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title = "Angular Velocity"
author = ["Thomas Dehaeze"]
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## Non-integrability of the angular velocity vector {#non-integrability-of-the-angular-velocity-vector}
The non-integrability of the angular velocity vector is well described in ([Legnani et al. 2012](#orga01afc4)).
> It is well known that the angular velocity vector is not the time derivative of any set of angular coordinates.
> In other words, it is impossible to define a set of three coordinates representing the 3D angular position of a body whose time derivative is equal to the angular velocity vector.
This is illustrated in Figure [1](#org4ad23f3).
<a id="org4ad23f3"></a>
{{< figure src="/ox-hugo/angular_nonintegrability.png" caption="Figure 1: Effect of different sequences of rotations of a rigid body. In both cases we get Rot(x)=0, Rot(y)=90deg and Rot(z)=90deg" >}}
## Bibliography {#bibliography}
<a id="orga01afc4"></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>.

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Here are some notes on the literature about the isotropy of parallel manipulators. Here are some notes on the literature about the isotropy of parallel manipulators.
## ([Tsai and Huang 2003](#org838fed0)) {#tsai-and-huang-2003--org838fed0} ## ([Tsai and Huang 2003](#org0724ed5)) {#tsai-and-huang-2003--org0724ed5}
## ([Fassi, Legnani, and Tosi 2005](#org2487ba1)) {#fassi-legnani-and-tosi-2005--org2487ba1} ## ([Fassi, Legnani, and Tosi 2005](#orgd63d03a)) {#fassi-legnani-and-tosi-2005--orgd63d03a}
## ([Bandyopadhyay and Ghosal 2008](#org89a27d1)) {#bandyopadhyay-and-ghosal-2008--org89a27d1} ## ([Bandyopadhyay and Ghosal 2008](#orgf824a14)) {#bandyopadhyay-and-ghosal-2008--orgf824a14}
Uses `mathematica` to inverse analytical Jacobian matrix and obtain conditions for isotropy. Uses `mathematica` to inverse analytical Jacobian matrix and obtain conditions for isotropy.
## ([Legnani et al. 2010](#orgbf26550)) {#legnani-et-al-dot-2010--orgbf26550} ## ([Legnani et al. 2010](#org513282b)) {#legnani-et-al-dot-2010--org513282b}
### Abstract {#abstract} ### 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. Conditions can be applied to the Stewart platform and isotropy points can be found.
## ([Tong et al. 2011](#org0e02b02)) {#tong-et-al-dot-2011--org0e02b02} ## ([Tong et al. 2011](#org6ea337f)) {#tong-et-al-dot-2011--org6ea337f}
A parallel manipulator consists of a movable platform, a fixed base, and six struts, each with a linear actuator. 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. 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. 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). This parallel manipulator with this kind of configurations are defined as generalized symmetric Gough-Stewart parallel manipulators (GSGSPMs).
<a id="orgc9be46b"></a> <a id="orgf6e6061"></a>
{{< figure src="/ox-hugo/tong11_architecture_gsgspm.png" caption="Figure 1: Architecture of a GSGSPM" >}} {{< figure src="/ox-hugo/tong11_architecture_gsgspm.png" caption="Figure 1: Architecture of a GSGSPM" >}}
@ -131,60 +131,79 @@ A compliance center exists consequentially for any GSGSPMs.
At the compliance center, a GSGSPM is uncoupled. At the compliance center, a GSGSPM is uncoupled.
## ([Legnani et al. 2012](#org1e949d1)) {#legnani-et-al-dot-2012--org1e949d1} ## ([Legnani et al. 2012](#org0747a45)) {#legnani-et-al-dot-2012--org0747a45}
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.
- Total decoupling occurs when the Jacobian is diagonal
- Partial decoupling is when the Jacobian is triangular
- Block decoupling is when the Jacobian is block diagonal
<a id="org0738057"></a>
{{< figure src="/ox-hugo/legnani12_isotropic_pkm.png" caption="Figure 2: An isotropic PKM" >}}
<summary>
The paper discusses the concepts of isotropy and decoupling in n-DoF PKM.
The role of different Jacobian matrices in the isotropy, decoupling and in general mobility analysis of manipulators is recalled.
It is highlighted how isotropy and decoupling may be achieved for pure translational manipulators in the whole workspace while rotational manipulators maybe decoupling in only one configuration.
</summary>
## ([Ding et al. 2014](#org1fe4a1a)) {#ding-et-al-dot-2014--org1fe4a1a} ## ([Ding et al. 2014](#orgbeecf44)) {#ding-et-al-dot-2014--orgbeecf44}
## ([Wu et al. 2018](#org7f6b447)) {#wu-et-al-dot-2018--org7f6b447} ## ([Wu et al. 2018](#orgcb2f4d0)) {#wu-et-al-dot-2018--orgcb2f4d0}
> Isotropy => J\*J' = a\*I Isotropy => J\*J' = a\*I
<!--quoteend--> - Stiffness isotropy = static isotropy
- velocity isotropy = kinematic isotropy
> - Stiffness isotropy = static 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))
> - velocity isotropy = kinematic isotropy
<!--quoteend--> Dynamic isotropy => same resonance frequency for all suspension modes.
> 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](#org0e02b02)) <a id="org171ed4c"></a>
<!--quoteend--> {{< figure src="/ox-hugo/wu18_stewart_picture.png" caption="Figure 3: Optimized Stewart platform" >}}
> Dynamic isotropy => same resonance frequency for all suspension modes
## ([Zhou et al. 2019](#org1f33e66)) {#zhou-et-al-dot-2019--org1f33e66} ## ([Yang et al. 2020](#orgda6537c)) {#yang-et-al-dot-2020--orgda6537c}
<summary>
This paper proposes a novel concept, namely _isotropic control_ to solve the problem of having identical performance in all DoF.
Dynamic equations of parallel mechanisms with base excitation are established and analyzed.
An isotropic control framework is then synthesized in modal space.
The multi-DoF system is transformed into multi identical single-DoF systems.
Under the framework of isotropic control, parallel mechanisms obtain an identical frequency response for all modes.
An identical corner frequency, active damping, and rate of low-frequency transmissibility are achieved for all modes.
</summary>
## ([Yang et al. 2020](#orgfee3e89)) {#yang-et-al-dot-2020--orgfee3e89} ## ([Kang et al. 2020](#org460918c)) {#kang-et-al-dot-2020--org460918c}
## ([Kang et al. 2020](#org0ed4d48)) {#kang-et-al-dot-2020--org0ed4d48}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org89a27d1"></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="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="org1fe4a1a"></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="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="org2487ba1"></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="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="org0ed4d48"></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="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="org1e949d1"></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="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="orgbf26550"></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="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="org0e02b02"></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="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="org838fed0"></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="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="org7f6b447"></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="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="orgfee3e89"></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="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="org1f33e66"></a>Zhou, Songlin, Jing Sun, Weixing Chen, Wei Li, and Feng Gao. 2019. “Method of Designing a Six-Axis Force Sensor for Stiffness Decoupling Based on Stewart Platform.” _Measurement_ 148 (nil):106966. <https://doi.org/10.1016/j.measurement.2019.106966>.

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Papers by J.E. McInroy: Papers by J.E. McInroy:
- ([O’Brien et al. 1998](#org320ae46)) - ([O’Brien et al. 1998](#org4f530c8))
- ([McInroy, O’Brien, and Neat 1999](#orgea92441)) - ([McInroy, O’Brien, and Neat 1999](#orgc2500da))
- ([McInroy 1999](#orgfb23e22)) - ([McInroy 1999](#orgf6f13f1))
- ([McInroy and Hamann 2000](#org90e178e)) - ([McInroy and Hamann 2000](#org353671a))
- ([Chen and McInroy 2000](#org418808c)) - ([Chen and McInroy 2000](#orgf235207))
- ([McInroy 2002](#orgaa607f0)) - ([McInroy 2002](#org5be7775))
- ([Li, Hamann, and McInroy 2001](#org7176257)) - ([Li, Hamann, and McInroy 2001](#orgdc37806))
- ([Lin and McInroy 2003](#org6427256)) - ([Lin and McInroy 2003](#orge3fb031))
- ([Jafari and McInroy 2003](#orga8add64)) - ([Jafari and McInroy 2003](#orge70feb2))
- ([Chen and McInroy 2004](#org3f36881)) - ([Chen and McInroy 2004](#org8ae8169))
Main advantage of flexure jointed Stewart platforms over conventional (long stroke) ones:
- Linear behavior
- Constant Jacobian matrices along all stroke
- No singularity
- Easier to decouple the dynamics that works for all the stroke
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org3f36881"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>. <a id="org8ae8169"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
<a id="org418808c"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="orgf235207"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="orga8add64"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>. <a id="orge70feb2"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
<a id="org6427256"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>. <a id="orge3fb031"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
<a id="org7176257"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>. <a id="orgdc37806"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
<a id="orgfb23e22"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>. <a id="orgf6f13f1"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="orgaa607f0"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>. <a id="org5be7775"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<a id="org90e178e"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>. <a id="org353671a"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
<a id="orgea92441"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>. <a id="orgc2500da"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
<a id="org320ae46"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>. <a id="org4f530c8"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.

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