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title = "Isotropy of Parallel Manipulator"
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author = ["Dehaeze Thomas"]
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: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
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Here are some notes on the literature about the isotropy of parallel manipulators.
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## (<a href="#citeproc_bib_item_9">Tsai and Huang 2003</a>) {#f86766}
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## (<a href="#citeproc_bib_item_4">Fassi, Legnani, and Tosi 2005</a>) {#0ac3c3}
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## (<a href="#citeproc_bib_item_2">Bandyopadhyay and Ghosal 2008</a>) {#2ab0b0}
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Uses `mathematica` to inverse analytical Jacobian matrix and obtain conditions for isotropy.
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## (<a href="#citeproc_bib_item_7">Legnani et al. 2010</a>) {#a75d91}
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### Abstract {#abstract}
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A manipulator exhibits an _isotropic behaviour_ when it has the same performances along all the directions of the working space.
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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).
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### Introduction {#introduction}
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**Kinetostatic** of parallel manipulator can be studied with the following equations:
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\begin{align}
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\dot{Q} &= J \dot{S} \\\\
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F\_s &= J^T F\_q \\\\
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J &= \frac{\partial Q}{\partial S}
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\end{align}
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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.
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### Isotropy {#isotropy}
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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.
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<div class="definition">
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- **Velocity isotropy**: A manipulator is isotropic with respect to the velocity, if it can perform the same velocity along all the directions.
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- **Force isotropy**: A manipulator is isotropic with respect to the force, if it can exert the same force along all the directions.
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- **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.
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- **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.
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</div>
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A 6-DoF spatial manipulator is isotropic with respect to velocity if:
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\begin{equation}
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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}
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\end{equation}
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The same condition holds for the force isotropy.
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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:
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\begin{equation}
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F\_s = \underbrace{J^T K\_q J}\_{K\_s} dS \quad K\_q = \diag(\dots,k\_i,\dots)
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\end{equation}
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where \\(k\_i\\) is the stiffness of the ith actuator.
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A general 6-DoF manipulator is **fully isotropic** with respect to stiffness if:
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\begin{equation}
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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
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\end{equation}
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In this case, it results:
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\begin{equation}
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F = k\_x dX, \quad T = k\_\phi d\phi
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\end{equation}
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where \\(k\_x\\) is the translation stiffness and \\(k\_\phi\\) is the rotation stiffness.
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This means that:
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- forces \\(F\\) applied to the TCP do not produce rotations \\(d\phi\\) but only translations \\(dX\\)
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- the translation is proportional to the force and parallel to it regardless to the force direction
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- torques \\(T\\) applied to the TCP do not produce translations \\(dx\\) but only rotations \\(d\phi\\)
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- the rotation is proportional to the torque and occurs around the same axis as the applied torque
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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\\).
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In this case the isotropy for velocity, force and stiffness are achieve simultaneously.
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A manipulator is **partially isotropic** if:
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\begin{equation}
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k\_{xx} = k\_{yy} \neq k\_{zz} \quad \text{and/or} \quad k\_{\alpha\alpha} = k\_{\beta\beta} \neq k\_{\gamma\gamma}
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\end{equation}
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### Point of isotropy {#point-of-isotropy}
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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.
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Then conditions are given to find an isotropic TCP.
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### Application to the Stewart platform {#application-to-the-stewart-platform}
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Conditions can be applied to the Stewart platform and isotropy points can be found.
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## (<a href="#citeproc_bib_item_8">Tong et al. 2011</a>) {#6febd5}
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A parallel manipulator consists of a movable platform, a fixed base, and six struts, each with a linear actuator.
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The struts are partitioned into two groups: the first group with strut 1,3,5 and the second group with strut 2,4,6.
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The attached points of each strut are uniformly spaced on the circumferences of two circles on the movable platform and the fixed base, respectively.
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The three struts in each group are rotational symmetry and repeat every 120 deg.
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This parallel manipulator with this kind of configurations are defined as generalized symmetric Gough-Stewart parallel manipulators (GSGSPMs).
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<a id="figure--fig:tong11-architecture-gsgspm"></a>
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{{< figure src="/ox-hugo/tong11_architecture_gsgspm.png" caption="<span class='figure-number'>Figure 1: </span>Architecture of a GSGSPM" >}}
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A compliance center exists consequentially for any GSGSPMs.
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At the compliance center, a GSGSPM is uncoupled.
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## (<a href="#citeproc_bib_item_6">Legnani et al. 2012</a>) {#633281}
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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.
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Decoupling may be referred to the end effector coordinate or to local kinetostatic properties related to the Jacobian.
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- Total decoupling occurs when the Jacobian is diagonal
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- Partial decoupling is when the Jacobian is triangular
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- Block decoupling is when the Jacobian is block diagonal
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<a id="figure--fig:legnani12-isotropic-pkm"></a>
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{{< figure src="/ox-hugo/legnani12_isotropic_pkm.png" caption="<span class='figure-number'>Figure 2: </span>An isotropic PKM" >}}
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<div class="sum">
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The paper discusses the concepts of isotropy and decoupling in n-DoF PKM.
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The role of different Jacobian matrices in the isotropy, decoupling and in general mobility analysis of manipulators is recalled.
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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.
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</div>
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## (<a href="#citeproc_bib_item_3">Ding et al. 2014</a>) {#623b74}
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## (<a href="#citeproc_bib_item_1">Afzali-Far 2016</a>) {#6e127c}
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> The problem of dynamic isotropy, as an optimal design solution for hexapods, is also addressed in this dissertation.
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> **Dynamic isotropy is a condition in which all eigenfrequencies of a robot are equal**.
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## (<a href="#citeproc_bib_item_10">Wu et al. 2018</a>) {#033041}
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Isotropy => J\*J' = a\*I
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- Stiffness isotropy = static isotropy
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- velocity isotropy = kinematic isotropy
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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**. (<a href="#citeproc_bib_item_8">Tong et al. 2011</a>)
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Dynamic isotropy => same resonance frequency for all suspension modes.
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<a id="figure--fig:wu18-stewart-picture"></a>
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{{< figure src="/ox-hugo/wu18_stewart_picture.png" caption="<span class='figure-number'>Figure 3: </span>Optimized Stewart platform" >}}
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## (<a href="#citeproc_bib_item_11">Yang et al. 2020</a>) {#e39296}
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<div class="sum">
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This paper proposes a novel concept, namely _isotropic control_ to solve the problem of having identical performance in all DoF.
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Dynamic equations of parallel mechanisms with base excitation are established and analyzed.
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An isotropic control framework is then synthesized in modal space.
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The multi-DoF system is transformed into multi identical single-DoF systems.
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Under the framework of isotropic control, parallel mechanisms obtain an identical frequency response for all modes.
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An identical corner frequency, active damping, and rate of low-frequency transmissibility are achieved for all modes.
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</div>
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## (<a href="#citeproc_bib_item_5">Kang et al. 2020</a>) {#0812ce}
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Bandyopadhyay, Sandipan, and Ashitava Ghosal. 2008. “An Algebraic Formulation of Kinematic Isotropy and Design of Isotropic 6-6 Stewart Platform Manipulators.” <i>Mechanism and Machine Theory</i> 43 (5): 591–616. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2007.05.003">10.1016/j.mechmachtheory.2007.05.003</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_3"></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.” <i>Journal of Dynamic Systems, Measurement, and Control</i> 136 (6). doi:<a href="https://doi.org/10.1115/1.4027945">10.1115/1.4027945</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_4"></a>Fassi, Irene, Giovanni Legnani, and Diego Tosi. 2005. “Geometrical Conditions for the Design of Partial or Full Isotropic Hexapods.” <i>Journal of Robotic Systems</i> 22 (10): 507–18. doi:<a href="https://doi.org/10.1002/rob.20074">10.1002/rob.20074</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_5"></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 <i>2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)</i>, 1350–55. IEEE.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_6"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58: 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_7"></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.” <i>Mechanism and Machine Theory</i> 45 (10). Elsevier: 1407–23.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_8"></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.” <i>Robotica</i> 30 (2): 305–14. doi:<a href="https://doi.org/10.1017/s0263574711000531">10.1017/s0263574711000531</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_9"></a>Tsai, K.Y., and K.D. Huang. 2003. “The Design of Isotropic 6-Dof Parallel Manipulators Using Isotropy Generators.” <i>Mechanism and Machine Theory</i> 38 (11): 1199–1214. doi:<a href="https://doi.org/10.1016/s0094-114x(03)00067-3">10.1016/s0094-114x(03)00067-3</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_10"></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.” <i>Robotics and Computer-Integrated Manufacturing</i> 49: 408–25. doi:<a href="https://doi.org/10.1016/j.rcim.2017.08.003">10.1016/j.rcim.2017.08.003</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_11"></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.” <i>IEEE/ASME Transactions on Mechatronics</i> 25 (4): 2027–34. doi:<a href="https://doi.org/10.1109/tmech.2020.2996641">10.1109/tmech.2020.2996641</a>.</div>
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</div>
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