Learning to manipulate a whip with simple primitive actions - A simulation study

doi:10.1016/j.isci.2023.107395

. 2023 Jul 14;26(8):107395.

doi: 10.1016/j.isci.2023.107395. eCollection 2023 Aug 18.

Learning to manipulate a whip with simple primitive actions - A simulation study

Moses C Nah¹, Aleksei Krotov², Marta Russo^{3

4}, Dagmar Sternad⁵, Neville Hogan^{1

6}

Affiliations

¹ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
² Department of Bioengineering, Northeastern University, Boston, MA 02115, USA.
³ Department of Biology, Northeastern University, Boston, MA 02115, USA.
⁴ Department of Neurology, Policlinico Tor Vergata and the Laboratory of Neuromotor Physiology, IRCCS Santa Lucia Foundation, Rome, Italy.
⁵ Department of Biology, Department of Electrical and Computer Engineering, Department of Physics, Institute of Experiential Robotics, Northeastern University, Boston, MA 02115, USA.
⁶ Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

PMID: 37554449
PMCID: PMC10405071
DOI: 10.1016/j.isci.2023.107395

Learning to manipulate a whip with simple primitive actions - A simulation study

Moses C Nah et al. iScience. 2023.

. 2023 Jul 14;26(8):107395.

doi: 10.1016/j.isci.2023.107395. eCollection 2023 Aug 18.

Authors

Moses C Nah¹, Aleksei Krotov², Marta Russo^{3

4}, Dagmar Sternad⁵, Neville Hogan^{1

6}

Affiliations

¹ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
² Department of Bioengineering, Northeastern University, Boston, MA 02115, USA.
³ Department of Biology, Northeastern University, Boston, MA 02115, USA.
⁴ Department of Neurology, Policlinico Tor Vergata and the Laboratory of Neuromotor Physiology, IRCCS Santa Lucia Foundation, Rome, Italy.
⁵ Department of Biology, Department of Electrical and Computer Engineering, Department of Physics, Institute of Experiential Robotics, Northeastern University, Boston, MA 02115, USA.
⁶ Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

PMID: 37554449
PMCID: PMC10405071
DOI: 10.1016/j.isci.2023.107395

Abstract

This simulation study investigated whether a 4-degrees-of-freedom (DOF) arm could strike a target with a 50-DOF whip using a motion profile similar to discrete human movements. The interactive dynamics of the multi-joint arm was modeled as a constant joint-space mechanical impedance, with values derived from experimental measurement. Targets at various locations could be hit with a single maximally smooth motion in joint-space coordinates. The arm movements that hit the targets were identified with fewer than 250 iterations. The optimal actions were essentially planar arm motions in extrinsic task-space coordinates, predominantly oriented along the most compliant direction of both task-space and joint-space mechanical impedances. Of the optimal movement parameters, striking a target was most sensitive to movement duration. This result suggests that the elementary actions observed in human motor behavior may support efficient motor control in interaction with a dynamically complex object.

Keywords: Engineering; Mechanical modeling; Robotics.

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Conflict of interest statement

The authors declare no competing interests.

Figures

**Figure 1**
Target locations Graphical depiction of the six target positions and the task-space coordinate frame of the simulation. Distance $R$ , which is the radius of a sphere centered at the shoulder of the upper limb (depicted as a white marker), is equal to the sum of the lengths of the upper limb and whip (i.e., $R = L_{1} + L_{2} + l \cdot N = 2.385 m$ ) (Table 3). The three nearby targets are separated from the three farther-away targets by a constant radial distance $d$ (0.4m). Target locations are shown in a spherical coordinate system (radius-azimuth-elevation): Target 1:( $R - d, 0^{\circ}, 0^{\circ}$ ), Target 2: $(R - d, 45^{\circ}, 0^{\circ})$ , Target 3: $(R - d, 45^{\circ}, 45^{\circ})$ , Target 4:( $R, 0^{\circ}, 0^{\circ}$ ), Target 5: $(R, 45^{\circ}, 0^{\circ})$ , Target 6: $(R, 45^{\circ}, 45^{\circ})$ .

**Figure 2**
Result of the optimization Minimum distance between the whip and target $L^{*}$ over iterations for targets 1 to 3 (top panel) and 4 to 6 (bottom panel). For the optimization of target 6, the occasional spikes are due to the DIRECT-L algorithm’s procedure.^,

**Figure 3**
Simulation results for targets 1, 2, and 3, shown in panels A, B, and C, respectively (**Left column**) Time-sequence of upper-limb and whip model. Three frames of the simulation were taken: at the start of the movement, at the moment when the whip hit the target, and a moment between the two. Black circles depict the shoulder, elbow and hand of the upper-limb model. Purple circles depict the point-masses of the whip model. Opacity of the color increases from the start to the end of the movement. (**Right Column**) Time-sequence of the corresponding optimal trajectories of elbow and end-effector. Four frames are plotted from the start to the end of the movement, i.e., for time $0 \leq t \leq D$ , where $D$ is the duration of the virtual (zero-torque) trajectory (Equation 3).

**Figure 4**
Simulation results for targets 4, 5, and 6, shown in panels A, B, and C, respectively (**Left column**) Time-sequence of upper-limb and whip model. Three frames of the simulation were taken: at the start of the movement, at the moment when the whip hit the target (for target 6, at the moment when the whip and the target were closest), and a moment between the two. Black circles depict the shoulder, elbow and hand of the upper-limb model. Purple circles depict the point-masses of the whip model. Opacity of the color increases from the start to the end of the movement. (**Right Column**) Time-sequence of the corresponding optimal trajectories of elbow and end-effector. Several frames are plotted from the start to the end of the movement, i.e., for time $0 \leq t \leq D$ , where $D$ is the duration of the virtual (zero-torque) trajectory (Equation 3).

**Figure 5**
Optimal end-effector movement for targets 1, 2, and 3, shown in panels A, B, and C, respectively (**Left Column**) The end-effector trajectory (Figure 3, right column) and its best-fit plane. The centroid $c$ and the normal vector $n$ of the best-fit plane are depicted respectively as a diamond symbol and an arrow originating from it. The square depicts the shoulder’s location. (**Right Column**) Corresponding weights of the eigenmovements, $w_{i}$ , i = 1, $\dots$ 4 of the optimal upper-limb movement vs. time (Equation 6). Values between the start and end of the movement are plotted, i.e., the width of the $x$ -axis is $D$ , the duration of the virtual (zero-torque) trajectory (Equation 3).

**Figure 6**
Optimal end-effector movement for targets 4, 5, and 6, shown in panels A, B, and C, respectively (**Left Column**) The end-effector trajectory (Figure 4, right column) and its best-fit plane. The centroid $c$ and the normal vector $n$ of the best-fit plane are depicted respectvely as a diamond symbol and an arrow originating from it. The square depicts the shoulder’s location. (**Right Column**) Corresponding weights of the eigenmovements, $w_{i}$ , i = 1, $\dots$ 4 of the optimal upper-limb movement vs. time (Equation 6). Values between the start and end of the movement are plotted, i.e., the width of the $x$ -axis is $D$ , the duration of the virtual (zero-torque) trajectory (Equation 3).

**Figure 7**
Sensitivity of task performance to the movement parameters for targets 1 to 5 Gray bars depict the upper and lower bounds of the search space of the DIRECT-L optimization algorithm. The upper (respectively lower) error bar depicts the upper (respectively lower) value of the error to miss the target. Dots within the error bar depict the optimal movement value (Table 1). T1 to T5 denote target 1 to target 5, respectively.

**Figure 8**
Square root of the sum of the squared weight values of the joint-space and task-space impedances T1 to T6 denote target 1 to target 6, respectively.

**Figure 9**
Task-space stiffness ellipsoids for targets 1, 2, and 3, shown in panels A, B, and C, respectively **(Left Column)** Time-sequence of upper-limb (black) and its stiffness ellipsoids. Four frames of the simulation were taken: at the start of the movement, at the end of the movement, and frames between the two. The iso-potential energy surface of the task-space impedance $K_{p}$ is plotted as an ellipsoid, i.e., the width of the ellipsoid is inversely proportional to the eigenvalue of $K_{p}$ . Red arrow depicts the eigenvector of $K_{p}$ with the smallest eigenvalue. Blue arrow depicts the velocity of the hand. **(Right Column)** Corresponding weights of the eigenvectors, $w_{i}$ , i = 1,2,3 of the stiffness ellipsoid vs. time (Equation 8). Values between the start and end of the movement are plotted, i.e., the width of the $x$ -axis is $D$ , the duration of the virtual (zero-torque) trajectory (Equation 3).

**Figure 10**
Task-space stiffness ellipsoids for targets 4, 5, and 6, shown in panels A, B, and C, respectively **(Left Column)** Time-sequence of upper-limb (black) and its stiffness ellipsoids. Four frames of the simulation were taken: at the start of the movement, at the end of the movement, and frames between the two. The iso-potential energy surface of the task-space impedance $K_{p}$ is plotted as an ellipsoid, i.e., the width of the ellipsoid is inversely proportional to the eigenvalue of $K_{p}$ . Red arrow depicts the eigenvector of $K_{p}$ with the smallest eigenvalue. Blue arrow depicts the velocity of the hand. **(Right Column)** Corresponding weights of the eigenvectors, $w_{i}$ , i = 1,2,3 of the stiffness ellipsoid vs. time (Equation 8). Values between the start and end of the movement are plotted, i.e., the width of the $x$ -axis is $D$ , the duration of the virtual (zero-torque) trajectory (Equation 3).

**Figure 11**
Model used for the simulation (A) Simulation model in its virtual configuration, rendered with the MuJoCo simulator. (B) Model of the upper limb and its model parameters. Rotational joints of shoulder (J1-J3), elbow (J4) and their axes of rotation are visualized as bullet shapes. Joints J1-J4 were equipped with torque actuators. (C) Sub-model of the whip and its model parameters. Each rotational joint is equipped with a linear rotational spring $k$ and rotational damper $b$ , visualized as bullet shapes.

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References

1. Bernstein B., Hall D.A., Trent H.M. On the dynamics of a bull whip. J. Acoust. Soc. Am. 1958;30:691–1115.
1. Goriely A., McMillen T. Shape of a cracking whip. Phys. Rev. Lett. 2002;88 - PubMed
1. McMillen T., Goriely A. Whip waves. Phys. Nonlinear Phenom. 2003;184:192–225.
1. Henrot C.C.I. Massachusetts Institute of Technology; 2016. Characterization of Whip Targeting Kinematics in Discrete and Rhythmic Tasks. Bachelor’s Thesis.
1. Krotov A., Russo M., Nah M., Hogan N., Sternad D. Motor control beyond reach—how humans hit a target with a whip. R. Soc. Open Sci. 2022;9 - PMC - PubMed

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[1] Bernstein B., Hall D.A., Trent H.M. On the dynamics of a bull whip. J. Acoust. Soc. Am. 1958;30:691–1115.

[2] Bernstein B., Hall D.A., Trent H.M. On the dynamics of a bull whip. J. Acoust. Soc. Am. 1958;30:691–1115.

[3] Goriely A., McMillen T. Shape of a cracking whip. Phys. Rev. Lett. 2002;88 - PubMed

[4] Goriely A., McMillen T. Shape of a cracking whip. Phys. Rev. Lett. 2002;88 - PubMed

[5] McMillen T., Goriely A. Whip waves. Phys. Nonlinear Phenom. 2003;184:192–225.

[6] McMillen T., Goriely A. Whip waves. Phys. Nonlinear Phenom. 2003;184:192–225.

[7] Henrot C.C.I. Massachusetts Institute of Technology; 2016. Characterization of Whip Targeting Kinematics in Discrete and Rhythmic Tasks. Bachelor’s Thesis.

[8] Henrot C.C.I. Massachusetts Institute of Technology; 2016. Characterization of Whip Targeting Kinematics in Discrete and Rhythmic Tasks. Bachelor’s Thesis.

[9] Krotov A., Russo M., Nah M., Hogan N., Sternad D. Motor control beyond reach—how humans hit a target with a whip. R. Soc. Open Sci. 2022;9 - PMC - PubMed

[10] Krotov A., Russo M., Nah M., Hogan N., Sternad D. Motor control beyond reach—how humans hit a target with a whip. R. Soc. Open Sci. 2022;9 - PMC - PubMed

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Learning to manipulate a whip with simple primitive actions - A simulation study

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Learning to manipulate a whip with simple primitive actions - A simulation study

Authors

Affiliations

Abstract

Conflict of interest statement

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Abstract

Conflict of interest statement

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