From b90a4e21ec1f719e928c3f9d15e2b25e0aeaa022 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 21 Sep 2021 16:00:48 +0200 Subject: [PATCH] Update Content - 2021-09-21 --- ...cinroy00_desig_contr_flexur_joint_hexap.md | 4 +- .../isotropy_of_parallel_manipulator.md | 190 ++++++++++++++++++ static/ox-hugo/tong11_architecture_gsgspm.png | Bin 0 -> 73118 bytes 3 files changed, 192 insertions(+), 2 deletions(-) create mode 100644 content/zettels/isotropy_of_parallel_manipulator.md create mode 100644 static/ox-hugo/tong11_architecture_gsgspm.png diff --git a/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md b/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md index 1baa134..1e14731 100644 --- a/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md +++ b/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} -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. . +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. . diff --git a/content/zettels/isotropy_of_parallel_manipulator.md b/content/zettels/isotropy_of_parallel_manipulator.md new file mode 100644 index 0000000..eb3d981 --- /dev/null +++ b/content/zettels/isotropy_of_parallel_manipulator.md @@ -0,0 +1,190 @@ ++++ +title = "Isotropy of Parallel Manipulator" +author = ["Thomas Dehaeze"] +draft = false ++++ + +Tags +: [Stewart Platforms]({{}}) + +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. + +
+
+ +- **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. + +
+ +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). + + + +{{< 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 + + + +> - 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](#org0e02b02)) + + + +> 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} + +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. . + +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. . + +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. . + +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. + +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. . + +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. + +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. . + +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. . + +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. . + +Yang, Xiaolong, Hongtao Wu, Yao Li, Shengzheng Kang, Bai Chen, Huimin Lu, Carman K. M. 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