https://doi.org/10.1109/tmech.2020.2996641>.