An Evolutionary Approach for the Optimal Design of the iCub mk.3 Parallel Wrist
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Presented by Raed Bsili in the IEEE-RAS 18th International Conference on Humanoid Robots (Humanoids 2018), 08/11/18, Beijing, China. The file should be open under `Slide Show` in order to visualize the mechanism animations. Notes have been added to help the reader understand with more details the scope of the talk.
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An Evolutionary Approach for the Optimal Design of the iCub mk.3 Parallel Wrist
- 1. An Evolutionary Approach for the Optimal Design of iCub mk.3 Parallel Wrist Raed Bsili, Giorgio Metta, Alberto Parmiggiani This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 1
- 2. Introduction This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 • 𝑃𝑖𝑡𝑐ℎ = ±56° 2 Differential (Cable-driven) Mechanism • (relatively) low Endurance • 𝑌𝑎𝑤 = ±38° RoM: Parallel Kinematics Mechanism iCub 2.5 Can we use it for iCub mk.3 wrist ? Advantages High Payload-to-weight ratio High Power-transmission Disadvantages • Limited workspace • Poor mechanism uniformity Disadvantages • (relatively) low Payload-to-weight ratio • (relatively) low Power-transmission
- 3. Candidate Mechanism: 2-DoF Parallel Wrist Design parameters set: 𝑿 = [𝒍 𝟏°, 𝒍 𝟐°, 𝒍 𝟑°, 𝜶°] Active joints (motorized) angles: 𝜽 = [𝜽 𝑳°, 𝜽 𝑹°] End-effector pose: 𝝓 = [𝝓 𝒑°, 𝝓 𝒚°] This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 3 Ogata and Hirose (IROS2004) Input Output Left and Right Pitch and Yaw 𝑢 𝐿 𝑣 𝐿 𝑣 𝑅 𝑢 𝑅 𝜃 𝐿 𝒍 𝟑 𝜶 𝒍 𝟐 𝒍 𝟏𝜃 𝑅 𝜙 𝑝 𝜙 𝑦 forearm Wrist End-Effector: hand interface x z y
- 4. Pure pitch Pure yaw Coupled Mechanism: Simulation This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 4
- 5. Performance: Workspace Number, 𝛀[−] This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 𝜽 𝑳, 𝜽 𝑹 ∈ [−𝑰, 𝑰] End-effector Workspace, W 𝑆𝐽 = 4𝐼2 where, 𝐼 = 65° 𝜃 𝐿° 𝜃 𝑅° 𝜙 𝑝° 𝜙 𝑦° 𝑿 = [𝒍 𝟏°, 𝒍 𝟐°, 𝒍 𝟑°, 𝜶°] 5 𝝓 = 𝒂𝒓𝒈𝒎𝒊𝒏 𝝓 𝒇 𝑲(𝜽, 𝝓) 3.) Generate the Workspace: 1.) For a given design parameters: 2.) Generate a square grid input: 4.) Compute the Workspace Number Ω 𝑋 = 𝑊 𝑑𝑊 𝑆𝐽 ∈ ℝ≥0 Kinematics model Actuated joints
- 6. Performance: Mechanism Uniformity, 𝚫 [−] This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 𝜙 = 𝐽 𝜃 ; 𝐽 𝜃, 𝜙 ∈ ℝ2×2 Local Isotropy Number 6 Global Isotropy Number Jacobian Isotropic Jacobian • Singular Jacobian ∆ 𝑋 = 𝑊 ∆ 𝐽 𝑑𝑊 𝑊 𝑑𝑊 ∈ 0,1 𝑿 = [𝒍 𝟏°, 𝒍 𝟐°, 𝒍 𝟑°, 𝜶°]: design parameters ∆ 𝐽 = 𝑚 𝑑𝑒𝑡(𝐽𝐽 𝑇) 𝑡𝑟(𝐽𝐽 𝑇)/𝑚 ∈ 0,1
- 7. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 Gradient-free Differential Evolution OPTIMIZATION Storn & Price (JGO1997) 𝒂𝒓𝒈𝒎𝒊𝒏 𝑿 𝒇 𝑿 = 𝟏 − (∆ 𝒎𝒊𝒏∆∆ 𝒎𝒂𝒙) 𝜴 𝑋 = 𝑙1°, 𝑙2°, 𝑙3°, 𝛼° ∈ ℝ4 𝒍 𝟏, 𝒍 𝟐, 𝒍 𝟑, 𝜶 ∈ [𝟗°, 𝟖𝟓°] • Bounds • Constraints 𝒈 𝑿 ≥ 𝟎 𝜴(𝑿) ≤ 𝟏 7 Performance: Optimization Find the best design parameters described by 𝑋 ∈ ℝ4, that maximize the mechanism uniformity and maximize its workspace width.
- 8. Results This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 𝐗 𝐨𝐩𝐭 = 𝐥 𝟏°, 𝐥 𝟐°, 𝐥 𝟑°, 𝛂° 𝐨𝐩𝐭 Mechanism Uniformity: (∆ 𝐦𝐢𝐧 , ∆ , ∆ 𝒎𝒂𝒙) Workspace Number 𝛀 Theoretical Range of Motion: ( 𝝓 𝒑 , 𝝓 𝒚 ) [33.6° , 83° , 32.7° , 10.7° ] (0.978 , 0.998 , 1) 0.5 ( ±65° , ±63.1° ) 𝐼 = 65 ° 8 Theoretical results
- 9. Results: Validation This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 𝜀 = ∆ 𝐶𝐴𝐷 − ∆ 9 Comparison between theoretical results and CAD-results.
- 10. Results: Consistency This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 𝜃 𝐿[°] 𝜃 𝑅[°] Y Z 10 Input on the active joints Output on the end-effector ( 3D points ! ) 2D Isotropy on the YZ-plane
- 11. Design Demonstration This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 Pure pitch, 𝜙 𝑝 = ±58° Pure yaw, 𝜙 𝑦 = ±50° Coupled pitch-yaw 11
- 12. Conclusion: iCub mk.3 & mk.2.5 wrists comparison This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 12 Pitch, 𝝓 𝒑 [°] Yaw, 𝝓 𝒚 [°] iCub mk.2.5 Wrist’s Range of Motion ±𝟓𝟔 ±𝟑𝟖 iCub mk.3 Parallel Wrist, Mechanical Range of Motion ±𝟓𝟖 ±𝟓𝟎 iCub mk.3 Parallel Wrist, Theoretical Range of Motion ±65 ±63.1 Higher Payload-to-Weight ratio: Wider workspace: More compact design Wrist Payload-to-Weight ratio [-] Maximum payload’s mass supported by the hand [kg] iCub mk. 2.5 Wrist ~ 𝟐. 𝟔 ~ 𝟎. 𝟔𝟔 iCub mk. 3 Wrist ~ 8 ~ 2 Better endurance
- 13. Thank You for Your Attention This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 13 r a e d . b s i l i @ i i t . i t • ( Please refer to our paper for more details on the “ kinematics and optimization” mathematical model! ) • ( Code will be published soon . . github: raedbsili1991 )
- 14. Annex I: (Kinematics) in which: and: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 14
- 15. Annex II: (Inverse Condition Number) 𝜅2 𝐽 = 1 𝐽 2∙ 𝐽−1 2 = 𝜎 𝑚𝑖𝑛 (𝐽) 𝜎 𝑚𝑎𝑥 (𝐽) ∈ [0,1] 𝜎 𝐽 : 𝑆𝑖𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐽𝑎𝑐𝑜𝑏𝑖𝑎𝑛 ∙ 2: Euclidean norm This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 15 ∙ 𝐹: Frobenius norm 𝐽 ∈ ℝ2×2
- 16. Annex II: (Inverse Condition Number) 𝜅2[−] 𝜅2 𝐽 = 1 𝐽 2∙ 𝐽−1 2 = 𝜎 𝑚𝑖𝑛 (𝐽) 𝜎 𝑚𝑎𝑥 (𝐽) ∈ [0,1] 𝜎 𝐽 : 𝑆𝑖𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐽𝑎𝑐𝑜𝑏𝑖𝑎𝑛 ∙ 2: Euclidean norm This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 676063 16
Editor's Notes
- • Let me give quick reminding on the current iCub wrist technology. • It is doing well, however, as any types of a manipulator, it has some drawbacks, among them the End, Power because of the cables and low P2R due to the differential mechanism. • A solution to overcome these drawbacks is to employ parallel kinematics mechanism. for the simple reasons is • However, themselves have some drawbacks, limited RoM, and poor mechanism uniformity. • So the motivation here, is, can we find a PKM that is able to overcome these drawbacks ? • A solution to maximize it and to reduce the share of the motive power required to drive the robot’s links is to employ parallel kinematic mechanisms (PKMs).
- •Here is our candidate mechanism, proposed by . ., it is consisting of a symmetric close-loops linkages in which all joints axis cross at a common point (CoR). • the mechanism behaviour is completely defined by 4 param sets, each describing a linkage bending angle. • A 2 actuated joints, left and right one. • And finally, as a consequences, 2 joints , pitch and yaw on the EE, describing the 2-DoF mechanism.
- • Here is a quick simulation to have a better feeling on how the mechanisms works. • In this study, we have developed an approach to optimize this mechanism. • but first, let me introduce the metrics we introduced to evaluate the wrist performance.
- The first metric is what we have called the workspace number, used to evaluate a mechanism workspace width. For a given X, generate a square grid input of actuated joints, consisting in generating all combination between the motorized joints, inside a range of +-65°. Generate the corresponding workspace by solving the forward kinematics. The proposed WN here is defined as the area spanned by the EE pitch/yaw angles, divided by the area spanned by the actuated joints square grid, which is basically the rectangle area. This metric is used to compare the workspace width between different parallel manipulators, independently from their scale.
- Jacobian is linear maping from joint space to the cartesian space. The iso number is described by the following algebraic equation, in which m denotes the EE DoF = 2 pitch and yaw. It is a bounded metric, when it is zero, singular position and the wrist loses 1 DoF, in the opposite case, when it is maximum, the Jacobian is perfectly-posed, and this is interpreted mathematically by having all Jacobian engeinvalues non-zeros and equal to each other. As the isotropy is a local property of the manipulator and it depends of its pose, we introduce here the GIN which is a metric describing how the manipulator Jacobian is isotropic over the whole workspace.
- - Op part, The performance optimisation has been described by the following sing-ob, bounded and constrained problem, - The constraints here is to respect the symmetry condition, and the bounds here is to have design solution realizable, from manufacturing POV. - this opt-problem has been solved using the grad-free DE algorithm.
- results, we have a design parameter set that has scored a maximum isotropic number, minimum 0.978, and almost 1 everywhere. It has scored as well significantly wide theoretical RoM, +65°, +63° on the pitch and the yaw. We have made preliminary design of the optimal set, and due to the small size of iCub forearm, we used 4B linkages to drive the active the joints.
- Left Gif: To validate the model of the optimal set we have found. square grid input on the motorized joints have been generated and imported by CAD model, and the Forward Kinematics are solved accordingly, we see here the hand (interfaced to the wrist), scanning and generating the workspace. Middle Gif: As a result, The 3D-position of the middle finger tip is saved during the workspace generation and plotted accordingly. We then have computed isotropy from the CAD model and compared it to our mathematical model. Right image: And as seen here, we can see there is a significantly excellent matching between the two.
- - Effect of an Isotropic (perfectly) posed Jacobian can be seen here by looking to these figures. - Well actually , For a constant step variation on the square grid input, we find a constant step variation on the EE, and this is constant over the whole workspace! There is no velocity amplification! This is a result of a having well-posed Jacobian (∆(𝐽)≈1). - Because, In the opposite case, when the Jacobian is ill-posed (when ∆(𝐽)→0) in a given EE pose, we would see a significant amplification of step variation (brutal increase of velocity).
- - This is the design simulation of the wrist on the iCub 3 forearm Interferences. - among design elements required us to constraint the Mechanical RoM to (+-58,+-50).
- To conclude this talk, The developed mathematical model with the introduced metrics allowed us designing a new wrist that has more compact design and better endurance than the previous iCub 2.5, gained mechanically 12° on the yaw and 2° on the pitch from RoM point of view. Finally, using the same actuators, we have increased the Payload-to-Weight ratio by roughly 3 times.