A Planar Parallel Device for Neurorehabilitation
<p>3D model of the prototype.</p> "> Figure 2
<p>Four configurations of the planar 5R parallel architecture. (<b>a</b>): “<math display="inline"><semantics> <mrow> <mo>+</mo> <mo>−</mo> </mrow> </semantics></math>”, (<b>b</b>): “<math display="inline"><semantics> <mrow> <mo>−</mo> <mo>−</mo> </mrow> </semantics></math>”, (<b>c</b>): “<math display="inline"><semantics> <mrow> <mo>−</mo> <mo>+</mo> </mrow> </semantics></math>”, (<b>d</b>): “<math display="inline"><semantics> <mrow> <mo>+</mo> <mo>−</mo> </mrow> </semantics></math>”, by denoting the convex (+) or the concave (−) configuration of the left and the right elbow joints, respectively.</p> "> Figure 3
<p>The planar 5R parallel mechanism.</p> "> Figure 4
<p>Forward kinematics scheme for up-configuration.</p> "> Figure 5
<p>(<b>a</b>) Sketch showing the total reachable workspace by combining the workspace of patients with minimum limb lengths and those with maximum limb lengths. (<b>b</b>) Sketch showing the placement of the rehabilitation device in accordance with the reachable workspace.</p> "> Figure 6
<p>Plot of the minimum singular value over the radial direction for different values of <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> "> Figure 7
<p>Conditioning index over the radial direction for different values of <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> "> Figure 8
<p>(<b>a</b>) Plot of conditioning number over the workspace. (<b>b</b>) Plot of the manipulability force ellipses over the workspace.</p> "> Figure 9
<p>3D model of the prototype.</p> "> Figure 10
<p>Control architecture for PLANarm2.</p> "> Figure 11
<p>Feedback loop of the velocity estimator.</p> "> Figure 12
<p>Bode diagram of the velocity estimator.</p> "> Figure 13
<p>Feedback loop of position controller.</p> "> Figure 14
<p>Step response of position controller.</p> "> Figure 15
<p>Schematic tunnel representation. On the left (<b>a</b>), the end-effector moves freely within the tunnel. On the right (<b>b</b>), a restoring force brings the end-effector back within the tunnel.</p> "> Figure 16
<p>Reference frame for tunnel control.</p> "> Figure 17
<p>Exponential filter response function with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>c</mi> <mi>u</mi> <mi>t</mi> <mi>o</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> Hz.</p> "> Figure 18
<p>Admittance frequency response function (FRF) with <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>e</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>.</p> "> Figure 19
<p>The top plot shows the normal distance from the desired trajectory against time, the middle plot shows the normal force applied on the end-effector against time and the bottom plot shows the force produced by the virtual spring.</p> "> Figure 20
<p>Placement of the markers on the prototype for the experimental procedure.</p> "> Figure 21
<p>Position of motion capture and encoder vs. time: <span class="html-italic">x</span>-position (<b>right</b>) and <span class="html-italic">y</span>-position (<b>left</b>).</p> ">
Abstract
:1. Introduction
2. Kinematics
2.1. Architecture
- in order to avoid the uncertainty singularity where is extended.
- and in order to have a manipulator with a surrounded workspace
2.2. Inverse Kinematics
2.3. Forward Kinematics
2.4. Jacobian
3. Mechanical Design
3.1. Workspace
3.2. Kinematic Optimization
- : The minimum singular value (minimum stiffness)The index corresponding to the minimum singular value is defined as:The greater the minimum singular value, the greater the minimum rigidity of the machine. The minimum singular value was plotted against the radial direction of the manipulator for different ratios of and the results are shown in Figure 6.
- : Conditioning number (Isotropy)The conditioning number is a measure of the isotropy of the manipulator from the rigidity point of view. It is defined as the ratio between the maximum and minimum singular values of the Jacobian:The closer this ratio is to 1, the more consistent the stiffness of the machine will be along the main directions. The conditioning over the workspace is radially symmetric; therefore, in order to understand the behaviour of the conditioning index when changing the length of the link, the conditioning index of the manipulator was plotted against the radial direction (y-direction) for different values of . The results are shown in Figure 7.
3.3. Kinetostatics
- Maximum torque:The robot target is 28 N, as the one of the MIT-MANUS [27], taken as a reference value for its considerable clinical exploitation. This force is translated to joints and on the basis of Equations (26)–(28).
- Maximum velocity:Considering common neurorehabilitation exercises, the maximum velocity required at the end-effector is assumed to be lower than 0.5 m/s in the Cartesian space. In fact, Krebs et al. states, with experiments, that the tangential velocities for circular movements performed by stroke patients is below 0.5 m/s [28]. They also present linear velocities for point-to-point movements lower than 0.25 m/s. The corresponding angular velocity on the joints depends on the configuration of the manipulator and it is maximal when the minimum singular value of the Jacobian is minimal. Accordingly, the maximum angular velocity needed on the actuated joints was calculated to be equal to 4 rad/s or 38 RPM.
4. Prototype
4.1. Description
4.2. Cost Estimation
5. Control
5.1. Low-Level Control
5.2. High-Level Control
5.2.1. Trajectory Controller
5.2.2. Admittance Controller
5.2.3. Tunnel Controller
6. Experimental Assessment
6.1. Admittance Controller Validation
6.2. Tunnel Controller Validation
6.3. Position Measurement Accuracy
6.3.1. Test Bench
6.3.2. Data Analysis
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Yamine, J.; Prini, A.; Nicora, M.L.; Dinon, T.; Giberti, H.; Malosio, M. A Planar Parallel Device for Neurorehabilitation. Robotics 2020, 9, 104. https://doi.org/10.3390/robotics9040104
Yamine J, Prini A, Nicora ML, Dinon T, Giberti H, Malosio M. A Planar Parallel Device for Neurorehabilitation. Robotics. 2020; 9(4):104. https://doi.org/10.3390/robotics9040104
Chicago/Turabian StyleYamine, Jawad, Alessio Prini, Matteo Lavit Nicora, Tito Dinon, Hermes Giberti, and Matteo Malosio. 2020. "A Planar Parallel Device for Neurorehabilitation" Robotics 9, no. 4: 104. https://doi.org/10.3390/robotics9040104