Update Content - 2021-09-21

content/article/mcinroy00_desig_contr_flexur_joint_hexap.md

@@ -9,7 +9,7 @@ Tags
 
 
 Reference
-: ([McInroy and Hamann 2000](#org0b673fd))
+: ([McInroy and Hamann 2000](#org04a7c92))
 
 Author(s)
 : McInroy, J., & Hamann, J.
@@ -21,4 +21,4 @@ Year
 
 ## Bibliography {#bibliography}
 
-<a id="org0b673fd"></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="org04a7c92"></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>.

content/zettels/isotropy_of_parallel_manipulator.md

@@ -0,0 +1,190 @@
++++
+title = "Isotropy of Parallel Manipulator"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}})
+
+Here are some notes on the literature about the isotropy of parallel manipulators.
+
+
+## ([Tsai and Huang 2003](#org838fed0)) {#tsai-and-huang-2003--org838fed0}
+
+
+## ([Fassi, Legnani, and Tosi 2005](#org2487ba1)) {#fassi-legnani-and-tosi-2005--org2487ba1}
+
+
+## ([Bandyopadhyay and Ghosal 2008](#org89a27d1)) {#bandyopadhyay-and-ghosal-2008--org89a27d1}
+
+Uses `mathematica` to inverse analytical Jacobian matrix and obtain conditions for isotropy.
+
+
+## ([Legnani et al. 2010](#orgbf26550)) {#legnani-et-al-dot-2010--orgbf26550}
+
+
+### Abstract {#abstract}
+
+A manipulator exhibits an _isotropic behaviour_ when it has the same performances along all the directions of the working space.
+
+The authors introduce the new concept of _Point of Isotropy_, showing how in some circumstances a non-isotropic manipulator may be transform into an isotropic one simply changing the location of its Tool Center Point (TCP).
+
+
+### Introduction {#introduction}
+
+**Kinetostatic** of parallel manipulator can be studied with the following equations:
+
+\begin{align}
+  \dot{Q} &= J \dot{S} \\\\\\
+  F\_s &= J^T F\_q \\\\\\
+  J   &= \frac{\partial Q}{\partial S}
+\end{align}
+
+where \\(J\\) is the Jacobian matrix which relates the "gripper" velocity \\(\dot{S}\\) with those of the actuators \\(\dot{Q}\\), as well as the forces \\(F\_q\\) exerted by the actuators with the forces/torques \\(F\_s\\) applied to the gripper.
+
+
+### Isotropy {#isotropy}
+
+A robot is called **isotropic** if at least in one point of the working space some of its kinetostatic properties are homogeneous with respect to all the directions.
+
+<div class="definition">
+  <div></div>
+
+-   **Velocity isotropy**: A manipulator is isotropic with respect to the velocity, if it can perform the same velocity along all the directions.
+-   **Force isotropy**: A manipulator is isotropic with respect to the force, if it can exert the same force along all the directions.
+-   **Stiffness isotropy**: A manipulator is isotropic with respect to the stiffness, if the deflection of the TCP produced by a force applied to it is always in the direction of the force and its magnitude is independent of the force direction.
+-   **Mass isotropy**: A manipulator is isotropic with respect to the equivalent gripper mass, if the acceleration of the TCP produced by a force applied to it is always in the direction of the force and its magnitude is independent of the force direction.
+
+</div>
+
+A 6-DoF spatial manipulator is isotropic with respect to velocity if:
+
+\begin{equation}
+J^T J = \diag(j\_{xx}, j\_{yy}, j\_{zz}, j\_{\alpha\alpha}, j\_{\beta\beta}, j\_{\gamma\gamma}) \quad \text{with} \quad j\_{xx}=j\_{yy}=j\_{zz} \quad \text{and} \quad j\_{\alpha\alpha}=j\_{\beta\beta}=j\_{\gamma\gamma}
+\end{equation}
+
+The same condition holds for the force isotropy.
+
+Assuming that the actuators are locked and that they are the only sources of compliance, the force \\(F\_s\\) to be applied to the end effector to produce a motion \\(dS\\) is:
+
+\begin{equation}
+F\_s = \underbrace{J^T K\_q J}\_{K\_s} dS \quad K\_q = \diag(\dots,k\_i,\dots)
+\end{equation}
+
+where \\(k\_i\\) is the stiffness of the ith actuator.
+A general 6-DoF manipulator is **fully isotropic** with respect to stiffness if:
+
+\begin{equation}
+K\_s = \diag(k\_{xx}, k\_{yy}, k\_{zz}, k\_{\alpha\alpha}, k\_{\beta\beta}, k\_{\gamma\gamma}) \quad \text{with} \quad k\_{xx}=k\_{yy}=k\_{zz}=k\_x \quad \text{and} \quad k\_{\alpha\alpha}=k\_{\beta\beta}=k\_{\gamma\gamma}=k\_\phi
+\end{equation}
+
+In this case, it results:
+
+\begin{equation}
+F = k\_x dX, \quad T = k\_\phi d\phi
+\end{equation}
+
+where \\(k\_x\\) is the translation stiffness and \\(k\_\phi\\) is the rotation stiffness.
+This means that:
+
+-   forces \\(F\\) applied to the TCP do not produce rotations \\(d\phi\\) but only translations \\(dX\\)
+-   the translation is proportional to the force and parallel to it regardless to the force direction
+-   torques \\(T\\) applied to the TCP do not produce translations \\(dx\\) but only rotations \\(d\phi\\)
+-   the rotation is proportional to the torque and occurs around the same axis as the applied torque
+
+In this special case in which all the actuators are identical to each other, and therefore have the same stiffness \\(k\\), we have \\(K\_s = kJ^TJ\\) and the condition number of the matrix \\(J^TJ\\) can be investigated instead of that of \\(J^T K\_q J\\).
+In this case the isotropy for velocity, force and stiffness are achieve simultaneously.
+
+A manipulator is **partially isotropic** if:
+
+\begin{equation}
+k\_{xx} = k\_{yy} \neq k\_{zz} \quad \text{and/or} \quad k\_{\alpha\alpha} = k\_{\beta\beta} \neq k\_{\gamma\gamma}
+\end{equation}
+
+
+### Point of isotropy {#point-of-isotropy}
+
+A parallel manipulator as a "point of isotropy" if it exists at least one point of its end effector for which the isotropy condition is achieved.
+
+Then conditions are given to find an isotropic TCP.
+
+
+### Application to the Stewart platform {#application-to-the-stewart-platform}
+
+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}
+
+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 attached points of each strut are uniformly spaced on the circumferences of two circles on the movable platform and the fixed base, respectively.
+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="orgc9be46b"></a>
+
+{{< figure src="/ox-hugo/tong11_architecture_gsgspm.png" caption="Figure 1: Architecture of a GSGSPM" >}}
+
+A compliance center exists consequentially for any GSGSPMs.
+At the compliance center, a GSGSPM is uncoupled.
+
+
+## ([Legnani et al. 2012](#org1e949d1)) {#legnani-et-al-dot-2012--org1e949d1}
+
+
+## ([Ding et al. 2014](#org1fe4a1a)) {#ding-et-al-dot-2014--org1fe4a1a}
+
+
+## ([Wu et al. 2018](#org7f6b447)) {#wu-et-al-dot-2018--org7f6b447}
+
+> Isotropy => J\*J' = a\*I
+
+<!--quoteend-->
+
+> -   Stiffness isotropy = static isotropy
+> -   velocity isotropy = kinematic isotropy
+
+<!--quoteend-->
+
+> 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))
+
+<!--quoteend-->
+
+> Dynamic isotropy => same resonance frequency for all suspension modes
+
+
+## ([Zhou et al. 2019](#org1f33e66)) {#zhou-et-al-dot-2019--org1f33e66}
+
+
+## ([Yang et al. 2020](#orgfee3e89)) {#yang-et-al-dot-2020--orgfee3e89}
+
+
+## ([Kang et al. 2020](#org0ed4d48)) {#kang-et-al-dot-2020--org0ed4d48}
+
+
+
+## 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="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="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="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="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="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="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="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="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="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="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>.